International Journal of Scientific & Engineering Research Volume 2, Issue 12, December-2011 1
ISSN 2229-5518
Clonal Selection Algorithm for Dynamic Economic
Dispatch with Nonsmooth Cost Functions
U. K. Rout, R. K. Swain, A. K. Barisal, R. C. Prusty
Abstract— This paper presents clonal selection algorithm to solve the Dynamic Economic Dispatch Problem (DEDP) of generating units considering valve point loading effects. It determines the optimal operation of units with predicted load demand over a certain period of time with an objective to minimize total production cost while the system is operating with ramp rate limits. This paper presents DED based on clonal selection technique for determination of the global or near global optimum dispatch solution. In the present case, load balance constraints, operating limits, valve point loading, ramp constraints and network loss coefficient are incorporated. Five unit test systems with non-linear characteristics of the generators are considered to illustrate the effectiveness of the proposed method. The feasibility of the proposed method is demonstrated and compared to those reported in the literature. The results are promising and show the effectiveness of the proposed method.
Index Terms— Clonal Selection Algorithm, Dynamic Economic load dispatch, Power generation operation and Control, Ramp rate limits
—————————— ——————————
HE dynamic load dispatch is an important optimization task of power system operation and control. The main objective of the dynamic economic dispatch is to minimize the cost of generation, subject to physical and operational con- straints. In traditional economic dispatch has quadratic cost function. In reality, a generating unit can not exhibits a convex fuel cost fuction.for a practical power system problem. There- fore, non-convex characteristics will come due to steam valve effect. Overwhelming literature, however, deals mostly with static economic dispatch, i.e. the horizon is divided into pe- riods and dispatch is optimized period-by-period. Accurate modeling of the DED problem will be improved when the valve point loadings in the generating units are taken into ac- count. Furthermore, they may generate multiple local opti- mum points in the solution space. Some traditional optimiza- tion methods have been applied to solve the DED problems. The traditional methods such as gradient projection method [1], lagrangian relaxation (LR) [6] and dynamic programmings are used to solve the DED problem. These methods were fac- ing problems to give optimal solution due to their non-linear and non-convex characteristic of generating unit. The stochas- tic search algorithm such as particle swarm optimization (PSO)[5,14,26], differential evolution (DE) [16,17], genetic algo- rithm (GA)[2,7], evolutionary programming (EP)[3,4,8,9], si- mulated annealing (SA)[10,15], and tabu search algorithm (TSA)[11] may prove to be very effective in solving non-linear economic dispatch problems without any restriction on the shape of the cost curves. Although these heuristic methods do not always guarantee discovering the global optimal solution in finite time, they often provide fast, reasonable and near
global optimal solutions. All of these methods are probabilistic rules to update their candidates’ positions in the solution space. The disadvantage of this method is requirement of large memory. So, the combination of the heuristic methods and deterministic method are required to solve the complex opti- mization problems. Heuristic methods are used for search purpose; to find near global optimum solution. Combining EP with sequential quadrature programming (SQP) [12] and PSO with SQP has been used to solve DED problem [13] because they can give near global optimal solution. It is very difficult to set the tuning parameters of optimization technique and exploring new ideas for solving the DED problem.
This paper proposes new optimization approaches, to solve dynamic dispatch problems using clonal selection principle. This is a very complex biological system, which accounts for resistance of a living body against harmful foreign antigens. It based on the principle of pattern recognition and clonal selec- tion principle, whereby clonal selection principle (invariable called AIS) is implemented on learning and memory acquisi- tion tasks. In AIS receptors presents on the antibody are re- sponsible for antigen antibody interaction. In this interaction, different antibody has different affinity towards an antigen and binding strength is directly proportional to this affinity. AIS effectively exploit these interactions and corresponding affinity by suitable mapping it to fitness (objective function) evaluation, constraints satisfaction or other relevant amenities of optimum research. [18]-[25].
————————————————
R.K. Swain, Professor in KIST, Bhubaneswar, Orissa, India.
