**A Robust Induction Motor Control using Sliding Mode Rotor Flux and Load Torque Observers**

Oscar Barambones, Patxi Alkorta, 1∗ 2 1 2

Jose M. Gonzalez de Duran and Jose A. Cortajarena

Additional information is available at the end of the chapter

### **Abstract**

A sliding mode position control for high-performance real-time applications of induction motors is developed in this work. The design also incorporates a sliding mode based flux and load torque observers in order to avoid this sensors, that increases the cost and reduces the reliability. Additionally, the proposed control scheme presents a low computational cost and therefore can be implemented easily in a real-time applications using a low cost DSP-processor.

The stability analysis of the controller under parameter uncertainties and load disturbances is provided using the Lyapunov stability theory. Finally simulated and experimental results show that the proposed controller with the proposed observer provides a good trajectory tracking and that this scheme is robust with respect to plant parameter variations and external load disturbances.

**Keywords**: Position Control, Sliding Mode Control, Robust Control, Induction Machines, Lyapunov Stability, Nonlinear Control

### **1. Introduction**

AC induction motors have been widely used in industrial applications such machine tools, steel mills and paper machines owing to their good performance provided by their solid architecture, low moment of inertia, low ripple of torque and high initiated torque. Some control techniques have been developed to regulate these induction motor servo drives in high-performance applications. One of the most popular technique is the indirect field oriented control method [1, 2].

The field-oriented technique guarantees the decoupling of torque and flux control commands of the induction motor, so that the induction motor can be controlled linearly as a separated excited D.C. motor. However, the control performance of the resulting linear system is

© 2015 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

still influenced by uncertainties, which usually are composed of unpredictable parameter variations, external load disturbances, and unmodelled and nonlinear dynamics.

In the last decades the proportional integral derivative (PID) controller has been widely used in the vector control of induction motors due to its good performance and its simple structure. However in some applications the PID controller may not meet the concerned robustness under parameter variations and external load disturbances. Therefore, many studies have been made on the motor drives in order to preserve the performance under these parameter variations and external load disturbance, such as nonlinear control, multivariable control, optimal control, H-∞ control and adaptive control [3]-[7], etc. However usually these controllers present a high computational cost and cannot be implemented over a low cost DSP processor to perform a real time control.

The sliding-mode control can offer many good properties, such as good performance against unmodelled dynamics, insensitivity to parameter variations, external disturbance rejection and fast dynamic response. These advantages of the sliding-mode control may be employed in the position and speed control of an AC servo system [8]. In [9] an integral sliding mode speed control for induction motor based on field oriented control theory is proposed. In the work of [10], an integrated sliding mode controller (SMC) based on space vector pulse width modulation method is proposed to achieve high-performance speed control of an induction motor. In this work using a field-oriented control principle, a flux SMC is first established to achieve fast direct flux control and then a speed SMC is presented to enhance speed control by the direct torque method. However in this work the performance of the proposed controller is not validated over a real induction motor.

Position control is often used in some applications of electrical drives like robotic systems, conveyor belts, etc. In these applications uncertainty and external disturbances are present and therefore a robust control system that maintain the desired control performance under this situations are frequently required [11]-[15].

The variable structure control strategy using the sliding-mode has also been focussed on many studies and research for the position control of the induction motors [16, 17].

The induction motor position control problem has been studied in [18] using a discrete time sliding mode control. The field oriented control theory is also used in order to decouple the flux and the electromagnetic torque. In this paper the authors calculates the rotor flux vector angular position using the slip estimates which is very sensitive to the rotor resistance variation. In contrast, a rotor flux sliding mode observer is proposed in our control scheme, in order to calculate an accurate value for the angular position of the rotor flux vector in the presence of system uncertainties.

On the other hand in the last decade remarkable efforts have been made to reduce the number of sensors in the control systems [19]-[22]. The sensors increases the cost and also reduces the reliability of the control system because this elements are generally expensive, delicate and difficult to instal.

In this chapter a robust approach for induction motor position control is presented. The proposed sliding mode control may overcome the system uncertainties and load disturbances that usually are present in the real systems. In the controller design, the field oriented control theory is used to simplify the system dynamical equations. Moreover, the proposed controller does not present a high computational cost and therefore can be implemented easily in a real-time applications using a low cost DSP-processor.

In this work a sliding mode flux observer is proposed in order to avoid the flux sensors. The estimated rotor flux is used to calculate the rotor flux vector angular position, whose value is essential in order to apply the field oriented control principle. A load torque estimation algorithm, based on a sliding mode observer, is also presented in order to improve the adaptive robust position control performance. Additionally, the overall control scheme does not involve a high computational cost and therefore can be implemented easily in a real time applications.

Moreover, the control scheme presented in this chapter is validated in a real test using a commercial induction motor of 7.5 kW in order to demonstrate the real performance of this controller. The experimental validation has been implemented using a control platform based on a DS1103 PPC Controller Board that has been designed and constructed in order to carry out the experimental validation of the proposed controller.

This manuscript is organized as follows. The sliding mode flux observer is developed in Section 2 and the sliding mode load torque observer is designed in Section 3. Then, the proposed sliding mode position control is presented in Section 4. In the Section 5, the experimental control platform is presented and some simulation and experimental results are carried out. Finally, concluding remarks are stated in Section 6.

### **2. Sliding mode observer for rotor flux estimator**

still influenced by uncertainties, which usually are composed of unpredictable parameter

In the last decades the proportional integral derivative (PID) controller has been widely used in the vector control of induction motors due to its good performance and its simple structure. However in some applications the PID controller may not meet the concerned robustness under parameter variations and external load disturbances. Therefore, many studies have been made on the motor drives in order to preserve the performance under these parameter variations and external load disturbance, such as nonlinear control, multivariable control, optimal control, H-∞ control and adaptive control [3]-[7], etc. However usually these controllers present a high computational cost and cannot be implemented over a low

The sliding-mode control can offer many good properties, such as good performance against unmodelled dynamics, insensitivity to parameter variations, external disturbance rejection and fast dynamic response. These advantages of the sliding-mode control may be employed in the position and speed control of an AC servo system [8]. In [9] an integral sliding mode speed control for induction motor based on field oriented control theory is proposed. In the work of [10], an integrated sliding mode controller (SMC) based on space vector pulse width modulation method is proposed to achieve high-performance speed control of an induction motor. In this work using a field-oriented control principle, a flux SMC is first established to achieve fast direct flux control and then a speed SMC is presented to enhance speed control by the direct torque method. However in this work the performance of the proposed

Position control is often used in some applications of electrical drives like robotic systems, conveyor belts, etc. In these applications uncertainty and external disturbances are present and therefore a robust control system that maintain the desired control performance under

The variable structure control strategy using the sliding-mode has also been focussed on

The induction motor position control problem has been studied in [18] using a discrete time sliding mode control. The field oriented control theory is also used in order to decouple the flux and the electromagnetic torque. In this paper the authors calculates the rotor flux vector angular position using the slip estimates which is very sensitive to the rotor resistance variation. In contrast, a rotor flux sliding mode observer is proposed in our control scheme, in order to calculate an accurate value for the angular position of the rotor flux vector in the

On the other hand in the last decade remarkable efforts have been made to reduce the number of sensors in the control systems [19]-[22]. The sensors increases the cost and also reduces the reliability of the control system because this elements are generally expensive, delicate

In this chapter a robust approach for induction motor position control is presented. The proposed sliding mode control may overcome the system uncertainties and load disturbances that usually are present in the real systems. In the controller design, the field oriented control theory is used to simplify the system dynamical equations. Moreover, the proposed controller does not present a high computational cost and therefore can be implemented easily in a

many studies and research for the position control of the induction motors [16, 17].

variations, external load disturbances, and unmodelled and nonlinear dynamics.

cost DSP processor to perform a real time control.

210 Induction Motors - Applications, Control and Fault Diagnostics

controller is not validated over a real induction motor.

real-time applications using a low cost DSP-processor.

this situations are frequently required [11]-[15].

presence of system uncertainties.

and difficult to instal.

Many schemes based on simplified motor models have been devised to estimate some internal variables of the induction motor from measured terminal quantities. This procedure is frequently used in order to avoid the presence of some sensors in the control scheme. In order to obtain an accurate dynamic representation of the motor, it is necessary to base the calculation on the coupled circuit equations of the motor.

Since the motor voltages and currents are measured in a stationary frame of reference, it is also convenient to express the induction motor dynamical equations in this stationary frame.

The system state space equations in the stationary reference frame can be written in the form [23]:

$$\begin{aligned} \dot{i}\_{ds} &= \frac{-1}{\sigma L\_{\rm s}} \left( R\_{\rm s} + \frac{L\_{m}^{2}}{L\_{r}^{2}} R\_{r} \right) i\_{ds} + \frac{L\_{m}}{\sigma L\_{\rm s} L\_{r}} \frac{1}{T\_{r}} \psi\_{dr} \\ &+ \frac{L\_{m}}{\sigma L\_{\rm s} L\_{r}} w\_{r} \psi\_{qr} + \frac{1}{\sigma L\_{\rm s}} V\_{ds} \\ \dot{i}\_{qs} &= \frac{-1}{\sigma L\_{\rm s}} \left( R\_{\rm s} + \frac{L\_{m}^{2}}{L\_{r}^{2}} R\_{r} \right) i\_{qs} - \frac{L\_{m}}{\sigma L\_{\rm s} L\_{r}} w\_{r} \psi\_{dr} \\ &+ \frac{L\_{m}}{\sigma L\_{\rm s} L\_{r}} \frac{1}{T\_{r}} \psi\_{qr} + \frac{1}{\sigma L\_{\rm s}} V\_{qs} \\ \dot{\psi}\_{dr} &= \frac{L\_{m}}{T\_{r}} i\_{ds} - \frac{1}{T\_{r}} \psi\_{dr} - w\_{r} \psi\_{qr} \\ \dot{\psi}\_{qr} &= \frac{L\_{m}}{T\_{r}} i\_{qs} + w\_{r} \psi\_{dr} - \frac{1}{T\_{r}} \psi\_{qr} \end{aligned} \tag{1}$$

where *Vds*, *Vqs* are stator voltages; *ids*, *iqs* are stator currents; *ψdr*, *ψqr* are rotor fluxes; *wr* is motor speed; *Rs*, *Rr* are stator and rotor resistances; *Ls*, *Lr* are stator and rotor inductances; *Lm*, is mutual inductance; *<sup>σ</sup>* <sup>=</sup> <sup>1</sup> <sup>−</sup> *<sup>L</sup>*<sup>2</sup> *m LsLr* is leakage coefficient; *Tr* <sup>=</sup> *Lr Rr* is rotor-time constant.

From singular perturbation theory [24], and based on the well-known induction motor model dynamics [23], the slow variables of the system are *ψdr*, *ψqr* and the fast variables are *ids*, *iqs*. Therefore, the corresponding singularly perturbed model of eqn.(1) is:

$$\begin{aligned} \dot{\varepsilon}\dot{i}\_{\rm ds} &= -L\_{m}\alpha\_{r}\dot{i}\_{\rm ds} + \alpha\_{r}\psi\_{dr} + \varpi\_{r}\psi\_{qr} + \frac{L\_{r}}{L\_{m}}\left(V\_{\rm ds} - R\_{s}\dot{i}\_{\rm ds}\right) \\ \dot{\varepsilon}\dot{i}\_{\rm qs} &= -L\_{m}\alpha\_{r}\dot{i}\_{\rm qs} - \varpi\_{r}\psi\_{dr} + \alpha\_{r}\psi\_{qr} + \frac{L\_{r}}{L\_{m}}\left(V\_{\rm qs} - R\_{s}\dot{i}\_{\rm qs}\right) \\ \dot{\psi}\_{dr} &= L\_{m}a\_{r}\dot{i}\_{\rm ds} - \alpha\_{r}\psi\_{dr} - \varpi\_{r}\psi\_{qr} \\ \dot{\psi}\_{qr} &= L\_{m}a\_{r}\dot{i}\_{\rm qs} + \varpi\_{r}\psi\_{dr} - \alpha\_{r}\psi\_{qr} \end{aligned} \tag{2}$$

where *<sup>ε</sup>* <sup>=</sup> *<sup>σ</sup>LsLr Lm* and *<sup>α</sup><sup>r</sup>* <sup>=</sup> <sup>1</sup> *Tr*

.

The proposed sliding mode observer is a copy of the original system model, which has corrector terms with switching functions based on the system outputs. Therefore, considering the measured stator currents as the system outputs, the corresponding sliding-mode-observer can be constructed as follows:

$$\begin{aligned} \dot{\epsilon}\_{\text{ds}}^{\hat{\scriptstyle}} &= -L\_m a\_{r} i\_{\text{ds}} + a\_{r} \hat{\mathbb{y}}\_{dr} + w\_{r} \hat{\mathbb{y}}\_{qr} + \frac{L\_r}{L\_m} \left( V\_{\text{ds}} - R\_{\text{s}} i\_{\text{ds}} \right) \\ &- k\_1 e\_{i\_{\text{d}}} + g\_{i\_{\text{d}}} \text{sgn}(e\_{\text{id}}) \end{aligned}$$

$$\begin{aligned} \dot{\epsilon}\_{\text{qs}}^{\hat{\scriptstyle}} &= -L\_m a\_{r} i\_{\text{qs}} - w\_{r} \hat{\mathbb{y}}\_{dr} + a\_{r} \hat{\mathbb{y}}\_{qr} + \frac{L\_r}{L\_m} \left( V\_{\text{qs}} - R\_{\text{s}} i\_{\text{qs}} \right) \\ &- k\_2 e\_{i\text{q}} + g\_{i\_{\text{q}}} \text{sgn}(e\_{\text{i}q}) \\ \dot{\psi}\_{dr} &= L\_m a\_{r} i\_{\text{d}s} - a\_{r} \hat{\psi}\_{dr} - w\_{r} \hat{\psi}\_{qr} + g\_{\text{y}q} \text{sgn}(e\_{\text{i}d}) \\ \dot{\psi}\_{qr} &= L\_m a\_{r} i\_{\text{qs}} + w\_{r} \hat{\psi}\_{dr} - a\_{r} \hat{\psi}\_{qr} + g\_{\text{y}q} \text{sgn}(e\_{\text{i}q}) \end{aligned} \tag{3}$$

where <sup>ˆ</sup>*<sup>i</sup>* and *<sup>ψ</sup>*<sup>ˆ</sup> are the estimations of *<sup>i</sup>* and *<sup>ψ</sup>*; *<sup>k</sup>*<sup>1</sup> and *<sup>k</sup>*<sup>2</sup> are positive constant gains; *gid* , *giq* , *<sup>g</sup>ψ<sup>d</sup>* and *<sup>g</sup>ψ<sup>q</sup>* are the observer gain matrix; *eid* <sup>=</sup> <sup>ˆ</sup>*ids* <sup>−</sup> *ids* and *eiq* <sup>=</sup> <sup>ˆ</sup>*iqs* <sup>−</sup> *iqs* are de current errors, and sgn() is the sign function.

Subtracting eqn.(2) from eqn.(3), the estimation error dynamics are:

$$\begin{aligned} \epsilon \dot{e}\_{i\_d} &= a\_r e\_{\psi\_d} + w\_r e\_{\psi\_q} - k\_1 e\_{i\_d} + g\_{i\_d} \operatorname{sgn}(e\_{id}) \\ \epsilon \dot{e}\_{i\_q} &= -w\_r e\_{\psi\_d} + a\_r e\_{\psi\_q} - k\_2 e\_{iq} + g\_{i\_q} \operatorname{sgn}(e\_{iq}) \\ \dot{e}\_{\Psi\_l} &= -a\_r e\_{\Psi\_l} - w\_r e\_{\psi\_l} + g\_{\psi\_d} \operatorname{sgn}(e\_{id}) \\ \dot{e}\_{\Psi\_l} &= w\_r e\_{\Psi\_l} - \alpha\_r e\_{\psi\_q} + g\_{\psi\_q} \operatorname{sgn}(e\_{iq}) \end{aligned} \tag{4}$$

where *eψ<sup>d</sup>* = *ψ*ˆ *dr* <sup>−</sup> *<sup>ψ</sup>dr*, *<sup>e</sup>ψ<sup>q</sup>* <sup>=</sup> *<sup>ψ</sup>*ˆ*qr* <sup>−</sup> *<sup>ψ</sup>qr*

where *Vds*, *Vqs* are stator voltages; *ids*, *iqs* are stator currents; *ψdr*, *ψqr* are rotor fluxes; *wr* is motor speed; *Rs*, *Rr* are stator and rotor resistances; *Ls*, *Lr* are stator and rotor inductances;

From singular perturbation theory [24], and based on the well-known induction motor model dynamics [23], the slow variables of the system are *ψdr*, *ψqr* and the fast variables are *ids*, *iqs*.

The proposed sliding mode observer is a copy of the original system model, which has corrector terms with switching functions based on the system outputs. Therefore, considering the measured stator currents as the system outputs, the corresponding

*dr* + *wrψ*ˆ*qr* +

*dr* + *αrψ*ˆ*qr* +

where <sup>ˆ</sup>*<sup>i</sup>* and *<sup>ψ</sup>*<sup>ˆ</sup> are the estimations of *<sup>i</sup>* and *<sup>ψ</sup>*; *<sup>k</sup>*<sup>1</sup> and *<sup>k</sup>*<sup>2</sup> are positive constant gains; *gid* , *giq* , *<sup>g</sup>ψ<sup>d</sup>* and *<sup>g</sup>ψ<sup>q</sup>* are the observer gain matrix; *eid* <sup>=</sup> <sup>ˆ</sup>*ids* <sup>−</sup> *ids* and *eiq* <sup>=</sup> <sup>ˆ</sup>*iqs* <sup>−</sup> *iqs* are de current errors,

*e*ˆ*id* = *αreψ<sup>d</sup>* + *wreψ<sup>q</sup>* − *k*1*eid* + *gid* sgn(*eid*)

*e*˙*ψ<sup>q</sup>* = *wreψ<sup>d</sup>* − *αreψ<sup>q</sup>* + *gψ<sup>q</sup>* sgn(*eiq*)

*e*ˆ*iq* = −*wreψ<sup>d</sup>* + *αreψ<sup>q</sup>* − *k*2*eiq* + *giq* sgn(*eiq*)

*dr* <sup>−</sup> *wrψ*ˆ*qr* <sup>+</sup> *<sup>g</sup>ψ<sup>d</sup>* sgn(*eid*)

*dr* <sup>−</sup> *<sup>α</sup>rψ*ˆ*qr* <sup>+</sup> *<sup>g</sup>ψ<sup>q</sup>* sgn(*eiq*)

is leakage coefficient; *Tr* <sup>=</sup> *Lr*

*Lr Lm*

*Lr Lm* 

*dr* = *Lmαrids* − *αrψdr* − *wrψqr* (2)

*Lr Lm*

*Lr Lm* 

−*k*2*eiq* + *giq* sgn(*eiq*) (3)

*e*˙*ψ<sup>d</sup>* = −*αreψ<sup>d</sup>* − *wreψ<sup>q</sup>* + *gψ<sup>d</sup>* sgn(*eid*) (4)

*Rr*

(*Vds* − *Rsids*)

*Vqs* − *Rsiqs*

(*Vds* − *Rsids*)

*Vqs* − *Rsiqs*

is rotor-time constant.

*m LsLr*

Therefore, the corresponding singularly perturbed model of eqn.(1) is:

*ids* = −*Lmαrids* + *αrψdr* + *wrψqr* +

*iqs* = −*Lmαriqs* − *wrψdr* + *αrψqr* +

*<sup>ψ</sup>*˙ *qr* <sup>=</sup> *Lmαriqs* <sup>+</sup> *wrψdr* <sup>−</sup> *<sup>α</sup>rψqr*

*Lm*, is mutual inductance; *<sup>σ</sup>* <sup>=</sup> <sup>1</sup> <sup>−</sup> *<sup>L</sup>*<sup>2</sup>

212 Induction Motors - Applications, Control and Fault Diagnostics

*ε*˙

*ε*˙

*ψ*˙

*ε*˙

*ε*˙

˙ *ψ*ˆ

˙

and sgn() is the sign function.

and *<sup>α</sup><sup>r</sup>* <sup>=</sup> <sup>1</sup>

*Tr* .

sliding-mode-observer can be constructed as follows:

<sup>ˆ</sup>*ids* <sup>=</sup> <sup>−</sup>*Lmαrids* <sup>+</sup> *<sup>α</sup>rψ*<sup>ˆ</sup>

<sup>ˆ</sup>*iqs* <sup>=</sup> <sup>−</sup>*Lmαriqs* <sup>−</sup> *wrψ*<sup>ˆ</sup>

*dr* <sup>=</sup> *Lmαrids* <sup>−</sup> *<sup>α</sup>rψ*<sup>ˆ</sup>

*ψ*ˆ*qr* = *Lmαriqs* + *wrψ*ˆ

Subtracting eqn.(2) from eqn.(3), the estimation error dynamics are:

*ε*˙

*ε*˙

−*k*1*eid* + *gid* sgn(*eid*)

where *<sup>ε</sup>* <sup>=</sup> *<sup>σ</sup>LsLr*

*Lm*

The previous equations can be expressed in matrix form as:

$$\begin{aligned} e\dot{e}\_i &= +Ae\_\psi + K\_i e\_i + G\_i \mathbf{Y}\_\varepsilon \\ \dot{e}\_\psi &= -Ae\_\psi + G\_\psi \mathbf{Y}\_\varepsilon \end{aligned} \tag{5}$$

where *A* = *αrI*<sup>2</sup> − *wrJ*2, *ei* = [*eid eiq* ] *<sup>T</sup>*, *e<sup>ψ</sup>* = [*eψ<sup>d</sup> eψ<sup>q</sup>* ] *<sup>T</sup>*, Υ*<sup>e</sup>* = [sgn(*eid*) sgn(*eiq*)]*T*, *Gi* = *gid* <sup>0</sup> 0 *giq* , *G<sup>ψ</sup>* = *gψ<sup>d</sup>* 0 0 *gψ<sup>q</sup> I*<sup>2</sup> = 1 0 0 1 , *J*<sup>2</sup> = 0 −1 1 0 , *Ki* = −*k*<sup>1</sup> 0 0 −*k*<sup>2</sup> 

Following the two-time-scale approach, the stability analysis of the above system can be considered determining the observer gains *Gi* and *Ki* of the fast subsystem or measured state variables (*ids*, *iqs*), to ensure the attractiveness of the sliding surface *ei* = 0 in the fast time scale. Thereafter, the observer gain *G<sup>ψ</sup>* of the slow subsystem or inaccessible state variables (*ψdr*, *ψqr*), are determined, such that the reduced-order system obtained when *ei* ∼= *e*˙*<sup>i</sup>* ∼= 0 is locally stable [24].

From singular perturbation theory, the fast reduced-order system of the observation errors can be obtained by introducing the new time variable *τ* = (*t* − *t*0)/*ε* and thereafter setting *ε* → 0 [24]. In the new time scale *τ*, taking into account that *dτ* = *dt*/*ε*, eqn.(5) becomes:

$$\begin{cases} \frac{d}{d\tau}e\_i = Ae\_\psi + K\_i e\_i + G\_i Y\_\varepsilon\\ \frac{d}{d\tau}e\_\psi = 0 \end{cases} \tag{6}$$

Therefore, if the observer gains *Gi* and *Ki* are adequately chosen, the sliding mode occurs in eqn.(6) along the manifold *ei* = [*eid eiq* ] *<sup>T</sup>* = 0.

The attractivity condition of the sliding surface *ei* = 0 given by:

$$e\_i^T \frac{de\_i}{d\tau} < 0 \tag{7}$$

is fulfilled with the following inequalities,

$$\left| g\_{i\_d} < - \left| a\_r e\_{\Psi\_d} + w\_r e\_{\Psi\_q} \right| - k\_1 \left| e\_{id} \right| \right. \tag{8}$$

$$\left| g\_{i\_{\theta}} < - \left| -w\_{r} e\_{\Psi\_{d}} + \alpha\_{r} e\_{\Psi\_{q}} \right| - k\_{2} \left| e\_{iq} \right| \right. \tag{9}$$

### **Proof:**

Let us define the following Lyapunov function candidate,

$$V = \frac{1}{2}e\_i^T e\_i$$

whose time derivative is,

$$\begin{split} \frac{dV}{d\tau} &= e\_i^T \frac{de\_i}{d\tau} \\ &= e\_i^T \left[ A e\_\Psi + K\_i e\_i + G\_i \mathbf{Y}\_\epsilon \right] \\ &= \begin{bmatrix} e\_{id} \left\{ g\_{i\_d} \text{sgn}(e\_{id}) + a\_r e\_{\Psi\_d} + \omega e\_{\Psi\_q} - k\_1 e\_{id} \right\} \\ e\_{iq} \left\{ g\_{i\_q} \text{sgn}(e\_{iq}) - \omega e\_{\Psi\_d} + a\_r e\_{\Psi\_q} - k\_2 e\_{iq} \right\} \end{bmatrix} \end{split} \tag{10}$$

Taking into account that all states and parameters of induction motor are bounded, then there exist sufficiently large negative numbers *gid* , *giq* , and positive numbers *k*<sup>1</sup> and *k*<sup>2</sup> so that the inequalities defined in eqn.(9) are verified and then the attractivity condition defined in eqn.(7) is fulfilled.

Then, once the currents trajectory reaches the sliding surface *ei* = 0, the observer error dynamics given by eqn.(6) behaves, in the sliding mode, as a reduced-order subsystem governed only by the rotor-flux error *eψ*, assuming that *ei* = *e*˙*<sup>i</sup>* = 0.

The slow error dynamics (when *ei* = 0 and *e*˙*<sup>i</sup>* = 0), can be obtained setting *ε* = 0 in the system equation presented in eqn.(5):

$$0 = \, +Ae\_{\psi} + G\_{l} \mathbf{Y}\_{\varepsilon}$$

$$\dot{e}\_{\psi} = \, -Ae\_{\psi} + G\_{\psi} \mathbf{Y}\_{\varepsilon} \tag{11}$$

In order to demonstrate de stability of the previous system, the following Lyapunov function candidate is proposed:

$$V = \frac{1}{2} e\_{\psi}^{T} e\_{\Psi} \tag{12}$$

The time derivative of the Lyapunov function candidate is:

$$\frac{dV}{dt} = \dot{e}\_{\psi}^{T} e\_{\psi} \tag{13}$$

From eqn.(11) it is deduced that:

**Proof:**

whose time derivative is,

eqn.(7) is fulfilled.

candidate is proposed:

system equation presented in eqn.(5):

Let us define the following Lyapunov function candidate,

*dV <sup>d</sup><sup>τ</sup>* <sup>=</sup> *<sup>e</sup><sup>T</sup> i dei dτ*

214 Induction Motors - Applications, Control and Fault Diagnostics

= *e<sup>T</sup>*

*eiq* 

governed only by the rotor-flux error *eψ*, assuming that *ei* = *e*˙*<sup>i</sup>* = 0.

The time derivative of the Lyapunov function candidate is:

= *eid*  *<sup>V</sup>* <sup>=</sup> <sup>1</sup> 2 *eT i ei*

*<sup>i</sup>* [*Ae<sup>ψ</sup>* + *Kiei* + *Gi*Υ*e*] (10)

 

*e*˙*<sup>ψ</sup>* = −*Ae<sup>ψ</sup>* + *Gψ*Υ*<sup>e</sup>* (11)

*<sup>ψ</sup>e<sup>ψ</sup>* (12)

*<sup>ψ</sup>e<sup>ψ</sup>* (13)

*gid* sgn(*eid*) + *αreψ<sup>d</sup>* + *weψ<sup>q</sup>* − *k*1*eid*

*giq* sgn(*eiq*) − *weψ<sup>d</sup>* + *αreψ<sup>q</sup>* − *k*2*eiq*

Taking into account that all states and parameters of induction motor are bounded, then there exist sufficiently large negative numbers *gid* , *giq* , and positive numbers *k*<sup>1</sup> and *k*<sup>2</sup> so that the inequalities defined in eqn.(9) are verified and then the attractivity condition defined in

Then, once the currents trajectory reaches the sliding surface *ei* = 0, the observer error dynamics given by eqn.(6) behaves, in the sliding mode, as a reduced-order subsystem

The slow error dynamics (when *ei* = 0 and *e*˙*<sup>i</sup>* = 0), can be obtained setting *ε* = 0 in the

0 = +*Ae<sup>ψ</sup>* + *Gi*Υ*<sup>e</sup>*

In order to demonstrate de stability of the previous system, the following Lyapunov function

*<sup>V</sup>* <sup>=</sup> <sup>1</sup> 2 *eT*

*dV dt* <sup>=</sup> *<sup>e</sup>*˙ *T*

$$e\_{\Psi} = -A^{-1} G\_{\text{i}} \mathbf{Y}\_{\text{\textdegree}} \tag{14}$$

$$
\dot{\varepsilon}\_{\Psi} = (G\_{\text{i}} + G\_{\text{\#}}) \mathbf{Y}\_{\text{\#}} \tag{15}
$$

Then from eqn.(13), (14) and (15)

*dV dt* <sup>=</sup> <sup>−</sup>Υ*<sup>T</sup> <sup>e</sup>* (*Gi* + *<sup>G</sup>ψ*)*<sup>T</sup> <sup>A</sup>*−<sup>1</sup>*Gi*Υ*<sup>e</sup>* <sup>=</sup> <sup>−</sup>Υ*<sup>T</sup> <sup>e</sup>* (*Gi* + *<sup>G</sup>ψ*)*<sup>T</sup> <sup>A</sup>*−<sup>1</sup>*Gi*Υ*<sup>e</sup>* <sup>=</sup> <sup>−</sup>Υ*<sup>T</sup> e* (*I*<sup>2</sup> + *GψG*−<sup>1</sup> *<sup>i</sup>* )*Gi T A*−<sup>1</sup>*Gi*Υ*<sup>e</sup>* <sup>=</sup> <sup>−</sup>Υ*<sup>T</sup> <sup>e</sup> <sup>G</sup><sup>T</sup> <sup>i</sup>* (*I*<sup>2</sup> <sup>+</sup> *<sup>G</sup>ψG*−<sup>1</sup> *<sup>i</sup>* )*<sup>T</sup> <sup>A</sup>*−<sup>1</sup>*Gi*Υ*<sup>e</sup>* <sup>=</sup> <sup>−</sup>Υ*<sup>T</sup> <sup>e</sup> <sup>G</sup><sup>T</sup> <sup>i</sup>* (*A*−1)*<sup>T</sup> <sup>A</sup>T*(*I*<sup>2</sup> <sup>+</sup> *<sup>G</sup>ψG*−<sup>1</sup> *<sup>i</sup>* )*<sup>T</sup> <sup>A</sup>*−<sup>1</sup>*Gi*Υ*<sup>e</sup>* <sup>=</sup> <sup>−</sup>(*A*−<sup>1</sup>*Gi*Υ*e*)*<sup>T</sup> <sup>A</sup>T*(*I*<sup>2</sup> <sup>+</sup> *<sup>G</sup>ψG*−<sup>1</sup> *<sup>i</sup>* )*<sup>T</sup> <sup>A</sup>*−<sup>1</sup>*Gi*Υ*<sup>e</sup>* <sup>=</sup> <sup>−</sup>*e<sup>T</sup> <sup>ψ</sup>AT*(*I*<sup>2</sup> <sup>+</sup> *<sup>G</sup>ψG*−<sup>1</sup> *<sup>i</sup>* )*Te<sup>ψ</sup>* <sup>=</sup> <sup>−</sup>*e<sup>T</sup> <sup>ψ</sup>*(*I*<sup>2</sup> <sup>+</sup> *<sup>G</sup>ψG*−<sup>1</sup> *<sup>i</sup>* )*A e<sup>ψ</sup>* (16)

To ensure that *V*˙ is negative definite the following sufficient condition can be requested:

$$(I\_2 + G\_\Psi G\_i^{-1})A \ge \varrho I\_2 \tag{17}$$

where is a positive constant

Solving the gain matrix *G<sup>ψ</sup>* in eqn.(17) yields:

$$\varrho\left(I\_2 + G\_{\Psi}G\_{\mathfrak{j}}^{-1}\right) \ge \varrho I\_2 A^{-1} \tag{18}$$

$$(I\_2 + G\_\Psi G\_i^{-1}) \ge \varrho A^{-1} \tag{19}$$

$$\text{G}\_{\text{\#}}\text{G}\_{i}^{-1} \ge \varrho A^{-1} - I\_{2} \tag{20}$$

$$\mathcal{G}\_{\psi} \le (\varrho A^{-1} - I\_2)\mathcal{G}\_i \tag{21}$$

Therefore, the time derivative of the Lyapunov function will be negative definite if the observer gain *G<sup>ψ</sup>* is chosen taking into account eqn.(21). As a result from eqn.(16) it is concluded that the equilibrium point (*e<sup>ψ</sup>* = 0) of the flux observer error dynamic given by eqn.(11) is exponentially stable; that is, the flux observer error converges to zero with exponential rate of convergence.

### **3. Load Torque Observer**

In the traditional sliding mode control schemes, the load torque should be known or should be measured using a torque sensors in order to compensate this load torque. On the other hand, the load torque could be also considered as a system uncertainty, but in this case the control system should be robust under all load torque values that would appear over the time and therefore the sliding gain should be adequately high in order to compensate the these load torque values. Obviously these high values for the sliding gain will increase the control activity and are undesirable in the real applications.

Therefore, when the load torque is unknown or is very variable over time, and the system has no torque sensors, a good solution could be the use of a load torque estimator. In this chapter a sliding mode observer is proposed in order to obtain the load torque applied to the induction motor without requiring the use of the load torque sensor.

The mechanical equation of an induction motor can be written as:

$$J\ddot{\theta}\_m + B\dot{\theta}\_m + T\_L = T\_\varepsilon \tag{22}$$

where *J* and *B* are the inertia constant and the viscous friction coefficient of the induction motor respectively; *TL* is the external load; *θ<sup>m</sup>* is the rotor mechanical position, which is related to the rotor electrical position, *θr*, by *θ<sup>m</sup>* = 2 *θr*/*p* where *p* is the pole numbers and *Te* denotes the generated torque of an induction motor, defined as [23]:

$$T\_{\ell} = \frac{3p}{4} \frac{L\_m}{L\_r} (\psi\_{dr}^{\ell} i\_{qs}^{\ell} - \psi\_{qr}^{\ell} i\_{ds}^{\ell}) \tag{23}$$

where *ψ<sup>e</sup> dr* and *<sup>ψ</sup><sup>e</sup> qr* are the rotor-flux linkages, with the subscript 'e' denoting that the quantity is referred to the synchronously rotating reference frame; *i e ds* and *i e qs* are the d-q stator current components, and *p* is the pole numbers.

The relation between the synchronously rotating reference frame and the stationary reference frame is performed by the so-called reverse Park's transformation:

$$
\begin{bmatrix} \mathbf{x}\_{\ell} \\ \mathbf{x}\_{b} \\ \mathbf{x}\_{\ell} \end{bmatrix} = \begin{bmatrix} \cos(\theta\_{\ell}) & -\sin(\theta\_{\ell}) \\ \cos(\theta\_{\ell} - 2\pi/3) & -\sin(\theta\_{\ell} - 2\pi/3) \\ \cos(\theta\_{\ell} + 2\pi/3) & -\sin(\theta\_{\ell} + 2\pi/3) \end{bmatrix} \begin{bmatrix} \mathbf{x}\_{d}^{\varepsilon} \\ \mathbf{x}\_{q}^{\varepsilon} \end{bmatrix} \tag{24}$$

where *θ<sup>e</sup>* is the angular position between the d-axis of the synchronously rotating reference frame and the a-axis of the stationary reference frame, and it is assumed that the quantities are balanced.

Using the field-orientation control principle, the current component *i e ds* is aligned in the direction of the rotor flux vector *ψ*¯*r*, and the current component *i e qs* is aligned in the perpendicular direction to it. At this condition, it is satisfied that:

$$
\psi\_{qr}^{\varepsilon} = 0, \qquad \psi\_{dr}^{\varepsilon} = |\bar{\psi}\_r| \tag{25}
$$

Taking into account the results presented in eqn.(25), the induction motor torque of eqn.(23) is simplified to:

$$T\_{\ell} = \frac{3p}{4} \frac{L\_m}{L\_r} \psi\_{dr}^{\ell} i\_{qs}^{\ell} = K\_T i\_{qs}^{\ell} \tag{26}$$

where *KT* is the torque constant, defined as follows:

$$K\_T = \frac{3p}{4} \frac{L\_m}{L\_r} \psi\_{dr}^{e^\*} \tag{27}$$

where *ψe*<sup>∗</sup> *dr* denotes the command rotor flux.

With the above mentioned proper field orientation, the rotor flux dynamics is given by [23]:

$$\frac{d\psi\_{dr}^{\varepsilon}}{dt} + \frac{\psi\_{dr}^{\varepsilon}}{T\_r} = \frac{L\_m}{T\_r} \dot{i}\_{ds}^{\varepsilon} \tag{28}$$

From the system mechanical equation eqn.(22) and the induction motor torque equation eqn.(26), the following dynamic equation is obtained:

$$
\dot{w}\_m = -\frac{B}{J} w\_m + \frac{K\_T}{J} \, \mathbf{i}\_{qs}^e - \frac{1}{J} T\_L \tag{29}
$$

where *wm* = ˙ *θm*

**3. Load Torque Observer**

216 Induction Motors - Applications, Control and Fault Diagnostics

where *ψ<sup>e</sup>*

are balanced.

*dr* and *<sup>ψ</sup><sup>e</sup>*

components, and *p* is the pole numbers.

 *xa xb xc*  =

control activity and are undesirable in the real applications.

induction motor without requiring the use of the load torque sensor. The mechanical equation of an induction motor can be written as:

> *J* ¨ *θ<sup>m</sup>* + *B* ˙

*Te* denotes the generated torque of an induction motor, defined as [23]:

*Te* <sup>=</sup> <sup>3</sup>*<sup>p</sup>* 4 *Lm Lr* (*ψ<sup>e</sup> dri e qs* <sup>−</sup> *<sup>ψ</sup><sup>e</sup> qri e*

is referred to the synchronously rotating reference frame; *i*

frame is performed by the so-called reverse Park's transformation:

Using the field-orientation control principle, the current component *i*

direction of the rotor flux vector *ψ*¯*r*, and the current component *i*

*ψe*

*qr* = 0, *<sup>ψ</sup><sup>e</sup>*

perpendicular direction to it. At this condition, it is satisfied that:

 

In the traditional sliding mode control schemes, the load torque should be known or should be measured using a torque sensors in order to compensate this load torque. On the other hand, the load torque could be also considered as a system uncertainty, but in this case the control system should be robust under all load torque values that would appear over the time and therefore the sliding gain should be adequately high in order to compensate the these load torque values. Obviously these high values for the sliding gain will increase the

Therefore, when the load torque is unknown or is very variable over time, and the system has no torque sensors, a good solution could be the use of a load torque estimator. In this chapter a sliding mode observer is proposed in order to obtain the load torque applied to the

where *J* and *B* are the inertia constant and the viscous friction coefficient of the induction motor respectively; *TL* is the external load; *θ<sup>m</sup>* is the rotor mechanical position, which is related to the rotor electrical position, *θr*, by *θ<sup>m</sup>* = 2 *θr*/*p* where *p* is the pole numbers and

The relation between the synchronously rotating reference frame and the stationary reference

cos(*θe*) − sin(*θe*) cos(*θ<sup>e</sup>* − 2*π*/3) − sin(*θ<sup>e</sup>* − 2*π*/3) cos(*θ<sup>e</sup>* + 2*π*/3) − sin(*θ<sup>e</sup>* + 2*π*/3)

where *θ<sup>e</sup>* is the angular position between the d-axis of the synchronously rotating reference frame and the a-axis of the stationary reference frame, and it is assumed that the quantities

*qr* are the rotor-flux linkages, with the subscript 'e' denoting that the quantity

*e ds* and *i e*

> *xe d xe q*

> > *e*

*e*

*dr* <sup>=</sup> <sup>|</sup>*ψ*¯*r*<sup>|</sup> (25)

*ds* is aligned in the

*qs* is aligned in the

*θ<sup>m</sup>* + *TL* = *Te* (22)

*ds*) (23)

*qs* are the d-q stator current

(24)

It is assumed that the load torque only changes at certain moments, and therefore the load torque can be considered as a quasi-constant signal:

$$
\dot{T}\_L = 0 \tag{30}
$$

Therefore, the system state space equations are:

$$\begin{aligned} \dot{w}\_m &= -\frac{B}{J} w\_m + \frac{K\_T}{J} i\_{q\text{s}}^e - \frac{1}{J} T\_L \\ \dot{T}\_L &= 0 \end{aligned} \tag{31}$$

Taking into account that the load torque *TL* is taken as a quasi-constant signal, the load torque can be considered the slow component of this system. Therefore, from singular perturbation theory [24], the stability of the above system can be demonstrated assuring the asymptotic stability of the fast component of this system (the rotor speed), and thereafter the convergence of the slow component (the load torque) for the reduced system, when the rotor speed estimation error is zero.

Then, from eqn.(31) the sliding-mode-observer can be constructed as:

$$
\begin{split}
\dot{w}\_m &= \frac{-B}{J}w\_m + \frac{K\_T}{J}\dot{\epsilon}\_{qs}^e - \frac{1}{J}\hat{T}\_L + k\_{w\_1}e\_w + h\_1 \operatorname{sgn}(e\_w) \\
\dot{\hat{T}}\_L &= -k\_{w\_2}e\_w - h\_2 \operatorname{sgn}(e\_w)
\end{split}
\tag{32}
$$

where *ew* = *wm* − *w*ˆ *<sup>m</sup>*, and *kw*<sup>1</sup> , *kw*<sup>2</sup> , *h*<sup>1</sup> and *h*<sup>2</sup> are a positive constants. Subtracting eqn. (32) from (31), the estimation error dynamic is obtained:

$$
\dot{e}\_{\overline{w}} = -\frac{1}{f} e\_T - k\_{w\_1} e\_w - h\_1 \operatorname{sgn}(e\_{\overline{w}})
$$

$$
\dot{e}\_T = k\_{\overline{w}\_2} e\_w + h\_2 \operatorname{sgn}(e\_{\overline{w}}) \tag{33}
$$

where *eT* <sup>=</sup> *TL* <sup>−</sup> *<sup>T</sup>*<sup>ˆ</sup> *L*

In order to demonstrate the stability of the fast component of the system the following Lyapunov function candidate is proposed:

$$V = \frac{1}{2}e\_w^2\tag{34}$$

The time derivative of this Lyapunov function candidate is:

$$
\dot{V} = \epsilon\_w \dot{\epsilon}\_w \tag{35}
$$

$$\epsilon = e\_w \left( -\frac{1}{J} e\_T - k\_{w\_1} e\_w - h\_1 \operatorname{sgn}(e\_w) \right) \tag{36}$$

$$=-h\_1|e\_{\overline{w}}| - k\_{\overline{w}\_1}e\_{\overline{w}}^2 - \frac{1}{J}e\_{\overline{w}}e\_T\tag{37}$$

To ensure that *V*˙ is negative definite the following sufficient condition can be requested:

$$\mathcal{M}\_1 \ge \left| \frac{1}{J} e\_T \right| - k\_{w\_1} |e\_w| + \eta\_w \tag{38}$$

where *η<sup>w</sup> >* 0

Therefore,

$$
\dot{V} \le -\eta\_{\text{\textquotedblleft}} |e\_{\text{w}}| \tag{39}
$$

From eqn.(39) it is deduced that the equilibrium point *ew* = 0 is asymptotically stable, and from this equation it can be also deduced that the maximum time in order to reach the equilibrium point *ew* = 0 is:

$$t\_{reach} \le \frac{e\_w(t=0)}{\eta\_w} \tag{40}$$

When the speed observation error reaches the equilibrium point, *ew* = 0 and *e*˙*<sup>w</sup>* = 0, and then from eqn.(33) it is obtained that the observer error dynamics behaves as the reduced-order subsystem presented below:

$$0 = -\frac{1}{f} e\_L - h\_1 \operatorname{sgn}(e\_w) \tag{41}$$

$$
\dot{e}\_T = h\_2 \operatorname{sgn}(e\_w) \tag{42}
$$

From the previous equations it is obtained:

Then, from eqn.(31) the sliding-mode-observer can be constructed as:

where *ew* = *wm* − *w*ˆ *<sup>m</sup>*, and *kw*<sup>1</sup> , *kw*<sup>2</sup> , *h*<sup>1</sup> and *h*<sup>2</sup> are a positive constants. Subtracting eqn. (32) from (31), the estimation error dynamic is obtained:

> *<sup>e</sup>*˙*<sup>w</sup>* <sup>=</sup> <sup>−</sup><sup>1</sup> *J*

The time derivative of this Lyapunov function candidate is:

= *ew* −1 *J*

= −*h*1|*ew*| − *kw*<sup>1</sup> *e*

*h*<sup>1</sup> ≥ 1 *J eT* 

*KT J i e qs* <sup>−</sup> <sup>1</sup> *J T*ˆ

*<sup>L</sup>* + *kw*<sup>1</sup> *ew* + *h*<sup>1</sup> sgn(*ew*)

*e*˙*<sup>T</sup>* = *kw*<sup>2</sup> *ew* + *h*<sup>2</sup> sgn(*ew*) (33)

*V*˙ = *ewe*˙*<sup>w</sup>* (35)

*eT* − *kw*<sup>1</sup> *ew* − *h*<sup>1</sup> sgn(*ew*)

2 *<sup>w</sup>* <sup>−</sup> <sup>1</sup> *J*

To ensure that *V*˙ is negative definite the following sufficient condition can be requested:

*<sup>w</sup>* (34)

*eweT* (37)

<sup>−</sup> *kw*<sup>1</sup> <sup>|</sup>*ew*<sup>|</sup> <sup>+</sup> *<sup>η</sup><sup>w</sup>* (38)

*<sup>V</sup>*˙ ≤ −*ηw*|*ew*<sup>|</sup> (39)

(36)

*<sup>L</sup>* = −*kw*<sup>2</sup> *ew* − *h*<sup>2</sup> sgn(*ew*) (32)

*eT* − *kw*<sup>1</sup> *ew* − *h*<sup>1</sup> sgn(*ew*)

In order to demonstrate the stability of the fast component of the system the following

*<sup>V</sup>* <sup>=</sup> <sup>1</sup> 2 *e* 2

*<sup>J</sup> wm* <sup>+</sup>

˙*w*<sup>ˆ</sup> *<sup>m</sup>* <sup>=</sup> <sup>−</sup>*<sup>B</sup>*

˙ *T*ˆ

218 Induction Motors - Applications, Control and Fault Diagnostics

*L*

Lyapunov function candidate is proposed:

where *eT* <sup>=</sup> *TL* <sup>−</sup> *<sup>T</sup>*<sup>ˆ</sup>

where *η<sup>w</sup> >* 0 Therefore,

$$
\dot{e}\_T = -\frac{1}{J} \frac{h\_2}{h\_1} e\_T \tag{43}
$$

The solution of the previous differential equation is:

$$e\_T(t) = \mathbb{C} \exp\left(-\frac{1}{J}\frac{h\_2}{h\_1}t\right) \tag{44}$$

Consequently, the load torque estimation error tends exponentially to zero.

Therefore, if the observer gains *h*1, *h*<sup>2</sup> *kw*<sup>1</sup> and *kw*<sup>2</sup> are adequately chosen, then the estimation error converges to zero. Consequently the estimated states *w*ˆ *<sup>m</sup>*, *T*ˆ *<sup>L</sup>* converges to the real states *wm*, *TL* as *t* tends to infinity. Hence, the load torque may be obtained from the states observer given by eqn.(32), that uses the rotor speed and the stator current in order to obtain the load torque applied to the induction motor.

### **4. Variable structure robust position control**

The mechanical equation of an induction motor presented in equation (29) can be rewritten as:

$$
\ddot{\theta}\_m + a \,\dot{\theta}\_m + \overline{f} = b \,\text{i}^e\_{q\text{s}} \tag{45}
$$

where the parameters are defined as:

$$a = \frac{B}{I}, \quad b = \frac{K\_T}{I}, \quad \vec{f} = \frac{T\_L}{I}; \tag{46}$$

Now, the previous mechanical equation (45) is considered with uncertainties as follows:

$$\ddot{\theta}\_{\rm m} = -(a + \triangle a)\dot{\theta}\_{\rm m} - (f + \triangle f) + (b + \triangle b)\dot{\imath}\_{\rm qs}^{\varepsilon} \tag{47}$$

where *<sup>f</sup>* <sup>=</sup> *<sup>T</sup>*<sup>ˆ</sup> *L <sup>J</sup>* , and the terms *<sup>a</sup>*, *<sup>b</sup>* and *<sup>f</sup>* represents the uncertainties of the terms *<sup>a</sup>*, *b* and *f* respectively.

It should be noted that the load torque *TL* has been replaced by the estimated load torque *T*ˆ *L* and the difference between the real and the estimated value is taken as an uncertainty.

Let us define the position tracking error as follows:

$$e(t) = \theta\_m(t) - \theta\_m^\*(t)\tag{48}$$

where *θ*∗ *<sup>m</sup>* is the rotor position command.

Taking the second derivative of the previous equation with respect to time yields:

$$
\ddot{\varepsilon}(t) = \ddot{\theta}\_m - \ddot{\theta}\_m^\* = \mathfrak{u}(t) + d(t) \tag{49}
$$

where the following terms have been collected in the signal *u*(*t*),

$$u(t) = b\,\dot{t}\_{q\bar{s}}^{\varepsilon}(t) - a\,\dot{\theta}\_m(t) - f(t) - \ddot{\theta}\_m^\*(t) \tag{50}$$

and the uncertainty terms have been collected in the signal *d*(*t*),

$$d(t) = -\triangle a \, w\_m(t) - \triangle f(t) + \triangle b \, \dot{r}\_{qs}^\varepsilon(t) \tag{51}$$

Now, the sliding variable *S*(*t*) is defined as:

$$S(t) = \dot{e}(t) + k\,e(t) + k\_i \int e(t) \,dt\tag{52}$$

where *k* and *ki* are a positive constant gains. Then, the sliding surface is defined as:

$$S(t) = \dot{e}(t) + k\,e(t) + k\_l \int e(t) \,dt = 0\tag{53}$$

The sliding mode controller is designed as:

Now, the previous mechanical equation (45) is considered with uncertainties as follows:

It should be noted that the load torque *TL* has been replaced by the estimated load torque *T*ˆ

and the difference between the real and the estimated value is taken as an uncertainty.

*e*(*t*) = *θm*(*t*) − *θ*<sup>∗</sup>

Taking the second derivative of the previous equation with respect to time yields:

*<sup>θ</sup><sup>m</sup>* <sup>−</sup> ¨ *θ*∗

*qs*(*t*) <sup>−</sup> *<sup>a</sup>* ˙

*<sup>d</sup>*(*t*) = <sup>−</sup>*a wm*(*t*) <sup>−</sup> *<sup>f</sup>*(*t*) + *b i<sup>e</sup>*

*S*(*t*) = *e*˙(*t*) + *k e*(*t*) + *ki*

*S*(*t*) = *e*˙(*t*) + *k e*(*t*) + *ki*

*<sup>θ</sup>m*(*t*) <sup>−</sup> *<sup>f</sup>*(*t*) <sup>−</sup> ¨

*θ*∗

*e*¨(*t*) = ¨

where the following terms have been collected in the signal *u*(*t*),

*u*(*t*) = *b i<sup>e</sup>*

and the uncertainty terms have been collected in the signal *d*(*t*),

*θ<sup>m</sup>* − (*f* + *f*)+(*b* + *b*)*i*

*<sup>J</sup>* , and the terms *<sup>a</sup>*, *<sup>b</sup>* and *<sup>f</sup>* represents the uncertainties of the terms *<sup>a</sup>*,

*e*

*<sup>m</sup>*(*t*) (48)

*<sup>m</sup>* = *u*(*t*) + *d*(*t*) (49)

*<sup>m</sup>*(*t*) (50)

*qs*(*t*) (51)

*e*(*t*) *dt* (52)

*e*(*t*) *dt* = 0 (53)

*qs* (47)

*L*

¨

220 Induction Motors - Applications, Control and Fault Diagnostics

Let us define the position tracking error as follows:

*<sup>m</sup>* is the rotor position command.

Now, the sliding variable *S*(*t*) is defined as:

where *k* and *ki* are a positive constant gains.

Then, the sliding surface is defined as:

where *<sup>f</sup>* <sup>=</sup> *<sup>T</sup>*<sup>ˆ</sup>

where *θ*∗

*b* and *f* respectively.

*L*

*<sup>θ</sup><sup>m</sup>* <sup>=</sup> <sup>−</sup>(*<sup>a</sup>* <sup>+</sup> *<sup>a</sup>*) ˙

$$
\mu(t) = -k\,\dot{e} - k\_\text{i}\,e - \beta \,\text{sgn}(S) \tag{54}
$$

where *k* and *ki* are the previously defined positive constant gains, *β* is the switching gain, *S* is the sliding variable defined in eqn. (52) and sgn(·) is the sign function.

**Assumption.** In order to obtain the position trajectory tracking, the gain *β* must be chosen so that *<sup>β</sup>* <sup>≥</sup> ¯*<sup>d</sup>* where ¯*<sup>d</sup>* <sup>≥</sup> sup*t*∈*R*0<sup>+</sup> <sup>|</sup>*d*(*t*)|. Note that this condition only implies that the system uncertainties are bounded magnitudes.

**Theorem.** Consider the induction motor given by equation (47), the control law (54) leads the rotor mechanical position *θm*(*t*) so that the position tracking error *e*(*t*) = *θm*(*t*) − *θ*<sup>∗</sup> *<sup>m</sup>*(*t*) tends to zero as the time tends to infinity.

**Proof:** Define the Lyapunov function candidate:

$$V(t) = \frac{1}{2} S(t) S(t) \tag{55}$$

Its time derivative is calculated as:

$$\begin{aligned} \dot{V}(t) &= \mathcal{S}(t)\dot{\mathcal{S}}(t) \\ &= \mathcal{S} \cdot [\ddot{\imath} + k\dot{\imath} + k\_{\dot{\imath}}e] \\ &= \mathcal{S} \cdot [\imath + d + k\dot{\imath} + k\_{\dot{\imath}}e] \\ &= \mathcal{S} \cdot [-k\dot{\imath} - k\_{\dot{\imath}}e - \beta \text{sgn}(\mathcal{S}) + d + k\dot{\imath} + k\_{\dot{\imath}}e] \\ &= \mathcal{S} \cdot [d - \beta \text{sgn}(\mathcal{S})] \\ &\leq -(\beta - |d|)|\mathcal{S}| \\ &\leq 0 \end{aligned} \tag{56}$$

It should be noted that the eqns. (52), (49) and (54) have been used in the proof.

Using the Lyapunov's direct method, since *V*(*t*) is clearly positive-definite, *V*˙ (*t*) is negative definite and *V*(*t*) tends to infinity as *S*(*t*) tends to infinity, then the equilibrium at the origin *S*(*t*) = 0 is globally asymptotically stable. Therefore *S*(*t*) tends to zero as the time *t* tends to infinity. Moreover, all trajectories starting off the sliding surface *S* = 0 must reach it in finite time and then they will remain on this surface. This system's behavior, once on the sliding surface is usually called *sliding mode*.

When the sliding mode occurs on the sliding surface (53), then *S*(*t*) = *S*˙(*t*) = 0, and therefore the dynamic behavior of the tracking problem (49) is equivalently governed by the following equation:

$$
\dot{S}(t) = 0 \quad \Rightarrow \quad \ddot{e}(t) + k\dot{e}(t) + k\_i e(t) = 0 \tag{57}
$$

Then, like *k* and *ki* are a positive constants, the tracking error *e*(*t*) and its derivatives *e*˙(*t*) and *e*¨(*t*) converges to zero exponentially.

It should be noted that, a typical motion under sliding mode control consists of a *reaching phase* during which trajectories starting off the sliding surface *S* = 0 move toward it and reach it in finite time, followed by *sliding phase* during which the motion will be confined to this surface and the system tracking error will be represented by the reduced-order model (57), where the tracking error tends to zero.

Finally, the torque current command, *i e*∗ *qs*(*t*), can be obtained directly substituting eqn. (54) in eqn. (50):

$$\dot{q}\_{qs}^{\varepsilon\*}(t) = \frac{1}{b} \left[ -k\,\dot{e} - k\_i \, e - \beta \,\text{sgn}(S) + a\,\dot{\theta}\_m + \ddot{\theta}\_m^\* + f(t) \right] \tag{58}$$

It should be noted that the current command is a bounded signal because all its components are bounded.

Therefore, the proposed variable structure position control resolves the position tracking problem for the induction motor in presence of some uncertainties in mechanical parameters and load torque.

