Simplified Reactive Torque Model Predictive Control of Induction Motor with Common Mode Voltage Suppression

Siyao CHU, Bin WANG, Xinwei NIU

Summary :

To reduce the common mode voltage (CMV), suppress the CMV spikes, and improve the steady-state performance, a simplified reactive torque model predictive control (RT-MPC) for induction motors (IMs) is proposed. The proposed prediction model can effectively reduce the complexity of the control algorithm with the direct torque control (DTC) based voltage vector (VV) preselection approach. In addition, the proposed CMV suppression strategy can restrict the CMV within ±V_dc/6, and does not require the exclusion of non-adjacent non-opposite VVs, thus resulting in the system showing good steady-state performance. The effectiveness of the proposed design has been tested and verified by the practical experiment. The proposed algorithm can reduce the execution time by an average of 26.33% compared to the major competitors.

• Publication: IEICE TRANSACTIONS on Electronics Vol.E107-C No.5 pp.132-140
• Publication Date: 2024/05/01
• Publicized: 2023/11/30
• Online ISSN: 1745-1353
• DOI: 10.1587/transele.2023ECP5034
• Type of Manuscript: PAPER
• Category: Electronic Instrumentation and Control

1.  Introduction

Thanks to the advantages of simple structure, high reliability and easy maintenance, induction motors (IMs) have widely attracted widespread attention. As two basic closed-loop control algorithms of IMs, field-oriented control (FOC) and direct torque control (DTC) have attracted more attention [1]. In addition, model predictive control (MPC) has been widely studied and applied in the field of motor control due to its simplicity and fast closed-loop response in recent years [2], [3].

In the field of AC motor control, model predictive torque control (MPTC) has been paid special attention as it provides straightforward implementation, fast torque dynamic response, and the consideration of other constraints and nonlinearities [4]. The authors in [4]-[7] proposed the reactive torque MPC (RT-MPC) method to improve the system control including the model predictive current control (MPCC) and the model predictive flux control (MPFC). Also, the computational intensive control algorithm will affect the system performance in MPC.  The existing computational burden reduction methods can generally be classified into two categories as reducing the complexity of prediction models and voltage vector (VV) preselection [5], [6], [8]-[11]. In [5] and [6], the torque and flux references are equivalently converted into a new flux reference or reference stator VV, thus reducing the complexity of prediction models.

The existing VV preselection methods can be divided into two categories as online calculation and lookup table (LUT) [8]-[11]. The candidate VVs are calculated based on the deadbeat principle for the online calculation method, which significantly reduces the computational burden and achieves better control performance [8]. LUT method includes different preselection methods by using DTC principle, deadbeat principle, and cost function [8]-[11].

In the two-level voltage source inverter (2L-VSI), the space vectors can be divided into two types, i.e., non-zero VV (NZVV) and zero VV (ZVV). Compared with NZVVs, the ZVV generates a higher common-mode voltage (CMV) amplitude, resulting in higher bearing current and shaft voltages, which causes damage to the motor and reduces the reliability of the overall motor drive system [12], [13]. Hardware-based and software-based solutions to suppress CMV have been widely used [14]-[25]. In the hardware-based method, filters and special topologies are the two main methods to suppress CMV, but this increases the hardware cost [14], [15]. In the software-based method, it can be divided into space vector pulse width modulation (PWM)-based method and MPC-based method [16]-[25].

MPC allows the consideration of other constraints and nonlinearities. Therefore, CMV can be suppressed by adding nonlinear or CMV terms with weighting factors to the cost function [17], [18]. Suppression of CMV by VV preselection is another MPC-based approach [19]-[26]. In [19]-[21], ZVV is abandoned, and only six NZVVs are adopted to restrict the CMV within \(\pm V_\mathrm{dc}/6\). In [20], a deadbeat control is proposed to reduce the candidate VVs. In [21], VV preselection considers both the reduction of switching loss and CMV.  However, the effect of deadtime is not considered in [19]-[21]. In [22], it is pointed out that non-adjacent non-opposite NZVV switching combinations lead to CMV spikes. Therefore, all non-adjacent non-opposite NZVV switching combinations are pre-excluded. As a result, the CMV is completely restricted within \(\pm V_\mathrm{dc}/6\). In [23], the influence of deadtime on the non-adjacent non-opposite NZVV switching combination is analyzed, and the VV preselection method is further improved to improve the control performance. However, VV preselection methods that exclude non-adjacent non-opposite NZVV still have a negative effect on steady-state performance.

