Complexity

Volume 2018, Article ID 7267593, 23 pages

https://doi.org/10.1155/2018/7267593

## Improved Firefly Algorithm: A Novel Method for Optimal Operation of Thermal Generating Units

^{1}Power System Optimization Research Group, Faculty of Electrical and Electronics Engineering, Ton Duc Thang University, Ho Chi Minh City 700000, Vietnam^{2}Department of Electromechanical and Electronic, Lac Hong University, Đồng Nai Province, Biên Hòa 810000, Vietnam^{3}Institute of Research and Development, Duy Tan University, Da Nang 55000, Vietnam^{4}Office of Science Research and Development, Lac Hong University, Biên Hòa 810000, Vietnam

Correspondence should be addressed to Le Van Dai; nv.ude.natyud@iadnavel

Received 8 November 2017; Revised 20 April 2018; Accepted 24 May 2018; Published 9 July 2018

Academic Editor: Michele Scarpiniti

Copyright © 2018 Thang Trung Nguyen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper presents a novel improved firefly algorithm (IFA) to deal the problem of the optimal operation of thermal generating units (OOTGU) with the purpose of reducing the total electricity generation fuel cost. The proposed IFA is developed based on combining three improvements. The first is to be based on the radius between two solutions, the second is updated step size for each considered solution based on different new equations, and the third is to slightly modify a formula producing new solutions by using normally distributed random numbers and canceling uniform random numbers of conventional firefly algorithm (FA). The effect of each proposed improvement on IFA is investigated by executing five benchmark functions and two different systems. The performance of IFA is investigated on six other study cases consisting of different types of objective function and complex level of constraints. The objective function considers single fuel with quadratic form and nonconvex form, and multifuels with the sum of several quadratic and nonconvex functions while a set of constraints taken into account are power loss, prohibited zone, ramp rate limit, spinning reserve, and all constraints in transmission power networks. The obtained results indicate the proposed improvements in terms of high optimal solution quality, stabilization of search ability, and fast convergence compared with FA. In addition, the comparisons with other methods also lead to a conclusion that the proposed method is a very promising optimization tool for systems with quadratic fuel cost function and with complicated constraints.

#### 1. Introduction

The optimal operation of thermal generating units (OOTGU) has been widely concerned in the power system operation field due to its significant important role. In fact, thermal units are using a huge amount of fossil fuel for generating electricity while primary fuels are expensive and will be exhausted in the near future. The objective of the OOTGU problem is to determine the generated active power of thermal generating units so that the total fuel cost can be reduced as much as possible; while all the constraints of thermal generating units and of transmission power networks are taken into account. The problem becomes more and more complicated since more and more complicated constraints of generator are considered such as ramp rate, prohibited zone, voltage limitations, and generation limits. In addition, constraints related to transmission power network are not simple conditions to be exactly met such as the active power spinning reserve, active and reactive power balance, limitation of transmission lines, limitations of transformer tap, and limitation of capacitor bank’s reactive power.

There have been a huge number of studies considering OOTGU as a main problem. The studies have applied conventional methods such as deterministic algorithms and new methods. Most deterministic methods are based on gradient, derivative, Lagrange optimization function while the new methods are based on metaheuristic algorithms, modified versions of metaheuristic algorithms, and combination of two different metaheuristic algorithms. In addition, the metaheuristic methods integrated to other methods dealing with constraints of problem are also effectively combined optimization tools.

Deterministic methods have been applied for the problem such as Lagrange optimization function-based method (LM) [1–3], dynamic programming (DP) [1], linear programming method (LPM) [4], Hierarchical method (HM) [5], Hopfield neural networks (HNN) [3, 6], improved Hopfield neural network method (IHNNM) [7], augmented Lagrange Hopfield network (ALHN) [8], and enhanced augmented Lagrange Hopfield network (EALHN) [9, 10]. Among the methods, LM was the first optimization tool for solving the problem of OOTGU with quadratic fuel cost functions and the balance constraint of generated power and required power such as load demand and power losses. The obtained results from the method were acceptable since it could deal with the constraint of active power balance constraint and maximum error was very small. DP also achieved results as good as those from LM. However, DP required a huge number of iterations for search process, especially for large-scale systems with large number of units. LPM and HM have applied approximation techniques to simplify the complex of constraints and nonsmooth fuel cost functions and then LM was acted as an optimization search algorithm. Consequently, the applicability of the two methods was restricted for complicated systems with many nonlinear constraints and nonsmooth fuel cost functions. HNN and its improved versions in [7–9] consisting of IHNNM, ALHN, and EALHN have become more effective than LM, DP, LPM, and HM in dealing with large-scale systems with faster search process and better optimal solutions. The method group is mainly based on energy function and Hopfield neural network. However, HNN variants did not use Lagrange optimization function but they established energy function at the beginning and then Hopfield neural network was applied. On the contrary, ALHN variants have constructed an augmented Lagrange optimization function first and then the function was converted into energy function with the present of inverse sigmoid function. The ALHN variants could overcome several drawbacks of HNN variants such as high oscillation, high error, and low optimal solutions for complicated systems. In general, these methods in the deterministic group have the same limitations of application for systems considering nonconvex fuel cost function and nondifferentiable functions.

