Three-Vector-Based Low Complexity Model Predictive Control for Soft Open Point
Three-Vector-Based Low Complexity Model Predictive Control for Soft Open Point
Wang, Zhengqi;Zhou, Haoyu;Huo, Qunhai;Hao, Sipeng
2022-01-05 00:00:00
Hindawi Mathematical Problems in Engineering Volume 2022, Article ID 1526676, 12 pages https://doi.org/10.1155/2022/1526676 Research Article Three-Vector-Based LowComplexity Model PredictiveControl for Soft Open Point 1 1 2 1 Zhengqi Wang , Haoyu Zhou , Qunhai Huo , and Sipeng Hao School of Electric Power Engineering, Nanjing Institute of Technology, Nanjing 211167, China Institute of Electrical Engineering, Chinese Academy of Sciences, Beijing 100190, China Correspondence should be addressed to Zhengqi Wang; wzqnjit@163.com Received 22 October 2021; Revised 25 November 2021; Accepted 10 December 2021; Published 5 January 2022 Academic Editor: Dazhong Ma Copyright © 2022 Zhengqi Wang et al. -is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Soft open point (SOP) can improve the flexibility and reliability of power supplies; thus, they are widely used in distribution network systems. Traditional single-vector model predictive control (SV-MPC) can quickly and flexibly control the power and current at both ports of the SOP. However, SV-MPC can only select one voltage vector in a sampling time, producing large current ripples, and power fluctuations. In order to solve the above problems, this paper proposes a three-vector-based low complexity model predictive control (TV-MPC). In the proposed control method, two effective voltage vectors and one zero voltage vector are selected in a sampling time. For the two-port SOP, methods are given to judge the sectors on both sides and select the voltage vectors. Furthermore, the calculation method of the distribution time is proposed as well. Finally, the effectiveness of the proposed method is verified by steady-state and dynamic-state simulation results compared with the SV-MPC. feedback linearization control in [9] can meet the re- 1. Introduction quirements of different operating modes. However, control applications will be difficult if a large number of PI coef- In recent years, more and more renewable energy sources such as solar energy and wind energy as well as new energy ficients need to be tuned or many controller parameters vehicles loads have been connected to the distribution need to be selected. network, which has had a serious impact on the stable and Model predictive control (MPC) is widely used in safe operation of the distribution network [1, 2]. -ere- power electronic converters due to its simple imple- fore, soft open point (SOP) is used more frequently in mentation and fast dynamic response. According to dif- distribution networks due to their advantages to flexibly ferent control purposes, MPC can be divided into model connect feeders of different voltage levels, improve the predictive current control (MPCC) and model predictive reliability of power supply, and continuously adjust power power control (MPPC). MPCC and MPPC are used in [3–5]. rectifiers, inverters, modular multilevel converters, in- -e two-port SOP can be regarded as an AC/DC/AC duction machines, and permanent-magnet synchronous converter composed of voltage source converters (VSCs) machines [10–14]. However, the single-vector MPC has [6]. -e power exchange at both ports can be realized only one voltage vector involved in the control during a through corresponding control. In [7], droop control is sampling time, and large current ripples and power fluc- adopted and proportional-integral (PI) controllers are tuations will be generated, resulting in unsatisfactory designed to realize the closed-loop control of the outer control effects. -erefore, how to improve power quality power loop and the inner current loop. In [8], the inner and reduce power fluctuations needs to be considered when current loop of the traditional PI double-loop control is MPC is used in converters. In order to improve the control replaced by a sliding mode control, which can reduce the PI performance, a variety of improved MPC methods have controllers. -e combination of sliding mode control and been proposed, such as implementing delay compensation, 2 Mathematical Problems in Engineering and has good dynamic characteristics. Finally, the article is or multistep prediction, improving the cost function, and increasing the number of voltage vectors in a sampling time summarized in Section 6. [15–19]. Among them, increasing the number of voltage vectors is the main way to improve the control perfor- 2. Mathematical Model of SOP mance. In [20], based on the dual-vector MPCC, a method of projecting the current error vector onto the active Figure 1 shows the topology of a two-port SOP, where the voltage vector is proposed, which minimizes the cost rectifier side and the inverter side are interconnected by the function and reduces the number of candidate voltage capacitance of the DC side. vectors. In [21], different sizes of virtual vectors are de- According to Kirchhoff’s law, the state equation of the fined, the obtained reference voltage vector and the two rectifier side and the inverter side in the motionless reference predicted vectors selected from the virtual vector are used framework can be obtained as for evaluation, and the virtual vector that minimizes the V i i V a a1 a1 a1N cost function is used in the next sampling time. -is ⎡ ⎢ ⎤ ⎥ ⎡ ⎢ ⎤ ⎥ ⎡ ⎢ ⎤ ⎥ ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ method greatly alleviates the computation burden and ⎢ ⎥ d ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ V ⎥ � L ⎢ i ⎥ + R ⎢ i ⎥ + ⎢ V ⎥, (1) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ b ⎥ s ⎢ b1 ⎥ s⎢ b1 ⎥ ⎢ b1N ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ avoids the weighting factor in predictive torque control. In ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ dt ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ [22], a three-vector MPCC is proposed for an open- V i i V c c1 c1 c1N winding linear permanent-magnet vernier motor. Based on the principle of deadbeat current control, an optimal e V i i a a2N a2 a2 vector is obtained, and the other two optimal vectors are ⎡ ⎢ ⎤ ⎥ ⎡ ⎢ ⎤ ⎥ ⎡ ⎢ ⎤ ⎥ ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ d ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ e ⎥ � ⎢ V ⎥ − L ⎢ i ⎥ − R ⎢ i ⎥, (2) obtained by cascading cost functions. In addition, the ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ b ⎦ ⎣ b2N ⎦ l ⎣ b2 ⎦ l⎣ b2 ⎦ dt switching frequency is reduced to improve the steady-state e V i i c c2N c2 c2 performance and suppress the zero-sequence current. In where V and i are the grid voltage and input current of [23], a combination of MPC and direct power control is x1 the rectifier side, respectively; e and i are the load back- used for the improved T-type three-level converter, and x x2 emf and output current of the inverter side, respectively; the switching states are grouped in advance. -e capacitor the subscripts x � a, b, c represents the three phases of the voltage is compared on the DC side to determine the optimal group of the switching state, which eliminates the SOP; L and R are the filter inductance and resistance; and s s L and R are the inverter inductance and resistance, need for midpoint voltage prediction and cost function l l respectively. calculations, while ensuring the neutral point voltage -e switching states of the SOP is defined as balance. In [24], the adaptive error correction strategy is applied to both the outer and inner prediction loops from the perspective of model errors, reducing the impact of S � , (3) errors caused by time delay, sampling error, and parameter mismatch on the control performance and effectively where S � 1 means that the upper switch of phase x is on and improving the output power quality compared to the the lower switch is off and S � 0 means that the upper switch traditional FCS-MPC. of phase x is off and the lower switch is on. Both sides of the traditional MPC strategy for the SOP -en, introducing the definition of the switching states only consider a single switch state control within a sampling into (1) and (2), the values of output voltages of each time, which results in large current ripples and power converter can be determined as fluctuations in the steady state. -e main contribution can be described as follows: V 2 −1 −1 S aiN a ⎡ ⎢ ⎤ ⎥ ⎡ ⎢ ⎤ ⎥⎡ ⎢ ⎤ ⎥ (1) A three-vector-based low complexity MPC scheme is ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ u ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ dc ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ proposed, which reduces the current ripple and ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ V ⎥ � ⎢ −1 2 −1 ⎥⎢ S ⎥, (4) ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎥⎢ ⎥ biN ⎥ ⎢ b ⎥ ⎢ ⎢ ⎥⎢ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ 3 ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ power fluctuation by increasing the number of ⎣ ⎦ ⎣ ⎦⎣ ⎦ switch states within a sampling time V −1 −1 2 S ciN c (2) Methods for judging sectors and selecting vectors are where u is the capacitor voltage in the DC link; V , V , dc aiN biN given, which reduce the