Detection of Stator and Rotor Asymmetries Faults in Wound Rotor Induction Machines: Modeling, Test and Real-Time Implementation

*Shahin Hedayati Kia*

## **Abstract**

This chapter deals with detection of stator and rotor asymmetries faults in wound rotor induction machines using rotor and stator currents signatures analysis. This is proposed as the experimental part of *fault diagnosis in electrical machines* course for master's degree students in electrical engineering at University of Picardie "Jules Verne". The aim is to demonstrate the main steps of real-time condition monitoring development for wound rotor induction machines. In this regard, the related parameters of classical model of wound rotor induction machine under study are initially estimated. Then, the latter model is validated through experiments in both healthy and faulty conditions at different levels of the load. Finally, an algorithm is implemented in a real-time data acquisition system for online detection of stator and rotor asymmetries faults. An experimental test bench based on a three-phase 90 W wound rotor induction machine and a real-time platform for hardware-in-the-loop test are utilized for validation of the proposed condition monitoring techniques.

**Keywords:** AC motor protection, asynchronous rotating machines, fault diagnosis, Fourier transform, hardware-in-the-loop, induction motors, monitoring, signal processing

## **1. Introduction**

Fault diagnosis of electrical machines is a very active topic of research and several books have been published, which detail new developed techniques for efficient condition monitoring of electrical machines. The run-to-break is an unplanned strategy of maintenance that needs to be avoided at the expense of high emergency repair cost. By means of preventive maintenance at regular intervals, which is commonly shorter than the expected time between failures, the maintenance actions can be planned in advance. Any potential breakdown in industrial systems can be predicted through the condition based maintenance (CBM) so called 'predictive maintenance' which gives a reasonable remaining useful life and leads consequently to the optimum time maintenance planning [1]. Since the electrical machines are the key components of the majority of industrial processes, it is essential to setup a CBM in order to minimize their downtime and consequently

increase their availability [2, 3]. Modeling and numerical simulations are the initial design stage of fault detection and diagnosis (FDD) systems [4]. For prototyping and testing both software-in-the-loop (S-i-L) and hardware-in-the-loop (H-i-L) realizations can be performed before the final stage of FDD system integration [4]. This leads to a better evaluation of FDD methods in all possible working condition scenarios which are sometimes hard to acquire in real practice using an experimental test bench. In this chapter, the illustration of these previous stages to Masters' degree students who attend to assimilate the ability of FDD technique development for electrical systems will be highlighted. The example of wound rotor induction machine (WRIM) is a good choice since WRIMs have been widely used in electrical power generation, particularly as doubly fed induction generators (DFIGs) in variable speed wind turbines. Moreover, the internal circuit parameters of a WRIM can be easily deduced using some basic experimental electrical circuit tests. The asymmetry fault in practice can be obtained by adding series resistance in one phase of stator and/or rotor winding which simplifies the evaluation of FDD methods through both numerical simulations and experiments. The state-of-the-art methods for FDD of asymmetries in WRIMs have been well detailed [5]. However, the implementation of FDD algorithms in real-time systems has been rarely investigated [6]. Recently, the H-i-L configuration is used for static eccentricity analysis in induction machines (IMs). However, the proposed model is exclusively validated using finite elements method (FEM). The real-time simulation results have been demonstrated the presence of fault-related frequency components in the stator current spectrum [3]. In this regard, introducing engineering students to FDD system design for electrical machines including its development stages is totally new in the literature [7–9]. The aim of this paper is to illustrate the main stages of FDD system design for the stator asymmetry fault (SAF) as well as the rotor asymmetry fault (RAF) in WRIMs. This is proposed as the experimental part of *fault diagnosis in electrical machines* course offered to the master's degree students in electrical engineering at University of Picardie "Jules Verne". Both stator and rotor windings asymmetries are investigated. Main emphasis is dedicated to signal-based techniques which are commonly used for detection of these specific defects. It is illustrated that the stator current is directly affected by the RAF whereas the SAF has a direct influence on the rotor current [5]. The fault diagnosis is commonly performed by computing the stator/rotor current Fourier transform to identify the fault-related frequency components in the spectrum in steady-state working condition. Once the validated WRIM model is implemented in a real-time platform for Hi-L test, the measured stator and rotor currents signals, provided by the real-time system, can be analyzed by the CompactRIO data acquisition system for evaluation of signal processing tools (SPTs) in all working condition scenarios of WRIM. An experimental test bench, based on a three-phase 90 W wound rotor induction machine and a real-time platform for H-i-L tests, are utilized for validation of the proposed condition monitoring techniques.

*λabcs*

*DOI: http://dx.doi.org/10.5772/intechopen.95236*

**r***<sup>s</sup>* ¼

*Detection of Stator and Rotor Asymmetries Faults in Wound Rotor Induction …*

**r***<sup>r</sup>* ¼

� 1 2

� 1 2

� 1 2

� 1 2

**L***<sup>s</sup>* ¼

**L***<sup>r</sup>* ¼

2 6 4

with

**37**

*<sup>λ</sup>abcr* � � <sup>¼</sup> **<sup>L</sup>***<sup>s</sup>* **<sup>L</sup>***sr* **L***T sr* **L***<sup>r</sup>* � � **i***abcs*

> 2 6 4

2 6 4

*Lls* <sup>þ</sup> *Lms* � <sup>1</sup>

*Lms* � <sup>1</sup>

*Lmr* � <sup>1</sup>

*Te* ¼ *iasLsr*f�*iar sin* ð Þ� *θ<sup>r</sup> ibr sin* ð Þ *θ<sup>r</sup>* þ 2*π=*3

*ibsLsr*f�*iar sin* ð Þ� *θ<sup>r</sup>* � 2*π=*3 *ibr sin* ð Þ *θ<sup>r</sup>*

*icsLsr*f�*iar sin* ð Þ� *θ<sup>r</sup>* þ 2*π=*3 *ibr sin* ð Þ *θ<sup>r</sup>* � 2*π=*3

*d*Ω*<sup>r</sup>*

*dθ<sup>r</sup>*

�*icr sin* ð Þg � *θ<sup>r</sup>* � 2*π=*3 *p*þ

�*icr sin* ð Þg � *θ<sup>r</sup>* þ 2*π=*3 *p*þ

*Te* � *Tl* ¼ *J*

Ω*<sup>r</sup>* ¼ 2 � *π* � *p* �

where *Lms*, *Lmr*, *Lls* and *Llr* are magnetizing and leakage stator and rotor inductances and *ras*, *rbs*, *rcs*, *rar*, *rbr* and *rcr* are stator and rotor phase resistances respectively. *Te* is the electromagnetic torque, *Tl* is the load torque, *J* is the total moment inertia, *f* is the viscous friction coefficient, *p* is the number of pole pairs, and *θ<sup>r</sup>* is the rotor angular speed. The estimation of WRIM model parameters, described by relations (1) and (2), is straightforward and can be performed through some basic electrical circuit tests. DC voltage–current experiments at rated working

�*icr sin* ð Þg � *θ<sup>r</sup> p*

*Llr* <sup>þ</sup> *Lmr* � <sup>1</sup>

*ras* 0 0 0 *rbs* 0 0 0 *rcs*

*rar* 0 0 0 *rbr* 0 0 0 *rcr*

2

2

2

*Lmr Lls* <sup>þ</sup> *Lmr* � <sup>1</sup>

2

**L***sr* ¼ *Lsr*� *cos*ð Þ *θ<sup>r</sup> cos*ð Þ *θ<sup>r</sup>* þ 2*π=*3 *cos*ð Þ *θ<sup>r</sup>* � 2*π=*3 *cos*ð Þ *θ<sup>r</sup>* � 2*π=*3 *cos*ð Þ *θ<sup>r</sup> cos*ð Þ *θ<sup>r</sup>* þ 2*π=*3 *cos*ð Þ *θ<sup>r</sup>* þ 2*π=*3 *cos*ð Þ *θ<sup>r</sup>* � 2*π=*3 *cos*ð Þ *θ<sup>r</sup>*

*Lms Lls* <sup>þ</sup> *Lms* � <sup>1</sup>

3 7

> 3 7

*Lms* � <sup>1</sup>

2 *Lms*

2 *Lms*

2 *Lmr*

2 *Lmr*

*Lmr Llr* þ *Lmr*

*Lms Lls* þ *Lms*

*Lmrs* � <sup>1</sup>

**<sup>i</sup>***abcr* � � (3)

<sup>5</sup> (4)

<sup>5</sup> (5)

(6)

(7)

(9)

> > 3 7

*dt* <sup>þ</sup> *<sup>f</sup>*Ω*<sup>r</sup>* (10)

*dt* (11)

<sup>5</sup> (8)

#### **2. Modeling of WRIM**

The model of WRIM in "*abc*" reference frame may be expressed as [10]:

$$\mathbf{v}\_{abcs} = \mathbf{r}\_s \mathbf{i}\_{abcs} + \frac{d}{dt} \lambda\_{abcs} \tag{1}$$

$$\mathbf{v}\_{abcr} = \mathbf{r}\_r \mathbf{i}\_{abcr} + \frac{d}{dt} \lambda\_{abcr} \tag{2}$$

*Detection of Stator and Rotor Asymmetries Faults in Wound Rotor Induction … DOI: http://dx.doi.org/10.5772/intechopen.95236*

$$
\begin{bmatrix}
\lambda\_{abcs} \\
\lambda\_{abcr}
\end{bmatrix} = \begin{bmatrix}
\mathbf{L}\_t & \mathbf{L}\_{sr} \\
\mathbf{L}\_{sr}^T & \mathbf{L}\_r
\end{bmatrix} \begin{bmatrix}
\mathbf{i}\_{abcs} \\
\mathbf{i}\_{abcr}
\end{bmatrix} \tag{3}
$$

$$\mathbf{r}\_s = \begin{bmatrix} r\_{ds} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & r\_{bs} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & r\_{cs} \end{bmatrix} \tag{4}$$

$$\mathbf{r}\_r = \begin{bmatrix} r\_{ar} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & r\_{br} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & r\_{cr} \end{bmatrix} \tag{5}$$

$$\mathbf{L}\_{s} = \begin{bmatrix} L\_{ls} + L\_{ms} & -\frac{1}{2}L\_{ms} & -\frac{1}{2}L\_{ms} \\ -\frac{1}{2}L\_{ms} & L\_{ls} + L\_{ms} & -\frac{1}{2}L\_{ms} \\ -\frac{1}{2}L\_{ms} & -\frac{1}{2}L\_{ms} & L\_{ls} + L\_{ms} \end{bmatrix} \tag{6}$$

$$\mathbf{L}\_{r} = \begin{bmatrix} \mathbf{L} & \mathbf{2} & \mathbf{2} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\\\ L\_{lr} + L\_{mr} & -\frac{1}{2}L\_{mrr} & -\frac{1}{2}L\_{mr} & \\ -\frac{1}{2}L\_{mr} & L\_{ls} + L\_{mr} & -\frac{1}{2}L\_{mr} & \\ -\frac{1}{2}L\_{mr} & -\frac{1}{2}L\_{mr} & L\_{lr} + L\_{mr} & \\ \end{bmatrix} \tag{7}$$

$$\mathbf{L}\_{tr} = L\_{tr} \times$$

$$\begin{bmatrix} \cos \left(\theta\_r \right) & \cos \left(\theta\_r + 2\pi/3 \right) & \cos \left(\theta\_r - 2\pi/3 \right) \\ \cos \left(\theta\_r - 2\pi/3 \right) & \cos \left(\theta\_r \right) & \cos \left(\theta\_r + 2\pi/3 \right) \\ \cos \left(\theta\_r + 2\pi/3 \right) & \cos \left(\theta\_r - 2\pi/3 \right) & \cos \left(\theta\_r \right) \end{bmatrix} \tag{8}$$

$$\mathbf{T} = \begin{bmatrix} \sin \theta\_r \sin \theta\_r \sin \left(\theta\_r \right) & \sin \left(\theta\_r + 2\pi/3 \right) \end{bmatrix}$$

$$\begin{aligned} T\_{\epsilon} &= i\_{at}L\_{sr} \{-i\_{ar}\sin\left(\theta\_{r}\right) - i\_{br}\sin\left(\theta\_{r} + 2\pi/3\right) \\ &- i\_{cr}\sin\left(\theta\_{r} - 2\pi/3\right)\} \times p + \\ &i\_{bs}L\_{sr} \{-i\_{ar}\sin\left(\theta\_{r} - 2\pi/3\right) - i\_{br}\sin\left(\theta\_{r}\right) \\ &- i\_{cr}\sin\left(\theta\_{r} + 2\pi/3\right)\} \times p + \\ &i\_{cs}L\_{sr} \{-i\_{ar}\sin\left(\theta\_{r} + 2\pi/3\right) - i\_{br}\sin\left(\theta\_{r} - 2\pi/3\right)\} \\ &- i\_{cr}\sin\left(\theta\_{r}\right)\} \times p \end{aligned} \tag{9}$$

$$T\_e - T\_l = f \frac{d\Omega\_r}{dt} + f\Omega\_r \tag{10}$$

with

increase their availability [2, 3]. Modeling and numerical simulations are the initial design stage of fault detection and diagnosis (FDD) systems [4]. For prototyping and testing both software-in-the-loop (S-i-L) and hardware-in-the-loop (H-i-L) realizations can be performed before the final stage of FDD system integration [4]. This leads to a better evaluation of FDD methods in all possible working condition scenarios which are sometimes hard to acquire in real practice using an experimental test bench. In this chapter, the illustration of these previous stages to Masters' degree students who attend to assimilate the ability of FDD technique development for electrical systems will be highlighted. The example of wound rotor induction machine (WRIM) is a good choice since WRIMs have been widely used in electrical power generation, particularly as doubly fed induction generators (DFIGs) in variable speed wind turbines. Moreover, the internal circuit parameters of a WRIM can be easily deduced using some basic experimental electrical circuit tests. The asymmetry fault in practice can be obtained by adding series resistance in one phase of stator and/or rotor winding which simplifies the evaluation of FDD methods through both numerical simulations and experiments. The state-of-the-art methods for FDD of asymmetries in WRIMs have been well detailed [5]. However, the implementation of FDD algorithms in real-time systems has been rarely investigated [6]. Recently, the H-i-L configuration is used for static eccentricity analysis in induction machines (IMs). However, the proposed model is exclusively validated using finite elements method (FEM). The real-time simulation results have been demonstrated the presence of fault-related frequency components in the stator current spectrum [3]. In this regard, introducing engineering students to FDD system design for electrical machines including its development stages is totally new in the literature [7–9]. The aim of this paper is to illustrate the main stages of FDD system design for the stator asymmetry fault (SAF) as well as the rotor asymmetry fault (RAF) in WRIMs. This is proposed as the experimental part of *fault diagnosis in electrical machines* course offered to the master's degree students in electrical engineering at University of Picardie "Jules Verne". Both stator and rotor windings asymmetries are investigated. Main emphasis is dedicated to signal-based techniques which are commonly used for detection of these specific defects. It is illustrated that the stator current is directly affected by the RAF whereas the SAF has a direct influence on the rotor current [5]. The fault diagnosis is commonly performed by computing the stator/rotor current Fourier transform to identify the fault-related frequency components in the spectrum in steady-state working condition. Once the validated WRIM model is implemented in a real-time platform for Hi-L test, the measured stator and rotor currents signals, provided by the real-time system, can be analyzed by the CompactRIO data acquisition system for evaluation of signal processing tools (SPTs) in all working condition scenarios of WRIM. An experimental test bench, based on a three-phase 90 W wound rotor induction machine and a real-time platform for H-i-L tests, are utilized for validation of the

*Emerging Electric Machines - Advances, Perspectives and Applications*

proposed condition monitoring techniques.

