**2. Rotor FOC**

In this study, a model based approach was selected for minimizing the induction motor losses (Lim & Nam 2004). This Loss Minimization Algorithm (LMA) was developed in d-q coordinates, considering an equivalent motor model in the synchronous reference frame, as described in section 3. In addition, as we will discuss in a later section, the induction motor controller is also based on rotor FOC. These reasons justify a brief review on induction motor rotor FOC. Figure 1 represents the basic concept of rotor FOC (based on Krishnan, 2001).

**Figure 1.** Rotor FOC principle for induction motors

Recall that, in the synchronous frame (e), the rotor magnetic flux (rd) is aligned with d axis, thus rd = r, rq =0. In the same reference frame, the stator current component ids is aligned with the rotor magnetic flux, controlling its value. On the other hand, iqs (shifted /2 electrical rad from ids) controls the motor electromagnetic torque:

$$\Psi\_r = \mathbf{L}\_m \mathbf{i}\_{ds} \left( \text{steady} - \text{state} \right) \tag{1}$$

$$\mathbf{T\_r(t)} = \mathbf{K\_r} \Psi\_r \mathbf{i\_{qs}} \tag{2}$$

From figure 1, it may be seen that ids and iqs are, respectively, the d and q components of the space vector **is** in the synchronous reference frame. This way, from the control philosophy perspective, ids and iqs regulation is implemented in this reference frame; however, from the control hardware perspective, ids and iqs must be considered in stator reference phasecoordinates (ia, ib, ic). To do that, it is mandatory to obtain ids and idq in the static d-q reference, which requires the information about s=e+T. The determination of e is the main issue, since T= arctg(iqs/ids); e calculation can be accomplished through slip and r (see figure 1) – indirect FOC.

Evaluation of an Energy Loss-Minimization Algorithm for EVs Based on Induction Motor 403

$$\text{Since}$$

402 Induction Motors – Modelling and Control

**Figure 1.** Rotor FOC principle for induction motors

figure 1) – indirect FOC.

electrical rad from ids) controls the motor electromagnetic torque:

**2. Rotor FOC** 

2001).

which maximizes the energy efficiency at given torque subject to voltage and currents limits. In Section 4 the developed EV non-causal simulation model (motor-to-wheel) is presented, while Section 5 includes the simulation results and its analysis for a set of standard driving cycles. Finally, Section 6 contains the main conclusions and some reference to future work.

In this study, a model based approach was selected for minimizing the induction motor losses (Lim & Nam 2004). This Loss Minimization Algorithm (LMA) was developed in d-q coordinates, considering an equivalent motor model in the synchronous reference frame, as described in section 3. In addition, as we will discuss in a later section, the induction motor controller is also based on rotor FOC. These reasons justify a brief review on induction motor rotor FOC. Figure 1 represents the basic concept of rotor FOC (based on Krishnan,

Recall that, in the synchronous frame (e), the rotor magnetic flux (rd) is aligned with d axis, thus rd = r, rq =0. In the same reference frame, the stator current component ids is aligned with the rotor magnetic flux, controlling its value. On the other hand, iqs (shifted /2

From figure 1, it may be seen that ids and iqs are, respectively, the d and q components of the space vector **is** in the synchronous reference frame. This way, from the control philosophy perspective, ids and iqs regulation is implemented in this reference frame; however, from the control hardware perspective, ids and iqs must be considered in stator reference phasecoordinates (ia, ib, ic). To do that, it is mandatory to obtain ids and idq in the static d-q reference, which requires the information about s=e+T. The determination of e is the main issue, since T= arctg(iqs/ids); e calculation can be accomplished through slip and r (see

r m ds L i steady state (1)

T t <sup>t</sup> *K i t r qs* (2)

$$\text{Since } \qquad o\square\_{\text{slip}} = o\square\_{\text{e}} - o\square\_{\text{r}} = \frac{L\_m}{L\_r \;/\ R\_r} \frac{\dot{l}\_{qs}}{\Psi\_r} \tag{3}$$

slip is given by:

$$
\partial\_{\text{slip}}\left(\mathbf{t}\right) = \partial\_{\text{slip}}\left(\mathbf{t}\_0\right) + \int\_{t\_0}^t \boldsymbol{\alpha}\_{s\text{ip}} dt \tag{4}
$$

Knowing the instantaneous rotor speed r, one have:

$$
\partial\_{\mathbf{r}} \left( \mathbf{t} \right) = \partial\_{\mathbf{r}} \left( \mathbf{t}\_0 \right) + \int\_{t\_0}^{t} \alpha\_r dt \tag{5}
$$

