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- Hindawi Publishing Corporation
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- Copyright © 2022 Iman Sayedi et al. This 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.
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- 10.1155/2022/5423532
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Abstract
Hindawi International Transactions on Electrical Energy Systems Volume 2022, Article ID 5423532, 16 pages https://doi.org/10.1155/2022/5423532 Research Article Optimal Load Distribution in DG Sources Using Model Predictive Control and the State Feedback Controller for Switching Control 1 2 1 Iman Sayedi, Mohammad H. Fatehi , and Mohsen Simab Department of Electrical Engineering, Marvdasht Branch, Islamic Azad University, Marvdasht, Iran Department of Electrical Engineering, Kazerun Branch, Islamic Azad University, Kazerun, Iran Correspondence should be addressed to Mohammad H. Fatehi; mh_fatehi@kau.ac.ir Received 20 December 2021; Revised 9 February 2022; Accepted 22 February 2022; Published 18 March 2022 Academic Editor: Ci Wei Gao Copyright © 2022 Iman Sayedi 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. ,e distributed energy management of interconnected microgrids, which is based on model predictive control (MPC), relies on the cooperation of all the agents (i.e., microgrids). Model predictive control or MPC is widely used in industrial applications as an effective tool for dealing with multivariable limited control problems. MPC uses an explicit system model to predict the future horizon of the system and its outputs. ,is predictability allows calculating the optimal order of inputs to minimize output errors over a limited horizon, which is effected by the limitations of the system. ,is study presents a distributed economic model predictive control method using the new state feedback controller to control the switching of interface converters and compensate for the unbalanced and nonlinear loads. In this model, the islanding mode and the reconnection of the grid are considered to improve the transient behavior of the system to achieve steady-state power distribution. It has been proposed that it could obtain better results in predictive control, utilizing similarity transform in the state matrix and its modification. First, this model is simulated on distributed generation sources with power-sharing and local loads using the state feedback controller in MATLAB Simpower. ,en, the performance of the proposed method is evaluated, confirming that it is more reliable than the FS-MPC and DSVM-MPC methods. main reason behind the wide range of different dynamic 1. Introduction operations probably relates to various factors, including the A microgrid is a small-scale power grid designed for sup- existence of zeroes outside the stable region, unstable poles, plying electricity to end users. Under certain conditions, and long delays with indefinite and variable times. microgrids have the ability to inject their excess energy into Furthermore, it can overcome the uncertainties of the the main grid [1]. ,e most important challenge when using plant parameters because of load demand fluctuations and microgrids involves maintaining the system’s security and the errors of the implementation. ,e new method has been stability. Most renewable energy-based distributed genera- built based on new simple frequency domain conditions and tion (DG) plants need voltage source inverters (VSIs) to the whale optimization algorithm (WOA). ,is method is connect to the microgrid [2]. Nowadays, choosing the ap- utilized to design a robust proportional-integral-derivative propriate method for controlling industrial processes is (PID) controller based on the WOA in order to enhance the highly important. Moreover, control methods in the in- damping characteristics of the wind energy conversion dustry must have several characteristics, such as ease of use system [4]. New two methods of artificial intelligence (AI) by the operator, simple configuration, and cost-effectiveness techniques are used to design the model predictive con- [3]. Although using proportional-integral-derivative (PID) trollers (MPCs) with superconducting magnetic energy controllers has become common in the industry, it should be storage (SMES) and capacitive energy storage (CES) for load noted that industrial processes involve a wide range of frequency control (LFC) [5]. ,e model predictive control different operations, limiting the use of such controllers. ,e (MPC) algorithm is a method for dealing with such complex 2 International Transactions on Electrical Energy Systems presents a number of important points for determining the industrial processes. Implementing the predictive control method for electric power converters can be difficult due to optimal values of the weighting coefficients. In addition, [10] introduces a predictive control method without using the large computational load required to instantly solve optimization equations. To mitigate this problem, a number weighting coefficients in the induction motor drive. On the of practical solutions have been considered, such as the out- other hand, [11] shows that using discrete space vector of-line calculation of optimization equations and solving modulation in the FS-MPC control method enables these equations by evaluating finite state switching. ,e employing virtual switching states in the control algorithm latter is known as the finite state model predictive control in addition to the real switching states. ,is new technique is (FS-MPC) since it works using a finite set of possible states called the discrete space vector modulation-model predictive control (DSVM-MPC), using which the required sampling for the electric power converter switching. So far, the FS- MPC control method has been used in various applications, frequency is reduced, while the switching frequency of the converter is stabilized [12]. In addition to the advantages of e.g., as rectifier, inverter, motor control, and uninterruptible power supply (UPS). Based on the results of the relevant the FS-MPC method, this control