D approach to the solution of multi-objective optimal power flow problem

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A modified MOEA/D approach to the solution of multi-objective optimal power flow problem

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Jingrui Zhang a,b,∗ , Qinghui Tang a , Po Li a , Daxiang Deng a , Yalin Chen a

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a b

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Department of Instrumental & Electrical Engineering, School of Aerospace Engineering, Xiamen University, Xiamen 361005, China Shenzhen Research Institute of Xiamen University, Shenzhen 518063, China

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a r t i c l e

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i n f o

a b s t r a c t

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Article history: Received 17 January 2016 Received in revised form 3 May 2016 Accepted 17 June 2016 Available online xxx

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Keywords: Multi-objective optimization Optimal power flow MOEA/D MOPSO NSGA-II

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1. Introduction

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This study presents a modified multi-objective evolutionary algorithm based decomposition (MOEA/D) approach to solve the optimal power flow (OPF) problem with multiple and competing objectives. The multi-objective OPF considers the total fuel cost, the emissions, the power losses and the voltage magnitude deviations as the objective functions. In the proposed MOEA/D, a modified Tchebycheff decomposition method is introduced as the decomposition approach in order to obtain uniformly distributed Pareto-Optimal solutions on each objective space. In addition, an efficiency mixed constraint handling mechanism is introduced to enhance the feasibility of the final Pareto solutions obtained. The mechanism employs both repair strategy and penalty function to handle the various complex constraints of the MOOPF problem. Furthermore, a fuzzy membership approach to select the best compromise solution from the obtained Pareto-Optimal solutions is also integrated. The standard IEEE 30-bus test system with seven different cases is considered to verify the performance of the proposed approach. The obtained results are compared with those in the literatures and the comparisons confirm the effectiveness and the performance of the proposed algorithm. © 2016 Elsevier B.V. All rights reserved.

The optimal power flow (OPF) problem plays one of the most important roles in the operation of modern power systems and has received a lot of interest over the years [1]. It involves the dispatching or setting for all the generator real powers, the generator bus voltages, the tap ratios of transformers and the reactive power generations of VAR sources. Simultaneously, the generator reactive powers, the load bus voltages and the power flow of network lines should also be calculated through the solving of OPF problems. The objective function of OPF problem was usually formulated as minimizing the total fuel cost in past years. However, voltage instability is now emerging as a new challenge for power system planning and operation [2] due to the continuous growth in the demand of electricity with unmatched generation and transmission capacity expansion. At the same time, insufficient reactive power sources produce large transmission losses. More and more concerns to environmental problem make it necessary to consider the emission as one of the objectives instead of constraints. In such

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∗ Corresponding author at: Department of Instrumental & Electrical Engineering, School of Aerospace Engineering, Xiamen University, Xiamen 361005, China. E-mail addresses: [email protected], [email protected] (J. Zhang).

situations, it is necessary to consider emissions, voltage magnitude deviations and transmission losses as part of the objective functions of the OPF problem. Hence, the OPF problem has become a multi-objective optimization problem. Q5 Multi-objective optimal power flow (MOOPF) is a large-scale highly constrained non-linear optimization problem. It is very difficult to solve the problem and the solving of it involves special efficiency solution techniques. In past decades, many heuristic methods have been proposed aiming at solving multi-objective OPF problems. These intelligent solution methods are usually divided into two kinds according to whether the preference information of a decision-maker is obtained before the solution. If the preference information can be obtained, then the MOOPF problem can be transformed into an optimization problem with only one objective function through weighted sum [3,4], fuzzy membership functions [5], etc. However, this type of solution just identifies one Pareto solution at one time. If the preference information changes, one has to run the solution approach once again with the changed preference information. If the preference information of a decision-maker can’t be obtained before the solution, one had to address the whole Pareto solutions directly. Then the decision-maker chooses one of the obtained Pareto candidates as the most preferred solution based on his/her preference information. This posterior approach which searches the whole Pareto optimal solutions before multi-criterion

http://dx.doi.org/10.1016/j.asoc.2016.06.022 1568-4946/© 2016 Elsevier B.V. All rights reserved.

Please cite this article in press as: J. Zhang, et al., A modified MOEA/D approach to the solution of multi-objective optimal power flow problem, Appl. Soft Comput. J. (2016), http://dx.doi.org/10.1016/j.asoc.2016.06.022

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decision [6] represents the main trend for solving multi-objective optimizations. This type of method tries to obtain a good representation of the Pareto Front (PF) to present to a decision maker. In recent years, the posterior multi-objective evolutionary algorithms (MOEAs) have been well investigated. Many novel and effective MOEAs have been proposed constantly. Among these methods, NSGA-II [7,8], multi-objective particle swarm optimization [9], immune algorithm [10,11], memetic algorithm [12] are popular developed and applied in many fields. The optimal power flow problem as one of the famous multi-objective optimizations also attracts many scholars. Some MOEAs such as multi-objective differential evolutionary [13–18], artificial bee colony algorithm [5,19–22], multi-objective adaptive immune algorithm [10], enhanced genetic algorithm [23], NSGA-II [7], multi-objective PSO [24,25], quasi-oppositional biogeography-based optimization [26], multi-objective harmony search algorithm [27], modified shuffle frog leaping algorithm [28,29], gravitational search algorithm [1,30–34], multi-objective modified imperialist competitive algorithm [35,36], multi-hive bee foraging algorithm [37], teaching-learning based optimization algorithm [2,38], multiobjective solution Q(␭) learning [39], etc., have been proposed aiming at the solution of MOOPF problems. However, the above methods have to do some more efforts in order to approach to the true Pareto-Optimal Front and obtain the diversity of the solutions. On the other hand, the preference information of a decision-maker is neither prior nor posterior in practical multi-objective optimization problems (MOPs). Reference point which consists of aspiration levels reflecting preference values for the objective functions is one of the most important ways to provide preference information. Also, it is a natural way of expressing the preference information as a target the decision-maker is hoping to obtain [6]. For this type of preference information expression, Zhang and Li [40] proposed a multi-objective evolutionary algorithm based on decomposition (MOEA/D) approach in 2007. This approach decomposes a multi-objective optimization problem (MOP) into a number of scalar single-objective optimization problems using uniformly distributed preference directions with the same reference point. In addition, the use of weight design methods and the neighbor information in the MOEA/D approach naturally guarantee the diversity of the obtained solutions. One can just focus on how to approach the true Pareto-Optimal Front within the approach. MOEA/D is a simple yet efficient MOEA and has been dedicated to knapsack problem [41–44], job shop scheduling [45], traveling salesman problem [46–49], test task scheduling problem [50], antenna array synthesis [49], wireless sensor networks [51], portfolio management [52] and reservoir flood control [6]. The experimental and practical results all show perfect performance of the MOEA/D. The MOEA/D’s successful solution to the famous multi-objective optimizations above will attract more and more applications in engineering and scientific areas. The aim of this work is to propose a modified MOEA/D approach to solve the MOOPF problem. To our best knowledge, this may be the first try of MOEA/D to the solution of MOOPF problem. In order to obtain uniformly distributed Pareto-Optimal solutions on each objective space, a modified Tchebycheff decomposition approach instead of traditional Tchebycheff decomposition one is introduced. To enhance the feasibility of the final Pareto-Optimal solutions obtained, an efficiency mixed constraint handling mechanism is employed. The mechanism employs both repair strategy and penalty function to handle the various complex constraints of the MOOPF problem. The modified MOEA/D approach also integrates a fuzzy membership approach to select the best compromise solution. The performance of comparison experiments with the popular NSGA-II and multi-objective particle swarm optimization

(MOPSO) on several test cases shows the superiority of the proposed modified MOEA/D approach. The rest of this paper is organized as follows. Section 2 formulates the general multi-objective optimization problem. Section 3 presents the mathematical formulation of the multi-objective OPF problem. Section 4 introduces the general framework of the modified MOEA/D. In Section 5, the implementation of the modified MOEA/D for the multi-objective OPF problem is present. Simulation studies and comparisons with other approaches are given in Section 6. Finally, conclusions are made in Section 7. 2. Multi-objective optimization problem In general, a multi-objective optimization problem consists of multiple conflicting objectives to be optimized simultaneously, while satisfying a range of equality and inequality constraints [30]. The multi-objective optimization problems have been usually for- Q6 mulated as follows.

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minimizeF = (f1 (x) , f2 (x) , · · ·, fM (x)) , x = [x1 , x2 , · · ·, xn ] subjecttogj (x) ≥ 0, j = 1, 2, · · ·, G hk (x) = 0, k = 1, 2, · · ·, H

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ximin ≤ xi ≤ ximax , i = 1, 2, · · ·, n where fi (x) represents the ith objective function of the optimization problem, x is a decision vector that consists of n decision variables, G and H represent the numbers of inequality constraints and equality constraints, respectively. In single objective optimizations, the optimal solution is usually unique, while the optimal solution for multi-objective optimizations is often a set due to the conflict between different objective functions. It is necessary to understand the concepts of Pareto-Dominance, Pareto-Optimal, Pareto-Optimal Set, and Pareto-Optimal Front in order to solve the optimization problem. The detailed definitions for these concepts please refer to [53]. Within the definitions, a multi-objective optimization problem can be seen as the method of looking for the Pareto-Optimal solutions or approaching the Pareto-Optimal Front (PF). 3. Multi-objective OPF problem The optimal power flow has been considered as one of the well-known multi-objective optimization problems. It aims at optimizing some selected objective function simultaneously through the optimal setting of control variables while satisfying various constraints. The control variables usually consist of the real power outputs from all the generators, the voltage magnitudes from all the generator buses, the tap ratios of transformers and the reactive power outputs of all VAR sources. The multi-objective optimal power flow problem is difficult to solve, because it must consider various complicated operation and system constraints. Some of the constraints are nonlinear and time coupling. It is very difficult to satisfy these various constraints. Here in the rest of this section, some of the popular objective functions and constraints considered for the multi-objective optimal power flow problem are reformulated. 3.1. Objective functions 3.1.1. Minimization of total fuel cost It is important to find an optimal operation scheme to generate electricity with minimum costs with the rise in fuel prices and the increased power loads. The total fuel cost ($/h) of generating units

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considering the valve loading effects can be expressed as follows [5]. fcost =

N g  i=1





2 min ai + bi Pgi + ci Pgi + |di × sin ei × Pgi − Pgi

  |

3.1.2. Minimization of total active power losses The series resistance and the shunt conductance of transmission lines will produce active power losses and more power losses will increase the generated power costs [20]. Here the total active power losses are considered as one of the objective functions, and it is expressed as follows.

