Artificial bee colony optimization for economic dispatch with valve point effect

Abstract

In recent years, various heuristic optimization methods have been proposed to solve economic dispatch (ED) problem in power systems. This paper presents the well-known power system ED problem solution considering valve-point effect by a new optimization algorithm called artificial bee colony (ABC). The proposed approach has been applied to various test systems with incremental fuel cost function, taking into account the valve-point effects. The results show that the proposed approach is efficient and robust when compared with other optimization algorithms reported in literature.

Keywords

artificial bee colony (ABC) algorithm / economic dispatch (ED) / valve-point effect / optimization

Introduction

In general, economic dispatch (ED) problem is one of the most important problems in the operation of power systems. The objective of the ED problem of electric power generation is to schedule the outputs of all generating units so as to meet the load demand at the minimum total operating fuel cost, subject to equality constraints on power balance and inequality constraints on power outputs. This makes the ED problem a large-scale highly nonlinear constrained optimization problem. Improvements in the scheduling of the generator power outputs can lead to very important fuel cost savings [ 1, 2]. Previous efforts on solving ED problems have been applied classical mathematical programming techniques such as interior point algorithm, linear programming and dual quadratic programming [ 3, 4]. In these mathematical techniques, the main assumption is that the fuel cost curve is considered as a monotonically increasing one.

However, when the problem is highly nonlinear or has non-smooth cost functions, some of these techniques may not be able to produce good solutions.

Approximately for the past 20 years, many researchers have used heuristic optimization techniques unlike conventional mathematical techniques in solving ED problems in power systems [ 5, 6]. Dakuo et al. [ 7] proposed a hybrid genetic algorithm approach based on differential evolution algorithm to solve the ED problem with valve-point effect. Coelho and Mariani [ 8] utilized the particle swarm approach based on quantum mechanics and harmonic oscillator potential to solve economic load dispatch with valve-point effects. Al-Sumait et al. [ 9] used pattern search method to solve this problem. Su and Lin [ 10] investigated to solve this problem using the Hopfield model. Zhang [ 11] proposed the quantum behaved particle swarm optimization algorithm for economic load dispatch in power system. Bhattacharya and Chattopadhyay [ 12] utilized the biogeography-based optimization algorithm to solve complex economic load dispatch problems. Hosseini et al. [ 13] used a novel mathematical-heuristic method for non-convex dynamic ED. Subramanian and Anandhakumar [ 14] attempted to provide a dynamic ED solution using composite cost function. And Hooshmand and Mohammadi [ 15] investigated the emission and the ED problem.

Artificial bee colony (ABC) optimization algorithms are formulated based on the natural foraging behavior of honey bees. ABC was first developed by Dr. Karaboga [ 16, 17]. Some artificial ideas are added to construct a robust ABC. Unlike classical search and optimization methods, ABC starts its search with a random set of solutions (colony size), instead of a single solution just like genetic algorithm (GA). Each population member is then evaluated for the given objective function and is assigned fitness. The best fits are entertained for the next generation while the others are discarded and compensated by a new set of random solutions in each generation. The only stopping criterion is the completion of maximum No. of cycles or generations. At the end of the cycles, the solution of the best fit is the desired solution.

The main objective of this study is to present the use of the ABC optimization technique to the subject of the ED in power systems. In this paper, the ABC method has been proposed to solve the ED problem with valve-point effect for 3, 13 and 40 unit test systems. In general terms, the contribution of this paper is the new efficient ABC approach for the ED problem with the valve-point effect. The results obtained with the proposed ABC approach were analyzed and compared with other optimization results reported in literature.

