DISCRETE GENETIC ALGORITHM AND REAL ANT COLONY

OPTIMIZATION FOR THE UNIT COMMITMENT PROBLEM

Guillaume Sandou

Supélec, Automatic Control Department – 3, rue Joliot Curie – 91192 Gif-sur-Yvette France

Keywords: Metaheuristics, unit commitment, ant colony, genetic algorithm, scheduling.

Abstract: In this paper, a cooperative metaheuristic for the solution of the Unit Commitment problem is presented.

This problem is known to be a large scale, mixed integer problem. Due to combinatorial complexity, the

exact solution is often intractable. Thus, a metaheuristic based method has to be used to compute a near

optimal solution with low computation times. A new approach is presented here. The main idea is to couple

a genetic algorithm to compute binary variables (on/off status of units), and an ant colony based algorithm

to compute real variables (produced powers). Finally, results show that the cooperative method leads to the

tractable computation of a satisfying solution for medium scale Unit Commitment problems.

1 INTRODUCTION

The Unit Commitment problem is a mixed integer

problem, referring to the optimal scheduling of

several production units, satisfying consumer’s

demand and technical constraints. Integer variables

are the on/off status of production units, and real

variables are produced powers. Numerous methods

have been applied; see (Sen and Kothari, 1998).

The first idea is to use an exact solution method:

exhaustive enumeration, “Branch and Bound” (Chen

and Wang, 1993), dynamic programming (Ouyang

and Shahidehpour, 1991). Due to temporal coupling

of constraints (time up / time down constraints), a

large temporal horizon is required, leading to a large

number of binary variables: exact methods suffer

from combinatorial complexity. Approximated

methods are required for tractable results.

Deterministic approximated methods have been

tested: priority lists in (Senjyu, et al., 2004) or expert

systems. Due to numerous constraints, this kind of

methods are often strongly suboptimal. Constraints

are considered by the Lagrangian relaxation method,

see (Zhai and Guan, 2002). Multi unit coupling

constraints are relaxed. As a result, the unit

Commitment problem is divided into several smaller

optimization problems. However, due to the non

convexity of the objective function, no guarantee

can be given on the duality gap and the actual

optimality of the solution. Further, an iterative

procedure has to be used: solution of the

optimization problems with fixed Lagrange

multipliers, updates of these multipliers, and so on.

This update can be performed with genetic

algorithms as in (Cheng, et al., 2000) or by

subgradient methods (Dotzauer, et al., 1999).

Stochastic approximated algorithms, called

metaheuristics are potentially interesting methods

for Unit Commitment as they are able to compute

near optimal solutions with low computation times.

A simulated annealing approach is used in (Yin Wa

Wong, 1998), tabu search is used in (Rajan and

Mohan, 2004) and genetic algorithms are used in

(Kasarlis, et al., 1996). Cooperative algorithms have

been developed to couple the advantages of several

optimization methods: genetic algorithms and

simulated annealing are used in (Cheng, et al.,

2002); simulated annealing and local search in

(Purushothama and Jenkins, 2003).

In (Serban and Sandou, 2007), a mixed ant

colony method has been proposed. The approach is

interesting, but, due to the positive feedback of ant

colony, may quickly converge to local minima. To

circumvent this problem, a new cooperative strategy

is defined in this paper. The idea is to use a

knowledge based genetic algorithm for binary

variables to achieve a deep exploration of the search

space, and simultaneously an ant colony based

algorithm for real variables.

The paper is organized as follows. In section 2,

the Unit Commitment problem is briefly called up.

256

Sandou G. (2007).

DISCRETE GENETIC ALGORITHM AND REAL ANT COLONY OPTIMIZATION FOR THE UNIT COMMITMENT PROBLEM.

In Proceedings of the Fourth International Conference on Informatics in Control, Automation and Robotics, pages 256-261

DOI: 10.5220/0001641602560261

Copyright

c

SciTePress

The cooperative metaheuristic ant colony/genetic

algorithm method is depicted in section 3. Both

algorithms are described, together with the definition

of a criterion guaranteeing feasibility of the solution.

Numerical results are given in section 4. Finally,

concluding remarks are drawn in section 5.

