DOI : 10.17577/IJERTV3IS061656
- Open Access
- Total Downloads : 524
- Authors : Sachin Kumar, Shivani Mehta, Dr. Y. S. Brar
- Paper ID : IJERTV3IS061656
- Volume & Issue : Volume 03, Issue 06 (June 2014)
- Published (First Online): 01-07-2014
- ISSN (Online) : 2278-0181
- Publisher Name : IJERT
- License:
This work is licensed under a Creative Commons Attribution 4.0 International License
Solution of Economic Load Dispatch Problem using Gravitational Search Algorithm with Valve Point Loading
Sachin Kumar#1, Shivani Mehta#2, Dr. Y. S. Brar#3
M.Tech Department of Electrical Engineering DAVIET Jalandhar#1 Assistant Professor Department of Electrical Engineering DAVIET Jalandhar # 2 Assistant Professor Department of Electrical Engineering GNDEC Ludhiana#3
Abstract : This paper describes gravitational search algorithm for solving the non convex Economic load Dispatch (ELD) problem with valve point effect. The main objective of economic load dispatch problem is to generate the required amount of power so that the total operating cost of system is minimized, while satisfying load demand and system equality and inequality Constraints. Different heuristic optimization methods have been proposed to solve this problem in previous study. So in this paper, gravitational search algorithm (GSA) based on law of gravity and mass interaction is proposed. This proposed approach has been tested on 3, 13, 40 unit systems. Simulation results of proposed approach are compared with some well-known heuristic search methods. The obtained results verify the efficiency of the proposed method with minimum computational time in solving various nonlinear functions.
Keyword : economic load dispatch, gravitational search algorithm, valve point effect.
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INTRODUCTION
The increasing energy demand and decreasing energy resources have necessitated the optimum use of available resources. Economic load dispatch is optimization scheme intends to find the generation outputs that minimize the total operating cost while satisfying several unit and system constraints. In main aim of economic load dispatch is to schedule the output of all generating units so as to meet total load demand at minimum fuel cost, also subject to equality and inequality constraints on power output. There are many methods developed for solving the economic load dispatch problems which are classified as classical and heuristic methods. In classical method, fuel cost curve is monotonically increasing one and it represented by quadratic function. Most of classical optimization techniques such as lambda iteration method, gradient method, Newtons method, linear programming, Interior point method and dynamic programming have been used to
solve the basic economic dispatch problem. But due to non convex and nonlinear behavior of ED problem and large number constraints, classical method cannot be execute well in solving the ED problems. So in order to overcome these non linear dispatch problems heuristic technique are developed. Many heuristic techniques like Hardiansyah[2] introduced Solving economic load dispatch problem with valve point effect using modified ABC algorithm. K.Senthil, K.Manikandan [3] proposed Economic Thermal Power Dispatch With Emission Constraint and Valve Point Effect Loading Using Improved Tabu Search Algorithm.
J.Jain, R.Singh [4] introduced Biogeographic-Based Optimization Algorithm for Load Dispatch in Power System. K. Meng, H. G. Wang, Z.Y. Dong, and K. P. Wong [7] proposed Quantum-Inspired Particle Swarm Optimization for Valve-Point Economic Load Dispatch. Chao-Lung Chiang proposed Improved Genetic Algorithm for Power Economic Dispatch of Units With Valve-Point Effects and Multiple Fuels.
