 Open Access
 Total Downloads : 1252
 Authors : V.Mahidhar, G.Sreenivasulu Reddy
 Paper ID : IJERTV1IS7519
 Volume & Issue : Volume 01, Issue 07 (September 2012)
 Published (First Online): 25092012
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Economic load dispatch with valvepoint effects and Ramp rates using New Approach in PSO
Economic load dispatch with valvepoint effects and Ramp rates using New Approach in PSO
V.Mahidhar1 G.Sreenivasulu reddy2
AbstractThis paper proposes a newly improved method of particle swarm optimization for solving economic dispatch problem with valve point effects and with ramp rates included .The new method of pso uses time varying acceleration coefficients .The proposed method is tested on a 13 unit system and 19 unit Indian systems. The result is compared with the traditional methods. This gave a good result for the observed systems. (Abstract)
Index termseconomic dispatch,valve point effects,Particle swarm optimization,time varying acceleration coefficient,ramp rates
NOMENCLATURE
a,b,c cost coeffiecients for quadratic cost function of unit i e,f cost coefficients of generator i reflecting valve point
B,B0,B00 loss coefficents of transmission lines Pd Demand of the system
Pl transmission losses
Iwt inertia weight
Pmin,Pmax generator limits of operation
The economic load dispatch problem has been extensively studied due to its importance in power system operation .This is a real time problem for properly allocating the real power
advantages and disadvantages in each and every method. In this paper a newly improved particle swarm optimization based on the time varying acceleration coefficients have been proposed. With the new method, the search ability of the PSO has been considerably enhanced in comparison with the previous methods like MSL. The proposed method has been tested with 13 unit system and the 19 unit Indian system and the results have been checked with the previous methods.
The remaining organization of the paper is as follows, section II addresses the formulation of the economic dispatch problem with valve point effects .The proposed method has been described in the section III .Numerical results are followed in section IV. Finally, the conclusion is given
II Problem formulation
The ED problem with the valve point effects is a nonconvex and non smooth problem with multiple minima due to taking into consideration of the ripples in the heat curve of the boilers. The model of the valve point effects have been proposed early by introducing sinusoidal function added to the quadratic fuel cost function[5] .The objective of the problem is to minimize the total cost of the thermal units while satisfying the power balance and generator limits including the ramp rates.
Mathematically, the problem is formulated as follows:
output among the online generating units so as the cost on the
= a P2 + b P + c + abs(e
sin f P
production of thermal power will get is minimized while
i gi i gi i
i i min gi
satisfying the unit and unit constraints .Many papers have been proposed for the efficient method of finding the least cost
Pgi (1a)
method for the economic load dispatch problem. At the early
Min F = n a P2 + b P
+ c + abs(e
sin f P
ages economic load dispatch was considered a quadratic
i=1
i gi
i gi i
i i min gi
objective function and the mathematical methods were used for it. The simplified model has been used since the mathematical methods require the fuel costs function to be differential. However, the inputoutput characteristics of the thermal generating units are actually more complicated due to the effect of valve point effects which includes a sine function in it. Therefore the practical ED problem is a nonconvex optimization problem subject to the constraints, which cannot be directly solved by the mathematical programming techniques. Hence more advanced techniques have to be incorporated in solving with the problem with multiple minima.
Recently the problem was extremely studied due to the advent
Pgi (1b)
Subject to

Power Balance
The total power generation from the online generating units must satisfy the load demand plus power loss.
=1
= + (2)
Where the power loss PL is calculated based on the power flows coefficients of Bmatrix as follows:
= + n B0i Pgi + B00 (3)
of advanced programming techniques like genetic algorithm, ant colony optimization, Macluarein series based lagrangian technique[13] and in PSO itself there are a variety of improvements which lead to the enhancement of the solution.
There are plenty of methods that have enhanced the result of the operation of economic load dispatch and there are several
=1 =1
i=1

