Water Utility Journal 7: 13-23, 2014. © 2014 E.W. Publications

Multi-objective artificial bee colony algorithm for long-term scheduling of hydropower system: A case study of China

X. Liao*, J. Zhou, S. Ouyang, R. Zhang and Y. Zhang School of Hydropower and Information Engineering, Huazhong University of Science and Technology, China * e- mail: liaoxiang@hust.edu.cn

Abstract: The multi-objective long-term economic dispatch in hydropower system is a complicated nonlinear optimization problem with a group of complex constraints which makes the optimization of conflict objectives through traditional methods a hard task. This paper is aim to present a novel multi-objective evolutionary algorithm named multi-objective artificial bee colony (MOABC) algorithm and compares the efficiency of MOABC and established algorithms in long-term cascaded hydropower system dispatch. The introduced modified employed phase improves the global optimal capability of MOABC algorithm, and a novel probability calculation method is employed to improve the search ability of onlooker bee phase. Moreover, a modified employed bees phase contributes to escape local extreme value. Additionally, a local search method based on chaos theory has been introduced. The update strategy of external archive set has been introduced. The performance of proposed MOABC has been demonstrated through a set of standard test functions. In order to verify the effectiveness of proposed algorithm further, the case of the world biggest hydropower system, Three Gorges Project (TGP), has been studied in this paper. Numerical results and comparisons demonstrate the effectiveness and efficiency of proposed algorithms which applying in the long-term scheduling of TGP hydropower systems in China. The results showed the proposed method have a better convergence ability and distribution of the Pareto front.

Key Words: hydropower scheduling; swarm intelligence; multi-objective; artificial bee colony

1. INTRODUCTION

The hydroelectricity consists of nearly 20% of the global power production in the new century (WCD 2000). Great potential energy of The Three Gorges Project (TGP), the biggest hydropower system in the world located in the Yangtze River, could be developed through improving the system operation strategies. In practice, rule curve which determining the action in real reservoir operation is extensively utilized. However, the scheduling of the hydropower system is a complicated optimization problem associated with a set of complex constraints containing hydraulic relationship, variation of reservoir storage, power output limit, water discharges limit, head variation and so on. Especially in multi-objective optimization of the cascaded hydropower system, the main hardship is to find the optimal solution from a holistic perspective by considering all objectives which are generally conflicting. Determining the operation polices of these complex characteristics above through traditional rule curves may achieve an alternative operation strategy but there is no guarantee that the scheduling result is optimal.

Deterministic optimization methods used in long-term scheduling of the hydropower system can be classified as mathematical programming and computational intelligence methods. In the past decades, many mathematical programming methods were applied in scheduling of hydropower system. Linear programming (LP) (Jacovkis et al. 1989; Ponnambalam et al. 1989), non-linear programming (NLP) (Barros et al. 2003), Lagrange relaxation (LR) (Guan et al. 1997), dynamic programming (DP) (Ferrero et al. 1998), are applied in optimization of reservoir operation polices. However, the traditional mathematical programming method faced various obstacles in practical implementation. The main drawback of LP is that all the objective functions and constraints must be linear. That means these mathematical models are not accurate because of the approximation. For

14 X. Liao et al.

NLP, the objective functions need to be continuous and differential, and the global optimality is also a problem in engineering application of NLP. The effects of optimization results of LR are influenced by the Lagrange multipliers updating strategy and initial solutions. When DP is applied in scheduling of a hydropower system, the decision variables need to be discretized. “Curse of dimensionality” occurred when the point of discretization is increasing.