The model of WRIM in "*abc*" reference frame may be expressed as [10]:

*d*

*d*

*dt <sup>λ</sup>abcs* (1)

*dt <sup>λ</sup>abcr* (2)

**v***abcs* ¼ **r***s***i***abcs* þ

**v***abcr* ¼ **r***r***i***abcr* þ

**2. Modeling of WRIM**

**36**

$$
\Delta \mathbf{2}\_r = \mathbf{2} \times \boldsymbol{\pi} \times \ p \times \frac{d\theta\_r}{dt} \tag{11}
$$

where *Lms*, *Lmr*, *Lls* and *Llr* are magnetizing and leakage stator and rotor inductances and *ras*, *rbs*, *rcs*, *rar*, *rbr* and *rcr* are stator and rotor phase resistances respectively. *Te* is the electromagnetic torque, *Tl* is the load torque, *J* is the total moment inertia, *f* is the viscous friction coefficient, *p* is the number of pole pairs, and *θ<sup>r</sup>* is the rotor angular speed. The estimation of WRIM model parameters, described by relations (1) and (2), is straightforward and can be performed through some basic electrical circuit tests. DC voltage–current experiments at rated working temperature of WRIM give an initial estimation of both stator phase resistances *ras*, *rbs*, and *rcs* (*ras* ≈ *rbs* ≈*rcs*) and rotor phase resistances *rar*, *rbr* and *rcr* (*rar* ≈*rbr* ≈*rcr*) respectively. The obtained values are commonly good enough for arranging the model and for studing the asymmetry fault in WRIMs. Knowing these previous resistances, the respective stator-related self-inductances i.e. *Las* ≈ *Lbs* ≈*Lcs* ¼ *Lms* þ *Lls* and mutual inductances i.e. *Labs* ≈*Lacs* ≈*Lbcs* ¼ �0*:*5 � *Lms* can be obtained according to the relations (12)–(15). An AC voltage source is necessary for providing rated voltages to the stator phase windings as it is depicted in **Figure 1**.

$$L\_{at} = \sqrt{\frac{\left(\frac{V\_{ar}}{I\_{ar}}\right)^2 - r\_{as}^2}{o\_s^2}}\tag{12}$$

where *Var max* is the voltage peak value obtained across one phase of the rotor

For development of FDD techniques, it is crucial to validate experimentally the proposed model of WRIM in healthy working condition at different levels of the load in both time and frequency domains. Accordingly, the parameters of "*abc*" reference frame model for a WRIM with electrical characteristics, shown in

**Table 1**, are estimated using (12)–(20) and listed in **Table 2**. **Figure 2** illustrates the realization of the model in Matlab/Simulink software using trapezoidal integration

Power 90 W Voltage 380 V Stator current 0.27 A Rotor speed 1430 rpm Pole pairs 2 Torque 0.6 N.m Rotor inertia 0.001 Kg*:*m<sup>2</sup>

*Rs* 79.13 Ω *Rr* 3.69 Ω *Ls* 2.82 H *Lr* 0.23 H *Lms* 2.20 H *Lmr* 0.22 H *Lsr* 0.67 H

*Electrical and mechanical characteristics of three-phase 90W WRIM.*

*Estimated parameters of three-phase 90W WRIM "abc" reference frame model.*

*Realization of WRIM "abc" reference frame model in Matlab/Simulink.*

winding when the stator is supplied by a voltage source and its current is *Ias*.

*Detection of Stator and Rotor Asymmetries Faults in Wound Rotor Induction …*

**3. Healthy working condition**

*DOI: http://dx.doi.org/10.5772/intechopen.95236*

**Table 1.**

**Table 2.**

**Figure 2.**

**39**

$$L\_{cs} \approx L\_{bs} \approx L\_{as} \tag{13}$$

$$L\_{abs} = \frac{V\_{bs}}{I\_{ds}\alpha\_s} \tag{14}$$

$$L\_{bcs} \approx L\_{acs} \approx L\_{abs} \tag{15}$$

Similarly, the respective rotor-related self-inductances i.e. *Lar* ≈*Lbr* ≈ *Lcr* ¼ *Lmr* þ *Llr* and mutual inductances i.e. *Labr* ≈*Lacr* ≈*Lbcr* ¼ �0*:*5 � *Lmr* can be evaluated according to the relations (16)–(19).

$$L\_{ar} = \sqrt{\frac{\left(\frac{V\_{ar}}{I\_{ar}}\right)^2 - r\_{ar}^2}{a\_s^2}}\tag{16}$$

$$L\_{cr} \approx L\_{br} \approx L\_{ar} \tag{17}$$

$$L\_{abr} = \frac{V\_{br}}{I\_{ar}a o\_s} \tag{18}$$

$$L\_{bcr} \approx L\_{acr} \approx L\_{abr} \tag{19}$$

The stator-rotor mutual inductance *Lsr* can be determined using (20).

$$L\_{tr} = \frac{V\_{ar-max}}{I\_{as}\alpha\_s} \tag{20}$$

**Figure 1.** *Scheme of experiments for estimation of WRIM 'abc' reference frame model parameters.*

*Detection of Stator and Rotor Asymmetries Faults in Wound Rotor Induction … DOI: http://dx.doi.org/10.5772/intechopen.95236*

where *Var max* is the voltage peak value obtained across one phase of the rotor winding when the stator is supplied by a voltage source and its current is *Ias*.

## **3. Healthy working condition**

For development of FDD techniques, it is crucial to validate experimentally the proposed model of WRIM in healthy working condition at different levels of the load in both time and frequency domains. Accordingly, the parameters of "*abc*" reference frame model for a WRIM with electrical characteristics, shown in **Table 1**, are estimated using (12)–(20) and listed in **Table 2**. **Figure 2** illustrates the realization of the model in Matlab/Simulink software using trapezoidal integration


#### **Table 1.**

temperature of WRIM give an initial estimation of both stator phase resistances *ras*, *rbs*, and *rcs* (*ras* ≈ *rbs* ≈*rcs*) and rotor phase resistances *rar*, *rbr* and *rcr* (*rar* ≈*rbr* ≈*rcr*) respectively. The obtained values are commonly good enough for arranging the model and for studing the asymmetry fault in WRIMs. Knowing these previous resistances, the respective stator-related self-inductances i.e. *Las* ≈ *Lbs* ≈*Lcs* ¼ *Lms* þ *Lls* and mutual inductances i.e. *Labs* ≈*Lacs* ≈*Lbcs* ¼ �0*:*5 � *Lms* can be obtained according to the relations (12)–(15). An AC voltage source is necessary for provid-

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *Vas Ias* � �<sup>2</sup>

> > *ω*2 *s*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *Var Iar* � �<sup>2</sup>

> *ω*2 *s*

� *r*<sup>2</sup> *ar*

*Labs* <sup>¼</sup> *Vbs Iasω<sup>s</sup>*

Similarly, the respective rotor-related self-inductances i.e. *Lar* ≈*Lbr* ≈ *Lcr* ¼ *Lmr* þ *Llr* and mutual inductances i.e. *Labr* ≈*Lacr* ≈*Lbcr* ¼ �0*:*5 � *Lmr* can be evalu-

> *Labr* <sup>¼</sup> *Vbr Iarω<sup>s</sup>*

*Lsr* <sup>¼</sup> *Var*� *max Iasω<sup>s</sup>*

The stator-rotor mutual inductance *Lsr* can be determined using (20).

*Scheme of experiments for estimation of WRIM 'abc' reference frame model parameters.*

� *r*<sup>2</sup> *as*

vuut (12)

*Lcs* ≈*Lbs* ≈*Las* (13)

*Lbcs* ≈*Lacs* ≈*Labs* (15)

vuut (16)

*Lcr* ≈*Lbr* ≈*Lar* (17)

*Lbcr* ≈*Lacr* ≈*Labr* (19)

(14)

(18)

(20)

ing rated voltages to the stator phase windings as it is depicted in **Figure 1**.

*Las* ¼

*Emerging Electric Machines - Advances, Perspectives and Applications*

*Lar* ¼

ated according to the relations (16)–(19).

**Figure 1.**

**38**

*Electrical and mechanical characteristics of three-phase 90W WRIM.*


#### **Table 2.**

*Estimated parameters of three-phase 90W WRIM "abc" reference frame model.*

**Figure 2.** *Realization of WRIM "abc" reference frame model in Matlab/Simulink.*

method. Two discrete-time integrators which are closely linked to the relations (1), (2) and (11) are utilized. The model is initially validated through experiment in time domain at different levels of the load. **Figure 3** depicts the results of numerical simulation and experiment at rated load of WRIM. This simple approach gives a general idea of WRIM modeling to the students who are not familiar with this technique. Besides, it is helpful at this stage to localize the main frequency components in both stator and rotor phase currents spectra. *fs* is the main frequency component in the stator phase current spectrum whereas *f Ir* is the main frequency component in the rotor phase current spectrum.

$$f\_{lr} = f\_s - p\frac{\Omega\_r}{60} \tag{21}$$

**4. RAF detection**

*DOI: http://dx.doi.org/10.5772/intechopen.95236*

**Figure 5.**

**Figure 6.**

**41**

*(c), (d) experiment.*

*(d) experiment.*

It is well known that any deviation from the normal operation of WRIM, resulted from an internal or external anomalies, may induce fault signatures in the electrical variables such as stator and rotor currents. It was illustrated that the stator current is directly affected by the RAF whereas the SAF has a direct influence on the rotor current [5, 11]. The fault diagnosis is commonly carried out by computing the stator/ rotor current Fourier transform to locate fault frequency components in the spectrum. An addition resistance *RRAF* ¼ 1Ω is included in one of the rotor phases to create the RAF. **Figure 5** illustrates the numerical simulation and experimental results of the stator and rotor phase currents in time domain. As it can be observed, it is quite difficult to detect the RAF through time domain analysis, particularly for small values of *RRAF*. If the rotor speed of WRIM is considered constant, the following unique frequency component will appear in the stator phase current spectrum [12]:

*Detection of Stator and Rotor Asymmetries Faults in Wound Rotor Induction …*

where *s* is the slip value. The RAF frequency-related component is well localized in both numerical simulation and experiment spectra of the stator phase current at rated slip value of WRIM (**Figure 6**). Furthermore, the fact that the stator phase

*RAF condition stator and rotor phase currents of WRIM in time domain (a), (b) numerical simulation (c),*

*RAF condition stator and rotor phase currents of WRIM in frequency domain (a), (b) numerical simulation*

current is directly affected by the RAF is well depicted in this last figure.

*f RAF* ¼ ð Þ 1 � 2*s fs* (22)

where *p* is the pole pairs and Ω*<sup>r</sup>* is the rotor mechanical speed. The stator and rotor currents spectra of numerical simulation and experiment at rated slip *sr* ¼ 0*:*047 are shown in **Figure 4**. The rotor and stator asymmetries can be performed simply by including an additional series resistance in one of the rotor and stator phases. This technique is the simplest way to familiarize students with fault detection methods in WRIMs which will be highlighted in next sections.

**Figure 3.**

*Healthy condition stator and rotor phase currents of WRIM in time domain (a), (b) numerical simulation (c), (d) experiment.*

#### **Figure 4.**

*Healthy condition stator and rotor phase currents of WRIM in frequency domain (a), (b) numerical simulation (c), (d) experiment.*

*Detection of Stator and Rotor Asymmetries Faults in Wound Rotor Induction … DOI: http://dx.doi.org/10.5772/intechopen.95236*

## **4. RAF detection**

method. Two discrete-time integrators which are closely linked to the relations (1), (2) and (11) are utilized. The model is initially validated through experiment in time domain at different levels of the load. **Figure 3** depicts the results of numerical simulation and experiment at rated load of WRIM. This simple approach gives a general idea of WRIM modeling to the students who are not familiar with this technique. Besides, it is helpful at this stage to localize the main frequency components in both stator and rotor phase currents spectra. *fs* is the main frequency component in the stator phase current spectrum whereas *f Ir* is the main frequency

*f Ir* ¼ *fs* � *p*

tion methods in WRIMs which will be highlighted in next sections.

where *p* is the pole pairs and Ω*<sup>r</sup>* is the rotor mechanical speed. The stator and rotor currents spectra of numerical simulation and experiment at rated slip *sr* ¼ 0*:*047 are shown in **Figure 4**. The rotor and stator asymmetries can be performed simply by including an additional series resistance in one of the rotor and stator phases. This technique is the simplest way to familiarize students with fault detec-

*Healthy condition stator and rotor phase currents of WRIM in time domain (a), (b) numerical simulation (c),*

*Healthy condition stator and rotor phase currents of WRIM in frequency domain (a), (b) numerical simulation*

Ω*r*

<sup>60</sup> (21)

component in the rotor phase current spectrum.

*Emerging Electric Machines - Advances, Perspectives and Applications*

**Figure 3.**

**Figure 4.**

**40**

*(c), (d) experiment.*

*(d) experiment.*

It is well known that any deviation from the normal operation of WRIM, resulted from an internal or external anomalies, may induce fault signatures in the electrical variables such as stator and rotor currents. It was illustrated that the stator current is directly affected by the RAF whereas the SAF has a direct influence on the rotor current [5, 11]. The fault diagnosis is commonly carried out by computing the stator/ rotor current Fourier transform to locate fault frequency components in the spectrum. An addition resistance *RRAF* ¼ 1Ω is included in one of the rotor phases to create the RAF. **Figure 5** illustrates the numerical simulation and experimental results of the stator and rotor phase currents in time domain. As it can be observed, it is quite difficult to detect the RAF through time domain analysis, particularly for small values of *RRAF*. If the rotor speed of WRIM is considered constant, the following unique frequency component will appear in the stator phase current spectrum [12]:

$$f\_{RAF} = (\mathbf{1} - \mathbf{2}) f\_s \tag{22}$$

where *s* is the slip value. The RAF frequency-related component is well localized in both numerical simulation and experiment spectra of the stator phase current at rated slip value of WRIM (**Figure 6**). Furthermore, the fact that the stator phase current is directly affected by the RAF is well depicted in this last figure.

**Figure 5.**

*RAF condition stator and rotor phase currents of WRIM in time domain (a), (b) numerical simulation (c), (d) experiment.*

**Figure 6.**

*RAF condition stator and rotor phase currents of WRIM in frequency domain (a), (b) numerical simulation (c), (d) experiment.*

## **5. SAF detection**

The frequency components in the rotor phase currents due to the SAF can be obtained as [13]:

$$f\_{\text{SAF},k'} = \left\{ \frac{k'}{p} (\mathbf{1} - \mathbf{s}) \pm \mathbf{1} \right\} f\_s \tag{23}$$

**6. Real-time RAF and SAF detections**

*DOI: http://dx.doi.org/10.5772/intechopen.95236*

(**Figure 9**).

