Accurate and Efficient Torque Control of an Interior Permanent Magnet Synchronous Motor in Electric Vehicles Based on Hall-Effect Sensors

Abstract

In this paper, an effective method to achieve accurate and efficient torque control of an interior permanent magnet synchronous motor (IPMSM) in electric vehicles, based on low-resolution Hall-effect sensors, is proposed. The high-resolution rotor position is estimated by a proportional integral (PI) regulator using the deviation between actual output power and reference output power. This method can compensate for the Hall position sensor mounting error, and estimate rotor position continuously and accurately. The permanent magnetic flux linkage is also estimated based on a current PI controller. Other important parameters, such as the d-axis and q-axis inductances, stator resistance, and energy loss, are measured offline by experiments. The measured parameters are saved as lookup tables which cover the entire current operating range at different current levels. Based on these accurate parameters, a maximum torque per ampere (MTPA) control strategy, combined with the feedforward parameter iteration method, can be achieved for accurate and efficient torque control. The effectiveness of the proposed method is verified by both simulation and experimental results.

1. Introduction

The interior permanent magnet synchronous motor (IPMSM) has been widely used in electric vehicles for its high efficiency and high power density. Accurate and efficient torque control is essential for automotive drive systems. However, it is difficult to achieve this target because of the varying parameters of the IPMSM [1].

Although the resolver can obtain high-resolution rotor position of the IPMSM, it is not suitable to be used in vehicles, considering its high cost, weight, and volume. Algorithms based on no position sensors [2] cannot estimate the rotor position accurately in all conditions, in addition to the high-volume computation. In fact, Hall-effect sensors can be used to realize low cost, weight, volume, and high-resolution rotor position. Considering the discrete position signals from Hall-effect sensors, it is quite necessary to conduct research on obtaining continuous rotor position signals.

The most popular method to estimate the rotor position is based on average motor speed [3]. When the motor speed varies frequently, this method may cause a very large error and failure of the control strategy. A vector-tracking observer was proposed to estimate the rotor positon in [4]. This method needs to acquire the stator voltages, thus increasing the cost and complexity of the system. An improved vector-tracking observer was proposed to eliminate the complexity of the system in [5]. A method based on back electromotive force (EMF) was proposed in [6]. Although this method can obtain accurate rotor speed, it cannot obtain accurate rotor position. A method based on flux estimation was proposed for a slotless permanent magnet synchronous motor (PMSM) in [7]. An improved square root unscented Kalman filter (SRUKF) was proposed to estimate rotor speed and position in [8]. Additionally, some studies used linear Hall sensors [9] to estimate rotor position instead of discrete Hall sensors.

To achieve accurate and efficient control, accurate parameters are essential. There are, mainly, two types of methods proposed to achieve parameter identification. One is based on offline measurements, and the other is based on real-time estimating arithmetic [1]. The real-time estimating arithmetic based on the PMSM model was analyzed in [10]. Some parameters must be assumed to be known since not all parameters are identifiable in the steady state. A method was proposed to estimate stator resistance and flux linkage by injecting short-period d-axis current into surface PMSM in [11]. However, it is not suitable for application in vehicles. Mechanical parameters of induction motors were estimated by using voltage sensors only in [12]. The variation of the d-axis and q-axis inductances measured offline [13], or calculated from the finite element [14] in consideration of the cross saturation were proposed. An analytical calculation method of d-axis and q-axis inductances for IPMSM based on winding function theory was proposed in [15]. An adaptive parameter estimator that can achieve maximum torque per ampere (MTPA) control was proposed in [16].

