This paper presents a new method combined in two meta-heuristic methods, the Genetic algorithm (GA) and micro-genetic algorithm (MGA), to solve the EPD ...
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ScienceDirect Energy Procedia 50 (2014) 937 – 944
The International Conference on Technologies and Materials for Renewable Energy, Environment and Sustainability, TMREES14
Solving Economic Dispatch Problem Using Hybrid GA-MGA R.L.Kherfane, M.Younes, N. Kherfane and F.Khodja Faculty of Technology Djillali Liabes University Sidi Bel Abbes, Algeria
Abstract This paper presents a new method combined in two meta-heuristic methods, the Genetic algorithm (GA) and micro-genetic algorithm (MGA), to solve the EPD (economic power dispatch) problem. The GA is applied to find the global optimum by against the MGA is used to search the local optimum. The problem is addressed by considering the injection of wind power into the electric network. The approach is applied to 30 buses and on a western part of the Algerian power network. The obtained results are compared with those in literature. © Ltd. This is an open article under the CC BY-NC-ND license © 2014 2014Elsevier The Authors. Published byaccess Elsevier Ltd. (http://creativecommons.org/licenses/by-nc-nd/3.0/). Selection and peer-review under responsibility of the Euro-Mediterranean Institute for Sustainable Development (EUMISD). Selection and peer-review under responsibility of the Euro-Mediterranean Institute for Sustainable Development (EUMISD)
Keywords— EPD (economic power dispatch), optimization, genetic algorithm, wind energy, hybrid approach.
1.
T
Introduction
he EPD (economic power dispatch) problem has been one of the most widely studied subjects in the power system community since Carpentier first published the concept in 1962 [1]. The EPD problem is a large-scale highly constrained nonlinear non-convex optimization problem [2]. To solve it, a number of conventional optimization techniques such as nonlinear programming (NLP) [3], quadratic programming (QP) [4], linear programming (LP) [5], and interior point methods [6], Newton-based method [7], mixed integer programming[8], dynamic programming [9], branch and bound [10] have been applied. All of these mathematical methods are fundamentally based on the convexity of objective function to find the global minimum. However, the EPD problem has the characteristics of high nonlinearity and no convexity.
1876-6102 © 2014 Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/). Selection and peer-review under responsibility of the Euro-Mediterranean Institute for Sustainable Development (EUMISD) doi:10.1016/j.egypro.2014.06.112
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Applications of conventional optimization techniques such as the gradient-based algorithms are not good enough to solve this problem, because it depends on the existence of the first and the second derivatives of the objective function and on the well computing of these derivatives in large search space. Therefore, conventional methods based on mathematical techniques cannot give a guarantee to find the global optimum. In addition, the performance of these traditional approaches also depends on the starting points and is likely to converge to local minimum or even diverge. Our problem is optimization of the fuel cost, while satisfying certain system constraints, for achieve a good design with different solutions. Our method is tested on the electrical network IEEE 30 bus and on the western part of the Algerian power network. Simulation results confirm the advantage of computation rapidity and solution accuracy of the proposed method. These results show great promise. 2.
Problem formulation 2.1. Minimization of fuel
The goal of conventional EPD problem is to find an optimal allocation of generating powers in a power system. The power balance constraint and the generating power constraints for all units should be satisfied. In other words, the EPD problem is to find the optimal combination of power generations which minimize the total fuel cost while satisfying the power balance equality constraint and several inequality constraints on the system [11]. The total fuel cost function is formulated as follows: Ng
f ( PG )
¦ f (P i
Gi
)
(1)
i 1
x Active power planning with smooth cost function
fi (PGi ) ai PGi2 bi PGi ci
(2)
Where f ( PG ) is the total production cost in $/hr; f i ( PGi ) is the fuel cost function of unit i in $/hr;
ai, bi, ci are the fuel cost coefficients of unit i; PGi is the real power output of unit i in MW; In minimizing total fuel cost the following constraints should be satisfied. x Economic dispatch considering valve-point loading Effect The valve-point loading is taken in consideration by adding a sine component to the cost of the generating units.
f ( PG )
Ng
2
^
¦ ai PGi bi PGi ci d i u sin ei u Pi
i 1
min
Pi
`
di and ei are the cost coefficients of the unit with valve point effects.
