# Robust Multiobjective Model Predictive Control with Computation Delay Compensation for Electric Vehicle Applications Using PMSM with Multilevel Inverter

Authors: Abdiddaim Katkout; Tamou Nasser; Ahmed Essadki
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

The Three-Level Neutral-Point-Clamped (3L-NPC) inverter fed Permanent Magnet Synchronous Motor (PMSM) drive is an attractive configuration for high performance Electric Vehicle (EV) applications. For such configuration, due to their high performances, the Finite-Control-Set Model Predictive Control (FCS-MPC) is a very attractive control solution. The FCS-MPC scheme is based on the prediction of the future behavior of the controlled variables using the dynamic model of PMSM and the discrete nature of the 3L-NPC inverter. However, the parametric uncertainties and time-varying parameters affect the FCS-MPC algorithm performances. In this paper, robust FCS-MPC controls based on “dynamic error correction” (DEC) and “modified revised prediction” (MRP) are proposed to improve the FCS-MPC robustness without affecting the controller performances and complexity. The proposed strategies are improved also by multiobjective (MO) algorithm optimization and computation delay compensation. The simulation results included prove the performance in robustness and efficiency of the proposed robust FCS-MPC-DEC.


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## Body

## 1. Introduction

The Electric Vehicle (EV) has received more attention as an alternative solution of energy crisis and environmental challenges [1]. In the next years, with the research efforts and the constant improvement of the electrical drive systems and control strategies, the penetration of EVs will increase exponentially [2]. The electrical drive part of EVs is constituted by electric machine, power converter, and drive train [3]. Currently, a great variety of electrical drive technologies can be used on EVs: two/multilevel power converter and induction machines (IM)/Permanent Magnet Synchronous Motor (PMSM) electric machine. For the electric machine part, the PMSM is considered as the potential candidates for EVs applications with potential high energy density, high efficiency, easier control scheme, robust operations, and reduced maintenance [4]. Generally, the two-level inverters are widely used to control the PMSM in EVs applications [5]. To deal with this structure, several control strategies are proposed: linear controls with the modulator (e.g., field-oriented control and direct torque control) and nonlinear controls without the modulator (e.g., switching table based) [6]. In different aspects, other classification include adaptive control (AC), sliding mode control (SMC), and active disturbance rejection control (ADRC) [7–10]. In the recent years, many Model Predictive Control (MPC) approaches are developed to improve the control performances for the power converter applications [11]. Based on the mathematical system model, MPC predict the future behavior of the system. The optimal control is obtained using predefined optimization criteria. The MPC presents several advantages: simple to apply in variety of processes and multiple-input-multiple-output (MIMO) systems, the system constraints and nonlinearities are considered in the control system. Furthermore, several control loops can be incorporated into one control law. Particularly, the Finite-Control-Set Model Predictive Control (FCS-MPC) is considered as attractive alternative predictive control [12]. Taking into account the discrete nature of the power converter, FSC-MPC uses the possible output voltage vectors to formulate the control algorithm and not require any external modulator stage. Furthermore, FCS-MPC provides good dynamic-state and steady-state performances [13]. In EVs’ applications, research efforts have been made to improve the efficiency of FCS-MPC algorithms. Due to the parameter variation and road conditions, the EVs’ process is considered as a time-varying system. The dynamic behavior of EVs, model uncertainties and inexactitude, unmodelled dynamics (motor and inverter), and unmeasurable parameters lead to a mismatch between the actual parameter values and their nominal values used in the prediction model and degrade the control robustness. There are many research studies accorded to robustness of the MPC controller [14]. A novel-simplified MPC scheme combined with the Lyapunov-based concept for interior PMSM is proposed in [15] to eliminate the flux-current mapping from the MPC formulation. Stumper et al. [15] propose a robust extension for predictive torque control (PTC) to improve the steady-state performances against parametric uncertainties for PMSM based on two-degree-of-freedom (2DoF) control. Then, the switching frequency and the ripples are eliminated without affecting the dynamics-state performances. In [16], the authors design a new discrete-time robust predictive current controller for PMSM. Using the static error correction (SEC) principle, the robustness of predictive current control of PMSM against the effect of the inaccurate of motor behavior is improved and implemented by a DSP model TMS320F2812 practically [17]. Nevertheless, the two-level inverters limit the set of control in eight possible combinations which lead to high output current/voltage total harmonic distortion (THD) and high switching losses. Furthermore, a high dv/dt derivative causes high stress in the semiconductor devices. To overcome these challenges, in the last few years, several multilevel inverter structures are recognized as a novel approach of high power converter performances [18]. Among them, the three-level neutral-point-clamped (3L-NPC) is the most interesting multilevel converter topology due to economy, complexity of control, and performances [19]. However, the computational burden will increase exponentially with the level of the power converter. This issue is addressed in [20] and the computational burden is reduced with 12.31%. The robustness of the FCS-MPC control for the multilevel converter fed PMSM high performances (e.g., EVs’ applications) is still an open research topic. In [6], based on one-horizon simple revised prediction (SRP), the system robustness against parameters variations is improved by reducing the control variables ripples. The SRP principle has been implemented on a fully FPGA-based platform (NI-cRIO 9082 system). Typically, the existing research works are based on mono-objective problem optimization, without computational delay compensation, and adopted for two-level power converters. The main contributions in this work are as follows:(i)
Improved the predictive control robustness against parameters variation and uncertainties using “dynamic error correction” (DEC) and “modified revised prediction” (MRP) principles without affecting the system complexity and control dynamic-state performances.(ii)
A multiobjective problem optimization: current tracking control, neutral-point voltage control, common-mode voltage control, and reduction of switching frequency. These control objectives are controlled using single loop and one control action without internal/external loops and additional tables.The main abovementioned contributions are adopted by the multilevel power converters structures with computational delay compensation based on the proposed modified two-step horizon of prediction approach.The remainder of the paper is structured as follows. The problem statement is given in Section2. The three principle elements in the MPC strategies are discussed. The prediction model is stated in Section 3. Cost function and optimization algorithm are illustrated in Section 4. In Section 5, the robustness of the proposed algorithms is verified by simulation using different scenarios.


