Permanent Magnet Flux Linkage Analysis and Maximum Torque per Ampere (MTPA) Control of High Saturation IPMSM

Abstract

The maximum torque per ampere (MTPA) control is significant for improving the efficiency of the interior permanent magnet synchronous motor (IPMSM). However, for the high saturation IPMSM, the change of the permanent magnet (PM) flux linkage is more complicated, which can cause the MTPA control to deviate from the optimal solution. Therefore, an improved MTPA control method for the high saturation IPMSM is proposed in this paper. Compared with other methods, the proposed method improves the conventional models of flux linkage and torque by analyzing the nonlinear variation of the PM flux linkage with the dq-axis currents. Subsequently, an expression suitable for the MTPA control of high saturated IPMSM is derived based on the improved models. The proposed parameter fitting models are then fitted using data from 11 operating points and incorporated into the MTPA optimization algorithm to obtain the MTPA curve. Finally, the effectiveness of the proposed method in enhancing the control accuracy of the MTPA angle is verified through simulations and experiments.

1. Introduction

Interior permanent magnet synchronous motor (IPMSM) has been an attractive choice in practical applications due to its excellent features of high efficiency and high power density, such as robotics, elevators, air conditioners, and compressors [1,2]. Meanwhile, due to the asymmetry of the dq-axis magnetic circuits, the reluctance torque and the magnet torque exist simultaneously in the electromagnetic torque of the IPMSM. For a given output torque, there are different combinations of the dq-axis currents [3]. In order to fully utilize the reluctance torque through the optimal combination of dq-axis currents, the maximum torque per ampere (MTPA) control has become a preferred control strategy of the IPMSM. The purpose of the MTPA control is to trace an optimal current angle under a given output torque to minimize the stator current amplitude, and the optimal current angle is known as the MTPA angle [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24].

The conventional method regards the parameters of the motor as constant values in the dq-axis model. An optimized equation is then used to calculate the MTPA angle for the different torque [4]. However, the inductances will change nonlinearly with the current, which affects the reluctance torque. The permanent magnet (PM) flux linkage will change with the temperature, which affects the magnet torque [5,6,7]. Hence, when the parameters of the motor are regarded as constants to implement the MTPA control, the real-time optimal distribution of the dq-axis currents cannot be achieved. For the problem of parameter uncertainty in MTPA control, many scholars have proposed various methods to identify the changing parameters under different operating loads. For instance, in Ref. [8], the finite element method is used to obtain the variation of parameters with current. In Ref. [9], the high frequency injection method is employed to identify the parameters during motor operation. Based on different parameter identification methods, MTPA control can be categorized into offline and online methods.

The offline methods are used to obtain the MTPA curve before the motor is operated in an actual environment. In Ref. [10], the finite element method is used to obtain the MTPA curve after the motor design is completed. Although the finite element method is accurate in theory, and the MTPA angle can be found, it requires knowing the detailed mechanical dimensions of the motor. Furthermore, some technical and assembly errors usually exist in the prototype manufacturing process, resulting in a mismatch between the actual and simulated values of the motor parameters [11]. The look-up table (LUT) method can solve the shortcomings of the finite element method [12,13,14]. In Ref. [14], the optimal dq-axis currents are measured corresponding to different working conditions through experiments; the data are saved to the controller storage space, and the optimal given values are accurately indexed according to different working conditions during the IPMSM operation. However, the LUT method requires a time-consuming experimental process and a large number of experimental data.