In recent decades, more and more scholars focus on the application of computational intelligent methods. The optimization methods based on computational intelligent such as genetic algorithm (Hincal et al. 2011), simulated annealing(Teegavarapu and Simonovic 2002), ant colony optimization (Jalali et al. 2007), particle swarm optimization (Afshar 2012; Kumar and Reddy 2007), honey bees mating optimization (Haddad et al. 2006), artificial bee colony (Liao et al. 2012) are applied in determining the reservoir operation rules. However, the methods mentioned above are designed to solve single objective problems. In order to handle the multi-objective problems, the common method is weighting each objective function and converting into one objective function (Mandal and Chakraborty 2011). Moreover, some objectives are treated as constraints to decrease the number of objective functions. However, the predetermined weighting factor could not reveal the relationship between the each objective functions. Actually, these methods mentioned above are single objective optimization problem. Thus, many multi-objective evolutionary algorithms (MOEA) such as non-dominated sorting genetic algorithm II (NSGA-II) (Deb et al. 2002), enhanced strength Pareto evolutionary algorithm (SPEA2) (Eckart et al. 2001), multi-objective particle swarm optimization algorithm (Tripathi et al. 2007) (MOPSO) and multi-objective differential evolution algorithm (MODE) (Xue et al. 2003) had been developed to handle the multi-objective optimization problems. MOEA have some advantages of handling conflicting objective functions because of the structure of algorithm. The goal of MOEA is to find a non-dominated solution set uniformly distributing on the true Pareto front. However, these methods mentioned above are still suffered from the premature convergence because of the evolutionary mechanism. Therefore, the development of new approach and improvement of existing method are necessary in order to solve the complicated constraint multi-objective problems.

Since the artificial bee colony algorithm (Karaboga 2005) had been developed in the past few years, the swarm intelligent technique drawing from the foraging behaviors of bee colony showed the great potential of solving the various complex optimization problem (Karaboga and Ozturk 2009; Ozturk et al. 2012). A Pareto-based discrete artificial bee colony (Li et al. 2011) was developed for solving the flexible job shop scheduling problem. However, it was designed for discrete optimization problem. Hence, we proposed a new multi-objective optimization technique named multi-objective artificial bee colony algorithm (MOABC) and verified the effectiveness of MOABC by a set of test functions. Finally, we applied the proposed method in the long-term scheduling of TGP and demonstrate the effectiveness and efficiency of MOABC.

The rest of paper is organized as follows: the section 2 introduces the long-term scheduling problem of the hydropower system. MOABC is described in detail and verified by test functions in section 3. MOABC is applying in the multi-objective long-term scheduling of TGP in section 4. Section 5 is the conclusion of our work.

2. PROBLEM FORMULATION

The multi-objective long-term economic dispatch in the hydropower system is aimed to utilize the potential energy of water in TGP and convert to the electrical energy. The manager of TGP always wants to maximize the generation benefit of the hydropower system. On the other hand, the power demand of grid must be satisfied through the operation of the hydropower system even in the dry season. Hence, maximize the firm output is another important task in the operation of the hydropower system. The firm output is defined as the minimum average monthly output of the hydropower system. In this paper, we formulate the multi-objective long-term scheduling problem as follows:

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2.1 Objective functions

When considering both generation benefits and firm output simultaneously, the optimization becomes multi-objective problems. Therefore, the problem can be formulated as follows:

1 , ,1 1

maxI T

i i i t i ti t

F C A Q H t= =

= ⋅ ⋅ ⋅ ⋅Δ∑∑ , (1)

2 , ,1

max{min }I

T

i i t i ti t

F A Q H∈Ω =

= ⋅ ⋅∑∑ , (2)

where 1F is to maximize the generation benefits over the whole scheduling period. 2F is to maximize the minimum total output during the scheduling period. I represents the number of hydro plants; T is the length of scheduling period; iC is the electricity price of hydro plant i ; iA is the power generation coefficient of hydro plant i ; IΩ is the set which includes all the hydropower unit; ,i tQ , ,i tH are the water discharge and net head of hydro plant i at time t , respectively.

2.2. Constraints

The objective functions mentioned above subject to the following constraints: Hydraulic connection:

, 1, 1, ,i t i t i t i tI Q S R t T− −= + + ∀ ∈ (3)

Water dynamic balance equilibrium:

, , 1 , 1 , ,( )i t i t i t i t i tV V I Q S t t T− −= + − − ⋅Δ ∀ ∈ (4)

Final water level limits:

, ,i start i endZ Z= (5)

Reservoir water level constraints:

min max, , ,i t i t i tZ Z Z t T≤ ≤ ∀ ∈ (6)

Water release constraints:

min max, , , ,( )i t i t i t i tQ Q S Q t T≤ + ≤ ∀ ∈ (7)

