LMA optimized solutions
This work addresses the problem of optimal selection of the flux level in induction motors used in electric vehicles (EVs). The basic function of a fully electric powertrain controller is to generate electric torque (force) which is required at any time by the driver. But, it is well-known that the flux level used in a controller for induction motors offers an extra degree of freedom that can be used to maximise energy efficiency. The induction motor is an efficient motor when working close to its rated operating point (Zeraoulia, Benbouzid et al. 2006). However, at light loads the efficiency is greatly reduced when magnetization flux is maintained at nominal value. In induction motor drives for EVs, where real operation conditions are significantly different from rated conditions, the energy saving control is crucial for improving the running distance per charge.
Due to the widespread use of induction motors, its efficiency optimization gave rise to a large number of research publications (Bazzi & Krein 2010). Algorithms for real-time implementation of loss-minimization methods are vital for designing intelligent and optimized EV controllers. Standard methods for induction motor control, including field-oriented control (FOC) or direct torque control (DTC), can be improved in efficiency by using loss minimization control. Basically, there are three different methods to improve the efficiency in induction motors: i) loss model based methods (which is considered in this work), ii) power measure based methods, also known as search controllers; and iii) hybrid controllers that combines the first two methods. The main goal of the present work is to investigate the potential benefits of loss minimization algorithms in EVs powered by induction motors. Accordingly, a detailed simulation case study will be provided which will show that, depending on the type of driving cycle, energy savings up to 12.5% can be achieved. The chapter is organized as follows: Section 2 reviews the basic concepts of rotor field oriented control (FOC). Section 3 introduces the loss minimization method based on a standard mathematical model of the induction motor and gives the value of the flux level 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.
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).
Recall that, in the synchronous frame (ωe), the rotor magnetic flux (Ψrd) is aligned with d axis, thus Ψrd = Ψr, Ψrq =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 π/2 electrical rad from ids) controls the motor electromagnetic torque:
From figure 1, it may be seen that ids and iqs are, respectively, the d and q components of the space vector isin the synchronous reference frame. This way, from the control philosophy perspective, idsand iqs regulation is implemented in this reference frame; however, from the control hardware perspective, ids and iqs must be considered in stator reference phase-coordinates (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 θs=θe+θT. The determination of θe is the main issue, since θT= arctg(iqs/ids); θe calculation can be accomplished through θslipand θr (see figure 1) – indirect FOC.
θslip is given by:
Knowing the instantaneous rotor speed ωr, one have:
From figure 1:
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):
Where:: d-axis stator current in the synchronous reference frame; : q-axis stator current in the synchronous reference frame;: electrical angular frequency;
Rs; Rr: stator and rotor resistances (respectively);
Rm: equivalent stator iron losses resistance;
Lr; Llr: rotor total inductance and rotor leakage inductance (respectively);
Lm: magnetizing inductance.
Note that Rd(ωe) and Rq(ωe) 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):
σ: 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 consideredωe.
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:
Idn : rated d-axis stator current
Te : electromagnetic torque (steady-state), considering rotor FOC;
[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:
where L(idse, iqse, λ) is the lagrangian associated to the problem, λ is the Lagrange multiplier, Ploss(idse, iqse) is the cost function and Te-Ktidseiqse is the restriction. Applying first-order optimal condition (15) gives the following equation system:
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 (ωn), which is defined by the interception of inequality restrictions border lines:
The calculated result is:
ωc is the boundary speed between constant power and power-speed (Pmec*ωe=constant) regions:
For region 1, the maximum torque is limited by Idn and Imax:
The maximum torque in region 2 is limited by Vmax and Imax:
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:
3.1.3. Optimal Idsgeneration
For the zone in the (idse, iqse) plane limited by restrictions (10), (11) and ieds≤Idn, optimal result (18) is valid, meaning that:
In the border lines of those restrictions, the previous relation can not be considered. So, for region 1, (18) is applied if:
The ieqs upper limit in (28) defines Tp1(see Figure 3):
Of course, for:
For region 2, (18) can be considered, until the voltage limit (Vmax) is achieved:
This way, Tp2 is given by the following expression:
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 ωe>ωc.
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).
