a novel genetic algorithm for dynamic economic dispatch of power generation

8/6/2019 A Novel Genetic Algorithm for Dynamic Economic Dispatch of Power Generation

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JOURNAL OF COMPUTING, VOLUME 3, ISSUE 4, APRIL 2011, ISSN 2151-9617

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A Novel Genetic Algorithm for DynamicEconomic Dispatch of Power Generation

S.Dinu

AbstractThis paper presents a new and efficient method for solving the dynamic economic dispatch (DED) problem. The

main goal of this problem consists in finding the optimal combination of power outputs over a certain period of time while

satisfying all system equality and inequality constraints. The proposed framework is based on a new Genetic Algorithm with

meiosis-specific features that provides efficient global and local search characteristics. The feasibility and the validity of the

proposed approach are evaluated through numerical simulation considering a five-generator system and the results are

compared with the solutions obtained from the literature. The simulation results reveal the superiority of the proposed technique

in solving the DED problem.

Index TermsDynamic Economic Dispatch (DED), Genetic Algorithm, total generation cost, constrained optimization.

1 INTRODUCTION

In power system operation, the expected load must bepredicted following daily, weekly and seasonal cycles andconsequently load increase or decrease in each generatingunit is required. To achieve this, optimum schedules ofthe generation units must be determined. Due to thecomplexity of the planning process, the scheduling prob-lem can be decomposed into different time horizons:- Short-term scheduling: with a planning period of 1 dayto 1 week and a time increment of one hour;- Medium term scheduling: when the planning horizon isup to one year;- Long-term scheduling: with a time horizon of 1-3 year.

Short-term generation scheduling consists of determin-

ing startup (which and when generating units should becommitted) and the generation levels for each of thecommitted unit over a period of one day to one week.This centralized resource scheduling problem involves infact two distinct tasks: the unit commitment problem andthe dispatch strategy.

The unit commitment (UC) decision determines whichof the available units will be turned on in each time pe-riod, taking into account a wide variety of parametersand technological aspects, while satisfying constraintsinvolved in the operation of the unit (start-up and shutdown costs, minimal operation point, etc.).

Economic Dispatch (ED) is one of the most importantfunctions of the energy management systems. The maingoal of ED of electric power generation consists in findingthe optimal combination of power outputs while satisfy-ing all system equality and inequality constraints. Al-though the basic objective is straightforward, the problemis extended in many ways when valve-point loading ef-fects, prohibited operating zones, fuel switching, active orreactive load, ramp limits of the generators and other

practical limitations are taken into consideration.The traditional ED problem is a static optimization

problem which attempts to minimize the cost of supply-ing power subject to constraints on static behavior of thegenerating units. Static economic dispatch is suitable foronly one period time-interval, while for multi-time inter-val it will be difficult to meet the demand.

Dynamic economic dispatch (DED) problem is an ex-tension of the static economic dispatch problem in whichthe ramp rate limits of the generators are taken into con-sideration. Adjusting the power output of generators inorder to balance the load variations can minimize systemlosses at all time and it is a fundamental function in pow-

er system operation.In the DED problem one usually divides the dispatchhorizon into a number of small time intervals over whichthe load is assumed to be constant and the system is con-sidered to be in temporal steady-state. Therefore, thenumber of decision variables will be the number of gene-rating units multiplied by the number of time intervals.Thus, for N units and T time intervals results a complexoptimization problem of size N*T. Its dynamic nature(large variation of load demand) and large dimensionalitymakes this problem a large-scale dynamic optimizationproblem, difficult to solve with conventional methods.

2 THEDYNAMICECONOMICDISPATCH:PROBLEMFORMULATION

Input-output characteristics of power generation unitsare the most important initial data for solving the prob-lem of optimal planning and operation of power plants.The widely used input-output characteristic of the ith ge-nerating unit is a quadratic function, as in [1]:

HT(PGi)=aiPGi2+biPGi+ci.........Btu/h (1)

where the suffix i stands for the unit number.HT is the heat input, PGi is the net output power and ai

S. Dinu is with the Constanta Maritime University, Romania


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, bi, and ci are the coefficients of the input - output charac-teristic.

