Abstract
This paper proposes a novel predictive strategy based on a model predictive control (MPC) for the interior permanent magnet synchronous motors (IPMSMs) driven by a threelevel simplified neutralpoint clamped inverter (3LSNPC) for electric vehicle applications (EVAs). Based on the prediction of the future behavior of the controlled variables, a predefined multiobjective cost function incorporates the control objectives which are evaluated for every sampling period to generate the optimal switching state applied directly to the inverter without the modulation stage. The control objectives in this paper are tracking current capacity, neutralpoint voltage balancing, commonmode voltage control, and switching frequency reduction. The principal concepts of the novel scheme are summarized as follows: first, the delay compensation based on the long horizon of prediction is adopted by a multilevel power converter structure. Second, based on the modified Lyapunov candidate function, both stability and recursive feasibility are ensured of the proposed predictive scheme. Third, the practicability of the realtime implementation is improved by the proposed “static voltage vector” (SVV) and “single state variation” (SSV) principles. Finally, the proposed concepts are implemented in the novel predictive control formulation as additional constraints without compromising the complexity and the good performances of the predictive controller. Therefore, only the switching states that guarantee the stability and the reduction of calculation burden criteria are considered in the evaluation of cost function. The proposed predictive scheme based on the “SVV” principle has demonstrated superior performance in simulation compared with the proposed scheme with the “SSV” principle. The computational burden and switching frequency rates are reduced by 35% and 56.22%, respectively.
1. Introduction
Currently, permanent magnet synchronous machines (PMSMs) are widespread in many electric vehicle applications (EVAs) [1, 2]. Depending on the magnet placement, two categories of the PMSM are defined: surfacemounted PMSM and interior PMSM [3]. Particularly, interior PMSM (IPMSM) demonstrates excellent properties, such as compact structure with small size and weight, higher power density, high torque to inertia ratio, wide speed range operations, low noise, and robustness [4]. Many control strategies for the IPMSM are proposed: linear control with a modulator and nonlinear controls without a modulator [5, 6]. In different aspects, other classifications include robust controllers (RC), adaptive controllers (AC), and intelligent controllers (IC) [7–10].
There are many research studies accorded to ensure a satisfactory control performance of IPMSMs. In [11], the authors propose an online maximum torque per ampere (MTPA) control based on the numerical optimization technique to avoid the large memory usage and the accurate interpolation algorithm required using the lookup table (LUT) methods. A deadbeatdirect torque and flux control (DBDTFC) of IPMSM drive for machine parameter variation is presented in [12]. The fluxweakening control strategy for the IPMSM with maximum torque per volt in VAs is presented in [13]. The desired dqoptimal current references are calculated analytically to achieve the operating point regarding the electrical limits. A comparative study on DTC of IPMSM for EVs between DTC combined with space vector modulation (DTCSVM) and deadbeat DTC (DBDTC) is presented in [14]. Nevertheless, these existing control strategies are limited by several challenges [15]:(i)Low dynamicstate performances and need of the internal loop(ii)Complex modulation stage and sensibility of the system variables(iii)Controlled variable ripples
To overcome these aforementioned issues, numerous predictive controls (PCs) are proposed to improve the control performances for the power converter applications [16]. Based on the dynamic system model, the PC predicts the future behavior of the controlled system. A predefined cost function incorporates the controlled variables which are minimized every sampling period to generate the optimal switching state [17]. In comparison with the existing control strategies, PC offers many advantages: easy application in a variety of processes and multipleinputmultipleoutput (MIMO) systems, system constraints and nonlinearities which can be incorporated directly into the control law, simple controller concept and implementation, and good dynamic and steadystate performances [18]. Furthermore, the computational delay and dead time can be compensated, and several control loops can be incorporated into one control law. The predictive controllers can be divided into two principal categories: Continuous control set MPC and finite control set MPC (FMPC). Particularly, FMPC is the most attractive predictive strategy for power converters [19]. Taking into account the discrete nature of the power converter, FMPC uses the available switching states to formulate the predictive problem algorithm without requiring any table of switches, external modulator stage, or regulator. Little has been published regarding FMPC of IPMSM for EVAs. Linear inequality matrices (LMIs) represent an effective tool to transform the problem of robust PC design into the form of a convex optimization problem [20]. LMIs are a powerful tool to deal with the many optimization problems in control theory, system identification, and signal processing. A robust predictive control with nonsymmetric constraints (NSC) using the LMI framework is presented in [20]. The method is based on three principles: timevarying vector of constraints, switching indicator function to determine the current boundaries, and support controller that certifies the robust stability. Ojaghi et al. [21] proposed an LMI approach to robust predictive control of nonlinear systems (NLS) with statedependent uncertainties. An LMI approach to mixedinteger predictive control of uncertain hybrid systems with binary and realvalued control inputs is proposed in [22].
