Fuzzified pso for multiobjective economic load dispatch problem

IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
__________________________________________________________________________________________
Volume: 02 Issue: 08 | Aug-2013, Available @ http://www.ijret.org 157
FUZZIFIED PSO FOR MULTIOBJECTIVE ECONOMIC LOAD
DISPATCH PROBLEM
N.Ramyasri1
, G. Srinivasulu Reddy2
1
Assistant Professor, 2
Associate Professor, Dept. of EEE, Narayana Engineering College, Nellore, AP – 524004
n.ramyasri10@gmail.com, gsmeghana@gmail.com
Abstract
Power system engineers are always striving hard to run the system with effective utilization of real and reactive powers generated by
the generating plants. Reactive power is used to provide better voltage profile as well as to reduce system losses. Membership
functions are written for fuel cost, losses, stability index and emission release. As minimization of real power loss over the
transmission lines, an attempt is made in this paper to optimize each objective individually using Fuzzy logic approach. In this paper
basic assumption is Decision Maker (DM) has imprecise or fuzzy goals of satisfying each of the objectives, the multi objective
problem is thus formulated as a fuzzy satisfaction maximization problem which is basically a min-max problem. The multi objective
problem is handled using the fuzzy decision satisfaction maximization technique which is an efficient technique to obtain trade off
solution in multi objective problems. The developed algorithm for Optimization of each objective is tested on IEEE 30 bus system.
Simulation results of IEEE 30 bus network are presented to show the effectiveness of the proposed method.
Keywords: Real power, Reactive power, losses, membership functions, fuzzy logic and trade off solution
----------------------------------------------------------------------***-----------------------------------------------------------------------
1 INTRODUCTION
The real power optimization sub-problem minimizes fuel cost
by controlling controllable generator outputs while keeping
the PV bus voltages unchanged. The system losses, stability
index and emission computed at this power dispatch are very
high compared with the results obtained when respective ones
are taken as objective. Similarly the reactive power sub-
problem deals with minimization of total transmission loss of
the system by controlling all the reactive power sources such
as taps, shunts etc. Thus when loss minimization is taken as
objective total system losses reduces but cost, emission and
stability indices are high. The emission dispatch sub problem
minimizes total emission output from the fossil fuel plants by
controlling the generator outputs. At this power output of
generator, the cost, total system losses and stability index are
high. Similarly Stability index sub problem minimizes the
index by controlling the PV bus voltages and thus improves
the system stability limit. But the cost, emission and system
losses are very high. Thus results of all the four sub problems
are conflicting with one other. This can be inferred from
previous chapters. In order to meet all the four objectives, we
need a compromised solution which minimizes fuel cost,
emission release, total transmission and losses and improved
stability limit. This trade-off solution is obtained using a fuzzy
decision satisfaction maximization method.
In this paper the data of four problems are fuzzified using
fuzzy Min-Max approach and then Particle swarm
optimization is used to determine the final trade off solution
from all these Fuzzified values. This method is tested on IEEE
30 bus system and the results are presented.
2. PROBLEM FORMULATION
Each particle consists of power generations of all units
excluding slack bus voltages, taps and shunts encoded in it.
The size of each particle is equal to sum of active power
generations, no of voltages excluding slack bus, number of
voltage, taps, and shunts.
Assuming the decision maker (DM) has imprecise or fuzzy
goals of satisfying each of the objectives, the multi objective
problem can be formulated as a fuzzy satisfaction
maximization problem which is basically a min-max problem.
Our task over here is to determine the compromise solution for
all the four optimization sub problems. Our goal is to
minimize G(X) = compromised solution of {G1(X1), G2 (X2),
G3(X3), G4(X4)}
While satisfying the set of constraints AX <B.
Where G1(X) is Fuel cost minimization sub problem.
G2(X) is Loss minimization sub problem.
G3(X) is emission minimization sub problem.
G4(X) is index minimization sub problem.
Let F1(Xi) be the fuel cost in $/hr for ith
control vector.
F2(Xi) be the losses in P.U for ith
control vector
F3(Xi) be the Stability index for ith
control vector
F4(Xi) be the Emission release in kg/hr for ith
control vector

__________________________________________________________________________________________
Let the individual optimal control vectors for the sub problems
be X1*, X2*, X3*, X4* respectively. We have to find out a
global optimal control vector X *such that
* * * * *
1 2 3 4
* * * * *
2 1 3 4
* * * * *
3 1 2 4
* * * * *
4 1 2 3
1 1 1X X (X , X , X )
2 2 3X X (X , X , X )
3 3 3X X (X , X , X )
4 4 4X X (X , X , X )
F F F
F F F
F F F
F F F
 