## 2. Problem Statement

The MPC strategy is based on the explicit use of the system model to select the optimal actionsabcopt=sa,sb,sc. Then, the performances of the MPC controller depend on the accuracy of the system model and its parameters. In this section, the effects of the variable system uncertainties, computational delay, and standard FCS-MPC principle are presented.


### 2.1. Impact of Variable System Uncertainties

The MPC controller established the optimal control action based on information about the dynamic behavior of the system. The system performances are affected with the accuracy of the information about the inductanceLdq (time-varying variable) and the permanent magnet flux linkage ψpm (unmeasurable variable). A variation in ψpm value only affects the feed-through matrix, Hdk; on the contrary, a variation in Ldq value affects both the system and control matrices, Adk and Bd (equation (15)).


### 2.2. Impact of Computational Delay

In the ideal case, due to negligibility of the calculation time, the optimal control actionsabcopt that minimizes the cost function f at instant k + 1 is applied at time k. Consequently, the controlled variable reaches their predicted value at k + 1 time. On the contrary, in the real case, a significant calculation time tk is required between the instants k when the system state xmk is measured and the generation of the optimal action sabcopt at k+tk. The computational delay affects the system performances and lead to the oscillation of the controlled variables around their references values [21].


### 2.3. Standard FCS-MPC

In this work, four control objectives (COs) are considered:(i)
(CO1) Fast and accurate reference current tracking to reduce the mechanical component stress of system and acoustic pollution(ii)
(CO2) Capacitor voltage balancing between upper and lower DC-link capacitors to reduce the sensibility of output inverter voltage to the ripples in DC-link voltage(iii)
(CO3) Limit the peak of common-mode voltage to reduce the overvoltage stress and increase the safety and reliability of electrical machine part of EVs(iv)
(CO4) Minimize the switching frequency to reduce the switching lossesThese control objectives are incorporated in a single cost function and can be designed as follows:(1)f=fi+fDC+fcmv+fNc,(2)fi=λiirefk+1−ipk+1,(3)fDC=λDCuzpk+1,(4)fcmv=λcmvucmvpk+1,(5)fNc=λNcNc,where fi, fDC, fcmv, and fNc are the cost functions for current tracking control, neutral-point voltage control, common-mode voltage control, and reduction of switching frequency, respectively. irefk+1 and ipk+1 represent the reference and predicted currents at instant k + 1, respectively. uzpk+1 and ucmvpk+1 denote predicted neutral point and common-mode voltages, respectively. λi, λDC, λcmv, and λNc are the weightings factors. It is worth mentioning that the weighting factors values have a direct influence on the controlled variables. The good performances of multiobjective optimization-based predictive control are guaranteed by the choice of the optimal weighting factors. In this work, the weighting factors λi, λDC, λcmv, and λNc are tuned using “trial-and-error” manner.In the cost function (5), Nc penalize the switching state variation as follows:(6)Nc=∑x=a,b,csxk+1−sxk.


## 2.1. Impact of Variable System Uncertainties

The MPC controller established the optimal control action based on information about the dynamic behavior of the system. The system performances are affected with the accuracy of the information about the inductanceLdq (time-varying variable) and the permanent magnet flux linkage ψpm (unmeasurable variable). A variation in ψpm value only affects the feed-through matrix, Hdk; on the contrary, a variation in Ldq value affects both the system and control matrices, Adk and Bd (equation (15)).


## 2.2. Impact of Computational Delay

In the ideal case, due to negligibility of the calculation time, the optimal control actionsabcopt that minimizes the cost function f at instant k + 1 is applied at time k. Consequently, the controlled variable reaches their predicted value at k + 1 time. On the contrary, in the real case, a significant calculation time tk is required between the instants k when the system state xmk is measured and the generation of the optimal action sabcopt at k+tk. The computational delay affects the system performances and lead to the oscillation of the controlled variables around their references values [21].


## 2.3. Standard FCS-MPC

In this work, four control objectives (COs) are considered:(i)
(CO1) Fast and accurate reference current tracking to reduce the mechanical component stress of system and acoustic pollution(ii)
(CO2) Capacitor voltage balancing between upper and lower DC-link capacitors to reduce the sensibility of output inverter voltage to the ripples in DC-link voltage(iii)
(CO3) Limit the peak of common-mode voltage to reduce the overvoltage stress and increase the safety and reliability of electrical machine part of EVs(iv)
(CO4) Minimize the switching frequency to reduce the switching lossesThese control objectives are incorporated in a single cost function and can be designed as follows:(1)f=fi+fDC+fcmv+fNc,(2)fi=λiirefk+1−ipk+1,(3)fDC=λDCuzpk+1,(4)fcmv=λcmvucmvpk+1,(5)fNc=λNcNc,where fi, fDC, fcmv, and fNc are the cost functions for current tracking control, neutral-point voltage control, common-mode voltage control, and reduction of switching frequency, respectively. irefk+1 and ipk+1 represent the reference and predicted currents at instant k + 1, respectively. uzpk+1 and ucmvpk+1 denote predicted neutral point and common-mode voltages, respectively. λi, λDC, λcmv, and λNc are the weightings factors. It is worth mentioning that the weighting factors values have a direct influence on the controlled variables. The good performances of multiobjective optimization-based predictive control are guaranteed by the choice of the optimal weighting factors. In this work, the weighting factors λi, λDC, λcmv, and λNc are tuned using “trial-and-error” manner.In the cost function (5), Nc penalize the switching state variation as follows:(6)Nc=∑x=a,b,csxk+1−sxk.


