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Abstract

A kind of flux-weakening control method based on speed loop structure-variable sliding mode controller is proposed for interior permanent magnet synchronous motor in electric vehicles. The method combines maximum torque per ampere with vector control strategy to control electric vehicle’s interior permanent magnet synchronous motor. During the flux-weakening control phase, the anti-windup integral controller is introduced into the current loop to prevent the current regulator from entering the saturated state. At the same time, in order to further improve the utilization rate of the direct current bus voltage and expand the flux-weakening regulating range, a space vector pulse-width modulation over-modulation unit is employed to contravariant the direct current bus voltage. Comparing with the conventional proportional–integral controller, the proposed sliding mode control algorithm shows that it has more reliable control performance. In addition, more prominent flux-weakening performance of the proposed flux-weakening method is illustrated by numerical simulation comparison.

Keywords

Interior permanent magnet synchronous motor structure-variable sliding mode control flux-weakening control anti-windup integral controller over-modulation

Introduction

Recently, the interior permanent magnet synchronous motor (IPMSM) has been widely used in high-performance applications due to its advantages, such as small volume, high efficiency, as well as high power density.^1,2 In the electric vehicle driving system, the motor speed should closely follow a specified reference trajectory, regardless of any load disturbance, parameter variation, and model uncertainty. Additionally, a wide speed range covers the constant torque, and the constant power region is desired.

For electric vehicles, there exist inevitable interferences such as current coupling, friction force, parameter variation, as well as load disturbance during the running process. Due to the presence of these interferences, it is difficult to describe IPMSM with accurate mathematical model, and it is hard to quickly suppress these disturbances with a linear control method. Structure-variable sliding mode control is a kind of nonlinear control method which could self-adjust in accordance with the current state of the system. It does not require accurate mathematical model and can purposefully force the system to move according to the scheduled sliding mode trajectory. The sliding mode trajectory can be set artificially in advance, and it is irrelevant to the object parameters and disturbances.

When IPMSM runs below the base speed, the maximum torque per ampere^3,4(MTPA) is commonly used for constant torque control. The MTPA is often implemented using look-up table method or curve fitting method. With poor real-time performance, and high memory cell consumption,⁵ the look-up table method has lower industrial popularity. The curve fitting algorithm is often used in engineering practice instead. In Dominguez⁶ and Huang et al.,⁷ the curve fitting algorithm is adopted to decompose the relationship among the quadrature-axis current, direct-axis current, and electromagnetic torque. The relationship between the quadrature-axis current $i_{q}$ and the electromagnetic torque $T_{e}$ is hard to be solved. Moreover, it is closely related to the rotor magnetic flux, the d-q axis inductance, and some other variable parameters. As a result, the curve fitting performance will be depressed by uncertainties of the parameters. In Jun-Jie et al.,⁸ they take the quadrature-axis current as the output of the speed loop controller, and then, the direct-axis current is decomposed according to the relationship of the d-q axis current in MTPA algorithm. But, the relationship between the electromagnetic torque and d-q axis current is not considered, so this method cannot coordinate current with torque. Inspired by Jun-Jie et al.,⁸ adopting the idea of feedback in this article, we regard the rotation speed error as the input of the speed loop sliding mode controller (SMC). In addition, we take the decomposed electromagnetic torque, which is implemented by sliding mode algorithm, as the output of the controller. After that, the quadrature-axis current is decomposed according to its relationship with the electromagnetic torque. And then, the direct-axis current is decomposed according to its relationship with the quadrature-axis current. By this way, not only the current is coordinated with torque but also the computation cost is significantly reduced.

When IPMSM runs above the base speed, the flux-weakening (FW) algorithm is usually employed to implement constant power control. Common methods for FW control strategy include formulation calculation method,⁹ gradient descent algorithm,¹⁰ look-up table method,¹¹ and negative $i_{d}$ current compensation method.¹² Negative $i_{d}$ current compensation method has been widely used in practice because it does not need motor parameters, and the algorithm is simple and reliable. In previous studies,^13–15 the negative $i_{d}$ current compensation method is used to study IPMSM FW control. In the above-mentioned documents, however, IPMSM running performance below the base speed is not analyzed specifically, and the system instability resulted from the voltage saturation of current regulator and the d-q axis current coupling is neglected during FW control phase.

