Multiobjective Grey Prediction Evolution Algorithm for Environmental/Economic Dispatch Problem

The environmental/economic dispatch (EED) problem, as one of the most important optimization problems in power systems operations, is a highly constrained, nonlinear, multiobjective optimization problem. Multiobjective evolutionary algorithms have become effective tools for solving the EED problem. To obtain higher quality Pareto solutions for EED as well as further improve the uniformity and diversity of the Pareto set, this paper proposes a novel multiobjective evolutionary algorithm, namely multiobjective grey prediction evolution algorithm (MOGPEA). The MOGPEA first develops a novel grey prediction evolution algorithm (GPEA) based on the even grey model (EGM(1,1)). Unlike other evolutionary algorithms, the GPEA considers the population series of evolutionary algorithms as a time series and uses the EGM(1,1) model to construct an exponential function as a reproduction operator for obtaining offspring. In addition, the MOGPEA adopts two learning strategies to improve the uniformity and diversity of the Pareto optimal solutions of the EED. One is a leader-updating strategy based on the maximum distance of each solution in an external archive, and the other is a leader-guiding strategy based on one solution of each external archive. To validate the effectiveness of the MOGPEA, a standard IEEE 30-bus 6-generator test system (with/without considering losses) is studied with fuel cost and emission as two conflicting objectives to be simultaneously optimized. The experimental results are compared with those obtained using a number of algorithms reported in the literature. The results reveal that the MOGPEA generates superior Pareto optimal solutions of the multiobjective EED problem. Matlab_Codes of this article can be found in https://github.com/Zhongbo-Hu/Prediction-Evolutionary-Algorithm-HOMEPAGE.


I. INTRODUCTION
The economic dispatch (ED) problem is a single objective optimization problem in power system operations [1], [2]. The purpose of the traditional ED is to meet the load demand in the most economical way. However, with increasing public awareness of environmental pollution, the clean air act has forced utilities to reduce the emission of SO 2 and NO x [3]. In these circumstances, environmental/economic dispatch The associate editor coordinating the review of this manuscript and approving it for publication was Qiu Ye Sun .
(EED) [4], [5] becomes pertinent and can not only bring great economic benefit but also reduce the pollutant emission.
The EED problem can be modeled as a multiobjective optimization problem with highly constrained and nonlinear. The fuel cost and the emission, as two conflicting objectives of the EED problem, are minimized in the conditions of satisfying the equality and inequality constraints. There have been many studies pertaining to the EED problem since it was proposed. Initially, the EED problem is generally converted into a single objective problem by considering the emission as a constraint or as a weighted function. The linear programming technique [6] is one of the representative of early approaches. The weighted sum method [7] is another usually used method, it transforms the objectives into a single objective problem by using a linear combination of different objectives. Although these conventional methods are easy to operate, there is a common problem that they all require multiple runs to achieve a trade-off between the two objectives.
In recent years, more and more multiobjective evolutionary algorithms (MOEAs) have been successfully used to solve the EED problem [8]. This kind of algorithms can find multiple trade-off solutions in a single run. They have thereby gradually become the main technique for solving the EED problem. According to the difference of the basic algorithms for constructing MOEAs, these multiobjective technologies can be divided into the following five categories.
The fourth type of MOEAs for the EED problem is based on hybrid approaches, e.g., the New Multiobjective Stochastic Search Technique (MOSST) [28], the Hybrid MOEA based on the techniques of PSO and DE (POS-DE) [29], the Modified NSGA-II, which integrated a Convergence Accelerator Operator (CAO) into the original NSGA-II (NSGA II-CAO) [30], the combination of DE and biogeographybased optimization (BBO) algorithm (DE-BBO) [31] and the Hybrid MultiObjective Differential Evolution/Tabu Search (MODE/TS) [32].
In order to obtain higher quality Paretp EED solutions as well as further improve the uniformity and diversity of the Pareto optimal set, this paper attempts to develop a novel multiobjective grey prediction evolutionary algorithm (MOGPEA) inspired by grey prediction theory.
Grey prediction theory, as an important embranchment of the grey system proposed by Deng in 1982 [52], is applicable to the problem of ''incomplete information and small sample size''. The grey prediction model (GM(1,1)) [53]- [56] is the core model of grey prediction theory. The accumulated generating operation (1-AGO) of the GM(1,1) can transform nonnegative discrete data sets into sequences with an approximate exponential law under specific conditions. This reduces the randomness of the original data. Then, the GM(1,1) uses the converted sequence to construct a prediction expression. If the time response sequence of the GM(1,1) is conducted by the solution of its whitenization differential equation, then the GM(1,1) is referred to as an even grey model (EGM(1,1)).
The comparison experiments on the standard IEEE 30-bus 6-generators systems show the effectiveness and superiority of the proposed MOGPEA, which can obtain higher quality, uniform and diverse EED solutions than many state-of-theart algorithms. The major contributions of this work are as follows: • Introduce a novel and competitive MOGPEA for the EED field. It first uses the EGM(1,1) to develop a grey prediction evolution algorithm (GPEA), and then designs two learning strategies to improve the uniformity and diversity of the Pareto optimal solutions of the EED.
• Introduce a novel evolution notion in the GPEA. Unlike other metaheuristics, the GPEA treats population sequences as a time series and then uses the EGM(1,1) model to forecast offspring (without any mutation and crossover operators).

