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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
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.
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 , and gradient descent approaches, suchas Lagrangian relaxation , 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  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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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:
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:
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)
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
Fig.2:Two-segment Cogeneration Unit FOR
3. BASIC EVOLUTIONARY PROGRAMMING
Evolutionary programming  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
i =+ +=+=
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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
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)
Create Initial Population
Mutate Childpopulationfrom Parent o ulation
Compete Parents and Childrento form next eneration
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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)
ikh2= [(h i,max hi,min)/D] [((fk fbest)/(fweak- fbest)) + (Sinit/3)
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 , 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 . 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 (...