(3)
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2.2. Problem constraints
x Active power balance equation For power balance an equality constraint should be satisfied, the generated power should be the same as total load demand added to the total line losses. It is represented as follows: Ng
¦ PGi
i 1
ND
¦ PDj
Pload PL
(4)
is the total system demand;
j 1
Ng
¦ PGi is the total system production;
i 1
Pload is the total load demand PL is the total transmission loss of the system in MW;
Ng is the number of generator units in the system; x Active power generation limits Generation constraints: Generator voltages, real power outputs and reactive power outputs are restricted by their lower and upper bounds as follows:
PGimin d PGi d PGimax
(5)
PGimin and PGimax are the minimum and maximum outputs of the i-th generation unit. 3.
Genetic Algorithm
The genetic algorithm is a search algorithm based on the mechanics of natural selection and natural genetics [12]. As summarized by Tomassini [13], the main idea is that in order for a population of individuals to adapt to some environment, it should behave like a natural system. This means that survival and reproduction of an individual is promoted by the elimination of useless or harmful traits and by rewarding useful behaviour. The genetic algorithm belongs to the family of evolutionary algorithms, along with genetic programming, evolution strategies, and evolutionary programming. Evolutionary algorithms can be considered as a broad class of stochastic optimization techniques. An evolutionary algorithm maintains a population of candidate solutions for the problem at hand. The population is then evolved by the iterative application of a set of stochastic operators. The set of operators usually consists of mutation, recombination, and selection or something very similar. Globally satisfactory, if sub-optimal, solutions to the problem are found in much the same way as populations in nature adapt to their surrounding environment. Using Tomassini’s terms, genetic algorithms (GAs) consider an optimization problem as the environment where feasible solutions are the individuals living in that environment. The degree of adaptation of an individual to its environment is the counterpart of the fitness function evaluated on a solution. Similarly, a set of feasible solutions takes the place of a population of organisms. An individual is a string of binary digits or some other set of symbols drawn from a finite set. Each encoded individual in the population may be viewed as a representation of a particular solution to a problem. In general, a genetic algorithm begins with a randomly generated set of individuals. Once the
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initial population has been created, the genetic algorithm enters a loop [13]. At the end of each iteration, a new population has been produced by applying a certain number of stochastic operators to the previous population. Each such iteration is known as a generation. The evolutionary cycle can be summarized as follows [12]: generation = 0 seed population while not (termination condition) do generation = generation + 1 calculate fitness selection crossover mutation end while 4.
Hybrid GA-MGA
The searching process starts with the genetic algorithm (GA) by initializing a group of random individuals (chromosomes). After a specific number of iterations to search for the best global position in the solution space, the calculation continues with the micro-genetic algorithm (MGA) for find the final solution (exact) near solution close. In this approach, the hybrid algorithm can quickly and accurately find an optimum solution. 5.