This paper proposes a simplified RT-MPC strategy with CMV suppression to reduce the CMV, suppress the CMV spikes, improve steady-state performance, and reduce the computational burden. In order to reduce the computational burden, the DTC-based VV preselection method is used to reduce the number of candidate VVs, and a simplified prediction model is proposed. In addition, a novel CMV suppression strategy is proposed to restrict CMV within \(\pm V_\mathrm{dc}/6\). The proposed CMV suppression strategy does not need to exclude non-adjacent non-opposite NZVVs, thus resulting in good control performance. The rest of this paper is organized into four sections. Section 2 presents the theoretical basis of CMV and the classic RT-MPC.  Section 3 presents the proposed simplified RT-MPC.  The simulation and experimental validations are provided in Sect. 4, and the conclusion after the comparative study is given in Sect. 5.

2.  Theoretical Basis of CMV and Traditional RT-MPC

2.1  Mathematical Model of IM

In the \(\alpha\)-\(\beta\) coordinate system, the mathematical model of IM driven by a 2L-VSI is given by [7]

\[\begin{align} & \boldsymbol{u}_{\mathrm{s}} =R_{\mathrm{s}}\boldsymbol{i}_{\mathrm{s}} +\frac{\mathrm{d}\boldsymbol{\psi}_{\mathrm{s}}}{\mathrm{d}t}, \tag{1} \\ & \boldsymbol{\psi}_{\mathrm{s}} =L_{\mathrm{s}} \boldsymbol{i}_{\mathrm{s}} +L_{\mathrm{m}} \boldsymbol{i}_{\mathrm{r}} =\boldsymbol{i}_{\mathrm{s}}/(\lambda L_{\mathrm{r}})+L_{\mathrm{m}}\boldsymbol{\psi}_{\mathrm{r}}/L_{\mathrm{r}}, \tag{2} \\ & \boldsymbol{\psi}_{\mathrm{r}} =L_{\mathrm{m}}\boldsymbol{i}_{\mathrm{s}} +L_{\mathrm{r}} \boldsymbol{i}_{\mathrm{r}} =-\boldsymbol{i}_{\mathrm{s}}/(\lambda L_{\mathrm{m}})+L_{\mathrm{r}}\boldsymbol{\psi}_{\mathrm{s}}/L_{\mathrm{m}}, \tag{3} \\ & \boldsymbol{i}_{\mathrm{s}}+\tau_{\sigma}\frac{\mathrm{d}\boldsymbol{i}_{\mathrm{s}}}{\mathrm{d}t}=\frac{k_{\mathrm{r}}}{R_{\sigma}} \left(\frac{1}{\tau_{\mathrm{r}}}-jn_{p}\omega_{\mathrm{m}}\right)\boldsymbol{\psi}_{\mathrm{r}}+\frac{\boldsymbol{u}_{\mathrm{s}}}{R_{\sigma}}, \tag{4} \end{align}\]

where \(\boldsymbol{u}_\mathrm{s}=[\boldsymbol{u}_{\mathrm{s}\alpha}\>\boldsymbol{u}_{\mathrm{s}\beta}]^{T}\) is the stator VV; \(\boldsymbol{i}_\mathrm{s}=[\boldsymbol{i}_{\mathrm{s}\alpha}\>\boldsymbol{i}_{\mathrm{s\beta}}]^\mathrm{T}\) and \(\boldsymbol{i}_\mathrm{r}=[\boldsymbol{i}_{\mathrm{r\alpha}}\>\boldsymbol{i}_{\mathrm{r\beta}}]^\mathrm{T}\) are the stator and rotor current vector; \(\boldsymbol{\psi}_\mathrm{s}=[\boldsymbol{\psi}_{\mathrm{s}\alpha}\>\boldsymbol{\psi}_{\mathrm{s}\beta}]^\mathrm{T}\) and \(\boldsymbol{\psi}_\mathrm{r}=[\boldsymbol{\psi}_{\mathrm{r\alpha}}\>\boldsymbol{\psi}_{\mathrm{r}\beta}]^\mathrm{T}\) are the stator and rotor flux vector; \(R_\mathrm{s}\), \(L_\mathrm{s}\), \(L_\mathrm{r}\), \(L_\mathrm{m}\) are the stator resistances, stator, rotor and mutual inductances, respectively; \(n_\mathrm{p}\) and \(\omega_\mathrm{m}\) are the number of pole pairs and rotor angular speed, respectively; \(\lambda =1/(L_\mathrm{s}L_\mathrm{r}-L_\mathrm{m}^{2})\), \(\tau_\mathrm{r}=L_\mathrm{r}/R_\mathrm{r}\), \(k_\mathrm{r}=L_\mathrm{m}/L_\mathrm{r}\), \(R_{\sigma}=R_\mathrm{s}+R_\mathrm{r}k_\mathrm{r}^{2}\), \(\tau_{\sigma}=\sigma L_\mathrm{s}/R_{\sigma}\); \(\sigma =1-L_\mathrm{m}^{2}/(L_\mathrm{s}L_\mathrm{r})\) is the leakage inductance coefficient; \(R_\mathrm{r}\) is the rotor resistance.

