Predictive Current Control for Switched Reluctance Motor Based on Local Linear Phase Voltage Model

Abstract

In this paper, a predictive phase current control (PCC) scheme based on a local linear phase voltage model for a switched reluctance motor is proposed. The current is controlled by regulating the average voltage through PWM, ensuring a fixed switching frequency. A linear model is proposed to approximate the relationship between the voltage and the current slope in a short period. By using the voltage and current slope information in the previous control cycle, the intercept and slope of the model can be identified online. In the previous control cycle, the phase voltage changes from zero to positive or negative DC-link voltage, and then the identified model is used to predict the average voltage required in the next PWM cycle for the actual current so as to accurately track its reference. The effectiveness of the proposed PCC was verified experimentally. The results demonstrate that the proposed control scheme can significantly reduce the current and torque ripples compared to hysteresis control with the same sampling rate. The proposed PCC is easy to implement, does not need to obtain the motor characteristics in advance and is not sensitive to the changes in characteristic parameters caused by motor aging, etc. It is relatively suitable for applications that need to accurately track the given current curve.

1. Introduction

The switched reluctance motor (SRM) has attracted increasing attention because of its simple and robust structure, absence of permanent magnets and windings in the rotor and high efficiency in a wide speed range. However, SRM has the main disadvantage of high torque ripple due to its double-salient structure [1,2,3]. In order to reduce the torque ripple, numerous solutions related to motor design or motor control have been proposed. Torque control methods can be divided into direct torque control and indirect torque control (ITC). In ITC, the torque is controlled by indirectly controlling the stator phase currents [4,5,6]. Among ITC methods, the torque sharing function (TSF) is commonly used, which distributes the total torque reference into two phase torque references in the commutation region. Based on the nonlinear phase torque characteristics, the phase torque references are converted to phase current references [7,8]. The final torque ripple suppression effect is highly dependent on the tracking performance of the phase current reference.

Hysteresis current control (HCC) is the most widely adopted method for controlling phase currents in SRM owing to its distinctive merits, such as simple structure, model independence, fast dynamic response, strong robustness, insensitivity to motor parameter variation and so on [9,10,11]. However, the switching frequency is unfixed in HCC due to the strong nonlinear magnetic characteristics of SRM, which will increase the difficulty of designing an electromagnetic interference (EMI) filter. In addition, in digital control systems, the control frequency is limited by hardware performance issues, such as the current sampling rate, processor speed and switching time of the power switches. The actual current may continuously deviate from its reference for about a complete control cycle, resulting in a large current ripple. High DC-link voltage and low incremental inductance will cause a larger current ripple. Improving the hardware performance contributes to a reduction in the current ripple, but it will increase the cost of the digital system.

A PI controller is also routinely used for motor current control. A PI controller controls the current by regulating the phase voltage with PWM, which has the advantage of a fixed switching frequency. However, the PI controller suffers from the difficulty in tuning gains due to the strong nonlinearity of SRM. It is hard to find only one set of PI parameters suitable for different speeds and loads simultaneously. Therefore, the variable-gain PI current controller has received research interest. In [12], the back electromotive force (EMF) was derived online by using a two-dimensional (2D) lookup table containing the partial derivative of flux linkage with respect to the position, and its effect on current control was decoupled by feedforward control. To save memory and computational time, the PI gains were simply adapted linearly with both the position and current based on the incremental inductance. In [13], the back EMF and incremental inductance were estimated online by using a pretrained B-spline neural network. The proportional gain and integral gain were linearly tuned with the incremental inductance and winding resistance, respectively. However, training the network was very time-consuming. In [14,15], a Fourier series was used for modeling the incremental inductance and decoupling the effect of the back EMF. In [16], phase winding was modeled by a resistance that varied with the rotor speed and the average reference current, and a pure integral controller was used for controlling the average current in a 120-electrical-degree-wide conducting region. In [17,18], the fixed PI gains most suitable for different operating points were selected based on a piecewise linear flux linkage model. In [19], fractional-order PI control was applied, and the PI parameters were optimized using ant colony optimization.