method also provides other benefits, including fixed switching frequency and low studies, this control method has high performance ability for the optimal operation of the whole system, while there is no sampling frequency [13]. However, due to the use of the need to fine-tune the controller parameters. Despite these same algorithm and its discrete nature, it includes a limited capabilities, this method has two major drawbacks that limit and discrete number of converter vector space points. its use in industrial systems. ,e first drawback is that to ,erefore, to account for a wide and variable range of op- achieve high performance, and a high sampling frequency is erating points for the converter, more points are needed, required, leading to costly hardware and the need for high which may lead to slow dynamic performance, low accuracy, and distortion in the output of the converter. ,e economic computation power. Moreover, in this method, the switching frequency is variable, which increases the values of costs for the MPC and the heuristic approach in the con- sidered price scenarios are reported in the economic costs of the converter’s output filter and the volume, weight, and total cost of the converter. When FS-MPC control designs prosumers equipped with production units, energy storage systems, and electric vehicles. To this purpose, the predictive are implemented in laboratory experiments, a large volume of calculations needs to be performed during each sampling control manages the available energy resources by exploiting period, which causes a significant delay in the activation of future information about energy prices, absorption and the actuator signal. ,erefore, if the delay caused by the production power profiles, and electric vehicle (EV) usage, measurement, calculations, and the operation of the actuator such as times of departure and arrival and predicted energy is not considered in the design of the controller, it can cause consumption [14]. In the model predictive control and state poor controller performance. In this regard, [6] describes the feedback controller, the chosen states of the system are compared with their reference quantities to generate the reason for this delay and the method for compensating for it, which creates an interval between the moment the current is converter switching. With the method proposed, in addition to the fact that we have maintained system dynamics, this measured and the moment the new switching state is ap- plied. During this interval, the previous switching state is similarity transform should be used to evaluate all systems. present at the converter’s output, which causes a difference Hence, the following state transform matrix is considered for between the load current and the reference current, in- this system, and better results in terms of overshoot, rise creasing the current’s ripple. To solve this problem, [6] uses time, settling time, and steady state have been obtained with the prediction of the next two samples instead of the pre- this method, which are shown in the results of this study. diction of the next sample. Moreover, [7] proposes a new ,is study proposes a state feedback control method and predictive control method, called fast predictive control. In discusses the optimal design using the iteration method to this method, the volume of the calculations will be signif- define all possible switching states along the predictor’s horizon for the controller, thus mitigating the drawbacks of icantly reduced, and it can be used in multilevel converters that have a large number of control vectors. Another dis- the previous methods. ,erefore, finding an economically applicable model with the best state-space model is very advantage of the FS-MPC method is the variable switching frequency [8]. ,e variable switching frequency creates a important. A comprehensive review of DC microgrids can wide range of harmonics at the output of the converter, be found in [15, 16]. which can cause resonance and make filter design more difficult. In [9], the average switching frequency is kept 2. Study of a Microgrid Containing a DG Source constant by adding corrective terms to the cost function. In addition, [9] modifies the current predictive control scheme Figure 1 depicts the single-line diagram of a power system in such a way that the switching frequency can become containing two DG sources. ,e microgrid is connected to somewhat independent of the sampling frequency. Con- the main grid at the PCC point. Two sources, i.e., DG1 and figuring and selecting the weighting coefficients in the FS- DG2, are directly connected to the microgrid using the CB-3 MPC method will be a major challenge, which has a sig- and CB-4 circuit breakers, respectively. Both DGs have local nificant impact on the system’s performance. Configuring loads, which may be nonlinear and unbalanced. In addition, these coefficients is more time-consuming than adjusting the the microgrid may have a joint load as well, which is as- parameters of the PI controller in classical current control sumed to be balanced and located at a long distance from the and adjusting the hysteresis bandwidth. Furthermore, [10] DGs. One of the tasks of the DGs is to correct the imbalance International Transactions on Electrical Energy Systems 3 Rs CB–5 Ls Utility Load Vs g P Q G G CB-1 P Q R G2 G2 P Q R D1 G1 G1 v D2 PCC g1 g2 CB-2 CB-3 CB-4 P Q LC LC P Q LC P Q L1 L1 L2 L2 p1 v p2 PCC1 i Balanced L1 P Q 1 1 P Q Common 2 2 2 Load Nonlinear Nonlinear VSC VSC And And Compensator Unbalanced Compensator Unbalanced Local Load 2 Local Load 1 DG-1 DG-2 dc1 2dc Figure 1: ,e single-line diagram for the microgrid and the electricity grid, including two sources of distributed generation [31]. and the nonlinearity of the local load. Several control block of this controller is shown in Figure 2. Here, in order schemes have been introduced in order to ensure the proper to achieve the desired solutions, the value of h is set to operation of the microgrid in the grid connection mode or 0.001. It should be noted that i1, i , and v signals are cf cf the islanding mode [17]. Moreover, several