An ant colony system approach for unit commitment problem

An ant colony system approach for unit commitment problem

Sishaj P. Simon, Narayana Prasad Padhy, R.S. Anand *

Department of Electrical Engineering, Indian Institute of Technology, Roorkee, Uttaranchal 247 667, India

Received 1 October 2004; received in revised form 19 October 2005; accepted 13 December 2005

Abstract

Ant colony system (ACS) model is more suitable for solving combinatorial optimization problem, so ACS has been applied to the hard

combinatorial Unit commitment problem (UCP). Here, a parallel can be drawn of ants finding the shortest path from source (nest) to its destination

(food) and solving UCP to obtain the minimum cost path (MCP) for scheduling of thermal units for the demand forecasted. Multi-stage decisions

give ant search a competitive edge over other conventional approaches like dynamic programming (DP) and branch and bound (BB) integer

programming techniques. Before the artificial ants starts finding the MCP, all possible combination of states satisfying the load demand with

spinning reserve constraint are selected for complete scheduling period which is called as the ant search space (ASS). Then the artificial ants are

allowed to explore the MCP in this search space. The proposed model has been demonstrated on a practical ten unit system and a brief study has

been performed with respect to generation cost, solution time and parameter settings on a numerical example with four unit system.

q 2006 Published by Elsevier Ltd.

Keywords: Combinatorial optimization; Ant colony system; Dynamic programming; Branch and bound

1. Introduction

The unit commitment problem is well known in power

industry and has the potential to save millions of dollars per

year in terms of economic operation. To determine the

optimum schedule of generating units (i.e. switching on and

off of N generating units over a period of time for the demand

forecasted to be served) by minimizing the over all cost of the

power generation while satisfying a set of system constraints is

the main objective of a hard combinatorial unit commitment

problem. To ‘commit’ a generating unit is to ‘turn it on’ that is

to bring the unit up to speed, synchronize it to the system, and

connect it so it can deliver power to the network. The problem

with ‘commit enough units and leave them on line’ is one of the

economics. It is quite expensive to run too many generating

units. A great deal of money can be saved by turning units off

(decommiting them) when they are not needed. The generic

UCP can be formulated as to minimise operational cost subject

to minimum up-time and down-time constraints, crew

constraints, ramp constraints, unit capability limits, deration

of units, unit status, generation constraints and reserve

0142-0615/$ - see front matter q 2006 Published by Elsevier Ltd.

doi:10.1016/j.ijepes.2005.12.004

* Corresponding author. Tel.: C91 1332 285590; fax: C91 1332 273560.

E-mail addresses: [email protected] (S.P. Simon), nppeefee@iitr.

ernet.in (N.P. Padhy), [email protected] (R.S. Anand).

constraints [1]. The exact solution of the UCP can be obtained

by complete enumeration, namely, dynamic or integer

programming. The drawbacks of these methods are the

enormous computational time which increases exponentially

with the number of units, and the large memory requirement.

Modifying the above-mentioned conventional techniques has

helped in improving the accuracy of the solution with respect to

well-timed decision-making [2]. So there is a balance between

the accuracy of the solution with respect to the solution time.

Traditional and conventional methodologies such as exhaus-

tive enumeration, priority listing, dynamic programming,

integer and linear programming, branch and bound method,

lagrangian relaxation, interior point optimization etc. are able

to solve UCP with success in varying degree. So efforts are

made recently by the application of simulated annealing,

hybrid methods, expert system, artificial neural network, fuzzy

system, genetic algorithms and swarm intelligent system for

obtaining the solution of UCP [3,4].

