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BASIC EVOLUTIONARY PROGRAMMING FOR STATIC DISPATCH OF COGENERATION

Kit Po Wong Cameron Algie

Artificial Intelligence and Power Systems Research GroupDepartment of Electrical and Electronic Engineering

The University of Western Australia

Abstract

This paper develops an Evolutionary Programming (EP) based algorithm for the combined power andheat dispatch (CHPD) problem for cogeneration systems. The CHPD problem is first formulatedmathematically. The EP approach is described and followed by the establishment of the components of

EP-CHPD algorithm. The algorithm is applied to a test case containing two cogeneration units. Theresults are presented and the performance of the algorithm is demonstrated.

1. INTRODUCTION

A non-utility company that generates significant

amounts of power is referred to as an Independent

Power Producer (IPP). IPPs primarily generate powerfor their own needs. In an environment of de-regulated

electricity supply industry and open network accesspolicy, IPPs can supply their excess generation capacityto host utilities, competitive power pools, and customersthrough a host network's grid.

Cogeneration, or combined heat and power (CHP)production, is the simultaneous production of electricityand useful heat. Some industrial processes have large

steam requirements as well as large power demands; IPPcogeneration is a viable option for these industries. Amodern IPP cogeneration unit typically consists of a gas

turbine (GT) thermal generator linked with a hot steamrecovery generator (HSRG). It is not unusual forcogeneration facilities to operate conventional power

and boiler (steam generation) units in addition to theircogeneration units.

Economic dispatch (ED) deals with minimising the totalfuel cost of a group of generators to meet a power

demand. The CHP dispatch (CHPD) problem is morecomplex than the conventional ED problem. Non-linearoptimisation methods, such as dual and quadratic

programming [1], and gradient descent approaches, suchas Lagrangian relaxation [2], have been applied to it.However, these methods cannot deal with discontinuous

and/or non-monotonic input/output models for generatorfuel characteristics. New methods capable of handlingthese fuel characteristic models are worth developing.

The advent of evolutionary computation and

evolutionary algorithms (EA) [3-5] has providedalternative approaches for solving conventional EDproblems. Simulated annealing, genetic, hybrid

genetic/simulated annealing algorithms and evolutionary

programming (EP) have been successfully applied tothis problem and the related optimal power flowproblem [6-8]. However, little work has been reported in

the literature on the application of EA to the CHPDproblem.

This paper adopts the EP methodology because it lendsitself to optimisation problems with continuous

variables, such as the generator loads of the CHPDproblem. EP does not depend on derivatives of theobjective function of the problem being solved,

accommodating discontinuous and non-monotonicfunctions. It simulates the mechanics of biologicalevolution over a number of iterations to find the globaloptimum.

This paper develops an EP-based algorithm for theCHPD problem of IPP cogeneration systems (EP-CHPD). The CHPD problem is first formulated

mathematically. The evolutionary process utilised by theEP approach is then described followed by theestablishment of the EP based algorithm for solving the

CHPD problem. The algorithm is applied to a previouslypublished numerical example [2] consisting of twocogeneration units, a boiler and a conventional

generator. The performance of the algorithm as thepopulation size is varied is assessed and the resultspresented.

2. COMBINED HEAT AND POWER DISPATCHPROBLEM

The problem of static dispatch determines the loads ofgenerators in a system that will meet a power demandduring a single scheduling period for the least cost. The

conventional economic dispatch (ED) problem dealswith one variable and class of generators, power loadsfor conventional thermal generators, meeting a single

power demand. The CHP dispatch problem (CHPD) isconsiderably more complicated, having two demands to


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+=+==

++n

iii 1

ibi

1

iici

1iti )h(f)h,p(f)p(f

meet, power and process heat (steam), up to threeclasses of generators, cogeneration units, thermalgenerators and boilers, and two types of variables, heat

and power loads from generators. The complexity isfurther increased by the non-separable nature of the

power and heat loads of cogeneration units. Theobjective function the CHPD problem seeks to minimiseis:

(1)

where:

fti, fciand fbiare the respective fuel characteristics of theconventional (thermal) generators, cogeneration unitsand boiler units.

piand hiare the power and heat loads for generator i.

i [1,2,...] denotes conventional generators, i

[+1,+2,...,] cogeneration units and i

[+1,+2,...,n] boiler units.