N L  k=1



Gk Vi2 + Vj2 − 2Vi Vj cos  ij



(2)

where NL is the total branch of the power systems considered, Gk is the conductance of the kth branch, Vi and Vj are the voltage magnitudes of terminal buses of branch k, and ij is phase angles difference between them.

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3.1.3. Minimization of total emission More and more countries or regions are concerned about environmental protection due to the increasingly serious problem of air pollutions. It is necessary to reduce the emissions of atmospheric pollutants caused by the thermal generation units [20]. The total emission (ton/h) of the atmospheric pollutants such as SOx and NOx caused by fossil-fueled thermal generation facilities can be expressed as:

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femission =

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N g  i=1

2 ˛i + ˇi Pgi + i Pgi + i ei Pgi



(3)

where ˛i , ˇi ,  i ,  i and i are the emission constant coefficients of the ith unit.

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3.1.4. Minimization of total voltage magnitude deviations Bus voltage is one of the most important indices for secure operation and voltage quality. The objectives without voltage indices may result in a feasible solution that has an unattractive voltage profile [20]. The objective function considered here is to minimize the voltage magnitude deviations (VMD) of all the load bus from 1 per unit (p.u.) [1]. The objective function can be defined as follows:

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fVMD =

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Npq  i=1



| Vi − Vreference |

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3.2. Constraints

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The MOOPF problem is a non-linear optimization problem which determines the optimal control variables for minimizing the some objectives subject to several equality and inequality constraints [27]. The selected equality and inequality constraints are formulated as follows. 3.2.1. Equality constraints The equality constraints of MOOPF problem are typical load flow equation and for each bus i, they can be described as follows. 0 = PGi − PDi − Vi

Nb



Nb



0 = QGi − QDi − Vi

V j=1 j

V j=1 j

Gij cos  ij + Bij sin  ij

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3.2.2. Inequality constraints The MOOPF is optimized subject to the following inequality constraints of the power systems. (1) Generator constraints: generator active power output Pg , generator reactive power output Qg , and generator voltage magnitude Vg are restricted by their lower and upper limits as follows. min ≤ P ≤ P max Pgi gi gi

Qgimin

≤ Qgi ≤

i = 1, 2, · · ·, Ng

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241

i = 1, 2, · · ·, Ng

(2) Transformer constraints: transformer taps are restricted by their minimum and maximum setting limits. imin ≤ i ≤ imax

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i = 1, 2, · · ·, Ng

Qgimax

min ≤ V ≤ V max Vgi gi gi

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i = 1, 2, · · ·, NT

where NT is the number of on-load voltage regulating transformer in the power systems. (3) Switchable VAR sources: the switchable VAR sources have restrictions as follows.

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Qcimin ≤ Qci ≤ Qcimax i = 1, 2, · · ·, NC

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where Nc is the number of switchable VAR sources.

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(4) Security constraints: these include the limits of voltage magnitudes on load bus and the limits of transmission line flow. VLimin ≤ VLi ≤ VLimax

i = 1, 2, · · ·, Npq

where Npq is the number of load bus. max SLi ≤ SLi

i = 1, 2, · · ·, NL

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253 254

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where NL is the number of transmission lines.

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4. General framework of MOEA/D

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(4)

where fVMD is the total voltage magnitude deviations; Npq is the total number of PQ bus and Vreference is the reference voltage magnitude in p.u., here Vreference = 1.0.

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where PGi andQGi are active and reactive power injection at bus i, PDi andQDi are active and reactive power demand at bus i and Nb is the total bus of the power system.

(1)

where ai , bi , ci , di and ei are the cost coefficients of the ith generator, min represents Pgi is the real power output of thermal unit i, and Pgi its minimum value.

floss =

3



Gij sin  ij − Bij cos  ij



i = 1, 2, . . ., Nb (5) i = 1, 2, . . ., Nb (6)

In this section, the general idea of MOEA/D algorithm used for addressing the multi-objective optimization problem is described. As indicated in [40], the objectives of many practical MOPs are often contradicting with each other, usually no point in feasible space can minimize all the objectives simultaneously. Actually, addressing all the non-dominated or non-inferior solutions has been becoming the purpose of the multi-objective optimization problem. In order to find the non-dominated solutions, MOEA/D decomposes a multi-objective optimization problem into a set of single objective optimization sub-problems using the weighted sum approach, the Tchebycheff approach and the boundary intersection approach [54]. These sub-problems are then optimized concurrently and collaboratively by evolving population of solutions using an evolutionary algorithm (EA). The sub-problem optimization uses the information, mainly from its neighboring sub-problem while the neighborhood relations among these sub-problems are defined based on the distances between their aggregation coefficient vectors.

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The approximation of the Parent-Optimal Front (PF) of a MOP can be decomposed into N scalar optimization sub-problems in MOEA/D. As described in Ref. [40], any decomposition approach can serve for converting approximation of the PF of MOP into a number of single objective optimization problems. Here the Tchebycheff approach is employed to describe the main framework of MOEA/D for the solution of MOPs. The jth single objective optimization subproblem is

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g te x|j , z ∗ = max

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1≤i≤m

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where j =



j





i |fi (x) − zi∗ | j

j

j

1 , 2 , · · ·, m

(7)

T . MOEA/D attempts to optimize these

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N scalar optimization sub-problems simultaneously instead of solving MOP problem directly in a single run. In Zhang’s approach, the j of  neighborhood

 is constituted of the T closest weight vectors in 1 , 2 , · · ·, N , and the neighborhood of the jth subproblem consists of all the sub-problems with the weight vectors from the neighborhood of j . The population is composed of the best solutions found so far for each sub-problem. Only the current solutions to its neighboring sub-problems are exploited for optimizing a subproblem in MOEA/D. The general main framework of MOEA/D is described as Fig. 1 shown.

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5. Modified MOEA/D approach to MOOPF problem

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The modified MOEA/D approach to the solution of the MOOPF problem is discussed and implemented in this section. The modified Tchebycheff decomposition approach, mixed constraints handling mechanism, and the fuzzy selection method of the best compromise solution are first introduced and then the detailed steps to the solution of MOOPF problem are present.

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5.1. Modified tchebycheff decomposition approach

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Fig. 1. The general main framework of MOEA/D.

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As we know, the measurement units and the value ranges of objective functions for the OPF problem are different. This condition will make each objective function has different weightiness and this will result in converging to some certain region. Zhang

and Li [40] have proposed the Tchebycheff approach to decrease the effects of different unit measurements and value ranges, but this is not enough. Here, we introduce a modified Tchebycheff approach to overcome the above lacks for the solution of MOOPF problem.

Table 1 Parameters setting for different cases. Method

Parameters

Case 1

MOEA/D

Population Size Niche Size Maximum Iteration F Crossover Probability Population Size Maximum Iteration Mutation Distribution Index Crossover Distribution Index Population Size Repository Size Maximum Iteration Grid Inflation Parameter Number of Grids per Dimension

100 20 500 0.5 0.5 – – – – – – – – –

NSGAII

MOPSO

Case 2 100 20 500 0.5 0.5 100 500 20 20 100 100 500 0.1 10

Case 3 100 20 500 0.5 0.5 100 500 20 20 100 100 500 0.1 10

Case 4 100 20 500 0.5 0.5 100 500 20 20 100 100 500 0.1 10

Case 5 300 20 500 0.5 0.5 300 500 20 20 300 300 500 0.1 10

Case 6

Case 7

300 20 500 0.5 0.5 300 500 20 20 300 300 500 0.1 10

300 20 500 0.5 0.5 300 500 20 20 300 300 500 0.1 10

Table 2

Q11 Obtained results when optimizing each single objective optimization independently. Optimizing Objective

Fuel Cost ($/h)

VMD (p.u.)

Power Loss (MW)

Emission (ton/h)

Min Fuel Cost Min VMD Min Power Loss Min Emission

799.0263 849.9159 967.0704 943.5519

1.7679 0.0936 2.0298 2.0196

8.6163 7.0740 2.8518 2.9800

0.3663 0.2835 0.2072 0.2047

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b

0.38 0.36 0.34 0.32 0.30 0.28 10.0 9.5 9.0 8.5 2.0 1.5 1.0 0.5

-50

Power loss (MW) Emission (ton/h)

0

Fuel cost ($/h)

820 810 800 -50

0

50 100 150 200 250 300 350 400 450 500 550

0

50 100 150 200 250 300 350 400 450 500 550

0.2 12 10 8 6 4 920

0.3

50 100 150 200 250 300 350 400 450 500 550

0

0.3

900

0.0

5

0.4

Fuel cost ($/h)

-50 0.40

VMD (p.u.)

