Application and performance comparison of variants of the firefly ...

3rd International Conference on Advances in Electrical, Electronics, Information, Communication and Bio-Informatics (AEEICB17)

Application and performance comparison of variants of the firefly algorithm to the economic load dispatch problem Fatma Sayed Moustafa

Ahmed El-Rafei

Department of Mathematics Faculty of Basic Sciences, German University in Cairo Egypt [email protected]

Department of Engineering Physics and Mathematics Faculty of Engineering, Ain Shams University Cairo, Egypt [email protected]

N.M. Badra

Almoataz Y. Abdelaziz

Department of Engineering Physics and Mathematics Faculty of Engineering, Ain Shams University Cairo, Egypt [email protected]

Department of Electrical Power and Machines Faculty of Engineering, Ain Shams University Cairo, Egypt [email protected]

Abstract—The Economic Load Dispatch problem is an optimization problem which minimizes cost such that the load demand is met and the generating equality and inequality constraints are satisfied. Previously, conventional techniques like linear programming and lambda iteration were applied to solve the economic dispatch problem given their simplicity. Nevertheless, they do not always converge to global optimum which gave rise to metaheuristic techniques such as evolutionary and bio-inspired swarm algorithms. Firefly algorithm is a swarm based recent metaheuristic that has a high convergence rate and short execution time compared to other metaheuristic techniques when solving the economic load dispatch problem. Given that the firefly algorithm has its shortcoming of getting trapped in local optima, many researchers have proposed modifications and hybrids that improve the performance of firefly algorithm to obtain optimum global solutions rapidly and efficiently. In this paper, three of these recent enhancements were adopted to solve the economic dispatch problem of six generating units. The performance of these variants was compared and analyzed. The results show high efficiency in achieving optimal results in less time than the original firefly algorithm. Keywords— Economic load dispatch, Firefly algorithm, Memetic firefly, Modified firefly, and Variable step size firefly

I.

Introduction

The main objective of electric power generating systems is to provide customers with reliable, economic, and high quality power supply. Due to the high competition in the industry of generating and supplying electricity and the large increase in power demand and fuel cost, the Economic Load Dispatch (ELD) plays a critical role in the efficient and economical operation of scheduled generating units. The ELD basically finds the optimum generation scheduling by minimizing fuel cost while considering the operational equality and inequality constraints and matching the load demand of the generating units. ELD is a highly non-convex nonlinear constrained optimization problem that requires intensive mathematical

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computations to minimize objective function while satisfying constraints. This task has proven to be costly and time consuming. Hence, the proper choice of an optimization technique is essential [1]. To solve the ELD problem, many mathematical based optimization techniques were used. The utilized deterministic conventional methods include, but not limited to: Lambda Iteration, Newton method, Linear Programming (LP), and Quadratic Programming (QP) [2-5]. These approaches are easy to use, have a simple mathematical model, and rapid search. However, they are local optimizers and cannot guarantee global optimum. Modern intelligent techniques have been proposed to overcome the deficiencies of these conventional methods. These intelligent techniques are general purpose, flexible, and depend on randomness and diversity. They search the solution space more meticulously and reach global optimal solutions, thus, overcoming the limitations of conventional methods. Artificial Intelligence techniques such as the Differential Evolution (DE), Genetic Algorithm (GA), Simulated Annealing (SA), Particle Swarm Optimization (PSO), Ant Colony optimization (ACO), and Firefly Algorithm (FFA) were used to solve the ELD problem [6-9]. FFA is a novel nature inspired promising metaheuristic method proposed by Yang [10]. FFA is a simple, robust, and evolving intelligent technique. Many researchers applied it to solve ELD problems showing the FFA advantages of obtaining accurate optimal solutions with higher convergence rate and less computational time compared to other techniques like GA, PSO, SA, DE, and others [11-12]. Although FFA has shown its supremacy, it can get trapped in local optima, it cannot memorize earlier obtained better solutions, and its parameters are decided initially and remained unchanged during the iterations. To overcome such shortcomings, some researchers have suggested modifications and enhancements to the original firefly algorithm creating new variants of the

