 Open Access
 Total Downloads : 655
 Authors : D. P. Dash, M. Basu, P. K. Chattopadhyay, R. N. Chakraborti
 Paper ID : IJERTV1IS6507
 Volume & Issue : Volume 01, Issue 06 (August 2012)
 Published (First Online): 30082012
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
A New Approach For Dynamic Economic Dispatch Using Hybrid Modified Particle Swarm Optimization
1 D. P DASH, 2 M. BASU, 3 J. PATTANAIK
1 Assoc.Professor, Electrical Engg. Dept O.E.C. Bhubaneswar
2 Professor, Power Engg. Dept., Jadavpur University
3 Research Scholar, Electrical Engg.Dept., Jadavpur University
ABSTRACT
This article presents a novel optimization approach to constrained dynamic economic dispatch (DED) problems using the hybrid particle swarm optimization (HPSO) technique. The proposed methodology easily takes care of different constraints like transmission losses, ramp rate limits and also uses for nonsmooth cost functions. To illustrate its efficiency and effectiveness, the developed HPSO approach is tested with different number of generating units and comparisons are performed with other approaches under consideration.
Convexity in the fuel cost function [3]. Accurate modeling of the DED problem will be improved when the valve point loadings in the generating units are taken into account. Previous efforts on solving DED problem have employed various mathematical programming methods and optimization techniques. Conventional method like Lagrangian relaxation [1], gradient projection method [2] and dynamic 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 methods often use approximations to limit complexity.

INTRODUCTION
The dynamic economic dispatch (DED) is an extension of the traditional economic dispatch problem used to determine the schedule of real time 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 contribute non
Recently, stochastic optimization techniques such as Genetic algorithm (GA) [45], evolutionary 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. 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 hybrid particle swarm optimization (HPSO) 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.
c. Generating unit ramp rate limits
Pit
Pi t 1
URi ,
i 1, 2, 3,….,N
(5)

DED PROBLEM FORMULATION
The objective of the DED is to schedule the
Pi t 1
Pit
DRi,
i 1,2,3,….,N
(6)
outputs economically over a certain period of
time under various system and operational constraints. The conventional DED problem
Where URi and
ramp down limits of
DRi
i th
are the rampup and
unit in MW. So the
minimizes the following incremental cost function associated to dispatchable units.
constraint given by Eq. (5) is modified as
follows:
M in F
T N
Fit
Pit
$ (1)
max Pi min , Pi t 1
DRi
min Pi max , Pi t 1
URi
(7)
t 1i 1
Where F is the total operating cost over the whole dispatch period, T is the no. of intervals in the scheduled horizon, N is the no. of generating

OVERVIEW OF PSO
The particle swarm optimization method
units and
Fit Pit
is the fuel cost in terms of its
conducts its search using a population of
real power output Pit at timet. Taking into
particles, corresponding to individuals. It starts with a random initialization of a population of
valvepoint effects, the fuel cost of the
i th
individuals in the search space and works on the
thermal generating unit is expressed as the sum of a quadratic and a sinusoidal function in the following form is given by
social behavior of the particles in the swarm, like birds flocking, fish schooling and the swarm theory. Therefore, it finds the global optimum by simply adjusting the trajectory of each individual
Fit Pit
a P 2
bi Pit
Ci ei sin fi Pi,min _ Pit
$ / h
towards its own best location and towards the
i it
(2)
best particle of the swarm at each generation of evolution. The position and the velocity of the
i th particle in the d dimensional search space can
Where
ai ,
bi ,
ci are cost coefficients and
be represented as X
, x ,……..,x
T and
ei , fi are constants from the valve point effect
i i1 i2 id
of the
i th
generating unit, subject to the
Vi i1 , vi2 ,……..,vid
T . Each particle has its
following equality and inequality constraints.
own best position (Pbest)
Pi t pi1 t , pi2 t ,……… pid t T

Real power balance
N
P _ P _ P 0
(3)
corresponding to the personal best objective value obtained so far at generation t . The global best particle (Gbest) is denoted by
it Dt Lt
t 1
P t p t , p t ,……… p t T . The new
Where t = 1, 2 T, is the total power demand at
g g1 g 2 gd
time t and
PLt
is the transmission power loss
velocity of each particle is calculated as follows:
at i th
interval in MW.
vIJ t 1
vij t
c1r1
pij t
xij t

Real power operating limits
c2 r2
p gi t
xij t
j 1,2,……..d.
(8)
Pt min
Pit
Pt max
(4)
Where
c1 and
c2 are constants of acceleration
Where
Pt min
and
Pt max
are respectively the
coefficients corresponding to cognitive and social behavior, is the inertia factor , n is the
minimum and maximum real power output of
population size ,
r1 and
r2 are two independent
i th
generator in MW.
random numbers. Thus, the position of each particle at each generation is updated according to the following equation:
xij t 1 xij t vij t 1
i 1,2,…….,n and j=1,2,.,d (9)


MODIFIED PSO :
In the conventional PSO method, the inertia weight is made constant for all the particles in a single generation, but the most important parameter that moves the current 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 according to their rank, between min and max as in the following form:
The concept of reinitialization is introduced in the proposed HPSO methodafter a specific 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.