New methods based on metaheuristic algorithms have been widely and successfully applied for the considered OOTGU problem even for very complicated systems with nonconvex fuel cost functions and constraints taking prohibited zones, active power spinning reserve, and ramp rate into account. Differential evolution variants have been applied for solving the problem in [11–13] consisting of conventional differential evolution (DE) [11], hybrid dynamic programming integer-coded (HDP-ICDE) [12], and colonial competitive differential evolution (CCDE) [13]. DE together with GA and PSO has been implemented for different cases with different constraints with the superiority of DE over genetic algorithm (GA) and particle swarm optimization (PSO) as a result. HDP-ICDE has used integer-coded differential evolution (ICDE) as an optimization tool for searching solutions meanwhile dynamic programming has been used as an evaluation tool for computing fitness function value of found solutions. The method could not overcome drawbacks of ICDE such as high number of solution evaluation and falling into local search. CCDE has changed the mutation operation by using different models based on so-far best solutions and replacement of worst solutions by more promising solutions that aim to improve the balance between local search and global search as well as to retain the best solutions. This method could show its superiority over DE and other versions of DE, but the test systems have been used and the result comparisons from the tests could not show its better performance than other methods because its cost and other methods’ cost were equal. The comparisons of search speed have not been carried out. The performance of cuckoo search algorithm (CSA) [14, 15] and improved cuckoo search algorithm (ICSA) [16] have been shown clearly since different types of constraints were considered and wide comparisons were implemented. However, the speed comparisons have been made only via execution time, leading to not good comparison criteria for conclusion. Different test systems considering constraints of transmission power network have been used to test the efficiency of PSO variants in [17–19]. PSO with pseudo-gradient and constriction factor (PG-CF-PSO) [17] has determined the best direction for updating velocity by using pseudo-gradient theory and used constriction factor to find good search zone. It has been demonstrated to be more effective than PSO via three IEEE systems with 30, 57, and 118 buses. New adaptive particle swarm optimization (NAPSO) [18] has integrated both mutation operation and adaptive PSO for avoiding local optimal solution. The method has used different fuzzy and self-adaptive techniques for updating parameter and solutions. Each proposed technique has been investigated by testing different PSO methods with each modification such as NAPSO without mutation (NAPSO1), NAPSO without fuzzy (NAPSO2), and NAPSO without self-adaptive (NAPSO3). These methods have had worse results than NAPSO but better results than standard PSO. Combination of particle swarm optimization and tabu search algorithm (PSO-TSA) [19] has used the ability of PSO for local search based on three different phases and TSA for global search by adjusting obtained solutions of PSO. The method has been compared to three difference implemented methods such as GA, PSO, and TSA, and it can be shown better objective function. In spite of the superiority over the three methods, PSO-TSA has been more complicated, included more control parameters, and used high number of produced new solutions. Different GA variants have been proposed for dealing with different set of constraints of the considered problem such as real-coded genetic algorithm (RCGA) [20], RCGA with arithmetic-average-bound crossover and wavelet (NRCGA) [20], hybrid real-coded genetic algorithm (HRCGA) [21], improved GA (IGA) [22], and IGA with updated multiplier (IGAMU) [22]. These methods have been developed by improving GA, using RCGA and other improved crossover operation, and combining improved version of GA and multiplier updating technique. Generally, more complicated modifications lead to more improvement of GA variants. However, there have been more drawbacks such as higher number of control parameters, difficult task of selection of such factors, and complicated implementation when combining two or more methods. The different versions of krill herd algorithm (KHA) [23] have been used for solving different cases of the considered problem such as KHA without using genetic operation (KHA1), KHA using crossover operation (KHA2), KHA using mutation operation (KHA3), and KHA using crossover operation and mutation operation (KHA4). Several disadvantages of these KHA methods have been pointed out in [24] such as low convergence speed and low success rate. Thus, opposition-based krill herd algorithm (OBKHA) [24] has been developed by applying learning technique based on opposition and it has been compared to KHA methods. As derived from the comparisons, OBKHA could be considered better than KHA methods about optimal solutions reflected via less minimum cost and stable search ability reflected via less average cost, less maximum cost, and less standard deviation; however, the convergence speed and the success rate have not been investigated since there were no comparisons of used population and used iterations. Biogeography-based optimization algorithm (BBOA) has been employed for minimizing cost and emission in [25] and for minimizing cost in [26]. BBOA has some characteristics in common with other metaheuristic algorithms consisting of mutation and selection like GA, DE, and evolution programming, but this method has much more number of control parameters such as the probability of habitat modification and mutation, the rate of mutation and immigration, and lower and upper boundaries of probability. The results reported in [26] could show good performance of BBOA via comparisons with several existing methods on four test cases, but the keywords for determining the best control parameters were missed. Clearly, the application of BBOA to real power system with complicated constraints in transmission power network was limited and studies about BBOA for complicated systems have not been carried so far. Grey wolf optimization algorithm (GWOA) [27, 28] has been applied for simple systems without complicated constraints and small number of units. Thus, the demonstration of the methods’ performance has not been persuasive. On the contrary, crisscross optimization algorithm (CSOA) [29] and exchange market algorithm (EMA) [30] have had more result comparisons in the implementation of large-scale systems and complicated constraints. The ant lion optimization algorithm (ALOA) [31] was also a new method with few applications, and the method in the study has not shown its potential search ability persuasively due to simple test employment. In addition, many new methods have been developed for solving the problem such as modified symbiotic organisms search algorithm (MSOSA) [32], mine blast algorithm (MBA) [33], clonal algorithm (CA) [34], mathematical programming algorithm (MPA) [35], improved quantum-inspired evolutionary algorithm (IQIEA) [36], cuckoo optimization algorithm (COA) [37], improved colliding bodies optimization algorithm (ICBOA) [38], flower pollination algorithm (FPA) [39], natural updated harmony search (NUHS) [40], lightning flash algorithm (LFA) [41, 42], moth swarm algorithm (MSA) [43], and orthogonal learning competitive swarm optimization algorithm (OLCSOA) [44]. Among the remaining methods, LFA, MSA, ICBOA, and OLCSOA were developed in the last two years and they were considered better than most existing methods for considered test cases. LFA and OLCSOA have been implemented for systems with different complex levels of thermal generating units such as nonsmooth fuel cost functions with valve-point loading effects, multifuel types, and prohibited zone meanwhile challenges of transmission power network constraints have not been taken into account. On the contrary, MSA has been focused on dealing with the constraint complex rather than the challenges of fuel cost function of thermal generating units. In general, metaheuristic algorithms have been widely and successfully applied for different test systems and their superiorities have been pointed out mainly based on fuel cost function comparisons. The performance of ICBOA has been measured by testing on different study cases with different types of fuel cost function and the consideration of all constraints of transmission power networks. ICBOA together with other methods such as artificial bee colony (ABC), DE, PSO, BBO, standard colliding bodies optimization algorithm (CBO), and enhanced CBO have been measured by testing on different study cases with different types of fuel cost function and the consideration of all constraints of transmission power networks. ICBO has been pointed out obtaining better optimal solutions than all methods, but the further comparisons of convergence speed have not been carried out and discussed.

