ABSTRACT
The utilization of renewable energy resources like wind energy for electrical power generation has undergone several developments in the recent past. The wind power generates electrical power without depleting the fossil fuel reserves and emission of harmful pollutants. The electrical power generation by wind energy is greatly affected by weather conditions. Hence to utilize the wind power into traditional grid certain security constraints are to be implemented. In this paper, economic dispatch (ED) problem using wind power generation with the minimization of the operational cost is discussed. A modified particle swarm optimization (PSO) technique employing Gaussian and Cauchy probability distribution function is developed for the solution of the ED problem. The developed models are tested on a standard IEEE 30 bus test system. From the analysis it is found that the proposed model is simple, reliable and consistent, hence it can be implemented for real time applications.
Key words:- Economic Dispatch problem, Wind Power Penetration, Particle Swarm Optimization, Gaussian and Cauchy Distribution
1.0 INTRODUCTION
Economic dispatch (ED) problem is a allocation real power generation levels to the generating units, so that the load system may be supplied most economically [1]. In conventional ED problem the thermal plants are used to produce the required real power. Optimisation techniques based on linear programming (LP) [2,3,9], non-linear programming (NLP) [4,6], quadratic programming (QP) [8,11] and interior point method (IPM) [5,7] are used to solve this ED problem. Many research have also been performed to solve ED problem using Heuristic techniques like Genetic Algorithm (GA) [11], Evolutionary Programming (EP)[12,13], Particle Swarm Optimization technique (PSO) [20].
In thermal power plants electrical energy is obtained by burning fossil fuels, which play vital role in imparting harmful pollutants like carbon-dioxide, sulphur dioxide into atmosphere which constitutes the Green House Gas (GHG) leading to global warming [25, 26]. Now, most countries have understood the effects of GHG and rigorous measures are taken to reduce the harmful pollutants thus leading to a increasing demand for clean air. The Wind energy offers a alternate power source that is free in the emission of carbon dioxide, the main component of GHG [27]. From the above discussions, it is found that in the near future wind power production is going to play a vital role in clean electricity production. Wind energy systems convert the wind's kinetic energy present in the wind into electrical energy. The economic feasibility of a wind energy generation is dependent on the availability of the wind resource [27]. The availability of the wind resource is unpredictable and hence certain security constraints are to be included in the economic dispatch problem. In 2008, Wang and Chanan Singh, introduced wind penetration model [14] for solving ED problem where, operational costs and security impacts were treated as conflicting objectives. In this paper, the model proposed by Wang and Singh is used to solve ED problem incorporating wind power using a modified particle swarm optimization (PSO) technique.
In 1995, the PSO technique was introduced by Kennedy and Eberhart [15], which was inspired by the social behaviours of animals such as fish schooling and bird flocking. PSO approach utilizes global and local exploration. From here, PSO has been used to solve electrical problems in the field of Optimal Power flow [18, 19, 22 â€" 24], Economic Dispatch [20, 25]and Minimization of Loss Power [21]. The results of these work shows the effectiveness and superiority of PSO over classical techniques and Genetic Algorithm. The other main advantage of using PSO algorithm is that it requires only few parameters to be tuned.
In 2008, Coelho and Lee [25] has employed chaotic and Gaussian function in the PSO algorithm to solve economic load dispatch problem. The proposed method was tested on 15 and 20 unit test systems and it revealed that the new method has outperformed the modern metaheuristic methods. Inspired by this technique, the Gaussian and Cauchy probability distribution technique has been employed in the PSO to solve the classical ED problem along with the inclusion of wind power generation. The proposed method has been tested on IEEE 30-bus test system. The simulation results reveal that the proposed PSO approaches developed using Gaussian and Cauchy distributions helps in diversifying and intensifying the search space of the particle’s swarm in PSO, thus preventing premature convergence to local minima and hence improving the performance of PSO.
2.0 PROBLEM FORMULATION
The objective function of the economic dispatch problem with wind power penetration model is given by,
(1)
Where,
N is the number of committed generators,
are the cost co-efficients of the generator and
is the real power output of generator.
is the wind power calculation based on linear membership function [14]
(2)
Where
is the average wind power,
is the penalty cost for not using all the available wind power,
the load demand,
is the transmission loss, and.
The economic dispatch problem described by (1) is subjected to the following practical constraints
Power Balance constraint
(3)
Generator Capacity Constraint
(4)
Where is the minimum and maximum power for generator respectively.
Wind power constraint
(5)
The transmission losses used in Eq. 3 is calculated using Kron’s formula.
