 Open Access
 Total Downloads : 552
 Authors : D. P. Dash, M. Basu, P. K. Chattopadhyay, R. N. Chakraborti
 Paper ID : IJERTV1IS6302
 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
HYBRIDIZATION OF ARTIFICIAL IMMUNE SYSTEM FOR OPTIMAL OPERATION OF POWER SYSTEM
D. P. Dasp, M. Basu2, P. K. Chattopadhyay3, R. N. Chakraborti3
1Electrical Engineering Department, Orissa Engg. College,Bhubaneswar 2Power Engineering Department, Jadavpur University, Kolkata 3Electrical Engineering Department, Jadavpur University, Kolkata
Abstract:
This paper presents Hybridization of Artificial Immune System (HAIS) and Sequential Quadratic Programming (SQP) based optimization technique to solve nonconvex economic load dispatch (NCELD) problem of thermal plants. The presented methodology can take care of economic dispatch problems involving constraints such as power balance, generator limits, and valve point loading. The HAIS method is tested with two power system cases consisting of 6 and 13 thermal units. Comparisons are performed with other approaches under consideration.
Keywords: Artificial Immune System, Economic load dispatch, Non convex cost function, Sequential quadratic programming

Introduction:
The objective of the economic load problem (ELD) is to schedule the committed generating unit outputs so as to meet the required load demand at minimum operating cost while satisfying all unit and system equality and inequality constraints [1]. In traditional EDPs, the cost function of each generator is approximately represented by a simple quadratic function and is solved using mathematical programming [2] based on several optimization techniques, such as dynamic programming [3], linear programming [4], homogenous linear programming [5], and nonlinear programming technique [6]. However, real inputoutput characteristics display higherorder
nonlinearities and discontinuities. Power plants usually have multiple valves that are used to control the power output of the unit. When steam admission valves in thermal units are first opened, a sudden increase in losses is observed. This leads to ripples in the cost function, which is known as the valvepoint loading. The ELD problem with valvepoint effects is represented as a non smooth optimization.
Recently, stochastic optimization techniques such as Genetic algorithm (GA) [78], evolutionary programming (EP) [910], simulated annealing (SA) [1112] and particle swarm optimization (PSO) [1315] have been given much attention by many researches due to their ability to seek for the near global optimal solution. However, all the previous works mentioned above neglected the nonsmooth characteristic of generator, which actually exist in the real power system.
In order to alleviate the aforementioned difficulties, this paper proposes a new optimization approach known as artificial immune system (AIS). AIS imitate the immunological ideas to develop some techniques used in various areas of research [16]. It works on the principle of pattern recognition (antibody and antigen) and clonal selection principle, whereby clonal selection principle called as AIS which is implemented to accomplish learning and memory acquisition tasks. AIS effectively exploit the interaction and corresponding affinity suitably mapping it to fitness evaluation, constraint satisfaction. AIS approach has been applied to solve economic load dispatch (ELD) problem [17].
This paper presents a novel optimization method based on Artificial Immune System Method (AISM) is applied to economic load dispatch in a practical power system. Under considering some nonlinear characteristics of a
order to achieve proper economic dispatch. To calculate this transmission loss B coefficients method is used. The transmission loss is represented by B coefficients as:
generator, such load demand, generators constraints, power loss and nonsmooth cost
N N
PL Pi Bij Pj
(3)
function. The proposed methodology emerges as a robust optimization technique for solving the ELD problem for different size power system.

Problem Statement:
The prime objective of the ELD problem is to determine the most economic loadings of generators to minimize the generation cost such that the load demands in the scheduling horizon can be met and simultaneously, the operating
i 1 j 1
where Pi and Pj are real power injections at ith
ij
and jth buses respectively and B are the loss
coefficients, which are constants under certain assumed operating conditions. For AISSQP method, transmission loss is not considered hence the power balance constraint of Eq. (2) is as follows:
N
constraints are satisfied. This constrained optimization problem can be written as:
Pi PD
i 1
0 (4)
Minimize F
N
Fi Pi
(1)
b. Real power operating limits
i 1 The generated output of each unit should remain between its minimum and maximum limits. The
where F is the total operating cost over the whole dispatch period, N is the no. of generating
following inequality constraint should be satisfied for each generator:
units and Fi Pi is the fuel cost in terms of its
Pmin P Pmax
(5)
real power output Pi . This minimization
i i i
problem is subjected to a variety of constraints
where Pmin and
Pimax are the minimum and
i
depending upon assumptions and practical implications like power balance constraints, generator output limits, transmission losses, ramp rate limits, etc. These constraints are discussed as follows:
maximum real power output respectively of generator.