 
 
 
(1)
The imprecise or fuzzy goal of the DM for each of the
objective functions is quantified by defining their
corresponding membership functions µi as a strictly
monotonically decreasing function with respect to the
objective function f where i=1 to 4. In case of a minimization
problem,
µi=0 or tends to zero, if fi > fi
max
and
µi = 1 or tends to 1,
if fi < fi
min
(2)
Where fi
max
and fi
min
are the unacceptable and desirable level
for respectively. In our proposed approach we have considered
a simple linear membership function for fi because none of the
objectives have very strict limits. The membership function, µi
for ith
objective is depicted in Fig. 1.1
Figure 1.1 Membership function for ith
objective
The membership function can be defined as
i imax
max
i i
i imin i imaxmax min
i i
i imin
0 F > F
F -F
μ = F F F
F -F
1 F < F
 
(3)
Using Eq. (3) the membership functions can be formulated as
2.1 Membership Function for Fuel Cost
1 1max
max
1 1
1 1min 1 1maxmax min
1 1
1 1min
0 F >F
F -F (X)
μ = F F F
F -F
1 F <F
 
(4)
Where F1max=max {F1(X1
*
), F1(X2
*
), F1(X3
*
), F1(X4
*
)}
F1min=min {F1(X1
*
), F1(X2
*
), F1(X3
*
), F1(X4
*
)}
2.2 Membership Function for Losses
2 2max
max
2 2
2 2
2 2min
0 F >F
F -F (X)
μ = F F F
F -F
1 F <F
 
(5)
Where F2max = max {F2(X1
*
), F2(X2
*
), F2(X3
*
), F2(X4
*
)}
F2min = min {F2(X1
*
), F2(X2
*
), F2(X3
*
), F2(X4
*
)}
2.3 Membership Function for Stability Index
3 3max
max
3 3
3 3
3 3min
0 F >F
F -F (X)
μ = F F F
F -F
1 F <F
 
(6)
*
), F3(X2
*
), F3(X3
*
), F3(X4
*
)}
F3min = min {F3(X1
*
), F3(X2
*
), F3(X3
*
), F3(X4
*
)}

__________________________________________________________________________________________
2.4 Membership Function for Emission Release
4 4max
max
4 4
4 4
4 4min
0 F >F
F -F (X)
μ = F F F
F -F
1 F <F
 
(7)
*
), F4(X2
*
), F4(X3
*
), F4(X4
*
)}
F4min = min {F4(X1
*
), F4(X2
*
), F4(X3
*
), F4(X4
*
)}
The maximum degree of overall satisfaction can be achieved
by maximizing a scalar λ, which is the intersection of the four
fuzzy membership functions.
Therefore objective function is maximization of λ = max (µ1,
µ2, µ3, µ4).
Where λ varies from 0 to 1.
3. PROPOSED ALGORITHM
The proposed solution strategy for the multi objective problem
is shown in the following algorithm
1. Reading the system data.
2. Reading the values of fixed cost, loss, index, emission for
each sub-problem.
3. Forming Ybus matrix and FLG matrix for Lindex
calculation.
4. Forming B1 sub matrix. Decompose B1 by Cholesky
decomposition.
5. Randomly population and velocities are initialized.
6. Set Pbest=0 and itercount=1.
7. Set particle count=1
8. Decoding the particle, then the Decoded particle gives the
values of power generations, voltage, magnitudes, tap
values and shunts.
9. Forming the Ybus and B2 sub matrix. Decompose B2 by
Cholesky decomposition
10. Run FDC load flow and compute loss.
11. Compute emission cost, fixed cost, and index values.
12. Fuzzifying the fuel cost, loss, emission and index
obtained in step (11) using equations from Eq. 4 to 7.
13. Calculating the evaluation value of each individual in the
population using Eq.(8). Compare each individual’s
evaluation value with its Pbest . If the evaluation value of
each individual is better than the previous Pbest, the
current value is set to be Pbest.
14. Incrementing the individual count by 1. If count <
population size go to step (8).
15. The best evaluation value among the Pbests is denoted as
gbest.
16. Modifying the member velocity V of each individual
according to
vi
k+1
=k*( w* vi
k
+ c1*rand1*(pbesti - xi) +
c2*rand2*(gbesti - xi))
xi
k+1
= xi + vi
k+1
17. Modifying the member position of each individual Pi
according to
Pi(k+1)
=Pi(k)
+Vi(k+1)
Pi(k+1)
must satisfy the constraints.
18. Incrementing iteration count by 1.If the number of
iterations reaches the maximum,
then go to Step 19, Otherwise, go to Step 7
19. The individual that generates the latest gbest, is the
required control vector for the final trade off solution.
Print the results
4. CASE STUDIES AND RESULTS
4. 1 IEEE 30 Bus System
25 independent runs are made for each sub problem and the
values of four factors considered at minimum value of each
sub problem (data) over 25 independent runs are determined.
These values for all sub problems are given in Table 1.1.
Table 1.1 results of various sub problems and final trade off solution for IEEE 30 bus system
Optimization Problem
Fuel Cost
($/hr)
Losses (
MW )
Stability
Index
Emission (
kg/hr)
Fuel cost minimization sub
problem results 806.498033 10.583711 0.272369 381.671279
Losses minimization sub problem
results
945.214704 4.328367 .272443 233.701959
Stability Index minimization sub
problem 897.142571 33.557655 0.162446 375.611008
Emission minimization sub problem 932.094511 4.404039 0.267070 229.144834
Final trade off solution
926.200519 4.634725 0.266459 242.142237