It should be pointed out that, as it is well known, the variable structure control signals may produce the so-called chattering phenomenon, caused by the discontinuity that appear in eqn.(58) across the sliding surface. Chattering is undesirable in practice, since it involves high control activity and further may excite high-frequency dynamics. However, in the induction motor system, this high frequency changes in the electromagnetic torque will be filtered by the mechanical system inertia. Nevertheless, in order to reduce the chattering effect, the control law can also be smoothed out. In this case a simple and easy solution (proposed in [25]) could be to replace the sign function by a tansigmoid function in order to avoid the discontinuity in the control signal.

### **5. Simulation and Experimental Results**

In this section the position regulation performance of the proposed sliding-mode field oriented control versus reference and load torque variations is analyzed by means of different simulation examples and real test using a commercial induction motor.

The block diagram of the proposed robust position control scheme is presented in Figure 1, and the function of the blocks that appear in this figure are explained below:

The block 'SMC Controller' represents the proposed sliding-mode controller, and it is implemented by equations (52) and (58). The block 'limiter' limits the current applied to the motor windings so that it remains within the limit value, being implemented by a saturation function. The block '*dq<sup>e</sup>* <sup>→</sup> *abc*' makes the conversion between the synchronously rotating

**Figure 1.** Block diagram of the proposed sliding-mode field oriented control

Figure 1: Block diagram of the proposed sliding-mode field oriented control It should be noted that the current command is a bounded signal because all its components are bounded. Therefore, the proposed variable structure position control resolves the position tracking problem for the induction motor in presence of some uncertainties in mechanical parameters and load torque. It should be pointed out that, as it is well known, the variable structure control signals may produce the so-called chattering phenomenon, caused by the discontinuity that appear in eqn.(58) across the sliding surface. Chattering is undesirable in practice, since it and stationary reference frames (Park's Transformation). The block 'Current Controller' consists of a SVPWM current control. The block 'SVPWM Inverter' is a six IGBT-diode bridge inverter with 540 V DC voltage source. The block 'Field Weakening' gives the flux command based on rotor speed, so that the PWM controller does not saturate. The block '*i e*∗ *ds* Calculation' provides the current reference *i e*∗ *ds* from the rotor flux reference through the equation (28). The block 'Flux Estimator' represents the proposed sliding mode flux estimator, and it is implemented by the eqn.(3). The block 'ˆ *θ<sup>e</sup>* Calculation' provides the angular position of the rotor flux vector. Finally, the block 'IM' represents the induction motor.

involves high control activity and further may excite high-frequency dynamics. However, in the induction motor system, this high frequency changes in the electromagnetic torque will be filtered by the mechanical system inertia. Nevertheless, in order to reduce the chattering effect, the control law can also be smoothed out. In this case a simple and easy In order to carry out the real experimental validation of the proposed control scheme, the control platform show in figure 2 is used . The block diagram of this experimental platform is shown in figure 3.

solution (proposed in [25]) could be to replace the sign function by a tansigmoid function in order to avoid the discontinuity in the control signal. 5 Simulation and Experimental Results In this section the position regulation performance of the proposed sliding-mode field oriented control versus reference and load torque variations is analyzed by means of different simulation examples and real test using a commercial induction motor. The block diagram of the proposed robust position control scheme is presented in Figure 1, and the function of the blocks that appear in this figure are explained below: This control platform allows to verify the real time performance of the induction motor controls in a real induction motor. The platform is formed by a PC with Windows XP in which it is installed MatLab7/Simulink R14 and ControlDesk 2.7 and the DS1103 Controller Board real time interface of dSpace. The power block is formed of a three-phase rectifier connected to 380 V/50 Hz AC electrical net and a capacitor bank of 27.200 *µF* in order to get a DC bus of 540 V. The platform also includes a three-phase IGBT/Diode bridge of 50A, and the M2AA 132M4 ABB induction motor of 7.5kW of die-cast aluminium squirrel-cage type and 1440 rpm, with the following parameters given by the manufacturer:

The block 'SMC Controller' represents the proposed sliding-mode controller, and it is

12


*<sup>S</sup>*˙(*t*) = <sup>0</sup> <sup>⇒</sup> *<sup>e</sup>*¨(*t*) + *<sup>k</sup> <sup>e</sup>*˙(*t*) + *kie*(*t*) = <sup>0</sup> (57)

*qs*(*t*), can be obtained directly substituting eqn. (54) in

(58)

*θ<sup>m</sup>* + ¨ *θ*∗ *<sup>m</sup>* + *f*(*t*)

Then, like *k* and *ki* are a positive constants, the tracking error *e*(*t*) and its derivatives *e*˙(*t*) and

It should be noted that, a typical motion under sliding mode control consists of a *reaching phase* during which trajectories starting off the sliding surface *S* = 0 move toward it and reach it in finite time, followed by *sliding phase* during which the motion will be confined to this surface and the system tracking error will be represented by the reduced-order model

<sup>−</sup>*<sup>k</sup> <sup>e</sup>*˙ <sup>−</sup> *ki <sup>e</sup>* <sup>−</sup> *<sup>β</sup>* sgn(*S*) + *<sup>a</sup>* ˙

It should be noted that the current command is a bounded signal because all its components

Therefore, the proposed variable structure position control resolves the position tracking problem for the induction motor in presence of some uncertainties in mechanical parameters

It should be pointed out that, as it is well known, the variable structure control signals may produce the so-called chattering phenomenon, caused by the discontinuity that appear in eqn.(58) across the sliding surface. Chattering is undesirable in practice, since it involves high control activity and further may excite high-frequency dynamics. However, in the induction motor system, this high frequency changes in the electromagnetic torque will be filtered by the mechanical system inertia. Nevertheless, in order to reduce the chattering effect, the control law can also be smoothed out. In this case a simple and easy solution (proposed in [25]) could be to replace the sign function by a tansigmoid function in order to avoid the

In this section the position regulation performance of the proposed sliding-mode field oriented control versus reference and load torque variations is analyzed by means of different

The block diagram of the proposed robust position control scheme is presented in Figure 1,

The block 'SMC Controller' represents the proposed sliding-mode controller, and it is implemented by equations (52) and (58). The block 'limiter' limits the current applied to the motor windings so that it remains within the limit value, being implemented by a saturation function. The block '*dq<sup>e</sup>* <sup>→</sup> *abc*' makes the conversion between the synchronously rotating

simulation examples and real test using a commercial induction motor.

and the function of the blocks that appear in this figure are explained below:

*e*∗

*e*¨(*t*) converges to zero exponentially.

(57), where the tracking error tends to zero.

222 Induction Motors - Applications, Control and Fault Diagnostics

Finally, the torque current command, *i*

*i e*∗ *qs*(*t*) = <sup>1</sup> *b* 

discontinuity in the control signal.

**5. Simulation and Experimental Results**

eqn. (50):

are bounded.

and load torque.


**Figure 2.** Induction motor experimental platform

• *αAl*, temperature coefficient of Aluminium, 0.0039*K*−<sup>1</sup>

The rotor position of this motor is measured using the G1BWGLDBI LTN incremental rotary encoder of 4096 square impulses per revolution. This pulses are quadruplicated in a decoder, giving a resolution of 16384 ppr which gives an angle resolution of 0.000385 rad (0.022 deg).

The platform also includes a 190U2 Unimotor synchronous AC servo motor of 10.6 kW connected to the induction motor to generate the load torque (controlled in torque). This servo motor is controlled by its VSI Unidrive inverter module.

The sample time used to realize the real implementation of the the position control is 100*µs*, and the processor used for the real tests is a floating point PowerPC at 1MHz, located in the real time DS1103 hardware of dSpace. This target incorporates the TMS320F240 DSP working as slave to generate the SVPWM pulses for the inverter. Finally, the position and currents control algorithms, the *θ<sup>e</sup>* angle and flux estimator, the SVPWM calculations, and the Park's transformations have been realized in C programming language in a unique S-Builder module of Simulink, in order to obtain a compact and portable code.

**Figure 3.** Block diagram of the induction motor experimental platform

**Figure 2.** Induction motor experimental platform

224 Induction Motors - Applications, Control and Fault Diagnostics

• *αAl*, temperature coefficient of Aluminium, 0.0039*K*−<sup>1</sup>

servo motor is controlled by its VSI Unidrive inverter module.

module of Simulink, in order to obtain a compact and portable code.

The rotor position of this motor is measured using the G1BWGLDBI LTN incremental rotary encoder of 4096 square impulses per revolution. This pulses are quadruplicated in a decoder, giving a resolution of 16384 ppr which gives an angle resolution of 0.000385 rad (0.022 deg). The platform also includes a 190U2 Unimotor synchronous AC servo motor of 10.6 kW connected to the induction motor to generate the load torque (controlled in torque). This

The sample time used to realize the real implementation of the the position control is 100*µs*, and the processor used for the real tests is a floating point PowerPC at 1MHz, located in the real time DS1103 hardware of dSpace. This target incorporates the TMS320F240 DSP working as slave to generate the SVPWM pulses for the inverter. Finally, the position and currents control algorithms, the *θ<sup>e</sup>* angle and flux estimator, the SVPWM calculations, and the Park's transformations have been realized in C programming language in a unique S-Builder In the experimental validation it is assumed that there is an uncertainty around 50% in the system mechanical parameters, that will be overcome by the proposed sliding mode control. The nominal value of the rotor flux is 1.01 Wb and it is obtained for a flux current command value of *i* ∗ *sd* = 8.61*A*. However, in some cases, for a very high rotor speed, the flux command should be reduced so that the PWM controller does not saturate.

On the other hand, the electromagnetic torque current command, *i* ∗ *sq*, has been limited to 30 A, in order to provide a protection against overcurrents in the induction motor's stator feed. Finally, the frequency of commutation of VSI module of the platform is limited to 8 kHz.

In this example the motor starts from a standstill state and it is required that the rotor position follows a position command, whose amplitude varies between 0 and 2*π rad*.

The system starts with an initial load torque *TL* = 0 *N*.*m*, and at time *t* = 0.1 *s*, the load torque steps from *TL* = 0 *N*.*m* to *TL* = 20 *N*.*m*, then at time *t* = 1 *s*, the load torque steps from *TL* = 20 *N*.*m* to *TL* = 40 *N*.*m* and finally at time *t* = 2 *s*, the load torque steps from *TL* = 40 *N*.*m* to *TL* = 60 *N*.*m*, which is a 20% above the nominal torque value (49 Nm).

In these examples the values for the controller parameters are: *k* = 46, *ki* = 160 and *β* = 20, the values for the flux observer parameters are: *gid* = −44.5, *giq* = −44.5, *gψ<sup>d</sup>* = −50, *gψ<sup>q</sup>* = −50, *k*<sup>1</sup> = 100 and *k*<sup>2</sup> = 100, and the values for the load torque observer parameters are: *kw*<sup>1</sup> = 25, *kw*<sup>2</sup> = 250 *h*<sup>1</sup> = 100 and *h*<sup>2</sup> = 100.

Figure 4 shows the simulation test of the proposed adaptive variable structure position control. The first graph shows the reference and the real rotor position, and the second graph shows the rotor position error. As it can be observed, after a transitory time in which the sliding gain is adapted, the rotor position tracks the desired position in spite of system uncertainties. Nevertheless, at time *t* = 1*s* and *t* = 2*s* a little position error can be observed. This error appears because there is a torque increment at this time, and then the controlled system lost the so called Ssliding modeŠ because the actual sliding gain is too small for ´ the new uncertainty introduced in the system due to the load torque increment. However, after a short time, the new load torque value is adapted and then the sliding gain value can compensate the system uncertainties, and hence the rotor position error is eliminated. The third graph shows the real and the estimated rotor flux. In this figure it can be observed that the proposed sliding mode observer provides an accurate and fast rotor flux estimation. The fourth graph shows the motor torque, the load torque and the estimated load torque. As it can be seen in this graph, after a transitory time, the load torque observer estimates the load torque value with a small estimation error. This figure also shows that the so-called chattering phenomenon appears in the motor torque. Although this high frequency changes in the torque will be reduced by the mechanical system inertia, they could cause undesirable vibrations in the real rotor, which may be a problem for certain systems. However, for the systems that do not support this chattering, it may be eliminated substituting the sign function by the saturation function in the control signal. The fifth graph shows the stator current *iA*. This graph shows that the current signal increases when the load torque increases in order to increment the motor torque. The sixth graph shows the time evolution of the sliding variable. In this figure it can be seen that the system reaches the sliding condition (*S*(*t*) = 0) at time *t* = 0.25*s*, but then the system lost this condition at time *t* = 1*s* and *t* = 2*s* due to the load torque increment which produces an increment in the system uncertainties.

Figure 5 shows the real test of the variable structure position control using the experimental platform. In this figure, a small noise can be observed in the signals due to the sensors used to make the real measurements in the system. The first graph shows the reference and the real rotor position. Like in the previous case (simulation test), the rotor position tracks the reference position in spite of system uncertainties. The second graph shows the rotor position error. In this experimental validation a small position error is obtained in the presence of a high load torque. It should be noted that this performance is not an easy task to achieve for an induction motor. The third graph shows the estimated rotor flux and the fourth graph shows the motor torque, the load torque and the estimated load torque. It can be noted that the proposed sliding mode observers also perform very well in a practice. The fifth graph shows the stator current *iA*, and finally the sixth graph shows the sliding variable *S*.

### **6. Conclusion**

In this chapter an induction motor position regulation using a sliding mode control for a real-time applications has been presented. In the design a field oriented vector control theory is employed in order to simplify the system dynamic equations.

Additionally, in order to avoid the flux sensors, because the flux sensors increase the cost and reduces the reliability, a rotor flux estimator is proposed. This flux estimator is a sliding mode observer and employs the measured stator voltages and currents in the stationary reference frame. The design incorporates also a load torque observer, based on sliding mode theory, in order to improve the controller performance.

In order to demonstrate the performance of the proposed design over a commercial induction motor of 7.5 kW, a new experimental platform has been designed and constructed in order to text the proposed robust controller in a real time application over a high power commercial induction motor

**Figure 4.** Position tracking simulation results

the new uncertainty introduced in the system due to the load torque increment. However, after a short time, the new load torque value is adapted and then the sliding gain value can compensate the system uncertainties, and hence the rotor position error is eliminated. The third graph shows the real and the estimated rotor flux. In this figure it can be observed that the proposed sliding mode observer provides an accurate and fast rotor flux estimation. The fourth graph shows the motor torque, the load torque and the estimated load torque. As it can be seen in this graph, after a transitory time, the load torque observer estimates the load torque value with a small estimation error. This figure also shows that the so-called chattering phenomenon appears in the motor torque. Although this high frequency changes in the torque will be reduced by the mechanical system inertia, they could cause undesirable vibrations in the real rotor, which may be a problem for certain systems. However, for the systems that do not support this chattering, it may be eliminated substituting the sign function by the saturation function in the control signal. The fifth graph shows the stator current *iA*. This graph shows that the current signal increases when the load torque increases in order to increment the motor torque. The sixth graph shows the time evolution of the sliding variable. In this figure it can be seen that the system reaches the sliding condition (*S*(*t*) = 0) at time *t* = 0.25*s*, but then the system lost this condition at time *t* = 1*s* and *t* = 2*s* due to the load torque increment which produces an increment in the system uncertainties. Figure 5 shows the real test of the variable structure position control using the experimental platform. In this figure, a small noise can be observed in the signals due to the sensors used to make the real measurements in the system. The first graph shows the reference and the real rotor position. Like in the previous case (simulation test), the rotor position tracks the reference position in spite of system uncertainties. The second graph shows the rotor position error. In this experimental validation a small position error is obtained in the presence of a high load torque. It should be noted that this performance is not an easy task to achieve for an induction motor. The third graph shows the estimated rotor flux and the fourth graph shows the motor torque, the load torque and the estimated load torque. It can be noted that the proposed sliding mode observers also perform very well in a practice. The fifth graph

226 Induction Motors - Applications, Control and Fault Diagnostics

shows the stator current *iA*, and finally the sixth graph shows the sliding variable *S*.

is employed in order to simplify the system dynamic equations.

theory, in order to improve the controller performance.

In this chapter an induction motor position regulation using a sliding mode control for a real-time applications has been presented. In the design a field oriented vector control theory

Additionally, in order to avoid the flux sensors, because the flux sensors increase the cost and reduces the reliability, a rotor flux estimator is proposed. This flux estimator is a sliding mode observer and employs the measured stator voltages and currents in the stationary reference frame. The design incorporates also a load torque observer, based on sliding mode

In order to demonstrate the performance of the proposed design over a commercial induction motor of 7.5 kW, a new experimental platform has been designed and constructed in order to text the proposed robust controller in a real time application over a high power commercial

**6. Conclusion**

induction motor

Due to the nature of the sliding mode control this control scheme is robust under system uncertainties and changes in the load torque applied to the induction motor. The closed loop stability of the presented design has been proved through Lyapunov stability theory.

Finally, by means of simulation and real examples, it has been confirmed that the proposed position control scheme presents a good performance in practice, and that the position tracking objective is achieved under parameter uncertainties and under load torque variations.

**Figure 5.** Position tracking experimental results

### **Acknowledgements**

The authors are very grateful to the Basque Government by the support of this work through the project S-PE12UN015 and S-PE13UN039 and to the UPV/EHU by its support through the projects GIU13/41 and UFI11/07.

### **Author details**

Oscar Barambones1∗, Patxi Alkorta2, Jose M. Gonzalez de Durana1 and Jose A. Cortajarena<sup>2</sup>


### **References**

0 0.5 1 1.5 2 2.5 3 3.5 4

reference exp real

exp

exp est

Te exp Tl est Tl real

iA exp

exp

Time (s)

0 0.5 1 1.5 2 2.5 3 3.5 4

Time (s)

0 0.5 1 1.5 2 2.5 3 3.5 4

Time (s)

<sup>0</sup> 0.5 <sup>1</sup> 1.5 <sup>2</sup> 2.5 <sup>3</sup> 3.5 <sup>4</sup> −20

Time (s)

0 0.5 1 1.5 2 2.5 3 3.5 4

Time (s)

0 0.5 1 1.5 2 2.5 3 3.5 4

Time (s)

The authors are very grateful to the Basque Government by the support of this work through the project S-PE12UN015 and S-PE13UN039 and to the UPV/EHU by its support through

0

−0.4 −0.2 0 0.2

−1 −0.5 0 0.5 1

−20

−30 −20 −10 0 10

Sliding Variable, s

**Figure 5.** Position tracking experimental results

the projects GIU13/41 and UFI11/07.

**Acknowledgements**

0

Stator current, iA (A)

20

Position Error (rad)

Rotor Flux alfa (Wb)

Te (Nm)

3.1416

Rotor Position (rad)

228 Induction Motors - Applications, Control and Fault Diagnostics

6.2832


## **An Optimized Hybrid Fuzzy-Fuzzy Controller for PWMdriven Variable Speed Drives**

Nordin Saad, Muawia A. Magzoub, Rosdiazli Ibrahim and Muhammad Irfan

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/61086

### **Abstract**

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This paper discusses the performance and the impact of disturbances onto a proposed hybrid fuzzy-fuzzy controller (HFFC) system to attain speed control of a variable speed induction motor (IM) drive. Notably, to design a scalar controller, the two fea‐ tures of field-oriented control (FOC), i.e., the frequency and current, are employed. Specifically, the features of fuzzy frequency and fuzzy current amplitude controls are exploited for the control of an induction motor in a closed-loop current amplitude in‐ put model; hence, with the combination of both controllers to form a hybrid control‐ ler. With respect to finding the rule base of a fuzzy controller, a genetic algorithm is employed to resolve the problem of an optimization that diminishes an objective func‐ tion, i.e., the Integrated Absolute Error (IAE) criterion. Furthermore, the principle of HFFC, for the purpose of overcoming the shortcoming of the FOC technique is estab‐ lished during the acceleration-deceleration stages to regulate the speed of the rotor us‐ ing the fuzzy frequency controller. On the other hand, during the steady-state stage, the fuzzy stator current magnitude controller is engaged. A simulation is conducted via MATLAB/Simulink to observe the performance of the controller. Thus, from a ser‐ ies of simulations and experimental tests, the controller shows to perform consistently well and possesses insensitive behavior towards the parameter deviations in the sys‐ tem, as well as robust to load and noise disturbances.

**Keywords:** Indirect field-oriented control (IFOC), hybrid fuzzy-fuzzy control (HFFC), hybrid fuzzy-PI control (HFPIC), disturbances, genetic algorithm (GA)

### **1. Introduction**

During the last forty years, induction motors have been largely utilized in applications that use variable speeds. In the industry, the term workhorse is used to refer to an induction motor.

© 2015 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

With the development in the field of silicon-rectifier devices, the variable speed induction motor drives techniques began to emerge in the late 1960s. At that time, the principal of speed control was only based on the steady state aspects of an induction machine. The rigorous research in this field has made possible the emergence of more techniques in industrial drives. One of the techniques that previously used for drives control is the V/f ratio that has been applied in an open-loop speed control of drives that normally need low dynamics. In addition, the slip frequency control technique is also effective for producing improved dynamics. Until the emergence of field-oriented controls (FOCs), this technique, based on all the high-per‐ formance induction motor (IM) drives, was considered as an industry standard for AC drives whose dynamics resemble DC motors [1–2, 19]. Hence, the invention of vector control and FOC are considered as the most significant factors in AC motor drives that boosted programs related to the development and research for improving control performance. Several benefits of process control can be obtained by adjusting the speed of the drive motors, such as varying speeds for its operation process, better control of speed variation, work efficiency, precise control for positioning, tension or torque, energy saving, and compensation of process varying variables [2].

Zadah, in 1965, described Fuzzy Logic as a novel kind of mathematical set approach consisting of a fuzzy set theory that is considered as a basic theory of the fuzzy logic. A fuzzy control system is established by applying a principle of fuzzy logic that consists of three phases: fuzzification, inference engine, and defuzzification. The inputs are converted into fuzzy sets in an initial phase. An inference engine defines the fuzzy rules in the next phase that links the outputs by using the sets of inputs via explicit rules. Finally, the conclusions are inferred by combining the outcomes of the fuzzy rules, which is then transformed into a sharp value from the fuzzy sets [3, 13].

In order to provide an effective means for a variable speed drive (VSD) control, several research studies exist that are based on control techniques and commercially available tools yielding a high degree of performance and reliability. Hence, a PLC-based hybrid-fuzzy control for pulsewidth modulation PWM driven VSD is examined that depends upon the s-domain transfer function in a scientifically presented model of an original plant by keeping the V/f ratio at a constant value as in [5]. Notably in [5], the optimizations of the controller's performance against the parameter variations and external disturbances are not fully considered. Hence, the disadvantages of the FOC method and the results gained from the simulation are overcame by implementing two stage controllers as explain in [6–8], though all the practical implemen‐ tations are quite satisfactory. Moreover, the satisfactory results are achieved by applying some controller algorithms for controlling the speed of an IM as in [9, 10].

Generally, by using simple solutions, fuzzy systems are capable of managing non-linear, complicated, and at times mathematically intangible dynamic systems [11]. Though, it is not an easy task to get an optimal set of fuzzy membership rules and functions, the designer needs to invest some times, skills, and experience for the tiresome fuzzy tuning exercise. Even though an iterative and heuristic process for transforming the membership functions to enhance performance has been recommended, in principle, there is no common method or rule for the fuzzy logic setup [11]. A lot of researchers have recently considered numerous intelligent schemes for the purpose of tuning the fuzzy system [12–17]. For instance, the genetic algorithm (GA) and neural network (NN) approach to optimize either the membership rules or functions have turned into a trend in the development of the fuzzy logic system [11]. The benefits of the GA approach, apart from involving less cost, it is easy to implement the procedures and requires a single objective to be assessed [22]. Figure 1 presents the basic configuration of GA for a fuzzy control system with fuzzifier, defuzzifier, and an inference engine.

**Figure 1.** Genetic algorithm for a fuzzy control system.

With the development in the field of silicon-rectifier devices, the variable speed induction motor drives techniques began to emerge in the late 1960s. At that time, the principal of speed control was only based on the steady state aspects of an induction machine. The rigorous research in this field has made possible the emergence of more techniques in industrial drives. One of the techniques that previously used for drives control is the V/f ratio that has been applied in an open-loop speed control of drives that normally need low dynamics. In addition, the slip frequency control technique is also effective for producing improved dynamics. Until the emergence of field-oriented controls (FOCs), this technique, based on all the high-per‐ formance induction motor (IM) drives, was considered as an industry standard for AC drives whose dynamics resemble DC motors [1–2, 19]. Hence, the invention of vector control and FOC are considered as the most significant factors in AC motor drives that boosted programs related to the development and research for improving control performance. Several benefits of process control can be obtained by adjusting the speed of the drive motors, such as varying speeds for its operation process, better control of speed variation, work efficiency, precise control for positioning, tension or torque, energy saving, and compensation of process varying

Zadah, in 1965, described Fuzzy Logic as a novel kind of mathematical set approach consisting of a fuzzy set theory that is considered as a basic theory of the fuzzy logic. A fuzzy control system is established by applying a principle of fuzzy logic that consists of three phases: fuzzification, inference engine, and defuzzification. The inputs are converted into fuzzy sets in an initial phase. An inference engine defines the fuzzy rules in the next phase that links the outputs by using the sets of inputs via explicit rules. Finally, the conclusions are inferred by combining the outcomes of the fuzzy rules, which is then transformed into a sharp value from

In order to provide an effective means for a variable speed drive (VSD) control, several research studies exist that are based on control techniques and commercially available tools yielding a high degree of performance and reliability. Hence, a PLC-based hybrid-fuzzy control for pulsewidth modulation PWM driven VSD is examined that depends upon the s-domain transfer function in a scientifically presented model of an original plant by keeping the V/f ratio at a constant value as in [5]. Notably in [5], the optimizations of the controller's performance against the parameter variations and external disturbances are not fully considered. Hence, the disadvantages of the FOC method and the results gained from the simulation are overcame by implementing two stage controllers as explain in [6–8], though all the practical implemen‐ tations are quite satisfactory. Moreover, the satisfactory results are achieved by applying some

Generally, by using simple solutions, fuzzy systems are capable of managing non-linear, complicated, and at times mathematically intangible dynamic systems [11]. Though, it is not an easy task to get an optimal set of fuzzy membership rules and functions, the designer needs to invest some times, skills, and experience for the tiresome fuzzy tuning exercise. Even though an iterative and heuristic process for transforming the membership functions to enhance performance has been recommended, in principle, there is no common method or rule for the fuzzy logic setup [11]. A lot of researchers have recently considered numerous intelligent schemes for the purpose of tuning the fuzzy system [12–17]. For instance, the genetic algorithm (GA) and neural network (NN) approach to optimize either the membership rules or functions

controller algorithms for controlling the speed of an IM as in [9, 10].

variables [2].

232 Induction Motors - Applications, Control and Fault Diagnostics

the fuzzy sets [3, 13].

A GA is employed in this research to attain the rules of the fuzzy inference system. However, the key aim of this study is to compare the performance of a fuzzy controller built on heuristics with a controller developed via the optimization technique.

Therefore, the best combination between the fuzzy input-output variables is needed to be discovered to enhance the inference rules of a fuzzy controller for a particular range of the fuzzy logic controller operation.

The FOC has two features [4, 6] that have been implemented in this work. Firstly, it is not able to do frequency control directly, due to the fact that the supply frequency changes during the period of acceleration-deceleration of the FOC while the slip frequency remains the same. Furthermore, in the presence of a torque command, the magnitude of a supply current magnitude remains stable.

The application of the FOC method has commonly been effective in achieving elevated performances in adjustable speed induction motor drives, however, it still suffers the following disadvantages [6]:


The simulation and modeling of an induction motor controller constructed using MATLAB/ Simulink and the examination of the performance of the controllers (i.e., hybrid fuzzy-fuzzy controller (HFFC), hybrid fuzzy-PI controller (HFPIC), and indirect field-oriented controller (IFOC)) on the system are discussed in this paper. Also, in this study, the objective is to enhance the performance of the controller using HFFC. The purpose and context of this study are outlined in Figure 2.

**Figure 2.** Overview of the study background.

The implementation of a fuzzy current amplitude controller on the induction motor model makes this work unique. This controller possesses the same supply features as FOC and insensitivity to the parameter variation for the motor and system robustness to noise and load disturbances are some of the advantages of this controller. Due to the fact that it provides better performance, the fuzzy current amplitude controller has been selected. While, the common structure of hybrid fuzzy-fuzzy controller is defined in Figure 3.

**Figure 3.** General structure of an HFFC.

### **2. Mathematical modeling**

The simulation and modeling of an induction motor controller constructed using MATLAB/ Simulink and the examination of the performance of the controllers (i.e., hybrid fuzzy-fuzzy controller (HFFC), hybrid fuzzy-PI controller (HFPIC), and indirect field-oriented controller (IFOC)) on the system are discussed in this paper. Also, in this study, the objective is to enhance the performance of the controller using HFFC. The purpose and context of this study are

> Application in variable speed drive

Hybrid Fuzzy-PI controller: + Insensitive to parameter variattion

+ robustness to noise and load disturbs

Comparison

The implementation of a fuzzy current amplitude controller on the induction motor model makes this work unique. This controller possesses the same supply features as FOC and insensitivity to the parameter variation for the motor and system robustness to noise and load disturbances are some of the advantages of this controller. Due to the fact that it provides better performance, the fuzzy current amplitude controller has been selected. While, the common

+

Frequency <sup>+</sup>

Current Amplitude Inverter

Induction Motor

+ reliable

Hybrid Fuzzy-Fuzzy controller - Need experience and skills

structure of hybrid fuzzy-fuzzy controller is defined in Figure 3.

Gain Gain

Controller Gain Saturation

Fuzzy Logic *<sup>m</sup>* Controller *e*

Fuzzy Logic

Intelligent Optimization Program GA

**Figure 2.** Overview of the study background.

+ -

**Figure 3.** General structure of an HFFC.

\* *m*

*m* Experimental Results

234 Induction Motors - Applications, Control and Fault Diagnostics


Speed Control

Induction Motor

Position Control

Controller Indirect Field-Oriented Control (IFOC): + Flux and torque are controlled + High Performance


outlined in Figure 2.

A higher order of mathematical equations that fall under one of the VSD control classifications can be used to model the dynamics of an induction motor. The steady-state model provides information about the performance of the induction motors in a steady state only. Table 1 provides the related parameters of IM. Figure 4 illustrates the IFOC model block of the proposed system for an IM.


**Table 1.** Electrical and mechanical parameters of the IM.

By employing a two-phase motor in a quadrature and direct axis, the dynamic model of the induction motor is developed. The description of the notations is tabulated in Table 2. The state-space model of an induction motor in a stationary reference frame can be derived with the help of the voltage and flux linkage relations of an induction motor in the reference frame that is randomly selected [11, 12].

**Figure 4.** Proposed IFOC model block.


**Table 2.** Nomenclatures.

The final state-space model of an induction motor with the controlled stator currents in a stationary frame can be written as shown in equations (1)–(6).

$$\frac{d\dot{\mathbf{i}}\_{\rm ds}^{\rm s}}{dt} = -\gamma \dot{\mathbf{i}}\_{\rm ds}^{\rm s} + \beta \alpha \rho\_r \dot{\lambda}\_{qr}^{\rm s} + \beta \alpha \dot{\lambda}\_{\rm dr}^{\rm s} + \beta\_1 V\_{\rm ds}^{\rm s} \tag{1}$$

$$\frac{d\dot{\mathbf{u}}\_{q\mathbf{s}}^{\ast}}{dt} = -\gamma \dot{\mathbf{r}}\_{q\mathbf{s}}^{\ast} - \beta \alpha \dot{\boldsymbol{\lambda}}\_{qr}^{\ast} + \beta \alpha \dot{\boldsymbol{\lambda}}\_{qr}^{\ast} + \beta\_1 \mathbf{V}\_{q\mathbf{s}}^{\ast} \tag{2}$$

An Optimized Hybrid Fuzzy-Fuzzy Controller for PWM-driven Variable Speed Drives http://dx.doi.org/10.5772/61086 237

$$\frac{d\mathcal{X}\_{dr}^{s}}{dt} = -\alpha \mathcal{X}\_{dr}^{s} - \alpha \iota\_{r} \mathcal{X}\_{qr}^{s} + \alpha L\_{M} \mathcal{i}\_{ds}^{s} \tag{3}$$

$$\frac{d\mathcal{X}\_{qr}^{s}}{dt} = -\alpha \mathcal{X}\_{qr}^{s} + \alpha \rho\_r \mathcal{X}\_{dr}^{s} + \alpha L\_M \mathcal{I}\_{qs}^{s} \tag{4}$$

$$\frac{d\phi\_m}{dt} = -\frac{B}{J}\phi o\_m + \frac{1}{J}(T\_c - T\_l) \quad o\_r = \frac{P}{2}\phi o\_m \tag{5}$$

$$T\_e = \left(\frac{\mathfrak{Z}}{2}\right) \left(\frac{P}{2}\right) \frac{L\_M}{Lr} \left(\dot{\mathfrak{i}}\_{qs}^s \mathcal{X}\_{dr}^s - \dot{\mathfrak{i}}\_{ds}^s \mathcal{X}\_{qr}^s\right) \tag{6}$$

Notably, the two features of the FOC that have been used in this research study are shown in Eq. (1) to (6). The first aspect that shows the supply frequency changes with the speed of a rotor [6] is given in Eq. (7) and (8).

$$
\alpha \alpha = \alpha\_r + \frac{P}{2}\alpha\_w \tag{7}
$$

where:

*i ds*, *i*

*i dr*, *i* frame

236 Induction Motors - Applications, Control and Fault Diagnostics

frame

frame

frame

frame

*J* The inertia of the rotor kgm<sup>2</sup> or Js<sup>2</sup>

2 *<sup>L</sup> <sup>s</sup> <sup>L</sup> <sup>r</sup>* , *<sup>β</sup>* <sup>=</sup> *<sup>L</sup> <sup>M</sup>*

*P* Number of pairs of poles

*ρ* Operator <sup>d</sup>

<sup>2</sup> , *<sup>σ</sup>* =1<sup>−</sup> *<sup>L</sup> <sup>M</sup>*

*<sup>γ</sup>* <sup>=</sup> *<sup>L</sup> <sup>M</sup>*

<sup>2</sup> *rr* <sup>+</sup> *<sup>L</sup> <sup>r</sup>* 2 *rs σ L <sup>s</sup> L <sup>r</sup>*

**Table 2.** Nomenclatures.

*ω* Synchronous speed or dominant frequency

dt

stationary frame can be written as shown in equations (1)–(6).

g

g

*s*

*dt*

*s*

*di*

*dt*

*L <sup>s</sup>*, *L <sup>r</sup>* Self-inductance of the stator and rotor, respectively

*L ls*, *L lr* Leakage resistance of the stator and rotor, respectively

*ωm*, *ω<sup>r</sup>* Mechanical and electrical angular rotor speed, respectively

*rs*, *rr* The resistance of a stator and rotor phase winding, respectively

*Te*, *Tl* Electromagnetic torque and Load torque reflected on the motor shaft, respectively

*Vds*, *Vqs* d- and q-axis stator voltage components, respectively, and expressed in stationary reference

*λds*, *λqs* d- and q-axis stator flux components, respectively, and expressed in stationary reference

*λdr*, *λqr* d- and q-axis rotor flux components, respectively, and expressed in stationary reference

*B* The damping constant that represents dissipation due to windage and friction

*<sup>σ</sup> <sup>L</sup> <sup>s</sup>* , *<sup>α</sup>* <sup>=</sup> *rr*

The final state-space model of an induction motor with the controlled stator currents in a

*ds s s ss ds r qr dr ds*

*qs s s ss qs r qr qr qs*

=- - + +

*i V*

*di i V*

=- + + +

 bw l bal

 bw l bal

*L r*

1

1

 b

(1)

(2)

 b

*<sup>σ</sup> <sup>L</sup> <sup>s</sup> <sup>L</sup> <sup>r</sup>* , *<sup>β</sup>*<sup>1</sup> <sup>=</sup> <sup>1</sup>

*L <sup>M</sup>* Magnetizing inductance

*qs* d- and q-axis stator current components, respectively, and expressed in stationary reference

*qr* d- and q-axis rotor current components, respectively, and expressed in stationary reference

$$\alpha \alpha\_r = \frac{\Im r\_r T\_l^{"\"}}{P \lambda\_{dr}^{"\"} }\tag{8}$$

If *T* \* is maintained constant during acceleration, then *ωr* is also constant. In Eq. (7), as *ω<sup>m</sup>* changes during acceleration, then *ω* has to be varied, so that Eq. (7) satisfies the first FOC feature. The second feature can be proven by substituting the conditions *λqr* =0 and *λdr* =*constant* in Eq. (3) [6, 21].

$$\left| \dot{\mathbf{i}}\_s \right| = \frac{2}{3} \sqrt{\dot{\mathbf{i}}\_{ds}^2 + \dot{\mathbf{i}}\_{qs}^2} \tag{9}$$

where:

$$\dot{\mathbf{u}}\_{\rm ds} = \frac{\mathcal{N}\_{\rm dr}^{\varepsilon}}{L\_{\rm M}} = constant \tag{10}$$

$$i\_{qs} = \frac{3L\_rT}{PL\_M\mathcal{L}\_{dr}^{\epsilon^\*}}\tag{11}$$

Eq. (9) will be a constant if the torque command *T* \* is a constant, which satisfies the second feature. The Figure 5 shows that the speed response may be divided into two stages. For the HFFC, Table 3 shows the relationship between the inputs and outputs.

**Figure 5.** Speed response stages.

### **3. Design of the controllers**

### **3.1. Fuzzy current amplitude controller**

In the stage of acceleration-deceleration, the stator current magnitude is regulated as the system is driven by the maximum permissible values of an inverter. During the final steadystate period, the speed of the rotor is controlled by adjusting the magnitude of the stator current, while the supply frequency is kept constant. In the presence of a constant supply frequency, the relationship between torque-current may be expressed as follows [21].

$$T = \left[\frac{3P}{2}\right] \left[\frac{r\_r L\_M^2 i\_s^2 \left[\rho o - \frac{P}{2} o\_w\right]}{r\_r^2 + \left[L\_M + L\_{tr}\right]^2 \left[\rho o - \frac{P}{2} o\_w\right]}\right] \tag{12}$$

The values for the current amplitude are depicted as follows:

$$\begin{vmatrix} \dot{i}\_s \end{vmatrix} = \begin{cases} 50.54 \ \left| \dot{i}\_s \right| \ge 50.54A & \text{when } \Delta o\_m \ne 0\\ \left| \dot{i}\_s \right| \left| \dot{i}\_s \right| < 50.54A & \text{when } \Delta o\_m = 0 \end{cases} \tag{13}$$

The main phase from every input-output data couples is to create a fuzzy rule to find out a degree of every data-value in each affiliated area of its corresponding fuzzy domain. Conse‐ quently, a variable is assigned to a region having a maximum value.


**Table 3.** HFFC relationship.

\* \*

feature. The Figure 5 shows that the speed response may be divided into two stages. For the

0 5 10 15 20 25 30 35 40

In the stage of acceleration-deceleration, the stator current magnitude is regulated as the system is driven by the maximum permissible values of an inverter. During the final steadystate period, the speed of the rotor is controlled by adjusting the magnitude of the stator current, while the supply frequency is kept constant. In the presence of a constant supply

frequency, the relationship between torque-current may be expressed as follows [21].

2 2

é ù é ù ê ú - ê ú é ù ë û <sup>=</sup> ê ú é ù ë û ++ - é ù

*r Ms m*

w

*<sup>P</sup> rL i*

*<sup>P</sup> r LL*

*r M lr m*

ë û ê ú ë û ë û

<sup>2</sup> <sup>2</sup> 3 2

2

 w (12)

 w

w

2

*<sup>P</sup> <sup>T</sup>*

**Acceleration period Steady-state period Time**

<sup>=</sup> (11)

**Fuzzy current amplitude control**

is a constant, which satisfies the second

3 *<sup>r</sup> qs e M dr*

*L T <sup>i</sup> PL* l

HFFC, Table 3 shows the relationship between the inputs and outputs.

**Fuzzy slip frequency control**

Eq. (9) will be a constant if the torque command *T* \*

238 Induction Motors - Applications, Control and Fault Diagnostics

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

**Figure 5.** Speed response stages.

**3. Design of the controllers**

**3.1. Fuzzy current amplitude controller**

**Speed (rad/sec)**

A truth or rule degree is assigned to a newly generated fresh rule from the input-output data couples. Therefore, a rule degree can be defined as an extension of assurance to relate the current- and voltage-related functions with an angle. Usually, a degree that is a creation of the membership function degree of every variable in its corresponding area is assigned in a formulated technique.

A compatible fuzzy rule is created by each training data set that is placed in the fuzzy rule base. Consequently, each input-output data couples is preserved and the rules are generated. A two-dimensional form that can be explored by the fuzzy reasoning tool is used to tabulate a knowledge base or fuzzy rule.

Figure 6 demonstrates the general structure of the fuzzy logic control (FLC), a combination of knowledge base, fuzzification, a defuzzification, and inference engine.

**Figure 6.** A general block diagram of FLC.

A speed error can be computed by comparing the reference speed and the speed signal feedback. The fuzzy knowledge base consists of membership functions of the inputs of the fuzzy controller including speed reference, error changing, and current amplitude/slip frequency outputs.


**Table 4.** Speed, current amplitude, and fuzzy linguistic values.

### **3.2. Membership functions**

Primarily, the fuzzy logic controller is used to convert the change of error variables and the crisp error into fuzzy variables, which is then plotted into the linguistic tags. All the member‐ ship functions and labels are connected with each other (Figures 7, 8, and 9), which comprises of two inputs and one output. The nine sets are formed by classifying the fuzzy sets that are outlined as follows:


**Table 5.** Classification of the fuzzy sets

The previously defined seven numbers of linguistics, along with both inputs and outputs, include membership functions. Eq. (9) shows the samples of current amplitude of the two stages as shown in Table 4.

**Figure 7.** Membership function of speed reference.

A speed error can be computed by comparing the reference speed and the speed signal feedback. The fuzzy knowledge base consists of membership functions of the inputs of the fuzzy controller including speed reference, error changing, and current amplitude/slip

**Deceleration** - -50.54 **NBB Steady-State** -120 -15.07 **NB**

**Acceleration** - 50.54 **PBB**

Primarily, the fuzzy logic controller is used to convert the change of error variables and the crisp error into fuzzy variables, which is then plotted into the linguistic tags. All the member‐ ship functions and labels are connected with each other (Figures 7, 8, and 9), which comprises of two inputs and one output. The nine sets are formed by classifying the fuzzy sets that are

The previously defined seven numbers of linguistics, along with both inputs and outputs, include membership functions. Eq. (9) shows the samples of current amplitude of the two

**Z**: Zero **PS**: Positive Small **PM**: Positive Medium **PB**: Positive Big **NS**: Negative Small **NM**: Negative Medium **NB**: Negative Big **PBB**: Positive Big Big **NBB**: Negative Big Big

**-** -80 -12.79 **NM -** -40 -11.214 **NS -** 0 0 **Z -** 40 11.214 **PS -** 80 12.79 **PM -** 120 15.07 **PB**

\* *A f <sup>A</sup>*

frequency outputs.

**Stages** *<sup>ω</sup><sup>m</sup>*

240 Induction Motors - Applications, Control and Fault Diagnostics

**Table 4.** Speed, current amplitude, and fuzzy linguistic values.

**3.2. Membership functions**

**Table 5.** Classification of the fuzzy sets

stages as shown in Table 4.

outlined as follows:

**Figure 8.** Membership function of speed error.

**Figure 9.** Membership functions of current amplitude.

**Figure 10.** Membership functions of slip frequency.

### **3.3. Rule base**

The fuzzy inputs can be plotted easily into the required output with the help of a useful tool, the rule base, as shown in Table 6.


**Table 6.** Rule matrix for fuzzy amplitude/slip controller.

### **3.4. Fuzzy frequency controller**


NBB NB NM NS Z PS PM PB PBB

f A

NBB NB NM NS Z PS PM PB PBB


f A, f<sup>r</sup>

0

0

0.2

0.4

0.6

Degree of membership

0.8

1

**Figure 9.** Membership functions of current amplitude.

242 Induction Motors - Applications, Control and Fault Diagnostics

**Figure 10.** Membership functions of slip frequency.

0.2

0.4

0.6

Degree of membership

0.8

1

A fuzzy frequency control is presented by using a frequency aspect of the field orientation principle. At a steady state phase, the torque command is a smaller value, whereas, the torque command becomes a larger value during the stage of acceleration-deceleration. The speed reference and rotor speed represent these values [4, 6, 11]. The following relations show a slip frequency at a steady-state.

$$\rho o\_r = \begin{cases} \frac{3r\_r}{P\lambda\_{dr}^{\hat{\epsilon}/2}} T\_{\text{acc}} & \text{when } \Delta o\_m \neq 0\\ \frac{3r\_r}{P\lambda\_{dr}^{\hat{\epsilon}/2}} \mu o\_m & \text{when } \Delta o\_m = 0 \end{cases} \tag{14}$$

So the slip frequency can be written as:

$$
\Delta o\_r = f\left(o\_{m'} \Delta o\_m\right) \tag{15}
$$

Eq. (14) shows the reference and speed error from the inputs of the fuzzy slip control. The membership functions of the output and input are depicted in Figures 7, 8, and 10. In addition, Table 6 shows the rule matrix. The samples of the slip frequency of the two stages can be obtained by Eq. (14), which is shown in Table 7.

### **3.5. Defuzzification**

The fuzzy control action is executed with the help of simulating the human decision process via the interference engine, from the understanding of the linguistic variable expressions and


**Table 7.** Speed, slip frequency, and fuzzy linguistic values.

the control rules. As a result, the knowledge base along with the inference engine is intercon‐ nected during the course of the control process. At first, through substituting the fuzzified inputs into rule base, the active rules are differentiated. Then, by employing one of the fuzzy reasoning methods, these rules are combined. The utmost distinctive fuzzy reasoning methods are the Max-Product and Min–Max. Since the Min-Max interference scheme is commonly used, it is also applied in this research. The fuzzy control actions are then regenerated with the help of defuzzification, which is deduced from the inference engine to a non-fuzzy control action. Thus, using the center of gravity method, the defuzzification is accomplished in the set of Eq. (16) to create a non-fuzzy control signal:

$$y = \frac{\sum\_{i=1}^{n} \mu(i)\mu\_{A,\alpha\_i}(y\_i)}{\sum\_{i=1}^{n} \mu\_{A,\alpha\_i}(y\_i)}\tag{16}$$

where, *μA*,*ω<sup>r</sup>* is the degree of membership function.

[NBB=1, NB=2, NM=3, NS=4, Z=5, PS=6, PM=7, PB=8, PBB=9]

Whereas, the linguistic standards of the antecedents relate to the entire values as follows:

[NB=1, NM=2, NS=3, Z=4, PS=5, PM=6, PB=7]

However, it is essential to recognize the values of the precedents, fuzzy operators in the rule base weights, and fuzzy rules to form the rule base of a fuzzy controller; in this situation, the antecedents 1 and 2 and the other stated values are positioned in a matrix inside the MATLAB functions, as can be seen in the matrix below [24]:

$$
\begin{bmatrix}
 \mathbf{Ant}\_1 \mathbf{Ant}\_2 \mathbf{Con}\_1 \mathbf{R} \mathbf{wRC} & \cdots & \mathbf{Ant}\_\gamma \mathbf{Ant}\_\gamma \mathbf{Con}\_\gamma \mathbf{R} \mathbf{wRC} \\
 \vdots & \ddots & \vdots \\
 \mathbf{Ant}\_3 \mathbf{Ant}\_3 \mathbf{Con}\_3 \mathbf{R} \mathbf{wRC} & \cdots & \mathbf{Ant}\_{21} \mathbf{Ant}\_{21} \mathbf{Con}\_{21} \mathbf{R} \mathbf{wRC}
\end{bmatrix}
$$

Where, *Antn*: antecedent *n*, *Antn*: antecedent *n*, *Conn*: consequent *n*, *Rw*: rule weight, and *RC*: rule connection. Finally, the input/output variables range and the membership function data are in the same MATLAB function. The MATLAB FIS (Fuzzy Inference System) file is the output of this function that relates to a structure in which all the information of fuzzy inference of the system is incorporated, which is utilized as a fuzzy controller in the feedback scheme of the Simulink library. The flowchart procedure is shown in Figure 11.

**Figure 11.** Flowchart procedure.

the control rules. As a result, the knowledge base along with the inference engine is intercon‐ nected during the course of the control process. At first, through substituting the fuzzified inputs into rule base, the active rules are differentiated. Then, by employing one of the fuzzy reasoning methods, these rules are combined. The utmost distinctive fuzzy reasoning methods are the Max-Product and Min–Max. Since the Min-Max interference scheme is commonly used, it is also applied in this research. The fuzzy control actions are then regenerated with the help of defuzzification, which is deduced from the inference engine to a non-fuzzy control action. Thus, using the center of gravity method, the defuzzification is accomplished in the set of Eq.

**Acceleration** - 16.80 **PBB**

**Deceleration** - -16.80 **NBB Steady-State** -120 -3.6328 **NB**

**-** -80 -2.42192 **NM -** -40 -1.21096 **NS -** 0 0 **Z -** 40 1.21096 **PS -** 80 2.42192 **PM -** 120 3.6328 **PB**

\* *<sup>ω</sup><sup>r</sup> <sup>f</sup> <sup>ω</sup><sup>r</sup>*

, 1 , 1

m

=

*A i i n*

m

*ui y*

*A i i*

Whereas, the linguistic standards of the antecedents relate to the entire values as follows:

However, it is essential to recognize the values of the precedents, fuzzy operators in the rule base weights, and fuzzy rules to form the rule base of a fuzzy controller; in this situation, the antecedents 1 and 2 and the other stated values are positioned in a matrix inside the MATLAB

*n*

= å

=

*y*

is the degree of membership function.

[NBB=1, NB=2, NM=3, NS=4, Z=5, PS=6, PM=7, PB=8, PBB=9]

[NB=1, NM=2, NS=3, Z=4, PS=5, PM=6, PB=7]

functions, as can be seen in the matrix below [24]:

() ( )

*r*

w

( ) *r*

<sup>å</sup> (16)

*y* w

(16) to create a non-fuzzy control signal:

**Table 7.** Speed, slip frequency, and fuzzy linguistic values.

**Stages** *<sup>ω</sup><sup>m</sup>*

244 Induction Motors - Applications, Control and Fault Diagnostics

where, *μA*,*ω<sup>r</sup>*

The parameters of the GA are set as below:


From the feedback scheme, the performance index is entertained and reverted to the genetic algorithm for the stability of the genetic process. Finally, the rule base attained is displayed as below:


### **4. GA-optimization method**

GAs are computational schemes, which on the basis of processes of natural evolution, utilize the operators who understand the process of heuristic search in a search space, in which it is presumed that the perfect solution for the optimization problem is available [22]. The process of GA is revealed in Figure 12 [21, 25].

**Figure 12.** The process flowchart of GA.

The objective function recommended in the optimization problem is the standard *IAE* (Integrated Absolute Error) that is described by [5, 23, 26]:

$$\text{IAE} = \int\_0^T \left| e\left(t\right) \right| dt \tag{17}$$

The objective function recommended in Eq. (17) that is reduced in the course of the optimiza‐ tion process reflects a good reaction to set point changes. Among the most significant points, i.e., the decay ratio, the settling time, the rise time, the overshoot, and the steady-state error, this objective function is also considered [23]. However, merely the rule base is attained with the assistance of the optimization technique.

The genetic algorithm is applied to a population of individuals (chromosome) in order to devise the rules where each of them encompasses a certain fuzzy controller. The antecedents conforming to the input linguistic variables are fixed in a group as a fragment of a MATLAB function [24]. During the assessment process, this function is utilized by the genetic algorithm and it gets as arguments in the consequents of a controller; the assessment is done in all the pre-established generations for each of the individuals of the population. Also, the binary code is implemented for simplicity. The values are:

[NBB NB NM NS Z PS PM PB PBB]

1 2 2 2 1 13 1 2 2 4 8 8 8 9 9 7 9 7 98

ë û

GAs are computational schemes, which on the basis of processes of natural evolution, utilize the operators who understand the process of heuristic search in a search space, in which it is presumed that the perfect solution for the optimization problem is available [22]. The process

The objective function recommended in the optimization problem is the standard *IAE*

( ) 0

The objective function recommended in Eq. (17) that is reduced in the course of the optimiza‐ tion process reflects a good reaction to set point changes. Among the most significant points, i.e., the decay ratio, the settling time, the rise time, the overshoot, and the steady-state error, this objective function is also considered [23]. However, merely the rule base is attained with

*T*

IAE

Decode Calculate fitness of individuals (evaluation 1,2,….,n)

MATLAB Fuction

· Selection · Crossover · Mutation

FIS Structure System Simulink

<sup>=</sup> *e t dt* ò (17)

Next generation

Values of 21 consequents

Encode Generate initial population (chromosome 1,2, ….,n)

(Integrated Absolute Error) that is described by [5, 23, 26]:

the assistance of the optimization technique.

**4. GA-optimization method**

246 Induction Motors - Applications, Control and Fault Diagnostics

of GA is revealed in Figure 12 [21, 25].

First Generation

**Figure 12.** The process flowchart of GA.

Start

Initial data: · Population size · Maximum range · Minimum range · Number of generation é ù ê ú

And the output language variables are coded in the order:

[0001 0010 0011 0100 0101 0110 0111 1000 1001]

A likely chromosome is shown in Table 5, which would be codified as:

[0001 0010 1001 0001 0100 1001 0001 0101 1001......1000 1001]

With the formation of a population of individuals (generated randomly), the genetic process starts in which every individual comprises the 21 consequents of a fuzzy controller in general. Since 21 consequents could be employed in the MATLAB function, to create the rules, it is essential to change the individual, codified as a binary chain, to complete numbers in the values from 1 to 9.

### **5. Outcomes and analysis**

A combination of the fuzzy frequency controller and fuzzy current amplitude controller is used to form an HFFC. This controller provides similar supply frequency as the FOC and is insensitive to the parameter variation for the motor and system robustness to noise and load disturbances, which are the advantages of this controller. A model of HFFC for an induction motor is produced by using the MATLAB/Simulink software, which is shown in the Figure 13. Table 1 shows the parameters that are chosen to perform the simulation study.

**Figure 13.** Simulink diagram of the HFFC.

An HFFC can be modeled by combining the fuzzy frequency controller and the fuzzy current amplitude controller. Throughout the final steady-state stage, the fuzzy frequency controller outputs the frequency that relates to the speed command. During the acceleration-deceleration stages, the fuzzy current amplitude controller outputs the maximum permitted current. The model of HFFC for the induction motor is built using MATLAB/Simulink as presented in Figure 14.

**Figure 14.** The model diagram of the HFFC.

### **6. Performance criteria**

Generally, following the reference and to cast off the disturbance are the two basic key objectives of control. The following performance criterion is employed for evaluating the efficiency of the controller in achieving the aims of control for complete speed control and the performances are identified for comparisons and investigation [5, 27]. The error (e), the difference between actual and reference value, is generally categorized into a number of quantities. One of the quantities to state the accumulative error magnitude is the IAE, whose formula is given in Eq. (17) [5, 23, 26]. The quadratic measure, i.e., integral of squared error (ISE), provides the error quantitative in quadratic mode. The ISE accrues the squared error. The ISE expression is presented as:

$$\text{ISE} = \int\_0^\top e^2(\mathbf{t})d\mathbf{t} \tag{18}$$

This criterion's main disadvantage is that it provides large weight if the error is large, such as, a poorly checked system. Other criteria are the integral of time weighted absolute error (ITAE) and integral of the time multiplied by the squared error (ITSE). The expression of ITAE and ITSE are articulated in Eq. (19) and Eq. (20).

$$\text{ITAE} = \underset{\bullet}{\text{ft}}\limits\_{0}^{\text{T}} e(t) \Big| \text{dt} \tag{19}$$

$$\text{ITSE} = \int\_0^\mathbf{T} t e^2 \left(\mathbf{t}\right) \mathbf{d}\mathbf{t} \tag{20}$$

The objective of control is to reduce every performance criteria's error. The Simulink model for calculating the performance indices is presented in Figure 15 [28].

**Figure 15.** Simulink model for computing the performance indices.

### **7. Simulation results**

An HFFC can be modeled by combining the fuzzy frequency controller and the fuzzy current amplitude controller. Throughout the final steady-state stage, the fuzzy frequency controller outputs the frequency that relates to the speed command. During the acceleration-deceleration stages, the fuzzy current amplitude controller outputs the maximum permitted current. The model of HFFC for the induction motor is built using MATLAB/Simulink as presented in

Generally, following the reference and to cast off the disturbance are the two basic key objectives of control. The following performance criterion is employed for evaluating the efficiency of the controller in achieving the aims of control for complete speed control and the performances are identified for comparisons and investigation [5, 27]. The error (e), the difference between actual and reference value, is generally categorized into a number of quantities. One of the quantities to state the accumulative error magnitude is the IAE, whose formula is given in Eq. (17) [5, 23, 26]. The quadratic measure, i.e., integral of squared error (ISE), provides the error quantitative in quadratic mode. The ISE accrues the squared error.

