Online efficiency optimization of IPMSM for electric vehicles

Received Mar 16, 2021 Revised Apr 30, 2021 Accepted May 9, 2021 This paper presents an online efficiency optimization method for the interior permanent magnet synchronous motor (IPMSM) drive system in an electric vehicle (EV). The proposed method considers accurately the total system losses including fundamental copper and iron losses, harmonic copper and iron losses, magnet loss, and inverter losses. Therefore, it has the capability to always guarantee maximum efficiency control. A highly trusted machine model is built using finite element analysis (FEA). This model considers accurately the magnetic saturation, spatial harmonics, and iron loss effect. The overall system efficiency is estimated online based on the accurate determination of system loss, and then the optimum current angle is defined online for the maximum efficiency per ampere (MEPA) control. A series of results is conducted to show the effectiveness and fidelity of proposed method. The results show the superior performance of proposed method over the conventional offline efficiency optimization methods.


INTRODUCTION
The electric vehicles (EVs) are the future of transportation. They have lots of advantages such as no emissions, low maintenance, low cost, and safety drive [1]- [3]. However, the batteries have limited capacities that affects the milage per charge. As most of battery power is consumed by the main drive system (motor + converter), it is very important to optimize their efficiencies to increase milage per battery charge.
At early days of transportation, the direct current machines (DCMs) were used because they have simple and linear controls. However, the existence of commutator and brushes cause lots of problems leading to lower speed ranges and lower efficiencies [4]. Recentely, due to the huge progress in field of power electronics and semi-conductors, the induction machines (IMs), synchronous reluctance machines (SynRMs), switched reluctance machines (SRMs), and interior permanent magnet synchronous machines (IPMSMs) are reported for EVs. Due to the presence of copper losses in IMs, they have relatively low efficiencies and low power factors which is disadvantageous for EV traction. The SynRMs have relatively high torque ripple. They also need high volt amper (VA) rating of inverters due to the poor power factor. The SRMs have hnherited high torque ripple due to the sequential commutation of coils, doubly salient structure, and deep magnetic saturation. They also poss complicated control algorithms [4]- [7]. The IPMSMs have the best performance to be used as the main drive in EVs. They offer not only high efficiency, wide speed range, high power density, but also small weight and size with low noise [8], [9]. Despite the high efficiency of the IPMSMs, much research has been conducted to improve motor efficiency. On one side, new design configurations for both the stator and rotor are developed to improve motor efficiency [5], [8]. Also, the converter design is investigated for better system efficiency [10], [11]. On the other side, new control techniques are introduced to improve system efficiency. First, the maximum torque per ampere (MTPA) is developed [12]- [15]. Then, the maximum efficiency per ampere (MEPA) is introduced [16]- [20].
The MEPA has better efficiency improvement compared to MTPA. This is mainly because the MTPA control reduces only the copper losses in PMSM [17], [18]. As a result, it does not guarantee the maximum system efficiency (motor + converter). On the contrary, the MEPA considers not only the copper losses in PMSM but also the iron losses, inverter losses, mechanical losses, and harmonic iron and copper losses. This, in turn, helps to accurately estimate and improve the overall system efficiency [17]- [20]. Therefore, the MEPA control has an increasing interest by the research community. Ni et al., Uddin and Rebeiro,and Yang et al. in [18]- [20], the fundamental iron losses are considered to maximize the system efficiency. However, harmonic iron losses are not considered which affect motor efficiency, especially in the high-speed and light load regions [16]. Balamurali et al and Ni et al. in [16]- [18], offline MEPA control methods are proposed. These methods require numerous repetitive experiments to generate the lookuptables. The detailed information of machine losses under all operation conditions is also required. Moreover, they are very time-consuming and require a huge computation burden. Furthermore, the high frequency machine losses are rarely considered, which results in an inevitable searching error [21], [22]. In addition, the generated lookup table does not consider the variation of stator resistance due to temperature, the variation of DC voltage, and the variation of machine parameters. The lookup table fits only a limited area of operation. To solve these problems, an online maximum efficiency per ampere (OMEPA) control method has been developed in this paper.
This paper presents an OMEPA control method for IPMSM in EVs. First, an accurate machine model that considers magnetic saturation, spatial harmonics, and iron loss effect is built. The finite element analysis (FEA) is employed to calculate the magnetic characteristics of IPMSM. The FEA considers accurately the saturation and spatial harmonics as well as cross-coupling. Second, the total system efficiency is calculated accurately by the consideration of detailed system losses including fundamental and harmonic iron losses, copper losses, as well as inverter losses. Third, an online searching algorithm for MEPA has been developed to derive the optimal current angle, and hence the optimal efficiency over the entire speed range.
The rest of the paper is organized is being as: Section 2 involves the detailed system and loss modelling. Section 3 gives the proposed online MEPA control. The results and discussions are in Section 4. Finally, Section 5 is the conclusion.