VOLUME 8, 2020
The rest of the paper is organized as follows. Section 2 describes related preliminaries including the mathematical model for the EED problem and some background material for the EGM(1,1). Section 3 provides the description of the GPEA based on the EGM(1,1). In addition, Section 4 introduces the MOGPEA extended for the multiobjective problem. A comprehensive experimental study for the MOGPEA is carried out in Section 5. The related experimental results are provided and discussed in Section 6. Finally, Section 7 draws some conclusions and future expectations.

A. FORMULATION OF THE EED PROBLEM
A classical EED problem is to simultaneously minimize competing fuel cost and emission objective functions while fulfilling certain system constraints. The components of the problem , including its objective functions and constraints , are as follows.

1) OBJECTIVE FUNCTIONS
Objective 1: Minimization of Fuel Cost: The total fuel cost F(P) can be represented as follows: where F(p) is the total fuel cost, i.e., the sum of the power generation costs of each generator in a thermal power plant , P i is the active output of the ith generator , N is the number of generators , and a i , b i , and c i are the coal consumption characteristic coefficients of the ith generator. Objective 2: Minimization of Emission: Let the total emission be E(P) as follows: where α i ,β i , γ i , η i , and δ i are coefficients of the ith generator emission characteristics. All the parameters are presented in Tab. 1.

Constraint 1: Power Balance Constraint:
The total power generation must cover the total load and the system network loss in the transmission lines as follows.
where N i=1 P i is the total power generation, P D is the system load, P L is the system network loss , which can be calculated as follows: where B ij , B 0i and B 00 are the transmission network power loss coefficients. The correlation parameters are shown in subsection 4.3.

Constraint 2: Generation Capacity Constraint:
The generating capacity of the unit itself is also constrained by the upper and lower limits of the output, which can be written as follows: where P min i and P max i are the minimum and maximum generation limit of ith generator, respectively.

3) PROBLEM FORMULATION
The EED problem is formulated as a constrained, multiobjective optimization problem and is given by the following: Subjected to : The even grey model (EGM(1,1)) is first proposed by professor Deng Julong, and it is the most influential and widely used grey prediction model at present [52]. The main steps of the EGM(1,1) are as follows. First, a data transformation process from a nonnegative discrete disordered data sequence into a approximate ordered sequence is implemented by the first-order accumulated generating operation (1-AGO). Second, an exponential function based on the transformed data sequence is constructed to predict the next value of the ordered sequence. Last, the prediction values of the original data are obtained by the inverse operation of the 1-AGO operator (1-IAGO). Assume that an original data sequence Definition 1 (1-AGO) [52]: X (1) is the sequence of the 1-AGO of X (0) : Definition 2 (EGM(1,1) Model) [52]: the sequence Z (1) is the mean sequence of the X (1) : the following equation is called even grey model EGM(1,1), here, parameter a is called as a grey developmental coefficient, and b is called as a gray control parameter. Then the time response function of Eq.10 is solved: Definition 3 (1-IAGO) [52]: The prediction value for the raw data sequence X (0) can be estimated by the first-order inverse accumulated generating operation the (1-IAGO).