From figure 1:

$$
\theta\_o \left( \mathbf{t} \right) = \theta\_{\text{slip}} \left( \mathbf{t} \right) + \theta\_r \left( \mathbf{t} \right) \tag{6}
$$

### **3. Loss minimization by selecting flux references**

The loss-minimization scheme demands the decrease or increase of the flux level depending on the torque. This means that the minimization algorithm selects the flux reference through the minimization of the copper and core losses while ensuring the desired torque requested by the driver. Different techniques for loss minimization in induction motor are presented in the literature (Bazzi & Krein 2010). Recently, (Lim & Nam 2004) proposed a LMA that features a major difference from previous works by taking into consideration the leakage inductance and the practical constrains on voltage and current in the high-speed region, which play a great role in EVs applications. This is an important difference from other works, like (Garcia et al., 1994), (Kioskeridis & Margaris, 1996), (Fernandez-Bernal et al., 2000), where leakage inductance are not considered (although similar motor loss models are included), leading to considerable result differences in the high-speed region. In addition, our work considers the optimization of both positive and negative torque generation with bounded constraints on both current and voltage.

### **3.1. The LMA method**

The implemented method is based on the conventional induction motor model where the iron losses are represented by an equivalent resistance (Rm) modelling the iron losses, placed in parallel with the magnetizing inductance (Lm). A simplification is then considered, allowing a partial decoupling between Rm and Lm: the iron losses are represented by separated circuits with dependent voltage sources (Vdme and Vqme). Figure 2 shows the complete equivalent model in the synchronous reference frame.

Considering steady state analysis with low slip values (s) – rotor iron losses may be neglected –, the total motor losses (copper and iron ones) are given by (Lim & Nam 2004):

$$P\_{\rm loss} = R\_d \left( \alpha \rho\_e \right) \mathbf{i}\_{\rm ds}^{\epsilon \ 2} + R\_q \left( \alpha \rho\_e \right) \mathbf{i}\_{q\rm s}^{\epsilon \ 2} \tag{7}$$

Evaluation of an Energy Loss-Minimization Algorithm for EVs Based on Induction Motor 405

equivalent resistors representing the total losses. Voltage and current constraints

2 22 ( ) ( ) <sup>e</sup> <sup>e</sup> max *e e*

2 22 max

<sup>2</sup> 1-L / ( ) <sup>m</sup> *L L s r*

The LMA´s goal is to achieve the optimal flux level that minimizes the motor total losses under voltage and current constraints. The motor rated flux level must also be taken into consideration, in order to avoid magnetic saturation. Moreover, the torque developed by the motor cannot be compromised by the LMA implementation. From the mathematical point of

e e min P (i ,i ) loss ds qs

s.t. : (10), (11)

e e T Ki i e t ds qs

2L Kp3 <sup>m</sup> t 2 L

r

 

An important observation is that voltage constraint depends on the considered e.

: induction machine leakage coeficient; Ls: stator total inductance;

Vmax; Imax: motor (or inverter) voltage and current limits, respectively;

*<sup>e</sup>*) are the direct (d) and quadrature (q) components of the

*Li Li V <sup>s</sup> ds s qs* (10)

*e e i iI ds qs* (11)

(12)

i I ds dn (13)

(14*)* 

Note that *Rd*(

Where:

Where:

*e*) and *Rq*(

view, the LMA algorithm consists in:

Idn : rated d-axis stator current

[p: pairs of magnetic poles]

<sup>e</sup>

Te : electromagnetic torque (steady-state), considering rotor FOC;

(mentioned before) are defined by (neglecting stator resistor drop):

$$R\_d \left( \alpha \rho\_e \right) = R\_s + \frac{\alpha\_e^2 L^2\_{\ \mu}}{R\_m} \tag{8}$$

$$R\_q \left( \phi\_e \right) = R\_s + \frac{R\_r L\_m^2}{L\_r^2} + \frac{\alpha\_e^2 L\_m^2 L\_{lr}^2}{R\_m L\_r^2} \tag{9}$$

Where:

*e ds i* : d-axis stator current in the synchronous reference frame;

*e qs i* : q-axis stator current in the synchronous reference frame;

*<sup>e</sup>* : electrical angular frequency;