VMD (p.u.)

a

Power loss (MW) Emission (ton/h)

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880 860 840 820 0.4

0.2 0.1 -50

Iteration

0

50 100 150 200 250 300 350 400 450 500 550

40 20

Power loss (MW)

0 980 960 940 920 900 880

60

VMD (p.u.)

VMD (p.u.)

860 40

-50

d Fuel cost ($/h)

-50 60

Power loss (MW)

20 0

Emission (ton/h)

Fuel cost ($/h)

Emission (ton/h)

c

Iteration

40 20 0 -50

0

50 100 150 200 250 300 350 400 450 500 550

0

50 100 150 200 250 300 350 400 450 500 550

0

50 100 150 200 250 300 350 400 450 500 550

950 900 850 40 30 20 10 30 0 20 10 0 30 20 10 0 -50

Iteration

Iteration

Fig. 2. The optimal convergence curves for different single objective optimizations.

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In the modified Tchebycheff approach, the jth single objective optimization sub-problem is

g

mte



j

x| , z

R



= max

1≤i≤m

j i

|fi (x) − zi∗ | |zimax − zimin | + ε

where j =



j

j

j

1 , 2 , · · ·, m

T

, z R is the set of reference points, zi∗

is the ideal value of the ith objective function, ε is a small positive value, and zimax and zimin are the maximal and minimal value of the

(8)

ith objective function, respectively. Here zimax and zimin are set as the maximum and minimum objective values of the reference points for simplification.

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Table 3 Comparisons of obtained objectives with other algorithms for test Case 2. Methods

TLBO [38] Min C

Cost Emission

Min E

801.9908 947.4392 0.3668 0.20503

MTLBO [38]

MSFLA [28]

Min C

Min C

Min E

801.8925 945.1965 0.3665 0.20493

MOPSO Min E

Min C

802.287 951.5106 0.3723 0.2056

NSGA-II Min E

801.33 953.44 0.3533 0.2054

MOEA/D

Min C

Min E

Min C

Min E

799.41 0.367

945.82 0.205

799.29

944.3 0.2048

0.3593

Table 4 Comparisons of obtained solutions using different methods for Case 2. Control Variables

MOEA/D

Min C Pg2 Pg5 Pg8 Pg11 Pg13 Vg1 Vg2 Vg5 Vg8 Vg11 Vg13 6−9 6−10 4−12 27−28 Qc10 Qc12 Qc15 Qc17 Qc20 Qc21 Qc23 Qc24 Qc29 Fuel cost Emission

49.874 21.784 21.317 12.386 12.000 1.1000 1.0861 1.0556 1.0677 1.0981 1.0937 1.0272 0.9057 1.0162 0.9775 1.600 1.243 3.242 4.362 4.270 5.000 3.760 4.653 2.689 799.29 0.3593

MOPSO

Min E 67.757 50.000 34.993 30.000 40.000 1.0864 1.0816 1.0564 1.0701 1.1000 1.0975 1.0104 0.9530 1.0030 0.9781 1.258 3.849 1.755 4.647 4.556 4.859 4.481 4.569 1.440 944.30 0.2048

NSGA-II

Comp

Min C

Min E

Comp

Min C

Min E

Comp

59.079 27.618 34.998 26.234 26.522 1.0996 1.0892 1.0622 1.0782 1.0967 1.1000 1.0454 0.9012 0.9929 0.9792 4.328 2.657 4.980 4.383 4.363 4.925 4.758 4.599 3.754 833.72 0.2438

46.685 20.451 20.235 14.827 16.718 1.0975 1.0775 1.0606 1.0662 1.0249 1.0374 1.0854 0.9526 1.0550 1.0173 4.073 3.092 4.108 2.531 4.107 2.202 2.514 3.834 3.260 801.33 0.3533

70.570 50.000 35.000 30.000 40.000 1.0893 1.0743 1.0645 1.0206 0.9953 1.0207 0.9609 1.1000 0.9063 1.0020 2.907 2.934 4.035 2.695 3.118 1.295 3.419 1.183 1.724 953.44 0.2054

58.112 31.146 31.503 24.920 25.244 1.1000 1.0803 1.0737 1.0649 1.0216 1.0421 1.0303 1.0405 0.9879 1.0144 4.149 3.105 3.895 2.631 3.991 2.161 2.544 2.534 3.369 833.86 0.2483

48.895 21.208 20.686 11.976 12.009 1.1000 1.0873 1.0630 1.0666 1.0997 1.0874 1.0262 0.9840 1.0077 0.9764 1.567 0.002 3.009 5.000 4.150 0.621 4.383 2.806 1.322 799.41 0.3670

67.747 50.000 35.000 30.000 40.000 1.0177 1.0112 0.9929 0.9958 1.0346 1.0898 0.9808 1.0113 1.0362 0.9603 0.064 2.759 1.589 2.478 0.594 0.798 3.749 4.525 3.362 945.82 0.2050

60.266 28.536 35.000 24.288 26.527 1.0770 1.0646 1.0100 1.0292 1.0541 1.0969 0.9944 0.9372 1.0294 0.9614 0.665 4.852 1.603 2.027 0.587 0.576 3.494 4.385 2.705 835.59 0.2449

Table 5 Comparisons of dominated solutions obtained by MOEA/D, NSGA-II and MOPSO. Test Case

Methods

Number of solution dominated by MOEA/D

Number of solution dominated by NSGA-II

Number of solution dominated by MOPSO

Case 2

MOEA/D NSGA-II MOPSO MOEA/D NSGA-II MOPSO MOEA/D NSGA-II MOPSO MOEA/D NSGA-II MOPSO MOEA/D NSGA-II MOPSO MOEA/D NSGA-II MOPSO

– 81 98 – 100 100 – 100 100 – 170 256 – 182 262 – 177 259

0 – 91 0 – 52 0 – 0 0 – 271 0 – 114 0 – 42

0 0 – 0 28 – 0 98 – 0 0 – 0 8 – 0 23 –

Case 3

Case 4

Case 5

Case 6

Case 7

321

322 323 324 325 326

5.2. Mixed constraints handling mechanism The MOOPF is a complicated constrained optimization problem. How to handle the complicated equality and inequality constraints is a difficult issue of a solution technique to the MOOPF problem. Here in our modified MOEA/D approach, a mixed constraints handling mechanism is incorporated. The mechanism employs both

repair strategy and penalty function approach to tackle the complicated constraints. The repair strategy deals with the constraints of control variables, i.e., the upper and bounder limits and the equality constraints. Within the repair strategy, the upper and bounder constraints of control variables can be satisfied through the resetting operation shown as Formula (9) during the solution procedures. The other constraints handled through the repair strat-

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0.38 MOEA/D NSGA-II MOPSO 0.36

0.34

Emission (ton/h)

0.32

0.3

0.28

0.26

0.24

0.22

0.2 780

800

820

840

860 880 Fuel cost ($/h)

900

920

940

960

Fig. 3. Pareto fronts obtained using MOEA/D, NSGA-II and MOPSO for Case 2.

334 335 336 337 338 339 340 341 342 343 344 345

346

egy are the equality constraints of active and reactive power flow at each bus. The repair to this equality constraints are performed through load flow calculation using Newton-Ralph method. Here we employ MATPOWER 4.1 [55], initially developed by Cornell University, to implement this calculation. The rest constraints of the MOOPF problem are handled through penalty function approach. In the penalty function approach, the constraint violations are formulated in p.u. firstly in order to balance the impacts of different types of constraints, and then they are multiplied by penalty coefficients and added to the objective function of the MOOPF problem. Hence, each objective function of the MOOPF problem is formulated as Formula (10).

x=

⎧ min x if x < xmin ⎪ ⎨ ⎪ ⎩

x

if xmin ≤ x ≤ xmax

xmax

if x > xmax

where fk is the kth origin objective function of the MOOPF problem, funvio (x) represents the constraint violation function with element x, and ␣, ˇ and  are penalty coefficients. 5.3. Best compromise solution based on fuzzy decision

ij =

347

⎧ 1 ⎪ ⎨ ⎪ ⎩

 

 

max Fj

 

 

− Fij / max Fj

 

− min Fj

Fij ≤ min Fj

 

min Fj

348

349

p

fk = fk + |˛ +|

NL

i=1

i=1

funvio (VLi ) | + |ˇ

funvio (SLi ) |

Ng i=1





(10)

354 355 356 357 358 359 360 361

 

≤ Fij ≤ max Fj

363

 

Fij ≥ max Fj

0

 

funvio Qgi |

352

362

(11)

Npq

351

353

To find out the best compromise solution among the final noninferior or non-dominated solutions is vital in decision making process after having obtained the Pareto-Optimal set. Here in this paper, a fuzzy membership approach is applied to obtain a satisfactory and best compromise solution over the trade-off curve. Due to imprecise nature of the decision maker’s judgment, each objective function of the ith solution is represented by a membership function uij defined as follow:

(9)

350

 

where min Fj and max Fj are the lower and upper bounds of the ith objective function, respectively. The higher membership function a solution has, the greater satisfaction it will get. For each

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Table 6 Comparisons of obtained solutions using different methods for Case 3. Control Variables

Pg2 Pg5 Pg8 Pg11 Pg13 Vg1 Vg2 Vg5 Vg8 Vg11 Vg13 6−9 6−10 4−12 27−28 Qc10 Qc12 Qc15 Qc17 Qc20 Qc21 Qc23 Qc24 Qc29 Fuel Cost Power loss