III. GREY PREDICTION EVOLUTION ALGORITHM (GPEA) BASED ON EGM(1, 1)
This section introduces a grey prediction evolution algorithm based on the EGM(1,1)(GPEA) [57], [58]. The algorithm includes an initialization operator, a reproduction operator and a selection operator. The GPEA is unique in that its reproduction operator which replaces the common mutation and crossover operator with the EGM(1,1) prediction. The process of the GPEA can be described as follows. , i = 1, 2, · · · , N and g = 1, 2, ..., g max , where g and g max are the current generation and the maximum number of generations respectively. The j th dimension of the i th individual is randomly produced according to the following: Here, rand(0, 1) is a random number with a uniform distribution between 0 and 1, lb j and ub j are the lower bound and upper bound of the j th dimension, respectively. Then, we distribute 3N individuals into three populations on average according to the fitness value of the individuals (from small to large). In detail, the top N individuals are divided into the first generation Y 0 (g = 0). Simultaneously, the middle N individuals are divided into the Y 1 (g = 1), and the bottom N individuals are divided into the Y 2 (g = 2). These three populations constitute an initial population series as a time series to predict the next generation population.

B. REPRODUCTION OPERATOR
In this section, a novel reproduction operator based on the EGM(1,1) model, called the egm11 reproduction operator, is proposed. The egm11 reproduction operator fits a exponential function by using successive three generations of a population sequence to forecast offspring. In addition, considering the calculating characteristics of the EGM(1,1) model, the egm11 reproduction operator is supplemented by a random perturbation and a linear fitting. The egm11 reproduction operator is shown as follows. Y g−2 , Y g−1 , and Y g , (g ≥ 2) denote three consecutive population series and three individuals y r1 , y r2 , and y r3 are randomly chosen from Y g−2 , Y g−1 , and Y g , respectively. Set Max y = max{|y r1,j − y r2,j |, |y r1,j − y r3,j |, |y r2,j − y r3,j |}, and Min y = min{|y r1,j − y r2,j |, |y r1,j − y r3,j |, |y r2,j − y r3,j |}. Then the j th dimension of the i th individual of the trial population U g is produced: here w is able to control the disturbance range, th ∈ [0.001, 0.1] is a preset value and used to control forecast, M is the maximum number of iterations and I is the current iteration number. Alg. 1 presents the pseudo code of the egm11 reproduction operator.

Algorithm 1 egm11 Reproduction Operator
Select parents; Parents are composed of three individuals from random select in Y g−2 , Y g−1 , and Y g , respectively.
The three individuals are assigned to series ; // linear prediction else u g i,j = y r3,j + w · Max y ; //random disturbance end end return U g ;

Algorithm 2
The Pseudo Code of MOGPEA Input: N : size of the population, D: dimension of the population, Na: maximum capacity of the archive, T max : maximum number of iterations, ux: upper limit of problem, lx: lower limit of problem Output: Op Initialization Initialize X 2 , X 1 , X 0 according to the formula (13); Op ← Ar and stop the algorithm ; return Op;

C. SELECTION OPERATOR
In order to select the better individual into the next generation, the GPEA carries out selection operation between the trial individual u g i and target individuals y g i . The individuals with a better fitness is selected to survive. This operation is described by the following expression.

IV. MULTIOBJECTIVE GREY PREDICTION EVOLUTION ALGORITHM (MOGPEA)
Based on the GPEA, this section develops a multiobjective grey prediction evolution algorithm (MOGPEA) for solving the EED problem. First, in order to improve the uniformity and diversity of the Pareto optimal solutions of the EED problem, a leader-updating strategy [50] and a leaderguiding strategy with learning characteristics are introduced to the MOGPEA. Second, the MOGPEA embeds an external archive strategy and a selection strategy based on dominance relation. This algorithm works as follows: Step 1 Initialization Step 1.1: Randomly initialize three generation population to form the initial populations chain.
Step 1.2: Evaluate the fitness of each generation population.
Step 1.3: Store all the nondominated solutions to external archive set.
Step 1.4: Maintain external archive by removing redundant solutions.
Step 2 Reproduction Step 2.1:Find the global best of each individual by leader updating strategy.
Step 2.2: Generate new individual by using the EGM(1,1) prediction or leader guiding strategy.
Step 2.3: Evaluate the fitness of new population.
Step 3 Selection Step 3.1: Select next generation using dominance relation.
Step 3.2: Store new nondominated solutions to external archive set.
Step 3.3: Maintain external archive by removing redundant solutions.
Step 3.4: The new population and the previous two generations form a new population chain.
Step 4 Stoping criteria: If the stop criteria is met, then stop. Otherwise, go back to Step 2. Here are the four strategies described above, and Alg . 2 represents the pseudo code of the MOGPEA.