In the past decade, model predictive control (MPC) has been used in controlling the current of SRM owing to its advantages of accurate tracking ability and fast response. In [20,21], two kinds of single-step MPC strategies, also known as deadbeat predictive control (DBPC) strategies, were proposed. One was based on offline flux linkage characteristics, while the other depended on offline incremental inductance and online estimation of back EMF. With the knowledge of the magnetic characteristics, the required phase voltage in the next digital time step for eliminating the current error can be directly calculated according to the phase voltage equation. These methods have the merit of a fixed switching frequency. Nevertheless, they all need additional memory to store the lookup tables to obtain flux linkage and incremental inductance and are susceptible to the error between the stored characteristics and the actual ones due to motor aging and uncertainties in mass production [22]. In [23,24], the current change rate of one phase in the jth (j = 1, 2, …) switching cycle from the start of the phase conduction was assumed to be the same as those of the other two phases in each electrical period. Under this assumption, the current change rate and hence the PWM duty cycles of the conducting phase could be predicted based on the logged data regarding the two previously conducted phases. This DBPC method has the advantages of model independence as well as accurate and fast-tracking ability at high speed. However, this method also requires additional memory to record historical data. At low speed, the required memory capability would be very large due to a large number of switching cycles and historical data in each electrical period. In addition, this assumption is too rigorous to meet the requirements for actual low-speed operation. The above DBPC schemes all have the drawback of vulnerability to measurement noise due to their large gain. In [25], an extended state observer was proposed to mitigate the effect of disturbances on DBPC. However, the setting of the observer parameters is not yet clear. In [26,27], stochastic MPC was applied for current tracking in an SRM drive by calculating the optimal duty cycles based on the inductance model, which was dynamically calibrated to suppress the effect of the measurement noise and model uncertainties. In [28], the authors applied a fixed-switching-frequency MPC method based on a modified power converter with only one current sensor to reduce the cost and volume of the SRM drive.

A predictive current control (PCC) for SRM drives is proposed in this paper in order to realize fast and accurate current tracking. The relationship between the phase voltage and current slope in a sufficiently short period is described by a linear model, which is identified online using the historical data of the average voltage and current slope during the last control cycle. The required average voltage during the next PWM cycle for eliminating the current error at the end of the cycle is then predicted based on the model. The practicability of the controller was examined experimentally. The results show that the proposed controller can significantly reduce the current and torque ripples compared with the hysteresis controller.

The main contributions of this paper can be summarized as follows:

(1)

A local linear phase voltage model that describes the relationship between the phase voltage and the phase current slope in a short period;

(2)

Predictive current control combined with the voltage model identification online. Compared to the DBPC control mentioned above, the proposed PCC does not require a priori knowledge of the motor parameters and is insensitive to the parameter variation. Compared to HCC, the proposed PCC can greatly reduce the current and torque ripples in a wide speed range without increasing the current sampling rate.

The rest of this paper is organized as follows: Section 2 describes the proposed PCC in detail. The experimental results using the proposed method and HCC are presented and analyzed in Section 3. Finally, conclusions are drawn in Section 4.

2. Predictive Current Control

In this paper, the classical asymmetric half-bridge circuit is used as the power converter. The converter for one phase is shown in Figure 1, where $U_{\mathrm{dc}}$ is the DC-link voltage, M₁ and M₂ are the upper and lower power switches, respectively, and D₁ and D₂ are the upper and lower power diodes, respectively. When the two switches are both turned on, positive DC-link voltage is applied to the phase winding. When the two switches are both turned off, the excitation current will freewheel through the diodes, resulting in negative DC-link voltage across the phase winding. When only one switch is turned on, zero voltage is applied.