**Figure 2.** Simplified motor equivalent model (Lim & Nam 2004)

Note that *Rd*(*e*) and *Rq*(*<sup>e</sup>*) are the direct (d) and quadrature (q) components of the equivalent resistors representing the total losses. Voltage and current constraints (mentioned before) are defined by (neglecting stator resistor drop):

$$(a\_{\mathfrak{e}}L\_{\mathfrak{s}}i\_{\mathrm{d}\mathfrak{s}}^{\mathcal{e}})^2 + (a\_{\mathfrak{e}}\sigma L\_{\mathfrak{s}}i\_{\mathfrak{q}\mathfrak{s}}^{\mathcal{e}})^2 \le V\_{\text{max}}^2 \tag{10}$$

$$i\_{\rm dis}^{\mathcal{e}} \, \mathcal{Z} + i\_{\rm qs}^{\mathcal{e}} \, \mathcal{Z} \le I\_{\rm max}^{\mathcal{2}} \,\tag{11}$$

Where:

404 Induction Motors – Modelling and Control

*<sup>e</sup>* : electrical angular frequency;


Where: *e ds*

> *e qs*

 *e e* 2 2 *loss d e ds q e qs PR iR i* 

*i* : d-axis stator current in the synchronous reference frame;

*i* : q-axis stator current in the synchronous reference frame;


**Figure 2.** Simplified motor equivalent model (Lim & Nam 2004)

*R R*

*qe s*


*R R*

*de s*

 

2 2 *e m*

*m L*

2 22 2 2 2 *r m e m lr*

*r mr*

*R* 

*RL L L*

*L RL* 

(7)

(8)

(9)

$$\sigma = 1 \text{-L}^2\_{\text{m}} / (\text{L}\_s \text{L}\_r) \tag{12}$$

: induction machine leakage coeficient; Ls: stator total inductance;

Vmax; Imax: motor (or inverter) voltage and current limits, respectively;

An important observation is that voltage constraint depends on the considered e.

The LMA´s goal is to achieve the optimal flux level that minimizes the motor total losses under voltage and current constraints. The motor rated flux level must also be taken into consideration, in order to avoid magnetic saturation. Moreover, the torque developed by the motor cannot be compromised by the LMA implementation. From the mathematical point of view, the LMA algorithm consists in:

$$\begin{array}{c} \min \mathbf{P}\_{\text{loss}} \left( \mathbf{i}\_{\text{ds}}^{\mathbf{e}}, \mathbf{i}\_{\text{qs}}^{\mathbf{e}} \right) \\\\ \text{s.t.:} (10), (11) \\\\ \mathbf{i}\_{\text{ds}}^{\mathbf{e}} \le \mathbf{I}\_{\text{dn}} \\\\ \mathbf{T}\_{\mathbf{e}} = \mathbf{K}\_{\text{t}} \mathbf{i}\_{\text{ds}}^{\mathbf{e}} \mathbf{i}\_{\text{qs}}^{\mathbf{e}} \end{array} \tag{13}$$

Where:

Idn : rated d-axis stator current

Te : electromagnetic torque (steady-state), considering rotor FOC;

$$\mathbf{K\_{t}} = \frac{3}{2} \mathbf{p} \frac{\mathbf{L\_{m}^{2}}}{\mathbf{L\_{r}}} \tag{14}$$

[p: pairs of magnetic poles]

### *3.1.1. Unconstrained optimization*

In the (*idse , iqse* ) domain, the optimal flux solution for the region inside the inequality restrictions is achieved through Lagrange multipliers method, since only one restriction is active – the torque one

For one restriction only, the general problem is formulated as follows:

$$\nabla L(\mathbf{i}\_{d\*}{}^{\epsilon}, \mathbf{i}\_{q\*}{}^{\epsilon}, \mathcal{X}) = \mathbf{0} \tag{15}$$

Evaluation of an Energy Loss-Minimization Algorithm for EVs Based on Induction Motor 407

i I ds dn ; <sup>e</sup> Te iqs K It dn

ds 2

*ds dn i I* (21)

 

 1/2 2 4 22 1/2 I (I 4T /K ) <sup>e</sup> max max e <sup>t</sup> iqs <sup>2</sup>

 1/2 2 4 424 2 2 1/2 V (V 4ω σ L T /K ) <sup>e</sup> max max e s e t ids <sup>2</sup> 2(<sup>ω</sup> L ) e s

 1/2 2 4 424 2 2 1/2 V (V 4ω σ L T /K ) <sup>e</sup> max max e s e <sup>t</sup> iqs <sup>2</sup> 2(<sup>ω</sup> L ) e s

(23)

(25)

