Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 427936 5 pageshttpdxdoiorg1011552013427936
Research ArticleEconomic Dispatch Using Parameter-Setting-FreeHarmony Search
Zong Woo Geem
Department of Energy and Information Technology Gachon University Seongnam 461-701 Republic of Korea
Correspondence should be addressed to Zong Woo Geem geemgachonackr
Received 5 February 2013 Revised 26 March 2013 Accepted 8 April 2013
Academic Editor Xin-She Yang
Copyright copy 2013 Zong Woo Geem This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Economic dispatch is one of the popular energy system optimization problems Recently it has been solved by variousphenomenon-mimicking metaheuristic algorithms such as genetic algorithm tabu search evolutionary programming particleswarm optimization harmony search honey bee mating optimization and firefly algorithm However those phenomenon-mimicking problems require a tedious and troublesome process of algorithm parameter value setting Without a proper parametersetting good results cannot be guaranteed Thus this study adopts a newly developed parameter-setting-free technique combinedwith the harmony search algorithm and applies it to the economic dispatch problem for the first time obtaining good resultsHopefully more researchers in energy system fields will adopt this user-friendly technique in their own problems in the future
1 Introduction
Economic dispatch (ED) is defined in the US Energy PolicyAct of 2005 as the operation of electrical generation facilitiesto produce energy at the least cost to reliably serve consumerswhile satisfying any operational limits of generation andtransmission facilities ED became a popular optimizationproblem in energy system field which has been tackled byvarious optimization techniques such as genetic algorithm(GA) [1] tabu search (TS) [2] evolutionary programming(EP) [3] particle swarm optimization (PSO) [4] harmonysearch (HS) [5] honey beemating optimization (HBMO) [6]and firefly algorithm (FA) [7]
As observed in the literature better results have beenobtained by phenomenon-mimicking metaheuristic algo-rithms rather than gradient-based mathematical techniquesIndeed the metaheuristic algorithm has advantages over themathematical technique in terms of several factors (1) theformer does not require complex derivative functions (2) theformer does not require a feasible starting solution vectorwhich is sensitive to the final solution quality and (3) theformer has more chance to find the global optimum
However the metaheuristic algorithm also has the weak-ness in the sense that it requires ldquoproper and appropriaterdquovalue setting for algorithm parameters [8] For example inGA only carefully chosen values for crossover and mutationrates can guarantee good final solution quality which is notan easy task for algorithmusers in practical fields who seldomknow how the algorithm exactly works
In order to overcome this troublesome parameter settingprocess researchers have proposed adaptive GA techniques[9] which adjust crossover and mutation rates adaptivelyinstead of using fixed rates to find good solutions withoutmanually setting the algorithm parameters This adaptivetechnique has been applied to various technical applicationssuch as environmental treatment [10] structural design [11]and sewer network design [12]
In energy system field the adaptive GA was also appliedto a reactive power dispatch optimization as early as 1998 [13]Afterwards however there have been seldom applicationsin major research databases using the adaptive techniqueThus this study intends to apply a newly developed adap-tive parameter-setting-free (PSF) technique [8] which iscombined with the HS algorithm to the economic dispatchproblem for the first time
2 Journal of Applied Mathematics
2 Economic Dispatch Problem
Theeconomic dispatch problem can be optimally formulatedThe objective function can be as follows
Min 119911 = sum119894
119862119894(119875119894) (1)
where 119862119894(sdot) is generation cost for generator 119894 and 119875
119894is
electrical power generated by generator 119894 Here 119862119894(sdot) can be
119894+10038161003816100381610038161003816119890119894times sin (119891
119894times (119875
min119894
minus 119875119894))10038161003816100381610038161003816
(2)
where 119886119894 119887119894 119888119894 119890119894 and 119891
119894are cost coefficients for generator 119894
The fourth term in the right-hand side of (2) represents valve-point effects
The above objective function is to be minimized whilesatisfying the following equality constraint
sum119894
119875119894= 119863 (3)
where 119863 is total load demand Also each generator shouldgenerate power between minimum and maximum limits asthe following inequality constraint
119875min119894
le 119875119894le 119875
max119894
(4)
3 Parameter-Setting-Free Technique
The parameter-setting-free harmony search (PSF-HS) algo-rithm was first proposed for optimizing the discrete-variableproblems such as structural design [14] water network design[15] and recreational magic square [8] PSF-HS was alsoapplied to a continuous-variable problem such as hydrologicparameter calibration [16]
However it was never applied to a continuous-variableproblem with technical constraints Thus this study firstapplies PSF-HS to the ED problem whose type is the con-tinuous-variable problemwith a technical constraint becauseits decision variable 119875
119894has the continuous value and it has
the equality constraint of total power demand as expressedin (3) Here the inequality constraint in (4) can be simplyconsidered as value rangeswithout using any penaltymethod
