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Open AccessArticle

Optimal P-Q Control of Grid-Connected Inverters in a Microgrid Based on Adaptive Population Extremal Optimization

1
School of Computer, South China Normal University, Guangzhou 510631, China
2
National-Local Joint Engineering Laboratory of Digitalize Electrical Design Technology, Wenzhou University, Wenzhou 325035, China
3
College of Electrical and Information Engineering, Hunan University, Changsha 410082, China
4
State Key Laboratory of Power Systems and Department of Electrical Engineering, Tsinghua University, Beijing 100084, China
*
Author to whom correspondence should be addressed.
Energies 2018, 11(8), 2107; https://doi.org/10.3390/en11082107
Received: 24 July 2018 / Revised: 6 August 2018 / Accepted: 9 August 2018 / Published: 13 August 2018
(This article belongs to the Section Electrical Power and Energy System)

Abstract

The optimal P-Q control issue of the active and reactive power for a microgrid in the grid-connected mode has attracted increasing interests recently. In this paper, an optimal active and reactive power control is developed for a three-phase grid-connected inverter in a microgrid by using an adaptive population-based extremal optimization algorithm (APEO). Firstly, the optimal P-Q control issue of grid-connected inverters in a microgrid is formulated as a constrained optimization problem, where six parameters of three decoupled PI controllers are real-coded as the decision variables, and the integral time absolute error (ITAE) between the output and referenced active power and the ITAE between the output and referenced reactive power are weighted as the objective function. Then, an effective and efficient APEO algorithm with an adaptive mutation operation is proposed for solving this constrained optimization problem. The simulation and experiments for a 3 kW three-phase grid-connected inverter under both nominal and variable reference active power values have shown that the proposed APEO-based P-Q control method outperforms the traditional Z-N empirical method, the adaptive genetic algorithm-based, and particle swarm optimization-based P-Q control methods.
Keywords: power control; grid-connected inverter; extremal optimization; design optimization; evolutionary algorithms power control; grid-connected inverter; extremal optimization; design optimization; evolutionary algorithms

