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Optimal power flow is an optimizing tool for operation and planning of modern power systems. This paper presents the solution of the optimal power flow using particle swarm optimization. The main goal of this paper is to verify the viability of using PSO problem composed by different objective functions. This OPF problem involves the optimization of various types of objective functions while satisfying a set of operational and physical constraints while keeping the power outputs of generators, bus voltages, shunt capacitors/reactors and transformers tap settings in their limits. The proposed PSO method is demonstrated and compared with Evolutionary Programming (EP) approach on the standard IEEE 14-bus system. The results show that the proposed PSO method is capable of obtaining higher quality solutions efficiently in OPF problem.
The fundamental mission of a power system is to provide consumers with sustained, reliable and cost-efficient electrical energy. In order to achieve this goal, system operators need to constantly adjust various controls such as generation outputs, transformer tap ratios, etc., to assure the continuous economic and secure system operations. This is a difficult task that relies highly on optimal power flow (OPF) function at power system control centers, the OPF procedure consists of using mathematical methodology to find the optimal operation of a power system under feasibility and security constraints. It has been consider as basic tool for determining secure and economic operating conditions of power systems.
Review of literature
The optimal power flow problem can be traced back early as early as 1920's when economic allocation of generation was the only concern. The economic operation of power system was achieved by dividing loads among available generator units such that their incremental generation costs are equal. This was a rather simple problem where only operating limits on real power generation were considered and the effect of system losses was either neglected or approximated by penalty factors calculated from loss formula or load flow Jacobian matrix.
As power system became increasingly large and complex, the security became an important issue, which requires more detailed system models. On the other hand, the evolution of digital computers made such detailed modelling become possible. In 1962, Carpentier for the first time established the Optimal Power Flow (OPF) problem on a rigorous mathematical base. He formulated it as a constrained nonlinear programming problem and derived its optimality conditions using Kuhn-Tucker theorem. In his formulation, the OPF problem is expressed in terms of all control and state variables,with both network and security constraints. The objective function can be total generation cost or transmission losses, depending on a specific application.
The first known Interior Point (IP) is usually attributed to Frisch in 1955, which is a logarithmic barrier method that was later extensively studied by Fiacco and McCormick to solve nonlinearly inequality constrained problem in 1960. In 1979 Khachiyan presented an ellipsoid method that would solve an LP problem in polynomial time. The greatest breakthrough in the Interior point method research field took place in 1984 by Karmarkar's method. After 1984, several variants of Karmarkar's Interior Point (IP) method have been proposed and implemented. Natural creatures sometimes behave as a swarm. One of the main streams of artificial life research is to examine how natural creatures behave as a swarm and reconfigure the swarm models inside a computer. Reynolds developed boid as a swarm model with simple rules and generated complicated swarm behavior by computer graphic animation . Boyd and Richerson examined the decision process of human beings and developed the concept of individual learning and cultural transmission . According to their examination, human beings make decisions using their own experiences and other persons' experiences.
A new optimization technique using an analogy of swarm behavior of natural creatures was started in the beginning of the 1990s. Dorigo developed ant colony optimization (ACO) based mainly on the social insect, especially ant, metaphor . Each individual exchanges information through pheromones implicitly in ACO. Eberhart and Kennedy developed particle swarm optimization (PSO) based on the analogy of swarms of birds and fish schooling . Each individual exchanges previous experiences in PSO. These research efforts are called swarm intelligence [5, 6]. S. M. Kumari, G. Priyanka and M. Sydulu et al., comparison of Genetic Algorithms and Particle Swarm Optimization for Optimal Power Flow Including FACTS devices are described. GA and hybrid particle swarm optimization is used for distribution state estimation . In [12,13] developed a method for Solving multi-objective optimal power flow using differential evolution algorithm. Optimal Power Flow for Steady State Security Enhancement using Enhanced Genetic Algorithm with FACTS Devices are proposed in [14,15]. The research efforts for loss minimization using Optimal Power Flow Based on Swarm Intelligences are given in [16,17]. Modified differential evolution algorithm for optimal power flow with non-smooth cost functions . This paper focuses on PSO as one of the swarm intelligence techniques.