Formulation of ED problem

The classical formulation of the standard ED problem is an optimization problem of determining the schedule of the fuel costs of real power outputs of generating units subject to the real power balanced with the total load demand as well as the limits on generators outputs. In mathematical terms the problem can be defined as

(1) \min F={\displaystyle \sum _{i=1}^N{F}_i{\left(P\right)}_i},

where F_i is the total fuel cost of the generator units, which is defined by

(2) F_i\left(P_i\right)=a_i\times P_i^2+b_i\times P_i+c_i,

where a_i, b_i and c_i are cost coefficients of generator i and P_i is subject to power balance constraints:

(3) D={\displaystyle \sum _{i=1}^N{P}_i}-P_{\mathrm{D}}-P_{\mathrm{L}}=0,

where P_D is the system load demand and P_L is the transmission loss. The generating capacity constraints of each generator must be between its minimum and maximum values.

(4) P_i^{\min }\le P_i\le P_i^{\max }\mathrm{for}i=1,\; 2,\cdots ,N.

The inclusion of valve-point loading effects makes the modeling of the incremental fuel cost function of the generators more practical. This increases the nonlinearity as well as number of local optima in the solution space. The incremental fuel cost function of the generating units with valve-point loadings are represented [ 2, 7, 8] as

(5) {\tilde{F}}_i\left(P_i\right)=F_i\left(P_i\right)+\left|e_i\times \sin \left(f_i\times \left(P_i^{\min }-P_i\right)\right)\right|,

(6) {\tilde{F}}_i\left(P_i\right)=a_i\times P_i^{\mathrm{2}}+b_i\times P_i+c_i+\left|e_i\times \sin \left(f_i\times \left(P_i^{\min }-P_i\right)\right)\right|,

where e_i and f_i are constants of the valve-point effect of generators. Therefore, the total fuel cost that must be minimized according to Eq. (7), which is modified [ 11] to

(7) \min F={\displaystyle \sum _{i=1}^N\widetilde{F_i}\left(P_i\right)},

where {\tilde{F}}_i is the cost function of generator ith ($/h) defined by Eq. (7). The system losses are ignored for all test systems considered in this study.

The fuel cost function curves without valve effects and with valve effects are shown in Fig. 1.

ABC foraging behavior

To find the optimal decision variables, to optimize an objective function and to satisfy the constraints, the variables are bounded to the limits. Equation (6) gives a function defined to take care of variable bounds [ 16].

Random solution generation

Food sources which are in their proximity are selected by the employed bees when they move to a new location. Each employed bee associated with a food source is responsible for nectar extraction from it.

P_i=P_{i\min }+\mathrm{rand}\left(0,1\right)\times \left(P_{i\max }-P_{i\min }\right),

\forall i\in \left\{\mathrm{1},\mathrm{2},\mathrm{...},n_{\mathrm{g}}\right\},

D={\displaystyle \sum _{i=1}^N{P}_i}-P_{\mathrm{D}}-P_{\mathrm{L}}=0,

where P_imin and P_imax are the lower and upper bounds of variable P_i. In Eq. (6) rand (0, 1) represents a random number between 0 and 1.

The solution is represented in a matrix form as

(8) {\mathit{X}}_i=\left[\begin{array}{lllllll}{P}_1\hfill & P_2\hfill & P_3\hfill & P_4\hfill & P_5\hfill & \cdots \hfill & P_{n_{\mathrm{g}}}\hfill \end{array}\right].

Similarly the food sources \left\{{\mathit{X}}_1,{\mathit{X}}_2,{\mathit{X}}_3,{\mathit{X}}_4,\mathrm{...},{\mathit{X}}_n\right\} is the set of all the randomly chosen solutions which satisfies all the defined constraints.

Evaluation of fitness of solutions

The food sources are ranked based on the quality and quantity of their nectar. Similarly, fitness is assigned to each solution, which represents the goodness of each solution.

(9) \mathrm{Fitness}\left(i\right)={\displaystyle \frac{\mathrm{1}}{1+\sum F_i}}\hspace{1em}\forall i\in \left\{1,2,3,\cdots ,n_{\mathrm{g}}\right\},

where {\displaystyle \sum F_i} represents the total fuel cost of generation.

Employed bee phase

Each solution is handled by an employed bee who searches for the food source in their neighborhood and if a better food source is found it discards its previous food source and starts exploring the new one until it finds a better food source.