MOEA/D

MOPSO

Min C

Min L

49.329 20.826 19.884 11.450 12.000 1.1000 1.0902 1.0626 1.0714 1.0991 1.0999 1.0341 0.9136 1.0002 0.9627 4.669 4.957 4.350 4.753 4.979 4.959 2.284 4.984 1.851 799.13 8.7851

80.000 50.000 35.000 29.978 40.000 1.1000 1.0964 1.0787 1.0852 1.0996 1.1000 1.0448 0.9234 0.9954 0.9727 4.849 4.834 4.940 5.000 4.908 4.958 3.675 5.000 2.788 967.04 2.8583

Min C

Min L

Comp

Min C

Min L

Comp

53.572 32.890 34.993 28.244 21.963 1.1000 1.0920 1.0690 1.0793 1.1000 1.0999 1.0617 0.9000 0.9953 0.9700 5.000 5.000 4.503 5.000 4.592 4.960 3.069 4.994 2.500 835.36 4.9099

47.434 21.369 27.868 13.784 13.549 1.0992 1.0859 1.0586 1.0668 1.0572 1.0646 0.9688 0.9870 1.0333 1.0039 4.147 2.707 2.620 1.455 2.825 2.547 3.360 2.482 2.020 800.96 8.1238

78.120 50.000 35.000 30.000 39.880 1.1000 1.0960 1.0762 1.0850 1.0741 1.0557 1.0476 1.0781 1.0562 1.0246 1.695 3.176 2.577 0.432 2.812 2.238 4.050 2.512 2.979 963.32 3.0858

57.245 36.271 33.879 22.312 26.475 1.0976 1.0910 1.0703 1.0804 1.0653 1.0560 1.0087 1.0278 1.0465 1.0165 3.047 3.099 2.343 0.359 2.702 2.378 3.892 2.710 2.471 843.63 4.9785

47.334 21.017 18.478 20.329 12.000 1.1000 1.0883 1.0646 1.0726 1.0748 1.0943 1.0759 0.9578 1.0391 1.0071 2.833 0.624 4.839 4.925 1.764 4.167 1.692 3.013 3.447 801.38 8.3504

80.000 49.923 35.000 20.360 34.348 1.1000 1.0957 1.0804 1.0821 1.0975 1.0958 1.0001 0.9931 1.0430 0.9988 3.771 1.379 2.334 1.294 2.707 2.520 1.975 4.927 4.917 936.97 3.4289

58.571 34.090 35.000 20.368 22.735 1.1000 1.0949 1.0760 1.0834 1.0965 1.0928 1.0044 1.0028 1.0439 0.9916 3.727 1.284 1.852 2.311 2.718 2.212 2.273 5.000 4.898 833.57 5.1990

Table 7 Comparisons of the obtained objectives with other methods for test Case 3. Methods

Objectives

Cost

Loss

NSMOGSA [1]

Min C Min L Comp Min C Min L Comp Min C Min L Comp Min C Min L Comp Min C Min L Comp Min C Min L Comp

799.6095 873.5107 819.2745 802.1046 928.5099 832.6709 801.714 875.0005 823.8875 800.96 963.32 843.63 801.38 936.97 833.57 799.13 967.04 835.36

7.9027 3.4925 5.1204 8.1412 3.5165 5.3143 8.1734 4.3571 5.7699 8.1238 3.0858 4.9785 8.3504 3.4289 5.199 8.7851 2.8583 4.9099

MOHS [27]

NSGA-II [27]

MOPSO

re-implemented NSGA-II MOEA/D

369 370

non-dominated solution, the normalized membership function can be formulated as:

Nobj

371

i =

j=1

372 373 374

375 376

377 378 379 380 381

ij

M Nobj i=1

NSGA-II

Comp

(12)

 j=1 ij

where M is the number of non-dominated solutions, and Nobj is the number of objective considered. The best compromise solution is the one who has the maximum value of i . 5.4. Detailed steps of modified MOEA/D approach to MOOPF problem The MOOPF problem involves the solving of control variables and the calculating of state variables. The popular control variables of a MOOPF problem are generator active power outputs, generator bus voltages, tap positions of tap changing transformers and reactive power outputs of shunt compensators [30]. The state vari-

ables are usually composed of generator reactive power outputs, generator bus phase angles, voltages and phase angles of load bus, active and reactive power outputs of slack bus, etc. The solution technique of a MOOPF problem needs to give all these control variables and state variables. All the control variables constitute an individual which represents one solution to the MOOPF problem. Any individual i can be defined as,

383 384 385 386 387 388



P2i , P3i , · · ·, PNi g , V1i , V2i , · · ·, VNi g , 1i , 2i , · · ·, i N , Q1i , Q2i , · · ·, QNi T C

Xi =

382



i = 1, 2, · · ·, N

To apply the modified MOEA/D algorithm to the presented MOOPF problem, the following steps should be done. Step 1: Define the input data. Select the optimized objective function considered, and read the power system data and their parameters, e.g., the grid data, the fuel cost coefficients and the emission parameters. Step 2: Set the parameters of the proposed modified MOEA/D algorithm. The parameters include population size, neighborhood size, external Pareto archive size, the maximum number of iterations, the mutation and crossover rate of the DE operator (Here we employ DE instead of genetic operator). Initialize the storage space for ideal and reference points of the algorithm, and set EP = ∅. Step 3: Initialize the evolution swarm; generate uniform spread of N weight vectors 1 , 2 , · · ·, N using simplex-latte design method, compute the Euclidean distances between any two weight vectors and then work out the T closest weight vectors to each weight vector.

 For each sub-problem i = 1, 2, · · ·, N, set B (i) = i1 , i2 , · · ·, iT , where i1 , i2 , · · ·, iT are the T closest weight vectors to i . The corresponding individuals are considered as members of the neighborhood of sub-problem i. Step 4: Perform load flow calculation using MATPOWER 4.1 for each individual, evaluate their fitness values according to the objective functions considered, and update the ideal point of each objective function and the reference points of the algorithm. Step 5: Set sub-problem i = 1. Step 6: Select three different individuals from the neighborhood of the ith sub-problem and generate a new individual xnew using DE

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9 MOEA/D NSGA-II MOPSO

8

Power loss (MW)

7

6

5

4

3

2 780

800

820

840

860

880 Fuel cost ($/h)

900

920

940

960

980

Fig. 4. Pareto fronts obtained using MOEA/D, NSGA-II and MOPSO for Case 3.

417 418 419 420 421 422 423

424

operator instead of GA operator with the employed mutation and crossover factors. Step 7: Perform load flow calculation over the new generated individual, evaluate its fitness values and update the reference points and the ideal point if any objective function of the individual is better than the current one.   Step 8: Calculate the g mte xnew |i , z R =



max j=1,2,···m

i

ij |

fj (xnew )−z ∗ z max −z min +ε j j

|

for the generated individual using g mte



|i , z R



the and the reference points. If the value of xnew is better than any current individuals of the neighborhood of 426 the ith sub-problem, then replace the current one and update its 427 corresponding fitness values. 428 Step 9: Remove from EP all the vectors dominated by xnew , and 429 add xnew to EP if no vectors in EP dominate xnew . 430 Step 10: i = i + 1, if i ≤ N, then go to Step 6. Otherwise, go to the 431 next step. 432 Step 11: If stopping criteria is satisfied, then output the Pareto 433 solutions of the problem, i.e., EP, and go to the next step. Otherwise, 434 go to Step 5. 435 Step 12: Upon the imprecise preference information of a 436 decision-maker, choose the best compromise solution from the EP 437 Q7 as Subsection 4.3 shown. 438 425

6. Simulation studies In order to verify the effectiveness and performance of the proposed modified MOEA/D approach for solving MOOPF problems, standard IEEE 30-bus power system is considered for simulation study. The system includes 6 generators, 41 transmission lines and 4 transformers with off-nominal tap ratio in the lines 6–9, 6–10, 4–12, and 27–28. The limits of generator buses and load buses are between 0.95–1.1 p.u., and 0.9–1.05 p.u., respectively. The lower and upper limits of transformer taps are 0.9 p.u. and 1.05 p.u., respectively. The detailed data of this system are given in [38].

6.1. Test systems and parameters setting The modified MOEA/D approach to the MOOPF problem is implemented in Matlab 2010 rb and the simulation is run on a Xeon(R) E3-1245 PC with 4GB RAM. To demonstrate the effectiveness of the proposed MOEA/D approach to the solution of MOOPF problems, seven different test cases have been considered as follows: Case 1. Minimization of fuel cost, emissions, power losses and voltage magnitude deviations independently using DE.

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440 441 442 443 444 445 446 447 448

449

450 451 452 453 454 455

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1 MOEA/D NSGA-II MOPSO 0.9

0.8

0.7

VMD (p.u.)

0.6

0.5

0.4

0.3

0.2

0.1 799

800

801

802

803 804 Fuel cost ($/h)

805

806

807

808

Fig. 5. Pareto fronts obtained using MOEA/D, NSGA-II and MOPSO for Case 4.

458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478

Case 2.

Minimization of fuel cost and emissions simultaneously.