The fuel cost characteristics (FCC) is calculated by mul-tiplying the fuel input with the corresponding fuel cost(K= constant) expressed in an arbitrary monetary unit(UM):

FCC(PGi) =K* HT(PGi) (2)

Thus, the FCC for the unit i can be written in the formof a quadratic polynomial similar to the heat input equa-tion:

FCC(PGi)=aiPGi2+biPGi+ci..........UM/h (3)

A common situation is that in which the generatingunits have multiple valves that are used to control thepower output of the units. Valve point effects are usuallymodeled by adding a sinusoid component to the basicquadratic fuel-cost characteristics:

FCC(PGi)=aiPGi2+biPGi+ci+|disin(ei(PGimin-PGi))| (4)

where ei andfi are the constants for the valve point load-ing effect of unit i and PGimin is the minimum limit of thegenerating output of the ith unit. If there are N generatorscommitted to the system and T intervals in the scheduledperiod, then the DED problem can therefore be stated as:

Minimize F= )(1 1

T

t

N

i

t

GiPFCC (5)

the total generation cost subject to given constraints:

(i) T1,...,t

,1

t

Loss

t

D

N

i

t

Gi PPP (6)

equality constraint: the system generation at time t equalsthe total system load at time t plus system losses at time t.

(ii) PGimin PGit PGimax , i=1,...,N (7)

where PGimin and PGimax are the minimum and maximumlimits of the generating output of the ith unit.

(iii) -DRi PGit - PGit-1 URi (8)

i=1,,N ; t=1,,Twhere URi and DRi are the ramp-up and ramp-downlimits of the ith unit.

Then the generating capacity constraints (7) are mod-ified as:

max(PGimin, PGit-1-DRi) PGit min(PGimax, PGit-1+URi) (9)

i=1,,N ; t=1,,TUsing the B-coefficient method, network losses are ex-

pressed using Georges formula [2]:

PLoss =

n

i

n

j

t

Gjij

t

Gi PBP1 1

(10)

where Bij = loss (or B-coefficients).Further constraints can be added depending on the studyrequirements: the use of multiple fuel types, prohibitedoperating zones, etc.

3 EVOLUTIONARYCOMPUTATIONINDEDPROBLEM

Evolutionary computation comprises a set of tech-niques (genetic algorithms, genetic programming, evolu-tionary programming and evolutionary strategies)inspired by the evolutionary processes which can beobserved in nature: reproduction, mutation,recombination, natural selection and survival of thefittest. Unlike conventional optimization methods, anevolutionary algorithm operates on a population apply-ing, over the generations, the principles of natural selec-

tion and survival of the fittest to produce better solu-tion. So it uses in the search process an entire population possible solutions to the problem - and not just onepoint in the search space. The algorithm performs specificoperations within a process of reproduction generated byspecific operators that are metaphorically linked withtheir biological correspondents: mutation, crossover, in-version [3]. The qualities of each individual are evaluatedby means of special evaluation function (fitness function).The new population (new solutions) selected on the basisof fitness function, which replaces the previous genera-tion, bound for optimal and provides the best solutionsfor the given issue (Fig. 1):

Fig. 1. The basic cycle in a evolutionary algorithm

In this paper, we propose an idea of using a GeneticAlgorithm (GA) with meiosis-specific features: duplica-tion and recombination with real valued representation

scheme for solution. GAs do not work on the real genera-tor outputs, but on string encodings of them.

3.1 Implementation and settings

A population of constant size PS consisting of arrays ofdecision variable vectors is given by:

Pkgen =[( PG11,..., PGn1),...,( PG1T,..., PGnT)] (11)

k=1,,PS ; gen=1,,GMAXwhere:PGit is the power output of the i-th generation unitat the time interval t;PS is the population size;


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GMAX is the maximum number of generations (first we setGMAX=1000).

In the initial population the output of (N-1) units canbe chosen arbitrary within their respective generatingcapacity constraints. The reference unit k (randomly se-lected) is constrained by the system power equation bal-ance, as in [4]. Thus, the dependent generation power PGkt

is computed from (6) as:

PGkt = PDt + PLosst -

N

kii

t

GiP1

, t=1,...,T (12)

The total operating cost for each chromosome is com-puted using (5). The reference unit limit constraint is ap-plied using a penalty factor that is proportional to theviolation and is zero in case of no violation.

