# A Simplified Finite Control Set Model Predictive Control for T-Type Three-Level Power Conversion System Based on LCL Filter

Authors: Ning Gao; Huaiyu Fan; Weimin Wu
Journal: Journal of Control Science and Engineering
 (2021)

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## Abstract

Finite control set model predictive control (FCSMPC) is a highly attractive and potential control method for grid-tied converters. However, there are several challenges when employing FCSMPC in an LCL filter-based T-type three-level power conversion system (PCS) for battery energy storage applications. These challenges mainly include the increasing complexity of control algorithm and excessive cost of additional sensors, which deteriorate the performance of PCS and limit the application of FCSMPC. In order to overcome these issues, this paper proposes a simplified FCSMPC algorithm to reduce the computation complexity. Furthermore, full-dimensional state observers are adopted and implemented to estimate the instantaneous values of grid-side current and capacitor voltage for purpose of removing unnecessary electrical sensors. The implementation of proposed FCSMPC algorithm is described step by step in detail. Simulation results are provided as a verification for the correctness of theoretical analysis. Finally, a three-phase T-type three-level PCS prototype rated at 2.30 kVA/110 V is built up. Experimental results extracted from the prototype can verify the effectiveness of the proposed control strategy.


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## Body

## 1. Introduction

Sustainable energy sources widely utilized in the power system leads to instability issues due to their undispatchable generating features [1, 2]. For this reason, the demand for energy storage system is growing remarkably for decades [3–5]. Battery energy storage system (BESS), highlighted by its fast response, economic acceptable, long cycle life, high density, and low installation requirement, is believed to be one of the best choices to provide a series of reliable electricity services among various kinds of energy storage techniques [6].PCS plays an important role as a specific bidirectional AC-DC converter in the BESS to interface battery units with the grid based on power electronics semiconductor techniques [7]. Three-phase two-level voltage source converter (2L-VSC) is commonly used in grid-tied converter applied in photovoltaic generation or wind turbines. However, its drawbacks, mainly including high voltage stress and poor harmonic performance, limit the further development of 2L-VSC especially in large-scale energy storage scenario. Three-level voltage source converter (3L-VSC) topologies are much more advantageous than 2L-VSC in this application for its beneficiations of lower output harmonics, higher efficiency, more compact structure, and reduced voltage changing rate [8]. A lot of 3L-VSC topologies have been reported and studied recently, such as neutral-point-clamped VSC [9], active NPC VSC [10], cascaded H-bridge topology [11], flying capacitor VSC [12], and T-type 3L-VSC [13]. T-type 3L-VSC is now receiving high priority to the development because six clamped diodes are removed compared to traditional NPC. On the other hand, T-type 3L-VSC owns less conduction loss, which means it is able to promote the efficiency if switching frequency is below 30 kHz [13]. Aside from the active components in the PCS, passive filter based on LCL structure tends to be adopted since it provides superior performance in high-order harmonics attenuation [14].It is significant to select a convenient control strategy for the PCS to obtain high-quality current injected to the grid. Various kinds of linear and nonlinear control methods are all verified choices to solve the control problem. Model predictive control (MPC) is an attractive model-based nonlinear optimal control strategy widely used in industrial areas [15, 16]. MPC applied in power electronics control area can be roughly divided into two categories: one is continuous control set MPC and the other is finite control set MPC. By comparing these two methods, FCSMPC uses the discrete nature of PCS which means the number of available switching states is finite. By evaluating them in terms of cost function and control restrictions, the most suitable switching state will be determined to be active later. As reported in [17], high performance of FCSMPC algorithm which is successfully implemented in an LCL-based grid-tied 2L-VSC has been confirmed by both simulation and experimental investigations.Although FCSMPC can also be adopted in a 3L-VSC, further improvement and extra strategy need to be studied because of the increased complexity of 3L-VSC and DC-link voltage balancing problem [18]. In practice, the calculation time of control algorithm will unavoidably become longer because repeating times for transferring predictive model and calculating cost function increase apparently along with the number of switching states, which is equal to 27 for 3L-VSC compared to 8 for 2L-VSC. In order to resolve this contradiction, researchers have proposed several methods to extenuate the computational burden of FCSMPC.In [19], a simplified computational method was proposed by introducing the Lyapunov principle into the design of sector distribution method based on space vector modulation technique for a nested neutral-point-clamped converter. By using the proposed method, the unwanted switching states can be eliminated and only 9 vectors, rather than for all the 64 vectors, need to be evaluated in total. In [20], an effective approach with equivalent transformation and specialized sector distribution was presented to reduce the running time without affecting the control performance. The proposed method can be used in various circuit topologies with multiple constraints. In [21], a simplified algorithm based on a new direct torque control switching table was presented. Based on this method, only three voltage vectors instead of eight are used for prediction and actuation. Moreover, the number of prediction vectors is reduced without any complex calculations.Removing the nonessential sensors is also an important topic to compare hardware cost and further exploit the merits of FCSMPC. In [22], a model calculation method with inverter voltage and current was proposed to estimate the capacitor voltage and grid current by ignoring the capacitor current. However, the assumption of ignoring capacitor current is only valid for the case that the impedance of the capacitor branch is much higher than the inductor branch. In [23], an effective method was presented to decrease the sensors of state variables. In the proposed method, the capacitor current loop was merged with the grid-injected current loop by transforming the block diagram; thus a kind of capacitor current sensor could be saved. Furthermore, full-order state observer [24] and Kalman filter [25] based on state-space model were investigated, which were both utilized to estimate the state variables of filter via the closed-loop system.Therefore, this paper proposes a simplified FCSMPC for a three-phase PCS based on T-type three-level topology to reduce its computational burden. Compared with the MPC strategy in [17], full-dimensional state observers are adopted for estimating the grid-side current and capacitor voltage within LCL filter for purpose of saving the unnecessary sensors. The remainder of this paper is organized as follows. Section 2 briefly introduces the system configuration and basic principle of FCSMPC. In Section 3, a detailed discrete predictive model of PCS is deduced. Based on this model, a description of the proposed simplified FCSMPC strategy is given from the view of reference generation, control set optimization, cost function, and state observers. Section 4 shows the simulation and experimental results for purpose of verifying the control algorithm in this paper. Finally, a brief conclusion is summarized in Section 5.