Other evolutionary computation (EC) techniques such as genetic algorithms (GAs), utilize multiple searching points in the solution space like PSO. Whereas GAs can treat combinatorial optimization problems, PSO was aimed to treat nonlinear optimization problems with continuous variables originally. Moreover, PSO has been expanded to handle combinatorial optimization problems and both discrete and continuous variables as well. Efficient treatment of mixed-integer nonlinear optimization problems (MINLPs) is one of the most difficult problems in practical optimization. Moreover, unlike other EC techniques, PSO can be realized with only a small program; namely, PSO can handle MINLPs with only a small program. This feature of PSO is one of its advantages compared with other optimization techniques.
BASIC PARTICLE SWARM OPTIMIZATION
Swarm behavior can be modeled with a few simple rules. Schools of fishes and swarms of birds can be modeled with such simple models. Namely, even if the behaviour rules of each individual (agent) are simple, the behavior of the swarm can be complicated. Reynolds utilized the following three vectors as simple rules in the researches on boid.
Step away from the nearest agent
Go toward the destination
Go to the center of the swarm.
The behaviour of each agent inside the swarm can be modelled with simple vectors. The research results are one of the basic backgrounds of PSO.
Boyd and Richardson examined the decision process of humans and developed the concept of individual learning and cultural transmission . According to their examination, people utilize two important kinds of information in decision process. The first one is their own experience; that is, they have tried the choices and know which state has been better so far, and they know how good it was. The second one is other people's experiences, i.e., they have knowledge of how the other agents around them have performed. Namely, they know which choices their neighbors have found most positive so far and how positive the best pattern of choices was. Each agent decides its decision using its own experiences and the experiences of others. The research results are also one of the basic background elements of PSO. According to the above background of PSO, Kennedy and Eberhart developed PSO through simulation of bird flocking in a two-dimensional space. The position of each agent is represented by its x, y axis position and also its velocity is expressed by vx (the velocity of x axis) and vy (the velocity of y axis). Modification of the agent position is realized by the position and velocity information. Bird flocking optimizes a certain objective function. Each agent knows its best value so far (pbest) and its x, y position. This information is an analogy of the personal experiences of each agent. Moreover, each agent knows the best value so far in the group (gbest) among pbests. This information is an analogy of the knowledge of how the other agents around them have performed. Each agent tries to modify its position using the following information:
The current positions (x, y)
The current velocities (vx, vy)
The distance between the current position and pbest
The distance between the current position and gbest.
This modification can be represented by the concept of velocity (modified value for the current
positions). Velocity of each agent can be modified by the following equation:
where k i v is velocity of agent i at iteration k, w is weighting function, c1 and c2 are weighting factors, rand1 and rand2 are random numbers between 0 and 1, k i s is current position of agent i at iteration k, pbesti is the pbest of agent i, and gbest is gbest of the group. Namely, velocity of an agent can be changed using three vectors such like boid. The velocity is usually limited to a certain maximum value. PSO using eqn. (1) is called the Gbest model.
The following weighting function is usually utilized in eqn. (1):
Where max w is the initial weight, min w is the final weight, itermax is maximum iteration number and iter is current iteration number.
The meanings of the right-hand side (RHS) of eqn. (1) can be explained as follows . The RHS of eqn. (1) consists of three terms (vectors). The first term is the previous velocity of the agent. The second and third terms are utilized to change the velocity of the agent. Without the second and third terms, the agent will keep on "flying" in the same direction until it hits the boundary. Namely, it tries to explore new areas and, therefore, the first term corresponds with diversification in the search procedure. On the other hand, without the first term, the velocity of the "flying" agent is only determined by using its current position and its best positions in history. Namely, the agents will try to converge to their pbests and/or gbest and, therefore, the terms correspond with intensification in the search procedure. As shown below, for example, max w and min w are set to 0.9 and 0.4. Therefore, at the beginning of the search procedure, diversification is heavily weighted, while intensification is heavily weighted at the end of the search procedure such like simulated annealing (SA). Namely, a certain velocity, which gradually gets close to pbests and gbest, can be calculated. PSO using eqns.(1) & (2) is called inertia weights approach (IWA).
Sk current searching point : modified searching point : current velocity : modified velocity : velocity based on pbest : velocity based on gbest The current position (searching point in the solution space) can be modified by the following equation (3):
Figure 1.shows a concept of modification of a searching point by PSO, and it shows a searching concept with agents in a solution space. Each agent changes its current position using the integration of vectors.