Researchers understood the optimization capabilities of the

behavior of ant colonies and in analysing found that ants are

capable of finding the shortest path from food sources to the

nest, which can be applied to different hard combinatorial

problems such as traveling salesman problem and quadratic

assignment problem [5,6]. Also multi-stage decision-making

ant search is a heuristic search technique applied in searching

the complete enumeration space called the ant search space,

which can solve the limitations of the multi-stage DP. In-Keun

Electrical Power and Energy Systems 28 (2006) 315–323

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S.P. Simon et al. / Electrical Power and Energy Systems 28 (2006) 315–323316

Yu et al. and N.S. Sisworahardjo and El-keib [7,8] have

attempted in applying ant algorithms to UCP and indicated the

applicability in terms of economy for a small system of few

generating units. Shyh-Jier Huang applied ACS techniques for

the enhancement of hydroelectric generation scheduling [9],

where ants are positioned on different nodes in the search space

of the hydro generation scheduling problem. Also in the work

of Shi Libao et al. [10], concept of random perturbation

behavior with a magnifying factor and mutation rate is

incorporated with the basic Ant System Model [5]. S.P.

Simon et al. has solved UCP using ACS with its exploration

and exploitation ability. It is also implemented by continuous

flowing ants and found that the solutions obtained are

economical [11]. This paper implements movement of ants in

the search space and also discusses the accuracy of the solution

with respect to the solution time. The proposed model is

compared with the conventional single stage DP and BB

integer programming approaches.

2. Background of ACS

MarcoDorigo and his colleague’s proposed ant algorithms in

the year 1991 as a multi-agent approach to difficult combinator-

ial optimization problems. Ant system (AS) originally a set of

three algorithms called ant-cycle, ant-density, and ant-quantity

was first proposed in Dorigo’s doctoral dissertation. The major

merit of AS, whose computational results were promising, was

to stimulate a number of researchers, to develop extensions and

improvements of its basic ideas so to produce more performing,

and often state-of-the-art, algorithms which has motivated to

apply ant algorithms to UCP [5].

Fig. 1. The above sketch shows how real ants find a shortest path. (A) Ants arrive at

The choice is random. (C) Since ants move at approximately constant speed, the ant

than those which choose the upper, longer, path. (D) Pheromone accumulates at a

proportional to the amount of pheromone deposited by ants.

2.1. ACS

An ant colony system (ACS) is a population based heuristic

algorithm on agents that simulate the natural behavior of ants

developing mechanisms of cooperation and learning which

enables the exploration of the positive feedback between

agents as a search mechanism [5,6,12–14].

2.2. Biological ACS

Social insects like ants, bees, wasps and termites work by

themselves in their simple tasks, independently of other

members of the colony. However, when they act as a

community, they are able to solve complex problems emerging

in their daily lives, by means of mutual cooperation. This

emergent behavior of a group of social insects is known

‘swarm intelligence’. An important and interesting behavior of

ant colonies is their foraging behavior and, in particular, how

ants can find shortest paths between food sources and their nest.

While walking from food sources to the nest and vice versa,

ants deposit on the ground a chemical substance called

pheromone, forming in this way a pheromone trail. The sketch

shown in Fig. 1 gives a general idea how real ants find a

shortest path. Ants can smell pheromone and, when choosing

their path, they tend to choose, in probability, paths marked by

strong pheromone concentrations. The pheromone trail allows

the ants to find their way back to the food by their nest mates.

The emergence of this shortest path selection behavior can be

explained in terms of autocatalysis (positive feedback) and

differential path length which uses a simple form of indirect

communication mediated by pheromone laying, known as

a decision point. (B) Some ants choose the upper path and some the lower path.

s which choose the lower, shorter, path reach the opposite decision point faster

higher rate on the shorter path. The number of dashed lines is approximately

S.P. Simon et al. / Electrical Power and Energy Systems 28 (2006) 315–323 317

‘stigmergy’ through the environment, either by physically

changing, or by depositing something on the environment [12].

2.3. Artificial ACS (AACS)

In AACS, the use of: (i) a colony of cooperating individuals,

(ii) an (artificial) pheromone trail for local stigmergetic

communication, (iii) a sequence of local moves to find shortest

paths, and (iv) a stochastic decision policy using local

information and no look ahead are the same as real ACS, but

artificial ants have also some characteristics which do not find

their counterpart in real ants [13,14]. They are:

1. Artificial ants live in a discrete world and their moves

consist of transitions from discrete states to discrete states.

2. Artificial ants have an internal state. This private state

contains the memory of the ant‘s past actions.

3. Artificial ants deposit an amount of pheromone, which is a

function of the quality of the solution found.