2.2  Definition of CMV

The topology of IM fed by 2L-VSI is shown in Fig. 1 (a). The CMV is defined as the potential between the load neutral point and the center of the dc bus [19]. In Fig. 1 (a), the CMV can be calculated based on

\[\begin{align} U_{\mathrm{CMV}} =\frac{u_{\mathrm{ao}}+u_{\mathrm{bo}}+u_{\mathrm{co}}}{3}=\frac{V_{\mathrm{dc}}}{6}(S_{\mathrm{a}}+S_{\mathrm{b}}+S_{\mathrm{c}}), \tag{5} \end{align}\]

where \(U_\mathrm{CMV}\) and \(V_\mathrm{dc}\) are the CMV of a 2L-VSI and the dc voltage, respectively; \(u_{{x}\mathrm{o}}\) and \(S_{x}(x\in \{\mathrm{a}, \mathrm{b}, \mathrm{c}\}\)) denote the voltage difference of phase \(x\) to o and the switching state of phase \(x\), respectively. In addition, \(S_{x}=1\) represents the upper insulated-gate bipolar transistor (IGBT) of phase \(x\) is turned on, and the lower IGBT of phase \(x\) is turned on when \(S_{x}=-1\).

According to equation (Eqn.) (6), the CMVs of eight VVs in Fig. 1 (b) can be calculated, as shown in Table I [19]. It can be seen that the CMV of NZVVs is \(\pm V_\mathrm{dc}/6\), while the ZVVs generate a larger absolute value of CMV.

Fig. 1  IM system. (a) Topology of the system. (b) Eight VVs.

Table 1  CMV of different VVs

2.3  Effects of Dead Time

Nowadays, scholars have proposed some control strategies to suppress CMV, such as the six NZVVs (6VV) strategy [19]. Unfortunately, due to the deadtime effects, there are still some CMV spikes under switching between non-adjacent non-opposite NZVVs [22].

Thus, the five NZVVs (5VV) strategy is proposed in [22] to eliminate the deadtime effects. In 5VV method, the non-adjacent non-opposite NZVV switching combinations are abandoned to suppress CMV spikes. In addition, the ZVV is replaced with two opposite VVs to improve the steady-state performance. Therefore, the candidate VVs of 5VV strategy include one ZVV and four NZVVs. However, the torque, stator flux, and current total harmonic distortion (THD) of the 5VV strategy will be increased due to the non-adjacent non-opposite NZVVs are abandoned. In [23], the causes of equivalent ZVV under deadtime are analyzed in detail, the results show that the three-phase current direction and non-adjacent non-opposite NZVV switching combinations cause the CMV spikes. Therefore, a new VV preselection strategy is proposed. Compared to 5VV strategy, the steady-state performance has been improved.

2.4  Traditional RT-MPC

The control diagram of RT-MPC is shown in Fig. 2, \(i_{x}(x\in \{\mathrm{a, b, c}\})\) is the current of phase \(x\), and \(\boldsymbol{V}_\mathrm{opt}\) is the optimal VV.

Fig. 2   Control diagram of RT-MPC.

The external loop controller generates torque reference \(T_\mathrm{e}^{\ \ast}\) and reactive torque reference \(T_\mathrm{R}^{\ \ast}\) based on proportional-integral (PI), and the difference between reference speed \(\omega_\mathrm{m}^{\ \ast}\), reference flux \(|\boldsymbol{\psi}_\mathrm{s}^{\ \ast}|\) and the corresponding actual speed and actual flux, respectively. For stator and rotor flux estimation, the basis state-space model of IM is given by

\[\begin{align} & \left\{\begin{array}{@{}l} \dfrac{\mathrm{d}\boldsymbol{i}_{\mathrm{s}}}{\mathrm{d}t} =A_{1}\boldsymbol{i}_{\mathrm{s}}+A_{2}\boldsymbol{\psi}_{\mathrm{s}}+B\boldsymbol{u}_{\mathrm{s}}, \\ \dfrac{\mathrm{d}\boldsymbol{\psi}_{\mathrm{s}}}{\mathrm{d}t}=A_{3}\boldsymbol{i}_{\mathrm{s}}+\boldsymbol{u}_{\mathrm{s}}, \end{array}\right. \tag{6} \\ & \left\{\begin{array}{@{}l} A_{1} = jn_{\mathrm{p}} \omega_{\mathrm{m}}-\lambda (R_{\mathrm{s}} L_{\mathrm{r}} +R_{\mathrm{r}} L_{\mathrm{s}}) , \\ A_{2} =\lambda (R_{\mathrm{r}}-jL_{\mathrm{r}} n_{\mathrm{p}} \omega_{\mathrm{m}}),\,A_{3} =-R_{\mathrm{s}},\,B=\lambda L_{\mathrm{r}}. \end{array}\right. \tag{7} \end{align}\]

First, the Heun’s method was used to discretize (6) to estimate the stator and rotor flux at \(k_\mathrm{th}\) instant, and then to predict the parameters at (\(k+1)_\mathrm{th}\) instant to compensate for the one-step delay in the digital implementation [5], [30].