The features of the searching procedure of PSO can be summarized as follows:
As shown in eqns. (1), (2), and (3), PSO can essentially handle continuous optimization problems
PSO utilizes several searching points, and the searching points gradually get close to the optimal point using their pbests and the gbest
The first term of the RHS of eqn. (1) corresponds with diversification in the search procedure. The second and third terms correspond with intensification in the search procedure. Namely, the method has a well-balanced mechanism to utilize diversification and intensification in the search procedure efficiently
The above concept is explained using only the x, y axis (two-dimensional space). However, the method can be easily applied to n-dimensional problems. Namely, PSO can handle continuous optimization problems with continuous state variables in an n-dimensional solution space.
Shi and Eberhart tried to examine the parameter selection of the above parameters [7, 8]. According to their examination, the following parameters are appropriate and the values do not depend on problems:
The values are also proved to be appropriate for power system problems [9, 10]. The basic PSO has been applied to a learning problem of neural networks and Schaffer  a famous benchmark function for GA, and the efficiency of the method has been observed .
MODELING OF OPTIMAL-POWER FLOW PROBLEM
The conventional formulation of the optimal-power-flow (OPF) problem determines the optimal settings of control variables such as real power generations, generator terminal voltages, transformer tap settings and phase-shifter angles while minimizing the objective function such as fuel cost as given in eqn.(4).
Where NG: No. of Generators PGi:
Active Power produced by generator i. ai, bi, ci:
Fuel cost coefficients of generator i.
The minimization problem of the objective function is subjected to the satisfaction of constraints from eqns. (5-9)
(i) Load-flow constraints:
RESULTS AND DISCUSSIONS s
This section presents the details of the study carried out on IEEE-30 bus and IEEE-14 bus test systems for testing the OPF methodology. The proposed algorithm was implemented in MATLAB computing environment with Pentium-IV, 2.66 GHz computer with 512 MB RAM. the proposed PSO based algorithm was applied to obtain the optimal-control variables in the IEEE 30 & 14 bus systems under base load conditions.
The upper and lower voltage limits at all the bus bars except slack were taken as 1.1 and 0.95 respectively. The slack bus bar voltage was fixed to a value 1.07p.u. Here the contingencies are not considered and the proposed PSO based algorithm was applied to find the optimal scheduling of the power system for the base case loading condition. The objective function in this case is minimization of total fuel cost. Generator active-power outputs, generator bus bar terminal voltages, transformer tap settings and shunt reactive power compensating elements were taken as optimization variables. The optimization variables are represented as floating point numbers in the population. The optimal values of control variables along with the real power generation of the slack bus bar generator are given in Table 1 & 2 for IEEE-30 & 14 bus systems respectively. The minimum cost obtained with the proposed PSO algorithm for IEEE-30 bus system is $800.966/h, which is less than the minimum generation cost of $803.1916/h obtained with interior point method. Also, it was found that all the state variables satisfy the lower and upper limits.
For comparison, the OPF problem was solved using an evolutionary programming method with the population size of 20 and 250 generations. All the solutions satisfy the constraints on reactive power generation limits and line flow limits. The convergence of generation cost is shown in Fig.2 & 3 for IEEE 30 & 14 bus systems respectively. From Fig. 2 & 3, it can be observed that the PSO took approximately 60 generations to reach the same production cost reached by EP. This shows that the proposed PSO algorithm occupies less computer space and takes less time to reach the optimal solution.
In this paper PSO based OPF algorithm has been validated with EP-OPF method using MATLAB software. It has been observed that optimal solution obtained by PSO-OPF is very close to that obtained by classical methods and it is clear that it is better than EP-OPF. So the proposed OPF methods are most suitable and valid for incorporating new objective functions and constraints. The algorithm is capable of determining the global optimum solution to the OPF problem in the presence of multiple local optima. This provides the opportunity to better model power system operations and therefore determine a more accurate operating state.
The performance of the developed OPF algorithms has been demonstrated by its application to the modified IEEE 30-bus and 14-bus test systems. The algorithms were accurately and reliably converged to the global optimum solution in each case. The PSO-algorithm is also capable of producing more favorable voltage profile while still maintaining a competitive cost.