Abstract
This chapter presents a solution for multi-objective Optimal Power Flow (OPF) problem via a genetic fuzzy formulation algorithm (GA-FMOPF). The OPF problem is formulated as a multiple objective problem subject to physical constraints. The objectives and constraints are modelled as fuzzy mathematical programming problems involving the minimization of the objective function with fuzzy parameters and uncertainties in set of constraints. So the method is capable of representing practical situations in power system operation where the limits on specific variables are soft and the small violations of these limits may be tolerable. Then, genetic algorithm is used in order to seek a feasible optimal solution to the environmental/economic dispatch problem. Illustrative examples are given to clarify the proposed method developed in this manuscript and the performance of this solution approach is evaluated by comparing its results with that of their existing methods.
TopIntroduction
Electric power systems are, perhaps, the most challenging industrial systems in term of task of planning and operating. These tasks demand acknowledge of the priorities and objectives involved. The basic requirement is to meet the demand for electric energy for the served area at the lowest possible cost. Another objective is to minimize the environmental impact of the operation. Continuity of service and reliability are major considerations. Safety for both personnel and equipment is a factor that may override some of the other objectives. The constraints include inequality ones which are the limits of control variables and state variables; and equality ones which are the power flow equations.
The goal of Optimal Power Flow (OPF), first introduced by Carpentier in 1962, is to find optimal settings of a given power system network that optimize the system objective functions such as total generation cost, system loss, emission of generating units, and load shedding while satisfying power flow equations, system security, and equipment operating limits. Different control variables, some of which are generators’ real power outputs and voltages, are manipulated to handle large-scale power systems in an effective and efficient manner (Parti, et al., 1983; Saadat, 1999; Naarayan, 2003). In its most general formulation, OPF is a nonlinear, non-convex, large-scale, static optimization problem, with both continuous and discrete control variables. The idea behind the combined economic and emission OPF is to compute the optimal generation for individual units of the power system by minimizing the fuel cost and emission levels simultaneously, subject to various system constraints.
A number of conventional optimization techniques have been utilized for solving the OPF problem, for instance: equal incremental cost method, sequential quadratic programming, decomposition method, Lagrangian relaxation method, and Newton methods (Kothari & Parmar, 2006). Although some of these techniques have good convergences’ characteristics, some of their major drawbacks are related to their convergence to local solution instead of global ones, if the initial guess is located within a local solution neighbourhood. The theoretical assumptions behind such algorithms may be not suitable for the OPF formulation. Optimization methods such as Simulated Annealing (SA) (Vasant, & Barsoum. 2010; Vasant, 2010; Aminian, Javid, Asghari, Gandomi, & Esmaeili, 2011), Evolutionary Programming (EP) (Tsoulos, & Vasant, 2010; Al-Obeidat, Belacel, Carretero, & Mahanti, 2011), Genetic Algorithms (GA) (Dieu, & Ongsakul, 2010) and Particle Swarm Optimizer (PSO) (Polprasert, Ongsakul, & Dieu, 2013; Dieu, & Schegner, 2012; Vo & Schegner, 2013) have been employed to overcome such drawbacks. Recent strategies have begun to emerge as a very promising alternative to overcome common difficulties of the previously-mentioned algorithms (Galiana & Conejo, 2009).
In crisp formulation of the OPF problem, operating parameters are assumed to be deterministic, but In real world OPF problems, input data or related parameters, such as market demand, capacity, and relevant operating costs, frequently are fuzzy owing to some information being incomplete or unobtainable (Galiana & Conejo, 2009). As example of constraints/objectives with Fuzziness we have: acceptable security risk, assessment of customer satisfaction, economic objectives, environmental objectives, equipment loading limits, normal operational limits, and so forth. Because of not enough information, none of these constraints or objectives is well defined. When deterministic constraints are not satisfied in the conventional mathematical programming, it is difficult to learn what kind of constraints is critical to what extent (Kothari & Singh Parmar, 2006; Galiana & Conejo, 2009).
