Hybrid PSO–SQP for economic dispatch with valve-point effect
Introduction
The primary objective of the economic dispatch problem (EDP) of electric power generation is to schedule the committed generating unit outputs so as to meet the required load demand at minimum operating cost while satisfying all unit and system equality and inequality constraints [1]. This makes the EDP a large-scale highly nonlinear constrained optimization problem. Improvements in scheduling the unit outputs can lead to significant cost savings. In traditional EDP, the cost function for each generator has been approximately represented by a single quadratic function and is solved using mathematical programming based on optimization techniques such as λ iteration method, gradient method, dynamic programming (DP) method, and so on [2]. These techniques require incremental fuel cost curves should be monotonically increasing to find global optimal solution. Whereas, the input output characteristics of large units are inherently highly nonlinear because of valve-point loadings, generating unit ramp rate limits, etc., and furthermore they may generate multiple local minimum points in the cost function.
For generating units, which actually having non-monotonically incremental fuel cost curves, the conventional method ignores or flattens out the portions of the incremental fuel cost curve that are not continuous or monotonically increasing. Hence, inaccurate dispatch result is induced. To obtain accurate dispatch results, approaches without restriction on the shape of incremental fuel cost functions are needed. Classical calculus-based techniques fail to address these types of problems satisfactorily. Unlike some traditional algorithms, dynamic programming [1] imposes no restrictions on the nature of the cost curves and therefore it can solve ELD problems with inherently nonlinear and discontinuous cost curves. This method, however, suffers from the “curse of dimensionality” or local optimality.
In this respect, stochastic search algorithms such as genetic algorithms (GAs) [3], evolutionary programming (EP) [1], [4] and simulated annealing (SA) [5], may prove to be very effective in solving nonlinear ELD problems without any restrictions on the shape of the cost curves. Although these heuristic methods do not always guarantee discovering the globally optimal solution in finite time, they often provide a fast and reasonable solution (sub-optimal near globally optimal). The SA method is a powerful optimization technique and it has the ability to find near global optimum solutions for the optimization problem. SA is applied in many power system problems. However, appropriate setting of the control parameters of the SA based algorithm is a difficult task and the speed of the algorithm is slow when applied to a real power system [6].
The evolutionary algorithms, EAs, (GA and EP) are search algorithms based on the simulated evolutionary process of natural selection, variation, and genetics. The evolutionary algorithms are more flexible and robust than conventional calculus-based methods. Both GA and EP can provide a near global solution [1]. However, the encoding and decoding schemes essential in the GA approach makes it to take longer time for convergence. EP differs from traditional GAs in two aspects: EP uses the control parameters (real values), but not their codings as in traditional GAs, and EP relies primarily on mutation and selection, but not crossover, as in traditional GAs. Hence, considerable computation time may be saved in EP. Although GA and EP seem to be good methods to solve optimization problems, when applied to problems consists of more number of local minima the solutions obtained from both methods are just near global optimum ones. And also GA and EP take long computation times in order to obtain the solutions for such problems [7]. Therefore, hybrid methods combining two or more optimization methods were introduced [7], [8], [9].
Particle swarm optimization (PSO) [10] is one of the modern heuristic algorithms under the EAs and gained lots of attention in various power system applications [10], [11], [12], [13]. PSO can be applied to nonlinear and non-continuous optimization problems with continuous variables. It has been developed through simulation of simplified social models. PSO is similar to the other evolutionary algorithms in that the system is initialized with a population of random solutions. However, each potential solution is also assigned a randomized velocity, and the potential solutions, call agents, corresponding to individuals. Each agent in PSO flies in the n-dimensional problem space with a velocity which is dynamically adjusted according to the flying experiences of its own and its colleagues.
Generally, the PSO is characterized as a simple heuristic of well balanced mechanism with flexibility to enhance and adapt to both global and local exploration abilities [13]. It is a stochastic search technique with reduced memory requirement, computationally effective and easier to implement compared to other EAs. PSO developed by Dr. Kennedy and Dr. Eberhart, shares some of the common features available in other EAs, except the selection procedure. Also, PSO will not follow survival of the fittest, the principle of other EAs. PSO when compared to EP has very fast converging characteristics; however it has a slow fine tuning ability of the solution. Also PSO has a more global searching ability at the beginning of the run and a local search near the end of the run. Therefore, while solving problems with more local optima, there are more possibilities for the PSO to explore local optima at the end of the run.