2 UNIT COMMITMENT

PROBLEM

The Unit Commitment problem is a classical large

scale mixed integer optimization problem.

Following notations are used throughout the paper:

N: length of time horizon,

n: (subscript) : time interval number n,

K: number of production unit,

k (superscript): production unit number k,

k

n

u : on/off status of production unit k during

time interval n (binary variable),

k

n

Q : power produced by production unit k

during time interval n (real variable).

2.1 Cost Function

The objective function is the sum of production,

start-up, and shut-down costs for all time intervals

and all units:

{}

∑∑

==

−

⎟

⎟

⎟

⎠

⎞

⎜

⎜

⎜

⎝

⎛

⎟

⎟

⎟

⎠

⎞

⎜

⎜

⎜

⎝

⎛

+

N

n

K

k

k

n

k

n

k

off/on

k

n

k

n

k

prod

Q,u

u,uc

u,Qc

k

n

k

n

11

1

)(

)(

min

,

(1)

where production cost of unit k can be expressed by:

,)(),(

01

2

2

k

n

kk

n

kk

n

kk

n

k

n

k

prod

uQQuQc

ααα

++=

(2)

start-up cost and shut-down costs are:

,)1()1(

)(

11

1

k

n

k

n

k

off

k

n

k

n

k

on

k

n

k

n

k

off/on

uucuuc

u,uc

−+−

=

−−

−

(3)

and

k

off

k

on

kkk

cc and,,,

012

ααα

are technical data of

production unit k.

2.2 Constraints

Constraints are:

Capacity constraints

k

n

kk

n

k

n

k

uQQuQ

maxmin

≤≤

(4)

Consumers’ demand satisfaction

dem

n

K

k

k

n

QQ ≥

∑

=1

(5)

Time up and time down constraints

⎟

⎠

⎞

⎜

⎝

⎛

===⇒

==

⎟

⎠

⎞

⎜

⎝

⎛

===⇒

==

−+

++

−

−+

++

−

0,,0,0

)0,1(

1,,1,1

)1,0(

1

21

1

1

21

1

k

Tn

k

n

k

n

k

n

k

n

k

Tn

k

n

k

n

k

n

k

n

k

down

k

up

uuu

uu

uuu

uu

(6)

Such constraints are temporally coupling

constraints which express dynamics on production

units.

k

down

k

up

kk

TT,Q,Q and

maxmin

are technical data.

3 COOPERATIVE

METAHEURISTIC SOLUTION

3.1 Algorithm Principles

As already mentionned, Unit Commitment is a large

scale mixed integer programming problem. Genetic

algorithm is a well known algorithm for

combinatorial optimization problems. In this study, a

specific criterion is defined (see section 3.2), based

on particular penalty functions to guarantee the

solution feasibility. Genetic algorithm is used to

compute binary variables and is depicted in section

3.3. Further, a stochastic algorithm is simultaneously

used to compute real variables, based on ant colony

optimization. It is presented in section 3.4.

3.2 Optimization Criterion

3.2.1 Criterion Expression

Consider that a feasible solution is known with a

cost c

f

. The following optimization criterion is

defined:

⎟

⎟

⎟

⎟

⎠

⎞

⎜

⎜

⎜

⎜

⎝

⎛

++

+

⎟

⎟

⎟

⎠

⎞

⎜

⎜

⎜

⎝

⎛

+

∑∑

==

−

=

=

=

)()).()1((

),(

),(

min

11

1/

,,1

,,1

),(

yByhc

uuc

uQc

f

N

n

K

k

k

n

k

n

k

offon

k

n

k

n

k

prod

Kk

Nn

Quy

k

n

k

n

ε

…

,

(7)

where:

ε

is a small positive real,

)( yh is a penalty function for non feasible

solutions,

DISCRETE GENETIC ALGORITHM AND REAL ANT COLONY OPTIMIZATION FOR THE UNIT COMMITMENT

PROBLEM

257

)( yB

is a boolean function (1 for non feasible

solutions and 0 for feasible ones).