Generaotor Limits
The real power output of unit i should be limited between its upper and lower bounds for safety operation represented by
Velocity of individual i at iteration k,
w inertia weight parameter,
c1, c2 acceleration coefficients,
rand1, rand2 random numbers between 0 and 1
Position of individual i at iteration k,
(4)
Best position of individual i at iteration k,
Where is lower bound and is the upper bound of operation of the generator

Ramp rate limits
In some of the previous approaches of ELD strategy, generators outputs were assumed to be handled instantaneously. Although in the practical case the output are constrained by the ramp up and down limits depending upon the nature of the generators power orientation, this scenario is
Best position of the group until iteration k
In this velocity updating process the acceleration coefficients c1and c2 and the inertia parameters are predefined. The random numbers rand1 and rand2 are uniformly generated in the range of [0,1].In general the inertia weight w is set according to the equation as follows:
= Ã— ( ) (8)
described mathematically as
,
are the inertia weight parameters,
1
1 (5)
K the iteration value,
ITmax the max. no. of iterations.
Where URi and DRi are the up and down ramp limits for generator i, respectively .Merging them with above equation (4), the power limits of the ith generator can be redefined as
This approach is called the inertia weight acceleration approach .Using the above equations the values will shift from pbest to gbest .Each particle is moved from current position to
max , 1
the nest position by the use of the modified velocity and the
,
modified position as shown below
min , , 1 + (6)
A.Particle swarm otimization
Particle swarm optimization (PSO) is a population based stochastic optimization technique developed by Kennedy and Elberhart in 1995[10], discovered through simplified social model simulation. It stimulates the behaviors of bird flocking involving the scenario of a group of birds randomly looking for food in an area. PSO is motivated from this scenario and is developed to solve complex optimization problems.
In the conventional PSO, suppose that the target problem has n dimensions and a population of particles, which encode solutions to the problem, move in the search space in an attempt to uncover better solutions. Each particle has a position vector of Xi and a velocity vector Vi. The position vector Xi and the velocity vector V I of the i th prticle in the
ndimensional search space can be represented as
+1 = + +1 (9)
B.Proposed Particle swarm optimization with Time varying acceleration coefficients
In this paper the proposed method is based on the improvements in PSO if operated with the time varying acceleration coefficients i.e PSOTVAC .In the implementation of the PSOTVAC for the ndimensional optimization problem
,the position and velocity of the particles are represented by
= [1, 2, ] and = 1, 2 . 3 respectively. The best previous particle is based on the evaluation of the fitness function Pbest,and the best among the all the particles is given by Gbest.The velocity and position of each particle in the next iteration for fitness evaluation is given as follows
1
2
+1 = +1 Ã— + Ã— 1 Ã— + Ã—
2 Ã— (10)
Xi = (xi1, xi2… xin) and Vi = (vi1, vi2, …, vin),respectively.
Each particle has a memory of the best position in the search space that it has found so far (Pbesti), and knows the best
location found to date by all the particles in the swarm (Gbest).
+1 = + +1 (11)
Where
= 1 1 Ã— + 1 (12)
1
Let Pbest = ( , , . , ) and
1
2
3
2 = 2 2 Ã—
+ 2 (13)
= 1
, , ,
2
max
be the best position of the individual i and all the individuals so far, respectively. At each step, the velocity of the I th
= ( ) Ã—
(14)
particle will be updated according to the following equation in the PSO algorithm:
+1 = Ã— + 1 Ã— 1 Ã— + 2
The upper and lower bounds for each particle xi are limited by the maximum and minimum limits of variable represented
by the particle ,respectively. The velocity of each particle is
Where,
Ã— 2 Ã— (7)
limited in [vimax,vimax] for i=1, 2,3.n, where the velocity of the particle for each element in the search space is determined by:
5 
0.00324 
7.74 
240 
150 
0.063 
60 
180 
6 
0.00324 
7.74 
240 
150 
0.063 
60 
180 
7 
0.00324 
7.74 
240 
150 
0.063 
60 
180 
8 
0.00324 
7.74 
240 
150 
0.063 
60 
180 
9 
0.00324 
7.74 
240 
150 
0.063 
60 
180 
10 
0.00284 
8.6 
126 
100 
0.084 
40 
120 
11 
0.00284 
8.6 
126 
100 
0.084 
40 
120 
12 
0.00284 
8.6 
126 
100 
0.084 
55 
120 
13 
0.00284 
8.6 
126 
100 
0.084 
55 
120 
= Ã— ( ) (15)
In this paper the value of the R is in the range from 0.01 to 0.05.