In this paper, an effective method to achieve accurate and efficient torque control of an IPMSM in electric vehicles, based on low-resolution Hall-effect sensors, is proposed. A variety of important parameters are estimated for torque control. To obtain the high-resolution rotor position, an estimation method based on the power closed-loop is proposed with low-resolution Hall-effect sensors. This method can compensate for the Hall position sensor mounting error, and estimate the rotor position continuously and accurately. The permanent magnetic flux linkage is estimated based on a current proportional integral (PI) controller. Other important parameters, such as the d-axis and q-axis inductances, stator resistance, and energy loss are measured offline by experiments. The measured parameters are saved as lookup tables which cover the entire current operating range at different current levels. Combined with the feedforward parameter iteration method, the MTPA control strategy can be achieved based on accurate parameters. As a result, it is an effective approach to realize accurate and efficient torque control.

MATLAB/SIMULINK models were built to analyze the proposed method. Furthermore, experiments were carried out to verify the method.

2. Parameter Estimation

2.1. Analysis of the IPMSM Energy Model

For IPMSM, the input energy comes from the inverter. According to the law of energy conservation, the input energy can be expressed as follows [17]:

$$E_{in}=E_{out}+E_{loss}$$

(1)

$E_{loss}$ can be expressed as follows:

$$E_{loss}=E_{Cu}+E_{Fe}+E_{Str}+E_M$$

(2)

From the point view of inverter, the input energy can be expressed as follows:

$$E_{in}={\displaystyle \int 3(v_q{i}_q+v_d{i}_d)/2dt}$$

(3)

and the output energy can be expressed as follows:

$$E_{out}={\displaystyle \int {\omega }_e{T}_e/p}dt$$

(4)

where ${\omega }_e$ can be obtained from Hall-effect sensors.

Considering the cost, the IPMSM used in vehicles has no torque transducer; hence the actual output energy cannot be obtained from Equation (4). However, if the input energy and the energy loss are known, the output energy can be obtained from Equation (1). The IPMSM energy model can be used to estimate the essential parameters, such as the permanent magnetic flux linkage [18], the difference of the q-axis and d-axis inductance [1]. In this paper, the energy model is used to estimate the high-resolution rotor position based on low-resolution Hall-effect sensors.

2.2. Rotor Position Estimation

2.2.1. Rotor Position Estimation Based on Average Motor Speed

The high-resolution rotor position can easily be estimated through a method based on average rotor speed, where the six sectors are classified according to the states of the Hall sensor’s signals [4]. Assuming that the rotor speed within a sector is constant and the average speed in present and previous sectors is uniform, the rotor speed can be approximated as follows [3]:

$${\omega }_e=\frac{{\theta }_1-{\theta }_0}{t_1-t_0}=\frac{d\theta }{t_1-t_0}$$

(5)

where $d\theta$ is the electrical degree between two sectors, $t_1-t_0$ is the time interval of the previous sector; then the rotor position can be estimated as follows:

$${\theta }^{*}={\theta }_1+{\omega }_{e}t={\theta }_1+\frac{d\theta }{t_1-t_0}t$$

(6)

Similarly, the high-resolution rotor position can be obtained, as shown in Figure 1.

${\theta }_1-{\theta }_0$ should be 60 degrees if the Hall sensors are aligned accurately. However, in the IPMSM, there is a certain error in the hall sensor installation. The back EMF can be used to avoid the error and eliminate the third harmonic. However, large estimation errors will occur when this method is used in vehicles where the motor speed changes frequently.

2.2.2. Rotor Position Estimation Based on the Power Closed-Loop Method

For improving the accuracy of estimated rotor position, a method based on the power closed-loop is proposed to estimate the rotor position, as shown in Figure 2.

The reference electromagnetic torque $T_e^{*}$ can be expressed as follows:

$$T_e^{*}=\frac{3}{2}p\left({\stackrel{\wedge }{\phi }}_f{i}_{oq}^{*}-dL{i}_{od}^{*}{i}_{oq}^{*}\right)$$

(7)

where:

$$\begin{array}{l}{i}_{od}^{*}=i_d^{*}-i_{cd}=\frac{1}{A}\left(i_d^{*}+\frac{{\omega }_e{L}_q{i}_q^{*}}{R_c}-\frac{{\omega }_e{}^2{L}_q{\phi }_f}{R_c{}^2}\right)\\ i_{oq}^{*}=i_q^{*}-i_{cq}=\frac{1}{A}\left(i_q^{*}-\frac{{\omega }_e\left({\phi }_f+L_d{i}_d^{*}\right)}{R_c}\right)\end{array}$$