> T 2 0

ISE (t)dt <sup>=</sup> *<sup>e</sup>*ò (18)

Figure 14.

**Figure 14.** The model diagram of the HFFC.

248 Induction Motors - Applications, Control and Fault Diagnostics

The ISE expression is presented as:

**6. Performance criteria**

By undertaking a simulation of an indirect rotor flux FOC, a new controller is matched with the FOC. As shown in Figure 16, the speed has augmented in 0 s from 0–120 rad/s; in 8 s period, the speed reduced to -120 rad/s; in the 12 s period, the speed has elevated to 50 rad/s; in the 16 s period, the speed has reduced to -120 rad/s; in the 18 s period, the speed has raised to 0 rad/ s; and increased to 120 rad/s in the 20 s.

**Figure 16.** Reference speed step change.

The simulation outcomes of the HFFC are presented in Figures 17, 18, 19 and 20. The two-stage control method provides a very fast speed. Due to the control of the current in a final steadystate stage, the oscillations of speed are completely eliminated at a final operating point. It is evident from the comparison of these results with an IFOC with PI-controller that the HFFC shows the two features of FOC controller, i.e., the current feature at the steady state stage and the frequency feature at accelerate-decelerate stage.

**Figure 17.** Fuzzy responses of the fuzzy controllers

**Figure 18.** Stator currents amplitude of HFFC.

<sup>0</sup> <sup>2</sup> <sup>4</sup> <sup>6</sup> <sup>8</sup> <sup>10</sup> <sup>12</sup> <sup>14</sup> <sup>16</sup> <sup>18</sup> <sup>20</sup> -150

Time (s)

Time (sec)

The simulation outcomes of the HFFC are presented in Figures 17, 18, 19 and 20. The two-stage control method provides a very fast speed. Due to the control of the current in a final steadystate stage, the oscillations of speed are completely eliminated at a final operating point. It is evident from the comparison of these results with an IFOC with PI-controller that the HFFC shows the two features of FOC controller, i.e., the current feature at the steady state stage and

> GA-hybrid fuzzy fuzzy controller (GA-HFFC) Hybrid fuzzy-PI controller (HFPIC) Hybrid fuzzy fuzzy controller (HFFC) Indirect field-oriented controller (IFOC)

0 2 4 6 8 10 12 14 16 18 20

Time (s)

Time (sec)



**Figure 17.** Fuzzy responses of the fuzzy controllers



0

Speed (rad/s)

50

100

150

**Figure 16.** Reference speed step change.

the frequency feature at accelerate-decelerate stage.


0

Speed (rad/s)

50

100

150

250 Induction Motors - Applications, Control and Fault Diagnostics

**Figure 19.** Torque response of HFFC.

The simulation outcomes on the efficiency of the controller centered on the performance measures are revealed in Table 7. A closer look at the overshoot, performance criteria, time rise, IAE, final steady-state value, ISE, ITAE, and ITSE is displayed to show reduced values. This validates the act of the modified controller with a better performance to effectively control the speed.

**Figure 20.** Slip frequency response of the fuzzy controllers.


**Table 8.** The performance index comparison of the system.

### **8. Effects of internal and external disturbances**

The mutual inductance and the rotor resistance are expected to be changed to 3 rr and 0.8 LM, respectively, in order to demonstrate the insensitivity of the HFFC to the variation of motor parameters. An insensitivity to the parameter variation shown by the speed response of a fuzzy controller is shown in Figure 21.

Additionally, the distributed random noises are added in the input current and the feedback speed to assess the effects of the noise of the input current and the noise of the speed sensor. The speed response to the current noise and with the measured speed illustrates that the HFFC possesses decent disturbance rejection (see Figure 22).

**Figure 21.** Speed response of HFFC with parameter variations.

0 2 4 6 8 10 12 14 16 18 20

HFPIC HFFC IFOC

Time (sec)

**No. Performance Index HFPIC HFFC IFOC GA-HFFC** Overshoot (%) 0.000 3.000 0.000 0.000 Rise time (sec) 0.4363 0.2803 0.6107 0.2648 Final steady-state value 120.0989 120.08483 120.0681 120.08391 IAE 289.3 194 158.28 186 ISE 3.409e+004 2.311e+004 1.223e+004 2.287e+004 ITAE 2544 1656 1457.6 1625 ITSE 2.691e+005 1.765e+005 1.31e+005 1.799e+005

The mutual inductance and the rotor resistance are expected to be changed to 3 rr and 0.8 LM, respectively, in order to demonstrate the insensitivity of the HFFC to the variation of motor parameters. An insensitivity to the parameter variation shown by the speed response of a fuzzy

Additionally, the distributed random noises are added in the input current and the feedback speed to assess the effects of the noise of the input current and the noise of the speed sensor. The speed response to the current noise and with the measured speed illustrates that the HFFC


**Figure 20.** Slip frequency response of the fuzzy controllers.

252 Induction Motors - Applications, Control and Fault Diagnostics

**Table 8.** The performance index comparison of the system.

controller is shown in Figure 21.

**8. Effects of internal and external disturbances**

possesses decent disturbance rejection (see Figure 22).

Slip frequency (rad/sec)

Furthermore, for studying the impact of magnetic saturation of the induction motor on the controller performance, in the induction motor model, Figures 23 and 24 demonstrate the simulation results of the torque and rotor speed responses. The flux upsurge is limited, owing to the magnetic saturation so that the torque oscillations are decreased considerably, but during the acceleration-deceleration stage, an extreme magnetic saturation will create a higher temperature rise and larger losses.

**Figure 22.** Speed response of HFFC with noise.

**Figure 23.** Speed response of HFFC with the effect of magnetic saturation.

Moreover, the effects of load torque variation on the HFFC system are analysed by the simulation. In the simulation, the control system encounters quick variations in the load torque: at *t* =2 *s*, the load increases from 0*%* to 100*%* of the rated torque, *Tl* , at *t* =7 *s*, the load decreases to 100*%* of *Tl* , and at *t* =10 *s* and 14 *s*, the load rises to 100*%* of *Tl* again, at *t* =19 *s*, the load declines to 100*%* of *Tl* .

**Figure 24.** Torque response of HFFC with effect of magnetic saturation.

The torque reaction of the fuzzy-fuzzy control system with a load torque variation is demon‐ strated in Figure 25. The above-mentioned simulation results illustrate that the greater variations can be created in a load torque by the fuzzy-fuzzy controller. However, the extracted outcomes of performance of the model are found to be corresponding precisely with the anticipations, when it is compared with an IFOC controller. The results also reveal that the HFFC performance is unresponsive to the parameter variation for the motor and the system strength to load and noise disturbances.

**Figure 25.** Torque response of HFFC with load changes.

### **9. The experimental results**

<sup>0</sup> <sup>2</sup> <sup>4</sup> <sup>6</sup> <sup>8</sup> <sup>10</sup> <sup>12</sup> <sup>14</sup> <sup>16</sup> <sup>18</sup> <sup>20</sup> -150

Moreover, the effects of load torque variation on the HFFC system are analysed by the simulation. In the simulation, the control system encounters quick variations in the load torque:

HFFC with magnetic saturation HFFC without magnetic saturation

<sup>0</sup> <sup>2</sup> <sup>4</sup> <sup>6</sup> <sup>8</sup> <sup>10</sup> <sup>12</sup> <sup>14</sup> <sup>16</sup> <sup>18</sup> <sup>20</sup> -500

Time (sec)

, at *t* =7 *s*, the load decreases

again, at *t* =19 *s*, the load

**Figure 23.** Speed response of HFFC with the effect of magnetic saturation.

**Figure 24.** Torque response of HFFC with effect of magnetic saturation.

.

254 Induction Motors - Applications, Control and Fault Diagnostics

at *t* =2 *s*, the load increases from 0*%* to 100*%* of the rated torque, *Tl*

, and at *t* =10 *s* and 14 *s*, the load rises to 100*%* of *Tl*

Time (sec)

HFFC with magnetic saturation HFFC without magnetic saturation


to 100*%* of *Tl*


Torque (N.m)

declines to 100*%* of *Tl*


0

Speed (rad/sec)

50

100

150

The following experiments are conducted to demonstrate and verify the operability of the proposed controller. There are several experiments that can be conducted, however the experimental results presented here are necessary for this purpose. The experimental rig constituting the induction motor speed controller system constructed comprises of the following equipment:


Figure 26 (a), (b), and (c) demonstrate the three-phase voltages waveforms *Va*, *Vb*, and *Vc*, respectively, acquired by DAQ. The actual voltages are 120° different phases from each other.

Figures 27 (a), (b), and (c) show the three-phase sinusoidal PWM for an inverter-fed induction motor in a TMS320F28335 eZdsp control card. These signals are the output from the digital signal processor (DSP) that have been applied to the inverter to control the induction motor. The frequency of switching has been set at 10 kHz. It is worth to point out that for this controller the operating frequency can be applied between 0 to 10 kHz.

**Figure 26.** (a) Va , (b) Vb and (c) Vc waveforms output.

In Figure 28 (a) and (b) are the output signals measured by the encoder at two different sampling rates: 5 kHz and 10 kHz, respectively, which demonstrate the different speeds that can be achieved. The encoder is providing signal to the controller that represents the feedback signals.

**Figure 27.** The three-phase PWM. (a) PWM1, (b) PWM2, and (c) PWM3.

**•** Current sensors

**•** Host PC

**•** LVDAM-EMS software for DAQ

256 Induction Motors - Applications, Control and Fault Diagnostics

Figure 26 (a), (b), and (c) demonstrate the three-phase voltages waveforms *Va*, *Vb*, and *Vc*, respectively, acquired by DAQ. The actual voltages are 120° different phases from each other.

Figures 27 (a), (b), and (c) show the three-phase sinusoidal PWM for an inverter-fed induction motor in a TMS320F28335 eZdsp control card. These signals are the output from the digital signal processor (DSP) that have been applied to the inverter to control the induction motor. The frequency of switching has been set at 10 kHz. It is worth to point out that for this controller

> (a) (b)

X -scale: 20 ms/div Y -scale: 50 V/div

Y

(c)

In Figure 28 (a) and (b) are the output signals measured by the encoder at two different sampling rates: 5 kHz and 10 kHz, respectively, which demonstrate the different speeds that can be achieved. The encoder is providing signal to the controller that represents the feedback

Y

X -scale: 20 ms/div Y -scale: 50 V/div

the operating frequency can be applied between 0 to 10 kHz.

X -scale: 20 ms/div Y -scale: 50 V/div

Y

**Figure 26.** (a) Va , (b) Vb and (c) Vc waveforms output.

signals.

**Figure 28.** Encoder output signals at different sampling rate (a) 5 kHz and (b) 10 kHz.

Figure 29 (a), (b), and (c) show the stator voltage and current in the acceleration and steadystate stages and speed response in the steady-state stage, respectively, looking at one of the three phases. Figure 29 (a) reveals the stator voltage that increases gradually to a constant value. For the stator current, as shown in Figure 29 (b), the current is initially at maximum value and gradually decreases to a constant value. These verify that the controller is perform‐ ing as expected during the acceleration and steady-state stages. Figure 29 (c) represents the speed at steady-state stage, which also proves that the controller is performing as expected, though some spikes are observed that can be related to the encoder used in this set-up. The spikes can be eliminated if a more sensitive encoder has been used.

**Figure 29.** (a) Stator voltage, (b) stator current, and (c) speed response outputs.

### **10. Conclusions**

The objective of this study is to elaborate and elucidate the effects and performance of internal and external disturbances for an established HFFC scheme to accordingly modify the speed of an induction motor. The fuzzy-fuzzy controller has been proven to be more effective as compared to a scalar controller due to the utilisation of the two aspects of the FOC. Besides, one of the key advantages of this controller includes the supply of the same FOC and frequency that is unresponsive to the parameter variation in the motor and system strength to noise and load disturbances. This study, under dynamic settings, produced a comprehensive evaluation and analysis of the three controllers, HFFC, IFOC, and HFPIC. The experimental results verify the performance of the proposed HFFC in controlling the IM variable speed drive. Therefore, for further enhancing the IM-VSD performance, the work to be considered includes the advancement in the augmentation of the controllers to improve the VSD performances.

### **Acknowledgements**

value and gradually decreases to a constant value. These verify that the controller is perform‐ ing as expected during the acceleration and steady-state stages. Figure 29 (c) represents the speed at steady-state stage, which also proves that the controller is performing as expected, though some spikes are observed that can be related to the encoder used in this set-up. The

> (a) (b)

X -scale: 50 ms/div Y -scale: 0.5 A/div

Y

(c)

The objective of this study is to elaborate and elucidate the effects and performance of internal and external disturbances for an established HFFC scheme to accordingly modify the speed of an induction motor. The fuzzy-fuzzy controller has been proven to be more effective as compared to a scalar controller due to the utilisation of the two aspects of the FOC. Besides, one of the key advantages of this controller includes the supply of the same FOC and frequency that is unresponsive to the parameter variation in the motor and system strength to noise and load disturbances. This study, under dynamic settings, produced a comprehensive evaluation and analysis of the three controllers, HFFC, IFOC, and HFPIC. The experimental results verify the performance of the proposed HFFC in controlling the IM variable speed drive. Therefore,

**Figure 29.** (a) Stator voltage, (b) stator current, and (c) speed response outputs.

Y

X -scale: 0.2 ms/div Y -scale: 500 r/min/div

spikes can be eliminated if a more sensitive encoder has been used.

258 Induction Motors - Applications, Control and Fault Diagnostics

X -scale: 50 ms/div Y -scale: 50 V/div

Y

**10. Conclusions**

The authors acknowledge the support from the Universiti Teknologi PETRONAS through the award of the Graduate Assistantship scheme and URIF.

### **Author details**

Nordin Saad\* , Muawia A. Magzoub, Rosdiazli Ibrahim and Muhammad Irfan

\*Address all correspondence to: nordiss@petronas.com.my

Department of Electrical and Electronic Engineering, Universiti Teknologi Petronas, Bandar Sri Iskandar, Perak, Malaysia

### **References**


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260 Induction Motors - Applications, Control and Fault Diagnostics


## **DTC-FPGA Drive for Induction Motors**

Rafael Rodríguez-Ponce, Fortino Mendoza-Mondragón, Moisés Martínez-Hernández and Marcelino Gutiérrez-Villalobos

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/60871

### **Abstract**

Direct torque control, or DTC, is an electrical motor strategy characterized for simplicity and high performance when controlling industrial machines such as induction motors. However, this technique is often accompanied by an unwanted deformation on the torque and flux signals denominated ripple, which can cause audible noise and vibration on the motor. Considerable research has been presented on this topic; nevertheless the original DTC algorithm is often modified to the point that it is as complex as other motor control strategies. To solve this problem, a novel architecture was designed in order to reduce the sampling period to a point where torque ripple is minimal, while maintaining the classical DTC control structure. In this work, the original DTC control strategy was implemented on a Virtex-5 field programmable gate array (FPGA). For the code, a two´s complement fixed-point format and a variable word-size approach was followed using very-high-speed integrated circuit hardware description language (VHDL). Results were validated using MATLAB/Simulink simulations and experimental tests on an induction motor. With this new architecture, the authors hope to provide guidelines and insights for future research on DTC drives for induction motors.

**Keywords:** Direct torque control, AC servo drive, Field programmable gate array

### **1. Introduction**

It is well known that one of the components most commonly found in any industrial or residential machine is the electric motor. Motors are used almost in any application where

© 2015 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

electricity must be converted to a mechanical motion of some kind. They come in a wide variety of sizes, ranging from very small, as the motors found inside a cellphone for creating vibration, to very large, as the ones used in wind tunnels for aircraft testing.

There are many types of electrical motors but one that has remained the favorite for almost any medium- to large-sized application is the AC induction motor (ACIM). The concept of this "sparkless" motor was first conceived by Nicola Tesla in the late nineteenth century. Although it was first designed as a polyphase structure that consisted of two stator phases in an orthogonal relationship, it has since been modified to a more common three-phase structure, which results in a more balanced operation of the motor voltages and currents [1]. The ACIM is rugged and highly reliable, can be manufactured at a low cost, and is almost maintenancefree, except for bearings and other external mechanical parts.

The ACIM is essentially a fixed-speed machine. However, most industrial applications require a motor in which torque or speed can easily be controlled. Therefore, several high-performance control strategies have been developed for AC motors; two of the most popular motor control methods are field oriented control (FOC) and direct torque control (DTC) [2]. Unlike FOC, DTC is characterized for its simplicity since it does not require PI regulators, coordinate transfor‐ mations, pulse width modulation (PWM) generators, or position encoders on the motor shaft [3]. In spite of its simplicity, DTC provides fast torque control in the steady state and under transient operating conditions with simple control structure [4].

One major disadvantage of DTC is that it has the distinct characteristic of ripple on torque and flux signals; this ripple is an unwanted deformation or "noise" on motor signals that can lead to audible noise and vibration in the motor [5]. It is possible to reduce ripple by reducing the sampling period [6]; for this reason, recent DTC drives have been implemented by using fast processing devices such as FPGAs [7].

The purpose of this chapter is to present the development of an ACIM drive on an FPGA; the original DTC strategy was implemented by using a fixed-point architecture on a Xilinx Virtex-5 development board and used on real-time experiments. By using an FPGA and a novel architecture, it was possible to reduce processing time to 1.6 μs, therefore reducing torque ripple to a minimum.

This chapter is organized as follows: Section 2 presents a mathematical model for the induction motor. Section 3 presents a simplified description of the DTC strategy. A detailed description of the FPGA-based induction motor drive is described in Section 4. Simula‐ tion and experimental results are presented in Section 5. Finally, the conclusion of this work is given in Section 6.

### **2. Induction motor mathematical model**

The cage rotor induction machine is widely used in industrial applications, such as belt conveyors, pumps, fans, cranes, etc. It presents great mechanical sturdiness and there is a good standardization between ACIM manufacturers worldwide. Nevertheless, the relative simplic‐ ity of the operation of the motor hides a great complexity, especially when it is aimed at controlling the performed electromechanical conversion [1].

There are a number of ACIM models; the model used for vector control design can be obtained by using the space vector theory. The 3-phase motor quantities, such as currents, voltages, and magnetic fluxes, are expressed in terms of complex space vectors. Such a model is valid for any instantaneous variation of voltage and current and adequately describes the performance of the machine under steady-state and transient operation. The motor is considered to be a 2 phase machine by using two orthogonal axes; with this model, the number of equations is reduced and the control design is simplified [8].

When describing a three-phase IM by a system of equations, the following simplifying assumptions are made [8]:

**•** The three-phase motor is symmetrical.

electricity must be converted to a mechanical motion of some kind. They come in a wide variety of sizes, ranging from very small, as the motors found inside a cellphone for creating vibration,

There are many types of electrical motors but one that has remained the favorite for almost any medium- to large-sized application is the AC induction motor (ACIM). The concept of this "sparkless" motor was first conceived by Nicola Tesla in the late nineteenth century. Although it was first designed as a polyphase structure that consisted of two stator phases in an orthogonal relationship, it has since been modified to a more common three-phase structure, which results in a more balanced operation of the motor voltages and currents [1]. The ACIM is rugged and highly reliable, can be manufactured at a low cost, and is almost maintenance-

The ACIM is essentially a fixed-speed machine. However, most industrial applications require a motor in which torque or speed can easily be controlled. Therefore, several high-performance control strategies have been developed for AC motors; two of the most popular motor control methods are field oriented control (FOC) and direct torque control (DTC) [2]. Unlike FOC, DTC is characterized for its simplicity since it does not require PI regulators, coordinate transfor‐ mations, pulse width modulation (PWM) generators, or position encoders on the motor shaft [3]. In spite of its simplicity, DTC provides fast torque control in the steady state and under

One major disadvantage of DTC is that it has the distinct characteristic of ripple on torque and flux signals; this ripple is an unwanted deformation or "noise" on motor signals that can lead to audible noise and vibration in the motor [5]. It is possible to reduce ripple by reducing the sampling period [6]; for this reason, recent DTC drives have been implemented by using fast

The purpose of this chapter is to present the development of an ACIM drive on an FPGA; the original DTC strategy was implemented by using a fixed-point architecture on a Xilinx Virtex-5 development board and used on real-time experiments. By using an FPGA and a novel architecture, it was possible to reduce processing time to 1.6 μs, therefore reducing torque

This chapter is organized as follows: Section 2 presents a mathematical model for the induction motor. Section 3 presents a simplified description of the DTC strategy. A detailed description of the FPGA-based induction motor drive is described in Section 4. Simula‐ tion and experimental results are presented in Section 5. Finally, the conclusion of this work

The cage rotor induction machine is widely used in industrial applications, such as belt conveyors, pumps, fans, cranes, etc. It presents great mechanical sturdiness and there is a good standardization between ACIM manufacturers worldwide. Nevertheless, the relative simplic‐

to very large, as the ones used in wind tunnels for aircraft testing.

264 Induction Motors - Applications, Control and Fault Diagnostics

free, except for bearings and other external mechanical parts.

transient operating conditions with simple control structure [4].

processing devices such as FPGAs [7].

**2. Induction motor mathematical model**

ripple to a minimum.

is given in Section 6.


Taking into consideration the earlier-stated assumptions, the following equations of the instantaneous stator phase voltage values can be written as follows (all variable descriptions are listed in Table 1):

$$
\Psi V\_A = \mathcal{R}\_s i\_A + \frac{d}{dt} \Psi\_A \tag{1}
$$

$$\Psi V\_{\rm B} = \mathcal{R}\_{\rm s} i\_{\rm B} + \frac{d}{dt} \Psi\_{\rm B} \tag{2}$$

$$\mathbf{V}\_{\text{C}} = \mathbf{R}\_{\text{s}} \mathbf{i}\_{\text{C}} + \frac{d}{dt} \boldsymbol{\Psi}\_{\text{C}} \tag{3}$$

A three-phase variable system can be uniquely described through a space vector, which is a complex term and time-dependent *k*(*t*) and a real homopolar component *k*0(*t*) as follows:

$$k(t) = \frac{2}{3} \left[ 1^\* k\_A + a^\* k\_B + a^{2\*} k\_C \right] \tag{4}$$

$$k\_0(t) = \frac{1}{3} \left[ k\_A + k\_B + k\_C \right] \tag{5}$$

The real axis direction coincides with that one of phase *A*. Usually, the neutral connection for a three-phase system is open, so that the homopolar component equals zero.

The ACIM model is given by the space vector form of the voltage equations. The system model defined in a two-phase stationary (α, *β*) coordinate system attached to the stator is expressed by the following equations:

**a.** The stator voltage differential equations:

$$\boldsymbol{V}\_{s\boldsymbol{\alpha}} = \mathbf{R}\_s \mathbf{i}\_{s\boldsymbol{\alpha}} + \frac{d}{dt} \boldsymbol{\Psi}\_{s\boldsymbol{\alpha}} \tag{6}$$

$$\mathbf{V}\_{s\rho} = \mathbf{R}\_s \mathbf{i}\_{s\rho} + \frac{d}{dt} \mathbf{\varPsi}\_{s\rho} \tag{7}$$

**b.** The rotor voltage differential equations:

$$\Psi V\_{ra} = 0 = \mathcal{R}\_r i\_{ra} + \frac{d}{dt} \Psi\_{ra} + o\Psi\_{r\beta} \tag{8}$$

$$V\_{r\beta} = 0 = R\_r i\_{r\beta} + \frac{d}{dt} \Psi\_{r\beta} + o\vartheta \Psi\_{ra} \tag{9}$$

**c.** The stator and rotor flux linkages expressed in terms of the stator and rotor current space vectors:

$$
\Psi\_{s\alpha} = L\_s \mathbf{i}\_{s\alpha} + L\_w \mathbf{i}\_{ra} \tag{10}
$$

$$
\Psi\_{s\rho} = L\_s \mathbf{i}\_{s\rho} + L\_m \mathbf{i}\_{r\rho} \tag{11}
$$

$$
\Psi\_{ra} = L\_r \mathbf{i}\_{ra} + L\_m \mathbf{i}\_{sa} \tag{12}
$$

$$
\Psi\_{r\rho} = L\_r \mathbf{i}\_{r\rho} + L\_m \mathbf{i}\_{s\rho} \tag{13}
$$

**d.** The electromagnetic torque expressed by utilizing space vector quantities:

0

by the following equations:

vectors:

**a.** The stator voltage differential equations:

266 Induction Motors - Applications, Control and Fault Diagnostics

**b.** The rotor voltage differential equations:

a three-phase system is open, so that the homopolar component equals zero.

<sup>1</sup> ( ) <sup>3</sup> *ABC kt k k k* = ++ é ù

The real axis direction coincides with that one of phase *A*. Usually, the neutral connection for

The ACIM model is given by the space vector form of the voltage equations. The system model defined in a two-phase stationary (α, *β*) coordinate system attached to the stator is expressed

> *s ss* Ψ*<sup>s</sup> <sup>d</sup> V Ri*

> *s ss* Ψ*<sup>s</sup> <sup>d</sup> V Ri*

0 *<sup>r</sup> r r r r*

0 *<sup>r</sup> r r r r*

Ψ*s ss mr Li L i* aa

Ψ*s ss mr Li L i* bb

Ψ*r rr ms Li L i* aa

Ψ*r rr ms Li L i* bb

 b

*dt*

**c.** The stator and rotor flux linkages expressed in terms of the stator and rotor current space

= = + Y +Y

 a *dt*

= = + Y +Y

 ab

 ba

> a

> > b

 a

 b w

w

aa

bb

*<sup>d</sup> V Ri*

*<sup>d</sup> V Ri*

a

b

*dt*

*dt*

 a

 b

ë û (5)

= + (6)

= + (7)

= + (10)

= + (11)

= + (12)

= + (13)

(8)

(9)

$$T\_c = \frac{3}{2} P \left( \Psi\_{sa} \dot{\mathbf{i}}\_{s\rho} - \Psi\_{s\rho} \dot{\mathbf{i}}\_{sa} \right) \tag{14}$$

The ACIM model is often used in vector control algorithms. The aim of vector control is to implement control schemes that produce high-dynamic performance and are similar to those used to control DC machines [2]. To achieve this, the reference frames may be aligned with the stator flux-linkage space vector, the space vector of the rotor current in the rotor reference frame, the rotor flux-linkage space vector, or the magnetizing space vector. The most popular reference frame is the reference frame attached to the rotor flux linkage space vector with direct axis (d) and quadrature axis (q) [8].

After transformation into d-q coordinates the motor model follows:

$$\boldsymbol{V}\_{sd} = \boldsymbol{R}\_{s}\boldsymbol{i}\_{sd} + \frac{d}{dt}\boldsymbol{\Psi}\_{sd} - \boldsymbol{\alpha}\_{s}\boldsymbol{\Psi}\_{sq} \tag{15}$$

$$\boldsymbol{V}\_{sq} = \boldsymbol{R}\_s \boldsymbol{i}\_{sq} + \frac{d}{dt} \boldsymbol{\Psi}\_{sq} - \alpha\_s \boldsymbol{\Psi}\_{sd} \tag{16}$$

$$
\Psi\_{rd} = 0 = R\_r i\_{rd} + \frac{d}{dt} \Psi\_{rd} - (\phi\_s - \alpha) \Psi\_{rq} \tag{17}
$$

$$\Psi\_{rq} = 0 = R\_r \dot{\mathbf{i}}\_{rq} + \frac{d}{dt} \Psi\_{rq} - (\alpha\_s - \alpha) \Psi\_{rd} \tag{18}$$

$$
\Psi \Psi\_{sd} = L\_s \mathbf{i}\_{sd} + L\_m \mathbf{i}\_{rd} \tag{19}
$$

$$
\Psi\_{sq} = L\_s i\_{sq} + L\_w i\_{rq} \tag{20}
$$

$$
\Psi\_{rd} = L\_r \dot{\mathbf{i}}\_{rd} + L\_m \dot{\mathbf{i}}\_{sd} \tag{21}
$$

$$
\Psi\_{rq} = L\_r i\_{rq} + L\_w i\_{sq} \tag{22}
$$

$$T\_e = \frac{3}{2} P \left( \Psi\_{sd} i\_{sq} - \Psi\_{sq} i\_{sd} \right) \tag{23}$$


**Table 1.** Variable description.

### **3. Classical DTC scheme**

The theory for the DTC control strategy was developed by Manfred Depenbrock as direct selfcontrol (DSC) and separately, as direct torque control (DTC) by Isao Takahashi and Toshihiko Noguchi, both in the mid-1980s, although the DTC innovation is usually credited to all three individuals [2]. A block diagram of the DTC strategy is shown in Figure 1.

The main objective of DTC is to estimate instantaneous values of torque and magnetic flux, based on motor current and voltage. Torque and flux vectors are controlled directly and independently by selecting the appropriate inverter voltage vector that will maintain torque and flux errors within the hysteresis comparator limits [3].

**Figure 1.** Direct torque control block diagram.

**Variable Description** *VA*, *VB*, *VC* Instantaneous values of the stator phase voltages

Ψ*A*, Ψ*B*, Ψ*<sup>C</sup>* Flux linkages of the stator phase windings

*α*, *β* Stator orthogonal coordinate system

*kA*, *kB*, *kC* Arbitrary phase variables *a*, *a* <sup>2</sup> Spatial operators *a* =*e <sup>j</sup>*2*π*/3

*Vs<sup>α</sup>*,*<sup>β</sup>* Stator voltages [V]

*<sup>s</sup>α*,*<sup>β</sup>* Stator currents [A] *Vr<sup>α</sup>*,*<sup>β</sup>* Rotor voltages [V]

*<sup>r</sup>α*,*<sup>β</sup>* Rotor currents [A]

Ψ*sα*,*<sup>β</sup>* Stator magnetic fluxes [Wb] Ψ*rα*,*<sup>β</sup>* Rotor magnetic fluxes [Wb] *Rs* Stator phase resistance [Ohm] *Rr* Rotor phase resistance [Ohm] *L <sup>s</sup>* Stator phase inductance [H] *L <sup>r</sup>* Rotor phase inductance [H]

*L <sup>m</sup>* Mutual (stator to rotor) inductance [H]

*P* Number of pole pairs

*Te* Electromagnetic torque [Nm]

individuals [2]. A block diagram of the DTC strategy is shown in Figure 1.

and flux errors within the hysteresis comparator limits [3].

*ω* / *ω<sup>s</sup>* Electrical rotor speed/synchronous speed [rad/s]

The theory for the DTC control strategy was developed by Manfred Depenbrock as direct selfcontrol (DSC) and separately, as direct torque control (DTC) by Isao Takahashi and Toshihiko Noguchi, both in the mid-1980s, although the DTC innovation is usually credited to all three

The main objective of DTC is to estimate instantaneous values of torque and magnetic flux, based on motor current and voltage. Torque and flux vectors are controlled directly and independently by selecting the appropriate inverter voltage vector that will maintain torque

*<sup>C</sup>* Instantaneous values of the stator phase currents

and *a* <sup>2</sup> =*e <sup>j</sup>*4*π*/3

*i <sup>A</sup>*, *i <sup>B</sup>*, *i*

268 Induction Motors - Applications, Control and Fault Diagnostics

*i*

*i*

**Table 1.** Variable description.

**3. Classical DTC scheme**

In order to estimate the motor torque and flux values, the instantaneous current (*i a*, *i <sup>b</sup>*) and DC bus voltage (*Vcd* ) signals are obtained from the ACIM as illustrated in Figure 1. These analog signals are converted to digital values by means of an analog to digital converter (ADC). The current and voltage signals, as well as the current state of the voltage source inverter (VSI) vector (*Sa*, *Sb*, *Sc*), are transformed from a 3-phase reference frame to a 2-phase reference frame (*α*, *β*), as follows:

**a.** The *α*, *β* current signals:

$$\mathbf{i}\_a = \mathbf{i}\_a \tag{24}$$

$$\dot{\mathbf{u}}\_{\rho} = \frac{\sqrt{3}}{3} \left( \dot{\mathbf{i}}\_{a} + 2\dot{\mathbf{i}}\_{b} \right) \tag{25}$$

**b.** The *α*, *β* voltage signals:

$$V\_{\alpha} = \frac{V\_{cd}}{3} \left(2\mathbf{S}\_{a} - \mathbf{S}\_{b} - \mathbf{S}\_{c}\right) \tag{26}$$

$$V\_{\rho} = \frac{\sqrt{3}}{3} V\_{cd} \left( \mathbf{S}\_b - \mathbf{S}\_c \right) \tag{27}$$

**c.** The *α*, *β* flux components:

$$
\sigma\_a = \sigma\_{a0} + \left(V\_a - R\_s i\_a\right) T\_s \tag{28}
$$

$$
\sigma\_{\rho} = \sigma\_{\rho 0} + \left(V\_{\rho} - R\_s i\_{\rho}\right) T\_s \tag{29}
$$

where

*Ts* – Sampling period [seconds]

*Rs* – Stator resistance [ohms]

*φα*0, *φβ*0 – Previous flux component value [Wb]

Based on this data, the flux magnitude and the electromagnetic torque are obtained as follows:

**a.** The stator magnetic flux magnitude:

$$
\sigma \varphi\_s = \sqrt{\left(\varphi\_\alpha^{\;\;2}\right) + \left(\varphi\_\beta^{\;\;2}\right)}\tag{30}
$$

**b.** The electromagnetic torque:

$$T\_c = \frac{3}{2} P \left( \varphi\_\alpha \mathbf{i}\_\beta - \varphi\_\beta \mathbf{i}\_\alpha \right) \tag{31}$$

In order to re-orient the flux vector *φs*, first it is necessary to determine where it is localized. For this reason, the flux vector circular trajectory is divided into six symmetrical sectors, as shown in Figure 2.

The angle *θs* can be calculated, based on the *α*, *β* flux components as follows:

$$\theta\_s = \tan^{-1} \frac{\varphi\_\rho}{\varphi\_a} \tag{32}$$

However, implementing Eq. (32) in an FPGA is complex and time consuming and is usually performed be means of the coordinate rotation digital computer (CORDIC) algorithm [9]. Instead, it is possible to determine the sector in which the flux vector is located, based on the signs of the flux components, as described in [3]. The sector can be determined by using Table 2 and Eq. (33).

**Figure 2.** Sectors of the flux vector circular trajectory.

**c.** The *α*, *β* flux components:

270 Induction Motors - Applications, Control and Fault Diagnostics

*Ts* – Sampling period [seconds]

*φα*0, *φβ*0 – Previous flux component value [Wb]

**a.** The stator magnetic flux magnitude:

**b.** The electromagnetic torque:

shown in Figure 2.

2 and Eq. (33).

*Rs* – Stator resistance [ohms]

where

j j aa

j j bb  a

 b

Based on this data, the flux magnitude and the electromagnetic torque are obtained as follows:

( ) ( ) 2 2

( ) <sup>3</sup> <sup>2</sup> *<sup>e</sup> T Pi i* ab

 = j

The angle *θs* can be calculated, based on the *α*, *β* flux components as follows:

q

 b

> ba

 j

In order to re-orient the flux vector *φs*, first it is necessary to determine where it is localized. For this reason, the flux vector circular trajectory is divided into six symmetrical sectors, as

> 1 *<sup>s</sup> tan*

b

j

j

However, implementing Eq. (32) in an FPGA is complex and time consuming and is usually performed be means of the coordinate rotation digital computer (CORDIC) algorithm [9]. Instead, it is possible to determine the sector in which the flux vector is located, based on the signs of the flux components, as described in [3]. The sector can be determined by using Table

a

 j

a

*s*

jj

 a

> b

=+- <sup>0</sup> (*V Ri T s s* ) (28)

=+- <sup>0</sup> (*V Ri T s s* ) (29)

= + (30)

(31)



**Table 2.** Stator flux space vector's sector.

$$\left|\boldsymbol{\varrho}\right|\_{\mathrm{ref}} = \sqrt{3}\left|\boldsymbol{\varrho}\_{\boldsymbol{\beta}}\right| - \left|\boldsymbol{\varrho}\_{a}\right|\tag{33}$$

For example, if both flux components are positive and the result of Eq. (33) is also positive, then the flux vector is located in sector 2. Instead, if the result of the equation is negative, the vector is located in sector 1.

The method described previously to determine the sector of the flux vector is easier to implement in a digital device and can be processed faster since it consists of a simple data table. As shown in Figure 1, the estimated magnetic flux and electromagnetic torque values are compared with the magnetic flux reference and the electromagnetic torque reference, respec‐ tively. The flux and torque errors (*eφ*, *eT* ) are delivered to the hysteresis controllers.

A two-level hysteresis controller is used to establish the limits of the flux error. For the torque error, a three-level hysteresis controller is used. The hysteresis controllers are shown in Figure 3.

**Figure 3.** Hysteresis controllers for (a) flux and for (b) torque.

The hysteresis controller output signals *ϕ* and *τ* are defined as follows:

$$
\phi = 1 \text{ for } e\_{\phi} > +L\_{\phi} \tag{34}
$$

$$
\phi = 0 \text{ for } e\_{\phi} < -L\_{\phi} \tag{35}
$$

$$
\tau = 1 \text{ for } e\_T > +L\_\text{r} \tag{36}
$$

$$
\tau = 0 \text{ for } e\_T = 0 \tag{37}
$$

$$
\tau = -1 \text{ for } e\_T < \text{ -}L\_r \tag{38}
$$

The digitized output variables *ϕ*, *τ* and the stator flux sector determine the appropriate voltage vector from the inverter switching table. Thus, the selection table generates pulses *Sa*, *Sb*, *Sc* to control the power switches in the inverter in order to generate six possible active vectors (v1-v6) and two zero vectors (v0, v7), as shown in Figure 4.

**Figure 4.** Voltage vectors based on eight possible inverter states.

As shown in Figure 1, the estimated magnetic flux and electromagnetic torque values are compared with the magnetic flux reference and the electromagnetic torque reference, respec‐

A two-level hysteresis controller is used to establish the limits of the flux error. For the torque error, a three-level hysteresis controller is used. The hysteresis controllers are shown

a) b)

DTᵉ

= >+ (34)

= <- (35)

= >+ (36)

= = *e* (37)

=- < (38)

t

*+L<sup>T</sup> -L<sup>T</sup> eT*

tively. The flux and torque errors (*eφ*, *eT* ) are delivered to the hysteresis controllers.

in Figure 3.

Dj<sup>s</sup>

**Figure 3.** Hysteresis controllers for (a) flux and for (b) torque.

The hysteresis controller output signals *ϕ* and *τ* are defined as follows:

f

f

t

t

t

vectors (v1-v6) and two zero vectors (v0, v7), as shown in Figure 4.

1 for *e L* j

0 for *e L* j

1 for *Te L*

0 for 0 *<sup>T</sup>*

1 for - *Te L*

The digitized output variables *ϕ*, *τ* and the stator flux sector determine the appropriate voltage vector from the inverter switching table. Thus, the selection table generates pulses *Sa*, *Sb*, *Sc* to control the power switches in the inverter in order to generate six possible active

 j

> j

t

> t

*-L*<sup>j</sup> *<sup>+</sup>L*<sup>j</sup> *e*<sup>j</sup>

f

272 Induction Motors - Applications, Control and Fault Diagnostics

For the stator flux vector laying in sector 1 (Figure 5), in order to increase its magnitude, voltage vectors v1, v2 or v6 can be selected. Conversely, a decrease can be obtained by selecting v3, v4 or v5. By applying one of the voltage vectors v0 or v7, the stator flux vector is not changed.

**Figure 5.** Selection of the optimum voltage vectors for the stator flux vector in sector 1.

For torque control, if the vector is moving as indicated in Figure 5, the torque can be increased by selecting vectors v2, v3 or v4. To decrease torque, vectors v1, v5 or v6 can be selected.

The above considerations allow the construction of the inverter switching table as presented in Table 3.


**Table 3.** Optimum switching table.

The optimal voltage vector is a vector such that, once applied to the VSI, will maintain the flux and torque signals within the hysteresis comparator limits [4]. The selected voltage vector is applied at the end of the sampling period.

### **4. DTC Digital Implementation**

In this section, a detailed description of a DTC drive for induction motors is presented. The drive was implemented on a Xilinx Virtex-5 FPGA based on two´s complement fixed-point architecture composed of 7 main blocks, which are described as follows and shown in Figure 6:


**g.** Global control block: a finite state machine (FSM) is included in this block and is in charge of the control of all the other DTC blocks.

**Figure 6.** DTC architecture on Xilinx Virtex5 FPGA.

The above considerations allow the construction of the inverter switching table as presented

The optimal voltage vector is a vector such that, once applied to the VSI, will maintain the flux and torque signals within the hysteresis comparator limits [4]. The selected voltage vector is

In this section, a detailed description of a DTC drive for induction motors is presented. The drive was implemented on a Xilinx Virtex-5 FPGA based on two´s complement fixed-point architecture composed of 7 main blocks, which are described as follows and shown in Figure

**a.** Conversion control block: this block controls 3 external 12-bit serial, ADCs that operate in parallel. The motor current and voltage signals are converted from serial to parallel. Since both signals are scaled versions of the original, in this block both values are restored

**b.** Torque and flux estimator: the real time electromagnetic torque and magnetic flux vectors

**d.** Reference comparison block: the real torque and flux estimated values are compared with

**e.** Hysteresis comparators: two-level and three-level hysteresis comparators are included in

**f.** Switching table: all the optimal voltage vectors for the inverter are contained in this block.

**c.** Flux sector detection block: the sector for the magnetic flux vector is detected.

are estimated based on motor current and voltage signals.

**Sector 1 2 3 4 5 6**

*τ* =1 v2(110) v3(010) v4(011) v5(001) v6(101) v1(100) *τ* =0 v7(111) v0(000) v7(111) v0(000) v7(111) v0(000) *τ* = −1 v6(101) v1(100) v2(110) v3(010) v4(011) v5(001)

*τ* =1 v3(010) v4(011) v5(001) v6(101) v1(100) v2(110) *τ* =0 v0(000) v7(111) v0(000) v7(111) v0(000) v7(111) *τ* = −1 v5(001) v6(101) v1(100) v2(110) v3(010) v4(011)

in Table 3.

*φ* =1

*φ* =0

6:

*φ τ*

**Table 3.** Optimum switching table.

to their real value.

this block.

torque and flux references.

applied at the end of the sampling period.

274 Induction Motors - Applications, Control and Fault Diagnostics

**4. DTC Digital Implementation**

One of the benefits of this DTC architecture presented is that it is completely generic; the data width can be modified depending on the application or the precision required, and all the DTC equations will adjust automatically. The flux data path has *n* bits while the torque data path can have *m* bits, as shown in Figure 6. The data paths can be extended for more precision, however this will also extend the sampling time. For this project, the flux data path was left at 20 bits and the torque data path was set to 23 bits in order to achieve a low sampling period of 1.6 μs.

### **4.1. Conversion control block**

The current signals (*ia, ib*) are first obtained by sensing two of the motor lines by means of coil sensors. The output is an AC signal that is amplified and added a DC offset, in order to have a positive only value between 0 and 3.3 V for the ADC. The signals are then converted by the ADC to a serial 12-bit value and then to a 12-bit parallel value. Finally, the offset value is subtracted and multiplied by a scaling factor in order to obtain the original current signal. This process is shown in Figure 7.

**Figure 7.** Current signal conversion process from current sensor to conversion control block.

The conversion process for the DC bus voltage signal is similar to the process described for the current signals, except that a current sensor is not used. Instead, by means of a resistive voltage divider, the voltage signal is reduced to a suitable value. Since the signal is always positive, there is no need to add an offset. The signal is only filtered and passed through several operational amplifier (OP AMP) stages, in order to isolate and adjust to a specific value between 0 and 3.3 V.

The DC voltage signal (*Vdc*) is converted to digital using a serial 12-bit ADC; the serial signal is converted to a parallel value and then multiplied by a scaling factor to restore it to the original DC value. The conversion process for the DC voltage signal is shown in Figure 8.

**Figure 8.** Voltage signal conversion process.

### **4.2. Torque and flux estimator**

The estimation block is the most important part of the DTC process, since the selection of the optimal voltage vector for the VSI depends on the accuracy of the magnetic flux vector [4].

The flux estimator was designed in VHDL to execute Eq. (24-31) presented previously in Section 3, where basically the stator flux is calculated based on stator currents and voltages; once flux stationary components are calculated, the stator flux can be obtained by adding both components squared and applying the square root operation.

Several equations are implemented in parallel, such as the voltage and current transformation to stationary coordinates and later the stator flux stationary components.

In FPGA implementation, word size is critical; a large word size reduces quantization errors but increases area and affects costs. On the contrary, a small word size affects precision, increasing control error and torque ripple [10]. Therefore, a fixed-point format with a variable word size was used in the implementation of the DTC equations.

The DTC architecture was designed for implementation on an FPGA with data words starting at 12 bits and increased according to the mathematical operations to avoid a loss in precision.

The estimator was divided in three stages as follows:

Stage 1 – In the first stage, the values of *ia*, *ib* and *Vcd,* and the previous inverter vector (*Sa*, *Sb*, *Sc*) are used to calculate the corresponding stationary components *iα*, *iβ*, *Vα* and V*β*. This stage is shown in Figure 9.

**Figure 9.** First stage of flux and torque estimator.

The conversion process for the DC bus voltage signal is similar to the process described for the current signals, except that a current sensor is not used. Instead, by means of a resistive voltage divider, the voltage signal is reduced to a suitable value. Since the signal is always positive, there is no need to add an offset. The signal is only filtered and passed through several operational amplifier (OP AMP) stages, in order to isolate and adjust to a specific value

The DC voltage signal (*Vdc*) is converted to digital using a serial 12-bit ADC; the serial signal is converted to a parallel value and then multiplied by a scaling factor to restore it to the original

> Control Signals

The estimation block is the most important part of the DTC process, since the selection of the optimal voltage vector for the VSI depends on the accuracy of the magnetic flux vector [4].

The flux estimator was designed in VHDL to execute Eq. (24-31) presented previously in Section 3, where basically the stator flux is calculated based on stator currents and voltages; once flux stationary components are calculated, the stator flux can be obtained by adding both

Several equations are implemented in parallel, such as the voltage and current transformation

In FPGA implementation, word size is critical; a large word size reduces quantization errors but increases area and affects costs. On the contrary, a small word size affects precision, increasing control error and torque ripple [10]. Therefore, a fixed-point format with a variable

The DTC architecture was designed for implementation on an FPGA with data words starting at 12 bits and increased according to the mathematical operations to avoid a loss in precision.

Stage 1 – In the first stage, the values of *ia*, *ib* and *Vcd,* and the previous inverter vector (*Sa*, *Sb*, *Sc*) are used to calculate the corresponding stationary components *iα*, *iβ*, *Vα* and V*β*. This stage

<sup>0</sup> 120V

A/D S P 7.5

3.3 0

x FACT

7.5

0

Conversion Block

DC value. The conversion process for the DC voltage signal is shown in Figure 8.

3.3V 0

OP AMP

components squared and applying the square root operation.

word size was used in the implementation of the DTC equations.

The estimator was divided in three stages as follows:

to stationary coordinates and later the stator flux stationary components.

between 0 and 3.3 V.

DC Bus Voltage

+120V

C1 R2

is shown in Figure 9.

R1

1.2V

276 Induction Motors - Applications, Control and Fault Diagnostics

**Figure 8.** Voltage signal conversion process.

**4.2. Torque and flux estimator**

In the previous figure, the fixed-point format is indicated in each vertical line and width adjustments are made when required. At the end of this stage, four 22-bit parallel registers restrict data flow until they receive a pulse from the estimator FSM; this assures that all values pass to the next stage at the same time.

Stage 2 – In this stage, flux stationary components *φα* and *φβ* are calculated based on data from stage 1 as described in (28) and (29). Both components are calculated based on the same equation, therefore a generic block was designed for this calculation and is used twice in parallel. The architecture for this stage is shown in Figure 10.

In this stage, both flux components are loaded to the register by a pulse from the estimator FSM, which serves as the previous flux value (*φα0* or *φβ0*) for the next calculation.

Stage 3 – In this last stage, the flux components are squared, added, and then the square root (SQRT) algorithm is applied as in Eq. (30); a special architecture was designed for the SQRT and will be described in detail later in section 4.8. The stator torque is calculated, by means of Eq. (31), in parallel with the flux equation. The architecture for this last stage is shown in Figure 11.

### **4.3. Flux sector detection block**

Based on the value and sign of the stationary flux components, the flux vector sector is determined by means of Eq.(33) and Table 2. The signs of the flux components are used to

**Figure 10.** Second stage of flux and torque estimator.

**Figure 11.** Last stage of flux and torque estimator.

determine the quadrant of the flux vector and the value of *φref* is used for selecting between the upper or lower sectors in that quadrant. The architecture for this block is shown in Figure 12.

**Figure 12.** Architecture of the sector detection block.

### **4.4. Reference comparison block**

Register

+ 2.34

2.34

**Figure 10.** Second stage of flux and torque estimator.

4.60 \*

> + 4.60 SQRT 2.30

Adjust 1.19 Register 1.19 *s*

**Figure 11.** Last stage of flux and torque estimator.

\*

Adjust 2.30

4.60

2.34

2.34

Adjust 2.30 7.15

*V*

278 Induction Motors - Applications, Control and Fault Diagnostics

7.15

Adjust 11.21 *i*


\*

\* 11.45

Adjust 2.34

*Rs* 4.6 *Ts* 0.24

0 0

11.21

*V i*

Adjust 11.21

7.15

7.15

\*

\* 11.45

Adjust 2.34


*Rs* 4.6 *Ts* 0.24

11.21

Register

Adjust

2.30

\* 6.2

Adjust 4.2

5.0 1.2

*P* 3/4

2.34

\* 9.45

+ 2.34

2.34

*i* 7.15

2.34 2.34

Adjust

2.30

Adjust Adjust 9.30 9.30

> - 9.30

> > \* 13.32

Register

4.19

Adjust 4.19

*Te*

7.15 2.34

*i*

\* 9.45

> In this DTC block, the estimated flux and torque values are subtracted from the corresponding reference values. The reference data may be entered by means of external slide switches or it can come from a user interface through the USB port. The structure of the USB interface is not discussed in this document. The structure of the comparison block is shown in Figure 13.

### **4.5. Hysteresis comparators**

A two-level comparator for flux and a three-level comparator for torque are implemented in this block. Both hysteresis comparators were designed as FSMs in order to provide fast transition from one to another state. The FSM for the hysteresis comparators are shown in Figure 14.

**Figure 13.** Reference comparison blocks for flux and torque.

**Figure 14.** FSMs for flux and torque hysteresis comparators.

### **4.6. Switching table**

The VSI optimal switching vectors listed in Table 3 are included in the switching table. A voltage vector is selected based on the hysteresis comparator values *ϕ* and *τ*, and on the flux vector sector. The table output is a 3-bit vector and its complement, which are fed to the VSI. The 3-bit vector is also sent back to the torque-flux estimator to obtain the next torque and flux values. The architecture for the switching table is shown in Figure 15.

### **4.7. Global control block**

In order to have a constant sampling period (*Ts*), a global FSM was used to control the data flow from one block to the next. Since there is a register at the output of every major block, the FSM sends a timed pulse to each one, depending on the selected width of the data path. The

**Figure 15.** Switching table architecture.

f = 0 f = 1 *e < L - e > L +*

j j j

**Figure 13.** Reference comparison blocks for flux and torque.

1.19

1.19 1.19

*s* \*

MUX

Switches USB

280 Induction Motors - Applications, Control and Fault Diagnostics

*s* \*


Register

*e*

1.19

1.19 *s*

Magnetic Flux Torque

1.19

j

values. The architecture for the switching table is shown in Figure 15.

*e < L -*

The VSI optimal switching vectors listed in Table 3 are included in the switching table. A voltage vector is selected based on the hysteresis comparator values *ϕ* and *τ*, and on the flux vector sector. The table output is a 3-bit vector and its complement, which are fed to the VSI. The 3-bit vector is also sent back to the torque-flux estimator to obtain the next torque and flux

In order to have a constant sampling period (*Ts*), a global FSM was used to control the data flow from one block to the next. Since there is a register at the output of every major block, the FSM sends a timed pulse to each one, depending on the selected width of the data path. The

Flux Torque

t = -1 t = 0

t t

*Te e > L - e < L + Te Te*

*e < L -*

*e > 0*

*Te Te*

t *Te*

4.19

4.19 4.19

*<sup>T</sup>*\* \* *<sup>ᵉ</sup> Tᵉ Tᵉ*

MUX

Switches USB


Register

*e*

*T*

4.19

4.19

4.19

*e > L +*

t

*e < 0*

t = 1

*Te Te* t t

*e > L +*

j

**4.6. Switching table**

**4.7. Global control block**

*e < L -*

j

*e > L +*

**Figure 14.** FSMs for flux and torque hysteresis comparators.

j j

ADC conversion and adjustment take a total of 600 ns and the estimation of torque and flux take 940 ns with the data path set to 20-bits for the flux and 23-bits for torque. The rest of the processes take only 20 ns each, giving a total of 1600 ns for the sampling period. The execution times are shown in Figure 16.

**Figure 16.** Execution times of each DTC block.

### **4.8. Square root algorithm**

In order to reduce current ripple to a minimum, the DTC algorithm must have a processing period as low as possible, and the square root is the calculation that usually takes the longest [4]. For this reason, a special architecture was devised in order to have an algorithm that could be scaled easily to any number of bits and could execute quickly [11].

A successive approximation register (SAR), similar to the used in commercial ADCs, was designed in order to arrive quickly to the result.

The whole algorithm is based on a square root calculation as follows:

$$y = \sqrt{x} \tag{39}$$

which can also be written as:

$$y^2 - \alpha = 0\tag{40}$$

In other words, if a certain number *y* is squared and *x* is subtracted, and the result is zero or very close to it, then *y* would be the square root of *x*. Hence, the main objective is to find *y* as fast as possible; this is where the SAR proved worthy.

The SAR is basically a register in which each bit is LOW and, bit by bit, is toggled to HIGH, starting from the most significant bit (MSB) down to the least significant bit (LSB) under the flowing conditions; if the result of Eq. (40) is:


The architecture used for the square root is shown in Figure 17.

**Figure 17.** Square root architecture.

The SAR is the main block of the SQRT architecture; a START pulse is received from the estimator FSM to initialize the SAR process. Once the result is found, it is loaded to a parallel register and passed on to the reference comparison block.

Despite the simplicity of the square root architecture presented, it proved to be a fast and precise algorithm that could be scaled easily to adapt to the generic nature of the torque and flux estimator. The only restriction is that it requires *x* having an even number of bits.

## **5. Results and discussion**

### **5.1. Simulation results**

A successive approximation register (SAR), similar to the used in commercial ADCs, was

In other words, if a certain number *y* is squared and *x* is subtracted, and the result is zero or very close to it, then *y* would be the square root of *x*. Hence, the main objective is to find *y* as

The SAR is basically a register in which each bit is LOW and, bit by bit, is toggled to HIGH, starting from the most significant bit (MSB) down to the least significant bit (LSB) under the

**•** Greater than zero, the HIGH bit is toggled back to LOW and the SAR proceeds to toggle the

**•** Less than zero, the HIGH bit is maintained and the SAR proceeds to toggle the next bit.

*y > x y = x*

*2 2*

*<sup>n</sup> <sup>n</sup> <sup>n</sup>*

**SAR**

*n*

**•** Equal to zero or if the SAR ran out of bits to toggle, the current value of *y* is the square root

*y x* = (39)

<sup>2</sup> *y x* - = 0 (40)

**Parallel Register**

*y y*

*n*

designed in order to arrive quickly to the result.

282 Induction Motors - Applications, Control and Fault Diagnostics

fast as possible; this is where the SAR proved worthy.

The architecture used for the square root is shown in Figure 17.

**Magnitude Comparator**

flowing conditions; if the result of Eq. (40) is:

of *x* and the process is terminated.

Multiplier

**\***

*2n*

*2n n* zeros

*x*

*y*

*2*

which can also be written as:

next bit.

*x*

*n*

Start

**Figure 17.** Square root architecture.

*n*

The whole algorithm is based on a square root calculation as follows:

The DTC architecture presented in Section 4 was first tested in MATLAB/Simulink for simulation studies and later was implemented on an FPGA-based induction motor drive for experimental verification.

The torque dynamic response is shown in Figure 18 with a sampling period of 1.6 μs and the hysteresis band reduced to 0.1 Nm.

**Figure 18.** Torque dynamic response in MATLAB/Simulink.

Thanks to the small sampling period, the torque ripple was reduced to a small value.

Similarly, the flux hysteresis band was reduced to 0.06 Wb and as a result, as shown in Figure 19, the flux locus is almost a perfect circle with very small ripple.

Consequently, due to the small sampling period and reduced torque ripple, the stator current signal appears almost as a perfect sinusoidal, as shown in Figure 20.