**Figure 9.**

**Figure 10.**

**43**

*Healthy condition H-i-L experimental results at rated load of WRIM.*

*Configuration of H-i-L test bench.*

The utilization of SPTs is the crucial stage of the RAF and the SAF detections in both steady-state and transient working conditions of WRIM. The developed methods can be classified in time, frequency and time-frequency/time-scale domains [2]. A brief review of the recent SPTs was mentioned in this topic of research [5]. Up to now, various experimental setups have been designed to evaluate the effectiveness of each SPT. They are mainly defined based upon the rated power of the installed electrical machine in the system. Furthermore, fault detection algorithms are commonly evaluated offline, whereas the new trends are mainly relied on the real-time FDD of electrical machines [6]. The concept of H-i-L is perfectly matched with such a development which is rarely studied [3]. In this regard, a real-time data acquisition system (CompactRIO data acquisition system) is used as a H-i-L with a high performance multi-core real-time platform in order to

*Detection of Stator and Rotor Asymmetries Faults in Wound Rotor Induction …*

analyze the performance of different kinds of SPTs in practical conditions

where *k*<sup>0</sup> ¼ 1, 2, 3, … . Taking only the fundamental frequency component into account with *<sup>k</sup>*<sup>0</sup> *<sup>p</sup>* ¼ 1, the relation (23) can be written as

*f SAF* ¼ ð Þ 2 � *s fs* (24)

An additional series resistance *RSAF* ¼ 10Ω is included in one of the stator phases to create the SAF. **Figure 7** illustrates the numerical simulation and experimental results of the stator and rotor phase currents at rated slip value of WRIM in time domain.

The SAF frequency-related component is well localized in both numerical simulation and experiment spectra of the rotor phase current at rated load of WRIM (**Figure 8**). Besides, it is well illustrated in **Figure 8**, where the rotor phase current is directly affected by the SAF [11].

**Figure 7.**

*SAF condition stator and rotor phase currents of WRIM in time domain (a), (b) numerical simulation (c), (d) experiment.*

#### **Figure 8.**

*SAF condition stator and rotor phase currents of WRIM in frequency domain (a), (b) numerical simulation (c), (d) experiment.*

*Detection of Stator and Rotor Asymmetries Faults in Wound Rotor Induction … DOI: http://dx.doi.org/10.5772/intechopen.95236*

## **6. Real-time RAF and SAF detections**

**5. SAF detection**

obtained as [13]:

account with *<sup>k</sup>*<sup>0</sup>

**Figure 7.**

*experiment.*

**Figure 8.**

**42**

*(c), (d) experiment.*

is directly affected by the SAF [11].

The frequency components in the rotor phase currents due to the SAF can be

where *k*<sup>0</sup> ¼ 1, 2, 3, … . Taking only the fundamental frequency component into

An additional series resistance *RSAF* ¼ 10Ω is included in one of the stator phases to create the SAF. **Figure 7** illustrates the numerical simulation and experimental results of the stator and rotor phase currents at rated slip value of WRIM in time domain. The SAF frequency-related component is well localized in both numerical simulation and experiment spectra of the rotor phase current at rated load of WRIM (**Figure 8**). Besides, it is well illustrated in **Figure 8**, where the rotor phase current

*SAF condition stator and rotor phase currents of WRIM in time domain (a), (b) numerical simulation (c), (d)*

*SAF condition stator and rotor phase currents of WRIM in frequency domain (a), (b) numerical simulation*

*<sup>p</sup>* ð Þ� <sup>1</sup> � *<sup>s</sup>* <sup>1</sup> 

*fs* (23)

*f SAF* ¼ ð Þ 2 � *s fs* (24)

*<sup>f</sup> SAF*,*k*<sup>0</sup> <sup>¼</sup> *<sup>k</sup>*<sup>0</sup>

*Emerging Electric Machines - Advances, Perspectives and Applications*

*<sup>p</sup>* ¼ 1, the relation (23) can be written as

The utilization of SPTs is the crucial stage of the RAF and the SAF detections in both steady-state and transient working conditions of WRIM. The developed methods can be classified in time, frequency and time-frequency/time-scale domains [2]. A brief review of the recent SPTs was mentioned in this topic of research [5]. Up to now, various experimental setups have been designed to evaluate the effectiveness of each SPT. They are mainly defined based upon the rated power of the installed electrical machine in the system. Furthermore, fault detection algorithms are commonly evaluated offline, whereas the new trends are mainly relied on the real-time FDD of electrical machines [6]. The concept of H-i-L is perfectly matched with such a development which is rarely studied [3]. In this regard, a real-time data acquisition system (CompactRIO data acquisition system) is used as a H-i-L with a high performance multi-core real-time platform in order to analyze the performance of different kinds of SPTs in practical conditions (**Figure 9**).

**Figure 9.** *Configuration of H-i-L test bench.*

**Figure 10.** *Healthy condition H-i-L experimental results at rated load of WRIM.*

This configuration is particularly attractive as it is totally independent of the type of the under study electrical machine and can be extended to any kind of fault for which an adapted model is well designed. Furthermore, there are more facilities to access the signatures which are commonly difficult to obtain without including high performance sensors in an experimental traditional test bench. The model of WRIM in "*abc*" reference frame, shown in **Figure 2**, is implemented in the real-time system with sampling time *Ts* <sup>¼</sup> <sup>10</sup>�4. The stator and rotor current signals, provided by multi I/O board of real-time system, are measured and analyzed by the CompactRIO data acquisition system at 5 kHz sampling frequency for 10s to detect the RAF and the SAF in steady-state working condition of WRIM.

frequency component 1ð Þ � 2*sr fs* (*sr* ¼ 0*:*047) is detected in the stator phase current spectrum at rated rotor speed of WRIM (**Figure 11**). The SAF reveals 2ð Þ � *sr fs* frequency component in the rotor phase current spectrum as it is shown in **Figure 12**. Besides, the electromagnetic torque is a good indicator of both the RAF and the SAF and can be used as an alternative signature for FDD design

*Detection of Stator and Rotor Asymmetries Faults in Wound Rotor Induction …*

This chapter presents for the first time the concept of H-i-L for fault diagnosis of WRIMs as a part of fault diagnosis of electrical machines course for master's degree students at University of Picardie "Jules Verne". The parameter of WRIM model in "*abc*" reference frame is estimated and validated through experiment at different levels of the load. The developed model is then implemented in a real-time system which is in the loop with a CompactRIO data acquisition platform. This configuration allows to evaluate the SPTs for real-time FDD design in all working conditions of WRIMs. Furthermore, this concept can be extended to condition monitoring of

any complex electromechanical system at development stage design.

MIS Lab. (EA4290), University of Picardie "Jules Verne", Amiens, France

© 2021 The Author(s). Licensee IntechOpen. 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,

\*Address all correspondence to: shdkia@u-picardie.fr

provided the original work is properly cited.

(**Figures 11** and **12**).

*DOI: http://dx.doi.org/10.5772/intechopen.95236*

**7. Conclusion**

**Abbreviations**

**Author details**

**45**

Shahin Hedayati Kia

RAF Rotor asymmetry fault SAF Stator asymmetry fault

IM Induction machine

SPT Signal processing tool FEM Finite element method S-i-L Software-in-the-loop H-i-L Hardware-in-the-loop

WRIM Wound rotor induction machine

DFIG Doubly fed induction generator FDD Fault detection and diagnosis CBM Condition based maintenance

The results of the analysis are illustrated in **Figures 10**–**12** for the healthy, the RAF and the SAF conditions respectively. The stator and the rotor currents in healthy condition at rated load of WRIM in both time and frequency domains are shown in **Figure 10**. As it would be expected, the main frequency components which are well identified in the spectra are *fs* and *f Ir* respectively. The fault-related

**Figure 11.** *RAF condition H-i-L experimental results at rated load of WRIM.*

**Figure 12.** *SAF condition H-i-L experimental results at rated load of WRIM.*

*Detection of Stator and Rotor Asymmetries Faults in Wound Rotor Induction … DOI: http://dx.doi.org/10.5772/intechopen.95236*

frequency component 1ð Þ � 2*sr fs* (*sr* ¼ 0*:*047) is detected in the stator phase current spectrum at rated rotor speed of WRIM (**Figure 11**). The SAF reveals 2ð Þ � *sr fs* frequency component in the rotor phase current spectrum as it is shown in **Figure 12**. Besides, the electromagnetic torque is a good indicator of both the RAF and the SAF and can be used as an alternative signature for FDD design (**Figures 11** and **12**).

## **7. Conclusion**

This configuration is particularly attractive as it is totally independent of the type of the under study electrical machine and can be extended to any kind of fault for which an adapted model is well designed. Furthermore, there are more facilities to access the signatures which are commonly difficult to obtain without including high performance sensors in an experimental traditional test bench. The model of WRIM in "*abc*" reference frame, shown in **Figure 2**, is implemented in the real-time system with sampling time *Ts* <sup>¼</sup> <sup>10</sup>�4. The stator and rotor current signals, provided by multi I/O board of real-time system, are measured and analyzed by the CompactRIO data acquisition system at 5 kHz sampling frequency for 10s to detect

The results of the analysis are illustrated in **Figures 10**–**12** for the healthy, the RAF and the SAF conditions respectively. The stator and the rotor currents in healthy condition at rated load of WRIM in both time and frequency domains are shown in **Figure 10**. As it would be expected, the main frequency components which are well identified in the spectra are *fs* and *f Ir* respectively. The fault-related

the RAF and the SAF in steady-state working condition of WRIM.

*Emerging Electric Machines - Advances, Perspectives and Applications*

*RAF condition H-i-L experimental results at rated load of WRIM.*

*SAF condition H-i-L experimental results at rated load of WRIM.*

**Figure 11.**

**Figure 12.**

**44**

This chapter presents for the first time the concept of H-i-L for fault diagnosis of WRIMs as a part of fault diagnosis of electrical machines course for master's degree students at University of Picardie "Jules Verne". The parameter of WRIM model in "*abc*" reference frame is estimated and validated through experiment at different levels of the load. The developed model is then implemented in a real-time system which is in the loop with a CompactRIO data acquisition platform. This configuration allows to evaluate the SPTs for real-time FDD design in all working conditions of WRIMs. Furthermore, this concept can be extended to condition monitoring of any complex electromechanical system at development stage design.

## **Abbreviations**


## **Author details**

Shahin Hedayati Kia MIS Lab. (EA4290), University of Picardie "Jules Verne", Amiens, France

\*Address all correspondence to: shdkia@u-picardie.fr

© 2021 The Author(s). Licensee IntechOpen. 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.

## **References**

[1] Randall RB. Vibration-Based Condition Monitoring: Industrial, Aerospace and Automotive Applications. Wiley; 2011.

[2] Frosini L. Monitoring and diagnostics of electrical machines and drives: A state of the art. In: 2019 IEEE workshop on electrical machines design, Control and Diagnosis (WEMDCD). vol. 1; 2019. p. 169–176.

[3] Sapena-Bano A, Riera-Guasp M, Martinez-Roman J, Pineda-Sanchez M, Puche-Panadero R, Perez-Cruz J. FEManalytical hybrid model for real time simulation of IMs under static eccentricity fault. In: 2019 IEEE 12th International Symposium on Diagnostics for Electrical Machines, Power Electronics and Drives (SDEMPED); 2019. p. 108–114.

[4] Isermann R. Fault diagnosis systems an introduction from fault detection to fault tolerance. SERBIULA (sistema Librum 20). 2006 01.

[5] Kia SH. Monitoring of wound rotor induction machines by means of discrete wavelet transform. Electric Power Components and Systems. 2018; 46(19–20):2021–2035.

[6] Monmasson E, Idkhajine L, Cirstea MN, Bahri I, Tisan A, Naouar MW. FPGAs in industrial control applications. IEEE Transactions on Industrial Informatics. 2011;7(2): 224–243.

[7] Gómez-de-Gabriel JM, Mandow A, Fernández-Lozano J, García-Cerezo A. Mobile robot lab project to introduce engineering students to fault diagnosis in mechatronic systems. IEEE Transactions on Education. 2015 Aug;58(3):187–193.

[8] Record P. Teaching the art of fault diagnosis in electronics by a virtual learning environment. IEEE

Transactions on Education. 2005 Aug; 48(3):375–381.

[9] Pagiatakis G, Dritsas L, Chatzarakis G, Todorov G, Stoev B. Introducing concepts and methodologies of fault detection into electrical engineering education: The induction machine example. In: 2017 IEEE Global Engineering Education Conference (EDUCON); 2017. p. 381–388.

[10] Krause P, Wasynczuk O, Sudhoff SD, Pekarek S. In: Induction motor drives. IEEE; 2013. Available from: https://ieeexplore.ieee.org/ document/6739387.

[11] Penman J, Sedding HG, Lloyd BA, Fink WT. Detection and location of interturn short circuits in the stator windings of operating motors. IEEE Transactions on Energy Conversion. 1994;9(4):652–658.

[12] Filippetti F, Franceschini G, Tassoni C, Vas P. AI techniques in induction machines diagnosis including the speed ripple effect. IEEE Transactions on Industry Applications. 1998;34(1):98–108.

[13] Shah D, Nandi S, Neti P. Stator inter-turn fault detection of doubly-fed induction generators using rotor current and search coil voltage signature analysis. In: 2007 IEEE Industry Applications Annual Meeting; 2007. p. 1948–1953.

**47**

**Chapter 4**

**Abstract**

Prospects for Increasing

the Dynamic Efficiency of

Asynchronous Double-Feed

Generators Using Structural

The chapter proposes to consider the problems of control of asynchronous machines with dual power supply, as a nonlinear structure, the transfer functions of which depend on the frequency of the stator voltage and the relative slip. The authors cite the results of research confirming the high efficiency of control of asynchronous electric motors, using cross-dynamic connections on the developed torque or a signal close to it (active component of the motor stator current). The proposed correction operates in a wide range of changes in the rotation and sliding speeds of the asynchronous electric generator. This is especially important for wind turbines that need to remain efficient at different speeds. As a justification, the results of experiments, modeling and industrial application of control algorithms with positive torque coupling are presented. Research results suggest that such

**Keywords:** asynchronous called double-feed machines (DFM), frequency regulation, dynamic positive feedbacks, active stator current, rotor current, signal spectrum,

The squirrel cage induction motors (SCIM), widely used in industry and power

engineering, are distinguished primarily by their high reliability and low cost. Asynchronous electric drive (AED) of mechanisms in which they are used, as a rule, do not require a significant range of speed and torque control, high control accuracy and fast transients. Even in drives with rather expensive frequency and voltage converters (FC), it is not easy to solve the problems of SCIM control. Research of

algorithms will improve the efficiency of wind power by 5–10%.

control systems of such drives continues at the present time.

Machines and Wind Power

Methods and Solutions

*Vladimir L. Kodkin, Alexandr S. Anikin* 

*and Alexandr A. Baldenkov*

parrying of step and harmonic moments

**1. Introduction**

## **Chapter 4**

**References**

p. 169–176.

[1] Randall RB. Vibration-Based Condition Monitoring: Industrial, Aerospace and Automotive Applications. Wiley; 2011.

[2] Frosini L. Monitoring and diagnostics of electrical machines and drives: A state of the art. In: 2019 IEEE workshop on electrical machines design, Control and Diagnosis (WEMDCD). vol. 1; 2019.

*Emerging Electric Machines - Advances, Perspectives and Applications*

Transactions on Education. 2005 Aug;

Chatzarakis G, Todorov G, Stoev B. Introducing concepts and methodologies

of fault detection into electrical engineering education: The induction machine example. In: 2017 IEEE Global Engineering Education Conference (EDUCON); 2017. p. 381–388.

[10] Krause P, Wasynczuk O,

document/6739387.

1994;9(4):652–658.

1998;34(1):98–108.

p. 1948–1953.

Sudhoff SD, Pekarek S. In: Induction motor drives. IEEE; 2013. Available from: https://ieeexplore.ieee.org/

[11] Penman J, Sedding HG, Lloyd BA, Fink WT. Detection and location of interturn short circuits in the stator windings of operating motors. IEEE Transactions on Energy Conversion.

