An Ant Colony Optimization (ACO) algorithm solution to Economic

An Ant Colony Optimization (ACO) algorithm solution to Economic Load Dispatch (ELD) problem.

Vlachos Aristidis

Department of Informatics University of Piraeus

80, Karaoli & Dimitriou Str. 18534 Piraeus – Greece

Abstract: This paper presents the solution of the Economic Load Dispatch (ELD) problem using an Ant Colony Optimization (ACO) algorithm: the Ant System with elitist strategy ( ). The idea of the elitist strategy in the context of the Ant System is to give extra emphasis to the best path found so far after every iteration. When the trail levels are updated, this path is treated as if a certain number of ants, namely the elitist ants, had chosen it. The is applied to sample ELD problem composed of six generators. The results of the are compared with those of the Genetic Algorithm (GA), Particle Swarm Optimization (PSO) and Gradient-Based approach.

eAS

eAS

eAS

Keywords: Economic Load Dispatch, Ant Colony Optimization, Ant System, Ant System with elitist strategy. 1 Introduction The Economic Load Dispatching (ELD) problem is one of key problems in power operation and planning. The ELD problem may be expressed by minimizing the total generation cost while satisfying an equality constraint and several inequality constraints.

The Economic Load Dispatch (ELD) problems have complex and non linear characteristics. To solve these problems, various salient mathematical approaches have been suggested for the past decades. Recently, heuristic optimization techniques, such as simulating annealing [1], tabu-search [2], genetic algorithms [3] and Particle Swarm Optimization (PSO) [4], are considered as realistic solution shcemes in power system optimization problems. More recently the focus has shifted to the use of Ant Colony Optimization (ACO) algorithms. In this paper, a cooperative agent algorithm, Ant System with elitist strategy, [5], which was inspired by the observation of the behavior of ant colonies, is investigated.

eAS

The main characteristics of are an additional positive feedback, distributed computation and the use of a constructive greedy heuristic. Additional feedback accounts for rapid discovery of good solutions, distributed computation avoids premature convergence and the greedy heuristic helps to find acceptable solutions.

The concept of elitism can also be found in genetic algorithms. In general, in a genetic algorithm the fittest individual of a generation (best solution of an iteration) will with positive probability not be included in the next generation, if the genetic operators selection, recombination and mutation are applied. In that case the genetic information of that individual will be lost. Therefore, the idea of elitism is to preserve the fittest individual of a generation. As a consequence, local search aspects (exploitation) become more important while global search aspects (exploration) become less important. This potential drawback is also valid for the ant system.

eAS

The rest of the paper is organized as follows: In section 2, we review the principal framework of the ACO approach, Ant System and Ant System with elitist strategy algorithm in details. In section 3, we present the formulation of Economic Load Dispatch problem. In section 4, we present the implementation of for ELD problems. The paper ends with case studies and conclusions.

eAS

2 Ant Colony Optimization (ACO) 2.1 Overview ACO [6], developed by Dorigo et al, is a discrete combinational optimization algorithm inspired by the foraging behavior of ants. These ants could manage to construct the shortest path from their

Proceedings of the 10th WSEAS International Conference on SYSTEMS, Vouliagmeni, Athens, Greece, July 10-12, 2006 (pp160-167)

colony to the feeding source through the use of pheromone trails. An ant leaves some quantity of pheromone on the ground when it travels and marks the path by a trail of this substance. The pheromone would evaporate at a certain rate as times goes on. The next ant will sense the pheromone remained in different paths and choose the one with a probability proportional to the amount of pheromone. Then the ant follows the path and leaves its own pheromone. This is a positive feedback process that favors the path along which more previous ants have traveled. Within ACO, the optimization problem is represented as a graph, , called construction graph, consisting of nodes and where is the set of edges.