## 2. System Configuration and Description

Figure1 shows the basic structure of PCS for BESS application. The battery pack which performs as the main electrochemical reaction-based energy storage component characterized by its high density and flexibility is coupled to the grid using PCS composed of a bidirectional three-phase T-type three-level AC-DC converter and LCL filter. Three-level power conversion topologies usually exhibit outstanding performances including total harmonic distortion (THD) and efficiency compared to traditional two-level converter, while high-order passive filter is able to attenuate high frequency harmonics more effectively than a single L filter.Figure 1
Structure of T-type three-level power conversion system for battery energy storage application.As general definitions of common variables,ia1, ib1, and ic1 represent the phase current flowing through inverter-side inductor L1a, L1b, L1c. ia2, ib2, and ic2 are phase current flowing through grid-side inductor L2a, L2b, and L2c. uca, ucb, and ucc represent the filtering capacitor voltages. ua, ub, and uc represent the output voltage of each phase legs, respectively. uga, ugb, and ugc are grid voltages. Voltage and current sensors are installed properly for signal acquisition. udc1 represents the voltage across the top-side capacitor, and udc2 represents voltage of bottom-side capacitor. Moreover, when analysing the instantaneous power control of PCS, the battery pack can be assumed as a stiff DC voltage source because its output voltage ubat varies slowly enough compared to the microsecond time-scale of PCS’s real-time control in most cases. The PCS is supervised by the superior energy management system, which decides the output feature of BESS by sending the active power reference p∗ and reactive power reference q∗ instructions to the PCS.Each phase leg of T-type 3L-VSC in the PCS depicted in Figure1 consists of four semiconductor switches named as Sx1 ∼ Sx4 (x = a, b, c), which can be divided into two groups: two main switches Sx1 and Sx4, and two auxiliary switches Sx2 and Sx3. Among these switches, driver signal of Sx1 is complementary to Sx3, and driver signal of Sx2 is complementary to Sx4, respectively. There are three output states in every phase leg: (1) “P” state: if Sx1 and Sx2 turn on simultaneously, in the same time Sx3 and Sx4 turn off. The output voltage of phase leg is equal to ubat/2 with reference to the neutral point (NP) of the DC-link upon the assumption that NP is balanced in steady state which means udc1 = udc2 = +ubat/2. (2) “0” state: if Sx2 and Sx3 turn on simultaneously, while Sx1 and Sx4 turn off in the same time, the midpoint of the phase leg connects to NP directly. The output voltage is zero in this state. (3) “N” state: if Sx3 and Sx4 turn on simultaneously, while Sx1 and Sx2 turn off in the same time. Similar to P state, the output voltage of phase leg is −ubat/2.Therefore, the PCS is able to generate totally 27 space vectors in stationary reference frame. The control set of PCS is defined to include all the combinations of switching states illustrated in Figure2. It is also named as space vector distribution diagram, which aims to describe their magnitudes and positions. According to the lengths of voltage vectors, there are 6 large, 6 medium, 12 small, and three zero vectors possibly generated by the PCS. It can be seen that small vectors, for instance, vectors P00 and 0NN, exert an identical effect on the output current of PCS. Therefore, these pairs of small vectors can be used to balance the neutral point voltage which will be discussed later. The detailed relations between space vector and switching status are depicted in Figure 3.Figure 2
Distribution diagram of space vectors inαβ reference frame.Figure 3
Control set definition for a T-type three-level PCS.The FCSMPC algorithm is needless to involve a modulation stage like traditional control method such as proportional-integrator control widely applied in grid-tied converter. Modified FCSMPC scheme in this paper is carried out in theαβ reference frame including six steps as follows:(1)
Signal regulation and acquisition, usingA/D converter of the digital processor to obtain the instantaneous value of ix1, ugx (x = a, b, c), udc1 and udc2. Mind that unlike conventional FCSMPC method, ucx and ix2 are not necessary to be sampled because observer technique is adopted afterwards.(2)
Calculate the references in the next sampling period according to the external referencesp∗ and q∗ commanded by the upstream supervisor system.(3)
Estimateucx and ix2 based on full-dimensional state observers.(4)
Construct the prediction model. This is a two-step process. First, make the vector search span smaller based on an approximate prediction model. Then, use an accurate prediction model to predict the exact values of state variables in the next instant for all possible vectors within the down-sized control set.(5)
Choose an appropriate cost function. Calculate the evaluation results for every possible vector.(6)
Find the optimized value of cost function and the corresponding vector. Finally, set the semiconductor switches to the desired states according to the optimization result.The beneficiations of proposed FCSMPC strategy compared to conventional method mainly include two aspects. First, the values ofucx and ix2 are estimated based on state observer in Step 2. As a result, the corresponding sensors and sampling circuits can be omitted for purpose of decreasing the overall hardware cost. Second, the attempting times decreases to no more than 7 after adopting the simplified calculation procedure proposed in this paper. As a comparison, 27 attempts must be made to find out the optimized vector in conventional FCSMPC implementations, which increase the calculation time-cost of processor. In Section 3, the implementation procedure of the proposed FCSMPC strategy will be introduced in detail.