[12] Filippetti F, Franceschini G, Tassoni C, Vas P. AI techniques in induction machines diagnosis including

Transactions on Industry Applications.

[13] Shah D, Nandi S, Neti P. Stator inter-turn fault detection of doubly-fed induction generators using rotor current

and search coil voltage signature analysis. In: 2007 IEEE Industry Applications Annual Meeting; 2007.

the speed ripple effect. IEEE

48(3):375–381.

[9] Pagiatakis G, Dritsas L,

[3] Sapena-Bano A, Riera-Guasp M, Martinez-Roman J, Pineda-Sanchez M, Puche-Panadero R, Perez-Cruz J. FEManalytical hybrid model for real time simulation of IMs under static eccentricity fault. In: 2019 IEEE 12th

[4] Isermann R. Fault diagnosis systems an introduction from fault detection to fault tolerance. SERBIULA (sistema

[5] Kia SH. Monitoring of wound rotor induction machines by means of discrete wavelet transform. Electric Power Components and Systems. 2018;

International Symposium on Diagnostics for Electrical Machines, Power Electronics and Drives (SDEMPED); 2019. p. 108–114.

Librum 20). 2006 01.

46(19–20):2021–2035.

224–243.

**46**

[6] Monmasson E, Idkhajine L, Cirstea MN, Bahri I, Tisan A, Naouar MW. FPGAs in industrial control applications. IEEE Transactions on Industrial Informatics. 2011;7(2):

[7] Gómez-de-Gabriel JM, Mandow A, Fernández-Lozano J, García-Cerezo A. Mobile robot lab project to introduce engineering students to fault diagnosis in mechatronic systems. IEEE Transactions on Education. 2015 Aug;58(3):187–193.

[8] Record P. Teaching the art of fault diagnosis in electronics by a virtual learning environment. IEEE

Prospects for Increasing the Dynamic Efficiency of Asynchronous Double-Feed Machines and Wind Power Generators Using Structural Methods and Solutions

*Vladimir L. Kodkin, Alexandr S. Anikin and Alexandr A. Baldenkov*

## **Abstract**

The chapter proposes to consider the problems of control of asynchronous machines with dual power supply, as a nonlinear structure, the transfer functions of which depend on the frequency of the stator voltage and the relative slip. The authors cite the results of research confirming the high efficiency of control of asynchronous electric motors, using cross-dynamic connections on the developed torque or a signal close to it (active component of the motor stator current). The proposed correction operates in a wide range of changes in the rotation and sliding speeds of the asynchronous electric generator. This is especially important for wind turbines that need to remain efficient at different speeds. As a justification, the results of experiments, modeling and industrial application of control algorithms with positive torque coupling are presented. Research results suggest that such algorithms will improve the efficiency of wind power by 5–10%.

**Keywords:** asynchronous called double-feed machines (DFM), frequency regulation, dynamic positive feedbacks, active stator current, rotor current, signal spectrum, parrying of step and harmonic moments

### **1. Introduction**

The squirrel cage induction motors (SCIM), widely used in industry and power engineering, are distinguished primarily by their high reliability and low cost. Asynchronous electric drive (AED) of mechanisms in which they are used, as a rule, do not require a significant range of speed and torque control, high control accuracy and fast transients. Even in drives with rather expensive frequency and voltage converters (FC), it is not easy to solve the problems of SCIM control. Research of control systems of such drives continues at the present time.

At the same time, in a number of units, wound rotor induction motors (WRIM) are widely used. The design of these motors allows connecting additional active resistances to the rotor and adjusting the ratio of active and reactive power. In this case, the stator and rotor currents, the rotation speed and the developed torque change at a constant rotation speed of the magnetic field. At the same time, the design of the engine becomes somewhat more complicated, and accordingly, its cost increases (slightly). It was this ability to regulate speed and torque that made WRIM the main electric motors in a number of mechanisms in the 60s and 70s before the widespread introduction of available FC. In a number of countries (for example, in Russia and the CIS), such drives are still used in hoisting and transport mechanisms. At the same time, the problems with the dynamics of the drive remain, in general, the same as for the SCIM.

The ability to adjust the WRIM operating mode from the rotor side not only in motor, but also in generator modes ensured the use of WRIM in the power industry, in those generating sets in which the rotation speed of driving machines cannot be sufficiently stable. At the same time, asynchronous machines with a phase rotor were called double-feed machines (DFM). Since the water flow rate in hydropower generators cannot be constant and it cannot be controlled by mechanical means with an accuracy of more than 1%, a voltage source is included in the rotor of the machine, which corrects the voltage parameters on the stator of the machine connected to the power grid.

In recent years, asynchronous dual-feed machines have become widely used in wind energy, which over the past few decades has emerged in a number of countries in a separate energy sector that successfully competes with traditional energy sources. DFM in wind turbines allow to generate electricity with the required parameters at different wind speeds, supplying energy directly to the network through the stator windings (**Figure 1**) [1].

At the same time, the problems with the regulation of the DFM, as an asynchronous electric machine, are fully manifested. They play a significant role in reducing the efficiency of wind turbines. A whole range of problems should be noted.

**Problem analysis**. The generally accepted mathematics for describing processes in AC electric machines plays a significant role. When describing the work of DFM [1, 2], vector equations and dependencies, traditional for AC machines, are used with a large number of assumptions and simplifications. One of the main ones is the neglect of the components of higher harmonics in the rotor and stator currents and voltages in the DFM. The DFM equations, like the equations describing all asynchronous and synchronous motors, do not take into account changes in the

**49**

**Figure 2.**

*vector control.*

*Prospects for Increasing the Dynamic Efficiency of Asynchronous Double-Feed Machines…*

frequency of voltages and currents at all. Naturally, these vector equations take into

*<sup>d</sup> U Ri j dt*

 Ψ = + +Ψ

 Ψ= + Ψ= +

Ψ

*<sup>d</sup> U Ri j p dt*

Eq. (1) are obtained from the equations of an alternating current electric machine under the assumption that the processes of changing currents and voltages in the DFM are sinusoidal signals of constant frequency. If we assume the change in this frequency, which occurs when regulating the speed and torque of the DFM, the original equations become so complicated that it will be impossible to analyze them

allow to reliably describe these processes and suggest effective correction.

=+ + −Ψ

2 2 22 2 1 11 2 2 22 1

( )

; ;  ω

;

;

(1)

*k*

ω

*m m*

*Li L i Li L i*

where U1, U2 – stator and rotor voltage vectors; Ψ1, Ψ2, i1, i2 – vectors of flux linkages of stator and rotor currents; R1, L1, R2, L2 – active resistances and inductance of stator and rotor; Lm – main inductance of the magnetizing circuit; ωk – angular velocity of rotation of the coordinate system; ω – angular speed of rotation of the

These methods of describing asynchronous electric drives with vector equations lead to a number of limitations in control devices. For example, an increase in the stator magnetic flux (ratio U\f) leads to a violation of stability and an increase in stator currents at low loads [3–6], therefore control algorithms in many inverters limit this parameter, reducing the possibility of accelerating transient processes. Another problem is the dependence of the drive dynamics on the stator voltage frequency [7–10]. Vector equations describing asynchronous electric drives do not

**Figure 2** shows the results of experiments with an asynchronous drive with vector control, closed by a speed signal with surges of a torque load. The processes of

*Diagrams of the speed change when parrying a stepped torque at different speeds of rotation in a drive with* 

1 1 11 1

*k*

ω

*DOI: http://dx.doi.org/10.5772/intechopen.96523*

account transient processes in a very simplified way [2]: Consider an example of such a description [1]:

rotor; *p* – number of pole pairs of the machine.

and select an effective correction based on them.

**Figure 1.** *Connection diagram DFM.*

*Prospects for Increasing the Dynamic Efficiency of Asynchronous Double-Feed Machines… DOI: http://dx.doi.org/10.5772/intechopen.96523*

frequency of voltages and currents at all. Naturally, these vector equations take into account transient processes in a very simplified way [2]:

Consider an example of such a description [1]:

*Emerging Electric Machines - Advances, Perspectives and Applications*

general, the same as for the SCIM.

nected to the power grid.

through the stator windings (**Figure 1**) [1].

At the same time, in a number of units, wound rotor induction motors (WRIM) are widely used. The design of these motors allows connecting additional active resistances to the rotor and adjusting the ratio of active and reactive power. In this case, the stator and rotor currents, the rotation speed and the developed torque change at a constant rotation speed of the magnetic field. At the same time, the design of the engine becomes somewhat more complicated, and accordingly, its cost increases (slightly). It was this ability to regulate speed and torque that made WRIM the main electric motors in a number of mechanisms in the 60s and 70s before the widespread introduction of available FC. In a number of countries (for example, in Russia and the CIS), such drives are still used in hoisting and transport mechanisms. At the same time, the problems with the dynamics of the drive remain, in

The ability to adjust the WRIM operating mode from the rotor side not only in motor, but also in generator modes ensured the use of WRIM in the power industry, in those generating sets in which the rotation speed of driving machines cannot be sufficiently stable. At the same time, asynchronous machines with a phase rotor were called double-feed machines (DFM). Since the water flow rate in hydropower generators cannot be constant and it cannot be controlled by mechanical means with an accuracy of more than 1%, a voltage source is included in the rotor of the machine, which corrects the voltage parameters on the stator of the machine con-

In recent years, asynchronous dual-feed machines have become widely used in wind energy, which over the past few decades has emerged in a number of countries in a separate energy sector that successfully competes with traditional energy sources. DFM in wind turbines allow to generate electricity with the required parameters at different wind speeds, supplying energy directly to the network

At the same time, the problems with the regulation of the DFM, as an asynchronous electric machine, are fully manifested. They play a significant role in reducing the efficiency of wind turbines. A whole range of problems should be noted.

**Problem analysis**. The generally accepted mathematics for describing processes in AC electric machines plays a significant role. When describing the work of DFM [1, 2], vector equations and dependencies, traditional for AC machines, are used with a large number of assumptions and simplifications. One of the main ones is the neglect of the components of higher harmonics in the rotor and stator currents and voltages in the DFM. The DFM equations, like the equations describing all asynchronous and synchronous motors, do not take into account changes in the

**48**

**Figure 1.**

*Connection diagram DFM.*

$$U\_1 = R\_1 i\_1 + \frac{d\Psi\_1}{dt} + j o\_k \Psi\_1;$$

$$\begin{cases} U\_2 = R\_2 i\_2 + \frac{d\Psi\_2}{dt} + j \left( o\_k - p o \right) \Psi\_2; \\ \Psi\_1 = L\_1 i\_1 + L\_m i\_2; \\ \Psi\_2 = L\_2 i\_2 + L\_m i\_1; \end{cases} \tag{1}$$

where U1, U2 – stator and rotor voltage vectors; Ψ1, Ψ2, i1, i2 – vectors of flux linkages of stator and rotor currents; R1, L1, R2, L2 – active resistances and inductance of stator and rotor; Lm – main inductance of the magnetizing circuit; ωk – angular velocity of rotation of the coordinate system; ω – angular speed of rotation of the rotor; *p* – number of pole pairs of the machine.

Eq. (1) are obtained from the equations of an alternating current electric machine under the assumption that the processes of changing currents and voltages in the DFM are sinusoidal signals of constant frequency. If we assume the change in this frequency, which occurs when regulating the speed and torque of the DFM, the original equations become so complicated that it will be impossible to analyze them and select an effective correction based on them.

These methods of describing asynchronous electric drives with vector equations lead to a number of limitations in control devices. For example, an increase in the stator magnetic flux (ratio U\f) leads to a violation of stability and an increase in stator currents at low loads [3–6], therefore control algorithms in many inverters limit this parameter, reducing the possibility of accelerating transient processes.

Another problem is the dependence of the drive dynamics on the stator voltage frequency [7–10]. Vector equations describing asynchronous electric drives do not allow to reliably describe these processes and suggest effective correction.

**Figure 2** shows the results of experiments with an asynchronous drive with vector control, closed by a speed signal with surges of a torque load. The processes of

#### **Figure 2.**

*Diagrams of the speed change when parrying a stepped torque at different speeds of rotation in a drive with vector control.*

parrying a torque load are different at different speeds of rotation and frequencies of stator voltage and are strongly "tightened". At even higher speeds, the stability of the drive [7–9, 11, 12] is impaired.

It should be noted that the traditional means and methods of regulation of asynchronous electric drives with frequency control (PID speed or torque controllers) with vector control or direct torque control work very poorly precisely when parrying "step" or harmonic torque loads. This is shown in articles [7–10, 13].

In DFM wind turbines, the wind works exactly as a moment load. In this case, the wind parameters are not stationary and difficult to predict.

There are many works devoted to the study of the parameters of wind flows, one of the time dependences is shown in **Figure 3**. As a rule, several ranges of speed variation can be distinguished in the wind speed - slow and faster.

The frequency of rapid variations in wind speed is from to too high for its effective "tracking" by powerful and large-sized electric drives of wind turbines [14–17]. Electric generators and wind turbines in general have significant inertia – tens and hundreds of seconds due to their very large dimensions.

The control systems of these installations should work as follows. DFM, as a generator, must convert mechanical wind energy, determined by the average wind speed, into electrical energy. And, as an electric drive, it must correct and smooth out wind gusts so that they do not "distort" the frequency and amplitude of the stator voltage. To solve this problem in the DFM must be the inverter connected to the rotor circuit. It can adjust the frequency and amplitude of the stator voltage precisely. At the same time, a change in the voltage in the rotor has a complex effect on the DFM, causing "its" transient processes.

In DFM, most often, rather traditional algorithms for asynchronous electric drives are used to optimize stationary modes - depending on the wind speed, the structure of the drive changes - the torque or speed of the motor is controlled by loops with a PID controller. At the same time, during the change in the wind speed, the restoration of the parameters of the DFM operation mode - the torque, the speed of rotation and, accordingly, the frequency of the stator voltage - occurs rather slowly, and in order not to violate the compliance with the requirements of the parameters of the stator voltage – frequency and amplitude, most often,

**51**

**Figure 4.**

*Prospects for Increasing the Dynamic Efficiency of Asynchronous Double-Feed Machines…*

energy transfer is blocked and the entire wind turbine is stopped. That is, the use of optimization methods is limited to modes of uniform power generation, i.e. they can be used only when the wind turbine is in a stationary state, with small and slow changes in the parameters of wind flows. In case of transient processes or in emer-

The reason that DFMs are disconnected from the grid during transient modes is the inability of the regulation system to track these changes in wind speed with minimal transients. The shutdown and protection devices receive considerable attention from developers and researchers. At the same time, the purpose of the protection devices is to preserve the operability of the equipment with "non-mode" parameters of the wind and power grid, and to reduce the time of inoperability of wind power units. But in all these devices, little attention is paid to the dynamics of installations in operating states. The modes and settings of the regulators remain standard, which means that they are quite ineffective in terms of dynamics. All this also reduces the efficiency of wind turbines to critically low values. At the same time, the "dynamic potential" of asynchronous electric drives is far from being exhausted and is not used in most drives because most often it is not required. But

The authors carried out research on the dynamics of asynchronous electric drives with frequency control, which are of undoubted interest for wind power. The research consisted of the development of theoretical provisions, bench experi-

Vector equations were replaced by continuous ones in a certain area of the multidimensional space formed by variable coordinates of the electric drive - rotation speed and mechanical moment and independent functions - stator voltage frequency and relative slip. As a result, a nonlinear transfer function was obtained that connects the developed mechanical torque and absolute slip - the difference between the stator voltage frequency and the engine speed. The formula for this function includes, as variables, the frequency of the stator voltage and the relative slip. The formula can be called the nonlinear transfer function or the dynamic Kloss formula. In articles [18–26] the conclusion of the proposed nonlinear transfer func-

> ( ) ( ) ( )

1 2

1

*Block diagram of the working section of the mechanical characteristics of an alternating current machine.*

*M Tp S W p*

ω

*k k*

2 1

 *Tp S* ′ <sup>+</sup> <sup>=</sup> +′ + 2

*k*

2 2 2

 β (2)

ments, simulation and application of industrial units in electric drives.

tion is given in sufficient detail, the result is as follows:

*DOI: http://dx.doi.org/10.5772/intechopen.96523*

not in the case of wind turbines.