2.1. Local Linear Phase Voltage Model

Mutual coupling between phases is usually ignored [29], and thus, the instantaneous voltage of phase A can be given as

$$v_{\mathrm{A}}=R_{\mathrm{s}}{i}_{\mathrm{A}}+L_{\mathrm{inc},\mathrm{A}}\frac{d{i}_{\mathrm{A}}}{dt}+i_{\mathrm{A}}\omega \frac{\partial L_{\mathrm{A}}}{\partial \theta }$$

(1)

where $i_{\mathrm{A}}$ is the phase current; $R_{\mathrm{s}}$, $L_{\mathrm{A}}$ and $L_{\mathrm{inc},\mathrm{A}}$ are the phase resistance, inductance and incremental inductance, respectively; $\theta$ and $\omega$ are the rotor position and speed, respectively; the first and third items on the right are the resistance voltage drop and back EMF, respectively. As can be seen from Figure 1, the instantaneous phase voltage can be $U_{\mathrm{dc}}$, $-U_{\mathrm{dc}}$ or $0$, depending on the switching states of the power switches.

It is observed that the phase current slope is nearly constant with a constant phase voltage in a short period. Therefore, we assume that the phase voltage can be regarded as a linear function related to the current slope in a short period, as expressed in Equation (2).

$$v_{\mathrm{A}}=P_{\mathrm{A}}\frac{d{i}_{\mathrm{A}}}{dt}+Q_{\mathrm{A}}$$

(2)

where $P_{\mathrm{A}}$ and $Q_{\mathrm{A}}$ are the slope and intercept of the linear function, respectively. $P_{\mathrm{A}}$ and $Q_{\mathrm{A}}$ can be expressed as Equations (3) and (4), respectively.

$$P_{\mathrm{A}}=L_{\mathrm{inc},\mathrm{A}}$$

(3)

$$Q_{\mathrm{A}}=R_{\mathrm{s}}{i}_{\mathrm{A}}+i_{\mathrm{A}}\omega \frac{\partial L_{\mathrm{A}}}{\partial \theta }$$

(4)

In a digital system, the average phase voltage in a short period can be derived as Equation (5).

$${\overline{v}}_{\mathrm{A}}=P_{\mathrm{A}}\frac{△i_{\mathrm{A}}}{△t}+Q_{\mathrm{A}}$$

(5)

2.2. Average Voltage Regulation

In this paper, the phase current is controlled by regulating the average phase voltage with PWM. The PWM generation mechanism for positive and negative average voltage is shown in Figure 2, where DM₁ and DM₂ are PWM drive signals for M₁ and M₂, respectively; CR₁ and CR₂ are the comparison values for DM₁ and DM₂, respectively; CTR, M_CTR and T_P denote PWM counter, maximum value of the counter and PWM cycle, respectively; E₁ indicates an event where the larger comparison value matches the down-counting PWM counter, and E₂ indicates an event where the larger comparison value matches the up-counting PWM counter.

The PWM signals are conventionally generated in center-aligned mode, where the PWM signals are symmetrical with respect to the point where counter underflows. The duty cycle depends on the amplitude relationship between the PWM counter and the corresponding comparison value. A positive average voltage is generated by applying positive DC-link voltage and zero voltage at different times of a PWM cycle, while a negative average voltage is generated by applying negative DC-link voltage and zero voltage at different times. An average voltage ranging from −U_dc to U_dc can also be directly implemented by applying positive and negative DC-link voltage at different times of a PWM cycle. However, the current ripple and switching losses will be larger.

The update instant of the comparison values is considered. It is recommended to update the comparison values right at the overflow of the PWM counter. Thus, positive and negative average voltage in a PWM cycle can be simply calculated by Equations (6) and (7), respectively. Otherwise, the average voltage calculation could be much more complicated for the reason that the average voltage is not only related to the updated comparison values but also affected by the nonupdated comparison values and the update instant.

$$\begin{array}{cc}{\overline{v}}_{\mathrm{A}}=U_{\mathrm{dc}}\frac{C{R}_2}{M_{\mathrm{CTR}}},& C{R}_1=0\end{array}$$

(6)

$$\begin{array}{cc}{\overline{v}}_{\mathrm{A}}=-U_{\mathrm{dc}}\frac{C{R}_1}{M_{\mathrm{CTR}}},& C{R}_2=0\end{array}$$

(7)

2.3. Predictive Current Control

Depending on the phase current reference detected at each E₂ event, the phase current control process in a rotor period is divided into the following three stages:

Stage I: In this stage, the target phase is in the non-conductive region with a zero-current reference, and thus, DM₁ and DM₂ are both disabled, and negative DC-link voltage is applied.

