Simulated annealing optimization for multiobjective economic dispatch solution
Irecom Laboratory, Department of Electrotechnics UDL University of Sidi Bel Abbes Sidi Bel Abbes, Algeria.
Emails: ziane_ismail2005@yahoo.fr ; farid.benhamida@yahoo.fr; graa_amel@yahoo.fr
^{* }Corresponding author, Phone: +213555835940
Abstract
This paper presents a multiobjective Simulated Annealing Optimization to solve a Dynamic Generation Dispatch problem. In this work, the problem is formulated as a multiobjective one with two competing functions, namely economic cost and emission functions, subject to different constraints. The inequality constraints considered are the generating unit capacity limits while the equality constraint is generationdemand balance. To show the advantages of the proposed algorithm, it has been applied for solving multiobjective EELD problems in a 3generator system with NO_{x} and SO_{2} emission. Results obtained with Simulated Annealing have been compared with other existing relevant approaches available in literatures. Experimental results show a proficiency of Simulated Annealing over other existing techniques in terms of robustness.
Keywords
Economic dispatch; Multiobjective optimization; Simulated annealing.
Introduction
The traditional economic dispatch is an optimization problem to find the most economical schedule of the generating units while satisfying load demands and load constraints.
The conventional economic dispatch (ED) cannot satisfy the environmental protection requirements, since it only considers minimizing the total fuel cost. The gaseous pollutants emitted by the power stations cause harmful effects with the human beings and the environment like the sulphur dioxide (SO_{2}), nitrogen oxide (NO_{x}), and the carbon dioxide (CO_{2}), etc. [1]. The optimization of production cost should not be the only objective but the reduction of emission must also be taken into account.
The ED problem can be handled as a multiobjective optimization problem that the objective functions are the total cost of electrical energy and the total emission function [2].
In general, multiobjective optimization problems are solved by reducing them to a scalar equivalent [3]. This is achieved by aggregating the objective functions into a single function [3].
Recently, multiobjective algorithms have also been used to solve the Dynamic Generation Dispatch problem. IBPVT approach [4], particle swarm optimization (PSO) [5], genetic algorithm (GA) [6], and linear programming [7]. New multiobjective stochastic search [8] has also been proposed to solve EED multiobjective problem by generating the Pareto optimal solution.
The objective of economic dispatch is to find the optimal combination of power generations that minimizes the total generation cost while satisfying an equality constraint and inequality constraints. On the other hand emission dispatch reduces the total emission from the system by an increase in the system operating cost.
Material and method
Economic dispatch and emission dispatch
The present formulation treats the EELD problem as a multiobjective mathematical programming problem which is concerned with the attempt to minimize each objective simultaneously. The equality and inequality constraints of the system must meanwhile, be satisfied. The following objectives and constraints are taken into account in the formulation of the EELD problem.
The economic dispatch problem can be modelled by:
_{} 
(1) 
where F_{T} is the total fuel cost; F_{T}(P_{i}) is the fuel cost of generating unit i; n is the no. of generator.
Fuel Cost Function: The fuel cost function of a generating unit is usually described by a quadratic function of power output P_{i} as:
_{} 
(2) 
where a_{i} ,b_{i} and c_{i} are the cost coefficient of unit i.
Emission Equation: The Emission equation kg/hr of a generating unit is usually described by a quadratic function of power output P_{i} as:
_{} 
(3) 
where d_{SO2i}, e_{SO2i} and f_{SO2i} are the SO_{2} emission coefficient of unit i.
Similarly, the emission dispatch problem for NO_{x} can be defined as the following optimization problem
_{} 
(4) 
where d_{NOxi}, e_{NOxi} and f_{NOxi} are the NO_{x} emission coefficient of unit i.
The emission dispatch problem for CO_{2} can be defined as the following optimization problem
_{} 
(5) 
where d_{CO2i} , e_{CO2i} and f_{CO2i}_{ }are the CO_{2} emission coefficient of unit i.
Transmission losses: The transmission losses P_{L} can be found using Bcoefficients
_{} 
(6) 
where B_{ij} is the transmission line coefficients.
Power Balance Constraints: The total supply must be equal to power demand
_{} 
(7) 
where P_{D} is the load demand.
Generator limit Constraints: The power generation of unit i should be between its minimum and maximum limits.
_{} 
(8) 
where P_{i min}_{ }is the minimum generation limit of unit i and P_{i} max is the maximum generation limit of unit i.
Multiobjective dispatch model
The general structure of multiobjective generation and emission dispatch problem is expressed as find:
_{} 
(9) 
Subject to:
_{} 
(10) 
The above mentioned multi objective optimization problem can be converted to a single objective optimization problem by introducing price penalty factors as follows:
_{} 
(11) 
where h_{SO2}, h_{NOx} and h_{CO}_{2} are price penalty factors for SO_{2}, NO_{x}, and CO_{2}, respectively, blending the emission costs with the normal fuel costs.
A simulated annealing algorithm for multiobjective dispatch model
The concept of simulated annealing was first introduced in the field of optimization in the early 1980’s by Kirkpatrick and independently by Cerny [9]. Simulated annealing is a robust, generalpurpose combinatorial optimization algorithm based on probabilistic methods which has been applied successfully too many areas such as VLSI circuit design, neuralnetworks, image processing, code design, capacitor placement in power systems, and economic load dispatch.