Case 3. Minimization of fuel cost and active power losses simultaneously. Case 4. Minimization of fuel cost and voltage magnitude deviations simultaneously. Case 5. Minimization of fuel cost, emissions and active power losses simultaneously. Case 6. Minimization of fuel cost, emissions and voltage magnitude deviations simultaneously. Case 7. Minimization of fuel cost, power losses, and voltage magnitude deviations simultaneously. In order to compare the efficiency on the MOOPF problem, the multi-objective PSO [9] and NSGA-II [8] are re-implemented in the same run environment. In the rest circumstance of the paper that does not cause ambiguity, we use MOPSO and NSGA-II to represent the re-implemented ones respectively. The obtained results and the comparisons of the proposed modified MOEA/D with other algorithms confirm the effectiveness and performance of the proposed algorithm. The parameters of the modified MOEA/D approach are tuned through trial and errors. The tuned parameters for different cases are shown in Table 1. The parameters employed in the re-

implemented NSGA-II and MOPSO approaches are also appended in the table. From this table we can see that most of the parameters in MOEA/D are set as the same values for different cases except for population sizes. The parameters are only tuned through several trials. The parameters for NSGA-II and MOPSO are set according with the references [8,9] except for the population size, maximum iteration and repository size which are set reference to MOEA/D. 6.2. Obtained optimal results 6.2.1. Case 1: minimizing each objective function independently using DE In order to analysis the relationships among different objective functions, we run the DE algorithm for each independent single objective optimization several times. The optimal objectives for different optimizations are shown in Table 2. The other corresponding objective values are also shown in this table. The convergence curves of the optimized objective and other corresponding objective values are shown in Fig. 2. In the figure, (a) represents the convergence curves of various objectives when optimizing the fuel cost, (b) represents the convergence curves when optimizing the voltage magnitude deviations, (c) represents the convergence curves when optimizing the power loss, and (d) represents the convergence curves when optimizing the emission. It is found from the

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486

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Table 8 Comparisons of solutions obtained using different methods for Case 4. Control Variables

MOEA/D

MOPSO

Min C Pg2 Pg5 Pg8 Pg11 Pg13 Vg1 Vg2 Vg5 Vg8 Vg11 Vg13 6−9 6−10 4−12 27−28 Qc10 Qc12 Qc15 Qc17 Qc20 Qc21 Qc23 Qc24 Qc29 Fuel Cost VMD

48.465 21.488 22.125 11.659 12.000 1.1000 1.0877 1.0646 1.0725 1.0676 1.0808 1.1000 0.9415 1.0849 1.0267 2.609 4.943 3.983 4.838 4.711 4.655 4.431 5.000 2.749 799.36 0.9271

Min VMD 48.405 21.724 22.890 12.050 12.000 1.0346 1.0220 1.0097 1.0078 1.0748 0.9890 1.0999 0.9125 0.9549 0.9604 3.830 4.992 4.937 2.851 5.000 4.996 5.000 4.994 1.434 803.83 0.1018

NSGA-II

Comp

Min C

Min VMD

Comp

Min C

Min VMD

Comp

48.825 21.303 20.252 11.928 12.016 1.0987 1.0793 1.0460 1.0509 1.0346 1.0223 1.1000 0.9573 1.0597 1.0134 3.386 5.000 3.870 2.636 4.989 4.980 4.600 4.986 2.521 799.99 0.3540

48.387 21.179 21.096 12.252 12.000 1.1000 1.0876 1.0621 1.0697 1.0823 1.0668 1.0347 1.0600 1.0593 1.0223 1.587 0.886 2.226 1.771 3.926 2.088 1.592 2.604 1.336 799.69 0.6006

48.796 19.031 25.006 13.478 14.629 1.0374 1.0293 1.0216 1.0093 1.0189 1.0044 0.9502 0.9673 0.9500 0.9655 1.917 0.965 2.361 2.116 1.803 4.596 4.321 2.788 3.350 805.53 0.1439

49.545 21.499 21.784 12.371 12.149 1.1000 1.0802 1.0496 1.0583 1.0637 1.0574 1.0254 1.0523 1.0661 1.0087 1.165 0.991 1.542 1.289 3.967 2.009 1.670 2.346 1.797 800.03 0.4422

48.346 20.187 23.305 11.561 12.002 1.1000 1.0869 1.0606 1.0689 1.0946 1.0879 1.0099 1.0199 1.0584 1.0085 2.335 0.056 3.542 1.568 0.950 0.017 3.821 0.946 1.831 799.76 0.8672

49.458 20.689 22.359 10.725 13.643 1.0127 0.9991 1.0176 1.0083 1.0597 1.0711 1.0002 0.9205 1.0796 0.9436 2.557 0.067 3.541 1.509 1.183 0.038 4.000 1.369 0.143 807.36 0.1514

48.437 20.314 23.005 11.368 12.000 1.1000 1.0823 1.0514 1.0566 1.0891 1.0446 1.0532 1.0038 1.0641 1.0064 2.388 0.055 3.537 1.626 0.970 0.053 3.629 1.509 0.951 800.06 0.4486

Table 9 Comparisons of obtained solutions using different methods for Case 5. Control Variables

MOEA/D Min C

Pg2 Pg5 Pg8 Pg11 Pg13 Vg1 Vg2 Vg5 Vg8 Vg11 Vg13 6−9 6−10 4−12 27−28 Qc10 Qc12 Qc15 Qc17 Qc20 Qc21 Qc23 Qc24 Qc29 Fuel Cost Emission Power Loss

501 502 503 504 505

506 507 508

MOPSO Min E

50.48 67.23 20.30 50.00 21.68 35.00 10.00 30.00 12.00 40.00 1.0996 1.1000 1.0886 1.0975 1.0646 1.0768 1.0675 1.0869 1.0998 1.0993 1.1000 1.0992 1.0819 1.0916 0.9000 0.9000 1.0060 0.9906 0.9684 0.9738 5.000 4.995 5.000 4.561 5.000 5.000 5.000 4.672 5.000 4.687 4.434 4.950 2.671 3.337 4.470 5.000 4.185 2.425 799.34 943.23 0.3689 0.2047 8.7769 2.9941

Min L

Comp

80.00 59.93 50.00 44.22 35.00 35.00 29.99 30.00 40.00 37.36 1.1000 1.1000 1.0971 1.0949 1.0794 1.0786 1.0863 1.0855 1.0995 1.1000 1.0999 1.0997 1.0557 1.0715 0.9078 0.9000 0.9912 0.9913 0.9745 0.9706 4.992 4.767 5.000 4.594 4.464 4.514 5.000 4.348 4.611 5.000 4.955 4.704 3.546 3.327 5.000 5.000 2.749 1.262 967.06 902.54 0.2072 0.2107 2.8544 3.4594

NSGA-II

Min C

Min E

Min L

Comp

47.52 21.30 20.63 13.47 12.14 1.1000 1.0888 1.0498 1.0772 1.0822 1.0464 1.0447 0.9982 1.0470 1.0199 0.497 1.859 2.978 3.110 4.944 1.253 1.360 2.593 4.844 800.09 0.3658 8.8476

71.53 49.99 35.00 30.00 40.00 1.0464 1.0355 1.0095 1.0169 1.0620 1.0866 0.9318 0.9899 1.0132 0.9455 3.202 0.448 3.376 1.815 0.559 2.703 0.367 1.284 1.364 951.70 0.2051 3.4514

70.82 47.23 34.13 29.99 39.09 1.0517 1.0287 0.9993 1.0066 1.0567 1.0857 0.9266 0.9859 1.0040 0.9318 3.064 0.569 3.365 1.178 0.244 2.941 0.500 1.163 1.188 958.16 0.2057 3.3886

63.02 48.45 42.06 21.35 33.66 20.82 30.00 11.79 34.36 12.02 1.0863 1.1000 1.0791 1.0877 1.0483 1.0619 1.0540 1.0687 1.0927 1.0994 1.0988 1.1000 0.9891 1.0657 1.0004 0.9001 1.0138 1.0273 0.9956 0.9747 4.001 2.899 1.667 4.913 2.319 3.110 2.589 1.933 1.774 4.864 3.520 0.876 0.073 4.637 1.561 2.437 3.002 2.192 891.48 799.27 0.2144 0.3678 3.9557 8.7149

figure that there really exists some certain relationships among these objectives considered, for example, there may be inverse relationships between emission and fuel cost, and there may be positive relationships between power loss and voltage magnitude deviations. 6.2.2. Case 2: minimizing fuel cost and emission simultaneously The present case considers minimizing the fuel cost and emission simultaneously. The comparisons of the optimal objectives

Min C

Min E

Min L

Comp

67.76 50.00 35.00 30.00 40.00 1.0553 1.0488 1.0264 1.0344 1.0897 1.0345 1.0403 0.9725 1.0210 0.9836 1.542 2.347 2.366 0.597 0.869 1.980 2.631 1.517 2.587 945.14 0.2049 3.4179

80.00 50.00 35.00 30.00 40.00 1.0577 1.0543 1.0336 1.0402 1.0949 1.0439 1.0272 0.9512 0.9980 0.9738 1.221 2.722 2.216 0.907 0.783 2.339 2.658 1.585 2.342 967.98 0.2073 3.2354

62.61 42.96 35.00 29.65 38.42 1.0978 1.0853 1.0581 1.0711 1.0376 1.0196 1.0636 1.0235 1.0836 1.0560 4.015 4.383 4.619 1.544 1.340 1.936 4.533 3.169 4.347 903.79 0.2103 3.7917

using the proposed methods with other algorithms in the past literature are shown in Table 3. The obtained optimal objectives using our re-implemented MOPSO and NSGA-II are also listed in the table. It is clearly seen from the table that the proposed MOEA/D obtains both the minimum fuel cost (Min C) and the minimum emission (Min E) The control variable values of the solutions with the minimum fuel cost, the minimum emission and their compromise (Comp) obtained using modified MOEA/D, NSGA-II and MOPSO are shown in Table 4. It is found obviously from this table that the

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MOEA/D NSGA-II MOPSO

10

9

8

Power loss (MW)

7

6

5

4

3

2 0.38 0.36 0.34

1000 0.32

950

0.3 0.28

900 0.26

850

0.24 800

0.22 Emission (ton/h)

0.2

750

Fuel cost ($/h)

Fig. 6. Pareto fronts obtained using MOEA/D, NSGA-II and MOPSO for Case 5.