Stage II: This stage begins at the start of the phase conduction when the detected current reference changes from zero to positive, and it ends at the next E₂ event. In this stage, both DM₁ and DM₂ are enabled.

Stage III: This stage occurs between stage II and stage I. In this stage, DM₁ and DM₂ are also enabled.

2.3.1. Predictive Current Control in Stage III

The principle is illustrated in Figure 3, where k indicates the current time index; CR_U and CR_L denote the upper and lower limits of the comparison values, respectively; E_2U represents the event in which CR_U matches the up-counting PWM counter. Phase current sampling is synchronized with E₁ and E₂ events. Calculation of the required average phase voltage and the corresponding comparison values in the next PWM cycle starts at the E₂ event. Comparison values are updated at the overflow of the counter.

We assume that the duration of two and a half PWM cycles is short enough, so the phase voltage in [k − 3, k + 4] can be approximated by the linear function shown in Equation (5). The phase voltages during [k − 3, k − 1], [k − 1, k] and [k, k + 4] can be derived as Equations (8)–(10), respectively.

$${\overline{v}}_{\mathrm{A},k-3,k-1}=P_{\mathrm{A}}\frac{i_{\mathrm{A},k-1}-i_{\mathrm{A},k-3}}{t_{\mathrm{A},k-1}-t_{\mathrm{A},k-3}}+Q_{\mathrm{A}}=P_{\mathrm{A}}\frac{△i_{\mathrm{A},k-3,k-1}}{△t_{\mathrm{A},k-3,k-1}}+Q_{\mathrm{A}}$$

(8)

$${\overline{v}}_{\mathrm{A},k-1,k}=P_{\mathrm{A}}\frac{i_{\mathrm{A},k}-i_{\mathrm{A},k-1}}{t_{\mathrm{A},k}-t_{\mathrm{A},k-1}}+Q_{\mathrm{A}}=P_{\mathrm{A}}\frac{△i_{\mathrm{A},k-1,k}}{△t_{\mathrm{A},k-1,k}}+Q_{\mathrm{A}}$$

(9)

$${\overline{v}}_{\mathrm{A},k,k+4}=P_{\mathrm{A}}\frac{i_{\mathrm{A},k+4}^{*}-i_{\mathrm{A},k}}{t_{\mathrm{A},k+4}-t_{\mathrm{A},k}}+Q_{\mathrm{A}}=P_{\mathrm{A}}\frac{△i_{\mathrm{A},k,k+4}}{△t_{\mathrm{A},k,k+4}}+Q_{\mathrm{A}}$$

(10)

In Equation (10), $i_{\mathrm{A},k+4}^{*}$ denotes the reference current at the time instant k + 4. In Equation (9), the voltage ${\overline{v}}_{\mathrm{A},k-1,k}$ can be either $U_{\mathrm{dc}}$ or $-U_{\mathrm{dc}}$ depending on the sign of the average voltage during the current PWM cycle ${\overline{v}}_{\mathrm{A},k-2,k+1}$.

${\overline{v}}_{\mathrm{A},k,k+4}$ should also satisfy Equation (11).

$$\begin{array}{ll}{\overline{v}}_{\mathrm{A},k,k+4}△t_{\mathrm{A},k,k+4}& ={\overline{v}}_{\mathrm{A},k,k+1}△t_{\mathrm{A},k,k+1}+{\overline{v}}_{\mathrm{A},k+1,k+4}{T}_{\mathrm{P}}\\ & ={\overline{v}}_{\mathrm{A},k+1,k+4}{T}_{\mathrm{P}}\end{array}$$