Recently, a heuristic technique called as gravitational search algorithm (GSA) is proposed. Gravitational search algorithm is inspired by law of gravitational and mass interaction. Gravitational search algorithm has been proposed by Rashedi et al. Gravitational search algorithm gives better performance than other optimization techniques. In this paper, Gravitational search algorithm is applied to non linear economic load dispatch problem with equality and inequality in power systems. The results obtained for proposed technique is compared with other optimized techniques
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ECONOMIC LOAD DISPATCH PROBLEM
FORMULATION
The main objective of economic load dispatch is to minimize operating cost of thermal power plant while satisfying the operating constraints and meeting the total demand of a power system. The ED problem is to minimize
the total fuel cost which can be defined mathematically as the sum of the cost function of each generator. The ED problem mathematically formulated with constraints as following
Min Where
turbines are used for generating unit then it exhibits large number variation in incremental fuel cost. The valve opening process produces large number of ripple like effect in heat curve and it looks like sine wave. These valve- point effects are illustrated in Fig. 1. Therefore cost function is modified as following
Min
=cost function of ith generator ($/hr)
, , = cost coefficients of ith generator
of ith generator n = number of generator
Subjected to following equality and equality constraints
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Power Balance Constraint
-
Generator Constraints
Generators
ELD problem with valve point effect
For more accurate and precise modeling of incremental fuel cost function, the above expression of incremental cost function is to be modified suitably. When multivalve steam
Where
, , = cost coefficients of ith generator
, , = fuel cost coefficients of the ith generating unit
reflecting valve-point effects
-
GRAVITATIONAL SEARCH ALGORITHM
Gravitational search algorithm is first introduced by Rashedi et al. in 2009.[1] This optimization algorithm is based on the gravitational law of physics. In the proposed algorithm, agents are considered as objects and their performance is measured by their masses. All these objects attract each other by the gravity force, and this force causes a global movement of all objects towards the objects with heavier masses [1]. Hence, masses cooperate using a direct form of communication, through gravitational force. The heavy masses which correspond to good solutions move more slowly than lighter ones, this guarantees the exploitation step of the algorithm. In GSA, each mass (agent) has four specifications: position, inertial mass, active gravitational mass, and passive gravitational mass. The position of the mass corresponds to a solution of the problem, and its gravitational and inertial masses are determined using a fitness function. In other words, each mass presents a solution, and the algorithm is navigated by properly adjusting the gravitational and inertia masses. By lapse of time, we expect that masses be attracted by the heaviest mass. This mass will present an optimum solution in the search space. GSA algorithm can be summarized by following steps.
Step 1 ) Set up number of masses/agents, N to be processed in GSA and initialize gravitational constant Go.
Step 2) Initialization of the GSA: For each ith mass, the agents are randomly generated in the range (0-1) and located between the maximum and the minimum operating limits of the generators. If there are N generating units, the ith particle is represented as
= ) where
i=1,2,3.N
The d-dimension of the ith particle is allocated a value of as given below to satisfy the constraints.
= + rand ( )
Step 3) Calculate the gravitational constant G(t) for iteration t
G(t)=exp(-)
Where is initial value gravitational constant choose randomly, is a user defined constant, t is current iteration and T is the total number of iteration.
Step 4) Evaluation of Fitness for All Agents in search space. (t) shows the fitness value of the ith agent at time t, and worst(t) and best(t)are defined as follows:- Best(t)=min( (t))
Worst(t)=max((t))
Where best(t) and worst(t) is best and worst fitness value of all agents respectively and
(t)= + ( )
Where, is penalty factor
Step 5) Evaluation of gravitational mass of each agent: In this step mass of each agent is updated. A heavier mass means more efficient agent. This means that better agents have higher attractions and walk more slowly. Therefore, gravitational mass is equal to
(t) =
(t)=
Step5) Evaluation of force between agents: In this step we compute the force acting in d-dimension on ith mass due to mass j at specific time t.
( )
Where, is the active gravitational mass related to jth agents, is the passive gravitational mass of ith agent,
is Euclidian distance between i and j agent
And
is a small constant
Step6) Determine the total force
In this step find out total force of agent i in dimension d
=
Where rand is random number and its value lies between (0, 1) and Vbest is the set of first V agents with the best fitness value and biggest mass.
Step 7 Calculate Acceleration and Velocity
By applying law of motion of physics, of ith agent in d-dimension at iteration t is shown as following:
=
Where is inertial mass of ith agent.
And velocity of ith agent in dimension d is equal to =×+
Where, vary in interval (0, 1) is previous velocity of an agent.
Step 8) Update the position of agent: Position of ith agent in d-dimension at iteration t could be calculated as
=
Step 9) In last step we repeat the 3 to 8 steps until the stop criteria reached.
