An Effective Model Predictive Control Method With Self-Balanced Capacitor Voltages for Single-Phase Three-Level Shunt Active Filters

This paper presents an effective model predictive control (MPC) method for single-phase three-level T-type inverter-based shunt active power filters (SAPFs). The SAPF using T-type inverter topology has not been reported in the literature yet. Contrary to most of the existing MPC methods, the proposed MPC method eliminates the need for using weighting factor and additional constraints required for balancing dc capacitor voltages in the cost function. The design of cost function is based on the energy function. Since the factor used in the formulation of the energy function does not have any adverse influence on the performance of the system, the cost function becomes weighting factor free. The weighting factor free based MPC brings simplicity in the practical implementation. The effectiveness of the proposed MPC method has been investigated in steady-state as well as dynamic transients caused by load changes. The theoretical considerations are verified through experimental studies performed on a 3 kVA system.


I. INTRODUCTION
The widespread use of renewable energy sources, distributed generation sources, plug-in electric vehicles, and powerelectronics loads such as consumer electronics, light emitting diode (LED) lights, electric drives, and diode rectifiers result in harmonic pollution in the power grid. Harmonic pollution deteriorates the power quality. Especially, the harmonic currents drawn by the aforementioned devices and loads increase losses, cause distortion in the grid voltage waveform, and cause interferences with the other devices connected at the point of common coupling (PCC). The power quality can be improved by using custom power devices such as static compensators [1], static VAR compensators [2], unified power quality controllers (UPQC) [3], dynamic voltage The associate editor coordinating the review of this manuscript and approving it for publication was Ahmed A. Zaki Diab . restorers (DVR) [4], [5], and active power filters [6], [7]. Among these devices, the shunt active power filters (SAPF) offer effective solutions for load current related problems. The main function of an SAPF is to achieve sinusoidal grid currents at unity power factor by injecting the required current harmonics at PCC. However, the control of SAPF is challenging due to the fast-changing nature of the filter current. In order to obtain good performance, a fast current controller should be designed so that the actual filter current tracks the reference filter current with maximum tracking accuracy. This means that the designed current loop should have a high control bandwidth. On the other hand, the inverter topology used in the SAPFs also plays an important role in improving the power quality.
With the aim of achieving the desired control objectives, many control methods have been proposed in continuoustime [8]- [12]. Traditional proportional-integral (PI) control VOLUME 9, 2021 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ method cannot offer satisfactory performance due to its limited control bandwidth [8]. Even though the linear quadratic regulator-based control method presented in [9] offers satisfactory performance, it involves too many control parameters, which should be tuned so as to obtain the desired performance. Lyapunov-function based control method offers fast response and assures global stability of the system at the expense of employing complicated controller in continuoustime [10]. Sliding mode control (SMC) approach exhibits fast current loop dynamics as well as simple implementation without parameter sensitivity [11]. However, it suffers from chattering, which leads to an uncontrollable switching frequency. Alternatively, the hysteresis current control method is also proposed which is robust to the parameter variations [12].
In the last decade, owing to the availability of powerful and cheap digital signal processors, numerous digital control techniques are proposed for SAPFs in discrete time. These digital control techniques include repetitive control [13]- [15], p-q theory-based control [16], deadbeat control [17], adaptive linear neural network based control [18], and predictive control [19]- [27]. The repetitive control method suppresses the periodic disturbances effectively, but its performance under non-periodic disturbances is not satisfactory. Even though the deadbeat control method has fast current loop dynamics, it is dependent on the system parameters, which cause performance degradation. In the last decade, the predictive control method has received considerable attention by many researchers due to its prominent advantages such as fast dynamic response and the prediction of future behavior of the controlled variables. The use of predicted variables results in obtaining the desired action determined by the preset optimization criterion. The first predictive control method applied to the control of SAPF was proposed in [19]. In [20], the properties of predictive control are combined with the artificial neural networks in generating the reference currents. The predictive control proposed in [21] is based on one-sample-period-ahead, which can predict the controlled variables one and two sampling periods in advance. Alternative to the classical predictive control method, model predictive control (MPC) has been introduced where a cost function is to be minimized [22], [23]. However, the use of weighting factor (WF) is essential in the cost function. The tuning of WF for a good performance is time consuming due to the lack of preset tuning procedure. Therefore, the tuning of WF is usually achieved by trial-and-error method. In addition, the tuned WF value is not unique and, therefore, it may not yield the same performance when the operating point of SAPF is changed. The finite control set MPC (FCS-MPC) method presented in [24] achieves suppression in the current ripple at the expense of using a modulator. Recently, a finite impulse response (FIR) based MPC is also introduced where the WF can be determined analytically [25]. In the aforementioned studies, the SAPF is based on two-level inverter topology.
In the last decade, the multilevel inverters (MIs) have received considerable attention by many researchers due to the prominent advantages compared to the two-level inverters. For this reason, the MIs are widely employed in many applications including the SAPFs [26]- [28]. In [26], a three-phase three-level SAPF is realized by using cascaded H-bridge (CHB) inverters. The SAPF was controlled by using SMC technique, which suffers from chattering. Also, the control of dc-link voltage is difficult. In [27] and [28], the FCS-MPC method was applied to control a single-phase neutral-point clamped (NPC) inverter based SAPF and threephase four-leg flying capacitor (FC) inverter based SAPF, respectively. The drawbacks of the NPC and FC inverter topologies are the clamping diode and flying capacitor requirements, which increase the cost and losses. On the other hand, both control methods do not need modulator. However, while the control method in [27] requires WF, the control method in [28] does not need WF at all. Among the MI topologies, the T-type inverter topology has fewer components with same number of levels [29]. However, the MPC method has not been analyzed yet in the case of three-level T-type inverter operating as SAPF.
In this paper, an effective MPC method is proposed for a single-phase three-level T-type inverter based SAPF. Unlike the existing MPC methods, the proposed control method does not employ WF in the cost function. In addition, the capacitor voltages are balanced without using an additional constraint in the cost function as mentioned in [30]. Therefore, a simplification is obtained in the design phase. The optimum switching states, which force the error variables to zero, are determined from the derivative of the energy function. The negative definiteness of the energy function assures the stability of the system. Extensive experimental results are presented to validate the proposed control approach.