SEQUENTIAL QUADRATIC PROGRAMMING (SQP):
Sequential quadratic programming (SQP) [13] is widely used to solve practical optimization
max
min
Ranki
(10)
problems. It outperforms every other nonlinear
i max
Total Population
programming method in terms of efficiency, accuracy and percentage of successful solutions.
So, from Eq. (9), shows that the best particle takes first rank, and the inertia weight for that particle is set to 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. (15), and if updated velocity goes beyond maximum velocity Vmax , than it is limited to
Vmax
The method closely mimics Newtons method for constrained optimization just as is done for unconstrained optimization. At each major iteration, an approximation is made of the Hessian of the Lagrange function using Broyden FletcherGoldfarbShanno (BFGS) quasi Newton updating method. This is then used to generate a quadratic programming subproblem whose solution is used to form a search direction for a line search procedure.
As the objective function to be minimized is
vij t 1
i vij t
c1r1 pij t
xij t
(11)
nonconvex, SQP requires a local minimum for
c2 r2
p gi t
xij t
an initial solution. In this paper, SQP is used as a local optimizer for finetunning the better region explored by AIS. Here, the formulation of SQP
vij t 1
sign vij t 1
min vij t 1 ,
V j max
(12)
subroutine is taken from [15].
For each iteration, a QP is solved to obtain the search direction which is used to update the
j 1,2,……,d
and i
1,2,……,n
control variables. QP problem can be described as follows
The new particle position is obtained by using Eq. (17), and if any particle position beyond the
Minimize the following
F d
1 d d
rang e specified, it is limited to its boundary using Eq. (18),
subject to
k k 2
k K k
xij t 1
xij t
vij t 1
gi k
g k d k 0
j 1,2,……,d;
i 1,2,…….n.
(13)
i 1,…., me
g
g d 0
xij t 1
min xij t 1 ,
rangej max ,
i k k k
xij t 1
max xij t 1 ,
range
j min
(14)
i me
1,…, m
where
k the Hessian matrix of the Lagrangian function at the k th iteration
d k the search direction at the k th iteration
k the real power vector at the k th iteration
g k constraints from (3) to (4)
me number of equality constraints

m number of constraints
L , F g
where is the vector of Lagrangian multiplier.
k is calculated using quasiNewton formula given by,
[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 re initialization concept and go to step (5.3).

Output the Gbest particle and its objective value.
qk qk
k Sk Sk k

solve the DED problem using the SQP
k 1
where
qk Sk Sk
k
Sk k 1 k
k Sk
method with the selected solution obtained from PSO.


SIMULATION RESULTS
qk L k 1 , k 1 L k , k 1
For each iteration of the QP subproblem the direction d k is calculated using the objective function. The solution obtained forms a new iterate given by,
k 1 k k dk
The step length value k is determined to produce a considerable reduction in an augmented Lagrangian merit function as follows
L , , F g g g
2
where is a nonnegative scalar. The procedure
The five unit system with nonsmooth fuel cost function is used to demonstrate the performance of the proposed HPSO. 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 $.

CONCLUSIONS
The paper has employed the HPSO 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
is repeated until the value of some tolerance value.
6. HPSO ALGORITHM
S k has reached
concluded that the HPSO 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
6.1) Initialize number of population of particles dimension d with random position velocities and get the input parameters such as range
problem and the large size problems will be solved by the proposed methodology.
Table1: Result for five unit system with 24 h load demand
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 
47.459 
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 
REFERENCES

G. P. Granelli, P. M Marannino, M. Montagna and A. 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 non smooth 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. 81 86, 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.

P. Attavriyanupp, H. Kita, T. Tanaka and J. Hasegawa, A Hybrid EP and SQP for Dynamic Economic Dispatch with Nonsmooth Fuel Cost function, IEEE Transactions on Power Systems, vol. 17, no. 2, pp. 411416, May 2002.

P. T. Boggs and J. W. Tolle, Sequential Quadratic Programming, Acta Numerica, no. 4, pp. 152, 1995.