This paper proposes an efficient method to solve the optimal power flow problem in power systems using Particle Swarm Optimization (PSO). The objective of the proposed method is to find the steady-state operating point which minimizes the fuel cost, while maintaining an acceptable system performance in terms of limits on generator power, line flow, and voltage. Three different inertia weights, a constant inertia weight (CIW), a time-varying inertia weight (TVIW), and global-local best inertia weight (GLbestIW), are considered with the particle swarm optimization algorithm to analyze the impact of inertia weight on the performance of PSO algorithm. The PSO algorithm is simulated for each of the method individually. It is observed that the PSO algorithm with the proposed inertia weight yields better results, both in terms of optimal solution and faster convergence. The proposed method has been tested on the standard IEEE 30 bus test system to prove its efficacy. The algorithm is computationally faster, in terms of the number of load flows executed, and provides better results than other heuristic techniques.
In the past two decades, the problem of optimal power flow (OPF) has received much attention. The OPF problem solution aims to optimize a selected objective function such as fuel cost via optimal adjustment of the power system control variables, while at the same time satisfying various equality and inequality constraints. The equality constraints are the power flow equations, and the inequality constraints are the limits on the control variables and the operating limits of power system dependent variables. Generally, the OPF problem is a large-scale highly constrained nonlinear nonconvex optimization problem. This is widely used in power system operation and planning. Many techniques such as nonlinear programming [
Nonlinear programming has many drawbacks such as algorithmic complexity. Linear programming methods are fast and reliable but require linearization of objective function as well as constraints with nonnegative variables. Quadratic programming is a special form of nonlinear programming which has some disadvantages associated with piecewise quadratic cost approximation. Newton-based method has a drawback of the convergence characteristics that are sensitive to initial conditions. The interior point method is computationally efficient but suffers from bad initial termination and optimality criteria. The problem of the OPF is highly nonlinear, where more than one local optimum exists. Hence the above-mentioned local optimization techniques are not suitable for such a problem. Therefore the conventional optimization methods are not able to identify the global optimum. Hence it becomes essential to develop optimization techniques that are efficient to overcome these drawbacks and handle such complexity.
Heuristic algorithms such as Genetic Algorithm (GA) [
A brief introduction has been provided in this section for the existing optimization techniques that have been applied to power system problems. The rest of the paper is arranged as follows. In Section
The optimal power flow problem is a nonlinear optimization problem with nonlinear objective function and nonlinear constraints. The OPF problem considered in this paper is to optimize the steady-state performance of a power system in terms of the total fuel cost while satisfying several equality and inequality constraints.
Mathematically, the OPF problem can be formulated as follows.
The fuel cost function is given as
Generator voltages, real power outputs, and reactive power outputs are restricted by their lower and upper limits as follows:
Transformer tap settings are bounded as follows:
Shunt VAR compensations are restricted by their limits as follows:
The constraints of the voltages at load buses and transmission line loadings are considered as follows:
PSO has been developed through simulation of simplified social models. The features of the method are as follows. The method is based on researches about swarms such as fish schooling and a flock of birds. It is based on a simple concept. Therefore, the computation time is short and it requires less memory. It was originally developed for nonlinear optimization problems with continuous variables. However, it is easily expanded to treat problems with discrete variables. Therefore, it is applicable for the OPF problem which is having both continuous and discrete variables.
The previous feature (c) is suitable for the OPF problem because it is practically efficient method which can handle both continuous and discrete variables. The previous features allow PSO to effectively handle the problem and it requires only short computation time.
According to the research results for a flock of birds, birds find food by flocking (not by each individual). The observation leads the assumption that all information is shared inside flocking. Moreover, according to observation of behavior of human groups, behavior of each individual (agent) is also based on behavior patterns authorized by the groups such as customs and other behavior patterns according to the experiences by each individual. PSO was developed through simulation of a simplified social system, and has been found to be robust in solving continuous nonlinear optimization problems. The PSO technique can generate a high-quality solution within shorter calculation time and stable convergence characteristic than other stochastic methods. Researchers have presented PSO solving techniques applied to OPF, economic dispatch problem, available transfer capability problem, reactive power optimization problem in the recent past. Many researches are still in progress for proving the potential of the PSO in solving complex power system operation problems.
