Improved Dynamic Harmony Search Optimization for Economic Dispatch Problems with Higher Order Cost Functions

This paper presents a modified harmony search algorithm with dynamically varying bandwidth, named improved dynamic harmony search algorithm (IDHSA) for economic load dispatch (ELD) problems with higher cost functions. The economic load dispatch problem aims to schedule power outputs of the generating units to meet the system load demand at min imum cost while satisfying the equality and inequality constraints. In the IDHS algorithm, the key difference from the conventional HS algorithm is that bandwidth (BW) operator changes dynamically at every iterat ion. The IDHS algorithm is tested with two different cases of power systems without and with transmission losses. The obtained results prove the accuracy and the effectiveness of the IDHS algorithm in determining the best solution compared than other optimization methods recently published in the literature.


Introduction
An electric power system consists of generation, transmission and distribution utilities to enhance electrical power to the consumers. Economic dispatch is the short-term determination of the optimal output power of generators to meet the system load and operate the generators at the lowest fuel cost [1]. The solution accuracy of economic dispatch problems is associated with the precision of the fuel cost curve parameters [2]. In the classical ED problem, the cost function of active power generation is approximated presented as a second-order polynomial. But because of the high nonlinearity and non-smooth of the real input-output characteristics of the generating units, another cost function model is used by applying cubic cost function in wh ich outputs enter linearly, as quadratics, and to the third degree [3]. This type of function is the most correct because it yields appropriately shaped average and marginal cost curves [4]. A third-order polynomial model is mo re p recisely to reflect the real response of thermal generators [5]. Several kinds of research prove that cubic cost function is more practical than a quadratic cost function to express the operating cost [6]. In the recent years, the solution of ED problem with higher-order cost function has captured the attention of various researchers using diverse algorithms include various mathematical procedures such as PSO algorithm suggested in [5], genetic algorithm (GA) in [6]. Al-sumait et al. imp lemented a pattern search (PS) algorith m in literature [7] to solve ED problems with a cubic fuel cost function. In reference [8], a firefly algorith m (FA) is applied to solve ED problems with CCF. Simu lated Annealing algorith m (SA) is proposed in [9], Quantum-PSO developed by F. Parvez Mahdi et al. in [10], a novel MVM OS technique in [11], Grasshopper Optimization Algorithm (GOA) is proposed in [12]. Modified Firefly Algorith m with Levy Flights and Derived Mutation (MFA-LF-DM) in [13] is developed to solve single-objective dynamic ED with CCF. Artificial Bee Colony (A BC) [14] is imp lemented for co mbined economic and emission dispatch problems. Gravitational Search Algorith m is provided in [15] to solve a multi-objective dispatch problem.
Harmony search (HS) algorithm is one of the most popular evolutionary algorith ms orig inally invented by Geem et al [16], wh ich draws inspiration fro m the musical process to attain an agreeable harmony. HS takes some major advantages compared with other metaheuristics, such as EP, GA, DE and tabu search (TS), wh ich is simple in concept and structure, converges quickly to the optimu m and easy to implement on optimizat ion problems [17,18]. In the last decade, the harmony search algorith m (HSA) has widely used to solve kinds of optimization problems [19,20]. In this work, to prove the effectiveness and the robustness of the proposed method, the cubic cost function economic d ispatch problems (CCFED) of two d ifferent cases grouped as lossless and lossy test systems have been solving and the results have been compared with some other optimization techniques reported in recent literature. The remainder of the paper is organized as fo llo ws: Section. 2 provides descriptions and formulat ions of ELD problems with cubic fuel cost functions. Whereas the harmony search algorithm is briefly discussed in section. 3. Sect ion. 4 includes an introduction to the IDHS algorith m and its implementation to CCFED problems. Simu lation, results, and analysis of the findings are presented in section.5. Finally, the conclusion of the work done in the paper is given in section 6.

Descriptions and Formulations of CCFED Problems
In the CCFED we seek a) to min imize the total cost of generating real power, which is written as [2,22]: Where, P i is the real power generation of unit i in (MW), N nu mber of generators in the network, a i , b i , c i and d i are the fuel cost coefficients of the unit i.

b) To minimize the transmission losses
Equation (2) is subject to the equality and inequality constraints. A. Equality constraints Where P T , P D and P L are the total power generated, power system demand and total power loss respectively.

B. Inequality constraints
The output power produced by generators has particularly limited by lower and higher powers. Equation (5) shows the relation of inequality constraint.