The basic HS algorithm manages a memory matrixnamed harmony memory as follows
Once thisHM is fully filled with randomly generated vectors(P1 PHMS) a new vector PNew is generated as follows
119875New119894
larr997888
119875min119894
le 119875119894le 119875max119894
wp 119877Random119875119894 (119896) isin 119875
1
119894 1198752119894 119875HMS
119894 wp 119877Memory
119875119894 (119896) + Δ wp 119877Pitch
(6)
where 119877Random is random selection rate 119877Memory is purememory consideration rate 119877Pitch is pure pitch adjustmentrate and Δ is pitch adjustment amount
If the newly generated vector PNew is better than theworst vector PWorst inHM those two vectors are swapped asfollows
PNewisin HM and PWorst
notin HM (7)
The basic HS algorithm performs (6) and (7) until a termina-tion criterion is satisfied
For PSF-HS one additionalmatrix named operation typematrix (OTM) is also managed as follows
[[[[[[
[
1199001
1=Random 1199001
2=Pitch sdot sdot sdot 1199001
119899=Memory
11990021=Memory 1199002
2=Memory sdot sdot sdot 1199002
119899=Pitch
sdot sdot sdot sdot sdot sdot sdot sdot sdot
119900HMS1
=Memory 119900HMS2
=Random sdot sdot sdot 119900HMS119899
=Memory
]]]]]]
]
(8)
OTM memorizes which operation (random selection mem-ory consideration and pitch adjustment) each value comesfrom For example if the value of 1198752
2in HM comes from
memory consideration operation the value of 11990022in OTM
is also set as ldquoMemoryrdquo This process happens when initialvectors are populated or when a new vector is inserted intoHM
Thus instead of using fixed algorithm parameter valuesPSF-HS can utilize adaptive parameter values by calculatingthem at each iteration as follows
119877119894Random =
119888119905 (119900119895
119894= Random 119895 = 1 2 HMS)
HMS
119894 = 1 2 119899
119877119894Memory =
119888119905 (119900119895
119894= Memory 119895 = 1 2 HMS)
HMS
119894 = 1 2 119899
119877119894Pitch =
119888119905 (119900119895
119894= Pitch 119895 = 1 2 HMS)
HMS
119894 = 1 2 119899
(9)
where 119888119905(sdot) is a function which counts specific elements thatsatisfy the condition
4 Numerical Example
The PSF-HS is applied to a popular bench-mark ED problemwith three generators The input data for the three-generatorproblem is shown in Table 1
When the total system demand is set to 850MW theoptimal solution is known as $823407 [2ndash4] which wasreplicated by using a popular gradient-based technique(generalized reduced gradient (GRG) method) which has
Journal of Applied Mathematics 3
Table 1 Data for three-generator example with valve-point loading
been also successfully applied to other energy optimizationproblems such as building chiller loading [17] combined heatand power ED [18] and hybrid renewable energy systemdesign [19] However the GRG method was able to obtainthe identical best solution only when it started with a vector(1198751= 300 119875
2= 150 119875
3= 400) Instead when different
starting vector (1198751= 600 119875
2= 200 119875
3= 400) was used
solution quality was worsened as $824141When PSF-HS was also applied to the problem it
obtained a near-optimal solution of $823447 after 100 runswhich has small discrepancy from the optimal solution($823407) by 0005 For the results from 100 runs max-imum and mean solutions are $842974 (24 discrepancy)and $829288 (07 discrepancy) respectively Here PSF-HS was performed using MS-Excel VBA environment withIntel CPU 33GHz Each run takes only one second in thiscomputing environment
Figure 1 shows the convergence history of power genera-tion cost for the case of the near-optimal solution $823447As seen in the figure PSF-HS closely approached to the near-optimal solution in early iterations
Table 2 shows the finalHM with HMS = 30 As observedin the table there are many similar vectors in HM becausePSF-HS tried local search instead of global search in latestage of computation
Figure 2 shows the history of random selection rate119877Random As observed in the figure all three parameters(1198771Random 1198772Random and 119877
3Random) started with highervalues (05) In less than 1000 iterations119877
1Random went up toaround 04 119877
2Random to around 05 and 1198773Random to around
08Then they abruptly wend down to less than 01 after 3000iterations
Figure 3 shows the history of pure memory considerationrate 119877Memory As observed in the figure all three parameters
(1198771Memory 1198772Memory and 1198773Memory) abruptly went up from
the starting point of 025 After 4000 iterations they becamemore than 08 and stayed
Figure 4 shows the history of pure pitch adjustmentrate 119877Pitch As observed in the figure all three parameters(1198771Pitch 1198772Pitch and 1198773Pitch) from the starting point of 025
monotonically stayed less than 03 except for one situationwhen 119877
3Pitch spiked near 3000 iterationsFurthermore the sensitivity analysis of initial parameter
values was performed While the original parameter set(119877Random = 05 119877Memory = 025 and 119877Pitch = 025)resulted in minimal solution of $824356 and average solu-tion of $828769 after 10 runs equal-valued parameter set(119877Random = 033 119877Memory = 033 and 119877Pitch = 033)resulted inminimal solution of $824212 and average solutionof $832211 memory-consideration-oriented parameter set(119877Random = 01 119877Memory = 07 and 119877Pitch = 02)resulted in minimal solution of $824134 and average solu-tion of $831445 random-selection-oriented parameter set(119877Random = 08 119877Memory = 01 and 119877Pitch = 01) resultedin minimal solution of $824129 and average solution of