Output limits:

min max, , , ,)i t i i t i t i tP A Q H P t T≤ ⋅ ⋅ ≤ ∀ ∈ (8)

where ,i tI , ,i tR are natural inflow and local inflow of hydro plant i at time t , respectively; 1,i tS − is abandoned flow of hydro plant 1i − at time t ; ,i tV is reservoir storage volume at the end of the

16 X. Liao et al.

period t . min,i tZ , max

,i tZ are minimum and maximum upriver water levels of hydro plant i at time t ; min

,i tQ , max,i tQ are minimum and maximum water discharges of hydro plant i at time t ; min

,i tP , max

,i tP are minimum and maximum outputs of hydro plant i at time t ; ,i startZ , ,i endZ are initial water level and final water level of the scheduling period.

3. MULTI-OBJECTIVE ARTIFICIAL BEE COLONY ALGORITHM

The multi-objective issue are different from the single objective optimization because the different elite solution keeping strategies and the comparison mechanisms of two solutions. In order to modify the artificial bee colony for adapting multi-objective problems, the following modifications are established.

3.1. Strategy of external archive set updating

Different from keeping one optimal solution in each cycle, external archives set Ω is used to keep the non-dominated solutions U in each computing cycle. The relationship between two solutions cannot be described simply as bigger or smaller and the final optimal result is not a single solution but a set of non-dominated solutions. Thus, the external archive set updating strategy can be concluded into three criterions:

i) If Ω is empty, solutions in U will be added into Ω directly. ii) If a solution u in U has not been dominated by any solutions in Ω , the solution u will be

added into Ω , the solutions in Ω which dominated by u will be deleted. Otherwise, the solution u will be abandoned.

iii) If the number of solutions in Ω is larger than the maximum size, the redundant solutions will be deleted according to evaluate the crowding distance metric (Deb et al. 2002) of each solution.

3.2. Modification of artificial bee colony algorithm

3.2.1. Employed bees phase

In proposed ABC algorithm, a parameter named modification rate (MR) is used to produce the new food sources. In order to control the probability of producing new food source, the following equation is used to determine the producing rate.

( ),,

n n n n n nn m m m m k mm n

m

v x x x if R MRv

x otherwiseφ⎧ = + − <

= ⎨⎩

(9)

where [0, ]n D∈ represents the dimension of individual in solution space (food resources), m is a serial number of a food source corresponding to a solution of problem. n

mv and nmx represent the

new and old food source, respectively. nmR is a random real number between [0,1] . n

mφ is a random real number distributed uniformly between [ 1,1]− which controls the effectiveness of distance between n

mx and nkx . Obviously, new food source is affected by the distribution of bee colony. MR

is a random number between [0,1] which controls the mutation probability of solutions. nmx and n

kx are chosen from the old population in original ABC algorithm, but in the proposed method, both the

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solutions are chosen from the external archives set. Because the solutions in external archives set have a better fitness compare to the old population, the convergence speed will more quickly.

3.2.2. Probability calculation

In order to calculate the probability parameter of onlooker bees phase, the original ABC used fitness value. The better fitness value, the higher possibilities of solutions to be chosen in onlooker bees phase, but in multi-objective problem, the fitness value of one objective is hard to describe the status of solution since the conflicting objectives. Hence, we use the crowding distance metric to measure the solutions in iterative computation. The probability is formulated as follow:

max

max

1 0.2, 0

0.8 0.2,

ii

ii

violation if violationviolation

pcdm otherwisecdm

⎧⎛ ⎞− × >⎪⎜ ⎟

⎪⎝ ⎠= ⎨⎪ + ×⎪⎩

(10)

where ip is the probability value of solution i . icdm represents the crowding distance metric of the solution i , maxcdm is the max crowding distance metric of the current population.

iviolation is the violation value of solution i , maxviolation is the max violation value of the solution in population.

3.2.3. Select mechanism

Different from single objective problem, dominance relationship had been used to determine which one is preferred between two of solutions in multi-objective optimization problems. In the proposed method, we chose the solution according to the following logic:

Compare two solutions, the solution which dominate the other will be added into the non-dominated set U . Both the two solutions will be added into U when they are non-dominated with each other.