(8)

$$A=1+\frac{{\omega }_e{}^2{L}_d{L}_q}{R_c{}^2}$$

(9)

The reference output power can be expressed as follows:

$$P_{out}{}^{*}=T_e^{*}{\omega }_e/p$$

(10)

For the current closed-loop control, the stator d-axis and q-axis current $i_d$ and $i_q$ can be expressed as follows:

$$i_d=i_d^{*},i_q=i_q^{*},(i_d,i_q)=\mathrm{ParkTrans}(\mathrm{ClarkeTrans}(i_u,i_v,i_w),{\theta }^{*})$$

(11)

The output voltage of the d-axis and q-axis can be expressed as follows:

$$({\mathrm{v}}_d,{\mathrm{v}}_q)=\mathrm{ParkTrans}(\mathrm{ClarkeTrans}({\mathrm{v}}_u,{\mathrm{v}}_v,{\mathrm{v}}_w),{\theta }^{*})$$

(12)

The input power from inverter to motor $P_{in}$ can be expressed as follows:

$$P_{in}=3(v_q{i}_q+v_d{i}_d)/2$$

(13)

The actual mechanical output power P_out can be expressed as follows:

$$P_{out}=P_{in}-P_{loss}$$

(14)

Assuming that the estimated rotor position is equal to the actual rotor position, then the actual electromagnetic torque should be equal to the reference electromagnetic torque, and the actual output power should be equal to the reference output power and vice versa. Therefore, the estimated rotor position can be adjusted to realize the equality of the actual output power and the reference output power. To achieve accurate torque control, a PI estimator with robust control is designed to obtain rotor position in this paper.

As shown in Equation (7), the permanent flux linkage and the difference of d-axis and q-axis inductances are used to calculate the reference electromagnetic torque. These parameters influence the accuracy of rotor position estimation significantly. In this paper, the permanent flux linkage is estimated online and difference of d-axis and q-axis inductances is measured offline by experiments.

2.3. Permanent Magnetic Flux Linkage Estimation

To realize accurate and efficient current control, the current PI controller with feedforward control voltage based on space vector pulse width modulation (SVPWM) is shown in Figure 3 [19].

For an IPMSM, it is needed to decouple the d-axis and q-axis voltages and current. The d-axis and q-axis voltages in the rotor rotating coordinate system can be expressed as follows:

$$\left[\begin{array}{c}{v}_d\\ v_q\end{array}\right]=\left[\begin{array}{cc}{R}_s& 0\\ 0& R_s\end{array}\right]\left[\begin{array}{c}{i}_d\\ i_q\end{array}\right]+\left[\begin{array}{cc}s& 0\\ 0& s\end{array}\right]\left[\begin{array}{c}{\phi }_d\\ {\phi }_q\end{array}\right]+\left[\begin{array}{cc}0& -{\omega }_e\\ {\omega }_e& 0\end{array}\right]\left[\begin{array}{c}{\phi }_d\\ {\phi }_q\end{array}\right]$$

(15)

The last term of Equation (15) is the back EMF. If the d-axis and q-axis flux linkages are accurate, it can be eliminated by decoupling control. The feedforward control voltages can be expressed as follows:

$$\left[\begin{array}{c}{v}_{d\_ff}\\ v_{q\_ff}\end{array}\right]=\left[\begin{array}{cc}0& -{\omega }_e\\ {\omega }_e& 0\end{array}\right]\left[\begin{array}{c}{\stackrel{\wedge }{\phi }}_d\\ {\stackrel{\wedge }{\phi }}_q\end{array}\right]$$

(16)

where $v_{d\_ff}$ and $v_{q\_ff}$ are the d-axis and q-axis feedforward control voltages, ${\stackrel{\wedge }{\phi }}_d$ and ${\stackrel{\wedge }{\phi }}_q$ are the estimated d-axis and q-axis flux linkages.