Second, the stator flux and stator current at (\(k+2)_\mathrm{th}\) instant under different candidate VVs are predicted according to the prediction model. Then, the reactive torque \(T_\mathrm{R}(k+2)\) and torque \(T_\mathrm{e}(k+2)\) under different VVs are obtained according to the predicted values [7]. The prediction model of RT-MPC is as follows

\[\begin{align} & \boldsymbol{\psi}_{\mathrm{s}} (k+2)=\boldsymbol{\psi}_{\mathrm{s}} (k+1)+T_{\mathrm{s}}\boldsymbol{V}_{i}-R_{\mathrm{s}} T_{\mathrm{s}} \boldsymbol{i}_{\mathrm{s}} (k+1), \tag{8} \\ & \begin{array}{@{}l} \boldsymbol{i}_{\mathrm{s}}(k+2)=(k_{\mathrm{r}} (1-jn_{p}\omega_{\mathrm{m}} \tau_{\mathrm{r}})\boldsymbol{\psi}_{\mathrm{r}}(k+1)+\tau_{\mathrm{r}}\boldsymbol{V}_{i}) \\ \phantom{\boldsymbol{i}_{\mathrm{s}}(k+2)=\ }T_{\mathrm{s}}/\tau_{\sigma}R_{\sigma}\tau_{\mathrm{r}}+(1-T_{\mathrm{s}}/\tau_{\sigma})\boldsymbol{i}_{\mathrm{s}}(k+1), \end{array} \tag{9} \\ & T_{\mathrm{e}} (k+2)=1.5n_{\mathrm{p}}\{\boldsymbol{\psi}_{\mathrm{s}} (k+2)\otimes \boldsymbol{i}_{\mathrm{s}}(k+2)\}, \tag{10} \\ & T_{\mathrm{R}} (k+2)=1.5n_{\mathrm{p}} \{\boldsymbol{\psi}_{\mathrm{s}} (k+2)\odot \boldsymbol{i}_{\mathrm{s}} (k+2)\}, \tag{11} \end{align}\]

where \(\boldsymbol{V}_{i}\) (\(i\in \{\mathrm{0, 1, 2, \ldots , 7}\}\)) represents the candidate VVs, \(\otimes\) and \(\odot\) are the external and inner product operators, respectively.

Third, a cost function containing reactive torque and torque is designed to evaluate the control performance of all candidate VVs, as shown in (12). It should be noted that since reactive torque and torque have the same dimension, the design of the weighting factors is avoided [7].

\[\begin{align} J=\left| T_{\mathrm{e}}^{\ \ast}-T_{\mathrm{e}}(k+2)\right| +\left| T_{\mathrm{R}}^{\ \ast}-T_{\mathrm{R}} (k+2)\right|, \tag{12} \end{align}\]

where \(J\) represents the cost function value. The predicted of torque and reactive torque under different switching states are evaluated by (12), and the one that can minimize the cost function is chosen and applied during the next control cycle.

3.  Proposed Simplified RT-MPC

3.1  Voltage Vectors Preselection

This paper uses the candidate VVs preselection method based on the DTC principle. The candidate VVs are selected using the present position of stator flux and torque deviation \(\varDelta T_\mathrm{e} = T_\mathrm{e}^{\ \ast}-T_\mathrm{e}\). The present position of stator flux \(\theta\) is estimated as

\[\begin{align} \theta =\arctan(\psi_{\mathrm{s\beta}}(k+1)/\psi_{\mathrm{s\alpha}} (k+1)), \tag{13} \end{align}\]

where \(\psi_{\mathrm{s}\alpha}(k+1)\), \(\psi_{\mathrm{s}\beta}(k+1)\) represents the real and imaginary parts of the stator flux \(\psi_\mathrm{s}(k+1)\), respectively.

In 2L-VSI, the space distribution of all VVs in the \(\alpha\)-\(\beta\) plane is divided into six sectors, and the sector number change periodically by an angle \(\pi/3\) rad steps, as shown in Fig. 3. Thus, the relationship of the sector number \(N\) and the stator flux angle is given by

\[\begin{align} N=\mathrm{ceil}((3\theta +\pi /2)/\pi), \tag{14} \end{align}\]

where ceil represents the function of rounding up.