Similarly, a mutant solution is generated for each solution using its randomly selected neighbor and the parameter to be changed.

\left\{{\mathit{X}}_1,{\mathit{X}}_2,{\mathit{X}}_3,{\mathit{X}}_4,\mathrm{...},{\mathit{X}}_n\right\}

is the solution set where each solution X_i is represented as {\mathit{X}}_i=\left[\begin{array}{lllllll}{P}_1\hfill & P_2\hfill & P_3\hfill & P_4\hfill & P_5\hfill & \cdots \hfill & P_{n_{\mathrm{g}}}\hfill \end{array}\right].

A random variable of all n_g variables is chosen and a neighbor of all n–1 neighbors is chosen randomly and a mutant solution is produced as

(10) {\mathit{X}}_{\mathrm{1mutant}}={\mathit{X}}_1\left(i\right)+\left[{\mathit{X}}_j\left(i\right)-{\mathit{X}}_1\left(i\right)\right]\times \left(2\times \mathrm{rand}-1\right),

where i and j is the randomly chosen parameter and the neighbor, respectively.

A greedy selection between the mutant and original solutions takes place resulting in the discard of the least fit solution. This process of selection is repeated for each solution. The solution whose mutant is less fit increases its trial and may lead to dissertation of the food source if the trial leads to a threshold limit.

Onlooker bee phase

The onlooker bees in the hive detect a food source by means of the information presented to them by the employed foragers. A food source is chosen with the probability which is proportional to its food quality. Different schemes can be used to calculate the probability values. For example

(11) \mathrm{Probability}\left(i\right)={\displaystyle \frac{\mathrm{Fitness}\left(i\right)}{\mathrm{sum}\left(\mathrm{Fitness}\right)}}, \mathrm{Probability}\left(i\right)={\displaystyle \frac{a\times \mathrm{Fitness}\left(i\right)}{\max \left(\mathrm{Fitness}\right)+b}}.

\mathrm{where}a+b=1.

A random number chosen which represents the expectancy of the onlooker bee is compared with the probability of a solution (food). If the solution meets the expectancy of the onlooker, then it moves to exploit the food source and becomes an employed bee and corresponding employed bee of food source retires. The new employed bee starts exploring the neighborhood and repeats the employed bee behavior.

If the expectancy is not reached, the onlooker chooses other food source (solution) with different expectancy until it becomes employed. The above procedure repeats while all the onlooker bees get employed to food source. The food source with the highest probability will be chosen maximum and the one with least probability is discarded more times.

Scout bee phase

The scout bee is to explore the search area and it is often represented by a randomly generated solution. It will replace an employed bee if its trials of mutation exceed a threshold limit.

The scout will encourage the exploration of unexplored area of the search space. The best solution and fitness values are memorized for every iteration. The above process is repeated for maximum number of iterations and the result at the end will ensure a global minimum or maximum.

ABC algorithm

The proposed ABC algorithm is summarized as follows:

Step 1.Read the line input data; initialize MaxIterC (maximum iteration count) and base case as the best solution;

Step 2.Construct initial bee population (solution) X_ij as each bee is formed by the open switches in the configuration and the number of employed bees are equal to onlooker bees;

Step 3.Evaluate the fitness value for each employed bee by using Eq. (9);

Step 4.Initialize cycle= 1;

Step 5. Generate a new population (solution) V_ij in the neighborhood of X_ij for employed bees using Eq. (13) and evaluate them;

Step 6.Apply the greedy selection process between X_i and V_i;

Step 7. Calculate the probability values P_i for the solutions X_i by means of their fitness values using Eq. (11);

Step 8.Produce the new populations V_i for the onlookers from the populations X_i, selected depending on P_i by applying roulette wheel selection process, and evaluate them;

Step 9.Apply the greedy selection process for the onlookers between X_i and V_i;

Step 10.Determine the abandoned solution, if exists, and replace it with a new randomly produced solution X_i for the scout bees using Eq. (6);

Step 11.Memorize the best solution achieved so far;

Step 12.cycle= cycle+ 1;

Step 13.If cycle<MIC, go to Step 5, otherwise go to Step 14;

Step 14. Stop.