4. Artificial ants timing in pheromone laying is problem-

dependent and often does not reflect real ant’s behavior. For

example, in many cases artificial ants update pheromone

trails only after having generated a solution.

Essentially, an ACS algorithm performs a loop applying two

basic procedures:

† a procedure specifying how ants construct or modify a

solution for the problem in hand;

† a procedure for updating the pheromone trail.

The construction or modification of a solution is performed

in a probabilistic way. The probability of adding a new term to

the solution under construction is in turn, a function of a

problem-dependent heuristic and the amount of pheromone

previously deposited in this trail. The pheromone trails are

updated considering the evaporation rate and the quality of the

current solution.

3. Problem formulation

The objective of the UCP is the minimisation of the total

generation costs (TGC) for the commitment schedules and is

defined as

TGCZXN

iZ1

XT

tZ1

FCitðPitÞCSTit CSDit$=h (1)

where NZtotal number of generator units; TZtotal number of

hours considered (1 day). FCit(Pit) fuel cost at tth hour ($/h):

they are represented by an input/output (I/O) curve that is

modeled with a polynomial curve (normally quadratic)

FCitðPitÞZ aiP2it CbiPit Cci$=h (2)

where aiZcost coefficient of generator ($/MW!MW); biZcost coefficient of generator ($/MW); ciZcost coefficient of

generator; PitZpower level at tth hour (MW); STitZstart-up

cost at tth hour ($/h); SDitZshut-down cost at tth hour ($/h).

The start-up cost is described by

STit ZTSitFit C 1KeðDitASitÞ� �

BSitFit CMSit (3)

where TSitZturbines start-up energy at ith hour (MBTu), FitZfuel input to the ith generator; DitZnumber of hours down at

tth hour; ASitZboiler cool-down coefficient at tth hour; BSitZboiler start-up energy at tth hour ($/h); MSitZstart-up

maintenance cost at tth hour ($/h).

Similarly the start-down cost is described by

SDit Z kPit (4)

where k is the proportional constant.

3.1. Subject to the following constraints

Power balanceXN

iZ1

Pit ZPDt CPRt (5)

Minimum up-time

0!Tiu%number of hours units Gi has been onKline (6)

Minimum down-time

0!Tid%number of hours units Gi has been offKline (7)

Maximum and minimum output limits on generators

Pminit %Pit%Pmax

it (8)

Ramp rate limits for unit generation changes

PitKPiðtK1Þ%URi as generation increases (9)

PiðtK1ÞKPit%DRi as generation decreases (10)

where PDtZdemand at tth hour; RtZspinning reserve at tth

hour; TiuZminimum up-time; TidZminimum down-time;

URiZramp-up rate limit of unit i (MW/h); DRiZramp-down

rate limit of unit i (MW/h).

Spinning reserve

A sufficient amount of spinning reserve expressed as a

percentage (20%) of total load demand should be maintained.

4. Implementation of ACS model

The ACS approach for solving UCP mainly consists of two

phases. In the first phase, all possible St states at the tth hour that

satisfies the load demand with spinning reserve are found and

continued for complete scheduling period of 24 h which

constitutes the ASS. The ASS, which involves multi-decision

states, is given in Fig. 2. In the initialisation part, the forecasted

load demands and other relevant problem data from the system

is taken for computation. Economic dispatch using Lagrange

multiplier method is used that calculates the generator output

and the production cost including system losses for each hour.

Exhaustive enumeration technique is used to find all possible

combinations of the generating units available. Once the ASS is

defined, the second phase involves the artificial ants allowed to

Fig. 2. An example of multi-decision search space.

Fig. 3. ACS for UCP flow chart.


pass continuously through this search space. Each ant starts its

journey from the initial condition termed as the starting node,

reaches the end stage. So it is a continuous flow of ants. Once an

ant reaches the end stage, a tour is completed and it calculates the

overall generation cost path. For each stage (tth hour), the ant

selects an operational cost (OC) calculated for all N generator

units and thereby minimizing the overall OC. This process is

continued till the time period becomes T and a tour is completed

for that particular ant. Whenever a tour is completed by an

individual ant and if the total generation cost found is lesser than

the minimum cost paths taken by the previous ants, the present

cost path is captured. This procedure is continued for all the

remaining ants available at the starting node, which enables to

trace the optimal path. The ant colony search mechanism can be

mainly divided into initialisation, transition strategy, phero-

mone update rule and parameter setting.