Generally, different topologies of power converters are available in the market. The aforementioned control strategies have been mainly limited to a twolevel power converter in a variety of IPMSM applications [23–25]. However, the limited switching state combinations available in the twolevel power converters (eight possible combinations) lead to high controlled variable ripples as well as voltage and current waveforms with a higher total harmonic distortion. Additionally, this results in higher switching losses and high switching device stress due to the high derivative [26].
To address the aforementioned issue and to fulfill the highperformance IPMSM application requirements (e.g., EVAs), numerous multilevel power converter structures are recognized: neutralpoint clamped (NPC), flying capacitor (FC), cascade Hbridge (CHB), and modular multilevel (MM) [27–29]. Several researchers have investigated the multilevel power converter, thanks to its benefits: low voltage applied to each component, less and low THD of the output voltage and current waveforms, and low switching loss using the redundancy of the switching states [30, 31]. In recent years, the NPC and Ttype are the most used multilevel power converter topologies in the industry [32–34]. The system complexity and calculation cost are increasing significantly with the number of switching devices. Maintaining the multilevel output, the 3LSNPC requires a few switching devices. A comparison in terms of semiconductor device number between 3LNPC, Ttype, and 3LSNPC is tabulated in Table 1 [35]. Needless to say, the main source of neutralpoint voltage fluctuations in NPC and Ttype topologies does not exist in the 3LSNPC topology.
In [33], the authors proposed a predictive control for a gridtie threelevel neutralpointclamped inverter with deadtime compensation. This predictive control is based on the orientation of grid voltage to predict the performance of control variables required for the system. Using the stability condition through a control Lyapunov function, the execution time is decreased by 26%. In [36], an asymptotically stable monoobjective predictive control of a gridconnected converter based on discrete space vector modulation is proposed. The presented approach combines the concept of discrete space vector modulation (DSVM) into the FMPC structure to improve the input current quality of the power converter. However, the additional virtual voltage vectors can cause an unacceptable computational burden to the controller. To alleviate this issue, the number of possible voltage vectors for the optimization process is reduced by a preselection technique. In this work, we propose to reduce computation burden by the proposed “static voltage vector” (SVV) and “single state variation” (SSV) principles. The computational burden is reduced by 35%. The aspect of reduction of switching frequency is introduced and decreased by 56.22%.
Literally, there is very little research, if any, performed on the predictive control of the multilevel 3LSNPC inverter. Furthermore, a little has been published regarding the feasibility and stability, delay compensation, and the computational burden reduction of IPMSM driven by a 3LSNPC inverter.
Taking into account the EVA requirements, the principal contributions of this paper are summarized as follows:(i)This paper proposes a novel and promising configuration of EVAs: FMPC predictive control strategy for IPMSM fed by the 3LSNPC inverter.(ii)A modified Lyapunov candidate function which guarantees the feasibility and stability by taking into consideration the discrete nature of the power converter is proposed. Both recursive feasibility and stability are embedded in MPC problem formulation as additional constraints.(iii)Multiobjective predictive problem formulation that includes four control objectives is considered: tracking current capacity, neutralpoint voltage balancing, commonmode voltage control, and switching frequency reduction.(iv)By using a twostep horizon of the prediction approach, the computational delay due to the calculation and communication time is compensated.(v)In order to guarantee the practicability of the realtime implementation and computational efficiency of the proposed predictive control, “static voltage vector” (SVV) and “single state variation” (SSV) principles are proposed and embedded in MPC problem formulation as additional constraints.