-
SIMULATION RESULTS
In order to demonstrate the performance of the proposed method, it is tested with 3 system tests with 3,13and 40 unit system are used to test the proposed approach for solving the ELD problem.
Parameters for proposed approach are shown in table 1which contains number of iteration T, gravitational , number of agents N, and user defined constant .
Table 1: Parameters used in GSA for different unit system
Parameters
3 unit system
13unit system
40unit system
10
10
10
N
10
20
50
100
100
100
T
100
1000
2000
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UNIT SYSTEMS DATA FOR 3 GENERATOR SYSTEMS [12]
The system consists of 3 thermal generating units. The total load demand on the system is 850 MW. The parameters of all thermal units are presented in Table 2.The obtained results for the 3-unit system using the GSA method are given in Table 3 and the results are compared with other methods reported in literature
TABLE 2: Cost coefficients and unit operating limits for 3 unit system
Units
Pmin
Pmax
a
b
c
e
f
1
100
600
0.001562
7.92
561
300
0.0315
2
50
200
0.004820
7.97
78
150
0.063
3
100
400
0.001940
7.85`
310
200
0.043
TABLE 3: Simulation Results and Its Comparison with GA
Generator
Generator output of GSA
Generator output of GA
1
414.7959
300.266900
2
133.1194
149.733100
3
302.0847
400.000000
Total demand
850 MW
850
Fuel
Cost($/h)
8197.7
8 237.071729
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UNIT SYSTEMS DATA FOR 13 GENERATOR SYSTEMS [13]
The system consists of 13 thermal generating units. The total load demand on the system is 2520 MW. The parameters of all thermal units are presented in Table 4
Figure 1. Convergence graph for 3-units with PD=850 MW
The obtained results for the 13-unit system using the GSA method are given in Table 5 and the results are compared with other methods reported in literature.
TABLE 4: Cost coefficients and unit operating limits for 13 unit system
un its
Pmin
Pmax
a
b
c
d
e
1
0
680
550
8.1
0.00028
300
0.035
2
0
360
309
8.1
0.00056
200
0.042
3
0
360
360
8.1
0.00056
200
0.042
4
60
200
307
7.74
0.00324
150
0.063
5
60
200
240
7.74
0.00324
150
0.063
6
60
200
240
7.74
0.00324
150
0.063
7
60
200
240
7.74
0.00324
150
0.063
8
60
200
240
7.74
0.00324
150
0.063
9
60
200
240
7.74
0.00324
150
0.063
10
40
120
126
8.6
0.00284
100
0.084
11
40
120
126
8.6
0.00284
100
0.084
12
55
120
126
8.6
0.00284
100
0.084
13
55
120
126
8.6
0.00284
100
0.084
TABLE 5: Simulation Results and Its Comparison with GA and TSA
Generator
Generator output of GSA
Generator output of GA
Generator output of TSA
1
652.5274
628.32
628.317
2
305.8146
356.80
299.206
3
360.0000
359.45
331.991
4
139.0306
159.73
159.733
5
123.8629
109.86
159.711
6
146.9523
159.73
159.744
7
154.6544
159.73
159.739
tr>
8
103.6574
159.73
159.742
9
159.8448
159.73
159.700
10
106.6390
76.92
40.009
11
95.5759
75
77.720
12
83.8732
60
92.378
13
103.5621
55
92.335
TOTAL
2520MW
2520
2520
FUEL
COST($/h)
24249
24400
24314.755
Figure2. Convergence graph for 13-units with PD=2520 MW
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UNIT SYSTEMS DATA FOR 40 GENERATOR SYSTEMS [11]
Generator
Generator output of GSA
Generator output of
PSO
1
110.2604
113.116
2
105.8822
113.010
3
96.5985
119.702
4
161.3755
81.647
5
76.0761
95.062
6
118.3619
139.209
7
277.7329
299.127
8
282.9290
287.491
9
255.8505