II. MODELING OF SAPF
The single-phase three-level SAPF using a T-type inverter is depicted in Fig. 1. Clearly, the SAPF has four switches per each leg. The state of each switch is defined as The three-level input voltage and neutral current can be defined in terms of the switching functions as follows By making use of different switching combinations, v xy can be generated as a five-level voltage with levels 0, ±V dc /2, and ±V dc .
The differential equations of the SAPF can be written as where v xy = uV dc , I o1 = ui c , u is the switching function. In the derivation of (7), it is assumed that V C1 = V dc /2. On the other hand, the capacitor currents in terms of the switching functions (S 1 and S 2 ) and grid current are obtained as The design of the passive components (filter inductance and dc capacitors) are calculated as follows [31], [32] where L max is the maximum value of filter inductance, i c is the high frequency ripple in compensation current (i c ), f s is switching frequency, I Cmax is the maximum compensation current, f is the grid frequency and V dc min is the minimum dc side voltage. Using parameters listed in Table 2 (see  Section IV), the maximum filter inductance is calculated as 2.62mH for 9kHz average switching frequency and 15% ripple ratio in i c . It is worth noting that the filter inductance (L) used in the experimental setup was selected to be 2mH. It should be noted that the minimum V dc should be larger than the peak value of grid voltage. Since the experimental tests are performed using 120V rms grid voltage, V dc min is selected to be 170V. (11), we obtain the values of C 1 and C 2 as 428µF. Since these values are not available, 470µF capacitors are used in the experimental setup.

III. ENERGY FUNCTION BASED MPC
The block diagram of the developed energy function based MPC technique is depicted in Fig. 2. The proposed approach consists of the following two steps.