To compensate for the shortcomings of offline methods, online methods can usually be used. Refs. [15,16,17] present the search-based methods to online search the optimal current angle corresponding to the minimum stator current by injecting a signal. Although the search-based methods do not depend on the dq-axis model, the response performance of the system is unsatisfactory, and the algorithm may fail in the process of torque variation, thus affecting the stability of the system. In Refs. [18,19,20,21,22], the parameters of the motor are estimated by the high frequency signal injection or advanced algorithm, and the estimated parameters are brought into the MTPA algorithm to obtain the MTPA angle. However, Ref. [23] has proved that although the parameter estimation method can compensate for the problem of parameter inaccuracy, each optimization iteration needs to update the parameters rather than directly consider the changes of parameters in the optimization algorithm. This problem will still lead to deviation in the calculated MTPA angle. In Refs. [23,24,25,26], the MTPA methods based on surface fitting are proposed. The proposed fitting models of the flux linkage and the inductance are integrated into the MTPA algorithm, which avoids updating parameters outside the MTPA algorithm.

However, the above methods all ignore the effect of the high saturation on the PM flux linkage. For the high saturation IPMSM, due to the restriction of weight and volume, its power density and saturation degree are much higher than those of commercial motors. Hence, in the actual operation process, the change of the motor parameters is more complicated than that of the low saturation motor [27,28,29]. Ref. [28] presents that the vector direction of the PM flux linkage will change with the saturation degree of the motor. Ref. [29] presents that the amplitude of the PM flux linkage will change nonlinearly with the dq-axis currents. These variations of the PM flux linkage will also cause the MTPA control to deviate from the optimal solution, while the above methods ignore these variations. Hence, the above methods are not applicable to the high saturation IPMSM.

To avoid these problems and develop an accurate MTPA control for the high saturation IPMSM, this paper proposes an improved MTPA control method based on a modified dq-axis model, and the organization is as follows. The conventional dq-axis model and the MTPA control method are described in Section 2. The nonlinear change of PM flux linkage with the current and the influence of PM flux linkage on torque are analyzed, and the proposed MTPA is explicated in Section 3. In Section 4 and Section 5, the simulation and experimental results are presented to verify the proposed method. Finally, the conclusions are given in Section 6.

2. Conventional IPMSM Model and MTPA Control

In the conventional dq-axis model of the IPMSM, the PM flux linkage and the inductance are usually regarded as constants, and the flux linkage equations are denoted as

$$\left[\begin{array}{l}{\psi }_d\\ {\psi }_q\end{array}\right]=\left[\begin{array}{cc}{L}_d& 0\\ 0& L_q\end{array}\right]\left[\begin{array}{c}{i}_d\\ i_q\end{array}\right]+\left[\begin{array}{c}{\psi }_f\\ 0\end{array}\right]$$

(1)

where ψ_d, ψ_q, i_d, i_q, L_d, and L_q are the d-axis and q-axis flux linkages, currents, and inductances, respectively; ψ_f is the PM flux linkage. The total torque of the motor is expressed by

$$T{e}_{total}=\frac{3}{2}p\left({\psi }_d{i}_q-{\psi }_q{i}_d\right)$$

(2)

where p is the pole pair; Te_total is the total torque. Substituting (1) into (2), the total torque can then be divided into the magnet torque caused by the PM flux linkage and the reluctance torque caused by the inductance difference.

$$T{e}_{total}=T{e}_{pm}+T{e}_{rel}=\frac{3}{2}p{i}_q{\psi }_f+\frac{3}{2}p\left(L_d-L_q\right)i_d{i}_q$$

(3)

where Te_pm is the magnet torque; Te_rel is the reluctance torque.

In addition, the relationships between the phase current and the dq-axis currents are shown in Figure 1a, and the equations are expressed as

$$\left[\begin{array}{c}{i}_d\\ i_q\end{array}\right]=\left[\begin{array}{cc}-i_s& 0\\ 0& i_s\end{array}\right]\left[\begin{array}{c}\sin \beta \\ \cos \beta \end{array}\right]$$

(4)

where i_s is the maximum phase current; β is the current angle between the phase current vector and the q-axis. The relationships between each torque and the current angle are shown in Figure 1b.