The heat (MWth) and power (MWe) demands to be met

by IPP cogeneration facilities are usually in the 10's tolow 100's. The electricity generated has to meet theplant's internal requirements, as well as the power it is

contracted to supply to the host network. Generatingpower for supply to the network has to include lossesthat will be incurred in transmission. Internal steam and

piped heated demands are used within a short distanceof steam generation; energy loss in this transport isnegligible. The schematic of IPP cogeneration system is

shown in Fig. 1. The balance of power and heatconstraints of the CHPD problem can be written as:

(2)

(3)

where:

Pint is on-site power demand (MWe),Putilis host utilities power demand (MWe),L is active power loss in transmission host network

(MWe), andH is heat (steam) demand (MWth).

The minimum and maximum generation capacity limitsfor boilers and conventional electricity units areexpressed in (4) and (5). Cogeneration units' heat and

power outputs are non-separable; one output will affectthe other output's feasible range (6) and (7). The feasibleoperating region (FOR) is usually bound by a one-

segment, though sometimes two-segment, irregularquadrilateral region. A two-segment cogeneration unit

Fig. 1:Schematic of an IPP Cogeneration System

FOR is shown in Fig. 2. The ranges of the variables inthe CHPD problem are:

pi,minpipi,max i 1,2, (4)

hi,minhihi,max i +1,+2,,n (5)

pi,min(hi) pipi,max(hi) i +1,+2,, (6)

hi,min(pi) hihi,max(pi) i +1,+2,, (7)

where:

pi,min and pi,max are the minimum and maximum powergeneration of generator i.hi,min and hi,max are the minimum and maximum heat

production of generator i.pi,min(hi), pi,max(hi), hi,min(pi) and hi,max(pi) are the linearinequalities that define the FOR of the cogeneration

units.

Fig.2:Two-segment Cogeneration Unit FOR

3. BASIC EVOLUTIONARY PROGRAMMING

Evolutionary programming [5] emulates natural

selection processes to find the global optimum ofcomplex minimisation and maximisation problems. A

population of individuals is evolved over a number ofgenerations, until the fittest individual is found. Eachindividual contains the variables, or genes , used to builda candidate solution for the problem. The solution datais used to assess the fitness of the individual.

Each generation a childpopulation is spawned, via gene

mutation, from the parent population. Gene mutationshifts continuous variable values along their feasible

0 hi(MWth/hr)

pi

(MWe/hr)LPPpp utilint

1

i

1

i +

+==

+=+

ii

Hhhn

1i

i

1ai

i =+ +=+=

+1

1

ON SITE

+1 n

L Putil

H

Pint

HEAT BUS

POWER BUS


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ranges. EP applies a mutation operation to all genes ofan individual.

All individuals of the child and parent populations arecompeted against a number of members from the other

population in a stochastic tournament. If a childindividual is fitter and has a better tournament scorethan a parent, it will take that parent's place in the nextgeneration. The tournament maintains some genetic

diversity by giving less fit genes a chance to survive viabetter tournament score.

The evolution process will terminate when the fitness ofbest solution has not improved over a significant numberof generations, or the maximum allowable number of

generations has been reached. Fig. 3 shows a flowdiagram of the EP process.

4. EP-CHPD ALGORITHM

Based on the basic EP technique in the last section, a

new algorithm for solving the CHPD problem isdeveloped below.

(a) Individual Representation: A real value vector,representing an individual, is shown as Fig. 4. Powerand heat load variables are pi and hi respectively.

Cogeneration units' heat and power loads must be set asa pair. For the dispatch problem is this paper, themajority of the power and heat demands are met by

cogeneration units and the remainders are supplied byconventional and boiler units. To achieve this, loads areset for all cogeneration units first, then all conventionalunits and lastly all boiler units.