A. DESIGN OF ENERGY FUNCTION BASED MPC
The main control objectives of the three-level SAPF are to regulate V dc to its reference, regulate capacitor voltages, obtain sinusoidal grid current at unity power factor, and guarantee stable operation under various load types. It is worth to note that when the regulation of V dc is accomplished, the control of capacitor voltages is achieved automatically. Also, the unity power factor operation occurs when the grid current is in phase with the grid voltage. In order to meet the latter objective, the grid voltage and reference grid current should be in the form of The sine wave template in i * g is obtained via a classical phase locked loop (PLL). The amplitude of reference grid current is produced by a proportional-integral (PI) regulator as follows where K p and K i are proportional and integral gains, respectively and V * dc is reference of V dc . From (6), it follows that the derivative of i * c can be written as where i * c denotes the reference of i c , which is defined as In (15), the measurement of i L is required. Unlike the classical MPC involving a cost function, the proposed MPC employs an energy function. It can be seen from Fig. 2 that there are two dc-side capacitors and one ac-side inductor, which have the ability to store energy in the SAPF. Therefore, the energy function in the continuous-time can be expressed in terms of capacitor voltage error (x 1 ) and inductor current error (x 2 ) as follows where the constants β 1 and β 2 should be positive and x 1 and x 2 are the error variables defined as follows It can be noticed that equation (16) is in the form of energy stored in inductor and capacitor. For instance, the first term in (16) gives the energy stored in capacitor while the second term is the energy stored in inductor. The details of formulating E(x) can be found in [33] and [34]. It is worth noting that inclusion of x 1 in E(x) has a self-balanced effect on the dc capacitor voltages. Now, taking the derivative of (17) and making use of (6), (8), (9), and (14) yieldṡ Equations (18) and (19) denote the dynamics of error variables in (17).
The energy function should satisfy the following condi- Clearly, the first two conditions are satisfied. The third condition (negative definiteness of E(x)) which guarantees the stability of the system should also be satisfied. It is worth to note that E(x) increases or decreases depending on the values of x 1 and x 2 . When the error variables converge to zero (x 1 = 0 and x 2 = 0), E(x) also tends to zero. Therefore, the main goal is to minimize E(x) and at the same time achievė Taking the derivative of (16) yieldṡ Now, substituting (4), (18) and (19) into (20) giveṡ (21) and assuming that Selecting β 1 = Cβ 2 /L eliminates the terms S 1 x 1 x 2 and S 2 x 1 x 2 which also reduces equation (22) tȯ The stability of the closed-loop system is assured iḟ E(x) < 0. The main aim in this study is to design MPC strategy by using (23). Since equation (23) Equation (24) can be used to select the optimum switching functions S 1 (k) and S 2 (k) such thatĖ x (k + 1) is negative. It is worth to note that the choice of β 2 /L has no effect on the performance of the controller provided that β 2 > 0. In order to verify this claim, the performance of the proposed energy function based MPC is investigated by using various β 2 values (see Fig. 11(b)). It can be observed that oncė E x (k + 1) < 0 is achieved, the system works successfully regardless of β 2 values. Therefore, unlike the classical MPC method whose performance depends on WF, the proposed MPC method is WF free that yields a simplification in the design of the controller. The error variables in (24) can be written as end for 8: return minimumĖ x (k + 1). 9: Choose switching states which yield minimumĖ x (k+1).
Applying the first order forward Euler approximation to (6), (8), and (9), the future values of i c (k), V C1 (k), and V C2 (k) at (k + 1) th sampling instant can be obtained as follows where T s is the sampling period and v xy (k) is given by On the other hand, the grid voltage, reference grid current and input voltage of rectifier at (k + 1) th sampling instant which are needed in (24) can be predicted as The steps of the proposed energy function based MPC method are given in the Algorithm. First, the system variables are measured to be used in the control algorithm. Then, the compensation current reference (i * c ) is computed. The predictions of e g (k +1), i * c (k +1) and v * xy (k +1) are calculated to be used in the energy function. Thereafter, the predicted values of the error variables (x 1 (k + 1) and x 2 (k + 1)) are calculated for each possible switching state. The derivative of energy function (Ė x (k + 1)) is evaluated using the predicted values in the previous steps. In the last step, the optimal switching vector is determined by minimizingĖ x (k + 1). The flowchart of the proposed MPC algorithm is given in Fig. 3.