## 3. System Dynamic

### 3.1. PMSM

The dynamic model of PMSM in thed-q rotating reference frame is given by the differential equation [22]:(7)diddt=−RsLdidt+LqLdωetiqt+1Ldusdt,(8)diqdt=−RsLqiqt+LdLqωetidt−1Lqωetψpm+1Ldusqt,(9)ωetdt=ZpJmTe−BvJmωet−ZpJmTL,where usdq is stator dq-voltages, V; idq is stator dq-currents, A; Rs is stator winding resistance, Ω; Ld and Lq are stator dq-inductances, H; ωe is electric rotatory frequency, rad/s; ψpm is permanent magnet flux, Wb; Te and TL electromagnetic and load torques, Nm; Zp is the number of pole pairs; Jm is moment of inertia, kg.m2; Bv is friction coefficient, Nm.s; and Ts sampling interval, s. The electromagnetic torque of PMSM is obtained by [23](10)Te=32Zpψpmiqt+Ld−Lqidtiqt.Remark 1.
Due to the uniform air-gap in surface mounted PMSM (SPMSM), the statordq-inductance are identical to each other: Ld = Lq = Ls. Opposed to SPMSM, the internal PMSM (IPMSM) is specified by the nonuniform air-gap flux. Thus, the stator q-inductance Lq is smaller than stator d-inductance Ld. The approximation Ld = Lq = Ls of SPMSM is considered in the rest of the paper.
Using thedq-current dynamic and the switching state of 3L-NPC inverter, the neutral-point voltage dynamic is [24](11)duzdt=14C2sa−sb−scid+3sb−sciq,where C=C1=C2 is the DC-link capacitor (Figure 1(a)).Figure 1
(a) Simplified electrical circuit of the system (b) Block diagram of standard FCS-MPC-ICF control scheme.
(a)(b)


### 3.2. 3L-NPC Inverter

The 3L-NPC inverter consists of three identical branches, each branch is composed with four switches and can be described by three switching states [1], [0], and [−1] (Table 1). Thus, there are 27 possible combinations of switching states corresponding to 19 inverter output voltages obtained in αβ-coordinate as follows [20]:(12)usαβ=UDC2sa−sb−sc+j3sb−sc,where UDC denotes the DC-link voltage.Table 1
Operating principle of 3L-NPC inverter.
Switching stateSwitchesOne-phase inverter voltagesx,x∈a,b,csx1sx2sx3sx4uxz,x∈A,B,C1ONONOFFOFFUdc/20ONOFFOFFON0−1OFFOFFONON−Udc/2Inabc coordinate, the common-mode voltage uCMV generated by the 3L-NPC inverter can be presented by [24](13)uCMV=uAZ−uBZ−uCZ3.uAZ is defined as the voltage between the phase (A) and the neutral-point voltage (Z) (Figure 1(a)). Table 2 shows the switching states and corresponding uCMV [25]. Simplified electrical circuit of 3L-NPC inverter-fed PMSM drives are presented in Figure 1(a).Table 2
uCMV corresponding for every switching state combination.
Switching state combinationucmv111Udc/2-1-1-1−Udc/2110, 011, 101Udc/30-1-1, -10-1, -1-10−Udc/3100, 010, 001, 11-1, -111, 1–11Udc/600-1, -100, 0–10, 1-1-1, -11-1, -1-11−Udc/6000, 10–1, 01-1, -110, -101 0–11, 1–100


## 3.1. PMSM

The dynamic model of PMSM in thed-q rotating reference frame is given by the differential equation [22]:(7)diddt=−RsLdidt+LqLdωetiqt+1Ldusdt,(8)diqdt=−RsLqiqt+LdLqωetidt−1Lqωetψpm+1Ldusqt,(9)ωetdt=ZpJmTe−BvJmωet−ZpJmTL,where usdq is stator dq-voltages, V; idq is stator dq-currents, A; Rs is stator winding resistance, Ω; Ld and Lq are stator dq-inductances, H; ωe is electric rotatory frequency, rad/s; ψpm is permanent magnet flux, Wb; Te and TL electromagnetic and load torques, Nm; Zp is the number of pole pairs; Jm is moment of inertia, kg.m2; Bv is friction coefficient, Nm.s; and Ts sampling interval, s. The electromagnetic torque of PMSM is obtained by [23](10)Te=32Zpψpmiqt+Ld−Lqidtiqt.Remark 1.
Due to the uniform air-gap in surface mounted PMSM (SPMSM), the statordq-inductance are identical to each other: Ld = Lq = Ls. Opposed to SPMSM, the internal PMSM (IPMSM) is specified by the nonuniform air-gap flux. Thus, the stator q-inductance Lq is smaller than stator d-inductance Ld. The approximation Ld = Lq = Ls of SPMSM is considered in the rest of the paper.
Using thedq-current dynamic and the switching state of 3L-NPC inverter, the neutral-point voltage dynamic is [24](11)duzdt=14C2sa−sb−scid+3sb−sciq,where C=C1=C2 is the DC-link capacitor (Figure 1(a)).Figure 1
(a) Simplified electrical circuit of the system (b) Block diagram of standard FCS-MPC-ICF control scheme.
(a)(b)


## 3.2. 3L-NPC Inverter

The 3L-NPC inverter consists of three identical branches, each branch is composed with four switches and can be described by three switching states [1], [0], and [−1] (Table 1). Thus, there are 27 possible combinations of switching states corresponding to 19 inverter output voltages obtained in αβ-coordinate as follows [20]:(12)usαβ=UDC2sa−sb−sc+j3sb−sc,where UDC denotes the DC-link voltage.Table 1
Operating principle of 3L-NPC inverter.
Switching stateSwitchesOne-phase inverter voltagesx,x∈a,b,csx1sx2sx3sx4uxz,x∈A,B,C1ONONOFFOFFUdc/20ONOFFOFFON0−1OFFOFFONON−Udc/2Inabc coordinate, the common-mode voltage uCMV generated by the 3L-NPC inverter can be presented by [24](13)uCMV=uAZ−uBZ−uCZ3.uAZ is defined as the voltage between the phase (A) and the neutral-point voltage (Z) (Figure 1(a)). Table 2 shows the switching states and corresponding uCMV [25]. Simplified electrical circuit of 3L-NPC inverter-fed PMSM drives are presented in Figure 1(a).Table 2
uCMV corresponding for every switching state combination.
Switching state combinationucmv111Udc/2-1-1-1−Udc/2110, 011, 101Udc/30-1-1, -10-1, -1-10−Udc/3100, 010, 001, 11-1, -111, 1–11Udc/600-1, -100, 0–10, 1-1-1, -11-1, -1-11−Udc/6000, 10–1, 01-1, -110, -101 0–11, 1–100