number of calculations and V are the output voltages of each side in phases a, b, ciN In Section 2, the mathematical model of the SOP is and c, respectively; and i � 1 means the rectifier side and i � 0 introduced and the power flow is analyzed. -en, the tra- means the inverter side. ditional MPC method for the SOP is introduced, and the In the system of the two-port SOP, eight VSC voltage control block diagram of the traditional method is given in vectors can be chosen, six of which are active voltage vectors Section 3. In Section 4, a three-vector-based low complexity (V , V , V , V , V , V ) and two of which are zero vectors 1 2 3 4 5 6 MPC method is presented, and methods for judging sectors (V , V ). -ese eight vectors are shown in Figure 2. 0 7 and selecting vectors at both sides are given. In Section 5, the According to the instantaneous reactive power theory, traditional method and the improved method are compared ignoring the line loss and the self-loss of the SOP, the output by simulations, which shows that the proposed method active power P and reactive power Q of each SOP port can be effectively reduces the current ripple and power fluctuation obtained as Mathematical Problems in Engineering 3 Rectifier side Inverter side S S S S S S 1 2 3 4 5 6 V R L e L R a s s a l l a1 a2 n b u b n dc c e S′ S′ S′ S′ S′ S′ 1 2 3 4 5 6 Figure 1: Two-port SOP equivalent circuit. 3. Traditional MPC Analysis V (010) V (110) 3 2 3.1. Outer-Loop Control Mode Selection. Each port of the SOP can work in different control modes, namely, PQ mode, II U Q mode, and U f mode. One port of the VSC should be dc dc needed to control the DC side voltage stability. When the III I SOP is working normally, its two ports generally work in PQ V (011) V (111) V (000) V (100) 4 7 0 1 mode and U Q mode. To ensure normal operation and dc meet load power supply requirements when a feeder fails, the U f mode is set and normally used in SOP systems with dc IV VI three or more ports. In this study, the outer loop of the rectifier side selects U Q mode, and the inner loop selects a direct power model dc predictive control (DPMPC). -e outer loop of the inverter V (001) V (101) 5 6 side selects a PQ mode, and the inner loop selects the MPCC to reduce the load current error. Figure 2: Voltage vector in the stationary αβ reference frame. 3.2. Analysis of DPMPC Method in Rectifier Side. By using ⎧ P � 1.5V i + V i ⎪ 1 α 1α β 1β the Clark transformation, Equation (1) can be converted to Q � 1.5V i − V i 1 β 1α α 1β the stationary αβ reference frame as follows: , (5) ⎪ di P � 1.5e i + e i ⎪ 2 α 2α β 2β 1αβ (7) L � V − V − R i , s αβ 1Nαβ s 1αβ ⎩ dt Q � 1.5 e i − e i 2 β 2α α 2β where i � [i , i ] is the input current vector; 1αβ 1α 1β where P and Q are the rectifier side active and reactive 1 1 V � [V , V ] are the grid voltage vectors of the rectifier αβ α β power, respectively, while P and Q are the inverter side 2 2 side; and V � [V , V ] are the voltage vectors of 1Nαβ 1Nα 1Nβ active and reactive power, respectively. V , V , i , and i 1α 1β α β the VSC. are the grid voltage and input current in the stationary αβ Using the forward Euler discretization method to predict reference frame of the rectifier side, respectively, while e , e , α β the current in the next control period of (7), the predicted i , and i are the load back-emfs and output currents in the 2α 2β current at the (k+1)th instant can be calculated as αβ reference frame of the inverter side, respectively. R T T When the SOP operates normally, the output active s s s i (k + 1) � 1 − i (k) + V (k) − V (k), 1αβ 1αβ αβ 1Nαβ power of the rectifier side P can be calculated by the the- L L s s s orem of conservation of power as (8) P � P + P , (6) s dc l where i (k) and V (k) are the input current and grid 1αβ αβ where P is the active power obtained by the DC side ca- dc voltage at the kth instant, respectively; V (k) is the 1Nαβ pacitor and P is the active power obtained by the load of the l voltage vector of the VSC in the rectifier side at the kth inverter side. instant; and T is the sampling time. s 4 Mathematical Problems in Engineering In the U Q mode, the rectifier side uses a part of the where i (k) � [i (k), i (k)] is the load current at the kth dc 2αβ 2α 2β power obtained from the grid to exchange with other ports, instant; e (k) � [e (k), e (k)] is the load back-emf in in- αβ α β and the other part is used to adjust the DC side voltage. -e verter side at the kth instant; and V (k) � [V (k), 2Nαβ 2Nα outer loop adopts PI control, and the active power reference V (k)] is the voltage vector of the VSC in the inverter 2Nβ value