Based on the existing research results, a kind of FW control system combined with the rotor field-oriented vector control algorithm is proposed in this article. The combination of the MTPA algorithm and the FW algorithm improves the performance and efficiency of the electric vehicles by regulating the current of the motor in the whole running process. To consider the constraint of the battery capacity of the electric vehicles and to improve the utilization rate of the direct current (DC) bus voltage of the inverter, a kind of space vector pulse-width modulation (SVPWM) over-modulation method is adopted. Meanwhile, a kind of feed-forward compensation method is introduced to realize the decoupling control and then improve the dynamic response of the current regulator. In order to depress motor’s internal and external interferences, and to achieve faster response, stronger adaptability, and higher control precision, a kind of SMC is designed to replace common proportional–integral (PI) controller. To prevent the voltage saturation of current controller, a kind of anti-windup integral controller is employed to improve the dynamic performance of the loop.

Model of IPMSM

The mathematical model of IPMSM in the d-q synchronously rotating reference frame can be obtained from the three-phase synchronous machine model. Due to the constant field produced by permanent magnets, the field variation is zero, and the following are assumed:

Saturation and losses of the core are negligible.

The induced electromotive force (EMF) is sinusoidal.

There is no dampener winding on the rotor.

Under these assumptions, the voltage and electromagnetic torque equations are

{\begin{matrix} u_{d} = R_{s} \cdot i_{d} + L_{d} \cdot p i_{d} - ω_{e} \cdot L_{q} \cdot i_{q} \\ u_{q} = R_{s} \cdot i_{q} + L_{q} \cdot p i_{q} + ω_{e} \cdot L_{d} \cdot i_{d} + ω_{e} \cdot ψ_{f} \end{matrix}

(1)

T_{e} = \frac{3}{2} n_{p} (ψ_{d} i_{q} - ψ_{q} i_{d}) = \frac{3}{2} n_{p} [ψ_{f} \cdot i_{q} + (L_{d} - L_{q}) \cdot i_{d} \cdot i_{q}]

(2)

In which, $R_{s}$ is the armature resistance; $p = d / dt$ is the differentiating operator; $u_{d}$ , $u_{q}$ , $i_{d}$ , $i_{q}$ , $L_{d}$ , and $L_{q}$ are the rotor d-q axis components of the armature voltage, current, and inductance, respectively. In $ω_{e} = n_{p} \cdot ω_{r}$ , $ω_{e}$ is the electrical angular velocity, $n_{p}$ is the number of pole-pairs, and $ω_{r}$ is the mechanical angular velocity of rotation. $ψ_{f}$ is the flux linkage due to rotor permanent magnet.

The torque balance expression of the three-phase IPMSM is presented as follows

\frac{J}{n_{p}} \cdot \frac{d ω_{e}}{dt} = T_{e} - T_{L}

(3)

where J is the rotational inertia, $T_{e}$ is the electromagnetic torque, and $T_{L}$ is the load torque.

The design of the SMC

Assuming that the speed error is e, $e = ω_{e}^{*} - ω_{e}$ . According to formula (3), we have

\begin{matrix} \overset{•}{e} = - \overset{•}{ω_{e}} = - \frac{n_{p}}{J} (T_{e} - T_{L}) \\ \overset{• •}{e} = - \overset{• •}{ω_{e}} = - \frac{n_{p}}{J} \cdot \overset{•}{T_{e}} \end{matrix}

(4)

where $\overset{•}{e}$ is the change rate of the speed error and $\overset{• •}{e}$ is the second derivative of the speed error. Commanding $A = n_{p} / J$ and $U = \overset{•}{T_{e}}$ , formula (4) can be rearranged as follows

[\begin{matrix} \overset{•}{e} \\ \overset{• •}{e} \end{matrix}] = [\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}] [\begin{matrix} e \\ \overset{•}{e} \end{matrix}] + [\begin{matrix} 0 \\ - A \end{matrix}] U