4.1. Initialisation

During initialisation the parameters such as the requisite

number of ants, the relative importance of the pheromone trail

a, relative importance of the visibility b, initial available

pheromone trail t0, a constant related to the quantity of the trail

laid by ants, evaporation factor r, tuning factor etc. have to be

fixed and taken care.

4.2. Transition strategy

The transition probability for the kth ant from one state i to

next state j for ACS model is given by

j Z

arg maxu2Jkif½tiuðtÞ�

a½hiu�bg

Jif q%q0

if qOq0

8><>: (11)

where tijZtrail intensity on edge (i,j); hijZ(1/Cij) called

heuristic function; CijZproduction cost occurred for that

particular stage; aZrelative importance of the trail, aR0;

bZrelative importance of the visibility, bR0; qZa random

variable uniformly distributed over [0,1]; q0Za tunable

parameter (0%q0%1); j2jki is a state that is randomly selected

according to probability,

PkiJðtÞZ

½tiuðtÞ�a½hiJ�

bPl2Jk

i½tilðtÞ�

a½hij�b

((12)

when q%q0 which correspond to an exploitation of the know

ledge available about the problem, that is the heuristic

knowledge about cost between states and the learned knowl-

edge memorized in the form of pheromone trails, whereas qOq0 favors more exploitation. Cutting exploration by tuning q0allows the activity of the system to concentrate on the best

solutions instead of letting it explores constantly. Here, ‘l’ is

the allowable states [12,14].

4.3. Pheromone trail update rule

In ACS, the global trail updating rule is applied only to the

edges belonging to the best tour since the beginning of the trail.

The updating rule is

tijðtÞ) ð1KrÞtijðtÞCrDtijðtÞ (13)

where (i,j)’s are the edgesbelonging toTC, the bestminimumcost

tour since the beginning of the trial, r is a parameter governing

pheromone decay, and change in pheromone is given by

DtijðtÞZ 1=CC (14)

where CC is the cost of TC. This procedure allows only the best

tour to be reinforced by a global update. Local updates are also

performed, so that other solutions can emerge. The local update is

performed as follows: when, while performing a tour, ant k is in

state i and selects a state J2Jki , the pheromone concentration of

Table 1

Parameter settings for the ACS model

a 0 0.1 0.5 1 2 5 7 10

Avg. TGC 26955.51 26953.43 26950.77 26951.41 26960.09 26955.82 26953.99 26952.22

b 0 1 2 5 7 10 15 20

Avg. TGC 27113.46 26958.22 26946.87 26957.00 26977.93 26975.96 26981.49 26986.40

r 0.1 0.3 0.5 0.7 0.9

Avg. TGC 26968.82 26958.27 26955.85 26961.35 26946.95

q0 0.1 0.3 0.5 0.7 0.9

Avg. TGC 27050.89 26978.91 26958.44 26936.03 26931.93

Avg. TGC, average total generation cost in $/day.


(i,j) is updated by the following formula:

tijðtÞ) ð1KrÞtijðtÞCrt0 (15)

wheret0 is the initial value of thepheromone trails. Theflowchart

of the ACS approach to UCP is shown in Fig. 3.

4.4. Parameter settings

Good convergence behavior can be obtained by proper

selection of parameters. The selection of parameters a,b,r,t0(initial pheromone value) and q0 affect directly or indirectly the

computation of the probability in the formula. Initially the

settings are done on a numerical four-unit system whose

generator characteristics and load demand is given in Appendix

A. It had been tested for each parameter taking several values

within a boundary limit, all the other being constant (default

Table 2

Transitional cost for four unit generating system

Period (h) Load demand

(MW)