Numerical results

To verify the feasibility and efficiency of the proposed algorithm, three tests were conducted for solving ED problem with valve-point effects, which are 3-, 13- and 40- unit systems ignoring the transmission loss, including valve-point loading.

The algorithm of this method was programmed in Matlab environment and run on a Pentium IV, 3 GHz personal computer with 1 GB random-access memory (RAM).

Test system 1: small system (3-unit system)

This test case study considering three thermal units of generation with effects of valve-point is given in Table 1 [ 18]. In this case, the load demand expected to be determined was P_D = 850 MW.

The simulation parameters for the proposed algorithm are: colony size (employed bees+ onlooker bees) = 20, food sources= 10, limit= 100, and max iterations= 500.

The results obtained for this case study are listed in Table 2, which shows that the ABC algorithm has approximately good solution for the power demand of 850 MW. The best fuel cost result obtained from the proposed ABC algorithm and other optimization algorithms are compared in Table 3. From Table 3 it is seen clearly that the GA and pattern search (PS) approaches did not meet the load demand.

A convergence characteristic of the ABC algorithm for the three generator systems shown in Fig. 2. Figure 3 shows the distribution of the generation cost of the best solution for each run in the test system of 3 units.

Test system 2: 13-unit system

This test case study considering the 13 thermal units of generation with effects of valve-point is given in Table 4 [ 8, 19].

The complexity and nonlinearity to solution procedure is increased. The required load demands to be met by all the 13 generating units are 1800 MW and 2520 MW.

The results obtained for this case study are given in Tables 5 and 6, which show that the simulation results obtained by the ABC algorithm for the best solution for power demand of 1800 MW and 2520 MW, respectively.

Simulation parameters: colony size= 200, food sources= 100, limit= 100, and max iterations= 1000.

The best fuel cost result obtained from the proposed ABC algorithm and other optimization algorithms are compared in Tables 7 and 8 for the load demand of 1800 MW and 2520 MW respectively. It appears that the proposed algorithm performs better as the problem becomes larger and more complex. Figures 4 and 5 show the convergence characteristic curves of the best case with valve point effect for the load demand of 1800 MW and 2520 MW respectively. Figures 6 and 7 show the distribution of the generation cost of the best solution value for 30 trails for the load demand of 1800 MW and 2520 MW respectively.

Test system 3: large system (40-unit system)

This test system consists of 40 generators with valve-point loading effects and has a total load demand of 10500 MW. The input data are given in Ref. [ 19]. The result obtained from the proposed ABC algorithm has been compared with new PSO with local random search (NPSO-LRS) [ 25], modified ED (MED) [ 26], and other methods. The best solutions are tabulated in Table 9 and the performance parameters are compared in Table 10. A convergence characteristic of the 40-generator systems in case of the ABC algorithm is demonstrated in Fig. 8. The distribution of the generation cost of the best solution for each run in the test system of 40-units is exhibited in Fig. 9.

Simulation parameters: colony size (employed bees+ onlooker bees) = 200, food sources= 100, limit= 100, and max iterations= 1000.

The comparison confirms the effectiveness, stable convergence characteristic, good computation efficiency and superiority of the proposed ABC algorithm over the other techniques in terms of solution quality.

Conclusions

The economic load dispatch problem with valve-point effects was studied using the ABC algorithm for various generator test system and the performance of the proposed approach was evaluated. Of all the evolutionary algorithms, the ABC algorithm is the best method to reach the near global optimal solution but at the cost of high computational time.

However good choice of the number of iterations, population size, employed and unemployed bees results in fast computation. The ABC can be modified using operators of fast computational algorithms to get a hybrid fast computational ABC.

A numerical simulation including comparative studies was presented to demonstrate the performance and applicability of the proposed method. The simulation results reveal the superiority of the proposed technique in solving the ED problem with valve point effects. Therefore, this approach could also be extended to other optimization and control problems of power systems.

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