(b) Initialisation :Initial generator loads are set within

their feasible ranges using uniformly distributed randomvariables. The intialisation process sets variables for onegeneration unit at a time. Modified heat and power

balance equations, which take into account previouslyassigned demand and the minimum feasible output ofup-coming generation units, are used to ensure thatconstraints (2) and (3) are satisfied.

The last power generation unit to be set in the dispatch

queue has all of the remaining power instead of random

setting. Likewise, the last steam production unit is set totake up all the remaining steam demand.

(c) Individual's Fitness:The fitness of an individual ismeasured by the "strength" of the solution it offers to a

problem. Provided its solution satisfies all constraints,the fitness value of an individual in EP-CHPD algorithmis calculated according to the expression below:

Fig. 3:Flowchart of the EP process

fk= Ck/ Cmax (8)

Where fkis the fitness of the kthindividual and Ckis the

total fuel cost of the k

th

individual's solution. Cmax, fuelcost when all generators are operating at full capacity,has been chosen as the normalisation factor. All fitnessscores will be in the range [0-1], with lower scores being

fitter. The fitness of global minimum can never be zero,as real generator fuel characteristics have fixedoperation costs incorporated in them.

(d) Mutation Operation: Each of the "m" individualsin the parent population spawns a child individual. Each

power and heat variable of the kth parent individual,respectively pik and hik, is mutated to a new variable,pi,m+kor hi,m+k respectively, in a child individual. The

operation (9), i [1,2,,], is used for power variables

and operation (10), i [+1,+2,,n], is used for heatvariables:

pi,m+k= pi,k+ N(0,ikp2) (9)

hi,m+k= hi,k+ N(0,ikh2) (10)

FINISH

StoppingRule

no

yes

START

Create Initial Population

Mutate Childpopulationfrom Parent o ulation

Compete Parents and Childrento form next eneration

pop1+ pop2

new pop1

new

pop1

pop1


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Cogeneration Units Conventional Units Boiler Units

p+1 h+1 p h p1 p h+1 hn

Fig. 4:Individual in EP-CHPD algorithm

In the above equations, N is a Gaussian distribution with

zero mean and standard deviation of ikp2 or ikh

2, forpower and heat loading respectively. The standarddeviations are evaluated according to:

ikp2= [(p i,max pi,min)/D] [((fk fbest)/(fweak- fbest)) + (Sinit/3)

g-1] (11)

ikh2= [(h i,max hi,min)/D] [((fk fbest)/(fweak- fbest)) + (Sinit/3)

g-1] (12)

The last term of (11) and (12) is an off-zero searchmechanism. It allows the strongest individual tocontinue searching in its local vicinity. The cooling

mutation of [8], g-1, is used to reduce Sinitto Sfinalas the

generation counter, g, approaches the last allowedgeneration, gmax. Sinitis the additional percentage of the

variable range that is to be searched either side of thecurrent best solution. It is expressed as a decimal andshould be small enough to not significantly affect the

rest of mutation operation. The geometric reductionconstant, , is set by the following equation:

= exp(ln[Sfinal/Sinit]/gmax) (13)

(e) Stopping Rule:The EP-CHPD algorithm terminates

when the maximum number of generations has beenreached.

(f) Range Clipping: The modified power balanceequations, used in the intialisation and mutation processto maintain heat and power balance, determine how

much power (and/or heat) is actually available, pAvail(hAvail), for a generator to dispatch. If pAvail (hAvail) isgreater than the maximum capacity of a generator the

random settings can select from the full range of values,[pi,min-pi,max] ([hi,min-hi,max]). Otherwise, the upper limit ofthe available range is "clipped" back, reducing the

allowable range of the variable, [p i,min-pAvail] ([hi,min-hAvail]).