3.0 Particle Swarm Optimization
3.1 Overview of PSO
The PSO method was introduced in 1995 by Kennedy and Eberhart [15]. The method is motivated by social behaviour of organisms such as fish schooling and bird flocking. PSO provides a population-based search procedure. Here individuals called as particles change their positions with time. These particles fly around in a multidimensional search space. During flight, each particle adjusts its position according to its own experience, and the experience of neighbouring particles. Thus each particle makes use of the best position encountered by itself and its neighbours. The direction of the particle is given by the set of particles neighbouring the particle and its past experience. Let and denote the particle position and its corresponding velocity in the search space. is the best previous position of the particle and is the best particle among all the particles in the group. The velocity and the position for each particle is calculated by using the following formulae
(6)
(7)
where and are the current position and velocity of the generation, is the inertia weight factor, and are acceleration constants, is the function that generates uniform random number in the range [0,1] and is the constriction factor introduced by Eberhart and Shi [16,17] to avoid the swarm from premature convergence and to ensure stability of the system
3.2 Algorithm to solve EDP using PSO
The process for implementing PSO is as follows:
a) Initialization of particles: - An initial parent population of size is generated randomly within the feasible range. The elements of each particle are real power output of committed generating units and the value of the membership function excluding the slack bus generator.
(8)
b) Evaluation of fitness function: The fitness function value of each particle in the initial population is computed and the maximum fitness value is stored.
c) Determination of pbest and gbest particles: Compare the evaluated fitness value of each particle with its pbest. If current value is better than pbest, then set the current location as the pbest location. Furthermore, if current value is better than gbest, then reset gbest to the current index in particle array.
d) Modification of member velocity: Change the member velocity of the each individual particle according to the equations (6). If , then and If , then .
e) Modification of member position: The member position in each particle is modified according to (7). If the evaluation value of each individual is better than the previous pbest, the current value is set to pbest. If the best pbest is better than gbest, the value is set to gbest.
f) If the number of iteration reaches maximum then go to next step. Else go to step b
g) The individual that generates the latest gbest is the optimal generation power of each unit with the minimum total generation cost.
The parameters of standard PSO includes: number of particles m, inertia weight w, acceleration constants and , maximum velocity. The inertia weight balances global and local explorations and it decreases linearly from 0.9 to 0.4 in each run. The constants and pulls each particle toward pbest and gbest positions.
4.0 MODIFIED PSO APPROACHES BASED ON GAUSSIAN AND CAUCHY DISTRIBUTION
Coelho and Krohling [25] proposed the use of truncated Gaussian and Cauchy probability distribution to generate random numbers for the velocity updating equation of PSO. In this paper, new approaches to PSO are proposed which are based on Gaussian probability distribution and Cauchy probability distribution. In this new approach, random numbers are generated using Gaussian probability function and/or Cauchy probability function in the interval [0,1].
The Gaussian distribution, also called normal distribution is an important family of continuous probability distributions. Each member of the family may be defined by two parameters, location and scale: the mean and the variance respectively. A standard normal distribution has zero mean and variance of one. Hence importance of the Gaussian distribution is due in part to the central limit theorem. Since a standard Gaussian distribution has zero mean and variance of value one, it helps in a faster convergence for local search.
This work proposes new PSO approaches with combination of Gaussian distribution and Cauchy distribution function. The modification to the conventional PSO (Model 1) proceeds as follows:
Model 2: Here the Gaussian distribution, is used to generate random numbers in the interval [0, 1], in the Cognitive part (Individual Thinking) of the particle. The modified velocity equation is given by
Model 3: Here the Gaussian distribution Gd, is used to generate random numbers in the interval [0,1], in the Social Part of the particle. The modified velocity equation is given by
Model 4: Here the Gaussian distribution Gd, is used to generate random numbers in the interval [0,1], in the Cognitive and Social Part. The modified velocity equation is given by
Model 5: Here the Cauchy distribution Cd, is used to generate random numbers in the interval [0,1], in the Cognitive Part. The modified velocity equation is given by
Model 6: Here the Cauchy distribution Cd, is used to generate random numbers in the interval [0,1], in the Social Part. The modified velocity equation is given by
Model 7: Here the Cauchy distribution Cd, is used to generate random numbers in the interval [0,1], in the Cognitive and Social Part. The modified velocity equation is given by
Model 8: Here the Gaussian distribution Gd, is used to generate random numbers in the interval [0,1], in the Social Part and Cauchy Distribution Cd, is used to generate random numbers in the interval [0,1] in the Cognitive Part. The modified velocity equation is given by
Model 9: Here the Cauchy distribution Cd, is used to generate random numbers in the interval [0,1], in the Social Part and Gaussian Distribution Gd, is used to generate random numbers in the interval [0,1] in the Cognitive Part. The modified velocity equation is given by
The above models that have been developed are implemented in the conventional PSO approach and their performances were studied.
5.0 Test System and Results
To assess the performance of the proposed PSO approaches in solving the Wind Power based ED problem a standard IEEE 30-bus test system is considered. The IEEE 30-bus system consists of 6 generator buses, with a total demand of 283.4 MW. Table 1 outs the system conficuration including the cost co-efficient and generator capacities. The available wind power is and the coefficient of penalty cost is set to 20 $/p.u
Table 1: System Configuration
Generator
G1
10
200
100
0.05
0.50
G2
10
150
120
0.05
0.60
G3
20
180
40
0.05
1.00
G4
10
100
60
0.05
1.20
G5
20
180
40
0.05
1.00
G6
10
150
100
0.05
0.60