1. Introduction

As an important distribution system, a microgrid integrates a variety of renewable and traditional distributed generations and different loads [1]. How to design optimal controllers to guarantee that the microgrid operates well in both islanded and grid-connected modes is one of the critical issues in the microgrid research area [2,3,4,5,6]. More specifically, it is important to control the voltage and frequency of each power converter connected to each distributed generation, called the VF control, in the islanded mode while it is necessary to regulate the output active and reactive powers of each distributed generation, called the P-Q control in the grid-connected mode. Some recent works have studied the optimal voltage control issue of the distribution systems in the presence of distributed energy resources [7,8]. This paper focuses on the optimal P-Q control issue of a microgrid in the grid-connected mode.
In the past decade, some P-Q control methods have been proposed for distributed generations [9,10,11,12,13,14,15]. Dai [9] developed an effective power flow control method for a distributed generation unit in the grid-connected mode by adopting a robust servomechanism voltage controller and a discrete-time sliding mode current controller on the basis of Newton–Raphson-based parameter estimation and feed forward control approaches. The research work [11] studied the control problem of active and reactive powers using a second-order generalized integrator in a single-phase grid-connected fuel cell system based on the boost inverter. In Reference [12], an individual-phase decoupled P-Q control method based on six control degrees was proposed for a three-phase voltage source converter. Adhikari and Li [13] proposed a P-Q control method with solar photovoltaic, maximum power point tracking (MPPT), and battery storage in the grid-connected mode. Adhikari et al. [14] proposed a two-selected control method using theP-Q control in the load-following mode while the P-V control was in the maximum power point tracking mode. Unfortunately, the design processes of multivariable parameters used in P-Q controllers in the above works rely on deeply empirical rules of the engineers, so the performance under dynamic loads variations often becomes poorer. In fact, the design issue of P-Q controllers is essentially optimally solving a constrained optimization problem, but there are only a few reported works concerning the design of P-Q controllers from the perspectives of constrained optimization. Al-Saedi et al. [16] presented a particle swarm optimization (PSO)-based P-Q control method in grid-connected operation under variable loads conditions. This seminal work has demonstrated the importance of PSO in the automatic tuning of P-Q control parameters for optimized operation during abrupt loads change, but two current control parameters in the designed control system lack the optimization process based on PSO, so it may be considered as an incomplete optimization process for designing P-Q controllers. On the other hand, some popular evolutionary and swarm algorithms have been applied successfully to the optimal control of power converters and power systems [17,18,19], which motivate us to design an effective and efficient optimization algorithm to the optimal P-Q control issue of three-phase grid-connected inverters in a microgrid.
As a novel optimization framework originally inspired by the far-from-equilibrium dynamics of self-organized criticality (SOC) [20], extremal optimization (EO) [21,22] is different from other evolutionary algorithms because it merely selects against the bad instead of favoring the good based on a uniform random or power-law probability distribution. According to the iterated mechanism, EO can be classified into two categories. The first one is the individual-based discrete EO such as the standard EO [21,22] and the self-organized optimization algorithm [23,24] for combinatorial optimization problems, where an individual often represents a discrete sequence, e.g., a cyclic permutation of cities for the traveling salesman problem, and individual-based iterated operations including selection and discrete mutations, e.g., the 2-Opt, 3-Opt, and Lin–Kernighan (LK) rules for TSP are adopted. In the binary-coded EO (BCEO) algorithm [25], a set of decision variables of a continuous optimization problem are coded as a binary string, and the power law-based selection and binary mutation operate on this binary string. The second is the population-based continuous EO by using population iterated operations, e.g., polynomial mutations and multi-non-uniform mutations. A variety of simulation and experimental results on different kinds of benchmark and real-world engineering optimization problems have shown that EO and its modified algorithms perform better than or at least the same as other popular evolutionary algorithms such as genetic algorithm (GA) and PSO [26]. In the past decade, some modified EO algorithms have been applied to the optimal design issue of PID and fractional-order PID controllers for complex control systems [27,28,29,30,31,32,33]. To the best of the authors’ knowledge, the applications of EO to the optimal P-Q control of power converters have never been reported.
Encouraged by the aforementioned analysis, a novel intelligent P-Q control method is proposed for three-phase grid-connected inverters in a microgrid by using an adaptive population-based extremal optimization (APEO). The proposed method formulates the optimal P-Q control issue of three-phase grid-connected inverters in a microgrid as a typical constrained optimization problem firstly, where six parameters of decoupled PI controllers are real-coded as the decision variables and the integral time absolute error (ITAE) between the output and referenced active power and the ITAE between the output and referenced reactive power are weighted as the objective function. Then, an effective and efficient APEO algorithm with an adaptive mutation operation is proposed to solve this optimal issue.
The major contributions of this work are described as follows:
(1)
To the best of the authors’ knowledge, the adaptive population-based extremal optimization is applied firstly to the optimal P-Q control issue of three-phase grid-connected inverters in a microgrid.
(2)
The superiority of the proposed method is demonstrated by both the simulation and experimental results for a three-phase grid-connected inverter in a microgrid. In fact, the previous reported PSO-based P-Q control method [16] was tested only using its simulation results.
(3)
In cases of both nominal and variable reference active power values, the proposed APEO-based P-Q control method can improve the performance of a three-phase grid-connected inverter in a microgrid compared to the traditional Z-N empirical method, the adaptive GA-based, and the PSO-based P-Q control methods.
The rest of this paper is structured as follows. Section 2 presents the preliminaries concerning grid-connected inverters and extremal optimization. In Section 3, an intelligent P-Q control method is designed for grid-connected inverters in a microgrid based on adaptive population EO. Section 4 gives the simulation results on a three-phase grid-connected inverter. Moreover, in order to further validate the superiority of the proposed APEO-based P-Q control method, the experimental results on a real 3 kW three-phase grid-connected inverter in a microgrid are presented in Section 5. Finally, the conclusion and open problems are given in Section 6.