This method is based on slow cooling of a material at state fusion, which leads it to a solid state with low energy.
The same basic principle can be used in an optimization algorithm. The objective function to be minimized can be considered as the system energy, while different combinations of the optimization are the configurations of the system given its degrees of freedom.
Analogy to physical annealing
The name simulated annealing comes from an analogy between combinatorial optimization and the physical process of annealing. In physical annealing a solid is cooled very slowly, starting from a high temperature, in order to achieve a state of minimum internal energy. It is cooled slowly so that thermal equilibrium is achieved at each temperature. Thermal equilibrium can be characterized by the Boltzmann distribution.
_{} 
(12) 
where X is a random variable indicating the current state, E_{x }is the energy of state x, k_{B} is Boltzmann’s constant, and T is temperature.
The evolution of the state of a solid in a heat bath toward thermal equilibrium can be efficiently simulated by a simple algorithm based on Monte Carlo techniques which was proposed by Metropolis [10] in 1953. The Metropolis algorithm takes the current state x, and generates a new state y by applying some small perturbation. The transition from state x to state y is then accepted with probability.
_{} 
(13) 
If accepted, y becomes the current state and the procedure is repeated. This acceptance rule is known as the Metropolis criterion. Given a particular combinatorial optimization problem let the solution x correspond to the current state of the solid, the cost function correspond to the energy of the current state, and the control parameter T correspond to the temperature of the solid. The simulated annealing algorithm consists simply of iterating the Metropolis algorithm for decreasing values of the artificial temperature parameter T.
Optimization problem 
Physical system 
solution x 
current state of the solid 
cost or objective value f(x) 
energy of current state 
control parameter T 
temperature 
optimal solution xopt 
ground state 
simulated annealing 
gradual cooling 
Some of the analogies between the thermal process of physical annealing and the artificial process of simulated annealing in a combinatorial optimization problem are summarized in Table 1.
Control parameters of AS algorithm
The algorithm of simulated annealing consists of operating parameters [11], [12], which should be well set in order to achieve its best performance. These are briefly mentioned in the following.
Initial temperature
At beginning, initial temperature must be set at a higher value, in order to get more probability of acceptance for nonoptimized solutions during the first stages of the algorithm. Too much higher selection of initial temperature makes an algorithm slow and computationally inefficient. On the other hand, very low initial temperature may not be capable of searching a minimum especially for multi model function. There is no particular way to find out proper initial temperature which is suitable for whole range of problems.
Final temperature
While working with SA algorithm generally the final temperature fall is set to zero degree Celsius. SA algorithm can take much longer time to execute the operation, if the decrement in the temperature is exponential in nature. Finally, the stopping criterion is selected, which can be either an appropriate low temperature or the value where the system get freeze at that temperature.
Temperature decrement
As initial and final temperatures have predefined values, it is essential to find the approach of transition from starting to its final temperature as the success of algorithm depends on it. The decrement of temperature at time “t” is
_{} 
(14) 
where d is a positive constant.
The temperature decrement can also be implemented using
_{} 
(15) 
where α, is a constant close to 1.
Iterations at each Temperature
To enhance efficiency of the algorithm, selection of proper number of iterations is another important factor. The realization of only iteration for each temperature and the fall in temperature should take place at a really slow rate which can be expressed as:
_{} 
(15) 
Generally, β have very small value.
Simulated annealing algorithm
The SA algorithm for dispatch problem is stepped as follows:
Step 1: Initialization of the values temperature T, parameter α and iterations number criterion. Find randomly, an initial feasible solution, which is assigned as the current solution Si and perform ELD in order to calculate the total cost, Fcost, with the preconditions (7) and (8) fulfilled.
Step 2: Set the iteration counter to μ = 1.
Step 3: Find a neighboring solution S_{j} through a random perturbation of the counter one and calculate the new total cost F_{cost}
Step 4: If the new solution is better, we accept it, if it is worse, we calculate the deviation of cost ΔS=S_{j}S_{i} and generate a random number uniformly distributed over_{}.
If _{}
Accept the new solution S_{j} to replace S_{i} .
Step 5: If the stopping criterion is not satisfied, reduce temperature using parameter α: T(t) = αT and go to Step 2.
The proposed SA algorithm has been tested on a 3generator system with NO_{x} and SO_{2} emission. The software was implemented by the MATLAB language. For conducting the test, the initial temperature is fixed at 0.4°C, alpha is fixed at 0.5 and max tries is 10000. The final temperature is 1e10°C.
Results and discussion
The generator cost coefficients, emission coefficients, generation limits, and loss coefficients of 3 units system are taken from [14]. The multiobjective EELD solution for the 3 unit system is solved using SA when system demand is 850 [MW].