As shown in Fig. 3, assuming \(\boldsymbol{\psi}_\mathrm{s}(k+1)\) is located in sector I.  For \(\varDelta T_\mathrm{e} > 0\), the VVs (\(\boldsymbol{V}_{2}\), \(\boldsymbol{V}_{3})\) satisfying torque increase (TI) are the candidate VVs. In addition, \(\boldsymbol{V}_{2}\) and \(\boldsymbol{V}_{3}\) also satisfied the possible condition of stator flux deviation \(\varDelta \psi_\mathrm{s} > 0\) or \(\varDelta \psi_\mathrm{s} < 0\), where \(\varDelta \psi_\mathrm{s} = |\boldsymbol{\psi}_\mathrm{s}^{\ \ast}|- |\boldsymbol{\psi}_\mathrm{s}|\), and \(|\boldsymbol{\psi}_\mathrm{s}|\) is the amplitude of \(\boldsymbol{\psi}_\mathrm{s}\). For \(\varDelta T_\mathrm{e}< 0\), \(\boldsymbol{V}_{5}\) and \(\boldsymbol{V}_{6}\) are the candidate VVs. When \(\varDelta T_\mathrm{e} = 0\), \(\boldsymbol{V}_{0}\) is the candidate VV.  Generally, it is necessary to employ the candidate VVs with the ZVV to reduce the torque and stator flux ripples. Hence, the total number of candidate VVs is three. With the same principle, the candidate VVs for all the sectors are shown in Table 2.

Fig. 3  Space distribution of all VVs of a 2L-VSI.

Table 2  Candidate VVs

3.2  Simplified Prediction Model

In the traditional RT-MPC, the prediction model includes the prediction of stator flux, stator current, torque, and reactive torque, and the prediction of stator current is complicated and has a significant computational burden. Thus, an improved prediction model is proposed in this paper to reduce the complexity of the prediction model.

Substituting (1) and (3) into (4) to eliminate the VV and the rotor flux term, the expression is as follows

\[\begin{align} \begin{array}{@{}l} \dfrac{\mathrm{d}\boldsymbol{i}_{\mathrm{s}}}{\mathrm{d}t}=(jn_{\mathrm{p}} \omega_{\mathrm{m}} -\lambda R_{\mathrm{r}} L_{\mathrm{s}})\boldsymbol{i}_{\mathrm{s}} +\lambda L_{\mathrm{r}} \dfrac{\mathrm{d}\boldsymbol{\psi}_{\mathrm{s}}}{\mathrm{d}t} \\ \phantom{\frac{\mathrm{d}\boldsymbol{i}_{\mathrm{s}}\ }{\mathrm{d}t}=}+\lambda (R_{\mathrm{r}} -jL_{\mathrm{r}} n_{\mathrm{p}} \omega_{\mathrm{m}} )\boldsymbol{\psi }_{\mathrm{s}}, \end{array} \tag{15} \end{align}\]

According to first-order discretization, (15) can be shown as follows

\[\begin{align} & \boldsymbol{i}_{\mathrm{s}} (k+2)=\boldsymbol{i}_{\mathrm{sk}} +\lambda L_{\mathrm{r}} \boldsymbol{\psi}_{\mathrm{s}} (k+2), \tag{16} \\ & \boldsymbol{i}_{\mathrm{sk}} =((jn_{\mathrm{p}} \omega_{\mathrm{m}}\!-\!\lambda R_{\mathrm{r}} L_{\mathrm{s}})\boldsymbol{i}_{\mathrm{s}}(k+1)\!+\!\lambda (R_{\mathrm{r}}\!-\!jL_{\mathrm{r}}n_{\mathrm{p}}\omega_{\mathrm{m}}) \nonumber\\ & \phantom{\boldsymbol{i}_{\mathrm{sk}}=\ }\boldsymbol{\psi}_{\mathrm{s}}(k\!+\!1))T_\mathrm{s}\!+\!\boldsymbol{i}_{\mathrm{s}}(k\!+\!1)\!-\!\lambda L_{\mathrm{r}} \boldsymbol{\psi}_{\mathrm{s}}(k+1). \tag{17} \end{align}\]

According to (10) and (11), since \(\boldsymbol{\psi}_\mathrm{s}(k+2) \otimes \boldsymbol{\psi}_\mathrm{s}(k+2)\)\(= 0\), \(\boldsymbol{\psi}_\mathrm{s}(k+2) \odot \boldsymbol{\psi}_\mathrm{s}(k+2) = |\boldsymbol{\psi}_\mathrm{s}(k+2)|^{2}\), the expressions of proposed prediction model for torque and reactive torque are demonstrated in Eqs. (18) and (19). Only the stator flux, torque, and reactive torque need to be predicted, which can effectively reduce the computational burden of the control algorithm.