(22)

2 2 1/2 T K I (I I ) <sup>t</sup> dn dn max m1 (24)

1/2 2 4 22 1/2 I (I 4T /K ) <sup>e</sup> max max e <sup>t</sup> <sup>i</sup>

*e*

(18)

e

**Zone Name Active Constraints Solution** 

 e i I ds dn

<sup>2</sup> ( )

e

*<sup>e</sup> L is ds*

For region 1, the maximum torque is limited by Idn and Imax:

The maximum torque in region 2 is limited by Vmax and Imax:

m2 <sup>t</sup> <sup>2</sup> <sup>1</sup> <sup>σ</sup>

 

e2 e2 2 i i I ds qs max

2 2 ( ) max

*<sup>e</sup> Li V e s qs*

max

*V*

n 2 2 2 2 1/2

*L I II*

c is the boundary speed between constant power and power-speed (Pmec\*e=constant)

*<sup>s</sup>* 2

22 2 2 1/2 <sup>2</sup> 1/2 [(V /( max e s max max max e s <sup>ω</sup> L )) I <sup>σ</sup> ] [I (V /(<sup>ω</sup> L )) ] T K

max <sup>c</sup> <sup>2</sup> max

*V I L* max 1 [ ( )] *<sup>s</sup> dn dn*

1/2 <sup>2</sup>

1

in Interior Points e e T Ki i e t ds qs

Limit e e T Ki i e t ds qs

Limit e e T Ki i e t ds qs

Limit e e T Ki i e t ds qs

0 LMA Operation

1 Max Torque

2 Max. Current

3 Max. Voltage

**Table 1.** LMA optimized solutions

The calculated result is:

regions:

$$\text{with:}\\\text{s.t.:}\\\qquad \qquad \qquad \qquad \text{L(i}\_{\text{ds}}{}^{\epsilon}, \text{i}\_{q\mu}{}^{\epsilon}, \text{\mathcal{A}})\\\text{=} \text{P}\_{\text{loss}}(\text{i}\_{\text{ds}}{}^{\epsilon}, \text{i}\_{q\mu}{}^{\epsilon}) + \text{\mathcal{X}}(\text{T}\_{e} - \text{K}\_{\text{r}} \text{i}\_{\text{ds}}{}^{\epsilon} \text{i}\_{q\mu}{}^{\epsilon})\tag{16}$$

where L(*idse , iqse* , ) is the lagrangian associated to the problem, is the Lagrange multiplier, Ploss(*idse , iqse* ) is the cost function and *Te*-*Ktidse iqse* is the restriction. Applying first-order optimal condition (15) gives the following equation system:

$$\frac{\partial \mathbf{L}}{\partial \mathbf{i}\_{\rm sq}} = 0 \qquad \frac{\partial \mathbf{L}}{\partial \mathbf{i}\_{\rm sq}} = 0 \qquad \frac{\partial \mathbf{L}}{\partial \lambda} = 0 \tag{17}$$

yielding

$$\mathbf{i}\_{ds}^{\epsilon} = \left(\frac{T\_{\epsilon}^{2}}{K\_{t}^{2}} \frac{R\_{q}(oo\_{\epsilon})}{R\_{d}(oo\_{\epsilon})}\right)^{1/4};\ \mathbf{i}\_{qs}^{\epsilon} = \left(\frac{T\_{\epsilon}^{2}}{K\_{t}^{2}} \frac{R\_{d}(oo\_{\epsilon})}{R\_{q}(oo\_{\epsilon})}\right)^{1/4} \tag{18}$$

### *3.1.2. Constrained optimization*

Previously, all the inequalities were considered inactive. In order to obtain the optimal solutions in each restriction boundary, the Lagrange multipliers method is applied for each inequality constraint activation (i.e. only "=" operator is valid), together with the torque one. This way, three non linear algebraic equation systems are defined for the inequality constraints. The optimal idse is given by these systems solutions, since it refers to regions on the border lines of the inequality restrictions.

Table 1 presents the solutions, in (idse, iqse) plane, for interior points (zone 0) and for inequality restriction borders (zones 1, 2 and 3).