518 519 520 521 522 523 524 525 526 527 528 529 530

531 532 533 534 535 536 537 538 539 540

MOEA/D obtains the minimum fuel cost and emission among these three approaches. The Pareto fronts obtained by MOEA/D, NSGA-II and MOPSO for this case have been illustrated in Fig. 3. It is also found from the figure that the MOEA/D obtains well distributed Pareto solutions over the Pareto front. This figure also shows that the modified MOEA/D method can efficiently explore better end Pareto-optimal solutions than MOPSO and NSGA-II do. The numbers of Pareto solutions dominated by the other two methods for this case are shown in Table 5. It is found from the table that no solutions obtained by MOEA/D are dominated by solutions obtained by NSGA-II and MOPSO. While there are 86 and 100 solutions obtained by NSGA-II and MOPSO are dominated by the members of the solutions obtained by MOEA/D, respectively. 6.2.3. Case 3: minimizing fuel cost and power loss simultaneously In this case, the total fuel cost and the active power loss are considered simultaneously as the multi-objective function of the OPF. The values of control variable for solutions with the minimum fuel cost (Min C), the minimum power loss (Min L) and the compromise (Comp) for this case are given in Table 6. This table also shows the corresponding fuel cost and power loss of these solutions obtained using MOEA/D, NSGA-II and MOPSO approaches. The comparisons of the optimal and compromise objectives by the proposed method with other algorithms published in the literature are shown

in Table 7. From this table, the similar conclusions of MOEA/DA obtaining the best results still can be get. The Pareto optimal solutions obtained by MOEA/D, NSGA-II and MOPSO are shown in Fig. 4. It is observed obviously from the figure that the MOEA/D obtains the well and diverse Pareto solutions. This figure also shows that the MOEA/D obtains the better Pareto front than other two methods in case of the MOOPF problem considering fuel cost and power losses. The numbers of solutions dominated by other two methods are also shown in Table 5 for this case. It is obviously seen from this table that no solutions obtained by MOEA/D will be dominated by the solutions obtained by the other two methods. 6.2.4. Case 4: minimizing fuel cost and voltage magnitude deviations In this case, the total fuel cost and voltage magnitude deviations are considered. The achieved optimal Pareto solutions using the proposed MOEA/D, NSGA-II and MOPSO are illustrated in Fig. 5. It is found from the comparisons that the MOEA/D and the MOPSO obtain better Pareto front than NSGA-II does. The solutions with the minimum fuel cost (Min C), the minimum voltage magnitude deviations (Min VMD) and their compromise (Comp) obtained by these three approaches are listed in Table 8. It is found from the table that MOEA/D obtains the minimum fuel cost and the minimum voltage magnitude deviations. It is also found from the tables that the com-

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Table 10 Comparisons of obtained solutions using different methods for Case 6. Control Variables

MOEA/D Min C

Pg2 Pg5 Pg8 Pg11 Pg13 Vg1 Vg2 Vg5 Vg8 Vg11 Vg13 6−9 6−10 4−12 27−28 Qc10 Qc12 Qc15 Qc17 Qc20 Qc21 Qc23 Qc24 Qc29 Fuel Cost Emission VMD

MOPSO Min E

Min VMD

52.36 67.37 76.77 22.39 50.00 50.00 20.67 35.00 30.07 10.41 29.97 27.37 12.00 40.00 39.96 1.0985 1.0902 1.0167 1.0854 1.0817 1.0135 1.0568 1.0696 1.0218 1.0685 1.0641 1.0096 1.0382 1.1000 1.0006 1.0634 1.0894 0.9909 1.0074 1.0791 1.0123 1.0228 0.9106 0.9000 1.0090 1.0116 0.9545 0.9820 0.9605 0.9663 3.977 4.780 4.926 0.000 0.000 3.150 4.304 3.307 5.000 1.683 3.223 1.283 3.031 4.819 4.920 4.950 4.108 5.000 4.962 4.584 4.917 4.124 3.948 5.000 2.217 5.000 3.884 800.25 943.81 951.41 0.3596 0.2048 0.2092 0.9520 1.6087 0.0939

NSGA-II

Comp

Min C

Min E

Min VMD

Comp

Min C

Min E

Min VMD

Comp

61.78 30.48 34.40 30.00 27.65 1.0276 1.0246 1.0102 1.0080 1.0168 0.9948 1.0182 0.9000 0.9472 0.9645 4.594 2.246 4.981 0.000 4.691 4.643 4.886 3.633 2.509 850.28 0.2332 0.1155

45.93 18.49 20.05 16.72 16.62 1.0868 1.0725 1.0432 1.0596 1.0854 1.0704 1.0220 0.9191 0.9731 0.9830 3.986 4.141 1.284 3.194 1.296 3.425 2.004 2.436 3.130 802.39 0.3565 1.1757

69.36 48.41 34.29 29.85 40.00 1.0448 1.0341 1.0086 1.0191 1.0454 1.0506 0.9567 0.9787 1.0225 0.9619 2.336 2.806 2.616 0.258 2.403 4.732 0.348 1.258 4.507 939.68 0.2059 0.3070

66.87 33.62 32.50 25.88 29.27 1.0487 1.0257 0.9975 1.0110 1.0380 1.0362 0.9667 0.9613 1.0320 0.9637 2.955 2.612 2.886 0.943 2.461 4.500 0.769 0.890 3.784 856.94 0.2309 0.2014

66.13 32.01 30.80 24.47 27.78 1.0480 1.0369 0.9910 1.0105 1.0400 1.0396 0.9835 0.9719 1.0336 0.9551 2.244 2.467 3.088 1.255 2.394 4.471 0.334 1.212 4.117 846.93 0.2386 0.2188

48.46 20.81 28.65 10.84 12.01 1.0861 1.0654 1.0230 1.0351 1.0421 1.0981 0.9720 0.9937 1.0623 0.9443 2.258 0.447 1.252 2.011 4.039 4.898 2.264 2.703 0.594 801.58 0.3512 0.7829

67.88 50.00 35.00 30.00 40.00 1.0524 1.0427 1.0100 1.0128 1.0355 1.0352 0.9628 1.0853 0.9480 0.9346 2.680 2.871 1.989 2.290 1.412 4.247 3.484 2.742 1.045 946.26 0.2050 0.2677

59.91 23.62 28.56 20.05 25.49 1.0289 1.0242 1.0144 1.0111 1.0433 0.9922 1.0474 0.9000 0.9436 0.9531 2.805 3.514 2.420 1.559 1.841 3.969 4.940 4.953 0.707 821.71 0.2715 0.1184

59.72 23.96 28.75 19.96 28.98 1.0317 1.0278 1.0133 1.0128 1.0433 0.9984 1.0473 0.9096 0.9384 0.9526 2.849 3.578 2.445 1.513 1.795 3.947 4.951 4.930 0.767 825.86 0.2648 0.1421

Min L

Min VMD

Comp

79.97 50.00 34.99 30.00 39.98 1.0707 1.0665 1.0440 1.0511 1.0809 1.0582 1.0268 1.0188 1.0371 1.0022 3.488 0.448 1.027 2.777 3.953 3.650 4.636 2.730 2.987 967.58 3.1320 0.5990

58.40 17.11 33.29 11.48 28.43 1.0432 1.0361 1.0221 1.0098 0.9929 0.9916 0.9806 0.9000 0.9306 0.9467 3.090 0.442 0.993 0.117 4.278 2.649 4.083 5.000 0.320 821.14 8.4726 0.1188

70.88 36.59 26.31 13.73 22.42 1.0450 1.0360 1.0156 1.0094 1.0393 1.0263 0.9199 1.0381 0.9802 0.9521 4.317 1.421 0.939 0.132 3.978 3.335 2.900 1.189 2.056 843.14 6.4917 0.1931

Table 11 Comparisons of obtained solutions using different methods for Case 7. Control Variables

MOEA/D Min C

Pg2 Pg5 Pg8 Pg11 Pg13 Vg1 Vg2 Vg5 Vg8 Vg11 Vg13 6−9 6−10 4−12 27−28 Qc10 Qc12 Qc15 Qc17 Qc20 Qc21 Qc23 Qc24 Qc29 Fuel Cost Power Loss VMD

MOPSO Min L

50.45 80.00 20.06 50.00 21.38 34.75 12.75 29.94 12.00 40.00 1.1000 1.0998 1.0893 1.0946 1.0709 1.0767 1.0766 1.0836 1.0692 1.1000 1.0958 1.1000 1.0923 1.0749 0.9000 0.9040 1.0705 1.0136 0.9922 0.9808 5.000 3.805 3.544 4.822 3.425 4.772 2.321 4.411 5.000 4.501 4.689 4.642 3.780 2.219 4.448 5.000 3.974 3.120 799.53 966.65 8.6715 2.8812 1.3732 1.8423