Artificial Immune System (AIS)
i th

Power balance constraint
The total generation should be equal to the total system power demand PD plus the transmission loss which is represented as follows:
N
Artificial immune system (AIS) [16] mimics the
biological principles of clone generation, proliferation and maturation. The main steps of AIS based on clonal selection principle are activation of antibodies, proliferation and differentiation on the encounter of cells with antigens, maturation by carrying out affinity maturation process, eliminating old antibodies to
Pi PD PL
i 1
0 (2)
maintain the diversity of antibodies and to avoid premature convergence, selection of those antibodies whose affinities with the antigen are greater.
where PD is the total power demand and PL is the transmission power loss in MW. The transmission loss must be taken into account in
In order to emulate AIS in optimization, the antibodies and affinity are taken as the feasible
solutions and the objective function respectively. Real number is used to represent the attributes of the antibodies. Initially, a population of random solutions is generated which represent a pool of antibodies. These antibodies undergo proliferation and maturation. The proliferation of antibodies is realized by cloning each member of the initial pool depending on their affinity.


Sequential Quadratic Programming:
k : the Hessian matrix of the Lagrangian function at the kth iteration
k
d : the search direction at the kth iteration
k
: the real power vector at the kth iteration g k : constraints from Eq. (4) to Eq. (5) me : number of equality constraints
m : number of constraints
Sequential quadratic programming (SQP) [17] is
L , F g
(9)
widely used to solve practical optimization problems. It outperforms every other nonlinear programming method in terms of efficiency, accuracy and percentage of successful solutions. At each major iteration, an approximation is
wher is the vector of Lagrangian multiplier.
k is calculated using quasiNewton formula given by:
made of the Hessian of the Lagrange function using BroydenFletcherGoldfarbShanno (BFGS) quasiNewton updating method. This is then used to generate a Quadratic Programming (QP) subproblem whose solution is used to form a search direction for a line search procedure. As the objective function to be minimized is nonconvex, SQP requires a local minimum for an initial solution. In this chapter, SQP is used as a local optimizer for finetuning
k 1 k
where
Sk k 1
qk L
qk qk qk Sk
k
k 1 , k 1
k Sk Sk k
Sk k Sk
L k , k 1
(10)
(11)
(12)
the better region explored by AIS. Here, the For each iteration of the QP subproblem the
formulation of SQP subroutine is taken from [17].
For each iteration, a QP is solved to obtain
direction d k is calculated using the objective function. The solution obtained forms a new iterate given by the following expression:
the search direction which is used to update the
control variables. QP problem can be described as follows:
k 1 k
k dk
(13)
Minimize the following:
1

Proposed Hybrid Algorithm:
The proposed hybrid method uses the property of the AIS, which can give a good solution even
F k dk
2 dk
K dk
(6)
when the problem has many local optimum solutions at the beginning and SQP, which has a
subject to the following constraints:
local search property that is used to obtain the final solution. Algorithms are as follows:
gi k
g k d k 0
i 1,…,me
(7)

Let pk
1 , 2 ,……,
i ,…….
, be
gi k g k d k 0
i m 1,…, m
(8)
the k th antibody of a population to be evolved and k 1,2,…, . The elements of pk are real
e
where
power outputs of the committed N generating units. The initial value of real power output of the ith unit is determined by setting
,
U
i ~ U min i
i min ,
max
i
i max , where i . denotes a uniform random
No. of clones = 5, SPM = 12, NIT = No. of iterations = 200. The optimal result obtained by the proposed AISSQP method is found to be
variable ranging over
min
,
i
max . Each
1820.6 $ which is so far the minimum one
among the other techniques as shown in Table 1
i
antibody should satisfy the constraints given by Eqs. (2.4) and (2.5).

As ED is a minimization problem, affinity is the inverse of the objective function and it is given by the following equation:
and Table 2.
7. Conclusions:
This paper employs the HAIS algorithm on constrained of economic dispatch problem.
Affinity
ai bi i
i 1
1
c
i
i 2 di sin ei
min
i
i
(14)
Comparative studies show significant improvement in fuel cost. The solution also has superior quality and very less fuel cost. According to the results, it can be concluded that the HAIS method can be effectively used to solve nonsmooth as well as smooth constrained

The antibodies are cloned directly
proportional to their affinities, giving rise to a temporary population of clones.

The clones undergo maturation process through hypermutation mechanism whose rate is inversely proportional to their affinities. Each mutated clone must satisfy the constraints given by Eqs. (4) and (5).

The affinities of the mutated clones are evaluated.

Aging operator eliminates those individuals which have more than generations from the current population.

Tournament selection is done to select a new
population of the same size as the initial population from the antibodies and mutated clones which are remained after application of aging operator. Select the solution which has the highest affinity value obtained so far.

Solve the ED problem using the SQP method with the selected solution obtained from AIS.
6. Simulations and Results:
The Hybrid AISSQP algorithms have been implemented for the solution of economic load dispatch with nonsmooth cost functions on MATLAB 7.0 platform on a 3.06 GHz, PentiumIV PC with 256 MB RAM. In case of
economic load dispatch problem. The proposed approach can be enriched by incorporating more realistic constraints to the problem and large size problems can be solved using this method.
References

Ross, D. W., and Kim, S., Dynamic economic dispatch of generation, IEEE Trans. Power Apparatus Syst., Vol. PAS99, No. 6, pp. 20602068, 1980.

Van Den Bosch, P. P. J., Optimal dynamic dispatch owing to spinningreserve and power rate limits, IEEE Trans. Power Apparatus Syst., Vol. PAS104, No. 12, pp. 33953401, 1985.