Assuming $v_{d\_fb}$ and $v_{q\_fb}$ are the outputs of the PI controller, the d-axis and q-axis voltages can be expressed as follows:

$$\left[\begin{array}{c}{v}_d\\ v_q\end{array}\right]=\left[\begin{array}{c}{v}_{d\_fb}\\ v_{q\_fb}\end{array}\right]+\left[\begin{array}{c}{v}_{d\_ff}\\ v_{q\_ff}\end{array}\right]$$

(17)

In steady circumstance, the actual current follows well to the reference current, and thus the differential term of Equation (15) can be eliminated:

$$\begin{array}{c}\left[\begin{array}{c}{v}_d\\ v_q\end{array}\right]=\left[\begin{array}{cc}{R}_s& 0\\ 0& R_s\end{array}\right]\left[\begin{array}{c}{i}_d\\ i_q\end{array}\right]+\left[\begin{array}{cc}0& -{\omega }_e\\ {\omega }_e& 0\end{array}\right]\left[\begin{array}{c}{\phi }_d\\ {\phi }_q\end{array}\right]\\ =\left[\begin{array}{c}{v}_{d\_fb}\\ v_{q\_fb}\end{array}\right]+\left[\begin{array}{cc}0& -{\omega }_e\\ {\omega }_e& 0\end{array}\right]\left[\begin{array}{c}{\stackrel{\wedge }{\phi }}_d\\ {\stackrel{\wedge }{\phi }}_q\end{array}\right]\end{array}$$

(18)

Furthermore:

$$\left[\begin{array}{c}{v}_{d\_fb}\\ v_{q\_fb}\end{array}\right]=\left[\begin{array}{cc}{R}_s& 0\\ 0& R_s\end{array}\right]\left[\begin{array}{c}{i}_d\\ i_q\end{array}\right]+\left[\begin{array}{cc}0& -{\omega }_e\\ {\omega }_e& 0\end{array}\right]\left[\begin{array}{c}{\phi }_d-{\stackrel{\wedge }{\phi }}_d\\ {\phi }_q-{\stackrel{\wedge }{\phi }}_q\end{array}\right]$$

(19)

As shown in Equation (19), the outputs of the current PI controller contain a voltage drop at stator resistance and an error of estimated flux linkage. If the variation of the stator resistance is negligible or can be compensated, the error of the estimated flux linkage can be obtained from the integration term of the PI controller, that is:

$$\left[\begin{array}{c}\Delta {\phi }_d\\ \Delta {\phi }_q\end{array}\right]=\frac{1}{{\omega }_e}\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]\left(\left[\begin{array}{c}{v}_{d\_fb\_i}\\ v_{q\_fb\_i}\end{array}\right]-\left[\begin{array}{cc}{R}_s& 0\\ 0& R_s\end{array}\right]\left[\begin{array}{c}{i}_d\\ i_q\end{array}\right]\right)$$

(20)

where $\Delta {\phi }_d$ and $\Delta {\phi }_q$ are the errors of d-axis and q-axis flux linkages between actual and estimated value ($\Delta {\phi }_d={\phi }_d-{\stackrel{\wedge }{\phi }}_d$, $\Delta {\phi }_q={\phi }_q-{\stackrel{\wedge }{\phi }}_q$), $v_{d\_fb\_i}$ and $v_{q\_fb\_i}$ are the integration terms of the PI controller.