The voltage and current limits (Vmax, Imax and Idn) lead naturally to three regions of operation referred to as constant torque (low-speed), constant power (midrange speed) and constant power-speed (high-speed), as defined in (Novotny & Lipo, 1996). The transition between constant torque region and power region is characterized by the rated speed (n), which is defined by the interception of inequality restrictions border lines:

$$(\left(o\_{\mathrm{n}}L\_{\mathrm{s}}\mathrm{i}\_{\mathrm{ds}}^{\epsilon}\right)^{2} + \left(o\_{\mathrm{n}}\sigma L\_{\mathrm{s}}\mathrm{i}\_{\mathrm{qs}}^{\epsilon}\right)^{2} = V\_{\mathrm{max}}^{2}\tag{19}$$

$$\begin{array}{rcl} \mathbf{i}\_{ds}^{e\ 2} & + & \mathbf{i}\_{qs}^{e\ 2} = \mathbf{I}\_{\max}^{2} \\ \end{array} \tag{20}$$


*e ds dn i I* (21)

**Table 1.** LMA optimized solutions

The calculated result is:

406 Induction Motors – Modelling and Control

*, iqse*

*, iqse*

*3.1.2. Constrained optimization* 

active – the torque one

In the (*idse*

where L(*idse*

*, iqse*

Ploss(*idse*

yielding

*3.1.1. Unconstrained optimization* 

) domain, the optimal flux solution for the region inside the inequality

(15)

(16)

(18*)* 

is the restriction. Applying first-order optimal

i i <sup>λ</sup> (17)

( ) ( )

*t q e*

 

1/4

*Li Li V* (19)

*i iI* (20)

restrictions is achieved through Lagrange multipliers method, since only one restriction is

 

, ) is the lagrangian associated to the problem, is the Lagrange multiplier,

2 2

*e e de*

*T R <sup>i</sup> K R*

*ds qs Li i*

*ds qs ds qs e t ds qs Li i i i T Ki i*

*iqse*

 L LL 0 0 0

;

Previously, all the inequalities were considered inactive. In order to obtain the optimal solutions in each restriction boundary, the Lagrange multipliers method is applied for each inequality constraint activation (i.e. only "=" operator is valid), together with the torque one. This way, three non linear algebraic equation systems are defined for the inequality constraints. The optimal idse is given by these systems solutions, since it refers to regions on

Table 1 presents the solutions, in (idse, iqse) plane, for interior points (zone 0) and for

The voltage and current limits (Vmax, Imax and Idn) lead naturally to three regions of operation referred to as constant torque (low-speed), constant power (midrange speed) and constant power-speed (high-speed), as defined in (Novotny & Lipo, 1996). The transition between constant torque region and power region is characterized by the rated speed (n), which is

> 2 2 2 nnmax ( ) ( ) *e e s ds s qs*

 

2 22 max *e e ds qs*

*qs*

e e sq sq

1/4 <sup>2</sup>

( ) ( )

2

defined by the interception of inequality restrictions border lines:

*K R*

*t d e*

 

*e q e e*

*T R*

For one restriction only, the general problem is formulated as follows:

with: loss ( , , )=P ( , )+ ( ) *e e e e e e*

( , ,) 0 *e e*

) is the cost function and *Te*-*Ktidse*

*ds*

*i*

the border lines of the inequality restrictions.

inequality restriction borders (zones 1, 2 and 3).

condition (15) gives the following equation system:

$$\rho o\_n = \frac{V\_{\text{max}}}{L\_s} \frac{1}{\left[I\_{dn}^2 + \sigma^2 (I\_{\text{max}}^2 - I\_{dn}^2)\right]^{1/2}} \tag{22}$$

c is the boundary speed between constant power and power-speed (Pmec\*e=constant) regions:

$$\rho\_c = \frac{V\_{\text{max}}}{I\_{\text{max}} L\_s} \left(\frac{\sigma^2 + 1}{2\sigma^2}\right)^{1/2} \tag{23}$$

For region 1, the maximum torque is limited by Idn and Imax:

$$\mathbf{T}\_{\rm m1} = \mathbf{K}\_{\rm t} \mathbf{I}\_{\rm dn} (\mathbf{I}\_{\rm max}^2 - \mathbf{I}\_{\rm dn}^2)^{1/2} \tag{24}$$

The maximum torque in region 2 is limited by Vmax and Imax:

$$\mathbf{T}\_{\rm m2} = \mathbf{K\_t} \frac{[\mathbf{(V\_{max} / (\mathbf{\dot{o}\_e} \mathbf{L\_s}))^2 - \mathbf{I\_{max}^2} \sigma^2]^{1/2} [\mathbf{I\_{max}^2} - (\mathbf{V\_{max} / (\mathbf{\dot{o}\_e} \mathbf{L\_s}))^2]^{1/2}}{1 - \sigma^2}}{1 - \sigma^2} \tag{25}$$