**Figure 19.** Flux locus in MATLAB/Simulink.

**Figure 20.** Stator current in MATLAB/Simulink.

### **5.2. Experimental results**

For experimental verification, the DTC strategy was implemented on a Xilinx Virtex-5 development board running at 100 MHz. The current and voltage signal conversions were done using Analog Devices AD7476A ADCs. For the motor power interface, a two-level VSI Fairchild Smart Power Module FNB41560 was used. All motor tests were done using a Texas Instruments 3-phase induction motor HVACIMTR. The motor parameters shown in Table 4 are the same used in simulations.


**Table 4.** Induction motor parameters.

time (s)

For experimental verification, the DTC strategy was implemented on a Xilinx Virtex-5 development board running at 100 MHz. The current and voltage signal conversions were done using Analog Devices AD7476A ADCs. For the motor power interface, a two-level VSI Fairchild Smart Power Module FNB41560 was used. All motor tests were done using a Texas

Current (Amps)



**Figure 20.** Stator current in MATLAB/Simulink.

PSIbeta (Wb)


**Figure 19.** Flux locus in MATLAB/Simulink.



0

0.05

0.1

0.15

0.2

284 Induction Motors - Applications, Control and Fault Diagnostics

**5.2. Experimental results**

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

PSIalfa (Wb)


Firstly, the torque dynamic response to a 2 Nm step is shown in Figure 21. A FUTEK torque sensor (TRS300) was used and the output analog signal was scaled, filtered, and displayed on a digital oscilloscope.

**Figure 21.** Experimental torque dynamic response.

To visualize the FPGA flux locus on the oscilloscope, two 16-bit digital to analog converters (AD5543) were used. The resulting image is shown in Figure 22, and as the flux simulation, it also appears as a perfect circle with reduced torque ripple.

**Figure 22.** Experimental flux locus.

Finally, the experimental stator current of one of the phases is shown in Figure 23.

**Figure 23.** Experimental stator current.

For comparison purposes, the sampling frequency was reduced to 100 KHz and, as expected, a large content of ripple was observed in the torque signal as shown in Figure 24. This increased ripple caused vibration and heating on the motor.

**Figure 24.** Experimental torque signal at reduced sampling frequency (100 KHz).

### **6. Conclusion**

To visualize the FPGA flux locus on the oscilloscope, two 16-bit digital to analog converters (AD5543) were used. The resulting image is shown in Figure 22, and as the flux simulation, it


time (15ms/div)

Finally, the experimental stator current of one of the phases is shown in Figure 23.

PSIalfa (Wb)

also appears as a perfect circle with reduced torque ripple.

0

PSIbeta (Wb)




0

Current (Amps)




**Figure 23.** Experimental stator current.

2

4

6

**Figure 22.** Experimental flux locus.

0.1

0.2

0.3

286 Induction Motors - Applications, Control and Fault Diagnostics

This chapter has presented an induction motor drive using classical DTC as the main control strategy. This technique was preferred over others due to its simplicity and high performance in motor control. Although DTC is characterized for presenting large ripple on flux and torque signals, it was possible to minimize it to a low value by reducing the sampling period to 1.6 μs. This reduction was achieved by implementation on an FPGA device and the application of a novel architecture for the square root algorithm in the torque/flux estimator. The DTC algorithm was designed based on a structural description and generic VHDL blocks, in order to make the controller easily re-scalable and completely independent of the FPGA technology. A Xilinx Virtex-5 FPGA development board running at 100 MHz was used for this project. The design coded in VHDL uses two´s complement fixed-point format and variable word size for all arithmetic calculations. The complete controller algorithm was simulated using doubleprecision on MATLAB/Simulink to compare with experimental results. The induction motor presented a smooth, vibration-free operation with a precise torque dynamic, which proves the validity of the presented torque algorithm.

### **Author details**

Rafael Rodríguez-Ponce1\*, Fortino Mendoza-Mondragón2 , Moisés Martínez-Hernández2 and Marcelino Gutiérrez-Villalobos2

\*Address all correspondence to: rrodriguez@upgto.edu.mx

1 Polytechnic University of Guanajuato, Robotics Engineering Department, Cortazar, Gua‐ najuato, México

2 Autonomous University of Querétaro, Automation and Control Department, Querétaro, México

### **References**


[10] Ferreira S., Haffner F., Pereira L.F., Moraes F. Design and Prototyping of Direct Tor‐ que Control of Induction Motors in FPGAs. In: 16th Symposium on Integrated Cir‐ cuits and System Design; 8-11 Sept.; Sao Paulo, Brazil. IEEE; 2003. p. 105-110. DOI: 10.1109/SBCCI.2003.1232814

**Author details**

najuato, México

**References**

México

Marcelino Gutiérrez-Villalobos2

288 Induction Motors - Applications, Control and Fault Diagnostics

2011. 274 p. DOI: 10.5772/600

DOI: 10.1109/TPEL.2012.2222675

Tech; 2011. 122 p. DOI: 10.5772/862

versity Press; 1998. 729 p.

DOI: 10.5772/636

TII.2013.2245908

Rafael Rodríguez-Ponce1\*, Fortino Mendoza-Mondragón2

\*Address all correspondence to: rrodriguez@upgto.edu.mx

1 Polytechnic University of Guanajuato, Robotics Engineering Department, Cortazar, Gua‐

2 Autonomous University of Querétaro, Automation and Control Department, Querétaro,

[1] Chomat M., editor. Electric Machines and Drives. 1st ed. Rijeka, Croatia: InTech;

[2] Vas P. Sensorless Vector and Direct Torque Control. 1st ed. New York: Oxford Uni‐

[3] Lamchich M.T., editor. Torque Control. 1st ed. Rijeka, Croatia: InTech; 2011. 304 p.

[4] Sutikno T., Idris N.R.N., Jidin A. A Review of Direct Torque Control for Induction Motors for Sustainable Reliability and Energy Efficient Drives. Renewable and Sus‐

[5] Jezernik K., Korelic J., Horvat R. PMSM Sliding Mode FPGA-Based Control for Tor‐ que Ripple Reduction. IEEE Transactions on Power Electronics. 2013; 28(7):3549-3556.

[6] Ahmad M., editor. Advances in Motor Torque Control. 1st ed. Rijeka, Croatia: In‐

[7] Bahri I., Idkhajine L., Monmasson E., El Amine Benkhelifa M. Hardware/Software Codesign Guidelines for System on Chip FPGA-Based Sensorless AC Drive Applica‐ tions. IEEE Transactions on Industrial Informatics. 2013; 9(4):2165-2176. DOI: 10.1109/

[8] Lepka J., Stekl P. 3-Phase AC Induction Motor Vector Control Using 56F80x, 56F8100

[9] Lis J., Kowalski C.T., Orlowska-Kowalska T. Sensorless DTC Control of the Induction Motor Using FPGA. IEEE International Symposium on Industrial Electronics; June

30th; Cambridge : IEEE; 2008. p. 1914-1919. DOI: 10.1109/ISIE.2008.4677287

or 56F8300 Device. Freescale Application Note. 2005; 1-68.

tainable Energy Reviews. 2014; 32:548-558. DOI: 10.1016/j.rser.2014.01.040

, Moisés Martínez-Hernández2

and

[11] Rodriguez R., Gomez R.A., Rodriguez J. Fast Square Root Calculation for DTC Mag‐ netic Flux Estimator. IEEE Latin America Transactions. 2014; 12(2):112-115. DOI: 10.1109/TLA.2014.6749526

## **Open-End Winding Induction Motor Drive Based on Indirect Matrix Converter**

Javier Riedemann, Rubén Peña and Ramón Blasco-Giménez

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/61157

### **Abstract**

Open-end winding induction machines fed from two standard two-level voltage source inverters (VSI) provide an attractive arrangement for AC drives. An alternative approach is to use a dual output indirect matrix converter (IMC). It is well known that IMC provides fully bidirectional power flow operation, with small input size filter requirements. Whilst a standard IMC consists of an AC–DC matrix converter input stage followed by a single VSI output stage, it is possible to replicate the VSI to produce multiple outputs. In this chapter, an open-end winding induction machine fed by an IMC with two output stages is presented. Different modulation strategies for the power converter are analyzed and discussed.

**Keywords:** Open-end winding, Electrical drive, Matrix converter, Pulse width mod‐ ulation (PWM)

### **1. Introduction**

An open-end winding induction machine, fed by two 2-level VSIs, offers several advantages when compared to a standard wye or delta connected induction machine drive. The main features of an open-end winding induction machine drive can be summarized as [1, 2]: equal power input from both sides of each winding, thus each VSI is rated at half the machine power rating; each phase stator current can be controlled independently; possibility to have twice the effective switching frequency (depending on the modulation strategy); extensibility to more

© 2015 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

phases, therefore multiphase induction machines can be considered if current reduction is required; possibility of reducing common-mode voltage; and certain degree of fault tolerance, as there is voltage space vector redundancy.

However, an open-end winding induction machine drive can have some drawbacks, such as [1]: possibility of zero sequence current flowing in the machine because of the occurrence of zero sequence voltage; increased conduction losses; more complex power converter require‐ ments, i.e., more power devices, circuit gate drives, etc.

To supply energy to an open-end winding machine, different power converter topologies have been developed; for instance, [3–6] propose an open-end winding induction machine drive based on two 2-level VSIs fed from isolated DC sources. This topology has the advantage of avoiding the circulation of zero sequence current; however, two isolation transformers are needed. On the other hand, [7–11] present a topology based on two 2-level VSIs fed by a single DC source. In this case, a zero sequence current could circulate in the machine windings (depending on the modulation strategy used), but just one transformer is needed, reducing the volume and cost of the drive.

Multilevel topologies for open-end winding AC drives are presented in [12–16] where different voltage levels can be achieved in the machine phase windings with certain power converter configurations, then reducing the output voltage distortion but increasing the system cost and complexity.

In the past decades, significant research effort has been focused on direct frequency changing power converters, such as the matrix converter (MC) [17] or the indirect matrix converter [18]. It is known that these power converter topologies offer a suitable solution for direct AC–AC conversion, achieving sinusoidal input and output currents, bidirectional power flow capa‐ bility and controllable input power factor, without using bulky energy storage elements [18]. Matrix converters have been utilized to supply open-end winding AC machines such as reported in [19–21].

In this chapter, the application of an IMC with two output stages to supply energy to an openend winding induction machine is described [22–24]. For evaluation purposes, simulations and experimental results are presented.

## **2. Power converter topologies for open-end winding induction machine drives**

An open-end winding induction machine drive can be supplied by different configurations of power converters. Some of the most common topologies will be reviewed in this chapter.

### **2.1. Two 2-level voltage source inverters fed by isolated DC sources**

This is the basic power converter for open-end winding AC drives. The circuit configuration is shown in Figure 1 where a standard two-level VSI is connected at each side of the machine stator winding [3]. The VSIs are supplied by isolated DC power sources.

**Figure 1.** Two 2-level VSIs fed by isolated DC sources for an open-end winding AC machine drive.

The voltage vectors for inverter 1 are shown in Table 1; the same space vectors are valid for inverter 2, but with superscript 2. As each VSI can produce eight voltage space vector locations independent of the other, there are 64 voltage vector combinations of the full converter, resulting in a vector locus similar to a three-level neutral point clamped (NPC) inverter [6].


**Table 1.** Switching states of the individual inverters

phases, therefore multiphase induction machines can be considered if current reduction is required; possibility of reducing common-mode voltage; and certain degree of fault tolerance,

However, an open-end winding induction machine drive can have some drawbacks, such as [1]: possibility of zero sequence current flowing in the machine because of the occurrence of zero sequence voltage; increased conduction losses; more complex power converter require‐

To supply energy to an open-end winding machine, different power converter topologies have been developed; for instance, [3–6] propose an open-end winding induction machine drive based on two 2-level VSIs fed from isolated DC sources. This topology has the advantage of avoiding the circulation of zero sequence current; however, two isolation transformers are needed. On the other hand, [7–11] present a topology based on two 2-level VSIs fed by a single DC source. In this case, a zero sequence current could circulate in the machine windings (depending on the modulation strategy used), but just one transformer is needed, reducing

Multilevel topologies for open-end winding AC drives are presented in [12–16] where different voltage levels can be achieved in the machine phase windings with certain power converter configurations, then reducing the output voltage distortion but increasing the system cost and

In the past decades, significant research effort has been focused on direct frequency changing power converters, such as the matrix converter (MC) [17] or the indirect matrix converter [18]. It is known that these power converter topologies offer a suitable solution for direct AC–AC conversion, achieving sinusoidal input and output currents, bidirectional power flow capa‐ bility and controllable input power factor, without using bulky energy storage elements [18]. Matrix converters have been utilized to supply open-end winding AC machines such as

In this chapter, the application of an IMC with two output stages to supply energy to an openend winding induction machine is described [22–24]. For evaluation purposes, simulations

**2. Power converter topologies for open-end winding induction machine**

An open-end winding induction machine drive can be supplied by different configurations of power converters. Some of the most common topologies will be reviewed in this chapter.

This is the basic power converter for open-end winding AC drives. The circuit configuration is shown in Figure 1 where a standard two-level VSI is connected at each side of the machine

**2.1. Two 2-level voltage source inverters fed by isolated DC sources**

stator winding [3]. The VSIs are supplied by isolated DC power sources.

as there is voltage space vector redundancy.

292 Induction Motors - Applications, Control and Fault Diagnostics

the volume and cost of the drive.

complexity.

reported in [19–21].

**drives**

and experimental results are presented.

ments, i.e., more power devices, circuit gate drives, etc.

Let *Vij* = *Vi* 1 *Vj* 2 with *i*, *j* =1...8 be the phase voltage vector combination of the dual-inverter output. A representation of the vector locations is shown in Figure 2 [6].

### **2.2. Two 2-level voltage source inverters fed by a single DC source**

This topology is basically the same described in Section 2.1, but now just one DC supply is considered for the drive, as shown in Figure 3. The disadvantage of this converter is that zero sequence current could circulate in the machine windings because of the generation of output zero sequence voltage; however, this issue can be addressed with an appropriate modulation strategy for the inverters [7].

### **2.3. Multilevel topologies**

Several multilevel power converters have been developed for open-end winding induction motor drives. For example, Figure 4a shows a three-level inverter [12] and Figure 4b shows a

**Figure 2.** Space vector locations of the dual-inverter scheme.

**Figure 3.** Two 2-level VSIs fed by a single DC source for an open-end winding AC machine drive.

five-level inverter [16]. The main advantage of the multilevel topologies is that the machine phase voltage presents lower voltage distortion, increasing the performance of the drive; but on the other hand, the complexity and cost of the system are also increased.

Open-End Winding Induction Motor Drive Based on Indirect Matrix Converter http://dx.doi.org/10.5772/61157 295

Figure 4. (a) Three-level inverter and (b) five-level inverter for open-end winding AC machine drives. **Figure 4.** (a) Three-level inverter and (b) five-level inverter for open-end winding AC machine drives.

#### **2.4 Direct power converters**  Modern direct power converters consider matrix converter and indirect matrix converters. A matrix **2.4. Direct power converters**

five-level inverter [16]. The main advantage of the multilevel topologies is that the machine phase voltage presents lower voltage distortion, increasing the performance of the drive; but

on the other hand, the complexity and cost of the system are also increased.

**Figure 3.** Two 2-level VSIs fed by a single DC source for an open-end winding AC machine drive.

**Figure 2.** Space vector locations of the dual-inverter scheme.

294 Induction Motors - Applications, Control and Fault Diagnostics

converter [17] is a direct frequency converter consisting of nine bidirectional switches (three switches per phase) allowing to connect any of the output terminals to any of the input voltages. For an openend winding induction motor drive, two MCs are required, connected in the arrangement shown in Figure 5. The main features of a matrix converter are: bidirectional power flow capability, sinusoidal input and output currents without bulky energy storage elements, and controllable input power factor. Modern direct power converters consider matrix converter and indirect matrix converters. A matrix converter [17] is a direct frequency converter consisting of nine bidirectional switches (three switches per phase) allowing to connect any of the output terminals to any of the input voltages. For an open-end winding induction motor drive, two MCs are required, connected

For a standard matrix converter, a total of 36 IGBTs and diodes are required in this topology.

in the arrangement shown in Figure 5. The main features of a matrix converter are: bidirectional power flow capability, sinusoidal input and output currents without bulky energy storage elements, and controllable input power factor. For a standard matrix converter, a total of 36 IGBTs and diodes are required in this topology.

**Figure 5.** Open-end winding induction motor drive based on matrix converters.

An indirect matrix converter [18] is also a direct frequency converter having the same features of an MC, but now a DC stage is clearly identified in the topology. The IMC consists of an input rectifier, an AC–DC matrix converter, built of six bidirectional switches; this rectifier produces the DC voltage to supply the converter output stage which is a standard two-level VSI. To supply an open-end winding AC machine, two output inverters are required as can be seen in Figure 6. Considering the six bidirectional switches of the input stage and the two output stages, a total of 24 discrete IGBTs and diodes are required in this topology.

**Figure 6.** Open-end winding induction motor drive based on indirect matrix converter.

### **3. Model of the open-end winding induction motor drive based on IMC**

The complete drive of Figure 6 can be modeled by state equations which describe the dynamic behavior of the system. The effects of power devices dead-times on zero sequence voltages are neglected. All the equations derived below are written in fixed *abc* coordinates.

The differential equations for the input side are:

$$\mathbf{v}\_s = R\_f \dot{\mathbf{i}}\_s + L\_f \frac{d\dot{\mathbf{i}}\_s}{dt} + \mathbf{v}\_l \tag{1}$$

$$\mathbf{i}\_s = \mathbf{C}\_f \frac{d\mathbf{v}\_t}{dt} + \mathbf{i}\_t \tag{2}$$

where

in the arrangement shown in Figure 5. The main features of a matrix converter are: bidirectional power flow capability, sinusoidal input and output currents without bulky energy storage elements, and controllable input power factor. For a standard matrix converter, a total of 36

An indirect matrix converter [18] is also a direct frequency converter having the same features of an MC, but now a DC stage is clearly identified in the topology. The IMC consists of an input rectifier, an AC–DC matrix converter, built of six bidirectional switches; this rectifier produces the DC voltage to supply the converter output stage which is a standard two-level VSI. To supply an open-end winding AC machine, two output inverters are required as can be seen in Figure 6. Considering the six bidirectional switches of the input stage and the two output

IGBTs and diodes are required in this topology.

296 Induction Motors - Applications, Control and Fault Diagnostics

**Figure 5.** Open-end winding induction motor drive based on matrix converters.

stages, a total of 24 discrete IGBTs and diodes are required in this topology.

**Figure 6.** Open-end winding induction motor drive based on indirect matrix converter.

$$\boldsymbol{\nu}\_{s} = \begin{bmatrix} \boldsymbol{\upsilon}\_{sa} & \boldsymbol{\upsilon}\_{sb} & \boldsymbol{\upsilon}\_{sc} \end{bmatrix}^{T}, \boldsymbol{\dot{\mathfrak{i}}\_{s}} = \begin{bmatrix} \boldsymbol{i}\_{sa} & \boldsymbol{i}\_{sb} & \boldsymbol{i}\_{sc} \end{bmatrix}^{T} \tag{3}$$

$$\boldsymbol{\nu}\_{t} = \begin{bmatrix} \boldsymbol{\upsilon}\_{ia} & \boldsymbol{\upsilon}\_{ib} & \boldsymbol{\upsilon}\_{ic} \end{bmatrix}^{T}, \mathbf{i}\_{t} = \begin{bmatrix} \mathbf{i}\_{ia} & \mathbf{i}\_{ib} & \mathbf{i}\_{ic} \end{bmatrix}^{T} \tag{4}$$

are the source voltage and current (3), and the rectifier input voltage and current (4).

The DC link voltage can be obtained as:

$$
\boldsymbol{\sigma}\_{\rm DC} = \mathbf{S}\_r^T \cdot \mathbf{v}\_l \tag{5}
$$

with the rectifier switching matrix:

$$\mathbf{S}\_r = \begin{bmatrix} \mathbf{S}\_{ap} - \mathbf{S}\_{au} \\ \mathbf{S}\_{bp} - \mathbf{S}\_{bn} \\ \mathbf{S}\_{cp} - \mathbf{S}\_{cu} \end{bmatrix} \tag{6}$$

where *Sxp*, *Sxn* ∈{0, 1} with *x* =*a*, *b*, *c*. The output pole voltage of Inverter 1 (*vo*1) and Inverter 2 (*vo*2), with respect to the negative DC link rail, are defined in (7).

$$
\boldsymbol{\nu}\_{o1} = \mathbf{S}\_{t1} \cdot \boldsymbol{\upsilon}\_{DC}, \\
\boldsymbol{\nu}\_{o2} = \mathbf{S}\_{t2} \cdot \boldsymbol{\upsilon}\_{DC} \tag{7}
$$

where the switching matrices of Inverter 1 (*Si*1) and Inverter 2 (*Si*2) are:

$$\mathbf{S}\_{I1} = \begin{bmatrix} \mathbf{S}\_{A1} \\ \mathbf{S}\_{B1} \\ \mathbf{S}\_{B1} \end{bmatrix} = \begin{bmatrix} \mathbf{S}\_{Ap1} - \mathbf{S}\_{An1} \\ \mathbf{S}\_{Ap1} - \mathbf{S}\_{Bn1} \\ \mathbf{S}\_{Cp1} - \mathbf{S}\_{Cn1} \end{bmatrix}, \mathbf{S}\_{I2} = \begin{bmatrix} \mathbf{S}\_{A2} \\ \mathbf{S}\_{B2} \\ \mathbf{S}\_{C2} \end{bmatrix} = \begin{bmatrix} \mathbf{S}\_{Ap2} - \mathbf{S}\_{An2} \\ \mathbf{S}\_{Bp2} - \mathbf{S}\_{Bn2} \\ \mathbf{S}\_{Cp2} - \mathbf{S}\_{Cn2} \end{bmatrix} \tag{8}$$

and *Sxpk* <sup>=</sup>*S*¯ *xnk* ∈{0, 1} with *x* =*a*, *b*, *c* , *k* =1, 2. The output phase voltage corresponds to the difference of both inverters pole voltages:

$$\boldsymbol{\nu}\_{\boldsymbol{p}h,o} = \begin{bmatrix} \boldsymbol{\upsilon}\_{\boldsymbol{p}h,o\boldsymbol{a}} & \boldsymbol{\upsilon}\_{\boldsymbol{p}h,o\boldsymbol{b}} & \boldsymbol{\upsilon}\_{\boldsymbol{p}h,o\boldsymbol{c}} \end{bmatrix}^{T} = \boldsymbol{\nu}\_{o1} - \boldsymbol{\nu}\_{o2} = \left(\mathbf{S}\_{t1} - \mathbf{S}\_{t2}\right)\boldsymbol{\upsilon}\_{\boldsymbol{DC}} \tag{9}$$

Considering a model of the AC machine with *Rs* the stator resistance, the output phase voltage can be written as:

$$\mathbf{w}\_{ph,o} = \mathbf{R}\_s \mathbf{i}\_o + \frac{d\Psi\_s \left(\mathbf{i}\_o, \mathbf{i}\_r, \theta\_r\right)}{dt} \tag{10}$$

where Ψ*s* is the stator flux linkage vector given by:

$$
\boldsymbol{\Psi}\_s = \begin{bmatrix} \boldsymbol{\Psi}\_{s\boldsymbol{a}} & \boldsymbol{\Psi}\_{s\boldsymbol{b}} & \boldsymbol{\Psi}\_{s\boldsymbol{c}} \end{bmatrix}^T \tag{11}
$$

The rotor angle is *θr* and the output (stator) current vector *io* and rotor current vector *ir* are given by (12):

$$\mathbf{i}\_o = \begin{bmatrix} \mathbf{i}\_{oa} & \mathbf{i}\_{ob} & \mathbf{i}\_{oc} \end{bmatrix}^T, \mathbf{i}\_r = \begin{bmatrix} \mathbf{i}\_{ra} & \mathbf{i}\_{rb} & \mathbf{i}\_{rc} \end{bmatrix}^T \tag{12}$$

As Ψ*s* is an implicit function of *t*, (10) can be rewritten by using the chain rule for the derivative:

$$\mathbf{w}\_{\rho\mathbf{i},o} = R\_s \mathbf{i}\_o + \frac{\partial \mathbf{V}\_s}{\partial \mathbf{i}\_o} \frac{d\mathbf{i}\_o}{dt} + \mathbf{\varvarphi}\_2 \left(t\right) \tag{13}$$

where

$$
\Psi \varphi\_2 \left( t \right) = \frac{\partial \Psi\_s}{\partial \dot{t}\_r} \frac{d\dot{t}\_r}{dt} + \frac{\partial \Psi\_s}{\partial \theta\_r} \frac{d\theta\_r}{dt} \tag{14}
$$

Assuming ∂Ψ*<sup>s</sup>* / ∂*io* is a bijective function of *t*, it can be defined as:

$$\frac{\partial \Psi\_s}{\partial \dot{t}\_o} = \Psi\_1^{-1}(t) \tag{15}$$

and (13) can be redefined by:

where the switching matrices of Inverter 1 (*Si*1) and Inverter 2 (*Si*2) are:

,

where Ψ*s* is the stator flux linkage vector given by:

*s*

= +

*R*

*ph o o*

and *Sxpk* <sup>=</sup>*S*¯

can be written as:

given by (12):

where

difference of both inverters pole voltages:

298 Induction Motors - Applications, Control and Fault Diagnostics

1 1 2 2 1 2

*i i S S* (8)

*xnk* ∈{0, 1} with *x* =*a*, *b*, *c* , *k* =1, 2. The output phase voltage corresponds to the

*ph oa ph ob ph oc DC* <sup>=</sup> é ù *vvv* =-= - *<sup>v</sup> ph o* ë û *oo i i <sup>v</sup> vv SS* (9)

1 2 1 1 2 2

1 1 1 12 2 2 2

*S S S S S S S SS S SS S S S S S S* é ù éù é ù - - é ù êú ê ú ê ú ê ú = =- = =- ê ú ê ú ê ú ê ú ë û êú ê ú - - ë û ëû ë û

, *A A Ap An Ap An B Bp Bn B Bp Bn C C Cp Cn Cp Cn*

, , , , 12 1 2 ( ) *<sup>T</sup>*

*d*

Considering a model of the AC machine with *Rs* the stator resistance, the output phase voltage

( )

*dt*

*i i*

**Ψ***s or*

ΨΨΨ *<sup>T</sup>*

The rotor angle is *θr* and the output (stator) current vector *io* and rotor current vector *ir* are

As Ψ*s* is an implicit function of *t*, (10) can be rewritten by using the chain rule for the derivative:

, 2 *<sup>s</sup>* ( ) *<sup>d</sup> R t dt*

*o*

*i*

<sup>2</sup> ( ) *<sup>r</sup>*

*r*

¶ ¶ = + ¶ ¶ **Ψ Ψ** *s s <sup>r</sup>*

*i*

¶ =+ + ¶ **Ψ***s o*

*<sup>i</sup> v i*

*ph o o*

*t*

*T T*

y

*r*

q

q

*<sup>ψ</sup> <sup>i</sup>* (14)

*d d*

*dt dt*

,

, , *<sup>r</sup>*

q

*v i* (10)

*sa sb sc* = é ù **Ψ***<sup>s</sup>* ë û (11)

*oa ob oc ra rb rc* = = é ùé ù *iii iii o r* ë ûë û *i i* (12)

(13)

$$\mathbf{w}\_{ph,o} = R\_s \dot{\mathbf{i}}\_o + \mathfrak{w}\_1^{-1} \left( t \right) \frac{d \dot{\mathbf{i}}\_o}{dt} + \mathfrak{w}\_2 \left( t \right) \tag{16}$$

The DC link current is:

$$\dot{\mathbf{u}}\_{\rm DC} = \left(\mathbf{S}\_{l1} + \mathbf{S}\_{l2}\right)^{\mathrm{T}} \dot{\mathbf{u}}\_{\bullet} \tag{17}$$

and the rectifier input current:

$$\dot{\mathbf{u}}\_{l} = \mathbf{S}\_{r} \dot{\mathbf{u}}\_{\text{DC}} \tag{18}$$

Taking into account (1) – (18), the state space model of the drive is given by (19) –(21):

$$\frac{d\dot{t}\_s}{dt} = -\frac{R\_f}{L\_f}\dot{t}\_s - \frac{1}{L\_f}\mathbf{v}\_i + \frac{1}{L\_f}\mathbf{v}\_s\tag{19}$$

$$\frac{d\boldsymbol{\nu}\_{l}}{dt} = \frac{1}{\mathbb{C}\_{f}}\boldsymbol{\dot{\boldsymbol{\nu}}\_{s}} - \frac{1}{\mathbb{C}\_{f}}\mathbb{S}\_{r} \left(\mathbb{S}\_{l1} + \mathbb{S}\_{l2}\right)^{T}\boldsymbol{\dot{\boldsymbol{\nu}}\_{o}}\tag{20}$$

$$\frac{d\dot{\mathbf{u}}\_o}{dt} = -\boldsymbol{\Psi}\_1(t)\boldsymbol{R}\_s\dot{\mathbf{i}}\_o - \boldsymbol{\Psi}\_1(t)\boldsymbol{\Psi}\_2(t) + \boldsymbol{\Psi}\_1(t)\left(\mathbf{S}\_{l1} - \mathbf{S}\_{l2}\right)\mathbf{S}\_r^T \cdot \boldsymbol{\mathbf{v}}\_l \tag{21}$$

### **4. Zero sequence voltage**

As mentioned before, the dual-inverter fed open-ended winding induction motor drive may suffer from zero sequence current caused by zero sequence voltage. This zero sequence voltage is produced because of the asymmetry of the instantaneous phase voltages applied to the machine windings (due to the voltage space vectors used). In general, zero sequence currents may give rise to increased RMS phase current, thus increasing the system losses, high current/ voltage THD, and machine over-heating and vibrations. The zero sequence voltage is given by [11]:

$$
\upsilon\_{zs} = \frac{\upsilon\_{A1A2} + \upsilon\_{B1B2} + \upsilon\_{C1C2}}{3} \tag{22}
$$

The zero sequence voltage contributions from the 64 space vector combinations are shown in Table 2. As can be noted, there are twenty voltage space vectors that do not produce zero sequence voltage; thus in order to avoid the circulation of zero sequence current in the machine windings, only these space vector combinations could be used in the modulation strategy for the dual inverter [9].


**Table 2.** Zero sequence voltage contributions from different space vector combinations

Moreover, from Table 2 and Figure 2, it can be noted that there are two different but equivalent sets of active vectors producing null zero sequence voltage (see Table 3), which could be used along with the zero vectors: *V*11, *V*22, *V*33, *V*44, *V*55, *V*66, *V*77, and *V*88.


**Table 3.** Active space vectors producing null zero sequence voltage

### **5. Common-mode voltage**

Conventional PWM inverters generate alternating common-mode voltages relative to ground which generate currents through the motor parasitic capacitances to the rotor iron [25]. These currents find their way via the motor bearings back to the grounded stator case. The so-called bearing currents have been found to be a major cause of premature bearing failure in PWM inverter motor drives [26].

may give rise to increased RMS phase current, thus increasing the system losses, high current/ voltage THD, and machine over-heating and vibrations. The zero sequence voltage is given

> 1 2 12 12 3 *AA BB CC*

The zero sequence voltage contributions from the 64 space vector combinations are shown in Table 2. As can be noted, there are twenty voltage space vectors that do not produce zero sequence voltage; thus in order to avoid the circulation of zero sequence current in the machine windings, only these space vector combinations could be used in the modulation strategy for

> *V*85 , *V*83 , *V*54 , *V*34 , *V*81 , *V*56 , *V*52 , *V*<sup>36</sup> *V*32 , *V*47 , *V*14 , *V*16 , *V*12 , *V*67 , *V*<sup>27</sup>

*V*88 , *V*55 , *V*53 , *V*35 , *V*33 , *V*44 , *V*51 , *V*31 , *V*46 , *V*<sup>42</sup> *V*15 , *V*13 , *V*64 , *V*24 , *V*11 , *V*66 , *V*62 , *V*26 , *V*22 , *V*<sup>77</sup>

> *V*58 , *V*38 , *V*45 , *V*43 , *V*18 , *V*65 , *V*25 , *V*<sup>63</sup> *V*23 , *V*74 , *V*41 , *V*61 , *V*21 , *V*76 , *V*<sup>72</sup>

+ + <sup>=</sup> (22)

*v vv*

*zs*

*V***zs Voltage vector combinations**

−*V* **DC** / 3 *V*84 , *V*86 , *V*82 , *V*57 , *V*37 , *V*<sup>17</sup>

+*V* **DC** / 3 *V*48 , *V*68 , *V*82 , *V*75 , *V*73 , *V*<sup>71</sup>

Moreover, from Table 2 and Figure 2, it can be noted that there are two different but equivalent sets of active vectors producing null zero sequence voltage (see Table 3), which could be used

**Set 1** *V*<sup>15</sup> *V*<sup>35</sup> *V*<sup>31</sup> *V*<sup>51</sup> *V*<sup>53</sup> *V*<sup>13</sup> **Set 2** *V*<sup>24</sup> *V*<sup>26</sup> *V*<sup>46</sup> *V*<sup>42</sup> *V*<sup>62</sup> *V*<sup>64</sup>

Conventional PWM inverters generate alternating common-mode voltages relative to ground which generate currents through the motor parasitic capacitances to the rotor iron [25]. These

+*V* **DC** / 2 *V*<sup>78</sup>

**Table 2.** Zero sequence voltage contributions from different space vector combinations

along with the zero vectors: *V*11, *V*22, *V*33, *V*44, *V*55, *V*66, *V*77, and *V*88.

**Table 3.** Active space vectors producing null zero sequence voltage

**5. Common-mode voltage**

−*V* **DC** / 2 *V*<sup>87</sup>

*v*

300 Induction Motors - Applications, Control and Fault Diagnostics

by [11]:

the dual inverter [9].

−*V* **DC** / 6

0

+*V* **DC** / 6

One of the main features of an open-end winding induction machine drive is the possibility of reducing the common-mode voltage by using certain space vector combinations of the dualinverter topology (Figure 2). In general, for an open-end winding machine, the common-mode voltage is given by [8]:

$$
\upsilon\_{cm} = \frac{1}{6} (\upsilon\_{A1} + \upsilon\_{B1} + \upsilon\_{C1} + \upsilon\_{A2} + \upsilon\_{B2} + \upsilon\_{C2}) \tag{23}
$$

where *vAi* , *vBi* , *vCi* , with *i* =1, 2, are the pole voltages of each inverter with respect to a common point of the drive (usually ground).

For the topology depicted in Figure 6, the common-mode voltage is given by:

$$
\boldsymbol{\upsilon}\_{cm} = \frac{1}{6} (\boldsymbol{\upsilon}\_{A1G} + \boldsymbol{\upsilon}\_{B1G} + \boldsymbol{\upsilon}\_{C1G} + \boldsymbol{\upsilon}\_{A2G} + \boldsymbol{\upsilon}\_{B2G} + \boldsymbol{\upsilon}\_{C2G}) \tag{24}
$$

where the common point *G* is the grounded neutral point of the source. These voltages can also be expressed as:

$$\begin{aligned} \boldsymbol{\upsilon}\_{\text{AlG}} &= \boldsymbol{\mathcal{S}}\_{A\text{pl}} \boldsymbol{\upsilon}\_{pG} + \boldsymbol{\mathcal{S}}\_{A\text{ml}} \boldsymbol{\upsilon}\_{\text{nG}} \\ \boldsymbol{\upsilon}\_{\text{B}\text{G}} &= \boldsymbol{\mathcal{S}}\_{B\text{pl}} \boldsymbol{\upsilon}\_{pG} + \boldsymbol{\mathcal{S}}\_{B\text{ml}} \boldsymbol{\upsilon}\_{\text{nG}} \\ \boldsymbol{\upsilon}\_{\text{C}\text{i}G} &= \boldsymbol{\mathcal{S}}\_{C\text{pl}} \boldsymbol{\upsilon}\_{pG} + \boldsymbol{\mathcal{S}}\_{C\text{nl}} \boldsymbol{\upsilon}\_{\text{nG}} \end{aligned} \tag{25}$$

where *vpG* and *vnG* are the voltages of the positive and negative rail of the DC link with respect to the grounded neutral point of the source, respectively; *Sxpi* , *Sxni* ∈{0, 1} with *x* = *A*, *B*, *C*, *i* =1, 2 are the switching functions of the inverter devices (0: switch opened, 1: switch closed) and *Sxni* =1−*Sxpi* (because of the complementary operation of the upper and lower switches of each inverter leg). Therefore,

$$\begin{split} \boldsymbol{\upsilon}\_{cm} &= \frac{1}{6} \Big[ \left( \mathbf{S}\_{A\rho 1} + \mathbf{S}\_{B\rho 1} + \mathbf{S}\_{\mathbb{C}\rho 1} + \mathbf{S}\_{A\rho 2} + \mathbf{S}\_{B\rho 2} + \mathbf{S}\_{\mathbb{C}\rho 2} \right) \boldsymbol{\upsilon}\_{pG} \\ &+ \left( \mathbf{S}\_{A\upsilon 1} + \mathbf{S}\_{B\upsilon 1} + \mathbf{S}\_{\mathbb{C}\upsilon 1} + \mathbf{S}\_{A\upsilon 2} + \mathbf{S}\_{B\upsilon 2} + \mathbf{S}\_{\mathbb{C}\upsilon 2} \right) \boldsymbol{\upsilon}\_{nG} \Big] \end{split} \tag{26}$$

Let *Nsw* =*SAp*<sup>1</sup> + *SBp*<sup>1</sup> + *SCp*<sup>1</sup> + *SAp*<sup>2</sup> + *SBp*<sup>2</sup> + *SCp*2, thus

$$
\upsilon\_{cm} = \frac{1}{6} \left[ N\_{sw} \upsilon\_{pG} + \left( 6 - N\_{sw} \right) \upsilon\_{nG} \right] \tag{27}
$$

where *Nsw* is the number of upper inverter switches closed. The squared RMS value of the common-mode voltage is:

$$\left| \upsilon\_{\textit{sw}\_{\textit{gas}}}^2 = \frac{1}{\text{36T}} \prod\_{0}^{T} N\_{\textit{sw}} \upsilon\_{\textit{p}\textit{G}} + \left( \mathsf{6} - N\_{\textit{sw}} \right) \upsilon\_{\textit{nG}} \right|^2 dt \tag{28}$$

where *T* is the period of *vpG* (equals the period of *vnG*). Further expansion yields:

$$\mathbf{G}\boldsymbol{\mathfrak{G}}\boldsymbol{v}\_{\text{ens}}^{2} = \boldsymbol{N}\_{\text{sw}}^{2}\frac{1}{T}\bigg[\boldsymbol{v}\_{\text{p}G}^{2}dt + 2\boldsymbol{N}\_{\text{sw}}\left(\boldsymbol{\mathfrak{G}} - \boldsymbol{N}\_{\text{sw}}\right)\frac{1}{T}\bigg[\boldsymbol{v}\_{\text{p}G}\boldsymbol{v}\_{\text{w}G}dt + \left(\boldsymbol{\mathfrak{G}} - \boldsymbol{N}\_{\text{sw}}\right)^{2}\frac{1}{T}\bigg[\boldsymbol{v}\_{\text{w}G}^{2}dt\tag{29}$$

The voltages of the DC link rails are given by:

$$\begin{aligned} \boldsymbol{\upsilon}\_{p\mathbf{G}} &= \mathbf{S}\_{ap}\boldsymbol{\upsilon}\_{ra} + \mathbf{S}\_{bp}\boldsymbol{\upsilon}\_{rb} + \mathbf{S}\_{cp}\boldsymbol{\upsilon}\_{rc} \\ \boldsymbol{\upsilon}\_{n\mathbf{G}} &= \mathbf{S}\_{an}\boldsymbol{\upsilon}\_{ra} + \mathbf{S}\_{bn}\boldsymbol{\upsilon}\_{rb} + \mathbf{S}\_{cn}\boldsymbol{\upsilon}\_{rc} \end{aligned} \tag{30}$$

where*vra*, *vrb*, and *vrc* are the converter input phase voltages and *Sxp*, *Sxn* with *x* =*a*, *b*, *c* are the switching functions of the rectifier. Accordingly, *vpG* and *vnG* will always be segments of different input phase voltages and

$$\left|\upsilon\_{\boldsymbol{\rho}\boldsymbol{G}}\left(t\right)\right| = \left|\upsilon\_{\boldsymbol{\alpha}\boldsymbol{G}}\left(t - t\_{\boldsymbol{o}}\right)\right|, \; t\_{\boldsymbol{o}} \in \mathsf{R} \tag{31}$$

thus

$$\oint\_{pG}^{T} v\_{pG}^{2} dt = \oint\_{0}^{T} v\_{nG}^{2} dt\tag{32}$$

Differentiating (29) with respect to *Nsw* and equating to zero, it can be found that *vcmRMS* 2 (and implicitly *vcmRMS* ) achieves a minimum value at *Nsw* =3, which means that in order to reduce the RMS common-mode voltage at the machine terminals, only three upper inverter switches should be closed at each switching period.

This can be further investigated by considering a virtual midpoint of the DC link as a reference point (see point 0 in Figure 6). Then, (24) can be rewritten as:

$$\boldsymbol{\upsilon}\_{cm} = \frac{1}{6} (\boldsymbol{\upsilon}\_{A10} + \boldsymbol{\upsilon}\_{B10} + \boldsymbol{\upsilon}\_{C10} + \boldsymbol{\upsilon}\_{A20} + \boldsymbol{\upsilon}\_{B20} + \boldsymbol{\upsilon}\_{C20}) + \boldsymbol{\upsilon}\_{0G} = \boldsymbol{\upsilon}\_{cm0} + \boldsymbol{\upsilon}\_{0G} \tag{33}$$

where the contributions of the input and output stages to the overall common-mode voltage have been separated (*v*0*<sup>G</sup>* and *vcm*0, respectively). The voltage *v*0*<sup>G</sup>* is the voltage between the reference point 0 and the grounded neutral point of the source. This voltage can be calculated as:

( ) <sup>1</sup> <sup>6</sup> <sup>6</sup> *cm sw pG sw nG v Nv N v* = +- é ù

( ) <sup>2</sup> <sup>2</sup>

*cm sw pG sw nG v N v N v dt*

2 2 22 2

*pG ap ra bp rb cp rc nG an ra bn rb cn rc*

where*vra*, *vrb*, and *vrc* are the converter input phase voltages and *Sxp*, *Sxn* with *x* =*a*, *b*, *c* are the switching functions of the rectifier. Accordingly, *vpG* and *vnG* will always be segments of

*v Sv Sv Sv v Sv Sv Sv* =++

> 2 2 0 0

Differentiating (29) with respect to *Nsw* and equating to zero, it can be found that *vcmRMS*

RMS common-mode voltage at the machine terminals, only three upper inverter switches

This can be further investigated by considering a virtual midpoint of the DC link as a reference

) achieves a minimum value at *Nsw* =3, which means that in order to reduce the

*T T*

*cm sw pG sw sw pG nG sw nG v N v dt N N v v dt N v dt T TT*

<sup>1</sup> <sup>6</sup>

where *T* is the period of *vpG* (equals the period of *vnG*). Further expansion yields:

1 11 <sup>36</sup> 2 6 <sup>6</sup> *RMS*

0

*T*

*T*

36 *RMS*

The voltages of the DC link rails are given by:

302 Induction Motors - Applications, Control and Fault Diagnostics

different input phase voltages and

should be closed at each switching period.

point (see point 0 in Figure 6). Then, (24) can be rewritten as:

thus

implicitly *vcmRMS*

common-mode voltage is:

where *Nsw* is the number of upper inverter switches closed. The squared RMS value of the

ë û (27)

= +- é ù òë û (28)

=++ (30)

( ) ( ) , *pG nG o o v t v tt t* =- ÎR (31)

*pG nG v dt v dt* <sup>=</sup> ò ò (32)

2 (and

( ) ( )

= +- + - ò òò (29)

0 00

*T TT*

$$
\boldsymbol{\upsilon}\_{0G} = \frac{1}{2} \left[ \left( \boldsymbol{\mathcal{S}}\_{ap} + \boldsymbol{\mathcal{S}}\_{an} \right) \boldsymbol{\upsilon}\_{ra} + \left( \boldsymbol{\mathcal{S}}\_{bp} + \boldsymbol{\mathcal{S}}\_{bn} \right) \boldsymbol{\upsilon}\_{rb} + \left( \boldsymbol{\mathcal{S}}\_{cp} + \boldsymbol{\mathcal{S}}\_{cn} \right) \boldsymbol{\upsilon}\_{nc} \right] \tag{34}
$$

It can be seen in (34) that *v*0*<sup>G</sup>* depends on the modulation of the input stage, which is totally defined by the duty cycles of the rectifier stage. On the other hand, the voltage *vcm*0 can be rewritten as:

$$
\boldsymbol{v}\_{cm0} = \frac{1}{6} \left[ \boldsymbol{N}\_{sw} \frac{\boldsymbol{v}\_{\rm DC}}{2} + \left( \boldsymbol{6} - \boldsymbol{N}\_{sw} \right) \left( \frac{\boldsymbol{v}\_{\rm DC}}{2} \right) \right] = \frac{1}{6} \left[ \boldsymbol{N}\_{sw} \boldsymbol{v}\_{\rm DC} - \boldsymbol{\Im} \boldsymbol{v}\_{\rm DC} \right] \tag{35}
$$

Therefore, it can be seen in (35) that by using *Nsw* =3, the contribution of the output inverters to the common-mode voltage is eliminated [8]. Table 4 shows the voltage space vector combinations of the dual-inverter topology which do not produce common-mode voltage.

As can be noted from Table 4 and Figure 2, there are larger and lower active vectors available which produce zero common-mode voltage. Any of them could be considered in the modu‐ lation strategy for the dual-inverter system depending on the machine voltage requirement. However, from Table 2 and Table 4, it can be appreciated that the space vectors which reduce the common-mode voltage are not the same vectors which reduce the zero sequence voltage; thus, if a common-mode voltage is required, a compensation should be done for the zero sequence voltage; in other case, large zero sequence current components will circulate in the machine windings; a type of compensation will be reviewed in the following section.



$$d\_{\boldsymbol{\gamma}}^{\boldsymbol{R}} = \frac{d\_{\boldsymbol{\gamma}}}{d\_{\boldsymbol{\gamma}} + d\_{\boldsymbol{\delta}}}, \ d\_{\boldsymbol{\delta}}^{\boldsymbol{R}} = \frac{d\_{\boldsymbol{\delta}}}{d\_{\boldsymbol{\gamma}} + d\_{\boldsymbol{\delta}}} \tag{36}$$

$$d\_{\times} = \sin\left(\frac{\pi}{3}\Big/-\theta\_{ref,i}\Big), \ d\_{\times} = \sin\left(\theta\_{ref,i}\right) \tag{37}$$

$$d\_{\gamma} = \cos\left(\theta\_{\text{ref},i}\right), \ d\_{\delta} = \cos\left(\pi\big\langle \begin{matrix} \pi\\ \heartsuit \end{matrix} - \theta\_{\text{ref},i} \right) \tag{38}$$

**Figure 9.** Transition between both rectifier modulation strategies.

#### **6.2. Modulation strategies for the output stages of the IMC**

### *6.2.1. Carrier-based modulation strategy*

In a PWM strategy based on a triangular carrier (SPWM), the duty cycles for each leg of inverter 1 are [29]:

$$d\_a = \frac{1}{2} \left( m(t) \cos \left( \frac{2\pi}{m\_f} k \right) + 1 \right) \tag{39}$$

$$d\_b = \frac{1}{2} \left( m \left( t \right) \cos \left( \frac{2\pi}{m\_f} k - \frac{2\pi}{3} \right) + 1 \right) \tag{40}$$

$$d\_c = \frac{1}{2} \left( m(t) \cos \left( \frac{2\pi}{m\_f} k + \frac{2\pi}{3} \right) + 1 \right) \tag{41}$$

In this case, It is necessary a variable modulation index given by *m*(*t*)=*mo*(*d<sup>γ</sup>* + *dδ*) to compensate the fluctuations of the DC link voltage, *mo* is the final modulation index (0≤*mo* ≤1), *mf* = *f <sup>s</sup>* / *f <sup>o</sup>*is the frequency index ( *f <sup>o</sup>*: output frequency, *f <sup>s</sup>*: switching frequency) and 0≤*k* ≤*mf* . Duty cycles of inverter 2 are obtained using (39)–(41), but considering a phase shift of 180° for the cosine functions. For implementation purposes, the duty cycles *da*, *db*, and *dc* are transformed into equivalent *α* −*β* −0 duty cycles. Thus, considering Figure 10, the *α* −*β* −0 duty cycles are:

$$d\_0 = 1 - d\_{\text{max}} \tag{42}$$

Open-End Winding Induction Motor Drive Based on Indirect Matrix Converter http://dx.doi.org/10.5772/61157 307

$$d\_{\alpha} = d\_{\max} - d\_{\text{mid}} \tag{43}$$

$$\mathcal{d}\_{\beta} = \mathcal{d}\_{\text{mid}} - \mathcal{d}\_{\text{min}} \tag{44}$$

$$d\_{\gamma} = d\_{\text{subu}}\tag{45}$$

**Figure 10.** A single inverter stage duty cycles.

**6.2. Modulation strategies for the output stages of the IMC**

**Figure 9.** Transition between both rectifier modulation strategies.

306 Induction Motors - Applications, Control and Fault Diagnostics

2 *<sup>a</sup>*

In a PWM strategy based on a triangular carrier (SPWM), the duty cycles for each leg of inverter

( ) 1 2 cos 1

*d mt k*

( ) 1 22

( ) 1 22

2 3 *<sup>c</sup>*

2 3 *<sup>b</sup>*

*d mt k*

*d mt k*

*f*

cos 1

cos 1

 p

 p

<sup>0</sup> 1 *max d d* = - (42)

*f*

*f*

In this case, It is necessary a variable modulation index given by *m*(*t*)=*mo*(*d<sup>γ</sup>* + *dδ*) to compensate the fluctuations of the DC link voltage, *mo* is the final modulation index (0≤*mo* ≤1), *mf* = *f <sup>s</sup>* / *f <sup>o</sup>*is the frequency index ( *f <sup>o</sup>*: output frequency, *f <sup>s</sup>*: switching frequency) and 0≤*k* ≤*mf* . Duty cycles of inverter 2 are obtained using (39)–(41), but considering a phase shift of 180° for the cosine functions. For implementation purposes, the duty cycles *da*, *db*, and *dc* are transformed into

*m* p

æ ö æ ö <sup>=</sup> ç ÷ ç ÷ + + ç ÷ è ø è ø

equivalent *α* −*β* −0 duty cycles. Thus, considering Figure 10, the *α* −*β* −0 duty cycles are:

*m* p

æ ö æ ö <sup>=</sup> ç ÷ ç ÷ - + ç ÷ è ø è ø

(39)

(40)

(41)

*m* p æ ö æ ö = + ç ÷ ç ÷ ç ÷ è ø è ø

*6.2.1. Carrier-based modulation strategy*

1 are [29]:

To obtain a correct balance of the input currents and the output voltages in a switching period, the modulation pattern should produce all combinations of the rectification and the inversion switching states [27], resulting in the following duty cycles for the active vectors:

$$\mathbf{d}\_{\alpha\gamma} = \mathbf{d}\_{\alpha}\mathbf{d}\_{\gamma}^{\mathbb{R}}, \mathbf{d}\_{\rho\gamma} = \mathbf{d}\_{\rho}\mathbf{d}\_{\gamma}^{\mathbb{R}}, \mathbf{d}\_{\alpha\delta} = \mathbf{d}\_{\alpha}\mathbf{d}\_{\delta}^{\mathbb{R}}, \mathbf{d}\_{\rho\delta} = \mathbf{d}\_{\rho}\mathbf{d}\_{\delta}^{\mathbb{R}}\tag{46}$$

The duty cycle corresponding to the switching state *SApSBpSCp* = 0 0 0 is:

$$d\_{00} = d\_{0, \text{tot}} - d\_{\text{min}} \tag{47}$$

and the combined zero vectors duty cycles are:

$$d\_{0\gamma} = d\_{00} d\_{\gamma}^R, d\_{0s} = d\_{00} d\_s^R \tag{48}$$

The output stages duty cycles are different for each inverter and are represented in Figure 11 [23] for inverter *i* (*i* =1, 2).

**Figure 11.** Inverters duty cycles.

### *6.2.2. Space vector modulation strategy for zero sequence reduction*

As mentioned in Section 4, in an open-end winding induction machine drive, the zero sequence voltage can be reduced by using the active space vectors given in Table 3. The zero vectors are mapped depending on the sector information [9]. The mapping is shown in Table 5.


**Table 5.** Mapping of zero vectors

The duty cycles for the output stages are calculated as:

$$d\_a = m\{t\} \sin\left(\frac{\pi}{3} \not\rightarrow \theta\_{n\circ,\rho}\right), \\ d\_\rho = m\{t\} \sin\left(\theta\_{n\circ,\rho}\right) a n d.d\_0 = 1 - d\_a - d\_\rho \tag{49}$$

where *m*(*t*)=*mo*(*d<sup>γ</sup>* + *dδ*) and 0≤*mo* ≤1. *θref* ,*<sup>o</sup>* is the angle of the output reference voltage space vector. As in the carrier-based modulation strategy, the duty cycles of the rectifier and the inverters should be combined; thus the active vector duty cycles are given in (46).

The combined zero vectors duty cycles are:

$$d\_{0\chi} = d\_0 d\_{\chi}^R \, d\_{0\delta} = d\_0 d\_{\delta}^R \tag{50}$$

Thus, the switching sequence, which is the same for both output stages, is shown in Figure 12, commutating the input stage with zero DC current [27].

**Figure 12.** Standard switching sequence for IMCs.

**Figure 11.** Inverters duty cycles.

308 Induction Motors - Applications, Control and Fault Diagnostics

**Table 5.** Mapping of zero vectors

a

*6.2.2. Space vector modulation strategy for zero sequence reduction*

The duty cycles for the output stages are calculated as:

p

The combined zero vectors duty cycles are:

q

As mentioned in Section 4, in an open-end winding induction machine drive, the zero sequence voltage can be reduced by using the active space vectors given in Table 3. The zero vectors are

**Sector I II III IV V VI Set 1 zero vectors** *V*<sup>55</sup> *V*<sup>33</sup> *V*<sup>11</sup> *V*<sup>55</sup> *V*<sup>33</sup> *V*<sup>11</sup> **Set 2 zero vectors** *V*<sup>22</sup> *V*<sup>66</sup> *V*<sup>44</sup> *V*<sup>22</sup> *V*<sup>66</sup> *V*<sup>44</sup>

> q

> > d

a b

= = (50)

=- - (49)

mapped depending on the sector information [9]. The mapping is shown in Table 5.

( ) ( , ) ( ) ( ) , 0 sin , sin 1 <sup>3</sup> *ref o ref o d mt d m t andd d d*

inverters should be combined; thus the active vector duty cycles are given in (46).

<sup>0000</sup> , *R R d dd d dd*

Thus, the switching sequence, which is the same for both output stages, is shown in Figure

 gd

g

12, commutating the input stage with zero DC current [27].

= -=

where *m*(*t*)=*mo*(*d<sup>γ</sup>* + *dδ*) and 0≤*mo* ≤1. *θref* ,*<sup>o</sup>* is the angle of the output reference voltage space vector. As in the carrier-based modulation strategy, the duty cycles of the rectifier and the

b

### *6.2.3. Space vector modulation strategy for common-mode voltage reduction*

If only the voltage space vectors shown in Table 4 are used in the modulation strategy for the output stages of the IMC, the contribution of the dual-inverter to the overall common-mode voltage can be eliminated. The duty cycles for the IMC outputs are calculated as in (49) – (50). However, as mentioned in Section 5, the space vectors producing null *vcm*<sup>0</sup> are not the same vectors producing null zero sequence voltage. Hence, as in this SVM strategy, the vectors used to modulate the converter output stages will eliminate the common-mode voltage; a compen‐ sation must be performed to avoid the circulation of zero sequence current in the machine.

The average zero sequence voltage within a sampling interval can be eliminated by forcing the zero sequence volt-seconds to zero [11] by applying the null voltage vectors with unequal times. Accordingly, the standard switching sequence used in the IMC [27] is modified in order to reduce/eliminate the average zero sequence voltage within a sampling period. For the modulation strategy presented, the duty cycles for both output VSIs are the same, which can be noted in Figure 13.