Substituting (4) into (3), the torque Equation (2) can be expressed as

$$T{e}_{total}=\frac{3}{2}p({\psi }_f{i}_s\cos \beta -\frac{1}{2}{L}_{∆}{i}_s^{2}sin2\beta ), L_{∆}=L_d-L_q$$

(5)

The purpose of the MTPA control is to trace a current angle to make the ratio of total torque and current maximize, which is expressed as

$$\underset{{\beta }_{}}{\max }\frac{T{e}_{total}}{i_s}$$

(6)

To find an appropriate current angle β from (6), make the partial derivative of the total torque expression (5) with respect to the current angle be zero ∂Te_total/∂β = 0.

$$\frac{\partial {\psi }_f}{\partial \beta }cos\beta -{\psi }_f\sin \beta -\frac{1}{2}\frac{\partial L_{∆}}{\partial \beta }{i}_s\sin 2\beta -L_{∆}{i}_s\cos 2\beta =0$$

(7)

The conventional MTPA control method regards the inductance and the PM flux linkage as constants. Hence, the first and third terms on the left side of (7) are zero, and the current angle can be obtained by

$$\beta ={\sin }^{-1}\frac{{\psi }_f\sqrt{{\psi }_f^2+8L_{∆}^2{i}_s^2}}{4L∆i_s}$$

(8)

However, the inductance and the PM flux linkage will change with different operating conditions of the motor, resulting in a significant error between the MTPA angle calculated by Equation (8) and the actual MTPA angle. Moreover, in the conventional dq-axis model, the change of the PM flux linkage with saturation is not considered; this will further increase the error.

3. PM Flux Linkage Analysis and Proposed MTPA Control

3.1. Proposed IPMSM Model and Fitting Models

For the high saturation IPMSM, the magnetic density of the motor is high, which will lead to the PM flux linkage change nonlinearly with the dq-axis currents. When i_d remains unchanged, the core is gradually magnetized with the increase of i_q, which will cause the vector direction of the PM flux linkage ${\psi }_f^{\prime }$ to gradually shift to the negative direction of the q-axis, and the q-axis component of the PM flux linkage ${\psi }_{\mathit{fq}}^{\prime }$ will be generated, as shown in Figure 2 (superscript′ indicates that magnetic saturation is considered).

With the increase of i_q, the saturation of the core increases, and the inclination degree of PM flux linkage also increases, which causes the d-axis component of the PM flux linkage ${\psi }_{\mathit{fd}}^{\prime }$ to decrease and ${\psi }_{\mathit{fq}}^{\prime }$ to increase along the negative direction of the q-axis, as shown in Figure 3a. Meanwhile, when i_q remains unchanged, with the decrease of i_d, due to the influence of the demagnetization current, desaturation will occur in the d-axis direction, which causes ${\psi }_{\mathit{fd}}^{\prime }$ to increase and ${\psi }_{\mathit{fq}}^{\prime }$ to decrease along the negative direction of the q-axis, as shown in Figure 3b.

Therefore, not only the inductance changes but also the change of PM flux linkage should be considered in the flux linkage equations, and (1) should be rewritten as

$$\left[\begin{array}{c}{\psi }_d^{\prime }\\ {\psi }_q^{\prime }\end{array}\right]=\left[\begin{array}{cc}{L}_d^{\prime }& 0\\ 0& L_q^{\prime }\end{array}\right]\left[\begin{array}{c}{i}_d\\ i_q\end{array}\right]+\left[\begin{array}{c}{\psi }_{fd}^{\prime }\\ {\psi }_{fq}^{\prime }\end{array}\right]$$

(9)

where ${\psi }_{fd}^{\prime }$ and ${\psi }_{fq}^{\prime }$ are the dq-axis components of the PM flux linkage, respectively. Then, the magnet torque can also be divided into the dq-axis components, and (3) is rewritten as

$$\begin{array}{c}T{e}_{\mathit{total}}^{\prime }=T{e}_{\mathit{pm}}^{\prime }+T{e}_{\mathit{rel}}^{\prime }=T{e}_{\mathit{pmd}}^{\prime }+T{e}_{\mathit{pmq}}^{\prime }+T{e}_{\mathit{rel}}^{\prime }\\ =\frac{3}{2}p{\psi }_{\mathit{fd}}^{\prime }{i}_q-\frac{3}{2}p{\psi }_{\mathit{fq}}^{\prime }{i}_d+\frac{3}{2}p{L}_{∆}^{\prime }{i}_d{i}_q\end{array}$$