In region 3, the maximum torque is limited by Vmax, but the current is smaller than Imax. So, the current limit does not interfere with Tm3:

$$\mathbf{T\_{m3}} = \mathbf{K\_t} \left( \frac{\mathbf{V\_{max}}}{\alpha\_\mathbf{e} \mathbf{L\_s}} \right)^2 \frac{1}{2\sigma} \tag{26}$$

Evaluation of an Energy Loss-Minimization Algorithm for EVs Based on Induction Motor 409

As stated before, only the voltage limit must be considered for region 3, which means that

Figure 3 presents the paths for Ids\* generation in ids; iqs coordinates (origin-Tp-Tm), considering the three described operation regions. Quadrants I and II are represented, in order to consider both motor and braking modes (optimal ids paths for quadrant II are

It is clear the linear evolution in the three regions (given by (27)), while in region 3, only voltage limit must be considered, since Imax is not reached. After that, in region 1, Idn imposes the optimal path. In region 2 both current and voltage limits (i.e. Imax and Vmax) restrict Ids

In order to get some insight on LMA main features, a first set of results is presented in

Tp3= Tp2. Of course, for this region one must consider e> c.

**Figure 3.** Ids\* paths for e1 (blue), e2 (green) and e3 (gray)

optimal path, while in region 3, Imax most probably is not reached.

**3.2. Optimal Ids generation for the simulated induction motor** 

figures 4-6, based on an induction motor, with the following parameters:

[Rs; Rr] (Ω) [0,399; 0,3538] [Ls; Lr] (H) [59,3; 60,4]\*10-3 [ls; lr] (H) [2,7; 3,8]\*10-3 Lm (H) 56,6\*10-3 Rm (Ω) 350 J(kg m2) 0,089

**Table 2.** Induction Motor Parameters (9 kW; 60 Hz; 4 poles; 1750 rpm)

symmetric to quadrant I paths).

### *3.1.3. Optimal Ids generation*

For the zone in the (idse, iqse) plane limited by restrictions (10), (11) and iedsIdn, optimal result (18) is valid, meaning that:

$$\dot{\boldsymbol{\alpha}}\_{ds}^{\epsilon} = \left(\frac{\boldsymbol{R}\_q(\boldsymbol{\alpha}\_{\epsilon})}{\boldsymbol{R}\_d(\boldsymbol{\alpha}\_{\epsilon})}\right)^{1/2} \dot{\boldsymbol{\alpha}}\_{qs}^{\epsilon} \tag{27}$$

In the border lines of those restrictions, the previous relation can not be considered. So, for region 1, (18) is applied if:

$$\dot{\alpha}\_{q\*}^{\circ} \le \left(\frac{R\_d(o\nu\_{\circ})}{R\_q(o\nu\_{\circ})}\right)^{1/2} \ast I\_{du} \tag{28}$$

The ieqs upper limit in (28) defines Tp1(see Figure 3):

$$\mathbf{T\_{p1}} = \mathbf{K\_t} \left(\frac{\text{Rd(o\_e)}}{\text{Rq(o\_e)}}\right)^{1/2} \mathbf{I\_{dn}}^2 \tag{29}$$

$$\text{Of course, for: } \left(\frac{\text{Rd(o}\_{\mathbf{e}})}{\text{Rq(o}\_{\mathbf{e}})}\right)^{1/2} \, ^\ast \mathbf{I}\_{\text{dn}} < \mathbf{i}\_{\text{qs}}^\mathbf{e} \le \left(\mathbf{l}\_{\text{max}}^2 - \mathbf{l}\_{\text{dn}}^2\right)^{1/2} \to \mathbf{i}\_{\text{ds}}^\mathbf{e} = \mathbf{l}\_{\text{dn}} \tag{30}$$

For region 2, (18) can be considered, until the voltage limit (Vmax) is achieved:

$$\mathrm{Li}\_{\mathrm{qs}}^{\mathrm{e}} \le \frac{\mathrm{V}\_{\mathrm{max}} \,/\,\mathrm{(\mathrm{o}\_{\mathrm{e}}} \mathrm{L}\_{\mathrm{s}} \mathrm{o} \)}{\left[ \mathrm{o}^{-2} + \mathrm{R}\_{\mathrm{d}} \mathrm{(\mathrm{o}\_{\mathrm{e}})} / \mathrm{R}\_{\mathrm{q}} (\mathrm{o}\_{\mathrm{e}}) \right]^{1/2}} \left( \frac{\mathrm{Rd} (\mathrm{o}\_{\mathrm{e}})}{\mathrm{R} (\mathrm{o}\_{\mathrm{e}})} \right)^{1/2} \tag{31}$$