2. Problem Formulation

Figure 1 shows the circuit diagram and the corresponding P-Q control scheme for a three-phase grid-connected inverter in a microgrid [16,34]. Here, Vdc is the DC voltage provided by a distribution generation unit, Cd and Cf are the capacitance of the DC side and the LC filter, respectively, Lf represents the equivalent inductance of the LC filter, and Rf is the equivalent resistance of the LC filter. The P-Q control scheme consists of the following key operations: the grid-side voltage, a current and phase detector, an inverter-side voltage and current detector, active and reactive power calculation, an active power PI controller, a reactive power PI controller, a current PI controller, abc/dq and dq/abc transformations, and space vector pulse width modulation (SVPWM).
The block diagram of decoupled P-Q controllers for a three-phase grid-connected inverter in a microgrid is presented in Figure 2. The active and reactive powers of the inverter denoted as P and Q, respectively, are computed as follows:
P = 1.5 × ( v o d × i o d + v o q × i o q )
Q = 1.5 × ( v o q × i o d v o d × i o q )
where vod and voq are the d-coordinate and q-coordinate of the grid-side voltage, respectively, and iod and ioq are the d-coordinate and q-coordinate of the grid-side current, respectively.
The transfer functions of active and reactive power PI controllers denoted as GP(s) and GQ(s) are defined as follows:
G P ( s ) = i d r ( s ) P r e f ( s ) P ( s ) = K p 1 + K i 1 s
Q = 1.5 × ( v o q × i o d v o d × i o q )
where Pref and Qref are the reference values of active and reactive powers, respectively; idr and iqr1 are the output values of the active and reactive PI controllers, respectively; Kp1 and Ki1 are the proportional and integral parameters of the active PI controller, respectively; Kp2 and Ki2 are the proportional and integral parameters of reactive PI controller, respectively.
The reference voltage signals in the dq frame of SVPWM is defined as follows:
[ V d o V q o ] = [ K p 3 ω L f ω L f K p 3 ] [ i d i q ] + [ K p 3 0 0 K p 3 ] [ i d r i q r ] + [ K i 3 0 0 K i 3 ] [ x d x q ] + [ v o d v o q ]
x d = ( i d r i d ) / s
x q = ( i q r i q ) / s
where id and iq are the d-coordinate and q-coordinate of the inverter-side current, respectively; Kp3 and Ki3 are the proportional and integral parameters of the current PI controller, respectively; w is the angular frequency; and iqr is the reference input of the q-coordinate of the current PI controller, i.e., iqr = w × Cf × vodiqr1. Note that idr is also the reference input of the d-coordinate of the current PI controller.
In order to manage the active and reactive power of each distributed generation in a microgrid under the grid-connected mode, the design issue of an optimal P-Q controller with six parameters, including Kp1, Ki1, Kp2, Ki2, Kp3, and Ki3, can be formulated as a typical constrained optimization problem. More specifically, six parameters of decoupled PI controllers are real-coded as decision variables and the integral time absolute error (ITAE) between the output and referenced active power and the ITAE between the output and referenced reactive power are weighted as a minimized function. The complete formulation of the optimal P-Q control issue for a three-phase grid-connected inverter is as follows:
min F ( x ) = w 1 0 T max t | P r e f P | d t + w 2 0 T max t | Q r e f Q | d t       x = ( K p 1 , K i 1 , K p 2 , K i 2 , K p 3 , K i 3 )       s . t .   Equations   ( 1 ) ( 7 )         l 1 K p 1 u 1         l 2 K i 1 u 2         l 3 K p 2 u 3         l 4 K i 2 u 4         l 5 K p 3 u 5         l 6 K i 3 u 6
where Pref and Qref are the referenced active and reactive powers, respectively; w1 and w2 are the weighted coefficients; Tmax is the maximum time of the time window; l1, l2, l3, l4, l5, and l6 are the lower limits of Kp1, Ki1, Kp2, Ki2, Kp3, and Ki3, respectively; and u1, u2, u3, u4, u5, and u6 are the upper limits of Kp1, Ki1, Kp2, Ki2, Kp3, and Ki3, respectively.