step 1. Choose the parameters of the PSOTVAC including the number of particles Np, maximum no. of iterations ITmax, initial value of social and cognitive acceleration factors c1i and c2i and the final value of social and cognitive acceleration factors c1f and c2f .
step 2. Generate particles for all generators defined in the data including the position and velocity .For each particle calculate the fuel cost using (1a) and determine the F Pbest which is the minimum value of the particles and also determine the value of the Gbest which is the min of the total determined Pbest (i.e)Fgbest=min(Fpbest)
step 3. Set Pbest to xi for each and Gbest to the position of the particle corresponding to the FPbest .Set number of iterations to 1.
i
step 4. Calculate the velocity (vik)and position (x k)for each particle (10) and (11) respectively.Note that the obtained position and velocity of the particle should lie in their lower limits and upper limits which are determined after ramp rates are considered .
pbest
0
URi
DRi
420
335
360
280
250
290
280
250
290
120
80
130
150
80
130
130
80
130
160
80
130
140
80
130
140
80
130
100
120
120
80
120
120
80
120
120
75
120
120
step 5. Evaluate the fitness with the updated position of the paticle using (1a).Compare the F(k) to the Fk1 to obtain the best fitness function up to the current
pbest
iteration Fk for each particle .Pick up the position
Pbest k corresponding to Fk for each particle.
i pbest
pbest
Determine the global best fitness function Fk and
corresponding position Gbestk.
step 6. If k<ITmax,k=k+1 and return to step5. Otherwise, stop.

The proposed method has been measured on a 13unit system, data taken from [14]with the load demand of 1800 Mw and an Indian system in the southern power grid that consist of 19 generating stations in the state of tamilnadu where theare are a large number of thermal power plants.The algorithm is coded in matlab platform and run for 100 trials for each test case on a 2.1GHz PC with 1.5GB of RAM. In all cases the parameter are selected as max=0.9,min=o.4,c1i=c2f=2.5,c2i=c1f=0.2.The other parameters for the test are R=1,and Np=200 and 400.
A.13unit system
S.No
Ai
Bi
Ci
Ei
Fi
Pmin
Pmax
1
0.00028
8.1
550
300
0.035
0
680
2
0.00056
8.1
309
200
0.042
0
360
3
0.00056
8.1
307
150
0.042
0
360
4
0.00324
7.74
240
150
0.063
60
180
Solution for the 13unit system for Load demand of 1800Mw The problem is solved for taking into account of 100 trails
are considered and the best and worst cases ar studied for the
100 trials for different population size and the values are found as in the table.
B.19unit Indian system
Data of the 19 unit Indian system is as follows:
Ai
Bi
Ci
Ei
Fi
0.0097
6.8
119
90
0.72
0.0055
4
90
79
0.05
0.0055
4
45
0
0
0.0025
0.85
0
0
0
0
5.28
0.891
0
0
0.008
3.5
110
0
0
0.006
5.439
21
0
0
0.0075
6
88
50
0.52
0.0085
6
55
0
0
0.009
5.2
90
0
0
0.0045
1.6
65
0
0
0.0025
0.85
78
58
0.02
0
2.55
49
0
0
0.0045
1.6
85
0
0
0.0065
4.7
80
92
0.75
0.0045
1.4
90
0
0
0.0025
0.85
10
0
0
0.0045
1.6
25
0
0
0.008
5.5
90
0
0
Population size
Best case
Worst case
400
17989.84$/h
18333.45$/h
200
17994.32$/h
18645.37$/h
Ramp rates of the generators
0
URi
DRi
250
95
150
300
138
180
300
100
200
20
5
12
120
80
90
300
100
150
130
70
100
600
400
500
500
200
300
30
10
15
100
55
85
50
25
25
120
80
90
50
40
45
125
95
105
55
25
40
55
25
40
150
80
100
550
100
150
Solution for the Pd=3750Mw is tested for 100 trials independently and the value of the best and the worst costs are found and are here tabulated below.
Population size
Best case
Worst case
200
26110.33$/h
27639.57$/h
400
26075.20$/h
27216.36$/h
The results of the both systems are tabulated as below for plant wise power output and also the best and worst performance are also given care of.
C.SIMULATION RESULTS