In the novel predictive control in this paper, the aforementioned contributions are explicitly included in the predictive problem formulation without affecting the controller performances and simplicity.
Section 2 presents the preliminaries of principal concepts of this paper. Section 3 describes the system dynamics. The FMPC and proposed predictive control formulation are illustrated in Sections 4 and 5, respectively. In Section 6, detailed simulation verification studies are carried to show the efficiency of the proposed predictive strategies. Finally, Section 7 draws the conclusion. The nomenclature of this paper is summarized in Table 2.
2. Preliminaries
The FMPC is a very attractive solution for controlling power converters and electrical drives. Nonetheless, stability, computational delay, and computational burden reduction are still open issues to be investigated. In this section, a statement about the three aspects is presented.
2.1. Stability
In conventional predictive problem formulation, the stability in the closed loop is not taken into account. It is worth mentioning that the stator reference current cannot be obtained by applying the voltage vectors resulting from the application of available 21 switching states in the 3LSNPC. Consequently, the equilibrium point is not achieved by some reference values. Accordingly, the optimization loop of the predictive control aims to generate the current value bounded around the reference value, the reason why we focus on practical stability. In this work, the 3LSNPC multilevel inverter structure is designed to produce the desired output voltage by multiple available switching states instead of limited switching state combinations in twolevel inverters [37].
In spite of good dynamic and steadystate performances of the conventional FMPC predictive strategy, the stability, still an open issue, is to be studied. A notable tool to overcome this issue is Lyapunov stability theory [38]. To guarantee stability in the predictive problem formulation, the cost function is considered as Lyapunov candidate function (LCF) between two sampling periods. The stability analysis focuses on showing that such cost function satisfies the conditions of practical LCF [39].
2.2. Impact of Computational Delay
The FMPC algorithm consists of the following steps [40]:(i)Measurement of controlled variables(ii)Prediction of future values of controlled variables for the combination of each switching state available in the 3LSNPC inverter(iii)Evaluation of predefined cost function for every sampling period(iv)Selection and application of the optimal switching state that minimizes the cost function
The operating cycle of the FMPC is illustrated in Figure 1. The predicted, reference, and measured values of stator current are presented by dashed black lines, dashed blue lines, and solid black lines, respectively. In the perfect case, due to the negligibility of the calculation time , FMPC algorithm steps are accomplished spontaneously at instance k to generate the optimal switching state , Figure 1(a). Consequently, the stator current reaches its reference value at the k + 1 instance [41]. In the real case, a significant computational delay is required for the operating cycle and communication time, Figure 1(b). The delay between the measurements and the generation of can cause some problems if not considered. Consequently, the previous optimal switching state will continue to be applied during and stator current which will oscillate around its reference , increasing the current undulation. The computational delay affects the system performances and increases the difficulty of its controllability.
(a)
(b)
(c)
To compensate the computational delay, we propose using a long horizon of prediction approach as shown in Figure 1(c). Using the optimal switching state and controlled variable values at the k instance, the system state values at the k + 1 instance are predicted; denotes the switching states available in the inverter. For each predicted value at the k + 1 instance, future values are predicted at k + 2. Accordingly, the optimal switching state that minimizes the cost function at the k + 2 instance is applied directly to the inverter without the modulation stage at the k + 1 instance [40].
2.3. Computational Efficiency
To compensate the computational delay for the 3LSNPC inverter, using a twostep horizon of prediction, a discrete set of possible trajectories of the switching states has to be enumerated for the evaluation of the cost function, Figure 2(a). This proposition led to a large number of computational burden, which makes the real challenge to the actual digital system implementation [42]. To reduce the number of realtime evaluations, this paper proposes a modified twostep horizon prediction using a proposed “static voltage vector” (SVV), Figure 2(b), and “single state variation” (SSV), Figure 2(c), principles.