Tabu Search[13] 
NSGAII[14] 
OBBO[15] 
BBO[15] 
BB_BC[16] 
SA 
P_{1} 
435.69 
436.366 
435.1981 
435.1966 
434.5152 
435.1972 
P_{2} 
298.828 
298.187 
299.9697 
299.9723 
300.7308 
299.9632 
P_{3} 
131.28 
131.228 
130.6604 
130.6600 
130.6044 
130.6684 
Losses(MW) 
15.798 
15.781 
15.8282 
15.8289 
15.8505 
15.8288 
Fuel cost ($/h) 
8344.598 
8344.606 
8344.5875 
8344.5927 
8344.5952 
8344. 5927 
SO_{2} Emission (ton/h) 
9.02146 
9.02083 
9.021948 
9.02195 
9.02261 
9.0220 
NO_{2} Emission (ton/h) 
0.09863 
0.09860 
0.098686 
0.098686 
0.09871 
0.0987 

Tabu Search[13] 
NSGAII[14] 
OBBO[15] 
BBO[15] 
BB_BC[16] 
SA 
P_{1} 
549.247 
541.308 
552.1113 
552.1111 
552.7414 
552.1083 
P_{2} 
234.582 
223.249 
219.4441 
219.4441 
219.0790 
219.4484 
P_{3} 
81.893 
99.919 
92.9597 
92.96053 
92.6958 
92.9592 
Losses(MW) 
15.722 
14.476 
14.5151 
14.5158 
14.5164 
14.5159 
Fuel cost ($/h) 
8403.485 
8387.518 
8396.4616 
8396.4665 
8397.023 
8396.4800 
SO_{2} Emission (ton/h) 
8.974 
8.96655 
8.965931 
8.965937 
8.965936 
8.9659 
NO_{2} Emission (ton/h) 
0.09740 
0.09637 
0.096817 
0.096817 
0.09684 
0.0968 

Tabu Search[13] 
NSGAII[14] 
OBBO[15] 
BBO[15] 
BB_BC[16] 
SA 
P_{1} 
502.914 
505.81 
508.5800 
508.5813 
508.291 
508.5594 
P_{2} 
254.294 
252.951 
250.4423 
250.4433 
250.600 
250.4430 
P_{3} 
108.592 
106.023 
105.7228 
105.7212 
105.854 
105.7433 
Losses(MW) 
15.8 
14.784 
14.7451 
14.7459 
14.747 
14.7457 
Fuel cost ($/h) 
8371.143 
8363.627 
8365.1088 
8365.1146 
8364.953 
8365.1022 
SO_{2} Emission (ton/h) 
8.874 
8.96655 
8.973662 
8.973667 
8.965936 
8.9737 
NO_{X} Emission (ton/h) 
0.0958 
0.09593 
0.095923 
0.095923 
0.09592 
0.0959 