A swarm consists of a set of particles moving within the search space, each representing a potential solution (fitness). In a physical
Each individual moves from the current position to the next one by the modified velocity in (
The parameters
The search mechanism of the PSO using the modified velocity and position of the individual
Search mechanism of PSO.
The general PSO algorithm is presented below.
The technique is initialized with a population of random solutions or particles and then searches the optima by updating generations. Each individual particle
In every iteration, each particle is updated by the following two best values. The first one is the personal best position
After finding the two best values, each particle updates its velocity and current position. The velocity of the particle is updated according to its own previous best position and the previous best position of its companions which is given in (
The acceleration coefficients control the distance moved by a particle in the iteration. The inertia weight controls the convergence behavior of PSO. Initially the inertia weight was considered as a constant value. However, experimental results indicated that it is better to initially set the inertia weight to larger value and gradually reduce it to get refined solutions. A new inertia weight which is neither set to a constant value nor set as a linearly decreasing time-varying function is used in this paper and appears in (
The initial populations are generated randomly, and it is a set of
It is an apparently disorganized population of moving particles that tend to cluster together while each particle seems to be moving in a random direction.
From the earlier research performed by Eberhart and Shi [
The range in which the algorithm computes the optimal control variables is called search space. The algorithm will search for the optimal solution in the search space between 0 and 1. When any of the optimal control values of any particle exceed the searching space, the value will be reinitialized. In this paper, the lower and upper boundaries are set to 0 and 1.
This refers to the maximum number of generations allowed for the fitness value to converge with the optimal solution. In this paper, the maximum generation is set as 200.
The conventional PSO algorithm initially used a constant value for the inertia weight.
In order to improve the performance of the PSO, the time-varying inertia weight was proposed in [
The GLbestIW method is proposed in [
Figure
Flow chart for the proposed technique.
The proposed PSO algorithm was implemented using MATLAB 7.0 software. PSO parameters are selected as shown in Table
PSO parameters.
Population size | 20 |
---|---|
Generations | 200 |
Acceleration coefficients | 2 |
Inertia weight | As proposed in ( |
Number of load flows | |
(Population | 4000 |
Stopping criteria | (i) When the difference between the results of the two consecutive iterations is |
(ii) The number of iterations reaches 200 |
The proposed algorithm is implemented and tested on a standard IEEE 30 bus test system as shown in Figure
System description.
S.no. | Variables | IEEE 30 bus test system |
---|---|---|
Number of buses | 30 | |
Number of branches | 41 | |
Number of generators | 6 | |
Number of generator buses | 6 | |
Number of shunts | 9 | |
Number of tap-changing transformers | 4 |
IEEE 30 bus test system.
The limits for different variables are given in Table
Limits for the different variables for IEEE 30 bus test system.
S.no. | Description | Units | Variable type | Lower limit | Upper limit |
---|---|---|---|---|---|
Generator bus voltage | p. | Continuous | 0.95 | 1.05 | |
Load bus voltage | p. | Continuous | 0.95 | 1.10 | |
Transformer taps | p. | Discrete | 0.90 | 1.10 | |
Shunt capacitor | p. | Discrete | 0.0 | 0.05 |
Generator cost coefficients for IEEE 30 bus test system.
0.00375 | 0.0175 | 0.0625 | 0.0083 | 0.025 | 0.025 | |
2.0 | 1.75 | 1.0 | 3.25 | 3.0 | 3.0 | |
0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
State Variable Constraints for IEEE 30 bus test system.
Bus | ||||||
---|---|---|---|---|---|---|
1 | 50 | 200 | −20 | 200 | 0.95 | 1.10 |
2 | 20 | 80 | −20 | 100 | 0.95 | 1.10 |
5 | 15 | 50 | −15 | 80 | 0.95 | 1.10 |
8 | 10 | 35 | −15 | 60 | 0.95 | 1.10 |
11 | 10 | 30 | −10 | 50 | 0.95 | 1.10 |
13 | 12 | 40 | −15 | 60 | 0.95 | 1.10 |
The three methods listed in Table
Various Methods.
