 Open Access
 Total Downloads : 169
 Authors : D. P. Dash, C. Chaudhury, K. C. Meher
 Paper ID : IJERTV3IS070521
 Volume & Issue : Volume 03, Issue 07 (July 2014)
 Published (First Online): 22072014
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
A New Solution for Dynamic Economic Dispatch With Valve Point Loading Using Modified Particle Swarm Optimization
1 D. P Dash, 2 C. Chaudhury,3 K. C. Meher,
1 Professor, Electrical Engg. Dept., Orissa engineering College, Bhubaneswar
2, 3 Asst. professor Electrical Engg. Dept., Orissa engineering College, Bhubaneswar
Abstract: This article presents a novel optimization approach to constrained dynamic economic dispatch (DED) problems using the modified particle swarm optimization (MPSO) technique. The proposed methodology easily takes care of different constraints like transmission losses, ramp rate limits and generating limit and power balance limit. To illustrate its efficiency and effectiveness, the developed MPSO approach is tested with different number of generating units and comparisons are performed with other approaches under consideration.

INTRODUCTION
The dynamic economic dispatch (DED) is an extension of the traditional economic dispatch problem used to determine the schedule of realtime control of power system operation so as to meet the load demand at the minimum operating cost under various system and operational constraints. DED procedure follows the dynamic connection by handling the ramp rate limits of generating units and by modifying the steady state cost to include the extra fuel consumption. The DED problem is not only the most accurate
formulation of the economic dispatch problem (EDP).
Most of the literature addresses DED problem with convex cost function [12]. However, in reality, large steam turbines have steam admission valves, which
However, all the previous work mentioned above neglected the nonsmooth characteristic of generator, which actually exist in the real power system.
This paper presents a novel optimization method based on modified particle swarm optimization (MPSO) algorithm applied to dynamic economic dispatch in a practical power system while considering some nonlinear characteristics of a generator such as ramp rate limits, generators constraints, power loss and nonsmooth cost function. The proposed methodology emerges as a robust optimization technique for solving the DED problem for different size power system.

DED PROBLEM FORMULATION
The objective of the DED is to schedule the outputs economically over a certain period of time under various system and operational constraints. The conventional DED problem
Minimizes the following incremental cost function associated to dispatchable units.
T N
M in F Fit Pit $ (1)
t 1i1
Where F is the total operating cost over the whole dispatch period, T is the no. of intervals in the scheduled horizon, N
contribute non convexity in the fuel cost function of the
generating units [3]. Accurate modeling of the DED
is the no. of generating units and
Fit
Pit
is the fuel cost
problem will be improved when the valve point loadings in
in terms of its real power output
Pit
at timet. Taking into
the generating units are taken into account. Previous efforts
valvepoint effects, the fuel cost of the
i th
thermal
on solving DED problem have employed various mathematical programming methods and optimization techniques. Conventional method like Lagrangian
generating unit is expressed as the sum of a quadratic and a sinusoidal function in the following form is given by
relaxation [1], gradient projection method [2] and dynamic
F P a P2 b P C e sinf P
P
$ / h
programming etc, when used to solve DED problem suffer from myopia for nonlinear, discontinuous search space, leading them to a less desirable performance and these
it it
i it
i it i i
i i,min it
(2)
methods often use approximations to limit complexity.
Where ai , bi , ci are cost coefficients and ei , fi
Recently, stochastic optimization techniques such as Genetic algorithm (GA) [45], evolutionary
are constants from the valve point effect of the
i th
programming (EP) [67], simulated annealing (SA) [89] and particle swarm optimization (PSO) [1012] have been given much attention by many researches due to their ability to seek for the near global optimal solution.
generating unit, subject to the following equality and
inequality constraints.

Real power balance
inertia factor , n is the population size ,
r1 and
r2 are two
N
Pit _ PDt _ PLt 0
t 1
(3)
independent random numbers. Thus, the position of each particle at each generation is updated according to the following equation:
Where t = 1, 2 T, is the total power demand at time t
and
PLt is the transmission power loss at
i th
interval in
xij
t 1 xij
t vij
t 1
MW.