NSGA-II

Min VMD

Comp

Min C

Min L

Min VMD

Comp

50.67 15.00 27.90 15.71 12.00 1.0393 1.0197 1.0208 1.0098 1.0130 1.0001 1.0185 0.9000 0.9586 0.9638 5.000 0.495 4.941 1.109 5.000 2.248 5.000 4.826 2.197 808.12 10.0173 0.0998

46.84 30.85 34.33 29.74 21.85 1.0365 1.0267 1.0011 1.0073 1.0506 1.0173 1.0827 0.9053 0.9927 0.9706 3.421 1.767 3.454 3.356 4.832 5.000 4.428 3.723 3.561 831.81 5.9926 0.1355

41.21 21.68 29.28 15.25 12.92 1.0723 1.0575 1.0344 1.0411 1.0249 1.0250 1.0615 0.9608 1.0472 1.0035 3.148 1.676 4.629 1.386 1.636 3.416 3.854 4.514 2.161 803.36 8.5513 0.3726

77.07 49.64 35.00 30.00 35.56 1.0973 1.0948 1.0696 1.0779 0.9958 1.0362 1.0978 1.0065 1.1000 1.0529 3.857 1.133 2.822 3.460 0.424 4.459 4.866 3.560 3.238 949.29 3.2734 0.6686

54.00 34.13 25.56 21.37 22.95 1.0381 1.0213 1.0012 1.0083 1.0609 1.0215 1.0156 0.9170 1.0097 0.9593 2.456 2.720 3.903 1.300 2.717 2.191 3.297 4.647 2.115 827.82 6.5929 0.1588

53.97 48.43 31.92 21.06 30.24 20.76 21.00 12.23 22.11 12.00 1.0472 1.1000 1.0317 1.0878 1.0100 1.0557 1.0175 1.0686 1.0512 1.0893 1.0229 1.0950 1.0294 1.0221 0.9323 0.9656 1.0194 1.0619 0.9712 1.0047 3.032 0.782 2.062 3.758 4.258 2.559 1.309 1.325 2.418 5.000 2.465 0.815 3.360 4.220 4.244 1.783 2.301 3.054 824.62 799.48 6.4223 8.7776 0.1877 1.1036

Min C

Table 12 The statistical results of ID on different cases after 31 independent runs. Test Cases

MOEA/D

NSGA-II

Mean

Std

Mean

Case 2 Case 3 Case 4 Case 5 Case 6 Case 7

0.35 0.38 0.04 0.28 0.89 0.55

0.02 0.02 0.01 0.01 0.14 0.03

1.62 1.80 7.77 0.47 0.65 1.30

MOPSO Std 3.17 2.06 13.12 0.14 0.10 1.20

p

h

Mean

Std

p

h

1.17E-06 1.17E-06 1.17E-06 1.17E-06 4.53E-06 1.17E-06

1 1 1 1 1 1

0.63 0.83 0.10 1.00 0.70 0.97

0.16 0.08 0.02 0.25 0.11 0.10

1.17E-06 1.17E-06 1.17E-06 1.17E-06 7.54E-05 1.17E-06

1 1 1 1 1 1

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MOEA/D NSGA-II MOPSO

2 1.8 1.6 1.4

VMD (p.u.)

1.2 1 0.8 0.6 0.4 0.2 0 0.38 0.36 0.34

960 0.32

940 920

0.3 900

0.28

880

0.26

860

0.24

840

0.22 Emission (ton/h)

0.2

820 800

Fuel cost ($/h)

Fig. 7. Pareto fronts obtained using different methods for Case 6.

564 565 566 567 568

569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586

promise solution by MOEA/D dominate the compromise ones by NSGA-II and MOPSO. The numbers of solutions dominated by other two methods are also shown in Table 5 for this case. It is found from the table that there are no solution obtained by MOEA/D would be dominated by that obtained by NSGA-II and MOPSO.

6.2.5. Case 5–Case 7 To verify the effective of the MOEA/D approach deeply, the algorithm is also run upon the cases with three optimized objective function, i.e., minimum fuel cost, emission and power losses simultaneously, minimum fuel cost, emission and voltage magnitude deviations simultaneously, and minimum fuel cost, power losses and voltage magnitude deviations simultaneously. The Pareto solutions obtained by MOEA/D approach for these three test cases are also compared with that obtained by MOPSO and NSGA-II. The Pareto fronts obtained by the modified MOEA/D, NSGA-II and MOPSO for this three cases are illustrated in Figs. 6, 7 and 8, respectively. It is found from the figures that MOEA/D obtains the most distributed and approximate Pareto solutions among these three methods for all the three cases. The comparisons of obtained solutions with the minimum fuel cost (Min C), the minimum emission (Min E), the minimum power loss (Min L) and their compromise solution (Comp) obtained by different methods for Case 5 are shown in Table 9. It is clearly seen from the table that the MOEA/D

obtains the minimum emission and power loss and that the NSGAII obtains the minimum fuel cost. It is also found from this table that the compromise solutions achieved by these three methods can’t dominate each other. The minimum cost, minimum emission, minimum voltage magnitude deviations (Min VMD) solutions achieved by MOEA/D, NSGA-II and MOPSO for Case 6 are listed in Table 10. It is clearly seen from the table that the modified MOEA/DA gains all the minimum values of these three objectives for this case. The comparisons of obtained solutions among different methods for Case 7 are shown in Table 11. It is clearly observed from this table that the MOEA/D obtains the minimum power losses and the minimum voltage magnitude deviations and that the NSGA-II obtains the minimum total fuel cost.

6.3. Analysis of parameters To demonstrate the parameters’ impact on the algorithm’s efficiency, the sensitivity analysis is performed for different cases. As a sample, here we only present the results of test Case 2. The ParetoOptimal solutions obtained by the modified MOEA/D approach with different population sizes, niche sizes and maximum iterations are shown in Figs. 9, 10 and 11 , respectively. Generally speaking, population sizes have positive effects to the optimal solutions. However, we can observe clearly from Fig. 9 that there are only slight differ-

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587 588 589 590 591 592 593 594 595 596 597 598 599

600

601 602 603 604 605 606 607 608

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MOEA/D NSGA-II MOPSO

2 1.8 1.6 1.4

VMD (p.u.)

1.2 1 0.8 0.6 0.4 0.2 0 11 10 9

1000 8

950

7 6

900 5

850

4

800

3 Power loss (MW)

2

750

Fuel cost ($/h)

Fig. 8. Pareto fronts obtained using different methods for Case 7. 9

7.2

6.9 6.85

6.8

8

30 40 50 60 70 80 90 100

7 6.95

7

6.8 804

6.6

806

808

810

6.4 7

6.2 6

Power loss (MW)

795

800

805

810

815

820

825

8303.8

3.3

6

3.25 3.2

3.6

3.15 3.1

3.4

918 5

920

922

924

3.2 3 4.4

4

2.8

4.2

910

915

920

925

930

935

940

945

4 4.1 3.8 3 3.6

4

3.9 3.4 3.2 2 780

868 860

870 865 800

872 870

874 875

880 820

885

890 840

895 860

880 Fuel cost ($/h)

900

920

940

960

980

Fig. 9. Pareto front obtained by MOEA/D using different population sizes with maximum iteration of 500 and niche size of 20.

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100 generation 200 generation 300 generation 400 generation 500 generation

6.2 6 8

5.8 5.6 5.4

7 5.2

5

Power loss (MW)

5

4.8 805

6

810

815

820

825

830

835

840

4.6 4.4 4.2

5 4 3.8

4.2 4

830

835

840

845

850

855

860

865

4 3.8 3.6

3

3.4 3.2 3 890

2 780

895

900 800

905

910 820

915

920

925

840

860

880 Fuel cost ($/h)

900

920

940

960

980

Fig. 10. Pareto front obtained by MOEA/D using different maximum iterations with population size of 100 and niche size of 20.

9 5 10 15 30 40 50 70 100

4.5

5.6

4.45 8

4.4

5.4

4.35 4.3

5.2 7

849

850

851

852

853

854

855

856

Power loss (MW)

5

4.8

6

4.6 5 4.4 820

825

830

835

840

845

850

855

4

3

2 780

800

820

840

860

880 Fuel cost ($/h)

900

920

940

960

980

Fig. 11. Pareto front obtained by MOEA/D using different niche sizes with population size of 100 and maximum iteration of 500.