Figure 3 History of Pure Memory Consideration Rate
$827240 It appeared that the initial parameter values are notvery sensitive to final solution quality
Especially when the results frommemory-consideration-oriented parameter set (119877Random = 01 119877Memory = 07and 119877Pitch = 02) and those from random-selection-orientedparameter set (119877Random = 08 119877Memory = 01 and 119877Pitch =
01) were statistically compared although their variances aredifferent based on 119865-test (119901 = 004) their averages are notsignificantly different based on 119905-test (119901 = 016)
5 Conclusions
This study applied PSF-HS to the ED problem for the firsttime obtaining a good solution which is very close tothe best solution ever found While existing metaheuristicalgorithms require carefully chosen algorithm parameters
PSF-HS did not require that tedious process Thus theresurely exists a tradeoff between original HS and PSF-HSAlso it should be noted that PSF-HS respectively considersindividual algorithm parameters for each variable which ismore efficient way than using lumped parameters for allvariables
For future study the structure of PSF-HS should beimproved to do better performance Also it can be applied tolarge-scale real-world problems to test scalability Also otherresearchers are expected to apply this novel technique to theirown energy-related problems
Acknowledgment
This work was supported by the Gachon University ResearchFund of 2013 (GCU-2013-R114)
References
[1] D C Walters and G B Sheble ldquoGenetic algorithm solution ofeconomic dispatchwith value point loadingrdquo IEEE Transactionson Power Systems vol 8 no 3 pp 1325ndash1332 1993
[2] W-M Lin F-S Cheng and M-T Tsay ldquoAn improved tabusearch for economic dispatch with multiple minimardquo IEEETransactions on Power Systems vol 17 no 1 pp 108ndash112 2002
[3] N Sinha R Chakrabarti and P K Chattopadhyay ldquoEvolution-ary programming techniques for economic load dispatchrdquo IEEETransactions on Evolutionary Computation vol 7 no 1 pp 83ndash94 2003
[4] J-B Park K-S Lee J-R Shin and K Y Lee ldquoA particleswarm optimization for economic dispatch with nonsmoothcost functionsrdquo IEEE Transactions on Power Systems vol 20 no1 pp 34ndash42 2005
[5] B K Panigrahi V R Pandi S Das Z Cui and R SharmaldquoEconomic load dispatch using population-variance harmonysearch algorithmrdquo Transactions of the Institute of Measurementand Control vol 34 no 6 pp 746ndash754 2012
Journal of Applied Mathematics 5
[6] T Niknam H D Mojarrad H Z Meymand and B B FirouzildquoA new honey bee mating optimization algorithm for non-smooth economic dispatchrdquo Energy vol 36 no 2 pp 896ndash9082011
[7] X-S Yang S S S Hosseini and A H Gandomi ldquoFireflyAlgorithm for solving non-convex economic dispatch problemswith valve loading effectrdquo Applied Soft Computing vol 12 no 3pp 1180ndash1186 2012
[8] Z W Geem and K-B Sim ldquoParameter-setting-free harmonysearch algorithmrdquo Applied Mathematics and Computation vol217 no 8 pp 3881ndash3889 2010
[9] M Srinivas and L M Patnaik ldquoAdaptive probabilities ofcrossover and mutation in genetic algorithmsrdquo IEEE Transac-tions on Systems Man and Cybernetics vol 24 no 4 pp 656ndash667 1994
[10] M S Gibbs H R Maier and G C Dandy ldquoComparisonof genetic algorithm parameter setting methods for chlorineinjection optimizationrdquo Journal of Water Resources Planningand Management vol 136 no 2 pp 288ndash291 2010
[11] S Bekiroglu T Dede and Y Ayvaz ldquoImplementation ofdifferent encoding types on structural optimization based onadaptive genetic algorithmrdquo Finite Elements in Analysis andDesign vol 45 no 11 pp 826ndash835 2009
[12] A Haghighi and A E Bakhshipour ldquoOptimization of sewernetworks using an adaptive genetic algorithmrdquoWater ResourcesManagement vol 26 no 12 pp 3441ndash3456 2012
[13] Q H Wu Y J Cao and J Y Wen ldquoOptimal reactive powerdispatch using an adaptive genetic algorithmrdquo InternationalJournal of Electrical Power and Energy Systems vol 20 no 8pp 563ndash569 1998
[14] O Hasanebi F Erdal and M P Saka ldquoAdaptive harmonysearchmethod for structural optimizationrdquo Journal of StructuralEngineering vol 136 no 4 pp 419ndash431 2010
[15] Z W Geem and Y H Cho ldquoOptimal design of water distribu-tion networks using parameter-setting-free harmony search fortwomajor parametersrdquo Journal ofWater Resources Planning andManagement vol 137 no 4 pp 377ndash380 2011
[16] Z W Geem ldquoParameter estimation of the nonlinear musk-ingum model using parameter-setting-free harmony searchrdquoJournal of Hydrologic Engineering vol 16 no 8 pp 684ndash6882011
[17] Z W Geem ldquoSolution quality improvement in chiller loadingoptimizationrdquo Applied Thermal Engineering vol 31 no 10 pp1848ndash1851 2011
[18] Z W Geem and Y H Cho ldquoHandling non-convex heat-powerfeasible region in combinedheat andpower economic dispatchrdquoInternational Journal of Electrical Power amp Energy Systems vol34 no 1 pp 171ndash173 2012
[19] Z W Geem ldquoSize optimization for a hybrid photovoltaic-wind energy systemrdquo International Journal of Electrical Poweramp Energy Systems vol 42 no 1 pp 448ndash451 2012
119894+10038161003816100381610038161003816119890119894times sin (119891
119894times (119875
min119894
minus 119875119894))10038161003816100381610038161003816
(2)
where 119886119894 119887119894 119888119894 119890119894 and 119891
119894are cost coefficients for generator 119894
The fourth term in the right-hand side of (2) represents valve-point effects
The above objective function is to be minimized whilesatisfying the following equality constraint
sum119894