Real power operating limits
Pt min Pit Pt max
(4)
i 1,2,…….,n and j=1,2,.,d (9)

MODIFIED PSO ALGORITHM
In the conventional PSO method, the inertia weight is made constant for all the particles in a single generation,
Where
Pt min
and
Pt max
are respectively the minimum
but the most important parameter that moves the current
and maximum real power output of


Generating unit ramp rate limits
i th
generator in MW.
position towards the optimum position is the inertia weight . In modified PSO, the particle position is adjusted such that the highly fitted particle (best particle) moves slowly when compared to the lowly fitted particle. This can be achieved by selecting different values for each particle
Pit Pi t 1 URi ,
i 1, 2, 3,….,N
(5)
according to their rank, between min and max as in the
following form:
Pi t 1 Pit DRi,
i 1,2,3,….,N
(6)

max min Ranki
(10)
Where
URi and
DRi are the rampup and ramp down
i max
Total Population
limits of
i th
unit in MW. So the constraint given by Eq.
(5) is modified as follows:
max P , P DR min P
, P UR
So, from Eq. (9), shows that the best particle takes first rank, and the inertia weight for that particle is set to
i min
i t 1 i
i max
i t 1
i
(7)
minimum value while for the lowest particle takes the maximum inertia weight, which particle move a high velocity. The velocity of each particle is updated using Eq.
3. OVERVIEW OF PSO
(15), and if updated velocity goes beyond maximum
The particle swarm optimization method conducts its
velocityVmax , than it is limited to
Vmax
search using a population of particles, corresponding to
vij t 1 i vij t c1r1 pij t xij t
individuals. It starts with a random initialization of a population of individuals in the search space and works on the social behavior of the particles in the swarm, like birds

c2 r2 pgi
t xij
t
(11)
flocking, fish schooling and the swarm theory. Therefore, it finds the global optimum by simply adjusting the trajectory of each individual towards its own best location and towards the best particle of the swarm at each generation of
vij t 1 signvij t 1 minvij t 1,
j 1,2,……,d and i 1,2,……,n
V j max
(12)
volution. The position and the velocity of the i th particle
in the d dimensional search space can be represented as
X i xi1, xi2 ,……..x, id T and Vi vi1 , vi2 ,……..v, id T .
The new particle position is obtained by using Eq. (17), and if any particle position beyond the rang e specified, it
is limited to its boundary using Eq. (18),
Each particle has its own best position (Pbest)
Pi t pi1 t, pi2 t,………., pid tT corresponding to the
xij
t 1 xij
t vij
t 1
(13)
personal best objective value obtained so far at generation
t . The global best particle (Gbest) is denoted by
j 1,2,……,d;
i 1,2,…….n.
Pg t pg1 t, pg 2 t,………., pgd tT . The new velocity
xij
t 1 minxij
t 1,
range
j max ,
of each particle is calculated as follows:
vIJ t 1 vij t c1r1 pij t xij t
xij t 1 maxxij t 1, range j min
(14)
c2 r2 pgi t xij t j 1,2,……..d.
(8)
The concept of reinitialization is introduced in the proposed MPSO method after a specific
Where
c1 and
c2 are constants of acceleration coefficients
corresponding to cognitive and social behavior, is the
number of generations if there is no improvement in the convergence of the algorithm. At the end of the method the specific generation is reinitialized with new randomly generated individuals. The number of new individuals is selected from k least individuals of the original population, where k is the percentage of total population to be changed. This reinitialization of population is performed after checking the change in the Fbest value in each and every specific generation.

MPSO ALGORITHM

Initialize number of population of particles dimension d with random position velocities and get the input parameters such as range [min, max] for each variable, c1, c2 and iteration counter. Set iteration counter = 0.

Increment iteration counter by one.

Find out the fitness function of all particles in the population and update the objective function.

If stopping criterion is reached than go to step (5.9).
Otherwise continue.

Evaluate the inertia factor according to Eq. (10).

Update the velocity given in Eq. (11) and correct it using Eq. (12).

Update the position of each particle using Eq. (13) and if the new position goes out of range, set it to boundary value using Eq. (14).

For every 5 generations, the {Fbest, new value} is compared with the {Fbest, old value}. If there is no change, then use the reinitialization concept and go to step (5.3).

Output the Gbest particle and its objective value.


