 Open Access
 Total Downloads : 508
 Authors : S. Khandualo, A. K Barisal, P. K Pradhan, P. K. Patro
 Paper ID : IJERTV3IS10124
 Volume & Issue : Volume 03, Issue 01 (January 2014)
 Published (First Online): 08012014
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
A Gravitational Search Algorithm for Solving Economic Load Dispatch Problem
S. Khandualo 1 , A. K Barisal,
Asst Manager (El) CHEP, Elect Engg Dept., Chiplima, OHPC Ltd, VSSUT,Burla, DistSambalpur,Odisha,India Sambalpur,Odisha

K Pradhan, P. K. Patro
Mechanical Engg Dept, Manager (El) , VSSUT,Burla, CHEP,Chiplima,OHPCLtd DistSambalpur,Odisha,India, Distsambalpur,Odisha,
AbstractThis paper presents the application of a new optimization algorithm i.e Gravitational Search Algorithm (GSA) based on the law of gravity and mass interactions for solving the Economic load dispatch problem of a power system. The Economic load dispatch is the dispatch of available electricity generation resources to supply the load and losses in the transmission links in such a manner that total cost of production of thermal generation is minimized satisfying the constraints in the system. The GSA technique is applied to a six generator twentysix bus test system and a twenty generator test system to illustrate the effectiveness of the proposed algorithm. Numerical results show that the proposed algorithm is capable of finding very nearly global solutions and achieves cheaper generation schedule in comparison to the other published methods.
Keywords Economic Dispatch, Gravitational Search Algorithm, Prohibited Operating Zones

INTRODUCTION (Heading 1)
Economic load dispatch has become a vital task for proper operation and planning of power system. The Main objective of Economic Load Dispatch problem is to minimize the total
from [12] and Test systemII adopted from [4] and the simulation results are compared to that of Bio geography based optimization [11], Hopfield Modeling framework [4], and Lambda Iteration Method [ 4], Intelligent Water Drop Algorithms[12], etc The results show the superiority of the proposed algorithm in solving the complex optimization problem in terms of minimization of cost, minimization of loss and computational time.

ECONOMIC LOAD DISPATCH PROBLEM FORMULATION
The main objective of Economic load dispatch problem is to minimize the total generation cost by economic loading of generators such that the operational and network constraints are satisfied.
Objective function is to minimize
m
FTotal Fi Pi …………………………………………………..(1)
i1
system operating cost represented by the fuel cost required for the system thermal generation while satisfying all units and system operational constraints i.e equality and inequality
constraints such as load balance constraint, ramp rate limits,
The cost function of ith unit Fi Pi
polynomial and is expressed as
is a quadratic
multi fuel options, prohibited operation zones etc , . Conventional methods based on Lagrangian multiplier, gradient search techniques [1], Dynamic Programming, [1] require models of thermal plants to be represented as piecewise linear or polynomial approximations of monotonically increasing nature. But such an approximation may lead to suboptimal solution resulting in huge loss of revenue over the time. Methods based on Dynamic Programming, Lamda Iteration methods, Gradient Search methods [1] to solve the Economic Load Dispatch problems was found that it provides solution but it will fail to obtaining solution feasibility and become more complex. Stochastic search algorithms like Tabu Search[2], Genetic Algorithm [3,8,12], Evolutionary Programming [5], Particle Swarm Optimization [7] have been proved to be very exciting in solving complex power systems problems, but these heuristic methods do not always guarantee the globally optimal solution. In this paper a new population based search algorithm called Gravitational search algorithm [6,10] is applied to two different test systems ,Test SystemI adopted
F P a b P c P2 ……………………………….(2)
i i i i i i i
where ai , bi , and ci are fuel cost coefficients of ith unit and m is the total number of committed units.
The solution to the problem must satisfy the operational and network constraints in the system.
The constraints are given below:
A. Power Balance Constraint
m
The total generation Pi
i1
should be equal to the total
D. Prohibited Operating Zones
Prohibited operating zone means the unit is prohibited from generation due to some technical fault in the machine such as vibration in the shaft bearing, or steam valve operation etc.
system demand PD and total transmission loss Ploss ., i.e
m
Pi PD Ploss ………………………………… (3)
i1
The transmission loss is represented as
With prohibited zones, the unit has a fuel cost curve of discontinuous in nature [12].
The additional constraints for Units with prohibited operating zones are
Pmin
P Pl …………………………………………………..(7
m m m
i i i,1
)
PL Pi Bij Pj Pi Bio Boo …… . (4)
i1 j
i1
Pu P Pl
, j 2,3,…n
Bij : The Transmission loss coefficient.
i, j1 i
i, j
i …………………………….(8)
Pl P Pmax …………………………………………………(9)
The template is used to format your paper and style the text. All margins, column widths, line spaces, and text fonts are
i,ni
i i,
i, j
prescribed; please do not alter them. You may note
Where j is the number of prohibited operating zones of
peculiarities. For example, the head margin in this template measures proportionately more than is customary. This
unit i , Pl
is the lower limit of jth prohibited unit. And
measurement and others are deliberate, using specifications
u
P
i, j 1
is the upper limit of j 1th prohibited operating zone
that anticipate your paper as one part of the entire proceedings, and not as an independent document. Please do not revise any of the current designations.
B. Generator perating limits
The output generation of each unit must be within minimum and maximum limit of generation. The optimized result must satisfy the following inequality constraint i.e
of ith unit. ni is the total number of prohibited operating zone of unit i .