control schemes measurable. have been introduced in order to ensure the proper oper- In [22], the authors present an adaptive optimization ation of the microgrid in the grid connection mode or the method for NMPC, where an extended Kalman filter (EKF) stand-alone mode [18]. A comprehensive review of dc is used for estimating the state variables. In [2], a system is microgrids can be found in [19–21]. investigated where the superheated steam temperature (SST) In the DG grid connection mode, the microgrid shares a is the main variable that must be controlled. To do so, several percentage of its local load with the main power grid, while categories of cascading PIDs have been used to control this the joint load is fully supplied by the main grid. During the variable. In [3], to adapt the model, neural networks are process of islanding, each DG source supplies its own local presented for controlling the model predictive control. In load, while the joint load is shared between the DG sources. [23], a fuzzy method based on Lyapunov fragment functions ,e mix powers drawn by the local loads include P + is used in the model predictive control. In [24], in order to L1 jQ and P + jQ . Moreover, the joint load draws the i make the system smart, generalized predictive control L1 L2 L2 LC current from the grid and it consumes the mix power (GPC) and a neural network model are used. In this method, P + jQ . a nonlinear neural network is used to extract the linear LC LC model of the system. In [25], horizon optimization is utilized Local loads at points PCC1 and PCC2 are connected to the DG sources with voltages of v and v , respectively. for predictive control using a genetic algorithm. In [26], the p1 p2 ,e real and reactive powers supplied by the DGs are prediction of the next two samples is used instead of pre- denoted by P1, Q1, P2, and Q2, respectively. It is assumed dicting the next sample. Moreover, [27] proposes a new that the microgrid is mainly resistive, and at the distribution predictive control method, called fast predictive control. In level, the impedances of the lines are denoted by R and this method, the volume of calculations will be significantly D1 R . reduced, and it can be used in multilevel converters that have D2 ,e main power supply is shown by V , while the feeder a large number of control vectors. In addition, [28] modifies inductance resistors are denoted by RS and LS, respectively. the current predictive control scheme in such a way that the ,e main power grid injects PG and QG powers into the switching frequency can become somewhat independent microgrid, and PS-PG and QS-QG powers supply the main from the sampling frequency. Increasing the sampling fre- quency improves the performance of FS-MPC, while re- load of the power grid.CB1 can disconnect the microgrid from the main power grid. ducing the ripple of the output current. However, increasing In the next sections, we will examine the performance the sampling frequency also increases the switching fre- of the system and the compensator in both grid con- quency, which ultimately leads to increased losses [29]. ,is nection and islanding modes. In all the modes mentioned study presents a predictive control method using the state in this section for the grid, a state feedback controller is feedback controller to control the switching interface con- used with a quadratic linear regulator to control the grid verters and compensate for the unbalanced and nonlinear and correct the unbalanced loads. ,e general diagram loads. 4 International Transactions on Electrical Energy Systems i (K) ref Minimization S (k) 3-phases of Cost I (K+1) Predictive Load Function Model Figure 2: ,e diagram of the finite state model predictive control for three-phase inverters [8]. When FS-MPC control schemes are implemented in 3. The Model Predictive Control (MPC) Method laboratory experiments, a large volume of calculations is ,e MPC method is an optimization problem where the cost performed in each sampling period, which causes a sig- function is minimized. Using the system model and the nificant delay in the activation of the actuator signal. values of the variables until time K, the state values are ,erefore, if the delay caused by the measurement, the predicted until time horizon K + N. Moreover, through the calculations, and the activation of the actuator is not con- optimization of the cost function, the first component of the sidered in the controller design, it can cause poor controller command sequence is applied at moment K+1. ,ese steps performance. In this regard, [26] describes the reason for are repeated for the next time steps. ,e cost function in- this delay and how to compensate for it. cludes the control objectives of the system, and its common terms include variables that need to follow a reference value [11]. 3.2. $e DSVM-MPC Controller. ,e main idea behind the According to equation (1), controlling these variables DSVM-MPC control method involves using other points in will be a function of the error between the predicted value the vector space. In the SVM modulation method, in ad- and its reference value. dition to the eight main switching states for the three-phase As can be seen in equation (1), this function can be the converter, other states can also be applied in the form of a size, the square, or the integral over a sampling time interval. linear combination of these base states. ,ese new states are called virtual states. However, because the cost function is k+1 calculated per each vector, it will only be possible to use a ∗ p (1) g � x (t) − x (t)dt. limited number of points in the vector space. In Figure 3, the real points (circles) and virtual points (squares) are shown for the switching of a three-phase converter [21] (Figure 3). 