SYSTEM MODELING
The system modeling involves the modelling of IPMSM and the detailed analysis of system losses including copper losses, fundamental iron losses, harmonic iron losses, magnet loss, and inverter losses.

Modeling of IPMSM
The modeling of an IPMSM can be represented by (1).
where (v d , v q ), (i d , i q ), and (λ d , λ q ) are d and q-axis components of voltage, current, and flux-linkage, respectively. R s is the stator resistance. ω e and ω are the electrical and mechanical angular speeds. T e is the motor torque, p is the pole pairs, J is the inertia, B is the frictional coefficient, and T L is the load torque.
To consider the effect of magnetic saturation and spatial harmonics, λ d , λ q , and T e are calculated as functions of i d , i q and rotor position θ i as illustrated by (2). The relationships and T e (i d , i q , θ i ) are calculated using the finite element analysis (FEA). These data are obtained via FEA by varying i d from -20 to 0 A in steps of 1A, i q is changed from from 0 to 20 A in steps of 1A, and θ i is varied from 0˚ (d-axis) to 72˚ (q-axis for 12 slots, 10 poles IPMSM) insteps of 1˚ (mech. degree). Then, the average fluxlinkages and torque are obtained according to (3). These results are shown in Figure 1  By performing inverses of (3), i d and i q are determined by (4). The IPMSM model can be built as illustrated in Figure 3. The model is built based on (1)- (4). The inputs are the v d and v q . The voltages are employed to estimate λ d and λ q using (1). Then, armature currents (i da , i qa ) are calculated using flux inverse model in (4). The iron losses are represented by currents i dfe and i qfe . The motor electromagnetic torque is estimated as a function of i da and i qa currents. Noting that the model involves not only magnetic saturation and spatial harmonic effects but also iron loss effect.  Figure 3. The schematic of IPMSM model

Loss modeling
This section describes the total system losses including motor and inverter losses. The total system losses (P loss ) are copper losses (P cu ), fundamental iron losses (P Fe, f ), harmonic iron losses (P Fe, h ), magnet loss (P Mag ), and inverter losses (P inv ). The total system losses (P loss ) are given by (5). = + , + ,ℎ + + (5)

Copper losses
The fundamental copper loss of an IPMSM can be estimated by (6). The harmonics copper losses can be ignored as the employed switching frequency of inverter is 10kHz [16].

Iron losses
The Bertotti iron loss formula is a widely used method to evaluate the iron loss in electric machines. It calculates iron losses per unit volume is being as [23].
= √3 where f is the frequency. The coefficients k h , k c , and k e are for hysteresis, eddy current, and additional losses, respectively. V t and V y are the total volumes of stator tooth and yoke, respectively. The A tc and A yc are the equivalent areas of stator tooth and yoke, respectively. h t and h y are the heights of stator tooth and yoke, respectively. Q is the number of slots. A t and A y are the physical areas of stator tooth and yoke, respectively. D s is the outer diameter of stator. is the material conductivity. k d is the lamination thickness. N T is the number of phase turns. K w is the winding factor. ϕ d and ϕ q are the d-and q-axis fluxes. α i is the pole arc factor. Without the consideration of excess loss, the equivalent iron loss resistance R c is given as R c = ⍵ e 2 /k 1 .

The harmonic iron loss
It is calculated as a function of ripple voltage rms, ∆ 2 , DC link voltage V dc , and modulation index m as given by (8) where V m is the peak output voltage. K h, eddy is the eddy current loss coefficient, it is taken as 2.3mW/V 2 [24].

