[1] scheduling of cogen

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Scheduling of cogeneration plants considering electricity wheelingusing enhanced immune algorithm

Sung-Ling Chen*, Ming-Tong Tsay, Hong-Jey Gow

Department of Electrical Engineering, Cheng-Shiu University, 840 Cherng-ching Road, Neau-song County, Kaohsiung 833, Taiwan, ROC

Received 12 May 2003; revised 13 July 2004; accepted 27 July 2004

Abstract

A new method based on immune algorithm (IA) is presented to solve the scheduling of cogeneration plants in a deregulated market. Theobjective function includes fuel cost, population cost, and electricity wheeling cost, subjective to the use of mixed fuels, operational limits,

emissions constraints, and transmission line flow constraints. Enhanced immune algorithm (EIA) is proposed by an improved crossover and

mutation mechanism with a competition and auto-adjust scheme to avoid prematurity. Table lists with heuristic rules are also employed in the

searching process to enhance the performance. EIA is also compared with the original IA. Test results verify that EIA can offer an efficient

way for cogeneration plants to solve the problem of economic dispatch, environmental protection, and electricity wheeling.

q 2004 Elsevier Ltd. All rights reserved.

Keywords: Immune algorithm; Cogeneration plants; Deregulated market

1. Introduction

Deregulating the power market creates competition andtrading mechanism for independent power producers (IPPs).

IPPs will use the existing transmission to sell power at a

market price. IPPs are also seeking chances to be connected

with the lower price of existing transmission for obtaining

their profits. The existing transmission owner could be

considered as a third party to provide the electricity wheeling

for seller/buyers. The electricity wheeling cost will affect the

bid price of IPPs under the deregulated environments.

Cogeneration systems have now been extensively

utilized by the industry. Cogeneration systems offer a

reliable, efficient, and economic means to supply both

thermal and electrical energy. If the electric powergenerated from the cogeneration systems is much more

than the consumption of the factory, the additional electric

power can be transmitted to the buyer. For a more effective

operation, efficient strategies have been developed in [18].

The model of cogeneration for a small industry is proposed

in [1]. An economic dispatching scheme with power

purchase facilities is presented in [25]. A non-linear

programming model with time-of-use (TOU) rates con-

sidered on operation is used in [6,7]. In [8] is proposed theevolutionary programming (EP) technique to solve a

multitime-interval scheduling in the daily operation of a

two-cogeneration system. However, these papers did not

involve the electricity wheeling problem. On the other hand,

it is imperative for cogeneration systems to operate

effectively according to the overall system schedule.

Various fuels, such as fuel oil (FO), liquid nature gas

(LNG), and coal are available at various cost bases. The

optimal operating strategy determines the optimal distri-

bution among the in-plant generation, fuels dispatch, and

electricity wheeling price while satisfying the overall

system constraints and transmission line constraints.Conventional operations are getting more and more

pressure to take the emission cost into consideration. In

recent years, rigid environmental regulation [911] forced

the utility planners to consider emission control as an

important objective. Various factors, such as cogeneration

unit types, time-schedules of load, fuel mix, billing rates,

and emission constraints are all required to be taken into

account. Considering the emission cost, a trade-off between

economy and environment need to be considered in

0142-0615/$ - see front matter q 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijepes.2004.07.008

Electrical Power and Energy Systems 27 (2005) 3138

www.elsevier.com/locate/ijepes

* Corresponding author.

E-mail address: [email protected] (S.-L. Chen).
http://www.elsevier.com/locate/ijepeshttp://www.elsevier.com/locate/ijepes


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the optimization process. It will become a more complicated

nonlinear problem when the emission cost is introduced

with mixed fuels. An efficient and reliable technique is

needed.

It is complicated to perform the economic dispatch of

cogeneration systems, especiallyfinding the best strategy in a

world of uncertainty. Conventional methods, such as linearprogramming, nonlinear programming, dynamic program-

ming,and mix-integer programming techniques,etc. become

more difficult to solve.Recently, newalgorithms based on the

artificial intelligence (AI) have been developed, such as

simulated annealing (SA) [12], genetic algorithm (GA)

[13,14], evolutionary programming [15,16], and immune

algorithm (IA) [1719]. Solution strategies proposed by most

AI algorithms need to consider a large solution space.