1) LEADER-UPDATING STRATEGY
To improve the uniformity and diversity of the nondominated solutions in the Pareto front, the MOGPEA first uses the maximum distance to measure the sparsity of the nondominated solutions, and then designs a leader-updating strategy based on the maximum distance. As shown in Fig. 1, the maximum distance of the ith point (nondominated solution) is calculated: After calculating the maximum distance of all nondominant solutions, MOGPEA adopts the roulette wheel selection mechanism to select the leader of the archive. In other words, the i solution can be the leader as long as its maximum distance is the greatest. Alg. 3. describes the leader-updating strategy.

2) LEADER-GUIDING STRATEGY
This section firstly introduces the sparse direction in the leader-guiding strategy. The sparse direction of the i solution is defined as follows: such as in Fig. 1, since ld A is greater than ud A , so l A = 1. While l B = −1 for the Bth solution. Next, the leader-guiding strategy based on the sparse direction is used to update the individual. The update formula is as follows The above formula indicates that X L moves towards the sparse direction to generate a new individual. This can increase the uniformity of the Pareto front.

Algorithm 3 Leader-Updating
Input: Ar(t) Output: X L Nt = |Ar(t)| ; for i = 1 to Nt do calculate d i ; end for i = 1 to N do index i = Roulette_wheel_selection(d i ); X Li (t) = Ar index i (t); end return X L ; VOLUME 8, 2020

3) SELECTION STRATEGY BASED ON DOMINANCE RELATION
As we all know, greedy selection is a common strategy for single objective optimization problems. However, in multiobjective problem, the selection strategy based on dominance relation is used to select the promising solutions into the next generation. This kind of selection strategy can improve the global search capability further. In the strategy, the trial individual T i can enter the next generation when T i dominates the target individual P i (T i ≺ P i ). When two individuals do not dominate each other, there is a half chance that each individual will go on to the next generation. The selection strategy based on dominance relation can be described as follows.

4) EXTERNAL ARCHIVE MAINTENANCE STRATEGY
At present, there are many external archive maintenance strategies. The most famous one is the fast non-dominated sort in NSGA-II. However, this method may delete several connected solutions with smaller crowding distances through calculating the crowding distance. This can lead to the remaining solutions too sparse. To avoid the above problems, in this paper, a cyclic crowded sorting algorithm [59] is used to pick out individuals. The pseudo code of the cyclic crowded sorting is shown in Alg. 4.

5) COMPLEXITY ANALYSIS OF THE MOGPEA
This section analyzes the complexity of MOGPEA. The introduction of two learning strategy and cyclic crowded sorting method of the MOGPEA will consume storage space and increase the time complexity. The complexity of the leader updating strategy is O(N ). The complexity of the leader guiding strategy is O(N ). The computational complexity of cyclic crowded sorting is O(Na), where Na is the current capacity of the archive and Na > N . Therefore, the final time complexity of the MOGPEA in one generation is O(Na). In addition, the space complexity of cyclic crowded sorting mechanism is O(Na). Overall, the consumption of the time and space is very small.

V. THE MOGPEA IMPLEMENTATION
In this section, the MOGPEA first introduces a constrainthandling strategy to deal with the equality and inequality constraints of the EED problem. Second, a fuzzy set mechanism is used to extract the best compromise solution from the final external archive. Last, the design of experiments and the setting of parameters are described.

A. CONSTRAINT HANDLING
In addition to high-dimensional and multiple objectives, high constraints are another difficult problem to deal with for the EED problem. For the inequality constraint of the EED problem, it is very easy to deal with the over-limited values by simply setting it to the corresponding boundary value. On the contrary, for the equality constraints, it becomes very complicated since the strong coupling between variables. In order to better solve the equality constraints and avoid consuming too much time, this paper employs a special constraint handling method to deal with the power balance constraint of the EED problem. A constraint violation threshold σ is set in advance and σ = 1e − 12. The constraint process is as follows: Step 1: For each infeasible solution x, set k is a random integer from 1 to D.
Step 2: Calculate the violation V (x): If V (x) > σ , then go to Step 3; otherwise, go to Step 4.
Step 3: Adjust x to make it satisfy the constraint: If the new x i,k violates the inequality constraint, and then it will be addressed by the inequality constraint method. Let k = mod(k, D) + 1, and go to Step 2.
Step 4: End the constraint handling process.