3rd International Conference on Advances in Electrical, Electronics, Information, Communication and Bio-Informatics (AEEICB17) Pi= Power output from generator (i) firefly algorithm [13]. Modified versions of the FFA were implemented to solve the ELD problem and have shown their Pimin = Minimum permitted power output by generator (i) efficiency [14-16]. Variants of firefly algorithm such as Pimax = Maximum permitted power output by generator (i) modified FFA (MFA) was applied on mechanical engineering design problems, memetic FFA (MFFA) was applied to solve III.Firefly Algorithm combinatorial problems and variable step size FFA (VSSFA) Firefly Algorithm is a new population based swarm was used to solve sixteen benchmark functions and the results intelligence metaheuristic method that is developed by Yang demonstrated their optimizing capabilities [17-19]. To the best in 2008 [10]. It is inspired by the bioluminescence behavior of of our knowledge, these methods were not applied to the ELD fireflies at night. The luminosity of fireflies attracts problem. In this paper, these three variants are applied to solve prospective mates and scare off predators in its neighborhood. the nonlinear and non-convex ELD problem. A comparative The three main rules to construct the main algorithm are: study was carried out on the solution of ELD problem using 1) Fireflies are attracted to the brighter ones independent of those recent variants and the classical firefly algorithm for their gender different test cases. Efficiency was evaluated by comparing 2) The objective function’s values determine the brightness of best solutions obtained in terms of execution time, fuel cost fireflies and power loss. 3) Brightness and attractiveness are directly proportional and they are inversely proportional to distance, such that: II. Problem Formulation 1 I∝β∝ r The objective of the nonlinear ELD optimization problem The factors affecting the algorithm are light intensity, is to minimize cost while satisfying the load demand and other attractiveness, distance, and movement and are given by: operational system equality and inequality constraints [1]. 2 (7) Light intensity I r =I0 e-γr 2 A. Objective function- cost function Attractiveness β r =β0 e-γr (8) Ng where I and β are the initial light intensity and initial 0 Minimize FT =F Pi = ∑i=1 Fi (Pi ) $/hr (1) 0 brightness, respectively. This distance between firefly i and firefly j is represented as: (2) F (P )=a +b P +c P 2 $/hr i

i

i

i i

i i

rij = xi -xj = ∑dk=1 xik -xjk

where Fт: Total Quadratic cost function; it could be also a cubic function Pi: Real power generated Ng: Number of generation busses ai, bi, ci : Fuel cost coefficients for ith unit B. Constraints The objective function must be minimized while considering the following constraints: 1) Equality constraint- Energy balance equation N

∑i=1g Pi =PD +PL N

(3)

N

PL = ∑i=1g ∑j=1g Pi Bij Pj

(4) where PD = Load demand PL= Power transmission losses Bij= Loss coefficients (constants) Pi, Pj = Active power injection at the ith and jth generators In some cases, power losses are neglected and the active power balance equation becomes: Ng ∑i=1 Pi =PD (5) 2) Inequality constraint- Generating limits Generated active power should lie between minimum and maximum operational values (6) Pi min ≤Pi ≤Pi max where

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2

(9)

The formula that controls the fireflies’ movement is given by: -γrij 2 t t t xt+1 xj -xi +α rand-0.5 (10) i =xi +β0 e where t is the number of current iteration, α∈ 0,1 is the randomization parameter and γ∈[0,∞) is the coefficient of absorption. The first term is the current position of the firefly i, the second term is due to attraction towards a brighter firefly j and the last term represents the random walk of the firefly. A. Variants of Firefly Algorithm Since FFA was developed, it has become a popular optimizer. In recent studies, major improvements and changes were applied to the FFA to enhance its performance. In this paper, the three variants of FFA described in the following subsections are applied to solve the ELD problem and comparisons were made with each other and the original FFA. B. Modified Firefly Algorithm Three new modifications were suggested in [17]: adding memory, newborn fireflies and updating formula. 1) Adding memory This modification is done via 2 approaches. The first approach is to transfer the best solutions obtained (m₁ high rank fireflies) in each iteration to the next iteration. The second approach is to make a copy of m₂ high rank fireflies in each iteration then replacing the worst solutions obtained (m₂ low

3rd International Conference on Advances in Electrical, Electronics, Information, Communication and Bio-Informatics (AEEICB17) rank fireflies) in each new iteration by the high rank ones copied from the previous iteration. 2) Newborn fireflies In each iteration, k of the low rank fireflies are replaced by newly generated ones using the same random initialization procedure used to initialize the population of fireflies. 3) Updating formula The fireflies approached the brighter firefly in a step wise strategy. Each firefly moves towards each brighter firefly one at a time. So the firefly ends up moving in an irregular path to update its location. Fireflies 1 to 6 are sorted according to their objective function, 1 being the brightest and 6 having the least brightness. During the first iteration, firefly 6 moves towards firefly 1 and updates its position. In the second iteration, firefly 6 moves towards firefly 2 and updates its position for the second time. The previous step is repeated for the third, fourth and fifth iterations, in each time firefly 6 moves along a different path approaching the firefly brighter than itself as shown in Fig. 1. For each firefly j with ( j-1) brighter fireflies, the firefly has to make (j-1) steps to update its position. Such approach is inefficient, time consuming and decreases the performance of the algorithm. Therefore, the suggested approach is to get a point to represent the overall distribution of brighter fireflies. Many representations could be possible, but the one implemented is the average position of the brighter fireflies. Therefore, the fireflies update their position in one step as shown in Fig. 2. The representative point Pi is the average of the coordinates of fireflies brighter than firefly i is given below: Pi =

1 i-1

∑i-1 k=1 xk

(11)

ri = Pi -xi (12) and thus changing the updating formula with a scaled random movement to be: t t t xt+1 (13) i =xi +β Pi -xi +αε 2 -γri with =β0 e , ε= rand-0.5 * U-L where U and L are the upper and lower bounds of the search space, respectively.