(5)

The sliding mode surface is designed as $s = ce + \overset{•}{e}$ , where c must meet the Hurwitz condition (c > 0). The Lyapunov’s function is designed as $V = (1 / 2) s^{2}$ , so

\overset{•}{V} = s \overset{•}{s} = s (c \overset{•}{e} + \overset{• •}{e}) = s (c \overset{•}{e} - AU)

(6)

According to the Lyapunov’s stability criterion, when V is positive definite and it has the continuous first-order partial derivative, as well as its first-order partial derivative is negative semi-definite, then the system is stable. In order to ensure $s \overset{•}{s} < 0$ , the sliding mode control law is designed as follows

U = \frac{1}{A} (α | e | sgn (s) + β \cdot s + c \overset{•}{e}) (α > 0, β > 0)

Combining equations (5) and (6) yields the following

\begin{matrix} \overset{•}{V} = s \overset{•}{s} \\ = s [c \overset{•}{e} - A \cdot \frac{1}{A} (α | e | sgn (s) + β \cdot s + c \overset{•}{e})] \\ = s [c \overset{•}{e} - α | e | sgn (s) - β s - c \overset{•}{e}] \\ = s (- α | e | sgn (s) - β s) \\ = - α | e | s \cdot sgn (s) - β s^{2} \\ = - α | e | \cdot | s | - β \cdot s^{2} \leq 0 \end{matrix}

(7)

The above analysis shows that the designed control law can guarantee that the system gets to the sliding mode surface gradually and steadily. The design of the SMC is depicted in Figure 1.

Figure 1.

The design of SMC controller.

Constant torque control strategy

Using vector control method, the running regions of IPMSM include constant torque area and constant power area. The most common control strategy for constant torque area is MTPA, which can reduce the input current and cut the switching loss of the inverter by coordinating torque with current and meanwhile satisfy the required torque output. Thus, the MTPA algorithm is more suitable for IPMSM than other traditional method. Setting $β$ as the phase angle between the space vector of armature current and the direct axis, the following formula holds

{\begin{matrix} i_{d} = i_{s} \cos β \\ i_{q} = i_{s} \sin β \end{matrix}

(8)

Substituting formula (8) into formula (2) yields the following

T_{e} = \frac{3}{2} n_{p} ψ_{f} i_{s} \sin β + \frac{3}{4} n_{p} (L_{d} - L_{q}) i_{s}^{2} \sin 2 β

(9)

According to formula (9), the relationship between the unit current electromagnetic torque and the current phase angle $β$ can be defined as

f (β) = \frac{T_{e}}{i} = \frac{3}{2} n_{p} ψ_{f} \sin β + \frac{3}{4} n_{p} (L_{d} - L_{q}) i_{s} \sin 2 β

(10)

The amplitude of $i_{s}$ is supposed to remain unchanged; when the electromagnetic torque $T_{e}$ in formula (10) reaches its maximum value, the condition of $\partial f (β) / \partial β = 0$ must be met, so

\begin{matrix} (L_{d} - L_{q}) i_{s} \cos 2 β + ψ_{f} \cos β = 0 \\ (L_{d} - L_{q}) i_{s} (2 \cos^{2} β - 1) + ψ_{f} \cos β = 0 \end{matrix}

From the formula above, we have

\cos β = \frac{- ψ_{f} + \sqrt{ψ_{f}^{2} + 8 {(L_{d} - L_{q})}^{2} i_{s}^{2}}}{4 (L_{d} - L_{q}) i_{s}}

Substituting the formula above into formula (8), yields

i_{d} = \frac{- ψ_{f} + \sqrt{ψ_{f}^{2} + 8 {(L_{d} - L_{q})}^{2} i_{s}^{2}}}{4 (L_{d} - L_{q})}

(11)