Key Terms in this Chapter
Optimization: Is the process of making something better. An engineer or scientist conjures up a new idea and optimization improves on that idea. Optimization consists in trying variations on an initial concept and using the information gained to improve on the idea. Optimization is everywhere, from engineering design to financial markets, from our daily activity to planning our holidays, and computer sciences to industrial applications. We always intend to maximize or minimize something.
GA-FMOPF: Applied genetic fuzzy formulation algorithm for multi-objective optimal power flow problem. Genetic algorithm (GA) is employed as an optimization tool to solve the reformulated problem (The two conflicting objectives, generation cost, and environmental pollution are minimized simultaneously).
Nonlinear Programming: A very general class of optimization whose objective and/or constraints are nonlinear functions. Most practical problems are nonlinear. Krush-Kuhn-Tucker conditions are linked with the optimality of nonlinear optimization, though there is no universality in nonlinear optimization. Global optimality can only be guaranteed for a few special class of optimization, such as convex optimization.
Linear Programming: A special class of mathematical programming whose objectives and constraints are all linear function. Simplex method, developed by George Dantzig in 1947 is one of the most powerful algorithms for linear programming.
Fuzzy Mathematical Programming: It has been applied firstly by Bellman and Zadeh in 1970s when they established the basic concepts of fuzzy goals, fuzzy constraints, and fuzzy decisions to the mathematical programming problem.
Economic Load Dispatch Problem: The economic dispatch problem is a non linear programming optimization problem. Were the main objective is to determine the optimal schedule of online generating units so as to meet the demand power at minimum operating cost under various system and operating constraints. The importance of economic dispatch is to get maximum usable power using minimum resources. Conventionally, the economic dispatch problem of a power system is solved in the environment of unit commitment and real time operation plants by assuming that each of the dispatchable on-line units can be regulated continuously between its minimum generation limit, P min , and its maximum generation limit, P max .
Multiobjective Optimization: Multiobjective optimization (MOO) instead of combining all the objectives to single objective, objectives are solved simultaneously and hence retain the significance of each objective. MOO has been available for about two decades, and its application in real world problems is continuously increasing. In a Multiobjective optimization environment the main challenge is to minimize the distance of the generated solutions to the Pareto set and to maximize the diversity of the developed Pareto set. A good Pareto set may be obtained by appropriate guiding of the search process through careful design of reproduction operators and fitness assignment strategies. The objective of Multiobjective is to determine the best approximation to this Pareto optimal set.
Metaheuristic Algorithms: A class of stochastic algorithms using a combination of randomization and local search. They are often based on learning from nature or biological systems. Popularly algorithms include genetic algorithms, particle swarm optimization, ant algorithms, and bee algorithms. Metaheuristic algorithms are usually designed for global optimization.
Genetic Algorithm: Is an optimization technique based on Darwin’s survival of the fittest hypothesis. It was developed after original work by John Holland in 1970s. In a GA, given a problem for which a closed-form solution is unidentified, or impossible to obtain with classical methods, an initial randomly generated population of possible solution is created. Its characteristics are then used in an equivalent string of genes that will be later recombined with genes from other individuals. Each solution is assimilated to an individual, who is evaluated and classified in relation with its closeness to the best, yet still unknown, solution to the problem. As in a biological system submitted to external constraints, the fittest members of the initial population are given better chances of reproducing and transmitting part of their genetic heritage to the next generation. A new population, or second generation, is then created by recombination of parental genes. It is expected that some members of this new population will have acquired the best characteristics of both parents and, being better adapted to the environmental conditions, will provide an improved solution to the problem. After it has replaced the original population, the new group is submitted to the same evaluation procedure, and later generates its own offspring’s. The process is repeated many times, until all members of a given generation share the same genetic heritage. From then on, there are virtually no differences between individuals. The members of these final generations, who are often quite different from their ancestors, possess genetic information that corresponds to the best solution to the optimization problem.
Mathematical Programming: How to achieving the best outcome of objective function in a function domain that can be constrained or not, in wich the aim is to find the best of all possible solutions. More formally, find a solution in the feasible region which has the minimum (or maximum) value of the objective function.