Essay: application of PSO to ELD problem

Abstract- This paper describes a successful adaptation of the particle swarm optimisation (PSO) algorithm to solve types of economic dispatch (ED) problems in power systems. Economic load dispatch is a non linear optimization problem which is of great importance in power systems[1]. Economic load dispatch (ELD) is the scheduling of generators to minimize the total operating cost depending on equality and inequality constraints. The transmission line loss also is to be kept as minimum as possible. The study is carried out for three unit test system system for without loss and with loss cases.
Key words- particle swarm optimization, Economic dispatch, Piece wise quadratic cost function
INTRODUCTION
The primary objective of the economic load dispatch is to allocate the generating units so that the system load may be supplied entirely and most economically satisfying the constraints. The economic dispatch (ED) is a constrained optimization problem and the nature of the problem is to find the most economical schedule of the generating units while satisfying load demand and operational constraints. The problem has been tackled by many researchers in the past.[2]
Classical optimization methods are highly sensitive to starting points and frequently converge to local optimum solution or diverge together. Linear programming methods are fast and reliable, but the main disadvantage is associated with the piecewise linear cost approximation. Non-linear programming methods have a problem of convergence and algorithm complexity. The premature and slow convergence of GA degrades its performance and reduces its search capability. The simulated annealing method is a powerful optimization technique and it has the ability to find near global optimum solutions for the optimization problem. In consequence, conventional techniques become very complicated when dealing with such increasingly complex dynamic system to solve economic dispatch problems, and are further limited by their lack of robustness and efficiency in a number of practical applications. In this paper, a brief survey covering recent implementation of soft computing techniques in ED problem is presented[3].
PROBLEM FORMULATION
2.1 Economic Dispatch[4]
The complicatedness of ELD may be expressed by minimizing the fuel cost of generating units under some constraints. The fuel cost curve is understood as a quadratic function of the active power output from the generating units. The Fuel Cost (FC) function of generating unit is usually described by a quadratic function of power output Pi as:
(2.1)
where ai,bi, ci are the cost coefficients of the unit i.
??i, bi, ci : fuel cost coefficients of the ith generating unit
N : number of thermal units Subjected to
1. Power balance constraint
PD+PL=??Pi (2.2)
2. Generating capacity limits
Pimin ‘ Pi ‘ Pimax (2.3)
Where
PD = total system demand (MW)
PL = total transmission network loss (MW)
Pi min = minimum power output limit of ith generator (MW)
Pi max = maximum power output limit of ith generator (MW)
PL can be calculated by
(2.4)
Where Bij’s are the elements of loss coefficient matrix B.
Particle Swarm Optimization Concept
Particle swarm optimization (PSO) is a method for performing numerical optimization without clear knowledge of the gradient of the problem to be optimized. PSO is originally attributed to Kennedy, Eberhart and Shi and was first planned for simulating social behaviour. The algorithm was simplified and it was experiential to be performing optimization. The intelligence.. EPSO optimizes a problem by maintaining a population of candidate solutions called particles and stirring these particles around in the search-space according to simple formulae. The travels of the particles are guided by the best found positions in the search-space, which are repeatedly updated as better positions are found by the particles.
PSO parameter
(i) Number of Particles
The typical range of the number of particles is 20-40.Actually for most of the problems 10 particles is large enough to get good results. For some difficult or special problems, one can try 100 or 200 particles as well.
(ii) Dimension of Particles
Dimension of particles is determined by the problem to be optimized.
(iii) Maximum Velocity
Vmax determines the maximum change that one particle can take during each iteration.
(iv) Acceleration Constants
The learning factors c1 and c2 determine the impact of the personal best Pbest and the global best Gbest, respectively. If c1 > c2, the particle has the tendency to converge to the best position found by itself Pbesti rather than the best position found by the population Gbest, and vice versa. Most implementations use a setting with c1 = c2 = 2 [27’31]. [5]
(v) Stopping Condition
The maximum numbers of iterations that PSO executes or the minimum error requirement are the stopping conditions.
(vi) Inertia Weight
A large value of inertia weight encourages global search while a small value facilitates local exploitation. Therefore, the inertia weight is crucial for the search behaviour of the PSO, and a good balance between exploration and exploitation can be obtained using a dynamical inertia weight. Experimental results show that it is favourable to start with a large inertia weight in the early search stage in order to improve exploration of the search space, and gradually reduce the inertia to achieve more refined solutions in the final search stage to significantly improve its performance.[6]
where Wmax and Wmin are the initial and final values of the inertia weight, respectively, and ITERmaxis the maximum iteration number. Typically, parameters Wmax and Wmin are set to 0.9 and 0.4, respectively
Basic PSO algorithm[7]
Fig.1 flow chart of pso
Implementation of PSO algorithm to ELD problem
Step 1 :
‘ The power of a particle of each unit and its velocity are randomly generated for the number of particles set and are checked whether they are within the specified limits.