## 3. Implementation Procedure of Proposed FCSMPC

### 3.1. Reference Generation

The control method of PCS can be carried out inαβ reference frame based on the application of Clark transform. In most cases, active and reactive power are major instructions given by the upstream monitoring system. After receiving these instructions, they should be transferred to instantaneous voltage and current references in αβ frame first. For a three-phase system, active power P and reactive Q are expressed as(1)p=32ugα×igα+ugβ×igβ,q=32ugβ×igα−ugα×igβ.Hence, the reference of grid-injected current can be inversely deduced according to equation (2). The superscript of p∗ and q∗ denotes that those variables are reference signals.(2)igα∗k+1igβ∗k+1=23ugα2k+1+ugβ2k+1ugαk+1ugβk+1ugβk+1−ugαk+1p∗q∗.It is easy to deduce the reference value ofucx and ix1 (x = αα, β) using the following equations:(3)ucα∗ucβ∗=ugαugβ+ωresL2−igβ∗igα∗,(4)iα1∗iβ1∗=1−ωres2L2Ciα2∗iβ2∗.Unfortunately,ugα (k + 1) and ugβ (k + 1) are not explicit variables in the k-th instant. It will lead to a delay for calculating the reference value if ugα (k) and ugβ (k) are directly substituted into equation (2) for an alternative choice. This delay will degrade the performance of PCS. To overcome this drawback, it is advisable to compensate the unit delay of grid voltage sampling conveniently based on predicting the voltage in the next instant using Lagrange extrapolation theorem [26]. ugx (k + 1) calculated based on three-order Lagrange extrapolation is expressed as(5)ugxk+1=3ugxk−3ugxk−1+ugxk−2x=α,β.


### 3.2. Discrete-Time Prediction Model of LCL Filter-Based PCS

After accomplishing the reference calculation step, establishing the prediction model is also an important part for the control algorithm. It is apparent that theα-axis subsystem and ß-axis subsystem are fully decoupled. Therefore, the model of the PCS based on LCL filter can be described by the state-space model given in the following equation:(6)x˙α=Axα+Biuα+Bgugα,x˙β=Axβ+Biuβ+Bgugβ,where the state spaces xα and xβ are selected as(7)xα=iα1iα2ucαT,xβ=iβ1iβ2ucβT.Expressions of matrixA, input matrix Bi, and Bg are given in the appendix. As the control method is usually implemented in a digital signal processor, the continuous-time state-space model needs to be converted to a discrete-time model by calculating the state transition equation first. By substituting sampling step time Ts into the continuous model, the PCS can be modelled as equation (8). The prediction step will be done using this discrete-time (8)xαk+1=Adxk+Biduαk+Bgdugαk,xβk+1=Adxβk+Biduβk+Bgdugβk.MatricesAd, Bid, and Bgd are also given in the appendix. Furthermore, DC-link voltage balance is another important point in a three-level converter. Control algorithm should be adopted to guarantee the voltage balance of neutral point [27]. The difference between udc1 and udc2 should be considered to eliminate its negative impact on the power quality, especially for even-order harmonic performance of PCS. udc1 and udc2 can be expressed as follows:(9)Cdcdudc1dt=idc1,Cdcdudc2dt=idc2.Cdc represents the capacitance of individual side capacitor. The difference of udc1 and udc2, denoted as Δudc, varies according to(10)CdcdΔudcdt=Cdcdudc1−udc2dt=idc1−idc2.The difference betweenidc1 and idc2 is denoted as ineu. Then, we can deduce the relationship between Δudc and ineu as(11)Δudc=∫idc1−idc2Cdt=∫ineuCdt.At last,ineu can be obtained according to the switch state Sa, Sb, and Sc.(12)ineu=1−Saia+1−Sbib+1−Scic=−Saia−Sbib−Scic,where Sx equals 1 when phase x (x = a, b, c) of the converter outputs positive voltage in “P” state, Sx equals 0 in “0” state, and Sx equals −1 in “N” state. For example, if operation state of converter is “P0N” as shown in Figure 4(a), ineu equals −ia1−ic1 = +ib1, if ib1 > 0, udc1 will increase, and udc2 will decrease; in contrast, udc1 will increase and udc2 will decrease if ib1 < 0. Figures 4(b) and 4(c) also illustrate the corresponding conditions when PCS generates “P00” and “0NN” vectors. It can be seen that P00 and 0NN vectors exert an opposite effect on Δudc.Figure 4
Effects of different vectors on neutral point current. (a) P0N case. (b) P00 case. (c) 0NN case.
(a)(b)(c)Finally, we can get the following equation to predictΔudc in the next period as(13)Δudck+1=Δudck−TsCdcSakiak+Sbkibk+Sckick.