119875119894= 119863 (3)
where 119863 is total load demand Also each generator shouldgenerate power between minimum and maximum limits asthe following inequality constraint
119875min119894
le 119875119894le 119875
max119894
(4)
3 Parameter-Setting-Free Technique
The parameter-setting-free harmony search (PSF-HS) algo-rithm was first proposed for optimizing the discrete-variableproblems such as structural design [14] water network design[15] and recreational magic square [8] PSF-HS was alsoapplied to a continuous-variable problem such as hydrologicparameter calibration [16]
However it was never applied to a continuous-variableproblem with technical constraints Thus this study firstapplies PSF-HS to the ED problem whose type is the con-tinuous-variable problemwith a technical constraint becauseits decision variable 119875
119894has the continuous value and it has
the equality constraint of total power demand as expressedin (3) Here the inequality constraint in (4) can be simplyconsidered as value rangeswithout using any penaltymethod
The basic HS algorithm manages a memory matrixnamed harmony memory as follows
Once thisHM is fully filled with randomly generated vectors(P1 PHMS) a new vector PNew is generated as follows
119875New119894
larr997888
119875min119894
le 119875119894le 119875max119894
wp 119877Random119875119894 (119896) isin 119875
1
119894 1198752119894 119875HMS
119894 wp 119877Memory
119875119894 (119896) + Δ wp 119877Pitch
(6)
where 119877Random is random selection rate 119877Memory is purememory consideration rate 119877Pitch is pure pitch adjustmentrate and Δ is pitch adjustment amount
If the newly generated vector PNew is better than theworst vector PWorst inHM those two vectors are swapped asfollows
PNewisin HM and PWorst
notin HM (7)
The basic HS algorithm performs (6) and (7) until a termina-tion criterion is satisfied
For PSF-HS one additionalmatrix named operation typematrix (OTM) is also managed as follows
[[[[[[
[
1199001
1=Random 1199001
2=Pitch sdot sdot sdot 1199001
119899=Memory
11990021=Memory 1199002
2=Memory sdot sdot sdot 1199002
119899=Pitch
sdot sdot sdot sdot sdot sdot sdot sdot sdot
119900HMS1
=Memory 119900HMS2
=Random sdot sdot sdot 119900HMS119899
=Memory
]]]]]]
]
(8)
OTM memorizes which operation (random selection mem-ory consideration and pitch adjustment) each value comesfrom For example if the value of 1198752
2in HM comes from
memory consideration operation the value of 11990022in OTM
is also set as ldquoMemoryrdquo This process happens when initialvectors are populated or when a new vector is inserted intoHM
Thus instead of using fixed algorithm parameter valuesPSF-HS can utilize adaptive parameter values by calculatingthem at each iteration as follows
119877119894Random =
119888119905 (119900119895
119894= Random 119895 = 1 2 HMS)
HMS
119894 = 1 2 119899
119877119894Memory =
119888119905 (119900119895
119894= Memory 119895 = 1 2 HMS)
HMS
119894 = 1 2 119899
119877119894Pitch =
119888119905 (119900119895
119894= Pitch 119895 = 1 2 HMS)
HMS
119894 = 1 2 119899
(9)
where 119888119905(sdot) is a function which counts specific elements thatsatisfy the condition
4 Numerical Example
The PSF-HS is applied to a popular bench-mark ED problemwith three generators The input data for the three-generatorproblem is shown in Table 1
When the total system demand is set to 850MW theoptimal solution is known as $823407 [2ndash4] which wasreplicated by using a popular gradient-based technique(generalized reduced gradient (GRG) method) which has
Journal of Applied Mathematics 3
Table 1 Data for three-generator example with valve-point loading
been also successfully applied to other energy optimizationproblems such as building chiller loading [17] combined heatand power ED [18] and hybrid renewable energy systemdesign [19] However the GRG method was able to obtainthe identical best solution only when it started with a vector(1198751= 300 119875
2= 150 119875
3= 400) Instead when different
starting vector (1198751= 600 119875
2= 200 119875
3= 400) was used
solution quality was worsened as $824141When PSF-HS was also applied to the problem it
obtained a near-optimal solution of $823447 after 100 runswhich has small discrepancy from the optimal solution($823407) by 0005 For the results from 100 runs max-imum and mean solutions are $842974 (24 discrepancy)and $829288 (07 discrepancy) respectively Here PSF-HS was performed using MS-Excel VBA environment withIntel CPU 33GHz Each run takes only one second in thiscomputing environment
Figure 1 shows the convergence history of power genera-tion cost for the case of the near-optimal solution $823447As seen in the figure PSF-HS closely approached to the near-optimal solution in early iterations
Table 2 shows the finalHM with HMS = 30 As observedin the table there are many similar vectors in HM becausePSF-HS tried local search instead of global search in latestage of computation
Figure 2 shows the history of random selection rate119877Random As observed in the figure all three parameters(1198771Random 1198772Random and 119877
3Random) started with highervalues (05) In less than 1000 iterations119877
1Random went up toaround 04 119877
2Random to around 05 and 1198773Random to around
08Then they abruptly wend down to less than 01 after 3000iterations
Figure 3 shows the history of pure memory considerationrate 119877Memory As observed in the figure all three parameters
(1198771Memory 1198772Memory and 1198773Memory) abruptly went up from
the starting point of 025 After 4000 iterations they becamemore than 08 and stayed