Fig. 2. Schematic diagram that depicts the representative point P which is the average position of fireflies brighter than firefly 6. The update of position is done in one step.

C. Memetic Firefly Algorithm Memetic algorithm (MA) balances between exploration and exploitation of search space. The exploration helps in discovering new search space (random walk) and the exploitation focuses on the vicinity of promising solutions and hence reaching optimum solutions more efficiently. Thus, it combines the benefits of population based and local search techniques. Memetic firefly algorithm (MFFA) enjoys the advantages of FFA over other techniques and balance of MA. MFFA is designed to solve high dimensional problems and multiple strategies are introduced. Parameters α and β are finetuned to avoid the inefficient search by high exploration and premature convergence by high exploitation [18]. α is adjusted dynamically by: 1

1 t

α=α * U-L (14) 9000 where α initially is set to 0.2, t is the number of current iteration, U is the upper bound of the search space and L is the lower bound of the search space. β changes in the range [0.2,1] using this formula: 2 (15) β=βmin +(β0 -βmin )e-γrij where βmin =0.2 and β0 =1 and thus changing the updating formula with a scaled random movement to be: t t t (16) xt+1 i =xi +β xj -xi +α rand-0.5 * U-L where U and L are the upper and lower bounds of search space, respectively. D. Variable Step Size Firefly Algorithm

Fig. 1. Schematic diagram that represents the step wise path that firefly 6 move along to update its position approaching the brighter fireflies 1-5. The update of position is done in 5 steps.

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In the original FFA, the parameters are set fixed. The step size α controlling randomness is fixed affecting the balance between diversification and intensification. To reach the optimum solution effectively, parameter α should decrease dynamically with the iterations. Since at the beginning of the iteration process, a large α is needed to allow for the exploration of new search space and to prevent getting stuck in local optima. It is suitable however to have a smaller α as the iterations go on to have less exploration and more

3rd International Conference on Advances in Electrical, Electronics, Information, Communication and Bio-Informatics (AEEICB17) 1.40 0.17 0.15 0.19 0.26 0.22 exploitation. This facilitates the convergence towards the global 0.17 0.60 0.13 0.16 0.15 0.20 optimum solution. 0.15 0.13 0.65 0.17 0.24 0.19 -4 B=10 × Variable Step Size Firefly Algorithm (VSSFA) adopts a strategy 0.19 0.16 0.17 0.71 0.30 0.25 for adjusting the step α with the iterations [19]. The formula for the 0.26 0.15 0.24 0.30 0.69 0.32 step size α is a function in the kth iteration and MaxGeneration which 0.22 0.20 0.19 0.25 0.32 0.85 is the total number of iterations. It is given by the following formula: A. Transmission losses not included 0.4 α k = (17) Tables III to V demonstrate best, mean, and standard deviation for 6 0.005* k-MaxGeneration 1+exp units lossless system for 600, 700 and 800 MW. Fuel cost and time of execution were compared in tables VI and VII. IV. Test and Results FFA and its variants were implemented to solve ELD problem with 3 and 6 thermal units on MATLAB® Version 2014a on an Intel® Core™ i7-3770, 3.4GHz, and 8GB RAM 32-bit OS computer. However, because of space limitations and similarity in results, only the six units system is presented. The parameters for FFA and its variants are given in Table I. The fuel cost and loss coefficients for six units system are given in table II [20]. In each case study, a reliability analysis of 20 independent runs was carried out on FFA and its variants to get best and mean cost, and the standard deviation (SD). Their efficiency was assessed by comparing execution time, the best solution and power loss. TABLE I. PARAMETERS FOR FIREFLY ALGORITHM AND ITS VARIANTS Values for different algorithms Parameters FFA

MFA

VSSFA

MFFA

MaxGeneration n α

150 25 0.2

150 25 0.2

150 25 Varies with iterations

γ β0 βmin m1 m2 k

1 1 NA NA NA NA

1 1 NA 0 1 1

1 1 NA NA NA NA

150 25 Initially α=0.2 but decreases with iterations 1 1 0.2 NA NA NA

TABLE II. PARAMETERS OF 6 UNITS SYSTEM AND LOSS COEFFICIENT B Pmax a b c Unit Pmin 1 10 125 756.79886 38.53973 0.1524 2 10 150 451.32513 46.15916 0.10587 3 35 225 1049.9977 40.39655 0.02803 4 35 210 1243.5311 38.30553 0.03546 5 130 325 1658.5596 36.32782 0.02111 6 125 315 1356.6592 38.27041 0.01799