GRAVITATIONAL SEARCH ALGORITHM
A. Abbreviations and Acronyms
GSA is one of the recent additions to heuristic algorithms was developed by Rashedi et al. in 2009 [6]. GSA is followed by the physical law of gravity and the law of motion. In the
Pi min
Pi Pi max ……………………………………………………..(5)
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
Pi is the power output of ith unit,
Pi min and
Pi max are the
global movement of all objects towards the objects with
minimum and maximum real power output of ith generating unit.
The cost function Fi Pi becomes discontinuous when following factors are considered
C. Valve point loadings
heavier masses. 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.
Consider a system f m masses, in which the position of the ith mass by
P P ,…Pd …, Pn ,i 1,2,…m … .. . (10)
i i i i
Large steam turbine generators are having a number of steam admission valves that are opened in sequence to obtain ever
Where
Pid
represents the position of ith mass in the dth
increasing output of the unit. When the valve is first opened, the throttling losses increases rapidly and the incremental heat rate rises suddenly. This causes a ripple in the heat rate curves, the cost
function is no longer a quadratic function but a combination of
dimension. At a specific time t, a gravitational force on mass
M pi t M aj t
i from mass j is given by
F d t Gt Pd t Pd t … (11)
sinusoidal function and quadratic function i.e represented as ij
/ Rij t j i
F P a
b P c P2
. i i
i i i
i i ……………………………… .(6)
Where
M pi
is the passive gravitational mass related to agent
ei sin fi Pi min Pi
i , M aj is the active gravitational mass related to agent
j . Gt is the Gravitational constant at time t, is a small constant and Rij t is Euclidian distance between two agents i and j .
compared with other published work, result reveals the superiority of the proposed algorithm.

Test SystemI
Rij t  Pi t, Pj t2
(12)
The system contains six thermal units, 26buses, and 46
The total force acting on the agent i in the dimension d is calculated as follows
transmission lines [12]. The load demand is 1263MW. The load balance constraint, generation limit constraint and
m
d
d
prohibited operating zone constraints are considered in the
Fi t
rand j Fij t
j1, ji
(13)
system. The input data for test system1 are furnished in TableI, TableII and the BCoefficients taken from [12].
Where rand j is a random number in the interval [0,1]
The acceleration of the agent i at time t , and in direction dth ,
i i ii
ad t F d t/ M t, . (14)

Test SystemII:
The Test SystemII consists of Twenty Thermal units and supplies a total load demand of P =2500MW, B coefficient
M ii is the inertia mass of ith agent.
Velocity of a particle is a function of its current velocity added to its current acceleration. Therefore the next position and next velocity of an agent can be calculated as
D
matrix is adopted from [4].
Uniti
ai
($/h)
bi
($/h)
ci
($/h)
Pimin
(MW)
Pimax
(MW)
1
240
7
0.007
100
500
2
200
10
0.0095
50
200
3
220
8.5
0.009
80
300
4
200
11
0.009
50
150
5
220
10.5
0.008
50
200
6
120
12
0.0075
50
120
TABLE I. INPUT DATA OF TEST SYSTEMI
vd t 1 rand Pvd t ad t
… (15)
i i i i
i
Pid t 1 Pid t vd t 1
.. (16)
The Gravitational constant G, is initialized at the beginning and will be decreased with time to control the search accuracy.
i.e G is a function of the initial value G0 and time t .
Gt G0eiteration/ max imumiteration , is a constant, iteration is the current iteration and maximum iteration is the maximum number of iterations given.
The masses of the agents are calculated using fitness evaluation. A heavier mass means a more efficient agent. This means that better agents have higher attractions and moves
Units
ai ($/h)
bi ($/h)
ci ($/h)
Pimin (MW)
Pimax (MW)
1
1000
18.19
0.00068
150
600
2
970
19.26
0.00071
50
200
3
600
19.8
0.0065
50
200
4
700
19.1
0.005
50
200
5
420
18.1
0.00738
50
160
6
360
19.26
0.00612
20
100
7
490
17.14
0.0079
25
125
8
660
18.92
0.00813
50
150
9
765
18.27
0.00522
50
200
10
770
18.92
0.00573
30
150
11
800
16.69
0.0048
100
300
12
970
16.76
0.0031
150
500
13
900
17.36
0.0085
40
160
14
700
18.7
0.00511
20
130
15
450
18.7
0.00398
25
185
16
370
14.26
0.0712
20
80
17
480
19.14
0.0089
30
85
18
680
18.92
0.00713
30
120
19
700
18.47
0.00622
40
120
20
850
19.79
0.00773
30
100
more slowly.
TABLE II. INPUT DATA OF TEST SYSTEMII
fit i t is the fitness value of agent i at time t , bestt and
worstt represents the strongest and weakness agent according to their fitness value.
For a minimization problem, bestt min j 1,…m fit j t
and worstt max fit t
(17)
j 1,…m j
The gravitational and inertial masses are updating by the following equations
mi t fiti t worstt/ bestt worstt
(18)
m
M i t
mi t / m j t
j1