) , ( LCG =

C L

An ant is a simple agent which iteratively solves a define problem. Intermediate solutions of the problem can be considered as states s . At each step ant is moving from state to state under action with probability

. The agents make local changes in the environment marking it with pheromones. All paths are adjusted at each step by increasing the amount of pheromones on the path which is the best solution and decreasing it on the others. The higher probability of the agent’s moves corresponds to the states with greater pheromone amount. The amount of pheromones at each step is calculated as follows:

k rs 1 r +s

ta

) 1 r ,r ,r ( +saskT

) 1 r ,r () 1 r ,r ( · ) 1 () 1 r ,r ( +∆++−=+ ssssss ττρτ (1)

The total pheromone amount deposited by ants at the current step is:

) 1 r ,r (1

) 1 r ,r ( +∆∑=

=+∆ sskn

kss ττ (2)

The probability that the ant will choose transition from state to state is determined by the roulette-wheel method:

thk

rs 1 r +s

) i ,r (

) 1 r ,r () 1 r ,r ,r (

ss

rk

i

sssaskT

τ

τ

∑

Ν∈

+=+ (3)

rk

Ν is the feasible neighborhood of ant when

located at node

k

r . The most important component of Ant

Colony Optimization is the management of pheromone trails which are used, in conjunction with an objective function, for constructing new solutions. Pheromone levels give a measure of how desirable it is to insert a given element in a solution.

Pheromone trails are used for exploration and exploitation. Exploration is the ability of the algorithms to search broadly through the search space of the problems and exploitation is the ability of the algorithms to search locally around good solutions that have been previously found.

The mathematical formulations of the ACO algorithms presented in this paper, namely Ant System (AS) and Ant System with elitist strategy ( ), are given in the following sections. eAS 2.2 Ant System (AS) The Ant System [7], is a general-purpose meta-heuristic algorithm that could be used to solve various combinatorial optimization problems. The characteristics of an Ant System algorithm are as follows: • It is a natural algorithm since it is based on

behaviour of ants in establishing paths from their colony to feeding sources and back ;

• It is parallel since all the ants (agents) in the Ant System move simultaneously, independently and without a supervisor;

• It is cooperative since each agent chooses a path on the basis of information (pheromone trails) laid by the agents which have previously selected the same path;

• It is robust since it can be applied with only minimal changes to other combinatorial optimization problems.

Solutions in AS are constructed as follows: At each construction step, an ant goes from node to node with probability that is computed as:

k ij

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧Ω∈

∑ Ω∈=

otherwise 0

]ih[ · )](ih[h

]ij[ · )](ij[jif

nt

nt

ijk

Pβατ

βατ

(4)


)(ij tτ is the memoirist information. This kind of information is modified during the run of the algorithm. It only depends on two factors: The number of ants that followed that edge ( , ) and the goodness of the path found by those ants. The memoirist information,

i j

)(ij tτ , is the concentration

of pheromone associated with edge ( i , ) in iteration .

jt

ijn is the heuristic information. It depends only on the problem that we want to solve. Usually, this information is calculated before the beginning of the run of the algorithm. It represents the a priori quality of the edge. Given n-city Traveling Salesman Problem (TSP) with distances , called visibility, is the inverse of the distance

ijd ijn

ij

1ij d

n = .

α and β are two adjustable parameters that control the relative weight of trail intensity )(ij tτ ,

and visibility . ijnΩ is the set of the nodes, that has not been visited yet. At the end of an iteration (i.e. each ant has generated a solution) the pheromone on each edge is updated. The pheromone updating equation is given [8]:

)(ij)(ij · ) 1 () 1 (ij ttt ττρτ ∆+−=+ (5)

] 1 ,0 (∈ρ is the evaporation rate. )(ij tτ∆ is a function of the solutions found at

iteration t and is given by

)(1

)(ij tijkn

kt ττ ∆∑

==∆ (6)

where: n is the number of ants.

)(tijk

τ∆ is the pheromone addition laid on edge ( ,

) by ant, at the end of iteration . This is given by:

i

j thk t

⎪⎪⎩

⎪⎪⎨

⎧

=∆

otherwise 0

) j i, ( edgeon k travelsant k)(

ifLQ

tijk

τ (7)

where: Q is the quality of pheromone laid by an ant per tour. is the tour length of ant . kL k 2.3 Ant System with elitist strategy ( ) eASThe quality of the solutions produced by the Ant System could be improved using the so-called elitist ants [9]. The idea of the elitist strategy in the context of the Ant System is to give extra emphasis to the best path found so far after every iteration. When the trail levels are updated this path is treated as if a certain number of ants, namely the elitist ants, had chosen that path. The Elitist Ant concept is almost the same as the Ant-Cycle approach [10]. The only difference introduced is the addition of an element into summation (in equation 5):

)()(ij)t(ij · ) 1 () 1 t(ij tije

et τττρτ ∆+∆+−=+ (8)

⎪⎪⎩

⎪⎪⎨

⎧

=∆

otherwise 0

foundsolution best the

ofpart is ) j i, ( edge

)(best)(

iftL

Q

tije

τ (9)

where: e is the number of elitist ants.