(a)
(b)
(c)
3. System Dynamics
3.1. IPMSM
A typical dqcoordinate rotating reference framebased mathematical model in the time domain of IPMSM drive systems is given by [24]
The parameters of the IPMSM dynamic model are defined in Table 3.
The electromagnetic torque of the IPMSM can be calculated as [43]
Remark 1. Due to the nonuniform airgap flux in the IPMSM, the stator qinductance is smaller than the stator dinductance . Opposed to the IPMSM, surfacemounted PMSM is specified by uniform airgap, and the stator dqinductances are identical to each other: [3].
In Figure 3(a), with the assumption of constant DCbus voltage , equal DClink capacitors , and [44],where .
(a)
(b)
3.2. 3LSNPC Inverter
The 3LSNPC inverter consists of a dual Buck converter integrated with a conventional twolevel inverter [45]. The 3LSNPC inverter is supplied by twoseriesconnected capacities and . The dual Buck converter consists of four transistors with neutral point Z to connect different voltage levels, Table 4, to the input of the twolevel inverter. The combinations of the switching states of 3LSNPC can be calculated by combinations, where and are the switching states of the twolevel inverter and dual Buck converter, respectively. The combinations of switching states are devised to 18 active and 14 zero voltage vectors leading to 21 inverter output voltages obtained in abccoordinate as described in Table 5 [46]. Three groups of voltage vectors are defined: zero voltage vectors , small voltage vectors , and large voltage vectors .
The topology of the IPMSM fed by 3LSNPC inverter configuration is depicted in Figure 3(a). In the abccoordinate, the commonmode voltage measured from the stator winding neutral to neutral point Z is expressed as [47]where is the phase voltage from an output terminal to neutral point Z.
4. FMPC Formulation
Generally, FMPC generates the optimal switching state by optimizing a predefined cost function, for each possible switching state available in the inverter, that describes the desired system behavior until a predefined control horizon. The multiobjective cost function considers the tracking error between the predicted and reference values of controlled variables allowing simultaneous control of all of them. FMPC is a receding horizon approach, namely, its optimization problem is formulated and resolved every sampling period updating the actual system state data.
In general, the cost function consists of two parts: the predicted control errors over the prediction horizon N and the penalty term. Typically, the following quadratic cost function form is used [48]:where is a vector of error between the predicted and reference values of the controlled variable at instant . is the vector of increments of the future values of the manipulated variable (control inputs) at instant . and represent the control and prediction horizons, respectively. Matrices and denote weighting coefficients. Then, the optimal switching state is
4.1. Cost Function Selection
Cost function designing is one of the most important stages in the definition of any MPC strategy. In this work, the purposes of the proposed predictive control are as follows:(i)Reducing the mechanical component stress of the IPMSM by fast and accurate reference current tracking(ii)Limiting the output 3LSNPC inverter voltage sensibility to the ripples in DClink voltage by capacitor voltage balancing(iii)Increasing the safety and reliability of the IPMSM by limiting the peak of commonmode voltage(iv)Reducing the switching losses by minimizing the switching frequency
When matrices and in equation (7) are diagonal, the aforementioned control objectives are included in a single cost function of the FMPC given in the following:
The parameters of FMPC cost function are defined in Table 6.
4.2. Weighting Coefficient Design
Cost function (9) synthesizes multiple controlled variables with different natures (e.g., current and voltage), order and magnitude, and effect. Accordingly, the tradeoff between the four controlled variables is defined by the weighting coefficients , , , and. .