RESULTS AND DISCUSSION
(19)
The proposed gravitational search algorithm was tested on two test systems having six units with 26 bus test system and twenty unit test systems. The GSA was programmed in MATLAB environment and executed on a 2.30GHz Pentium III processor with 4GB RAM. The simulation results were
TABLE III. COMPARATIVE RESULTS OF TEST SYSTEMI
Units (MW)
GA
PSO [13 ]
BBO [11]
IWD [12 ]
Proposed GSA
1
474.80
447.497
447.399
450.13
462.6177
2
178.63
173.322
173.239
173.62
184.9532
3
262.20
263.474
263.316
260.61
256.6655
4
134.28
139.059
138.000
139.49
133.0794
5
151.90
165.476
165.410
159.7
152.3787
6
74.181
87.128
87.0797
90.51
85.3885
Total Gen
1276.0
1276.01
1275.44
1274.05
1275.083
Ploss
13.021
12.9584
12.446
12.05
12.083
Total Gen Cost
( $/h)
15459
15450
15446.0
15439
15373.83
CPU time per iteration
(s)
0.22
0.06
0.0638
0.0254
0.00807
Fig.1.Convergence characteristics of GSA on Test SystemI
Fig.2: Convergence characteristics of GSA on Test SystemII
TABLE IV. OPTIMAL SOLUTION OF TEST SYSTEMII
Unit Generation (MW)
Lambda iteration method
Hopfield Model [4]
BBO [11]
Proposed GSA
1
512.780
512.7804
513.089
570.7216
2
169.103
169.1035
173.353
174.6018
3
126.889
126.8897
126.923
98.8116
4
102.865
102.8656
103.329
87.7545
5
113.683
113.6836
113.774
123.2827
6
73.571
73.5709
73.0669
71.0592
7
115.287
115.2876
114.984
99.7516
8
116.399
116.3994
116.423
98
9
100.406
100.4063
100.694
122.1304
10
106.026
106.0267
99.9997
106.9398
11
150.239
150.2395
148.977
154.3317
12
292.764
292.7647
294.020
298.2807
13
119.115
119.1155
119.575
120.3959
14
30.834
30.8342
30.5478
29.8224
15
115.805
115.8056
116.454
115.6996
16
36.2545
36.2545
36.2278
37
17
66.859
66.859
66.8594
61.7785
18
87.972
87.972
88.5470
85.0294
19
100.803
100.8033
100.980
84.2168
20
54.305
54.305
54.2725
51.4542
Total Gen
2591.96
2591.967
2592.10
2591.062
Ploss
91.967
91.9669
92.101
91.0624
Total Gen Cost( $/h)
62456.6
62456.63
62456.7
62314.94
CPU time in Sec
33.757
6.355
0.02928
2.525
The convergence characteristics of the proposed method on Test systemI and Test systemII are provided in Figs. 1 and 2, respectively. The results obtained by the proposed GSA method on Test System1 are compared with genetic algorithm (GA), particle swarm optimization (PSO), biogeography based optimization (BBO) and intelligent water drop algorithm (IWD) methods in Table III. The GSA method provides cheaper generation schedule in comparison to all other above methods in less execution time. Similarly in Table IV, the proposed GSA outperforms to other established methods in terms of quality of solution and execution time. Moreover, GSA method performs better to other methods reported in literature in both of the test systems considered for study.

CONCLUSION

A new algorithm called Gravitational Search Algorithm was developed and demonstrated on two test systems to solve the economic load dispatch of two different test systems. Results show that GSA based algorithms are more capable of finding highly nearglobal solutions than lamda iteration method, Hopfield model, BBO etc. The optimal cost obtained by the GSA is quite cheaper than the other published work for the system adopted. In future, attempts can be made to apply the hybrid gravitational search algorithm to large thermal system in conjunction with hydro, wind energy by incorporating emission, spinning reserve and reliability constraints.
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