Economic Load Dispatch by Grey Wolf Optimization
Rajat Rana1, Er Ankush Sharma2
1Research Scholar, 2Assistant Professor
Department of Electrical Engineering
RPIIT Technical Campus, Karnal, Haryana (INDIA)
,
Abstract— Economic Load Dispatch aims at distributing the load demand between various generation stations in a system such that the total cost of generation is minimum. This is of vital importance since it not only reduces the operation cost of the generation utility but also helps in conserving fast dwindling energy resources. Modern day power systems are large interconnected systems with a large number of generator units each having its own cost curve. In this paper an effective and reliable Grey Wolf Optimization (GWO) technique is proposed for the economic load dispatch problem.
Keywords: GWO, APD, Economic load dispatch, Power Demand
1. INTRODUCTION
The main aim of electric supply utility has been identified as to provide the smooth electrical energy to the consumers. While doing so, it should be ensured that the electrical power is generated with minimum cost. Hence in order to achieve an economic operation of the system, the total demand must be appropriately shared among the units. This will minimize the total generation cost for the system with the voltage level maintained at the safe operating limits. Thereby, fulfilling the main objective. Economic dispatch is defined as the process of allocating generation levels to the generating units in the mix so that the system load is fully supplied in the most economic way. The method of economic dispatch for generating units at different loads must have total fuel cost at the minimum point. There are many conventional methods that are in use to solve economic dispatch problem such as Lagrange multiplier method, Lambda iteration method and Newton Raphson method. In the conventional methods, it is difficult to solve the optimal economic problem if the load is changed. Grey Wolf Optimization (GWO) technique is proposed for the economic load dispatch problem.
2. PROBLEM FORMULATION
The ED problem may be stated as to curtail the fuel cost of generator units under several constraints.Mathematically,it may expressed as:-
A) Economic dispatch problem (Minimization Of Fuel Cost)
Fuel cost model
Fi (Pi) = aiPi2 + biPi + ci ……….. (1)
Min F = i=1NFiPi
=i=1N aiPi2+biPi+ci………...(2)
Subjected to following constraints-
i=1NPi-Pd-Pl=0……… ……(3)
and Pimin ≤ Pi ≤ Pimax …….…(4)
where,
ai , bi , ci are cost coefficients of generator i
Pi = real power output of ith generator
Pd= total load demand
Pl = transmission losses
B = coefficient of transmission losses
Pl = i=1Nj=1NPi Bij Pj+ i=1NBoi Pi+Boo……………………………….. (5)
3. Grey Wolf Optimization
Grey wolf is a new population based method which is introduced in 2014 by Mirjalili et al. GWO algorithm is inspired by grey wolves. The technique follows the social hierarchy and hunting path of grey wolves.
The follow-up of grey wolf hunting are as follows:
1) Tracking, chasing, and approaching the victim.
2) Pursuing, encircling, and harassing the victim until it stops moving.
3) Attack towards the victim.
· Social hierarchy
For modeling the social hierarchy of wolves until designing GWO, the fittest solution is considered as the alpha (α). Accordingly, the second and third best solutions are named beta (β) and delta (∆) respectively. The rest of the candidate solutions are considered to be omega (w). The x wolves follow these three wolves.
· Encircling prey
Next, for designing encircling behavior, some equations are considered:
D= (C.Xp(t) - X(t))
X(t+1)=Xp(t) -A.D
where t is the current iteration, A and C are coefficient vectors, Xp(t) represents the position vector of the victim.
The vectors A and C can be calculated as below:
X=2.a.r1-a
C=2.r2
Where, a include are linearly decreased from 2 to 0 over the course of iterations and r1 and r2 are random vectors in the range [0, 1]
· Hunting
In GWO, the first three best solutions obtained are stored so far and push the other search agents (including the omegas) to update their positions due to the position of the best search agents. The following equations are modelled.