3. The Proposed Method

In this section, an intelligent P-Q control method is presented for three-phase grid-connected inverters in a microgrid by using an adaptive population-based extremal optimization algorithm (APEO). The key idea behind the proposed method is firstly formulating the optimal P-Q control of grid-connected inverters in a microgrid as a typical constrained optimization problem shown as Equation (8). Then, an effective and efficient APEO algorithm with an adaptive mutation operation is designed to solve this optimization problem.
The detailed steps of the proposed algorithm are described as follows:
Input: The model of a three-phase grid-connected inverter with P-Q controllers, a sampling period Ts, the lower limits constraints (l1, l2, l3, l4, l5, l6) and upper limits constraints (u1, u2, u3, u4, u5, u6) of the P-Q control parameters (Kp1, Ki1, Kp2, Ki2, Kp3, Ki3), the weight coefficients w1 and w2 used for evaluating the objective function, a population size of N, the maximum number of iterations Imax, and the shape parameter b of the adaptive mutation operation.
Output: The best solution Sbest (the best control parameters Kpo1, Kio1, Kpo2, Kio2, Kpo3, Kio3), the corresponding global fitness Fbest, the real-time curve of the active and reactive power, and the current waveform of the transformer.
Step1: Generate a random initial real-coded population PI = {S1, S2, …, SN}, where the population size N is generally set as an even number, each solution Si represents a real-coded sequence of six control parameters including Kp1, Ki1, Kp2, Ki2, Kp3, Ki3, and set P = PI. More specifically, the generated process of Si is S i = L + R i ( U L ) ,   i = 1 , 2 , , N , where L = (l1, l2, l3, l4, l5, l6) and U = (u1, u2, u3, u4, u5, u6), and Ri is a set of randomly generated values between 0 and 1.
Step 2: Evaluate the fitness {Fi, i = 1, 2, …, N} of each solution Si in population P by means of Equation (8) firstly. Then rank all the solutions {Si, i = 1, 2, …, N} in ascending order of the fitness values {Fi, i = 1, 2, …, N}, which means obtaining a permutation П of the labels i such that. Set the best fitness of the current iteration as Fbest = FП(1) and the corresponding best solution Sbest = SП(1).
Step3: Select the solutions whose fitness ranks are from П(1) to П(N/2) to replace those solution whose fitness rank from П(1+N/2) to П(N), and set the intermediate population PM = {SM1, SM2, …, SMN}, where SMj = SM(j+N/2) = SП(j), j = 1, 2, …, N/2.
Step4: Generate a new population PN = {SN1, SN2, …, SNN} from PM by adopting a multi-non-uniform mutation (MNUM). To be more specific, SNi is computed by the following equations:
S N i = { S M i + ( U S M i ) A ( t ) ,   if   r < 0.5   and   L S N i U ; S M i + ( S M i L ) A ( t ) ,   if   r 0.5   and   L S N i U ; S M i ,   if   S N i < L   or   S N i > U ;
A ( t ) = [ r 1 ( 1 t I max ) ] b
where t is defined as the current number of iterations, r and r1 are the randomly generated numbers between 0 and 1, and b represents the shape parameter, which adjusts the dynamics of the mutation operation.
Step 5: Set SNN = Sbest and accept P = PN unconditionally.
Step 6: Repeat step 2 to step 5 until some stopping criterion, e.g., the maximum number of iterations Imax, is satisfied.
Step 7: Output the best control parameters solution Sbest = (Kpo1, Kio1, Kpo2, Kio2, Kpo3, Kio3), the corresponding global fitness Fbest, the real-time curve of the active and reactive power, and the current waveform of loads.
Figure 3 presents the flowchart of APEO-based P-Q controllers design algorithm for three-phase grid-connected inverters. From the design perspective of the evolutionary algorithm, the proposed APEO-based P-Q controller design algorithm for three-phase grid-connected inverters has only the selection and mutation operations. Moreover, the proposed APEO method has fewer adjustable parameters than the adaptive genetic algorithm (AGA) [35] and the particle swarm optimization (PSO) [16]. More specifically, except for the maximum number of iterations Imax and the population size N, only one shape parameter b needs to be tuned in the proposed APEO algorithm. However, two additional adjustable parameters in AGA and five additional parameters in PSO need be tuned for a specific practical three-phase grid-connected inverter. In this sense, the proposed APEO-based P-Q controller design algorithm can be considered as being simpler than the AGA and PSO-based P-Q control methods. In addition, the superiority of the proposed APEO method to the AGA and PSO-based P-Q algorithms will be verified by the simulation and experimental results on a three-phase grid-connected inverter in a microgrid under both nominal and variable reference active power values in the next two sections.