## 4. FCS-MPCs Algorithms

### 4.1. Standard FCS-MPC with Improved Cost Function (FCS-MPC-ICF)

Considering a sampling periodTs, the discrete state space model of PMSM is obtained by employing the forward Euler method for equations (7) and (8):(14)xmk+1=Adkxmk+Bdumk+Hdk,where xmk=imdimqT and umk=usdusqT. Adk, Bd, and Hdk are the system, input, and feed-through matrices. They are given by(15)Adk=1−RsTsLsTsωek−Tsωek1−RsTsLs,Bd=TsLs00TsLs,Hdk=0−TsψpmkLsωek.The predictive reference current is calculated using the Lagrange method [20]:(16)irefk+1=3irefk−3irefk−1+irefk−2.The evolution of neutral-point voltage is defined as follows:(17)uzpk+1=uzk+Ts4C2sak−sbk−sckiαk+3Ts4C2sbk−sckiβk.To compensate the computational delay, according to long horizon of prediction approach, we propose to use two-step horizon of prediction. In the 3L-NPC inverter case, combination of 27 switching states are available, a discrete set of272=729 possible trajectories of the switching states is required to evaluate the cost function led to a large number of cost computations which makes the real challenge to the actual digital system implementation. To reduce the number of real time evaluation, this paper proposes to use a modified two-step horizon prediction by applying the same voltage vector in the two-step horizon. The improved cost function fdcom is evaluated at time k + 2 to generate the optimal switching state at time k + 1:(18)fdcom=fidcom+fDCdcom+fcmvdcom+fNcdcom,(19)fidcom=λiirefk+1−ipk+1,(20)fDCdcom=λDCuzpk+1,(21)fcmvdcom=λcmvucmvpk+1,(22)fNcdcom=λNcNcdcom,(23)Ncdcom=∑x=a,b,csxk+2−sxk.The reference currentirefk+2 is extrapolated by using the second-order Lagrange extrapolation [20]:(24)irefk+2=6irefk−8irefk−1+3irefk−2.The evolution ofxm and uzp at instant k + 2 are expressed based on equations (14) and (17):(25)xmk+2=Adk+1xmk+1+Bdumk+1+Hdvk+1,(26)uzpk+2=uzk+1+Ts4C2Sak+1−Sbk+1−Sck+1iαk+1+3Ts4C2Sbk+1−Sck+1iβk+1.In order to guarantee the stability, a Lyapunov candidate function (LCF) can be used to ensure that the stator currentidq are maintained according to their reference value idqref as follows [26]:(27)Vi˜d,i˜q=12Υdi˜d2+12ϒqi˜q2.With(28)ΔV=Vi˜dpk+1,i˜qpk+1−Vi˜dk,i˜qk,where Υd and Υq are the positive gains, i˜dq=i˜d,i˜q is the state error of the stator current defined as follows:(29)i˜dq=idmkiqmk−idrefkiqrefk.The optimal switching statesabcopt is given by the following equation:(30)sabcopt=argminsabcoptk∈−1,0,13,sabcoptk+1∈−1,0,13fdcom,subject toVk>ϱv&ΔV<0∨Vk<ϱv&Vk+1<ϱvsabck=sabck+1.With the aim to ensure the stability of the closed loop, two cases are considered [26]: the first one corresponds to Vk>ϱv and the second one Vk<ϱv. If Vk>ϱv only switches obtaining ΔV<0 are considered and if Vk>ϱv only switches assuring Vk+1<ϱv are considered. Hence, ϱv can be used as a stabilizing constraint, which guarantees feasibility and stability for the optimization problem.Any parameters variation or uncertainties compensation is included in the standard FCS-MPC-ICF algorithm control formulation. A simplified block diagram of standard FCS-MPC-ICF is presented in Figure1(b). It is worth to be mentioned that the “error correction” and “revised prediction” principles are based on the same approach: modified prediction equation to take into account the time-varying behavior of the system. EC is based on the contribution of the controlled parameter error in the previous sampling period in the formulation of optimization problem. On the contrary, RP improves the robustness against the variations of controlled variables taking their different roles into account. These principles are developed in this work based on “dynamic error correction” and “modified revised prediction,” and they are implemented separately in robust FCS-MPC-DEC and robust FCS-MPC-MRP controls. In both proposed robust predictive controls, the parameters variation and uncertainties are explicitly included in the algorithms formulation.


### 4.2. Robust FCS-MPC-DEC

The future behavior of the controlled system is predicted using equation (12) for the 27 possible combinations in the 3L-NPC inverter for each prediction horizon. The accuracy of the prediction is directly affected by the quality of the parameter information. The EC is based on the contribution of the weighted error between the predicted and measured controlled variable values in the previous sampling time ηdqk−1, in the prediction equation. In [15], the author used the previous current error without considering the optimal switching state combination applied in the next sampling interval. Due to the variation of the switching state applied in every sampling time, the direct consideration of the previous error is not suitable (Figure 2). Siami et al. [17] proposed to use a static weighted error for each switching state combination j(j = 1, …, 8). In this study, we proposed to use a dynamic DEC for every given switching combinations j(j = 1, …, 27). The weighted error is calculated for each possible switching state combination. The modified predicted state is(31)idq,jp,modk+1=idq,jpk+1+σ1ηdq,jk,where σ1 is a weighting factor between 0 and 1 and ηdq,j is the discrete current error defined as follows:(32)ηdq,jk=id,jpkiq,jpk−id,jmkiq,jmk.Figure 2
Behavior of standard FCS-MPC-ICF at normal conditions. (a) The stator currentia. (b) The output voltage Ua. (c) Fast Fourier Transform of the current. (d) Fast Fourier Transform of the output voltage. (e) Performance of the neutral-point voltage Uz. (f) Performance of common-mode voltage Ucmv.
(a)(b)(c)


### 4.3. Robust FCS-MPC-MRP

#### 4.3.1. Robustness againstLs Variation

To consider the mismatch between the actual value of the inductance and their nominal value used in the controller, the system state is modified as follows:(33)xmmodk=σ2xmk+1−σ2xpk,where xm and xp are the measured and predicted values of the system state, respectively, σ2 denotes a weighting factor between 0 and 1 according to the BIBO stability criteria (in Appendix). The tunable parameter will add a “weighted average” process to the measured and predicted values of the system. Therefore, the modified state of xmmod, which takes the benefits of this (“weighted average” process), will also add a smoothing and filtering effect to the system states whenever with or without parameter variations. Consequently, improved control variable quality is predictable.