Extensive numerical computationis oftenrequired especially

when the load flow technique has to be used. On the other

hand, conventional methods may be faster, they are very

often limited by the problem structure and may diverge or

could lead to a local minimum. This paper presents an

enhanced immune algorithm (EIA)-based immune algori-

thms to solve theeconomicdispatch of cogeneration systems.

IA is based on the operation of human immune

system. The immune system is a basic bio-defense system

against viruses and disease-causing organisms. This complex

defense mechanism combines genes to deal with the

inbreaking antigens. Using this heuristic algorithm, IA has

advantages and some characteristics to show a better

performance than many other algorithms [1719]. IA takes

advantage of the conventional GA and tabu search, but the

choice of a proper probability for crossover and mutation is

still a dilemma. This paper proposed EIA which further

improves IA. In order to avoid prematurity, the crossover and

mutation mechanism is refined by a competition and auto-adjust scheme. Crossover and mutation were combined in

one step in EIA, and a competition mechanism was imple-

mented to automatically determine the choice of either one.

Numerical examples are also provided to show its

effectiveness.

2. Cogeneration systems

Fig. 1 shows the diagram of a cogeneration system with

common high and medium pressure steam headers. There

are M cogeneration systems at different buses. The system

has k back-pressure turbines for power generation. Fuel

co-firing can be obtained through different rows. The fuels

including FO, LNG, and coal were used in each boiler for

producing high-pressure steam. It is required to assess the

economic and operational benefits through optimal steam

loading allocation among boilers and turbines. Models for

operation evaluation, the I/O curves of boiler and turbine,

construction of multi-fuel unit model, and emission cost are

all needed.

Fig. 1. The diagram of the cogeneration system.

S.-L. Chen et al. / Electrical Power and Energy Systems 27 (2005) 313832


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2.1. I/O cost curve of boilers

By using the regression technique, the coefficients of I/O

curve can be calculated from the operational records of

boilers. Assuming that the I/O curve of a boiler has a 3rd

order polynomial, the R2 is equal to 0.999. R2 is used as a

measure of the accuracy of future predictions and the bestvalue is 1. In this paper, the I/O curve of boilers is defined as

follows

FbijMbijZA0CA1MbijCA2M2bijCA3M

3bij (1)

where

Fbij(Mbij), enthalpy of the jth boiler at ith bus (MBTU/H);

Mbij, steam output of the jth boiler at ith bus (T/H);

A0, A1, A2, A3, coefficients of the I/O operation curve.

With mixed fuel used, a proper modification will be

needed. The dual-fueled unit model was formulated in [20].

Eq. (2) is used to represent a unit burning three fuelssimultaneously. We have

FbTiMbiZFb1iMbil1i Ch1=2l2iCh1=3l3i (2)

where

FbTi(Mbi), total enthalpy at ith bus (MBTU/H);

Fb1i(Mbi), enthalpy of fuel 1 at ith bus (MBTU/H);

h1/2, the efficiency ratio of fuel 1/fuel 2;

h1/3, the efficiency ratio of fuel 1/fuel 3;

l1i, l2i, l3i, the mixed ratio of fuel 1, 2, and 3 at ith bus.

With l1iCl

2iCl

3iZ

1, the I/O cost curve of boilers canbe described by

FBCTiZFbTiMbiBCTi (3)

BCTiZBC1l1iCBC2l2CBC3l3i (4)

where

FBCTi, the total operation cost of the boilers at ith bus

(NT$);

BCTi, the total cost of fuel m ixture at ith bus

(NT$/MBTU);

BC1, BC2, BC3, the cost of fuel 1, 2, and 3 (NT$/MBTU).

2.2. I/O operation curve of steam turbines

For back-pressure turbine generator, the power equation

for turbine j can be formulated by [21]

PgijtZK0iCK1jMmijCK2jM2mij iZ 1; 2;.;M

jZ 1; 2;.; k(5)

where Mmij is the medium pressure extraction flow of

turbine j at ith bus; Pgij is the generated electric power from

turbine j; K0j, K1j, and K2j are coefficients of turbine j which

can then be found by curve fitting technique with field data.

2.3. Emission model

Two primary emissions are sulfur dioxide (SOx) and

nitrogen oxides (NOx). Emission models may be defined asthe amount of fuel consumed or as a function of boiler

steam. In this paper, the emissions are modeled as a function

of fuel enthalpy dependent on the emission factor [11].