(11)

where ${\overline{v}}_{\mathrm{A},k+1,k+4}$ is the average voltage required in the next PWM cycle. Combining Equations (8)–(11), ${\overline{v}}_{\mathrm{A},k+1,k+4}$ can be predicted by

$$\begin{array}{l}{\overline{v}}_{\mathrm{A},k+1,k+4}=f_{\mathrm{P}}△t_{\mathrm{A},k,k+4}{\overline{v}}_{\mathrm{A},k-1,k}+\\ \frac{f_{\mathrm{P}}△t_{\mathrm{A},k-3,k-1}\left({\overline{v}}_{\mathrm{A},k-3,k-1}-{\overline{v}}_{\mathrm{A},k-1,k}\right)\left(△i_{\mathrm{A},k,k+4}△t_{\mathrm{A},k-1,k}-△i_{\mathrm{A},k-1,k}△t_{\mathrm{A},k,k+4}\right)}{△i_{\mathrm{A},k-3,k-1}△t_{\mathrm{A},k-1,k}-△i_{\mathrm{A},k-1,k}△t_{\mathrm{A},k-3,k-1}}\end{array}$$

(12)

where $f_{\mathrm{P}}$ is the PWM frequency. As can be seen from Figure 3, ${\overline{v}}_{\mathrm{A},k-3,k-1}$ is fixed to zero. Thus, Equation (12) can be simplified to Equation (13).

$$\begin{array}{l}{\overline{v}}_{\mathrm{A},k+1,k+4}=f_{\mathrm{P}}{\overline{v}}_{\mathrm{A},k-1,k}△t_{\mathrm{A},k,k+4}-\\ \frac{f_{\mathrm{P}}{\overline{v}}_{\mathrm{A},k-1,k}△t_{\mathrm{A},k-3,k-1}\left(△i_{\mathrm{A},k,k+4}△t_{\mathrm{A},k-1,k}-△i_{\mathrm{A},k-1,k}△t_{\mathrm{A},k,k+4}\right)}{△i_{\mathrm{A},k-3,k-1}△t_{\mathrm{A},k-1,k}-△i_{\mathrm{A},k-1,k}△t_{\mathrm{A},k-3,k-1}}\end{array}$$

(13)

According to Figure 2 and Figure 3, $\Delta t_{\mathrm{A},k-3,k-1}$, $\Delta t_{\mathrm{A},k-1,k}$ and $\Delta t_{\mathrm{A},k,k+4}$ can be obtained as Equations (14)–(16), respectively.

$$\begin{array}{ll}△t_{\mathrm{A},k-3,k-1}& = △t_{\mathrm{A},k-3,k-2} +△t_{\mathrm{A},k-2,k-1}\\ & =0.5\left(T_{\mathrm{P}}-\frac{\left|{\overline{v}}_{\mathrm{A},k-5,k-2}\right|}{U_{\mathrm{dc}}}\right)+0.5\left(T_{\mathrm{P}}-\frac{\left|{\overline{v}}_{\mathrm{A},k-2,k+1}\right|}{U_{\mathrm{dc}}}\right)\\ & =T_{\mathrm{P}}-\frac{0.5}{U_{\mathrm{dc}}}\left(\left|{\overline{v}}_{\mathrm{A},k-5,k-2}\right|+\left|{\overline{v}}_{\mathrm{A},k-2,k+1}\right|\right)\end{array}$$

(14)

$$△t_{\mathrm{A},k-1,k}=\frac{\left|{\overline{v}}_{\mathrm{A},k-2,k+1}\right|T_{\mathrm{P}}}{U_{\mathrm{dc}}}$$

(15)

$$\begin{array}{ll}△t_{\mathrm{A},k,k+4}& = △t_{\mathrm{A},k,k+1}+T_{\mathrm{P}}\\ & =0.5\left(T_{\mathrm{P}}-\frac{\left|{\overline{v}}_{\mathrm{A},k-2,k+1}\right|}{U_{\mathrm{dc}}}\right)+T_{\mathrm{P}}\\ & =1.5T_{\mathrm{P}}-\frac{\left|{\overline{v}}_{\mathrm{A},k-2,k+1}\right|}{2U_{\mathrm{dc}}}\end{array}$$

(16)

where ${\overline{v}}_{\mathrm{A},k-5,k-2}$ and ${\overline{v}}_{\mathrm{A},k-2,k+1}$ denote the average voltages in the previous and current PWM cycles.