3.3. Chaos local search strategy

In order to improve the convergence effectiveness, the Chaos local search had been adopted in MOABC. The detail procedure of chaos local search is described as follow:

STEP 1. Logistic map has been used to generate the chaos sequence. The logistic map is formulated as follow:

1 (1 )p p psk sk skC C Cµ+ = ⋅ − (11)

where µ is a control parameter, pskC is chaos variable of k th non-dominated solution in Ω after

p times iteration. (0,1)pskC ∈ . Once the iteration number p is increasing, the p

skC reflect the chaotic and dynamic characteristic when 4µ = and 0 {0.25,0.5,0.75}skC ∉ .

STEP 2. We generate the new solution by the following equations.

min max min( )p pk skx x C x x= + − (12)

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where pkx is new solution generated by

pskC .

STEP 3. Generate the new solution as follow:

(1 )new p bestk k c c kx x M M x= ⋅ + − ⋅ (13)

where cM is chaos mutation rate between [0,1] which control the mutation of bestkx .

STEP 4. Compare the dominance relationship between newkx and

bestkx , and keep the better one in

Ω .

STEP 5. Go to STEP 2 until the chaos iteration number achieved the maximum number.

3.4 Framework of MOABC

The framework of MOABC is shown as Fig.1.

Start

Initialize food sources and computation condition

Employed bees phase

Onlooker bees phase

Output the result

Cycle=1

Cycle>maxcycle?

Y

Cycle=Cycle+1

Scout bee phase

Update the external archive set

Chaos local search

N

Figure 1. The framework of MOABC.

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3.5 Numerical test of MOABC

In order to verify the effectiveness of MOABC, a set of famous test functions had been tested in this paper. The parameters setting of ZDT test functions are listed as follow: MR=0.02, max generation number is 500, the artificial bee colony size is 100, external archive sets is set to 50.

(a) ZDT 1 (b) ZDT 2

(c) ZDT 3 (d) ZDT 4

(e) ZDT 6

Figure 2. Optimization results of MOABC in ZDT test functions.

The ZDT test functions demonstrate the effectiveness of MOABC algorithm in Fig.2. We can clearly see from the Fig.2 that all the solutions calculated by MOABC are distributed approximate

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

F2

F1

True Pareto

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

F2

F1

True Pareto

-0.9

-0.4

0.1

0.6

1.1

0 0.2 0.4 0.6 0.8

F2

F1

true Pareto

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

F2

F1

True Pareto front

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

F2

F1

True Pareto front

20 X. Liao et al.

uniformly on the true Pareto front. The algorithm has an acceptable performance of distribution and convergence ability according the Fig.2.

4. CASE STUDY: MULTI-OBJECTIVE LONG-TERM SCHEDULING OF THREE GORGES PROJECT

4.1. Parameters setting of TGP hydropower system

The MOABC algorithm had been proposed to solve the multi-objective long-term economic dispatch of TGP cascaded reservoirs which containing Three Gorges Dam and Gezhou Dam. In order to test the effectiveness of proposed method in real engineering scene, the inflow sequences of flood and dry seasons had been used as the computation conditions. For convenience, the local inflow had not been considered in the mathematical model. The scheduling period is a water year and the interval is a month. The two different water inflow sequences of TGP hydropower system are listed in Table 1 and the boundary conditions are listed in Table 2.

Table 1. Inflows of TGP hydropower system.

month Water inflow (m3/s) month Water inflow (m3/s) flood dry flood dry

1 5590 4720 7 47400 35400 2 5130 4310 8 45600 33200 3 6610 5070 9 44400 31600 4 11300 7990 10 29600 22400 5 19100 13900 11 15400 12000 6 28700 21800 12 7940 6620

Table 2. Boundary conditions for TGP hydropower systems.

parameter Three Gorges Gezhou Dam Hydro plant discharge range (m3/s) [4500,98800] [4500,86000] Power generation coefficient 8.8 8.5 Upriver water level range (m) [145,175] 65 Power generation range (MW) [4990,1822.6] [946,288.6] Water head range (m) [61,113] [6,27.8] Initial water level (m) 175 65

The multi-objective differential evolutionary (Zhou et al. 2011) (MODE) algorithm had been

applied to solve the scheduling problem of TGP for demonstrating the effectiveness of MOABC. For both methods, the population size is set to 50, the external archive set size is set to 30 and the max iteration number is set to 2000. In proposed method, the limit number is set to 50. And MR=0.01. In MODE, the parameter F is set to 0.25 and cross rate (CR) is set to 0.15.