Dynamic Programming

Unit status Transition c

($/h)!103

1 410 0111 0.8643

2 500 0111 1.0796

3 575 0111 1.2770

4 650 1111 1.7834

5 555 1111 1.1879

6 450 1110 0.9632

7 400 1110 0.8504

8 445 1111 0.9353

9 535 1111 1.1395

10 600 1111 1.3009

11 540 1111 1.1515

12 495 1111 1.0460

13 450 1111 0.9461

14 516 1111 1.0946

15 585 1111 1.2626

16 625 1111 1.3660

17 530 1111 1.1276

18 465 1111 0.9788

19 405 1111 0.8512

20 492 1111 1.0392

21 568 1111 1.2200

22 610 1111 1.3267

23 550 1111 1.1757

24 483 1111 1.0188

Total cost $/day)!104 2.698640

Time taken (s) 2.04

settings; in each experiment only one of the values is changed),

over 10 simulations for each setting is performed in order to

achieve some statistical information about the average

evolution. The range of interval considered for each parameters

are a{0 10}, b{0 10}, r{0.1 0.9} and q0{0%q0%1}. The initial

trail level is set as t0Z1. The number of ants allowed to pass

through the search space is taken as 50. The results based on

parameter settings for the ACS model is presented in Table 1.

ACS shows for b a monotonic decrease of the cost up to bZ2.

After this value the average cost start to increase. The test on the

other parameters shows that a has an optimum around 0.2 and r

should be set as high as possible. The tuning factor when set to

high value performed well which indicates that the model

incorporates more exploitation of the knowledge available

about the problem. The results obtained in this experiment are

consistent according to the level of understanding of this model:

Branch and bound Ant colony system

ost Unit status Transition cost

($/h)!103Unit status Transition cost

($/h)!103

1111 1.2116 1111 1.2116

1111 1.0575 1111 1.0575

1111 1.2374 1111 1.2374

1111 1.4334 1111 1.4334

1111 1.1879 1111 1.1879

1110 0.9632 1110 0.9632

1110 0.8504 1110 0.8504

1111 0.9353 1111 0.9353

1111 1.1395 1111 1.1395

1111 1.3009 1111 1.3009

1111 1.1515 1111 1.1515

1111 1.0460 1111 1.0460

1111 0.9461 1111 0.9461

1111 1.0946 1111 1.0946

1111 1.2626 1111 1.2626

1111 1.3660 1111 1.3660

1111 1.1276 1111 1.1276

1111 0.9788 1111 0.9788

1111 0.8512 1111 0.8512

1111 1.0392 1111 1.0392

1111 1.2200 1111 1.2200

1111 1.3267 1111 1.3267

1111 1.1757 1111 1.1757

1111 1.0188 1111 1.0188

2.692194 2.692194

77.28 3.42

Table 3

Transitional cost for 10 unit generating system

Period (h) Load demand

(MW)