(10)

where T$e_{\mathit{pmd}}^{\prime }$ and T$e_{\mathit{pmq}}^{\prime }$ are the dq-axis components of the magnet torque, respectively. The relationships between each torque and current angle are shown in Figure 4.

Figure 4 presents that when the current angle is 0 to 90 degrees (i_d ≤ 0, i_q ≥ 0), T$e_{\mathit{pmq}}^{\prime }$ will cause T$e_{\mathit{total}}^{\prime }$ to decrease, while the MTPA control needs to trace the appropriate current angle according to the change of T$e_{\mathit{total}}^{\prime }$ (6). Therefore, in the actual MTPA control, the influence of T$e_{\mathit{pmq}}^{\prime }$ on T$e_{\mathit{total}}^{\prime }$ cannot be ignored. It means that for the MTPA control of high saturation IPMSM, it is necessary to consider the changes of the PM flux linkage amplitude and vector direction.

Substituting (4) into (10), make the partial derivative of T$e_{\mathit{total}}^{\prime }$ with respect to β be zero.

$$\begin{array}{ll}\frac{\partial T{e}_{\mathit{total}}^{\prime }}{\partial \beta }& =\frac{\partial {\psi }_{fd}^{\prime }}{\partial \beta }cos\beta -{\psi }_{fd}^{\prime }\sin \beta -\frac{1}{2}\frac{\partial L_{∆}^{\prime }}{\partial \beta }{i}_s\sin 2\beta \\ & -L_{∆}^{\prime }{i}_s\cos 2\beta +\frac{\partial {\psi }_{fq}^{\prime }}{\partial \beta }\sin \beta +{\psi }_{fq}^{\prime }\cos \beta =0\end{array}$$

(11)

According to ref [23], ∂${\psi }_{fd}^{\prime }$/∂β, ∂${\psi }_{fq}^{\prime }$/∂β, and ∂$L_{∆}^{\prime }$/∂β cannot be ignored in the MTPA algorithm (11). Thus, this paper proposes the fitting models of ${\psi }_{fd}^{\prime }$, ${\psi }_{fq}^{\prime }$, and $L_{\Delta }^{\prime }$ in regard to the dq-axis currents, respectively.

$$\begin{array}{c}{L}_{∆}^{\prime }=a_0+a_1{i}_d+a_2{i}_q+a_3{i}_d{i}_q\hfill \\ {\psi }_{fd}^{\prime }=b_0+b_1{i}_d+b_2{i}_q+b_3{i}_d{i}_q+b_4{i}_q^2, {\psi }_{fq}^{\prime }=c_0{i}_q+c_1{i}_d{i}_q+c_2{i}_q^2\hfill \end{array}$$

(12)

where a_i, b_i, and c_i are constant coefficients (i = 0, 1, 2, 3, 4).

3.2. Determination of the Coefficients and Implementation of the MTPA Control

This paper only considers the motoring mode of the IPMSM. Assuming that the dq-axis currents ranges are −i ≤ i_d ≤ 0 and i ≥ i_q ≥ 0, plot four quarter circles with amplitudes of 1/4, 1/2, 3/4, and 1 of the current i, respectively. Divide the quarter circle with radius i into four sectors equally, and select points from 1 to 11. Point 1 is on a circle with a radius of 0.25i; points 2 and 3 are on a circle with a radius of 0.5i; points 4, 5, and 6 are on a circle with a radius of 0.75i; points 7, 8, 9, 10 and 11 are on a circle with a radius of i; and each point is on the boundary of the sector. The specific coordinates of each point are shown in Figure 5.