This way, Tp2 is given by the following expression:

$$\mathbf{T\_{p2} = K\_t} \frac{\mathbf{V\_{max}^2} [\mathbf{R\_{d}(o\_e)} \, ^\ast \mathbf{R\_{d}(o\_e)}]^{1/2}}{\left[\sigma^2 \mathbf{R\_{d}(o\_e)} + \mathbf{R\_{q}(o\_e)}\right] \left(o\_e \mathbf{I\_{s}}\right)^2} \tag{32}$$

Above this limit, ieds (and ieqs) is given by zone 3 solution (table 1).

As stated before, only the voltage limit must be considered for region 3, which means that Tp3= Tp2. Of course, for this region one must consider e> c.

Figure 3 presents the paths for Ids\* generation in ids; iqs coordinates (origin-Tp-Tm), considering the three described operation regions. Quadrants I and II are represented, in order to consider both motor and braking modes (optimal ids paths for quadrant II are symmetric to quadrant I paths).

**Figure 3.** Ids\* paths for e1 (blue), e2 (green) and e3 (gray)

408 Induction Motors – Modelling and Control

*3.1.3. Optimal Ids generation* 

(18) is valid, meaning that:

region 1, (18) is applied if:

Of course, for:

The ieqs upper limit in (28) defines Tp1(see Figure 3):

This way, Tp2 is given by the following expression:

Above this limit, ieds (and ieqs) is given by zone 3 solution (table 1).

the current limit does not interfere with Tm3:

In region 3, the maximum torque is limited by Vmax, but the current is smaller than Imax. So,

 

Vmax <sup>1</sup> T Kt m3 <sup>ω</sup> L 2<sup>σ</sup> e s

For the zone in the (idse, iqse) plane limited by restrictions (10), (11) and iedsIdn, optimal result

 

In the border lines of those restrictions, the previous relation can not be considered. So, for

<sup>2</sup> 1/2 Rd(<sup>ω</sup> ) <sup>e</sup> TK I p1 <sup>t</sup> Rq(<sup>ω</sup> dn ) <sup>e</sup>

1/2 Rd(ωe) e2 2 e 1/2 \* I i (I I ) i I Rq(<sup>ω</sup> dn qs max dn ds dn ) <sup>e</sup> (30)

1/2

 V /(<sup>ω</sup> <sup>L</sup> <sup>σ</sup>) Rd(<sup>ω</sup> ) <sup>e</sup> max e s e iqs <sup>2</sup> 1/2 Rq(<sup>ω</sup> ) [<sup>σ</sup> R (<sup>ω</sup> )/R (<sup>ω</sup> )] <sup>e</sup> <sup>d</sup> e qe

> <sup>2</sup> 1/2 V [R ( max e e <sup>ω</sup> )\*R (<sup>ω</sup> )] d d T K p2 <sup>t</sup> 2 2 [<sup>σ</sup> R (<sup>ω</sup> ) R(<sup>ω</sup> )](<sup>ω</sup> L ) <sup>d</sup> e q e es

For region 2, (18) can be considered, until the voltage limit (Vmax) is achieved:

 

*R i i R*

*e d e qs dn q e <sup>R</sup> i I R* 

1/2 ( ) ( ) *e e q e ds qs d e*

> 1/2 ( ) \* ( )

2

(26)

(27)

(28)

(29)

(31)

(32)

It is clear the linear evolution in the three regions (given by (27)), while in region 3, only voltage limit must be considered, since Imax is not reached. After that, in region 1, Idn imposes the optimal path. In region 2 both current and voltage limits (i.e. Imax and Vmax) restrict Ids optimal path, while in region 3, Imax most probably is not reached.

## **3.2. Optimal Ids generation for the simulated induction motor**

In order to get some insight on LMA main features, a first set of results is presented in figures 4-6, based on an induction motor, with the following parameters:

> [Rs; Rr] (Ω) [0,399; 0,3538] [Ls; Lr] (H) [59,3; 60,4]\*10-3 [ls; lr] (H) [2,7; 3,8]\*10-3 Lm (H) 56,6\*10-3 Rm (Ω) 350 J(kg m2) 0,089

**Table 2.** Induction Motor Parameters (9 kW; 60 Hz; 4 poles; 1750 rpm)

Figure 4 represents the optimal Id generation for conventional approach, i.e. constant flux +field weakening (CF+FW), and the LMA approach.