Dynamic programming Branch and bound Ant colony system

Unit status Transition cost



($/h)!103

1 1170 1111101001 2.6381 1111111001 2.7258 1111111101 2.8496

2 1250 1111111001 2.7191 1111111001 2.6061 1111111101 2.6062

3 1380 1111111001 2.8894 1111111011 2.9817 1111111101 2.8870

4 1570 1111111101 3.4260 1111111111 3.4098 1111111111 3.3968

5 1690 1111111101 3.6074 1111111111 3.5787 1111111111 3.5787

6 1820 1111111101 3.9488 1111111111 3.9064 1111111111 3.9064

7 1910 1111111111 4.2474 1111111111 4.1464 1111111111 4.1464

8 1940 1111111111 4.2297 1111111111 4.2297 1111111111 2.2297

9 1990 1111111111 4.3782 1111111111 4.3782 1111111111 4.3782

10 1990 1111111111 4.3782 1111111111 4.3782 1111111111 4.3782

11 1970 1111111111 4.3171 1111111111 4.3171 1111111111 4.3171

12 1940 1111111111 4.2297 1111111111 4.2297 1111111111 4.2297

13 1910 1111111111 4.1464 1111111111 4.1464 1111111111 4.1464

14 1830 1111111111 3.9325 1111111111 3.9325 1111111111 3.9325

15 1870 1111111111 4.0384 1111111111 4.0384 1111111111 4.0384

16 1830 1111111111 3.9325 1111111111 3.9325 1111111111 3.9325

17 1690 1111111111 3.5787 1111111111 3.5787 1111111111 3.5787

18 1510 1111111111 3.1609 1111111111 3.1609 1111111111 3.1609

19 1420 1111111111 2.9685 1101111111 2.9962 1101111111 2.9962

20 1310 1111111111 2.7349 1101111111 2.7217 1101111111 2.7217

21 1260 1111111111 2.6332 1101111111 2.6143 1101111111 2.6143

22 1210 1111111111 2.5332 1101111111 2.5087 1101111111 2.5087

23 1250 1111111111 2.6129 1101111111 2.5930 1101111111 2.5930

24 1140 1101111111 2.3941 1101111111 2.3641 1101111111 2.3641

Total cost $/day!104 8.365240 8.347525 8.349142

Time taken (s) 13.05 383.56 112.54


a high value forameans that trail is very important and therefore

ants tend to choose edges chosen by other ants in the past. This is

true until the value of b becomes very high: in this case even if

there is high amount of trail on an edge, an ant always has a high

probability of choosing another state which is of low cost. High

values of b and/or low values of a make the algorithm very

similar to a stochastic multi-greedy algorithm. The control

parameters selected for theACSmodel is shown as the darkened

in Table 1. The final combination of parameters (a,b,r,q0) thatprovided the best results are (0.5, 2, 0.9, 0.9) [15]. The same

parameter settings are initialised for the practical ten unit system

data given in Appendix A.

Fig. 4. Convergence of total generation cost (four unit system).

5. Dynamic programming

Dynamic programming is a methodical procedure, which

systematically evaluates a large number of possible decisions in a

multi-step problem. A subset of possible decisions is associated

with each sequential problem step and a single one must be

selected, i.e. a single decisionmust bemade in each problem step.

There is a cost associatedwith eachpossible decision and this cost

may be affected by the decision made in the preceding step.

Additional costs, termed ‘transition costs’, may be incurred in

going from a decision in one problem step to a decision in the

following problem step over what is termed as a ‘transition path’.

The objective is to make a decision in each problem step, which

minimizes the total cost for all the decision made [16].

6. Branch and bound integer programming

The simplest method of solving an integer optimization

problem involves enumerating all the available paths, discarding

the infeasible paths, evaluating the objective cost function at all

feasible paths, and identifying the path that has the best optimum.

Although such an exhaustive search in the solution space is

simple to implement, it will be computationally expensive even

for a moderate-size problems. The branch and boundmethod can

be considered as a refined enumeration method in which most of

the non-promising paths are discarded without checking them

thereby restricting the branching of paths [17,18].

Fig. 5. Comparison of transitional cost (four unit system).Fig. 7. Total generation cost taken by each ant (ten unit system).

Fig. 8. Convergence of total generation cost (ten unit system).


7. Results and discussion

The commitment schedules and the transitional cost at each

stage obtainedusingDP,BBandACS for four and ten unit system

given in Appendix A is tabulated in Tables 2 and 3. The results

observed for the four-unit system shows that the ACS model

performed better than DP and BB. The solution time using DP

approachis theminimumbut the totalgenerationcost ishighwhen

comparedwith other approaches.Multi-stage decisionmakes the

ACSadvantageousoverother approaches. So the proposedmodel

cannot guarantee better solution at each and every hour, perhaps it

has been concluded that total generation cost so obtained by the

proposed model is quite encouraging against conventional

techniques. The evolution of the convergence of minimum cost

path can be understood fromFigs. 4 and 5, i.e. after the passing of

minimum number of ants through the ant search space and

completing their tours, convergence of the cost finally becomes

stagnant. Even after the stagnant situation, ants are still trying to

explore if any of the available minimum cost paths may exist,

which indicates the strength of the ACSmodel. This can be under

stood by the oscillations seen in Figs. 6 and 7. It is alsoobserved in

Figs. 8 and 9, that the transitional cost, even though at one stage

may be higher for ACS approach with respect to DP and BB, the

ants can still foresee and pick out the stages afterwards which can

be of low cost path and thus maintaining the overall reduction of

generation cost. Thus, the proposed model searches the optimal

Fig. 6. Total generation cost taken by each ant (four unit system).

path (24 h schedule) instead of individual solution state. The test

results indicate that the ants can absorb the information from the

experiencegained from its fellowants toget the best resultswhich

is almost near global optimum. The best optimum solution is got

by the branch and bound method, however ACS method has

helped in achievingnearglobal optimumsolution and the solution

time is less when compared with the branch and bound method.