__________________________________________________________________________________________
System generation =288.034725, total load 283.4 MW. The
control variables are given in table 1.2 and 1.3.and Final bus
voltages, power generation, line index values are given in
table1.4.
Table 1.2 Positions of tap changing transformers
S.No From-To buses Tap value
1 6-9 0.9
2 6-10 1.1
3 4-12 0.975
4 28-27 0.9875
Table 1.3 Shunt suceptance values
S.No
From-To
buses
Shunt suceptance
1 10 1.009617
2 24 1.0
Figure 1.2 Gbest particles (λ) Vs Iterations
Table 1.4 Final bus voltages, Power generations, Lindex
S.No Voltage Pgen Qgen Lindex
1. 1.000000 0.712626 -0.231329 0.000000
2. 1.003158 0.729346 -0.027326 0.000000
3. 0.991421 0.500045 0.130286 0.000000
4. 0.995914 0.349738 0.633051 0.000000
5. 1.013388 0.222398 0.008802 0.000000
6. 1.000000 0.365994 0.674902 0.000000
7. 1.004216 -0.000000 -0.000000 0.001359
8. 0.998299 -0.000001 -0.000000 0.004968
9. 1.065043 0.000114 -0.000000 0.091909
10. 1.042844 0.000594 -0.000001 0.111362
11. 1.065043 0.000000 0.000000 0.091909
12. 1.055699 -0.000930 -0.000005 0.114096
13. 1.045549 -0.000019 -0.000007 0.122494
14. 1.040786 -0.000015 0.000001 0.120544
0 10 20 30 40 50 60 70 80 90 100
0.8
0.82
0.84
0.86
0.88
0.9
0.92
0.94
0.96
0.98
1
Gbest particle Vs iterations
Iterations
Gbest

__________________________________________________________________________________________
15. 1.036565 -0.000020 0.000002 0.118529
16. 1.027964 0.000961 0.000004 0.110391
17. 1.033049 0.000009 0.000002 0.113342
18. 1.025878 -0.000003 0.000001 0.126912
19. 1.023364 -0.000002 -0.000000 0.128017
20. 1.027458 0.000004 -0.000001 0.124354
21. 1.027112 0.000018 -0.000005 0.115108
22. 1.026608 -0.000500 -0.000001 0.114624
23. 1.026413 -0.000004 -0.000000 0.118209
24. 1.021240 0.000003 -0.000002 0.114837
25. 1.043109 -0.000003 0.000000 0.099710
26. 1.025885 -0.000002 -0.000000 0.105356
27. 1.064945 0.000067 -0.000003 0.089225
28. 0.992422 -0.000060 0.000000 0.012565
29. 1.045961 -0.000002 0.000000 0.104411
30. 1.034977 -0.000008 0.000001 0.118026
CONCLUSIONS
The final trade off solution is obtained. 25 independent runs
are made for each sub problem and the optimal values of four
objectives considered at minimum value of each sub problem
over 25 independent runs are determined. In this work basic
assumption made is that the decision maker (DM) has
imprecise or fuzzy goals of satisfying each of the objectives,
the multi objective problem is thus formulated as a fuzzy
satisfaction maximization problem which is basically a min-
max problem. The multi objective problem is handled using
the fuzzy decision satisfaction maximization technique which
is an efficient technique to obtain trade off solution in multi
objective problems.
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