B. FUZZY-BASED THEORY FOR THE COMPROMISE SOLUTION
In practical applications, the decision maker only usually needs one solution in the final external archive. This solution is called the best compromise solution, and to some extent it satisfies all objectives. This paper extracts the best compromise solution by using a fuzzy-based mechanism [34] and Fig. 2 illustrates a fuzzy-based membership function. The membership function µ ik of the kth objective of the ith solution is calculated in the following way:  (26) Here, M denotes the number of objective functions (M = 2 in this paper), and Nt is the number of nondominated solutions. The best compromise solution is the solution for which µ k is the largest.

C. EXPERIMENTAL DESIGN AND PARAMETER SELECTION
The MOGPEA is tested on the standard IEEE 30-bus 6-generator system (as shown in Fig.3). The fuel cost coefficient, emission coefficient, and generation limit are referenced in [22] and given in Tab. 2. The transmission loss coefficients are given in Tab. 1. The load demand is 2.834 MW. The simulation program is written in MATLAB and run at 1.6GHz Intel Pentium core i7 processor with 4GB-RAM. The source codes of this algorithm can be found in https://github.com/Zhongbo-Hu/Prediction-Evolutionary-Algorithm-HOMEPAGE.
In order to investigate the effectiveness of the MOGPEA for solving the EED problems, two different cases are studied as follows:   • Case 1: the transmission loss of power balance constraint is not considered.
• Case 2: the transmission loss of power balance constraint is considered.
Here, the number of population N = 50 and the capacity of the archive Na = 50. Stopping criterion for two case are taken as 100 and 200 maximum number of iterations, respectively. Thirty independent runs of the MOGPEA are carried out to collect the statistical results. In addition, the parameters of several compared algorithms are given below. MOPSO [17]: the inertia weight is 0.7, the personal learning coefficient is 1.4, the global learning coefficient is 1.4, the number of grids per dimension is 7. NSGA-II [11]: the crossover percentage is 0.7 and the mutation percentage is 0.7. PESA-II [60] : the crossover percentage is 0.7 and the mutation percentage is 0.7.

VI. RESULTS AND DISCUSSIONS
First, a simple multi-objective unconstrained test function is used to verify the effectiveness of the two learning strategies.  Second, experimental result is obtained on the IEEE 30-bus 6-generator test system.

A. EFFECTIVENESS ANALYSIS OF TWO LEARNING STRATEGIES
A unconstrained multiobjective function is firstly used to verify the effectiveness of the two learning strategies. The function is described as follows: Next, the original MOGPEA (MOGPEA1), the original MOGPEA + strategy 2 (MOGPEA2) and the original MOG-PEA + strategy 1 + strategy 2 (MOGPEA3) are verified on this test function. Here, N = 40, D = 2, Na = 35, and T max = 30. Fig. 4 shows the Pareto fronts obtained by the above three algorithms. From the figure, we can infer two conclusions: Fig. 4, the uniformity of the Pareto front of MOG-PEA3 is better than that of MOGPEA2, which is better than that of MOGPEA1. Therefore, we can obtain conclusion 1: the strategy 1 and strategy 2 can improve the uniformity of the Pareto front.
• From the third graph in Fig. 4, the coverage of the extreme solutions of the Pareto front for the MOG-PEA3 marked by two red circles is more widespread than that of the MOGPEA2 marked by two green diamonds. It is more widespread than that of the MOG-PEA1 marked by two blue squares. From the above, we can obtain conclusion 2: the strategy 1 and strategy 2 can increase the diversity of the Pareto front.

B. IEEE 30-BUS 6-GENERATOR TEST SYSTEM
This section conducts two experiments in the IEEE 30-bus 6-generator test system to evaluate the performance of the MOGPEA. One experiment is to compare the extreme solutions and compromise solutions of the MOGPEA with other famous algorithms. Another is to use some evaluation indicator to test solution quality, such as SP, HV and CM.