Substituting the relationship $i_{s}^{2} = i_{d}^{2} + i_{q}^{2}$ into formula (11) yields the following

i_{d} = \frac{- ψ_{f} + \sqrt{ψ_{f}^{2} + 4 {(L_{d} - L_{q})}^{2} i_{q}^{2}}}{2 (L_{d} - L_{q})}

(12)

If equation (12) is substituted into formula (2), then

T_{e} = \frac{3}{4} n_{p} i_{q} [\sqrt{ψ_{f}^{2} + 4 {(L_{d} - L_{q})}^{2} i_{q}^{2}} + ψ_{f}]

(13)

If $n_{p}$ is set to be 4 (the number of pole-pairs is selected as 4 in simulations), we can get the following formula

36 {(L_{d} - L_{q})}^{2} i_{q}^{4} + 6 ψ_{f} T_{e} i_{q} - T_{e}^{2} = 0

(14)

Use equations (12) and (14) to build simulation model. The simulation design of MTPA is shown in Figure 2.

Figure 2.

The design of MTPA.

Constant power control strategy

The design of the FW system

The armature voltage of the motor cannot exceed its rated value, and the internal back electromotive force (BEMF) must be lower than the rated value. The BEMF is proportional to the product of the air-gap flux and the speed, and then, to raise the speed, the air-gap flux must be reduced. When an IPMSM is fed by an inverter, the armature current and voltage may not exceed the maximum inverter current and the voltage. These constraints can be expressed as

\begin{matrix} Us = \sqrt{U_{d}^{2} + U_{q}^{2}} \leq U s_{max} = {(\frac{U_{dc}}{\sqrt{3}})}^{2} \\ Is = \sqrt{i_{d}^{2} + i_{q}^{2}} \leq I s_{max} \end{matrix}

(15)

where $U s_{max}$ and $I s_{max}$ are the maximum voltage and current of the inverter, Us and $I_{s}$ are the terminal voltage and current of the inverter, and $U_{dc}$ is the DC bus voltage. The relationship among the difference value of $\sqrt{U_{d}^{2} + U_{q}^{2}} / U_{dc}$ and $1 / \sqrt{3}$ , running region, and control strategy can be summarized in Table 1.

Table 1.

The relationship table.

Difference	Operating region	Control strategy
$1 / \sqrt{3} - \sqrt{U_{d}^{2} + U_{q}^{2}} / U_{dc} \geq 0$	Constant torque	MTPA
$1 / \sqrt{3} - \sqrt{U_{d}^{2} + U_{q}^{2}} / U_{dc} \leq 0$	Constant power	FW

MTPA: maximum torque per ampere; FW: flux weakening.

Neglecting the armature resistance component in equation (1), the steady-state voltage equation can be expressed as

{\begin{matrix} u_{d} = - ω_{e} \cdot L_{q} \cdot i_{q} \\ u_{q} = ω_{e} \cdot L_{d} \cdot i_{d} + ω_{e} \cdot ψ_{f} \end{matrix}

(16)

Substituting equation (16) into equation (15) yields

\begin{array}{l} U s = \sqrt{{(- ω_{e} \cdot L_{q} i_{q})}^{2} + (ω_{e} \cdot L_{d} i_{d} + ω_{e} \cdot ψ_{f})} \\ = ω_{e} \cdot \sqrt{{(L_{q} i_{q})}^{2} + {(L_{d} i_{d} + ψ_{f})}^{2}} \end{array}

(17)

From equation (17), it is obvious that when the terminal voltage value Us has reached its upper limit value $U s_{max}$ , the speed $ω_{e}$ can be increased only by adjusting the d-q axis current. The quadrature-axis current $i_{q}$ should be reduced, and the direct-axis demagnetizing current should be increased to guarantee that the direct-axis current $i_{d}$ is reduced simultaneously.