As can be seen from Equations (8) and (9), $\Delta t_{\mathrm{A},k-3,k-1}$ and $\Delta t_{\mathrm{A},k-1,k}$ must be positive. Therefore, upper and lower limits are set for the comparison values. Thus, the average voltage in each PWM cycle should satisfy Equation (17) or (18). The upper and lower limits should satisfy Equation (19).

$$U_{\mathrm{dc}}\frac{C{R}_L}{M_{\mathrm{CTR}}}\le {\overline{v}}_{\mathrm{A}}\le U_{\mathrm{dc}}\frac{C{R}_U}{M_{\mathrm{CTR}}}$$

(17)

$$-U_{\mathrm{dc}}\frac{C{R}_U}{M_{\mathrm{CTR}}}\le {\overline{v}}_{\mathrm{A}}\le -U_{\mathrm{dc}}\frac{C{R}_L}{M_{\mathrm{CTR}}}$$

(18)

$$0<C{R}_L<C{R}_U<M_{\mathrm{CTR}}$$

(19)

Considering the inevitable calculation time of the required comparison values, the upper limit CR_U should also meet

$$0.5T_{\mathrm{P}}\left(1-\frac{C{R}_{\mathrm{U}}}{M_{\mathrm{CTR}}}\right)>T_{\mathrm{cal}}$$

(20)

where T_cal denotes the calculation time. Thus, the calculation will end before the next overflow of the PWM counter when the comparison values update, and the average voltage and comparison values can have a simple directly proportional relationship, as mentioned above. The block diagram and flowchart of the proposed PCC in stage III are illustrated in Figure 4 and Figure 5, respectively. In Figure 4, $abs$ and $sgn$ denote the absolute value function and the signum function, respectively.

2.3.2. Current Control in Stage II

Since there is a no-current region before and in stage II, as shown in Figure 6, the predictive current control in stage III is not applicable here. Therefore, a maximum positive average voltage is applied in the PWM cycle after the start of stage II to quickly establish the current. The above-mentioned current control is still not strictly applicable by the end of stage II due to the presence of the no-current region between the last $E_1$ and $E_2$ events. However, the current during stage II and the following PWM cycle is small, and the phase conduction in SRM usually starts in the region close to the unaligned rotor position, where the inductance is insensitive to the rotor position and current. Thus, the resistive voltage drop and back EMF are small, as can be seen from Equations (3) and (4). Therefore, the influence of zero voltage on current variation can be neglected, and the average voltage in the next PWM cycle can also be predicted using Equation (13) by the end of stage II.

For ease of reference, symbols in the formulas of Section 2 are summarized in

Table 1. List of symbols in formulas of Section 2.

Symbol	Meaning
$U_{\mathrm{dc}}$	DC-link voltage
$v,i$	Voltage and current of a phase winding
$R_{\mathrm{s}},L,L_{\mathrm{inc}}$	Resistance, inductance and incremental inductance of a phase winding
$\theta ,\omega$	Rotor position and speed
$P,Q$	Slope and intercept of the local linear phase voltage model
$T_{\mathrm{P}},f_{\mathrm{P}}$	PWM cycle and frequency
$M_{\mathrm{CTR}}$	Maximum value of the PWM counter
$C{R}_1,C{R}_2$	Comparison values for the PWM drive signals of the upper and lower power switches in phase A converter
$C{R}_{\mathrm{U}},C{R}_{\mathrm{L}}$	$\mathrm{Upper}\mathrm{and}\mathrm{lower}\mathrm{limits}\mathrm{of}\mathrm{the}\mathrm{comparison}\mathrm{values}C{R}_1$$\mathrm{and}C{R}_{\mathrm{L}}$
$T_{\mathrm{cal}}$	Calculation time of the comparison values
k, A	Subscripts denoting current time index and quantities related to phase A
*, ¯	Superscripts denoting referenced and average quantities
Δ	Triangle on the left of other symbols denoting incremental quantities
${\overline{v}}_{\mathrm{A},sv,ev}$	$\mathrm{Average}\mathrm{voltage}\mathrm{during}\mathrm{the}\mathrm{time}\mathrm{interval}\mathrm{from}\mathrm{the}\mathrm{time}\mathrm{index}sv$$\mathrm{to}\mathrm{the}\mathrm{time}\mathrm{index}ev$
$\Delta i_{\mathrm{A},si,ei}$	$\mathrm{Difference}\mathrm{between}\mathrm{phase}\mathrm{A}\mathrm{currents}\mathrm{at}\mathrm{time}\mathrm{indexes}si$$\mathrm{and}ei$
$\Delta t_{\mathrm{A},st,et}$	$\mathrm{Time}\mathrm{difference}\mathrm{between}\mathrm{time}\mathrm{indexes}st$$\mathrm{and}et$