**2. Theoretical provisions**

gency situations, the methods are not applicable.

**Figure 3.** *Example of a graph of the change in wind speed.*

#### *Prospects for Increasing the Dynamic Efficiency of Asynchronous Double-Feed Machines… DOI: http://dx.doi.org/10.5772/intechopen.96523*

energy transfer is blocked and the entire wind turbine is stopped. That is, the use of optimization methods is limited to modes of uniform power generation, i.e. they can be used only when the wind turbine is in a stationary state, with small and slow changes in the parameters of wind flows. In case of transient processes or in emergency situations, the methods are not applicable.

The reason that DFMs are disconnected from the grid during transient modes is the inability of the regulation system to track these changes in wind speed with minimal transients. The shutdown and protection devices receive considerable attention from developers and researchers. At the same time, the purpose of the protection devices is to preserve the operability of the equipment with "non-mode" parameters of the wind and power grid, and to reduce the time of inoperability of wind power units. But in all these devices, little attention is paid to the dynamics of installations in operating states. The modes and settings of the regulators remain standard, which means that they are quite ineffective in terms of dynamics. All this also reduces the efficiency of wind turbines to critically low values. At the same time, the "dynamic potential" of asynchronous electric drives is far from being exhausted and is not used in most drives because most often it is not required. But not in the case of wind turbines.

The authors carried out research on the dynamics of asynchronous electric drives with frequency control, which are of undoubted interest for wind power. The research consisted of the development of theoretical provisions, bench experiments, simulation and application of industrial units in electric drives.

## **2. Theoretical provisions**

*Emerging Electric Machines - Advances, Perspectives and Applications*

the wind parameters are not stationary and difficult to predict.

variation can be distinguished in the wind speed - slow and faster.

tens and hundreds of seconds due to their very large dimensions.

on the DFM, causing "its" transient processes.

the drive [7–9, 11, 12] is impaired.

parrying a torque load are different at different speeds of rotation and frequencies of stator voltage and are strongly "tightened". At even higher speeds, the stability of

It should be noted that the traditional means and methods of regulation of asynchronous electric drives with frequency control (PID speed or torque controllers) with vector control or direct torque control work very poorly precisely when parrying "step" or harmonic torque loads. This is shown in articles [7–10, 13].

In DFM wind turbines, the wind works exactly as a moment load. In this case,

There are many works devoted to the study of the parameters of wind flows, one of the time dependences is shown in **Figure 3**. As a rule, several ranges of speed

The frequency of rapid variations in wind speed is from to too high for its effective "tracking" by powerful and large-sized electric drives of wind turbines [14–17]. Electric generators and wind turbines in general have significant inertia –

The control systems of these installations should work as follows. DFM, as a generator, must convert mechanical wind energy, determined by the average wind speed, into electrical energy. And, as an electric drive, it must correct and smooth out wind gusts so that they do not "distort" the frequency and amplitude of the stator voltage. To solve this problem in the DFM must be the inverter connected to the rotor circuit. It can adjust the frequency and amplitude of the stator voltage precisely. At the same time, a change in the voltage in the rotor has a complex effect

In DFM, most often, rather traditional algorithms for asynchronous electric drives are used to optimize stationary modes - depending on the wind speed, the structure of the drive changes - the torque or speed of the motor is controlled by loops with a PID controller. At the same time, during the change in the wind speed, the restoration of the parameters of the DFM operation mode - the torque, the speed of rotation and, accordingly, the frequency of the stator voltage - occurs rather slowly, and in order not to violate the compliance with the requirements of the parameters of the stator voltage – frequency and amplitude, most often,

**50**

**Figure 3.**

*Example of a graph of the change in wind speed.*

Vector equations were replaced by continuous ones in a certain area of the multidimensional space formed by variable coordinates of the electric drive - rotation speed and mechanical moment and independent functions - stator voltage frequency and relative slip. As a result, a nonlinear transfer function was obtained that connects the developed mechanical torque and absolute slip - the difference between the stator voltage frequency and the engine speed. The formula for this function includes, as variables, the frequency of the stator voltage and the relative slip. The formula can be called the nonlinear transfer function or the dynamic Kloss formula. In articles [18–26] the conclusion of the proposed nonlinear transfer function is given in sufficient detail, the result is as follows:

$$\mathcal{W}(\boldsymbol{p}) = \frac{2\mathcal{M}\_k \left(\boldsymbol{T}\_z^\prime \boldsymbol{p} + \mathbf{1}\right) \mathbf{S}\_k}{o\_1 \left[\left(\mathbf{1} + \boldsymbol{T}\_z^\prime \boldsymbol{p}\right)^2 \boldsymbol{S}\_k^2 + \boldsymbol{\beta}^2\right]} \tag{2}$$

**Figure 4.**

*Block diagram of the working section of the mechanical characteristics of an alternating current machine.*

**Figure 5.** *Block diagram of the corrected electric drive.*

where, ω1 is the stator voltage frequency, β is the relative slip, depending on the drive load.

This transfer function corresponds to the block diagram shown in **Figure 4**.

In works [18–27] it is shown how it is possible to linearize the specified transfer function, that is, to exclude the dependence of the transfer function and the dynamics of the asynchronous drive on the frequency of the stator voltage and slip by positive feedback on the developed torque. The structural diagram will take the form (**Figure 5**).

The transfer function of the correcting link, which is necessary in positive feedback to maintain the stability of the drive, is as follows:

$$\mathbf{W}\_{\text{DPF}} = \frac{\alpha\_1 \boldsymbol{\beta}^2}{2\mathbf{M}\_k \mathbf{S}\_k \left(\boldsymbol{T}\_2^\prime \boldsymbol{p} + \mathbf{1}\right)}\tag{3}$$

**53**

torque.

*Prospects for Increasing the Dynamic Efficiency of Asynchronous Double-Feed Machines…*

This made it possible to formulate a hypothesis that the identification of an induction motor with a frequency converter by a nonlinear transfer function is more accurate than vector equations, which is confirmed by the choice of a more

It should be noted that experiments with parrying the moment disturbance at a constant frequency of the stator voltage with minimal transients in the speed of rotation of the engine and in the torque is the most desirable process for the DFM of

In this case, the corrected drive will ensure the "bringing" of all variable coordinates of the DFM to the required "zone, where the rotor frequency converter will equalize the frequency of the stator voltage to the specified value with high

In asynchronous electric drives using mass-produced frequency converters, it is quite problematic to introduce a positive torque connection. As experiments have shown [8–10, 18–27], it can be replaced by a connection according to the active component of the stator current, which is measured by almost all known frequency

In powerful and not cheap drives of wind power systems, it is advisable to install sensors for direct measurement of the mechanical moment and speed sensors, and magnetic flux sensors in the DFM. In this case, the complexity and uniqueness of each high-power wind turbine allows the application of solutions with a high cost,

As mentioned above, analytical expressions describing asynchronous electric drives have significant errors. Therefore, decisive importance in assessing the cor-

The test bench (**Figure 6**), On which the research was carried out, contains - a load asynchronous squirrel-cage motor (M1) and a working electric motor with a phase rotor (M2) operating on one shaft, frequency converters (FC1, FC2) that control motors, rotor current sensors of the working electric motor, and a common

The order of experiments is formulated according to the transfer function of the

The signal supplied to the input of the frequency converter U2 of the working WRIM drives the drive to a certain speed of rotation (and the corresponding frequency of the stator voltage). This determines the parameters of the transfer

The signal generator SG sends a periodic signal of a certain frequency to the input of the frequency converter U1 of the load SCIM, which creates a load torque with an amplitude of 10% of the nominal value, which determines the range of variation of the transfer function parameter, which depends on slip – formula (2). The same input receives a step signal at the level of 100% of the nominal mechanical

The purpose of the experiment is to provide evidence of the effectiveness of the drive control method. The control system of the drive must ensure maximum parrying of any moment disturbance, that is, the better the drive maintains the rotation speed (the less it deviates from the set speed), the more efficient its control.

For studies of the DFM of wind power plants, the reactions of control systems to

rectness and effectiveness should be given to experimental research.

shaft speed sensor (BR1) and a periodic reference signal generator (SG1).

function, depending on the frequency of the stator voltage.

Various operating modes of electric drives were investigated.

moment loads are of greatest interest.

*DOI: http://dx.doi.org/10.5772/intechopen.96523*

converters widely used in industry.

**3. Bench experiments**

electric drive (2)-(4).

but at the same time with high efficiency.

wind turbines.

accuracy.

effective correction selected for this transfer function.

Equivalent transfer function of the drive with this connection will take the form:

$$\mathcal{W}\_{\text{eq}} = \frac{2\mathcal{M}\_k \mathcal{S}\_k \left( T\_z' p + \mathbf{1} \right)}{o o\_1 \left[ \left( \mathbf{1} + T\_z' p \right)^2 \mathcal{S}\_k^2 \right]} = \frac{2\mathcal{M}\_k}{o o\_1 \mathcal{S}\_k \left( \mathbf{1} + T\_z' p \right)} \tag{4}$$

The resulting transfer function of an asynchronous electric motor with parameters depending on the frequency of the stator voltage and slip is an "incorrect" expression from the point of view of the exact mathematics of functional transformations (Laplace transforms). But Eqs. (2) and (3) can describe the dynamics of transient processes with small frequency changes and slip (which change little the value of W (p) with significantly smaller errors than vector equations, which are accurate not only at a constant frequency, but also with the mandatory sinusoidality of currents and voltages in the stator and rotor of an induction motor [11, 15, 23–25, 28–30].

The method using nonlinear transfer functions turns out to be more accurate than the generally accepted apparatus of vector equations. Especially important for the DFM wind turbine is the fact that the efficiency of the choice of correction is also significantly higher for the processes of parrying moment disturbances in asynchronous electric drives with frequency control [21, 22, 27, 28, 31–33].

Thus, the proposed positive dynamic connection by the torque of the engine or its analogue significantly reduces the time of the transient parrying processes and their maximum values. Experiments investigating the response of the drive to load surges with various methods of WRIM control have fully confirmed this.

*Prospects for Increasing the Dynamic Efficiency of Asynchronous Double-Feed Machines… DOI: http://dx.doi.org/10.5772/intechopen.96523*

This made it possible to formulate a hypothesis that the identification of an induction motor with a frequency converter by a nonlinear transfer function is more accurate than vector equations, which is confirmed by the choice of a more effective correction selected for this transfer function.

It should be noted that experiments with parrying the moment disturbance at a constant frequency of the stator voltage with minimal transients in the speed of rotation of the engine and in the torque is the most desirable process for the DFM of wind turbines.

In this case, the corrected drive will ensure the "bringing" of all variable coordinates of the DFM to the required "zone, where the rotor frequency converter will equalize the frequency of the stator voltage to the specified value with high accuracy.

In asynchronous electric drives using mass-produced frequency converters, it is quite problematic to introduce a positive torque connection. As experiments have shown [8–10, 18–27], it can be replaced by a connection according to the active component of the stator current, which is measured by almost all known frequency converters widely used in industry.

In powerful and not cheap drives of wind power systems, it is advisable to install sensors for direct measurement of the mechanical moment and speed sensors, and magnetic flux sensors in the DFM. In this case, the complexity and uniqueness of each high-power wind turbine allows the application of solutions with a high cost, but at the same time with high efficiency.

As mentioned above, analytical expressions describing asynchronous electric drives have significant errors. Therefore, decisive importance in assessing the correctness and effectiveness should be given to experimental research.

### **3. Bench experiments**

*Emerging Electric Machines - Advances, Perspectives and Applications*

where, ω1 is the stator voltage frequency, β is the relative slip, depending on the

This transfer function corresponds to the block diagram shown in **Figure 4**. In works [18–27] it is shown how it is possible to linearize the specified transfer function, that is, to exclude the dependence of the transfer function and the dynamics of the asynchronous drive on the frequency of the stator voltage and slip by positive feedback on the developed torque. The structural diagram will take the

The transfer function of the correcting link, which is necessary in positive

( ) *DPF MS Tp k k* ω β

′ <sup>=</sup> <sup>+</sup>

Equivalent transfer function of the drive with this connection will take the form:

( )

<sup>+</sup> <sup>=</sup> <sup>=</sup> + ′ <sup>+</sup> 2

2 1 2 1 1

*MS Tp <sup>M</sup> <sup>W</sup>*

depending on the frequency of the stator voltage and slip is an "incorrect" expression from the point of view of the exact mathematics of functional transformations (Laplace transforms). But Eqs. (2) and (3) can describe the dynamics of transient processes with small frequency changes and slip (which change little the value of W (p) with significantly smaller errors than vector equations, which are accurate not only at a constant frequency, but also with the mandatory sinusoidality of currents and voltages in the stator and rotor of an induction motor [11, 15, 23–25, 28–30]. The method using nonlinear transfer functions turns out to be more accurate than the generally accepted apparatus of vector equations. Especially important for the DFM wind turbine is the fact that the efficiency of the choice of correction is also significantly higher for the processes of parrying moment disturbances in asynchronous electric drives with frequency control [21, 22, 27, 28, 31–33].

′

*Tp S*

2 1

2 1 2

( ) ( ) *k k k*

<sup>2</sup> <sup>2</sup> 1 2 1 2

The resulting transfer function of an asynchronous electric motor with parameters

Thus, the proposed positive dynamic connection by the torque of the engine or its analogue significantly reduces the time of the transient parrying processes and their maximum values. Experiments investigating the response of the drive to load

surges with various methods of WRIM control have fully confirmed this.

*k k*

ω

*S Tp*

′

(3)

(4)

feedback to maintain the stability of the drive, is as follows:

*eqv*

ω

W

**52**

drive load.

*Block diagram of the corrected electric drive.*

**Figure 5.**

form (**Figure 5**).

The test bench (**Figure 6**), On which the research was carried out, contains - a load asynchronous squirrel-cage motor (M1) and a working electric motor with a phase rotor (M2) operating on one shaft, frequency converters (FC1, FC2) that control motors, rotor current sensors of the working electric motor, and a common shaft speed sensor (BR1) and a periodic reference signal generator (SG1).

The order of experiments is formulated according to the transfer function of the electric drive (2)-(4).

The signal supplied to the input of the frequency converter U2 of the working WRIM drives the drive to a certain speed of rotation (and the corresponding frequency of the stator voltage). This determines the parameters of the transfer function, depending on the frequency of the stator voltage.

The signal generator SG sends a periodic signal of a certain frequency to the input of the frequency converter U1 of the load SCIM, which creates a load torque with an amplitude of 10% of the nominal value, which determines the range of variation of the transfer function parameter, which depends on slip – formula (2). The same input receives a step signal at the level of 100% of the nominal mechanical torque.