Then, the estimated permanent magnetic flux linkage can be expressed as follows:

$${\stackrel{\wedge }{\phi }}_f={\phi }_d+\Delta {\phi }_d-L_d{i}_d=(v_q-R_s{i}_q)/{\omega }_e-L_d{i}_d$$

(21)

As shown in Equation (21), the accuracy of the estimated permanent magnetic flux linkage is affected by the q-axis voltage $(v_q)$, the d-axis and q-axis inductances $(L_d,L_q)$, the stator resistance $(R_s)$, and the d-axis and q-axis current $(i_d,i_q)$. $v_q$ is equivalent to the reference voltage, which includes the output lag compensation [20] and dead time compensation [21]. In this paper, $L_d,L_q$, and $R_s$ are measured offline by experiments. The measured parameters are saved as lookup tables which cover the entire current operating range at different current levels. Thus, $L_d,L_q$, and $R_s$ can be considered to be accurate. In fact, $R_s$ is approximately accurate with temperature compensation, especially at a high speed when the back EMF is much greater than the voltage drop at stator resistance. $i_d,i_q$ can be obtained from current sensors and Hall-effect sensors. Considering the rotor electrical angular speed $({\omega }_e)$ is in the denominator, the estimation should be avoided at too low speed.

3. Torque Control Strategy

3.1. Feedforward Parameter Iteration Method

A variety of parameters are needed for obtaining the target current. The rotor position and permanent magnetic flux linkage can be estimated, as mentioned above. Other important parameters, such as the d-axis and q-axis inductances, stator resistance, and energy loss, are measured offline by experiments. The measured parameters are saved as lookup tables which cover the entire current operating range at different current levels. The parameters are obtained according to the present input current, and then the target current is calculated according to control strategies. Therefore, the parameters do not match the target current well [22].

Assuming the responding time from zero to maximum torque $T_{e\max }$ is $t_{respond}$, the largest change in value of the electromagnetic torque in a control period can be expressed as follows:

$$T_{e\_change}=\frac{T_{e\max }}{2t_{respond}f}$$

(22)

The largest change in value of the q-axis current in a control period can be expressed as follows:

$$i_{q\_change}=\frac{T_{e\max }}{3p{\phi }_f{t}_{respond}f}$$

(23)

In fact, the torque demand varies frequently for the IPMSM used in vehicles, and the target current is calculated according to the actual torque demand. It is difficult to define an equation to calculate the mismatch between the parameters and current. For a given torque demand, starting with present parameters (the d-axis and q-axis inductances, stator resistance, and energy loss), one set of d-axis and q-axis current is obtained, which is calculated by control strategies (such as MTPA). The obtained current is used for a new set of motor parameter calculation (using parameter lookup tables). Another set of d-axis and q-axis current is calculated with the new parameters. This recursive calculation is carried out until the differences between the new output current and the previous one have reached a defined minimum error [22]. The feedforward parameter iteration method follows five steps:

(1)

Obtain the present motor parameters (the d-axis and q-axis inductances, stator resistance, and energy loss) from lookup tables;

(2)

Calculate one set of d-axis and q-axis current using control strategies;

(3)

Calculate the motor parameters with the current from step 2 using lookup tables;

(4)

Calculate another set of d-axis and q-axis current with the new set of motor parameters from step 3 using control strategies.

(5)

If the current error is within tolerance between the current and the previous step, the iteration is complete. Otherwise, return to step 2.

3.2. Maximum Torque Per Ampere Control Strategy

The MTPA control strategy is an efficiency optimization algorithm which only considers the copper loss. The aim of the optimization algorithm is to realize the minimum copper loss for a given torque. Since the copper loss is proportional to the square of stator current, it is essential to find the minimum stator current for a given torque. The MTPA control strategy can be expressed as follows:

$$\begin{array}{l}{i}_{cd}=0\\ i_{cq}=0\\ i_{od}=i_d-i_{cd}=i_d\\ i_{oq}=i_q-i_{cq}=i_q\end{array}$$

(24)

$$T_e=\frac{3}{2}p\left({\phi }_f{i}_q+\left(L_d-L_q\right)i_d{i}_q\right)$$

(25)

$$\min \left\{P_{Cu}=\frac{3}{2}{R}_s\left(i_d{}^2+i_q{}^2\right)=\frac{3}{2}{R}_s\left[i_d{}^2+{\left(\frac{2T_e}{3p\left({\phi }_f+\left(L_d-L_q\right)i_d\right)}\right)}^2\right]\right\}$$