The classification of the cost functions is depended on the controlled variables incorporated in its formulation. Three kinds of cost functions are defined [48]:(i)Cost function without weighting coefficients: one type of variable is controlled without using the weighting coefficients.(ii)Cost function with secondary terms: primary goal and secondary constraints are defined to accomplish a proper system behavior and control performances. In this case, the weighting coefficient of the second term is forced to zero (). The procedure of tuning the weighting coefficients is converted to the cost function without weighting coefficient procedure. Based on a steadystate error and total harmonic distortion (THD) criteria, the system behavior without the weighting coefficients is evaluated to fulfill the control requirements. Finally, by evaluating the system behavior, the simulation started with and increases progressively until it defined a good tradeoff between the primary goal and secondary constraints. A common approach to select the optimal weighting coefficient is branch and bound technique.(iii)Cost function with equally important terms: in this case, several variables have equal importance to control the system which are incorporated in the cost function. Firstly, the cost function is normalized. Accordingly, all terms will be similarly important (). The second and third steps are the same as cost function with secondary terms; is considered instead of zero value.
4.2.1. Optimization Algorithm
The discretestate space model of the IPMSM can be obtained by forward Euler’s method using equations (1) and (2) as follows [43]:where and are the stator current and output voltage vectors, respectively, , , and denote system, input, and feedthrough matrices, and is the sampling period.
The future values of reference current are extrapolated using the Lagrange method [49]:
By using equation (5) and actual information of at the k instance, the predicted value of the neutralpoint voltage at the k + 1 instance is obtained as
To compensate the computational delay, according to the long horizon of the prediction approach, we propose to use a twostep horizon of prediction, Section 2.2 and Figure 1. The improved cost function with computational delay compensation of improved FMPC (IFMPC) is evaluated at time to generate the optimal switching state applied at time :
By using the secondorder Lagrange extrapolation, the reference current at the k + 2 instance is [49]
Based on equations (10) and (12), the predicted system state and neutralpoint voltage data at the k + 2 instance are expressed aswhere is the stator current in coordinates.
4.3. Stability: Modified Lyapunov Function Constraint
In order to avoid overcurrent/voltage in the power converter and IPMSM, the stator current reference is limited by maximum permissible current and safety factor (to reduce the voltage and provide robustness against model uncertainties) [50]:
By taking into account constraints (16) and (17) and the improved cost function with delay compensation (13), optimization problem (8) is converted to
and denote the state and control input constraint sets, respectively. The improved cost function (18a) synthetizes the control objectives for the twostep horizon of prediction. This cost function is evaluated by taking the system constraints into account. The dynamic of the system is fulfilled by (18b) using the measured system state at the k instance. The state and control input constraint sets are implemented by (18c) and (18d), respectively.
The conventional predictive control does not guarantee stability. Accordingly, the modified Lyapunov candidate function (LCF) is proposed to ensure that the neighborhood of the reference signal set is reached in finite time.
Definition 1. For all , , there exists a preliminary LCF , , , and :where is a tuning parameter used to compromise the controlled variables.
Proof 1. For every , there exists an admissible input such that , i.e., : is the convex control set and is defined as the convex hull of [38]: is the sum of the feedback controller .
Definition 2. Considering parameter, is a sublevel set of the LCF:Due to the discrete nature of the 3LSNPC inverter, the controlled variable cannot always achieve its reference value, namely, the capacity of the controlled system state towards the origin by feasible inputs is limited.
Proposition 1. To describe how fast the LCF decreases beyond a certain value, we define
has a direct effect on the LCF performances; if is fixed as a large value, LCF decreases faster, and the algorithm robustness against the model uncertainties is improved. Accordingly, equation (18c) is implemented in the predictive problem formulation as a stabilizing constraint to ensure the recursive feasibility in the problem optimization by the finitetime convergence of the system states to the set .
In equation (23), the dynamic of the current error is defined as the difference between the reference and actual stator current:
The proposed modified LCF in this work is defined as
Proposition 2. The timevarying function is proposed as
is a time instant when the modified LCF constraint suits the standard one. Conditioned by the constraint in equation (18c) is relaxed, the recursive feasibility of the proposed control is guaranteed regardless of the cost function.
By considering (19), the value of the LCF can be decreased robustly for all . Consequently, any state is driven towards the set. In addition, for all there exists when . Consequently, there exists a sequence such that
The FMPC system is said to be global and robust asymptotically set stabilizable.
Proposition 3. Using the specific structure of the and sets, the largest set is
Therefore, the modified LCF iswithwhere and are the positive gains.