\[\begin{align} & T_{\mathrm{e}}(k+2)=1.5n_{\mathrm{p}}\{\boldsymbol{\psi}_{\mathrm{s}}(k+2)\otimes \boldsymbol{i}_{\mathrm{sk}}\}, \tag{18} \\ & T_{\mathrm{R}}(\!k\!+\!2\!)\!=\!1.5n_{\mathrm{p}}\left\{\boldsymbol{\psi}_{\mathrm{s}}(\!k\!+\!2\!) \odot \boldsymbol{i}_{\mathrm{sk}}\!+\!\lambda L_{\mathrm{r}}\left|{\boldsymbol{\psi}_{\mathrm{s}}(\!k\!+\!2\!)}\right|^{2}\right\}.\!\! \tag{19} \end{align}\]

3.3  CMV Reduction Strategy

Among the eight basic VVs generated by 2L-VSI, ZVV has a higher CMV.  To reduce CMV, and ensure steady-state performance of IM, ZVV is often replaced with two NZVVs placed in opposite directions with respect to each other, as shown in Fig. 4. In addition, the non-adjacent non-opposite NZVV switching combinations should be abandoned to suppress CMV spikes at deadtime [22]. However, it has a negative influence on steady-state performance [23]. A novel CMV suppression strategy is proposed to improve the steady-state performance, reduce CMV and suppress CMV spikes. The CMV suppression strategy is as follows:

Fig. 4  Replacement method of ZVV proposed in [22].

First, the optimal VV can be obtained from the three candidate VVs in Table 2, the simplified prediction model and the cost function of RT-MPC.  Then, the optimal VV is divided into three categories according to the switching state of the previous control cycle: ZVV, non-adjacent non-opposite NZVV, and adjacent or opposite NZVV.  The principle of distinction is shown in Fig. 5, opt and \(old\) are the subscript \(i\) of VV \(\boldsymbol{V}_{i}\).

Fig. 5  Three types of optimal VVs.

Second, it is processed according to different types of optimal VVs, for the ZVV of case 1, utilizing the switching sequence shown in Fig. 4. When it is an opposite or adjacent NZVV of case 3, the optimal VV is used as the output in one control cycle. For the optimal VV in case 2, discarding the optimal VV may result in a poor steady-state performance but will produce CMV spikes during application. Therefore, a control strategy is proposed to suppress CMV spikes and improve steady-state performance.

The proposed control strategy inserts intermediate VV between non-adjacent non-opposite NZVVs. For example, assuming that the optimal VV at the previous control cycle is \(\boldsymbol{V}_{2}\). In the present cycle, \(\boldsymbol{V}_{0}\), \(\boldsymbol{V}_{3}\), and \(\boldsymbol{V}_{4}\) are the candidate VVs, and \(\boldsymbol{V}_{4}\) is the optimal VV.  Therefore, \(\boldsymbol{V}_{3}\) is inserted between \(\boldsymbol{V}_{2}\) and \(\boldsymbol{V}_{4}\). Thus, the optimal VV of the present control cycle consists of \(\boldsymbol{V}_{3}\) and \(\boldsymbol{V}_{4}\). The duty cycle of \(\boldsymbol{V}_{3}\) can be a fixed value or obtained by using duty cycle calculation methods such as modulation MPC [27]-[29]. In this case, the steady-state performance of the controller is improved while the CMV is suppressed. The proposed CMV reduction strategy is summarized in Table 3, \(\boldsymbol{V}_{old}\), \(\boldsymbol{V}_\mathrm{opt}\), \(\boldsymbol{V}_{y}\), \(\boldsymbol{V}_{x}\) are the previous, present control cycle optimal VV and the two candidate NZVVs, respectively.

Table 3   Switching sequences of different types of optimal VVs

3.4  Proposed Control Flow

The control diagram of the simplified RT-MPC method is shown in Fig. 6. For clarity, the control flow is summarized in the following steps:

Step 1: Measurement: sample the three-phase current and rotor speed, then calculate the stator current in \(\alpha\)-\(\beta\) frame.

Step 2: Apply: apply the optimal VV \(\boldsymbol{V}_\mathrm{opt}\) of the previous control cycle.

Step 3: Estimate and delay compensation: estimate stator and rotor flux respectively. Then, predict stator current, stator, and rotor flux based on \(\boldsymbol{V}_\mathrm{opt}\).

Step 4: VV preselection: candidate VVs are selected according to (13), (14), torque deviation, and Table 2.

Step 5: Prediction: stator flux, torque, and reactive torque, are predicted by (8), (18), (19).

Step 6: Cost function evaluation: the cost function (12) is used to evaluate the predicted torque and reactive torque values, and the optimal closed-loop action is obtained.

Step 7: CMV reduction: the optimal VV is classified according to Fig. 5, then the switching sequence is obtained according to Table 3.