292.316
10
198.4792
279.273
11
194.7330
169.766
12
261.4072
94.344
13
302.8148
214.871
14
363.7843
304.790
15
325.7610
304.563
16
382.4561
304.302
17
470.1274
489.173
18
451.5342
491.336
19
478.0455
510.880
20
500.7619
511.474
21
529.9021
524.814
22
515.3287
524.775
23
529.2006
525.563
24
518.1049
522.712
25
489.4889
503.211
26
513.8339
524.199
27
10.6119
10.082
28
10.2303
10.663
29
12.8966
10.418
30
92.6348
94.244
31
187.9979
189.377
32
176.9925
189.796
33
184.4834
189.813
34
146.4241
199.797
35
172.6954
199.284
36
183.6914
198.165
37
101.0808
109.291
38
104.7847
109.087
39
90.2306
109.909
40
514.4148
512.348
Total demand
10500 MW
10500
Total cost
121940.0
122624.35
The system consists of 40 thermal generating units. The total load demand on the system is 10500 MW. The parameters of all thermal units are presented in Table 6.The obtained results for the 40-unit system using the GSA method are given in Table 7 and the results are compared with other methods reported in literature
TABLE 6: Cost coefficients and unit operating limits for 40 unit system
TABLE 7: Simulation Results and Its Comparison with PSO
Unit
Pmin
Pmax
a
b
c
d
e
1
36
114
94.705
6.73
0.00690
100
0.084
2
36
114
94.705
6.73
0.00690
100
0.084
3
60
120
309.540
7.07
0.02028
100
0.084
4
80
190
369.030
8.18
0.00942
150
0.063
5
47
97
148.890
5.35
0.01140
120
0.077
6
68
140
222.330
8.05
0.01142
100
0.084
7
110
300
278.710
8.03
0.00357
200
0.042
8
135
300
391.980
6.99
0.00492
200
0.042
9
135
300
255.760
6.60
0.00573
200
0.042
10
130
300
72.820
12.90
0.00605
200
0.042
11
94
375
635.200
12.90
0.00515
200
0.042
12
94
375
654.690
12.80
0.00569
200
0.042
13
125
500
913.400
12.50
0.00421
300
0.035
14
125
500
1760.4
8.84
0.00752
300
0.035
15
125
500
1728.3
9.15
0.00708
300
0.035
16
125
500
1728.3
9.15
0.00708
300
0.035
17
220
500
647.85
7.97
0.00313
300
0.035
18
220
500
649.690
7.95
0.00313
300
0.035
19
242
550
647.830
7.97
0.00313
300
0.035
20
242
550
647.810
7.97
0.0313
300
0.035
21
254
550
785.960
6.63
0.00298
300
0.035
22
254
550
785.960
6.63
0.00298
300
0.035
23
254
550
794.530
6.66
0.00284
300
0.035
24
254
550
794.530
6.66
0.00284
300
0.035
25
254
550
801.32
7.10
0.00277
300
0.035
26
254
550
801.32
7.10
0.00277
300
0.035
27
10
150
1055.1
3.33
0.52124
120
0.077
28
10
150
1055.1
3.33
0.52124
120
0.077
29
10
150
1055.1
3.33
0.52124
120
0.077
30
47
97
148.89
5.35
0.0114
120
0.077
31
60
190
222.92
6.43
0.00160
150
0.0063
32
60
190
222.92
6.43
0.0016
150
0.0063
33
60
190
222.92
6.43
0.0016
150
0.0063
34
90
200
107.87
8.95
0.0001
200
0.042
35
90
200
116.58
8.62
0.0001
200
0.042
36
90
200
116.58
8.62
0.0001
200
0.042
37
25
110
307.45
5.88
0.0161
80
0.098
38
25
110
307.45
5.88
0.0161
80
0.098
39
25
110
307.45
5.88
0.0161
80
0.098
40
242
550
647.830
7.97
0.0031
300
0.035
Figure 3. Convergence graph for 40-units with PD=10500MW
-
-
CONCLUSION
In this paper, a novel approach based on the Newtons laws of gravity and mass interaction GSA has been presented and applied to economic power dispatch optimization problem with valve point effect. Here this technique is applied to 3, 13, 40 unit system and effectiveness of GSA was tested. From the simulation results, it can be seen that GSA has better convergence rate and also less number of iteration when it is compared to other methods.
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