#### 4.3.2. Robustness againstψpm Variation

IntroducingLs=Ls0+ΔLs and ψpm=ψpm0+Δψpm, where Ls0 and ψpm0 are the nominal values and ΔLs and Δψpm are the parametric uncertainties, and using equation (14), the relationship between the measured and the predicted current values is(34)iqmk=Ls0Lsiqmk+ΔLsLsTsωek−1idpk−1+1LsTsωek−1Δψpm.In the steady-state SPMSM operationid=0, the error between the predicted and measured q-axis current is(35)ηqmk=iqmk−iqpk≈1LsTsωek−1Δψpm.The permanent magnet flux uncertainties will contribute into the current error. To compensate the effect of the permanent magnet flux uncertainties into the equation prediction equation (35), a modified permanent magnet flux is proposed as follows:(36)ψpmmodk=ψpmk+σ3∑ηqmk,where σ3 is the weighted factor.The proposed control algorithm for standard FCS-MPC-ICF, robust FCS-MPC-DEC, and robust FCS-MPC-MRP are described in Algorithms1–3, respectively. Algorithm 1 shows the detailed instructions, equations, and parameters used in the simulation of standard FCS-MPC-ICF. The algorithms 1 and 2 can be summarized in the following steps: measurements, calculation, and extrapolation of references and prediction of control variables and minimization of the improved cost function. Both Algorithms 3 and 4 illustrated the specific instructions and parameters used in proposed robust FCS-MPC-DEC and robust FCS-MPC-MRP (e.g., modified system state and current error vector).Algorithm 1: Standard FCS-MPC-ICF.
Readidqrefk and extrapolate idqref=k+1 and idqrefk+2 by using equations (16) and (24);Measureidqmk, ωek and uzk;% Initialize the values ofsabcopt and fdcomsabcopt=1; fdcom=zeros1:27;Fori = 1 to 27 doCalculateusαβ based on equation (12);Predictidqpk+1 and idqpk+2 from equations (14) and (25);Evaluate the candidate Lyapunov function:V(k + 1), V(k), ΔV using equations (27) and (28);if ((V(k) > ϵv) & (ΔV < 0)) ∨ ((V(k) < ϵv) & (V(k + 1) <ϵv))  thenPredictuzpk+1 and uzpk+2 based on equations (17) and (26);Calculateducmvpi from Table 2;CalculateNcdcom from equation (23);Compute the cost functionsfidcom, fDCdcom, fcmvdcom and fNcdcom based on equations (19)–(22);Evaluate the cost functionfdcom by using equation (18);EndEndSelect and applied the optimal switching statesabcopt=minfdcom;Algorithm 2: Robust FCS-MPC-DEC.
% Initialize the values ofsabcopt, fdcom and current error ηdq,j;sabcopt=1; fdcom=zeros1:27; ηdq,j=zeros2:27;Fori = 1 to 27 doCalculate the current errorηdq,j:,i based on equation (32);Predict the modifiedidq,jp,modk+1 and idq,jp,modk+2 using the currenterrorηdq,j:,i and equations (31) and (25);EndSelect and applied the optimal switching stateηdq,jsabcopt=minfdcom;Algorithm 3: Robust FCS-MPC-MRP against Ls variation.
% Initialize the values ofsabcopt and fdcomsabcopt=1; fdcom = zeros(1 : 27);Fori = 1 to 27 doPredict the modifiedxmmodk+1 and xmmodk+2 using equation (33) and (25);EndSelect and applied the optimal switching statesabcopt=minfdcom;Algorithm 4: Robust FCS-MPC-MRP against ψpm variation.
% Initialize the values ofsabcopt, fdcom and ∑ηqmsabcopt=1; fdcom=zeros1:27; ∑ηqm=0;Fori = 1 to 27 doCalculate the current errorηqm and the summation ∑ηqm based on equation (36);Calculate the modified permanent magnet fluxψpmmod using the equation (37);EndRemark 2.
The factorsσ1, σ2, and σ3 are tuned in a “trial-and-error” manner.


## 4.1. Standard FCS-MPC with Improved Cost Function (FCS-MPC-ICF)

Considering a sampling periodTs, the discrete state space model of PMSM is obtained by employing the forward Euler method for equations (7) and (8):(14)xmk+1=Adkxmk+Bdumk+Hdk,where xmk=imdimqT and umk=usdusqT. Adk, Bd, and Hdk are the system, input, and feed-through matrices. They are given by(15)Adk=1−RsTsLsTsωek−Tsωek1−RsTsLs,Bd=TsLs00TsLs,Hdk=0−TsψpmkLsωek.The predictive reference current is calculated using the Lagrange method [20]:(16)irefk+1=3irefk−3irefk−1+irefk−2.The evolution of neutral-point voltage is defined as follows:(17)uzpk+1=uzk+Ts4C2sak−sbk−sckiαk+3Ts4C2sbk−sckiβk.To compensate the computational delay, according to long horizon of prediction approach, we propose to use two-step horizon of prediction. In the 3L-NPC inverter case, combination of 27 switching states are available, a discrete set of272=729 possible trajectories of the switching states is required to evaluate the cost function led to a large number of cost computations which makes the real challenge to the actual digital system implementation. To reduce the number of real time evaluation, this paper proposes to use a modified two-step horizon prediction by applying the same voltage vector in the two-step horizon. The improved cost function fdcom is evaluated at time k + 2 to generate the optimal switching state at time k + 1:(18)fdcom=fidcom+fDCdcom+fcmvdcom+fNcdcom,(19)fidcom=λiirefk+1−ipk+1,(20)fDCdcom=λDCuzpk+1,(21)fcmvdcom=λcmvucmvpk+1,(22)fNcdcom=λNcNcdcom,(23)Ncdcom=∑x=a,b,csxk+2−sxk.The reference currentirefk+2 is extrapolated by using the second-order Lagrange extrapolation [20]:(24)irefk+2=6irefk−8irefk−1+3irefk−2.The evolution ofxm and uzp at instant k + 2 are expressed based on equations (14) and (17):(25)xmk+2=Adk+1xmk+1+Bdumk+1+Hdvk+1,(26)uzpk+2=uzk+1+Ts4C2Sak+1−Sbk+1−Sck+1iαk+1+3Ts4C2Sbk+1−Sck+1iβk+1.In order to guarantee the stability, a Lyapunov candidate function (LCF) can be used to ensure that the stator currentidq are maintained according to their reference value idqref as follows [26]:(27)Vi˜d,i˜q=12Υdi˜d2+12ϒqi˜q2.With(28)ΔV=Vi˜dpk+1,i˜qpk+1−Vi˜dk,i˜qk,where Υd and Υq are the positive gains, i˜dq=i˜d,i˜q is the state error of the stator current defined as follows:(29)i˜dq=idmkiqmk−idrefkiqrefk.The optimal switching statesabcopt is given by the following equation:(30)sabcopt=argminsabcoptk∈−1,0,13,sabcoptk+1∈−1,0,13fdcom,subject toVk>ϱv&ΔV<0∨Vk<ϱv&Vk+1<ϱvsabck=sabck+1.With the aim to ensure the stability of the closed loop, two cases are considered [26]: the first one corresponds to Vk>ϱv and the second one Vk<ϱv. If Vk>ϱv only switches obtaining ΔV<0 are considered and if Vk>ϱv only switches assuring Vk+1<ϱv are considered. Hence, ϱv can be used as a stabilizing constraint, which guarantees feasibility and stability for the optimization problem.Any parameters variation or uncertainties compensation is included in the standard FCS-MPC-ICF algorithm control formulation. A simplified block diagram of standard FCS-MPC-ICF is presented in Figure1(b). It is worth to be mentioned that the “error correction” and “revised prediction” principles are based on the same approach: modified prediction equation to take into account the time-varying behavior of the system. EC is based on the contribution of the controlled parameter error in the previous sampling period in the formulation of optimization problem. On the contrary, RP improves the robustness against the variations of controlled variables taking their different roles into account. These principles are developed in this work based on “dynamic error correction” and “modified revised prediction,” and they are implemented separately in robust FCS-MPC-DEC and robust FCS-MPC-MRP controls. In both proposed robust predictive controls, the parameters variation and uncertainties are explicitly included in the algorithms formulation.