Simulation Results
The proposed GA-MGA approach based on global and local search is developed in the Matlab programming language using 7.04 version and tested on two networks: 5.1. Network 1
Standard IEEE 30 bus, bus data, generator data and the minimum and maximum limits for the control variables are given in table 1. Test system has six generators at the buses 1, 2, 5, 8, 11 and 13; and four transformers with off-nominal tap ratio at lines 6–9, 6–10, 4–12 and 28–27. In addition, buses 10, 12, 15, 17, 20, 21, 23, 24 and 29 were selected as shunt VAR compensation buses [11]. The total system demand is 283.4 MW. The maximum and minimum voltages of all load buses are considered to be 1.05–0.95 in p.u. The proposed approach has been applied to solve different cases: Table 1. Generators parameters of the IEEE 30 Bus
Bus
min pGi (MW)
max pGi (MW)
Cost coefficients
ai
bi
PG1 PG2 PG5 PG8 PG11
50 20 15 10 10
200 80 50 35 30
0.00375 0.01750 0.06250 0.00834 0.02500
2.00 1.75 1.00 3.25 3.00
ci
PG13
12
40
0.02500 3.00 0.00
0.00 0.00 0.00 0.00 0.00
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x Case 1: quadratic fuel cost minimization In this case the objective function is a quadratic form (eq 6). The fuel cost minimization with the hybrid method decreased to 800.486983 $/h compared with each method alone (Table 2). And more the results obtained from the GA-MGA are compared with other methods reported in the literature. The results of this comparison are shown in Table 3. It can be seen that the minimum total active power obtained by this method is less than the methods [14, 15, 16, 17, 18]. Table 2. Optimization results of GA-MGA approach for IEEE 30 bus
PG1 (MW) PG2 (MW) PG3 (MW) PG8 (MW) PG11(MW) PG13(MW) Fuel cost ($/h) Real loss (MW) T(S)
GA
MGA
GA-MGA
175.2708 54.00526 19.45330 12.77906 16.12452 15.11279
180.1945 48.13875 21.56847 12.30196 15.35001 15.13691
183.7565 46.7759 21.2341 15.7282 10.7189 14.4090
803.2197
801.9229
800.4869
9.3457
9.2906
9.2226
0.89
0.63
0.41
Table 3. Optimization results of GA-MGA approach for IEEE 30 referred from different literatures
176.0 48.76 21.56 22.05 12.44 12.00
GAF OPF [15] 174.96 50.353 21.451 21.176 12.667 12.11
802.2
802
802.38
804.0
802.62
801.21
800.4869
9.45
9.3230
9.4390
8.84
9.3900
9.0300
9.2226
/
/
/
/
/
/
0.41
Tabu search [14] PG1 PG2 PG3 PG8 PG11 PG13 Fuel cost ($/h) Real loss (MW) T(s)
GA OPF [15]
RGA [16]
EP [17]
GA fuzzy [18]
GA-MGA
175.64 48.941 21.176 22.647 12.431 12.00
174.0 46.8 22.0 23.9 11.0 14.5
173.84 49.998 21.386 22.63 12.928 12.00
178.17 45.16 20.65 21.29 15.16 12.00
183.7565 46.7759 21.2341 15.7282 10.7189 14.4090
x Case2: Quadratic Cost Curve with Sine Components In this case, the valve-point loading effects was studied, see the characteristics in table 4. The best results of the proposed approach compared with other methods are illustrated in Table 5 which shows clearly the superiority of the proposed approach. Table 4. Generator cost coef¿cient with valve point effect for modi¿ed IEEE 30-bus system. PG1 PG2 PG5 PG8 PG11 PG13
ai
bi
ci
di
ei
0.0016 0.01 0.0625 0.00834 0.025 0.025
2 2.5 1.00 3.25 3.00 3.00
150 25 0.00 0.00 0.00 0.00
50 40 0.00 0.00 0.00 0.00
0.063 0.098 0.00 0.00 0.00 0.00
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Table 5. Optimization results of GA-MGA approach with valve point effect for IEEE 30 FPSO GA MGA PSO [18] GA-MGA [18] PG1 199.34 199.66 199.78 199.78 199.73 PG2 20.03 20.14 20.24 20.00 20.00 PG3 23.13 18.70 21.60 25.42 18.49 PG8 22.78 17.18 19.91 22.43 24.29 PG11 13.97 10.31 14.22 13.37 16.74 PG13 14.56 27.77 18.13 12.94 10.55 14.57 Fuel cost 923.07 928.56 923.72 923.54 922.77 ($/h) Real loss 10.41 10.360 10.480 10.55 10.4200 (MW) 1.51 0.89 / / 0.62 T(S)
5.2. Network 2
The approach has also been tested on the electrical network in western Algeria. It consists of 2 generators supposed connected to a wind farm, with a capacity of 10 MW, on an area of 30 hectares. The site where the wind farm is located is Kabertene, 73 km north of the city of Adrar, in the Algerian Sahara. The fuel cost (in Nm3/h) equations for the two generators are:
f1 ( PG1 )
0.85PG21 150 PG1 2000
f 2 ( PG 2 ) 1.7 PG22 250 PG 2 3000 Subject to:
30 d PGi d 510 10 d PGi d 70 Pload
505MW We considered two cases:
x Case 1: The application of the methods GA , MGA and GA-MGA on objective function. The results obtained from the GA-MGA are compared with other methods reported in the literature [19]; this comparison is shown in Tables 6 and 7.