**Figure 13.** Modified switching sequence for the IMC with two outputs.

Taking into account that the same space vectors sequence applied in *γ <sup>R</sup>* interval is applied in the *δ <sup>R</sup>* interval but in reverse order, then the value of *x*, which causes the cancellation of the zero sequence volt-seconds, is calculated at every sampling period to satisfy [11]:

$$\left(\upsilon\_{z\ast}\mathbf{x}\left(0\boldsymbol{\upchi}+0\boldsymbol{\updelta}\right)+\upsilon\_{z\ast}\left(\alpha\boldsymbol{\upgamma}+a\boldsymbol{\updelta}\right)+\upsilon\_{z\ast}\left(\beta\boldsymbol{\upgamma}+\beta\boldsymbol{\updelta}\right)+\upsilon\_{z\ast}\left(1-\mathbf{x}\right)\left(0\boldsymbol{\upgamma}+0\boldsymbol{\updelta}\right)=0\tag{51}$$

where *vzsk* with *k* =1, 2, 3, 4, is the zero sequence voltage value at intervals *x*0*γ*, *αγ*, *βγ*, and (1− *x*)0*γ*, respectively. The value of *vzsk* can be calculated by using (22) and considering the space vectors used in the modulation strategy.

### **7. Open-end winding induction machine drive**

In this section, the application of a two-output IMC supplying an open-end winding induc‐ tion machine will be presented. The performance of the drive by using some of the modula‐ tion strategies discussed above will be shown and analyzed via simulation and experimental results. The simulations are carried out in a PSIM/MATLAB simulation platform. On the other hand, experimental results are obtained using the setup shown in Figure 14. The IMC has been designed and built at the Power Electronics, Machines and Control lab facilities, University of Nottingham, UK. A six-pole induction machine rated at 7.5 kW is used. A DSP board, based on the TMS320C6713 processor, is used as the control platform. The calculation of duty cycles is carried out on the DSP among several other tasks. An FPGA interface board, designed at Nottingham University, is used to implement the modulation strategies and data acquisition. Communication between the DSP and a PC is achieved using a DSK6713HPI (Host Port Interface) daughter card. The converter input stage uses SK60GM123 modules and the output stages use SK35GD126 modules. The switching frequency is 10 kHz and voltages and cur‐ rents are also sampled at 10 kHz. The load used in the experimental system is a DC generator, supplying a resistive load, coupled with the induction motor shaft.

**Figure 14.** Experimental setup.

### **7.1. SVM strategy for zero sequence voltage reduction and changing DC voltage**

The modulation strategies, presented in Section 6.1 and 6.2.2, are used to implement a standard feedforward vector control scheme [30] for the machine currents (Figure 15). Simulation and experimental results are shown in Sections 7.1.1 and 7.1.2, respectively.

**Figure 15.** Feedforward vector control scheme of induction machine.

### *7.1.1. Simulation results*

where *vzsk* with *k* =1, 2, 3, 4, is the zero sequence voltage value at intervals *x*0*γ*, *αγ*, *βγ*, and (1− *x*)0*γ*, respectively. The value of *vzsk* can be calculated by using (22) and considering the

In this section, the application of a two-output IMC supplying an open-end winding induc‐ tion machine will be presented. The performance of the drive by using some of the modula‐ tion strategies discussed above will be shown and analyzed via simulation and experimental results. The simulations are carried out in a PSIM/MATLAB simulation platform. On the other hand, experimental results are obtained using the setup shown in Figure 14. The IMC has been designed and built at the Power Electronics, Machines and Control lab facilities, University of Nottingham, UK. A six-pole induction machine rated at 7.5 kW is used. A DSP board, based on the TMS320C6713 processor, is used as the control platform. The calculation of duty cycles is carried out on the DSP among several other tasks. An FPGA interface board, designed at Nottingham University, is used to implement the modulation strategies and data acquisition. Communication between the DSP and a PC is achieved using a DSK6713HPI (Host Port Interface) daughter card. The converter input stage uses SK60GM123 modules and the output stages use SK35GD126 modules. The switching frequency is 10 kHz and voltages and cur‐ rents are also sampled at 10 kHz. The load used in the experimental system is a DC generator,

space vectors used in the modulation strategy.

310 Induction Motors - Applications, Control and Fault Diagnostics

**7. Open-end winding induction machine drive**

supplying a resistive load, coupled with the induction motor shaft.

**7.1. SVM strategy for zero sequence voltage reduction and changing DC voltage**

experimental results are shown in Sections 7.1.1 and 7.1.2, respectively.

The modulation strategies, presented in Section 6.1 and 6.2.2, are used to implement a standard feedforward vector control scheme [30] for the machine currents (Figure 15). Simulation and

**Figure 14.** Experimental setup.

The performance of the vector control scheme is verified by applying step changes in the *dq*axis reference currents. Figure 16 shows the waveforms obtained by applying a step change in the *q*-axis reference current from 7.7 to 10 A at *t* =0.16 *s*, while *d-*axis current is kept constant at 6 A. Figure 16a shows *dq* reference currents and their corresponding responses. A good tracking of the reference currents is obtained. The motor currents and phase *a* voltage are shown in Figure 16b. The step change in *q*-axis current is followed with changes in the magnitude and phase of the instantaneous machine currents. Moreover, a transition between reduced virtual DC voltage to maximum virtual DC voltage can be noticed in the output phase voltage of Figure 16b (bottom).

Figure 17a shows the performance of the control scheme when a step change from 6 to 8 A is applied in *d*-axis reference current while *q-*axis current is kept constant at 7.7 A. The motor currents and phase- *a* voltage are shown in Figure 17b. Again, the transition between both modulation strategies for the input rectifier can be noticed.

Figure 18 (top) shows the converter input phase voltage and current; the unity input displace‐ ment factor can be noted. Figure 18 (bottom) shows the output zero sequence voltage that has been obtained from the voltage across the load in each phase and then applying (22). As can be seen, the zero sequence voltage has been eliminated due to the modulation strategy used for the output stages.

Figure 16. *q*-axis current step change. (a) Motor *q*-axis current (top) and *d-*axis current (bottom). (b) Motor currents (top) and phase voltage (bottom). **Figure 16.** *q*-axis current step change. (a) Motor *q*-axis current (top) and *d-*axis current (bottom). (b) Motor currents (top) and phase voltage (bottom).

Figure 17a shows the performance of the control scheme when a step change from 6 to 8 A is applied

a) b) Motor currents (top) and phase voltage (bottom). **Figure 17.** *d*-axis current step change. (a) Motor *q*-axis current (top) and *d-*axis current (bottom). (b) Motor currents (top) and phase voltage (bottom).

Figure 17. *d*-axis current step change. (a) Motor *q*-axis current (top) and *d-*axis current (bottom). (b)

Figure 17. *d*-axis current step change. (a) Motor *q*-axis current (top) and *d-*axis current (bottom). (b)

Figure 18 (top) shows the converter input phase voltage and current; the unity input displacement

#### *7.1.2. Experimental results* Motor currents (top) and phase voltage (bottom). factor can be noted. Figure 18 (bottom) shows the output zero sequence voltage that has been obtained from the voltage across the load in each phase and then applying (22). As can be seen, the

**7.1.2 Experimental results**

applied.

The *dq*-axis currents are shown in Figure 19a. As the speed controller saturates when a step change in the speed reference takes place, a step change in *q*-axis current reference is applied. The *d*-axis current reference is kept constant at 6 A. A good performance of the control scheme can be appreciated agreeing with the simulated results. The instantaneous motor currents and phase *a* voltage are shown in Figure 19b. Again, the transition between both rectifier modu‐ lation strategies can be noted in the output phase voltage (Figure 19b bottom) when the change in *q*-axis reference current is applied. Figure 18 (top) shows the converter input phase voltage and current; the unity input displacement factor can be noted. Figure 18 (bottom) shows the output zero sequence voltage that has been obtained from the voltage across the load in each phase and then applying (22). As can be seen, the zero sequence voltage has been eliminated due to the modulation strategy used for the output stages. zero sequence voltage has been eliminated due to the modulation strategy used for the output stages.

The *dq*-axis currents are shown in Figure 19a. As the speed controller saturates when a step change in the speed reference takes place, a step change in *q*-axis current reference is applied. The *d*-axis current reference is kept constant at 6 A. A good performance of the control scheme can be appreciated agreeing with the simulated results. The motor instantaneous currents and phase *a* voltage are shown in Figure 19b. Again, the transition between both rectifier modulation strategies can be noted in the output phase voltage (Figure 19b bottom) when the change in *q*-axis reference current is

Figure 18. Input phase voltage and current (top) and zero sequence voltage (bottom).

Open-End Winding Induction Motor Drive Based on Indirect Matrix Converter http://dx.doi.org/10.5772/61157 313

**Figure 18.** Input phase voltage and current (top) and zero sequence voltage (bottom).

*7.1.2. Experimental results*

(top) and phase voltage (bottom).

**7.1.2 Experimental results**

applied.

the input rectifier can be noticed.

(top) and phase voltage (bottom).

in *q*-axis reference current is applied.

Motor currents (top) and phase voltage (bottom).

Motor currents (top) and phase voltage (bottom).

Motor currents (top) and phase voltage (bottom).

312 Induction Motors - Applications, Control and Fault Diagnostics

The *dq*-axis currents are shown in Figure 19a. As the speed controller saturates when a step change in the speed reference takes place, a step change in *q*-axis current reference is applied. The *d*-axis current reference is kept constant at 6 A. A good performance of the control scheme can be appreciated agreeing with the simulated results. The instantaneous motor currents and phase *a* voltage are shown in Figure 19b. Again, the transition between both rectifier modu‐ lation strategies can be noted in the output phase voltage (Figure 19b bottom) when the change

Figure 18. Input phase voltage and current (top) and zero sequence voltage (bottom).

The *dq*-axis currents are shown in Figure 19a. As the speed controller saturates when a step change in the speed reference takes place, a step change in *q*-axis current reference is applied. The *d*-axis current reference is kept constant at 6 A. A good performance of the control scheme can be appreciated agreeing with the simulated results. The motor instantaneous currents and phase *a* voltage are shown in Figure 19b. Again, the transition between both rectifier modulation strategies can be noted in the output phase voltage (Figure 19b bottom) when the change in *q*-axis reference current is

Figure 18 (top) shows the converter input phase voltage and current; the unity input displacement factor can be noted. Figure 18 (bottom) shows the output zero sequence voltage that has been obtained from the voltage across the load in each phase and then applying (22). As can be seen, the zero sequence voltage has been eliminated due to the modulation strategy used for the output stages.

a) b) Figure 17. *d*-axis current step change. (a) Motor *q*-axis current (top) and *d-*axis current (bottom). (b)

**Figure 17.** *d*-axis current step change. (a) Motor *q*-axis current (top) and *d-*axis current (bottom). (b) Motor currents

Figure 18 (top) shows the converter input phase voltage and current; the unity input displacement factor can be noted. Figure 18 (bottom) shows the output zero sequence voltage that has been obtained from the voltage across the load in each phase and then applying (22). As can be seen, the zero sequence voltage has been eliminated due to the modulation strategy used for the output stages.

a) b) Figure 17. *d*-axis current step change. (a) Motor *q*-axis current (top) and *d-*axis current (bottom). (b)

a) b) Figure 16. *q*-axis current step change. (a) Motor *q*-axis current (top) and *d-*axis current (bottom). (b)

Figure 17a shows the performance of the control scheme when a step change from 6 to 8 A is applied in *d*-axis reference current while *q-*axis current is kept constant at 7.7 A. The motor currents and phase-ܽ voltage are shown in Figure 17b. Again, the transition between both modulation strategies for

**Figure 16.** *q*-axis current step change. (a) Motor *q*-axis current (top) and *d-*axis current (bottom). (b) Motor currents

Motor currents (top) and phase voltage (bottom). **Figure 19.** *q*-axis current step change. (a) Motor *q*-axis current (top) and *d-*axis current (bottom). (b) Motor currents (top) and phase voltage (bottom).

Figure 19. *q*-axis current step change. (a) Motor *q*-axis current (top) and *d-*axis current (bottom). (b)

Figure 20a shows the *dq* machine currents when a step change in *d*-axis reference current is applied

a) b) Figure 20. *q*-axis current step change. (a) Motor *q*-axis current (top) and *d-*axis current (bottom). (b)

Finally, Figure 21 (top) shows the input phase voltage and current. The zero sequence voltage shown in Figure 21 (bottom) is not exactly zero, but this is probably due to the measurement procedure because not all of the channels are sampled at the same time and because in Figure 18 the input

Motor currents (top) and phase voltage (bottom).

switches are ideal.

Figure 20a shows the *dq* machine currents when a step change in *d*-axis reference current is applied while *q-*axis current is kept constant at 7.7 A. The motor currents and phase- *a* voltage are shown in Figure 20b. A good correspondence with the simulation results can be noted. while *q-*axis current is kept constant at 7.7 A. The motor currents and phase-ܽ voltage are shown in Figure 20b. A good correspondence with the simulation results can be noted.

Motor currents (top) and phase voltage (bottom). **Figure 20.** *q*-axis current step change. (a) Motor *q*-axis current (top) and *d-*axis current (bottom). (b) Motor currents (top) and phase voltage (bottom).

Figure 20. *q*-axis current step change. (a) Motor *q*-axis current (top) and *d-*axis current (bottom). (b)

Finally, Figure 21 (top) shows the input phase voltage and current. The zero sequence voltage shown

Finally, Figure 21 (top) shows the input phase voltage and current. The zero sequence voltage shown in Figure 21 (bottom) is not exactly zero, but this is probably due to the measurement procedure because not all of the channels are sampled at the same time and because in Figure 18 the input switches are ideal. in Figure 21 (bottom) is not exactly zero, but this is probably due to the measurement procedure because not all of the channels are sampled at the same time and because in Figure 18 the input switches are ideal.

The machine currents for 25 Hz operation are shown in Figure 22b (top), while Figure 22b (bottom)

**Figure 21.** Input rectifier voltage and current (top) and zero sequence voltage (bottom). shows machine currents for 50 Hz operation.

### **7.2. SVM strategy for common-mode voltage reduction and zero sequence voltage compensation**

Simulation and experimental results for the modulation strategy presented in Section 6.2.2 will be shown in this section. The rectifier is modulated to maximize the DC link voltage (Figure 7) and the full drive is tested in open-loop *v/f* operation.

### *7.2.1. Simulation results*

Hz output (top) and 50 Hz output (bottom).

(bottom).

Finally, Figure 21 (top) shows the input phase voltage and current. The zero sequence voltage shown in Figure 21 (bottom) is not exactly zero, but this is probably due to the measurement procedure because not all of the channels are sampled at the same time and because in Figure

**Figure 20.** *q*-axis current step change. (a) Motor *q*-axis current (top) and *d-*axis current (bottom). (b) Motor currents

Finally, Figure 21 (top) shows the input phase voltage and current. The zero sequence voltage shown in Figure 21 (bottom) is not exactly zero, but this is probably due to the measurement procedure because not all of the channels are sampled at the same time and because in Figure 18 the input

a) b) Figure 20. *q*-axis current step change. (a) Motor *q*-axis current (top) and *d-*axis current (bottom). (b)

**Figure 21.** Input rectifier voltage and current (top) and zero sequence voltage (bottom).

Figure 21. Input rectifier voltage and current (top) and zero sequence voltage (bottom).

**7.2 SVM strategy for common-mode voltage reduction and zero sequence voltage compensation**  Simulation and experimental results for the modulation strategy presented in Section 6.2.2 will be shown in this section. The rectifier is modulated to maximize the DC link voltage (Figure 7) and the

The DC link voltage and phase-ܽ machine voltage are shown in Figure 22a, top and bottom, respectively. The reference output voltage and frequency were set to 150 V and 50 Hz, respectively. The machine currents for 25 Hz operation are shown in Figure 22b (top), while Figure 22b (bottom)

18 the input switches are ideal.

(top) and phase voltage (bottom).

switches are ideal.

Motor currents (top) and phase voltage (bottom).

314 Induction Motors - Applications, Control and Fault Diagnostics

full drive is tested in open-loop *v/f* operation.

shows machine currents for 50 Hz operation.

**7.2.1 Simulation results** 

The DC link voltage and phase- *a* machine voltage are shown in Figure 22a, top and bottom, respectively. The reference output voltage and frequency were set to 150 V and 50 Hz, respectively. The machine currents for 25 Hz operation are shown in Figure 22b (top), while Figure 22b (bottom) shows machine currents for 50 Hz operation.

Small disturbances, occurring every 60, can be noted in the motor currents shown in Figure 22b. **Figure 22.** (a) DC link voltage (top) and output phase voltage (bottom). (b) Machine currents for 25 Hz output (top) and 50 Hz output (bottom).

These current disturbances are due to the application of zero voltage vectors to machine windings, see PWM pattern in Figure 13, aiming to reduce the zero sequence voltage. During the application of zero voltage vectors, each machine phase winding is supplied with a voltage of െܸ or ܸ. When െܸ voltage is applied to the machine windings, the current decreases according to the zero vector duty cycle. Figure 23 shows the current disturbance along with the corresponding DC link voltage and output phase voltage. Small disturbances, occurring every 60°, can be noted in the motor currents shown in Figure 22b. These current disturbances are due to the application of zero voltage vectors to machine windings, see PWM pattern in Figure 13, aiming to reduce the zero sequence voltage. During the application of zero voltage vectors, each machine phase winding is supplied with a voltage of −*V DC* or +*V DC*. When −*V DC* voltage is applied to the machine windings, the current decreases according to the zero vector duty cycle. Figure 23 shows the current disturbance along with the corresponding DC link voltage and output phase voltage.

The input (supply) currents are shown in Figure 24 (top) while Figure 24 (bottom) shows the converter input phase voltage (blue) and current (green) for an output reference of 150 V and 50 Hz. The unity displacement factor is evident in Figure 24 (bottom).

Figure 23. Phase a machine current (top), DC link voltage (middle), and machine phase-ܽ voltage

**Current Harmonic RMS Value (A)**  Fundamental 14.400 2nd 0.250 3rd 0.184 4th 0.112 5th 0.025

The low-order harmonics of the machine currents are presented in Table 6.

The low-order harmonics of the machine currents are presented in Table 6.

**Figure 23.** Phase a machine current (top), DC link voltage (middle), and machine phase- *a* voltage (bottom).


**Table 6.** Harmonic content of the machine currents

Figure 25a shows the common-mode voltage separated into *v*0*<sup>G</sup>* and *vcm*<sup>0</sup> as defined in (33)– (35). Due to absence of the reference point 0 in the real (and also simulated) converter, the common-mode voltages *v*0*<sup>G</sup>* and *vcm*0 shown in Figure 25a top and bottom, respectively, are obtained as:

$$\boldsymbol{\upsilon}\_{\rm 0G} = \boldsymbol{\upsilon}\_{\rm nG} + \frac{\boldsymbol{\upsilon}\_{\rm DC}}{2} \tag{52}$$

$$\boldsymbol{\upsilon}\_{\rm cm0} = \frac{1}{6} (\boldsymbol{\upsilon}\_{A1n} + \boldsymbol{\upsilon}\_{B1n} + \boldsymbol{\upsilon}\_{C1n} + \boldsymbol{\upsilon}\_{A2n} + \boldsymbol{\upsilon}\_{B2n} + \boldsymbol{\upsilon}\_{C2n}) - \frac{\boldsymbol{\upsilon}\_{\rm DC}}{2}$$

**Figure 24.** Input currents (top) and input phase voltage and current (bottom).

where *n* is the negative rail of the DC link.

It can be seen in the simulation results that the contribution of the output inverters to the common-mode voltage is completely eliminated due to the modulation strategy used. Figure 25b shows the zero sequence voltage (top) and its frequency spectrum (bottom). It can be noted that the low-order zero sequence harmonics are reduced because of the asymmetry of the null vector duty cycles used in the switching sequence for each output stage.

### *7.2.2. Experimental results*

**Figure 23.** Phase a machine current (top), DC link voltage (middle), and machine phase- *a* voltage (bottom).

**Table 6.** Harmonic content of the machine currents

316 Induction Motors - Applications, Control and Fault Diagnostics

1

obtained as:

**Current Harmonic RMS Value (A)** Fundamental 14.400 2nd 0.250 3rd 0.184 4th 0.112 5th 0.025 6th 0.815

Figure 25a shows the common-mode voltage separated into *v*0*<sup>G</sup>* and *vcm*<sup>0</sup> as defined in (33)– (35). Due to absence of the reference point 0 in the real (and also simulated) converter, the common-mode voltages *v*0*<sup>G</sup>* and *vcm*0 shown in Figure 25a top and bottom, respectively, are

( )

= +++ ++ -

6 2

2

*DC*

(52)

*v*

*DC*

*v*

0 111 22 2

*cm A n B n C n A n B n C n*

*v vvvv vv*

*G nG*

= +

*v v*

0

Figure 26a (top) shows the DC link voltage while Figure 26a (bottom) shows the voltage across the machine phase- *a* winding. The output phase voltage presents a fundamental component of 141 V, 50 Hz, slightly less than the voltage reference (150 V) because of the device voltage drops. As can be seen, the modulation strategy used results in a bipolar pulse width modulated voltage at the converter output. The machine currents for 25 Hz and 50 Hz operation are shown in Figure 26b (top and bottom), respectively. The reference output voltages are set to 75 and 150 V, respectively. Correspondence between the simulation and the experimental result can be observed.

In Figure 26b, the effect of the zero voltage vectors in the PWM pattern shown in Figure 13 is also observed. The supply currents are shown in Figure 27 (top), again with good correspond‐ ence with the simulation study. Figure 27 (bottom) shows the input phase voltage and current. the switching sequence for each output stage.

���� � <sup>1</sup>

where� is the negative rail of the DC link.

Figure 24. Input currents (top) and input phase voltage and current (bottom).

Figure 25a shows the common-mode voltage separated into ��� and ���� as defined in (33)–(35). Due to absence of the reference point 0 in the real (and also simulated) converter, the common-mode

> ��� 2

<sup>6</sup> ����� � ���� � ���� � ���� � ���� � ����� � ���

It can be seen in the simulation results that the contribution of the output inverters to the common-

zero sequence harmonics are reduced because of the asymmetry of the null vector duty cycles used in

(52)

2

voltages ��� and ���� shown in Figure 25a top and bottom, respectively, are obtained as:

��� � ��� �

Figure 25. (a) Common-mode voltages ��� (top) and ���� (bottom). (b) Zero sequence voltage (top) and its frequency spectrum (bottom). **Figure 25.** (a) Common-mode voltages *v*0*G* (top) and *vcm*0 (bottom). (b) Zero sequence voltage (top) and its frequency spectrum (bottom).

Figure 28a shows the common-mode voltages *v*0*<sup>G</sup>* (top) and *vcm*<sup>0</sup> (bottom). The voltage *v*0*<sup>G</sup>* follows very closely the simulation results shown in Figure 25. The voltage *vcm*0 is not exactly zero, but this is probably due to the measurement procedure because not all of the channels are sampled at the same time and because in Figure 25 the input switches are ideal. Finally, Figure 28b shows the zero sequence voltage (top) and its frequency spectrum (bottom), agreeing closely with the simulation results. **7.2.2 Experimental results** Figure 26a (top) shows the DC link voltage while Figure 26a (bottom) shows the voltage across the machine phase-� winding. The output phase voltage presents a fundamental component of 141 V, 50 Hz, slightly less than the voltage reference (150 V) because of the device voltage drops. As can be seen, the modulation strategy used results in a bipolar pulse width modulated voltage at the converter output. The machine currents for 25 Hz and 50 Hz operation are shown in Figure 26b (top and bottom), respectively. The reference output voltages are set to 75 and 150 V, respectively.

Correspondence between the simulation and the experimental result can be observed.

Hz output (top) and 50 Hz output (bottom). In Figure 26b, the effect of the zero voltage vectors in the PWM pattern shown in Figure 13 is also observed. The supply currents are shown in Figure 27 (top), again with good correspondence with the **Figure 26.** (a) DC link voltage (top) and output phase voltage (bottom). (b) Machine currents for 25 Hz output (top) and 50 Hz output (bottom).

Figure 28a shows the common-mode voltages ��� (top) and ���� (bottom). The voltage ��� follows very closely the simulation results shown in Figure 25. The voltage ���� is not exactly zero, but this is probably due to the measurement procedure because not all of the channels are sampled at the same time and because in Figure 25 the input switches are ideal. Finally, Figure 28b shows the zero sequence voltage (top) and its frequency spectrum (bottom), agreeing closely with the simulation

simulation study. Figure 27 (bottom) shows the input phase voltage and current.

Figure 27. Input currents (top) and input phase voltage and current (bottom).

results.

Figure 26. (a) DC link voltage (top) and output phase voltage (bottom). (b) Machine currents for 25

is probably due to the measurement procedure because not all of the channels are sampled at the same

sequence voltage (top) and its frequency spectrum (bottom), agreeing closely with the simulation

in Figure 26b (top and bottom), respectively. The reference output voltages are set to 75 and 150 V, respectively.

Figure 26. (a) DC link voltage (top) and output phase voltage (bottom). (b) Machine currents for 25 Hz output (top) and 50 Hz output

In Figure 26b, the effect of the zero voltage vectors in the PWM pattern shown in Figure 13 is also observed. The supply

Correspondence between the simulation and the experimental result can be observed.

Figure 27. Input currents (top) and input phase voltage and current (bottom). Figure 28a shows the common-mode voltages ݒீ) top) and ݒ) bottom). The voltage ݒீ follows very closely the simulation **Figure 27.** Input currents (top) and input phase voltage and current (bottom). time and because in Figure 25 the input switches are ideal. Finally, Figure 28b shows the zero

machine connection have been shown and briefly discussed. Because of the advantages of indirect matrix converter, emphasis has been done in the application of this converter to control the machine. Therefore, an indirect matrix converter with two outputs stages has been proposed to supply energy to the open-end winding induction motor. This topology has been thoroughly modeled and pulse **Figure 28.** a) Common-mode voltages *v*0*G* (top) and *vcm*0. (bottom). (b) Zero sequence voltage (top) and its frequency spectrum (bottom).

width modulation strategies for the input and output stages of the proposed topology have been shown. Issues such as zero sequence voltage and common-mode voltage, presented in the dual-inverter configuration, have been analyzed and strategies to eliminate

Figure 28. (a) Common-mode voltages ݒீ) top) and ݒ) bottom). (b) Zero sequence voltage (top)

#### and/or reduce such effects have been presented. The control and modulation strategies have been simulated and experimentally tested in a prototype rig. Results for open and closed-loop operation of the open-end winding topology based on IMC have been shown. **8. Conclusion 8. Conclusion**

and its frequency spectrum (bottom).

results.

motor drive. Electric Machines (ICEM), 2010 XIX Int Conf 2010;1(1):1–6, 6–8.

Electron, IEEE Transac 2013;28(5):2427,2436.

2011 IEEE 8th Int Conf 2011;1:140,144.

Figure 28a shows the common-mode voltages *v*0*<sup>G</sup>* (top) and *vcm*<sup>0</sup> (bottom). The voltage *v*0*<sup>G</sup>* follows very closely the simulation results shown in Figure 25. The voltage *vcm*0 is not exactly zero, but this is probably due to the measurement procedure because not all of the channels are sampled at the same time and because in Figure 25 the input switches are ideal. Finally, Figure 28b shows the zero sequence voltage (top) and its frequency spectrum (bottom),

**Figure 25.** (a) Common-mode voltages *v*0*G* (top) and *vcm*0 (bottom). (b) Zero sequence voltage (top) and its frequency

Figure 26a (top) shows the DC link voltage while Figure 26a (bottom) shows the voltage across the machine phase-� winding. The output phase voltage presents a fundamental component of 141 V, 50 Hz, slightly less than the voltage reference (150 V) because of the device voltage drops. As can be seen, the modulation strategy used results in a bipolar pulse width modulated voltage at the converter output. The machine currents for 25 Hz and 50 Hz operation are shown in Figure 26b (top and bottom), respectively. The reference output voltages are set to 75 and 150 V, respectively.

Correspondence between the simulation and the experimental result can be observed.

a) b) Figure 25. (a) Common-mode voltages ��� (top) and ���� (bottom). (b) Zero sequence voltage (top)

Figure 24. Input currents (top) and input phase voltage and current (bottom).

���� � <sup>1</sup>

where� is the negative rail of the DC link.

the switching sequence for each output stage.

318 Induction Motors - Applications, Control and Fault Diagnostics

Figure 25a shows the common-mode voltage separated into ��� and ���� as defined in (33)–(35). Due to absence of the reference point 0 in the real (and also simulated) converter, the common-mode

> ��� 2

<sup>6</sup> ����� � ���� � ���� � ���� � ���� � ����� � ���

It can be seen in the simulation results that the contribution of the output inverters to the commonmode voltage is completely eliminated due to the modulation strategy used. Figure 25b shows the zero sequence voltage (top) and its frequency spectrum (bottom). It can be noted that the low-order zero sequence harmonics are reduced because of the asymmetry of the null vector duty cycles used in

(52)

(bottom).

(bottom).

**1. Conclusion**

**Acknowledgements**

acknowledged. **References**

shows the input phase voltage and current.

2

voltages ��� and ���� shown in Figure 25a top and bottom, respectively, are obtained as:

��� � ��� �

a) b) Figure 26. (a) DC link voltage (top) and output phase voltage (bottom). (b) Machine currents for 25

In Figure 26b, the effect of the zero voltage vectors in the PWM pattern shown in Figure 13 is also observed. The supply currents are shown in Figure 27 (top), again with good correspondence with the

**Figure 26.** (a) DC link voltage (top) and output phase voltage (bottom). (b) Machine currents for 25 Hz output (top)

Figure 28a shows the common-mode voltages ��� (top) and ���� (bottom). The voltage ��� follows very closely the simulation results shown in Figure 25. The voltage ���� is not exactly zero, but this is probably due to the measurement procedure because not all of the channels are sampled at the same time and because in Figure 25 the input switches are ideal. Finally, Figure 28b shows the zero sequence voltage (top) and its frequency spectrum (bottom), agreeing closely with the simulation

simulation study. Figure 27 (bottom) shows the input phase voltage and current.

Figure 27. Input currents (top) and input phase voltage and current (bottom).

agreeing closely with the simulation results.

and its frequency spectrum (bottom).

**7.2.2 Experimental results**

spectrum (bottom).

Hz output (top) and 50 Hz output (bottom).

and 50 Hz output (bottom).

results.

This work was funded by Fondecyt Chile under Grant 1151325. The financial support of CONICYT/FONDAP/15110019 is also The open-end winding induction machine has been presented in this chapter. Different power converter topologies for this type of machine connection have been shown and briefly discussed. Because of the advantages of indirect matrix converter, emphasis has been done in

[1] Wang Y, Panda D, Lipo,TA, Pan D. Open-winding power conversion systems fed by half-controlled converters. Power

[2] Wang Y, Lipo TA, Pan D. Robust operation of double-output AC machine drive. Power Electron ECCE Asia (ICPE & ECCE),

[3] Ramachandrasekhar K, Mohan S, Srinivas S. An improved PWM for a dual two-level inverter fed open-end winding induction

the application of this converter to control the machine. Therefore, an indirect matrix converter with two outputs stages has been proposed to supply energy to the open-end winding induction motor. This topology has been thoroughly modeled and pulse width modulation strategies for the input and output stages of the proposed topology have been shown. Issues such as zero sequence voltage and common-mode voltage, presented in the dual-inverter configuration, have been analyzed and strategies to eliminate and/or reduce such effects have been presented. The control and modulation strategies have been simulated and experimen‐ tally tested in a prototype rig. Results for open and closed-loop operation of the open-end winding topology based on IMC have been shown.

### **Acknowledgements**

This work was funded by Fondecyt Chile under Grant 1151325. The financial support of CONICYT/FONDAP/15110019 is also acknowledged.

### **Author details**

Javier Riedemann1 , Rubén Peña2 and Ramón Blasco-Giménez3

1 Department of Electrical and Electronic Engineering, University of Bío-Bío, Concepción, Chile

2 Department of Electrical Engineering, University of Concepción, Concepción, Chile

3 Department of System Engineering and Control, Polytechnic University of Valencia, Valencia, Spain

### **References**


[4] Mohapatra KK, Gopakumar K, Somasekhar VT, Umanand L. A harmonic elimina‐ tion and suppression scheme for an open-end winding induction motor drive. Ind Electron, IEEE Transac 2003;50(6):1187–98.

the application of this converter to control the machine. Therefore, an indirect matrix converter with two outputs stages has been proposed to supply energy to the open-end winding induction motor. This topology has been thoroughly modeled and pulse width modulation strategies for the input and output stages of the proposed topology have been shown. Issues such as zero sequence voltage and common-mode voltage, presented in the dual-inverter configuration, have been analyzed and strategies to eliminate and/or reduce such effects have been presented. The control and modulation strategies have been simulated and experimen‐ tally tested in a prototype rig. Results for open and closed-loop operation of the open-end

This work was funded by Fondecyt Chile under Grant 1151325. The financial support of

and Ramón Blasco-Giménez3

1 Department of Electrical and Electronic Engineering, University of Bío-Bío, Concepción,

[1] Wang Y, Panda D, Lipo,TA, Pan D. Open-winding power conversion systems fed by half-controlled converters. Power Electron, IEEE Transac 2013;28(5):2427,2436.

[2] Wang Y, Lipo TA, Pan D. Robust operation of double-output AC machine drive. Power Electron ECCE Asia (ICPE & ECCE), 2011 IEEE 8th Int Conf 2011;1:140,144.

[3] Ramachandrasekhar K, Mohan S, Srinivas S. An improved PWM for a dual two-level inverter fed open-end winding induction motor drive. Electric Machines (ICEM),

2 Department of Electrical Engineering, University of Concepción, Concepción, Chile

3 Department of System Engineering and Control, Polytechnic University of Valencia,

winding topology based on IMC have been shown.

320 Induction Motors - Applications, Control and Fault Diagnostics

CONICYT/FONDAP/15110019 is also acknowledged.

, Rubén Peña2

2010 XIX Int Conf 2010;1(1):1–6, 6–8.

**Acknowledgements**

**Author details**

Javier Riedemann1

Valencia, Spain

**References**

Chile


[14] Somasekhar VT, Baiju MR, Mohapatra KK, Gopakumar K. A multilevel inverter sys‐ tem for an induction motor with open-end windings. IECON 02 [Ind Electron Soc

[15] Lakshminarayanan S, Mondal G, Gopakumar K. Multilevel inverter with 18-sided polygonal voltage space vector for an open-end winding induction motor drive. EU‐

[16] Baiju MR, Gopakumar K, Mohapatra KK, Somasekhar VT, Umanand L. Five-level in‐ verter voltage-space phasor generation for an open-end winding induction motor

[17] Wheeler PW, Rodriguez J, Clare JC, Empringham L, Weinstein A. Matrix converters:

[18] Kolar JW, Friedli T, Rodriguez J, Wheeler PW. Review of three-phase PWM AC–AC

[19] Mohapatra KK, Mohan N,. Open-end winding induction motor driven With matrix converter for common-mode elimination. Power Electron Drives Energy Syst2006.

[20] Gupta RK, Somani A, Mohapatra KK, Mohan N. Space vector PWM for a direct ma‐ trix converter based open-end winding ac drives with enhanced capabilities. Appl Power Electron Conf Exposition (APEC), 2010 Twenty-Fifth Ann IEEE 2010;1:901–8.

[21] Pitic CI, Klumpner C. A new matrix converter-voltage source inverter hybrid ar‐ rangement for an adjustable speed-open winding induction motor drive with im‐ proved performance. Power Electron Machines Drives, 2008. PEMD 2008. 4th IET

[22] Riedemann J, Pena R, Cardenas R, Clare J, Wheeler P, Rivera M. Common mode volt‐ age and zero sequence current reduction in an open-end load fed by a two output indirect matrix converter. Power Electron Applic (EPE), 2013 15th Eur Conf

[23] Javier R, Ruben P, Roberto C, Jon C, Pat W, Marco R. Switching strategies for an indi‐ rect matrix converter fed open-end load. Ind Electron (ISIE), 2013 IEEE Int Symp

[24] Riedemann J, Pena R, Cardenas R, Clare J, Wheeler P, Blasco-Gimenez R. Control strategy of a dual-inverter system for an open-end winding induction machine based on indirect matrix converter. Power Electron Applic (EPE'14-ECCE Europe), 2014

[25] Chen S. Bearing current, EMI and soft switching in induction motor drives: a system‐ atic analysis, design and evaluation, Ph.D. dissertation, Univ. Wisconsin, Madison,

ROCON, 2007. Int Conf Computer as a Tool. 2007;1:1810–7.

drive. Electric Power Applic IEE Proc 2003;150(5):531,538.

a technology review. Ind Electron IEEE Transac 2002;49(2):276,288.

converter topologies. Ind Electron IEEE Transac 2011;58(11):4988,5006.

IEEE 2002 28th Ann Conf 2002;2:973 –8.

322 Induction Motors - Applications, Control and Fault Diagnostics

PEDES '06. Int Conf 2006;1:1–6.

Conf 2008;1:60–65.

2013;1:1,9.

2013;1:1,6.

1995.

16th Eur Conf 2014;1:1,8.

**Multiphase Induction Motors**

## **Open-Phase Fault Operation on Multiphase Induction Motor Drives**

Hugo Guzman, Ignacio Gonzalez, Federico Barrero and Mario Durán

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/60810

### **Abstract**

Multiphase machines have been recognized in the last few years like an attractive alterna‐ tive to conventional three-phase ones. This is due to their usefulness in a niche of applica‐ tions where the reduction in the total power per phase and, mainly, the high overall system reliability and the ability of using the multiphase machine in faulty conditions are required. Electric vehicle and railway traction, all-electric ships, more-electric aircraft or wind power generation systems are examples of up-to-date real applications using multi‐ phase machines, most of them taking advantage of the ability of continuing the operation in faulty conditions. Between the available multiphase machines, symmetrical five-phase induction machines are probably one of the most frequently considered multiphase ma‐ chines in recent research. However, other multiphase machines have also been used in the last few years due to the development of more powerful microprocessors. This chap‐ ter analyzes the behavior of generic n-phase machines (being n any odd number higher than 3) in faulty operation (considering the most common faulty operation, i.e. the openphase fault). The obtained results will be then particularized to the 5-phase case, where some simulation and experimental results will be presented to show the behavior of the entire system in healthy and faulty conditions. The chapter will be organized as follows: First, the different faults in a multiphase machine are analyzed. Fault conditions are de‐ tailed and explained, and the interest of a multiphase machine in the management of faults is stated. The effect of the open-phase fault operation in the machine model is then studied. A generic n-phase machine is considered, being n any odd number greater than three. The analysis is afterwards particularized to the 5-phase machine, where the openphase fault condition is managed using different control methods and the obtained re‐ sults are compared. Finally, the conclusions are presented in the last section of the chapter.

**Keywords:** Multiphase drives, Fault-tolerance, Predictive control techniques, Resonant controllers

© 2015 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### **1. Introduction**

Multiphase machines have been recognized in the last few years as an attractive alternative to conventional three-phase ones. This is due to their usefulness in a niche of applications where the reduction in the total power per phase and, the high overall system reliability and the ability of using the multiphase machine in faulty conditions are required. Electric vehicle and railway traction, all-electric ships, more-electric aircraft or wind power generation systems are examples of up-to-date real applications using multiphase machines, most of them taking advantage of the ability of continuing the operation in faulty conditions. Among the available multiphase machines, symmetrical five-phase induction machines are probably one of the most frequently considered multiphase machines in recent research. However, other multi‐ phase machines have also been used in the last few years due to the development of more powerful microprocessors. This chapter analyzes the behavior of generic *n*-phase machines (*n* being any odd number higher than 3) in faulty operation (considering the most common faulty operation, i.e. the open-phase fault). The obtained results will be then particularized to the 5-phase case, where some simulation and experimental results will be presented to show the behavior of the entire system in healthy and faulty conditions.

The chapter will be organized as follows:

First, the different faults in a multiphase machine are analyzed. Fault conditions are detailed and explained, and the interest of a multiphase machine in the management of faults is stated. The effect of the open-phase fault operation in the machine model is then studied. A generic *n*-phase machine is considered, *n* being any odd number greater than three. The analysis is afterwards particularized to the 5-phase machine, where the open-phase fault condition is managed using different control methods and the obtained results are compared. Finally, the conclusions are presented in the last section of the chapter.

### **2. Faults in electromechanical multiphase drives**

An electrical drive is an electromagnetic equipment subject to different electrical and mechan‐ ical faults which, depending on its nature and on the special characteristics of the system, may result in abnormal operation or shut down. In order to increase the use of electrical drives in safety-critical and high-demand applications, the development of cost-effective, robust and reliable systems is imperative. This issue has recently become one of the latest challenges in the field of electrical drives design [1]. Therefore, fault-tolerance, which can be defined as the ability to ensure proper speed or torque reference tracking in the electrical drive under abnormal conditions, has been considered in three-phase electrical drives taking into account different design and research approaches, including redundant equipment and over-dimen‐ sioned designs, leading to effective and viable fault standing but costly solutions. Faulttolerance in three-phase drives for different types of faults is a viable and mature research field, where the drive performance and the control capability is ensured at the expense of extra equipment [1]. However, this is not the case in the multiphase drives area in spite of the higher number of phases that the multiphase machine possesses, which favors its higher faulttolerance capability compared with conventional three-phase drives. Multiphase drives do not need extra electrical equipment to manage post-fault operation, requiring only proper postfault control techniques in order to continue operating [2]. Therefore, they are ideal for traction and aerospace applications for security reasons or in offshore wind farms where corrective maintenance can be difficult under bad weather conditions [3-6]. multiphase drives area in spite of the higher number of phases that the multiphase machine possesses, which favors its higher fault-tolerance capability compared with conventional three-phase drives. Multiphase drives do not need extra electrical equipment to manage post-fault operation, requiring only proper post-fault control techniques in order to continue operating [2]. Therefore, they are ideal for traction and aerospace applications for security reasons or in offshore wind farms where corrective maintenance can be difficult under bad

ensured at the expense of extra equipment [1]. However, this is not the case in the

Figure 1. Types of faults on a five-phase drive. **Figure 1.** Types of faults on a five-phase drive.

weather conditions [3-6].

**1. Introduction**

Multiphase machines have been recognized in the last few years as an attractive alternative to conventional three-phase ones. This is due to their usefulness in a niche of applications where the reduction in the total power per phase and, the high overall system reliability and the ability of using the multiphase machine in faulty conditions are required. Electric vehicle and railway traction, all-electric ships, more-electric aircraft or wind power generation systems are examples of up-to-date real applications using multiphase machines, most of them taking advantage of the ability of continuing the operation in faulty conditions. Among the available multiphase machines, symmetrical five-phase induction machines are probably one of the most frequently considered multiphase machines in recent research. However, other multi‐ phase machines have also been used in the last few years due to the development of more powerful microprocessors. This chapter analyzes the behavior of generic *n*-phase machines (*n* being any odd number higher than 3) in faulty operation (considering the most common faulty operation, i.e. the open-phase fault). The obtained results will be then particularized to the 5-phase case, where some simulation and experimental results will be presented to show

First, the different faults in a multiphase machine are analyzed. Fault conditions are detailed and explained, and the interest of a multiphase machine in the management of faults is stated. The effect of the open-phase fault operation in the machine model is then studied. A generic *n*-phase machine is considered, *n* being any odd number greater than three. The analysis is afterwards particularized to the 5-phase machine, where the open-phase fault condition is managed using different control methods and the obtained results are compared. Finally, the

An electrical drive is an electromagnetic equipment subject to different electrical and mechan‐ ical faults which, depending on its nature and on the special characteristics of the system, may result in abnormal operation or shut down. In order to increase the use of electrical drives in safety-critical and high-demand applications, the development of cost-effective, robust and reliable systems is imperative. This issue has recently become one of the latest challenges in the field of electrical drives design [1]. Therefore, fault-tolerance, which can be defined as the ability to ensure proper speed or torque reference tracking in the electrical drive under abnormal conditions, has been considered in three-phase electrical drives taking into account different design and research approaches, including redundant equipment and over-dimen‐ sioned designs, leading to effective and viable fault standing but costly solutions. Faulttolerance in three-phase drives for different types of faults is a viable and mature research field, where the drive performance and the control capability is ensured at the expense of extra equipment [1]. However, this is not the case in the multiphase drives area in spite of the higher

the behavior of the entire system in healthy and faulty conditions.

conclusions are presented in the last section of the chapter.

**2. Faults in electromechanical multiphase drives**

The chapter will be organized as follows:

328 Induction Motors - Applications, Control and Fault Diagnostics

Faults in an electromechanical drive can be also classified depending on the nature (electrical or mechanical), the location or the effect they have on the overall system (notice that different types of faults can result in the same abnormal machine behavior). The most common classification of faults in electrical drives defines three main groups of faults that can appear in the electrical drive. The power converter, electronic sensors (current, temperature, speed and voltage) and the electrical machine focus the main faults in an electrical drives, as shown in Figure 1. These faults are detailed hereafter. Faults in an electromechanical drive can be also classified depending on the nature (electrical or mechanical), the location or the effect they have on the overall system (notice that different types of faults can result in the same abnormal machine behavior). The most common classification of faults in electrical drives defines three main groups of faults that can appear in the electrical drive. The power converter, electronic sensors (current, temperature, speed and voltage) and the electrical machine focus the main faults in an electrical drives, as shown in Figure 1. These faults are detailed hereafter.


manufacturing problems, dynamic stress from shaft torque, environmental conditions and fatigued mechanical parts [10-11].


Statistically, the most common faults in electrical machines are the bearing failures, stator winding faults, broken rotor bar, shaft and coupling faults, cracked rotor end-rings, and airgap eccentricity [7-9], leading to unbalanced stator currents and voltages, the appearance of specific harmonics in the phase currents, overall torque oscillation and reduction, machines vibration, noise, overheating and efficiency reduction [10- 11].


remain in a constant ON or OFF state. As a result, the power converter may either lose a complete phase (also termed open-phase fault) or may physically maintain the number of phases and current flow but lose specific control capabilities on either one or both of the semiconductor of a certain phase. Thus, the configuration of the electrical drive varies, and the post-fault electrical drive may be regarded as an entire different system [26].

manufacturing problems, dynamic stress from shaft torque, environmental conditions

**4.** Air-gap irregularities due to static or dynamic eccentricity problems. Eccentricity is caused by manufacturing and constructive errors that generate an unequal air gap between the stator and the rotor, leading to unbalanced radial forces and possible rotorstator contact [10]. Static eccentricity appears when the position of the air gap inequality is fixed, whereas dynamic eccentricity happens when the rotor center is not properly aligned at the rotation center and the position of the air gap does not rotate equally. **5.** Bearing faults, which are mainly caused by assembling errors (misalignment of bearings)

**6.** Bent shaft faults, which are similar to dynamic eccentricity faults [10]. These faults appear when force unbalance or machine-load misalignment happens, resulting in machine

Statistically, the most common faults in electrical machines are the bearing failures, stator winding faults, broken rotor bar, shaft and coupling faults, cracked rotor end-rings, and airgap eccentricity [7-9], leading to unbalanced stator currents and voltages, the appearance of specific harmonics in the phase currents, overall torque oscillation and reduction, machines

**1.** Sensor Faults. Electrical drives commonly include speed, voltage and current sensors for control and protection purposes (Figure 1). In the multiphase drive case, standard FOC and predictive control techniques require a speed and at least *n* −1 (for an *n*-phase drive) current measurements in order to ensure proper control behavior. In case of abnormal sensor operation, inexistent or nonaccurate signals can downgrade the system perform‐ ance or result in a complete drive failure [17-20]. Sensor faults have been mainly analyzed for three-phase drives, and recent works have also addressed this type of faults for the multiphase case [21-23]. Notice that depending on the faulty sensor (i.e. DC-link voltage, current or speed), the effect in conventional three-phase or multiphase drive is mainly the same. In any case, the analysis of these kinds of faults mainly focuses on handling only one faulty sensor due to the small probability of fault in more than one sensor [24], which would include current and speed sensor faults, which are the most critical in electrical drive applications. The main reason for this is that high-performance drives are based on speed and current closed-loop controllers and consequently on speed and current sensors. Any variation or systematic error on the measured quantities may result in instantaneous power demanding control actions, subjecting the whole system to possible electrical stress

**2.** Power Converter Faults. The most common types of faults in electrical drives are those associated to the power converter [25]. Power converter faults are presented graphically in Figure 1, and can be further classified as single short-circuit switch fault, single opencircuit switch fault, phase-leg short-circuit fault, phase-leg open-circuit fault or opencircuit line fault [2]. These types of faults are mainly due to the burn out of the semiconductor or due to the semiconductor driver failure, forcing the semiconductor to

and fatigued mechanical parts [10-11].

330 Induction Motors - Applications, Control and Fault Diagnostics

that result in bearing vibration forced into the shaft [16].

vibration, noise, overheating and efficiency reduction [10- 11].

vibration and further machine failure [11].

[17].

The phase redundancy that multiphase drives possess allows managing faulty operation without the need of extra equipment, depending on the specific electrical machine configura‐ tion. Postfault control techniques exploit extra degrees of freedom of the multiphase system to maintain a circular Magneto-Motive Force (MMF) and achieve the desired speed or torque references. Depending on the type of fault and the electrical drive characteristics, different postfault control strategies, drive configuration and electrical machine winding connections are adopted under postfault operation. For instance, in the case of short-circuit faults, the proposed fault management strategies are based on controlling the available four healthy phases in a five-phase drive, maintaining operation at the expense of higher stator phase winding losses and torque ripple [27]. Nonetheless, this increase in torque ripple is managed in a dual three-phase drive [13] maintaining postfault operation with one three-phase drive in short-circuit and compensating the braking torque with the healthy three-phase drive [28]. The inclusion of auxiliary semiconductors in the electrical machine windings, in order to change from a short-circuit fault to an open-circuit or open-phase fault, was also addressed in [29], where ripple-free output torque was obtained with the appropriate control of the remaining four healthy phases. As a result, the multiphase electrical drive is able to manage different types of faults but at the expense of extra electronic equipment, like in the conventional threephase case. Different winding connections have also been considered for single and phase short circuit faults for a dual three-phase machine, assessing the effect of the harmonics obtained in the machines losses and torque, and evaluating its performance under different working conditions [30]. A similar approach has also been followed for open-phase and openline faults, where different drive topologies or machine winding connections have been considered. In one study [3], a six-phase drive was designed in order to independently control each phase of a three-phase machine under different types of faults and its viability was stated emulating an open-circuit line fault. Five-phase machines considering penta- and star-type winding connections are also compared in another study [30], where fundamental and thirdharmonic components are used to control the post-fault operation of the electrical drive. The available torque is increased, while torque ripple and losses are reduced. It is concluded that penta-winding connection results in improved fault-tolerance capabilities due to the higher number of open-circuit phases it can withstand (three open-phase faults in a five-phase drive).

But fault management does not include only postfault control techniques. It is divided in four different states namely, fault occurrence, fault detection, fault isolation and, finally, postfault control or the fault-tolerant control operation. Different fault detection and fault isolation techniques have been proposed based on the specific characteristics of the electrical drive to ensure proper postfault behavior. Then, a proper postfault control method is implemented to maintain correct reference tracking. This book chapter will be only focused on the postfault controller, and fault-detection and fault-isolation techniques will not be addressed.

### **3. Analysis of an open-phase fault in multiphase drives with odd number of phases**

The most common fault, the open-phase fault, is studied in this section. The ability of a multiphase machine managing the fault operation lies in the greater number of phases and in the greater number of independent variables that model the system. The model of the multi‐ phase machine is analyzed. The analysis is done for a generic multiphase machine. Then, the modeling equations of a*n n*-phase multiphase drive under an open-phase fault operation is presented, emphasizing their effect in the healthy model to understand the imposed con‐ straints for the design of postfault control techniques.