4.2. Results discussion

We applied the proposed method to solve the scheduling problem of TGP and compared with the established method. The results are shown in Fig. 3. All the non-dominated solutions which generated by inflows in dry season had been listed in the Table 3. Through the results, it can be clearly seen that the non-dominated front generated by MOABC has a better convergence effectiveness compared to MODE in the flood and dry water inflow conditions. Furthermore, the

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results generated by MOABC with the inflows in dry season are obviously better than by MODE. That shows the potential of MOABC method when the inflows are insufficient. In addition, the solutions distributed approximate uniformly on the non-dominated front. That means the distribution strategy of external archive set had been verified. From the Table 3 we can see that all the solutions are non-dominated. The results testify the effectiveness of external archive set updating strategy in another aspect.

From the results we discussed above, we can conclude that the proposed MOABC is an effective technology to solve the multi-objective problem. The performance of proposed method had been satisfied by applying in long-term scheduling of TGP.

(a) flood (b) dry

Figure 3. Comparison results between MOABC and MODE.

Table 3. Non-dominated solutions of dry season

Index Power generation Firm output Index Power generation Firm output 1 1171.942 756.055 16 1177.867 696.114 2 1172.387 753.405 17 1178.197 691.569 3 1172.871 751.368 18 1178.567 686.810 4 1173.317 747.291 19 1178.972 680.517 5 1173.757 743.149 20 1179.365 673.443 6 1174.189 739.514 21 1179.734 667.118 7 1174.671 734.207 22 1180.082 659.995 8 1175.120 729.672 23 1180.470 652.838 9 1175.512 726.248 24 1180.811 647.950 10 1175.970 720.247 25 1181.175 639.231 11 1176.322 717.683 26 1181.480 631.721 12 1176.749 710.518 27 1181.834 622.563 13 1177.154 706.580 28 1182.132 615.085 14 1177.502 701.260 29 1182.440 604.029 15 1171.942 756.055 30 1182.723 595.487

590

640

690

740

790

840

890

1255 1260 1265 1270 1275 1280

firm

out

put

power generation

MOABC MODE

590

610

630

650

670

690

710

730

750

770

1165 1170 1175 1180 1185

firm

out

put

power generation

MOABC MODE

22 X. Liao et al.

5. CONCLUSION

This paper presents a novel multi-objective evolutionary algorithm named multi-objective artificial bee colony (MOABC) algorithm and compares the efficiency of MOABC and established algorithms in long-term cascaded hydropower system dispatch. A set of test functions and the long-term scheduling problem of Three Gorges Project have been used to verify the effectiveness of proposed method. The results showed that the proposed method has a better convergence ability and distribution of the Pareto front comparing to the established method.

ACKNOWLEDGEMENTS

This work was supported by the National Natural Science Foundation of China (No. 51239004), and the Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20100142110012). The authors also thank for the insightful comments and suggestions of anonymous reviewers.

REFERENCES

Afshar, M. H. (2012). "Large scale reservoir operation by Constrained Particle Swarm Optimization algorithms." Journal of Hydro-Environment Research, 6(1), 75-87.

Barros, M. T. L., Tsai, F. T. C., Yang, S. L., Lopes, J. E. G., and Yeh, W. W. G. (2003). "Optimization of large-scale hydropower system operations." Journal of Water Resources Planning and Management-Asce, 129(3), 178-188.

Deb, K., Pratap, A., Agarwal, S., and Meyarivan, T. (2002). "A fast and elitist multiobjective genetic algorithm: NSGA-II." Ieee Transactions on Evolutionary Computation, 6(2), 182-197.