With this criterion, any unfeasible solution has a

higher cost than the feasible known solution: any

unconstrained optimization algorithm can solve the

problem. Thus, an elitist genetic algorithm can be

used. The definition of criterion (7) only supposed

that a feasible solution is known. It can be easily

computed using a simple priority list. This is a very

suboptimal solution, but the quality of this first

feasible solution is not crucial, as the criterion can

be updated when new feasible solutions are known.

3.2.2 Penalty Expression

The following variables are added:

)1(

)1(

1

1

k

n

k

n

k

n

k

n

k

n

k

n

uu

uu

−=

−=

−

−

ε

δ

(8)

With these variables, time-up and time-down

constraints are expressed by linear expressions:

k

n

k

up

T

j

k

jn

k

Tn

k

n

k

n

Tu

uu

up

k

k

up

δ

δ

≥⇔

==⇒=

∑

−

=

+

−+

+

1

0

1

1

)1,,1(1 …

(9)

k

n

k

down

T

j

k

jn

k

Tn

k

n

k

n

Tu

uu

down

k

k

down

ε

ε

≥−⇔

==⇒=

∑

−

=

+

−+

+

1

0

1

1

)1(

)0,,0(1

…

(10)

Capacity constraints and consumers’ demands

satisfaction are linear. All constraints can be

expressed by a linear equation,

cc

BxA ≤ , where x is

Tk

n

k

n

k

n

k

n

K,,k;N,,n;,,Q,u )11( …… ==

εδ

, leading

to a high tractability of the boolean and the penalty

functions computation.

3.3 Genetic Algorithm for on/off

Variables

3.3.1 Algorithm Principles

Genetic algorithm is a well known optimization

method. Fig. 1 and 2 represent classical cross-over

and mutation operators for Unit Commitment

problem.

0

0

111

110

0

1

01000

N périods

1

0

0 0 1

1 1 0

0

0

0 1 0 1 1

0

0

111

110

0

0

01000

1

0

0 0 1

1 1 0

0

1

0 1 0 1 1

Children

Parents

2 random points

K

units

mask

Figure 1: Classical crossing over operator.

0

0

111

110

0

1

01000

0

0

1 0 1

1 1 0

0

1

0 1 0 0 0

Parent Child

Figure 2: Classical mutation operator.

Individuals are in a matrix form. Crossover

operator create 2 potentially low cost children from

2 parents by merging their variables (or genes). The

mutation operator allows the introduction of new

genes in the population by randomly changing one

of the variables. Finally, the selection operator is

performed with a classical biased roulette method.

3.3.2 Knowledge Based Operators

It has been observed that the genetic algorithm can

be more efficient by using some knowledge based

operators. New genetic operators are added,

considering properties of the problem. The first

operator is a “selective mutation operator”. Consider

unit scheduling of fig. 3. Due to time-up and time-

down constraints, a classical mutation leads very

often to an infeasible solution. To increase the

probability of reaching a new feasible point, a

“selective mutation operator” is introduced: this

operator detects switching times and allows a

random mutation only for these genes.

1 1 1 1 1 1 0 0 0 0

Switching times :

Authorized

m

utat

i

o

n

s

Figure 3: Selective mutation operator.

ICINCO 2007 - International Conference on Informatics in Control, Automation and Robotics

258

0 1 1 1 0

1 0 0 1 1

N

K

Figure 4: Exchange operator.

The second operator is an exchange operator,

introduced by (Kasarlis, et al., 1996). Some

production units are profitable or have larger

capacities. It may be interesting to exchange a part

of the planning of two production units (see fig. 4).

Finally, all-on and all-off operator are

introduced. Consider fig. 5. If the unit has a time

down constraints of two hours, it may be difficult to

go from feasible point a) to feasible point b). The

all-on (resp. all-off) operator randomly select two

time intervals and a production unit and switch on

(resp. switch off) the production unit between these

time intervals: the idea is to increase the probability

of “crossing the infeasible space”.

« All On »

2

=

down

T

1 1 1 1 1 1 1 0 0 1

a)

b)

Figure 5: All-on operator.