)(best tL is the tour length of best solution found. 3 Formulation of Economic Load

Dispatch (ELD) problem The objective of an ELD problem is to minimize the total fuel cost at thermal plants, [11], while satisfying an equality constraint and several inequality constraints. The objective function Ψ can be described as follows:

NGiPFNG

i ..., ,1 ), i (i

1=∑

==Ψ (10)

) i (i PF is the individual generation production

cost in terms of its real generation power subject to the constraint of equality in real power balance:

iP

0D-i1

=∑=

PPNG

i (11)


the network loss is not considered in this paper for simplicity, and the inequality of real power limitation on the generator outputs:

max iimin i PPP ≤≤ (12)

iP is the fuel cost of the generator. is the number of generators in the system. is the total current system load demand.

thi NG

DP

In equation (8) the generator cost curves are quadratic polynomial:

2Gi · iGi · ii) i (i PcPbaPF ++= (13)

ia and are the cost coefficients of the

generator. ib ic thi

4 Implementation of for ELD problems

eAS

4.1 Expression approach For every generator the area of its power limits is divided in discrete values. The division can be done in various ways. In this paper we can divide all fragments in equal number of sub-fragments. So far, every generator we have done did not have a continuous fragment of power but discrete definite set depending on the separation that has taken place. The algorithm works like this: each one ant starts from the first generator and selects a power level (Fig. 1). Power Levels 1 2

Generato

Fig. 1: Movement of Generators an

Then, it goes to the next and chooses another power level for that generator and this is repeated until it reaches the last generator (Fig.1). At the end the total cost is calculated in order to decide whether the solution is satisfactory or not. Let us suppose that an ant is in generator and must choose a power level for it, according to the probability distribution called random-proportional rule:

k ij

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧∈

∑

∈=

otherwise 0

)(j ]ij[ · )](ij[

)(j

]ij[ · )](ij[

)(

tik

jifnt

tjk

j

nt

tijk

P

βατ

βατ

(12)

where: j is a power level that is randomly chosen

according to probability equation (12).

ik

j is )(k ttabuN − – set of power levels where

ant can go from level . k iktabu is the tabu list from ant in iteration . k t

α , β are parameters that control the relative

weight of trail intensity, ijτ and visibility, . ijnIn the classical problem TSP, the visibility, , is defined as the inverse of the distance between two cities . So, it could be also used here as the inverse of cost for the particular power level

ijn

) , ( ji

)i(i

1ij PF

n = . ijτ is the pheromone quantity that is

found in the edge that connects every generator with power level. An extra number of ants, the elitist ants, amplify the best cost, , found up to the current

iteration

bestCost

t , with quantity best

· Cost

Qe and so the

elitist pheromone update rule is given:


Nrs

an Ant Betweend Power Levels

…

G

∑=

∆+∆+−=+n

it

ije

etijk

1)( · )()t(ij · ) 1 () 1 t(ij τττρτ (13)

where:

] 1 ,0 (∈ρ is the evaporation rate.

)(tijk

τ∆ is defined as in equation (6).

)(tije

τ∆ is defined as follows:

⎪⎪⎩

⎪⎪⎨

⎧

∉

∈=∆

bestT ) j i, ( 0

bestT ) j i, ( best)(

if

ifCost

Q

tije

τ (14)

where:

bestCost is the best solution during the current iteration t .

bestT is the list that keeps track for the best solution. Further down, we present the steps of the algorithm used for the ELD solution. 4.2 The algorithm • Step 1: Define (discrete) power for every

generator. For every generator and for every power level we calculate the visibility

)(1

iiij PF

n = . Define the pheromone, giving

it a large value, in all edges that connect every generator with the power level respectively. Define the number of ants and the number of iterations.