3.1. $e Finite State Model Predictive Control (FS-MPC). ,is control method also offers benefits such as a fixed ,is method uses the discrete nature of electronic power switching frequency and a low sampling frequency. How- converters in such a way that all converter voltage vectors are ever, due to using a similar algorithm and its discrete nature, tested in the cost function, and the vector that minimizes the it covers a limited and discrete number of points in the cost function is selected [20]. converter’s vector space [19]. Figure 2 shows the finite state model predictive control (FS-MPC). ,e algorithm for solving this method includes the following steps [26]: 3.3. $e Proposed DE-MPC Controller. ,e microgrid shown in Figure 1 is considered. It can be observed that when A : Load current measurement; entering the is landing mode and reconnecting the grid, the B : Prediction of the load current in the next samples for system’s response is not highly satisfactory since it takes a all possible switching states according to the following long time for the power distribution to reach a steady state. equation: To solve this problem, we first used a PSO algorithm and PID r × T T L S S control, and the issue was largely mitigated. In this section, (2) i (k + 1) � 1 − i(k) + v (k) − vs . p S i L L we try to design a distributed economic model predictive controller (DEMPC) to improve the transient state of the ,is equation is obtained by discretizing the voltage system. equation. ,e model predictive controller method is designed and C: ,e cost function is evaluated for each prediction. implemented based on the following three steps: In this method, the cost function is expressed as the (1) A model is used to predict the behavior of the control error between the reference current and the predicted variables for the next time step. current for each of the possible switching states ((3)). (2) A cost function is determined, including control objectives and the expected behavior of the g[n] � i − i S + i − i S . (3) αref αp i βref βp i system. D: ,e switching state that minimizes the cost function (3) ,e appropriate command is extracted by mini- is selected. mizing the value of the cost function. International Transactions on Electrical Energy Systems 5 V 1 Converter control region V (1,0,0) Converter real state V (1,0.5,0) Converter virtual state V (0.75,0,0) V (0.4,0,0) Figure 3: ,e vector space for switching of a power converter [21]. moment K + 2. ,ese types of calculations are called re- ,e model used for the prediction is a discrete-time model that can be represented in the form of state equations ceding horizon strategies. Figure 4 shows the performance of according to equations (4) and (5): the DE-MPC method. Using the system model and the values of the variables until time K, the state values until time x(k + 1) � Ax(k) + Bu(k), (4) horizon K + N are predicted. In addition, by optimizing the cost function, the first component of the command sequence y(k) � Cx(k) + Du(k). (5) is applied at moment K + 1. ,ese steps are repeated for the next time steps. In these equations, the vector x(k) denotes the cur- In this method, the number of switching states is defined rent values of the state variables, x(k + 1) is the future to control the microgrid, and based on this number and the prediction value for the state variables, u(k) denotes the proposed cost function, the switching operation is per- current values of the input variables, and y(k) is the formed. Furthermore, Figure 5 shows the state feedback vector of the current output values. In the next step, the controller diagram. Moreover, the proposed controller cost function must be determined. According to equation equations are explained in Section 3.4. (6), in this function, the reference values, values of the Among constraints that exist in this system, we could future states, and the future control commands are mention constraints in switching control S, which is con- considered. sidered in the equation below. ,e control signal, which is J � f(x(k), u(k), ....., u(k + N)). (6) defined by predictive control, is continuous. ,e variable S is denoted that to u signal. As it is mentioned, this signal takes It is noted that the constraints of the state equations are one of values +1 or −1. ,e principle of switching is pre- considered as follows. sented below: S.t: u ≤ u(k + i) ≤ u ; i � 0,. . ., N -1. min max u ∆u ≤ ∆u(k + i) ≤ ∆u ; i � 0,. . ., N -1. If u (k)> h then u � +1, min max u c (7) Y (k + i) ≤ y(k + i) ≤ y (k + i); i � 1,. . ., N min max p else If u (k)< − h then u � −1, ,e quantities of the prediction horizon are usually considered to be twofold the control horizon. ,at is in the in which h is a very small number. Choosing the h value simulation, N � 10 and N � 5 should be considered in this p u determines switching frequency, in a way that reference regard. values are tracked. After several sequences, when we ,e DE-MPC control method is an optimization achieved a more accurate model, we could choose this h problem in which the cost function is minimized. In this value even smaller. optimization, the system model and control objectives are ,e single-phase equivalent circuit of converted is shown considered for K+1 to K + N time steps. ,e result of this in Figure 6. Using this figure and in the presence of the LCL optimization is N consecutive commands. ,e first com- filter, the state vector in conventional methods is considered ponent of this command sequence is applied at moment as below: K + 1. Similarly, during this time, using the new measure- ment values, the optimization is performed for the next i i v x � . (8) f 1 cf moment, and the appropriate command is selected for the 6 International Transactions on Electrical Energy Systems Switching U (K-1) Sequence Z (t) є R U (t) є R S (K ) opt ref PI MPC Microgrid Algorithm - x (t) єR V V i i cf c cf l Figure 4: Principles of the operation of the predictive method. x = Λx + Γ uc + Γ v 1 2 PCC PCC i P ,Q 1 1 1 1ref v cf cf cf cfref K R i f f cf f C v + v cfref cf _ _ u,v dc1 Figure 5: ,e state feedback controller diagram block. P ,Q 1 1 L L f 1 cf C v PCC uV f cf dc Figure 6: Single-phase equivalent circuit VSC (LCL filter) [32]. Figure 6 shows the block diagram of the implemented state variables, it will be obtained the following descriptions control method. Using Figure 6 and considering selected for system state space: −R −1 ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ V ⎢ L L ⎥ ⎢ ⎥ ⎢ f f ⎥ dc . ⎢ ⎥ 0 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ i ⎢ ⎥ i ⎢ ⎥ ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ f ⎢ ⎥ ⎢ L ⎥ ⎢ ⎥ ⎢ ⎥ f ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ f ⎥ ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ 1 ⎥ ⎢ . ⎥ ⎢ 1 ⎥ ⎥ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ • ⎢ ⎥ ⎢ ⎥⎢ ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ ⎥⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ − ⎥ ⎢ ⎥ ⎢ ⎥⎢ i ⎥ ⎢ ⎥ ⎢ ⎥ x � ⎢ ⎥ � ⎢ 0 0 ⎥⎢ ⎥ + ⎢ ⎥u + ⎢ ⎥v . (9) ⎢ ⎥ ⎢ ⎥⎢ 1 ⎥ ⎢ ⎥ ⎢ ⎥ 1 ⎢ i ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ PCC ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 1 ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ L ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ L ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ 1 ⎥ ⎢ ⎥ 1 ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ . ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ v ⎣ ⎦ ⎢ ⎥ ⎢ ⎥ v ⎢ ⎥ cf ⎢ ⎥ ⎢ ⎥ cf ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0 ⎢ ⎥ ⎢ 1 −1 ⎥ ⎢ ⎥ 0 ⎣ ⎦ C C f f International Transactions on Electrical Energy Systems 7 −1 Hence, the following state vectors are given that by x _ � C AC x + C Bu + C Cv � Λx + Γ u + Γ v . p p c p pcc 1 c 2 PCC equation (13). (13) dc ,e control rule is defined as follows: ⎢ ⎤ ⎥ ⎡ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ L ⎥ ⎢ ⎥ ⎢ f ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ u (k) � −Kx (k) − x (k). (14) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ c 1 ref ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ B � ⎢ ⎥, ⎢ ⎥ 1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ In the above equation, K is a gain matrix, and x is the ⎢ ⎥ ⎢ ⎥ ref ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ reference vector. ,e gain matrix can be obtained using the ⎣ ⎦ DE-MPC control method and the proposed switching method. ,e control rule of equation (13) includes switching ⎢ ⎥ ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ control, which is discussed in detail below. ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ Assuming complete control over u, a quadratic optimal ⎢ ⎥ ⎢ 1 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ − ⎥, B � ⎢ ⎥ ⎢ ⎥ linear steady state can be designed for this problem. As noted 2 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ L ⎥ ⎢ ⎥ ⎢ ⎥ 1 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ (10) earlier, the control rule is as follows: ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ 0 u � −Kx (k) − x (k). (15) 1 ref −R ,e control law discussed so far is for the system in −1 ⎢ 0 ⎥ ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ which the DGs have an output inductor. Alternatively, when ⎢ ⎥ ⎢ ⎥ ⎢ L L ⎢ ⎥ ⎢ f f ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ the DGs do not have an output inductance, the inductance ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ L1 is removed and the output filter is a simple LC filter. ,e ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ system state is then modified follows: ⎢ ⎥ ⎢ ⎥ A � ⎢ 0 0 ⎥. ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ L ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 1 ⎥ T ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ x � i · v . (16) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 2 cf cf ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ∗ ⎢ 1 −1 ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ With respect to Figure 6, the reference for v v and given cf ∗ ∗ C C V and Ø , the current phasor through the capacitor C is f f f given by equation (14). In the following, it has been proposed that it could be ∗ ∗ I � wC V∗∠ δ + 90 . (17) obtained better results in predictive control, utilizing sim- cf f 1 ilarity transform in the state matrix and its modification. ∗ ,e reference i is obtained from the instantaneous cf With this method, in addition to the fact that it has ∗ value of I . As we have seen, it is much easier to predict the cf maintained system dynamics, we can use this similarity reference values from the capacitor voltage (v ) and then to cf transform to evaluate all systems. predict its current (i ) than to predict the current i and δ is c f angle of V , respectively. It is evident in equation (11) that reference for all elements of the states is required for state 3.4. $e Proposed Method. Figure 6 depicts a block diagram feedback. Since V and δ are obtained from the droop of the implemented control method. In the model predictive equation, the references for the capacitor voltage and current control and state feedback controller, the chosen states of the are given by v � V sin(wt + δ) and i � Vѡ C sin(ѡt + cref f cref system are compared with their reference quantities to δ +90). generate the converter switching. It is easy to generate a In the above equation, x (k) denotes reference vectors. ref reference for the output voltage v and current i from cf c ,e DE-MPC controller minimizes the following perfor- power flow conditions. However, the same cannot be said mance index: about the reference for the i . To facilitate this, we define the new state vectors as follows: j � x − x Qx − x + ρu Rudt. (18) ref ref i i v x � . (11) cf l cf ,e index given in equation (18) must be minimized to obtain the optimal control rule, u, by solving the steady-state ,erefore, we will have the following conversion matrix: equation. In equation (18), the weighting matrix, Q, is a definite or 1 −1 0 semidefinite positive matrix, which is real and symmetric, ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ x � ⎢ 0 1 0 ⎥x � C x . (12) ⎢ ⎥ while the penalty control matrix, R, is a definite positive ⎢ ⎥ 1 p 1 ⎢ ⎥ ⎣ ⎦ matrix, which is real and symmetric. Moreover, ρ is a 0 0 1 constant positive number. According to Bryson’s rule, the initial choice for R and Q matrices is possible in the form of ,e transformed state equations are obtained by com- diagonal matrices as follows: bining equations (11) and (12) as follows: 8 International Transactions on Electrical Energy Systems di i [k + 1] − i [k] α,β α,β α,β Q � , i ∈ {1, 2, . . . , l}, (24) ii ≈ . maximum acceptable value for