of this port P is as follows: side at the kth instant. ref In order to adjust the load current, a cost function is used P � k u − u + k u − u dt + Reei , to find the best switching state: ref p dcref dc i dcref dc 2ref (9) g � i − i (k + 1) + i − i (k + 1) . (14) 2αref 2α 2βref 2β where k and k are the proportional and integral coefficients of p i Figure 3 shows the traditional SOP MPC block diagram. PI control, respectively; u and u are the reference voltage dcref dc At each sampling time, Equations (12) and (14) are used to and sampling voltage of the DC side, respectively; and i is 2ref calculate the cost function values of all switch states on both the conjugate load reference current of the inverter side. -e sides. -e switch state that minimizes the cost function is first two items in (9) calculate the power required to adjust the selected to drive the SOP. DC side voltage, and the active power obtained by the load on the inverter side is calculated by the last item. 4. Three-Vector-Based Low Complexity MPC -e voltage vectors and the predicted current vectors can Method for SOP be used to calculate the predicted instantaneous input active power and reactive power as follows: In the traditional MPC, the VSCs on both sides adopt a single- vector predictive control, the vector selection is very limited, ⎧ ⎪ P (k + 1) � 1.5V (k + 1)i (k + 1) + V (k + 1)i (k + 1) 1 α 1α β 1β and the voltage vector output by the two ports cannot reach the Q (k + 1) � 1.5 V (k + 1)i (k + 1) − V (k + 1)i (k + 1) entire spatial circle track range. As a result, the current ripple 1 β 1α α 1β content and power fluctuations on both sides become larger. In (10) order to improve the accuracy of control, it is necessary to increase the sampling frequency of the system, thereby in- In (10), when the sampling time is small, it can be assumed that V (k + 1) ≈ V (k), but if the sampling time is not small creasing the computational burden of the processor. Another αβ αβ problem is that the switching frequency is not fixed. In order to enough to ignore the change in the grid voltage within the sampling time, vector angle compensation can be used to reduce the current ripples and power fluctuations while fixing the switching frequency, a three-vector-based low complexity calculate the grid voltage at the (k+1)th instant as follows: MPC is used on both sides of the port in this study. jωT V (k + 1) � V (k)e , (11) αβ αβ where ω is the grid voltage pulsation. 4.1. Cost Function Design. In a three-vector-based MPC, cost -en, Equation (12) can be used as a cost function to functions (12) and (14) can be improved as follows: evaluate the prediction error generated by each switch, 2 2 (15) f � P − P (k + 1) + Q − Q (k + 1) , ref 1 ref 1 which is described as f � P − P (k + 1) + Q − Q (k + 1) . (12) 2 ref 1 ref 1 (16) g � i − i (k + 1) + i − i (k + 1) . 2αref 2α 2βref 2β In DPMPC, all eight voltage vectors are substituted into In these improved cost functions, the error between the (8) to predict the current at the (k+1)th instant, the predicted predicted value and the reference value is compared, which powers are substituted into (12) to calculate the error, and is conducive to choosing the best switch value. the switch with the smallest cost function is selected as the Using adjacent effective vectors for vector synthesis, six optimal switch and is applied to the SOP rectifier side at the vector combinations can be obtained: (V , V , V ), 1 2 0,7 next moment. (V , V , V ), (V , V , V ), (V , V , V ), (V , V , V ), 2 3 0,7 3 4 0,7 4 5 0,7 5 6 0,7 and (V , V , V ). 6 1 0,7 In a three-vector-based MPC, since three vectors are 3.3. Analysis of MPCC Method in Inverter Side. MPCC is synthesized in one sampling time, an additional cost used in the inverter side to compare the predicted load function needs to be designed to comprehensively evaluate current with the reference load current and select the voltage the errors of the three vectors in the sampling time. In this with the smallest cost function to apply in the next control study, the weighted root means that the square error is used period. to design a new cost function: Transforming (2) to the stationary αβ reference frame and adopting the forward Euler discretization method, the 2 2 2 2 2 2 2 (17) ω � e d + e d + e d , 1 1 2 2 0 0 predicted load current at the (k+1)th instant is as follows: where e , e , and e are the cost function values corresponding 1 2 0 R T T l s s i (k + 1) � 1 − i (k) + V (k) − e (k) , to the two effective voltage vectors and zero vectors, respec- 2αβ 2αβ 2Nαβ αβ L L l l tively; and d , d , and d are the operating times