4. Simulation Results

4.1. Test for Benchmark Functions

Here, five benchmark functions shown in Table 1 are chosen from the literature [36] to illustrate the superiority of the proposed APEO algorithm to other popular optimization algorithms, such as the genetic algorithm (GA) [36], PSO [36], the original population-based extremal optimization version with lévy mutation termed as PEO [37], the hybrid PSO-EO algorithm [36], and the real-coded PEO with polynomial mutation termed as RPEO-PLM [38]. For the fair comparison, the parameters settings of GA, PSO, PEO, PSO-EO, and RPEO-PLM algorithms are the same as those in Reference [36,38]. The shape parameter b used in the APEO algorithm is set as b = 5. The number of maximum iterations is set the same as that in References [36,38], which is 20,000 for F1, F2, and F3; 10,000 for F4;and 100,000 for F5. Similarly, the popular size is set as follows: N = 10 for F1 and F3, and N = 30 for the other three functions. Each algorithm has been implemented by 20 independent runs for each function.
Table 2 compares the performance of APEO, GA, PSO, PEO, PSO-EO, and RPEO-PLM for the five test functions. Note that their performance is evaluated by the statistical results including the best values, average values, worst values, and the standard deviation of the 20 optimized fitness. The best performance is in bold. Clearly, APEO achieves the best performance for F1, F2, and F5, and the same performance as PSO, PSO-EO, and RPEO-PLM yet better than GA and PSO for F3 and F4. As a consequence, APEO performs better than or at least competitive with GA, PSO, PEO, PSO-EO, and RPEO-PLM for the five test functions. In other words, APEO can be considered as more suitable than the other popular optimization algorithms for the optimal P-Q control issue of three-phase grid-connected inverters in a microgrid.

4.2. Simulation Study for P-Q Control of Three-Phase Grid-Connected Inverter

In order to demonstrate the effectiveness of the proposed APEO-based P-Q controllers design method, this section presents the simulation results for a 3 kW three-phase grid-connected inverter in a microgrid. The six control parameters of P-Q controllers are tuned by traditional Z-N empirical method [39], AGA [35], PSO [16], and APEO. The system parameters for a three-phase grid-connected inverter are as follows: Vdc = 320 V, Cd = 1120 μF, Rf = 0.15 Ω, Lf = 2.5 mH, Cf = 45 μF. The lower and upper limits of the six parameters used in three decoupled PI controllers are set as l1 = 0.01, l2 = 30, l3= 0.01, l4 = 0.00001, l5 = 0.00001, l6 = 0.00001, u1 = 0.03, u2 = 50, u3 = 0.03, u4 = 10, u5 = 25, u6 = 500. The sampling time Ts is set as 2 × 10−6 s and the weights parameters w1 and w2 are set as 1 and 1, respectively.
The adjustable parameters of AGA, PSO, and APEO for the optimal design of P-Q controllers used in the following simulations are shown in Table 3. It should be noted that all the simulations have been run on the MATLAB2012b software on a 2.50 GHz PC with an i7-6500U processor running on 8 GB of RAM.

4.2.1. Case 1: Under Nominal Condition

In the first case, the reference values of the active and reactive powers for the above described three-phase grid-connected inverter are set as Pref = 2500 W and Qref = 0 Var. Here, each evolutionary algorithm for the parameters optimization of the P-Q controllers has been implemented for 30 independent runs. Table 4 presents the statistical results of AGA, PSO, and APEO such as the minimum (fmin), median (fmedian), maximum (fmax), mean (fmean), and standard deviation (fsd) values of the final global fitness obtained by the30 independent runs. It is clear that the APEO-based P-Q control method is better than the AGA and PSO-based P-Q controllers in terms of all the performance indices.
In order to illustrate the convergence characteristics of the proposed method, Figure 4 presents the comparative convergence process of an independent run associated with the fmin value shown in Table 4. Clearly, although the best fitness Fbsest of APEO is worse than that of AGA and PSO at the beginning because APEO starts its optimization process from a completely random solution, APEO outperforms AGA and PSO after six iterations. The premature convergence of AGA and PSO for the P-Q controller design is very obvious because their best fitness values have not been improved since the third and fourth iteration in AGA and PSO, respectively. Furthermore, Figure 5 presents the evolutionary process of the six P-Q controller parameters in APEO. In conclusion, APEO is better able to explore the problem space of the P-Q controller for a three-phase grid-connected inverter than AGA and PSO.
Table 5 presents the best P-Q controller parameters and the corresponding best performance values, including the fmin value, the settling time of the active and reactive powers denoted as tsP and tsQ, respectively, obtained by the traditional Z-N empirical method, AGA, PSO, and APEO. The active and reactive powers under different control parameters obtained by different methods when the DG unit is connected to the grid are compared in Figure 6. It is clear that the tsP and tsQ obtained by APEO are the least among the four methods. In other words, the response of active and reactive powers obtained by APEO is faster than those by the other methods.

4.2.2. Case 2: Robustness Test

In order to test the robustness against the variable reference values Pref of different methods, this subsection presents the comparison of the dynamic response of the active and reactive powers under different control parameters obtained by the Z-N empirical method, AGA, PSO, and APEO under the variable reference values of the active power. Here, the variable conditions of the reference active power values are set as the Pref value increases suddenly from 500 W to 2500 W at 0.10 s while Pref decreases suddenly from 2500 W to 500 W at 0.20 s. The dynamic response of the active and reactive powers are compared in Figure 7. The overshoots and settling times of the active and reactive powers obtained by APEO are all the least among the four methods under Pref variance.