Fig. 6   Control diagram of the simplified RT-MPC.

4.  Experimental Verification and Result Analysis

A test bench of the IM control system is established to verify the validity of the proposed simplified RT-MPC strategy, as shown in Fig. 7. The algorithm uses DSP28379D as the core control device on the experimental platform. The IM uses a 4-pole motor with a rated voltage 380 V, rated power 1.5 kW, and rated shaft speed 1400 r/min; the parameters of IM are listed in Table 4. The IM is driven by IGBT module (SKM50GB123D), and the DC bus voltage is about 540 V.  The RT-MPC [8], 6VV strategy [19], 5VV strategy [22], and the proposed simplified RT-MPC are evaluated under 20 kHz operating conditions. In addition, the duty cycle of the inserted VV in case 2 of simplified RT-MPC is 0.5. It is noted that only the simplified RT-MPC uses the simplified prediction model, while other methods use the prediction model in reference [7].

Fig. 7  IM experiment platform.

Table 4  IM parameters

4.1  Execution Time Comparison

The execution time of four control strategies is shown in Fig. 8. The execution time of cost function optimization depends on the number of candidate VVs and the complexity of the prediction model, which is the most important to the overall execution time. RT-MPC, 6VV strategy, 5VV strategy, and simplified RT-MPC rely on 7, 6, 5, and 3 iterations respectively, and the simplified prediction model reduces the execution time of a single prediction. As a result, compared with RT-MPC, 6VV strategy and 5VV strategy, the execution time of the proposed simplified RT-MPC was reduced by approximately 31%, 25%, and 23%, respectively. An average of 26.33% performance improvement is reached based on the proposed control model.

Fig. 8  Execution time of four control strategies.

4.2  Control Performance Comparison

The steady-state performance of RT-MPC, 6VV strategy, 5VV strategy, and simplified RT-MPC under different conditions are tested, as shown in Figs. 9, 10 and 11. In Figs. 9, 10 and 11, the IM operates at 200 r/min, 800 r/min, and 1400 r/min with 10 Nm load, respectively. From top to bottom are stator flux, torque, stator current, and the amplitude of CMV.  In order to quantitatively evaluate the steady-state performance of four control strategies, the torque and stator flux ripple \(T_{\mathrm{e\_ripple}}\), \(\psi_{\mathrm{s\_ripple}}\) are as follows

\[\begin{align} & T_{\mathrm{e\_ripple}} =\sqrt{\frac{1}{n}\sum_{x=1}^n(T_{\mathrm{e}}(x)-T_{\mathrm{e}}^{\ \ast})^{2}}, \tag{20} \\ & \psi_{\mathrm{s\_ripple}} =\sqrt{\frac{1}{n}\sum_{x=1}^n(\psi_{\mathrm{s}}(x)-\left|\psi_{\mathrm{s}}^{\ \ast}\right|)^{2}}, \tag{21} \end{align}\]

where \(T_\mathrm{e}(x)\) and \(\psi_\mathrm{s}(x)\) are the measured torque and stator flux, respectively, and \(n\) represents the number of samplings.

Fig. 9  Steady-state performance comparison under 200 r/min, 10 Nm load. (a) RT-MPC.  (b) 6VV strategy. (c) 5VV strategy. (d) Simplified RT-MPC.

Fig. 10  Steady-state performance comparison under 800 r/min, 10 Nm load. (a) RT-MPC.  (b) 6VV strategy. (c) 5VV strategy. (d) Simplified RT-MPC.

Fig. 11  Steady-state performance comparison under 1400 r/min, 10 Nm load. (a) RT-MPC.  (b) 6VV strategy. (c) 5VV strategy. (d) Simplified RT-MPC.

As shown in Figs. 9 (a), 10 (a), and 11 (a), RT-MPC uses 6 NZVVs and 1 ZVV, thus the flux, torque ripple, and current THD are lowest, and CMV root mean square (rms) values is highest among the four control strategies. The CMV rms values under different speeds are 244 V,199.1 V, and 140.1 V respectively. When the rotor speed increases, the frequency of NZVV as the optimal VV increases, resulting in the CMV rms decreases. In addition, ZVV \(\boldsymbol{V}_{0}\) and deadtime effects result in CMV with an amplitude of \(\pm V_\mathrm{dc}/2\) [22]. In the 6VV strategy in Figs. 9 (b), 10 (b), and 11 (b), the steady-state performance is decreased due to ZVV elimination. In addition, the effect of deadtime results in CMV spikes with an amplitude of \(\pm V_\mathrm{dc}/2\), and the CMV rms values are 91.7 V, 92.4 V, and 92.8 V.