Eckart, Z., Marco, L., and Lothar, T. (2001). "SPEA2: Improving the Strength Pareto Evolutionary Algorithm." Ferrero, R. W., Rivera, J. F., and Shahidehpour, S. M. (1998). "Dynamic programming two-stage algorithm for long-term

hydrothermal scheduling of multireservoir systems." Ieee Transactions on Power Systems, 13(4), 1534-1540. Guan, X. H., Ni, E. N., Li, R. H., and Luh, P. B. (1997). "An optimization-based algorithm for scheduling hydrothermal power

systems with cascaded reservoirs and discrete hydro constraints." Ieee Transactions on Power Systems, 12(4), 1775-1780. Haddad, O. B., Afshar, A., and Marino, M. A. (2006). "Honey-bees mating optimization (HBMO) algorithm: A new heuristic

approach for water resources optimization." Water Resources Management, 20(5), 661-680. Hincal, O., Altan-Sakarya, A. B., and Ger, A. M. (2011). "Optimization of Multireservoir Systems by Genetic Algorithm." Water

Resources Management, 25(5), 1465-1487. Jacovkis, P. M., Gradowczyk, H., Freisztav, A. M., and Tabak, E. G. (1989). "A linear programming approach to water-resources

optimization." Zeitschrift für Operations Research, 33(5), 341-362. Jalali, M. R., Afshar, A., and Marino, M. A. (2007). "Multi-colony ant algorithm for continuous multi-reservoir operation

optimization problem." Water Resources Management, 21(9), 1429-1447. Karaboga, D. (2005). "An idea based on honey bee swarm for numerical optimization." Technical Report-TR06, Erciyes University,

Engineering Faculty, Computer Engineering Department. Karaboga, D., and Ozturk, C. (2009). "Neural Networks Training by Artificial Bee Colony Algorithm on Pattern Classification."

Neural Network World, 19(3), 279-292. Kumar, D. N., and Reddy, M. J. (2007). "Multipurpose reservoir operation using particle swarm optimization." Journal of Water

Resources Planning and Management-Asce, 133(3), 192-201. Li, J. Q., Pan, Q. K., and Gao, K. Z. (2011). "Pareto-based discrete artificial bee colony algorithm for multi-objective flexible job

shop scheduling problems." International Journal of Advanced Manufacturing Technology, 55(9-12), 1159-1169. Liao, X., Zhou, J., Zhang, R., and Zhang, Y. (2012). "An adaptive artificial bee colony algorithm for long-term economic dispatch in

cascaded hydropower systems." International Journal of Electrical Power & Energy Systems, 43(1), 1340-1345. Mandal, K. K., and Chakraborty, N. (2011). "Short-term combined economic emission scheduling of hydrothermal systems with

cascaded reservoirs using particle swarm optimization technique." Applied Soft Computing, 11(1), 1295-1302. Ozturk, C., Karaboga, D., and Gorkemli, B. (2012). "Artificial bee colony algorithm for dynamic deployment of wireless sensor

networks." Turkish Journal of Electrical Engineering and Computer Sciences, 20(2), 255-262. Ponnambalam, K., Vannelli, A., and Unny, T. E. (1989). "An application of Karmarkar's interior-point linear programming algorithm

for multi-reservoir operations optimization." Stochastic Hydrology and Hydraulics, 3(1), 17-29. Teegavarapu, R. S. V., and Simonovic, S. P. (2002). "Optimal operation of reservoir systems using simulated annealing." Water

Resources Management, 16(5), 401-428. Tripathi, P. K., Bandyopadhyay, S., and Pal, S. K. (2007). "Multi-Objective Particle Swarm Optimization with time variant inertia

and acceleration coefficients." Information Sciences, 177(22), 5033-5049. WCD. (2000). "Dams and development: a new framework for decision-making." Available via DIALOG,

http://www.dams.org/docs/report/wcdreport.pdf.

Water Utility Journal 7 (2014) 23

Xue, F., Sanderson, A. C., and Graves, R. J. "Pareto-based multi-objective differential evolution." Evolutionary Computation, 2003. CEC '03. The 2003 Congress on, 862-869 Vol.2.

Zhou, J. Z., Lu, Y. L., Qin, H., Wang, Y., and Zhang, Y. C. (2011). "Environmental/economic dispatch problem of power system by using an enhanced multi-objective differential evolution algorithm." Energy Conversion and Management, 52(2), 1175-1183.