## 4.2. Robust FCS-MPC-DEC

The future behavior of the controlled system is predicted using equation (12) for the 27 possible combinations in the 3L-NPC inverter for each prediction horizon. The accuracy of the prediction is directly affected by the quality of the parameter information. The EC is based on the contribution of the weighted error between the predicted and measured controlled variable values in the previous sampling time ηdqk−1, in the prediction equation. In [15], the author used the previous current error without considering the optimal switching state combination applied in the next sampling interval. Due to the variation of the switching state applied in every sampling time, the direct consideration of the previous error is not suitable (Figure 2). Siami et al. [17] proposed to use a static weighted error for each switching state combination j(j = 1, …, 8). In this study, we proposed to use a dynamic DEC for every given switching combinations j(j = 1, …, 27). The weighted error is calculated for each possible switching state combination. The modified predicted state is(31)idq,jp,modk+1=idq,jpk+1+σ1ηdq,jk,where σ1 is a weighting factor between 0 and 1 and ηdq,j is the discrete current error defined as follows:(32)ηdq,jk=id,jpkiq,jpk−id,jmkiq,jmk.Figure 2
Behavior of standard FCS-MPC-ICF at normal conditions. (a) The stator currentia. (b) The output voltage Ua. (c) Fast Fourier Transform of the current. (d) Fast Fourier Transform of the output voltage. (e) Performance of the neutral-point voltage Uz. (f) Performance of common-mode voltage Ucmv.
(a)(b)(c)


## 4.3. Robust FCS-MPC-MRP

### 4.3.1. Robustness againstLs Variation

To consider the mismatch between the actual value of the inductance and their nominal value used in the controller, the system state is modified as follows:(33)xmmodk=σ2xmk+1−σ2xpk,where xm and xp are the measured and predicted values of the system state, respectively, σ2 denotes a weighting factor between 0 and 1 according to the BIBO stability criteria (in Appendix). The tunable parameter will add a “weighted average” process to the measured and predicted values of the system. Therefore, the modified state of xmmod, which takes the benefits of this (“weighted average” process), will also add a smoothing and filtering effect to the system states whenever with or without parameter variations. Consequently, improved control variable quality is predictable.


### 4.3.2. Robustness againstψpm Variation

IntroducingLs=Ls0+ΔLs and ψpm=ψpm0+Δψpm, where Ls0 and ψpm0 are the nominal values and ΔLs and Δψpm are the parametric uncertainties, and using equation (14), the relationship between the measured and the predicted current values is(34)iqmk=Ls0Lsiqmk+ΔLsLsTsωek−1idpk−1+1LsTsωek−1Δψpm.In the steady-state SPMSM operationid=0, the error between the predicted and measured q-axis current is(35)ηqmk=iqmk−iqpk≈1LsTsωek−1Δψpm.The permanent magnet flux uncertainties will contribute into the current error. To compensate the effect of the permanent magnet flux uncertainties into the equation prediction equation (35), a modified permanent magnet flux is proposed as follows:(36)ψpmmodk=ψpmk+σ3∑ηqmk,where σ3 is the weighted factor.The proposed control algorithm for standard FCS-MPC-ICF, robust FCS-MPC-DEC, and robust FCS-MPC-MRP are described in Algorithms1–3, respectively. Algorithm 1 shows the detailed instructions, equations, and parameters used in the simulation of standard FCS-MPC-ICF. The algorithms 1 and 2 can be summarized in the following steps: measurements, calculation, and extrapolation of references and prediction of control variables and minimization of the improved cost function. Both Algorithms 3 and 4 illustrated the specific instructions and parameters used in proposed robust FCS-MPC-DEC and robust FCS-MPC-MRP (e.g., modified system state and current error vector).Algorithm 1: Standard FCS-MPC-ICF.
Readidqrefk and extrapolate idqref=k+1 and idqrefk+2 by using equations (16) and (24);Measureidqmk, ωek and uzk;% Initialize the values ofsabcopt and fdcomsabcopt=1; fdcom=zeros1:27;Fori = 1 to 27 doCalculateusαβ based on equation (12);Predictidqpk+1 and idqpk+2 from equations (14) and (25);Evaluate the candidate Lyapunov function:V(k + 1), V(k), ΔV using equations (27) and (28);if ((V(k) > ϵv) & (ΔV < 0)) ∨ ((V(k) < ϵv) & (V(k + 1) <ϵv))  thenPredictuzpk+1 and uzpk+2 based on equations (17) and (26);Calculateducmvpi from Table 2;CalculateNcdcom from equation (23);Compute the cost functionsfidcom, fDCdcom, fcmvdcom and fNcdcom based on equations (19)–(22);Evaluate the cost functionfdcom by using equation (18);EndEndSelect and applied the optimal switching statesabcopt=minfdcom;Algorithm 2: Robust FCS-MPC-DEC.
% Initialize the values ofsabcopt, fdcom and current error ηdq,j;sabcopt=1; fdcom=zeros1:27; ηdq,j=zeros2:27;Fori = 1 to 27 doCalculate the current errorηdq,j:,i based on equation (32);Predict the modifiedidq,jp,modk+1 and idq,jp,modk+2 using the currenterrorηdq,j:,i and equations (31) and (25);EndSelect and applied the optimal switching stateηdq,jsabcopt=minfdcom;Algorithm 3: Robust FCS-MPC-MRP against Ls variation.
% Initialize the values ofsabcopt and fdcomsabcopt=1; fdcom = zeros(1 : 27);Fori = 1 to 27 doPredict the modifiedxmmodk+1 and xmmodk+2 using equation (33) and (25);EndSelect and applied the optimal switching statesabcopt=minfdcom;Algorithm 4: Robust FCS-MPC-MRP against ψpm variation.
% Initialize the values ofsabcopt, fdcom and ∑ηqmsabcopt=1; fdcom=zeros1:27; ∑ηqm=0;Fori = 1 to 27 doCalculate the current errorηqm and the summation ∑ηqm based on equation (36);Calculate the modified permanent magnet fluxψpmmod using the equation (37);EndRemark 2.
The factorsσ1, σ2, and σ3 are tuned in a “trial-and-error” manner.