Unlike the RT-MPC and 6VV strategies, the CMV of 5VV and simplified RT-MPC strategies can be restricted to \(\pm V_\mathrm{dc}/6\), as shown in Figs. 9 (c), (d), Figs. 10 (c), (d), and Figs. 11 (c), (d). The CMV rms values of both methods are about 90 V.  The experimental results verify that the proposed simplified RT-MPC can effectively reduce CMV and eliminate CMV spikes caused by deadtime effect. The flux, torque ripple and current THD of four control strategies at different speeds are shown in Figs. 12 (a), (b), and (c). In the VV preselection stage, the 6VV strategy directly excludes ZVV, and the 5VV strategy excludes non-adjacent non-opposite NZVV switching combinations, thus deteriorating steady-state performance. As shown in Fig. 12, the stator flux, torque ripple, and current THD of simplified RT-MPC are better than those of 6VV and 5VV strategies. Therefore, the simplified RT-MPC effectively reduces the CMV, suppress CMV spikes, and provide better steady-state performance.

Fig. 12  Rms of different indicators. (a) Flux. (b) Toque. (c) THD.

The dynamic performance of the four control strategies is tested. Figure 13 presents the dynamic response results when the reference speed changes from 1400 r/min to \(-1400\) r/min, and the dynamic response results of external load disturbance under rated speed are shown in Fig. 14. Although the four control strategies have different VV preselection principles, they have almost the similar dynamic response, which indicates that the proposed method can still have good dynamic performance after reducing the candidate VVs. Figure 13 also shows that the simplified RT-MPC method has good control performance under fast-speed step change. In addition, the CMV of 5VV and simplified RT-MPC strategies can be restricted to \(\pm V_\mathrm{dc}/6\) under rated speed reversal and an external load disturbance, as shown in Figs. 13 (c), (d), and Figs. 14 (c), (d). The above experimental results also show that the simplified prediction model has almost no negative effect on the RT-MPC algorithm.

Fig. 13  Rated speed reversal condition comparison. (a) RT-MPC.  (b) 6VV strategy. (c) 5VV strategy. (d) Simplified RT-MPC.

Fig. 14  External load disturbance under rated speed. (a) RT-MPC.  (b) 6VV strategy. (c) 5VV strategy. (d) Simplified RT-MPC.

4.3  Parameter Sensitivity Experiment

In order to validate the robustness against motor parameter deviation based on the simplified RT-MPC method, the stator, rotor resistance, and inductance mismatch waveforms are shown in Figs. 15 (a), (b), (c), and (d), respectively. \(R\_\mathrm{s}\), \(R\_\mathrm{r}\), \(L\_\mathrm{s}\), and \(L\_\mathrm{r}\) are the parameters of stator, rotor resistance, and inductance in the control algorithm respectively. Compared with the rotor side parameter mismatch, the stator side parameter mismatch has more influence on the steady-state performance. The stator current and torque increase with the increase of the stator resistance, after a transient process for a period of time, and then return to a steady state. When the stator inductance is increased, the stator current and torque return to the steady state after a short transient process. When the parameters on the rotor side are mismatched, there is almost no negative effect on steady-state performance. In addition, the CMV of the simplified RT-MPC strategy can be restricted to \(\pm V_\mathrm{dc}/6\) under IM parameters mismatch.

Fig. 15  Experimental waveforms of robustness against parameter sensitivity of simplified RT-MPC.  (a) \(R_\mathrm{s}\). (b) \(R_\mathrm{r}\). (c) \(L_\mathrm{s}\). (d) \(L_\mathrm{r}\).

5.  Conclusion

In this paper, a new simplified RT-MPC strategy is proposed to reduce CMV, torque, stator flux ripple, current THD, and suppress CMV spikes. To reduce the execution time of proposed method, the DTC-based VV preselection method and a novel simplified prediction model are used to reduce the complexity of RT-MPC strategy. The experimental results show that the simplified prediction model has no negative effect on RT-MPC algorithm. In addition, a new CMV suppression strategy considering deadtime effects is proposed to restrict the CMV spikes and improve the steady-state performance. The experimental results show that compared to the RT-MPC and 6VV strategy, the simplified RT-MPC can restrict CMV within \(\pm V_\mathrm{dc}/6\) and reduce execution time. Compared to 5VV strategy, the simplified RT-MPC has better steady-state performance and lower execution time. In addition, the experimental results also show that simplified RT-MPC has good parameters robustness.

Acknowledgements

This work was supported in part by Shanghai Municipal Science and Technology Commission of China (Grant No.19DZ2254800), and by Shanghai Science and Technology Innovation Action Plan (No.21DZ1205500).

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Authors

Siyao CHU
  Shanghai Ocean University

Bin WANG
  Shanghai Ocean University

Xinwei NIU
  School of Science, Engineering, and Technology

Keyword

induction motor,  simplified reactive torque model predictive control,  prediction model,  common mode voltage,  steady-state performance