Application of the invasive weed optimization algorithm to economic dispatch problems

Abstract

In this paper the invasive weed optimization algorithm has been applied to a variety of economic dispatch (ED) problems. The ED problem is concerned with minimizing the fuel cost by optimally loading the electrical generators which are committed to supply a given demand. Some involve prohibited operating zones, transmission losses and valve point loading. In general, they are non-linear non-convex optimization problems which cannot be directly solved by conventional methods. In this work the invasive weed algorithm, a meta-heuristic method inspired by the proliferation of weeds, has been applied to four numerical examples and has resulted in promising solutions compared to published results.

Keywords

economic dispatch (ED) / invasive weed optimization / non-convexity / prohibited operating zones (POZ) / valve point loading / meta-heuristic

Introduction

The economic dispatch (ED) problem is one concerned with obtaining the minimum cost of fuel consumed by a set of generators which are committed to supply a given demand for electricity. The cost functions are usually expressed as quadratic functions of power. Non-linearity and non-convexity of this optimization problem is introduced when there are prohibited operating zones (POZ), multiple fuels with different cost functions, valve point loading and so on. Traditionally classical methods such as lambda iteration and dynamic programming were applied. Due to limitations of these algorithms, meta-heuristic methods were introduced in the last decade of the last century and were found to be promising. POZ are due to unfavorable operating regions in the generators where if operation is carried out there will be excessive vibrations of the shaft and are hence to be avoided. Thus there is a break in continuity in the feasible operating region, rendering the non-applicability of classical methods. One of the earliest approaches in solving the POZ problem was presented in Ref. [1] and solved by decomposition of the discontinuous feasible region into several continuous feasible regions and then applying the traditional lambda iteration method. An improvement is suggested in Ref. [2] where initially the POZ were ignored and if the solution happened to lie within them, the feasible regions closest to the solution were taken depending on the incremental cost on either side. This considerably minimizes the search within feasible regions. A genetic algorithm approach was proposed in Ref. [3]. Evolutionary Programming for the POZ problem was introduced in Ref. [4]. A zoom feature in dynamic programming was used in Ref. [5] to solve ED problems with transmission losses. Firefly algorithm (FA) and particle swarm optimization (PSO) have been used for solving POZ problems in Refs. [6] and [7]. Large steam turbine generators whose input and output characteristics are not always smooth have a number of steam admission valves that are opened in sequence to increase the output. This valve point effect introduces ripples in the heat rate curves, thereby making it non convex. Such non convex problems have been considered in Ref. [8].

In this paper the invasive weed optimization problem has been applied to the ED problem with POZ, transmission losses and valve point loading. This algorithm was inspired by the behavior of weed colonization and distribution. It was introduced by Mehrabian and Lucas [9] in 2006. It was used to solve the optimal control problem in Ref. [10], optimal placement of piezo electric actuators on smart structures [11], time modulated linear array antenna synthesis [12], multi-objective optimization [13] and optimization of printed Yagi antenna [14]. It was used in hybrid with PSO for fast and global optimization in Ref. [15] and for training feed forward neural networks in Ref. [16]. In Ref. [17] a modified invasive weed optimization with dual mutation technique was applied to the dynamic ED problem.

Mathematical formulation of ED problems

The ED problems dealt with here involve POZ, transmission losses and valve point loading. The general formulation of the ED problem is

(1)min{\displaystyle \sum _{i=\mathrm{1}}^{N_G}{a}_i{P}_{\mathrm{G}i}^{\mathrm{2}}+b_i{P}_{\mathrm{G}i}+c_i},

(2)\mathrm{s}.\mathrm{t}.{\displaystyle \sum _{i=\mathrm{1}}^{N_G}{P}_{\mathrm{G}i}}=P_{\mathrm{D}}+P_{\mathrm{L}},

(3)P_{\mathrm{G}i\min }\le P_{\mathrm{G}i}\le P_{Gi\max }(i=\mathrm{1}to{N}_{\mathrm{G}}),

where P_Gi is the power generated by the ith generator; P_D, the load demand; P_L, the transmission losses; P_Gimin, the minimum limit of the ith generator; P_Gimax, the maximum limit of the ith generator; N_G, the number of generators; and a_i, b_i&c_i, the generator cost coefficients.

For units with POZ, the additional constraints are

(4)P_{\mathrm{G}i\min }\le P_{\mathrm{G}i}\le P_{\mathrm{G}i}^{\mathrm{1}},

(5)P_{\mathrm{G}i,k-\mathrm{1}}^{\mathrm{u}}\le P_{\mathrm{G}i}\le P_{\mathrm{G}i,k}^{\mathrm{l}}(k=\mathrm{2}to{z}_{\mathrm{G}i}),

(6)P_{\mathrm{G}i,Z_{\mathrm{G}i}}^{\mathrm{u}}\le P_{\mathrm{G}i}\le P_{\mathrm{G}i\max },

where P_{\mathrm{G}i,k}^{\mathrm{l}} and P_{\mathrm{G}i,k}^{\mathrm{u}} are the lower and upper limits of the kth prohibited zone.

For problems involving transmission losses, the total loss is given by

(7)P_{\mathrm{L}}={\displaystyle \sum _{i=\mathrm{1}}^{N_{\mathrm{G}i}}{\displaystyle \sum _{j=\mathrm{1}}^{N_{\mathrm{G}i}}{P}_{\mathrm{G}i}{B}_{ij}{P}_{\mathrm{G}j}}},

where B_ij is the ijth element of the B(-loss) coefficient matrix.