The firefly algorithm (FA) is also a population-based metaheuristic algorithm similar to PSO, DE, CSA, and so on built by Yang in 2008 for solving optimization problems [45]. The configuration of FA consists of three procedures of updating the distance between two considered fireflies, updating a step size, and updating new solutions. FA has shown its superiority over other traditional algorithms consisting of GA and PSO [46] by the comparisons of different benchmark optimization functions. In 2011, Yang and his coworkers applied the algorithm for solving the considered OOTGU problem [47] and they have stated that the method has been efficient for the problem by carrying out result comparisons via testing on three different systems. However, the performance of the method has not been demonstrated for large-scale systems with large number of generators and highly complicated constraints in transmission networks. Thus, there were several improvement versions of the method for dealing with the problem and the comparisons with it were also done for evaluation. An improved firefly algorithm (IFA) has been proposed by Kazemzadeh-Parsi in [48] by applying three different modifications. The first modification was to use the better solutions of the previous iteration for replacing the worst solutions in the current iteration. The second modification was to replace low solutions by randomly produced solutions, and the third modification was to move the worse solutions only to one representative solution instead of all better solutions. The representative solution is determined by using the average solution of all better solutions. The memetic firefly algorithm (MFA) in [49] has focused on the balance of exploration acting as global search and exploitation acting as local search by using adaptive attractiveness *β* and adaptive step parameter *α* with respect to the change of iteration in formula updating new solutions for each lower quality solution. Another improved FA with using adaptive step parameter *α* (ASPFA) was proposed in [50] by suggesting an adaptive formula for updating step parameter *α* based on current iteration and the maximum number of iterations. The chaotic firefly algorithm (CFA) was developed in [51] for solving the considered OOTGU problem with different test cases of fuel cost function and constraints. The method has used chaotic distribution to produce the values of attractiveness *β* and step parameter *α* instead of using random distribution in FA. Moustafa et al. [52] have implemented conventional FA, IFA, MFA, and ASPFA on 3-unit system and 6-unit system for comparison. The result comparisons have indicated that IFA with three modifications was the best one and FA was the worst one among four FA variants. However, there was no comparison between these methods with others. On the contrary, CFA in [51] has been tested on many tests with more complicated objective functions considering valve-point effects and multifuels, and more complicated constraints considering POZ, ramp rate, and active power spinning reserve. CFA has been demonstrated to be more effective than many methods such as PSO, ABC, DE, and GA variants. Another modified firefly algorithm (*α*-MFA) based on the modification on distance between two fireflies and the modification on generation of step parameter *α* was proposed in [53] for dealing with OOTGU problem with 3-, 6-, 13-, and 15-unit systems considering valve-point loading effects, ramp rate constraints, and POZ constraints. The five improved versions of FA have been demonstrated to be better than conventional FA and other methods. However, the superiority of the methods was not shown clearly since it has not been implemented for complicated systems taking all constraints of transmission power networks and generators into consideration.