Method name | Description |
---|---|
PSO-1 | PSO with constant inertia weight (CIW) |
PSO-2 | PSO with time-varying inertia weight (TVIW) |
PSO-3 | PSO with proposed global-local best inertia weight (GLbestIW) |
The average deviation which gives the average of the absolute deviation of the fitness value from their mean is also tabulated. Added to these analyses,
Statistical Analyses of fitness value in 100th iteration.
Stat.test | Average | SD | AVEDEV | |||
---|---|---|---|---|---|---|
PSO-1 | 803.8812 | 0.2562 | 0.0757 | Method no. | Best method | |
PSO-2 | 802.4946 | 0.1188 | 0.0716 | 1 and 2 | .97192 | 2 |
PSO-3 | 801.8441 | 0.0002 | 0.0002 | 2 and 3 | .00000 | 3 |
Statistical Analyses of fitness value in 200th iteration.
Stat.test | Average | SD | AVEDEV | |||
---|---|---|---|---|---|---|
PSO-1 | 803.8449 | 0.00794 | 0.00248 | Method no. | Best method | |
PSO-2 | 802.8438 | 0.0008 | 0.0003 | 1 and 2 | .98799 | 2 |
PSO-3 | 801.8438 | 0.0001 | 0.0001 | 2 and 3 | 1.00000 | 3 |
ANOVA test for the different PSO methods.
Table
Comparison of different PSO methods.
S. no. | Number of trails | Method | Min. cost | Max. cost | Average |
---|---|---|---|---|---|
1 | CIW | 802.959 | 822.351 | 809.587 | |
TVIW | 802.741 | 824.391 | 809.741 | ||
GLBestIW | 801.113 | 816.277 | 807.828 | ||
100 | CIW | 802.843 | 804.921 | 803.881 | |
TVIW | 802.543 | 802.551 | 802.494 | ||
GLBestIW | 801.843 | 801.845 | 801.844 | ||
200 | CIW | 802.843 | 804.913 | 803.844 | |
TVIW | 802.543 | 802.852 | 802.843 | ||
GLBestIW | 801.843 | 801.845 | 801.843 |
Generator output.
Unit power output | CIW | TVIW | GLBestIW |
---|---|---|---|
175.73 | 176.23 | 176.72 | |
48.83 | 48.94 | 48.96 | |
21.47 | 21.42 | 21.52 | |
21.65 | 21.34 | 21.57 | |
12.09 | 12.23 | 12.37 | |
12 | 12 | 12.02 | |
Total power Output (MW) | 291.771 | 292.16 | 293.16 |
Cost ($/h) | 802.843 | 802.543 | 801.843 |
Table
Performance comparison.
Parameter | Matpower | CPSO | GLBestIW PSO |
---|---|---|---|
176.2 | 179.2 | 176.74 | |
48.79 | 48.3 | 48.8 | |
21.48 | 20.92 | 21.47 | |
22.07 | 20.56 | 21.64 | |
12.19 | 11.57 | 12.14 | |
12.00 | 12.48 | 12.00 | |
Cost ($/hr) | 802.1 | 802.0 | 801.84 |
Figure
Comparison graph for the different PSO methods.
This paper presents a GLbestIW-based PSO technique for the solution of optimal power flow problem in a power system. The results of study on the impact of inertia weight for improving the performance of the PSO to obtain the optimal power flow solution are presented and discussed. The OPF problem considered in this paper is to minimize the fuel cost and determine the control strategy with continuous and discrete control variables, such as generator bus voltages, transformer tap positions, and reactive power installations. The performance of the proposed GLbestIW-based PSO has been validated on the standard IEEE 30 bus test system. It is shown through different trials that the GLbestIW PSO outperforms other methods in terms of high quality solution, consistency, faster convergence, and accuracy.