1) COMPARISON OF EXTREME SOLUTIONS AND COMPROMISE SOLUTIONS
First, the original GPEA is carried out to search for the extreme solutions of the two objective of the EED problem  in two cases respectively. Tab. 3 shows the obtained best extreme solution for two cases, and Fig. 5 and 6 give the convergence of two objectives for two cases. As can be observed from Tab. 3, the optimal values of fuel cost objective for Case 1 and Case 2 are 600.111417 $/h and 605.998378 $/h and the optimal values of the emission objective are 0.194203 ton/h and 0.194179 ton/h for Case 1 and Case 2. Next, the proposed MOGPEA is implemented to simultaneously optimize both objectives of the EED problem, and the results of extreme solutions and compromise solutions for two cases are discussed below.
Case 1: Applying the MOGPEA to Case 1, the Pareto front is displayed in Fig. 7. The figure clearly indicates that the solutions are well-distributed and almost cover the entire Pareto front of the problem. Tab. 4 and 5 compare the best extreme solutions of the MOGPEA for fuel cost and emission with results reported in the literatures that were obtained by using NSGA [10], NPGA [14], SPEA [34], MOPSO [17], BB-MOPSO [22], FMOEP [36], MBFA [37], NSGA-II [11], MOACSA [44] and SMODE [27]. In addition, to compare the compromise solutions of different algorithms, the average satisfactory degree (ASD) [22] is calculated. Tab. 6 compares the results obtained by the MOGPEA for the compromise solution and the ASD with results reported in the literatures that were obtained by using NSGA [10], NPGA [14], SPEA [34], FCPSO [20], BB-MOPSO [22] and MOCDOA [50]. The bold values in Tab. 4-6 are the best obtained results.
As seen from Tab. 4 and 5, 600.11 $/h and 0.1942 ton/h are the two best extreme values for the fuel cost and emission, respectively. In Tab. 4, although the MOGPEA generates the same best cost value as BB-MOPSO, NSGA-II, MOACSA and SMODE, the obtained corresponding emission by the MOGPEA is 0.2219 ton/h, which is the best among the four compared algorithms. In addition, the MOGPEA is better than the other six comparison algorithms in terms of the best cost (including NSGA, NPGA, SPEA, MOPSO, FMOEP and MBFA). In Tab. 5, it is clear that the proposed MOG-PEA gives minimum emission of 0.1942 ton/h, which is equal to the emission obtained from the SPEA, MOPSO, BB-MOPSO, FMOEP, MBFA, NSGA-II, MOACSA and SMODE algorithms, and it performs better than NSGA and NPGA for the problem. In Tab. 6, according to the ASD, the MOGPEA attains the best ASD value among seven algorithms (equal to 0.7677).In other words, the proposed MOG-PEA gives the best compromise solution of 609.54 $/h and 0.2009 ton/h, which are significantly better than the solutions given by the other algorithms. All of these results demonstrate the potential and effectiveness of the MOGEPA to solve the EED problem.   above results obtained by the MOGPEA are very competitive among all comparison algorithms, which demonstrates the effectiveness of the MOGPEA in solving EED problems.