5. Experimental Results

IN order to further demonstrate the superiority of the proposed APEO-based P-Q control method compared to the traditional Z-N method, AGA, and PSO-based P-Q controllers design method, this section presents the experimental results on a real 3 kW three-phase grid-connected inverter. The system parameters of the three-phase grid-connected inverter are the same as those in the simulation studies. Figure 8 shows the experimental platform for the P-Q control of a three-phase grid-connected inverter in a microgrid.
Two experiments have been designed to compare the performance obtained by the traditional Z-N method, AGA, and PSO-based P-Q controllers design methods when the reference value of the active power Pref increases suddenly from 500 W to 2500 W and decreases suddenly from 2500 W to 500 W. The P-Q controller parameters are the same as those shown in Table 3. The experimental dynamic response of the active and reactive powers obtained by the Z-N method, AGA, PSO, and APEO are shown in Figure 9, Figure 10, Figure 11 and Figure 12, respectively. It is evident that the fluctuation of the experimental active and reactive power obtained by APEO is the least while the fluctuation obtained by Z-N method is the worst. In other words, the transient performance including the overshoots and settling time of the active and reactive powers obtained by APEO are all the best. On the other hand, the experimental results have also indicated that the P-Q control performance obtained by an evolutionary algorithm such as AGA, PSO, and APEO is obviously better than the traditional Z-N empirical method.
The above experimental results of active and reactive power responses obtained by different methods are slightly worse than the corresponding results, but the superiority of the APEO-based P-Q control method to other methods is still demonstrated by these experimental results. The difference between the simulation and experimental results is due to the ideal characteristics of the three-phase transformer used in the simulations. In fact, the experimental P-Q control performance of a real three-phase grid-connected inverter in a microgrid is further improved by adopting more effective evolutionary algorithms and other advanced control structure, e.g., model predictive control.

6. Conclusions and Open Problems

This paper presents a novel APEO-based intelligent decoupled P-Q control method for the optimal P-Q control issue of three-phase grid-connected inverters in a microgrid. The key ideas behind this proposed APEO-based P-Q control method include encoding six parameters of three decoupled PI controllers in a P-Q controller as the real-coded decision variables, evaluating the control performance of a P-Q controller by considering the integral time absolute error (ITAE) between the output and referenced active power and the ITAE between the output and referenced reactive power, and updating the population by using selection and MNUM operations. The APEO-based P-Q control method is simpler than the existing popular evolutionary algorithms such as AGA-based [35] and PSO-based P-Q control algorithms [16] because of the fewer adjustable parameters and simpler operations in the population-based iterated optimization mechanism of the APEO-based P-Q control method. Furthermore, the simulation and experimental results for a 3 kW three-phase grid-connected inverter in a microgrid have demonstrated that the proposed APEO-based P-Q controller is superior to the traditional Z-N empirical method [39], AGA-based P-Q controllers [35], and PSO-based [16] P-Q controllers in terms of the control performances under nominal and variable control objective conditions. As a consequence, the proposed APEO-based P-Q controller design method can be considered as a promising intelligent P-Q control method for the optimal P-Q control issue of power converters in practical engineering systems. Of course, the P-Q controllers of three-phase grid-connected inverters can be optimized by other theoretical PID methods [40]. However, the P-Q control performance of three-phase grid-connected inverters can be further improved by adopting multi-objective evolutionary algorithms. Additionally, in future, how to extend the basic idea of the APEO-based P-Q controller to more complex power converters and power systems will be studied.

Author Contributions

M.-R.C. presented the problem formulation and designed the main algorithm; H.W. implemented the experiments and analyzed the experimental results; G.-Q.Z. proposed the novel idea behind the proposed method; Y.-X.D. analyzed the simulation results; D.-Q.B. designed the experimental platform. All authors approved the final manuscript.

Funding

This work was partially supported by Zhejiang Provincial Natural Science Foundation of China (Nos. LY16F030011 and LZ16E050002), and National Natural Science Foundation of China (Nos. 51207112 and 61373158).

Acknowledgments

The authors gratefully acknowledge the helpful comments and suggestions of editors and anonymous reviewers.

Conflicts of Interest

The authors declare no conflict of interest.