Figure 3 shows the principle of the FW control. $θ$ , which ranges from $- π / 2$ to 0, is determined by the difference value between $1 / \sqrt{3}$ and $\sqrt{U_{d}^{2} + U_{q}^{2}} / U_{dc}$ through the PI controller. (When the system enters the FW phase, $1 / \sqrt{3} - \sqrt{U_{d}^{2} + U_{q}^{2}} / U_{dc} \leq 0$ , so $θ$ must be negative. In addition, $θ$ is expressed in radian system under polar coordinates and it lies in the forth quadrant, so it ranges from $- π / 2$ to 0.) $θ$ is then used to redistribute $i_{q}$ . $| i_{q} | \cdot \sin θ$ is the direct-axis demagnetization current, direction of which is opposite to the $i_{d}$ . Define

{\begin{matrix} i'_{d} = | i_{q} | \cdot \sin θ + i_{d} \\ i'_{q} = i_{q} \cdot \cos θ \end{matrix}

(18)

Figure 3.

The FW module schematic.

By this way, we get the d-q axis current $i'_{d}$ and $i'_{q}$ which are smaller than the original d-q axis current $i_{d}$ and $i_{q}$ , respectively, and then, the speed $ω_{e}$ will increase with the decrease in the d-q axis current. Finally, we take the $i'_{d}$ and $i'_{q}$ as the input of the current loop controller.

Meanwhile, in order to ensure $I'_{s} = \sqrt{{i'_{d}}^{2} + {i'_{q}}^{2}} \leq I s'_{max}$ , we have

| i'_{q} | \leq \sqrt{{(I s'_{max})}^{2} - {i'_{d}}^{2}}

(19)

where the $| i'_{q} |$ denotes the amplitude of $i'_{q}$ .

The anti-windup integral controller

In this article, the back calculation and tracking (BCT) method is employed to compensate the integral windup. Figure 4 is the block diagram of the BCT scheme, where $k_{a}$ denotes the anti-windup gain. Once the controller output exceeds the actuator limits, a feedback signal is generated from the difference between the saturated and unsaturated control signals and is used to reduce the integrator input.

Figure 4.

The block diagram of the anti-windup integral scheme.

The d-q axis current decoupling

From the equation (1), we have

{\begin{matrix} \frac{d i_{d}}{dt} = \frac{1}{L_{d}} (U_{d} + ω_{e} L_{q} i_{q} - R_{s} i_{d}) \\ \frac{d i_{q}}{dt} = \frac{1}{L_{q}} (U_{q} - ω_{e} L_{d} i_{d} - ω_{e} ψ_{f} - R_{s} i_{q}) \end{matrix}

(20)

It is obvious that the d-q axis currents $i_{d}$ and $i_{q}$ are cross-coupled by $ω_{e} L_{q} i_{q}$ and $ω_{e} L_{d} i_{d}$ , respectively. If $L_{d}$ and $L_{q}$ are larger, the cross-coupling terms $ω_{e} L_{q} i_{q}$ and $ω_{e} L_{d} i_{d}$ will worsen the current and torque responses at high speed. In order to control $i_{d}$ and $i_{q}$ independently, we introduce a kind of feed-forward method to compensate the coupling terms $ω_{e} L_{q} i_{q}$ and $ω_{e} L_{d} i_{d}$ . The voltage feed-forward compensation formulas are defined as

{\begin{matrix} U'_{d} = U_{d} + ω_{e} L_{q} i_{q} \\ U'_{q} = U_{q} - ω_{e} L_{d} i_{d} - ω_{e} ψ_{f} \end{matrix}

(21)

where $U'_{d}$ and $U'_{q}$ are the d axis and q axis voltage components after compensating, respectively.

Substituting equation (21) into equation (20), we have

{\begin{matrix} \frac{d i_{d}}{dt} = \frac{1}{L_{d}} (U'_{d} - R_{s} i_{d}) \\ \frac{d i_{q}}{dt} = \frac{1}{L_{q}} (U'_{q} - R_{s} i_{q}) \end{matrix}

(22)

By this way, the d-q axis current cross-coupling is eliminated in equation (22) and Figure 5.

Figure 5.

The design of d-q axis current decoupling unit.