### 3.3. Simplified Optimization Strategy

For conventional FCSMPC strategy, 27 different vectors need to be substituted separately to the aforementioned mathematical model. However, only a minor part of the complete control set is possible to be the candidate vectors and should participate in the cost function calculation step in one specific control period. In order to reduce the computation burden, most vectors far from the optimized vector can be excluded easily by a rough estimation before the accurate prediction model is executed.For this reason, if the capacitors in the LCL filter are assumed to be nonexisted first, a rough estimation of output vector can be deduced in the same way as continuous control state model predictive control in a singleL filter (assuming L = L1 + L2) based converter. In this case, the approximately estimated output voltage of PCS, denoted as uαE and uβE, can be obtained as given in equation (14).(14)uαEk+1=L×iα2∗k+1−iα1k+Ts×ugαkTs,uβEk+1=L×iβ2∗k+1−iβ1k+Ts×ugβkTs.Apparently, the values ofuαE and uβE are very likely unequal to any vector that the converter can generate directly. However, the optimized vector must locate in the vicinity of uαE and uβE. That is to say, only the vectors near (uαE, uβE) are possible candidates for the next switch period. The other vectors can be discarded in the further optimization process. For example, if uαE = 0.5ubat, uβE = 0.1ubat, the optimized vector will only be selected from the subset of PNN, P0N PPN and 00P according to Figure 2. Therefore, how to judge which vectors are the nearest ones to the (uαE, uβE) as fast as possible is essential to simplify the algorithm to reduce the computation burden of digital controller. For this purpose, intermediate integers a, b, and c are defined as(15)a=signuαE,b=sign3uβE−uαE,c=sign−3uβE−uαE.Function sign (y) returns the polarity of input number y. If y > 0, sign (y) equals 1; otherwise, sign (x) returns 0. After that, the origin of αβ plane can be referenced to one of the six vertices of small vectors. Based on this method, complicated three-level modulation can be reduced to a two-level one. We can get the relationship between modified uαm, uβm, and original signals as(16)uαm=uαE−uα0,uβm=uβE−uβ0.From the above equations,uα0 and uβ0 can be obtained:(17)uα0=13a−16b−16c,uβ0=36b−c.After redefining the virtual origin point, the vector location in one hexagon should be deduced. In order to achieve this goal, auxiliary integersd, e, and f are defined as(18)d=signuβm,e=sign3uαm−uβm,f=sign−3uαm−uβm.On the basis of equation (18), a new integer N is defined as(19)N=d+2e+4f.By calculating the value ofa, b, c, and N, the location of (uαm, uβm) in a hexagon can be deduced as shown in Figure 5. Meanwhile, Table 1 lists the possible candidate vectors for exhaustive conditions.Figure 5
An index diagram from auxiliary integer values to the location of point (uαE, uβE).Table 1
Selection guide for candidate vectors based ona, b, c, and N.
[a, b, c, N]Candidate vector1001P0N P00 0NN PP0 00N1002PNN PN0 P00 0NN1003PNN P0N P00 0NN1004P00 0NN P0P 0N0 (PPP 000 NNN)1005P00 0NN PP0 0NN (PPP 000 NNN)1006PN0 P00 0NN P0P 0N01101PPN 0PN PP0 00N1102P0N P00 0NN PP0 00N1103P0N PPN PP0 00N1104PP0 00N 0P0 N0N (PPP 000 NNN)1105PP0 00N 0PN 0P0 N0N1106P00 0NN PP0 00N (PPP 000 NNN)01010PN NPN 0P0 N0N0102PP0 00N 0P0 N0N (PPP 000 NNN)01030PN PP0 00N 0P0 N0N0104NP0 0P0 N0N 0PP N000105NPN NP0 0P0 N0N01060P0 N0N 0PP N00 (PPP 000 NNN)0111NP0 0P0 N0N 0PP N0001120PP N00 00P NN0 (PPP 000 NNN)01130P0 N0N 0PP N00 (PPP 000 NNN)0114NPP N0P 0PP N000115NP0 NPP 0PP N000116N0P 0PP N00 00P NN000110PP N00 00P NN0 (PPP 000 NNN)00120NP 00P NN0 P0P 0N0001300P NN0 P0P 0N0 (PPP 000 NNN)0014N0P NNP 00P NN00015N0P 0PP N00 00P NN00016NNP 0NP 00P NN01011P00 0NN P0P 0N0 (PPP 000 NNN)1012PNP PN0 P0P 0N01013PN0 P00 0NN P0P 0N010140NP 00P NN0 P0P 0N0101500P NN0 P0P 0N0 (PPP 000 NNN)10160NP PNP P0P 0N0Then, the respective vectors are substituted to the accurate predictive model given from equation (6) to equation (13), aiming to choose the best vector which will become active in the next period. After using this method, only 5–7 vectors are necessary to be considered in the accurate calculation step. This is helpful to largely decrease the calculation time consumption of the PCS compared to traditional model predictive control algorithm. Note that, in fact, it is needless to calculate all three zero vectors in prediction. If the possibly chosen vector in the next period is zero vector, these zero vectors impact the grid current and DC voltage in the same way. Therefore, the optimized vector is chosen to be the one which can decrease the switching frequency to the lowest. For example, if the last state is “PP0”, “PPP” will be chosen since only Sc1 and Sc3 participate in switching over process in this case.


### 3.4. Full-Dimensional State Observer

To avoid the additional nonessential sensors, full-dimensional state observers are adopted. The configuration of continuous-time based state observer forα-axis components is shown in Figure 6. The ß-axis observer has the same structure as α-axis. The α-axis state observer is modelled in continuous-time domain as(20)x^˙α=Ax^α+Biuα+Bgugα+KCxα−x^α.Figure 6
Structure ofα-axis full-dimensional state observer.Becauseiα1 and iβ1 are indispensable variables directly probed by current sensors in PCS control, they are chosen to construct the feedback loop of state observer by setting output matrix C to [1 0 0]T. In order to verify the feasibility of state observer, the observability should be certified first. As can be seen in equation (21), the rank of observability matrix equals 3, which is full ranked, indicating that it is possible to estimate all the state variables in the α-axis subsystem. Based on the same principle, the variables in the ß-axis subsystem are also estimated in the same way as α-axis subsystem.(21)rankCCACA2T=3.MatrixK performs as the feedback gain matrix to calibrate the observed values to the real values in the physical system. By adjusting K, the eigenvalues of observer can be arbitrarily designed.(22)K=K1K2K3T.The response of observation errorΔxα will vary according to the calibration parameters setting.(23)Δx^αt=eA−KCtΔx^α0.To guarantee the convergence of observation error, the eigenvalue of A − KC is very important. The characteristic polynomial of A − KC is given by(24)detsI−A+KC=s3+K1s2+L1+L2−L2CK3L1L2Cs+K1L1+K2L2L1L2C=0.The root of characteristic polynomial should all locate in the left half complex plane to avoid an unstable condition; otherwise, the observation error will diverge to infinity. By applying Routh criterion, we can deduce the parameterK1 ∼ K3 should satisfy the rule as follows:(25)K1>0,K1L1+K2L2>0,K3<L1+L2L2C,K1−K1K3C>K2.After the feedback parameters are determined, it is easy to implement the state observer in a digital controller by utilizing a general discretization solution such as backward-Euler method.