The purpose of the experiment is to provide evidence of the effectiveness of the drive control method. The control system of the drive must ensure maximum parrying of any moment disturbance, that is, the better the drive maintains the rotation speed (the less it deviates from the set speed), the more efficient its control.

Various operating modes of electric drives were investigated.

For studies of the DFM of wind power plants, the reactions of control systems to moment loads are of greatest interest.

**Figure 7** shows the diagrams of changes in the speed of rotation of the engine with a "step" load with various control methods - open scalar control, vector control with speed feedback, and in a drive with torque (current) coupling. Transient time and maximum speed deviation for stepped load torque – minimum in an electric drive with DPF.

**55**

smallest.

**Figure 8.**

**spectrum**

electric drives.

stator current) are obvious.

analysis of velocity diagrams.

*Prospects for Increasing the Dynamic Efficiency of Asynchronous Double-Feed Machines…*

In **Figure 8** diagrams obtained in the course of experiments with variable load harmonic change in torque with a frequency of 1 Hz, − in the example, the ampli-

Advantages of a drive with positive torque coupling (or its close analogue - active

The amplitude of deviations from the set speed for a periodic disturbing

**4. Analysis of the efficiency of asynchronous drives in the rotor current** 

In a number of modes, according to the speed diagrams, it is rather difficult to assess the degree of advantage of this or that control method for asynchronous

A technique for assessing the dynamics of the drive by the frequency of the rotor current is proposed and developed, which is of undoubted interest for DFM. Experiments have shown that this estimate is much more convincing than the

Since the frequency of the rotor current is precisely determined by the slip in the motor, the rotor current spectra characterize the control efficiency of the drive. This

**Figures 9**–**11** shows diagrams and spectra of the rotor currents with load surges

with different control methods: DPF drive (**Figure 9**), open-loop scalar drive (**Figure 10**), closed loop scalar drive (**Figure 11**). Particular attention should be paid to the nature of the stator current when using a dynamic positive connection

torque with a frequency of 1 Hz in an electric drive with DPF is also the

tude of change is 10% of the nominal torque value.

*Drive response to harmonically varying load torque.*

technique is discussed in detail in [18–21, 23–27, 34].

for the motor torque (for the active stator current).

*DOI: http://dx.doi.org/10.5772/intechopen.96523*

**Figure 7.** *Drive response to "step" load surge*

*Prospects for Increasing the Dynamic Efficiency of Asynchronous Double-Feed Machines… DOI: http://dx.doi.org/10.5772/intechopen.96523*

**Figure 8.**

*Emerging Electric Machines - Advances, Perspectives and Applications*

**Figure 7** shows the diagrams of changes in the speed of rotation of the engine with a "step" load with various control methods - open scalar control, vector control with speed feedback, and in a drive with torque (current) coupling. Transient time and maximum speed deviation for stepped load torque – minimum in an electric

**54**

**Figure 7.**

*Drive response to "step" load surge*

drive with DPF.

*Electrical circuit of the experimental stand.*

**Figure 6.**

*Drive response to harmonically varying load torque.*

In **Figure 8** diagrams obtained in the course of experiments with variable load harmonic change in torque with a frequency of 1 Hz, − in the example, the amplitude of change is 10% of the nominal torque value.

The amplitude of deviations from the set speed for a periodic disturbing torque with a frequency of 1 Hz in an electric drive with DPF is also the smallest.

Advantages of a drive with positive torque coupling (or its close analogue - active stator current) are obvious.

## **4. Analysis of the efficiency of asynchronous drives in the rotor current spectrum**

In a number of modes, according to the speed diagrams, it is rather difficult to assess the degree of advantage of this or that control method for asynchronous electric drives.

A technique for assessing the dynamics of the drive by the frequency of the rotor current is proposed and developed, which is of undoubted interest for DFM. Experiments have shown that this estimate is much more convincing than the analysis of velocity diagrams.

Since the frequency of the rotor current is precisely determined by the slip in the motor, the rotor current spectra characterize the control efficiency of the drive. This technique is discussed in detail in [18–21, 23–27, 34].

**Figures 9**–**11** shows diagrams and spectra of the rotor currents with load surges with different control methods: DPF drive (**Figure 9**), open-loop scalar drive (**Figure 10**), closed loop scalar drive (**Figure 11**). Particular attention should be paid to the nature of the stator current when using a dynamic positive connection for the motor torque (for the active stator current).

#### **Figure 9.**

#### **Figure 10.**

*Diagram and spectrum of the rotor current with vector control.*

#### **Figure 11.**

*Diagram and spectrum of the rotor current for scalar control.*

Experiments with load surges have unambiguously shown a significant advantage of this scheme.

**57**

**Figure 12.**

*The frequency of the stator voltage is 10 Hz (a) and 20 Hz (b).*

*Prospects for Increasing the Dynamic Efficiency of Asynchronous Double-Feed Machines…*

• −1,7 Hz in an asynchronous drive with a positive torque coupling.

• −3,5 Hz in an asynchronous drive with a positive torque coupling.

than in drives with scalar or vector, speed-closed controls. (Position P1).

rent does not lead to a significant increase in magnetic induction.

That is, an electric drive with a dynamic positive connection in terms of the torque developed by the drive (or an analogue of this signal - the active component of the stator current) requires significantly less slip to create the required torque

Detailed drive experiments with this feedback have shown the nature of this efficiency. Since the transfer function (2) depends on the frequency of the stator voltage, the experiments were carried out at several different speeds of rotation of

The experiments were carried out at five operating speeds (31,42 rps, 62,83 rps, 94,25 rps, 125,67 rps, 157,08 rps) which corresponds to the frequencies of the stator voltage - 10 Hz, 20 Hz, 30 Hz, 40 Hz, 50 Hz. Diagrams of processes in speed, rotor currents and their spectra are shown in **Figures 12**–**14**. In all cases, under load, rotor current distortions and the third harmonic are observed, compared with the main

It is known from the theory of automatic control that odd harmonics in periodic signals of a closed automatic control system (ACS) arise in the presence of symmetric static nonlinearities. In AED, such a static nonlinearity is the magnetization curve of the stator and rotor. This nonlinearity is characterized by saturation regions in which an increase in the magnetic field strength proportional to the cur-

This testifies to the saturation of the magnetic structure of the motor, which occurs only under load and ensures high efficiency of torque generation under load with a smaller mismatch between the synchronous speed and the rotor speed. It can

• 8,75 Hz in an asynchronous drive with a vector control and.

• both with load- 4,75 Hz in an asynchronous drive with scalar control,

*DOI: http://dx.doi.org/10.5772/intechopen.96523*

the engine and, accordingly, these frequencies.

one in the rotor current spectra.

Basic frequencies of the rotor current.


*Prospects for Increasing the Dynamic Efficiency of Asynchronous Double-Feed Machines… DOI: http://dx.doi.org/10.5772/intechopen.96523*


That is, an electric drive with a dynamic positive connection in terms of the torque developed by the drive (or an analogue of this signal - the active component of the stator current) requires significantly less slip to create the required torque than in drives with scalar or vector, speed-closed controls. (Position P1).

Detailed drive experiments with this feedback have shown the nature of this efficiency. Since the transfer function (2) depends on the frequency of the stator voltage, the experiments were carried out at several different speeds of rotation of the engine and, accordingly, these frequencies.

The experiments were carried out at five operating speeds (31,42 rps, 62,83 rps, 94,25 rps, 125,67 rps, 157,08 rps) which corresponds to the frequencies of the stator voltage - 10 Hz, 20 Hz, 30 Hz, 40 Hz, 50 Hz. Diagrams of processes in speed, rotor currents and their spectra are shown in **Figures 12**–**14**. In all cases, under load, rotor current distortions and the third harmonic are observed, compared with the main one in the rotor current spectra.

It is known from the theory of automatic control that odd harmonics in periodic signals of a closed automatic control system (ACS) arise in the presence of symmetric static nonlinearities. In AED, such a static nonlinearity is the magnetization curve of the stator and rotor. This nonlinearity is characterized by saturation regions in which an increase in the magnetic field strength proportional to the current does not lead to a significant increase in magnetic induction.

This testifies to the saturation of the magnetic structure of the motor, which occurs only under load and ensures high efficiency of torque generation under load with a smaller mismatch between the synchronous speed and the rotor speed. It can

**Figure 12.** *The frequency of the stator voltage is 10 Hz (a) and 20 Hz (b).*

*Emerging Electric Machines - Advances, Perspectives and Applications*

*Diagram and spectrum of the rotor current for scalar control with positive feedback.*

Experiments with load surges have unambiguously shown a significant advan-

• no load −1,7 Hz in an asynchronous drive with scalar control,

• 2,1 Hz in an asynchronous drive with a vector control and.

**56**

tage of this scheme.

**Figure 11.**

**Figure 10.**

**Figure 9.**

Basic frequencies of the rotor current.

*Diagram and spectrum of the rotor current for scalar control.*

*Diagram and spectrum of the rotor current with vector control.*

**Figure 13.** *The frequency of the stator voltage is 30 Hz (a) and 40 Hz (b).*

#### **Figure 14.**

*The frequency of the stator voltage is 50 Hz.*

be assumed that sections 1 (**Figure 15**) of the rotor current, characterized by rapid rise and fall, correspond to precisely these sections of the saturation.

With changes in load, the frequencies of the 1st and 3rd harmonics also changed, but the ratio of frequencies did not practically change. The distribution of frequencies and amplitudes of the rotor current depending on different frequencies of the supply voltage is presented in **Table 1**.

**59**

**Figure 15.**

**Table 1.**

*Saturation sections characteristic.*

*Prospects for Increasing the Dynamic Efficiency of Asynchronous Double-Feed Machines…*

1.These experiments show that the proposed corrections force the magnetic flux under load - that is, they make it possible to obtain the maximum stator magnetic flux - the maximum possible drive efficiency and at the same time

*Numerical values of the frequencies and amplitudes of the prevailing harmonics of rotor currents in* 

*experiments to parry the load at different speeds of rotation (***Figures 12***–***14***).*

2.According to the frequency of the rotor current without load and under load, it follows that at all speeds of rotation, the parrying of the load occurs at the same absolute slip - 3 Hz, i.e. an initially non-linear asynchronous electric drive is linearized by this connection more accurately than by vector control

In general, the experiments carried out with the stand shown in **Figure 6** showed that an electric drive with a torque correction (or an active component of the stator current) has advantages in the dynamics of almost all possible modes -

the drive maintains the stability of the processes in all modes.

**ω1, Hz No load With load**

**1st harm. 3rd harm. 1st harm. 3rd harm** *f,* **Hz** *A***, V** *f***, Hz** *A,***V** *f,* **Hz** *A,* **V** *f,* **Hz** *A,***V**

 1,5 0,39 4,25 0,019 3,75 1,24 11,5 0,23 1,5 0,42 4,75 0,037 3,75 1,32 11 0,21 1,75 0,65 5,25 0,039 3,5 1,0 10,25 0,15 1,75 0,66 5,5 0,042 3,25 1,35 9,75 0,16 1,75 0,52 5.5 0,036 3.25 0,96 9,5 0,18

with a PID speed controller (**Figure 11**).

parrying load surges – stepwise and periodic.

*DOI: http://dx.doi.org/10.5772/intechopen.96523*

Experiments have shown two very important results of a positive torque relationship.

*Prospects for Increasing the Dynamic Efficiency of Asynchronous Double-Feed Machines… DOI: http://dx.doi.org/10.5772/intechopen.96523*

**Figure 15.** *Saturation sections characteristic.*


#### **Table 1.**

*Emerging Electric Machines - Advances, Perspectives and Applications*

*The frequency of the stator voltage is 30 Hz (a) and 40 Hz (b).*

be assumed that sections 1 (**Figure 15**) of the rotor current, characterized by rapid

Experiments have shown two very important results of a positive torque

With changes in load, the frequencies of the 1st and 3rd harmonics also changed, but the ratio of frequencies did not practically change. The distribution of frequencies and amplitudes of the rotor current depending on different frequencies of the

rise and fall, correspond to precisely these sections of the saturation.

supply voltage is presented in **Table 1**.

*The frequency of the stator voltage is 50 Hz.*

**58**

relationship.

**Figure 14.**

**Figure 13.**

*Numerical values of the frequencies and amplitudes of the prevailing harmonics of rotor currents in experiments to parry the load at different speeds of rotation (***Figures 12***–***14***).*


In general, the experiments carried out with the stand shown in **Figure 6** showed that an electric drive with a torque correction (or an active component of the stator current) has advantages in the dynamics of almost all possible modes parrying load surges – stepwise and periodic.

## **5. Modeling**

Simulations have also confirmed the effectiveness of the dynamics of corrective link actuators. So in the model (**Figure 16**) with load surges, the parrying efficiency in schemes with DPF is much higher. This is proved by both the processes in speed (**Figures 17**–**19(a)**) and the rotor current spectra (**Figures 17**–**19(c)** and **Table 2**).

**Figures 17**–**19** shows diagrams of simulated drive acceleration processes from zero to speed corresponding to the frequencies of the stator voltage 10, 20, 30, 40, 50 Hz. Load surges are modeled at steady-state speeds. The diagrams of the speed (diagram a), of the mechanical torque, developed by the motor (diagram b) and the rotor current (diagram c) are displayed. It is obvious that in all modes the frequency of the rotor current in the drive model closed in speed (**Figure 17**) is significantly higher than in a drive with stator current feedback (**Figure 19**).

The values of the fundamental frequencies of the rotor current in the models of the electric drive with different control algorithms is presented in **Table 2**.

In these diagrams, the processes in the speeds of rotation do not differ significantly, but the processes in the rotor currents have significantly different frequencies, which shows the high efficiency of the proposed method for evaluating the

**Figure 16.**

*Model of an asynchronous electric drive with vector control in Matlab Simulink.*

**61**

**Figure 18.**

**Figure 19.**

**Table 2.**

*Processes in a scalar drive model.*

*Prospects for Increasing the Dynamic Efficiency of Asynchronous Double-Feed Machines…*

method of controlling asynchronous electric machines by rotor currents. It should be expected that the method will be useful in milking machines of wind power plants.

*The fundamental frequencies of the rotor currents of the processes in the model shown in* **Figures 17***–***19***.*

SC 5,12 2,98 2,66 2,56 2,66 SVC 10,64 12,81 10,62 11,63 11,63 SC with DPF 1,90 1,71 1,71 1,75 2,06

**10 20 30 40 50**

The correction was introduced into the electric drives of the transport line, which were experiencing moment loads. According to the conditions of the

**6. Implementation in production mechanisms**

*Processes in the drive model with torque feedback (active stator current).*

**Control algorithm Fundamental frequency, Hz**

*DOI: http://dx.doi.org/10.5772/intechopen.96523*

**Figure 17.** *Processes in a vector control drive model.*

*Prospects for Increasing the Dynamic Efficiency of Asynchronous Double-Feed Machines… DOI: http://dx.doi.org/10.5772/intechopen.96523*

**Figure 18.** *Processes in a scalar drive model.*

*Emerging Electric Machines - Advances, Perspectives and Applications*

higher than in a drive with stator current feedback (**Figure 19**).

*Model of an asynchronous electric drive with vector control in Matlab Simulink.*

the electric drive with different control algorithms is presented in **Table 2**.