At first, PGiminand PGimaxare redefined:

PGimin(t)= max(PGimin, PGit-1 -DRi ) (13)

PGimax(t)= min(PGimax , PGit-1 +URi)PGimin =

)min(

,1max

t

GiTtP

(14)

PGimax =)max(

,1min

t

GiTtP

, i=1,...,N

Then, the reference unit constraint penalty is calcu-lated, as in [4]:

(1-PGk/ PGkmin)Fmax, if PGk < PGkminPT= (PGk/ PGkmax -1)Fmax, if PGk >PGkmax (15)

0, else

Where: )(

1 1

max

max

T

t

N

i

GiPFCCF

One can observe that for the rest of the (N-1) units, theconstraint (9) is automatically satisfied. The resultedaugmented cost function for each individual is:

FT = F+PT (16)

Because GA is designed for the solution of a minimiza-tion problem, the fitness function is calculated as the in-verse of the function FT:

fitness = 1/ FT (17)

Once the individuals of current population are eva-luated according to their fitness, the individuals that willbe the parents of the next generation are selected accord-ing to the desired selection scheme. This study uses theproportional (roulette wheel) selection. Next, the selectedindividuals are paired off randomly to give rise to newoffsprings. The reproduction of the individuals in thisstudy is inspired by the organic mechanism of a meioticcell division. In this context, the term "meiosis" refers tothe process whereby a nucleus divides by two divisions(meiosis I and meiosis II) into four gametes. Meiosishalves the number of chromosomes before sexual repro-

duction, thereby ensuring that chromosome number doesnot double with each generation. Before meiosis, eachchromosome is replicated, forming two sisters "chroma-tids" that remain linked together. The two sister chroma-tids forming each homolog are then separated during thesecond meiotic division. The implemented crossover isarithmetic crossover. The probability of crossover is pc, sothat an average of pc x 100% chromosomes undergoescrossover. Fertilization (putting together two gametesresulted from meiosis) is done by randomly combininggametes from the gene pool: two of the gametes from thefour that have been formed are then selected randomly toform two new offsprings. The scheme of the designed GAfor the DED problem is given below:

begint:= 0;initialize population P(0) randomly;evaluate P(0)while t Gmax//roulette wheel selection

for all member of population

r:=random[0,1]; k:=0; partial_sum:=0repeatk:=k+1;partial_sum:=partial_sum +fitness(k);until(r randi)offspring1[i] offspring1mut[i]offspring2[i] offspring2mut[i]

endifrepeatevaluate P(t+1)repeat; end.


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The next genetic operator, mutation, is a mechanismfor extending the search on the new areas of search space.Mutation modifies the genotype, and thus the phenotype,by random altering of bits values inside chromosomewith given probability.

Beside the non-uniform mutation operator [5], we pro-pose a new mutation operator defined as follows:

Ifgen

kP = [(1

1GP ,...,1

GnP ),...,(T

GP 1 ,...,T

GnP )] and PGit is se-

lected at random for mutation, the resultant PGit(mut) is

),2

)(max(min

minmax

GiGiGit

Gi PPP

genfP

ifr0,5 (18)

PGit, otherwise.

where f(gen)=

)1(max1

G

gen

r

(19)

r is a randomly generated number in interval (0,1);gen is the current generation; is a system parameter determining the degree of depen-dency on the iteration number. In this study we set =3.

As can be seen from (19), the amplitude of the changesdecreases as one approach the maximum number of gen-erations. Thus, this mutation operator performs globalsearch during the initial generations and local search inthe later generations. Moreover, the local searching abilityof the algorithm is improved, as well as the algorithmsefficiency. For both mutation operators we have set themutation rate at 0.05.

3.2 Application study

The effectiveness of the proposed approach is tested ona DED problem reported by [6] for a five-generator sys-tem.

Unit data, generation limits and load pattern of unitssystem are taken from the ones above. Also B-coefficientmatrix is given. The scheduling horizon is chosen as oneday with 24 intervals of one hour each.

TABLE1GENERATING UNIT DATA

Unit1 Unit2 Unit3 Unit4 Unit5

ai($/h) 25 60 100 120 40

bi($/MWh) 2.0 1.8 2.1 2.0 1.8

ci($/MW2h) 0.0080 0.0030 0.0012 0.0010 0.0015

di($/h) 100 140 160 180 200

ei(1/MW) 0.042 0.040 0.038 0.037 0.035

PGimin(MW) 10 20 30 40 50

PGimax(MW) 75 125 175 250 300

UR(MW/h) 30 30 40 50 50

DR(MW/h) 30 30 40 50 50

TABLE 2

LOAD DEMAND FOR 24 HOURS

TABLE 3B-COEFFICIENTS

The best generation over the scheduled period is givenin Table 4. It was obtained for the GA parameters set atthe values:

- size of population PS: 50- crossover rate pc=0.78- total number of generations GMAX=500GA parameters were selected following parameter

sensitivity analysis. From the results, one observes thatthe system generation satisfies the system load plus thesystem losses in each interval.