### 3.5. Cost Function

Finally, the cost function should be chosen properly to finish the whole algorithm. This is the vital step in FCSMPC. In order to approach the control of both output current and NP balance, the cost function in this paper is chosen to be(26)J=εi12+λi2εi22+λuεu2+λudcΔUdck+12,where λi1, λi2, λu, and λudc represent the weighing factors to adjust the priority of control objective. εi1, εi2, and εu represent the errors between reference and predicted value in the next instant. Their expressions are given by(27)εi1=iα1∗k+1−iα1k+12+iβ1∗k+1−iβ1k+12,εi2=iα2∗k+1−iα2k+12+iβ2∗k+1−iβ2k+12,εu=ucα∗k+1−ucαk+12+ucβ∗k+1−ucβk+12.By substituting the predictive results in the cost function, the appropriate vector corresponding to the minimum value ofJ can be selected to be sent out by the PCS to form the optimized grid-injected current.


## 3.1. Reference Generation

The control method of PCS can be carried out inαβ reference frame based on the application of Clark transform. In most cases, active and reactive power are major instructions given by the upstream monitoring system. After receiving these instructions, they should be transferred to instantaneous voltage and current references in αβ frame first. For a three-phase system, active power P and reactive Q are expressed as(1)p=32ugα×igα+ugβ×igβ,q=32ugβ×igα−ugα×igβ.Hence, the reference of grid-injected current can be inversely deduced according to equation (2). The superscript of p∗ and q∗ denotes that those variables are reference signals.(2)igα∗k+1igβ∗k+1=23ugα2k+1+ugβ2k+1ugαk+1ugβk+1ugβk+1−ugαk+1p∗q∗.It is easy to deduce the reference value ofucx and ix1 (x = αα, β) using the following equations:(3)ucα∗ucβ∗=ugαugβ+ωresL2−igβ∗igα∗,(4)iα1∗iβ1∗=1−ωres2L2Ciα2∗iβ2∗.Unfortunately,ugα (k + 1) and ugβ (k + 1) are not explicit variables in the k-th instant. It will lead to a delay for calculating the reference value if ugα (k) and ugβ (k) are directly substituted into equation (2) for an alternative choice. This delay will degrade the performance of PCS. To overcome this drawback, it is advisable to compensate the unit delay of grid voltage sampling conveniently based on predicting the voltage in the next instant using Lagrange extrapolation theorem [26]. ugx (k + 1) calculated based on three-order Lagrange extrapolation is expressed as(5)ugxk+1=3ugxk−3ugxk−1+ugxk−2x=α,β.


## 3.2. Discrete-Time Prediction Model of LCL Filter-Based PCS

After accomplishing the reference calculation step, establishing the prediction model is also an important part for the control algorithm. It is apparent that theα-axis subsystem and ß-axis subsystem are fully decoupled. Therefore, the model of the PCS based on LCL filter can be described by the state-space model given in the following equation:(6)x˙α=Axα+Biuα+Bgugα,x˙β=Axβ+Biuβ+Bgugβ,where the state spaces xα and xβ are selected as(7)xα=iα1iα2ucαT,xβ=iβ1iβ2ucβT.Expressions of matrixA, input matrix Bi, and Bg are given in the appendix. As the control method is usually implemented in a digital signal processor, the continuous-time state-space model needs to be converted to a discrete-time model by calculating the state transition equation first. By substituting sampling step time Ts into the continuous model, the PCS can be modelled as equation (8). The prediction step will be done using this discrete-time (8)xαk+1=Adxk+Biduαk+Bgdugαk,xβk+1=Adxβk+Biduβk+Bgdugβk.MatricesAd, Bid, and Bgd are also given in the appendix. Furthermore, DC-link voltage balance is another important point in a three-level converter. Control algorithm should be adopted to guarantee the voltage balance of neutral point [27]. The difference between udc1 and udc2 should be considered to eliminate its negative impact on the power quality, especially for even-order harmonic performance of PCS. udc1 and udc2 can be expressed as follows:(9)Cdcdudc1dt=idc1,Cdcdudc2dt=idc2.Cdc represents the capacitance of individual side capacitor. The difference of udc1 and udc2, denoted as Δudc, varies according to(10)CdcdΔudcdt=Cdcdudc1−udc2dt=idc1−idc2.The difference betweenidc1 and idc2 is denoted as ineu. Then, we can deduce the relationship between Δudc and ineu as(11)Δudc=∫idc1−idc2Cdt=∫ineuCdt.At last,ineu can be obtained according to the switch state Sa, Sb, and Sc.(12)ineu=1−Saia+1−Sbib+1−Scic=−Saia−Sbib−Scic,where Sx equals 1 when phase x (x = a, b, c) of the converter outputs positive voltage in “P” state, Sx equals 0 in “0” state, and Sx equals −1 in “N” state. For example, if operation state of converter is “P0N” as shown in Figure 4(a), ineu equals −ia1−ic1 = +ib1, if ib1 > 0, udc1 will increase, and udc2 will decrease; in contrast, udc1 will increase and udc2 will decrease if ib1 < 0. Figures 4(b) and 4(c) also illustrate the corresponding conditions when PCS generates “P00” and “0NN” vectors. It can be seen that P00 and 0NN vectors exert an opposite effect on Δudc.Figure 4
Effects of different vectors on neutral point current. (a) P0N case. (b) P00 case. (c) 0NN case.
(a)(b)(c)Finally, we can get the following equation to predictΔudc in the next period as(13)Δudck+1=Δudck−TsCdcSakiak+Sbkibk+Sckick.