Figure 4 shows the history of pure pitch adjustmentrate 119877Pitch As observed in the figure all three parameters(1198771Pitch 1198772Pitch and 1198773Pitch) from the starting point of 025
monotonically stayed less than 03 except for one situationwhen 119877
3Pitch spiked near 3000 iterationsFurthermore the sensitivity analysis of initial parameter
values was performed While the original parameter set(119877Random = 05 119877Memory = 025 and 119877Pitch = 025)resulted in minimal solution of $824356 and average solu-tion of $828769 after 10 runs equal-valued parameter set(119877Random = 033 119877Memory = 033 and 119877Pitch = 033)resulted inminimal solution of $824212 and average solutionof $832211 memory-consideration-oriented parameter set(119877Random = 01 119877Memory = 07 and 119877Pitch = 02)resulted in minimal solution of $824134 and average solu-tion of $831445 random-selection-oriented parameter set(119877Random = 08 119877Memory = 01 and 119877Pitch = 01) resultedin minimal solution of $824129 and average solution of
Figure 3 History of Pure Memory Consideration Rate
$827240 It appeared that the initial parameter values are notvery sensitive to final solution quality
Especially when the results frommemory-consideration-oriented parameter set (119877Random = 01 119877Memory = 07and 119877Pitch = 02) and those from random-selection-orientedparameter set (119877Random = 08 119877Memory = 01 and 119877Pitch =
01) were statistically compared although their variances aredifferent based on 119865-test (119901 = 004) their averages are notsignificantly different based on 119905-test (119901 = 016)
5 Conclusions
This study applied PSF-HS to the ED problem for the firsttime obtaining a good solution which is very close tothe best solution ever found While existing metaheuristicalgorithms require carefully chosen algorithm parameters
PSF-HS did not require that tedious process Thus theresurely exists a tradeoff between original HS and PSF-HSAlso it should be noted that PSF-HS respectively considersindividual algorithm parameters for each variable which ismore efficient way than using lumped parameters for allvariables
For future study the structure of PSF-HS should beimproved to do better performance Also it can be applied tolarge-scale real-world problems to test scalability Also otherresearchers are expected to apply this novel technique to theirown energy-related problems
Acknowledgment
This work was supported by the Gachon University ResearchFund of 2013 (GCU-2013-R114)
References
[1] D C Walters and G B Sheble ldquoGenetic algorithm solution ofeconomic dispatchwith value point loadingrdquo IEEE Transactionson Power Systems vol 8 no 3 pp 1325ndash1332 1993
[2] W-M Lin F-S Cheng and M-T Tsay ldquoAn improved tabusearch for economic dispatch with multiple minimardquo IEEETransactions on Power Systems vol 17 no 1 pp 108ndash112 2002
[3] N Sinha R Chakrabarti and P K Chattopadhyay ldquoEvolution-ary programming techniques for economic load dispatchrdquo IEEETransactions on Evolutionary Computation vol 7 no 1 pp 83ndash94 2003
[4] J-B Park K-S Lee J-R Shin and K Y Lee ldquoA particleswarm optimization for economic dispatch with nonsmoothcost functionsrdquo IEEE Transactions on Power Systems vol 20 no1 pp 34ndash42 2005
[5] B K Panigrahi V R Pandi S Das Z Cui and R SharmaldquoEconomic load dispatch using population-variance harmonysearch algorithmrdquo Transactions of the Institute of Measurementand Control vol 34 no 6 pp 746ndash754 2012
Journal of Applied Mathematics 5
[6] T Niknam H D Mojarrad H Z Meymand and B B FirouzildquoA new honey bee mating optimization algorithm for non-smooth economic dispatchrdquo Energy vol 36 no 2 pp 896ndash9082011
[7] X-S Yang S S S Hosseini and A H Gandomi ldquoFireflyAlgorithm for solving non-convex economic dispatch problemswith valve loading effectrdquo Applied Soft Computing vol 12 no 3pp 1180ndash1186 2012
[8] Z W Geem and K-B Sim ldquoParameter-setting-free harmonysearch algorithmrdquo Applied Mathematics and Computation vol217 no 8 pp 3881ndash3889 2010
[9] M Srinivas and L M Patnaik ldquoAdaptive probabilities ofcrossover and mutation in genetic algorithmsrdquo IEEE Transac-tions on Systems Man and Cybernetics vol 24 no 4 pp 656ndash667 1994
[10] M S Gibbs H R Maier and G C Dandy ldquoComparisonof genetic algorithm parameter setting methods for chlorineinjection optimizationrdquo Journal of Water Resources Planningand Management vol 136 no 2 pp 288ndash291 2010
[11] S Bekiroglu T Dede and Y Ayvaz ldquoImplementation ofdifferent encoding types on structural optimization based onadaptive genetic algorithmrdquo Finite Elements in Analysis andDesign vol 45 no 11 pp 826ndash835 2009
[12] A Haghighi and A E Bakhshipour ldquoOptimization of sewernetworks using an adaptive genetic algorithmrdquoWater ResourcesManagement vol 26 no 12 pp 3441ndash3456 2012
[13] Q H Wu Y J Cao and J Y Wen ldquoOptimal reactive powerdispatch using an adaptive genetic algorithmrdquo InternationalJournal of Electrical Power and Energy Systems vol 20 no 8pp 563ndash569 1998
[14] O Hasanebi F Erdal and M P Saka ldquoAdaptive harmonysearchmethod for structural optimizationrdquo Journal of StructuralEngineering vol 136 no 4 pp 419ndash431 2010
[15] Z W Geem and Y H Cho ldquoOptimal design of water distribu-tion networks using parameter-setting-free harmony search fortwomajor parametersrdquo Journal ofWater Resources Planning andManagement vol 137 no 4 pp 377ndash380 2011
[16] Z W Geem ldquoParameter estimation of the nonlinear musk-ingum model using parameter-setting-free harmony searchrdquoJournal of Hydrologic Engineering vol 16 no 8 pp 684ndash6882011