IV. EXPERIMENTAL VERIFICATION
The effectiveness of the proposed MPC method is verified by experimental studies by implementing the block diagram shown in Fig. 2. In the experimental studies, a full-wave diode-bridge rectifier is used as the nonlinear load. The experimental studies have been carried out on the prototype shown in Fig. 4 where the power grid is emulated via a regenerative  grid simulator (Chroma 61860). The proposed MPC method was implemented by using OPAL-RT OP5600. The pulse width modulation (PWM) signals generated by OPAL-RT are applied to drive the switches in the T-type inverter. The system and control parameters used in the experimental studies are listed in Table 2. Fig. 5 shows the waveforms of grid voltage (e g ), grid current (i g ), load current (i L ), filter current (i c ), filter current reference (i * c ), dc-link voltage (V dc ), dc-link voltage reference (V * dc ) and capacitor voltages (V C1 , V C2 ). Despite the highly distorted load current, the grid current is sinusoidal and in phase with the grid voltage. Hence, the unity power factor operation is achieved. The filter current and its reference are overlapped, which implies that the controller has almost zero tracking error. It is obvious that the control of the dc-link and capacitor voltages is achieved at 250V and 125V, respectively.    are suppressed as shown in Fig. 6(b). Clearly, only the 3 rd , 5 th , and 7 th harmonic components are discernible with small magnitudes. Majority of odd harmonic components such as 9 th , 11 th , 13 th , 15 th , and other components are suppressed effectively. As a consequence of this fact, the THD of grid current is measured to be 2.7%. The effectiveness of the proposed controller on the operation of SAPF can be understood if the measured THD values are considered. Comparing the measured THD values of load and grid currents, one can see that the proposed SAPF operates with high performance. Fig. 7 shows operation of the SAPF for a step change in the nonlinear load current. It is worth noting that all variables are regulated before the step change occurs. When the load current increases (dc load resistance has been changed from 25 to 12.5 ), the dc-link voltage exhibits small undershoot and tracks its reference successfully as shown in Fig. 7(a). The capacitor voltages behave similar to the dc-link voltage. The grid current is also increased in response to the load current increment. On the other hand, when the load current is decreased (dc load resistance has been changed from 12.5 to 25 ), the dc-link voltage gives rise to small overshoot and tracks its reference successfully after the transient is over as shown in Fig. 7(b). Similarly, the capacitor voltages are also regulated in this case. Finally, it can be seen that the grid current is decreased in response to the decrement in load current.  Fig. 8 shows the waveforms of grid current, PI output, capacitor voltage error and filter current error corresponding to the load current changes in Fig. 8. Apparently, the grid current increases so as to track its reference when the load current is increased as shown in Fig. 8(a). Similarly, as a result of decrement in the load current, the grid current decreases as well and tracks its reference as shown in Fig. 8(b). In both cases, the error variables are completely zero, which indicate that the proposed controller acts very fast in tracking the references. Fig. 9 shows the waveforms of grid voltage, grid current, load current (i L ), filter current, dc-link voltage and capacitor voltages for start-up. Initially, the system is started with SAPF disabled. In this case, the grid current is equal to the load current and, eventually, the filter current is zero. The dclink and capacitor voltages are also zero. When the SAPF is enabled, the filter current is not zero anymore, which implies that the grid current and load current are not equal. In this case, the filter current is added with the load current at the point of common coupling such that the grid current becomes sinusoidal and in phase with grid voltage. On the other hand, dc-link and capacitor voltages gradually converge to 250V and 125V, respectively.   capacitor voltages is accomplished and the grid current is sinusoidal, the unity power factor is not satisfied completely as shown in Fig. 10(a). The spectrum of grid current is presented in Fig. 10(b). The THD of grid current is measured as 3.80%. Comparing the THDs of grid current obtained by proposed MPC and classical MPC, one can see that the proposed MPC method yields smaller THD value as shown in Fig. 6(b). Fig. 11 shows the waveforms obtained by the classical and proposed MPC methods for a step variation in WF and β 2 , respectively. It is worth noting that the control of dc-link and capacitor voltages and reactive power compensation together with the unity power factor are satisfied before the step change in WF as shown in Fig. 11(a). However, when the WF FIGURE 11. Waveforms of grid-voltage (e g ), grid current (i g ), load current (i L ), filter current (i c ), dc-link voltage (V dc ), and capacitor voltages (V C 1 and V C 2 ) obtained by: (a) Classical MPC for a step change in WF from 1 to 10, (b) Proposed MPC for a step change in β 2 from 1 to 10.