## 3.3. Simplified Optimization Strategy

For conventional FCSMPC strategy, 27 different vectors need to be substituted separately to the aforementioned mathematical model. However, only a minor part of the complete control set is possible to be the candidate vectors and should participate in the cost function calculation step in one specific control period. In order to reduce the computation burden, most vectors far from the optimized vector can be excluded easily by a rough estimation before the accurate prediction model is executed.For this reason, if the capacitors in the LCL filter are assumed to be nonexisted first, a rough estimation of output vector can be deduced in the same way as continuous control state model predictive control in a singleL filter (assuming L = L1 + L2) based converter. In this case, the approximately estimated output voltage of PCS, denoted as uαE and uβE, can be obtained as given in equation (14).(14)uαEk+1=L×iα2∗k+1−iα1k+Ts×ugαkTs,uβEk+1=L×iβ2∗k+1−iβ1k+Ts×ugβkTs.Apparently, the values ofuαE and uβE are very likely unequal to any vector that the converter can generate directly. However, the optimized vector must locate in the vicinity of uαE and uβE. That is to say, only the vectors near (uαE, uβE) are possible candidates for the next switch period. The other vectors can be discarded in the further optimization process. For example, if uαE = 0.5ubat, uβE = 0.1ubat, the optimized vector will only be selected from the subset of PNN, P0N PPN and 00P according to Figure 2. Therefore, how to judge which vectors are the nearest ones to the (uαE, uβE) as fast as possible is essential to simplify the algorithm to reduce the computation burden of digital controller. For this purpose, intermediate integers a, b, and c are defined as(15)a=signuαE,b=sign3uβE−uαE,c=sign−3uβE−uαE.Function sign (y) returns the polarity of input number y. If y > 0, sign (y) equals 1; otherwise, sign (x) returns 0. After that, the origin of αβ plane can be referenced to one of the six vertices of small vectors. Based on this method, complicated three-level modulation can be reduced to a two-level one. We can get the relationship between modified uαm, uβm, and original signals as(16)uαm=uαE−uα0,uβm=uβE−uβ0.From the above equations,uα0 and uβ0 can be obtained:(17)uα0=13a−16b−16c,uβ0=36b−c.After redefining the virtual origin point, the vector location in one hexagon should be deduced. In order to achieve this goal, auxiliary integersd, e, and f are defined as(18)d=signuβm,e=sign3uαm−uβm,f=sign−3uαm−uβm.On the basis of equation (18), a new integer N is defined as(19)N=d+2e+4f.By calculating the value ofa, b, c, and N, the location of (uαm, uβm) in a hexagon can be deduced as shown in Figure 5. Meanwhile, Table 1 lists the possible candidate vectors for exhaustive conditions.Figure 5
An index diagram from auxiliary integer values to the location of point (uαE, uβE).Table 1
Selection guide for candidate vectors based ona, b, c, and N.
[a, b, c, N]Candidate vector1001P0N P00 0NN PP0 00N1002PNN PN0 P00 0NN1003PNN P0N P00 0NN1004P00 0NN P0P 0N0 (PPP 000 NNN)1005P00 0NN PP0 0NN (PPP 000 NNN)1006PN0 P00 0NN P0P 0N01101PPN 0PN PP0 00N1102P0N P00 0NN PP0 00N1103P0N PPN PP0 00N1104PP0 00N 0P0 N0N (PPP 000 NNN)1105PP0 00N 0PN 0P0 N0N1106P00 0NN PP0 00N (PPP 000 NNN)01010PN NPN 0P0 N0N0102PP0 00N 0P0 N0N (PPP 000 NNN)01030PN PP0 00N 0P0 N0N0104NP0 0P0 N0N 0PP N000105NPN NP0 0P0 N0N01060P0 N0N 0PP N00 (PPP 000 NNN)0111NP0 0P0 N0N 0PP N0001120PP N00 00P NN0 (PPP 000 NNN)01130P0 N0N 0PP N00 (PPP 000 NNN)0114NPP N0P 0PP N000115NP0 NPP 0PP N000116N0P 0PP N00 00P NN000110PP N00 00P NN0 (PPP 000 NNN)00120NP 00P NN0 P0P 0N0001300P NN0 P0P 0N0 (PPP 000 NNN)0014N0P NNP 00P NN00015N0P 0PP N00 00P NN00016NNP 0NP 00P NN01011P00 0NN P0P 0N0 (PPP 000 NNN)1012PNP PN0 P0P 0N01013PN0 P00 0NN P0P 0N010140NP 00P NN0 P0P 0N0101500P NN0 P0P 0N0 (PPP 000 NNN)10160NP PNP P0P 0N0Then, the respective vectors are substituted to the accurate predictive model given from equation (6) to equation (13), aiming to choose the best vector which will become active in the next period. After using this method, only 5–7 vectors are necessary to be considered in the accurate calculation step. This is helpful to largely decrease the calculation time consumption of the PCS compared to traditional model predictive control algorithm. Note that, in fact, it is needless to calculate all three zero vectors in prediction. If the possibly chosen vector in the next period is zero vector, these zero vectors impact the grid current and DC voltage in the same way. Therefore, the optimized vector is chosen to be the one which can decrease the switching frequency to the lowest. For example, if the last state is “PP0”, “PPP” will be chosen since only Sc1 and Sc3 participate in switching over process in this case.