Pgimin ‘ Pgi ‘ Pgimax
Vimin ‘ Vi ‘ Vimax
Step 2 :
‘ Each set of solution in the search space should satisfy the following equation.
Step 3 :
The pbest values of particles which satisfy the equality constraint are utilized in cost evaluation function F.
Ft=
To calculate total power generation cost, where ai, bi, ci are constants for ith generator. Identify the set of pbest values of particles which provide minimum cost. This set of pbest values (best evalued among pbest) are known as gbest values of generation.
Step 4 :
The member velocity v of each individual Pg is updated according to the velocity update equation with respect to pbest and gbest value determined on random basis.
C1 and C2 : acceleration constants.
w: inertia weight factor
Step 5 :
The velocity components constraint according in the limits from the following condition are checked.
Vmin = -0.5 * Pgimin
Vmax = -0.5 * Pgimax
Step 6 :
The new position of each particle is modified as
Step 7 :
When number of iterations reach maximum, identify the iteration which provides minimum power generation cost and determine te corresponding contribution of power generation by all units.
EXAMPLE AND RESULT
To verify the feasibility of the proposed classical PSO method three unit test system is taken for without transmission loss and with transmission loss cases.
A. Case-1 3-unit system
The system contains 3 thermal units[1]. The data is given below
F1 = 0.00524P12 + 8.66 P1 + 328.12 Rs/Hr
F2 = 0.00608P22 + 10.05 P2 + 136.92 Rs/Hr
F3 = 0.00592P32 + 9.75 P3 + 59.15 Rs/Hr
240 MW ‘ P1 ‘ 90 MW
238 MW ‘ P2 ‘ 85 MW
100 MW ‘ P3 ‘ 20 MW
B-Coefficient Matrix:
B = [0.000134 0.0000176 0.000183
0.0000176 0.000153 0.000282
0.000183 0.000282 0.00162 ]
the corresponding loads is given as 300MW and 450 MW respectively [25]
Table-1 Optimal scheduling of generators for 3-unit system without losses by PSO
Load Demand
(MW) Pg1
(MW) Pg2
(MW) Pg3
(MW) Fuel cost
(Rs/hr)
300 161.076541 317.49659 155.573291 2768.637755
450 152.768436 182.873239 39.238932 4402.969639
(i). Simulation Results of 3 Unit without Loss with 450 MW load
Figure 2-Graph between G-best solutions and Cost in R/hr for a load of 450 mw
We can evaluate these results obtained from PSO method with conventional method. This comparison is shown in the below Table.
Table 2-Comparison of different methods for 3-unit system loss neglected
Power demand
(MW) Fuel Cost (Rs/hr)
Conventional Method PSO Method
300 2768.657850 2768.637755
450 4402.989732 4402.969639
From the above table we can see that PSO method is providing better results.
(ii) Three-Unit Thermal System with Transmission Losses
When the above system is tested for a load demand of 300 MW and 450 MW [25] using the proposed PSO method including transmission losses which can be calculated with the help of loss matrix Bmn provided in section then the results.
Table-3 Optimal scheduling of generators for 3-unit system with losses by PSO
Load Demand
MW Pg1
MW Pg2
MW Pg3
MW Fuel cost
Rs/hr
300 120.458556 87.893538 23.134332 2814.102989
450 161.957099 175.374458 66.783262 4248.895225
(ii). Simulation Results of 3 Unit with Loss with 450 MW
Figure3-Graph between G-best solutions and Cost in R/hr for a load of 450 mw
From the above simulation results we can compare the results with Conventional Method and are shown in the below Table 4.
Table 4- Solution of different methods including losses ‘ 3-unit system
Power demand
(MW) Fuel Cost (Rs/hr)
Conventional method PSO Method
300 2815.023402 2814.102989
450 4249.784023 4248.895225
CONCLUSIONS
We can illustrate important conclusions on the basis of the work done. Some important conclusions are given below.
The selection of parameters c1, c2 and W is very much important in PSO method. It is assured in various research papers that the good results are obtained when c1 = 2.0 and c2= 2.0 and W value is varied from 0.9 to 0.4 for both cases loss neglected and loss included.
We can see from Table 2 and Table 4 that Classical PSO gives better result than Conventional Method.
REFERENCES:
[1]Aiello, M. A., and Leuzzi, F. (2010), ‘Waste Tyrerubberized concrete: Properties at fresh and hardened state.’ Journal of Waste Management, ELSEVIER, 30,1696-1704.
[2]Batayneh, M. K., Marie, I., and Asi, I. (2008), ‘Promoting the use of crumb rubber concrete in developing countries.’ Journal of Waste Management, ELSEVIER, 28, 2171-2176.
[3]Egyptian Code Committee 203, (2003), ‘Experimental guide for testing of concrete materials.’ Part 3 of the Egyptian code of practice for the design and construction of reinforced concrete structures.
[4]Eldin, N. N., and Senouci, A. B. (1993), ‘Rubber-Tyreparticles as concrete aggregate.’Journal of Material inCivil Engineering, ASCE, 5(4), 478-496.