Considering the modified LCF (29) and (30) in improved cost function (18b), the following modified IFMPC (MIFMPC) is proposed:
It can be shown that, in predictive problem formulation (31a), the modified LCF stability criteria constraint is imposed with (31e). With the aim to ensure the stability of the closed loop, the switching states which violate the constraints are discarded, while the remaining states are compared according to the cost function. Two cases are considered: the first one corresponds to and the second one . If , only switches obtaining are considered, and if , only switches assuring are considered. Hence, can be used as a stabilizing constraint that guarantees recursive feasibility and stability for the optimization problem.
A simplified algorithm of the proposed MIFMPC is summarized in Algorithm 1.

5. Proposed Computationally Efficient MIFMPC Algorithms
As mentioned in Section 2.3, an amount of iterations is required to evaluate the improved cost function (31a) which makes a challenge in MIFMPC algorithm implementation. To overcome this issue, in this work, the MIFMPC in (31a) is developed with two proposed computationefficient principles: “static voltage vector” (SVV), Figure 2(b), and “single state variation” (SSV), Figure 2(c). It is worth to state that the “SVV” and “SSV” principles are based on the approach of limiting the evaluation of cost function (31a) in case of a long horizon of a prediction, and they are applied separately to the proposed computationally efficient MIFMPC (CMIFMPC) strategy.
5.1. Computationally Efficient MIFMPCSVV
In order to reduce the computational burden, the proposed static voltage vector (SVV) principle is based on the evaluation of (32a) using fixed vector voltage during the twostep horizon of prediction instead of different voltage vectors, Figure 2(b). In computationally efficient MIFMPCSVV (CMIFMPCSVV) formulation in (32a), the “SVV” principle is implemented using constraint (32f):
The simplified algorithm of the proposed CMIFMPCSVV is presented in Algorithm 2. The two loops depicted in the predictive control formulation presented in Algorithm 1 are embarked in a single loop in Algorithm 2.

5.2. Computationally Efficient MIFMPCSSV
To avoid the high computational burden in twostep horizon of prediction in (31a) formulation of the proposed MIFMPC strategy, the single state variation (SSV) principle is proposed, namely, in the first stage of prediction, the 21 system states are predicted. In the second stage, the transition of the switching state is limited to the combinations of inputs that have only one state variation in k + 2, Figure 2(c). For example, if the vector voltage is considered in the first stage, the voltage vectors considered in the second stage are , , , , and , Table 5. The “SSV” principle constraint is implemented in proposed CMIFMPCSSV formulation (33a) using constraint (33f):
The simplified algorithm of the proposed CMIFMPCSSV is illustrated in Algorithm 3.

Remark 2. The weighting coefficients , , , and are tuned in a “trialanderror” manner.
6. Simulation and Comparisons
In order to verify the performances of the proposed predictive controls, a comparative study based on numerical simulations of proposed CMIFMPCSVV and proposed CMIFMPCSSV strategies for IPMSM fed by the 3LSNPC multilevel inverter is presented in this section. The proposed system and control schemes which are presented in Figure 3 are carried out using Matlab/Simulink software and SimPowerSystems toolbox with parameters detailed in Table 7. Figures 4–6 show the simulation results of the behavior of the controlled variables.
(a)
(b)
(c)
(d)
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Remark 3. In this section and for clarity purposes, both proposed CMIFMPCSVV and CMIFMPCSSV predictive strategies are indicated by SVV and SVV abbreviations in the simulation figures, respectively.
6.1. SteadyState Performances
The reference, predicted, and measured stator current values are illustrated in Figure 4(a). Both measured currents and can accurately track their reference values, and good tracking performance can be reached by both proposed predictive algorithms. The behavior of proposed predictive strategies is illustrated in Figure 4(b). The current error obtained with the proposed CMIFMPCSVV is less than the error obtained by using CMIFMPCSSV, , where .