[17] Z W Geem ldquoSolution quality improvement in chiller loadingoptimizationrdquo Applied Thermal Engineering vol 31 no 10 pp1848ndash1851 2011
[18] Z W Geem and Y H Cho ldquoHandling non-convex heat-powerfeasible region in combinedheat andpower economic dispatchrdquoInternational Journal of Electrical Power amp Energy Systems vol34 no 1 pp 171ndash173 2012
[19] Z W Geem ldquoSize optimization for a hybrid photovoltaic-wind energy systemrdquo International Journal of Electrical Poweramp Energy Systems vol 42 no 1 pp 448ndash451 2012
been also successfully applied to other energy optimizationproblems such as building chiller loading [17] combined heatand power ED [18] and hybrid renewable energy systemdesign [19] However the GRG method was able to obtainthe identical best solution only when it started with a vector(1198751= 300 119875
2= 150 119875
3= 400) Instead when different
starting vector (1198751= 600 119875
2= 200 119875
3= 400) was used
solution quality was worsened as $824141When PSF-HS was also applied to the problem it
obtained a near-optimal solution of $823447 after 100 runswhich has small discrepancy from the optimal solution($823407) by 0005 For the results from 100 runs max-imum and mean solutions are $842974 (24 discrepancy)and $829288 (07 discrepancy) respectively Here PSF-HS was performed using MS-Excel VBA environment withIntel CPU 33GHz Each run takes only one second in thiscomputing environment
Figure 1 shows the convergence history of power genera-tion cost for the case of the near-optimal solution $823447As seen in the figure PSF-HS closely approached to the near-optimal solution in early iterations
Table 2 shows the finalHM with HMS = 30 As observedin the table there are many similar vectors in HM becausePSF-HS tried local search instead of global search in latestage of computation
Figure 2 shows the history of random selection rate119877Random As observed in the figure all three parameters(1198771Random 1198772Random and 119877
3Random) started with highervalues (05) In less than 1000 iterations119877
1Random went up toaround 04 119877
2Random to around 05 and 1198773Random to around
08Then they abruptly wend down to less than 01 after 3000iterations
Figure 3 shows the history of pure memory considerationrate 119877Memory As observed in the figure all three parameters
(1198771Memory 1198772Memory and 1198773Memory) abruptly went up from
the starting point of 025 After 4000 iterations they becamemore than 08 and stayed
Figure 4 shows the history of pure pitch adjustmentrate 119877Pitch As observed in the figure all three parameters(1198771Pitch 1198772Pitch and 1198773Pitch) from the starting point of 025
monotonically stayed less than 03 except for one situationwhen 119877
3Pitch spiked near 3000 iterationsFurthermore the sensitivity analysis of initial parameter
values was performed While the original parameter set(119877Random = 05 119877Memory = 025 and 119877Pitch = 025)resulted in minimal solution of $824356 and average solu-tion of $828769 after 10 runs equal-valued parameter set(119877Random = 033 119877Memory = 033 and 119877Pitch = 033)resulted inminimal solution of $824212 and average solutionof $832211 memory-consideration-oriented parameter set(119877Random = 01 119877Memory = 07 and 119877Pitch = 02)resulted in minimal solution of $824134 and average solu-tion of $831445 random-selection-oriented parameter set(119877Random = 08 119877Memory = 01 and 119877Pitch = 01) resultedin minimal solution of $824129 and average solution of
Figure 3 History of Pure Memory Consideration Rate
$827240 It appeared that the initial parameter values are notvery sensitive to final solution quality
Especially when the results frommemory-consideration-oriented parameter set (119877Random = 01 119877Memory = 07and 119877Pitch = 02) and those from random-selection-orientedparameter set (119877Random = 08 119877Memory = 01 and 119877Pitch =
01) were statistically compared although their variances aredifferent based on 119865-test (119901 = 004) their averages are notsignificantly different based on 119905-test (119901 = 016)
5 Conclusions
This study applied PSF-HS to the ED problem for the firsttime obtaining a good solution which is very close tothe best solution ever found While existing metaheuristicalgorithms require carefully chosen algorithm parameters
PSF-HS did not require that tedious process Thus theresurely exists a tradeoff between original HS and PSF-HSAlso it should be noted that PSF-HS respectively considersindividual algorithm parameters for each variable which ismore efficient way than using lumped parameters for allvariables
For future study the structure of PSF-HS should beimproved to do better performance Also it can be applied tolarge-scale real-world problems to test scalability Also otherresearchers are expected to apply this novel technique to theirown energy-related problems
Acknowledgment
This work was supported by the Gachon University ResearchFund of 2013 (GCU-2013-R114)
References
[1] D C Walters and G B Sheble ldquoGenetic algorithm solution ofeconomic dispatchwith value point loadingrdquo IEEE Transactionson Power Systems vol 8 no 3 pp 1325ndash1332 1993
[2] W-M Lin F-S Cheng and M-T Tsay ldquoAn improved tabusearch for economic dispatch with multiple minimardquo IEEETransactions on Power Systems vol 17 no 1 pp 108ndash112 2002
[3] N Sinha R Chakrabarti and P K Chattopadhyay ldquoEvolution-ary programming techniques for economic load dispatchrdquo IEEETransactions on Evolutionary Computation vol 7 no 1 pp 83ndash94 2003