Table 13 The statistical results of HD on different cases after 31 independent runs. Test Cases

MOEA/D Mean

Case 2 Case 3 Case 4 Case 5 Case 6 Case 7

8.05 3.23 1.82 3.45 16.48 3.76

NSGA-II Std 0.41 1.22 0.77 0.58 3.34 1.14

MOPSO

Mean

Std

p

h

Mean

Std

p

h

19.40 26.61 11.08 5.14 3.31 9.96

28.84 25.66 15.15 6.07 1.17 14.74

9.06E-01 2.31E-05 2.52E-05 5.83E-01 1.17E-06 6.81E-01

0 1 1 0 1 0

9.45 9.21 1.39 22.07 10.46 6.58

8.22 4.80 1.44 10.06 5.40 1.34

1.17E-01 3.75E-06 6.84E-03 1.17E-06 1.97E-04 7.89E-06

0 1 1 1 1 1

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Table 14 The statistical results of HV on the different cases after 31 independent runs. Test Cases

MOEA/D Mean

Case 2 Case 3 Case 4 Case 5 Case 6 Case 7

24.01 901.95 86.14 119.73 55.89 2258.79

NSGA-II Std 0.03 1.08 0.49 0.07 0.14 5.46

MOPSO

Mean

Std

p

h

Mean

Std

p

h

22.45 812.31 71.45 112.43 52.47 2045.16

2.06 56.45 19.09 3.55 1.05 35.46

1.17E-06 1.17E-06 1.30E-06 1.17E-06 1.17E-06 1.17E-06

1 1 1 1 1 1

23.29 856.57 84.50 109.02 52.04 2008.16

0.15 8.63 0.52 1.21 0.75 39.13

1.17E-06 1.17E-06 1.17E-06 1.17E-06 1.17E-06 1.17E-06

1 1 1 1 1 1

Table 15 The statistical results of Purity on the different cases after 31 independent runs. Test Cases

Case 2 Case 3 Case 4 Case 5 Case 6 Case 7

MOEA/D

NSGA-II

MOPSO

Mean

Std

Mean

Std

p

h

Mean

Std

p

h

0.45 0.44 0.15 0.59 0.53 0.60

0.24 0.24 0.13 0.14 0.14 0.08

0.01 0.00 0.00 0.04 0.19 0.05

0.03 0.00 0.00 0.05 0.11 0.03

1.17E-06 1.17E-06 2.54E-06 1.17E-06 1.29E-06 1.17E-06

1 1 1 1 1 1

0.00 0.00 0.00 0.00 0.00 0.00

0.01 0.00 0.00 0.00 0.00 0.00

1.17E-06 1.17E-06 2.54E-06 1.17E-06 1.17E-06 1.17E-06

1 1 1 1 1 1

Fig. 12. Box plots of the three algorithms on metric ID for different cases in 31 runs.

609 610 611 612 613 614 615 616 617 618

619

620 621 622

ences among these different parameters of population size except for 30. Here we surmise that the slight differences may benefit from another important parameter, i.e., the maximum number of generation (500 iterations employed here). It is observed from Fig. 10 that the iteration of 100 provided the worst Pareto solutions and that the larger number of maximum iterations the MOEA/D has the better Pareto solutions it will obtain. From Fig. 11 and other two figures, we can observe that different niche sizes produced nearly equal solutions and hence, the niche size provides the minimum influence to the solution quality. 6.4. Statistical analysis In order to compare the performance among the proposed modified MOEA/D, NSGA-II and MOPSO, some necessary statistical tests should be done. On the performances of multi-objective

evolutionary algorithms, two aspects should be considered – convergence and diversity. In terms of convergence, the obtained non-dominated solutions should be close to the true ParetoOptimal Front. In terms of diversity, these solutions should be distributed as uniformly and diversely as possible. However, convergence and diversity can hardly be measured by a single metric. Hence, four different metrics are employed here. (1) Distance from reference set (ID ). This metric was proposed by Czyzak and Jaszkiewicz [56] and it is used to indicate the closeness of the obtained solutions to the true Pareto-Optimal Front. (2) Hausdorff distance (HD ). This metric was proposed by Oliver et al. and it describes the degree of similarity between the obtained solutions and the true Pareto-Optimal Front [57]. (3) Hypevolume (HV ). This metric was proposed by Zizler et al. [58] and it calculates the volume in the objective domain covered by the obtained non-dominated solutions. It is used to evaluate the broadness of the non-dominated

Please cite this article in press as: J. Zhang, et al., A modified MOEA/D approach to the solution of multi-objective optimal power flow problem, Appl. Soft Comput. J. (2016), http://dx.doi.org/10.1016/j.asoc.2016.06.022

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Fig. 13. Box plots of the three algorithms on metric HD for different cases in 31 runs.

Fig. 14. Box plots of the three algorithms on metric HV for different cases in 31 runs.

639 640 641 642 643 644 645 646 647 648 649 650

individuals. (4) Purity (Purity). This metric was proposed by Bandyopadhyay et al. [59] and it is mainly used to compare the quality and the convergence of results. As shown in Section 3, multi-objective optimal power flow is a difficult complicated constrained non-linear optimization problem. No one knows where the real Pareto-Optimal solutions are and what the true Pareto-Optimal Front is. It is difficult to find the true Pareto-Optimal Front in MOOPF problems and the PF itself may be not uniformly distributed. Here in this paper the non-dominated solutions obtained by the three algorithms in 31 runs are collected together, and those solutions remained non-dominated in this set are used as the reference set for each test case. The reference set

are used in the performance metrics instead of the true ParetoOptimal Front. While in the metric of HV, the reference point is set through all the worst values in each objective domain. Detailed description of the metrics of ID , HD and HV please refer to [53,60] while the Purity please refer to [12]. The following tables show the experimental results of the proposed modified MOEA/D, NSGA-II and MOPSO in 31 independent runs for test Case 2–Case 7. The mean and standard deviation values of each metric are included in the tables. The Wilcoxon signed rank test has also been carried out at a significance level of 0.05 between the proposed modified MOEA/D and other two algorithms. The statistical results of ID , HD , HV and Purity and of Wilcoxon signed rank test in 31 independent

Please cite this article in press as: J. Zhang, et al., A modified MOEA/D approach to the solution of multi-objective optimal power flow problem, Appl. Soft Comput. J. (2016), http://dx.doi.org/10.1016/j.asoc.2016.06.022

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Fig. 15. Box plots of the three algorithms on metric Purity for different cases in 31 runs.

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runs are shown in Tables 12, 13, 14, 15 respectively. The corresponding box plots of the three algorithms on these metrics for different cases are depicted in Figs. 12 , 13, 14, and 15 respectively. In the following analysis, some notations are used for clarity: (1) For the Wilcoxon signed rank test, p is the probability of a hypothesis of equal median for two paired samples, and h is the result of the test. If the median of the difference between the proposed MOEA/D and another compared algorithm is zero, h = 0; Otherwise, a significant difference, then h = 1. (2) The best results of average values and standard deviations are indicated in bold in each test case. Table 12 shows the statistical results of performance indicator ID among the proposed modified MOEA/D, NSGA-II and MOPSO on test Case 2–test Case 7. For the ID , MOEA/D is superior to NSGAII and MOPSO on 5 test cases. ID indicates the closeness between obtained PF and the reference set. The smaller the ID , the closer it approaches to the true PF. In addition, the median differences between MOEA/D and other two algorithms are significant on all the six test cases. Table 13 shows the statistics results of performance indicator HD among the three algorithms on different test cases. For the HD , MOEA/D is superior to NSGA-II and MOPSO on 4 test cases. Hausdorff distance indicates the maximum mismatch between the obtained PF and the true PF. The smaller the HD , the closer it approaches to the true PF. In addition, the median differences between MOEA/D and NSGA-II are significant on 3 test cases. The median differences between MOEA/D and MOPSO are significant on 5 test cases. Table 14 shows the statistics results of performance indicator HV among the three algorithms on different cases. As seen in the table, the means values of MOEA/D about HV are always the largest. In addition the standard deviations of MOEA/D about HV are always the least. Because we have been regarding the worst value in each objective domain as the reference point, the larger the HV , the closer it approaches to the true PF. Moreover, as to the Wilcoxon signed rank test, on all test cases, the median of the differences between MOEA/D and the other two compared algorithms are significant. Table 15 shows the statistics results of performance indicator Purity among the three algorithms on different cases. As can be

seen from Table 15, MOEA/D performs significantly better than NSGA-II and MOPSO on all considered cases. In addition, the median differences between MOEA/D and NSGA-II, MOPSO are significant. For a more visual comparison of the three algorithms, the box plots of all the experimental results for different cases are given in Figs. 12–15. 7. Conclusions The multi-objective OPF has become one of the most popular optimization problems in the modern power systems. In this article, multi-objectives, such as the fuel cost, the emission impacts, the power losses and the voltage magnitude deviations are considered to form different OPF problems with complicated constraints. Aiming at the solution of MOOPF problem, a modified MOEA/D approach has been successfully proposed and implemented. In the proposed approach, a modified decomposition method is employed to obtain uniformly distributed Pareto solutions on each objective space. The modified MOEA/D also employs a mixed constraints handling mechanism to enhance the feasibility of the solutions. The penalty function approach and the repair strategy are both employed to handle the complicated constraints of the optimization problem. A fuzzy membership approach is also applied to obtain a satisfactory and best compromise solution over the obtained Pareto-Optimal solutions. The MOEA/D approach has been test on the IEEE 30-bus test power system with different cases to reveal its efficacy over the other optimization techniques reported in the past literature. The experimental results on test cases of optimal power flow problems indicate that the proposed algorithm is able to find well distributed Pareto-Optimal solutions that spread evenly on or approach to the true Pareto-optimal Front. Furthermore, two other popular MOEAs, i.e., NSGA-II and MOPSO are also re-implemented on the same test systems in order to further compare the performance of the proposed MOEA/D. It is observed from the comparisons that the MOEA/D approach can yield the optimal Pareto-Optimal solutions of the test system. The obtained results also confirm that the MOEA/D can be effectively used to solve multi-objective optimal power flow problems and ought to find growing attention in the near future. Then the param-

Please cite this article in press as: J. Zhang, et al., A modified MOEA/D approach to the solution of multi-objective optimal power flow problem, Appl. Soft Comput. J. (2016), http://dx.doi.org/10.1016/j.asoc.2016.06.022

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eter analysis and the statistical analysis are also performed. All the analysis show the proposed MOEA/D can obtain the competitive solutions among these three methods. For future work, the authors will try to reduce the population size through some novel weight design methods and improve the solution speed. Some test on the larger scale power systems will be another focus in future.

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The authors gratefully acknowledge the support of National Natural Science Foundation of China (61403321) and Natural Science Foundation of Guangdong Province, China (2014A030310003).

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