.

3. Experimental Verification

In order to prove the superiority of the proposed PCC over the conventional HCC, experiments were conducted based on the test bench shown in Figure 7. The parameters of the experimental SRM are listed in

Table 2. Parameters of the experimental switched reluctance motor.

Parameter Name	Value	Parameter Name	Value
Rated power	1 kW	Number of phases	3
Rated voltage	72 V	Number of stator poles	12
Rated speed	4300 rpm	Number of rotor poles	8

. A magnetic powder brake was used as the load machine. A three-phase asymmetric half-bridge circuit was used as the power converter. MOSFETs were selected as the power switches. A 72 V battery was used as the DC power supply. The STM32F103RCT6 microcontroller (MCU) was used to realize the control algorithms.

The block diagrams corresponding to the proposed PCC and conventional HCC are given in Figure 8, where $LUTi$, $En$ and $T^{*}$ denote the lookup table storing the current–position–torque characteristics, the PWM enable signals and the total torque reference, respectively. Cosine TSF is used to generate the phase reference currents [30]. By ignoring the variation of the total torque reference and the rotor speed in three PWM cycles, the reference currents $i_{\mathrm{A},k+4}^{*}$, $i_{\mathrm{B},k+4}^{*}$ and $i_{\mathrm{C},k+4}^{*}$ at the overflow after next are predicted, as illustrated in Figure 8a. The prediction of these three reference currents starts at the previous overflow.

For the proposed PCC, the PWM frequency was set to 10 kHz; the upper and lower limits of comparison values were set to 80% and 20% of the maximum value of the PWM counter, respectively. For conventional HCC, the current sampling and control frequency were both set to 20 kHz to make a fair comparison; the hysteresis band was set to 0.5 A. The reason for the PWM frequency selection is explained in Section 3.3 below.

To quantify the overall current-tracking performance, the current root-mean-square error (RMSE) is defined as

$$i_{\mathrm{A},\mathrm{RMSE}}=\sqrt{\frac{1}{N}{\displaystyle \sum _{j=1}^N{i}_{\mathrm{A},\mathrm{err}}{}^2}}=\sqrt{\frac{1}{N}{\displaystyle \sum _{j=1}^N{\left[i_{\mathrm{A}}^{*}\left(j\right)-i_{\mathrm{A}}\left(j\right)\right]}^2}}$$

(21)

where N is the number of data points.

Similarly, to measure the torque ripple, the torque RMSE is defined as

$$T_{\mathrm{A},\mathrm{RMSE}}=\sqrt{\frac{1}{N}{\displaystyle \sum _{j=1}^N{T}_{\mathrm{A},\mathrm{err}}{}^2}}=\sqrt{\frac{1}{N}{\displaystyle \sum _{j=1}^N{\left[T_{\mathrm{A}}^{*}\left(j\right)-T_{\mathrm{A}}\left(j\right)\right]}^2}}$$

(22)

3.1. Experimental Results under Heavy Load

In order to evaluate the heavy load performance at different speeds, experimental waveforms for both control strategies under a 5 Nm load at speeds of 100, 400 and 700 rpm are presented in Figure 9, Figure 10 and Figure 11, respectively. The quantitative comparison is shown in Figure 12, where the current and torque RMSEs of the two strategies are given under a 5 Nm load at speeds of 100, 250, 400, 550 and 700 rpm.