The SVPWM over-modulation unit

Comparing with sinusoidal pulse-width modulation (SPWM) method, SVPWM can degrade the switching loss and prevent the current distortion; therefore, it is widely used. But traditional SVPWM modulation method only has finite utilization rate of the inverter DC bus voltage, and it narrows the running region and degrades control performance of the motor. Considering the limit of the battery capacity, a kind of SVPWM over-modulation method is adopted to further improve the utilization rate of the inverter DC bus voltage.

Over-modulation has two parts: over-modulation I zone and over-modulation II zone. As shown in Figure 6(a), in over-modulation I zone, reference voltage vector $U_{ref}$ is partly located inside the regular hexagon and partly located outside the regular hexagon. As shown in Figure 6(b), in over-modulation II zone, $U_{ref}$ is entirely located outside the regular hexagon. Only the output voltage vectors which located inside the regular hexagon can realize the linear modulation.

Figure 6.

(a) Over-modulation I zone and (b) over-modulation II zone.

In the first sector, for example, as shown in Figure 7, $U_{s 1}$ is the voltage vector whose amplitude exceeds the linear modulation region and $U_{s 2}$ is the voltage vector whose amplitude exceeds the regular hexagon boundary.

Figure 7.

Over-modulation principle diagram.

When $U_{s 2}$ is $| OH |$ , $| OE | + | EH | \cdot \cos (π / 3) = | OF |$ . At the same time, $| OE | / | OF | = T_{x} / T_{s}, | EH | / | OF | = T_{y} / T_{s}$ , where $T_{s}$ is the sampling period, $T_{x}$ is the action time of $U_{1}$ , and $T_{y}$ is the action time of $U_{2}$ . So, we can use the formula above to judge whether the voltage vector located inside the polygon CDLJHF or not. At the same time, the following conditions must be met when over-modulation occurs

{\begin{matrix} T_{x} + T_{y} / 2 < T_{y} \\ T_{x} + T_{y} > T_{s} \end{matrix} (U_{s 1} or U_{s 2} lies below the straight line OO')

{\begin{matrix} T_{x} + T_{y} / 2 < T_{s} \\ T_{x} / 2 + T_{y} < T_{s} \end{matrix} (U_{s 1} or U_{s 2} lies above the straight line OO')

When over-modulation occurs, we should recalculate the action time of $U_{1}$ and $U_{2}$ ; $T'_{x}$ and $T'_{y}$ are defined as the new action time of $U_{1}$ and $U_{2}$ , respectively, and the expressions of $T'_{x}$ and $T'_{y}$ are as follows

{\begin{matrix} T'_{x} = [T_{x} / (T_{x} + T_{y})] \cdot T_{s} \\ T'_{y} = [T_{y} / (T_{x} + T_{y})] \cdot T_{s} \end{matrix}

When $U_{s 1}$ or $U_{s 2}$ located outside the polygon CDLJHF, if $T_{x} > T_{y}$ , we output $U_{1}$ and if $T_{y} > T_{x}$ we output $U_{2}$ . So, the flowchart of the over-modulation implementation can be summed up as shown in Figure 8.

Figure 8.

Flowchart of over-modulation implementation.

The principle diagram of the FW control system

The complete control scheme of the proposed FW system in this article is presented in Figure 9. The electromagnetic torque $T_{e}$ reference is determined by the SMC speed controller according to the speed error. It is then used to decompose the d-q axis current $i_{d}$ and $i_{q}$ by the MTPA algorithm. After that, the d-q axis currents and voltages and the DC bus voltage are fed to the FW module, whose outputs are the d-q axis current reference. The d-q axis voltage references are generated from the anti-windup integral controller. By inversing PARK and over-modulation unit, three phase control voltages $U_{a}$ , $U_{b}$ , and $U_{c}$ are obtained to drive the inverter and the motor. (PARK: The transformation from the two-phase α-β stationary reference frame to the two-phased-q synchronously rotating reference frame. Inversing PARK: The transformation from the two-phase d-q synchronously rotating reference frame to the two-phase α-β stationary reference frame).

Figure 9.

The FW control schematic.

The simulation analysis

Parameters in the simulation model are presented in Table 2.

Table 2.

The simulation parameters.