z dt T i s In addition, by substituting equation (24) into (21), the R � , j ∈ 1, 2, . . . , m . { } jj 2 future value of the load current vector is obtained as follows: maximum acceptable value for u T L (19) i[k + 1] � v [k] − i [k]R − . (25) α,β α,β α,β L T In the above equation, l is the number of control outputs Equation (25) is used in the controller block for pre- and m is the number of inputs. Moreover, z is called the dicting the future current values based on the measured controlled output, which is related to the signal we want to voltage vector. To select the voltage vector and control the minimize in the shortest possible time. In this method, the current, the predicted current is evaluated using the fol- output voltage is indirectly regulated by controlling the lowing cost function: inductor’s current. ,e reason behind selecting the current ∗ ∗ instead of the voltage involves the presence of more ripples g[k + 1] � i [k + 1] − i [k + 1] + i [k + 1] − i [k + 1] . α α β β in the current, which increases the predictions in the pro- (26) posed controller. In this way, more accurate data about the future can be predicted, increasing the robustness of the In this equation, i∗ α, β[k+1] is the estimate of the proposed controller. reference current vector in the next horizon. For grids with a To do so, an optimal objective function is specified that sufficiently small sampling time, it can be assumed that this determines the order of switching. ,e order of switching current is equal to its previous value, i.e., i∗ α, β[k+1] i∗ α, along the prediction horizon will be according to equation β[k]. However, for large sampling times, the future value of (20): the reference current needs to be extrapolated. To make the values of the three-phase current vectors independent from U(k) � [u(k)u(k + 1) . . . u(k + N − 1)] . (20) each other, the voltage vectors can be first obtained from the values of the switching signals and the voltage of the DC link In this equation, U is the optimal switching state whose capacitors. first element, i.e., u(k), which is applied to the circuit, while its other elements are applied in subsequent time steps. v � v S + v S + v S , aN c1 1a c2 2a c3 3a In this method, one of the control goals of the objective v � v S + v S + v S , (27) bN c1 1b c2 2b c3 3b function involves reducing the difference between the v � v S + v S + v S . current and its reference value according to equation (21): cN c1 1c c2 2c c3 3c i (k) � i (k) − i (K + 1) . (21) ,e shared-mode voltage is obtained as follows: Lerr Lref L v + v + v aN bN cN v � v � . (28) In this way, the cost function can be expressed as NO cm equation (22): ,e cost function in the proposed algorithm is to k+N−1 minimize the voltage drop. j(k) � i (j | k). (22) Lerr ,e objective function that should be minimized is j�k defined as follows: ,e optimal switching state is achieved by minimizing sim the cost function as follows: F � t. P − P + Q − Q dt, i � L , G , 1, L , G , 2. i iss i iss 1 1 2 2 U (k) � arg(min J(k)). (23) (29) ,e reason for using these arguments in the proposed In the above equation, P , PL , PG , P , Pl , and 1ss 1ss 1ss 2ss 2ss equation is that it can better minimize the cost function. PG are the final (stable) values obtained for the active 2ss powers in Table 1. Moreover, Q , QL , QG , Q , QL , Optimal switching in equation (23) is performed using the 1ss 1ss 1ss 2ss 2ss iteration method. All possible switching states are defined and QG are the final (stable) values of the reactive powers. 2ss In fact, this objective function is defined in such a way for the controller along the N horizon, which is shown by U(k). that the values of the powers have the least deviation from the final values or the values of their steady states, while ,erefore, there will be 2 switching states. Matrices (20) and (21) are calculated for all the switching states, while the reaching this steady state in the shortest possible time. ,e model-based predictive control process algorithm for con- cost function equation (22) is also calculated. Finally, the switching state with the lowest j value is trolling the current in the selected system is as follows: selected and applied to the switch. (i) Applying V[k]; ,e existing differential approximation, i.e., di/dt, can be (ii) V[k-1] � v[k]; considered as follows as a simple step-forward Euler equation: (iii) Measuring currents ic, ia, and ib; International Transactions on Electrical Energy Systems 9 Table 1: ,e numerical results of the joint load sharing by the distributed generation source by selecting optimal parameters using the LQR controller (employing the PSO method). Initial Intermediate amount, MW(in the islanding Final amount Recovery time (second) Active power Amount (MW) mode) (MW) (After reconnecting the grid) P 1.1 1 1.1 0.01 L1 P 0.88 −0.15 0.88 0.02 G1 P 0.22 1.15 0.22 0.02 Figure 7 P 1.1 0.97 1.1 0.01 L2 P 0.55 −0.3 0.55 0.02 G2 P 0.55 1.27 0.55 0.02 Voltage drop Figure 8 In PCC2% in PCC1% 2.7% 3.1% Figure 14 shows the convergence diagram of the ob- (iv) Calculating the current vector; jective function. In this case, in order to compare the Predicting the next horizon’s current i[k + 1]; proposed method with DSVM-MPC, the convergence dia- Calculating the cost function gi; gram of DE-MPC is first simulated using 100 iterations. It Is gi< gopt? shows the convergence diagram of the objective function. Yes: g � g , I � I . According to the convergence diagram, it can be concluded opt j opt j that the diagram of the proposed controller cost function is No: Restart. lower than the DSVM method. In terms of economy, the better performance can be seen than the other controllers studied. ,e reasons for choosing PSO include simplicity in 4. The Simulation Results implementation and a successful track record in these works. It should be noted that the effectiveness of an algorithm ,is section presents the optimal distribution of the joint load using the distributed generation source, which is separately largely depends on the type and structure of the problem, but simulated using the LQR and PID methods. ,is simulation the fact that PSO has been