To overcome this drawback a hybrid method that integrates the PSO with a gradient search algorithm called SQP [14] is proposed in this paper. In the beginning of the run PSO has more possibilities to explore a large space and therefore the agents are freer to move and sit on various valleys. The best value of all the agents will be taken as the initial starting point for the SQP and will be fine tuned. Thus, the possibility of exploring a global minimum in problems with more local optima is increased. The search will continue until a termination criterion is satisfied. To validate the performance of the proposed approach three economic dispatch problems with incremental fuel cost functions taking into account the valve-point loading effects were tested and the results obtained were compared with those obtained using PSO, hybrid EP–SQP technique and other techniques reported in recent literatures [1], [6].
Section snippets
EDP formulation
The classic EDP minimizes the following incremental fuel cost function associated to dispatchable units [16]:The inclusion of valve-point loading effects makes the modeling of the incremental fuel cost function of the generators more practical. This increases the non-linearity as well as number of local optima in the solution space. Also the solution procedure can easily trap in the local optima in the vicinity of optimal value. The incremental fuel cost function of the generating
Particle swarm optimization [10]
Understanding the emergence and evolution of biological and social order has been a fundamental goal of evolutionary theory. Here, we discuss another type of biological system–social system, more specifically, the collective behaviors of simple individuals interacting with their environment and each other. PSO is one of the modern heuristic algorithms developed by Kennedy and Eberhart [10]. It has been developed through simulation of simplified social models. Compared to other evolutionary
Sequential quadratic programming (SQP) [14]
SQP method seems to be the best nonlinear programming methods for constrained optimization. It outperforms every other nonlinear programming method in terms of efficiency, accuracy, and percentage of successful solutions, over a large number of test problems. The method resembles closely to Newton’s method for constrained optimization just as is done for unconstrained optimization. At each iteration an approximation is made of the Hessian of the Lagrangian function using a BFGS quasi-Newton
Solution methodology
The pseudo code of the proposed solution methodology that integrates the PSO with SQP for EDP with valve-point effects can be summarized as follows:
Step 1: get the data for the system.
Step 2: initialize randomly the searching points, velocities of the agents of PSO and count t.
Step 3: do.
Step 4: evaluate the objective function and update the inertia weight and count t.
Step 5: identify the Gbest(t) of the current run t.
Step 6: is Gbest(t) < Gbest(t−1).
Step 7: solve the EDP using the SQP method
Simulation results
The proposed PSO–SQP approach was tested with three test cases of EDP with valve-point effects. To simulate the valve-point loading effects of generating units, a recurring sinusoid component is added with the quadratic cost function. The software was written in MATLAB 6.1 and executed on a Pentium II 500 MHz personal computer. Hereinafter, the results represent the average of 30 runs of the proposed method for all the three test cases. In the following section, optimal range of inertia weight
Discussion
Traditionally to solve the EDP effectively, conventional techniques need the incremental fuel cost curves to be of featured monotonically increasing and continuous. But practically the generating units, which actually having non-monotonically incremental fuel cost curves when solved, the conventional method ignores or flattens out the portions of the incremental fuel cost curve that are non-continuous or non-monotonically increasing. To obtain accurate dispatch results, technique without
Conclusion
An approach by integrating the PSO with the SQP for solving the EDP with valve-point effects is presented. PSO with inertia weight has the ability of global exploration at the beginning of the run and has a local exploration at the ending of the run. Thus, there are possibilities for it to miss the global optimum point while in the beginning of the run, and will fine tune the point at the end of the run which may not be the global optimum. To overcome this, SQP method is integrated and used to
References (18)
- et al.
Solution of Economic Load Dispatch using real coded Hybrid Stochastic Search
Int. J. Electr. Power Energy Syst.
(1999) - et al.
An enhanced Lagrangian neural network for the ELD problems with piecewise quadratic cost functions and nonlinear constraints
Electr. Power Syst. Res.
(2002) Optimal power flow using particle swarm optimization
Int. J. Electr. Power Energy Syst.
(2002)- et al.
Evolutionary programming techniques for economic load dispatch
IEEE Trans. Evol. Comput.
(2003) - D. Srinivasan, F. Wen, C.S. Chang, A.C. Liew, A survey of evolutionary computing in power systems, IEEE Proc. (1996)...
- et al.
Genetic algorithm solution of economic dispatch with valve-point loadings
IEEE Trans. Power Syst.
(1993) - et al.
Evolutionary programming based economic dispatch for units with non-smooth incremental fuel cost functions
IEEE Trans. Power Syst.
(1996) - et al.
Genetic and genetic/simulated-annealing approaches to economic dispatch
IEEE Proc. Gener. Trans. Distrib.
(1994) - et al.
A hybrid EP and SQP for dynamic economic dispatch with nonsmooth incremental fuel cost function
IEEE Trans. Power Syst.
(2002)
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