The magnet loss
The magnet loss depends on magnet volume (V M ), magnet width (b M ), magnet resistivity ( M ), and the maximum value of flux density (B m ).

The inverter losses
The inverter has both conduction and switching losses [25]. The conduction loss of one insulated gate bipolar transistor (IGBT) (P co-IGBT ) and conduction loss for one diode (P co-diode ) in a 2-level VSI can be defined by (10). Besides, the switching losses of one IGBT (P SW-IGBT ) and switching loss for one diode (P SWdiode ) in a 2-level VSI can be defined by (11) [26]. The total inverter losses (P inv ) can be estimated by (12).
where cosθ is the power factor, V ceo and V Do depict the threshold voltages for IGBT and diode, respectively, R o and R D represent the resistances of IGBT and diode, respectively, f s is the switching frequency, e on and e off are the required amount of energy to turn-on and turn-off the IGBT, respectively, e rr is the required amount of energy to turn-off the diode, I m is the peak magnitude of current, V nom and I nom are the nominal voltage and current of loss measurements, respectively.

THE PROPOSED ONLINE EFFICIENCY OPTIMIZATION METHOD
The proposed OMEPA control determines the current angle ( ) that maximizes the system efficiency. The flowchart of searching algorithm is illustrated in Figure 4. The procedure is achieved online for a known maximum current (I m ), and motor speed (⍵). For each operating point, the current angle is changed in small steps (Δ ). The d-and q-axis current components are estimated by (13). Then, the total system losses (P loss ) are estimated using (5)- (12). After that, the system efficiency (ƞ) is updated by (13) considering the voltage and current constraints in (14). The online efficiency calculation should also consider the variations in temperature, DC voltage, and motor parameters. − Measure DC voltage (V dc ), speed (ω), and temperature. − The winding resistance is updated as a function of temperature using (15). − For each current angle , the d-and q-axis current components are calculated using (13). − The d-and q-axis flux linkages are updated using (16). − The d-and q-axis voltage components are updated using (17). − The system losses are estimated using (5)- (12). − The online system efficiency is updated using (13) considering the voltage and current constraints in (14). − If ≥ max , determine angle that corresponds to maximum efficiency. Else, is changed by Δ , then go to point 2.  (17).
Enter the desired speed and peak current β = βinitial Calculate the direct and quadrature currents (id , iq ) Calculate the total system losses (fundamental and harmonic motor losses + inverter losses) using (5 ) Update torque and efficiency modeling considering saturation and cross saturation and iron losses.
where V max and I max are the maximum permissible phase voltage and phase current, respectively. R o depicts the winding resistance at temperature T o . T is the present temperature of windings. α is the thermal coefficient of copper. L d and L q are d-axis and q-axis inductances. λ PM is the PM flux linkage.

RESULTS AND DISSCUSSION
The results include a steady state comparison between proposed OMEPA control method and offline maximum efficiency per ampere-lookup table (MEPA-LUT) control method reported in Ni et al [18]. Moreover, the dynamic behavior of OMEPA is presented and analyzed. The influence of DC voltage variation, temperature variation, and the variation of motor parameters is also investigated.

The steady state comparision
It is worthy noted that, the MEPA-LUT has an inherited capability to accurately determine the system efficiency as it runs motor model at each change in current angle ( ). It also considers the update of reference voltages after the PI controller which improves the estimation of system losses. However, this method considers certain operating conditions. Any change of theses condition could lead to unaccepted performance. If the proposed OMEPA method can produce a high efficiency as MEPA-LUT control method, this means one thing that the proposed OMEPA can accurately estimate the system efficiency without the need to run motor model. It updates the flux and voltages after the variation of using (16), (17). Figure 5 (a), and Figure 5 (b) shows the steady state comparison between proposed OMEPA and conventional MEPA-LUT methods under two speeds of 1000 and 1750r/min, respectively. for each speed (1000 and 1750r/min), the maximum current is changed from 2A to 12A. Then, the best current angle is defined by each control method (OMEPA and MEPA-LUT). After that, the corresponding system efficiencies are calculated and compared. As noted, the two efficiencies are very close. This in turn verifies the calculation accuracy of system efficiency using the proposed OMEPA. The lower part of Figure 5 shows efficiency error (η Error ). This error is defined by η Error = η OMEPA -η MEPA-LUT . Positive error means higher system efficiency by the proposed OMEPA and vice versa. As noted, the efficiency error (η Error ) is almost positive despite its small values. This happens because the proposed OMEPA method could search with smaller angle resolutions (Δ ) compared to MEPA-LUT method. Figure 6 shows the dynamic performance of the proposed OMEPA under sudden changes in load torque at the rated speed of 1500r/min. The load is changes as illustrated by Figure 6 (a). As a result, the maximum current (I m ) is also changing correspondingly with load torque see Figure 6 (b). The proposed OMEPA determines the best current angle ( ) that corresponding to maximum efficiency see Figure 6 (c). As noted, the proposed OMEPA updates each control period which reflects the fast dynamic performance in Figure 6 (c). The variation of also varies both the d-and q-axis current as shown in Figure 6 (d), Figure 6 (e), respectively. Theses affects the system losses and hence the system efficiency as seen in Figure 6 (f).