Simulations have also confirmed the effectiveness of the dynamics of corrective link actuators. So in the model (**Figure 16**) with load surges, the parrying efficiency in schemes with DPF is much higher. This is proved by both the processes in speed (**Figures 17**–**19(a)**) and the rotor current spectra (**Figures 17**–**19(c)** and **Table 2**). **Figures 17**–**19** shows diagrams of simulated drive acceleration processes from zero to speed corresponding to the frequencies of the stator voltage 10, 20, 30, 40, 50 Hz. Load surges are modeled at steady-state speeds. The diagrams of the speed (diagram a), of the mechanical torque, developed by the motor (diagram b) and the rotor current (diagram c) are displayed. It is obvious that in all modes the frequency of the rotor current in the drive model closed in speed (**Figure 17**) is significantly

The values of the fundamental frequencies of the rotor current in the models of

In these diagrams, the processes in the speeds of rotation do not differ significantly, but the processes in the rotor currents have significantly different frequencies, which shows the high efficiency of the proposed method for evaluating the

**5. Modeling**

**60**

**Figure 17.**

*Processes in a vector control drive model.*

**Figure 16.**

#### **Figure 19.**

*Processes in the drive model with torque feedback (active stator current).*


#### **Table 2.**

*The fundamental frequencies of the rotor currents of the processes in the model shown in* **Figures 17***–***19***.*

method of controlling asynchronous electric machines by rotor currents. It should be expected that the method will be useful in milking machines of wind power plants.

## **6. Implementation in production mechanisms**

The correction was introduced into the electric drives of the transport line, which were experiencing moment loads. According to the conditions of the

#### *Emerging Electric Machines - Advances, Perspectives and Applications*

technology, transient processes during the "capture" of the transported workpieces had to be minimized. At the same time, it was important to leave asynchronous drives of mechanisms, without transferring these mechanisms to expensive precision drives. Also, several control methods were tested, according to transient processes, shown in **Figures 20**–**22**. A clear advantage in the quality of working out disturbances - behind a drive with torque coupling (active component of the stator current).

**Figure 20.** *Scalar controlled electric drive (SC).*

**Figure 21.** *Vector controlled electric drive (SVC).*

**63**

*Prospects for Increasing the Dynamic Efficiency of Asynchronous Double-Feed Machines…*

Under production conditions, it was possible to achieve a significant reduction

According to the simplified vector equations of the DFM during transient processes in the drive, it is quite simple to correct the frequency of the stator voltage by changes in the voltage supplied to the DFM rotor within small limits. As follows from the vector equations, it is sufficient to take into account changes in the rotation speed of the DFM. However, in reality, a wind power plant is a multi-connected structure with complex mutually influencing transient processes that are caused by dynamic gusts of wind and transients in the wind turbines themselves. Vector equations that greatly simplify the description of these complexes do not allow obtaining

The identification of processes in asynchronous electric motors with a wound rotor by a nonlinear transfer function proposed in previous studies describes the processes in SCIM, WRIM and DFM much more accurately and efficiently.

Multidimensional transfer functions will make it possible to describe transient processes in such complex control systems as DFM or induction motors with a

Numerous experiments have shown that cross-connections compensate for the influence of these factors - the influence of variable loads, different frequencies of stator voltage (in this case, it is invariable) is much more effective than other known

In addition to transient processes, transfer functions will make it possible to describe reactions to variable disturbing factors - periodic reference signals and moment disturbances are described in the works, and in this case, these are the

The dynamic drive will fend off gusts of wind and keep the rotor speed in the range without significant dips, and the stator frequency loop will perform precise control as now. The network mismatch time will decrease and the wind utilization rate will increase. Thus, the DPF correction improves the overloading capacity of the asynchronous machine, and with a dynamic load this improvement is more active than with a static load, which makes the application of this solution in TIR

Experiments and simulations have shown that in a system with DFM, cross-links in torque and speed also linearize the DFM electric drive system, as in asynchronous electric drives with moment disturbances, providing minimal transient processes that do not interfere with the operation of the system when the wind changes. It should be expected that an asynchronous electric drive with linearizing couplings, effectively

Thus, it can be assumed that the correction has significant prospects when it is

Thus, the advantage of the method of controlling asynchronous electric motors with dynamic torque coupling, confirmed in all experiments and simulations, suggests that for machines with double power supply, a similar algorithm for controlling a frequency converter from the rotor side of the machine will improve their dynamic characteristics and the final efficiency by 5–10% with minimal capital investment.

working with moment disturbances, will also be effective in wind turbines.

used in electric drives based on DFM in wind turbines.

*DOI: http://dx.doi.org/10.5772/intechopen.96523*

**7. The discussion of the results**

control methods.

even more effective.

**8. Conclusion**

in currents in all modes and times of the transient process.

an effective control structure in transient processes.

wound rotor and effective methods of their correction.

effects of wind and regulation from the rotor.

**Figure 22.** *Stator current feedback electric drive (DPF).*

Under production conditions, it was possible to achieve a significant reduction in currents in all modes and times of the transient process.

## **7. The discussion of the results**

*Emerging Electric Machines - Advances, Perspectives and Applications*

current).

**Figure 20.**

**Figure 21.**

*Scalar controlled electric drive (SC).*

*Vector controlled electric drive (SVC).*

*Stator current feedback electric drive (DPF).*

technology, transient processes during the "capture" of the transported workpieces had to be minimized. At the same time, it was important to leave asynchronous drives of mechanisms, without transferring these mechanisms to expensive precision drives. Also, several control methods were tested, according to transient processes, shown in **Figures 20**–**22**. A clear advantage in the quality of working out disturbances - behind a drive with torque coupling (active component of the stator

**62**

**Figure 22.**

According to the simplified vector equations of the DFM during transient processes in the drive, it is quite simple to correct the frequency of the stator voltage by changes in the voltage supplied to the DFM rotor within small limits. As follows from the vector equations, it is sufficient to take into account changes in the rotation speed of the DFM. However, in reality, a wind power plant is a multi-connected structure with complex mutually influencing transient processes that are caused by dynamic gusts of wind and transients in the wind turbines themselves. Vector equations that greatly simplify the description of these complexes do not allow obtaining an effective control structure in transient processes.

The identification of processes in asynchronous electric motors with a wound rotor by a nonlinear transfer function proposed in previous studies describes the processes in SCIM, WRIM and DFM much more accurately and efficiently.

Multidimensional transfer functions will make it possible to describe transient processes in such complex control systems as DFM or induction motors with a wound rotor and effective methods of their correction.

Numerous experiments have shown that cross-connections compensate for the influence of these factors - the influence of variable loads, different frequencies of stator voltage (in this case, it is invariable) is much more effective than other known control methods.

In addition to transient processes, transfer functions will make it possible to describe reactions to variable disturbing factors - periodic reference signals and moment disturbances are described in the works, and in this case, these are the effects of wind and regulation from the rotor.

The dynamic drive will fend off gusts of wind and keep the rotor speed in the range without significant dips, and the stator frequency loop will perform precise control as now. The network mismatch time will decrease and the wind utilization rate will increase. Thus, the DPF correction improves the overloading capacity of the asynchronous machine, and with a dynamic load this improvement is more active than with a static load, which makes the application of this solution in TIR even more effective.

## **8. Conclusion**

Experiments and simulations have shown that in a system with DFM, cross-links in torque and speed also linearize the DFM electric drive system, as in asynchronous electric drives with moment disturbances, providing minimal transient processes that do not interfere with the operation of the system when the wind changes. It should be expected that an asynchronous electric drive with linearizing couplings, effectively working with moment disturbances, will also be effective in wind turbines.

Thus, it can be assumed that the correction has significant prospects when it is used in electric drives based on DFM in wind turbines.

Thus, the advantage of the method of controlling asynchronous electric motors with dynamic torque coupling, confirmed in all experiments and simulations, suggests that for machines with double power supply, a similar algorithm for controlling a frequency converter from the rotor side of the machine will improve their dynamic characteristics and the final efficiency by 5–10% with minimal capital investment.

*Emerging Electric Machines - Advances, Perspectives and Applications*

## **Author details**

Vladimir L. Kodkin\*, Alexandr S. Anikin and Alexandr A. Baldenkov South Ural State University, Chelyabinsk, Russian Federation

\*Address all correspondence to: kodkina2@mail.ru

© 2021 The Author(s). Licensee IntechOpen. 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.

**65**

*Prospects for Increasing the Dynamic Efficiency of Asynchronous Double-Feed Machines…*

Anikin A.S., Shmarin Y.A. Russian Electrical Engineering. 2014. 85. No. 10. P. 641-644. DOI : 10.3103/

S1068371214100101

[8] Experimental research of asynchronous electric drive with positive dynamic feedback on stator current Kodkin V.L., Anikin A.S., Baldenkov A.A. 2017 International Conference on Industrial Engineering, Applications and Manufacturing (ICIEAM) - Proceedings 2017 DOI: 10.1109/ICIEAM.2017.8076179

[9] Dynamic Load Disturbance Correction for Alternative Current Electric Drives V.L. Kodkin, A.S. Anikin, Y.A. Shmarin 2nd International Conference on Industrial Engineering, Applications and Manufacturing (ICIEAM); South Ural State University Chelyabinsk - Proceedings 2016 DOI: 10.1109/SIBCON.2015.7146978

[10] Spectral Analysis of Rotor Currents in Frequency-controlled Electric Drives Kodkin V.L., Anikin A.S., Baldenkov A.A. 2nd International Conference on Automation, Mechanical and Electrical Engineering (AMEE 2017) - Proceedings 2017 DOI: doi:10.2991/amee-17.2017.26

[11] Popov, V. M. [Hyperstability of Dynamic Systems], Springer-Verlag,

[12] Kodkin. V.L. Methods of optimizing the speed and accuracy of optical complex guidance systems based on equivalence of automatic control system domain of attraction and unconditional stability of their equivalent circuits/ V.L. Kodkin// Proceedings of SPIE - The International Society for Optical

Berlin (1973)

Engineering. – 2016

[13] Pat. No. 2412526 Russian Federation, IPC Н02Р 23/00. The device for frequency control of

*DOI: http://dx.doi.org/10.5772/intechopen.96523*

"Structural model, model study of the dynamics of an electric drive with a double-feed machine and direct torque control" Scientific and Technical Bulletin of Information Technologies, Mechanics and Optics, no. 4 (80), 2012,

[2] Lyapin A.S. Model study of the double fed machine with current control. Scientific and Technical Journal of Information Technologies,

10.17586/2226-1494-2016-16-4-731-737

[3] Pat. No. 2336624 Russian Federation,

[5] PROBLEMS OF IMPLEMENTATION

OF FREQUENCY CONTROL IN MINING INDUSTRY / V.L. Kodkin, A.S. Anikin, М.А. Malcher // Bulletin of the South Ural State University Series: Energy. - Vol. 18. - No.37.–2012.– P.67-71.

[6] FREQUENCY CONTROL OF ASYNCHRONOUS ELECTRIC

SIBCON.2015.7146978

[7] EFFECTIVE FREQUENCY CONTROL FOR INDUCTION ELECTRIC DRIVES UNDER OVERLOADING Kodkin V.L.,

DRIVES IN TRANSPORT Kodkin V.L., Anikin A.S. In the collection: 2015 International Siberian Conference on Control and Communications, SIBCON 2015 - Proceedings 2015. С. 7146978. WOS:000380571600016 DOI : 10.1109/

Mechanics and Optics, 2016, vol. 16, no. 4, pp. 731-737. doi:

IPC Н02Р 27/06. Direct Motor Speed Controller / V.L. Kodkin, E.R. Khaybakov. - No. 2006106477/09; Claimed. 02.03.2006; Publ. 02.03.2006,

[4] Malcher, M.A. Problems of introduction of frequency regulation in the mining industry / М.А. Malcher, A.S. Anikin // Mining equipment and electromechanics. - 2011. - №4.

Bul. № 29. – 11p.

[1] Lyapin Anatoly Sergeevich.

**References**

pp. 60-64.

*Prospects for Increasing the Dynamic Efficiency of Asynchronous Double-Feed Machines… DOI: http://dx.doi.org/10.5772/intechopen.96523*

## **References**

*Emerging Electric Machines - Advances, Perspectives and Applications*

**64**

**Author details**

Vladimir L. Kodkin\*, Alexandr S. Anikin and Alexandr A. Baldenkov

© 2021 The Author(s). Licensee IntechOpen. 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,

South Ural State University, Chelyabinsk, Russian Federation

\*Address all correspondence to: kodkina2@mail.ru

provided the original work is properly cited.

[1] Lyapin Anatoly Sergeevich. "Structural model, model study of the dynamics of an electric drive with a double-feed machine and direct torque control" Scientific and Technical Bulletin of Information Technologies, Mechanics and Optics, no. 4 (80), 2012, pp. 60-64.

[2] Lyapin A.S. Model study of the double fed machine with current control. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2016, vol. 16, no. 4, pp. 731-737. doi: 10.17586/2226-1494-2016-16-4-731-737

[3] Pat. No. 2336624 Russian Federation, IPC Н02Р 27/06. Direct Motor Speed Controller / V.L. Kodkin, E.R. Khaybakov. - No. 2006106477/09; Claimed. 02.03.2006; Publ. 02.03.2006, Bul. № 29. – 11p.

[4] Malcher, M.A. Problems of introduction of frequency regulation in the mining industry / М.А. Malcher, A.S. Anikin // Mining equipment and electromechanics. - 2011. - №4.

[5] PROBLEMS OF IMPLEMENTATION OF FREQUENCY CONTROL IN MINING INDUSTRY / V.L. Kodkin, A.S. Anikin, М.А. Malcher // Bulletin of the South Ural State University Series: Energy. - Vol. 18. - No.37.–2012.– P.67-71.

[6] FREQUENCY CONTROL OF ASYNCHRONOUS ELECTRIC DRIVES IN TRANSPORT Kodkin V.L., Anikin A.S. In the collection: 2015 International Siberian Conference on Control and Communications, SIBCON 2015 - Proceedings 2015. С. 7146978. WOS:000380571600016 DOI : 10.1109/ SIBCON.2015.7146978

[7] EFFECTIVE FREQUENCY CONTROL FOR INDUCTION ELECTRIC DRIVES UNDER OVERLOADING Kodkin V.L.,

Anikin A.S., Shmarin Y.A. Russian Electrical Engineering. 2014. 85. No. 10. P. 641-644. DOI : 10.3103/ S1068371214100101

[8] Experimental research of asynchronous electric drive with positive dynamic feedback on stator current Kodkin V.L., Anikin A.S., Baldenkov A.A. 2017 International Conference on Industrial Engineering, Applications and Manufacturing (ICIEAM) - Proceedings 2017 DOI: 10.1109/ICIEAM.2017.8076179

[9] Dynamic Load Disturbance Correction for Alternative Current Electric Drives V.L. Kodkin, A.S. Anikin, Y.A. Shmarin 2nd International Conference on Industrial Engineering, Applications and Manufacturing (ICIEAM); South Ural State University Chelyabinsk - Proceedings 2016 DOI: 10.1109/SIBCON.2015.7146978

[10] Spectral Analysis of Rotor Currents in Frequency-controlled Electric Drives Kodkin V.L., Anikin A.S., Baldenkov A.A. 2nd International Conference on Automation, Mechanical and Electrical Engineering (AMEE 2017) - Proceedings 2017 DOI: doi:10.2991/amee-17.2017.26

[11] Popov, V. M. [Hyperstability of Dynamic Systems], Springer-Verlag, Berlin (1973)

[12] Kodkin. V.L. Methods of optimizing the speed and accuracy of optical complex guidance systems based on equivalence of automatic control system domain of attraction and unconditional stability of their equivalent circuits/ V.L. Kodkin// Proceedings of SPIE - The International Society for Optical Engineering. – 2016

[13] Pat. No. 2412526 Russian Federation, IPC Н02Р 23/00. The device for frequency control of

an asynchronous electric drive / V.L. Kodkin, A.S. Anikin. - No. 2010108563/07; Claimed. 09/03/2010; Publ. 20.02.2011, Bul. № 5-12 p.