A.K. Barisal and R.C. Prusty are with the Department of Electrical Engi-
neering VSSUT, Burla, India. Email:a_barisal@rediffmail.com
This work suggests a methodlogy using clonal selection al- gorithm to obtain the optimal generation dispatch solutions for dynamic dispatch problem in deregulated power system. The proposed methodology has been applied to five units test system to show its effectiveness and applicability. The results obtained from the proposed methodology are compared with
IJSER © 2011
International Journal of Scientific & Engineering Research Volume 2, Issue 12, December-2011 2
ISSN 2229-5518
the reported results given in literature..
The objective of the dynamic economic dispatch is to minimize the generator output economically over a certain period of time under various systems and operational constraints. The problem is formulated as follows.
system. Learning and memory are the main characteristics system. Learning and memory are the main characteristics of immune system.The natural immune system is parallel adap- tive system. The antibodies are produced by lymphocytes through clonal proliferation. In order to initiate clonal concept in optimization, the affinity concept is transferred to fitness or objective function evaluation and constraint satisfaction.
Start
T N
Min Fit pit
t 1 i 1
(1)
Randomly generate real number in
the feasible range
Where F Total operating cost over whole dispatch periods
T Numbers of hour in the time horizon;
N Number of dispatchable units;
Yes
Constraints
Violation
Fit (Pit ) The fuel cost in terms of its real power output
Pit at
Enter into initial populat ion
a time t. Taking into account the valve point-point effects, the
fuel cost function of the thermal generating unit is expressed as the sum of a quadratics and sinusoidal functions.
Fit (Pit ) ai Pi
bi Pi ci | ei (sin( f i (Pi min Pi )) |
(2)
Population=full? No
ai , bi , ci , ei and
f i are constants of fuel cost function of unit
Evaluation of Fitness
i subject to the following.Real power balance constraint
N
Clonal proliferation based on
Fitn ess
Pit PDt Plt
i 1
0 ; t 1,2,....T
(3)
Mutation
Where PDt
is the total assumed load demand during at a time
Constraints
t; Plt the transmission loss at a time t. Real power operating
Violation
limits
Pit min
Pit Pit max i 1,2,....T
t 1,2,....T
(4)
Evaluate the fitness of Clones
Where
P i min and
Pi max are the minimum and maximum real
power outputs of ith generator, respectively. Generator unit ramp rate limits
Pit Pi (t 1) URi i 1,...n
Pi (t 1) Pit DRi i 1,...n
(5) (6)
No
Population=full?
Tournament Selection
No
Convergence?
URi
and
DRi are ramp up and ramp down rate limits of ith
Stop
generator respectively and these are expressed in MW/h.
The artificial immune system is a very intricate biological sys- tem, which accounts for resistance of living body against for- eign antigens. It works on the principle of pattern recognition
Fig. 1. Implementation of Clonal Selection Algorithm
Here, antigen represents constraints and antibody-antigen interaction refers to constraints satisfaction, i.e., higher the satisfaction of constraints more is the affinity.
Fig. 1.shows the procedures to implement the clonal selection
IJSER © 2011
International Journal of Scientific & Engineering Research Volume 2, Issue 12, December-2011 3
ISSN 2229-5518
algorithm through a flow-chart. The algorithm starts with the random generation of real numbers to check for constraint violation. In case of any constraints violation, random data are generated again and again. This process is repeated iteratively until a deliberate fixed size of population is attained. When the population becomes full then each antibody is evaluated and clones are generated. The number of clones generated per antibody is dependent on the fitness values; i.e. larger the number of clones generated for the antibodies higher the fit- ness value. The mutation rate is adaptive which is similar to evolutionary programming [24]. Consequently, clones with higher fitness are made liable to undergo mutation to a lesser extent as compared to those with lower fitness. This is re- peated till all the clones from the temporary clonal population are endured to mutation. Finally, tournament selection is done to select same number of muted clones as existed in the initial population. This completes one generation of the clonal selec- tion algorithm. The convergence parameter is set when the best solutions of each generation cease to change. Thereby, stopping criteria is taken when either the satisfied conver- gence level is reached or the maximum number of generations
tribution of the variables ranging in the interval of
(Pit min, Pit max ) . Initialize the interaction count.
The affinity value of each antibody in the population set is eva- luated using the equation (2).