[4] J-B Park K-S Lee J-R Shin and K Y Lee ldquoA particleswarm optimization for economic dispatch with nonsmoothcost functionsrdquo IEEE Transactions on Power Systems vol 20 no1 pp 34ndash42 2005
[5] B K Panigrahi V R Pandi S Das Z Cui and R SharmaldquoEconomic load dispatch using population-variance harmonysearch algorithmrdquo Transactions of the Institute of Measurementand Control vol 34 no 6 pp 746ndash754 2012
Journal of Applied Mathematics 5
[6] T Niknam H D Mojarrad H Z Meymand and B B FirouzildquoA new honey bee mating optimization algorithm for non-smooth economic dispatchrdquo Energy vol 36 no 2 pp 896ndash9082011
[7] X-S Yang S S S Hosseini and A H Gandomi ldquoFireflyAlgorithm for solving non-convex economic dispatch problemswith valve loading effectrdquo Applied Soft Computing vol 12 no 3pp 1180ndash1186 2012
[8] Z W Geem and K-B Sim ldquoParameter-setting-free harmonysearch algorithmrdquo Applied Mathematics and Computation vol217 no 8 pp 3881ndash3889 2010
[9] M Srinivas and L M Patnaik ldquoAdaptive probabilities ofcrossover and mutation in genetic algorithmsrdquo IEEE Transac-tions on Systems Man and Cybernetics vol 24 no 4 pp 656ndash667 1994
[10] M S Gibbs H R Maier and G C Dandy ldquoComparisonof genetic algorithm parameter setting methods for chlorineinjection optimizationrdquo Journal of Water Resources Planningand Management vol 136 no 2 pp 288ndash291 2010
[11] S Bekiroglu T Dede and Y Ayvaz ldquoImplementation ofdifferent encoding types on structural optimization based onadaptive genetic algorithmrdquo Finite Elements in Analysis andDesign vol 45 no 11 pp 826ndash835 2009
[12] A Haghighi and A E Bakhshipour ldquoOptimization of sewernetworks using an adaptive genetic algorithmrdquoWater ResourcesManagement vol 26 no 12 pp 3441ndash3456 2012
[13] Q H Wu Y J Cao and J Y Wen ldquoOptimal reactive powerdispatch using an adaptive genetic algorithmrdquo InternationalJournal of Electrical Power and Energy Systems vol 20 no 8pp 563ndash569 1998
[14] O Hasanebi F Erdal and M P Saka ldquoAdaptive harmonysearchmethod for structural optimizationrdquo Journal of StructuralEngineering vol 136 no 4 pp 419ndash431 2010
[15] Z W Geem and Y H Cho ldquoOptimal design of water distribu-tion networks using parameter-setting-free harmony search fortwomajor parametersrdquo Journal ofWater Resources Planning andManagement vol 137 no 4 pp 377ndash380 2011
[16] Z W Geem ldquoParameter estimation of the nonlinear musk-ingum model using parameter-setting-free harmony searchrdquoJournal of Hydrologic Engineering vol 16 no 8 pp 684ndash6882011
[17] Z W Geem ldquoSolution quality improvement in chiller loadingoptimizationrdquo Applied Thermal Engineering vol 31 no 10 pp1848ndash1851 2011
[18] Z W Geem and Y H Cho ldquoHandling non-convex heat-powerfeasible region in combinedheat andpower economic dispatchrdquoInternational Journal of Electrical Power amp Energy Systems vol34 no 1 pp 171ndash173 2012
[19] Z W Geem ldquoSize optimization for a hybrid photovoltaic-wind energy systemrdquo International Journal of Electrical Poweramp Energy Systems vol 42 no 1 pp 448ndash451 2012
Figure 3 History of Pure Memory Consideration Rate
$827240 It appeared that the initial parameter values are notvery sensitive to final solution quality
Especially when the results frommemory-consideration-oriented parameter set (119877Random = 01 119877Memory = 07and 119877Pitch = 02) and those from random-selection-orientedparameter set (119877Random = 08 119877Memory = 01 and 119877Pitch =
01) were statistically compared although their variances aredifferent based on 119865-test (119901 = 004) their averages are notsignificantly different based on 119905-test (119901 = 016)
5 Conclusions
This study applied PSF-HS to the ED problem for the firsttime obtaining a good solution which is very close tothe best solution ever found While existing metaheuristicalgorithms require carefully chosen algorithm parameters
PSF-HS did not require that tedious process Thus theresurely exists a tradeoff between original HS and PSF-HSAlso it should be noted that PSF-HS respectively considersindividual algorithm parameters for each variable which ismore efficient way than using lumped parameters for allvariables
For future study the structure of PSF-HS should beimproved to do better performance Also it can be applied tolarge-scale real-world problems to test scalability Also otherresearchers are expected to apply this novel technique to theirown energy-related problems
Acknowledgment
This work was supported by the Gachon University ResearchFund of 2013 (GCU-2013-R114)
References
[1] D C Walters and G B Sheble ldquoGenetic algorithm solution ofeconomic dispatchwith value point loadingrdquo IEEE Transactionson Power Systems vol 8 no 3 pp 1325ndash1332 1993
[2] W-M Lin F-S Cheng and M-T Tsay ldquoAn improved tabusearch for economic dispatch with multiple minimardquo IEEETransactions on Power Systems vol 17 no 1 pp 108ndash112 2002
[3] N Sinha R Chakrabarti and P K Chattopadhyay ldquoEvolution-ary programming techniques for economic load dispatchrdquo IEEETransactions on Evolutionary Computation vol 7 no 1 pp 83ndash94 2003
[4] J-B Park K-S Lee J-R Shin and K Y Lee ldquoA particleswarm optimization for economic dispatch with nonsmoothcost functionsrdquo IEEE Transactions on Power Systems vol 20 no1 pp 34ndash42 2005
[5] B K Panigrahi V R Pandi S Das Z Cui and R SharmaldquoEconomic load dispatch using population-variance harmonysearch algorithmrdquo Transactions of the Institute of Measurementand Control vol 34 no 6 pp 746ndash754 2012
Journal of Applied Mathematics 5