A. STEADY-STATE AND DYNAMIC RESPONSE TESTS
value is changed from 1 to 10, the capacitor voltages deviate from 125 V and become unbalanced. If the system is operated with WF = 10 for a long time, it would adversely affect the operation of SAPF since the control of dc-link voltage will be lost. This can be considered as the main disadvantage of the classical MPC that is dependent on the WF value. In literature, no procedure for tuning the WF for optimum performance is reported. The tuning of WF is usually achieved by trial-and-error method. On the other hand, changing β 2 value from 1 to 10 has no effect on the operation of the system as shown in Fig. 11(b). Since the proposed MPC does not require the WF, it brings a simplification in the design of the controller.   the WF value variations. This result clearly shows the disadvantage of classical MPC in which WF tuning is essential to obtain good performance. Since there is no preset rule for tuning WF, the design of WF becomes challenging in some applications. However, the THD is not affected when the value of β 2 is increased. Contrary to the classical MPC, the coefficient (β 2 ) employed in the cost function of the proposed MPC has no effect on the THD of grid current.
The proposed energy function based MPC method is also compared with five control methods presented in [27] and [35]- [38]. The comparison is based on the type of controller, weighting factor necessity, number of weighting factors, number of required sensors, additional controller necessity for controlling dc capacitor voltages, and reference filter current generation method as shown in Table 3.
Obviously, the proposed energy based MPC method is beneficial in terms of weighting factor free structure, reduced controller complexity as it does not need additional control loop for dc capacitor voltage regulation and reference filter current generation. The proposed energy function based MPC method possesses weighting factor free structure, which offers simplicity in the design. Opposed to [27] and [35]- [37], the proposed method in this study and [38] does not need an additional controller for regulating dc capacitor voltages. However, the control method in [38] needs weighting factor and requires one extra sensor compared to the proposed method here. On the other hand, contrary to the filter current reference generation in the proposed method, the methods in [27] and [35]- [37] require many computations, additional transformations, low pass filter and high pass filter.

C. ROBUSTNESS ANALYSIS
Since the proposed control method is dependent on the system parameters, the robustness of the control approach to parameter variations is essential. For comparison purposes, the robustness of the classical MPC method is also investigated. The results that are obtained under the same operating conditions in Fig. 5 and Fig. 10(a) are presented in Fig. 13. The inductance parameter (L) in the control software was varied from -25% to 25% by gradual increments in L. As clearly seen in Fig. 13, the lowest THD values are obtained by both control methods when L in the control software is equal to the actual L in the experimental setup. Clearly, the THD values of both methods are increased when the mismatch in L increases. The maximum THD value is measured when the mismatch in L is −25%. However, the THD value obtained by the proposed control method is always smaller than the classical MPC method for the same mismatch range.

V. CONCLUSION
An energy function based MPC method is proposed for single-phase three-level T-type inverter based SAPFs. Unlike the existing MPC methods, the proposed MPC method eliminates the need for using weighting factor in the cost function. The design of the cost function in the proposed MPC is based on the negative definiteness of the rate of change of energy function. Furthermore, the proposed MPC method is able to stabilize the dc capacitor voltages without using additional constraints in the cost function. Among the MPC methods developed for multilevel SAPFs, the proposed MPC has the simplest structure. The performance of the proposed MPC method has been verified under steady-state and transients caused by the load variations. Also, the performances of the proposed and classical MPC methods are compared in terms of grid current THD and influence of WF variations. It is pointed out that the performance of the proposed MPC and grid current THD are not affected from the variations in WF value.  FREDE BLAABJERG (Fellow, IEEE) received the Ph.D. degree in electrical engineering from Aalborg University, in 1995. He was with ABB-Scandia, Randers, Denmark, from 1987 to 1988. He became an assistant professor, in 1992; an associate professor, in 1996; and a full professor of power electronics and drives, in 1998. Since 2017, he became a villum investigator. In 2017, he became a Honoris Causa at University Politehnica Timisoara (UPT), Romania. He is also a Honoris Causa at Tallinn Technical University (TTU), Estonia. His current research interests include power electronics and its applications, such as in wind turbines, PV systems, reliability, harmonics, and adjustable speed drives. He has published more than 600 journal articles in the fields of power electronics and its applications. He is the coauthor of four monographs and an editor of ten books in power electronics and its applications.