Basic Harmony Search Algorithm HSA
The basic HS algorithm works as follows [16-18]: Step 1   Where f(x) is the objective function, xi is the solution vector of the HMS, LB i and UB i are the lower and upper values of x i . Step 2. Harmony memory in itializat ion. HM is a set of decision variables [17], the init ial harmony memory is randomly generated by using (8): Where j= 1, 2, ..., HMS and rand () is random nu mber, uniformly distributed within the range [0, 1].
Mathematically, the HM matrix can be represented by the following expression: x f x f x f

(9)
Step 3. Improvise a new harmony as follows: is generated based on the main HS operators HMCR, PAR and BW. Such operators are considered in production of a new harmony as the following [18]: A new solution vector x i based on the disturbance principle can be generated as follows: Step 4. Update the HM vector, ) , , , Step 5. (Checking the stopping criterion): Repeat steps 3 and 4 until the stopping criterion (normally, when the maximu m number of improvisations is reached).
A simplified flowchart of the HS method is demonstrated in Fig1.

Improved Dynamic Harmony
Search Algorithms (IDHSA)

The Motivation of the IDHS Algorithm
To enhance the behavior and the convergence speed of the original HSA, many variants of the HS method based on parameters setting have been studied with incorporating some modifications to the main operators of HSA (i.e., PAR, HM CR, and BW), such as the variations proposed in the literature [25][26][27][28][29]. The PAR and BW parameters, deciding the precision of the solution. If we choose a small BW and low PAR values, the convergence speed of HSA becomes slowly and the fine-tuning of solution vectors will increase. Furthermore, if we choose a bigger PAR value with a wide BW value in the early generation, HSA will converge quickly to the best solution. Therefore, is very necessary that PAR and BW were dynamically adjusting at every iteration.
In [30], Kalivarapu et al have developed a new modification of harmony search algorith m, with bandwidth varying dynamically at every iteration to improve the performance and co mp lexity of existing HSA. The principle behind th is idea is to use a wider BW to search in the entire domain and dynamically adjust the BW towards the optimal solution.

New Parameter of BW
In this subsection, the modified HS algorith m with dynamic BW operator will be studied. Dynamic BW (DBW) is represented as a decreasing function of the current generation and number of maximu m iterations specified for the problem, i.e. in the initial stages of this algorith m, BW is dynamically mod ified by maintaining a higher value and gradually decreasing by a low value to ensure close convergence of the optimal solution [30].
The selection of termination condition depends on the desired precision level of solution [31]. According to the criteria mentioned above, we find that the equation of a low-pass filter is approximately close to the requirement.
The new BW equation can be expressed as follows: Where, h and γ are the constant parameters depend on the limit values of BW. To satisfy the above-mentioned criteria, the exponent α must be greater than 1. 'i' and 'Nmax' are the current iteration and total number of iterations.
According to [30], a reasonably fair result can be obtained when minimu m value of BW (BW min) should be very small, generally ~ 0.1% of the range of decision variables. While, BW max is assumed to 5~10% of the range of decision variables. The constant γ is evaluated with the logarithmic decremental function as follow [30]: Where b 1 and b 2 are the constants and their numerical values are experimentally evaluated in [30,31], the best values of b 1 and b 2 that gives a minimu m BW value ensures precise and fast convergence towards the optimal solution are considered to be 50 and 100, respectively (best values).
For the problems having low decision variables (≤ 8 variables), Eq. (13) is more efficient to compute dynamic BW. For highly co mplex optimizat ion problems, the best solution is found by modified (13) to a discontinuous adaptative dynamic BW function [30]. The proposed DADBW function is expressed as: Where, 'pivot' decides the optimu m point where the BW changes (generally N max /2).

Implementation of IDHSA to CCFED Problems
Based on the corresponding improvement methods, the proposed IDHS algorithm process is shows below: Step 1. Specify the generator cost coefficients (a i , b i , c i and d i ,), total number of generator units (N), specify P max and P min of all generators and load demand P D .
Initialize the parameters of IDHS algorith m (HMS, HMCR, PAR, BW max , BW min , NI, N max and Pivot).
Step 2. In itialize HM matrix with size (HMS × N). The initial HM is randomly generating as follow: To not violate the generator limits and not to work at the prohibited zone after the generation of each solution vector, the total generation of units is compared to load demand. Fig. 2 shows the process of verifying the equality constraint used in the IDHS algorithm. Th is process is iterated for other units until the ε is zero.
Step 3. Calculate the fitness value for each harmony vector in the HM matrix using (5). The HM matrix is represented as follow:

(17)
Step 4. Generate a new harmony solution as described in step 3 (section 3), with bandwidth (BW) is dynamically updating using (15). The new harmony imp rovisation process is depicted by the flowchart shown in Fig.3.
Step 5. Update the HM vector and calculate the fitness value f(P G ), Step 6. If the maximu m nu mber of improvisations is reached, go to Step 7; otherwise, repeat steps 4-5.
Step 7. Print the optimal value of real power generation of generators and total cost of generation. Fig. 4 shows and explains the detailed phases of simu lations to solve the CCFED p roblem using IDHSA. After the generation of each harmony (solution vector), the total generation of units is compared to load demand P D .

Results and Discussion
To evaluate the efficiency of the proposed IDHS algorith m, the above algorith m is first applied to a CCFED problem with a 3-units system (Wollenberg's power network) [32] and 5-units systems [5] and then to Liang's power network [24]. It should be noted that the tow test systems are lossless in the first case while the power losses are considered in the second case. For both grouped cases studied, the algorithm of IDHS is run 20 independent trials; the software proposed was imp lemented in Matlab (R2017a) and executed on a PC with an Intel (R) Core i3-3110 CPU processor (2.40 GHz), 4 Go RAM and MS Windows 10. For conducting the test, the parameters of IDHS algorithm for all test systems are given as fo llo ws: HMS = 5, HM CR = 0.85, PAR = 0.3, BW min=0.01, BW max = 0.5, NI = 50 and the max tries (Nmax) is set at 3x104. The required data with cubic cost function coefficients of all test systems are given in Tables 1, 2 and 3.

Case 1: Lossless Test Systems
The IDHS algorithm is applied to solve the CCFED problem with 3-generator [32] and 5-generator systems [5] for the load power system respectively as 2500 (MW) and 1800 (MW).
The best generation schedule of 3 generating units along with the total cost obtained by proposed IDHS over 20 runs are co mpared with GA [5], PSO [5], SA [9], and FA [8] in Table 4.
The optimal total cost obtained by the proposed IDHS algorith m is 22729.3019 $/h. In this case study, the results without losses are compared with as seen from table 4, that the proposed IDHS is better co mpared to GA [5], PSO [5] and SA [9] in terms of provided better solution, with the exception of FA [8] generated the best solution of 22728 $/h. The computational time of IDHSA is faster (0.258 s). Fig. 5 shows the convergence speed of the IDHS algorith m to CCFED minimizat ion for Wollenberg's power system [32]. IDHS has been trapped into local optimu m at about 349 iterations.  The optimal power output of three generating units obtained by each algorithm is offered in Fig. 6. The best dispatch and best fuel cost solution obtained by IDHS are compared with those given by GA [5], PSO [5], and FA [8] in Table 5. Notably, the optimal cost value obtained is 18610 $/h, it is achieved by FA [8], wh ile the best cost obtained by the IDHS algorith m is 18610.375 $/h, which is the comparatively less than other methods. Also, the computational time of IDHSA is very fast (0.2736 s).   Fig. 7 shows the behavior of the IDHSA convergence visualized in terms of fuel cost against 3.104 generations for the load demand of 1800 MW. The IDHS algorithm converges quickly to the best cost value; at about 813 iterations. The best generation schedule of five generating units obtained by every algorithm is illustrated in Fig. 8.

Case 2: Lossless Test Systems
The second case study covers three thermal units [24] with an overall load demand of 1400 MW, with considering transmission losses. In case, the IDHS algorith m is applied to solve the CCFED problem for Liang's network system [24].
The obtained results with losses by the proposed IDHS algorith m are co mpared with the best solutions of SA [9], DP [24] and TLBO [33] methods in Table 6. We can see fro m Tab le 6 that the min imu m fuel cost of the proposed IDHS (6640.2776 $/h) is co mparatively lower than any other methods except for the TLBO [33] (6639.13 $/h) performs significantly better than IDHS, but the results are competitive. IDHS algorith m gives less power losses (43.3984 MW) co mpared to the mentioned methods while satisfying the equality and inequality constraints. The computational time g ives by the IDHS algorithm is (17.9345 s). As observed from Tables 4-6, the power output of each unit lies within the min imu m and maximu m generator capacity limits and satisfies the power balance constraint. Fig. 9 shows the best generation schedule of three generating units [24] obtained by comparative algorith ms. Fig. 10 clearly shows that power losses represent only 3.0998% of the total power generation as with the cost of losses.