Modified shuffled frog leaping algorithm with genetic algorithm crossover for solving economic load dispatch problem with valve-point effect
Graphical abstract
Introduction
Economic load dispatch (ELD) is a familiar problem pertaining to the allocation of the amount of power to be generated by different units in the system on an optimum economic basis. The generated power has to meet the load demand and transmission losses. This implies that the dispatch at the true minimum cost requires that we take the network losses into account. Also for the secure operation of the power system, the generators must dispatch in such a way so that the transmission capacity limits are not exceeded.
Many researchers are involved in tackling the ELD problem for significant economical benefit. Conventional methods such as lambda iteration method, and gradient based method [1] are used to solve the ELD problem by changing the fuel cost curve in a piecewise linear function or monotonically increasing function. These methods ignore the portions of incremental cost curve that are not continuous or monotonically increasing. But input–output characteristics of modern units are inherently non-linear because of ramp rate limits, valve point loadings, etc. So in the classical method, the fuel cost curve is approximated according to their requirement but use of such approximation may lead to huge loss of revenue over the time. Dynamic programming, proposed in [2], is a method to solve non-linear and discontinuous ELD problem but simulation time increases rapidly with respect to system size in this method.
Other than classical methods, different artificial intelligence based methods have been successfully utilized to compute ELD problems. These methods are evolutionary programming [3], [4], particle swarm optimization [5], tabu search [6], differential evolution [7], biography based optimization [8], genetic algorithm [9], [10], artificial neural network [11], intelligent water drop algorithm [12], etc. However the cost curve of a generator containing discontinuities like valve point loading is more realistically denoted as segmented piecewise non-linear function [13] rather than a single quadratic function. Some studies have been done for this type of ELD problem with valve point effect such as novel niche quantum genetic algorithm [14], hybrid quantum mechanics inspired particle swarm optimization [15], combining of chaotic differential evolution and quadratic programming [16], biogeography based optimization [17], hybrid solution methodology integrating particle swarm optimization (PSO) algorithm [18], [19], [20], enhanced bee swarm optimization method [21], enhanced adaptive particle swarm optimization (EAPSO) algorithm [22], the sequential quadratic programming (SQP) method [23] and artificial bee colony algorithm [24]. A new soft computing technique like [25] proposed a Modified Honey Bee Mating Optimization (MHBMO) to solve the dynamic optimal power flow (DOPF) problem considering the valve-point effects. Another new approach called Taguchi method [26] that involves the use of orthogonal arrays in estimating the gradient of the cost function is employed to solve the economic dispatch problem with non-smooth cost functions. Each and every method has its own limitations. Neural network suffers from excessive iterations, resulting in huge calculations as well as more processing time. Genetic algorithm has a disadvantage of premature convergence, and due to that its performance degrades and its search capability is reduced. In PSO, the algorithm progresses slowly and due to its inability to adjust the velocity step size it may be difficult to continue the search at a finer grain. For multi modal function PSO sometimes fails to reach global optimal point. However, regulation of the control parameters of these hybrid methods is a challenging and complicated task and enhancement is done using a mutation operator. DE has been found to yield better and faster solution, satisfying all the constraints, both for uni-modal and multi-modal systems by using its different crossover strategies. However with increase in system complexity and size, the DE method is unable to map its entire array of unknown variables together in an efficient way. A new technique of a θ-multi-objective-teaching–learning-based optimization algorithm [27] and modified teaching–learning algorithm [28] are employed to solve the dynamic economic emission dispatch problem where problem formulation is much more complicated. In [29] a new self-adaptive modified firefly algorithm is suggested to solve the ELD problem. Although this algorithm is completely suitable for tuning off its parameters, it suffers from the difficulties of trapping in local optima.