3.4 Continuous Ant Colony

Optimization for Produced Powers

Ant colony optimization was firstly introduced by

Marco Dorigo. Ants’ behaviour has been used as a

metaphor to design algorithms for combinatorial

optimization problems such as the Travelling

Salesman Problem (Dorigo, et al. 1997). Extensions

for continuous search spaces have been proposed by

(Socha and Dorigo, 2006) and have been used in

(Serban and Sandou, 2007) in a pure ant colony

solution of Unit Commitment. Results are here

called up. For each binary solution,

),,1;,,1;( KkNnuU

k

n

…… === , real values Q =

),,(),,1;,,1;(

1 KNk

n

xxKkNnQ ……… === have

to be associated. To compute these real variables, a

matrix

T of s real solutions, called “archive matrix

of solutions”, is stored:

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

⎡

=

KN

s

i

sss

KN

j

i

jjj

KNi

KNi

xxxx

xxxx

xxxx

xxxx

……

…

…

21

21

22

2

2

1

2

11

2

1

1

1

T .

(11)

These solutions are evaluated with respect to the

objective function (1). Costs are stored in

H:

[

]

T

sj

ffff ……

21

=H , with

()

.

u,uc

u,Qc

Q,Uff

N

n

K

k

k

n

k

n

k

off/on

k

n

k

n

k

prod

jjj

∑∑

==

−

⎟

⎟

⎟

⎠

⎞

⎜

⎜

⎜

⎝

⎛

⎟

⎟

⎟

⎠

⎞

⎜

⎜

⎜

⎝

⎛

+

==

11

1

)(

)(

(12)

The solutions are sorted according to their costs.

From these costs, weights are defined according to

the ranks of the solutions in the matrix. For the

solution with rank r, the weight is defined by:

()

.

2

1

22

2

2

1

sq

r

r

e

πqs

ω

−

−

=

(13)

Finally, a discrete probability distribution is

defined from these weights:

∑

=

=

s

j

jrr

p

1

/

ωω

(14)

q is a tuning parameter of the algorithm. To

compute a new real solution, the following

procedure is performed:

a “model ant”, say l, is chosen, according to

this discrete probability distribution (14).

Each real variable

i

new

x

, i = 1,…,KN, is

chosen with a Gaussian probability whose

mean and standard deviation is computed by:

.

xx

s

x

s

m

i

l

i

m

i

new

i

l

i

new

⎪

⎩

⎪

⎨

⎧

−

−

=

=

∑

=1

1

ξ

σ

μ

(15)

ξ is also a tuning parameter of the algorithm.

When real variables have been chosen, consumer’s

demands (5) may not be fulfilled. Furthermore, the

selection of produced powers may lead to

overproduction. To get rid of these problems, the

following improvement procedure is used:

Select

k

n

Q with the previous algorithm.

If

dem

n

K

k

k

n

k

n

QuQ >

∑

=1

(resp. < ), then

randomly choose, if possible, one of switched

on units, and decrease (resp. increase) the

corresponding produced power until

DISCRETE GENETIC ALGORITHM AND REAL ANT COLONY OPTIMIZATION FOR THE UNIT COMMITMENT

PROBLEM

259

dem

n

K

k

k

n

k

n

QuQ =

∑

=1

. If it is not sufficient,

choose several production units, if possible.

When all new solutions have been computed, the

best new solutions are stored in matrix

T, replacing

solutions whose costs were too high. This is an

analogy with physical evaporation of pheromone.

4 NUMERICAL RESULTS

4.1 Algorithm Implementation

The proposed cooperative method has been tested

with Matlab 6.5 with a Pentium IV 2.5 GHz. When

the stochastic cooperative algorithm is completed, a

final local search is performed: binary values are set

to their final values, and a real optimization based on

Semi Definite Programming is performed to solve

this particular economic dispatch problem. As

stochastic algorithms are considered, 70 tests are

performed, and statistical data about the results are

given. Optimization horizon is 24 hours with a

sampling time of one hour.

4.2 “4 unit” Academic Example

A 4 unit case is considered (see table 1).

Table 1: Characteristics for the “4 unit case”.