In the first place, the *n*-phase one neutral induction machine model is studied. The machine can be modeled by a set of stator and rotor phase voltage equilibrium equations referred to a fixed reference frame linked to the stator as follows:

$$
\mathbf{p}\left[\boldsymbol{V}\_{s}\right] = \left[\boldsymbol{R}\_{s}\right]\left[\boldsymbol{I}\_{s}\right] + \frac{d}{dt}\left[\boldsymbol{\mathcal{A}}\_{s}\right] = \left[\boldsymbol{R}\_{s}\right]\left[\boldsymbol{I}\_{s}\right] + \mathbf{p}\left(\left[\boldsymbol{L}\_{ss}\right]\left[\boldsymbol{I}\_{s}\right] + \left[\boldsymbol{L}\_{sr}\left(\boldsymbol{\theta}\right)\right]\left[\boldsymbol{I}\_{r}\right]\right) \tag{1}
$$

$$
\begin{bmatrix} V\_r \end{bmatrix} = \begin{bmatrix} \mathcal{R}\_r \end{bmatrix} \begin{bmatrix} I\_r \end{bmatrix} + \frac{d}{dt} \begin{bmatrix} \mathcal{A}\_r \end{bmatrix} = \begin{bmatrix} \mathcal{R}\_r \end{bmatrix} \begin{bmatrix} I\_r \end{bmatrix} + \mathbf{p} \cdot \left( \begin{bmatrix} L\_m \end{bmatrix} \begin{bmatrix} I\_r \end{bmatrix} + \begin{bmatrix} L\_m(\theta) \end{bmatrix} \begin{bmatrix} I\_s \end{bmatrix} \right) \tag{2}
$$

Where *θ* represents the rotor electrical angular position with respect to the stator, and rotates at the rotor electrical velocity *ωr*. The voltage, current and flux matrices are given by (3)-(8). Notice that the voltage rotor components (4) are equal to zero.

$$\begin{bmatrix} V\_s \\ \end{bmatrix} = \begin{bmatrix} \upsilon\_{as}\upsilon\_{bs}\upsilon\_{cs}\upsilon\_{ds}\upsilon\_{as}\cdots\upsilon\_{ns} \end{bmatrix}^T \tag{3}$$

$$\begin{bmatrix} V\_r \end{bmatrix} = \begin{bmatrix} \upsilon\_{ar}\upsilon\_{br}\upsilon\_{cr}\upsilon\_{dr}\upsilon\_{ar}\cdots\upsilon\_{nr} \end{bmatrix}^T \tag{4}$$

$$\mathbb{E}\left[\mathcal{A}\_{s}\right] = \left[\mathcal{A}\_{as}\mathcal{A}\_{bs}\mathcal{A}\_{cs}\mathcal{A}\_{ds}\mathcal{A}\_{cs}\cdots\mathcal{A}\_{ns}\right]^{T} \tag{5}$$

$$\left[\begin{array}{c} \mathcal{Z}\_r \\ \end{array}\right] = \left[\begin{array}{c} \mathcal{Z}\_{ar}\mathcal{Z}\_{br}\mathcal{Z}\_{cr}\mathcal{Z}\_{dr}\mathcal{Z}\_{dr}\cdots\mathcal{Z}\_{nr} \\ \end{array}\right]^T \tag{6}$$

$$\begin{bmatrix} \begin{bmatrix} I\_s \end{bmatrix} = \begin{bmatrix} \dot{\imath}\_{as} \ \dot{\imath}\_{bs} \dot{\imath}\_{cs} \ \dot{\imath}\_{ds} \dot{\imath}\_{cs} \cdots \dot{\imath}\_{us} \end{bmatrix}^T \tag{7}$$

$$\begin{bmatrix} I\_r \end{bmatrix} = \begin{bmatrix} i\_{ar} i\_{br} i\_{cr} i\_{dr} i\_{dr} \cdots i\_{nr} \end{bmatrix}^T \tag{8}$$

The rotor and stator resistance and inductance matrices are defined as follows:

#### Open-Phase Fault Operation on Multiphase Induction Motor Drives http://dx.doi.org/10.5772/60810 333

$$
\begin{bmatrix} R\_s \end{bmatrix} = \begin{bmatrix} R\_s & 0 & 0 & 0 & \cdots & 0 \\ 0 & R\_s & 0 & 0 & \cdots & 0 \\ 0 & 0 & R\_s & 0 & \cdots & 0 \\ 0 & 0 & 0 & R\_s & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & 0 & \cdots & R\_s \end{bmatrix} \begin{bmatrix} R\_r \end{bmatrix} = \begin{bmatrix} R\_r & 0 & 0 & 0 & \cdots & 0 \\ 0 & R\_r & 0 & 0 & \cdots & 0 \\ 0 & 0 & R\_r & 0 & \cdots & 0 \\ 0 & 0 & 0 & R\_r & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & 0 & \cdots & R\_r \end{bmatrix} \tag{9}
$$

$$
\begin{bmatrix} L\_\omega \end{bmatrix} = L\_i \begin{bmatrix} I\_n \end{bmatrix} + L\_{\text{ins}} \begin{bmatrix} \Lambda \left(\mathfrak{\theta}\right)\_n \end{bmatrix} \tag{10}
$$

$$
\begin{bmatrix} L\_{rr} \ \end{bmatrix} = L\_{lr} \cdot \begin{bmatrix} I\_n \ \end{bmatrix} + L\_{nr} \cdot \begin{bmatrix} \Lambda \left( \mathcal{G} \right)\_n \ \end{bmatrix} \tag{11}
$$

$$
\begin{bmatrix}
1 & \cos(\mathfrak{z}) & \cos(2\mathfrak{z}) & \cos(3\mathfrak{z}) & \cdots & \cos((n-1)\mathfrak{z}) \\
\cos((n-1)\mathfrak{z}) & 1 & \cos(\mathfrak{z}) & \cos(2\mathfrak{z}) & \cdots & \cos((n-2)\mathfrak{z}) \\
\cos((n-2)\mathfrak{z}) & \cos((n-1)\mathfrak{z}) & 1 & \cos(\mathfrak{z}) & \cdots & \cos((n-3)\mathfrak{z}) \\
\cos((n-3)\mathfrak{z}) & \cos((n-2)\mathfrak{z}) & \cos((n-1)\mathfrak{z}) & 1 & \cdots & \cos((n-4)\mathfrak{z}) \\
\vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\
\cos(2\mathfrak{z}) & \cos(3\mathfrak{z}) & \cos(4\mathfrak{z}) & \cos(5\mathfrak{z}) & \cdots & \cos(\mathfrak{z}) \\
\cos(\mathfrak{z}) & \cos(2\mathfrak{z}) & \cos(3\mathfrak{z}) & \cos(4\mathfrak{z}) & \cdots & 1 \\
\end{bmatrix}
\tag{12}
$$

Due to the machine symmetry, the stator-rotor (*L msr*) and rotor-stator (*L mrs*) mutual induc‐ tances are given by:

$$L\_{msr} = L\_{ms} = L\_{ms} \to L\_{ms} = k\_w^2 \cdot L\_{nr} \tag{13}$$

Making possible to conclude that:

**3. Analysis of an open-phase fault in multiphase drives with odd number**

The most common fault, the open-phase fault, is studied in this section. The ability of a multiphase machine managing the fault operation lies in the greater number of phases and in the greater number of independent variables that model the system. The model of the multi‐ phase machine is analyzed. The analysis is done for a generic multiphase machine. Then, the modeling equations of a*n n*-phase multiphase drive under an open-phase fault operation is presented, emphasizing their effect in the healthy model to understand the imposed con‐

In the first place, the *n*-phase one neutral induction machine model is studied. The machine can be modeled by a set of stator and rotor phase voltage equilibrium equations referred to a

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The rotor and stator resistance and inductance matrices are defined as follows:

Where *θ* represents the rotor electrical angular position with respect to the stator, and rotates at the rotor electrical velocity *ωr*. The voltage, current and flux matrices are given by (3)-(8).

 q

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straints for the design of postfault control techniques.

fixed reference frame linked to the stator as follows:

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·

332 Induction Motors - Applications, Control and Fault Diagnostics

·

**of phases**

$$\mathbb{E}\left[\boldsymbol{L}\_{\circ}\left(\boldsymbol{\theta}\right)\right] = \left[\boldsymbol{L}\_{\circ \circ}\left(\boldsymbol{\theta}\right)\right]^{T} \xrightarrow{} \mathbb{E}\left[\boldsymbol{L}\_{\circ \circ}\left(\boldsymbol{\theta}\right)\right] = \boldsymbol{L}\_{\circ \circ}\left[\boldsymbol{\Psi}\left(\boldsymbol{\theta}\right)\right] \tag{14}$$

$$
\begin{bmatrix}
\Psi(\boldsymbol{\theta}) \\
\hline
\begin{bmatrix}
\Psi(\boldsymbol{\theta})
\end{bmatrix} = 
\begin{bmatrix}
\cos(\boldsymbol{\Lambda}\_{1}) & \cos(\boldsymbol{\Lambda}\_{2}) & \cos(\boldsymbol{\Lambda}\_{3}) & \cos(\boldsymbol{\Lambda}\_{4}) & \cdots & \cos(\boldsymbol{\Lambda}\_{n}) \\
\cos(\boldsymbol{\Lambda}\_{n}) & \cos(\boldsymbol{\Lambda}\_{1}) & \cos(\boldsymbol{\Lambda}\_{2}) & \cos(\boldsymbol{\Lambda}\_{3}) & \cdots & \cos(\boldsymbol{\Lambda}\_{(n-1)}) \\
\cos(\boldsymbol{\Lambda}\_{(n-1)}) & \cos(\boldsymbol{\Lambda}\_{n}) & \cos(\boldsymbol{\Lambda}\_{1}) & \cos(\boldsymbol{\Lambda}\_{2}) & \cdots & \cos(\boldsymbol{\Lambda}\_{(n-2)}) \\
\cos(\boldsymbol{\Lambda}\_{(n-2)}) & \cos(\boldsymbol{\Lambda}\_{(n-1)}) & \cos(\boldsymbol{\Lambda}\_{n}) & \cos(\boldsymbol{\Lambda}\_{1}) & \cdots & \cos(\boldsymbol{\Lambda}\_{(n-3)}) \\
\vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\
\cos(\boldsymbol{\Lambda}\_{2}) & \cos(\boldsymbol{\Lambda}\_{3}) & \cos(\boldsymbol{\Lambda}\_{4}) & \cos(\boldsymbol{\Lambda}\_{5}) & \cdots & \cos(\boldsymbol{\Lambda}\_{1})
\end{bmatrix} \tag{15}
$$

Notice that [*In*] is the identity matrix of order *n*, *∆<sup>i</sup>* angles are defined as: *∆<sup>i</sup> =θ + (i−1)ϑ,* being *i=*{1,2,3, ,*n*}, *L ls* and *L lr* are the stator and rotor leakage inductances, and *ϑ* is the angle between phase windings.

Depending on the working state of the electrical drive and the number of phases it possesses, different transformation matrices can be used in order to describe the machine's electrical parameters in an *α* - *β* - *x* - *y* - *z* reference frame. For instance, for normal operation the traditional Clarke transformation (16) is used. (16)

$$
\begin{bmatrix}1&\cos(\mathfrak{z})&\cos(2\mathfrak{z})&\cos(3\mathfrak{z})&\cdots&\cos((n-1)\mathfrak{z})\\0&\sin(\mathfrak{s})&\sin(2\mathfrak{z})&\sin(3\mathfrak{z})&\cdots&\sin((n-1)\mathfrak{z})\\1&\cos(2\mathfrak{s})&\cos(4\mathfrak{s})&\cos(6\mathfrak{s})&\cdots&\cos(2(n-1)\mathfrak{z})\\0&\sin(2\mathfrak{s})&\sin(4\mathfrak{s})&\sin(6\mathfrak{s})&\cdots&\sin(2(n-1)\mathfrak{z})\\1&\cos(3\mathfrak{s})&\cos(6\mathfrak{s})&\cos(9\mathfrak{s})&\cdots&\cos(3(n-1)\mathfrak{z})\\0&\sin(3\mathfrak{s})&\sin(6\mathfrak{s})&\sin(9\mathfrak{s})&\cdots&\sin(3(n-1)\mathfrak{z})\\0&\sin(4\mathfrak{s})&\cos(8\mathfrak{s})&\cos(12\mathfrak{s})&\cdots&\cos(4(n-1)\mathfrak{z})\\0&\sin(4\mathfrak{s})&\sin(8\mathfrak{s})&\sin(12\mathfrak{s})&\cdots&\sin(4(n-1)\mathfrak{z})\\\vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\1&\cos\left(\frac{n-1}{2}\mathfrak{g}\right)&\cos\left(2\frac{n-1}{2}\mathfrak{g}\right)&\cos\left(3\frac{n-1}{2}\mathfrak{g}\right)&\cdots&\cos\left((n-1)\frac{n-1}{2}\mathfrak{g}\right)\\0&\sin\left(\frac{n-1}{2}\mathfrak{g}\right)&\sin\left(2\frac{n-1}{2}\mathfrak{g}\right)&\sin\left(3\frac{n-1}{2}\mathfrak{g}\right)&\cdots&\sin\left((n-1)\frac{n-1}{2}\mathfrak{g}\right)\\\frac{1}{\mathcal{Z}}&\frac{1}{\mathcal{Z}}&\frac{1}{\mathcal{Z}}&\frac{1}{\mathcal{Z}}&\cdots&\frac{1}{\mathcal{Z}}\end{bmatrix}
$$

In order to eliminate the time dependence of the coupling inductances and divide the model in a set of different independent-orthogonal equations, the Clarke transformation is applied to the machine model. The stator and rotor voltage, current and flux components in the *α*1 - *β*<sup>1</sup> - *α*2 - *β*2 - … - *zn* reference frame can be calculated by:

$$
\begin{bmatrix} \upsilon\_{sa\_1} \\ \upsilon\_{s\_1 \boldsymbol{\alpha}\_1} \\ \upsilon\_{s\_2 \boldsymbol{\alpha}\_2} \\ \vdots \\ \upsilon\_{s\_n \boldsymbol{\alpha}\_2} \\ \upsilon\_{sz\_n} \end{bmatrix} = \left[ T\_n \right] \* \left[ V\_s \right] \begin{bmatrix} \dot{I}\_{sa\_1} \\ \dot{I}\_{s\_1 \boldsymbol{\alpha}\_1} \\ \dot{I}\_{s\_2 \boldsymbol{\alpha}\_2} \\ \dot{I}\_{s\_2 \boldsymbol{\alpha}\_2} \\ \vdots \\ \dot{I}\_{sz\_n} \end{bmatrix} = \left[ T\_n \right] \* \left[ I\_s \right] \begin{bmatrix} \dot{\mathcal{A}}\_{sa\_1} \\ \dot{\mathcal{A}}\_{s\_1 \boldsymbol{\alpha}\_1} \\ \dot{\mathcal{A}}\_{s\_2 \boldsymbol{\alpha}\_2} \\ \dot{\mathcal{A}}\_{s\_2 \boldsymbol{\alpha}\_2} \\ \vdots \\ \dot{\mathcal{A}}\_{sz\_n} \end{bmatrix} = \left[ T\_n \right] \* \left[ \mathcal{A}\_s \right] \tag{17}
$$

$$
\begin{bmatrix} \boldsymbol{\upsilon}\_{r\alpha\_1} \\ \boldsymbol{\upsilon}\_{r\rho\_1} \\ \boldsymbol{\upsilon}\_{r\rho\_2} \\ \boldsymbol{\upsilon}\_{r\rho\_2} \\ \vdots \\ \boldsymbol{\upsilon}\_{r\alpha\_n} \end{bmatrix} = \begin{bmatrix} \boldsymbol{\dot{\iota}}\_{r\alpha\_1} \\ \boldsymbol{\dot{\iota}}\_{r\rho\_1} \\ \boldsymbol{\dot{\iota}}\_{r\rho\_2} \\ \boldsymbol{\dot{\iota}}\_{r\rho\_2} \\ \vdots \\ \boldsymbol{\dot{\iota}}\_{r\alpha\_n} \end{bmatrix} = \begin{bmatrix} \boldsymbol{\dot{\mathcal{A}}}\_{r\alpha\_1} \\ \boldsymbol{\dot{\mathcal{A}}}\_{r\rho\_1} \\ \boldsymbol{\dot{\mathcal{A}}}\_{r\rho\_2} \\ \boldsymbol{\dot{\mathcal{A}}}\_{r\rho\_2} \\ \vdots \\ \boldsymbol{\dot{\iota}}\_{r\alpha\_n} \end{bmatrix} = \begin{bmatrix} \boldsymbol{\dot{\mathcal{A}}}\_{r\alpha\_1} \\ \boldsymbol{\dot{\mathcal{A}}}\_{r\rho\_1} \\ \boldsymbol{\dot{\mathcal{A}}}\_{r\rho\_2} \\ \vdots \\ \boldsymbol{\dot{\iota}}\_{r\alpha\_n} \end{bmatrix} = \begin{bmatrix} \boldsymbol{T}\_n \end{bmatrix} \* \begin{bmatrix} \boldsymbol{\mathcal{A}}\_r \end{bmatrix} \tag{18}
$$

Multiplying the transformation matrix *Tn* with the stator and rotor phase voltage equations, (1) and (2), we get:

Notice that [*In*] is the identity matrix of order *n*, *∆<sup>i</sup>* angles are defined as: *∆<sup>i</sup>*

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In order to eliminate the time dependence of the coupling inductances and divide the model in a set of different independent-orthogonal equations, the Clarke transformation is applied to the machine model. The stator and rotor voltage, current and flux components in the *α*1 - *β*<sup>1</sup>

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334 Induction Motors - Applications, Control and Fault Diagnostics

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Depending on the working state of the electrical drive and the number of phases it possesses, different transformation matrices can be used in order to describe the machine's electrical parameters in an *α* - *β* - *x* - *y* - *z* reference frame. For instance, for normal operation the

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$$\begin{aligned} \left[\begin{array}{l} \left[\boldsymbol{T}\_{n}\right] \left\{\boldsymbol{V}\_{s}\right\} = \left[\boldsymbol{T}\_{n}\right] \left[\boldsymbol{R}\_{s}\right] \left\{\boldsymbol{T}\_{n}\right\}^{-1} \left\{\boldsymbol{T}\_{n}\right\} \left[\boldsymbol{I}\_{s}\right] \\ + p \cdot \left[\boldsymbol{T}\_{n}\right] \left[\boldsymbol{L}\_{ss}\right] \left[\boldsymbol{T}\_{n}\right]^{-1} \left\{\boldsymbol{T}\_{n}\right\} \left[\boldsymbol{I}\_{s}\right] \\ + p \cdot \left[\boldsymbol{T}\_{n}\right] \left[\boldsymbol{L}\_{ss} \left(\boldsymbol{\theta}\right)\right] \left[\boldsymbol{T}\_{n}\right]^{-1} \left\{\boldsymbol{T}\_{n}\right\} \left[\boldsymbol{I}\_{s}\right] \end{aligned} \tag{19}$$

$$\begin{aligned} \left[\begin{bmatrix} 0 \end{bmatrix} \right] &= \left[ \begin{bmatrix} T\_n \end{bmatrix} \right] \left[ \begin{matrix} R\_r \end{matrix} \right] \left[ \begin{matrix} T\_n \end{matrix} \right]^{-1} \left[ T\_n \right] \left[ \begin{matrix} I\_r \end{matrix} \right] \\ &+ p \left[ \begin{matrix} T\_n \end{matrix} \right] \left[ \begin{matrix} L\_n \end{matrix} \right] \left[ \begin{matrix} T\_n \end{matrix} \right]^{-1} \left[ T\_n \right] \left[ \begin{matrix} I\_r \end{matrix} \right] \\ &+ p \left[ \begin{matrix} T\_n \end{matrix} \right] \left[ \begin{matrix} L\_n \end{matrix} \left( \theta \right) \right] \left[ \begin{matrix} T\_n \end{matrix} \right]^{-1} \left[ T\_n \right] \left[ \begin{matrix} I\_s \end{matrix} \right] \end{aligned} \tag{20}$$

When an open-phase fault occurs in phase "*i*", the stator windings become an unbalanced system, the faulty phase current is now zero (*iis*=0), leading to a modification in the machine equations.

Due to the fact that the machine has no longer symmetrical stator windings, the back-emf terms are no longer mutually canceled, consequently the sum of the phase voltages are no longer zero.

$$\sum \begin{bmatrix} Vs \ \end{bmatrix} \neq 0 \tag{21}$$

Even though the faulty phase stator current will be zero, the corresponding phase voltage with respect to the neutral machine point will have an equivalent voltage value equal to the backemf (22).

$$
\boldsymbol{w}\_{\rm is} = \boldsymbol{R}\_{\rm s} \boldsymbol{i}\_{\rm is} + \boldsymbol{p} \cdot \boldsymbol{\mathcal{A}}\_{\rm is} = \boldsymbol{p} \cdot \boldsymbol{\mathcal{A}}\_{\rm is} = \text{BackEmf}\_{\rm i} \tag{22}
$$

Taking this into account the new voltage, current and flux matrices are:

$$\begin{bmatrix} V\_s \\ \end{bmatrix} = \begin{bmatrix} \upsilon\_{as}\upsilon\_{bs}\upsilon\_{cs} \cdots -BackEMF\_i \cdots \upsilon\_{ns} \end{bmatrix}^T \tag{23}$$

$$\begin{bmatrix} \mathbf{O} \end{bmatrix} = \begin{bmatrix} \boldsymbol{\upsilon}\_{ar} \boldsymbol{\upsilon}\_{br} \boldsymbol{\upsilon}\_{cr} \boldsymbol{\upsilon}\_{dr} \boldsymbol{\upsilon}\_{cr} \cdots \boldsymbol{\upsilon}\_{nr} \end{bmatrix}^{\mathrm{T}} \tag{24}$$

$$\begin{bmatrix} \boldsymbol{\mathcal{Z}}\_{s} \end{bmatrix} = \begin{bmatrix} \boldsymbol{\mathcal{Z}}\_{as}\boldsymbol{\mathcal{Z}}\_{bs}\boldsymbol{\mathcal{Z}}\_{cs}\boldsymbol{\mathcal{Z}}\_{ds}\boldsymbol{\mathcal{Z}}\_{cs}\cdots\boldsymbol{\mathcal{Z}}\_{ns} \end{bmatrix}^{T} \tag{25}$$

$$\begin{bmatrix} \mathcal{Z}\_r \end{bmatrix} = \begin{bmatrix} \mathcal{Z}\_{ar}\mathcal{Z}\_{br}\mathcal{Z}\_{cr}\mathcal{Z}\_{dr}\mathcal{Z}\_{cr}\cdots\mathcal{Z}\_{nr} \end{bmatrix}^T \tag{26}$$

$$\begin{bmatrix} I\_s \end{bmatrix} = \begin{bmatrix} i\_{as} \ i\_{bs} i\_{cs} \cdots 0 \cdots i\_{ns} \end{bmatrix}^T \tag{27}$$

$$\begin{bmatrix} I\_r \end{bmatrix} = \begin{bmatrix} i\_{ar} i\_{br} i\_{cr} i\_{dr} i\_{dr} \cdots i\_{nr} \end{bmatrix}^T \tag{28}$$

Notice that the rotor components remain the same as in normal operation, due to the fact that in postfault operation the machine rotor maintains a symmetrical winding distribution.

The absence of the stator phase results in a loss in one degree of freedom. Depending on the position of the faulty phase the transformation matrix (16) is modified, making it no longer possible to generate the same number of orthogonal sub-systems, leading to the removal of one or more of the generated components (29).

( ) ( ) ( ) (( ) ) ( ) ( ) ( ) (( ) ) ( ) ( ) ( ) ( ( ) ) ( ) ( ) ( ) ( ( ) ) ( ) ( ) ( ) ( ( ) ) ( ) ( ) ( ) ( ( ) ) 1 cos cos 2 cos 3 0 cos 1 0 sin sin 2 sin 3 0 sin 1 1 cos 2 cos 4 cos 6 0 cos 2 1 0 sin 2 sin 4 sin 6 0 sin 2 1 1 cos 3 cos 6 cos 9 0 cos 3 1 0 sin 3 sin 6 sin 9 0 sin 3 1 <sup>2</sup> <sup>0</sup> 00 0 0 0 0 0 <sup>1</sup> 1 cos JJJ J JJJ J JJ J J JJ J J JJ J J JJ J J - - - - - é ù = ë û - L L L L L L L L L L L L M M M M ML M L L M M M M MO M *nUF n n n n n n <sup>T</sup> <sup>n</sup> <sup>n</sup>* ( ) ( ) 1 1 <sup>1</sup> cos 2 cos 3 0 cos 1 222 2 <sup>111</sup> <sup>1</sup> 0 sin sin 2 sin 3 0 sin 1 222 2 11 1 1 <sup>1</sup> <sup>0</sup> 22 2 2 2 JJJ J JJJ J <sup>é</sup> <sup>ù</sup> <sup>ê</sup> <sup>ú</sup> <sup>ê</sup> <sup>ú</sup> <sup>ê</sup> <sup>ú</sup> <sup>ê</sup> <sup>ú</sup> <sup>ê</sup> <sup>ú</sup> <sup>ê</sup> <sup>ú</sup> <sup>ê</sup> <sup>ú</sup> <sup>ê</sup> <sup>ú</sup> <sup>ê</sup> <sup>ú</sup> <sup>ê</sup> <sup>ú</sup> <sup>ê</sup> <sup>ú</sup> <sup>ê</sup> <sup>ú</sup> <sup>ê</sup> <sup>ú</sup> <sup>ê</sup> <sup>ú</sup> <sup>ê</sup> <sup>ú</sup> <sup>ê</sup> <sup>ú</sup> <sup>ê</sup> æ öæ öæ ö æ ö - - - <sup>ú</sup> ç ÷ç ÷ç ÷ ç ÷ - <sup>ê</sup> <sup>ú</sup> è øè øè ø è ø <sup>ê</sup> <sup>ú</sup> ê æ öæ öæ ö --- <sup>æ</sup> - ö ú ç ÷ç ÷ç ÷ ç ÷ - <sup>ê</sup> <sup>ú</sup> è øè øè ø è ø <sup>ê</sup> <sup>ú</sup> <sup>ê</sup> <sup>ú</sup> <sup>ê</sup> <sup>ú</sup> <sup>ë</sup> <sup>û</sup> L L L L L L *n n <sup>n</sup> <sup>n</sup> nnn <sup>n</sup> <sup>n</sup>* (29)

Consequently, the machine coupling inductance matrices (10-14) for the stator-rotor compo‐ nents need to be arranged considering the absence of the faulty phase.

éùé = - ù ëûë L L û

éù é 0 = ù ëû ë L û

lllll

= ù ëû ë L û

lllll

= ù ëû ë L û

éù é = 0 ù ëû ë L L û

é ù <sup>=</sup> [ ] ë û <sup>L</sup> *<sup>T</sup>*

Notice that the rotor components remain the same as in normal operation, due to the fact that in postfault operation the machine rotor maintains a symmetrical winding distribution.

The absence of the stator phase results in a loss in one degree of freedom. Depending on the position of the faulty phase the transformation matrix (16) is modified, making it no longer possible to generate the same number of orthogonal sub-systems, leading to the removal of

> 1 cos cos 2 cos 3 0 cos 1 0 sin sin 2 sin 3 0 sin 1 1 cos 2 cos 4 cos 6 0 cos 2 1 0 sin 2 sin 4 sin 6 0 sin 2 1 1 cos 3 cos 6 cos 9 0 cos 3 1 0 sin 3 sin 6 sin 9 0 sin 3 1

00 0 0 0 0

M M M M MO M

11 1 1 <sup>1</sup> <sup>0</sup> 22 2 2 2

M M M M ML M

<sup>2</sup> <sup>0</sup>

JJJ

JJJ

JJJ

JJJ

JJ

JJ

JJ

JJ

( ) ( ) ( ) (( ) ) ( ) ( ) ( ) (( ) ) ( ) ( ) ( ) ( ( ) ) ( ) ( ) ( ) ( ( ) ) ( ) ( ) ( ) ( ( ) ) ( ) ( ) ( ) ( ( ) )

 J

 J

 J

 J

*<sup>n</sup>* ( )

*nnn <sup>n</sup> <sup>n</sup>*

<sup>é</sup> <sup>ù</sup> <sup>ê</sup> <sup>ú</sup> <sup>ê</sup> <sup>ú</sup> <sup>ê</sup> <sup>ú</sup> <sup>ê</sup> <sup>ú</sup> <sup>ê</sup> <sup>ú</sup> <sup>ê</sup> <sup>ú</sup> <sup>ê</sup> <sup>ú</sup> <sup>ê</sup> <sup>ú</sup> <sup>ê</sup> <sup>ú</sup> <sup>ê</sup> <sup>ú</sup> <sup>ê</sup> <sup>ú</sup> <sup>ê</sup> <sup>ú</sup> <sup>ê</sup> <sup>ú</sup> <sup>ê</sup> <sup>ú</sup> <sup>ê</sup> <sup>ú</sup> <sup>ê</sup> <sup>ú</sup> <sup>ê</sup> æ öæ öæ ö æ ö - - - <sup>ú</sup> ç ÷ç ÷ç ÷ ç ÷ - <sup>ê</sup> <sup>ú</sup> è øè øè ø è ø <sup>ê</sup> <sup>ú</sup> ê æ öæ öæ ö --- <sup>æ</sup> - ö ú ç ÷ç ÷ç ÷ ç ÷ - <sup>ê</sup> <sup>ú</sup> è øè øè ø è ø <sup>ê</sup> <sup>ú</sup> <sup>ê</sup> <sup>ú</sup> <sup>ê</sup> <sup>ú</sup> <sup>ë</sup> <sup>û</sup>

<sup>111</sup> <sup>1</sup> 0 sin sin 2 sin 3 0 sin 1 222 2

222 2

1 1 <sup>1</sup> cos 2 cos 3 0 cos 1

*n n <sup>n</sup> <sup>n</sup>*

éù é l

336 Induction Motors - Applications, Control and Fault Diagnostics

éù é l

one or more of the generated components (29).

<sup>1</sup> 1 cos


é ù = ë û

*<sup>T</sup> <sup>n</sup>*

*nUF*

*T V v v v BackEMF v s as bs cs i ns* (23)

*ar br cr dr er nr vvvvv v* (24)

*s as bs cs ds es ns* (25)

*r ar br cr dr er nr* (26)

*s as bs cs ns I i ii i* (27)

*r ar br cr dr er nr I iiiii i* (28)

J


*n n n n n n*

( )

J

J

J

J

J

J

(29)

J

0

L L

L L

L L

L L

L L L L L L L L L L L L

*T*

*T*

*T*

*T*

 l

 l

$$
\begin{bmatrix}
\Lambda(\mathcal{G})\_{\text{u}\text{H}} \\
\hline
\end{bmatrix} = \begin{bmatrix}
1 & \cos(\mathfrak{z}) & \cos(2\mathfrak{z}) & \cdots & 0 & \cdots & \cos((n-1)\mathfrak{z}) \\
\cos((n-1)\mathfrak{z}) & 1 & \cos(\mathfrak{z}) & \cdots & 0 & \cdots & \cos((n-2)\mathfrak{z}) \\
\cos((n-2)\mathfrak{z}) & \cos((n-1)\mathfrak{z}) & 1 & \cdots & 0 & \cdots & \cos((n-3)\mathfrak{z}) \\
\cos((n-3)\mathfrak{z}) & \cos((n-2)\mathfrak{z}) & \cos((n-1)\mathfrak{z}) & \cdots & 0 & \cdots & \cos((n-4)\mathfrak{z}) \\
\vdots & \vdots & \vdots & \vdots & 0 & \ddots & \vdots \\
0 & 0 & 0 & 0 & 0 & \cdots & 0 \\
\vdots & \vdots & \vdots & \vdots & 0 & \vdots & \vdots \\
\cos(\mathfrak{z}) & \cos(2\mathfrak{z}) & \cos(3\mathfrak{z}) & \cdots & 0 & \cdots & 1 \\
\end{bmatrix}
\tag{30}
$$

The equations (1) and (2) need to be multiplied by the new Clarke transformation matrix *Tnuf* , to express the stator and rotor voltage, current and flux components in the *α*1 - *β*1 - *α*2 *β*2 - … - *zn* reference frame in post-fault situation:

$$\begin{aligned} \left[\begin{array}{c} T\_{nu'} \end{array}\right] \left[\begin{array}{c} V\_s \end{array}\right] &= \left[\begin{array}{c} T\_{nu'} \end{array}\right] \left[\begin{array}{c} R\_s \end{array}\right] \left[\begin{array}{c} T\_{nu'} \end{array}\right]^{-1} \left[\begin{array}{c} T\_{nu'} \end{array}\right] \left[\begin{array}{c} I\_s \end{array}\right] \\ &+ p \cdot \left[\begin{array}{c} T\_{nu'} \end{array}\right] \left[\begin{array}{c} L\_{su'} \end{array}\right] \left[\begin{array}{c} T\_{nu'} \end{array}\right]^{-1} \cdot \left[\begin{array}{c} T\_{su'} \end{array}\right] \left[\begin{array}{c} I\_s \end{array}\right] \\ &+ p \cdot \left[\begin{array}{c} T\_{nu'} \end{array}\right] \left[\begin{array}{c} \mathcal{L}\_{su'} \end{array}\right] \left[\begin{array}{c} T\_{nu'} \end{array}\right]^{-1} \cdot \left[\begin{array}{c} T\_{nu'} \end{array}\right] \left[\begin{array}{c} I\_r \end{array}\right] \end{aligned} \tag{31}$$

$$\begin{aligned} \begin{bmatrix} \mathbf{O} \end{bmatrix} &= \begin{bmatrix} T\_{nuf} \end{bmatrix} \begin{bmatrix} \mathbf{R}\_r \end{bmatrix} \begin{bmatrix} T\_{nuf} \end{bmatrix}^{-1} \begin{bmatrix} T\_{nuf} \end{bmatrix} \begin{bmatrix} I\_r \end{bmatrix} \\ &+ p \begin{bmatrix} T\_{nuf} \end{bmatrix} \begin{bmatrix} \mathbf{L}\_{nr} \end{bmatrix} \begin{bmatrix} T\_{nuf} \end{bmatrix}^{-1} \begin{bmatrix} T\_{nuf} \end{bmatrix} \begin{bmatrix} I\_r \end{bmatrix} \\ &+ p \begin{bmatrix} T\_{nuf} \end{bmatrix} \begin{bmatrix} \mathbf{L}\_{nr} \left( \boldsymbol{\theta} \right) \end{bmatrix} \begin{bmatrix} T\_{nuf} \end{bmatrix}^{-1} \begin{bmatrix} T\_{nuf} \end{bmatrix} \begin{bmatrix} I\_s \end{bmatrix} \end{aligned} \tag{32}$$

The equations (31) and (32) depict the stator and rotor voltage vector equations in the *α*1 - *β*1 *α*2 - *β*2 - … - *zn* reference frame in postfault situation, when an open-phase fault occurs in a multiphase drive with odd number of phases. In the next section these equations are particu‐ larized for a five-phase machine.

### **4. Open-phase fault operation in five-phase drives**

The case study presented in this chapter is a 5-phase induction machine with symmetrical and distributed windings. The *n*-phase mathematical model presented in the previous section must be first particularized for the 5-phase case to understand the system behavior in the faulty situation and to predict the effect of the selected control actions on the post-fault controlled system. Two different post-fault control strategies for the open-phase fault management will be presented. The first one is based on linear Proportional Resonant (PR) current controllers and the field oriented control technique. The second one is also based on the field oriented control method but combined with a Predictive Current Control (PCC) technique. Both control methods can be applied during postfault operation, and will be described in this section, along with the criteria that can be used to generate the current references in the drive during the fault. These criteria differ from those established in healthy operation, and constitute one of the bases of the postfault operation of the drive.

### **4.1. A. Five-phase induction machine modeling under an open phase fault**

The general *n*-phase induction machine model introduced before can be particularized for the 5-phase case. Taking also into account that the faulty phase is '*a*', which can be made without any lack of generality, from now on it can be assumed that *i as* = 0. The stator/rotor resistance, inductance and coupling general matrices can be obtained as follows:

$$\begin{bmatrix} \ R\_s \ \end{bmatrix} = R\_s \cdot \begin{bmatrix} I\_4 \ \end{bmatrix} \tag{33}$$

$$\left\lfloor R\_r \right\rfloor = R\_r \cdot \left\lfloor I\_4 \right\rfloor \tag{34}$$

$$L\_{ss} = L\_{ls} \begin{bmatrix} I\_4 \end{bmatrix} + L\_m \begin{bmatrix} \Lambda(\mathcal{G}) \end{bmatrix} \tag{35}$$

$$L\_{rr} = L\_{lr} \left[ I\_4 \right] + L\_{nl} \left[ \Lambda(\mathcal{G}) \right] \tag{36}$$

$$
\begin{bmatrix}
\Lambda\begin{bmatrix}
\mathcal{A}\end{bmatrix}\_{\boldsymbol{u}}
\end{bmatrix} = 
\begin{bmatrix}
1 & \cos(\mathcal{A}) & \cos(2\mathcal{A}) & \cos(3\mathcal{A}) & \cos(4\mathcal{A}) \\
\cos(4\mathcal{A}) & 1 & \cos(\mathcal{A}) & \cos(2\mathcal{A}) & \cos(3\mathcal{A}) \\
\cos(3\mathcal{A}) & \cos(4\mathcal{A}) & 1 & \cos(\mathcal{A}) & \cos(2\mathcal{A}) \\
\cos(2\mathcal{A}) & \cos(3\mathcal{A}) & \cos(4\mathcal{A}) & 1 & \cos(\mathcal{A}) \\
\cos(\mathcal{A}) & \cos(2\mathcal{A}) & \cos(3\mathcal{A}) & \cos(4\mathcal{A}) & 1
\end{bmatrix}
\tag{37}
$$

$$
\begin{bmatrix}
\begin{bmatrix}
\cos(\Lambda\_{1}) & \cos(\Lambda\_{2}) & \cos(\Lambda\_{3}) & \cos(\Lambda\_{4}) & \cos(\Lambda\_{5}) \\
\cos(\Lambda\_{5}) & \cos(\Lambda\_{1}) & \cos(\Lambda\_{2}) & \cos(\Lambda\_{3}) & \cos(\Lambda\_{4}) \\
\cos(\Lambda\_{4}) & \cos(\Lambda\_{5}) & \cos(\Lambda\_{1}) & \cos(\Lambda\_{2}) & \cos(\Lambda\_{3}) \\
\cos(\Lambda\_{3}) & \cos(\Lambda\_{4}) & \cos(\Lambda\_{5}) & \cos(\Lambda\_{1}) & \cos(\Lambda\_{2}) \\
\cos(\Lambda\_{3}) & \cos(\Lambda\_{4}) & \cos(\Lambda\_{4}) & \cos(\Lambda\_{5}) & \cos(\Lambda\_{1})
\end{bmatrix}
\tag{38}
$$

where *L ls* and *L lr* are the stator and rotor leakage inductances, *M* is the mutual inductance of the machine *M* =5*L <sup>m</sup>* / 2, and the stator and rotor inductances are defined as *L <sup>s</sup>* = *M* + *L ls* and *L <sup>r</sup>* = *M* + *L lr*, respectively.

system. Two different post-fault control strategies for the open-phase fault management will be presented. The first one is based on linear Proportional Resonant (PR) current controllers and the field oriented control technique. The second one is also based on the field oriented control method but combined with a Predictive Current Control (PCC) technique. Both control methods can be applied during postfault operation, and will be described in this section, along with the criteria that can be used to generate the current references in the drive during the fault. These criteria differ from those established in healthy operation, and constitute one of

The general *n*-phase induction machine model introduced before can be particularized for the 5-phase case. Taking also into account that the faulty phase is '*a*', which can be made without

<sup>4</sup> = +L · · () éù é ù

<sup>4</sup> = +L · · () éù é ù

J

JJJ

Ψ( ) cos cos cos cos cos

é ù = DDDDD ë û

J

J

JJ

( ) ( ) ( ) ( )

cos cos cos cos cos cos cos cos cos cos

cos cos cos cos cos cos cos cos cos cos

J

J

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

JJJ

ë û

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

DDDDD ê D D D D Dú ë û

é ù DDDDD ê ú DDDDD

12345 51234 45123 34512 23451

 J

1 cos cos 2 cos 3 cos 4 cos 4 1 cos cos 2 cos 3 cos 3 cos 4 1 cos cos 2 cos 2 cos 3 cos 4 1 cos cos cos 2 cos 3 cos 4 1

é ù ê ú

( ) ( ) ( ) ( )

*<sup>n</sup>* (37)

*as* = 0. The stator/rotor resistance,

<sup>4</sup> é ù = ·[ ] ë û *R RI s s* (33)

<sup>4</sup> é ù = ·[ ] ë û *R RI r r* (34)

*ss ls* ëû ë û *<sup>m</sup> L LI L* (35)

*rr lr* ëû ë û *<sup>m</sup> L LI L* (36)

JJ

 J

 J

(38)

JJJ

 J

**4.1. A. Five-phase induction machine modeling under an open phase fault**

the bases of the postfault operation of the drive.

338 Induction Motors - Applications, Control and Fault Diagnostics

( )

J

é ù L = ë û

q

any lack of generality, from now on it can be assumed that *i*

inductance and coupling general matrices can be obtained as follows:

The 5-phase case is characterized by the transformation matrix (*T*5). The stator and rotor phase variables can be mapped to a set of four independent variables divided in two orthogonal stationary planes (namely *α*-*β* and *x*-*y* subspaces) and a zero sequence component (*z* compo‐ nent). Notice that the distributed windings' characteristic of the five-phase machine deter‐ mines that the torque production is only dependent of the *α* - *β* components, while *x* - *y* components only generate motor losses. This particularization for the 5-phase case can be summarized in the following equations:

$$
\begin{bmatrix}
\dot{\mathbf{i}}\_{as} & \dot{\mathbf{i}}\_{\beta s} & \dot{\mathbf{i}}\_{as} & \dot{\mathbf{i}}\_{ys} & \dot{\mathbf{i}}\_{ss}
\end{bmatrix}^T = \begin{bmatrix}
T\_g \\
\end{bmatrix} \begin{bmatrix}
\dot{\mathbf{i}}\_{as} & \dot{\mathbf{i}}\_{bs} & \dot{\mathbf{i}}\_{cs} & \dot{\mathbf{i}}\_{ds} & \dot{\mathbf{i}}\_{as}
\end{bmatrix}^T \tag{39}
$$

$$
\begin{bmatrix} \boldsymbol{\upsilon}\_{as} & \boldsymbol{\upsilon}\_{\rho\_{\rm bs}} & \boldsymbol{\upsilon}\_{\rm us} & \boldsymbol{\upsilon}\_{ys} & \boldsymbol{\upsilon}\_{\rm us} \end{bmatrix}^{\mathrm{T}} = \begin{bmatrix} \boldsymbol{T}\_{\rm b} \end{bmatrix} \begin{bmatrix} \boldsymbol{\upsilon}\_{as} & \boldsymbol{\upsilon}\_{\rm bs} & \boldsymbol{\upsilon}\_{\rm cs} & \boldsymbol{\upsilon}\_{\rm bs} & \boldsymbol{\upsilon}\_{\rm cs} \end{bmatrix}^{\mathrm{T}} \tag{40}
$$

$$
\begin{array}{c|cccc}
 & 1 & \cos(\mathfrak{g}) & \cos(2\mathfrak{g}) & \cos(3\mathfrak{g}) & \cos(4\mathfrak{g}) \\
 & \text{ $0$ } & \sin(\mathfrak{g}) & \sin(2\mathfrak{g}) & \sin(3\mathfrak{g}) & \sin(4\mathfrak{g}) \\
\text{ $1$ } & \cos(2\mathfrak{g}) & \cos(4\mathfrak{g}) & \cos(\mathfrak{g}) & \cos(3\mathfrak{g}) \\
 & \text{ $0$ } & \sin(2\mathfrak{g}) & \sin(4\mathfrak{g}) & \sin(3\mathfrak{g}) & \sin(3\mathfrak{g}) \\
\text{ $1/2$ } & \text{ $1/2$ } & \text{ $1/2$ } & \text{ $1/2$ } & \text{ $1/2$ } \\
\end{array}
\tag{41}
$$

While the traditional Clarke transformation matrix (*T5*) is applied in healthy state, a modified matrix can be used under open-phase postfault operation in order to have a reduced-order subset of equations. If the reduced-order Clarke transformation matrix remains orthogonal as in (*T5*), the asymmetries lead to noncircular *α* - *β* current components. In order to compensate for the stator/rotor impedance asymmetries appearing in postfault situation, a new nonor‐ thogonal transformation matrix that will be named (*TPCC*) is used here [31]. When the fault appears, it is no longer possible to define four independent variables in the system because a fixed relationship exists between *α* and *x* current components, being *i sx* = −*i s*.

$$
\begin{bmatrix} T\_{\text{pcc}} \end{bmatrix} = \frac{2}{5} \begin{bmatrix} \cos(\mathfrak{H}) - 1 & \cos(2\mathfrak{H}) - 1 & \cos(3\mathfrak{H}) - 1 & \cos(4\mathfrak{H}) - 1 \\ \sin(\mathfrak{H}) & \sin 2\mathfrak{H} & \sin(3\mathfrak{H}) & \sin(4\mathfrak{H}) \\ \sin(2\mathfrak{H}) & \sin(4\mathfrak{H}) & \sin(6\mathfrak{H}) & \sin(8\mathfrak{H}) \\ 1 & 1 & 1 & 1 \end{bmatrix} \tag{42}
$$

In a similar way, the coordinate transformation can be applied to the machine voltage equations. The stator phase voltages in normal operation (*Vpre*) depend on the switching state of every leg of the power converter (*Si* ), being *Si* =0 if the lower switch is ON and the upper switch is OFF, and *Si* =1 if the opposite occurs.

$$
\begin{bmatrix} v\_{as} \\ v\_{bs} \\ v\_{cs} \\ v\_{ds} \\ v\_{cs} \end{bmatrix} = \frac{V\_{DC}}{5} \begin{bmatrix} 4 & -1 & -1 & -1 & -1 \\ -1 & 4 & -1 & -1 & -1 \\ -1 & -1 & 4 & -1 & -1 \\ -1 & -1 & -1 & 4 & -1 \\ -1 & -1 & -1 & -1 & 4 \end{bmatrix} \begin{bmatrix} S\_a \\ S\_b \\ S\_c \\ S\_d \\ v\_c \end{bmatrix} = V\_{pre} \tag{43}
$$

During prefault operation the five-phase drive possesses 2<sup>5</sup> =32 switching states and the sum of the healthy phase voltages is zero (∑*vin* =0). However, if an open-phase fault appears, the available switching states are reduced to 2<sup>4</sup> =16 and the faulty phase current is zero. Nonethe‐ less, the faulty phase voltage is not null since there is a back-emf induced in the faulty phase, leading to an asymmetric effect in the machine modeling [31]. Taking this into account, the phase voltage of the faulty phase ('*a*') is given by:

$$
\omega\_{as} = \mathbf{R}\_s \cdot \mathbf{i}\_{as} + \frac{d}{dt} \mathcal{A}\_{as} = \frac{d}{dt} \mathcal{A}\_{as} = \text{BackEmf}\_a \tag{44}
$$

Consequently, the back-emf term can be expressed as (45), estimating the stator flux term in (44) and considering the transformation matrix (*T5*) and *i sx* = −*i s*.

$$\begin{aligned} \dot{\lambda}\_{as} &= \dot{\lambda}\_{as} + \dot{\lambda}\_{\infty} = \mathbf{L}\_{s}\mathbf{i}\_{as} + \mathbf{L}\_{m}\mathbf{i}\_{ar} + \mathbf{L}\_{ls}\mathbf{i}\_{as} \\ \mathbf{B}ackEmf\_{a} &= \frac{d}{dt} \Big[ \mathbf{L}\_{s}\mathbf{i}\_{as} + \mathbf{L}\_{m}\mathbf{i}\_{ar} - \mathbf{L}\_{ls}\mathbf{i}\_{as} \Big] \end{aligned} \tag{45}$$

As a result, the stator phase voltage matrix (43) must be modified considering the faulty phase back-emf in the phase voltage equilibrium equations and must guarantee sinusoidal flux [4]. Consequently, taking into account the faulty phase voltage and the absence of current in the open-phase, the stator phase voltage matrix can be written as in (46).

$$
\begin{bmatrix} v\_{bs} \\ v\_{cs} \\ v\_{ds} \\ v\_{ds} \\ v\_{sc} \end{bmatrix} = \frac{V\_{DC}}{4} \begin{bmatrix} 3 & -1 & -1 & -1 \\ -1 & 3 & -1 & -1 \\ -1 & -1 & 3 & -1 \\ -1 & -1 & -1 & 3 \end{bmatrix} \begin{bmatrix} S\_b \\ S\_c \\ S\_d \\ S\_d \\ S\_e \end{bmatrix} - \frac{L\_m \cdot \frac{di\_{as}}{dt} + L\_m \cdot \frac{di\_{ar}}{dt}}{4} \begin{bmatrix} I\_4 \end{bmatrix} \tag{46}
$$

where [*I4*] is the identity matrix of order 4 and the second term on the right hand side is the counter electromotive force (45).

### **4.2. B. Implemented fault-tolerant control methods**

of every leg of the power converter (*Si*

switch is OFF, and *Si* =1 if the opposite occurs.

340 Induction Motors - Applications, Control and Fault Diagnostics

5

*V*

phase voltage of the faulty phase ('*a*') is given by:

*DC*

4 1111 14 1 1 1 · · 1 14 1 1

> 1 1 14 1 1 1 1 14

During prefault operation the five-phase drive possesses 2<sup>5</sup> =32 switching states and the sum of the healthy phase voltages is zero (∑*vin* =0). However, if an open-phase fault appears, the available switching states are reduced to 2<sup>4</sup> =16 and the faulty phase current is zero. Nonethe‐ less, the faulty phase voltage is not null since there is a back-emf induced in the faulty phase, leading to an asymmetric effect in the machine modeling [31]. Taking this into account, the

> l*as s as as as a*

Consequently, the back-emf term can be expressed as (45), estimating the stator flux term in

 aa

*as s xs s s m r ls xs*

=+= + +

*<sup>d</sup> BackEmf L i L i L i dt*

a

 aa

= +- é ù ë û

*Li L i L i*

*a s s m r ls s*

As a result, the stator phase voltage matrix (43) must be modified considering the faulty phase back-emf in the phase voltage equilibrium equations and must guarantee sinusoidal flux [4]. Consequently, taking into account the faulty phase voltage and the absence of current in the

· · 13 1 1 ·· · 4 4 1 13 1

é ù é ù ê ú é ù --- ê ú ê ú ê ú ê ú <sup>+</sup> - -- ê ú ê ú <sup>=</sup> ê ú - é ù

*di di v S L L <sup>V</sup> dt dt v SI*

*s r bs <sup>b</sup> m m DC*

ë û ê ú ê ú -- - ê ú ê ú ê ú ê ú --- ê ú ë û ê ú ë û ë û

a

*d d v Ri BackEmf dt dt* (44)

*s*.

4

 a

*cs c pre*

é ù é ù ---- é ù ê ú ê ú ê ú - --- ê ú ê ú ê ú = = -- -- ê ú ê ú ê ú ê ú --- - ê ú ê ú ê ú ---- ë û ë û ë û

*v S V v S v v*

*ds d es e*

> =+ = = · l

(44) and considering the transformation matrix (*T5*) and *i sx* = −*i*

a

open-phase, the stator phase voltage matrix can be written as in (46).

3 111

1 1 13

*cs c ds d es e*

*v S v S*

lll

*as a bs b*

*v S v S*

), being *Si* =0 if the lower switch is ON and the upper

(43)

(45)

(46)

In what follows, the two implemented open-phase fault-tolerant controllers are presented. Different control criteria can be implemented depending on the overall electrical drive aim. However, only field oriented control methods have been recently used to manage postfault (open-phase type) operations. The inner current controllers of the field oriented controller have been implemented using linear or predictive control techniques. Both methods require a redefinition of the stator current references in the postfault operation to ensure minimum copper losses, a minimum derating strategy or minimum torque ripples in the drive [4, 31-35]. The maximum achievable *α* - *β* currents in the electrical drive vary depending on the selected control criteria. In general, the minimum copper loss criteria is used in applications where efficiency is of special interest and, consequently, Joules losses need to be minimized, while the minimum derating or the minimum torque ripple strategies are preferred when the faulty electrical drive must provide the maximum achievable torque or ensure smooth, vibrationfree operation, respectively. From the postfault operation control performance and controller perspective, all the techniques behave in a similar way in the multiphase drive. In our case, the minimum copper loss criteria will be used for comparison purposes for the sake of simplicity.

### **5. Minimum copper loss criteria**

The minimum copper loss (MCL) criterion focuses on reducing the drive losses. The *α* - *β* stator current references are then calculated in order to ensure proper torque/flux control while imposing a rotating circle-shaped MMF and maintaining the amplitudes of the phase currents bellow the rated values of the drive (these maximum values are established by the power semiconductors of converter and the stator windings of the electrical machine). As a result, the drive needs to be derated in such a way that the remaining healthy phases do not exceed their nominal current value (*In*) and the maximum reference currents in the *α* - *β* subspace are [36]:

$$\mathbf{i}\_{\alpha s}^{\text{max}} = \mathbf{0}.681 \mathbf{3} \cdot \mathbf{I}\_n \mathbf{i}\_{\beta s}^{\text{max}} = -\mathbf{0}.681 \mathbf{3} \cdot \mathbf{I}\_n \tag{47}$$

The non-torque contributing *y* -current reference is set to zero (*i y* \* =0) in order to minimize Joules losses, whereas the *x* -current component is not anymore an independent variable for the controller (it is inherently fixed to the *α* -current component after the fault occurrence). Notice that the procedure to manage the postfault operation effectively minimizes the electrical drive losses, at the expense of reducing the maximum obtainable postfault torque and generating unequal peaks of the phase currents [36, 37].

### **6. Predictive Current Control (PCC)**

The first fault-tolerant control scheme is the PCC method, based on Finite-Control Set (FCS) Model-Based Predictive Techniques [38]. An accurate discrete system model is required in order to predict the machines' operation for every VSI state. The implemented controller is based on an outer PI-based closed-loop speed control and an inner fault-tolerant PCC method, as shown in Figure 2. During every sampling period (*k*), the speed and the stator currents of the machine are measured. Then, stator currents are mapped into the stationary *α - β* - *x* - *y* subspaces by means of the modified Clarke transformation (*TPCC*) for postfault operation. The postfault available voltage vectors (24 =16) are used afterwards to predict the stator currents evolution for the next sampling period (*k+1*). These current references are finally evaluated in a cost function (*J*) to determine which voltage vector produces the minimum values of *J*. This voltage vector is referred as the optimum switching state (*Si optimum*(*k* + 1)) to be applied in the power converter of the electrical drive to minimize the cost function (equivalent to the control law). Notice that different cost functions can be defined in order to include different control criteria. This can be easily done by setting weighting factors in the definition of the cost function, as shown in (48) where the *A*, *B*, *C* and *D* terms multiply errors between the reference (*i si* \* ) and the predicted (*i* ^ *si* ) stator currents in the *α - β - x* - *y* reference frame (49)-(50).

$$J = A \left| \overline{\dot{i}\_{s\alpha}} \right| + B \left| \overline{\dot{i}\_{s\beta}} \right| + C \left| \overline{\dot{i}\_{s\beta}} \right| + D \left| \overline{\dot{i}\_{s\beta}} \right| \tag{48}$$

$$\overline{\dot{i}\_{sa}} = \dot{i}\_{sa}^\* \left(k+1\right) - \hat{i}\_{sa} \left(k+1\right),\\ \overline{\dot{i}\_{s\rho}} = \dot{i}\_{s\rho}^\* \left(k+1\right) - \hat{i}\_{s\rho} \left(k+1\right) \tag{49}$$

$$\overline{\dot{i}\_{sx}} = \stackrel{\circ}{i\_{sx}} \left(k+1\right) - \stackrel{\circ}{i\_{sx}} \left(k+1\right), \overline{\dot{i}\_{sy}} = \stackrel{\circ}{i\_{sy}} \left(k+1\right) - \stackrel{\circ}{i\_{sy}} \left(k+1\right) \tag{50}$$

The main control criterion in healthy operation is to maintain a desired electrical torque, while ensuring sinusoidal stator current references in phase coordinates (*a-b-c-d-e*). This objective is met under normal drive operation by setting a constant circular stator current reference vector in the *α* - *β* plane and a zero reference stator current vector in the *x* - *y* plane. In postfault operation, the *x* -axis stator current is inherently fixed to the *α* -axis stator current. Then, the *α* - *β* stator current references can be set following a circular trajectory but with a derating factor in its maximum value, while the *y*-axis stator current is now controlled to be null.

Successively, the *α-β* current components are mapped in the rotating *d*-*q* reference frame by means of the Park rotating transformation (51) and the field-oriented control position estimator (52).

$$
\begin{bmatrix} \dot{i}\_{sa} \\ \dot{i}\_{s\theta} \end{bmatrix} = \begin{bmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{bmatrix} \begin{bmatrix} \dot{i}\_{sd} \\ \dot{i}\_{sq} \end{bmatrix} \tag{51}
$$

$$\theta = \int \left( \alpha\_r + \frac{\dot{i}\_{sq}^\*}{\tau\_r \cdot \dot{i}\_{sd}^\*} \right) dt \tag{52}$$

The implementation of PCC techniques for multiphase fault-tolerant drives requires the same control scheme for pre- and postfault operation, as long as the following considerations are addressed after the fault occurrence detection: • The weight of the x - y currents has to be changed in the cost function. The x current


need to be modified as in (46) and the Clarke transformation of (42).

Figure 2. Postfault controller based on the PCC technique. **Figure 2.** Postfault controller based on the PCC technique.

**6. Predictive Current Control (PCC)**

342 Induction Motors - Applications, Control and Fault Diagnostics

(*i si* \*

(52).

) and the predicted (*i*

voltage vector is referred as the optimum switching state (*Si*

^ *si*

aa

The first fault-tolerant control scheme is the PCC method, based on Finite-Control Set (FCS) Model-Based Predictive Techniques [38]. An accurate discrete system model is required in order to predict the machines' operation for every VSI state. The implemented controller is based on an outer PI-based closed-loop speed control and an inner fault-tolerant PCC method, as shown in Figure 2. During every sampling period (*k*), the speed and the stator currents of the machine are measured. Then, stator currents are mapped into the stationary *α - β* - *x* - *y* subspaces by means of the modified Clarke transformation (*TPCC*) for postfault operation. The postfault available voltage vectors (24 =16) are used afterwards to predict the stator currents evolution for the next sampling period (*k+1*). These current references are finally evaluated in a cost function (*J*) to determine which voltage vector produces the minimum values of *J*. This

power converter of the electrical drive to minimize the cost function (equivalent to the control law). Notice that different cost functions can be defined in order to include different control criteria. This can be easily done by setting weighting factors in the definition of the cost function, as shown in (48) where the *A*, *B*, *C* and *D* terms multiply errors between the reference

abb

= +- + = +- + *ss s ss s*

 a

( ) ( ) ( ) ( ) \* \* 1 1 , ˆ 1 1 ˆ

The main control criterion in healthy operation is to maintain a desired electrical torque, while ensuring sinusoidal stator current references in phase coordinates (*a-b-c-d-e*). This objective is met under normal drive operation by setting a constant circular stator current reference vector in the *α* - *β* plane and a zero reference stator current vector in the *x* - *y* plane. In postfault operation, the *x* -axis stator current is inherently fixed to the *α* -axis stator current. Then, the *α* - *β* stator current references can be set following a circular trajectory but with a derating factor in its maximum value, while the *y*-axis stator current is now controlled to be null.

Successively, the *α-β* current components are mapped in the rotating *d*-*q* reference frame by means of the Park rotating transformation (51) and the field-oriented control position estimator

cos sin( ) ( ) · sin( ) cos( )

é ù é ù - é ù ê ú <sup>=</sup> ê ú ê ú ê ú ë û ë û ê ú ë û *s sd s sq i i*

 q

 q

q

q

a

b

 b b

) stator currents in the *α - β - x* - *y* reference frame (49)-(50).

*i ik ik i ik ik* (49)

( ) ( ) ( ) ( ) \* \* 1 1 , 1 1 ˆ ˆ = +- + = +- + *sx sx sx sy sy sy i ik ik i ik ik* (50)

= +++ *sss sy J Ai Bi Ci Di* (48)

 b

*<sup>i</sup> <sup>i</sup>* (51)

*optimum*(*k* + 1)) to be applied in the

### **7. Proportional Resonant Control (PR)**

7. Proportional Resonant Control (PR)

x-y Proportional Resonant (PR) regulators, as it is explained in [37]. The control technique is detailed in Figure 3. It is based on a rotor flux controller, where the speed and flux control are implemented in a rotor-flux-oriented reference frame (d-q coordinates) using PI The second open-phase fault-tolerant implemented control scheme is based on stator current *x*-*y* Proportional Resonant (PR) regulators, as it is explained in [37]. The control techni‐

regulators. For simplicity, the d-current reference is set to a constant value while the qcurrent reference is obtained from the speed error and a PI controller. The phase currents of the machine can then be mapped in the stationary α-β-x-y planes using the classic Clarke transformation and a position estimator. In order to improve the controllers' performance,

The second open-phase fault-tolerant implemented control scheme is based on stator current

que is detailed in Figure 3. It is based on a rotor flux controller, where the speed and flux control are implemented in a rotor-flux-oriented reference frame (*d*-*q* coordinates) using PI regulators. For simplicity, the *d*-current reference is set to a constant value while the *q*current reference is obtained from the speed error and a PI controller. The phase currents of the machine can then be mapped in the stationary *α-β-x-y* planes using the classic Clarke transformation and a position estimator. In order to improve the controllers' perform‐ ance, two feedforward terms *ed* and *eq*, which depend on the machine model (it is used a rotor-flux estimator based on the speed measurement and the *d*-current component [37]), are included in the control loop:

$$\mathbf{e}\_d = \sigma \cdot \mathbf{L}\_s \cdot \mathbf{i}\_{sq}^" \cdot \boldsymbol{\alpha}\_e \tag{53}$$

$$e\_q = L\_s \cdot \frac{\mathcal{A}\_r^\*}{L\_m} \cdot o\_s \tag{54}$$

$$\frac{d}{dt}\mathcal{\lambda}\_r + \left(\frac{1}{\tau\_r} - j\alpha\_m \right) \mathcal{\lambda}\_r = \frac{L\_m}{\tau\_r} \dot{\mathfrak{i}}\_{sd} \tag{55}$$

Traditional PI regulators are capable of following the constant *x*-*y* current references under normal operation. However, PR regulators are required under postfault operation to appro‐ priately track the oscillating *x*-*y* reference current components [39], where the *x*-current component is forced to track the stator current in the *α*-current component and the *y*-current reference is set depending on the postfault control method.