The platform used for the implementation of this proposed

approach is onPentium IV3 GHz, 500 MBofRAMand has been

simulated in the MATLAB environment (Table 3).

Fig. 9. Comparison of transitional cost (ten unit system).


8. Conclusions

A new commitment schedule by ant algorithm approach has

been presented. Since ant algorithms are more suitable for

combinatorial optimization problem and their potential in finding

near global optimum solution, they are very well suited for UCP.

Not only the minimum cost path is determined but also based on

pheromone deposition other related features such as pheromone

updating, optimal control parameters such as the tuning factor,

relative importance of the trail have been discussed. Further

extensions and improvements with this proposedmodel based on

exploring capabilities of the artificial ants can positively enhance

the efficiency of solving UCP. The effectiveness of this proposed

method has been demonstrated on a four and 10 unit test systems

and the results obtained are quite encouraging and indicate the

viability to deal with future unit commitment problems.

Acknowledgements

The authors are thankful to the Head, Department of

Electrical Engineering, Indian Institute of Technology,

Roorkee for providing the required computational and

experimental facilities. The financial assistance provided by

the Ministry of Human Resources and Development, New

Delhi, India is gratefully acknowledged.

Appendix A

Table A1 Numerical four unit systems

Unit (no.) Max.

(MW)

Min.

(MW)

Ramp level

(MW/h)

Minimum

up-time (h)

Minimum

down-time (h)

Shut-down

cost ($)

Start-up cost Cold start

(h)

Initial unit

status

Hot ($) Cold ($)

1 80 25 16 4 2 80 150 350 4 K5

2 250 60 50 5 3 110 170 400 5 8

3 300 75 60 5 4 300 500 1100 5 8

4 60 20 12 1 1 0 0 0.02 0 K6

Fuel cost equations

C1Z25C1:5000P1C0:00396P21

C2Z72C1:3500P2C0:00261P21

C3Z49C1:2643P3C0:00289P23

C4Z15C1:4000P4C0:00510P24

Unit characteristics (four unit system). Initial unit status: hours off (K) line or on (C) line

Table A2 ten unit system

Unit (no.) Max.

(MW)

Min. (MW) Ramp level

(MW/h)

Minimum

up-time (h)

Minimum

down-time (h)

Shut-down

cost ($)

Start-up cost Cold Start

(h)

Initial unit

status

Hot ($) Cold ($)

1 200 80 40 3 2 50 70 176 3 4

2 320 120 64 4 2 60 74 187 4 5

3 150 50 30 3 2 30 50 113 3 5

4 520 250 104 5 3 85 110 267 5 7

5 280 80 56 4 2 52 72 180 3 5

6 150 50 30 3 2 30 40 113 2 K3

7 120 30 24 3 2 25 35 94 2 K3

8 110 30 22 3 2 32 45 114 1 K3

9 80 20 16 0 0 28 40 101 0 K1

10 60 20 12 0 0 20 30 85 0 K1

Fuel cost equations

C1Z82:00C1:2136P1C0:00148P21

C2Z49:00C1:2643P2C0:00289P22

C3Z100:0C1:3285P3C0:00135P23

C4Z105:0C1:3954P4C0:00127P24

C5Z72:00C1:3500P5C0:00261P25

C6Z29:00C1:5400P6C0:00212P26

C7Z32:00C1:4000P7C0:00382P27

C8Z40:00C1:3500P8C0:00393P28

C9Z25:00C1:5000P9C0:00396P29

C10Z15:00C1:4000P10C0:0051P210

Unit characteristics (ten unit system) Initial unit status: hours off (K) line or on (C) line

and Energy Systems 28 (2006) 315–323 323

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An ant colony system approach for unit commitment problem