Points 4, 7, and 8 are selected to occupy more weight in the desaturation area of the motor, and points 6, 10, and 11 are selected to occupy more weight in the saturation area of the motor. The inductance parameters of 11 points can be firstly obtained by parameter estimation. Then, ${\psi }_{fd}^{\prime }$ and ${\psi }_{fq}^{\prime }$ at each point can be calculated by using the steady-state voltage Equation (13) according to the monitored dq-axis voltage.

$${\psi }_{fd}^{\prime }=\frac{v_q-R_s{i}_q}{w_e}-L_d^{\prime }{i}_d,{\psi }_{fq}^{\prime }=\frac{R_s{i}_d-v_d}{w_e}-L_q^{\prime }{i}_q$$

(13)

where R_s is the stator resistance; v_d and v_q are the dq-axis voltage, respectively; w_e is the electrical speed.

Finally, the coefficients in (12) can be determined based on the data of the 11 operating points in Figure 5 and the proposed optimization algorithm (14).

$$\begin{array}{c}\mathrm{Minimize}:\\ {\displaystyle \sum _{i=1}^{11}{(L_{∆}^{\ast }-L_{∆}^{\wedge })}^2+{\displaystyle \sum _{i=1}^{11}{({\psi }_{fd}^{\ast }-{\psi }_{fd}^{\wedge })}^2}+{\displaystyle \sum _{i=1}^{11}{({\psi }_{fq}^{\ast }-{\psi }_{fq}^{\wedge })}^2}}\end{array}$$

(14)

where $L_{∆}^{*}$, ${\psi }_{fd}^{*}$, and ${\psi }_{fq}^{*}$ are the actual value; $L_{∆}^{\wedge }$, ${\psi }_{fd}^{\wedge }$ and ${\psi }_{fq}^{\wedge }$ are the fitted value.

After determining the coefficients of the fitting models, the total torque expression without the PM flux linkage and the inductance can be expressed as

$$T{e}_{\mathit{total}}^{\prime }=\frac{3}{2}p(b_0{i}_q+k{i}_d{i}_q+b_2{i}_q^2+m{i}_d{i}_q^2+b_4{i}_q^3+n{i}_d^2{i}_q+a_3{i}_d^2{i}_q^2)$$

(15)

where

$$k=b_1+a_0+c_0,m=b_3+a_2+c_2,n=a_1+c_1$$

(16)

Substituting (4) into (15), make the partial derivative of T$e_{\mathit{total}}^{\prime }$ with respect to β be zero.

$$\begin{array}{ll}\frac{\partial T{e}_{\mathit{total}}^{\prime }}{\partial \beta }& =k{i}_s{\sin }^2\beta -b_0\sin \beta -k{i}_s{\cos }^2\beta -2b_2{i}_s\cos \beta \sin \beta \\ & - m{i}_s^2{\cos }^3\beta +2m{i}_s^2{\sin }^2\beta \cos \beta -3b_4{i}_s^2{\cos }^2\beta \sin \beta \\ & + 2n{i}_s^2{\cos }^2\beta \sin \beta -n{i}_s^2{\sin }^3\beta +2a_3{i}_s^3\sin \beta {\cos }^3\beta \\ & - 2a_3{i}_s^3{\sin }^3\beta \cos \beta =0\end{array}$$

(17)

It can be seen from (17) that the equation has only two variables (i_s, β), which can be solved by numerical solution or nonlinear equation solver. For real-time applications, the proposed algorithm can also be utilized to generate LUT in advance to improve the dynamics of the control system. Compared with the conventional LUT method, the proposed method can generate the LUT offline only by using a PC without sampling the entire working range of the motor, which significantly reduces the workload. Eventually, the dq-axis currents trajectories can be obtained by substituting the current angle β into (4). This method not only considers the changes of the inductance in the optimization process but also considers the changes of the PM flux linkage in the high saturation state. The control block diagram is shown in Figure 6.