(g) Overshoot adjustment: This is a standard feature

for the standard ED problem where variables only havea maximum and a minimum limit. When a freshly

mutated variable goes beyond a generator's limit, it isadjusted to value of the limit it overshot. Overshootadjustment helps speed up solution time in cases where

conventional and/or boiler units' optimal settings are attheir limits. However, overshoot adjustment forcogeneration units is computationally expensive and

complex to implement properly. Depending on thelocation of the optimal setting of the cogeneration unit,overshoot correction can impede the evolution processfor little or no improvement of the optimal solution.

(h) Cogeneration Constraint Relaxation: Rather than

implementing an overshoot adjustment mechanism forcogeneration units, EP-CHPD relaxes cogeneration FORlimits to the maximum and minimum possible heat and

power for each unit during mutation. The heat andpower balances are also partially relaxed, with the totalpower and heat of each individual only having to be

"greater than or equal", as opposed to "exactly equal", to

the heat and power demands. The best individual cannever be an impossible solution, as rigorous boundary

checks are always applied to candidates before they areallowed to be the "best candidate" and be retained. Thepartial relaxation of heat and power balance does not

affect the best result in the later generations of theevolutionary process, as over supplying solutions havehigher costs than solutions that meet demands exactly.

5. APPLICATION EXAMPLE

The developed EP-CHPD algorithm has been validated

by applying it to a test system [2]. The test system is anexample of a single area cogeneration system. Itcontains one conventional electricity unit, one boilerunit and two cogeneration units. Conventional and boiler

units operating ranges and their cost characteristics aregiven in (14) and (17). The cogeneration unit's FORdiagrams are given in Fig. 5 and the cost characteristics

are given in (15) and (16). In Fig. 5, the relaxedconstraints and optimal load points are also shown. Thepower and heat demands of the system are 200 MWeand 115 MWth respectively.

fti(p1) = 50p1 0 p1150 (14)

fc2(p2,h2) = 2650 + 14.5p2+ 0.0345p22+ 4.2 h2+ 0.03h2

2+ 0.031p2h2 (15)

fc3(p3,h3) = 1250 + 34.5p3+ 0.0435p32+

0.6 h3+ 0.027h32+ 0.011p3h3 (16)

fbi(h4) = 23.4h4 0 h42695.2 (17)


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Fig. 5:Cogeneration Unit's FOR

Conventional "1" Cogen "2" Cogen "3" Boiler "4" Totals

H 40 75 0 115 MWthP 0 160 40 200 MWeOptimal

Total Cost $9257.10

H 39.39 75.61 0 115 MWth

EP-CHPDBest

P 0 159.31 40.69 200 MWe

Total Cost $9262.59Table 1: Summary of Optimal and EP-CHPD Best Results

During the validation study, the following EP-CHPDparameter settings have been used in (11)-(13): D = 12,

Sinit= 0.05, Sfinal= 0.02, gmax= 100. Cases for populationsizes between 10 and 100 have been executed for 50trials each. The optimal solution [2] and the bestsolution from EP-CHPD are summarised in Table 1.

The optimal setting for cogeneration unit 3 lies in lower

right hand corner of its FOR boundary, as seen in Fig. 5.EP-CHPD obtained p3and h3 settings very close to thispoint, but was unable to search down to the intersection

of the two bounds. However, the cost of EP-CHPD'sbest solution is only 0.06% greater than the optimalsolution. At least 10 trials per population size returned a

solution with a best cost less than $9270 and all trialsreturned at least one best cost less than $9266, less than

0.1% error. Though EP-CHPD is unable to find theabsolute optimum solution, it can guarantee finding asolution that is practically optimal.

The computational efficiency of the EP-CHPDalgorithm can be indicated by the generation/ populationsize and evaluation/ population size charts. Evaluation is

defined as the product of the population size and the

generation at which the optimal solution is found. Dueto their random nature, the evaluation of EP-CHPD

solving a particular case is averaged over a number oftrials. Some trials do not succeed and they are excludedfrom the evaluation determination.