## 4.3.1. Robustness againstLs Variation

To consider the mismatch between the actual value of the inductance and their nominal value used in the controller, the system state is modified as follows:(33)xmmodk=σ2xmk+1−σ2xpk,where xm and xp are the measured and predicted values of the system state, respectively, σ2 denotes a weighting factor between 0 and 1 according to the BIBO stability criteria (in Appendix). The tunable parameter will add a “weighted average” process to the measured and predicted values of the system. Therefore, the modified state of xmmod, which takes the benefits of this (“weighted average” process), will also add a smoothing and filtering effect to the system states whenever with or without parameter variations. Consequently, improved control variable quality is predictable.


## 4.3.2. Robustness againstψpm Variation

IntroducingLs=Ls0+ΔLs and ψpm=ψpm0+Δψpm, where Ls0 and ψpm0 are the nominal values and ΔLs and Δψpm are the parametric uncertainties, and using equation (14), the relationship between the measured and the predicted current values is(34)iqmk=Ls0Lsiqmk+ΔLsLsTsωek−1idpk−1+1LsTsωek−1Δψpm.In the steady-state SPMSM operationid=0, the error between the predicted and measured q-axis current is(35)ηqmk=iqmk−iqpk≈1LsTsωek−1Δψpm.The permanent magnet flux uncertainties will contribute into the current error. To compensate the effect of the permanent magnet flux uncertainties into the equation prediction equation (35), a modified permanent magnet flux is proposed as follows:(36)ψpmmodk=ψpmk+σ3∑ηqmk,where σ3 is the weighted factor.The proposed control algorithm for standard FCS-MPC-ICF, robust FCS-MPC-DEC, and robust FCS-MPC-MRP are described in Algorithms1–3, respectively. Algorithm 1 shows the detailed instructions, equations, and parameters used in the simulation of standard FCS-MPC-ICF. The algorithms 1 and 2 can be summarized in the following steps: measurements, calculation, and extrapolation of references and prediction of control variables and minimization of the improved cost function. Both Algorithms 3 and 4 illustrated the specific instructions and parameters used in proposed robust FCS-MPC-DEC and robust FCS-MPC-MRP (e.g., modified system state and current error vector).Algorithm 1: Standard FCS-MPC-ICF.
Readidqrefk and extrapolate idqref=k+1 and idqrefk+2 by using equations (16) and (24);Measureidqmk, ωek and uzk;% Initialize the values ofsabcopt and fdcomsabcopt=1; fdcom=zeros1:27;Fori = 1 to 27 doCalculateusαβ based on equation (12);Predictidqpk+1 and idqpk+2 from equations (14) and (25);Evaluate the candidate Lyapunov function:V(k + 1), V(k), ΔV using equations (27) and (28);if ((V(k) > ϵv) & (ΔV < 0)) ∨ ((V(k) < ϵv) & (V(k + 1) <ϵv))  thenPredictuzpk+1 and uzpk+2 based on equations (17) and (26);Calculateducmvpi from Table 2;CalculateNcdcom from equation (23);Compute the cost functionsfidcom, fDCdcom, fcmvdcom and fNcdcom based on equations (19)–(22);Evaluate the cost functionfdcom by using equation (18);EndEndSelect and applied the optimal switching statesabcopt=minfdcom;Algorithm 2: Robust FCS-MPC-DEC.
% Initialize the values ofsabcopt, fdcom and current error ηdq,j;sabcopt=1; fdcom=zeros1:27; ηdq,j=zeros2:27;Fori = 1 to 27 doCalculate the current errorηdq,j:,i based on equation (32);Predict the modifiedidq,jp,modk+1 and idq,jp,modk+2 using the currenterrorηdq,j:,i and equations (31) and (25);EndSelect and applied the optimal switching stateηdq,jsabcopt=minfdcom;Algorithm 3: Robust FCS-MPC-MRP against Ls variation.
% Initialize the values ofsabcopt and fdcomsabcopt=1; fdcom = zeros(1 : 27);Fori = 1 to 27 doPredict the modifiedxmmodk+1 and xmmodk+2 using equation (33) and (25);EndSelect and applied the optimal switching statesabcopt=minfdcom;Algorithm 4: Robust FCS-MPC-MRP against ψpm variation.
% Initialize the values ofsabcopt, fdcom and ∑ηqmsabcopt=1; fdcom=zeros1:27; ∑ηqm=0;Fori = 1 to 27 doCalculate the current errorηqm and the summation ∑ηqm based on equation (36);Calculate the modified permanent magnet fluxψpmmod using the equation (37);EndRemark 2.
The factorsσ1, σ2, and σ3 are tuned in a “trial-and-error” manner.