4. Simulation Analysis and Verification

To verify the accuracy of the proposed method, this paper establishes a finite element model (FEM) of the high saturation IPMSM, as shown in Figure 7. The main parameters of the model are listed in

Table 1. Key parameters of IPMSM.

Parameters	Quantity	Parameters	Quantity
Rated power	300 kW	Pole pairs	3
Rated current	470 A	PM flux linkage	0.332 Wb
Rated speed	3000 r/min	Stator resistance	0.005 Ω
Rated torque	955 Nm	d-axis inductance	0.4723 mH
Rated frequency	150 Hz	q-axis inductance	1.0228 mH

, including the dq-axis inductance and the PM flux linkage when the dq-axis currents are zero.

For the high saturation IPMSM at the different load conditions, the stator core is usually in the high saturation state. Especially, the maximum magnetic density (MMD) of stator teeth is generally greater than 1.8 T. To verify the saturation degree of stator teeth, the motor speed is set to the rated speed. Then, the MMD of the stator teeth is simulated at the different load conditions, as shown in Figure 8a.

Figure 8a presents that the MMD of the stator teeth exceeds 1.8 T for 78% of the rated current range. When i_d = −275 A and i_q = 605 A, the IPMSM can operate at the rated torque with the minimum current. Meanwhile, the MMD of the stator teeth is 2.05 T, as shown in Figure 8b. Thus, Figure 8 indicates that the saturation of the IPMSM model is high. In the high saturation state, the inductance and the PM flux linkage will change nonlinearly with the dq-axis currents, as shown in Figure 9.

According to the analysis in Section 3, the variation of the inductance and PM flux linkage will directly affect the amplitude of the reluctance and magnet torque, respectively, as shown in Figure 10.

Figure 10 presents that for the high saturation IPMSM, the total torque is composed of three components. Each torque component significantly impacts the total torque, and all of them change nonlinearly with the change of the dq-axis current. For this reason, the variation of these three torque components should be considered simultaneously in the MTPA optimization algorithm. According to the coordinates of 11 working points in Figure 5, the simulated inductance and the PM flux linkage are substituted into (14), and the coefficients of fitting models are obtained, as shown in

Table 2. Results of coefficient fitting based on simulation.

Coefficient	ai	bi	ci
i = 0	−5.55 × 10−4	0.342	−3.91 × 10−4
i = 1	−1.12 × 10−8	−1.32 × 10−4	−1.23 × 10−7
i = 2	6.23 × 10−7	−4.37 × 10−5	3.43 × 10−7
i = 3	−1.46 × 10−10	−1.34 × 10−7	0
i = 4	0	−5.41 × 10−8	0

.

Then, the coefficients are substituted into (17), and the equation results are directly substituted into (4) by numerical solution, as shown in Figure 6. Finally, compared with other methods, (1) the conventional method substitutes the constant parameters at no-load condition into (8); (2) the parameter estimation method [22] substitutes the variable parameters at different conditions into (8); (3) the conventional fitting method [23]. The MTPA curves obtained by different methods are shown in Figure 11. It is obvious from Figure 11 that the proposed method is closer to the actual MTPA curve than other methods, which verifies the validity of the analysis and the method proposed in this paper.

5. Experimental Results

A 300 kW high saturation IPMSM has been prototyped to verify the proposed approach, and the design parameters of the prototype are given in Table 1. The experiment platform is shown in Figure 12a. The prototype is driven by a traction motor at 3000 rpm and operates in the torque control mode. Meanwhile, a torque sensor (ATESTEO: DF3) is also installed to measure the torque at different load conditions. In addition, in order to reduce the influence of temperature on parameters, so as to better verify the influence of high saturation on parameter, the cooling mode of the prototype is water-cooling, the temperature of cooling water is 30 °C, and the water flow is 60 L/min. The Hi-Techniques’ high-speed logger is utilized to observe and reserve the experimental information in real time, and the PC software is used to export and plot the experimental data, as shown in Figure 12b.