successfully used in a wide range was performed in MATLAB Simpower environment in the of scientific fields shows its good potential for solving op- presence of two distributed generation sources. In addition, by timization problems. Hence, one of the main weaknesses of optimally adjusting the parameters of each controller using the previous methods is convergence to local optimums and the PSO algorithm, the reliability of the microgrid substation has lack of a robust global search. As the formulations of classic increased. In the following, the proposed predictive control PSO indicate, the best approach to increase the efficiency of method, i.e., DE-MPC, is compared with PID controllers using this algorithm is to adjust the coefficients so as to improve its local and global searches. In the improved algorithm, the PSO. Moreover, FS-MPC and DSVM-MPC are simulated using a single distributed generation source, and the results are inertia weight (w � 0.4, w � 0.9) and the acceleration min max compared with each other. coefficients and population number (c1 � 0.2 and c2 � 2.3, Figure 7 shows the simulated model of the system in population � 50) are obtained by the parameters mentioned MATLAB Simpower. Suppose that at time t � 0.5s, the im- [16, 30]. pedance of the joint load is reduced by half of the initial value. ,e simulations results demonstrated the good per- In this section, the active power and voltage DGs are formance of best so far quantities’ optimization with the obtained in Figures 8(a) and 8(b). In Figure 9 as can be seen proposed algorithm in solving the model predictive control from these figures and Table 1, in this case, after recon- cost function. ,e notable features of the developed al- necting the microgrid to the main grid, the power values gorithm include fast convergence and the progress of rapidly reach their steady state; according to Figure 9, the search based on the rotational motion of the system during fluctuation of power in PCC1 and PCC2 point voltage is optimization. reduced. In addition, to compare the performance of two In Figure 15, the active power distribution is calcu- methods, in Table 2, the numerical results of the PID lated for one DG in the presence of the proposed con- controller using the PSO algorithm for two DGs are ob- troller. Furthermore, this method can be a suitable tained, and in Figures 10 and 11 it that the active power alternative for the mentioned controllers. Figure 15 dedication in DG-1 and DG-2. shows the actual power allocation, and Figure 16 It can be illustrated the active power distribution with the shows the PCCI voltage point with the DE-MPC con- PID controller and three-phase voltage waveform at the PCC1 troller. Table 3 presents the numerical results obtained point by PSO algorithm for one DG in Figures 12 and 13 from these diagrams. respectively. Furthermore, Table 3 presents the PID controller According to Figures 15 and 16, the information pre- simulation results for one DG. In order to better compare the sented in Table 3, it can be observed that in the presence of proposed method (DEMPC) by PID controller using PSO the DE-MPC controller, the transient response of the system algorithm, the numerical results for one DG are evaluated. is significantly improved, and after connecting the grid, the 10 International Transactions on Electrical Energy Systems Feeder Impedance Rsc Ls Vsa Utility Load + CB-5 A a a Vsb B b C c c P Q G G Vsc CB-1 RD2 RD1 P ,Q P ,Q G1 G1 G2 G2 lg1a lg2a lg1b PCC lg2b lg1c lg2c CB-3 CB-4 P Q nonlinear load2 CB-2 L2 L2 P Q L1 L1 nonlinear load1 p1 v p2 P Q P Q 1 1 2 2 Commom Load Unbalance load2 Unbalance load1 VSC1 VSC2 Figure 7: ,e simulated model of the system in MATLABSimpower. Suppose that at time t � 0.5 (s), the impedance of the joint load is reduced by half of the initial value. 1.5 PL1 PL2 P2 0.5 0.5 P1 PG1 PG2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 time (s) time (s) (a) (b) Figure 8: Dedication of active power in (a) DG-1 and (b) DG-2. power distribution returns to its original state even in less Comparing the results in Table 3 and Table 4 with the time compared to PID and PSO. In addition, for non- results of the power allocation in the islanding mode using complex model predictive control to define optimization the proposed DE-MPC controller, it can be seen that when functions and solve them, the constrained linear functions using the DE-MPC controller, the transient response of the such as fminprog and nonlinear constrained functions such system is greatly improved, and after reconnecting the as fmincon can be used in the MATLAB software. Also, if the microgrid to the main power grid (in less than 1 second), the problem is unconstrained by linear functions, the fmin- power distribution rapidly reaches its steady state, and there search function is used. will be no power fluctuations. I1c Real Power (MW) c C b B I1b I1a A a A a C B b B b B Real Power (MW) C c C c A I2a a A I2b b B I2c c C International Transactions on Electrical Energy Systems 11 Voltage droop 3.1% 6 6 Voltage droop 2.7% -6 -6 0.35 0.4 0.45 0.35 0.4 0.45 time ( s) time (s) Vp1a Vp2a Vp1b Vp2b Vp1c Vp2c (a) (b) Figure 9: (a) PCC1 and (b) PCC2 point voltages (KV) (by selecting optimal values for R and Q parameters). Table 2: ,e numerical results of joint power allocation by the distributed generation sources in the presence of a PID controller (using the PSO method). Active Initial Final amount Recovery time (second) Intermediate amount, MW (in the islanding mode) power Amount (MW) (MW) (After reconnecting the grid) P 1.1 1 1.1 0.01 L1 P 0.88 −0.16 0.88 0.03 G1 P 0.22 1.16 0.22 0.03 Figure 9 P 1.1 0.98 1.1 0.01 L2 P 0.55 −0.3 0.55 0.02 G2 P 0.55 1.27 0.55 0.02 Voltage drop Figure 10 In PCC1%in PCC2% 2.7%3.1% 1.5 PL1 PL2 P2 0.5 0.5 P1 PG1 PG2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 time (s) time (s) (a) (b) Figure 10: Dedication of active power in DG-1 and DG-2. Real Power (MW) Voltage (KV) Real Power (MW) Voltage (KV) 12 International Transactions on Electrical Energy Systems Voltage droop 3.1% Voltage droop 2.7% 6 -6 -6 0.35 0.4 0.45 0.35 0.4 0.45 time ( s) time (s) Vp1a Vp2a Vp1b Vp2b Vp1c Vp2c (a) (b) Figure 11: PCC1 and PCC2 (KV) point voltages (including the PID controller). PL1 0.5 P1 PG1 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 time (s) Figure 12: DG-1 active power distribution with the PID controller. 