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Figure 1. The P-Q control scheme of a three-phase grid-connected inverter in a microgrid.
Figure 1. The P-Q control scheme of a three-phase grid-connected inverter in a microgrid.
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Figure 2. The block diagram of decoupled P-Q controllers for a three-phase grid-connected inverter in a microgrid.
Figure 2. The block diagram of decoupled P-Q controllers for a three-phase grid-connected inverter in a microgrid.
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Figure 3. The flowchart of APEO-based P-Q controllers’ design algorithm for three-phase grid-connected inverters.
Figure 3. The flowchart of APEO-based P-Q controllers’ design algorithm for three-phase grid-connected inverters.
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Figure 4. The comparison of the convergence process of APEO, PSO, and AGA for P-Q controllers.
Figure 4. The comparison of the convergence process of APEO, PSO, and AGA for P-Q controllers.
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Figure 5. The evolutionary process of the P-Q controller parameters in APEO.
Figure 5. The evolutionary process of the P-Q controller parameters in APEO.
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Figure 6. The comparison of active (a) and reactive (b) power obtained by different methods when the distributed generation (DG) unit is connected to the grid.
Figure 6. The comparison of active (a) and reactive (b) power obtained by different methods when the distributed generation (DG) unit is connected to the grid.
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Figure 7. The dynamic response of the active (a) and reactive (b) powers under different control parameters obtained by different methods when the set value of the active power Pref increases suddenly from 0.5 kW to 2.5 kW at 0.10 s and decreases suddenly from 2.5 kW to 0.5 kW at 0.20 s.
Figure 7. The dynamic response of the active (a) and reactive (b) powers under different control parameters obtained by different methods when the set value of the active power Pref increases suddenly from 0.5 kW to 2.5 kW at 0.10 s and decreases suddenly from 2.5 kW to 0.5 kW at 0.20 s.
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Figure 8. The experimental platform of the P-Q control for three-phase grid-connected inverters in a microgrid.
Figure 8. The experimental platform of the P-Q control for three-phase grid-connected inverters in a microgrid.
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Figure 9. The experimental dynamic response of the active and reactive powers obtained by the Z-N empirical method when the reference value of the active power Pref increases suddenly from 0.5 kW to 2.5 kW and decreases suddenly from 2.5 kW to 0.5 kW.
Figure 9. The experimental dynamic response of the active and reactive powers obtained by the Z-N empirical method when the reference value of the active power Pref increases suddenly from 0.5 kW to 2.5 kW and decreases suddenly from 2.5 kW to 0.5 kW.
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Figure 10. The experimental dynamic response of the active and reactive powers obtained by AGA when the reference value of the active power Pref increases suddenly from 0.5 kW to 2.5 kW and decreases suddenly from 2.5 kW to 0.5 kW.
Figure 10. The experimental dynamic response of the active and reactive powers obtained by AGA when the reference value of the active power Pref increases suddenly from 0.5 kW to 2.5 kW and decreases suddenly from 2.5 kW to 0.5 kW.
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Figure 11. The experimental dynamic response of the active and reactive powers obtained by PSO when the reference value of the active power Pref increases suddenly from 0.5 kW to 2.5 kW and decreases suddenly from 2.5 kW to 0.5 kW.
Figure 11. The experimental dynamic response of the active and reactive powers obtained by PSO when the reference value of the active power Pref increases suddenly from 0.5 kW to 2.5 kW and decreases suddenly from 2.5 kW to 0.5 kW.