Because EP-CHPD is unable to find the exact optimumsolution, a modified version of the evaluation system

was used. Trials that failed to improve upon the bestsolution of the initial population, all of which were inthe range $9900-$10500, were cut from the data. The

cost of the best solution at each generation was averagedover the remaining sets of trial data. The generation atwhich the average cost went below $9275.60, or + 0.2%

error, for the first time, was used as the generation ofnear-optimal solution. $9275.60 also approximately

corresponds to cogeneration units 2 and 3 load settingsbeing 1 MWe and 1 MWth off optimal set tings.

Fig. 6 shows when population size is small, a large

number of generations is needed by the algorithm to finda near-optimal solution. When the population sizeincreases, the number of generations needed decreases.

This continues until population size of 70, at which

Cogenerat ion Uni t 2

0

250

0 200h 2

p 2

FOR Bounds Re la xed B ou nd s

Cogenerat ion Uni t 3

0

25 0

0 200h 3

p 3

Optimal Load Point

p = 247

p = 81

h = 180

215

98.8

104.815.9 32.4 75

p = 40h = 135.6

= 125.8110.2

44

40

160


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point only about 25 generations are needed. Apopulation size of 14 or greater is required to guaranteea near-optimal solution within 100 generations.

Fig. 6:Generation of Near-Optimal Solution against

Population size

Fig. 7:Evaluation against Population Size, with Trend

Lines.

The data in Fig. 7, showing evaluation against

population size, has been split into two segments,population sizes 14-30 and population sizes 30-100.

The occurrence rate of unsuccessful trials was at most 2

out of 50 trials for population size between 30-100. Forthe smaller population sizes of 14-30 there are about 3-5unsuccessful trials out of 50. Population sizes of 10 and

13 were tried, with respectively 8 and 7 unsuccessful

trials per 50; the failure rates, above 10%, are too highto consider using populations less than 14.

The actual minimum evaluation is found whenpopulation size is 15. Due to the higher incidence of

unsuccessful trials for population sizes less than 30, theevaluation for population size of 30 has been found to bethe reliable minimum.

6. CONCLUSIONS

An algorithm, EP-CHPD, based on evolutionary

programming for solving the combined heat and powerdispatch problem for cogeneration systems has beendeveloped. Random initialisation and constraint

relaxation for the feasible operating region of thecogeneration units have also been developed andincorporated into the algorithm. The algorithm has been

validated on a single area cogeneration system's test case

and its performance, both in terms of solution accuracyand computational efficiency, has been investigated andillustrated. The new algorithm is very promising.

7. REFERENCES

[1] F.J. Rooijers and R.A.M. van Amerongen, "StaticEconomic Dispatch for Co-Generation Systems", IEEE

Transaction On Power Systems (Trans-PWRS), Vol. 9,

No. 3, Aug 1994, p. 1392-98.

[2] Tao Guo, M.I. Henwood and M. van Ooijen, "AnAlgorithm for Heat and Power Dispatch", IEEE Trans-

PWRS, Vol. 11, No. 4, Nov 1996, p. 1778-84.

[3] D.E. Goldberg, Genetic Algorithms in Search,Optimisation and Machine Learning, Addison-Wesley,

1989.

[4] Z. Michalewicz, Genetic algorithms + data structures =evolution programs, 3rd rev. extended ed., Springer-

Verlag, 1996.

[5] D.B. Fogel, Evolutionary Computation: Toward a new

philosphy in machine intelligence, IEEE Press, 1995.

[6] K.P. Wong and C.C. Fung, "Simulated Annealing BasedEconomic Dispatch Algorithm", IEE Proceedings-C, Vol.

40, No. 6, Nov 1993, p. 509-15.

[7] K.P. Wong and C.C. Fung, "Genetic andGenetic/Simulated-Annealing Approaches to EconomicDispatch", IEE Proceedings On Generation, Transmission

and Distribution, Vol. 141 No. 5, Sep 1994, p. 507-13.

[8] K.P. Wong and J. Yuryevich, "Evolutionary-Programming-Based Algorithm for Environmentally-

Constrained Economic Dispatch", IEEE Trans-PWRS,

Vol. 13 No. 2, May 1998, p. 301-6.

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