The PR controller is implemented using two PI regulators in two different reference frames to track positive and negative stator current sequences [37], one rotating in the direction of the field-oriented reference frame (*ω*1) and the other in the opposite direction (−*ω*1). These PI regulators are capable of appropriately following the current references with nonoscillating terms. When their outputs are summed, and the action of the PR control is generated, the controller is capable of effectively driving to zero the total tracking error.

The main advantage of implementing linear controllers in open-phase fault-tolerant drives is that the asymmetry in the impedance terms in the *α*-*β* plane does not affect the controller performance. Then, there is no need to consider the back-emf of the faulty phase in the voltage equilibrium equations, and the same electrical drive model can be used for control purposes in normal or abnormal operation. Nonetheless, it must be considered that due to the low bandwidth that PI regulators possess, the parameters of the utilized PI must be tuned for different operating points in pre- or postfault operating conditions, increasing the complexity of the implemented controller.

Figure 3. Postfault controller based on PR technique. **Figure 3.** Postfault controller based on PR technique.

que is detailed in Figure 3. It is based on a rotor flux controller, where the speed and flux control are implemented in a rotor-flux-oriented reference frame (*d*-*q* coordinates) using PI regulators. For simplicity, the *d*-current reference is set to a constant value while the *q*current reference is obtained from the speed error and a PI controller. The phase currents of the machine can then be mapped in the stationary *α-β-x-y* planes using the classic Clarke transformation and a position estimator. In order to improve the controllers' perform‐ ance, two feedforward terms *ed* and *eq*, which depend on the machine model (it is used a rotor-flux estimator based on the speed measurement and the *d*-current component [37]),

\*

\* l= ××  w

w *r qs e m*

> t

= *<sup>m</sup>*

×× ×=*d s sq e e Li* (53)

*<sup>d</sup> <sup>L</sup> j i dt* (55)

*<sup>L</sup>* (54)

s

*e L*

1

t

controller is capable of effectively driving to zero the total tracking error.

 wl

*r m r sd r r*

Traditional PI regulators are capable of following the constant *x*-*y* current references under normal operation. However, PR regulators are required under postfault operation to appro‐ priately track the oscillating *x*-*y* reference current components [39], where the *x*-current component is forced to track the stator current in the *α*-current component and the *y*-current

The PR controller is implemented using two PI regulators in two different reference frames to track positive and negative stator current sequences [37], one rotating in the direction of the field-oriented reference frame (*ω*1) and the other in the opposite direction (−*ω*1). These PI regulators are capable of appropriately following the current references with nonoscillating terms. When their outputs are summed, and the action of the PR control is generated, the

The main advantage of implementing linear controllers in open-phase fault-tolerant drives is that the asymmetry in the impedance terms in the *α*-*β* plane does not affect the controller performance. Then, there is no need to consider the back-emf of the faulty phase in the voltage equilibrium equations, and the same electrical drive model can be used for control purposes in normal or abnormal operation. Nonetheless, it must be considered that due to the low bandwidth that PI regulators possess, the parameters of the utilized PI must be tuned for different operating points in pre- or postfault operating conditions, increasing the complexity

æ ö + - ç ÷ è ø

l

reference is set depending on the postfault control method.

are included in the control loop:

344 Induction Motors - Applications, Control and Fault Diagnostics

of the implemented controller.

#### In this section, simulation and experimental results will be presented to show the behavior **8. Experimental and simulation results**

8. Experimental and simulation results

distributed windings. Simulation results were obtained using the mathematical model of the machine and a Matlab & Simulink based simulation tool described in [40], whereas the experimentation was done using an electrical drive test-bench designed and implemented in a lab. To start with, the test bench will be described. 8.1. A. Simulation Environment and Test-bench In this section, simulation and experimental results will be presented to show the behavior in healthy and faulty states of the five-phase induction machine with symmetrical and distributed windings. Simulation results were obtained using the mathematical model of the machine and a Matlab & Simulink based simulation tool described in [40], whereas the experimentation was done using an electrical drive test-bench designed and implemented in a lab. To start with, the test bench will be described.

in healthy and faulty states of the five-phase induction machine with symmetrical and

#### The developed Matlab & Simulink simulation environments are shown in Figure 4. Each **8.1. A. Simulation Environment and Test-bench**

simulation model is composed of three main parts: the controller algorithm (PR and PCC based, respectively), the voltage source converter and the five-phase induction machine model. Depending on the selected postfault control criteria, appropriate current references must be provided to the controller. The minimum copper loss is used during postfault operation and, consequently, the y-current reference is set to zero. The developed Matlab & Simulink simulation environments are shown in Figure 4. Each simulation model is composed of three main parts: the controller algorithm (PR and PCC based, respectively), the voltage source converter and the five-phase induction machine model. Depending on the selected postfault control criteria, appropriate current references must be provided to the controller. The minimum copper loss is used during postfault operation and, consequently, the *y*-current reference is set to zero.

The experimental test-bench is shown in Figure 5. The five-phase machine was built based on a conventional three-phase induction machine (IM) that has been rewound to obtain a symmetrical five-phase induction motor with distributed windings. This five-phase machine is driven by two conventional SEMIKRON (SKS22F) three-phase two-level voltage source inverters (VSI's), connected to an independent external DC power supply as the DC-Link. The IM is mechanically connected to a DC motor, which can provide a programmable mechanical load torque to the five-phase drive. The rotational speed is measured by means of an incre‐ mental encoder from the manufacturer Hohner with reference 10-11657-2500, coupled to the

(lower figure).

Figure 4. Developed MATLAB/Simulink model, including both PCC (upper figure) and PR controllers **Figure 4.** Developed MATLAB/Simulink model, including both PCC (upper figure) and PR controllers (lower figure).

shaft. For control purposes, four phase hall-effect current sensors are used to measure the stator phase currents. The control actions are performed using a DSP-based Electronic Control Unit (ECU) connected to a personal computer (this PC acts as a Human Interface Unit which manages the entire test bench) using a standard RS232 cable. The user of the system can program the control algorithm using the Texas Instruments proprietary software called Code Composer Studio. This software runs in the DSP and configures the ECU's internal peripherals, the communication protocol and the data acquisition system. The experimental test-bench is shown in Figure 5. The five-phase machine was built based on a conventional three-phase induction machine (IM) that has been rewound to obtain a symmetrical five-phase induction motor with distributed windings. This five-phase machine is driven by two conventional SEMIKRON (SKS22F) three-phase two-level voltage source inverters (VSI's), connected to an independent external DC power supply as the DC-Link. The IM is mechanically connected to a DC motor, which can provide a programmable mechanical load torque to the five-phase drive. The rotational speed is measured by means

The PCC and PR control strategies are implemented in the DSP to analyze and compare the behavior of the real system. Regardless of the control strategy, the experimental tests that follow are performed setting a constant *d*-axis stator current reference of 0.57 A for constantflux operation, while the *q*-axis stator current reference is obtained from the PI-based speed controller (Figure 2 and Figure 3). The VSI's DC-link voltage was set to 300 V. The fixed of an incremental encoder from the manufacturer Hohner with reference 10-11657-2500, coupled to the shaft. For control purposes, four phase hall-effect current sensors are used to measure the stator phase currents. The control actions are performed using a DSP-based Electronic Control Unit (ECU) connected to a personal computer (this PC acts as a Human Interface Unit which manages the entire test bench) using a standard RS232 cable. The user of the system can program the control algorithm using the Texas Instruments proprietary software called Code Composer Studio. This software runs in the DSP and configures the ECU's internal peripherals, the communication protocol and the data acquisition system.

The PCC and PR control strategies are implemented in the DSP to analyze and compare the

postfault operation of the multiphase drive considers always an open-phase fault in leg 'a'.

switching and sampling frequency for PR is set to 2.5 kHz, whereas the sampling period for PCC is set to 0.1 ms, providing around 2.5 kHz of average switching frequency. The postfault operation of the multiphase drive considers always an open-phase fault in leg '*a*'. flux operation, while the q-axis stator current reference is obtained from the PI-based speed controller (Figure 2 and Figure 3). The VSI's DC-link voltage was set to 300 V. The fixed switching and sampling frequency for PR is set to 2.5 kHz, whereas the sampling period for PCC is set to 0.1 ms, providing around 2.5 kHz of average switching frequency. The

Figure 5. Experimental test bench. **Figure 5.** Experimental test bench.

shaft. For control purposes, four phase hall-effect current sensors are used to measure the stator phase currents. The control actions are performed using a DSP-based Electronic Control Unit (ECU) connected to a personal computer (this PC acts as a Human Interface Unit which manages the entire test bench) using a standard RS232 cable. The user of the system can program the control algorithm using the Texas Instruments proprietary software called Code Composer Studio. This software runs in the DSP and configures the ECU's internal peripherals,

The experimental test-bench is shown in Figure 5. The five-phase machine was built based on a conventional three-phase induction machine (IM) that has been rewound to obtain a symmetrical five-phase induction motor with distributed windings. This five-phase machine is driven by two conventional SEMIKRON (SKS22F) three-phase two-level voltage source inverters (VSI's), connected to an independent external DC power supply as the DC-Link. The IM is mechanically connected to a DC motor, which can provide a programmable mechanical load torque to the five-phase drive. The rotational speed is measured by means of an incremental encoder from the manufacturer Hohner with reference 10-11657-2500, coupled to the shaft. For control purposes, four phase hall-effect current sensors are used to measure the stator phase currents. The control actions are performed using a DSP-based Electronic Control Unit (ECU) connected to a personal computer (this PC acts as a Human Interface Unit which manages the entire test bench) using a standard RS232 cable. The user

Figure 4. Developed MATLAB/Simulink model, including both PCC (upper figure) and PR controllers

**Figure 4.** Developed MATLAB/Simulink model, including both PCC (upper figure) and PR controllers (lower figure).

The PCC and PR control strategies are implemented in the DSP to analyze and compare the behavior of the real system. Regardless of the control strategy, the experimental tests that follow are performed setting a constant *d*-axis stator current reference of 0.57 A for constantflux operation, while the *q*-axis stator current reference is obtained from the PI-based speed controller (Figure 2 and Figure 3). The VSI's DC-link voltage was set to 300 V. The fixed

the communication protocol and the data acquisition system.

346 Induction Motors - Applications, Control and Fault Diagnostics

(lower figure).

#### 8.2. B. Steady-state performance in postfault operation **8.2. B. Steady-state performance in postfault operation**

The steady-state performance can be easily studied using the simulation environment. First, the postfault model of the system, and the PCC and PR controllers are implemented using aforementioned Matlab & Simulink environments. The behavior of the system is evaluated The steady-state performance can be easily studied using the simulation environment. First, the postfault model of the system, and the PCC and PR controllers are implemented using aforementioned Matlab & Simulink environments. The behavior of the system is evaluated driving the motor at a reference speed of 500 rpm and applying a load torque of 56% of the nominal one (*Tn*). The obtained stator phase currents are shown in steady-state in Figures 6 and 7 for PCC and PR controllers, respectively. The minimum copper loss operation is applied. Then, phase currents possess unequal peak values, with phases *b-e* equal in magnitude and higher than those of phases *c*-*d*. It is observed that the fault-tolerant PCC produces higher current ripple than the PR control method, even though the sampling frequency of the predictive controller is set at four times the value of the PR method. This is due to the intrinsic property of FCS predictive controllers, where the switching frequency is not fixed and depends

on the electrical drive operating point. The behavior of the entire system offers faster response and lower switching frequency using PCC than PR-based controllers. The *α* - *β* current vector describes a circular trajectory and the *x-y* terms present the same behavior using both postfault controllers, being the *x*-current term fixed to –*α* and the *y*-current term null. controllers. The α - β current vector describes a circular trajectory and the x-y terms present the same behavior using both postfault controllers, being the x-current term fixed to –α and the y-current term null. controllers. The α - β current vector describes a circular trajectory and the x-y terms present the same behavior using both postfault controllers, being the x-current term fixed to –α and the y-current term null.

driving the motor at a reference speed of 500 rpm and applying a load torque of 56% of the nominal one ( <sup>n</sup> T ). The obtained stator phase currents are shown in steady-state in Figures 6

driving the motor at a reference speed of 500 rpm and applying a load torque of 56% of the nominal one ( <sup>n</sup> T ). The obtained stator phase currents are shown in steady-state in Figures 6

and 7 for PCC and PR controllers, respectively. The minimum copper loss operation is applied. Then, phase currents possess unequal peak values, with phases b-e equal in magnitude and higher than those of phases c-d. It is observed that the fault-tolerant PCC produces higher current ripple than the PR control method, even though the sampling frequency of the predictive controller is set at four times the value of the PR method. This is

and 7 for PCC and PR controllers, respectively. The minimum copper loss operation is applied. Then, phase currents possess unequal peak values, with phases b-e equal in magnitude and higher than those of phases c-d. It is observed that the fault-tolerant PCC produces higher current ripple than the PR control method, even though the sampling frequency of the predictive controller is set at four times the value of the PR method. This is due to the intrinsic property of FCS predictive controllers, where the switching frequency is

system offers faster response and lower switching frequency using PCC than PR-based

system offers faster response and lower switching frequency using PCC than PR-based

Figure 6. Phase current evolution in different subspaces using the PCC controller and the minimum **Figure 6.** Phase current evolution in different subspaces using the PCC controller and the minimum copper loss criteri‐ on. copper loss criterion.

Figure 6. Phase current evolution in different subspaces using the PCC controller and the minimum

Time [s] -4 -2 <sup>0</sup> <sup>2</sup> <sup>4</sup> -4 **Figure 7.** Phase current evolution in different subspaces using the PR controller and the minimum copper loss criteri‐ on.


i x s [A]

1.55 1.6 1.65 1.7 1.75


controllers. The α - β current vector describes a circular trajectory and the x-y terms present the same behavior using both postfault controllers, being the x-current term fixed to –α and controllers. The α - β current vector describes a circular trajectory and the x-y terms present the same behavior using both postfault controllers, being the x-current term fixed to –α and The same steady-state test is performed experimentally using the real test bench. The experi‐ mentally obtained results in postfault situation are presented in Figure 8. Notice that simula‐ tion and experimental results agree, and the fault-tolerant system using the PCC controller provides higher current ripple than using the PR control method. Nonetheless, both controllers appropriately track the current references in all subspaces, producing a circular trajectory similar to the one obtained in healthy operation. The same steady-state test is performed experimentally using the real test bench. The experimentally obtained results in postfault situation are presented in Figure 8. Notice that simulation and experimental results agree, and the fault-tolerant system using the PCC controller provides higher current ripple than using the PR control method. Nonetheless, both controllers appropriately track the current references in all subspaces, producing a

circular trajectory similar to the one obtained in healthy operation.

Figure 8. Stator phase current evolution in different subspaces using the PCC (left side) and the PR (right side) controller and the minimum copper loss (MCL) criterion. **Figure 8.** Stator phase current evolution in different subspaces using the PCC (left side) and the PR (right side) control‐ ler and the minimum copper loss (MCL) criterion.

#### 8.3. C. Dynamic operation: From pre- to postfault operation **8.3. C. Dynamic operation: From pre- to postfault operation**

copper loss criterion.

on the electrical drive operating point. The behavior of the entire system offers faster response and lower switching frequency using PCC than PR-based controllers. The *α* - *β* current vector describes a circular trajectory and the *x-y* terms present the same behavior using both postfault

cs <sup>i</sup>

cs <sup>i</sup>

ds <sup>i</sup>

ds <sup>i</sup> es

es

i

β s [A]



i <sup>α</sup> s vs iβ<sup>s</sup>

i <sup>α</sup> s [A]



i x s [A]

i

Figure 6. Phase current evolution in different subspaces using the PCC controller and the minimum

i

y s [A]

y s [A]

0

2

4

i

β s [A]

0

2

4


i xs vs iys


i x s vs iy s


i x s [A] i <sup>α</sup> s vs iβ<sup>s</sup>

i <sup>α</sup> s [A]

i x s vs iy s


i x s [A]



i

i

y s [A]

y s [A]

0


i x s [A]

2

4

i

i

β s [A]

β s [A]

0


i <sup>α</sup> s vs iβ<sup>s</sup>

i <sup>α</sup> s [A]

2

4

i <sup>α</sup> s vs iβ<sup>s</sup>

i <sup>α</sup> s [A]

i xs vs iys

driving the motor at a reference speed of 500 rpm and applying a load torque of 56% of the nominal one ( <sup>n</sup> T ). The obtained stator phase currents are shown in steady-state in Figures 6

driving the motor at a reference speed of 500 rpm and applying a load torque of 56% of the nominal one ( <sup>n</sup> T ). The obtained stator phase currents are shown in steady-state in Figures 6

and 7 for PCC and PR controllers, respectively. The minimum copper loss operation is applied. Then, phase currents possess unequal peak values, with phases b-e equal in magnitude and higher than those of phases c-d. It is observed that the fault-tolerant PCC produces higher current ripple than the PR control method, even though the sampling frequency of the predictive controller is set at four times the value of the PR method. This is due to the intrinsic property of FCS predictive controllers, where the switching frequency is not fixed and depends on the electrical drive operating point. The behavior of the entire system offers faster response and lower switching frequency using PCC than PR-based

and 7 for PCC and PR controllers, respectively. The minimum copper loss operation is applied. Then, phase currents possess unequal peak values, with phases b-e equal in magnitude and higher than those of phases c-d. It is observed that the fault-tolerant PCC produces higher current ripple than the PR control method, even though the sampling frequency of the predictive controller is set at four times the value of the PR method. This is due to the intrinsic property of FCS predictive controllers, where the switching frequency is not fixed and depends on the electrical drive operating point. The behavior of the entire system offers faster response and lower switching frequency using PCC than PR-based

controllers, being the *x*-current term fixed to –*α* and the *y*-current term null.

bs <sup>i</sup>

bs <sup>i</sup>

1.55 1.6 1.65 1.7 1.75

1.55 1.6 1.65 1.7 1.75

Time [s]

Time [s]

1.55 1.6 1.65 1.7 1.75

1.55 1.6 1.65 1.7 1.75

Time [s]

bs <sup>i</sup>

bs <sup>i</sup>

cs <sup>i</sup>

**Figure 7.** Phase current evolution in different subspaces using the PR controller and the minimum copper loss criteri‐

cs <sup>i</sup>

**Figure 6.** Phase current evolution in different subspaces using the PCC controller and the minimum copper loss criteri‐

ds <sup>i</sup>

ds <sup>i</sup> es

es

Time [s]

the y-current term null.

348 Induction Motors - Applications, Control and Fault Diagnostics

the y-current term null.

i as <sup>i</sup>

i as <sup>i</sup>



on.


on.


Current [A]

Current [A]

copper loss criterion.

copper loss criterion.

i as <sup>i</sup>

i as <sup>i</sup>

Current [A]

Current [A]

Figure 6. Phase current evolution in different subspaces using the PCC controller and the minimum The pre- and postfault operations are now analyzed and compared. In order to provide a more realistic insight, tests have been conducted considering a fault detection delay. Consequently, a delay between the fault occurrence and the control action is observed. The pre- and postfault operations are now analyzed and compared. In order to provide a more realistic insight, tests have been conducted considering a fault detection delay. Consequently, a delay between the fault occurrence and the control action is observed.

> The results provided in Figure 9 show the pre- to postfault transition with a fault detection delay of 40 ms between the fault occurrence in phase 'a' at t s = 0.2 and the control software reconfiguration. The results obtained when the PCC is implemented are presented in the left column whereas results obtained with PR are shown in the right side. The speed reference is The results provided in Figure 9 show the pre- to postfault transition with a fault detection delay of 40 ms between the fault occurrence in phase '*a*' at *t* =0.2 *s* and the control software reconfiguration. The results obtained when the PCC is implemented are presented in the left column whereas results obtained with PR are shown in the right side. The speed reference is set to 500 rpm, as in previous tests, while a constant load torque of (0.56*Tn*) is demanded.

> set to 500 rpm, as in previous tests, while a constant load torque of ( 0.56 <sup>n</sup> T ) is demanded. In the case of PCC, the q -current waveform clearly indicates that the control is completely lost during the fault detection delay (Figure 9, second row), and as a result a speed drop is observed (Figure 9, first row). Notice that the β-current component is not affected during the fault detection delay because the faulty phase 'a' does not contribute to the β component (Figure 9, third row). Conversely, the α and x stator current components are both driven to zero (Figure 9, third and fourth rows), causing torque oscillations. This abnormal operation In the case of PCC, the *q* -current waveform clearly indicates that the control is completely lost during the fault detection delay (Figure 9, second row), and as a result a speed drop is observed (Figure 9, first row). Notice that the *β*-current component is not affected during the fault detection delay because the faulty phase '*a*' does not contribute to the *β* component (Figure 9, third row). Conversely, the *α* and *x* stator current components are both driven to zero (Figure 9, third and fourth rows), causing torque oscillations. This abnormal operation is observed during the fault detection delay due to the absence of an accurate system model for the PCC to provide an adequate control. After the fault detection delay, the control scheme is recon‐ figured and a more accurate system model is considered. As a result, the *α*-current reference is immediately tracked (Figure 9, third row), the *x*-current becomes sinusoidal (*i <sup>x</sup>* = −*i <sup>α</sup>s*) and

Figure 9. Transition from pre- to postfault operation considering fault detection delay. The motor is driven at 500 rpm with a constant load torque 0.56 <sup>n</sup> T . The minimum copper loss strategy is used in postfault operation, and PCC (left plots) and PR-based (right figures) controllers are applied. The fault occurs at t = 0.2 s but it is detected 40 ms after its occurrence. The speed response and q-current component in pre- and postfault situations are shown in rows (a) and (b), while the zoomed-in postfault **Figure 9.** Transition from pre- to postfault operation considering fault detection delay. The motor is driven at 500 rpm with a constant load torque 0.56*Tn*. The minimum copper loss strategy is used in postfault operation, and PCC (left plots) and PR-based (right figures) controllers are applied. The fault occurs at t = 0.2 s but it is detected 40 ms after its occurrence. The speed response and *q*-current component in pre- and postfault situations are shown in rows (a) and (b), while the zoomed-in postfault *α*-*β* and *x*-*y* currents are presented in rows (c) and (d), respectively.

α-β and x-y currents are presented in rows (c) and (d), respectively.

the *y*-current is null according to the minimum copper loss criterion (Figure 9, fourth row). On the other hand, the *q*-current waveform when PR control is implemented shows a slight drop in the moment when the phase is open (Figure 9, second row), but the control action is maintained during the fault detection delay and the motor speed is only slightly affected (Figure 9, first row). Thus, the effect of the delay and the control reconfiguration is noticeably less severe in the case of PR compared to PCC. This can be explained by the fact that the prefault PR control scheme is essentially similar to the postfault scheme except for the transition from The transition from pre- to postfault is also tested under low speed operation (Figure 10). The fault detection delay is considered equal to 200 ms and an instantaneous control

reconfiguration of the system after the fault occurrence is not considered. As in previous tests, the fault occurs at t = 0.2s and constant load and speed references are maintained from

The machine is driven at 50 rpm, and a 56% of the nominal torque is applied during the test.

PI to PR in the *x* - *y* current controllers. Once the postfault current references have been properly tracked, PR control can effectively provide the reference torque and regulate the speed, but an important current oscillation appears at double the fundamental frequency due to some negative sequence current that cannot be regulated by the *d* - *q* controllers. The speed is, however, not affected, so the system is regulated with minimum modifications in postfault situation. This value of torque matches with the maximum quantity that the minimum copper loss criteria can manage in a postfault situation. As it is observed, the speed reference tracking is slightly affected after the fault occurrence with PR controller (Figure 10, first row); however, this effect is much more noticeable when the PCC controller is implemented (Figure 10, first row). Despite this considerable drop of speed, the system reaches the reference speed using the PCC controller sooner than when using the PR technique. Then, PCC controllers present

again faster responses compared with the PR controllers.

pre- to postfault operation.

Figure 10. Transition from pre- to postfault operation considering fault detection delay. The motor is driven at 50 rpm with a constant load torque 0.56 <sup>n</sup> T . The minimum copper loss strategy is used in postfault operation, and PCC (left plots) and PR-based (right figures) controllers are applied. The fault **Figure 10.** Transition from pre- to postfault operation considering fault detection delay. The motor is driven at 50 rpm with a constant load torque 0.56*Tn*. The minimum copper loss strategy is used in postfault operation, and PCC (left plots) and PR-based (right figures) controllers are applied. The fault occurs at t = 0.2 s but it is detected 200 ms after its occurrence. The speed response and *q*-current component in pre- and postfault situations are shown in rows (a) and (b), while the zoomed-in postfault *α*-*β* and *x*-*y* currents are presented in rows (c) and (d), respectively.

the *y*-current is null according to the minimum copper loss criterion (Figure 9, fourth row). On the other hand, the *q*-current waveform when PR control is implemented shows a slight drop in the moment when the phase is open (Figure 9, second row), but the control action is maintained during the fault detection delay and the motor speed is only slightly affected (Figure 9, first row). Thus, the effect of the delay and the control reconfiguration is noticeably less severe in the case of PR compared to PCC. This can be explained by the fact that the prefault PR control scheme is essentially similar to the postfault scheme except for the transition from

The transition from pre- to postfault is also tested under low speed operation (Figure 10). The fault detection delay is considered equal to 200 ms and an instantaneous control

Figure 9. Transition from pre- to postfault operation considering fault detection delay. The motor is driven at 500 rpm with a constant load torque 0.56 <sup>n</sup> T . The minimum copper loss strategy is used in postfault operation, and PCC (left plots) and PR-based (right figures) controllers are applied. The fault occurs at t = 0.2 s but it is detected 40 ms after its occurrence. The speed response and q-current component in pre- and postfault situations are shown in rows (a) and (b), while the zoomed-in postfault

**Figure 9.** Transition from pre- to postfault operation considering fault detection delay. The motor is driven at 500 rpm with a constant load torque 0.56*Tn*. The minimum copper loss strategy is used in postfault operation, and PCC (left plots) and PR-based (right figures) controllers are applied. The fault occurs at t = 0.2 s but it is detected 40 ms after its occurrence. The speed response and *q*-current component in pre- and postfault situations are shown in rows (a) and

<sup>0</sup> 0.2 0.4 0.6 0.8 <sup>1</sup> 1.2 -3

<sup>0</sup> 0.2 0.4 0.6 0.8 <sup>1</sup> 1.2 <sup>400</sup>

Time [s]

i sq i\*sq

ys i\*xys

<sup>β</sup><sup>s</sup> i\*αβ<sup>s</sup>

ωm ω\* m

Time [s]

i <sup>α</sup><sup>s</sup> <sup>i</sup>

0.15 0.2 0.25 0.3 0.35 0.4 -4

Time [s]

0.15 0.2 0.25 0.3 0.35 0.4 -4

i xs <sup>i</sup>

Time [s]

Current [A]

Current [A]

Speed [RPM]

ωm ω\* m

> i sq i\*sq

<sup>β</sup><sup>s</sup> i\*αβ<sup>s</sup>

ys i\*xys

<sup>0</sup> 0.2 0.4 0.6 0.8 <sup>1</sup> 1.2 <sup>400</sup>

350 Induction Motors - Applications, Control and Fault Diagnostics

Time [s]

<sup>0</sup> 0.2 0.4 0.6 0.8 <sup>1</sup> 1.2 -3

Time [s]

i <sup>α</sup><sup>s</sup> <sup>i</sup>

0.15 0.2 0.25 0.3 0.35 0.4 -4

Time [s]

i xs <sup>i</sup>

0.15 0.2 0.25 0.3 0.35 0.4 -4

Time [s]

α-β and x-y currents are presented in rows (c) and (d), respectively.

(b), while the zoomed-in postfault *α*-*β* and *x*-*y* currents are presented in rows (c) and (d), respectively.

Current [A]

Current [A]

Current [A]

Speed [RPM]

Current [A]

The transition from pre- to postfault is also tested under low speed operation (Figure 10). The fault detection delay is considered equal to 200 ms and an instantaneous control reconfigura‐ tion of the system after the fault occurrence is not considered. As in previous tests, the fault occurs at *t* = 0.2s and constant load and speed references are maintained from pre- to postfault operation.

The machine is driven at 50 rpm, and a 56% of the nominal torque is applied during the test. This value of torque matches with the maximum quantity that the minimum copper loss criteria can manage in a postfault situation. As it is observed, the speed reference tracking is slightly affected after the fault occurrence with PR controller (Figure 10, first row); however, this effect is much more noticeable when the PCC controller is implemented (Figure 10, first row). Despite this considerable drop of speed, the system reaches the reference speed using the PCC controller sooner than when using the PR technique. Then, PCC controllers present again faster responses compared with the PR controllers.

### **9. Conclusions**

This chapter focuses on the management of open-phase faults in multiphase electrical drives. First of all, the different types of faults that appear in conventional and multiphase drives are presented. The ability to continue operating in the event of a fault, which is one of the main advantages of multiphase drives compared to standard three-phase ones, is discussed next. The open-phase fault being the most common type of fault, it is next analyzed in a generic multiphase drive with an odd number of phases. The analysis is particularized for one of the most common multiphase drives, the five-phase induction machine with symmetrical and distributed windings. The considered open-circuit is located in phase 'a', but the result is general due to the spatial symmetry of stator windings. Two recently proposed controllers based on the field oriented control technique, the PR and PCC-based methods, are described as alternatives to manage the pre- and postfault operation with a minimum cost in the redesign and performance of the controllers. Both methods must share the strategy to operate in postfault operation, which must change the limits of the impressed stator currents to guarantee the safety operation of the entire system. This is the case of the minimum copper loss criterion, described in the document and applied with PCC and PR techniques to study the performance of a five-phase IM using simulation and experimental results. These results not only show the behavior of the system in steady and transient states, but also compare the ability of predictive and linear controllers to manage the fault appearance. Provided results show that speed control in postfault operation is viable using either PCC or PR control methods, with nearly similar performance. Speed response of the predictive technique is faster than using a PR controller at the expense of a higher steady-state current ripple. Additionally, PCC proves to be more affected in the transition from pre- to postfault modes of operation because the high depend‐ ence on the model accuracy provides less robustness during the unavoidable fault detection delay. Both control methods, however, ensure safe operation within the postfault current ratings, and proper postfault current reference tracking.

### **Author details**

The transition from pre- to postfault is also tested under low speed operation (Figure 10). The fault detection delay is considered equal to 200 ms and an instantaneous control reconfigura‐ tion of the system after the fault occurrence is not considered. As in previous tests, the fault occurs at *t* = 0.2s and constant load and speed references are maintained from pre- to postfault

The machine is driven at 50 rpm, and a 56% of the nominal torque is applied during the test. This value of torque matches with the maximum quantity that the minimum copper loss criteria can manage in a postfault situation. As it is observed, the speed reference tracking is slightly affected after the fault occurrence with PR controller (Figure 10, first row); however, this effect is much more noticeable when the PCC controller is implemented (Figure 10, first row). Despite this considerable drop of speed, the system reaches the reference speed using the PCC controller sooner than when using the PR technique. Then, PCC controllers present

This chapter focuses on the management of open-phase faults in multiphase electrical drives. First of all, the different types of faults that appear in conventional and multiphase drives are presented. The ability to continue operating in the event of a fault, which is one of the main advantages of multiphase drives compared to standard three-phase ones, is discussed next. The open-phase fault being the most common type of fault, it is next analyzed in a generic multiphase drive with an odd number of phases. The analysis is particularized for one of the most common multiphase drives, the five-phase induction machine with symmetrical and distributed windings. The considered open-circuit is located in phase 'a', but the result is general due to the spatial symmetry of stator windings. Two recently proposed controllers based on the field oriented control technique, the PR and PCC-based methods, are described as alternatives to manage the pre- and postfault operation with a minimum cost in the redesign and performance of the controllers. Both methods must share the strategy to operate in postfault operation, which must change the limits of the impressed stator currents to guarantee the safety operation of the entire system. This is the case of the minimum copper loss criterion, described in the document and applied with PCC and PR techniques to study the performance of a five-phase IM using simulation and experimental results. These results not only show the behavior of the system in steady and transient states, but also compare the ability of predictive and linear controllers to manage the fault appearance. Provided results show that speed control in postfault operation is viable using either PCC or PR control methods, with nearly similar performance. Speed response of the predictive technique is faster than using a PR controller at the expense of a higher steady-state current ripple. Additionally, PCC proves to be more affected in the transition from pre- to postfault modes of operation because the high depend‐ ence on the model accuracy provides less robustness during the unavoidable fault detection delay. Both control methods, however, ensure safe operation within the postfault current

again faster responses compared with the PR controllers.

352 Induction Motors - Applications, Control and Fault Diagnostics

ratings, and proper postfault current reference tracking.

operation.

**9. Conclusions**

Hugo Guzman1\*, Ignacio Gonzalez2 , Federico Barrero2 and Mario Durán1

\*Address all correspondence to: hugguzjim@uma.es

1 Universidad de Málaga, Spain

2 Universidad de Sevilla, Spain

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### **Reduced-order Observer Analysis in MBPC Techniques Applied to the Six-phase Induction Motor Drives Reduced-Order Observer Analysis in MBPC Techniques Applied to the Six-Phase Induction Motor Drives**

Raúl Gregor, Jorge Rodas, Derlis Gregor and Federico Barrero Raúl Gregor, Jorge Rodas, Derlis Gregor and Federico Barrero

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

### **Abstract**

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tion Control Conference (EPE-PEMC 2012).

Model-based predictive control techniques have been recently applied with success in power electronics, particularly in the fields of current control applied to AC multiphase electrical drives. In AC electrical drives control, most of state variables (i.e., rotor currents, rotor fluxes, etc.) cannot be measured, so they must be estimated. As a result of this issue, this chapter proposes a comparative study of reduced-order observers used to estimate the rotor currents in an model-based predictive current control applied to the six-phase induction motor. The proposed control techniques are evaluated using the Luenberger observer and the optimal estimator based on Kalman filter. Different operation modes are analyzed and are further compared in terms of statistical parameters of performance (i.e., covariance, standard deviation, mean square error, etc.). The effectiveness of proposed methods is verified by a set of comparative experiments obtained by using a six-phase induction motor system experimental setup.

**Keywords**: Model-based predictive control (MBPC), Kalman filter (KF), Luenberger observer (LO), Six-phase induction motor (SpIM)

### **1. Introduction**

While the first variable speed drives back to the late 1960s, multiphase drives have only gained the special attention of the research community during the past few years in comparison with the traditional three-phase scheme for various applications especially in those where high reliability and fault tolerance are needed, as cases of ship propulsion, locomotive traction, electric and hybrid electric vehicles, more-electric aircraft,

the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2015 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

©2012 prezimena autora, kod vise prvi et al., licensee InTech. This is an open access chapter distributed under

and high-power industrial applications - and recently in wind energy applications [1]-[3]. Different types of multiphase machines have been recently developed mainly for high-power applications where the increase of the number of phases enables reduction of power per phase, which leads to a reduction of the power per inverter leg. Often the multiphase machines can be classified according to the phase numbers in 5-phase [4]-[8], 6-phase [9]-[11], 9-phase [13]-[14], 12-phase [15]-[17], and 18-phase [18] and by the spatial distribution of the phases within the stator winding symmetrically or asymmetrically. The six-phase induction motor (SpIM) fed by two sets of voltage source inverters was investigated since 1993. Because of the configuration of induction motor having two sets of balanced windings, with phase shift of 30 electrical degrees, six harmonic torque pulsations produced by two sets of windings, respectively, are antiphase and therefore can be completely eliminated. Nowadays, numerous control strategies such as direct torque control (DTC), model-based predictive control (MBPC), and vector control have been developed for SpIM. The DTC technique has the advantages of low machine parameter dependence and fast dynamic torque response. Moreover, the main advantage of the MBPC technique is it focuses on flexibility to define different control criteria, changing only a cost function, a reason why this control technique has been recently applied to the SpIM [19]. MBPC is a control theory developed at the end of the 1970s but has been recently introduced as a viable alternative in power converters and drives. Various control schemes based on MBPC, including current, flux and torque, speed, and sensorless speed control, have been recently reported. Developed schemes have demonstrated good performance in the current and torque control of conventional drives, at the expense of a high computational burden. It is a more flexible control scheme than DTC, and it also provides faster torque response than the field-oriented control (FOC). The interest in predictive control approach and multiphase drives has grown during the last few years, when the development of modern microelectronics devices has removed the computational barriers in their implementation. However, predictive control techniques have been only proved as a viable alternative to conventional controllers in the current regulation of the multiphase power converter. Predictive torque control (PTC), as a variation of the predictive current control methods, has been recently analyzed as an alternative to classic DTC at a theoretical level [20].

In this work, the predictive model of the SpIM is obtained from the vector space decomposition (VSD) approach using the state-space representation method where the two state variables are the stator and rotor currents. As the rotor currents are not measurable parameters, these must be estimated. This chapter hence focuses in the efficiency analysis of the MBPC techniques using the Luenberger Observer (LO) and the optimal estimator based on Kalman Filter (KF). The chapter provides a background material about model-based predictive current control applied to SpIM and includes experimental results by using an experimental setup based on a digital signal controller (DSC). Finally, the main results are discussed in the conclusion section.

### **2. The SpIM mathematical model**

The asymmetrical SpIM with two sets of three-phase stator windings spatially shifted by 30 electrical degrees and isolated neutral points as seen on Figure 1 (a) is one of the most widely discussed topologies. The asymmetrical SpIM is a continuous system which can be described by a set of differential equations. The model can be simplified by using the VSD theory introduced introduced in [21], [22], [26] which enables to transform the

**Figure 1.** Asymmetrical SpIM feed topology and winding configuration

2 Induction Motor

theoretical level [20].

discussed in the conclusion section.

**2. The SpIM mathematical model**

and high-power industrial applications - and recently in wind energy applications [1]-[3]. Different types of multiphase machines have been recently developed mainly for high-power applications where the increase of the number of phases enables reduction of power per phase, which leads to a reduction of the power per inverter leg. Often the multiphase machines can be classified according to the phase numbers in 5-phase [4]-[8], 6-phase [9]-[11], 9-phase [13]-[14], 12-phase [15]-[17], and 18-phase [18] and by the spatial distribution of the phases within the stator winding symmetrically or asymmetrically. The six-phase induction motor (SpIM) fed by two sets of voltage source inverters was investigated since 1993. Because of the configuration of induction motor having two sets of balanced windings, with phase shift of 30 electrical degrees, six harmonic torque pulsations produced by two sets of windings, respectively, are antiphase and therefore can be completely eliminated. Nowadays, numerous control strategies such as direct torque control (DTC), model-based predictive control (MBPC), and vector control have been developed for SpIM. The DTC technique has the advantages of low machine parameter dependence and fast dynamic torque response. Moreover, the main advantage of the MBPC technique is it focuses on flexibility to define different control criteria, changing only a cost function, a reason why this control technique has been recently applied to the SpIM [19]. MBPC is a control theory developed at the end of the 1970s but has been recently introduced as a viable alternative in power converters and drives. Various control schemes based on MBPC, including current, flux and torque, speed, and sensorless speed control, have been recently reported. Developed schemes have demonstrated good performance in the current and torque control of conventional drives, at the expense of a high computational burden. It is a more flexible control scheme than DTC, and it also provides faster torque response than the field-oriented control (FOC). The interest in predictive control approach and multiphase drives has grown during the last few years, when the development of modern microelectronics devices has removed the computational barriers in their implementation. However, predictive control techniques have been only proved as a viable alternative to conventional controllers in the current regulation of the multiphase power converter. Predictive torque control (PTC), as a variation of the predictive current control methods, has been recently analyzed as an alternative to classic DTC at a

In this work, the predictive model of the SpIM is obtained from the vector space decomposition (VSD) approach using the state-space representation method where the two state variables are the stator and rotor currents. As the rotor currents are not measurable parameters, these must be estimated. This chapter hence focuses in the efficiency analysis of the MBPC techniques using the Luenberger Observer (LO) and the optimal estimator based on Kalman Filter (KF). The chapter provides a background material about model-based predictive current control applied to SpIM and includes experimental results by using an experimental setup based on a digital signal controller (DSC). Finally, the main results are

The asymmetrical SpIM with two sets of three-phase stator windings spatially shifted by 30 electrical degrees and isolated neutral points as seen on Figure 1 (a) is one of the most widely discussed topologies. The asymmetrical SpIM is a continuous system which can be described by a set of differential equations. The model can be simplified by using the VSD theory introduced introduced in [21], [22], [26] which enables to transform the original six-dimensional space of the motor model into three two-dimensional orthogonal subspaces in stationary reference frames (*α* − *β*), (*x* − *y*), and (*z*<sup>1</sup> − *z*2) by means of a 6 × 6 transformation matrix using an amplitude-invariant criterion. This matrix, namely, **T**, is defined as:

$$\mathbf{T} = \frac{1}{3} \begin{bmatrix} 1 & \frac{\sqrt{3}}{2} & -\frac{1}{2} & -\frac{\sqrt{3}}{2} & -\frac{1}{2} & 0\\ 0 & \frac{1}{2} & \frac{\sqrt{3}}{2} & \frac{1}{2} & -\frac{\sqrt{3}}{2} & -1\\ 1 - \frac{\sqrt{3}}{2} & -\frac{1}{2} & \frac{\sqrt{3}}{2} & -\frac{1}{2} & 0\\ 0 & \frac{1}{2} & -\frac{\sqrt{3}}{2} & \frac{1}{2} & \frac{\sqrt{3}}{2} & -1\\ 1 & 0 & 1 & 0 & 1 & 0\\ 0 & 1 & 0 & 1 & 0 & 1 \end{bmatrix} . \tag{1}$$

It is worth remarking that, according to the VSD approach, the electromechanical energy conversion variables are mapped in the (*α* − *β*) subspace, meanwhile the current components in the (*x* − *y*) subspace represent supply harmonics of order 6*n* ± 1 (*n* = 1, 3, 5, ...) and only produce losses. The voltage vectors in the (*z*<sup>1</sup> − *z*2) subspace are zero due to the isolated neutral points configuration [23]. Moreover, the SpIM is supplied by a 2-level 12-pulse IGBT based VSC and a Dc-Link (**VDc**), as shown in Figure 1 (b).

The VSC has a discrete nature with a total number of 2<sup>6</sup> = 64 different switching state vectors defined by six switching functions corresponding to the six inverter legs (*Sa*, *Sd*, *Sb*, *Se*, *Sc*, *Sf*), where *Sa*−*<sup>f</sup>* ∈ {0, 1}. The different switching state vectors and the **VDc** voltage define the phase voltages which can in turn be mapped to the (*α* − *β*) − (*x* − *y*) space according to the VSD approach [24]. To represent the stationary reference frame (*α* − *β*) in the dynamic reference frame (*d* − *q*), a rotation transformation can be used. This transformation matrix, namely, **T***dq* is represented as:

$$\mathbf{T}\_{dq} = \begin{bmatrix} \cos\left(\theta\_r\right) & \sin\left(\theta\_r\right) \\ -\sin\left(\theta\_r\right)\cos\left(\theta\_r\right) \end{bmatrix} \tag{2}$$

where *θ<sup>r</sup>* is the rotor angular position referred to the stator as shown in Figure 1 (a).

From the VSD approach, the following conclusions should be emphasized:


The VSI with isolated neutrals is depicted in Figure 1 (b), being the gating signal represented by [*Sa*, ..., *Sf* ] and their complementary values by [*Sa*, ..., *Sf* ], where **S***<sup>i</sup>* ∈ {0, 1}. The discrete nature of the VSI defines the phase voltages which can be mapped in the (*α* − *β*) − (*x* − *y*) according to the VSD approach. Figure 2 shows the active vectors in the (*α* − *β*) and (*x* − *y*) subspaces, where each switching vector state is identified using the switching function by two octal numbers corresponding to the binary numbers [*SaSbSc*] and [*SdSeSf* ], respectively. Stator voltages are related to the input control signals through the VSI model. An ideal inverter converts gating signals into stator voltages that can be projected to (*α* − *β*) and (*x* − *y*) subspaces and gathered in a row vector **U***αβxys* computed as

$$\mathbf{U}\_{\text{a\\$xyz}} = \begin{bmatrix} u\_{\text{a\\$}\prime} \ u\_{\text{f\\$}\prime} \ u\_{\text{x\\$}\prime} \ u\_{\text{y\\$}\prime} \ 0 \ 0 \end{bmatrix}^T = V d c \mathbf{T} \,\mathbf{M} \,\tag{3}$$

where (*T*) indicates the transposed matrix and **M** represents the model of the VSI that can be expressed as function to the switching vectors as follows:

$$\mathbf{M} = \frac{1}{3} \begin{bmatrix} 2 & 0 & -1 & 0 & -1 & 0 \\ 0 & 2 & 0 & -1 & 0 & -1 \\ -1 & 0 & 2 & 0 & -1 & 0 \\ 0 & -1 & 0 & 2 & 0 & -1 \\ -1 & 0 & -1 & 0 & 2 & 0 \\ 0 & -1 & 0 & -1 & 0 & 2 \end{bmatrix} \mathbf{S}^{T}. \tag{4}$$

As shown in Figure 2, the 64 possible voltage vectors lead to only 49 different vectors in the (*α* − *β*) and (*x* − *y*) subspaces. Applying the transformation matrix, the mathematical model

**Figure 2.** Voltage space vectors and switching states in the (*α* − *β*) and (*x* − *y*) subspaces for a six-phase asymmetrical VSI

of the SpIM can be written using the state-space (SS) representation as follows:

4 Induction Motor

the machine.

**T***dq* =

From the VSD approach, the following conclusions should be emphasized:

1 (*n* = 1, 2, 3, ...) are represented in this subspace.

 *cos*(*θr*) *sin* (*θr*) −*sin* (*θr*) *cos*(*θr*)

1. The electromechanical energy conversion variables are mapped to the (*α* − *β*) subspace. Therefore, the fundamental supply component and the supply harmonics of order 12*n* ±

2. The current components in the (*x* − *y*) subspace do not contribute to the air-gap flux and are limited only by the stator resistance and stator leakage inductance. These components represent the supply harmonics of the order 6*n* ± 1 (*n* = 1, 3, 5, ...) and only produce

3. The voltage vectors in the (*z*<sup>1</sup> − *z*2) are zero due to the separated neutral configuration of

The VSI with isolated neutrals is depicted in Figure 1 (b), being the gating signal represented by [*Sa*, ..., *Sf* ] and their complementary values by [*Sa*, ..., *Sf* ], where **S***<sup>i</sup>* ∈ {0, 1}. The discrete nature of the VSI defines the phase voltages which can be mapped in the (*α* − *β*) − (*x* − *y*) according to the VSD approach. Figure 2 shows the active vectors in the (*α* − *β*) and (*x* − *y*) subspaces, where each switching vector state is identified using the switching function by two octal numbers corresponding to the binary numbers [*SaSbSc*] and [*SdSeSf* ], respectively. Stator voltages are related to the input control signals through the VSI model. An ideal inverter converts gating signals into stator voltages that can be projected to (*α* − *β*) and

*uαs*, *uβs*, *uxs*, *uys*, 0, 0

where (*T*) indicates the transposed matrix and **M** represents the model of the VSI that can be

2 0 −1 0 −1 0 020 −1 0 −1 −10 2 0 −1 0 0 −10 2 0 −1 −1 0 −10 2 0 0 −1 0 −10 2

As shown in Figure 2, the 64 possible voltage vectors lead to only 49 different vectors in the (*α* − *β*) and (*x* − *y*) subspaces. Applying the transformation matrix, the mathematical model

*T*

 

where *θ<sup>r</sup>* is the rotor angular position referred to the stator as shown in Figure 1 (a).

losses, so consequently they should be controlled to be as small as possible.

(*x* − *y*) subspaces and gathered in a row vector **U***αβxys* computed as

**U***αβxys* =

expressed as function to the switching vectors as follows:

**<sup>M</sup>** <sup>=</sup> <sup>1</sup> 3   , (2)

= *Vdc* **T M**, (3)

**S***T*. (4)

$$[\mathbf{u}]\_{a\beta} = [\mathbf{G}] \frac{d}{dt} \begin{bmatrix} \mathbf{x} \end{bmatrix}\_{a\beta} + [\mathbf{F}] \begin{bmatrix} \mathbf{x} \end{bmatrix}\_{a\beta} \tag{5}$$

where [**u**]*αβ* <sup>=</sup> *<sup>u</sup>α<sup>s</sup> <sup>u</sup>β<sup>s</sup>* 0 0 *<sup>T</sup>* represents the input vector, [**x**]*αβ* <sup>=</sup> *iα<sup>s</sup> iβ<sup>s</sup> iα<sup>r</sup> iβ<sup>r</sup> <sup>T</sup>* denotes the state vector, and [**F**] and [**G**] are matrices that define the dynamics of the drive that for the particular case of the SpIM are represented as follows:

$$\begin{aligned} [\mathbf{F}] = \begin{bmatrix} R\_s & 0 & 0 & 0 \\ 0 & R\_s & 0 & 0 \\ 0 & \omega\_r \ L\_m & R\_r & \omega\_r \ L\_r \\ -\omega\_r \ L\_m & 0 & -\omega\_r \ L\_r & R\_r \end{bmatrix} \end{aligned} \tag{6}$$

$$\begin{aligned} [\mathbf{G}] = \begin{bmatrix} L\_{\mathbf{s}} & 0 & L\_{m} & 0 \\ 0 & L\_{\mathbf{s}} & 0 & L\_{m} \\ L\_{m} & 0 & L\_{r} & 0 \\ 0 & L\_{m} & 0 & L\_{r} \end{bmatrix} \; \prime \end{aligned} \tag{7}$$

where *Rs* and *Rr* are the stator and rotor resistance, *ω<sup>r</sup>* is the rotor angular speed, and *Ls* = *Lls* + 3 *Lm*, *Lr* = *Llr* + 3 *Lm*, and *Lm* are the stator, rotor, and magnetizing inductances, respectively. For a machine with *P* pairs of poles, the mechanical part of the drive is given by the following equations:

$$T\_{\varepsilon} = 3\frac{P}{2} \left( \psi\_{\beta r} i\_{ar} - \psi\_{ar} i\_{\beta r} \right) \,, \tag{8}$$

$$J\_i \frac{d}{dt} \omega\_r + B\_i \omega\_r = \frac{P}{2} \left( T\_\varepsilon - T\_L \right) \,. \tag{9}$$

where *TL* denotes the load torque, *Ji* the inertia, *ψαβ<sup>r</sup>* the rotor flux, and *Bi* the friction coefficient.

**Figure 3.** Scheme of the experimental setup

The equations in (*x* − *y*) subspace do not link to the rotor side and consequently do not contribute to the air-gap flux; however, they are an important source of Joule losses. Using the SS representation, these equations can be written as:

$$[\mathbf{u}]\_{xy} = \begin{bmatrix} L\_{ls} & 0 \\ 0 & L\_{ls} \end{bmatrix} \frac{d}{dt} \begin{bmatrix} \mathbf{i} \end{bmatrix}\_{xy} + \begin{bmatrix} R\_s & 0 \\ 0 & R\_s \end{bmatrix} \begin{bmatrix} \mathbf{i} \end{bmatrix}\_{xy \ \prime} \tag{10}$$

where *Lls* represents the stator leakage inductance.

### **3. SpIM parameter identification**

A commercial three-phase induction machine with three pairs of poles, 72 slots, and 15 kW of rated power has been rewound to obtain an asymmetrical six-phase winding (configured with two isolated neutral points) with the same pairs of poles and power with the original three-phase machine. Conventional test (blocked rotor and no-load tests) procedures have been applied to determine experimentally the electrical and mechanical parameters of the SpIM. The obtained values are shown in Table 1.

Two three-phase VSC modules manufactured by Semikron SKS 35F B6U+E1CIF+B6CI21V series are used to generate the six-phase stator voltages and to obtain the experimental results. A hardware timer based on the LM555 device operating in monostable mode is implemented to control the internal pre-charge circuit of both the SKS 35F modules. The Dc-Link voltage is V*Dc* = 585 V. The implementation of the control system is based on the DSC TMS320LF28335 manufactured by Texas Instruments and the MSK28335 board from Technosoft which has 12 pulse-width modulation (PWM) outputs. The PWM is configured with a 10 kHz of switching frequency. Stator currents are measured by using Hall effect sensors (LA-55P from LEM). The analog-to-digital (A/D) converter peripherals of the MSK28335 board with 16 parallel channels are used to capture all the measured signals. On


**Table 1.** Electrical and mechanical parameters

6 Induction Motor

**Figure 3.** Scheme of the experimental setup

the SS representation, these equations can be written as:

[**u**]*xy* =

where *Lls* represents the stator leakage inductance.

SpIM. The obtained values are shown in Table 1.

**3. SpIM parameter identification**

 *Lls* 0 0 *Lls*

The equations in (*x* − *y*) subspace do not link to the rotor side and consequently do not contribute to the air-gap flux; however, they are an important source of Joule losses. Using

*dt* [**i**]*xy* <sup>+</sup>

A commercial three-phase induction machine with three pairs of poles, 72 slots, and 15 kW of rated power has been rewound to obtain an asymmetrical six-phase winding (configured with two isolated neutral points) with the same pairs of poles and power with the original three-phase machine. Conventional test (blocked rotor and no-load tests) procedures have been applied to determine experimentally the electrical and mechanical parameters of the

Two three-phase VSC modules manufactured by Semikron SKS 35F B6U+E1CIF+B6CI21V series are used to generate the six-phase stator voltages and to obtain the experimental results. A hardware timer based on the LM555 device operating in monostable mode is implemented to control the internal pre-charge circuit of both the SKS 35F modules. The Dc-Link voltage is V*Dc* = 585 V. The implementation of the control system is based on the DSC TMS320LF28335 manufactured by Texas Instruments and the MSK28335 board from Technosoft which has 12 pulse-width modulation (PWM) outputs. The PWM is configured with a 10 kHz of switching frequency. Stator currents are measured by using Hall effect sensors (LA-55P from LEM). The analog-to-digital (A/D) converter peripherals of the MSK28335 board with 16 parallel channels are used to capture all the measured signals. On

 *Rs* 0 0 *Rs* [**i**]*xy* , (10)

*d*

Six-phase Induction Motor

the other hand, the mechanical speed is measured by employing a Hengstler RI 58-O digital incremental encoder with a resolution of 10,000 pulses per revolution and the enhanced quadrature encoder pulse (eQEP) peripheral of the DSC. To preserve the system integrity, input, and output, digital outputs of the control board are galvanically isolated by means of a Texas Instruments ISO7230CDW isolator. Figure 3 shows a picture of the different parts of the experimental test bench. In order to validate the electrical and mechanical parameters, a PLL software implementation is used to calculate the stator current angle (*θ*). Finally, the angle is used to calculate the stator current in dynamic reference frame (*ids* − *iqs*) using the transformation matrix shown in Eq. (2). Statistical parameters of performance (taking as reference the experimental evolution of stator currents in dynamic reference frame) are quantifiable for two different implementations: the SpIM model based on MatLab/Simulink simulation environment and a real SpIM using the experimental setup.