Dα= (C1.Xα(t) - X(t))
Dβ= (C2.Xβ (t) - X(t))
D∆= (C3.X∆ (t) - X(t))
X1=Xα - A1. Dα
X2=Xβ - A2. Dβ
X3=X∆ - A3. D∆
X(t+1)=X1+X2+X3/3;
The final position would be in a random position within a circle which is defined by the positions of alpha, beta, and delta in the search space
4. SIMULATION RESULTS
4.1 Algorithms Numerical Settings
GWO: Population size =50, coefficient a=[0-2],Iterations=500
4.2 APD Formulation using gwo
Variables
Power Generation (PG) and cost coefficients (a,b,c) of units with fitness function as fuel cost, quadratic in nature and valve point effect .
Constraints
Equality Constraints: Power Generation-Power Demand=0(PG=Pd)
In-Equality Constraints: Power Generation should be between minimum and maximum limit of power generation.
Stopping Criteria
It is the maximum number of iteration for optimum solution.
4.3 Test System Data
To check the effectiveness of GWO for APD problems, two different case studies are taken.
Table 4.3.1:-Three generating unit system data
Unit / a($/MW2 ) / b($/MW) / c($ ) / PGmin(MW) / PGmax(MW)1 / 0.008 / 7 / 200 / 10 / 85
2 / 0.009 / 6.3 / 180 / 10 / 80
3 / 0.007 / 6.8 / 140 / 10 / 70
Table 4.3.2:- Six generating unit system data
Unit / a($/MW2 / b($/MW) / c($ ) / PGmin(MW) / PGmax(MW)1 / 0.007 / 7 / 240 / 100 / 500
2 / 0.005 / 10 / 200 / 50 / 200
3 / 0.009 / 8.5 / 220 / 80 / 300
4 / 0.009 / 11 / 200 / 50 / 150
5 / 0.0080 / 10.5 / 220 / 50 / 200
0.0075 / 12 / 120 / 50 / 120
4.4 Simulation and Numerical Result
Proposed technique is tested on different benchmarks for simulation. Comparative analysis is demonstrated with other optimization techniques
Table 4.4.1:-Results comparison with other techniques on three unit system
(Power Demand-150 MW)
Parameters / GWO / CS / ABC / FAPG1(MW) / 23.7073 / 33.490 / 33.049 / 32.729
PG2(MW) / 83.7653 / 64.116 / 61.764 / 63.843
PG3(MW) / 44.5295 / 55.126 / 57.872 / 56.151
Cost($/hr) / 1526.7 / 1600.46 / 1600.51 / 1600.47
Fig. 4.4.1: Comparison of Total Cost of GWO with other techniques on three unit System
Table 4.4.2: Result comparison of different technique for six unit system
Techniques / GWO / CS / ABC / FA / PSO / SFL / BFO / HSCost($/hr) / 8352.015 / 8356.06 / 8372.27 / 8388.45 / 8401.45 / 8419.78 / 8428.69 / 8398.06
Fig 4.4.2 Power Generation compariosn cost of GWO with other techniques on six unit SystemFig. 4.4.3: Convergence of Fitness Function with Iteration and Feasible Search Space Boundary on Six Unit System
5. Conclusion
In this work, potential of GWO is explored for solving APD problems considering valve point effect.The efficiency and effectiveness of the proposed technique is benchmarked for different test cases consisting of three, six and generating units with high non-linearity. The results of the GWO compared with that of other intelligence optimization algorithms in terms of operating cost of generators and power generation. Wide contrasting simulation results are observed with the other swarm, nature and bio inspired algorithms. GWO results in minimum operating cost, minimum standard deviation among best, mean and worst solution showing good explorability, fast convergence with iteration leads to robustness and good solution quality.
6. Future Scope
In future,the proposed technique can be effectively hybridized with other optimization techniques to solve convex and non-convex APD problems with incorporation of multi area objectives and constraints related to tie line, emission.
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