The fittest antigen will be cloned (reproduced) independently and proportionally to their antigenic affinities, the higher the antigenic affinity, the higher the clones generated for each of the selected antigens. The equation for adaptive cloning process was developed based on the fittest antibody will pro- duce more clones compared to weaker ones. Equation (7) is implemented to determine the number of clones is generated according the affinity measures or the fitness.
f i
is exhausted.
Number of clonesi= 1 i 20
×200 (7)
f i
f i =Fitness value
i 1
The main objective of dynamic dispatch problem is to obtain the amount of real power to be generated by each committed genera- tors, while achieving a minimum generation cost within the con- straints. The section provides the solution methodology to the above-mentioned dynamic problems through clonal selection
technique.
f i =sum of fitness in a population
Real number was used to represents the attributes of the anti- bodies. Each antibody attribute will be in a form of pair of real
valued vector (Pi ,i ), i 1,....N g , Where i
a strategy
For T intervals in the generation scheduling horizon, there is T
parameter [21]. Each antibody will go through the mutation
process according to the expression given by equation 8 and 9.
dispatched for the n generators. An array of control variable vec- tors can be shown as
' ( j) ( j) exp(
(N (0,1) N j (0,1))
(8)
P1 1P12....P1T
P ' ( j) P( j) ' ( j)N
(0,1)
(9)
i i j
P2 1P22....P2T
Where
N (0,1) is a normally distributed random number with
S=
P 1,2,...M
zero mean and standard deviation equal to one.
N j (0,1)
is a
random number generated anew for every i. and j. The factors
Pn1Pn 2 ....PnT
= ((2(n) 1/2) 1/2)-1 and ’=((2n) 1/2)-1 are commonly known as
'
Where Pit is the real power output of generator i at interval t
learning rates. An offspring Pi
mutation.
is calculated using gaussian
For the complete M population, the candidate solution of each individual (generating unit’s power output) is randomly initia- lized within the feasible range in such a way that is should satisfy the constraint given by Eq. (4). A component of a candidate is
n=No of population for an antibody.
In implementation, it was assumed that the highest affinities were sorted in an ascending order. In selection, the offspring produced by mutation process will be sorted and calculate the
initialized as
Pit ~ U ( Pit min, Pit max ) ,where U is the uniform dis-
best value from the offspring.
IJSER © 2011
International Journal of Scientific & Engineering Research Volume 2, Issue 12, December-2011 4
ISSN 2229-5518
There are various criteria available to stop a stochastic optimi- zation algorithm. Some examples are tolerance, number of function evaluations and number of iterations. In this paper, maximum number of iterations is chosen as the stopping crite- rion, when there is no significant improvement in the solution.
clones is 720. The total production cost of this proposed me- thod is 43446.224$. Figure 2 shows the convergence characte- ristic of clonal selection algorithm for five-unit test system. Table 2 shows the comparison results for different methods.
4
x 10
If the stopping criterion is not satisfied, the above procedure is
repeated from clone with incremented iteration.