### **3.1. Digital PLL implementation**

Figure 4 (a) shows that the dynamic performance of the proposed PLL is highly influenced by the compensator G(*z*). Considering that the reference signal is the stator current in *d* axis and since the loop gain includes an integral term, *θ* must track the constant component of the reference signal with zero steady-state error. However, to ensure zero steady-state error, the loop gain must include at least two integrators. Therefore, G(*z*) must include at least one integral term, that is, one pole at *z* = 1. The other poles and zeros of G(*z*) are determined mainly by the closed-loop bandwidth requirements of the PLL and stability indices such as phase margin and gain margin, according with the procedure described in [25]. Due to the fact that G(*z*) is controllable, the transfer function can be expressed into controllable canonical form as follows:

$$\mathbf{x}\_{(k+1|k)} = [\mathbf{F}]\_{5 \times 5} \left[ \mathbf{x}\_{(k|k)} \right]\_{5 \times 1} + [\mathbf{D}]\_{5 \times 1} \left[ e\_{(k|k)} \right]\_{5 \times 1'} \tag{11}$$

$$
\boldsymbol{\omega}^\*\_{\boldsymbol{\alpha}}(\boldsymbol{k}|\boldsymbol{k}) = [\mathbf{C}]\_{1 \times \dots \times} [\mathbf{x}\_{(\boldsymbol{k}|\boldsymbol{k})}]\_{5 \times 1} \boldsymbol{\prime} \tag{12}
$$

(a) Block diagram of a PLL with special design of the compensator

(b) Representation of G(*z*) transfer function on controllable canonical form

**Figure 4.** PLL software implementation block diagram

where the matrix [**F**]5×<sup>5</sup> and the vectors [**D**]5×<sup>1</sup> and [**C**]1×<sup>5</sup> define the dynamics of the PLL compensator [G(*z*)], which for the set of state variables shown in Figure 4 (b) are as follows:

$$[\mathbf{F}]\_{5\times5} = \begin{bmatrix} 2.5 \ -2.2 \ 0.9 \ -0.2 \ 0.01 \\ 1 \ 0 \ 0 \ 0 \ 0 \ 0 \\ 0 \ 1 \ 0 \ 0 \ 0 \\ 0 \ 0 \ 0 \ 1 \ 0 \ 0 \\ 0 \ 0 \ 0 \ 0 \ 1 \ 0 \end{bmatrix},\tag{13}$$

$$\begin{bmatrix} \mathbf{D} \end{bmatrix}\_{5 \times 1} = \begin{bmatrix} 1 \ 0 \ 0 \ 0 \ 0 \ 0 \end{bmatrix}^T,\tag{14}$$

$$\mathbf{[C]}\_{1\times5} = \begin{bmatrix} 1.7 \ -5.7 \ 8.1 \ -5.8 \ 1.6 \end{bmatrix}.\tag{15}$$

This state-space realization is called controllable canonical form because the resulting model is guaranteed to be controllable. Since the control enters a chain of integrators, it has the ability to move every state as shown in Figure 4 (b).

The proposed PLL architecture has been implemented by using the TMS320LF28335 DSC, considering floating-point arithmetic and 10 kHz sampling frequency. The PLL algorithm is executed as an interrupt service routine (ISR), which is triggered by one of the general-purpose timer circuits available on chip. The same timer also triggers the acquisition of input signals, simultaneously with the sampling interrupt. As the on-chip A/D converters have a fast conversion rate (approximately 106-ns conversion time), input data are made available at the beginning of the ISR with negligible time delay. The current components in

**Figure 5.** Stator current angle evolution obtained experimentally by using the proposed PLL with special design of the compensator

stationary references frame (*α*-*β*) are calculated at each sampling time from the measured phase stator currents (*ibs*, *ics*, *ids*, *if s*) by using Eq. (1), immediately after performing A/D conversion.

Figure 5 shows the stator current angle evolution obtained experimentally by using the proposed PLL architecture, when the SpIM is fed with electrical frequency voltages (*fe*) of 40 Hz. It can be seen that the angle evolves from 0 to 2*π* during a single period of the stator current wave. It is also possible to observe that the result is satisfactory even when the stator currents in stationary reference frame are distorted due to electrical noise.

### **3.2. SpIM parameter validation**

8 Induction Motor

(a) Block diagram of a PLL with special design of the compensator

(b) Representation of G(*z*) transfer function on controllable canonical form

where the matrix [**F**]5×<sup>5</sup> and the vectors [**D**]5×<sup>1</sup> and [**C**]1×<sup>5</sup> define the dynamics of the PLL compensator [G(*z*)], which for the set of state variables shown in Figure 4 (b) are as follows:

> 2.5 −2.2 0.9 −0.2 0.01 1000 0 0100 0 0010 0 0001 0

1.7 <sup>−</sup>5.7 8.1 <sup>−</sup>5.8 1.6

This state-space realization is called controllable canonical form because the resulting model is guaranteed to be controllable. Since the control enters a chain of integrators, it has the

The proposed PLL architecture has been implemented by using the TMS320LF28335 DSC, considering floating-point arithmetic and 10 kHz sampling frequency. The PLL algorithm is executed as an interrupt service routine (ISR), which is triggered by one of the general-purpose timer circuits available on chip. The same timer also triggers the acquisition of input signals, simultaneously with the sampling interrupt. As the on-chip A/D converters have a fast conversion rate (approximately 106-ns conversion time), input data are made available at the beginning of the ISR with negligible time delay. The current components in

 

10000 *<sup>T</sup>* , (14)

, (13)

. (15)

**Figure 4.** PLL software implementation block diagram

[**F**]5×<sup>5</sup> <sup>=</sup>

[**C**]1×<sup>5</sup> <sup>=</sup>

ability to move every state as shown in Figure 4 (b).

 

[**D**]5×<sup>1</sup> <sup>=</sup>

SpIM electrical and mechanical parameters have been analyzed and validated using the experimental setup as well as a SpIM MatLab/Simulink model where a fourth-order Runge-Kutta numerical integration method has been applied to compute the evolution of the state variables step by step in the time domain. Table 1 shows the electrical and mechanical parameters of the asymmetrical SpIM which have been considered during the simulation. The validation of the measured parameters has been evaluated under no-load conditions.

Figure 6 shows the stator current start-up characteristics when a VSC supplied with 585 V of Dc-Link is considered and when a sinusoidal modulation index of 0.275 and 40 Hz of frequency is applied. Figure 6 (a) shows the *i<sup>β</sup>* current evolution of the SpIM provided by the MatLab/Simulink model. In this case, the VSC, the PWM scheme, and the AC motor are simulated within the MatLab/Simulink model. The stator current evolution is compared with the *id* current obtained using the experimental setup in order to verify the analogy between the MatLab/Simulink model simulation results and the experimental results especially with respect to the time constants associated with the SpIM (start-up current, speed, steady-state current, etc.). It can be seen that the time constant converges to the value obtained experimentally both in transient and steady-state conditions where it is possible to quantify a steady-state current of approximately 2 A. Moreover, Figure 6 (b) shows the results obtained experimentally. These results have been compared with the *id* current obtained experimentally. It can be seen that the start-up current evolution converges to a common value for the MatLab/Simulink-based simulations as well as for the experimental

MatLab/Simulink simulation environment experimental tests

**Figure 6.** Stator current start-up characteristics

**Figure 7.** Transient rotor speed evolution

setup, with a start-up transient of approximately 1.15 s. After 1.5 s, the reference frequency is changed from 40 to 50 Hz, while the modulation index is kept constant at 0.275.

Statistical performance parameters such as the covariance, the standard deviation (SD), and the mean square error (MSE) are used in order to evaluate the accuracy of the parameters, taking as reference the results obtained through simulations, as well as those obtained by means of experimental tests. The envelope of the fundamental frequency component of the


**Table 2.** Performance analysis.

10 Induction Motor

8

setup, with a start-up transient of approximately 1.15 s. After 1.5 s, the reference frequency

Statistical performance parameters such as the covariance, the standard deviation (SD), and the mean square error (MSE) are used in order to evaluate the accuracy of the parameters, taking as reference the results obtained through simulations, as well as those obtained by means of experimental tests. The envelope of the fundamental frequency component of the

is changed from 40 to 50 Hz, while the modulation index is kept constant at 0.275.

experimental tests

8

(b) Stator current (*iβ*) current obtained by

(a) Stator current (*iβ*) obtained by using the MatLab/Simulink simulation environment

> , ,

**Figure 6.** Stator current start-up characteristics

**Figure 7.** Transient rotor speed evolution

stator currents in stationary reference frame can be calculated using the Hilbert transform (HT) method. This envelope detection method involves creating the analytic signal of the stator current using the HT. An analytic signal is a complex signal, where the real part (*iαs*) is considered the original signal and the imaginary part (*jiβs*) is the HT of the original signal. A discrete-time analytic signal (¯*h*(*k*)) can be defined as follows:

$$\hbar(k) = \mathfrak{i}\_{\rm as}(k) + j\mathfrak{i}\_{\rm \beta \rm s}(k), \tag{16}$$

while the envelope of the signal can be determined by computing the modulus of the analytic signal from the following equation:

$$|\hbar(k)| = \sqrt{\left[\sum\_{i=0}^{n} i\_{\rm as}\left(k\right)\right]^2 + \left[\sum\_{i=0}^{n} i\_{\rm fs}\left(k\right)\right]^2}.\tag{17}$$

Using the above equation, it is possible to determine the envelope evolution of the stator current, which is used to evaluate those aforementioned statistical performance parameters. This analysis enables to determine the degree of dispersion of the envelope (of the stator current) with respect to the value obtained experimentally through the PLL software implementation (which is shown in red color in Figure 6). The statistic relationship between the curves (*iqs* and stator current envelope) and the MSE has been analyzed under steady-state conditions. Table 2 details the obtained results for the two different SpIM implementation methods considered in Figure 6. Notice that the obtained performance results are similar for both cases (MatLab/Simulink model and experimental). Moreover, Figure 7 shows the rotor speed evolution for the two cases analyzed before. It can be seen that the results provided by the MatLab/Simulink model in steady state converge to the values obtained experimentally using a motor having three pairs of poles and 50 Hz of nominal frequency (close to 1,000 rpm).

Further analysis has been done to validate the parameters under different test conditions. For example, a change in the modulation index from 0.275 to 0.481 was considered at *t* = 1.5 s, while a constant voltage frequency of 40 Hz was considered. Figure 8 (a) shows the trajectory of the *iα<sup>s</sup> vs*. *iβ<sup>s</sup>* as well as (*ids* − *iqs*) current evolution considering at least four current periods in steady-state operation, where it is also possible to observe the effect of the change of the modulation index in the reference voltages. Figure 8 (a) shows the results obtained using the MatLab/Simulink model, and Figure 8 (b) shows the experimental results. As in the previous case, it can be seen that the simulated current converges to values equivalent to those obtained experimentally and exhibiting similar dynamic behavior. Finally, Figure 8 (c)

(a) *iα<sup>s</sup> vs*. *iβ<sup>s</sup>* current obtained by using the MatLab/Simulink simulation environment (b) *iα<sup>s</sup> vs*. *iβ<sup>s</sup>* current obtained by experimental tests

#### **Figure 8.** Dynamic reference frame characteristics

shows the stator current evolution in the dynamic reference frame (*d* − *q*) obtained by means of Eq. (2) using the angle values calculated by the PLL software implementation. It can be seen that the steady-state current values converge to those values shown in **Figure 8 (a)** and Figure 8 (b) before and after applying the change in the modulation index from 0.275 to 0.481, being these values close to 2 and 4 A, respectively. These results validate the SpIM electrical and mechanical parameters shown in Table 1.

### **4. Predictive model**

Assuming the mathematical model expressed by Eq. (5) and using the state variables defined by the vector [**x**]*αβ*, the derivative of states can be defined as follows:

$$\begin{aligned} \dot{\mathbf{x}}\_1 &= c\_3 \left( R\_r \mathbf{x}\_3 + \omega\_r \mathbf{x}\_4 L\_r + \omega\_r \mathbf{x}\_2 L\_m \right) + c\_2 \left( \mathbf{u}\_{\rm as} - R\_s \mathbf{x}\_1 \right), \\ \dot{\mathbf{x}}\_2 &= c\_3 \left( R\_r \mathbf{x}\_4 - \omega\_r \mathbf{x}\_3 L\_r - \omega\_r \mathbf{x}\_1 L\_m \right) + c\_2 \left( \mathbf{u}\_{\rm f\bar{s}} - R\_s \mathbf{x}\_2 \right), \\ \dot{\mathbf{x}}\_3 &= c\_4 \left( -R\_r \mathbf{x}\_3 - \omega\_r \mathbf{x}\_4 L\_r - \omega\_r \mathbf{x}\_2 L\_m \right) + c\_3 \left( -\mathbf{u}\_{\rm as} + R\_s \mathbf{x}\_1 \right), \\ \dot{\mathbf{x}}\_4 &= c\_4 \left( -R\_r \mathbf{x}\_4 + \omega\_r \mathbf{x}\_3 L\_r + \omega\_r \mathbf{x}\_1 L\_m \right) + c\_3 \left( -\mathbf{u}\_{\rm f\bar{s}} + R\_s \mathbf{x}\_2 \right), \end{aligned} \tag{18}$$

where *ci* (*i* = 1, 2, 3, 4) are constants defined as:

12 Induction Motor

*i<sup>s</sup>* <sup>=</sup> 3.98

*i<sup>s</sup>* <sup>=</sup> 3.94

*i<sup>s</sup>* <sup>=</sup> 1.94

(b) *iα<sup>s</sup> vs*. *iβ<sup>s</sup>* current obtained by experimental

*i*

tests

*<sup>q</sup>* <sup>=</sup> 3.96 *<sup>i</sup>*

(c) Stator current evolution in dynamic

shows the stator current evolution in the dynamic reference frame (*d* − *q*) obtained by means of Eq. (2) using the angle values calculated by the PLL software implementation. It can be seen that the steady-state current values converge to those values shown in **Figure 8 (a)** and Figure 8 (b) before and after applying the change in the modulation index from 0.275 to 0.481, being these values close to 2 and 4 A, respectively. These results validate the SpIM electrical

Assuming the mathematical model expressed by Eq. (5) and using the state variables defined

<sup>1</sup> = *c*<sup>3</sup> (*Rrx*<sup>3</sup> + *ωrx*4*Lr* + *ωrx*2*Lm*) + *c*<sup>2</sup> (*uα<sup>s</sup>* − *Rsx*1),

<sup>3</sup> = *c*<sup>4</sup> (−*Rrx*<sup>3</sup> − *ωrx*4*Lr* − *ωrx*2*Lm*) + *c*<sup>3</sup> (−*uα<sup>s</sup>* + *Rsx*1),

*uβ<sup>s</sup>* − *Rsx*<sup>2</sup>

−*uβ<sup>s</sup>* + *Rsx*<sup>2</sup>

 ,

> ,

(18)

*<sup>q</sup>* = 1.94

reference frame

by the vector [**x**]*αβ*, the derivative of states can be defined as follows:

<sup>2</sup> = *c*<sup>3</sup> (*Rrx*<sup>4</sup> − *ωrx*3*Lr* − *ωrx*1*Lm*) + *c*<sup>2</sup>

<sup>4</sup> = *c*<sup>4</sup> (−*Rrx*<sup>4</sup> + *ωrx*3*Lr* + *ωrx*1*Lm*) + *c*<sup>3</sup>

**Figure 8.** Dynamic reference frame characteristics

and mechanical parameters shown in Table 1.

*x*˙

*x*˙

*x*˙

*x*˙

**4. Predictive model**

*i<sup>s</sup>* <sup>=</sup> 1.94

(a) *iα<sup>s</sup> vs*. *iβ<sup>s</sup>* current obtained by using the MatLab/Simulink simulation environment

$$\mathbf{c}\_{1} = \mathbf{L}\_{\mathbf{s}} \mathbf{L}\_{r} - \mathbf{L}\_{m\prime}^{2} \ c\_{2} = \frac{\mathbf{L}\_{r}}{c\_{1}}, \ c\_{3} = \frac{\mathbf{L}\_{m}}{c\_{1}}, \ c\_{4} = \frac{\mathbf{L}\_{\mathbf{s}}}{c\_{1}}.\tag{19}$$

This set of differential equations can be represented in the state-space form as follows:

$$\begin{aligned} \dot{\mathbf{X}}(t) &= f\left(\mathbf{X}(t), \mathbf{U}(t)\right), \\ \mathbf{Y}(t) &= \mathbf{C}\mathbf{X}(t), \end{aligned} \tag{20}$$

with state vector **X**(*t*) = [*x*1, *x*2, *x*3, *x*4] *<sup>T</sup>*, input vector **<sup>U</sup>**(*t*) = *uαs*, *uβ<sup>s</sup>* , and output vector **Y**(*t*) = [*x*1, *x*2] *<sup>T</sup>*. The components of the vectorial function *f* and matrix **C** are obtained in a straightforward manner from Eq. (18) and the definitions of state and output vector.

The continuous time model represented by Eq. (20) can be discretized in order to be used for the predictive controller using the forward difference approximation method of the first derived, also known as the forward Euler method. Thus, a prediction of the future next-sample state **<sup>X</sup>**ˆ(*<sup>k</sup>* <sup>+</sup> <sup>1</sup>|*k*) is expressed as:

$$
\hat{\mathbf{X}}(k+1|k) = \mathbf{X}(k) + T\_m f\left(\mathbf{X}(k), \mathbf{U}(k)\right), \tag{21}
$$

where (*k*) is the current sample and *Tm* the sampling time. In Eq. (21), currents and voltages of the stator and the mechanical speed are measurable variables; however, the rotor currents cannot be measured directly. This difficulty can be overcome by means of estimating the rotor current using the reduced-order estimator concept. Figure 9 shows the proposed predictive current control technique for the asymmetrical SpIM.

### **4.1. The Estimator Based on the State Variables (SV)**

The state variables evolution in discrete time can be represented using the following equations:

$$
\begin{bmatrix}
\hat{\mathbf{X}}\_d(k+1) \\
\hat{\mathbf{X}}\_b(k+1)
\end{bmatrix} = \begin{bmatrix}
\overline{\mathbf{A}}\_{11} \ \overline{\mathbf{A}}\_{12} \\
\overline{\mathbf{A}}\_{21} \ \overline{\mathbf{A}}\_{22}
\end{bmatrix} \begin{bmatrix}
\mathbf{X}\_d(k) \\
\mathbf{X}\_b(k)
\end{bmatrix} + \begin{bmatrix}
\overline{\mathbf{B}}\_1 \\
\overline{\mathbf{B}}\_2
\end{bmatrix} \mathbf{U}\_{a\bar{\mathbf{B}}\bar{\mathbf{A}}},
$$

$$
\mathbf{Y}(k) = \begin{bmatrix}
\overline{\mathbf{I}} \ \overline{\mathbf{0}}
\end{bmatrix} \begin{bmatrix}
\mathbf{X}\_d(k) \\
\mathbf{X}\_b(k)
\end{bmatrix},
\tag{22}
$$

where **X***<sup>a</sup>* = *iαs*(*k*) *iβs*(*k*) *<sup>T</sup>* is the vector directly measured which is **<sup>Y</sup>**, **<sup>X</sup>***<sup>b</sup>* <sup>=</sup> *iαr*(*k*) *iβr*(*k*) *T* is the remaining portion to be estimated, **I** represents the identity matrix, and **A** and **B** are matrices whose components are obtained in the following equations:

**Figure 9.** Proposed predictive current control technique for the asymmetrical SpIM

$$
\overline{\mathbf{A}} = \begin{bmatrix}
(1 - T\_m c\_2 R\_5) & T\_m c\_3 L\_m \omega\_r & \vdots & T\_m c\_3 R\_7 & T\_m c\_3 L\_r \omega\_r \\
\dots & \dots & \dots & \dots & \dots & \dots \\
& & \dots & \dots & \dots & \dots \\
& & T\_m c\_3 R\_8 & -T\_m c\_4 L\_m \omega\_r & \vdots & (1 - T\_m c\_4 R\_7) & -T\_m c\_4 L\_r \omega\_r \\
& T\_m c\_4 L\_m \omega\_r & T\_m c\_3 R\_8 & \vdots & T\_m c\_4 L\_r \omega\_r & (1 - T\_m c\_4 R\_7) \\
& & & & & \\
\mathbf{E} = \begin{bmatrix}
T\_m c\_2 & 0 \\
0 & T\_m c\_2 \\
\dots & \dots & \dots \\
0 & -T\_m c\_3
\end{bmatrix}.
\end{bmatrix},
(23)$$

The prediction of the stator currents can be calculated as follows:

$$\hat{\mathbf{i}}\_{\rm as}(k+1|k) = (1 - T\_{\rm m}c\_{2}\mathbf{R}\_{\rm s})\mathbf{i}\_{\rm as}(k) + T\_{\rm m}c\_{3}L\_{\rm m}\omega\_{\rm r}(k)\mathbf{i}\_{\rm f8s}(k) + T\_{\rm m}c\_{2}\mathbf{u}\_{\rm as}(k) + T\_{\rm m}c\_{3}\mathbf{\tilde{j}}\_{\rm as}(k), \tag{24}$$

where *ξαs*(*k*) = *Rriαr*(*k*) + *Lrωr*(*k*)*iβr*(*k*) .

On the other hand, the quadrature current can be written as follows:

$$\begin{aligned} \hat{\mathbf{i}}\_{\rm f\mathbf{s}}(k+1|k) &= -T\_{\rm m}c\_{\rm 3}\mathbf{L}\_{\rm \rm ff}\boldsymbol{\omega}\_{\rm f}(k)\mathbf{i}\_{\rm f\mathbf{s}}(k) + (1 - T\_{\rm m}c\_{2}\mathbf{R}\_{\rm s})\mathbf{i}\_{\rm f\mathbf{s}}(k) + T\_{\rm m}c\_{2}\boldsymbol{\omega}\_{\rm f\mathbf{s}}(k) + T\_{\rm m}c\_{3}\mathbf{\overline{\xi}}\_{\rm f\mathbf{s}}(k), \\\\ \text{where } \mathsf{\mathsf{T}}\_{\rm f\mathbf{s}}(k) &= \left(R\_{\rm r}\mathbf{i}\_{\rm f\mathbf{r}}(k) + L\_{\rm r}\boldsymbol{\omega}\_{\rm r}\mathbf{i}\_{\rm ar}(k)\right). \end{aligned} \tag{25}$$

It can be seen from the above equations that the prediction of the stator currents has a measurable (*m*(*k*) = *mα*(*k*), *mβ*(*k*) ) and unmeasured (*e*(*k*) = *eα*(*k*),*eβ*(*k*) ) parts. Assuming this, the prediction equations can be rewritten as follows:

$$
\hat{I}\_{as}(k+1|k) = m\_a(k) + e\_a(k), \tag{26}
$$

$$
\hat{I}\_{\beta\mathfrak{s}}(k+1|k) = m\_{\beta}(k) + e\_{\beta}(k), \tag{27}
$$

where

Sensor

 

,

(23)

(25)

**Six-phase Induction Motor**

. *Tmc*3*Rr Tmc*3*Lrω<sup>r</sup>*

. −*Tmc*3*Lrω<sup>r</sup> Tmc*3*Rr*

. (1 − *Tmc*4*Rr*) −*Tmc*4*Lrω<sup>r</sup>*

. *Tmc*4*Lrω<sup>r</sup>* (1 − *Tmc*4*Rr*)

14 Induction Motor

**cost function**

**A** =

**B** =

where *ξαs*(*k*) =

where *ξβs*(*k*) =

 

 

**Figure 9.** Proposed predictive current control technique for the asymmetrical SpIM

(1 − *Tmc*2*Rs*) *Tmc*3*Lmω<sup>r</sup>*

<sup>−</sup>*Tmc*3*Lmω<sup>r</sup>* (<sup>1</sup> <sup>−</sup> *Tmc*2*Rs*) .

*Tmc*3*Rs* −*Tmc*4*Lmω<sup>r</sup>*

 .

*Tmc*4*Lmω<sup>r</sup> Tmc*3*Rs*

The prediction of the stator currents can be calculated as follows:

On the other hand, the quadrature current can be written as follows:

*Rriαr*(*k*) + *Lrωr*(*k*)*iβr*(*k*)

*Rriβr*(*k*) + *Lrωriαr*(*k*)

*Tmc*<sup>2</sup> 0 0 *Tmc*<sup>2</sup> ··· ··· −*Tmc*<sup>3</sup> 0 0 −*Tmc*<sup>3</sup> . .

.

··· ··· ··· ··· ···

. .

. .

<sup>ˆ</sup>*iαs*(*<sup>k</sup>* <sup>+</sup> <sup>1</sup>|*k*) =(<sup>1</sup> <sup>−</sup> *Tmc*2*Rs*)*iαs*(*k*) + *Tmc*3*Lmωr*(*k*)*iβs*(*k*) + *Tmc*2*uαs*(*k*) + *Tmc*3*ξαs*(*k*), (24)

 .

 .

<sup>ˆ</sup>*iβs*(*<sup>k</sup>* <sup>+</sup> <sup>1</sup>|*k*) = <sup>−</sup> *Tmc*3*Lmωr*(*k*)*iαs*(*k*)+(<sup>1</sup> <sup>−</sup> *Tmc*2*Rs*)*iβs*(*k*) + *Tmc*2*uβs*(*k*) + *Tmc*3*ξβs*(*k*),

$$m\_{\rm at}(k) = (1 - T\_{\rm m}c\_{2}R\_{\rm s})i\_{\rm as}(k) + T\_{\rm m}c\_{3}L\_{\rm m}\omega\_{\rm r}(k)i\_{\rm f\&}(k) + T\_{\rm m}c\_{2}u\_{\rm as}(k),\tag{28}$$

$$m\_{\mathfrak{F}}(k) = -T\_{\mathfrak{m}}c\_{3}L\_{\mathfrak{m}}\omega\_{\mathfrak{r}}(k)i\_{\text{as}}(k) + (1 - T\_{\mathfrak{m}}c\_{2}\mathbb{R}\_{\mathfrak{s}})i\_{\text{\tilde{\rho}s}}(k) + T\_{\mathfrak{m}}c\_{2}\mathbb{1}\mu\_{\tilde{\rho}\mathbf{s}}(k),\tag{29}$$

$$
\mathfrak{e}\_{\mathfrak{A}}(k) = T\_m \mathfrak{c}\_{\mathfrak{J}} \mathfrak{f}\_{\mathfrak{A}s}(k), \tag{30}
$$

$$
\varepsilon\_{\beta}(k) = T\_m c\_3 \mathfrak{J}\_{\beta \mathfrak{s}}(k). \tag{31}
$$

Analyzing Eqs. (26) and (27), which establish a prediction of the stator currents in the (*α* − *β*) subspace for a (*k* + 1) sampling time using the measurements of the (*k*) sampling time, it can be noted that the term *m*(*k*) contains measurable variables, such as stator currents, rotor speed, and the stator voltages, while the term *e*(*k*) contains unmeasurable variables of the asymmetrical SpIM, for this particular case are the rotor currents in the (*α* − *β*) subspace. Consequently, to solve the equations, it is necessary to obtain an estimate of the value of *e*ˆ(*k*|*k*), since the rotor currents are not measurable states of the system. This can be solved using the following equations:

$$\mathfrak{e}\_{\mathfrak{a}}(k|k) = \mathfrak{e}\_{\mathfrak{a}}(k-1) = i\_{\mathfrak{a}s}(k) - m\_{\mathfrak{a}}(k-1),\tag{32}$$

$$\mathfrak{e}\_{\beta}(k|k) = \mathfrak{e}\_{\beta}(k-1) = i\_{\beta s}(k) - m\_{\beta}(k-1). \tag{33}$$

Considering null initial conditions *e*ˆ*α*(0) = 0 and *e*ˆ*β*(0) = 0, the estimated portion that represented the rotor currents can be calculated from a recursive formula given by:

$$
\hat{e}\_{\mathfrak{a}}(k|k) = \hat{e}\_{\mathfrak{a}}(k-1) + (i\_{\mathfrak{a}s}(k) - \hat{i}\_{\mathfrak{a}s}(k-1)),
\tag{34}
$$

$$
\hat{e}\_{\beta}(k|k) = \hat{e}\_{\beta}(k-1) + (i\_{\beta s}(k) - \hat{i}\_{\beta s}(k-1)).\tag{35}
$$

### **4.2. The estimator based on a Luenberger Observer**

The dynamics of the unmeasured part of the state vector defined by Eq. (22) is described as:

$$\mathbf{X}\_b(k+1) = \overline{\mathbf{A}}\_{22}\mathbf{X}\_b(k) + \overline{\mathbf{A}}\_{21}\mathbf{X}\_b(k) + \overline{\mathbf{B}}\_2\mathbf{U}\_{a\beta\varsigma} \tag{36}$$

where the last two terms are known and can be considered as an input for the **X***<sup>b</sup>* dynamics. The **X***<sup>a</sup>* part may be expressed as:

$$\mathbf{X}\_{a}(k+1) - \overline{\mathbf{A}}\_{11}\mathbf{X}\_{a}(k) - \overline{\mathbf{B}}\_{1}\mathbf{U}\_{a\beta s} = \overline{\mathbf{A}}\_{12}\mathbf{X}\_{b}(k). \tag{37}$$

Note that Eq. (37) represents a relationship between a measured quantity on the left and the unknown state vector on the right. Assuming this, Eq. (36) can be rewritten as follows:

$$
\begin{aligned}
\hat{\mathbf{X}}\_b(k+1) &= (\overline{\mathbf{A}}\_{22} - \mathbf{K}\_l \overline{\mathbf{A}}\_{12}) \hat{\mathbf{X}}\_b(k) + \mathbf{K}\_l \mathbf{Y}(k+1) + \\
(\overline{\mathbf{A}}\_{21} - \mathbf{K}\_l \overline{\mathbf{A}}\_{11}) \mathbf{Y}(k) &+ (\overline{\mathbf{B}}\_2 - \mathbf{K}\_l \overline{\mathbf{B}}\_1) \mathbf{U}\_{a\beta\bar{s}}(k),
\end{aligned} \tag{38}
$$

where **K***<sup>l</sup>* is the Luenberger gain matrix. Therefore, Eqs. (37) and (38) describe the dynamics of the reduced-order estimators for Luenberger observer [27].

### **4.3. The Estimator Based on a Kalman Filter**

Considering uncorrelated process and measurement of Gaussian noises, Eq. (22) can be also written as follows:

$$\begin{aligned} \hat{\mathbf{X}}(k+1|k) &= \overline{\mathbf{A}}\mathbf{X}(k) + \overline{\mathbf{B}}\mathbf{U}(k) + \mathbf{H}\boldsymbol{\sigma}(k), \\ \mathbf{Y}(k) &= \mathbf{C}\mathbf{X}(k) + \boldsymbol{\nu}(k), \end{aligned} \tag{39}$$

where **H** is the noise weight matrix, (*k*) is the noise matrix of the system model (process noise), and *ν*(*k*) is the matrix noise of measurement. The covariance matrices *R* and *R<sup>ν</sup>* of these noises are defined as:

$$\begin{aligned} R\_{\mathcal{O}} &= \textit{cov}(\mathcal{O}) = E \left\{ \boldsymbol{\mathcal{O}} \cdot \boldsymbol{\mathcal{O}}^{T} \right\}\_{\prime} \\ R\_{\boldsymbol{\nu}} &= \textit{cov}(\boldsymbol{\nu}) = E \left\{ \boldsymbol{\nu} \cdot \boldsymbol{\nu}^{T} \right\}\_{\prime} \end{aligned} \tag{40}$$

where *E* {.} denotes the expected value. Thus, the dynamics of the reduced-order estimator equations are:

$$\begin{aligned} \hat{\mathbf{X}}\_b(k+1|k) &= (\overline{\mathbf{A}}\_{22} - \mathbf{K}\_k \overline{\mathbf{A}}\_{12}) \hat{\mathbf{X}}\_b(k) + \mathbf{K}\_k \mathbf{Y}(k+1) + \\ & (\overline{\mathbf{A}}\_{21} - \mathbf{K}\_k \overline{\mathbf{A}}\_{11}) \mathbf{Y}(k) + (\overline{\mathbf{B}}\_2 - \mathbf{K}\_k \overline{\mathbf{B}}\_1) \mathbf{U}\_{a\beta\hat{s}}(k), \end{aligned} \tag{41}$$

where **K***<sup>k</sup>* represents the KF gain matrix that is calculated at each sampling time in a recursive manner from the covariance of the noises as:

**<sup>K</sup>***k*(*k*) = **<sup>Γ</sup>**(*k*) · **<sup>C</sup>***TR*−<sup>1</sup> *<sup>ν</sup>* , (42)

where **Γ** is the covariance of the new estimation, as a function of the old covariance estimation (*ϕ*) as follows:

$$\Gamma(k) = \boldsymbol{\varrho}(k) - \boldsymbol{\varrho}(k) \cdot \mathbf{C}^T (\mathbf{C} \cdot \boldsymbol{\varrho}(k) \cdot \mathbf{C}^T + \mathbb{R}\_{\boldsymbol{\nu}})^{-1} \cdot \mathbf{C} \cdot \boldsymbol{\varrho}(k). \tag{43}$$

From the state equation, which includes the process noise, it is possible to obtain a correction of the covariance of the estimated state as:

$$\boldsymbol{\varrho}(k+1) = \mathbf{A}\boldsymbol{\Gamma}(k) \cdot \mathbf{A}^T + \mathbf{H}\boldsymbol{R}\_{\mathcal{Q}} \cdot \mathbf{H}^T;\tag{44}$$

this completes the required relations for the optimal state estimation. Thus, **K***<sup>k</sup>* provides the minimum estimation errors, given a knowledge of the process noise magnitude (*R*), the measurement noise magnitude (*Rν*), and the covariance initial condition (*ϕ*(0)) [28].

### **4.4. Cost function**

16 Induction Motor

**4.2. The estimator based on a Luenberger Observer**

of the reduced-order estimators for Luenberger observer [27].

**4.3. The Estimator Based on a Kalman Filter**

written as follows:

these noises are defined as:

equations are:

The **X***<sup>a</sup>* part may be expressed as:

The dynamics of the unmeasured part of the state vector defined by Eq. (22) is described as:

where the last two terms are known and can be considered as an input for the **X***<sup>b</sup>* dynamics.

Note that Eq. (37) represents a relationship between a measured quantity on the left and the unknown state vector on the right. Assuming this, Eq. (36) can be rewritten as follows:

**<sup>X</sup>**<sup>ˆ</sup> *<sup>b</sup>*(*<sup>k</sup>* <sup>+</sup> <sup>1</sup>)=(**A**<sup>22</sup> <sup>−</sup> **<sup>K</sup>***l***A**12)**X**<sup>ˆ</sup> *<sup>b</sup>*(*k*) + **<sup>K</sup>***l***Y**(*<sup>k</sup>* <sup>+</sup> <sup>1</sup>)+

where **K***<sup>l</sup>* is the Luenberger gain matrix. Therefore, Eqs. (37) and (38) describe the dynamics

Considering uncorrelated process and measurement of Gaussian noises, Eq. (22) can be also

**<sup>X</sup>**ˆ(*<sup>k</sup>* <sup>+</sup> <sup>1</sup>|*k*) = **AX**(*k*) + **BU**(*k*) + **<sup>H</sup>**(*k*),

where **H** is the noise weight matrix, (*k*) is the noise matrix of the system model (process noise), and *ν*(*k*) is the matrix noise of measurement. The covariance matrices *R* and *R<sup>ν</sup>* of

where *E* {.} denotes the expected value. Thus, the dynamics of the reduced-order estimator

**<sup>X</sup>**<sup>ˆ</sup> *<sup>b</sup>*(*<sup>k</sup>* <sup>+</sup> <sup>1</sup>|*k*)=(**A**<sup>22</sup> <sup>−</sup> **<sup>K</sup>***k***A**12)**X**<sup>ˆ</sup> *<sup>b</sup>*(*k*) + **<sup>K</sup>***k***Y**(*<sup>k</sup>* <sup>+</sup> <sup>1</sup>) +

 · *<sup>T</sup>* ,

 *<sup>ν</sup>* · *<sup>ν</sup><sup>T</sup>* ,

*R* = *cov*() = *E*

*R<sup>ν</sup>* = *cov*(*ν*) = *E*

**X***b*(*k* + 1) = **A**22**X***b*(*k*) + **A**21**X***a*(*k*) + **B**2**U***αβs*, (36)

**X***a*(*k* + 1) − **A**11**X***a*(*k*) − **B**1**U***αβ<sup>s</sup>* = **A**12**X***b*(*k*). (37)

(**A**<sup>21</sup> − **K***l***A**11)**Y**(*k*)+(**B**<sup>2</sup> − **K***l***B**1)**U***αβs*(*k*), (38)

**<sup>Y</sup>**(*k*) = **CX**(*k*) + *<sup>ν</sup>*(*k*), (39)

(**A**<sup>21</sup> − **K***k***A**11)**Y**(*k*)+(**B**<sup>2</sup> − **K***k***B**1)**U***αβs*(*k*), (41)

(40)

The cost function should include all aspects to be optimized. In the current predictive control applied to the asymmetrical six-phase induction motor, the most important features to be optimized are the tracking errors of the stator currents in the (*α* − *β*) subspace for a next sampling time, since these variables are related to the electromechanical conversion. To minimize the prediction errors at each sampling time *k*, it is enough to utilize a simple term as:

$$J = \left\| \begin{array}{l} \mathbb{\hat{e}}\_{\text{ias}}(k+1|k) \parallel \text{\textquotedblleft} + \parallel \mathbb{\hat{e}}\_{\text{f\ $\textquotedblleft}}(k+1|k) \parallel \text{\textquotedblright} \end{array} \right\|^{2} \leftrightarrow \begin{cases} \mathbb{\hat{e}}\_{\text{ias}}(k+1|k) = \mathrm{i}\_{\text{as}}^{\*}(k+1) - \mathrm{i}\_{\text{as}}^{\*}(k+1|k),\\ \mathbb{\hat{e}}\_{\text{f\$ \textquotedblleft}}(k+1|k) = \mathrm{i}\_{\text{f\ $\textquotedblleft}}^{\*}(k+1) - \mathrm{i}\_{\text{f\$ \textquotedblleft}}^{\*}(k+1|k), \end{cases} \end{array} \right\} \tag{45}$$

where . denotes the vector modulus, *i* ∗ *<sup>s</sup>* is a vector containing the reference for the stator currents, and <sup>ˆ</sup>*is*(*<sup>k</sup>* <sup>+</sup> <sup>1</sup>|*k*) is the prediction of the stator currents calculated from measured and estimated states and the voltage vector *Uαβs*(*k*). Figure 10 (a) shows all projections of the stator current predictions calculated from the prediction model. The current control selects the control vector that minimizes the cost function at each sampling time. Figure 10 (b) shows the selection of the optimal vector based on a minimization of prediction errors.

More complicated cost functions can be devised, for instance, to minimize harmonic content, VSI switching losses, torque and flux, and/or active and reactive power. Also, in multiphase drives, stator current can be decomposed in subspaces in different ways. An appropriate decomposition allows to put more emphasis on harmonic reduction as will be shown in the case study for a six-phase motor drive [29,30]. The most relevant cost functions are shown in Table 3. The superscript (∗) denotes the reference value, and the terms involved in each cost function are detailed in Table 4.

(a) Projection of the stator current prediction in stationary reference frame (*α* − *β*) (b) Evaluation of the cost function (*J*) and selection of the optimal vector (*Sopt*)



**Table 3.** Possible cost functions in function to the controlled variables


**Table 4.** Description of the terms involved in each cost function of Table 3

### **4.5. Optimizer**

The optimization is done by exhaustive search over all possible realizations of the control actions. However, for electrical machines, some combinations of gating signals produce the same stator voltages, as shown in Figure 2. This means that, for prediction purposes, they are equivalent. This reduces the effective number of gating combinations to *<sup>ε</sup>* <sup>=</sup> <sup>2</sup>*<sup>φ</sup>* <sup>−</sup> *<sup>r</sup>*, *<sup>r</sup>* being the number of redundant configurations and *φ* the phase numbers of the machine. For the particular case of the SpIM, assuming the previous consideration, the search space for the optimal solution are 49 different vectors (48 active and 1 null). For a generic multiphase machine, the optimization algorithm produces the optimum gating signal combination (*Sopt*) using the estimator based on the state variables as follows:

**Algorithm 1** Optimization algorithm for state variable method

18 Induction Motor

*î*0 -0(*k+*1|*k*)

*i \** (*k+*1)

*î*1-1(*k+*1|*k*) *î*5-1(*k+*1|*k*)

*iα* [A]

Currents (*α*-*β*) and harmonic (*x* − *y*) ||*i*

Torque and flux |*T*<sup>∗</sup>

Currents (*α*-*β*) and voltage balance ||*i*

Currents (*α*-*β*) and VSI switching losses ||*i*

**Table 3.** Possible cost functions in function to the controlled variables Variable description

Active and reactive power |*Qin*| + |*P*<sup>∗</sup>

(a) Projection of the stator current prediction in

*î*5-5(*k+*1|*k*)

*î*6-4(*k+*1|*k*)

*i*(*k*)

*î*6-6(*k*+1|*k*) → *Sopt*(*k*+1|*k*)

*î*4 -5(*k+*1|*k*)

**Figure 10.** Minimization of tracking error in stator currents in stationary reference frame (*α* − *β*)

Controlled variables Cost functions (*J*)

*i<sup>α</sup>* Measured *α* current *i<sup>β</sup>* Measured *β* current *ix* Measured *x* current *iy* Measured *y* current *Qin* Reactive power *Pin* Active power *Te* Torque

*ψ<sup>s</sup>* Flux of the stator *λ* Weighting factor

*Ns* Number of switches

**Table 4.** Description of the terms involved in each cost function of Table 3

*î*4-4(*k+*1|*k*)

*iβ* [A]

> ∗ *<sup>α</sup>* − *iα*| + |*i*

> ∗ *<sup>α</sup>* − *iα*| + |*i*

> ∗ *<sup>α</sup>* − *iα*| + |*i*

*Vc*1, *Vc*<sup>2</sup> Voltages on each capacitor (VSI balanced)

The optimization is done by exhaustive search over all possible realizations of the control actions. However, for electrical machines, some combinations of gating signals produce the same stator voltages, as shown in Figure 2. This means that, for prediction purposes, they are equivalent. This reduces the effective number of gating combinations to *<sup>ε</sup>* <sup>=</sup> <sup>2</sup>*<sup>φ</sup>* <sup>−</sup> *<sup>r</sup>*, *<sup>r</sup>*

*êiβs*(*k+*1|*k*) *J*

*i \** (*k+*1)

optimal vector (*Sopt*)

∗

*in* − *Pin*|

∗

∗

*<sup>e</sup>* − *Te*| + *λ*||*ψ*<sup>∗</sup>

*<sup>β</sup>* <sup>−</sup> *<sup>i</sup>β*||<sup>2</sup> <sup>+</sup> *<sup>λ</sup>*||*<sup>i</sup>*

*<sup>s</sup>* |−|*ψs*||

*<sup>β</sup>* − *iβ*|| + *λNs*

*<sup>β</sup>* − *iβ*| + *λ*|*Vc*<sup>1</sup> − *Vc*2|

*êiαs*(*k+*1|*k*) *î*6-4(*k+*1|*k*)

∗ *<sup>y</sup>* − *iy*||

*i*(*k*)

*iα* [A]

∗ *<sup>x</sup>* − *ix*| + |*i*

(b) Evaluation of the cost function (*J*) and selection of the

*î*6-6(*k*+1|*k*) → *Sopt*(*k*+1|*k*)

*î*2-6(*k+*1|*k*)

*î*2-2(*k+*1|*k*)

*î*3 -2(*k+*1|*k*)

*î*3 -3(*k+*1|*k*)

**4.5. Optimizer**

*iβ* [A]

*î*1 -3(*k+*1|*k*)

stationary reference frame (*α* − *β*)

*Jo* := ∞, *i* := 1. **while** *i* ≤ *ε* **do S***<sup>i</sup>* ← **S***i*,*<sup>j</sup>* ∀ *j* = 1, ..., *φ*. **comment:** Compute stator voltages. Eq. (3). **comment:** Compute the prediction of the states. Eq. (22). **comment:** Compute the cost function. Eq. (45). **if** *J < Jo* **then** *Jo* <sup>←</sup> *<sup>J</sup>*, **<sup>S</sup>***opt* <sup>←</sup> **<sup>S</sup>***i*. **end if** *i* := *i* + 1. **end while**

Algorithms 2 and 3 show the pseudocode for the particular case of the proposed estimation methods, the Luenberger observer and Kalman filter, respectively.

**Algorithm 2** Proposed algorithm for Luenberger observer method

```
comment: Optimization algorithm.
Jo := ∞, i := 1
while i ≤ ε do
  Si ← Si,j ∀ j = 1, ..., φ
  Compute stator voltages. Eq. (3).
  Compute the prediction of the measurement states. Eqs. (36)-(37) assuming null initial
  conditions Xb(0) = 0.
  Compute the cost function. Eq. (45).
  if J < Jo then
     Jo ← J, Sopt ← Si
  end if
  i := i + 1
end while
Compute the prediction for Xˆ b(k + 1) by using Eq. (38).
```
### **5. Simulation results and discussion**

A MatLab/Simulink simulation environment has been designed to analyze the efficiency of the proposed reduced-order observer applied to the model-based predictive current control of the SpIM considering the electrical and mechanical parameters that are shown in Table 1. Numerical integration using fourth-order Runge-Kutta algorithm has been applied to compute the evolution of the state variables step by step in the time domain. A detailed block diagram of the proposed predictive current control technique is provided in Figure 9.

```
Algorithm 3 Proposed algorithm for Kalman Filter method
```
Compute the covariance matrix. Eq. (43). Compute the Kalman Filter gain matrix. Eq. (42). **comment:** Optimization algorithm. *Jo* := ∞, *i* := 1 **while** *i* ≤ *ε* **do S***<sup>i</sup>* ← **S***i*,*<sup>j</sup>* ∀ *j* = 1, ..., *f* Compute stator voltages. Eq. (3). Compute the prediction of the measurement state. Eq. (39). Compute the cost function. Eq. (45). **if** *J < Jo* **then** *Jo* <sup>←</sup> *<sup>J</sup>*, **<sup>S</sup>***opt* <sup>←</sup> **<sup>S</sup>***<sup>i</sup>* **end if** *i* := *i* + 1 **end while** Compute the correction for the covariance matrix. Eq. (44).

The reduced-order observer efficiency has been analyzed by performing parametric simulations considering a 10 kHz of sampling frequency and non-ideal conditions assuming that the control system has measurement (*Rv*) and process (*Rw*) noises. Figure 11 (a) (top) shows the obtained parametric simulation results for the particular case of the estimator based on the SV technique when are considered a constant frequency reference of 50 Hz with 15 A of reference current in stationary reference frame (*iαs*) and different levels of measurement and process noises (from 0 to 0.16) under varying load torque conditions (from 0 to 20 N·m). It can be seen in this figure the evolution of the MSE (measured between the reference and simulated currents) when the load torque and the measurement and process noises simultaneously increase. This behavior is associated with uncertainties in the estimation of the stator current due to the method based on the state variables. It can be observed that the MSE increases in direct proportion under varying load torque conditions (from 0.25 to 0.65 A). Figure 11 (a) (middle) shows the stator current tracking characteristic, where the following parameters are considered: *Rv* = *Rw* = 0.08 and *TL* = 10 N·m. The references and simulated and prediction currents are represented in red, black, and green colors, respectively. According to Table 5, under these operating conditions, the performance of the MBPC based on the SV method in terms of MSE*iα<sup>s</sup>* and THD*iα<sup>s</sup>* can be quantified in 0.69 A and 6 %, respectively. On the other hand, Figure 11 (a) (bottom) shows the rotor current evolution, calculated according to the SV methods under the same conditions described above.

Moreover, Figure 11 (b) (top) and Figure 11 (c) (top) show the parametric simulation of the MBPC technique for cases based on the LO and KF estimators, respectively. It can be noted in these graphs that the proposed MBPC methods based on the LO and KF estimators introduce improvements quantified with respect to the MSE, mainly when these control algorithms are compared with the MBPC method based on state variables. It can be concluded from these graphs that the MBPC based on the LO and KF estimators exhibits low sensitivity (in terms of MSE) to change of the load torque, and the performance is related with the measurement and process noise levels. Under the same test conditions considered above, the MSE measured between the reference and simulated currents in alpha axis are 0.47 A and 0.48 A for

**Figure 11.** Performance analysis considering a 15 A and 50 Hz of reference current

20 Induction Motor

*Jo* := ∞, *i* := 1 **while** *i* ≤ *ε* **do**

**if** *J < Jo* **then**

**end if** *i* := *i* + 1 **end while**

described above.

**Algorithm 3** Proposed algorithm for Kalman Filter method

Compute the prediction of the measurement state. Eq. (39).

Compute the correction for the covariance matrix. Eq. (44).

The reduced-order observer efficiency has been analyzed by performing parametric simulations considering a 10 kHz of sampling frequency and non-ideal conditions assuming that the control system has measurement (*Rv*) and process (*Rw*) noises. Figure 11 (a) (top) shows the obtained parametric simulation results for the particular case of the estimator based on the SV technique when are considered a constant frequency reference of 50 Hz with 15 A of reference current in stationary reference frame (*iαs*) and different levels of measurement and process noises (from 0 to 0.16) under varying load torque conditions (from 0 to 20 N·m). It can be seen in this figure the evolution of the MSE (measured between the reference and simulated currents) when the load torque and the measurement and process noises simultaneously increase. This behavior is associated with uncertainties in the estimation of the stator current due to the method based on the state variables. It can be observed that the MSE increases in direct proportion under varying load torque conditions (from 0.25 to 0.65 A). Figure 11 (a) (middle) shows the stator current tracking characteristic, where the following parameters are considered: *Rv* = *Rw* = 0.08 and *TL* = 10 N·m. The references and simulated and prediction currents are represented in red, black, and green colors, respectively. According to Table 5, under these operating conditions, the performance of the MBPC based on the SV method in terms of MSE*iα<sup>s</sup>* and THD*iα<sup>s</sup>* can be quantified in 0.69 A and 6 %, respectively. On the other hand, Figure 11 (a) (bottom) shows the rotor current evolution, calculated according to the SV methods under the same conditions

Moreover, Figure 11 (b) (top) and Figure 11 (c) (top) show the parametric simulation of the MBPC technique for cases based on the LO and KF estimators, respectively. It can be noted in these graphs that the proposed MBPC methods based on the LO and KF estimators introduce improvements quantified with respect to the MSE, mainly when these control algorithms are compared with the MBPC method based on state variables. It can be concluded from these graphs that the MBPC based on the LO and KF estimators exhibits low sensitivity (in terms of MSE) to change of the load torque, and the performance is related with the measurement and process noise levels. Under the same test conditions considered above, the MSE measured between the reference and simulated currents in alpha axis are 0.47 A and 0.48 A for

Compute the Kalman Filter gain matrix. Eq. (42).

Compute the covariance matrix. Eq. (43).

**comment:** Optimization algorithm.

Compute stator voltages. Eq. (3).

Compute the cost function. Eq. (45).

**S***<sup>i</sup>* ← **S***i*,*<sup>j</sup>* ∀ *j* = 1, ..., *f*

*Jo* <sup>←</sup> *<sup>J</sup>*, **<sup>S</sup>***opt* <sup>←</sup> **<sup>S</sup>***<sup>i</sup>*

**Figure 12.** Performance analysis considering a 15 A of reference current and *TL* = 0

Figure 11 (b) (middle) and Figure 11 (c) (middle), respectively. Figure 11 (b) (bottom) and Figure 11 (c) (bottom) show the rotor current estimated, for the cases based on the LO and KF estimators, respectively.

The performance of the MBPC based on reduced-order estimators has been evaluated considering a 15 A of reference current with no-load condition and different levels of measurement and process noises (from 0 to 0.16) under varying reference frequencies (from 30 to 50 Hz). Figure 12 (top) shows the performance analysis in terms of MSE, where it is possible to observe from the parametric simulation that the three control methods evaluated has low sensitivity to the frequency variation when are considered no-load conditions. It can be seen that the efficiency strongly depends on the measurement and process noise levels, as in the previous case. Figure 12 (middle) shows the stator current tracking characteristic, where the following parameters are considered: *Rv* = *Rw* = 0.12 and 45 Hz of reference frequency. Finally, Figure 12 (bottom) shows the rotor current estimated, for the case of study.


**Table 5.** Performance analysis

A similar analysis was performed for the case of beta current component, obtaining similar results as shown in Table 5. These simulation results substantiate the expected performance of the proposed algorithms based on reduced-order observers.

### **6. Conclusion**

In this chapter, an efficiency analysis of two reduced-order observers for rotor current estimator applied to the model-based predictive current control of the SpIM has been presented. The electrical and mechanical parameters of the SpIM have been measured and validated experimentally using an experimental setup. Real 15 kW SpIM parameters have been used to perform simulations using a MatLab/Simulink simulation environment. The simulation results obtained by different operation points under no-load and full-load conditions as well as different measurement and process noises have shown an increase in the efficiency of the proposed current control methods (based on the Kalman filter and Luenberger observer) measured with respect to the mean squared error of the stator currents in stationary reference frame, especially when they are compared with the control method based on state variables. Furthermore, the optimal estimator based on the Kalman filter achieves better performance than the Luenberger observer in terms of THD, mainly because it takes into account the effects of the noises in the control structure, recalculating the state feedback matrix at each sampling time recursively given the covariance of the new estimation as a function of the old covariance estimation. These results show that the experimental implementation of these control techniques are feasible and can be applied to the SpIM to increase the efficiency of the MBPC technique.

### **Acknowledgment**

22 Induction Motor

study.

**Table 5.** Performance analysis

**6. Conclusion**

The performance of the MBPC based on reduced-order estimators has been evaluated considering a 15 A of reference current with no-load condition and different levels of measurement and process noises (from 0 to 0.16) under varying reference frequencies (from 30 to 50 Hz). Figure 12 (top) shows the performance analysis in terms of MSE, where it is possible to observe from the parametric simulation that the three control methods evaluated has low sensitivity to the frequency variation when are considered no-load conditions. It can be seen that the efficiency strongly depends on the measurement and process noise levels, as in the previous case. Figure 12 (middle) shows the stator current tracking characteristic, where the following parameters are considered: *Rv* = *Rw* = 0.12 and 45 Hz of reference frequency. Finally, Figure 12 (bottom) shows the rotor current estimated, for the case of

> State variables 0.6965 0.6571 6.00% 6.06% Luenberger observer 0.4799 0.4971 4.39% 4.62% Kalman filter 0.4802 0.5084 4.29% 4.37%

> State variables 0.7798 0.7702 7.44% 7.17% Luenberger observer 0.5622 0.5951 5.13% 5.18% Kalman filter 0.5897 0.5208 5.10% 5.20%

A similar analysis was performed for the case of beta current component, obtaining similar results as shown in Table 5. These simulation results substantiate the expected performance

In this chapter, an efficiency analysis of two reduced-order observers for rotor current estimator applied to the model-based predictive current control of the SpIM has been presented. The electrical and mechanical parameters of the SpIM have been measured and validated experimentally using an experimental setup. Real 15 kW SpIM parameters have been used to perform simulations using a MatLab/Simulink simulation environment. The simulation results obtained by different operation points under no-load and full-load conditions as well as different measurement and process noises have shown an increase in the efficiency of the proposed current control methods (based on the Kalman filter and Luenberger observer) measured with respect to the mean squared error of the stator currents in stationary reference frame, especially when they are compared with the control method based on state variables. Furthermore, the optimal estimator based on the Kalman filter achieves better performance than the Luenberger observer in terms of THD, mainly because it takes into account the effects of the noises in the control structure, recalculating the state feedback matrix at each sampling time recursively given the covariance of the new estimation

of the proposed algorithms based on reduced-order observers.

Figure 11 analysis MSE*iα<sup>s</sup>* MSE*iβ<sup>s</sup>* THD*iα<sup>s</sup>* THD*iβ<sup>s</sup>*

Figure 12 analysis MSE*iα<sup>s</sup>* MSE*iβ<sup>s</sup>* THD*iα<sup>s</sup>* THD*iβ<sup>s</sup>* The authors would like to thank the Paraguayan Government for the economical support they provided by means of a CONACYT grant project 14-INV-101 – Desarrollo y análisis de eficiencia de nuevos algoritmos de control enfocados al generador hexafásico en aplicaciones de energía eólica. In addition, they wish to express their gratitude to the reviewers for their helpful comments and suggestions.

### **Author details**

Raúl Gregor1∗, Jorge Rodas1, Derlis Gregor2 and Federico Barrero<sup>3</sup>

\*Address all correspondence to: gregor.raul@gmail.com

1 Facultad de Ingeniería, Universidad Nacional de Asuncion, Department of Power and Control Systems, Asuncion, Paraguay

2 Facultad de Ingeniería, Universidad Nacional de Asuncion, Department of Computer Science, Asuncion, Paraguay

3 Escuela Superior de Ingenieros, Universidad de Sevilla, Department of Electronic Engineering, Sevilla, España

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## *Edited by Raul Igmar Gregor Recalde*

AC motors play a major role in modern industrial applications. Squirrel-cage induction motors (SCIMs) are probably the most frequently used when compared to other AC motors because of their low cost, ruggedness, and low maintenance. The material presented in this book is organized into four sections, covering the applications and structural properties of induction motors (IMs), fault detection and diagnostics, control strategies, and the more recently developed topology based on the multiphase (more than three phases) induction motors. This material should be of specific interest to engineers and researchers who are engaged in the modeling, design, and implementation of control algorithms applied to induction motors and, more generally, to readers broadly interested in nonlinear control, health condition monitoring, and fault diagnosis.

Photo by Nordroden / iStock

Induction Motors - Applications, Control and Fault Diagnostics

Induction Motors

Applications, Control and Fault Diagnostics

*Edited by Raul Igmar Gregor Recalde*