NSGAII[14] 
OBBO[15] 
OBBO [15] 
BB_BC[16] 
SA 
P_{1} 
496.328 
507.11971 
507.11954 
442.893 
442.1752 
P_{2} 
260.426 
251.64200 
251.64262 
305.503 
305.8545 
P_{3} 
108.144 
106.00030 
106.00042 
117.546 
117.9238 
Losses(MW) 
14.898 
14.76251 
14.76258 
15.94 
15.9535 
Fuel cost ($/h) 
8358.896 
8364.30627 
8364.31126 
8345.813 
8345.7469 
SO_{2} Emission (ton/h) 
8.97870 
8.974195 
8.974201 
9.01602 
9.0166 
NO_{X} Emission (ton/h) 
0.09599 
0.0959248 
0.0959248 
0.09776 
0.0978 
Total cost ($/h) 
31234.99029 
31226.40958 
31226.41898 
25035.140 
25035.1214 
The obtained results for minimum emission of SO_{2} are presented in Table 3, and the obtained results for minimum emission of NO_{x} are presented in Table 4. The best compromise solution for the precedent methods are presented in Table 5.
The best fuel cost is obtained by SA (8344. 5927 $/h). The best SO_{2} emission output is 8.9659 ton/h. The best NO_{x} emission output is 0.0959 ton/h. The best compromise solution is 25035.1214 $/h. The same test system was taken to solve the dynamic generation dispatch problem using the proposed SA method. The power demands for the 5 time intervals are given in Table 6.
Table 6. Best compromise solution.
Load 
550 
600 
700 
800 
850 
P_{1} 
307.1310 
329.3138 
374.0705 
419.3406 
442.1703 
P_{2} 
160.3247 
184.6185 
233.1539 
281.6376 
305.8578 
P_{3} 
88.6302 
93.4363 
103.1426 
112.9674 
117.9255 
Losses(MW) 
6.0859 
7.3686 
10.3669 
13.9456 
15.9536 
Fuel cost ($/h) 
5581.4799 
6028.7064 
6939.2480 
7871.4463 
8345.7467 
SO_{2} Emission (kg/h) 
6.0071 
6.4941 
7.4855 
8.5004 
9.0166 
NO_{X} Emission (kg/h) 
0.0880 
0.0882 
0.0902 
0.0947 
0.0978 
Total cost ($/h) 
18760.5958 
19660.5278 
21633.8869 
23841.8787 
25035.1214 
Time interval 
P_{D} 
Best compromise solution 
Best fuel cost 
Percent change (%) 

Fuel cost 
SO_{2} emission 
NO_{x} emission 
Fuel cost 
SO_{2} emission 
NO_{x} emission 
Fuel cost 
SO_{2} emission 
NO_{x} emission 

1 
550 
5581.4799 
6.0071 
0.0880 
5575.7533 
6.0283 
0.0889 
0.1026 
0.3529 
1.0227 
2 
600 
6028.7064 
6.4941 
0.0882 
6024.9087 
6.514 
0.0889 
0.0630 
0.3064 
0.7937 
3 
700 
6939.248 
7.4855 
0.0902 
6937.9036 
7.5012 
0.0909 
0.0194 
0.2097 
0.7761 
4 
800 
7871.4463 
8.5004 
0.0947 
7870.6929 
8.5097 
0.0954 
0.0096 
0.1094 
0.7392 
5 
850 
8345.7467 
9.0166 
0.0978 
8344.5927 
9.0219 
0.0987 
0.0138 
0.0588 
0.9202 
Time interval 
P_{D} 
Best compromise solution 
Best SO_{2} emission 
Percent change (%) 

Fuel cost 
SO_{2} emission 
NO_{x} emission 
Fuel cost 
SO_{2} emission 
NO_{x} emission 
Fuel cost 
SO_{2} emission 
NO_{x} emission 