2) COMPARISON OF SOLUTION QUALITY
Three commonly used multiobjective performance metrics are used to evaluate the quality of the solution obtained by MOGEPA. In addition, the solution quality of the MOGPEA is compared with that of MOPSO [17], NSGA-II [11] and PESA-II [60]. This section only considers the solutions for Case 2.
(1) Spacing (SP): The spacing (SP) [61], proposed by Schott, is adopted to evaluate the uniformity of the Pareto optimal set found. The calculation of the SP is as follows: where d i is the Euclidean distance between two consecutive solutions in the nondominated solution set andd is the mean values of all d i . The SP values are closely related to the uniformity of the solution set, that is, the smaller the value, the more VOLUME 8, 2020   uniform it is. When SP value is equal to 0, the obtained nondominated solution is equidistant. Fig. 8 shows the SP value of the MOGPEA for 30 runs on Case 2. The obtained The SP values of four algorithms are compared in Tab. 9. It can be observed that, compared with the MOPSO, NSGA-II and PESA-II, the average performance of the MOGPEA is the best. To more clearly and intuitively demonstrate the uniformity of the obtained nondominated solution, Fig. 9 shows the Pareto front of the MOGPEA and the other three algorithms. From Fig. 9, the uniformity of the MOGPEA is clearly superior to that of MOPSO, NSGA-II and PESA-II. All in all, the uniformity of the obtained solutions by the MOGPEA is very competitive compared with other three algorithms.
(2) Hypervolume (HV): The hypervolume (HV) [25], proposed by Zitler and Thiele, is a comprehensive performance index that can be used to evaluate both convergence and diversity. The larger the HV value, the better the comprehensive performance of the algorithm. The definition of HV is as follows: Here, v i is the hypervolume of the reference point and the ith solution in the solution set. The reference point is same for the four algorithms. Fig. 10  (3) C-metric (CM): When the true Pareto front of multiobjective problem is not known, CM [62] is often used to evaluate the quality of the obtained solutions. Let S 1 , S 2 S be the two solution sets obtained by two different algorithms. The CM is defined by the following formula.   CM (S 1 , S 2 ) = |{a 2 ∈ S 2 , ∃a 1 ∈ S 1 : a 1 ≺ a 2 }| |S 2 | CM (S 1 , S 2 )=0 means that none of the solutions in S 2 are dominated by the solutions in S 1 . CM (S 1 , S 2 )=1 means that all of the solutions of S 1 dominate or are equal to some solutions of S 2 . Tab. 11 shows the comparison results among the four algorithms in terms of CM. From Tab. 11, nearly 12% and 10% of the solutions obtained by the NSGA-II and PESA-II respectively, are dominated by those of the MOG-PEA. However, 96%, 22% and 6% of the solutions obtained by the MOGPEA are dominated by those of the MOPSO, NSGA-II and PESA-II, respectively. Thus, the coverage of the MOGPEA is better than that of the PESA-II. However, it not as good as that of the MOPSO and NSGA-II.
According to the above analysis, it can be concluded that the MOGPEA has a better performance for Case 2 in terms of the uniformity and diversity of the solutions. However, it may not be very competitive for the convergence of solutions. This result can be attributed to the collective efforts of the two learning strategies proposed. On the one hand, the update strategy of the leader based on the maximum distance is able to assign the leader to the location with sparse solutions, while the leader-guiding strategy proposed is able to make the leader move in the direction of its sparse neighborhoods. Therefore, their collective efforts ensure the uniformity and    diversity of solutions in the external archive set. On the other hand, the high exploration capability of the proposed reproduction operator makes the convergence rate of our algorithm slow. All in all, the MOGPEA has good uniformity and diversity . However, there is still room to improve its convergence.

VII. CONCLUSION
This paper first proposes a grey prediction evolution algorithm (GPEA) by resorting to the even grey model (EGM(1,1)). The GPEA differs from most of metaheuristics in that its reproduction operator does not make use of any mutation and crossover operators but rather considers the consecutive three population series of evolutionary algorithms as time series and uses the EGM(1,1) model to construct an exponential function for obtaining offspring. To solve the environmental/economic dispatch (EED) problem , which is a constrained multiobjective optimization prob-lem with conflicting fuel cost and emission objectives, a multiobjective grey prediction evolution algorithm (MOGPEA) is developed in which two learning strategies are introduced for improving the uniformity and diversity of the obtained Pareto optimal solutions. One is a leader-updating strategy based on the maximum distance to measure the degree of sparseness of the solutions, and the other is a leader-guiding strategy based on the sparse mark to search the area around a leader. Furthermore, the constraints of the EED problem are solved using a special function processing strategy, a selection strategy based on the dominance relation replaces greedy selection of the GPEA, and a cyclic crowded sorting method maintains the external archive.
A standard IEEE 30-bus 6-generator test system is used to verify the effectiveness of the MOGPEA. Two cases in this system have been considered. The extreme solutions and compromise solutions of the MOGPEA are compared with that of state-of-the-art algorithms. The compared results exhibit that the MOGPEA has a good compromise solution and highly diverse Pareto optimal solutions. Three metrics (SP, HV, CM) all show that MOGPEA yields solutions exhibiting better diversity and uniformity compared with the MOPSO, NSGA-II and PESA-II. The experimental results show that the MOGPEA is efficient and competitive for solving the multiobjective EED problem. This paper demonstrates that the novel GPEA can obtain competitive solutions for the EED problem, and the two learning strategies (i.e., Leader-updating strategy and Leader-guiding strategy) have good effects on improving the uniformity and diversity of the Pareto front. In the future, we will use the algorithm and the two learning strategies to investigate more realistic dynamic EED (DEED) problems.