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 Idsgeneration 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|
Figure 4 represents the optimal Id generation for conventional approach, i.e. constant flux +field weakening (CF+FW), and the LMA approach.
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.
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.
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 theconstant 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.
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).
4. Simulation model
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 was built, which is represented in figure 7.
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.
4.1. Drive cycle+vehicle model
The drive cycles plus the vehicle and mechanical transmission models were implemented with the QuasiStatic Simulation Toolbox (QSS TB), based on Matlab/Simulink, developed by (Guzzella, Amstutz, 2005). The QSS TB library integrates a set of several elements, such as driving cycles, vehicle dynamics, internal combustion engine, electrical motor and mechanical transmisson. Batteries, supercapacitor and fuel cell are also included.
Essentially, it considers a backward (wheel-to-engine) quasi-satic causal model which, based on driving cycle speeds (at discrete times), calculates accelerations and determines the necessary forces, based on the vehicle features and an eventual mechanical transmission. The implemented model includes the QSS TB elements depicted in figure 8.
The load power demanded to the induction motor (Tload*ωr) considers the drive cycle, vehicle dynamics (rolling and aerodynamic resistance, only in the plane) and also a mechanical transmission with a fixed gear ratio. The vehicle dynamics is modelled by the following equation:
Mt - vehicle mass + equivalent mass of rotating parts;
v(t) – vehicle instantaneous longitudinal speed;
Fd(t) – instantaneous driving force;
g – gravity acceleration;
Cr, Cw – rolling friction coefficient, aerodynamic drag coefficient;
ρ, A – air density; vehicle´s cross section.
Besides the inertia force, associated to vehicle displacement, the inertia of rotating parts (i.e., kinetic energy stored on it caused by rotational movement) should also be considered, since it is the motor(s) who supply it. This is considered in the “equivalent mass of rotating parts” Mt term (see Table 3). It should be noted that driving cycle block output speed (v) and acceleration (dv) are discrete values. The time step size default value is 1 s; however, in order to increase simulation accuracy, its value was fixed in 0,01 s.
|Total vehicle’s mass (kg)||350|
|Rotating mass (%)||5|
|Vehicle´s cross section (m2)||1,5|
|Wheel diameter (m)||0,3|
|Aerodynamic drag coefficient||0,3|
|Rolling friction coefficient||0,008|
|Idling losses by friction (W)||10|
|Minimum wheel speed beyond which losses are generated (rad/s)||1|
4.2. Rotor flux setpoint generation
Figure 9 presents the developed LMA block set. Rd and Rq are inputs for the block regions “we<wn” and “we>wn”. Basically, these two elements generate Ids*, according to 3.1.3. As it was described, for zones in the (ids; iqs) plane limited by restrictions (10), (11) and ids≤Idn, equation (27) is applied. For the border lines, the three defined regions must be considered: in region 1, Ids* is restricted to its maximum allowable value (Idn); for regions 2 and 3, only voltage limit is considered in Ids* generation restriction. Since Tp2= Tp3, the same block can be used for generating Ids* in these two regions.
Since the flux level should not decrease below a minimum value (Id_min), in order to guarantee that Id_min≤Ids≤ Idn, two saturation blocks are placed at “we<wn” and “we>wn” outputs.
Figure 10 shows the interior of “we<wn” block.
As it can be seen, Ids* is generated by (27), while Ids<Idn; after that Ids*=Idn. It should be noted that the absolute value of Iqs must be used, in order to consider both motor and braking modes.
The block “we>wn” is represented in figure 11. Equation (27) regulates Ids* generation until (31) is no longer true (notice that the absolute value of iqs is compared to the product of “Vmax restriction” by (Rd/Rq)1/2). After that, Ids* is given by zone 3 solution in table 1 (s3)– “Id* for Vmax restriction border” block. It also should be pointed that when a load point overcomes the voltage limit, the result given by (s3) is a complex value. In order to deal with this issue, for these situations Ids is taken from the conventional flux regulator.
In contrast with the LMA, the conventional flux regulation (depicted as block (3b) in Figure 7) generates a Ids setpoint according to the following strategy:
4.3. Rotor indirect FOC
Figure 12 shows the block structure for indirect FOC.