PG1 PG2 Fuel cost (Nm3/h) Real loss (MW) T(S)
Table 6. Comparison of different methods referred from different literatures. Fletcher Sonelgaz* FletcherGA[19] MGA GA-MGA [19] [19] Reeves [19] 466.64 469.93 450.95 465.94 474.544761 449.940322 54.25 49.98 69.00 55 45.364954 69.998818 278649
279940
271764
278319
282441.9847
270400.304021
15.94
15.94
15.94
15.94
14.9098
14.9391
0.05
0.01
0.05
/
0.034
0.023
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* Sonelgaz: Algerian Electricity and Gas Board. x case 2: Here, the GA-MGA method is applied on objective function, for improve the cost of fuel, taking into account the injection of wind energy (Pw=10 MW) into the network. Table 7. Comparison of the EPD control variables for different cases in Algerian network with wind energy PG1 PG2 PG3 Fuel cost (Nm3/h) Real loss (MW) T(S)
25% Pw 447.9193 69.7047 2.50000
50% Pw 445.5800 69.3089 5.0000
75% Pw 445.2494 67.0567 7.5000
100% Pw 441.9930 67.9222 10.0000
269647.49
266106.1714
264790.5266
262181.44
15.1240
14.8889
14.8061
14.9152
0.20
0.189
0.17
0.15
From the results of Table 8, we can notice that the injection of the wind power in the network has helped in reducing the cost for example by injecting 25% of the total wind energy, we save 752.8080 Nm3/h of fuel cost, with 50% PW we save 4294.1 Nm3/h, with the injection of 75% PW we save 5609.8 Nm3/h, with the injection of 100% wind-energy we save 8218.9 Nm3/h. 6.
Conclusion
This paper proposes a hybrid method which has two meta-heuristics, the genetic algorithm (GA) and micro-genetic algorithm (MGA). This approach is employed to solve EPD (Economic power dispatch) problem including fuel cost, which should be minimized while satisfying certain system constraints. The proposed approach has been tested on IEEE 30-bus and on the western part of the Algerian power network to verify its efficiency. This approach appears to be very efficient in particular for its fast convergence. Its effectiveness is demonstrated by comparing its performance with other evolutionary optimization algorithms .The computational results reveal that the GA-MGA approach has excellent convergence characteristics and is superior to other optimization algorithms. Also, the results confirm its great potential in handling the problem of EPD. And more the problem is addressed by considering the injection of wind power into the electric network, the results show that the wind turbine power has contributed significantly to improving the multi-objective function and the minimization of fuel cost. References [1] Carpentier J., Contribution a l’étude du dispatching économique. Bulletin de la Société Française des Electriciens, 1962, vol. 3, pp. 431–447. [2] Vanderbei, J.R and Shanno,F.D. , An interior-point algorithm for nonconvex nonlinear programming. Comput. Optim. Appl., 1999. vol. 13, pp. 231–252. [3] Bottero,M.H., Caliana,E.D.and A. R., Fahmideh Vojdani., Economic dispatch using the reduced hessian. IEEE Trans on Power Appra. Syst., 1982. vol. 101, pp. 3679-3688. [4] Reid, G. E., and Hasdorf, L., Economic dispatch using quadratic programming. IEEE Trans on Power Appara. Syst., 1973. vol. 92, pp. 2015-2023. [5] Stott, B., and Hobson, E., Power system security control calculation using linear programming, IEEE Trans on Power Appara. Syst., 1978. vol. 97, pp. 1713-1731. [6] Momoh, J. A., and Zhu, J. Z., Improved interior point method for OPF problems. IEEE Trans on Power System, 1999. vol. 14, pp.1114-1120.
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