ESMbZgSFbMb (6)

ENMbZgNFbMb (7)

And the emission model with mixed fuels can be

formulated by

ESij$ZgS1jl1ijCgS2jl2ijCgS3jl3ijFbTijMbij;l1ij;l2ij;l3ij

(8)

ENij$ZgN1jl1ijCgN2jl2ijCgN3jl3ijFbTijMbij;l1ij;l2ij;l3ij(9)

where

ESij($), the amount of pollutant SOx for the jth boiler at

the ith bus (T/h);

ENij($), the amount of pollutant NOx for the jth boiler at

the ith bus (T/h);

FbTij($), total enthalpy for the jth boiler at the ith bus

(T/h);

gS1j, gN1j, the emission factor of SOx and NOx with oil

for the jth boiler (T/Mbtu);

gS2j, gN2j, the emission factor of SOx and NOx with LNG


gS3j, gN3j, the emission factor of SOx and NOx with coal


l1ij, l2ij, l3ij, the mixed ratio of fuel 1, 2, and 3 for the jth

boiler at ith bus.

3. Problem formulation

The optimization needs to meet the steam demand of in-

plant process and electricity. We have the objective function

including fuel cost, emission cost, and electricity wheelingcost formulated by

Min obj_f

Z

XMiZ1

XkjZ1

BCijl1ij; l2ij; l3ijFbijMbij; l1ij; l2ij; l3ij

(

CCS

XkjZ1

ESijMbij; l1ij; l2ij; l3ij

CCN

XkjZ1

ENijMbij; l1ij; l2ij; l3ijCWUC!PGi

)10

S.-L. Chen et al. / Electrical Power and Energy Systems 27 (2005) 3138 33


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CS and CN are the charged pollution emission fee for SOxand NOx. WUC is the wheeling unit cost (NT$/MWh). PG iis the total wheeling electricity at bus i.

The constraints considered are described as follows:

(1) Steam balance for boilers, turbine and industrial process

areXkjZ1

MbijKDhi KXkjZ1

MhijZ 0 iZ1; 2;.;M (11)

XkjZ1

MhijKXkjZ1

MmijZ0 iZ1; 2;.;M (12)

XkjZ1

MmijKDmiZ 0 iZ1; 2;.;M (13)

where Mhij is the high pressure injection flows of turbine

j at ith bus. Dhi and Dmi are the high and medium

pressure steam demands at ith bus.(2) The total fuel proportion in each boiler must sum up to 1

l1ijtCl2ijtCl3ijtZ1:0 iZ1;.;M

jZ1; 2;.; k(14)

(3) Power balance in the power system

XMiZ1

XkjZ1

PgijKILOADi

!Z

XMiZ1

PGi (15)

ILOADi is the internal load of cogeneration systems at

ith bus.

(4) Operation constraints for boilers, steam turbine, power

generation, emission control:

Mminbij %Mbij%M

maxbij iZ1; 2;.;M

jZ1; 2;.; k(16)

Mminhij %Mhij%M

maxhij iZ1; 2;.;M

jZ1; 2;.; k(17)

Mminmij %Mmij%M

maxmij iZ 1; 2;.;M

jZ1; 2;.; k(18)

Pmingij %Pgij%P

maxgij iZ1; 2;.;M jZ1; 2;.; k

(19)

S$ZXkjZ1

ESijMbij; l1ij; l2ij; l3ij%Slimit

iZ1; 2;.;M

(20)

N$ZXk

jZ1

ENijMbij; l1ij; l2ij; l3ij%Nlimit

iZ1; 2;.;M

(21)

jf[j%fmax[ (22)

Mminbij ; Mmaxbij , lower and upper limits of flows for

boiler j;

Mminhij ; Mmaxhij , lower and upper limits of high pressure

injection flows for turbine j;

Mminmij ; Mmaxmij , lower and upper limits of medium

pressure extraction flows for turbine j;

Pmingij ; Pmaxgij , lower and upper limits of the generatedelectric power for turbine j;

Slimit, Nlimit, the total permissible emissions for SOxand NOx;

f[f[Zf0[Ca[iDPi, flow on line [ after the congera-

tion plants on bus i;

f0[ , flow on line [ before the congeration plants on

bus i;

a[i, the generation shift factors;

DPi, change in generation at bus i.

4. Solution algorithm

IA is a search algorithm based on the mechanism of

nature selection and genetics. Antigens and antibodies can

be viewed as objective functions and feasible solutions,

respectively. The process of genetic structure is similar to

GA, including crossover, mutation, and reproduction. For

the efficiency of a performance, IA is modified by improved

crossover and mutation mechanism. It is called an EIA. EIA

was developed as follows.