Equations (1) and (2) are the basis of “Iq reference” block. Equations (3)-(6) are implemented in “θe calculation” block (notice that ωe=ωslip + ωr). The bottom block considers the coordinates change of instantaneous stator currents, from phase domain to d-q synchronous frame. To do so, the following well known coordinate transformation matrix is applied:
The “Current Control” block generates stator reference voltage (Vsdq*) in synchronous frame (through PI´s current ids and iqs controllers), which is applied to the motor model, in phase coordinates, in order to make the real instantaneous stator currents to achieve the reference values.
4.4. Induction motor model
Figure 13 presents the induction motor model considered in simulations, which also includes motor iron losses.
When comparing this model to the one considered in LMA (figure 2), the major differences are in parallel (magnetizing) branch. Since core losses currents are not considered in the major circuit, it is expectable that the voltages (Vedm and Veqm) on the independent sources are larger compared to the parallel branch voltages in the equivalent model of figure 13. Since core losses are given by ((Vedm)2+(Veqm)2)/Rm, it seems plausible to admit that the core losses in LMA model are higher than the ones in figure 13 model.
5. Simulation results and analysis
An important note is that simulation results were extracted through block 5 (see figure 7), where Puis obtained directly through Te*ωr, based on the drive cycle reference values. In block 3a, Pu is achieved considering Pab-plosses (note that Pab=usaisa+usbisb+uscisc, i.e. the sum ofinstantaneous power of motor phases a, b, c – see figure 14). Motor losses considered byLMA are based on equation (7). There are some differences in Pu values when block 5 or block 3a are considered, which seems to put in evidence the issue mentioned in 4.4.
For each drive cycle, results are presented following the same pattern: the first figure includes the main results for conventional flux regulation and LMA. The second figure
represents the load torque demanded by the drive cycle, while in the third one the motor limits and working points imposed by the drive cycle are illustrated, together with the most significant LMA´s efficiency gain zones. Finally, a table with LMA and conventional flux regulation energy performances is also presented.
From a general perspective, these results confirm the main LMA features, described in section 3.2 visible differences from conventional flux regulation occur for low load torque, particularly for relative low speeds. This agrees to the fact that in regions where Ids has a large regulation flexibility, LMA and conventional flux regulation have clearly different performances.
|Motor losses (kJ)||83,7||45,0|
|Energy efficiency (%)||72,6||83,1|
In almost 50% of the ECE-R15 drive cycle duration, motor speed is between 0 and 2000 rpm, with the motor torque among -13 Nm and 16 Nm (aprox.) – see figure 16. So, LMA inclusion allows significant loss reductions (table 5): with LMA, total losses are about 20% of Eu(energy supplied by the motor); without LMA, goes up to 38% of Eu. As expected, the main Ids differences occur for n<2000 rpm, particularly for low torques (with LMA, smaller Ids values are clearly visible). In a similar way, LMA performance in braking modes brings good results, since demanded torque has always low values. It should be pointed that when the vehicle is immobilized (Iqs=0), LMA performance leads to very significant results (figure 15), since Ids is regulated to its minimum value, while with conventional flux regulation, Ids has its maximum value. In this case, motor iron losses are much higher when compared to the ones with LMA.
From an energy perspective, although LMA acts directly on the iron losses (since it regulates Ids), it has also an impact in motor copper losses (as mentioned in section 3.2). Although for a given torque value, Iqs with LMA is higher than with conventional flux regulation (since Ids is smaller), a better equilibrium between Ids and Iqs is achieved with LMA. Since copper losses are also dependent on Ids2 and Iqs2, the motor efficiency has a clear improvement in this drive cycle scenario, which may be seen from figure 17. Nevertheless, efficiency values are relative low, which is no surprise if one take into consideration the efficiency maps (figure 5) together with figure 17 (notice the efficiency (power) curve gains and cycle working points, particularly for n<2000 rpm ).