Q

min

(MW)

Q

max

(MW)

α

0

(€)

α

1

c

on

(€)

c

off

(€)

T

down

(h)

T

up

(h)

1 10 40 25 2.6 10 2 2 4

2 10 40 25 7.9 10 2 2 4

3 10 40 25 13.1 10 2 3 3

4 10 40 25 18.3 10 2 3 3

At time 0, all units are switched off and can be

switched on. Note that linear costs have been chosen

(α

2

= 0). For this relative small scale cases, and for

linear costs, an exact solution has been computed by

“Branch and Bound”. Consumer’s demand is

depicted in fig. 6. This demand can be fulfilled by 2

production units (see 2 units limit in fig. 6), except

for hour number 9. Because of time up constraints

this unit will be switched on for 3 hours. The

optimal solution is given in fig. 7.

The corresponding optimization problem is made

of 96 binary variables (24 hours and 4 units) and 96

real variables. Table 2 shows results of optimization.

Statistical results are given: best case, mean, number

of success (a test is successful if the best solution is

found).

The following parameters were used:

Figure 6: Consumer’s demand.

Figure 7: Optimal solution for “4 unit case”.

Genetic population size: 50,

Cross-over rate: 70%,

Mutation rate: 5%,

Knowledge based operators rate: 10%,

Archive matrix size: s = 20,

Tuning parameters 8.0;1 =

=

ξ

q .

Results show that interesesting solutions can be

computed with relatively low computation times.

Table 2: Optimization results “4 unit case”.

Case Best Mean Nb.

Success

Time

100 iter.

8778 €

(+0%)

9449 €

(+7.6%)

8/70 22 s

200 iter.

8778 €

(+0%)

9004 €

(+2.6%)

32/70 45 s

500 iter.

8778 €

(+0%)

8922 €

(+1.6%)

45/70 115 s

4.3 Medium Scale Case

A “10 unit” case is now considered (see table 3).

This a medium scale case. Low start up and start

down costs have been considered, leading to the

possibility of guessing the optimal solution. The

corresponding optimal cost is 29795 €. For a 24 hour

optimization, this problem is made of 240 binary

ICINCO 2007 - International Conference on Informatics in Control, Automation and Robotics

260

optimization variables and 240 real variables.

Results for the cooperative method are given in table

4. As in previous examples, 70 tests are performed

and statistical results are given (best case, mean).

The same values were used for parameters.

Table 3: Characteristics for the “10 unit case”.

Q

min

MW

Q

max

MW

α

0

€

α

1

c

on

€

c

off

€

T

dow

n

h

T

up

h

1 10 40 25 2.6 10 2 2 4

2 10 40 25 5.2 10 2 2 4

3 10 40 25 7.9 10 2 3 6

4 10 40 25 10.5 10 2 3 6

5 10 40 25 13.1 10 2 3 4

6 10 40 25 15.7 10 2 3 4

7 10 40 25 18.3 10 2 3 4

8 10 40 25 21.0 10 2 3 4

9 10 40 25 23.6 10 2 3 4

10 10 40 25 26.2 10 2 3 4

Results show the viability of the cooperative

method to solve mixed integer optimization

problems. Low computation times are observed,

even for this medium scale case.

Table 4: Optimization results “10 unit case”.

Best Mean Time

500 iter.

30210 €

(+1.4%)

32695 €

(+9.7%)

275 s

1000 iter.

29851 €

(+0.2%)

32138 €

(+7.8%)

550 s

5 CONCLUSION

In this paper, a cooperative method ant

colony/genetic algorithm for Unit Commitment

solution has been proposed. The main idea is to use

a genetic algorithm with knowledge based operators

to compute binary variables and a real ant colony

algorithm to compute real variables. To guarantee

the feasibility of the final solution, a criterion has

also been defined. Finally, the proposed method

leads to near optimal solutions, with guarantees of

feasibility and with low computation times.

Some dedicated methods are able to find better

solutions than the proposed cooperative algorithm,

and can consider larger scale cases. However, this

cooperative method seems to be promising and the

study has proven its viability.

Forthcoming works deal with the use of such a

cooperative metaheuristic method to solve generic

non linear mixed integer optimization problems, as

the use of the method does not require any structural

property of the problem.

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DISCRETE GENETIC ALGORITHM AND REAL ANT COLONY OPTIMIZATION FOR THE UNIT COMMITMENT

PROBLEM

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