4.1. Encoding

The coding scheme can be illustrated in Fig. 2, where

each antibody indicates a combination of the steam of

boilers and the ratio of fuels. The antibody is encoded as

Fig. 2. Chromosome string of the antibody.



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a chromosome string, which is produced by Eq. (24). When

EIA search is terminated, the chromosome will then be

decoded.

D2BfdMiliKMmini l

mini =resolieg (23)

resoliZ Mmax

ilmax

iKMmin

ilmin

i=2bitK1

D2B, decimal to binary conversion; bit, the number of bits

in a chromosome.

4.2. Affinity and diversity evaluation

Immune system generates different antibodies according

to the affinity recognition between antigens and antibodies

or between two antibodies. There are two classes of affinity

in IA. One is the affinity between antigens and antibodies. It

represents the combination intensity between antigen and an

antibody. The other one is the affinity between two

antibodies; it shows the similarity between two antibodies.From information theory, the entropy can be applied to

measure diversity of antibodies. It can be computed by

EkNZK

XNiZ1

Pik log10 Pik (24)

where Nis the number of antibodies, Pik is the probability of

the ith allele coming out of the kth allele. For example, if all

alleles at the kth antibody are the same, Ek(N)Z0. Thus the

total diversity of the kth antibody is

ENZ1

MX

M

kZ1

EkN (25)

where M is the number of gene of the kth antibody.

Two affinity forms must be taken into account in the

proposed EIA. One is the affinity between two antibodies,

i.e. jth and kth. It can be calculated by

AffbjkZ 1CE2K1 (26)

where E(2) is the diversity of the jth and kth antibody only.

Note that if two antibodies are the same, (Affb)jkis equal to1.

(Affb)jkis set between zero and one. Another one is applied to

investigate the affinity between antibodies and antigens with

Affgk

Z 1CObj_fkK1

(27)

Obj_fk is the objective function for the kth antibody. The

affinity score of each antibody is obtained by calculating the

objective function and taking (11)(22) into account. If one

or more variables violate their limits, the corresponding

chromosome will be put into the tabu list to avoid generating

the same infeasible solution again.

4.3. Antibody recomposition

New chromosomes will be obtained from crossover and

mutation. Crossover is a structured recombination operation

by exchanging genes of two parent antibodies. Mutation is

the occasional random alteration of genes. An improved

crossover and mutation (ICM) scheme is proposed in this

paper.

4.3.1. Simple crossover and mutation scheme (SCM)

The crossover process randomly (uniform distribution)selects two parents to exchange genes with a crossover

rate Pc. The location of the gene within the chromosome

is called locus. The crossover point is also randomly

chosen from the loci. If one or both offspring is infeasible,

another mate will be chosen again for crossover. The

mutation process randomly (uniform distribution) selects

one parent with a mutation rate Pm. We could randomly

select a locus to mutate. If the offspring is infeasible,

another parent will be chosen until a feasible solution can

be obtained.

4.3.2. Improved crossover and mutation scheme (ICM)Crossover generally executes before mutation through-

out searching process. In original IA, a higher crossover rate

allows the exploration of solution space around the parent

solution. The mutation rate controls the rate at which new

genes are introduced, and explores new solution territory. If

it is too low, the solution might settle at a local optimum. On

the contrary, a high rate could generate too many

possibilities. The offspring lose their resemblance to the

parents; the algorithm will not learn from the past and could

become unstable. It is a dilemma to choose suitable

crossover and mutation rate for SCM. The ICM scheme is

thus proposed to avoid the difficulty.

(i) Randomly select two parents, and generate offsprings

by introducing CP(g) with

(a) If Rand!CP(g), use mutation;

(b) If RandOCP(g), use crossover.

where

Rand, the uniform random number in (0,1);

CP, 0.1%CP%0.95, the control parameter with initial

value set to 0.5;

g, the current generation number.

The offspring will be generated until all parents are

processed. Fig. 3 shows the initial relationship of

crossover and mutation in ICM. If the best current

solution comes from crossover, there is a more likelihood

for crossover to generate better offsprings for the next

population. On the contrary, there is a more likelihood for

mutation to generate better offsprings. If the best solution

remains the same, the operation of crossover or mutation

needs to hold back. The probability of crossover and

mutation sums to one.