Recently, a new meta-heuristic algorithm called shuffled frog leaping algorithm (SFLA) is introduced [30], [31]. It aims to model and mimic the behavior of frogs searching for food laid on stones randomly located in a pond. It combines the advantages of the gene based memetic algorithm (MA) and the social behavior-based particle swarm optimization (PSO) algorithm and has found applications in areas such as optimizing bridge-deck repairs [32], materialized views selection [33], bi-criteria permutation flow shop scheduling problem [34], reservoir flood control operation [35], and a mixed-model assembly line sequencing problem [36]. In the field of electrical research the SFLA technique is extensively used in optimal reactive power flow [37], [38], combined economic emission dispatch [39], and unit commitment [40]. The simple SFLA technique is improved using GA crossover in local iteration as modified shuffled frog leaping algorithm (MSFLA) and is used in ELD problems without non-convex characteristic [41]. Another type of MSFLA is used in the ELD problem with valve-point effects as a non-smooth optimization problem having complex and non-convex characteristics with heavy equality and inequality constraints [42] where modification is done only by shuffling the frog in local iteration.
This paper contributes a new hybrid algorithm in an evolutionary algorithm family to solve the ELD problem with valve-point effects. The proposed hybrid algorithm combines shuffled frog leaping algorithm (SFLA) and genetic algorithm (GA) that chooses genes (features) related to classification. It is named as modified shuffled frog leaping algorithm (MSFLA) with GA crossover which is used to solve a non-smooth optimization problem having complex and non-convex characteristics with heavy equality and inequality constraints. This makes the challenge of finding the global optimum difficult. The motivation behind choosing GA crossover in SFLA is to get a better solution using crossover between two frogs (taking the best one and the worst one according to the fitness value), getting two new offsprings with their different fitness values. As the operation is performed between the best and the worst frog, two offsprings with obviously better fitness values compared to the worst frog (parent frog) are produced. Hence the range of fitness values is obviously decreasing with iterations, helping to find the optimum result with the minimum number of iterations, in minimum time. In the case of SFLA the frogs are shuffled within the different memeplexes but no new frogs with better fitness values are generated. Moreover this hybrid approach presents an accurate result with the minimum number of iterations compared to other algorithms because in MSFLA there are two types of search involved – local and global search, while in the case of other algorithms (GA, PSO, ACO), there is only one type of search or iteration (i.e. global search). As two types of search techniques are involved in MSFLA, it works as a divide and conquer method. Thus it gives a better outcome compared to other optimization methods. In order to check the applicability of the proposed method, four power systems are considered, including the IEEE 30-bus standard test system, the 203-bus, 264-line, 23-generator Eastern Indian power grid system, and systems of 13 and 40 generating units. The results obtained through the combined approach are analyzed and compared with those reported in recent literature.
A brief description and mathematical formulation of the ELD problem is discussed in Section 2. The concept of SFLA is discussed in Section 3 and modification of respective algorithm is done in Section 4 while the implementation is represented in Section 5. Simulation studies are discussed in Section 6 and conclusion is drawn in Section 7.
Section snippets
Objective function
The objective of ELD operation is to minimize the generation cost which mainly comprises fuel cost at thermal power plants subjected to the operating constraints of a power system. The fuel cost function is represented in Eq. (1) where α, β, γ are respective cost co-efficients and is the generated power of unit i. This study considers the valve-point effects as a complementary component of objective function. Therefore the objective function is described as the superposition of
Principle of shuffled frog leaping algorithm (SFLA)
SFLA is a kind of post-heuristic computing technique based on swarm intelligence put forward by Eusuff and Lansey [30], [31]. SFLA mimics the memetic evolution of a group of frogs. It is based on evolution of memes and a global exchange of information among the frog population. It combines the benefits of gene based memetic algorithm and social behavior based particle swarm optimization algorithm with such characteristics as simple concept, fewer parameters, prompt formation, great capability
Proposed modified shuffled frog leaping algorithm with GA crossover
The proposed modified shuffled frog leaping algorithm (MSFLA) with genetic algorithm (GA) crossover is designed based on the same framework of shuffled frog leaping algorithm (SFLA). Fig. 3 outlines the flow chart of the proposed algorithm.