1 
550 
5581.4799 
6.0071 
0.0880 
5621.4302 
5.9771 
0.0925 
0.7158 
0.4994 
5.1136 
2 
600 
6028.7064 
6.4941 
0.0882 
6073.1566 
6.4619 
0.0922 
0.7373 
0.4958 
4.5351 
3 
700 
6939.248 
7.4855 
0.0902 
6987.6191 
7.4476 
0.0922 
0.6971 
0.5063 
2.2173 
4 
800 
7871.4463 
8.5004 
0.0947 
7921.8504 
8.4545 
0.0947 
0.6403 
0.5400 
0.0000 
5 
850 
8345.7467 
9.0166 
0.0978 
8396.4286 
8.9659 
0.0968 
0.6073 
0.5623 
1.0225 
Time Interval 
P_{D} 
Best compromise solution 
Best NO_{x} emission 
Percent change (%) 

Fuel cost 
SO_{2} emission 
NO_{x} emission 
Fuel cost 
SO_{2} emission 
NO_{x} emission 
Fuel cost 
SO_{2} emission 
NO_{x} emission 

1 
550 
5581.4799 
6.0071 
0.0880 
5583.1452 
6.0061 
0.0880 
0.0298 
0.0166 
0.0000 
2 
600 
6028.7064 
6.4941 
0.0882 
6033.0901 
6.4862 
0.0881 
0.0727 
0.1216 
0.1134 
3 
700 
6939.248 
7.4855 
0.0902 
6949.3417 
7.4638 
0.0897 
0.1455 
0.2899 
0.5543 
4 
800 
7871.4463 
8.5004 
0.0947 
7887.6146 
8.4645 
0.0933 
0.2054 
0.4223 
1.4784 
5 
850 
8345.7467 
9.0166 
0.0978 
8365.1352 
8.9737 
0.0959 
0.2323 
0.4758 
1.9427 
Figure 1. Best compromise solution for fuel cost, SO_{2} and NO_{x} emission for the 5 time intervals.
In Figure 1, the black axis presents the fuel cost, the blue axis presents the NO_{x} emission and the red axis presents the SO_{2} emission for the 5 time intervals. The Table 7 presents a percent change between the best compromise solution and the best fuel cost for the 5 time intervals. We note that the results from the best compromise solution give the best NO_{x} and SO_{2} emission. The Tables 8 and 9 present a percent change between the best compromise solution and the best SO_{2} emission, the best compromise solution and the best NO_{x} emission, for the 5 time interval.
In Figure 1, the load demand augmentation makes increasing in the fuel cost and the gas emission outputs (SO_{2} and NO_{x}). In Figure 2, the best compromise solution can give a better minimum NO_{x} emission output than the best fuel cost and the best SO_{2} emission solutions.
Figure 2. NO_{x} emission for the 4 solution cases (best fuel cost, best SO_{2 }emission, best NO_{x} emission and best compromise solution).
In contrast, from an environmental perspective, minimizing the harmful impacts of the gaseous emission is an imperative objective. These two conflicting objectives in particular are considered in both the single and multiobjective optimization of power system operation treated in this work. In general, optimization methods are classified into two main categories; deterministic and heuristic. In this thesis, one of the recently introduced heuristic optimization methods, SA, has been discussed, developed and employed to treat the considered economic and environmental optimization problems of power system operation. Various types of optimization functions were considered including single and multiobjective. The dynamic generation dispatch problem has been solved considering the transmission power losses. By these simulated results, SA method provides superior result than previously reported methods.
Conclusions
In this paper, Simulated Annealing Optimization (SA) has been proposed. In order to prove the effectiveness of the algorithm it is applied to EELD problem with three generating units. The results obtained by proposed method were compared with Tabu Search, NSGAII, OBBO, BBO, and BB_BC. The best fuel cost is obtained by SA (8344. 5927 $/h). The best SO_{2} emission output is 8.9659 ton/h. The best NO_{x} emission output is 0.0959 ton/h. The best compromise solution can create compatibility between production costs and the environment.
The comparison shows that the Simulated Annealing performs better than above mentioned methods. SA has superior features, including quality of solution, stable convergence characteristics and better computational efficiency. Therefore, this result shows that SA optimization is a promising technique for solving complicated problems in power system.
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