## 3.4. Full-Dimensional State Observer

To avoid the additional nonessential sensors, full-dimensional state observers are adopted. The configuration of continuous-time based state observer forα-axis components is shown in Figure 6. The ß-axis observer has the same structure as α-axis. The α-axis state observer is modelled in continuous-time domain as(20)x^˙α=Ax^α+Biuα+Bgugα+KCxα−x^α.Figure 6
Structure ofα-axis full-dimensional state observer.Becauseiα1 and iβ1 are indispensable variables directly probed by current sensors in PCS control, they are chosen to construct the feedback loop of state observer by setting output matrix C to [1 0 0]T. In order to verify the feasibility of state observer, the observability should be certified first. As can be seen in equation (21), the rank of observability matrix equals 3, which is full ranked, indicating that it is possible to estimate all the state variables in the α-axis subsystem. Based on the same principle, the variables in the ß-axis subsystem are also estimated in the same way as α-axis subsystem.(21)rankCCACA2T=3.MatrixK performs as the feedback gain matrix to calibrate the observed values to the real values in the physical system. By adjusting K, the eigenvalues of observer can be arbitrarily designed.(22)K=K1K2K3T.The response of observation errorΔxα will vary according to the calibration parameters setting.(23)Δx^αt=eA−KCtΔx^α0.To guarantee the convergence of observation error, the eigenvalue of A − KC is very important. The characteristic polynomial of A − KC is given by(24)detsI−A+KC=s3+K1s2+L1+L2−L2CK3L1L2Cs+K1L1+K2L2L1L2C=0.The root of characteristic polynomial should all locate in the left half complex plane to avoid an unstable condition; otherwise, the observation error will diverge to infinity. By applying Routh criterion, we can deduce the parameterK1 ∼ K3 should satisfy the rule as follows:(25)K1>0,K1L1+K2L2>0,K3<L1+L2L2C,K1−K1K3C>K2.After the feedback parameters are determined, it is easy to implement the state observer in a digital controller by utilizing a general discretization solution such as backward-Euler method.


## 3.5. Cost Function

Finally, the cost function should be chosen properly to finish the whole algorithm. This is the vital step in FCSMPC. In order to approach the control of both output current and NP balance, the cost function in this paper is chosen to be(26)J=εi12+λi2εi22+λuεu2+λudcΔUdck+12,where λi1, λi2, λu, and λudc represent the weighing factors to adjust the priority of control objective. εi1, εi2, and εu represent the errors between reference and predicted value in the next instant. Their expressions are given by(27)εi1=iα1∗k+1−iα1k+12+iβ1∗k+1−iβ1k+12,εi2=iα2∗k+1−iα2k+12+iβ2∗k+1−iβ2k+12,εu=ucα∗k+1−ucαk+12+ucβ∗k+1−ucβk+12.By substituting the predictive results in the cost function, the appropriate vector corresponding to the minimum value ofJ can be selected to be sent out by the PCS to form the optimized grid-injected current.