(26)

For an IPMSM, $L_d\ne L_q$, Equation (26) has no explicit solutions for a given torque. If the control parameter is the q-axis current [23] or the amplitude of stator current [24], the explicit solution of $i_d$ can be expressed by $i_q$ as follows:

$$i_d=\frac{{\phi }_f-\sqrt{{\phi }_f{}^2+4{\left(L_q-L_d\right)}^2{i}_q{}^2}}{2\left(L_q-L_d\right)}$$

(27)

or the explicit solution of $i_d$ can be expressed by $i_s$ as follows:

$$i_d=\frac{{\phi }_f-\sqrt{{\phi }_f{}^2+8{\left(L_q-L_d\right)}^2{i}_s{}^2}}{4\left(L_q-L_d\right)}$$

(28)

normalizing Equation (25), it can be expressed as follows:

$$T_{en}=i_{qn}\left(1-i_{dn}\right)$$

(29)

where

$$\begin{array}{l}{i}_b={\phi }_f/\left(L_q-L_d\right),T_{eb}=1.5p{\phi }_f{i}_b\\ i_{dn}=i_d/i_b,i_{qn}=i_q/i_b,T_{en}=T_e/T_{eb}\end{array}$$

(30)

Equation (29) shows that the output torque depends on the d-axis and q-axis current. For a given $T_{en}$, $i_{qn}$ and $i_{dn}$ have no unique solution. Substituting Equation (29) into Equation (26), the MTPA control strategy can be expressed as follows:

$$\min \left\{P_{Cu}=\frac{3}{2}{R}_s{i}_b{}^2\left[i_{dn}{}^2+\frac{T_{en}{}^2}{{(1-i_{dn})}^2}\right]\right\}$$

(31)

For a given IPMSM, $R_s$ and $i_b$ are certain, and the MTPA control strategy can be expressed as follows:

$$\min \left\{i_{dn}{}^2+\frac{T_{en}{}^2}{{(1-i_{dn})}^2}\right\}$$

(32)

solving Equation (32):

$$\begin{array}{l}{T}_{en}=\sqrt{i_{dn}{(i_{dn}-1)}^3}\\ T_{en}=\frac{i_{qn}}{2}(1+\sqrt{1+4i_{qn}^2})\end{array}$$

(33)

Although the relationship between the normalized torque and stator current can be obtained, Equation (33) has no explicit solutions either. However, i_dn and i_qn can be obtained from the lookup tables, as shown in Figure 4. One of the current can be obtained from the lookup tables. The other current can be obtained from the lookup tables directly or calculated by the first current.

The MTPA control strategy only needs two lookup tables, which are independent of the motor parameters, hence it is widely used in the IPMSM for its simplicity and small amount of calculations.

The IPMSM control system with the MTPA control strategy is shown in Figure 5.

4. Simulation and Experimental Results

4.1. Simulation Results

MATLAB/SIMULINK models were developed to examine the proposed method. A 75-kW IPMSM for a new energy vehicle application is used in this paper, and its parameters are shown in

Table 1. Parameters of the interior permanent magnet synchronous motor (IPMSM).

Parameter	Symbol	Value
Number of pole pairs	p	6
Stator resistance	Rs	4.23 mΩ
Magnet flux linkage	ϕf	0.1039 Wb
d-axis inductance	Ld	0.171 mH
q-axis inductance	Lq	0.391 mH
DC linkage voltage	Vdc	288 V
Maximum speed	nb	4000 Rpm
Peak current	Ipk	570 Apk
Rated power	Pr	75 kW
Peak torque	Tmax	540 Nm
Coefficient of RSac to RsDC	μ	0.1
Hysteresis current coefficient	Kh	0.6637
Eddy current coefficient	Ke	0.00084
Stray loss coefficient	cStr	2.56 × 10−9
Mechanical friction torque	Tfric	5.24
Windage torque coefficient	cwind	3.35 × 10−3

[1].