TABLE4BEST GENERATION OVER THE PERIOD

Hour P1

(MW)

P2

(MW)

P3

(MW)

P4

(MW)

P5

(MW)

PLoss

1 18,84 98,29 31,25 116,12 150,21 4,71

2 21,93 102,68 42,28 111,09 162,15 5,13

3 26,18 105,12 65,19 113,21 170,50 5,204 31,45 105,79 83,11 115,19 199,70 5,24

5 36,84 103,18 84,29 114,20 225,39 5,90

6 29,15 106,20 82,15 129,78 267,67 6,957 28,10 104,09 89,20 150,15 261,89 7,43

8 26,03 107,18 85,30 160,29 283,85 8,65

9 27,15 106,11 120,90 163,15 282,67 9,98

10 35,98 108,24 120,79 161,29 287,81 10,11

11 40,16 111,07 129,25 155,43 296,37 12,28

12 37,09 109,12 138,98 168,15 299,63 12,9713 33,15 110,29 120,15 179,98 271,46 11,03

14 36,29 112,03 118,75 165,23 268,61 10,91

15 35,79 112,09 116,98 144,15 255,06 10,0716 30,12 107,03 114,05 129,80 207,53 8,53

17 27,98 108,12 113,90 118,65 197,47 8,12

18 26,12 107,02 113,65 127,98 242,01 8,78

19 28,75 107,85 125,13 129,11 273,24 10,0820 29,11 106,95 148,25 138,15 292,50 10,96

21 33,68 104,19 143,15 120,13 288,60 9,75

22 30,97 102,98 128,79 111,18 239,11 8,03

23 26,84 99,45 107,48 103,12 196,21 6,10

24 23,15 98,79 82,15 100,87 163,16 5,12

The results of the proposed method are compared withthat of the Simulated Annealing method (SA) [7] andMaclaurin Series Based Lagrangian Method (MSL) [6].

Hour(h)

Load(MW)

Hour(h)

Load(MW)

Hour(h)

Load(MW)

Hour(h)

Load(MW)

1 410 7 626 13 704 19 654

2 435 8 654 14 690 20 704

3 475 9 690 15 654 21 680

4 530 10 704 16 580 22 605

5 558 11 720 17 558 23 527

6 608 12 740 18 608 24 463

0.000049 0.000014 0.000015 0.000015 0.000020

0.000014 0.000045 0.000016 0.000020 0.000018

0.000015 0.000016 0.000039 0.000010 0.000012

0.000015 0.000020 0.000010 0.000040 0.000014

0.000020 0.000018 0.000012 0.000014 0.000035


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The best total cost obtained by the proposed algorithmis 46930 compared to 47356 of the SA method and 49216of the MSL method.

Further more, it can be observed from Table 1 that Unit1 and Unit 2 are the most ramp-rate limited units.

By comparing their schedules obtained with the cur-rent solution and that of [6] one notices that the proposedmethod is going to flat these schedules (Fig. 2).

Fig. 2. Compared schedules for Unit 1 and Unit 2

The sensitivity analysis was performed on GA parame-

ters to determine their influence on the algorithms per-

formance.

First, analysis was performed for 10 values of popula-

tion size (PS) in the range [10, 100] with an increment of

10, while the value for crossover was set at 0.7, in accor-

dance with previous results in GA applications.

The results obtained after performing 50 independent

runs for each case (Fig. 3) indicate that the performance of

the algorithm was improved when the population size

increased from 10 to 50, but no significant improvements

were observed when the population was further in-

creased.

A large number of studies which explore the interac-

tion among different GA parameters showed that in gen-

eral GAs will work well with high crossover & low muta-

tion probability [8], [9], [10], [11].

Fig. 3. Total fuel cost for different population size

Maintaining the obtained population of 50, the cros-sover rate was changed from 0.5 to 0.9 with an incrementof 0.01 and 50 independent tests were performed for eachcase. The obtained results are shown in Fig. 4. The best

option for crossover rate is located very close to 0.8. Afterthis value, the performance decreases with the increase incrossover rate.

Fig. 4. Total fuel cost for different crossover rate

For further analysis regarding the total number of gen-erations, a population size of 50 and a crossover probabil-ity of 0.78 were chosen. Fig. 5 shows the the minimum oftotal fuel cost in each generation: cost decreases propor-tionally as generation advances. From the results one canobserve an intense decrease of cost in early generations,where individuals are far from the optimum.

The algorithm finds an optimal solution within lessthan 1000 generations that leads to less computationaltime for the solution process.

Another test was carried out regarding the new pro-posed mutation operator. The results obtained previouslywith non-uniform mutation were compared in terms ofsolution quality with those obtained for this mutationoperator.