Granelli, G. P., Marannino, P., Montagna, M., and Silvestri, A., Fast and efficient gradient projection algorithm for dynamic generation dispatching, IEE Proc. Generat. Transm. Distrib.,Vol. 136, No. 5, pp. 295302, 1989.

Hindi, K. S., and Ab Ghani, M. R., Dynamic economic dispatch for large scale power systems: A Lagrangian relaxation approach, Elect. Power Syst. Res., Vol. 13, No. 1, pp. 51 56, 1991.

Lee, F. N., Lemonidis, L., and Liu, K.C.,
Pricebased ramprate model for dynamic dispatch and unit commitment, IEEE Trans. Power Syst., Vol. 9, No. 3, pp. 12331242,
August 1994.
AIS population size ( )
is taken as 50. The

Travers, D. L., and Kaye, R. J., Dynamic
following set of values is chosen after thorough
dispatch by constructive dynamic
investigation. N P
= population size = 50,
NC =
programming, IEEE Trans. Power Syst., Vol. 13, No. 1, pp. 7278, February 1998.

Han, X. S., Gooi, H. B., and Kirschen, D. S.,
Dynamic economic dispatch: Feasible and optimal solutions, IEEE Trans. Power Syst., Vol. 16, No. 1, pp. 2228, February 2001.

Wong, K. P., and Fung, C. C., Simulated annealing based economic dispatch algorithm, IEEProc. Generat. Transm. Distrib., Vol. 140, No. 6, pp. 509515, 1993.

Walter, D. C., and Sheble, G. B., Genetic algorithm solution of economic dispatch with valve point loading, IEEE Trans. Power Syst., Vol. 8, pp. 13251332, August 1993.

Cheng, P. H., and Chang, H. C., Large scale economic dispatch by genetic algorithm, IEEE Trans. Power Syst., Vol. 10, No. 4, pp. 1919 1926, November 1995.

Gaing, Z.L., Particle swarm optimization to solving the economic dispatch considering the generator constraints, IEEE Trans. Power Syst., Vol. 18, No. 3, pp. 11871195, August 2003.

Panigrahi, B. K., Yadav, S. R., Agrawal, S., and Tiwari, M. K., A clonal algorithm to solve
economic load dispatch, Elect. Power Syst. Res., 2006.

Attavriyanupp, P., Kita, H., Tanaka, T., and Hasegawa, J., A hybrid EP and SQP for dynamic economic dispatch with nonsmooth fuel cost function, IEEE Trans. Power Syst., Vol. 17, No. 2, pp. 411416, May 2002.

de Castro, L. N., and Von Zuben, F. J.,
Artificial immune systems: Part IBasic theory and applications, Technical Report TR DCA 01/99, December 1999.

de Castro, L. N., and Zuben, F. J., Learning and optimization using through the clonal selection principle, IEEE Trans. Evolut. Comput., Vol. 6, No. 3, pp. 239251, 2002.

Cutello, V., Morelli, G., Nicosia, G., and Pavone, M., Immune algorithms with aging operators for the string folding problem and the protein folding problem, EvoCOP 2005, LNCS, Vol. 3448, pp. 8090, 2005.

Boggs, P. T., and Tolle, J. W., Sequential quadratic programming, Acta Numer., No. 4, pp. 152, 1995.
Table 1: Optimal olution for 6unit system by Proposed Method
Output (MW) 
HAIS (proposed ) 
SAPSO 
SOHPSO 
PSO 
P1 
448.3546 
446.71 
438.21 
447.4970 
P2 
174.0576 
173.01 
172.58 
173.3221 
P3 
264.0924 
265.00 
257.42 
263.4745 
P4 
139.7022 
139.00 
141.09 
139.0954 
P5 
166.2479 
165.23 
179.37 
165.4761 
P6 
83.0011 
86.78 
86.88 
87.1280 
Total output 
1275.4547 
1275.7 
1275.55 
1276. 01 
Ploss (MW) 
12.4547 
12.733 
12.32 
12. 9584 
Total Gen. Cost ($) 
15443.264 
15447 
15446.02 
15,450 
Table 2: Optimal solution for 13unit system by Proposed Method
Output (MW) 
EP 
AIS 
AISSQP 
P1 
537.0592 
628.3184 
624.6347 
P2 
75.9977 
360.0000 
348.5615 
P3 
291.5525 
299.1993 
224.3853 
P4 
160.3258 
159.7330 
112.4184 
P5 
60.0000 
109.8666 
60.0000 
P6 
106.2951 
60.0000 
60.0000 
P7 
161.3267 
60.0000 
60.0000 
P8 
108.5179 
60.0004 
60.0000 
P9 
61.8441 
60.0002 
60.0000 
P10 
46.9510 
40.0000 
40.0000 
P11 
63.5265 
40.0000 
40.0000 
P12 
56.0699 
55.0000 
55.0000 
P13 
70.5337 
67.8822 
55.0000 
Total output (MW) 
1800 
2000 
1800 
Total gen. cost ($) 
1843.6 
1986.0 
1820.6 