[14] Pat. №155 351 Russian Federation, IPC F03D 7/04, F03D 1/00, F03D 3/00. Adaptive combined device for regulating the rotor speed of the wind power plant / E.V. Solomin, A.S. Anikin., E.A. Sirotkin, E.E. Solomin, A.A. Sirotkin, S.V. Kozlov - No. 2014154564/06; Claimed. 12/31/2014; Publ. 10/10/2015 Bul. № 28-5 p.

[15] Vorob'ev, N.N. Theory of series. //4 ed., Revised. And additional. - Moscow: Science, 1979, − 408 p.

[16] Meshcheryakov VN, and Bezdenezhnykh DV. "Flux linkage observer for a double feed machine controlled by stator and rotor circuits" Vestnik Voronezh State Technical University, vol. 6, no. 11, 2010, pp. 170-173.

[17] Mazalov Andrey Andreevich. "Adaptive wind turbine with a double-feed AC machine" Izvestia of the Southern Federal University. Engineering Sciences, vol. 126, no. 1, 2012, pp. 26-33.

[18] Kodkin, V.L The dynamics identification of asynchronous electric drives via frequency response / V.L. Kodkin, A.S. Anikin, A.A. Baldenkov //International Journal of Power Electronics and Drive Systems.–2019.– Vol. 10 No. 1.– P.66-73

[19] Kodkin, V. Families of Frequency Characteristics, as a Basis for the Identification of Asynchronous Electric Drives / V. Kodkin, A. Anikin, A. Baldenkov //2018 International Russian Automation Conference (RusAutoCon).–2018

[20] Usoltsev, A.A. Vector control of asynchronous motors. Tutorial. - Spb .: ITMO, 2002-120 p. http://servomotors. ru/documentation/frequency\_control\_ of\_asynchronous\_motors/chastupr.pdf

[21] Kodkin, V.L Analysis of Stability of Electric Drives as Non-linear Systems According to Popov Criterion Adjusted to Amplitude and Phase Frequency Characteristics of Its Elements, A.S. Anikin, A.A. Baldenkov //2nd International Conference on Applied Mathematics, Simulation and Modelling (AMSM 2017) - Proceedings 2017 P.7-14, DOI: 10.12783/dtetr/amsm2017/14810

[22] Kodkin, V.L The analysis of the quality of the frequency control of induction motor carried out on the basis of the processes in the rotor circuit / V.L. Kodkin, A.S. Anikin, A.A. Baldenkov //Journal of Physics: Conference Series.–2018.–Vol. 944(1)

[23] Kodkin, V. Stabilization of the stator and rotor flux linkage of the induction motor in the asynchronous electric drives with frequency regulation / V. Kodkin, A. Anikin, A. Baldenkov //International Journal of Power Electronics and Drive Systems.– 2020.– Vol. 11 No. 1.– P.213-219 DOI: http://doi. org/10.11591/ijpeds.v11.i1.pp213-219

[24] Kodkin, V. Structural correction of nonlinear dynamics of frequencycontrolled induction motor drives / V. Kodkin, A. Anikin, A. Baldenkov //International Journal of Power Electronics and Drive Systems.–2020.– Vol. 11 No. 1.– P.220-227 DOI: http://doi. org/10.11591/ijpeds.v11.i1.pp220-227

[25] Kodkin, V. Experimental study of the VFD's speed stabilization (retention) efficiency under torque disturbances / V. Kodkin, A. Anikin, //International Journal of Power Electronics and Drive Systems.– 2021.– Vol. 12 No. 1

[26] Kodkin, V. The experimental identification method of the dynamic efficiency for frequency regulation algorithms of AEDs / V. Kodkin, A.

**67**

view/6757/6394

*Prospects for Increasing the Dynamic Efficiency of Asynchronous Double-Feed Machines…*

(IJPEDS), Vol. 5, No. 3, February 2015, pp. 336~343 https://www.iaescore.com/ journals/index.php/IJPEDS/article/

[33] Kodkin, V.L Performance identification of the asynchronous electric drives by the spectrum of rotor currents / V.L. Kodkin, A.S. Anikin, A.A. Baldenkov //International Journal of Power Electronics and Drive Systems.–2019.–Vol. 10 No. 1.– P.211-218 DOI: http://doi.org/10.11591/ijpeds.v10.

[34] Pat. No. 2599529 Russian Federation, IPC H02P 25/02. The device for frequency control of an asynchronous electric drive / V.L. Kodkin, A.S. Anikin., Ya.A. Shmarin, A.A. Baldenkov - № 2014151549/07; Claimed. 11/17/2015; Publ. 10.10.2016,

view/5068/4681

i1.pp211-218

Bul. № 28-12 p.

*DOI: http://dx.doi.org/10.5772/intechopen.96523*

Anikin, //International Journal of Power Electronics and Drive Systems.– 2021.–

electromagnetic loads for machines with double power supply taking into account saturation and higher harmonics" SPbPU Scientific and Technical Bulletin. Natural and engineering sciences, no. 1

[28] Yahya Ahmed Alamri, Nik Rumzi Nik Idris, Ibrahim Mohd. Alsofyani, Tole Sutikno Improved Stator Flux Estimation for Direct Torque Control of Induction Motor Drives // International Journal of Power Electronics and Drive System (IJPEDS), Vol. 7, No. 4, December 2016, pp. 1049~1060 https:// www.iaescore.com/journals/index.php/

IJPEDS/article/view/5822/5589

[29] Bonar D.D., Khoury M.J. Real Infinite Series / The Mathematical Association of America, 2006. — 274 p.

[30] Loday-Richaud M. Divergent Series, Summability and Resurgence II. Simple and Multiple Summability / Springer International Publishing, 2016. — 272 p.

[31] Srinivas Gangishetti, Tarakalyani Sandipamu Different Control Schemes for Sensor Less Vector Control of

Induction Motor // International Journal of Power Electronics and Drive System (IJPEDS), Vol. 8, No. 2, June 2017, pp. 712~721. https://www.iaescore.com/ journals/index.php /IJPEDS/article/

[32] Md. Rashedul Islam, Md. Maruful Islam, Md. Kamal Hossain, Pintu Kumar Sadhu Performance Analysis of a DTC and SVM Based Field-Orientation Control Induction Motor Drive // International Journal of Power Electronics and Drive System

[27] Boguslavsky Ilya Zelikovich, Danilevich Yanush Bronislavovich, Popov Viktor Vasilievich, and Rogachevsky Vladimir Samuilovich. "Features of the calculation of

(166), 2013, pp. 67-73.

Vol. 12 No. 1

*Prospects for Increasing the Dynamic Efficiency of Asynchronous Double-Feed Machines… DOI: http://dx.doi.org/10.5772/intechopen.96523*

Anikin, //International Journal of Power Electronics and Drive Systems.– 2021.– Vol. 12 No. 1

*Emerging Electric Machines - Advances, Perspectives and Applications*

ru/documentation/frequency\_control\_ of\_asynchronous\_motors/chastupr.pdf

[21] Kodkin, V.L Analysis of Stability of Electric Drives as Non-linear Systems According to Popov Criterion Adjusted to Amplitude and Phase Frequency Characteristics of Its Elements, A.S. Anikin, A.A. Baldenkov //2nd International Conference on Applied Mathematics, Simulation and Modelling (AMSM 2017) - Proceedings 2017 P.7-14, DOI: 10.12783/dtetr/amsm2017/14810

[22] Kodkin, V.L The analysis of the quality of the frequency control of induction motor carried out on the basis of the processes in the rotor circuit / V.L. Kodkin, A.S. Anikin, A.A. Baldenkov //Journal of Physics: Conference Series.–2018.–Vol. 944(1)

[23] Kodkin, V. Stabilization of the stator and rotor flux linkage of the induction motor in the asynchronous electric drives with frequency regulation / V. Kodkin, A. Anikin, A. Baldenkov //International Journal of Power Electronics and Drive Systems.– 2020.– Vol. 11 No. 1.– P.213-219 DOI: http://doi. org/10.11591/ijpeds.v11.i1.pp213-219

[24] Kodkin, V. Structural correction of nonlinear dynamics of frequencycontrolled induction motor drives / V. Kodkin, A. Anikin, A. Baldenkov //International Journal of Power Electronics and Drive Systems.–2020.– Vol. 11 No. 1.– P.220-227 DOI: http://doi. org/10.11591/ijpeds.v11.i1.pp220-227

[25] Kodkin, V. Experimental study of the VFD's speed stabilization (retention) efficiency under torque disturbances / V. Kodkin, A. Anikin, //International Journal of Power Electronics and Drive Systems.– 2021.–

[26] Kodkin, V. The experimental identification method of the dynamic efficiency for frequency regulation algorithms of AEDs / V. Kodkin, A.

Vol. 12 No. 1

an asynchronous electric drive / V.L. Kodkin, A.S. Anikin. - No. 2010108563/07; Claimed. 09/03/2010; Publ. 20.02.2011, Bul. № 5-12 p.

[14] Pat. №155 351 Russian Federation, IPC F03D 7/04, F03D 1/00, F03D 3/00. Adaptive combined device for regulating the rotor speed of the wind power plant / E.V. Solomin, A.S. Anikin., E.A. Sirotkin, E.E. Solomin, A.A. Sirotkin, S.V. Kozlov - No. 2014154564/06; Claimed. 12/31/2014; Publ. 10/10/2015 Bul. № 28-5 p.

[15] Vorob'ev, N.N. Theory of series. //4 ed., Revised. And additional. - Moscow:

Science, 1979, − 408 p.

170-173.

2012, pp. 26-33.

Vol. 10 No. 1.– P.66-73

(RusAutoCon).–2018

[16] Meshcheryakov VN, and Bezdenezhnykh DV. "Flux linkage observer for a double feed machine controlled by stator and rotor circuits" Vestnik Voronezh State Technical University, vol. 6, no. 11, 2010, pp.

[17] Mazalov Andrey Andreevich. "Adaptive wind turbine with a double-feed AC machine" Izvestia of the Southern Federal University. Engineering Sciences, vol. 126, no. 1,

[18] Kodkin, V.L The dynamics

identification of asynchronous electric drives via frequency response / V.L. Kodkin, A.S. Anikin, A.A. Baldenkov //International Journal of Power Electronics and Drive Systems.–2019.–

[19] Kodkin, V. Families of Frequency Characteristics, as a Basis for the Identification of Asynchronous Electric Drives / V. Kodkin, A. Anikin, A. Baldenkov //2018 International Russian Automation Conference

[20] Usoltsev, A.A. Vector control of asynchronous motors. Tutorial. - Spb .: ITMO, 2002-120 p. http://servomotors.

**66**

[27] Boguslavsky Ilya Zelikovich, Danilevich Yanush Bronislavovich, Popov Viktor Vasilievich, and Rogachevsky Vladimir Samuilovich. "Features of the calculation of electromagnetic loads for machines with double power supply taking into account saturation and higher harmonics" SPbPU Scientific and Technical Bulletin. Natural and engineering sciences, no. 1 (166), 2013, pp. 67-73.

[28] Yahya Ahmed Alamri, Nik Rumzi Nik Idris, Ibrahim Mohd. Alsofyani, Tole Sutikno Improved Stator Flux Estimation for Direct Torque Control of Induction Motor Drives // International Journal of Power Electronics and Drive System (IJPEDS), Vol. 7, No. 4, December 2016, pp. 1049~1060 https:// www.iaescore.com/journals/index.php/ IJPEDS/article/view/5822/5589

[29] Bonar D.D., Khoury M.J. Real Infinite Series / The Mathematical Association of America, 2006. — 274 p.

[30] Loday-Richaud M. Divergent Series, Summability and Resurgence II. Simple and Multiple Summability / Springer International Publishing, 2016. — 272 p.

[31] Srinivas Gangishetti, Tarakalyani Sandipamu Different Control Schemes for Sensor Less Vector Control of Induction Motor // International Journal of Power Electronics and Drive System (IJPEDS), Vol. 8, No. 2, June 2017, pp. 712~721. https://www.iaescore.com/ journals/index.php /IJPEDS/article/ view/6757/6394

[32] Md. Rashedul Islam, Md. Maruful Islam, Md. Kamal Hossain, Pintu Kumar Sadhu Performance Analysis of a DTC and SVM Based Field-Orientation Control Induction Motor Drive // International Journal of Power Electronics and Drive System

(IJPEDS), Vol. 5, No. 3, February 2015, pp. 336~343 https://www.iaescore.com/ journals/index.php/IJPEDS/article/ view/5068/4681

[33] Kodkin, V.L Performance identification of the asynchronous electric drives by the spectrum of rotor currents / V.L. Kodkin, A.S. Anikin, A.A. Baldenkov //International Journal of Power Electronics and Drive Systems.–2019.–Vol. 10 No. 1.– P.211-218 DOI: http://doi.org/10.11591/ijpeds.v10. i1.pp211-218

[34] Pat. No. 2599529 Russian Federation, IPC H02P 25/02. The device for frequency control of an asynchronous electric drive / V.L. Kodkin, A.S. Anikin., Ya.A. Shmarin, A.A. Baldenkov - № 2014151549/07; Claimed. 11/17/2015; Publ. 10.10.2016, Bul. № 28-12 p.

**Chapter 5**

**Abstract**

addressing these challenges.

**1. Introduction**

**69**

Very Low Voltage and High

Efficiency Motorisation for

This chapter details the design of a new innovative solid bar winding for electrical machines (either motors or generators) dedicated to the electric propulsion. The goal of this new winding technique is to enhance the performance by better utilizing the stator slot and increasing the copper fill factor to higher than 75%, and also to reduce the inactive copper at the end-windings. Accordingly, many advantages arise from the application of this solid bar winding: higher torque-to-weight ratio, better thermal behavior, lower rotor losses, higher efficiency, higher reliability and lower cogging torque. However, the solid bar has its inherent constraints, which should be considered with care when designing an electric motor: the AC copper losses and the manufacturing process. The suggested winding technique aims at

**Keywords:** solid bar winding, permanent magnet synchronous machine, high performance electric motor, high power-to-weight ratio, electric propulsion,

The need for a higher competitive electrical machines, mainly in terms of power

The opportunities for achieving a big improvement against the state of the art

The following expression provides the basic relationship of the sizing electromagnetic power of electric motors (rotational movement). This expression highlight some, but not all, obvious paths to follow in order to improve the performance.

are very limited and challenging due to the very small degree of freedom.

density and efficiency, is increasing especially in embedded applications (aerospace, Vertical Take-Off and Landing, Electric Vehicle, etc.); these performances are a key differentiator between competitors. As a rule of thumb, nowadays, a good power-to-weight ratio of PM electric motors is around 3 kW/kg (EMAG + mechanical packaging) [1]. Nevertheless, higher values have been proclaimed by many companies and star-ups, but for experimental prototypes where the maturity of the product is still questionable. The definition of the power to weight ratio is still versatile and controversial because, on one hand, the estimation of the total motor weight relies on many parts where some of them are not always considered in the calculation: EMAG, mechanics, coolant weight (in some cases is shared with the system), cables, power electronic, etc., on the other hand, the flexible definition of

AC copper loss, low voltage winding, high slot fill factor

the output power (continuous or transient).

Electric Vehicles

*Daniel Matt and Nadhem Boubaker*

## **Chapter 5**
