Economic non-convex optimization problem with various practical

Economic load dispatch (ELD) plays a vital role in the power system operation. It minimizes total generation cost, subject to load demand and other equality and inequality constraints. The ELD problem can be broadly categorized as convex/smooth and non-convex/ non-smooth. In convex problem cost of generator is represented by a single quadratic function. It can be solved by traditional optimization methods based on mathematical programming techniques. However, the ELD problem with valve point loading effects is represented as a non-convex optimization problem with various practical constraints; this makes it a challenging problem that cannot be handled effectively by classical optimization methods.
Due to the high economic benefits, a wide variety of optimization techniques have been adopted to solve non-convex ELD problems. Some of these techniques are based on conventional optimization methods, whereas others on evolutionary optimization methods. Lambda (?) iteration 1, 2, gradient method 1 and base point factors method 1, 2 are a number of examples for such conventional techniques. These methods have limitation on cost curves nature. In addition, due to several local minima in the non-convex ELD problems, these methods have oscillatory problems leading to high computational time 3. But, due to no restriction on the cost curve, dynamic programming 4 can solve non-convex ELD problems. But it is computationally extensive, and suffers from the problem of dimensionality 5.
Recently, metaheuristic techniques such as evolutionary programming (EP) 6, genetic algorithm (GA) 7, tabu search 8, simulated annealing (SA) 9, harmony search (HS) 10, differential evolution (DE) 11, bacteria foraging optimization (BFO) 12, ant colony optimization (ACO) 13, artificial bee colony (ABC) 14 and particle swarm optimization (PSO) 15-31 have been successfully applied to solve ELD problems due to their ability to handle complex optimization problems. Also, some hybridization of these methods has been used to solve kind of ELD problems effectively 32, 33. Despite it a lot of progresses have been made on metaheuristic methods for the ELD problems. There is still much room to improve on their searching capability, stability and convergence characteristics.
Among all evolutionary algorithms (EAs), PSO (developed by Kennedy and Eberhart in 1995) 34 is widely used for solving ELD problems due to its, few parameters to adjust, ease to understand, ease to implement and computationally efficiency. Also, PSO algorithm has flexibility to enhance both local exploitation and global exploration abilities 17. Despite of these advantages, traditional PSO suffers from premature convergence, especially for problems with multiple local optimums 18, 19. To overcome this problem, many variants of PSO were proposed for solving ELD problems 15-31. Some of these methods concentrate on control parameters of the traditional PSO like acceleration coefficients and inertia weight. Consider for instance, in 17, 18, 19, 26, 27, 29, 30 the time varying acceleration coefficients approach is proposed. In these approaches, during the iterative process, value of acceleration coefficients (c_1 and c_2) varies for the cognitive and social component. The results of mentioned methods indicated that by proper tuning of these coefficients (c_1 and c_2), the particles are guided towards optimum solution. Moreover, a large number of inertia weight (w) settings were proposed 35 for the classical PSO. These are classified in four main groups: constant, random, time varying and adaptive inertia weights. In order to balance local and global search abilities of the PSO, time varying inertia weight has been found more suitable 35, 36.
Moreover, as shown by Kennedy et al. 38, Clerc and Kennedy 39 and Trelea 40, performance of the PSO greatly depends on its parameters. The acceleration coefficients of cognitive and social component guide the particles in traditional PSO to the optimum point. The cognitive acceleration coefficient controls local search ability and social acceleration coefficient wanders the particles around search space and controls global search ability. Tuning relative values of the cognitive and social acceleration coefficient plays an important role in solution quality of the PSO. Also, inertia weight plays a key role in the PSO because it is a crucial tool to balance the exploration and exploitation. General speaking, in order to achieve a better accuracy and higher speed many researchers have tried to modify the control parameter of the PSO.
This paper proposes a modified time varying PSO algorithm (called MTVPSO) for solving ELD problems with or without valve point loading effects. In MTVPSO, novel time varying cognitive and social acceleration coefficients and gradually decreased inertia weight are introduced in velocity update equation of the PSO algorithm. By using these novel parameters, the proposed MTVPSO leads to a proper balance between the cognitive and social components and avoids risk of being trapped into local optima. Numerical and graphical results confirm that MTVPSO is quite a permissible algorithm for solving ELD problems.
The remainder of paper is organized as follows. Section 2, gives a brief on the ELD problem considering various constraints. Section 3, describes the proposed MTVPSO algorithm. Implementation of the proposed MTVPSO to solve ELD problems is provided in Section 4. Section 5, presents simulation results and analysis. Finally, conclusion with future research work is given in Section 6.


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