0

5

10

15

20

25

30

35

40

45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24hour (h)

P(MW)

Schedule f rom reference schedule f rom the p roposed app roach

0

20

40

60

80

100

120

140

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24hour (h)

P(MW)

schedule from ref erence schedule from the proposed approach

Unit 1 schedulng

Unit 2 scheduling


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Fig. 5. Total fuel cost in each generation

The results of this comparison are summarized on Fig-

ures 6-7 and in Table 5. Fig. 6 shows the best objectivevalue (minimum fuel cost) obtained by the 100 runs per-formed when using the proposed algorithm for non-uniform mutation operator, while Fig. 7 shows the bestobjective value obtained by the 100 runs performed whenusing the new proposed mutation operator.

Fig. 6. Best objective values for the non-uniform muta-tion operator

Fig. 7. Best objective values for the new proposed mu-tation operator

Table 5 summarizes the minimum cost, maximum cost,average cost and standard deviation obtained after 100

independent runs in both cases. The difference betweenmax and min objective values expresses the search rangeof the algorithms. St. Dev denotes standard deviation,which expresses the searching capacity of each algorithm.As Table 5, the new mutation operator provides a smallermean value and a smaller standard deviation.

TABLE 5PERFORMANCES OF THE PROPOSED ALGORITHM

4CONCLUSION

This paper presents a new and efficient method forsolving the dynamic economic dispatch problem. The

specific algorithm used in this study has proven to be avaluable help in dealing with this problem. Throughits proposed enhancements (meiosis-specific featuresand the new mutation operator), the procedure de-monstrates superior global and local search characte-ristics, even with a small population size.

A numerical simulation including comparative stu-dies has been presented to demonstrate the perfor-mance and applicability of the proposed method. Thesimulation results reveal the superiority of the pro-posed technique in solving the DED problem. There-fore this approach could also be extended to other op-timization and control problems of power systems.

In addition, sensitivity analysis has been carried out tostudy the effects of the various genetic parameters onthe convergence behavior of the proposed approach.

This analyze of the influence of the changing para-meters may have further consequences on designingother evolutionary algorithms for this problem.

REFERENCES

[1] D.P. Kothari and J.S.Dhillon, Power System Optimization. Pren-

tice Hall, pp. 444-445, 2004

[2] E.E. George, Intrasystem transmission lased AIEE Trans.

(Electrical Engineering), vol. 62, pp. 153-158, 1943

[3] D.B. Fogel, Evolutionary computation: Toward a new philosophy of

machine intelligence. Piscataway, NJ: IEEE Press, 2000.[4] N. Amjady and H. Nasiri-Rad, Economic dispatch using an

efficient real-coded genetic algorithm, IET Generation, Trans-

mission, Distribution, vol. 3(3), pp. 266278, 2009

[5] Z. Michalewicz, T. Logan and S. Swaminathan, Evolutionary

operators for continuous convex parameter space Proceedings

of Third Annual Conference on Evolutionary Programming , pp.84-

97, 1994

[6] S. Hemamalini and S.P. Simon, Dynamic Economic Dispatch

with Valve-Point Effect Using Maclaurin Series Based Lagran-

gian Method International Journal of Computer Applications, vol.

17, pp. 71-78, 2010

[7] B.K. Panigrahi, V.R. Pandi and S. Das, Adaptive particle

Results obtained with non-uniform mutation operator

100 runs Average St. Dev

[46926.04 , 46934.12] 46930.49 2.025

Results obtained with the new proposed mutation operator

100 runs Average St. Dev

[46927.11 , 46932.74] 46929.83 1.517


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swarm optimization approach for static and dynamic economic

load dispatch, Energy conversion and management, vol. 49, pp.

1407-1415, 2008

[8] K. Metaxiotis, Intelligent Information Systems and Knowledge

Management for Energy: Applications for Decision Support, Usage,

and Environmental Protection. IGI Global Publishing, pp. 135-139,

2009

[9] Y.L. Kwang and M.A. El-Sharkawi, Modern Heuristic Optimiza-

tion Techniques: Theory and Applications to Power Systems. IEEE

Press Series on Power Engineering, pp. 337-349, 2008

[10] K.P. Dahal, K.C. Tan and P.I. Cowling, Evolutionary Scheduling.

Springer Berlin/Heidelberg, pp. 317-330, 2007

[11] A. Pereira-Neto, C. Unsihuay and O.R. Saavedra, Efficient

evolutionary strategy optimization procedure to solve the non-

convex economic dispatch problem with generator constraints

IEE proc.- Gener. Transm. and Distrib. 152, (5), pp. 653-660, 2005

a novel genetic algorithm for dynamic economic dispatch of power generation

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