[6] T Niknam H D Mojarrad H Z Meymand and B B FirouzildquoA new honey bee mating optimization algorithm for non-smooth economic dispatchrdquo Energy vol 36 no 2 pp 896ndash9082011
[7] X-S Yang S S S Hosseini and A H Gandomi ldquoFireflyAlgorithm for solving non-convex economic dispatch problemswith valve loading effectrdquo Applied Soft Computing vol 12 no 3pp 1180ndash1186 2012
[8] Z W Geem and K-B Sim ldquoParameter-setting-free harmonysearch algorithmrdquo Applied Mathematics and Computation vol217 no 8 pp 3881ndash3889 2010
[9] M Srinivas and L M Patnaik ldquoAdaptive probabilities ofcrossover and mutation in genetic algorithmsrdquo IEEE Transac-tions on Systems Man and Cybernetics vol 24 no 4 pp 656ndash667 1994
[10] M S Gibbs H R Maier and G C Dandy ldquoComparisonof genetic algorithm parameter setting methods for chlorineinjection optimizationrdquo Journal of Water Resources Planningand Management vol 136 no 2 pp 288ndash291 2010
[11] S Bekiroglu T Dede and Y Ayvaz ldquoImplementation ofdifferent encoding types on structural optimization based onadaptive genetic algorithmrdquo Finite Elements in Analysis andDesign vol 45 no 11 pp 826ndash835 2009
[12] A Haghighi and A E Bakhshipour ldquoOptimization of sewernetworks using an adaptive genetic algorithmrdquoWater ResourcesManagement vol 26 no 12 pp 3441ndash3456 2012
[13] Q H Wu Y J Cao and J Y Wen ldquoOptimal reactive powerdispatch using an adaptive genetic algorithmrdquo InternationalJournal of Electrical Power and Energy Systems vol 20 no 8pp 563ndash569 1998
[14] O Hasanebi F Erdal and M P Saka ldquoAdaptive harmonysearchmethod for structural optimizationrdquo Journal of StructuralEngineering vol 136 no 4 pp 419ndash431 2010
[15] Z W Geem and Y H Cho ldquoOptimal design of water distribu-tion networks using parameter-setting-free harmony search fortwomajor parametersrdquo Journal ofWater Resources Planning andManagement vol 137 no 4 pp 377ndash380 2011
[16] Z W Geem ldquoParameter estimation of the nonlinear musk-ingum model using parameter-setting-free harmony searchrdquoJournal of Hydrologic Engineering vol 16 no 8 pp 684ndash6882011
[17] Z W Geem ldquoSolution quality improvement in chiller loadingoptimizationrdquo Applied Thermal Engineering vol 31 no 10 pp1848ndash1851 2011
[18] Z W Geem and Y H Cho ldquoHandling non-convex heat-powerfeasible region in combinedheat andpower economic dispatchrdquoInternational Journal of Electrical Power amp Energy Systems vol34 no 1 pp 171ndash173 2012
[19] Z W Geem ldquoSize optimization for a hybrid photovoltaic-wind energy systemrdquo International Journal of Electrical Poweramp Energy Systems vol 42 no 1 pp 448ndash451 2012
[6] T Niknam H D Mojarrad H Z Meymand and B B FirouzildquoA new honey bee mating optimization algorithm for non-smooth economic dispatchrdquo Energy vol 36 no 2 pp 896ndash9082011
[7] X-S Yang S S S Hosseini and A H Gandomi ldquoFireflyAlgorithm for solving non-convex economic dispatch problemswith valve loading effectrdquo Applied Soft Computing vol 12 no 3pp 1180ndash1186 2012
[8] Z W Geem and K-B Sim ldquoParameter-setting-free harmonysearch algorithmrdquo Applied Mathematics and Computation vol217 no 8 pp 3881ndash3889 2010
[9] M Srinivas and L M Patnaik ldquoAdaptive probabilities ofcrossover and mutation in genetic algorithmsrdquo IEEE Transac-tions on Systems Man and Cybernetics vol 24 no 4 pp 656ndash667 1994
[10] M S Gibbs H R Maier and G C Dandy ldquoComparisonof genetic algorithm parameter setting methods for chlorineinjection optimizationrdquo Journal of Water Resources Planningand Management vol 136 no 2 pp 288ndash291 2010
[11] S Bekiroglu T Dede and Y Ayvaz ldquoImplementation ofdifferent encoding types on structural optimization based onadaptive genetic algorithmrdquo Finite Elements in Analysis andDesign vol 45 no 11 pp 826ndash835 2009
[12] A Haghighi and A E Bakhshipour ldquoOptimization of sewernetworks using an adaptive genetic algorithmrdquoWater ResourcesManagement vol 26 no 12 pp 3441ndash3456 2012
[13] Q H Wu Y J Cao and J Y Wen ldquoOptimal reactive powerdispatch using an adaptive genetic algorithmrdquo InternationalJournal of Electrical Power and Energy Systems vol 20 no 8pp 563ndash569 1998
[14] O Hasanebi F Erdal and M P Saka ldquoAdaptive harmonysearchmethod for structural optimizationrdquo Journal of StructuralEngineering vol 136 no 4 pp 419ndash431 2010
[15] Z W Geem and Y H Cho ldquoOptimal design of water distribu-tion networks using parameter-setting-free harmony search fortwomajor parametersrdquo Journal ofWater Resources Planning andManagement vol 137 no 4 pp 377ndash380 2011
[16] Z W Geem ldquoParameter estimation of the nonlinear musk-ingum model using parameter-setting-free harmony searchrdquoJournal of Hydrologic Engineering vol 16 no 8 pp 684ndash6882011
[17] Z W Geem ldquoSolution quality improvement in chiller loadingoptimizationrdquo Applied Thermal Engineering vol 31 no 10 pp1848ndash1851 2011
[18] Z W Geem and Y H Cho ldquoHandling non-convex heat-powerfeasible region in combinedheat andpower economic dispatchrdquoInternational Journal of Electrical Power amp Energy Systems vol34 no 1 pp 171ndash173 2012
[19] Z W Geem ldquoSize optimization for a hybrid photovoltaic-wind energy systemrdquo International Journal of Electrical Poweramp Energy Systems vol 42 no 1 pp 448ndash451 2012
![Multi-objective parameter SFB configuration of machine 823 ... fileobjective machine learning are [1{3]. Although the single-objective case of algo-rithm con guration and hyper-parameter](https://static.documents.pub/doc/80x56/5e1f52429c1e085430562916/multi-objective-parameter-sfb-configuration-of-machine-823-machine-learning.jpg)