With the aim to better study the steadystate performances of the proposed predictive controls, we propose to use the tracking error [40]:
Q determined the number of simulation iterations during the simulation. and are measured and reference controlled variables of , respectively. The calculated tracking error of the controlled variables , , and is depicted in Table 8. From these results, the proposed CMIFMPCSVV obtains accurate controlled variable tracking ability better than the proposed CMIFMPCSSV predictive strategy.
The behavior of output inverter voltage, commonmode voltage, and neutralpoint voltage is depicted in Figures 5(c)–5(f) and 6, respectively. Figure 6 shows the good ability to balance the neutralpoint voltage by the proposed CMIFMPCSVV which is better than the proposed CMIFMPCSSV.
6.2. DynamicState Performances
The dynamicstate performances of proposed CMIFMPCSVV and proposed CMIFMPCSSV predictive controls are illustrated in Table 9. As shown in Figure 4(a), the proposed predictive strategies present good dynamicstate performances with settling times of 1.1 ms and 2.5 ms, respectively, and without overshoot and undershoot. The proposed “static voltage vector” (SVV) principle significantly improves the dynamic performance of the predictive strategy compared with the proposed “single state variation” (SSV) principle.
The tic/toc Matlab command returns the execution time taken to run the proposed predictive algorithms. , , and are the maximum, average, and minimum execution time required by iteration for the proposed predictive strategies which are presented in Table 10, respectively. The average of the proposed CMIFMPCSVV control is 17.421 μs, in contrast to the CMIFMPCSSV strategy of 21.38 μs. It is noted that the computation time of the proposed CMIFMPCSVV is reduced by to of the proposed CMIFMPCSSV; 21 and 60 are the iteration numbers in the second stage of prediction in Figures 2(b) and 2(c).
6.3. Total Harmonic Distortion (THD)
To evaluate the effect of the proposed predictive controls on the quality of stator current and output inverter voltage, the Powergui toolbox in Simulink is used to calculate the THD of both of them, Figures 4(c), 4(d), 5(c), and 5(d). The results illustrated in Table 11 indicate that the THD of the proposed CMIFMPCSVV is better than the proposed CMIFMPCSSV control.
6.4. Switching Frequency
The FMPC predictive strategy generates the optimal switching states with a variableswitching frequency procedure. To analyse the effect of the two proposed predictive strategies on the switching frequency, we propose to use the average switching number per semiconductor factor [49]:
The average switching number per semiconductor factor of proposed CMIFMPCSVV and proposed CMIFMPCSSV predictive strategies is listed in Table 12. The proposed CMIFMPCSVV reduces the switching frequency by to of the proposed CMIFMPCSSV control.
7. Conclusion and Future Work
In this paper, a novel scheme of predictive control has been proposed and applied to the interior permanent magnet synchronous motor (IPMSM) driven by a threelevel simplified neutralpoint clamped inverter (3LSNPC) for electric vehicle applications (EVAs). The main advantages of the novel scheme are summarized as follows: firstly, using a twostep horizon of prediction, the computational delay is compensated. The delay compensation is adopted with a 3LSNPC inverter structure. Secondly, the stability and feasibility of the proposed predictive control are guaranteed using a modified control Lyapunov function. Thirdly, the computational burden is reduced by applying the proposed “static voltage vector” (SVV) and “single state variation” (SSV) principles. Fourthly, the aforementioned concepts are combined in the predictive problem formulation as additional constraints without sacrificing the simplicity and performances of the controller structure. The simulation results demonstrated that the proposed CMIFMPCSVV outperforms the proposed CMIFMPCSSV in terms of: good tracking performance, low tracking error of controlled variables, and reduced computational burden and switching frequency by 35% and 56.22%, respectively. The conservatism introduced by using a conventional Lyapunov function constraint can be alleviated by using the proposed modified Lyapunov function constraint which enables nonmonotone convergence to the terminal set and therefore can lead to better performance. The main challenge in this study is the selection of suitable weighting factors in the cost function to achieve the optimal balance between the objectives. To avoid the time and effort consuming simulations in weighting factor selection processes, the future work proposes to use an Artificial Neural Network (ANN) to select the optimal weighting factors in the MPC cost function.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.