Hour P1 P2 P3 P4 P5
1 17..587 98.797 31..374 125.2 140.47
2 10.802 98.478 64.773 125.22 139.61
3 12.077 98.947 103.68 125.11 139.81
4 10.711 98.52 112.75 173.11 140.83
5 10.545 96.88 108.92 209.17 139.14
6 38.336 102.59 112.78 210.26 151.65
5.2
5.1
5
4.9
4.8
4.7
4.6
4.5
4.4
4.3
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Iterations
7 12.907 98.522 112.55 210.04 199.96
8 12.379 98.481 112.99 210.12 228.91
9 39.893 103.79 113.71 210.73 231.56
10 64.216 98.298 112.55 209.6 229.44
11 74.548 100.32 114.12 210.54 231.07
12 74.584 99.335 113.53 234.15 229.56
13 64.774 97.845 112.88 209.17 229.54
14 46.19 99.501 112.59 210.21 231.11
15 16.659 99.09 112.97 206.27 227.62
16 11.014 86.428 104.11 157.42 227.78
17 10.492 88.48 111.9 125.21 228.22
18 38.967 103.6 116.44 126.02 230.52
19 47.613 98.732 112.04 174.14 230.24
20 64.243 98.474 112.61 208.88 229.97
21 38.517 98.575 112.87 209.8 229.92
22 10.723 98.077 111.66 162.04 230.16
23 10.605 98.113 112.62 124.56 186.96
24 | 10.998 | 81.325 | 112.24 | 124.26 | 138.64 |
A clonal selection algorithm for the DED problem described above has been applied to five-unit test systems with non- smooth fuel cost function to demonstrate the performance of the proposed method. The five-unit test system [17] with non- smooth cost functions used to demonstrate the performance of the proposed method. The optimal generation dispatch with ramp rate limits is shown in Table 1. The simulations were carried out on a PC with a Pentium IV 2.8-GHZ processor. The software was developed using the MATLAB 7.01. The best solution obtained through the proposed method is compared to those reported in the recent literature. The number of clones will depend the fitness value. In order to obtain best perfor- mance the mummer of population is taken 20. The number of
Figure 2 Convergence characteristics of 5 units test system
SA [15] 47356 351.98
PSO[26] 50124 258.00
DE[17] 43213 376
Clonal 43446.22 242
An efficient algorithm based on one of the soft computing tools namely clonal selection algorithm is proposed in this paper to solve dynamic economic dispatch problem. The pre- sented evaluation fuction model and optimally selected clones have enhanced the performance of the clonal selection algo- rithm. The effectiveness of the presented algorithm using
5unit test system has compared with reported in literature. The solution quality, reliability and computational efficiency show the superiority of the presented clonal selection algo- rithm. This algorithm can also be extended to solve DED, by inclusion of more inequality constraints such as line flows, voltage constraints, spinning reserves, etc. to obtain a more accurate dispatch solution for practical power system prob- lems.
The authors wish to thank to Veer Surendra Sai University of Technology, Burla, Orissa, India for providing the facilities to cary out this research work.
IJSER © 2011
International Journal of Scientific & Engineering Research Volume 2, Issue 12, December-2011 5
ISSN 2229-5518
[1] G.P. Granelli, P.Marannino, M.Montagna, and A. Silvestri,“Fast and
efficient gradient projection algorithm for dynamic generation dis-
patching,”IEE Proc. Generat.Transm. Distrib, vol. 136, No.5, pp.295-
302, September 1989.
[2] F.Li, R.Morgan, and D.Williams, “Hybrid genetic approaches to ramping rate constrained dynamic economic dispatch,” Elect. Power Syst. Res., Vol. 43, No. 2, pp. 97-103, November 1997.
[3] X.S.Han, H.B. Gooi, and D.S.Kirchen, ―Dynamic economic dispatch:
Feasible and optimal solutions,‖ IEEE Trans. Power syst. Vol. 16,No.
1,pp. 22-28,February 2001.
[4] P. Attaviriyanupap,H. Kita, E. Tanaka, and J. Hasegawa, ―A fuzzy- optimization approach to dynamic economic dispatch considering uncertainties,‖ IEEE Trans. Power Syst.,Vol. 19,No. 3, pp. 1299-
1307,August 2004.
[5] Z.-L. Gaing, ―Particle swarm optimization to solving the economic dispatch considering the generation constraints,‖ IEEE Trans. Power Syst., Vol. 18,No. 3,pp. 1187-1195,August 2003.
[6] K.S. Hindi, and M.R. Ab Ghani., ―Dynamic economic dispatch for large-scale power systems: A Lagrangian relaxation approach,‖ Elect. Power Energy Syst., Vol. 13, No. 1, pp. 51-56, February 1991.
[7] D.C. Walters, and G.B. Sheble,‖Genetic algorithm solution of eco-
nomic dispatch with valve point loadings,‖ IEEE Trans. Power Syst.,
Vol.8, No. 3,pp. 1325-1331,august 1993.
[8] H.T. Yang, P.C. Yang, and C.L. Huang, ― Evolutionary programming based economic dispatch for units with non smooth incremental fuel cost functions,‖ IEEE Trans. Power Syst., Vol. 11, No. 1, pp. 112-118, February 1996.