Effect of DC voltage variation
In most EVs, the battery voltage (DC voltage) changes depending on its state of charge (SOC). The voltage variation affects system performance especially at high speeds. Figure 7 shows the simulation results under variation of DC voltage at 2000r/min. The DC voltage is changed from 200V (rated voltage) to 170V at 0.1sec. As seen in Figure 7 (a), Figure 7 (b) and Figure 7 (c), for the convention MEPA-LUT method, from 0 to 0.1sec, the system tracks accurately the commanded reference currents and produces the required torque under the rated voltage. When the voltage decreases to 170V at 0.1sec, the system fails to track the currents and hence fails to produce the required torque. This happens because the LUT are generated under rated voltage of 200V and does not consider voltage variations. Hence, the reference d-and q-axis currents need to be corrected with the variation of DC voltage. On the contrary, for the proposed OMEPA method, the system tracks accurately the commanded reference current despite voltage variations, it also produces the required torque Figure 7

Effect of temperature and motor parameter variations
The IPMSMs in EVs have temperature sensors that are installed in stator windings. These sensors provide instantaneous temperature variations not only for protection issues but also for control ones. The winding resistance of IPMSM changes with the temperature as illustrated by (15). The change of stator resistance will affect the amount of copper losses. Hence, it affects the accuracy of efficiency calculation for both the online and offline calculations.
The offline MEPA control methods assumes constant stator resistance. When the temperature increases, the resistance will also increase, and hence the copper losses. Therefore, the offline MEPA method will deviate from the MEPA operation. In order to include the effect of temperature within the offline MEPA control, several experimental measurements are required which means cost, effort, and time. On the contrary, once the temperature is measured by the temperature sensors, the stator resistance is updated using (15) within the control algorithm of the proposed OMEPA. This is a very simple implementation method that compensates for the resistance variation due to the temperature. Figure 8 (a), and Figure 8 (b) shows the effect of temperature on both copper loss and efficiency, respectively. The amount of copper losses considering temperature effect (P cu-CTE ) is greater than that without considering temperature effect (P cu-WCTE ) as illustrated by Figure 8 (a). The corresponding system efficiency at 10A is shown in Figure 8 (b), as seen, the proposed OMEPA has higher system efficiency as it considers the real amount of copper losses due to temperature variations. If the motor parameters changes, they can be considered within the control algorithm of OMEPA by the same way the temperature is considered. Online parameter estimation methods can be used to estimate the accurate motor parameters, then, theses parameters can be used very simply within the OMEPA control.
(a) (b) Figure 8. Effect of temperature variation on copper losses and efficiency, (a) copper losses, and (b) system efficiency at 1000r/min and at 100⁰C

CONCLUSION
An online maximum efficiency control method for IPMSM in EVs is introduced in this paper. The system efficiency is calculated online based on the detailed system loss models. A searching algorithm for the current angle that corresponds to maximum efficiency point is developed. The proposed OMEPA control offers high dynamic performance as it updates the current angle at each control period. It also offers higher flexibility as it can consider any instantaneous variations of DC voltage (battery voltage), temperature, and motor parameters. The proposed OMEPA control method is independent in tracking the maximum efficiency point as well as robust to DC voltage variations.