## 4. Simulation and Experimental Results

In order to evaluate the feasibility of PCS applied in BESS based on the simplified control algorithm proposed in this paper, simulation study is performed based on MATLAB/Simulink. Then, experimental research studies are also carried out. The diagram of simulation model is shown in Figure7. Its main electric parameters are listed in Table 2.Figure 7
Simulation model diagram of PCS based on MATLAB/Simulink.Table 2
Main electric parameters for the simulation model of PCS.
ParameterValueUnitRated power2300VANominal total DC-link voltage360VdcNominal grid voltage (phase to phase)110VrmsNominal grid frequency50.0HzSampling frequency30.0kHzInductance of L13.6mHInductance of L21.2mHFiltering capacitance ofC3.3μFDC-link capacitance ofCdc4.7mFFirst, the steady state performance of PCS is verified by setting active power reference to 2300 W. Meanwhile, reactive power reference is forced to be zero to achieve unity power operation. The simulation waveforms are shown in Figures8(a)–8(c). It can be seen that the output voltage per one phase leg of 3L-VSC has three states and line voltage has five states according to Figure 8(a). The grid-injected current possesses the same frequency and phase angle with grid voltage, which certifies that power factor (PF) is equal to 1.0. The three-phase waveforms of grid-side voltage and current are illustrated in Figure 8(b). The amplitude of symmetric grid current is 10 A. This corresponds exactly to the power reference. Moreover, it is possible to adjust the power factor by setting reactive power to different values. In Figure 8(c), the grid voltage is leading to current by approximately 25° after setting reactive power reference to 1100 Var. PF is 0.9 in this case. Power factor can accurately follow the command. As a supplement, the similar test is carried out for battery-charging operation. Figures 9(a)–9(c) show the simulation waveforms.Figure 8
Simulation results of PCS in full load battery-discharging condition. (a) Waveforms ofuga, iga, ua, and uab (PF = 1.0). (b) Waveforms of three-phase grid voltage and current (PF = 1.0). (c) Waveforms of three-phase grid voltage and current (PF = 0.9).
(a)(b)(c)Figure 9
Simulation results of PCS in full load battery-charging condition. (a) Waveforms ofuga, iga, ua, and uab (PF = 1.0). (b) Waveforms of three-phase grid voltage and current (PF = 1.0). (c) Waveforms of three-phase grid voltage and current (PF = 0.9).
(a)(b)(c)The dynamic performance of the proposed control strategy is tested by executing a full power transition between charging and discharging conditions. Figure10(a) shows the simulated transient response when PCS changes from discharging operation to charging operation at rated power. Furthermore, Figure 10(b) shows the simulated transient response when PCS changes from charging to discharging condition. It can be seen that both dynamic processes end in about 5 ms, which can meet the responding speed requirement in most situations.Figure 10
Simulation results of PCS in dynamic Process. (a) Waveforms of grid current both in dq and abc reference frame for transition from full load discharging to charging operation. (b) Waveforms of grid current both in dq and abc reference frame for transition from full load charging to discharging operation.
(a)(b)For the three-level energy conversion topology with split dc-link capacitors, the balance of dc-link is very important. Therefore, the effect of the DC voltage equalization control implemented integrated within FCSMPC is also tested based on the simulation model. Figure11 shows the simulation results to make a comparison of control algorithm with and without NP balance strategy. At the beginning of simulation, the balance algorithm is shut down by setting λudc to zero until 200 ms. After activating the balance algorithm, we can find that the diverging DC voltages udc1 and udc2 start to converge again. Finally, udc1 approximately equals udc2. On the basis of simulation results, we can conclude that the DC-link is well balanced.Figure 11
Simulation results of DC-link voltage with and without NP balance algorithm.Then, the performance of state observer is further tested. Figure12(a) compares the actual and estimated value of igα and igβ in discharging situation. Meanwhile, Figure 12(b) compares the estimated and actual value of filter capacitor’s voltage in αβ reference frame. Figures 12(c) and 12(d) give similar simulation results during charging operation state. On the basis of these simulation results, the convergence of observer can be confirmed. Because of the numerical similarity between actual and estimated values, it is feasible to replace the actual values with the estimated values extracted from the observer.Figure 12
Simulation results of full-dimensional state observers. (a) Waveforms of actual and estimated grid current when PCS is discharging. (b) Waveforms of actual and estimated capacitor voltage when PCS is discharging. (c) Waveforms of actual and estimated grid current when PCS is charging. (d) Waveforms of actual and estimated capacitor voltage when PCS is charging.
(a)(b)(c)(d)Finally, in some specific applications, PCS needs to deal with the issue to connect into unbalanced grid. Therefore, it is necessary to verify the ability of the proposed control method to resist the negative impact of nonideal grid. Figures13(a) and 13(b) show the simulation results for PCS when it is connected to unbalanced grid. Both discharging and charging conditions are taken into account in the simulation. We can see that the PCS works normally in this case. Power quality of the grid-injected current is acceptable even if three-phase grid voltage is asymmetric.Figure 13
Simulation results of PCS connected to a nonideal grid. (a) Three-phase grid voltage and current in battery-discharging condition. (b) Three-phase grid voltage and current in battery-charging condition.
(a)(b)For purpose of further verifying the correctness of theoretical analysis and control algorithm, a T-type 3L PCS experimental platform is constructed in the laboratory and a series of tests are carried out based on the prototype. The parameters of prototype are identical to the simulation model which has been given in Table2. Figure 14 shows the photo of prototype. IKW40T120 IGBTs are chosen to be the main switches (Sx1 and Sx4). FGA40N65SMD IGBTs are chosen to be the auxiliary switches (Sx2 and Sx3). Driving circuits of these semiconductor switches are based on 1EDI20I12AF. TMS320F28335 performs as the main digital controller. DC terminals of the prototype are connected to the programmable DC power supply Chroma 62150H–600S. The grid voltage is emulated by programmable AC power supply Chroma 61830 and provided to the AC terminals of the prototype. A GWinstek MDO2204ES oscilloscope is used to capture the voltage and current signals.Figure 14
Photo of the experimental platform.First, the steady state performance of T-type 3L PCS is tested and verified. The waveforms obtained based on the prototype are shown in Figure15. Figures 15(a) and 15(b) show the testing results for nominal battery discharging and charging operation at rated power. We can see that the grid-injected current is mostly sinusoidal and maintains the same phase angle and frequency as grid voltage to guarantee a unity power factor operation. If the power factor is not equal to 1.0, the PCS is able to provide or absorb reactive power as required to/from the grid. Figures 15(c) and 15(d) depict the measured waveforms in such cases by setting commanded reactive power to ±1100 VA. It can be seen that an apparent phase difference (approximately 25°) arises between the voltage and current signals. Figure 15(e) shows the output line voltage of T-type 3L PCS. The line voltage has five levels. Therefore, it contains fewer voltage harmonics compared to 2L PCS. The experimental results in Figure 15 are evident to confirm the correctness and feasibility of the control algorithm discussed in this paper.Figure 15
Experimental results of the PCS prototype. (a) Grid voltage and current waveforms in battery-discharging condition (unity PF). (b) Grid voltage and current waveforms in battery-charging condition (unity PF). (c) Grid voltage and current waveforms in battery-discharging condition (|PF| = 0.9). (d) Grid voltage and current waveforms in battery-charging condition (|PF| = 0.9). (e) Output line voltage waveforms of T-type 3L PCS.
(a)(b)(c)(d)(e)After that, a series of experiments to verify the dynamic performance of PCS prototype based on the proposed control method are carried out. Figure16(a) shows the acquired grid voltage and current waveforms when PCS starts to discharge at rated power. Similarly, Figure 16(b) illustrates the measured grid voltage and current waveforms when PCS starts to charge at rated power. It can be deduced that the start-up process of PCS is stable and fast without significant oscillation or overshoot. Figure 16(c) depicts the transient response when PCS suddenly changes from discharging operation to charging operation both at rated power. Figure 16(d) shows the transient response for a reverse transition. It can be found the transient affairs end up within less than half a grid cycle, which is acceptable for most industrial energy storage applications. We can see that the experimental results are in good agreement with the simulation results, which can verify the correctness of the algorithm proposed in this paper.Figure 16
Experimental results of the PCS prototype. (a) Grid voltage and current waveforms for nominal discharging start-up test. (b) Grid voltage and current waveforms for nominal charging start-up test. (c) Transient waveforms when PCS changes from discharging to charging condition. (d) Transient waveforms when PCS changes from charging to discharging condition.
(a)(b)(c)(d)


## 5. Conclusions

This paper proposes a simplified FCSMPC strategy for T-type three-level PCS applied in BESS. The simplification is reflected in two aspects: (1) simplify the algorithm by a two-step predictive calculation to decrease the overall calculation complexity; (2) simplify the sensors and sampling circuits by adopting full-dimensional state observer to estimate capacitor voltage and grid current. Moreover, the observer is able to estimate the variables correctly, which makes it possible to save several sensors in the main circuit. According to the simulation and experimental results, we can conclude that the proposed control strategy is feasible and effective. The steady state and dynamic performance of PCS based on the modified control method is reliable for either charging or discharging operating conditions.



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*Source: 9919338-2021-06-01.xml*