[9] N. Sinha, R. Chakrabarti, and P.K. Chattopadhyay, ―Evolutionary programming techniques for economic load dispatch,‖ IEEE Trans. Evol. Computat., Vol. &, No. 1,pp. 83-94, February 2003.
[10] D.N. Simopoulos, D. Kavatza, and C.D. Vournas, ―Unit commitment by an enhanced simulated annealing algorithm,‖ IEEE Trans. Power Syst., Vol. 21, No. 1,pp. 68-76, February 2006.
[11] M.M. Lin, F.S. Cheng, and M.T. Tsay, ―An improved tabu search for economic dispatch with multiple minima,‖ IEEE Trans. Power Syst., Vol. 17, No.1, pp.108-112, February 2002.
[12] P. Attaviriyanupap, H. Kita, E. Tanaka, and J. Hasegawa, ―A hybrid EP and SQP for dynamic economic dispatch with non smooth fuel cost function,‖ IEEE Trans. Power Syst., Vol. 17, No. 2, pp. 411-416, May 2002.
[13] T.A.A Victoire, and A.E. Jeyakumar,―Deterministically guided PSO
for dynamic dispatch considering valve-point effect,‖ Elect. Power
Syst. Res., Vol. 73, No. 3, pp.313-322, March 2005.
[14] T.A.A Victoire, and A.E. Jeyakumar, ―Reserved constrained dynamic
dispatch of units with valve point effects,‖ Elect. Trans. Power Syst.,
Vol. 20, No. 3, pp.1273-1282, August 2005.
[15] C.K. Panigrahi, P.K. Chattopadhyay,R Chakrabarti, and M. Basu,
―Simulated annealing technique for dynamic economic dispatch,‖
Elect. Power Compon. Syst., Vol. 34, No. 5, pp. 577-586, May 2006.
[16] R. Storn, and K. Price, ―Differential evolution – a simple and efficient heuristic for global optimization over continuous space,‖ J. Global Optim, Vol. 11, pp.341-359, December 1997.
[17] R. Balamurugan,S.Subramanian, ―Differential Evolution-Based Dy-
namic Economic Dispatch of Generating Units with Valve–Point Ef-
fects,‖ Elect. Power Components and System,Vol 36,pp.828-843,2008. [18] L. N.de Castro and J. Timis, ―Artificial Immune System: A Novel Paradigm to Pattern Recognition,‖ In Artificial Neural Networks in Pattern Recognition, SOCO-2002, University of Paiselely, U.K.pp67-
84, 2002.
[19] D. Dasgupta and N.A. Okline, “Immunity-Based Survey,” in the Proceedings of IEEE international Conference on System, Man and Cybernetics, vol, pp.369-374, oct.1997.
[20] L.N.de Castro, F.J. Zuben, “Learning and Optimization using through the clonal Selection Principle,’’IEEE Trans. on Evolutionary Computation, vol.6 pp.239-251, June 2002.
[21] L.N.de Castro and F.J. Von Zuben, “Artificial Immune System: Part 1-
Basic Theory and Applications,’’ Technical report,TR-DCA 01/99
Dec.1999.
[22] F.M.Burnet, “The Clonal Selection Theory of Acquired Immunity,
Cambridge University Press Cambridge, U.K, 1959.
[23] B.K. Panigrahi, S.R. Yadav, S. Agrawal, and “M.K. Tiwari, “A clonal algorithm to solve economic load dispatch”, Elect. Power syst. Res., Vol 77, pp 1381-1389 2007.
[24] T.K.A Rahman, S.I Suliman, and I.Musurin, “Artificial Immune-Based Optimization Technique for Solving Economic Dispatch in Power System”, Springer-Verlag, pp 338-345, 2006.
[25] T.K.A Rahman, Z.M. Yasin, W.N.W. Abdullah, “Artificial Immune Based for Solving Economic Dispatch in Power system”, IEEE Na- tional Power and Energy Conference Proceedings, Malaysia 2004 pp
31-35.
[26] R.Chakrabarti, P. K.Chattopadhyay,M.Basu,C.K.Panigrahi, “Particle Swarm Optimization Technique for Dynamic EconomicDis- patch”,Vol.87,pp48-54,Sept.2006,IE(India)
IJSER © 2011