Results for 13 unit System Unit wise at two different
populations
Population of 200
Population of 400
Ramp rate considered
Best
Worst
Best
Worst
Pmin
Pmax
239.6204
291.961
278.8884
265.7837
100
300
434.0161
382.2182
434.4727
371.2344
120
438
225.5227
217.7803
239.768
187.9166
100
250
24.9674
24.83847
24.92365
24.54044
8
25
63.67815
63.72809
63.56116
63.73158
50
63.75
299.5455
280.499
293.6119
290.7806
150
300
63.75
63.11671
63.40492
63.75
50
63.75
438.3529
498.7605
438.3957
486.6361
100
500
447.101
564.4981
461.921
561.6225
200
600
39.96066
38.46988
39.44294
39.47403
15
40
149.9579
109.1706
142.992
149.8819
50
150
74.91513
74.75709
74.97589
75
25
75
63.74556
63.36901
63.75
63.60993
50
63.75
89.96228
89.97188
89.98735
89.83108
5
90
219.9863
154.043
212.6942
149.8625
20
220
79.95427
79.967
79.36067
79.91894
15
80
80
79.9947
79.9828
80
15
80
229.9124
206.9454
230
228.1075
50
230
485.0501
465.9277
437.8501
478.371
400
500
Best cost=26110.33$/h
Best cost=26075.20$/h
TableI
Population of 400
Population of 200
Ramp rate considered
Best
Worst
Best
Worst
Pmin
Pmax
419.045
419.0423
419.064
329.3065
60 680
234.4629
84.80964
234.4393
84.95515
10
360
160.0968
159.5928
159.6177
360
10
360
159.7404
159.7355
109.934
87.24056
60
180
109.8664
109.8687
109.9668
60.06077
60
180
109.8649
131.8278
109.8705
118.8122
60
180
109.8792
109.8732
159.7877
159.5289
60
180
159.7388
159.7366
159.7507
159.6766
60
180
109.8986
109.867
109.8772
60.28229
60
180
77.39096
93.44354
40.0948
90.2643
40
120
40.01582
114.8019
40.08994
77.36856
40
120
55.00009
55.00204
92.47863
92.51869
55
120
55
92.39894
55.027
119.9939
55
120
Best cost=17989.84$/h
Best Cost=17994.32$/h
From the above table it can be cleared that at high population levels we will get good results.
C.Results of other methods
Method
Cost obtained for 13unit system
MSL[4]
18158$/h
CPSO
18006$/h
B.Results for 19 unit System Unit wise at two different
populations
TableI
From the table below on the right side it is cleared that at .
high population we will have good results.

In this paper, the PSOTVAC method has been very efficiently implemented for a 13 unit system for solving economic load dispatch with valve point load effects and ramp rates included. With the improvements the search ability a solution quality has been considered improved in comparisons with other methods. The results comparisons from the test cases have shown the efficiency of the system in finding the solution for the nonconvex problem. Therefore the proposed method could be favorable for solving the complicated ED problems with nonconvex objective function.
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V.MahidharM.tech student of electrical power Engineering in Narayana Engineering College, received his B.E in Electrical and electronics from Arunai engineering college, Anna university
G.Sreenivasulu ReddyPresently working as a associate professor in Narayana Engineering college ,Nellore.Presently pursuing his PhD .