Firstly, the prototype is tested to find the actual MTPA curve. The detailed test steps are as follows. Set i_d to make the prototype operate at no-load condition. Then, as the load gradually increases, the voltage, current, and torque are recorded at different load conditions. Finally, gradually decrease i_d and repeat the previous steps. The measured torque within the rated current range is shown in Figure 13, and the MTPA curve is OB.

Due to certain technical and assembly errors in the prototype manufacturing process, the actual value and the simulated value are different. For instance, the minimum current at the rated torque is different from the simulation. When the IPMSM is operated at rated torque, the dq-axis currents are i_d = −350 A and i_q = 580 A, respectively. The recorded waveforms of the DC bus voltage, three-phase voltage, current, and torque are shown in Figure 14.

Then, the parameter estimation method is used to estimate the inductance parameters of the 11 operating points, and the dq-axis components of PM flux linkage can be calculated according to the monitored dq-axis voltage, as shown in Figure 15. According to the algorithm (14), the fitting coefficients of the prototype are listed in

Table 3. Results of coefficient fitting based on experiments.

Coefficient	ai	bi	ci
i = 0	−6.28 × 10−4	0.317	−3.34 × 10−4
i = 1	−1.13 × 10−7	−6.19 × 10−5	−9.24 × 10−8
i = 2	6.97 × 10−7	−1.1 × 10−4	2.54 × 10−7
i = 3	1.92 × 10−10	−2.3 × 10−7	0
i = 4	0	−8.23 × 10−9	0

.

Based on the fitting coefficients of the prototype, the offline numerical calculation is used to obtain the relationship between i_s and β, and generates a LUT for real-time applications, as shown in Figure 6. Eventually, the MTPA curves and MTPA angle errors of the different methods are shown in Figure 16.

Figure 16b presents that the error of the proposed method is relatively stable, and the error does not increase with the increase of the saturation degree. For example, when the motor is operated at the rated load, although the saturation degree of the motor is large, the error of the proposed method is still small. The errors of different methods at the rated load are listed in

Table 4. Comparison of different methods.

Methods	Errors at Rated Load
Proposed	0.6°
Simulation	−6.1°
Conventional	6.4°
Conventional fitting	4.3°
Parameter estimation	−4.8°

.

In the rated load range, the maximum errors caused by the parameter estimation and conventional fitting method are −4.8° and 5.1°, respectively. The main reason for these errors is that the influence of the high saturation on the PM flux linkage is not considered. Furthermore, the maximum error of the conventional method and the simulation are 10° and −6.1°, respectively. The maximum error of the proposed method is −1.8°. Therefore, the proposed method can achieve better accuracy compared with other methods. Meanwhile, these experimental results indicate that for the MTPA control of high saturation IPMSM, the nonlinear change of the PM flux linkage with the dq-axis current cannot be ignored.

6. Conclusions

The MTPA control is a regular method for the IPMSM to find an optimal current angle to minimize the stator current amplitude for given output torque. However, for the high saturation IPMSM, the change of the motor parameters is more complicated than that of the low saturation motor, and the nonlinear relationship between the PM flux linkage and the dq-axis currents becomes more distinct. When the conventional MTPA control method is used, the accuracy of the MTPA angle often fails to meet the requirements.

Thus, an improved MTPA control method is proposed in this paper. By analyzing the change of the PM flux linkage and the influence of the PM flux linkage on the total torque, the conventional flux linkage and torque models are improved. Based on the improved dq-axis models, an expression more suitable for MTPA control of high saturation IPMSM is derived. Compared to other methods, the proposed method takes into account the nonlinear variation of the PM flux linkage with magnetic saturation. Additionally, this method also considers the q-axis component generated by the PM flux linkage and the partial derivatives of the parameters in the MTPA optimization algorithm. Simulation and experimental results demonstrate that the proposed method can better follow the MTPA angle of the high saturation IPMSM at different load conditions.