(ii) IfFmin(g) is the minimum cost of the gth generation,

and Fmin(gK1)OFmin(g) comes from crossover, the control



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parameter CP(gC1) will decrease. We have

CPgC1ZCPgKD (28)

where D is the auto-adjust distance, it can be calculated by

DZjCPgKCPgK1j

2

Fig. 4 shows the variation of probability of crossover.

(iii) If Fmin(gK1)OFmin(g) comes from mutation,

CP(gC1) will increase. For Fmin(gK1)OFmin(g), we have

CPgC1ZCPgCD (29)

The variation of probability of mutation is illustrated in

Fig. 5.

(iv) IfFmin(gK1)ZFmin(g), the control parameter needs

to hold back.

If CP(g)OCP(gK1)

CPgC1ZCPgKD (30)

otherwise

CPgC1ZCPgCD (31)

where CP must be held between 0 and 1, and D* is also the

auto-adjust distance, it can be calculated by

DZ jCPgKCPgK1j

4.4. Stopping rule

The process of generating new antibody with the best

affinity between antibody and antigen will continue until the

affinity values are optimized or the maximum generation

number is reached. In our study, the stopping rule is set to300 generations.

5. Case study

In this paper, the IEEE 6 bus with some practical

modifications is used as examples to show the effect of the

proposed method. Two cogeneration systems with five

back-pressure steam turbines and five steam boilers are

located at buses 2 and 3. The data of two cogeneration

systems in IEEE 6-bus system are described in Table 1. Theoperating data for cogeneration systems was tested on the

Petroleum Company. The fuels used are FO, LNG, and coal.

The fuel consumption and steam generation were measured

in the field. By using the measurement data, curve fitting

method was used to get the I/O operation curves. Table 2

shows the coefficients of the I/O operation curve for boilers.

The associated coefficients for steam turbines are listed in

Table 3.

According to Eq. (2), the fuel mixture ratio is required to

get the I/O operation curve for a boiler that is simul-

taneously burning three fuels. Table 4 shows the efficiency

of boilers with various fuels.All facilities including generators, boilers, and steam

turbines have their capacity limitation. The rated limits can

be expressed in Tables 5 and 6.

Fig. 3. Probability map of crossover and mutation in ICM for CPZ0.5.

Fig. 4. Variation of probability of crossover.

Fig. 5. Variation of probability of mutation.

Table 1

Two cogeneration systems of IEEE 6-bus system

Cogeneration 1 Cogeneration 2

The number of units 5 back-pressure 5 back-pressure

Location bus 2 3

High-pressure demand (T/H) 40 40

Medium-pressure demand (T/H) 510 510

Internal load (MW) 5 10Wheeling unit cost (NT$/MWH) 92 92

SOx/NOx limited (T/H) 0.9/0.5 0.9/0.5

Table 2

The coefficients of the I/O operation curve of boilers

Unit no. A0i A1i A2i A3i

Boiler 1 K20.27984 3.47598 K0.00664 2.152!10K5

Boiler 2 523.56446 K17.82798 0.25662 K9.771!10K4

Boiler 3 K716.0406 25.69203 K0.23667 7.667!10K4

Boiler 4 27.23617 1.69342 0.00956 K3.189!10K5

Boiler 5 K14.45154 2.78372 K0.00288 6.2!10K6



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5.1. Simulation results

Table 7 shows the fuels dispatch of the boilers. The ratio

of fuel in each boiler summed up to 1 and each fuel must be

used in each boiler. In our study, coal only will be burned for

maximum economy, if the emission constraints are not

considered.

Table 8 shows the simulation results for two cogenera-

tion systems. From Table 8, all constraints including

emission constraints are same. The buyer at buses 2 and 3

consume 32.0616 and 28.7345MW, respectively.

Table 9 is the cost analysis of two cogeneration systems.

From Table 9, it is noted that wheeling/emission cost is

important on the overall economy of the cogeneration

systems. The total cost of operation dispatch for considering

wheeling/emission cost is higher without considering

wheeling/emission cost. It is inherently compromisedbetween the cost and pollution emission.

5.2. Convergence test

Fig. 6 illustrates the convergence characteristics of EIA

and IA. It shows the improvement of the EIA over IA. An

IBM PC with P-IV2.0 GHz CPU and 512MB SDRAM is

used in our test. Table 10 shows the maximum, minimum,

and average optimized cost with 100 random initial parents

tested. The population size of each trial is 20. Although the

solution improvement is subtle, it did show the capability of

EIA in exploring a more likely global optimum.

