Engineering Applications of Artificial Intelligence
A modified particle swarm optimization for economic dispatch with non-smooth cost functions
Introduction
Economic dispatch (ED) is one of the most important problems to be solved in the operation and planning of a power system (Wood and Wollenberg, 1996). The primary objective of the ED problem is to determine the optimal combination of power outputs of all generating units so that the required load demand at minimum operating cost is met while satisfying system equality and inequality constraints. In the traditional ED problem, the cost function for each generator has been approximately represented by a single quadratic function and is solved using mathematical programming based on the optimization techniques such as lambda-iteration method, gradient method, and dynamic programming method, etc. However many mathematical assumptions such as convex, quadratic, differentiable and linear objectives and constraints are required to simplify the problem.
The practical ED problem with ramp rate limits, prohibited operating zones, valve-point effects and multifuel options is represented as a non-smooth or non-convex optimization problem with equality and inequality constraints and this makes the problem of finding the global optimum difficult and cannot be solved easily by traditional methods.
A considerable amount of work has been adopted by researches to solve a practical ED problem by considering different non-convex cost functions using various heuristic approaches (Chen and Chang, 1995, Chiang, 2005, Jayabarathi and Sadasivam, 2000, Kennedy and Eberhart, 1995, Lee et al., 1998, Lin et al., 2001, Lin et al., 2002, Orero and Irving, 1996, Park et al., 2005, Park et al., 2007, Park et al., 1993, Selvakumar and Thanushkodi, 2007, Sinha et al., 2003, Walters and Sheble, 1993, Wong and Wong, 1994, Wong and Fung, 1993, Yang et al., 1996). This paper introduces a modified PSO (MPSO) and its solution to the non-convex ED problems. Two types of non-smooth ED problems; ED with ramp rate limits and prohibited operating zones and ED with combined valve-point loading effects and multifuel options will be considered.
The PSO has been proven to be very effective for static and dynamic optimization problems. But in some cases, it converges prematurely without finding local optimum. In PSO algorithm, it is possible for the inertia weight to drive all velocities to zero before the swarm manages to reach a local extremum. Thus, in this paper MPSO is introduced to address the issue of premature convergence to solutions that are not guaranteed to be local extrema.
To validate the results obtained by the MPSO, the problem is solved by PSO. Also, the GCPSO, which is introduced by Van den Bergh and Engelbrecht (2002) to address the issue of premature convergence of PSO, is applied to validate the results. Furthermore, the results obtained by MPSO, PSO and GCPSO are compared with those obtained by other approaches reported in the literature. To make a proper background, PSO, GCPSO and the proposed modified PSO (MPSO) are explained in the next Section.
Section snippets
PSO algorithm
The particle swarm optimizer is a population based optimization method that was introduced by Kennedy and Eberhart (1995). In PSO, each particle moves in the search space with a velocity according to its own previous best solution and its group’s previous best solution. The dimension of the search space can be any positive integer. Each particle updates its position and velocity with the following equations:where Xi(t)and Vi(t) are vectors representing the position and
Problem formulation
For convenience in solving the ED problem, the unit generation output is usually assumed to be adjusted smoothly and instantaneously. Practically, the operating range of all online units is restricted by their ramp rate limits by forcing the units to operate continually between two adjacent specific operation zones. In addition, the prohibited operating zones, valve-point effects and multifuel options must be taken into account. The traditional and practical ED is explained below.
Study systems
To assess the efficiency of the proposed MPSO, it has been applied to ED problem by considering three test systems having non-convex solution spaces. These data are given by Selvakumar and Thanushkodi (2007) and Gaing (2003) and are widely used as benchmarks in this field and have been used by many other research groups around the world for similar purposes.
The results obtained from the MPSO are compared with those of other methods reported in literature, i.e., the particle swarm optimization
Implementation of PSO, GCPSO and MPSO
In order to find the effectiveness and superiority of the MPSO, the test results are compared with the results obtained by other algorithms available in the literature. Therefore, to make the results comparable, the same number of population and iterations available in the literature are used in this paper. The implementation of PSO, GCPSO and MPSO for ED problem of the study systems are given below:
For the study system 1 with six generators, the goal of the optimization is to find the best
Conclusion
This paper presents an alternative approach to the non-smooth ED problem using a modified particle swarm optimization (MPSO). Two types of non-smooth ED problems; ED with ramp rate limits and prohibited operating zones and ED with combined valve-point loading effects and multifuel options are considered. For the ED problems with non-smooth cost functions, the MPSO has provided the global solution for the study systems. From the comparisons of the results obtained by MPSO and the results of
References (20)
- et al.