Item	Value	Unit
DC bus voltage $U_{dc}$	192	V
Carrier time T	0.0001	s
Direct-axis inductance $L_{d}$	0.2e − 3	H
Quadrature-axis inductance $L_{q}$	0.47e − 3	H
Armature resistance $R_{s}$	$0.025$	$Ω$
Rotor linkage $ψ_{f}$	0.062	W b
Rotational inertia J	1e − 2	kg m²
Viscous damping coefficient F	0	N m s
Number of pole-pairs $n_{p}$	4	−
Maximum voltage $U s_{max}$	111	V
Maximum current $I s_{max}$	150	A
Load torque $T_{L}$	5–10	N m

To verify the performance of the proposed FW control system, several simulation tests are implemented under different operating conditions. Figures 10 –12 represent the output value of the speed, the torque, and the angle of the SMC system, respectively, in this article and the SMC system of traditional lead angle at the speed from 600 to 4000 rad/s.

Figure 10.

The speed reference and measured value at 4000 rad/s.

Figure 11.

The electromagnetic torque output value at 4000 rad/s.

Figure 12.

The FW angle output value at 4000 rad/s.

Figures 13 –15 denote the output value of the speed, the torque, and the angle of the SMC system, respectively, in this article without the anti-windup integral controller and the over-modulation unit at the speed from 600 to 4000 rad/s.

Figure 13.

The speed reference and measured value at 4000 rad/s without the anti-windup integral controller.

Figure 14.

The electromagnetic torque output value at 4000 rad/s without the anti-windup integral controller.

Figure 15.

The FW angle output value at 4000 rad/s without the anti-windup integral controller.

Under the same operating conditions, the more negative angle system describes that the more the system’s terminal voltage exceeds the maximum voltage, the deeper the FW level. This indicates that the system has lower utilization to DC bus voltage and has worse FW performance.

As seen from Figures 10 to 12, under almost the same speed and torque respond, the absolute value of FW angle output of the SMC system in this article is smaller than that of the SMC system of traditional lead angle. The comparison demonstrates the effectiveness of the FW method in this article.

As shown from Figures 13 to 15, if the original system does not have the anti-windup integral controller, the absolute value of FW angle output will obviously become larger. Meanwhile, when the motor starts at 0 s and accelerates at 0.2 s, the system will output a bigger torque because the current regulator has entered the saturated state at these moments.

In addition, if the original system does not have the over-modulation unit, the system will yield a bigger absolute value of FW angle. Besides, its speed and torque respond will begin to fluctuate.

The experiments mentioned above have illustrated that the anti-windup integral controller and the over-modulation unit are necessary and helpful to improve the IPMSM FW performance. The ability to track signals precisely and instantaneously is necessary for motor control. For this purpose, the speed reference is suddenly decreased from 600 to −4000 rad/s at 0.2 s in Figures 16 and 17.

Figure 16.

The speed reference and measured value at −4000 rad/s.

Figure 17.

The electromagnetic torque output value at −4000 rad/s.

When a reverse speed reference signal leads to a reverse current, the SMC system of traditional lead angle becomes out of control. On the contrary, during the process when the speed forward accelerates from 0 to 600 rad/s and then reverses braking into 0 rad/s and again up to −4000 rad/s, the SMC system in this article can always run steadily. The results prove that the FW method in this article has more significant ability to regulate a negative current.

The ability to withstand disturbances is an important feature for IPMSM drive. For this purpose, the load torque $T_{L}$ is suddenly changed from 30 to −30 N m at 0.4 s in Figures 18 and 19.

Figure 18.

The speed reference and measured value from 30 to −30 N m.

Figure 19.

The electromagnetic torque output value from 30 to −30 N m.

As shown above, when a violent fluctuation of the load torque causes rapid current changes, the SMC system of lead angle becomes out of control. The measured speed and torque value of the proposed SMC system can always follow their reference value more smoothly and steadily than the PI system in this article. The results illustrate the validity of the proposed FW method and the SMC algorithm.