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Figure 12. The experimental dynamic response of the active and reactive powers obtained by APEO when the reference value of the active power Pref increases suddenly from 0.5 kW to 2.5 kW and decreases suddenly from 2.5 kW to 0.5 kW.
Figure 12. The experimental dynamic response of the active and reactive powers obtained by APEO when the reference value of the active power Pref increases suddenly from 0.5 kW to 2.5 kW and decreases suddenly from 2.5 kW to 0.5 kW.
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Table 1. The five benchmark functions.
Table 1. The five benchmark functions.
Function Function ExpressionSearch SpacenGlobal Optimum
Michalewicz F 1 = i = 1 n sin ( x i ) sin 2 m ( i x i 2 π ) , m = 10 (0, π)n10−9.66 min
Schwefel F 2 = i = 1 n x i sin ( x i ) (−500, 500)n30−12,569.487 min
Rastrigin F 3 = i = 1 n [ x i 2 10 cos ( 2 π x i ) + 10 ] (−5.12, 5.12)n300 min
Ackley F 4 = 20 + e 20 exp ( 0.2 1 n i = 1 n x i 2 ) exp ( i = 1 n cos ( 2 π x i ) n ) (−32.768, 32.768)n300 min
Rosenbrock F 5 = i = 1 n 1 [ 100 ( x i + 1 x i 2 ) 2 + ( x i 1 ) 2 ] (−30, 30)n300 min
Table 2. The comparative performance of APEO and the other popular optimization algorithms for test functions.
Table 2. The comparative performance of APEO and the other popular optimization algorithms for test functions.
Test FunctionAlgorithmBestAverageWorstStandard DeviationRank
F1APEO−9.66−9.66−9.661.45 × 10151
RPEO-PLM [38]−9.66−9.66−9.665.767 × 1052
PSO-EO [36]−9.66−9.66−9.662.15 × 1033
PSO [36]−9.66−9.52−9.060.176
GA [36]−9.66−9.62−9.500.064
PEO [37]−9.61−9.55−9.500.035
F2APEO−12,569.5−12,569.5−12,569.51.82 × 1051
RPEO-PLM [38]−12,569.5−12,569.5−12,569.51.052 × 1052
PSO-EO [36]−12,569.5−12,568.0−12,562.62.013
PSO [36]−9577.7−10,139.3−11,026.2625.75
GA [36]−9549.3−8846.0−8404.5481.06
PEO [37]−12,214.2−12,083.3−11,977.390.34
F3APEO00001
RPEO-PLM [38]00001
PSO-EO [36]00001
PSO [36]00001
GA [36]0.0460.0149.93 × 1040.0145
PEO [37]2.472.141.850.256
F4APEO−8.88 × 1016−8.88 × 1016−8.88 × 101601
RPEO-PLM [38]−8.88 × 1016−8.88 × 1016−8.88 × 101601
PSO-EO [36]−8.88 × 1016−8.88 × 1016−8.88 × 101601
PSO [36]−8.88 × 1016−8.88 × 1016−8.88 × 101601
GA [36]0.0940.0540.030.025
PEO [37]0.120.110.098.4 × 1036
F5APEO1.21 × 10194.47 × 10174.67 × 10161.15 × 10161
RPEO-PLM [38]3.050 × 10−108.360 × 1071.050 × 10−52.283 × 10−62
PSO-EO [36]9.99 × 1049.88 × 1049.54 × 1042.39 × 1053
PSO [36]26.826.025.40.595
GA [36]39.733.130.13.956
PEO [37]9.639.429.300.134
Table 3. The adjustable parameter settings of APEO, PSO, and AGA used for the optimal design of the P-Q controllers in a microgrid.
Table 3. The adjustable parameter settings of APEO, PSO, and AGA used for the optimal design of the P-Q controllers in a microgrid.
AlgorithmParameters Setting
AGA [35]Population size N = 30, Imax = 30, the crossover probability pc = 0.9, the mutation probability pm = 0.1 − 0.01 × n/N, where n = 1, 2,..., N.
PSO [16]Population size = 30, Imax = 30, inertia weight w = 0.6, the upper limit of velocity Vmax = 0.05, the lower limit of velocity Vmin = −0.05, acceleration factors c1 = 2.0, c2 = 2.0.
APEON = 30, Imax = 30, b = 0.1.
Table 4. The statistical performance of AGA, PSO, and APEO for designing P-Q controllers.
Table 4. The statistical performance of AGA, PSO, and APEO for designing P-Q controllers.
Algorithmfmaxfmedianfmeanfminfsd
AGA [35]0.26460.25890.25860.25310.0038
PSO [16]0.25320.24950.24940.24620.0023
APEO0.24340.24300.24310.24270.0002
Table 5. The parameters of the best-decoupled PI controllers and the corresponding best performance obtained by different P-Q control methods.
Table 5. The parameters of the best-decoupled PI controllers and the corresponding best performance obtained by different P-Q control methods.
AlgorithmKpo1Kio1Kpo2Kio2Kpo3Kio3FminTsP(s)TsQ(s)
Z-N method0.021931.40930.02922.804010.7959303.24780.68700.05010.0783
AGA0.024241.40780.02677.136524.7489429.252680.25310.04060.0582
PSO0.0299300.0310255000.24620.04500.0461
APEO0.028549.99470.02999.960024.9999499.96150.24270.03240.0375
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