Large-scale economic dispatch by genetic algorithm
IEEE Trans. Power Syst.
(1995) Improved genetic algorithm for power economic dispatch of units with valve-point effects and multiple fuels
IEEE Trans. Power Syst.
(2005)Particle swarm optimization to solving the economic dispatch considering the generator constraints
IEEE Trans. Power Syst.
(2003)- et al.
Evolutionary programming-based economic dispatch for units with multiple fuel options
Eur. Trans. Elect. Power
(2000) - Kennedy, J. and Eberhart, R., 1995. Particle swarm optimization. In: Proceeding of the IEEE International Conference on...
- et al.
Adaptive Hopfield neural network for economic load dispatch
IEEE Trans. Power Syst.
(1998) - et al.
Nonconvex economic dispatch by integrated artificial intelligence
IEEE Trans. Power Syst.
(2001) - et al.
An improved tabu search for economic dispatch with multiple minima
IEEE Trans. Power Syst.
(2002) - et al.
Economic dispatch of generators with prohibited operating zones: a genetic algorithm approach
Proc. Inst. Elect. Eng., Gen., Transm., Distrib.
(1996) - et al.
A hybrid particle swarm optimization employing crossover operation for economic dispatch problems with valve-point effects
Eng. Intelligent Syst. Electr. Eng. Commun.
(2007)
Cited by (72)
Algorithmic trading using combinational rule vector and deep reinforcement learning
2023, Applied Soft ComputingEvolutionary simplex adaptive Hooke-Jeeves algorithm for economic load dispatch problem considering valve point loading effects
2021, Ain Shams Engineering JournalSolving constrained optimal power flow with renewables using hybrid modified imperialist competitive algorithm and sequential quadratic programming
2019, Electric Power Systems ResearchSolving non-convex economic load dispatch problem via artificial cooperative search algorithm
2019, Expert Systems with ApplicationsCitation Excerpt :Metaheuristic approaches can consider ED constraints such as non-convex cost functions or non-smooth operating conditions (Gjorgiev & Čepin, 2013). These methods include genetic algorithm (GA) (Amjady & Nasiri-Rad, 2010; Elsayed, Sarker, & Essam, 2014; Li, Das, Pahwa, & Deb, 2013), tabu search (Whei-Min, Fu-Sheng, & Ming-Tong, 2002), particle swarm optimization (PSO) (Neyestani, Farsangi, & Nezamabadi-pour, 2010; Safari & Shayeghi, 2011; L. Wang & Singh, 2009), evolutionary programming (EP) (Sinha, Chakrabarti, & Chattopadhyay, 2003), ant colony (Pothiya, Ngamroo, & Kongprawechnon, 2010), differential evolution (DE) (Jiang, Zhou, Wang, & Zhang, 2013), bacterial foraging algorithm (BFA) (Farhat & El-Hawary, 2010), harmony search (HS) (Jeddi & Vahidinasab, 2014), firefly algorithm (FA) (Yang, Sadat Hosseini, & Gandomi, 2012), group search optimizer (GSO) (Zare, Haque, & Davoodi, 2012), biogeography-based optimization (BBO) (Bhattacharya & Chattopadhyay, 2010), backtracking search algorithm (BSA) (Modiri-Delshad & Rahim, 2014), synergic predator-prey optimization (SPPO) (Singh, Dhillon, & Kothari, 2016), seeker optimization algorithm (SOA) (Shaw, Mukherjee, & Ghoshal, 2012), artificial bee colony (ABC) (Aydın & Özyön, 2013), and colonial competitive differential evolution (CCDE) (Ghasemi, Taghizadeh, Ghavidel, & Abbasian, 2016). Furthermore, to address the high complexities of practical ED problems, two or more techniques can be combined with a hybrid method.
An efficient hybrid MPSO-GA algorithm for solving non-smooth/non-convex economic dispatch problem with practical constraints
2018, Ain Shams Engineering JournalSemidefinite programming solution of economic dispatch problem with non-smooth, non-convex cost functions
2018, Electric Power Systems Research