of the three 1 2 0 (13) voltage vectors in the sampling time T , respectively. s Mathematical Problems in Engineering 5 Vector anger αβ V (k+1) αβ compensation abc V (k) V (k) abc αβ (11) Power calculation (10) Current i (k) i (k+1) 1abc 1αβ αβ predictive (8) abc 7 i (k) 1αβ 7 7 Q (k+1) P (k+1) a1 Rectifier Cost function P side b1 + dcref u + dcref minimization PI − + (12) ref c1 load ref Q =0 ref dc Load power calculation (9) dc a2 i (k) 2α βref Inverter Cost function b2 side i (k+1) Minimization 2αβ c2 (14) αβ Current predictive (13) abc i (k) i (k) 2abc 2αβ αβ abc e (k) e (k) abc αβ Figure 3: Control block diagram of the traditional SOP model predictive control. Since the sum of squares of errors has been selected as where V and i are the reference voltage vectors of 2αβref 2Nαβref the cost function in (15) and (16), the cost function (17) is the VSC and the reference current, respectively. written as follows: After the reference voltage is obtained, the selection of the sector depends on the phase angle θ of the reference 2 2 2 ref (18) G � e d + e d + e d . 1 2 0 1 2 0 voltage. -e phase angle of the reference voltage in the stationary αβ reference frame can be calculated as follows: Taking (18) as the new cost function, the voltage vector time allocation obtained by minimizing the weighted root θ � arctanV /V . (20) ref 2Nβref 2Nαref mean square error is the optimal vector that makes the predicted value of the system closest to the reference value. -e sectors and vectors selected according to the phase angle θ are shown in Table 1. By judging the sectors, the ref two effective vectors and zero vectors selected on the inverter 4.2. Sector Judgment and Vector Selection. In order to im- side are determined, which effectively reduces the amount of prove the control accuracy and fix the switching frequency, calculation. before calculating the duty cycle, it is necessary to judge the On the rectifier side, power predictive control is adopted. reference vector sectors on both sides of the SOP to select the In order to achieve power tracking, the variables that affect port voltage vector. the value of the cost function (15) are analyzed, and the On the inverter side, current predictive control is derivative of (5) can be obtained as adopted. From the perspective of tracking the reference dV di dP dV di β 1β current, in this study, the inverter side adopts the principle ⎧ ⎪ 1 α 1α � i + V + i + V 1α α 1β β ⎪ dt dt dt dt dt of deadbeat control, which can be derived as follows: . (21) V � i (k) − i (k) + R i (k) + e (k), ⎪ 2Nαβref 2αβref 2αβ l 2αβ αβ dV di ⎪ dQ di dV β 1β 1 1α α � i + V − i − V 1α β 1β α dt dt dt dt dt (19) 6 Mathematical Problems in Engineering Table 1: Selection relationship among sectors, vectors, and phase dV ⎧ ⎪ � −ωV angles. ⎪ dt Phase angle Sector Vector selection . (22) ∘ ∘ ⎪ θ ∈ [0 , 60 ) I V , V , V ⎪ dV ref 1 2 7 ⎪ β ∘ ∘ � ωV θ ∈ [60 , 120 ) II V , V , V α ref 2 3 0 dt ∘ ∘ θ ∈ [120 , 180 ) III V , V , V ref 3 4 7 ∘ ∘ θ ∈ [180 , 240 ) IV V , V , V Substituting (22) and (7) into (21), (21) can be rewritten ref 4 5 0 ∘ ∘ θ ∈ [240 , 300 ) V V , V , V as ref 5 6 7 ∘ ∘ θ ∈ [300 , 360 ) VI V , V , V ref 6 1 0 For a three-phase balanced grid voltage, the differential of the grid voltage can be expressed as follows: V − V − R i ⎧ ⎪ dP V − V − R i β 1Nβ s 1β ⎪ 1 α 1Nα s 1α � V + ωi + V − ωi α 1β β 1α dt L L s s . (23) V − V − R i dQ V − V − R i ⎪ β 1Nβ s 1β 1 α 1Nα s 1α ⎪ � V + ωi − V − ωi β 1β α 1α dt L L s s Using the forward Euler method to discretize (23), the where f and f are the derivatives of active power P and p q 1 predictive control of the power in the next sampling period reactive power Q , respectively. can be obtained as For power reference values P and Q , the corre- 1ref 1ref sponding voltage vector V is an unknown quantity, 1Nref P (k + 1) � P (k) + f (k)T ⎧ ⎨ 1 1 p s which can be obtained from (24) as follows: , (24) Q (k + 1) � Q (k) + f (k)T 1 1 q s V − V − R i ⎧ ⎪ V − V − R i β 1Nβref s 1β ⎪ α 1Nαref s 1α P � P (k) + T V + ωi + V − ωi ref 1 s α 1β β 1α L L s s . (25) V − V − R i V − V − R i ⎪ β 1Nβref s 1β α 1Nαref s 1α ⎪ Q � Q (k) + T V + ωi − V − ωi ref 1 s β 1β α 1α L L s s After substituting (25) and (24) into the cost function effective voltage vectors that minimize the cost function f (15), (15) can be rewritten as must be in the same sector as the reference voltage vector V , ensuring that the two effective voltage vector 1Nref 2 2 f � P − P (k + 1) + Q − Q (k + 1) , ref 1 ref 1 combinations with the smallest cost function must be within the six voltage vector combinations listed in Section T 2 4.1. s 2 2 2 � V + V