For valve point loading, the objective function is given as

(8)min{\displaystyle \sum _{i=\mathrm{1}}^{N_{\mathrm{G}}}{a}_i{P}_{\mathrm{G}i}^{\mathrm{2}}+b_i{P}_{\mathrm{G}i}+c_i}+\left|e_i\sin (f_i(P_{\mathrm{G}i\min }-P_{\mathrm{G}i}))\right|,

where e_i and f_i are the additional generator constraints.

Overview of the invasive weed optimization algorithm

The invasive weed algorithm is based on the growth of weeds within an area. At first the weeds are spread throughout an area with uniform random distribution. As the weeds grow, they produce seeds in proportion to their fitness. The lowest and highest number of seeds are fixed arbitrarily with the least fit weed producing the lowest number of seeds and the most fit the highest number. The number of seeds produced by each weed varies linearly with the fitness. The seeds produce copies of the parent weeds. The weeds then shift their position randomly with a normal distribution of zero mean and a standard distribution which is large in the beginning but reduces as further generations are produced. There is a maximum population size. The total number is maintained constant after this size is reached by eliminating the weaker weeds. The optimization algorithm is as follows.

Step 1 Initialization: A number of candidates is generated randomly, uniformly distributed over the search space.

Step 2 Reproduction: The lowest and highest number of seeds are fixed arbitrarily. The fitness of each candidate is calculated. The least fit is assigned the lowest number of the seeds f_l and the most fit the highest number f_h. Each weed is assigned a number of seeds depending linearly on its fitness. The seeds produce copies of the parent.

Step 3 Relocation: The weeds move randomly around their present position. This random motion follows a normal distribution with zero mean and standard deviation which decreases over the generations as given by the formula below

(9)sd={\left({\displaystyle \frac{\mathrm{i}\mathrm{t}\mathrm{e}{\mathrm{r}}_{\max }-iter}{\mathrm{i}\mathrm{t}\mathrm{e}{\mathrm{r}}_{\max }}}\right)}^{\mathrm{p}\mathrm{o}\mathrm{w}}(\mathrm{s}{\mathrm{d}}_{\max }-\mathrm{s}{\mathrm{d}}_{\min })+\mathrm{s}{\mathrm{d}}_{\min },

where sd-standard deviation sd_max and sd_min are the limiting values of sd and pow is a nonlinear modulation index.

Step 4 Limiting the population size: If the total number exceeds a certain maximum number NP, the excess weeds of least fitness are eliminated mantaining the population constant at this number.

Step 5 Steps 1 to 4 are repeated over a fixed number of iterations (generations). After the fixed number is reached, the candidate with the highest fitness is declared the optimal.

Case studies

The proposed approach has been implemented to ED problems where the cost functions are with ① POZ, ② transmission losses, and ③ valve point loading effects. The results obtained are compared with those already published. Constraint violations are taken care of by including penalty terms to the objective function. The simulations were carried out in MATLAB R2008a run on a 2.40 GHz Intel (R) Core (TM) i3 processor with 1.86 GB of RAM.

Example 1

The data for this example has been taken from Ref. [2]. It contains 5 generators of which 3 generators have 2 POZs each. If during the computation, any of the solutions fall within a prohibited operatin zone, it is shifted to the boundary of the nearest operating zone [4]. The results obtained for this problem solved by the proposed algorithm is compared with λ-δ iterative method [2] in Table 1. From the table it is observed that both the methods give almost identical generation costs.

The convergence characteristics of IWO is illustrated in Fig. 1. It is found that the convergence is smooth and takes place after 70 iterations.

The parameters used in Example 1 are Number of weeds= 20; sd_max = 2; sd_min = 0.01; maximum population size NP= 30; nonlinear modulation index pow= 3; maximum number of seeds f_h = 5; minimum number of seeds f_l = 1.

Example 2

This example is taken from Ref. [2]. The system has 15 online units with a total demand of 2650 MW. Of these, 4 units have POZs. The results obtained with the proposed algorithm are compared with earlier published results in Table 2. The result obtained by the dynamic programming method [3] is $ 32506 while the proposed method yields $ 32507.98. These may be compared with the λ-δ iterative method [2], Deterministic crowding GA [3] and PSO [7]. The result obtained by IWO falls in between those obtained by GA and PSO. Convergence characteristics is demonstrated in Fig. 2. The unevenness in the convergence can be accounted for by the greater number of POZ compared to the previous example. With an increase in the number of generating units and POZs, the number of iterations required for convergence has increased considerably as compared with Example 1.

Example 3

This example data has been taken from Ref. [5]. This system has 3 generators in which the cost equation is cubic with B coefficients considered. The result obtained from the proposed algorithm is compared with dynamic programming [5] in Table 3. The results are almost identical. Figure 3 depicts the convergence characteristics.

Example 4

The data for this test system is obtained from Ref. [8]. It consists of 3 generating units incorporating valve point loading effects hence adding a sinusoidal term to the objective function. The results obtained by the proposed algorithm is compared with FA [6] in Table 4. As seen from Table 4, the solutions are identical in both methods. The convergence characteristics are displayed in Fig. 4.

Conclusions

The invasive weed optimization algorithm has been applied to 3 different kinds of ED problems which involve POZ, transmission line losses and valve point loading effects. A comparative study of the results indicate that it is as good as, if not better than other meta-heuristic and traditional algorithms. The results obtained are encouraging enough to investigate its application to other optimization problems.

References

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