This paper proposes a particle swarm optimization (PSO) method for solving the economic dispatch (ED) problem in power systems. Many nonlinear characteristics of the generator, such as ramp rate limits, prohibited operating zone, and nonsmooth cost functions are considered using the proposed method in practical generator operation. The feasibility of the proposed method is demonstrated for three different systems, and it is compared with the GA method in terms of the solution quality and computation efficiency. The experimental results show that the proposed PSO method was indeed capable of obtaining higher quality solutions efficiently in ED problems.

Particle swarm optimization to solving the economic dispatch considering the generator constraints

With the fast-paced changing technologies in the power industry, new power references addressing new technologies are coming to the market. So there is an urgent need to keep track of international experiences and activities taking place in the field of modern unit-commitment (UC) problem. This paper gives a bibliographical survey, mathematical formulations, and general backgrounds of research and developments in the field of UC problem for past 35 years based on more than 150 published articles. The collected literature has been divided into many sections, so that new researchers do not face any difficulty in carrying out research in the area of next-generation UC problem under both the regulated and deregulated power industry.

Unit commitment-a bibliographical survey

This paper proposes a new version of the classical particle swarm optimization (PSO), namely, new PSO (NPSO), to solve nonconvex economic dispatch problems. In the classical PSO, the movement of a particle is governed by three behaviors, namely, inertial, cognitive, and social. The cognitive behavior helps the particle to remember its previously visited best position. This paper proposes a split-up in the cognitive behavior. That is, the particle is made to remember its worst position also. This modification helps to explore the search space very effectively. In order to well exploit the promising solution region, a simple local random search (LRS) procedure is integrated with NPSO. The resultant NPSO-LRS algorithm is very effective in solving the nonconvex economic dispatch problems. To validate the proposed NPSO-LRS method, it is applied to three test systems having nonconvex solution spaces, and better results are obtained when compared with previous approaches

A New Particle Swarm Optimization Solution to Nonconvex Economic Dispatch Problems

This paper presents a new genetic approach for solving the economic dispatch problem in large-scale power systems. A new encoding technique is developed. The chromosome contains only an encoding of the normalized system incremental cost in this encoding technique. Therefore, the total number of bits of chromosome is entirely independent of the number of units. The salient feature makes the proposed genetic approach attractive in large and complex systems which other methodologies may fail to achieve. Moreover, the approach can take network losses, ramp rate limits, and prohibited zone avoidance into account because of genetic algorithm's flexibility. Numerical results on an actual utility system of up to 40 units show that the proposed approach is faster and more robust than the well-known lambda-iteration method in large-scale systems.

Large-scale economic dispatch by genetic algorithm

The growing costs of fuel and operation of power generating units warrant improvement of optimization methodologies for economic dispatch (ED) problems. The practical ED problems have non-convex objective functions with equality and inequality constraints that make it much harder to find the global optimum using any mathematical algorithms. Modern optimization algorithms are often meta-heuristic, and they are very promising in solving nonlinear programming problems. This paper presents a novel approach to determining the feasible optimal solution of the ED problems using the recently developed Firefly Algorithm (FA). Many nonlinear characteristics of power generators, and their operational constraints, such as generation limitations, prohibited operating zones, ramp rate limits, transmission loss, and nonlinear cost functions, were all contemplated for practical operation. To demonstrate the efficiency and applicability of the proposed method, we study four ED test systems having non-convex solution spaces and compared with some of the most recently published ED solution methods. The results of this study show that the proposed FA is able to find more economical loads than those determined by other methods. This algorithm is considered to be a promising alternative algorithm for solving the ED problems in practical power systems.

Firefly Algorithm for solving non-convex economic dispatch problems with valve loading effect

A method is described for solving the reserve constrained economic dispatch problem when some of the online generating units have prohibited operating zone(s). For a unit with prohibited zone(s), the zone(s) divide the operating region between the minimum generation limit (Pmin) and the maximum generation limit (Pmax) into disjoint convex subregions. These disjoint subregions form a nonconvex decisions space and the associated economic dispatch problem is thus a nonconvex optimization problem. As a result, the conventional Lagrangian relaxation (LR) approach (e.g. the lambda - delta iterative approach) cannot be applied directly. The method proposed decomposes the nonconvex decision space into a small number of subsets such that each of the associated dispatch problems is either infeasible or one that can be directly solved via the conventional LR approach. Based on the decomposition, the optimal solution is the least costly one among all the feasible solutions of the associated dispatch problems. Examples are also given to illustrate the proposed method. >

Reserve constrained economic dispatch with prohibited operating zones

The authors present the results of an investigation of the relative computational speeds and solution qualities of six different probabilistic production cost simulation methods for power systems. The six methods are: the piecewise linear approximation (PLA), segmentation, equivalent energy function (EEF), cumulant, mixture of normal approximation (MONA), and fast Fourier transform (FFT) methods. For energy calculations, considering both speed and accuracy, it is concluded that if accuracy is paramount then the EEF method with the MW interval chosen to be the maximum common divisor of the block capacities is preferred. If speed is paramount, then either MONA or cumulants is preferred. >

Comparison of probabilistic production cost simulation methods

A combined control variable and stratified sampling method is proposed for Monte Carlo production simulation. Using the production cost obtained from the load duration curve type simulation as the control variable, both the theory and a sample study suggest that the required (for an acceptable coefficient of variation) number of samples can be reduced by approximately a factor of 1000. In the example study, 8000 samples were replaced with 10 samples; in addition, the proposed method using the 10 samples produces an estimator of the mean production cost that has a much smaller variance than an estimator based on a traditional Monte Carlo study which uses 8000 samples. It is believed that this significant reduction in running time will enable chronological simulation to be used in a number of long range planning studies. The advantage of this method is additional insight into actual operation through chronological simulation as well as less bias (or systematic error). With this proposed method, it is believed that the computational time requirement of stochastic production cost simulation has been reduced to the point that its advantages outweigh its additional (over probabilistic simulation) running time. >

Sample size reduction in stochastic production simulation

Load uncertainty contributes to system operational risk; thus the study of system operating reserve or unit commitment risk requires a load model that includes the uncertainty in load as well as the variation with time. The authors propose a Gauss-Markov load model. This random process model includes both the time variation and the uncertainty in load. This load model is used to predict, conditional on what is known at a previous hour, the mean and the variance of the system hourly load. This mean and variance are required for a system operating reserve study and a broad spectrum of production simulations. The proposed load model is described, the model is justified, and the model is illustrated via an example. >

A Gauss-Markov load model for application in risk evaluation and production simulation

A stochastic outage capacity state model is presented for evaluating the random error in power system production cost which is estimated via the Baleriaux-Booth approach. The proposed model is thought to be the first model capable of evaluating the variance of production cost of a system of realistic size. The proposed model is used to evaluate the normalized standard random error of production cost for the IEEE Reliability Test System with an 168-hour simulation horizon. The result is about 13%. An extension approximation is suggested for estimating the normalized standard random error of production cost for a long simulation horizon (e.g. one year) from the results obtained from a short-term (e.g. a day, a week) simulation. Using this result the annual normalized standard random error is about 2.7%. >

A.M. Breipohl

Papers

Reserve constrained economic dispatch with prohibited operating zones

Comparison of probabilistic production cost simulation methods

Sample size reduction in stochastic production simulation

A Gauss-Markov load model for application in risk evaluation and production simulation

Evaluation of the variance of production cost using a stochastic outage capacity state model