## 5. Results and Analysis

In this section, the control performances of proposed robust FCS-MPC-DEC and robust FCS-MPC-MRP for 3L-NPC inverter fed PMSM for EVs’ applications are validated by simulations and compared with the control performance of standard FCS-MPC-ICF under different scenarios both with and without parameters variations and uncertainties. Using Matlab/Simulink software and SimPowerSystems toolbox, the simulations are carried with the parameters, as indicated in Table3.Table 3
System parameters.
ParameterValuePMSM power ratePN (W)1440PMSM pole pairsZp [1]2PMSM resistanceRsΩ2.89PMSM inductanceLs (mH)7PM fluxψ (Wb)0.125Moment of inertiaJm [kg.m2]10−5Friction coefficientBv (Nm.s)11.10−5DC capacitanceC1=C2=C (μF)1With the aim to study better the robustness of the proposed robust FCS-MPCs controls, we proposed to use the two performances variables.(i)
The mean absolute reference tracking error [21]:(37)e¯MARTEx=1N∑i=1Nxrefi−xmesi.N denotes the number of simulation iterations. xref and xmes are the reference and measured controlled variables x, respectively.(ii)
The average switching number per semiconductor:(38)f¯ASN=1N∑k=1Nnijk12,where nij is the switching numbers of power semiconductors of phase i∈a,b,c and order j∈1,2,3,4 defined as follows:(39)nij=sijk+1−sijk.The behavior of the standard FCS-MPC-ICF at the nominal conditions is illustrated in Figure3. In the test, the good dynamic-state performances in fast and accurate reference current tracking are demonstrated (Figure 3(a)). The steady-state performances of the FCS-MPC-ICF are validated with low mean absolute reference tracking error and THD of current 0.0091 and 3.29 (Figure 3(c)), respectively. The output voltage is shown in Figure 3(b) with THD of 22.79 (Figure 3(d)). The average switching number per semiconductor f¯ASN is 9834. As shown in Figure 3(e), the deviation of DC-link voltage uz is maintained around 6.66% of its nominal value. Figure 3(f) shows the evolution of ucmv and the overvoltage stresses of PMSM are reduced to an average value of 0.2.Figure 3
The evolution of current errorηdq of standard FCS-MPC-ICF. (a) sabcopt. (b) The current error ηd. (c) The current error ηq.
(a)(b)(c)(d)(e)(f)The evolution of current errorηdqk in function of applied optimal switching state sabcopt is illustrated in Figure 2. The application of dynamic DEC is motivated by the dynamic behavior of ηdq and its not constant for same sabcopt. Consequently, the proposition of fixed error values of each switching combination considered in static error correction is incorrect.Figure4 and Table 4 evaluates the robustness of standard FCS-MPC-ICF and proposed robust FCS-MPC-DEC and FCS-MPC-MRP controls against parameter system variations. The inductance Ls and the permanent magnet flux ψpm are varied to −50% and 200% of their nominal values. The performances of standard FCS-MPC-ICF are affected in presence of variations of Ls and ψpm, and the error and controlled variable ripples around their nominal values are highly increased. These control ripples run to the instability in case of high parameters variations (in case of 200% of parameter variation). In case of proposed robust FCS-MPC-DEC and FCS-MPC-MRP, the robustness against parameter variation is improved. The proposed robust FCS-MPC-DEC solution outperforms both the proposed robust FCS-MPC-MRP and standard FCS-MPC-ICF controls in terms of smaller e¯MARTEidq, uCMV, uz, f¯ASN, calculation time tc, and THD of current and output inverter voltage.Figure 4
The evolution of current errorηdq: (a) 0.5Ls, (b) 2Ls, (c) 0.5ψmg, and (d) 2ψmg; coral: standard FCS-MP-ICF, red: robust FCS-MPC-DEC, and blue: robust FCS-MPC-MRP.
(a)(b)(c)(d)Table 4
Comparison of the control performances under different controls.
ParameterVariationControle¯MARTEidqucmvuz (%)f¯ASNtcTHDidqusΔLs−50%Standard FCS-MPC-ICF0.48920.767.051.6010421.964.6329.06Robust FCS-MPC-DEC0.00980.236.689.8310314.703.6827.58Robust FCS-MPC-MRP0.01000.256.791.2210418.803.7328.26200%Standard FCS-MPC-ICF0.57404.288.311.8510439.416.3536.16Robust FCS-MPC-DEC0.12970.757.164.1910331.574.6835.23Robust FCS-MPC-MRP0.13060.767.337.7110333.204.7135.31Δψpm−50%Standard FCS-MPC-ICF0.31990.756.952.2810429.474.0728.32Robust FCS-MPC-DEC0.01240.286.811.9410325.663.4525.70Robust FCS-MPC-MRP0.01360.386.951.9410327.593.5926.07200%Standard FCS-MPC-ICF0.50372.438.232.3110447.806.0170.84Robust FCS-MPC-DEC0.09400.507.069.8310334.214.4532.55Robust FCS-MPC-MRP0.10100.687.231.1810436.694.5132.71


## 6. Conclusion

A robust model predictive control for multilevel inverter fed PMSM for electrical vehicle application is proposed in this paper. The computational delay is compensated using a proposed modified two-step horizon prediction. A proposed improved multiobjective cost functionfdcom incorporates the control objectives and evaluates every sampling interval to generate the optimal switching state applied directly to the inverter without modulation stage. The time-varying behavior of the electrical vehicle system affects the good dynamic and steady-state performances of the standard finite-control-set model predictive control with improved cost function (FCS-MPC-ICF) control. Simulation results under different scenarios of parameters variation approve the effectiveness of the proposed robust FCS-MPC with “dynamic error correction” (DEC) and “modified revised prediction” (MRP) principles in terms of control robustness. The proposed robust FCS-MPC-DEC solution outperforms both the proposed robust FCS-MPC-MRP and standard FCS-MPC-ICF controls.



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*Source: 7015865-2020-05-15.xml*