In the paper, we propose three modifications on the FA in order to tackle several disadvantages of FA such as premature convergence to local optimum solution and impossibility of jumping out of the search zone with many local optimum solutions. In the first modification, we propose a new formula to update radius between a considered firefly (a solution) to another firefly (another solution) with lower fitness function than the considered solution. The proposed radius based on and the best solution become more effective better than that based on and used in FA. In the second modification, we propose a new algorithm for producing new solutions of an old solution by suggesting two models for the updated step size. A large or a smaller updated step size will be used to diversify new solutions and to avoid converging to a local optimum and trapping into search zone with many local optimums. In the third modification, uniform distribution is replaced with normal distribution aiming at diversifying the search zone. As a result, the improvement of the new algorithm is highly considerable compared to FA. The application of each modification and three modifications is evaluated by testing on difference of four systems with other six cases. These test cases consider thermal generating units using single fuel and multifuels, convex and nonconvex objective function, and all constraints in real power systems. The main contributions of the paper can be summarized as follows: (i)Carry out three modifications on the new solution produced phase of conventional FA and point out the specific impact of each modification on the obtained results.(ii)The proposed method has few control parameters, and the parameters are easily tuned. The proposed method is more effective than FA, but the implementation of the proposed method is as simple as that of FA.(iii)Thoroughly present the application of the proposed IFA method for solving different systems with different constraints in optimal operation of power systems. The exact selection of control variables can help the proposed method satisfy all considered constraints.(iv)Develop basic application of the proposed method for optimization problem and lead to conclusion that the proposed method is effective for systems with complicated constraints but it works less effectively for systems considering nonconvex objective functions.

The rest of the paper is organized as follows. Section 2 describes the problem formulation. Section 3 presents the firefly algorithms. Section 4 shows the implementation of IFA for OOTGU problem. Section 5 presents the results of numerical simulations. Finally, the conclusions and future works are given in Section 6.

#### 2. Problem Formulation

##### 2.1. Objective Function

In OOTGU with single fuel, the fuel cost of each generating unit is expressed as a quadratic function of its power output. The objective of the problem is to minimize the total fuel cost of available units and can be written as follows [1]: where is the fuel cost function of thermal unit and it can be represented in a single quadratic form or piecewise form depending on the number of fuel cost options that thermal plants use. If the thermal units use only one fuel, its form is the second order equation as shown in (2) while the form of when using multiple fuels is a piecewise curve consisting of at least two second order equations. For the case of multifuel options, is shown in (3) [1].

In addition, once valve-point loading effects (VPLE) of thermal units are considered during the process of increase or decrease of power output, (2) and (3) will become more complicated models as follows [20].

For easy understanding, the case of single fuel option and the case of multifuel options neglecting and considering VPLE are depicted in Figures 1 and 2. It is clear that the curves with VPLE are much more complicated.