TABLE III. RELIABILITY ANALYSIS FOR POWER DEMAND =600MW FFA MFA VSSFA MFFA Best 31489 31447 31576 31481 Mean 31842.75 31452.95 31945.7 31620.6 SD 243.84008 2.9285348 244.08931 95.848784 TABLE IV. RELIABILITY ANALYSIS FOR POWER DEMAND =700MW FFA MFA VSSFA MFFA Best 36075 36006 36036 36021 Mean 36353.7 36010.3 36212.2 36114.6 SD 152.7413 2.5152168 75.797931 44.446775 TABLE V. RELIABILITY ANALYSIS FOR POWER DEMAND =800MW FFA MFA VSSFA MFFA Best 40739 40676 40701 40740 Mean 40982.05 40681.3 40886.6 40950.3 SD 121.88065 2.6969769 77.10171 110.85605 TABLE VI. COMPARISON OF COST FOR FIREFLY VARIANTS Power demand FFA MFA VSSFA MFFA 600 31489 31447 31576 31481 700 36075 36006 36036 36021 800 40739 40676 40701 40740 TABLE VII. COMPARISON OF TIME FOR FIREFLY VARIANTS Power demand FFA MFA VSSFA MFFA 600 0.519390 0.081707 0.572476 0.250614 700 0.545151 0.083877 0.598494 0.422341 800 0.545414 0.081805 0.540558 0.460001

From the above tables, it can be shown that not only MFA obtains the minimum cost for different power demands, but also has a gain in speed that outperforms that of the original FFA and the other two variants. B. Transmission losses included Tables VIII to X demonstrate best, mean, and standard deviation for 6 units loss included system for 600, 700 and 800 MW. Fuel cost, time of execution and power loss were compared in tables XI to XIII. TABLE VIII. RELIABILITY ANALYSIS FOR POWER DEMAND =600MW FFA MFA VSSFA MFFA Best 32122 32098 32159 32109 Mean 32373.45 32103.95 32364.75 32274.45 SD 159.54325 4.7069378 159.48894 103.44512 TABLE IX. RELIABILITY ANALYSIS FOR POWER DEMAND =700MW FFA MFA VSSFA MFFA Best 37004 36914 36960 36978 Mean 37317.3 36918.9 37165.5 37036.9 SD 186.88812 3.3229663 96.776193 35.982306

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3rd International Conference on Advances in Electrical, Electronics, Information, Communication and Bio-Informatics (AEEICB17) TABLE X. RELIABILITY ANALYSIS FOR POWER DEMAND =800MW FFA MFA VSSFA MFFA Best 41939 41898 41976 41930 Mean 42240.35 41901.211 420079.85 41989.05 SD 167.72574 2.3470773 68.854339 33.536273 TABLE XI. Power demand 600 700 800

COMPARING COST FOR FIREFLY VARIANTS FFA MFA VSSFA MFFA 32122 32098 32159 32109 37004 36914 36960 36978 41939 41898 41976 41930

[4]

TABLE XII. Power demand 600 700 800

COMPARING TIME FOR FIREFLY VARIANTS FFA MFA VSSFA MFFA 3.373328 0.495458 3.422343 1.127240 3.411830 0.512442 3.438569 2.675444 3.326030 0.513960 3.352762 1.076264

[6]

TABLE XIII. Power demand 600 700 800

COMPARING LOSS FOR FIREFLY VARIANTS FFA MFA VSSFA MFFA 14.7116 13.7368 13.9435 14.1097 19.5620 18.6507 18.8286 18.4684 24.5250 24.2210 24.7649 25.2604

Even with transmission losses included, MFA maintains its superiority in terms of obtaining minimum cost with the least execution time. It also minimizes the power loss as shown in the above tables. V. Conclusion In previous studies, firefly algorithm was applied to solve the economic dispatch problem and has succeeded in finding more accurate optimal solutions than other metaheuristic algorithms. In this paper, recent variants of firefly algorithm: modified firefly, memetic firefly, and variable step size firefly algorithms were applied to solve the ELD problem. A thorough comparison was carried out between those variants in terms of execution time, optimal cost and losses (in case of including transmission losses) when solving six thermal units systems. The results verify that the MFA is capable of yielding economical solutions with the least computational time and minimal losses compared to FFA, MFFA, and VSSFA. The MFFA demonstrated its success and potential in finding high quality solutions in a more reasonable time than the VSSFA and FFA. Since the ELD problem is a real-time application, the optimum results obtained by MFA with the significant gain in speed makes it a suitable choice to solve the ELD problem over FFA, VSSFA, and MFFA.

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