The ability to withstand the motor parameter variations is another important criterion of a control system. Experiments have been done in cases when the viscous damping coefficient F raises from 0 to 0.02 N m s in Figures 20 and 21, and the rotational inertia J increases from 1e − 2 to 6e − 2 kg m² in Figures 22 and 23.

Figure 20.

The speed reference and measured value from 0 to 0.02 N ms.

Figure 21.

The electromagnetic torque output value from 0 to 0.02 N ms.

Figure 22.

The speed reference and measured value from 1e⁻² to 6e⁻² kg m².

Figure 23.

The electromagnetic torque output value from 1e⁻² to 6 e⁻² kg m².

As shown above, with the increase in the friction and the rotational inertia, although other systems have tuned to give an optimum speed and torque respond, the SMC system in this article yields the most significant robust performance with smoother and smaller torque output. It manifests that the SMC algorithm and the FW method in this article contribute to enhance the system control performance.

When IPMSM runs above the base speed during FW control phase, the fluctuation of load torque and inductance parameters will influence the system stability and the FW properties. The relevant experiments have been shown in Figures 24 –29.

Figure 24.

The speed reference and measured value under different load torque reference at 4000 rad/s.

Figure 25.

The electromagnetic torque output value under different load torque reference at 4000 rad/s.

Figure 26.

The FW angle output value under different load torque reference at 4000 rad/s.

Figure 27.

The speed reference and measured value under different inductance parameters at 4000 rad/s.

Figure 28.

The electromagnetic torque output value under different inductance parameters at 4000 rad/s.

Figure 29.

The FW angle output value under different inductance parameters at 4000 rad/s.

Figures 24 –26 represent the output value of the speed, the torque, and the angle of the SMC system, respectively, in this article under different load torque reference at the speed from 600 to 4000 rad/s. As shown in Figures 24 –26, under the same operating conditions, the larger load torque reference results in the larger electromagnetic torque output value and the more negative FW angle. Furthermore, the more negative FW angle further indicates the deeper FW level and the worse FW performance. The simulation results illustrate that with the increase in the load torque reference, the system can be kept stable to a certain extent. But when the maximum load torque reference is increased to 25 N m, the response of speed, electromagnetic torque, and angle begin to deteriorate, and if the maximum load torque reference is larger, the system will tend to be out of control. In this case, we can increase the DC bus voltage $U_{dc}$ or the speed reference $ω_{r}^{*}$ to make the system restore stability.

Figures 27 –29 represent the output value of the speed, the torque, and the angle of the SMC system, respectively, in this article under different inductance parameters at the speed from 600 to 4000 rad/s.

Considering that the inductance parameters may vary significantly in some cases, in order to understand the impact of this change on the system stability, the new quadrature-axis and direct-axis inductance values are set as $L'_{q} = a \cdot L_{q}$ and $L'_{d} = a \cdot L_{d}$ , respectively. As shown in Figure 28, under the same operating conditions, when a = 2, the response of the electromagnetic torque can be kept stable, but when a = 5, the response of the electromagnetic torque begins to obviously fluctuate, and if a is larger, the system will tend to be out of control. As shown in Figure 29, with the increase in the inductance values, the FW performance becomes worse. The simulation results show that with the increase in the inductance parameters, the change will not damage the system stability to a certain extent, but if the inductance parameters are changed too much, the system stability will deteriorate.

Conclusion

Compared with the PI system, the SMC algorithm can improve the IPMSM running performance with more stable speed and torque output in case of motor’s internal and external interferences.

The anti-windup integral controller and the over-modulation unit contribute to strengthening the IPMSM FW properties with smoother speed respond, smaller torque, and absolute value of FW angle output.

When the speed reference, the load torque, the viscous damping coefficient, and the rotational inertia are changed suddenly, compared with other FW systems, the FW system proposed in this article with SMC controller has the most significant stability and powerful robustness to speed and torque respond.

Footnotes

Handling Editor: Crinela Pislaru

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This paper is supported by National Natural Science Foundation (61104027) and Open Fund of Engineering Research Center for Metallurgical Automation and Measurement Technology of Ministry of Education (MADT201702).

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