5 Voltage droop 2.5% -5 0.35 0.4 0.45 Time (s) Figure 13: ,ree-phase voltage at the PCC1 point. Table 3: ,e numerical results of the islanding mode in the presence of a PID controller (using the PSO method) and a distributed generation source. Active Initial amount Intermediate amount (in the islanding Final amount Recovery time (second) after reconnecting power (MW) mode) (MW) (MW) the grid (CB-1) PL1 0.7 0.64 0.7 0.03 Figure 11 PG1 0.63 −0.14 0.63 0.03 P1 0.07 0.76 0.07 0.04 Voltage drop Voltage range, KV Intermediate amount (in the islanding Figure 12 (%) PPC1 Initial amount mode) 2.5% 5 4.89 Voltage (KV) Voltage (KV) Real Power (MW) Voltage (KV) International Transactions on Electrical Energy Systems 13 best ofar 0 1020304050607080 90 100 Iteration DEMPC DSVM-MPC Figure 14: ,e convergence diagrams of DE-MPC and DSVM-MPC. 0.8 P1 PL1 0.6 0.4 t=0.4 Islanding t=1 reconnection 0.2 PG1 -0.2 0 0.5 1 1.5 time (s) Figure 15: ,e active power distribution of DG-1 using the DE-MPC controller with state feedback. Voltage droop 2.1% -5 0.35 0.4 0.45 Time (s) Figure 16: ,e three-phase voltage at the PCCI point using the DE-MPC controller with state feedback. Voltage (KV) Real Power (MW) cost function 14 International Transactions on Electrical Energy Systems Table 4: ,e numerical results for the islanding mode in the presence of the proposed DE-MPC controller. Active Initial amount Intermediate amount (in the Final amount Recovery time (second) after power (MW) islanding mode) (MW) (MW) reconnecting the grid (CB-1) PL1 0.7 0.68 0.7 0.02 Figure 14 PG1 0.63 −0.12 0.63 0.02 P1 0.07 0.71 0.07 0.03 Voltage drop Voltage range, KV Initial Intermediate amount (in the islanding (%) PPC1 amount mode) Proposed DE-MPC 2.1% 5 4.89 Figure 15 FS-MPC 2.7% 5 4.85 DSVM-MPC 2.5% 5 4.87 OV PL1 curve Rise time Settling time PID 3% 0.08s 0.25s Proposed DE-MPC 0 0.05s 0.1 s the state feedback controller. It has changed the inductor 5. Conclusions and capacitor in Figure 6 section for 2%, its results are In this study, a distributed model predictive controller was evaluated, and it is noticed that this system has good ro- proposed. A comparison of the results obtained in Tables 3 bustness against potential changes. Hence, the following and 4 revealed that the proposed distributed economic state transform matrix is considered for this system, and model predictive controller significantly improves the better results in terms of overshoot, rise time, settling time, transient response of the system and the power quality of the and steady state have been obtained with this method. As a grid compared to the LQR, PID, FS-MPC, and DVSM-MPC result, the recovery time for reswitching from the island methods. It is noteworthy that the proposed DE-MPC mode to the connection mode of the distributed generation controller has a better performance than other controllers in sources is reduced. Some areas for future work are the state terms of balancing and stabilizing the microgrid when it is feedback control strategy scheme that can be modified to connected to the main grid. By changing the predictive share power in microgrid with inertial and noninertial DG. control rule and replacing the current with the voltage, more Improvement in supplementary droop control for enhanced data are provided than in the previous case. ,erefore, in this system damping under weak operating conditions and way, a large volume of data becomes available for the protection of back-to-back converters in case of a fault in proposed predictive horizon. In addition, this study pro- utility or microgrid faults can be investigated. poses a novel PCC voltage compensation method for islanded microgrids by improving the power-sharing con- Abbreviations trol schemes among the DGs to compensate for the PCC voltage deviation caused by the droop control and the state FS-MPC: Finite set model predictive control feedback controller. A smooth transfer between the DE-MPC: Distributed economic model predictive islanding mode and grid connection mode assure a stable control operation of the system. ,e model predictive control ef- DSVM-MPC: Discrete space vector modulation-model ficacy is checked in the case of the LCL filter. ,e application predictive control is mainly aimed at the rural area where the unbalanced load GPC: Generalized predictive control is common and wireless communication is always desirable VSI: Voltage source inverters due to large network size. However, load sharing can be DG: Distributed generation made more accurate by incorporating the line impedance PI: Proportional integral. values in the power reference calculation. By switching the control action of the DGs from state feedback control in grid Data Availability connection mode to voltage control in islanding modes, a seamless transfer is achieved. A step-by-step matrix trans- ,e data used to support the finding of this study are formation control method is proposed for a smooth tran- available from the corresponding author upon request. sition during islanding and resynchronization. It must be Meanwhile, readers can contact us via e-mail: mh_fatehi@ mentioned that the proposed distributed economic model kau.ac.ir. predictive controller has more successfully performed in creating balance and stability in the microgrid in the grid Ethical Approval connection mode. In addition, this study proposes a novel PCC voltage compensation method for islanded microgrids ,e author’s approval of the manuscript should not be by improving the power-sharing control schemes among the submitted to more than one journal for simultaneous DGs to compensate for the PCC voltage deviation caused by consideration. International Transactions on Electrical Energy Systems 15 [14] F. 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Journal
International Transactions on Electrical Energy Systems – Hindawi Publishing Corporation
Published: Mar 18, 2022