It can be seen that over the tested speed range, the current- and torque-tracking errors of the proposed control scheme are significantly lower than those of HCC under heavy load. The current RMSE using the proposed PCC is 44.4% lower on average than that using HCC, and the torque RMSE using the proposed PCC is 48.6% lower on average. Moreover, the proposed control method has the advantage of maintaining a fixed switching frequency without increasing the phase current sampling rate. It can also be observed that under the proposed PCC, the peak inverse current ripple occurs immediately after the phase conduction. This is due to the application of the maximum average voltage (80% of the DC-link voltage in this case) in stage II. However, the peak inverse ripple occurs somewhere near the unaligned rotor position, where the inductance profiles are nearly flat, and consequently, the torque production ability is weak. Therefore, this inverse peak will not cause a large torque ripple, as can be seen from the torque-related waveforms in Figure 9, Figure 10 and Figure 11.

3.2. Experimental Results under Light Load

In order to evaluate the performance under light load at different speeds, the experimental waveforms under a 1.25 Nm load at speeds of 600, 1000 and 1400 rpm are presented in Figure 13, Figure 14 and Figure 15, respectively. A quantitative comparison under a 1.25 Nm load is shown in Figure 16. It can also be seen that both the current- and torque-tracking errors of the proposed control scheme are markedly lower than those of HCC under light load at different speeds. The current RMSE using the proposed PCC is 19.3% lower on average than that using HCC, and the torque RMSE using the proposed method is 55.9% lower on average. As can be seen from Figure 16, the current RMSE generally increases with the speed under the proposed PCC. This is mainly because the phase conduction time of each electrical period is shorter, and consequently, the peak inverse current ripple right after conduction will impose a larger impact on the overall current-tracking performance at high speed. However, the torque ripple is still only slightly affected by this peak inverse current ripple.

3.3. Computational Time

The execution time of the proposed PCC was measured by comparing the values of the PWM counter just before and after the execution. The CPU frequency was set to 72 MHz. All numbers with decimal places are represented by integers, which are derived by increasing the numbers in proportion to a certain extent, and thus, the computation time of the arithmetic operations can be greatly reduced when using an MCU with no floating-point unit. For example, a current of 0.01 A is represented by an integer of 1. The maximum execution time of only about 3.56 us for controlling single-phase current was obtained at stage III, and the execution time (nearly 0.931 us) was at the minimum in stage I. When two phases were conducted simultaneously during commutation, the maximum time for running the three-phase current control algorithm was nearly 8.042 us. Therefore, the proposed PCC can be implemented in a low-cost MCU.

Based on the relationship between the PWM cycle and the calculation time given in Equation (20), the PWM cycle should be greater than 80.42 us. With an allowance, the PWM cycle was set to 100 us, and accordingly, the PWM frequency was set to 10 kHz.

4. Conclusions

A predictive phase current control scheme is proposed in this paper. Current regulation is realized through PWM control to ensure a fixed switching frequency. A linear phase voltage model is introduced to fit the relationship between the voltage and the current slope in a short period. The parameters of the model are dynamically identified using the historical voltage and current information in the previous control cycle, where the voltage changes from zero to positive or negative DC-link voltage, and then the average voltage required in the next PWM cycle is predicted according to the model. Experimental results are provided to compare the performance of the proposed control scheme with the conventional hysteresis current control. The main conclusions are as follows:

(1)

Given the same sampling rate, the current and torque ripples using the proposed PCC are significantly lower than those using hysteresis control. The current RMSE using the proposed method is 32.6% lower on average and 48.1% lower at most, and the torque RMSE using the proposed method is 50.9% lower on average and 62.96% lower at most.

(2)

The execution time of the proposed control scheme is computationally efficient, so the proposed PCC can be implemented in a low-cost MCU.

(3)

The proposed PCC is easy to implement and does not need to know the motor characteristics in advance. It is insensitive to the characteristic changes caused by factors such as motor aging.

In summary, the proposed PCC is relatively suitable for tracking a given current profile. In future work, the proposed control scheme could be modified to be suitable for power converters with reduced current sensors.