Implementation of MSFLA with GA cross-over for ELD problem
This paper proposes the improved MSFLA with GA crossover to solve the ELD problem with non-convex valve point effect. The stages of the proposed algorithm and further procedures particularizing the modified SFLA (MSFLA) with genetic algorithm (GA) based crossover operation, including the design of global as well as local exploration strategies within each memeplex and crossover operation implementation, are explained here under.
Simulation result and analysis
This section employs four examples to illustrate the effectiveness of the proposed MSFLA with GA crossover with respect to the quality of the solution obtained. The following test cases are studied, analyzed and compared with other ELD methods.
Case I: generators number 13; with valve point effect.
Case II: generators number 40; with valve point effect.
Case III: generators number 6; IEEE 30 bus standard test system; without valve point effect.
Case IV: generators number 23; 203-bus 264-line
Conclusion
A hybrid approach by integrating the shuffled frog leaping algorithm (SFLA) with the Genetic Algorithm (GA) as modified shuffled frog leaping algorithm (MSFLA) for solving constraint economic dispatch problems considering valve point non-linearities of generators is presented. An optimal range of global iteration, local iteration and population size for the MSFLA is estimated to solve all the test cases considered in this paper. The feasibility of the MSFLA method is illustrated by conducting
Acknowledgments
The authors acknowledged the active support rendered by Elizabeth Loniello, Trainee Engineer, Volkert, Inc., USA and Shegil Attour, Graduate Student, New Jersey Institute of Technology, USA, in developing and modifying the English language of the text material presented in this paper.
References (43)
- et al.
Evolutionary programming based economic dispatch of generators with prohibited operating zones
Electrical Power System Research
(1999) - et al.
Environmentally constrained economic dispatch using the lagrangian relaxation method
IEEE Transactions on Power Systems
(1993) - et al.
A zoom feature for a dynamic programming solution to economic dispatch including transmission losses
IEEE Transactions on Power Systems
(1992) - et al.
Evolutionary programming techniques for economic load dispatch
IEEE Transactions on Evolutionary Computation
(2003) - et al.
A particle swarm optimization for economic dispatch with non-smooth cost functions
IEEE Transactions on Power Systems
(1993) - et al.
An improved tabu search for economic dispatch with multiple minima
IEEE Transactions on Power Systems
(2002) - et al.
Differential evolution for economic load dispatch problems
Electric Power Systems Research
(2008) - et al.
Biogeography-based optimization for different economic load dispatch problems
IEEE Transactions on Power Systems
(2010) - et al.
Large-scale economic dispatch by genetic algorithm
IEEE Transactions on Power Systems
(1995) - et al.
Economic dispatch using an efficient realcoded genetic algorithm
IET Generation, Transmission and Distribution
(2009)
New approach with a Hopfield modeling framework to economic dispatch
IEEE Transactions on Power Systems
An intelligent water drop algorithm for solving economic load dispatch problem
International Journal of Electrical and Electronics Engineering
Advanced engineered-conditioning genetic approach to power economic dispatch
IEE Proceedings – Generation, Transmission and Distribution
Economic load dispatch using a novel niche quantum genetic algorithm for units with valve-point effect
Solving economic load dispatch problem with valve-point effects using a hybrid quantum mechanics inspired particle swarm optimization
IET Generation, Transmission & Distribution
Combining of chaotic differential evolution and quadratic programming for economic dispatch optimization with valve-point effect
IEEE Transactions on Power Systems
Biogeography based optimization to solve economic load dispatch considering valve point effects
Quantum-inspired particle swarm optimization for valve-point economic load dispatch
IEEE Transactions on Power Systems
Closure to discussion on economic load dispatch – a comparative study on heuristic optimization techniques with an improved coordinated aggregation-based PSO
IEEE Transactions on Power Systems
Improved particle swarm optimisation for multi-objective optimal power flow considering the cost, loss, emission and voltage stability index
IET Generation, Transmission & Distribution
Enhanced bee swarm optimization algorithm for dynamic economic dispatch
IEEE Systems Journal
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