Table 3

The coefficients of steam turbines

Unit no. K0i K1i K2i

Gen 1 1.67419 0.02042 0.000216

Gen 2 K1.46151 0.08497 0.00008

Gen 3 K1.393504 0.093989 0.000037

Gen 4 K0.23472 0.054951 0.000299

Gen 5 K3.2978 0.138129 K0.00029

Table 4

The efficiency of boilers with various fuels

Boiler no. Fuel oil (%) LNG (%) Coal (%)

Boiler 1 88 90 86

Boiler 2 87 89 86

Boiler 3 89 91 87

Boiler 4 86 89 85

Boiler 5 87 89 85

Table 6

The limits of steam output

Unit no. Min (T/h) Max (T/h)

Mh1, Mm1 68.77 154.67

Mh2, Mm2 70.22 121.09

Mh3, Mm3 60.21 115.93

Mh4, Mm4 65.00 114.68

Mh5, Mm5 69.49 133.93

Table 5

Upper and lower limits of boiler flows and generator

Unit no. Min (T) Max (T) Unit no. Min

(MW)

Max

(MW)

Boiler 1 68 137.5 Gen 1 4.1 10

Boiler 2 52 120 Gen 2 4.9 10

Boiler 3 60 137.5 Gen 3 4.4 10Boiler 4 52 100 Gen 4 4.6 10

Boiler 5 127 250 Gen 5 4.9 10

Table 8

The simulation results for two cogeneration systems

Item Cogeneration 1

(bus-2)

Cogeneration 2

(bus-3)

Pg1 (MW) 7.6071 5.3268

Pg2 (MW) 6.0786 8.3301

Pg3 (MW) 8.4553 7.9039

Pg4 (MW) 5.0068 8.0302

Pg5 (MW) 9.9138 9.1435Total generation (MW) 37.0616 38.7345

Buyer (MW) 32.0616 28.7345

Mb1 (T/H) 69.4946 85.3920

Mb2 (T/H) 55.2571 69.0166

Mb3 (T/H) 119.8485 98.8636

Mb4 (T/H) 55.8006 52.0000

Mb5 (T/H) 249.5992 244.7278

Total emission of SOx (T/H) 0.8996 0.9000

Total emission of NOx (T/H) 0.4852 0.4910

Table 9

The cost analysis of two cogeneration systems

Cogeneration 1

(bus-2)

Cogeneration 2

(bus-3)

Total cost

Fuels cost 8.2007!104 9.0206!104 1.7221!105

Wheeling cost 3.4097!103 3.5636!103 6.9732!103

Emission cost 1.4819!104 1.4892!104 2.9711!104

Total cost 1.0024!105 1.0866!105 2.0890!105

Unit: New Taiwan Dollar (NT$).

Table 7

Fuel dispatch of the boilers

Cogeneration 1 Cogeneration 2

Oil LNG Coal Oil LNG Coal

Mb1 0.3177 0.0059 0.6764 0.0616 0.0039 0.9345

Mb2 0.0655 0.0010 0.9335 0.2571 0.0196 0.7234

Mb3 0.1945 0.0313 0.7742 0.0694 0.0626 0.8680Mb4 0.3803 0.0596 0.5601 0.1867 0.0078 0.8055

Mb5 0.0244 0.0049 0.9707 0.0616 0.0274 0.9110



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6. Conclusions

This paper presents an EIA to minimize the economic

dispatch of cogeneration systems in a competitive market.

The computation time and the operation cost by the EIAmethod have been compared with original IA method.

Results are found to be significantly improved. EIA is

superior to IA in two ways: one is the use of tabu lists to

avoid invalid searches, especially in applications with many

constraints and local optimum; another one is the automatic

regulation of the frequency of crossover and mutation

operations, particularly in applications sensitive to the

probabilistic rates. EIA has great potential to be further

applied to many ill-conditioned problems in power system

planning and operations.

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Fig. 6. The convergence characteristics of EIA and IA.

Table 10

The comparison between EIA and IA

EIA IA

Max. converged cost (NT$) 2.0891!105 2.1373!105

Min. converged cost (NT$) 2.0890!105 2.1358!105

Average converged cost (NT$) 2.0890!105 2.1367!105

CPU time (min:s) 8:02 08:26


[1] scheduling of cogen

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