5.2. Europe: City
For about 60% of total time of the “Europe: City” cycle, the vehicle speed is also below 2000 rpm, with the motor torque between -13 Nm and 16 Nm (aprox.). The vehicle is at rest for about 25% of the drive cycle duration. Basically, it puts the motor in the same (T,ω) working region as ECE-R15 (see figures 17 and 20). However, since it has a short time period (195 seg.), energy level demanded is much lower – the lowest one from the chosen drive cycle set. Similar relative energy losses are achieved: 20% for LMA and 36% without LMA of Eu (table 6). In both motor and braking modes, LMA most relevant results are in low speed – low torque region.
|Motor losses (kJ)||19,8||11,2|
|Energy efficiency (%)||73,6||83,2|
The slightly efficiency increase for this cycle (when compared to ECE-R15) may be associated to the relative decrease of vehicle resting period (about 33% in ECE-R15).
5.3.11. – Mode (Japan)
Drive cycle period where n<2000 rpm is relative short (<33%); the motor torque lies between 13 Nm and -10 Nm and the vehicle is immobilized a little less than 25% of the cycle duration. As expected, it´s in the initial resting time period and on the final 25 sec that Ids values generated by the LMA are significantly different from the conventional flux Idsvalues. In other words, the motor losses difference are attached to these time periods, particularly to the resting one (figure 21). On the other drive cycle periods, motor losses are very similar (notice that in some intervals, LMA losses are slightly larger. This unexpected result is most probably related to the issue discussed in section 4.4).
LMA total losses are about 14% of Eu, while conventional regulation losses are 16% of Eu. Although curve efficiency gains in figure 23 are referred to power efficiency, cycle working points somehow agree with efficiency energy gain achieved with LMA (table 7): for n> 2000 rpm there is a significant number of points between 1% and 5 % efficiency curves gain; also notice that some points are below 1% efficiency gain.
|Motor losses (kJ)||12,5||10,8|
|Energy efficiency (%)||86||87,6|
For this drive cycle, the time period for which n<2000 rpm is shorter then the previous cycles. Motor torque limit is now -20 Nm and 25 Nm (aprox), while maximum speed is 8000 rpm. Frequent accelerations, as well as its long time period (1840 sec), make this cycle the most energy demanding. At the same time, pushes the motor to its limits: figure 24 shows that motor exceeds its nominal power between [200-300] s and later in the interval [1500-1700] sec. However, this overload (whose maximum instantaneous power is about 11 kW) occurs for a small number of intervals, each one with a very short existence. This way, it´s reasonable to assume that motor is not under electric hazardous working conditions. From a mechanical perspective, maximum speed - about 4 times motor nominal speed – is reached for relative short intervals, so one may assume that the motor (and the vehicle) will be safe in this working conditions.
Due to high speeds and relative high torque demand (figure 25), LMA shows a relative performance closer to the conventional regulation. As expected, relevant differences for Id generation occur for relative low speed (basically, when the vehicle is at rest) and low torque values, e.g. intervals [50-200; 800-950] sec. – figures 25 and 26. With LMA and without it, motor losses are, respectively, 11,4% and 13,1% of Eu (table 8). Motor efficiency map (figure 5) explains the high efficiency values associated to this cycle, while the small efficiency gain achieved is according to figures 25 and 26.
|Motor losses (kJ)||225,1||194,6|
|Energy efficiency (%)||88,4||89,8|
As a final remark, it is interesting to note that Europe:city and ECE-R15 have similar efficiency levels; also the same fact can be seen for 11-Mode and FTP-75.
Induction motor drives for EVs are submitted to a large set of working conditions, quite different from rated ones. Motor energy saving is fundamental for improving EVs performances. Under the loss model based approach previously discussed (LMA), a set of simulation results was presented in this book chapter, aiming to improve the induction motor energy performance. Different standard driving cycle scenarios were considered in order to evaluate the chosen LMA features: compared to conventional flux regulation, the major improvements in motor efficiency are for low load torque, particularly for relative low speeds. These are the motor working points where its efficiency is tipically lower, which is an interesting LMA feature. This is in agreement to the fact that LMA action has a more significative impact on ECE-R15 and Europe:city efficiencies, as explained through figures 15,17 and 18, 20 analysis.
Due to LMA impact on iron losses (function of Id), a possibility to be considered in future works is the impact of LMA on motors with higher power rates and/or high efficiency level motors, where the relative weights of iron and copper losses are different.