The simulation model is shown in Figure 6.

The PWM frequency is set as 10 kHz and the PI current regulating frequency is set as 20 kHz. The dead time and the voltage drop of the power device are ignored in the simulation (in the actual application, dead time and voltage drop of the power device will be compensated, so the simplification is valid at this stage). Iron loss and mechanical frictional loss are also ignored in the simulation. To make the simulation more accurate, the stator resistance and the d-axis and q-axis inductances are measured offline. In fact, only the varying trends but not accurate values of these parameters are needed in the simulation, because they are just used to verify the effectiveness of proposed method.

The simulation results are shown in Figure 7, Figure 8 and Figure 9 at different conditions. As shown, the actual electromagnetic torque follows well with the reference electromagnetic torque based on the accurate estimated and measured parameters; hence the accurate torque control is achieved. The simulation results under any other condition also verify the effectiveness of the proposed method (except the zero-speed and zero-torque condition under which the d-q axis voltages and the reference power are zero. In that case, rotor position estimator and flux linkage estimator cannot function).

4.2. Experimental Results

The effectiveness of the proposed method was tested experimentally with the IPMSM as Table 1 shows. A dynamometer was used to emulate the load. A digital signal processor (DSP) was used to carry out the real-time algorithm. Six-pack insulated gate bipolar transistors (IGBT) were used as power switches. A LiFePO₄ battery pack (288 V/180 Ah) was used as a power source. The experimental bench is shown in Figure 10.

The experimental results are shown in Figure 11, Figure 12 and Figure 13.

As shown in Figure 11 and Figure 12, the minimum rotor position error is less than one electrical degree and the maximum rotor position error is less than five electrical degrees. However, the maximum rotor position error with the average speed based estimation method is around ten electrical degrees [4]. Furthermore, the output torque of motor is always accurate.

Figure 11 shows that the voltage of battery drops dramatically to 250 V under rated conditions, hence, the motor enters the flux-weakening region. Compared with the simulation results, q-axis current decreases, whereas d-axis current increases, and magnetic saturation is not so serious. Thus, the estimated flux linkage in the experiment is slightly greater than the simulation result.

Figure 12 shows almost the same results with the simulation except that the flux linkage decreases due to the increment of temperature. Under the maximum torque conditions, the reluctance torque is greater than the other conditions; hence less current is needed to obtain the same output torque. Therefore, the torque control is more efficient.

Figure 13 shows the efficiency of the motor drive system, including the motor and inverter. As shown, the maximum efficiency of the motor drive system can reach 0.94 around the rated conditions, and the minimum efficiency of the system is above 0.8. Thus, the efficient control has been realized.

5. Conclusions

In this paper, an effective method to achieve accurate and efficient torque control of an IPMSM in electric vehicles, based on low-resolution Hall-effect sensors, is proposed. Assuming that the d-axis and q-axis inductances, stator resistance, and energy loss are known (measured offline), the high-resolution rotor position and permanent flux linkage are estimated online based on the IPMSM energy model. Combined with the feedforward parameter iteration method, the MTPA control strategy is achieved based on accurate parameters. The simulation and experimental results show that, by using this method, accurate and efficient torque control can be realized.

By using low-resolution Hall-effect sensors instead of resolvers, electric vehicles can reduce cost, weight, and volume. Since the rotor position estimation is based on a power closed-loop, the IPMSM can output accurate torque even if the estimated rotor position has some errors.

Since the flux linkage and power calculations are related to rotor speed, the proposed method may have some drawbacks in the low-speed region. However, this method provides a relatively more effective way to achieve the efficient and accurate control of IPMSM, particularly when the IPMSM works under high-power, high-speed working conditions. Moreover, the method proposed in this paper can also be applied in other types of motors in electric vehicles, such as in-wheel motors, surface PMSM, and PM-assisted synchronous reluctance motors.