Evaluation of an Energy Loss-Minimization Algorithm for EVs Based on Induction Motor 411

constant torque zone), the LMA's efficiency gain is almost constant, while in the late case the

energy savings decrease in a smooth way to zero.

LMA - CF

0.01

0.05

0.1

0.25

0.17

0.3

**4. Simulation model** 

0.01

0.05

0.1

0.25

0 38 0.38

0.17

T [N.m]

0.3

was built, which is represented in figure 7.

(a) (b)

0.01

0.17

0.3

0.38

speed [rpm]

<sup>0</sup> <sup>1000</sup> <sup>2000</sup> <sup>3000</sup> <sup>4000</sup> <sup>5000</sup> <sup>0</sup>

0.2

0.05

0.1

**Figure 6.** LMA Efficiency gain (a) and relative loss differences (b) compared to (CF+FW)

torque hyperbolas (e.g. higher torques are provided by Id min as the speed grows).

Naturally, LMA acts directly on motor iron losses, since it regulates Ids. However, it has also an impact in motor copper losses, because it provides a better equilibrium between Ids and Iqs, particularly in regions where Ids regulation has wide limits. This can be seen in (7).

0.01

0.01

0.2

0.4

0.6 0.1

0.3

0.5

0.8

0.7

0

0 .1

0

0

4

0

.2

.0 1

T [N.m ]

0.01

0.01

0.1

0.3

0.2

0.4

0.1

0.7

0.3

0.5

0.2

Ploss(CF)-(Ploss(LMA))/Ploss(CF)

0.4

speed [rpm]

<sup>0</sup> <sup>1000</sup> <sup>2000</sup> <sup>3000</sup> <sup>4000</sup> <sup>5000</sup> <sup>0</sup>

0.6

As a side-note, when considering the plane surfaces in figure 4 (for Id max and Id min), interesting correlations can be made with figure 3, through (Id max, Id min) dashed lines and the

To evaluate the LMA´s contributions to the EV energy consumption reduction, and comparing it to the conventional flux regulation, a simulation study was performed with four diferent driving cycles: ECE-R15, Europe: City, 11-Mode (Japan) and FTP-75. Simulation with other drive-cycles was also implemented, but results achieved with these four give a wide overview of LMA´s features. For that purpose, a Matlab/Simulink model

Basically, Id\* is generated through (CF+FW) method or by the LMA – blocks (3a) and (3b), respectively. The induction motor is controlled by conventional rotor FOC (block 4); the motor model in block 5 is presented in section 4.4. The motor load and speed references are generated based on a particular drive cycle features (block 1), which includes the vehicle dynamic and mechanical transmission models. Finally, block 2 implements the speed controller (based on a proportional+integral(PI) control law) which generates the motor

torque reference. In the following sections, the main model blocks are discribed.

Inspecting these results one can find that the LMA influence on Id\* is mostly visible for low torques (T<20 N.m). It is interesting to note that in the high speed zone (>2000 rpm), LMA and conventional flux regulation tend to present closer Id\* values, as the speed increases. Also, for high torque values (above 30 N.m) both approaches have similar performances.

**Figure 4.** Ids\* generations: a) CF+FW; b) LMA

From the above analysis, it is expectable that the differences in the generation of Id\* lead to different efficiencies curves of the induction motor, which is, indeed, observed in the maps illustrated in Figure 5.

**Figure 5.** Induction motor efficiency maps: a) CF+FW; b) LMA

A complementary perspective is presented in figure 6. It can be seen that the main LMA influence region is below 15 N.m (about 30% of motor nominal torque), with a slight behavior difference, according to n<2000 rpm or n>2000rpm: in the former case (coincident with the constant torque zone), the LMA's efficiency gain is almost constant, while in the late case the energy savings decrease in a smooth way to zero.

**Figure 6.** LMA Efficiency gain (a) and relative loss differences (b) compared to (CF+FW)

Naturally, LMA acts directly on motor iron losses, since it regulates Ids. However, it has also an impact in motor copper losses, because it provides a better equilibrium between Ids and Iqs, particularly in regions where Ids regulation has wide limits. This can be seen in (7).

As a side-note, when considering the plane surfaces in figure 4 (for Id max and Id min), interesting correlations can be made with figure 3, through (Id max, Id min) dashed lines and the torque hyperbolas (e.g. higher torques are provided by Id min as the speed grows).