• Step 2: For every ant and for every generator select a power level based on random-proportional transition rule.

• Step 3: Calculate the cost for all ants based on the division of power levels and save the best.

• Step 4: Renew pheromone using the elitist pheromone update rule.

• Step 5: Repeat the procedure from step 2 until a specific number of iterations are completed.

5 Case studies The efficiency and effectiveness of the proposed

algorithm has been applied to ELD problem with six generators. The results obtained are compared with Particle Swarm Optimization (PSO) algorithm [4], Genetic Algorithm [12], and Gradient-Based approach [13].

eAS

Table 1 shows the cost functions and the related minimum / maximum operating points of six generators.

Table 1: Cost Coefficients of Test System

with Six Generators .

Unit ia ib ic min iP max iP

1P 0 200 37,5 0,50 2,00

2P 0 175 175 0,20 0,80

5P 0 100 625 0,15 0,50

8P 0 325 83,4 0,10 0,35

11P 0 300 250 0,10 0,30

13P 0 300 250 0,12 0,40 The following parameters have been chosen heuristically: • number of ants = 6, • a = 2 • p = 2 • e = 1 • p = 0,5 • 0τ = 0,000001 • number of iterations = 200 • Q = 1 Table 2 represents the number of iterations and the corresponding best fuel cost.

Table 2: Iterations and Corresponding

Iterations Best Fuel Cost [$ / h]

1 805,762925

5 663,613715

10 606,163915

20 535,703565

30 535,703565

40 535,703565

50 535,703565

60 535,703565

70 535,703565

80 535,703565

90 535,703565

100 535,703565

150 535,703565


200 535,703565

The lowest fuel cost as a function of the iteration number is shown in Fig. 2.

520 560 600 640 680 720 760 800

0 40 80 120 160 200

Iterations

Bes

t Fu

el C

ost

($ /

h)

)

Fig. 2: Finding the Lowest Fuel Cost by Using the Algorithm eAS

In Table 3, the results of application algorithm for six generators test system, are presented and compared with Genetic Algorithm, PSO algorithm and Gradient-Based approach.

eAS

Table 3: Simulation Results for Test System with Six Generators .

Genera

tor ia ib ic min iP max iP Initial results

1P 0 200 37,5 0,50 2,00 0,9921 1,9900

2P 0 175 175 0,20 0,80 0,8000 0,2000

5P 0 100 625 0,15 0,50 0,5000 0,1500

8P 0 325 83,4 0,10 0,35 0,2000 0,1000

11P 0 300 250 0,10 0,30 0,2000 0,1000

13P 0 300 250 0,12 0,40 0,9921 0,1200

Fuel cost ($ / h) using eAS 901,916 535,704

Fuel cost ($ / h) using PSO 800,410

Fuel cost ($ / h) using GA 800,805

Fuel cost ($ / h) using Gradient-Based approach 804,583

From Table 3 it appears that the best fuel cost accrues with the application of the algorithm. eASTable 4 shows the relationship between the changes of parameter (while the others remain constant) and the equivalent best fuel cost.

a

Table 4: Changes of Parameter and a

Corresponding best Fuel Cost

a Best Fuel Cost [$ / h]

1 584,809680

2 535,703565

3 515,044665

4 504,662815

5 504,662815

6 498,544665

7 491,037165

8 484,949,015

9 484,949,015

10 484,949,015

15 484,949,015

20 482,942715

The lowest fuel cost as a function of parameter is shown in Fig. 3.

a

480 500 520 540 560 580 600

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Parameter α

Bes

t Fu

el C

ost

Fig. 3: Lowest Fuel Cost as a Function

of Parameter . a Table 5 shows the relationship between the change of the elitist ant number and the equivalent best fuel cost. The algorithm was applied to the data below: eASnumber of generators = 6 number of ants = 6 number of iterations = 1000 • α = 1 • β = 1 • ρ = 0.5 • Ζο = 0.000001


• Q = 1

Table 5: Relationship between the e Number and Best Fuel Cost .