SIMULATION RESULTS
The five unit system with nonsmooth fuel cost function is used to demonstrate the performance of the proposed MPSO. We have used the same system data as done by Panigrahi et al. [8]. The load demand of the system is taken over 24 hour. The result of the proposed method is given in Table 1. The earlier reported result for the cost is 47356 $. For the present simulation, the cost is found to be 44568 $.
No. of
hours
Load
demand
PG1
(MW)
PG2
(MW)
PG3
(MW)
PG4
(MW)
PG5
(MW)
1
410
12.3675
104.4735
108.9301
38.4012
140.3918
2
435
42.4708
95.9732
113.6381
40.1022
138.6778
3
475
72.0578
96.6296
121.2753
43.9813
139.7721
4
530
45.0234
97.9622
116.7643
75.0224
179.8301
5
558
19.7435
105.2582
115.7942
89.9066
197.7502
6
608
41.9471
103.3492
116.6698
96.8961
215.1322
7
626
11.9462
89.4341
116.7647
171.8153
221.9615
8
654
23.6745
85.3441
117.9742
210.0113
228.9501
9
690
47.4359
98.0986
117.7644
208.0518
231.5196
10
704
64.1105
99.5385
116.6747
209.1853
229.5385
11
720
43.0118
101.5421
142.6332
210.1692
230.1596
12
740
39.7598
97.3598
164.9799
207.5818
229.3214
13
704
42.6758
96.5389
143.9599
208.9857
228.3597
14
690
48.6036
96.7388
118.7045
208.9947
220.6617
15
654
19.6033
95.3773
110.7656
200.6448
201.9233
16
580
11.1709
87.4059
112.8548
206.2245
191.5503
17
558
11.1801
97.6698
97.4321
207.5764
178.4851
18
608
23.5582
99.5398
113.6849
209.8158
156.1376
19
654
21.0434
100.5196
114.6753
211.1986
188.1579
20
704
49.4497
105.3416
114.4284
210.6318
196.1342
21
680
34,4159
103.6783
116.0614
210.8963
213.9615
22
605
11.7202
90.5406
108.5995
198.7053
215.0352
23
527
10.0035
62.1432
92.0095
160.9033
223.2762
24
463
10.0205
39.6943
83.0064
135.9715
225.6296
Table 1: Result for five unit system with 24 h load demand

CONCLUSIONS

The paper has employed the MPSO algorithm on constrained of dynamic economic dispatch problem. The proposed approach has produced comparable to or better than those generated by other algorithms, and the solution has superior quality and good convergence characteristics. From this limited comparative study, it can be conclued that the MPSO can be effectively used to solve nonsmooth as well as smooth constrained economic load dispatch problems. In the future, the work will can be made to incorporate more realistic constraints to the problem and the large size problems will be solved by the proposed methodology.
REFERENCES

G. P. Granelli, P. M Marannino, M. Montagna and

Silvestri, Fast and efficient gradient projection algorithm for dynamic generation dispatching, Proc. Inst. Elect. Engg. Gener. Transm. Distrib, vol.136, no. 5, pp. 295302, Sep.1989.


F. Li, R. Morganand and D. Williams, Hybrid genetic approaches to ramping rate constrained dynamic economic dispatch, Elect.Power Syst.Res., vol. 43, no. 2, pp. 97103, Nov. 1997.

X. S. Han, H. B. Gooi and Daniel S.Kirschen, Dynamic economic dispatch: feasible and optimal solutions, IEEE Trans. Power Syst., vol. 16, no. 1, pp. 2228, Feb.2001.

D. C. Walters and G. B. Sheble, Genetic algorithm solution of economic dispatch with valve point loading, IEEE Trans. Power Syst., vol. 8, no. 3, pp. 13251331, Aug. 1993.

C. C Fung, S. Y Chow, K. P Wong, Solving the economic dispatch with an integrated parallel genetic algorithm, In Proc. of International Conference on Power System Technology, Vol. 3, pp. 12571262, 2000.

N.Sinha, R.Chakrabarti and P.K.Chattopadhyay, Evolutionary programming techniques for economic load dispatch, IEEE Trans. Evol. Comput., vol. 7, no. 1, pp. 8394, Feb. 2003.

H. T Yang, P. C Yang, C. L Huang, Evolutionary programming based economic dispatch for units with nonsmooth fuel cost functions, IEEE Trans. power Syst., vol. 11, no. 1, pp.112118, 1996.

C. K. Panigrahi, P. K. Chatopadhyay, R. N. Chakrabarti and M. Basu, Simulated annealing technique for dynamic economic dispatch, Elect. Power Comp. Syst., vol. 34, no. 5, pp. 577586, May 2006.

K. P Wong, Y. W Wong, Genetic/simulated annealing approach to economic dispatch, In Proc. Inst. Elect. Engg. Gen. Transm. Distrib., vol. 141, no. 5, pp. 507513, 1994.

ZweLee Gaing, Particle swarm optimization to solving the economic dispatch considering the generator constraints, IEEE Trans. Power Syst.,
vol. 18, no. 3, pp. 11871195, Aug. 2003.

R. C Eberhart, Y. Shi, Particle swarm optimization: developments, application and resources, In Proc. Congress on evolutionary computation, IEEE, pp. 8186, 2001.

J. B Park, K. S Lee, J. R Shin, K.Y Lee, A particle swarm optimization for economic dispatch with nonsmooth cost functions, IEEE Trans. Power Syst., Vol. 20, no. 1, pp. 3442, 2005.