e Best Fuel Cost [$ / h]

2 611,341265

3 605,671265

4 604,354965

5 598,354965

6 590,860515

7 584,910515

8 591,298665

9 579,818665

10 579,818665

20 567,900515

30 562,510515

The lowest fuel cost as a function of the number of elitist ants is shown in Fig. 4.

e

560

580

600

620

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

Number of e

Bes

t Fu

el C

ost

Fig. 4: Lowest Fuel Cost as a Function of the Number e

From Fig. 3 we can see that if the number of elitist ants increase, we have best values of fuel cost. 6 Conclusion In this paper, we have studied the Economic Load Dispatch problem (ELD). In the literature, some heuristic approaches have been developed for deliving quality solutions to the ELD problem. In this paper, we used a specific ACO system called Ant System with elitist strategy ( ) for finding solutions to the ELD problem.

eAS

We have also compared the performance of the proposed algorithm on a set of the ELD problem instances with the GA approach, Particle Swarm Optimization (PSO) and Gradient-Based approach. Computational results have evinced that the algorithm yields solution values that are comparable to those of GA, PSO algorithms and Gradient-Based approach, and the proposed algorithm is very fast and able to solve very large instances.

eAS

References: [1] Worg, K.P., Fung, C.C., Simulated-

Annealing Based Economic Dispatch Algorithm, IEEE Proc. Pt. C, 1993, PP. 509-515.

[2] Whei-Min Lin, Fu-Sheng Cheng and Ming-Tong Tsay, 2002, An Improved Tabu Seacrh for Economic Dispatch with Multiple Minima, IEEE Trans. Power Syst, Vol 17, No. 1, PP. 108-112.

[3] Basckar, S., Subbaraj, P., Rao, M.V.C., 2003, Hybrid Real Coded Genetic Algorithm Solution to Economic Load Dispatch Problem, Computers and Electrical Engineering, 29, PP. 407-419.

[4] Abido, M.A., 2002, Optimal Power Flow Using Particle Swarm Optimization,

Electrical Power and Energy Systems, 24, PP. 563-571.

[5] Dorigo, M., Maniezzo, V. and Colorni, A., 1996, Ant System: Optimization by a Colony of Cooperating Agents. IEEE, Transaction on Systems, Man, and Cybernetics 26 (1), PP. 29-41.

[6] Dorigo, M., Di Caro, G. 1999. The Ant Colony Optimization Meta-Heuristic. In New Ideas Optimization, D. Corne, M. Dorigo and F. Glover Editors, Mc Graw-Hill.

[7] Bonabeau, E., Dorigo, M., Theraulaz, G., 1999 Swarm Intelligence, From Natural to Artificial Systems, a Volume in the Santa Fe Institute Studies in the Sciences of Complexity, Oxford University Press.


[8] Dorigo, M., Strutze, T., 2004. Ant Colony Optimization, MIT press.

[9] Panta, L., 2002. Modeling Transportation Problems Using Concepts of Swarm Intelligence and Soft Computing. Ph. D. Thesis, Virginia Polytechnic Institute and State University.

[10] Colorni, A., Dorigo, M., and Maniezzo, V., 1991. Distributed Optimization by Ant Colonies, in Proc. First Europe. Conf. Artificial Life, F. Varela and P. Bourgine, Eds. Paris, France, Elsevier PP. 134-142.

[11] Osman, M.S., Abo-Sinna, M.A., Mousa, A.A., 2004. A Solution to the Optimal Power

Flow Using Genetic Algorithm. Applied Mathematics and Computation 155, PP. 391-405.

[12] Lai L.L., Ma J.T., 1997. Improved Genetic Algorithms for Optimal Power Flow Under both Normal and Contingent Operation States. Int. J Elec. Pwr. Energy Syst, 19 (5), PP. 287-292.

[13] Lee, K., Park, Y., Ortiz, J.1985. A United Approach to Optimal Real and Reactive Power Dispatch. IEEE Trans. Pwr. Appar. Syst., 104 (5), PP. 1147-1153.


An Ant Colony Optimization (ACO) algorithm solution to Economic

Documents

Transcript of An Ant Colony Optimization (ACO) algorithm solution to Economic