Multicriteria optimization in load management and ... -
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Multicriteria optimization in load management and ... -

0XOWLFULWHULD2SWLPL]DWLRQLQ/RDG0DQDJHPHQWDQG(QHUJ\0DUNHW3UREOHPVP. EKEL M. HORTA J. PEREIRA Jr.Post Graduate Program inElectrical EngineeringPontifical Catholic University of Minas GeraisAv. Dom Jose Gaspar, 50030535-610, Belo Horizonte, MGBRAZILA. PRAKHOVNIK A. KHARCHENKOInstitute of Energy Savings andEnergy ManagementNational Technical University of the Ukraine“Kiev Polytechnic Institute”115, Borschagovskaya St., 03056, KievUKRAINE$EVWUDFW The problems of power and energy shortage (natural or associated with the economic feasibility ofload management) allocation are formulated within the framework of multicriteria optimization models. Thisallows one to realize a new technology of load management, which provides the consideration and minimizationof diverse consequences of power and energy shortage allocation as well as creation of incentive influences forconsumers. Analysis of multicriteria models is associated with decision making in a fuzzy environment and isbased on solving PD[PLQ problems. The results of the paper are of a universal character, applicable to energymarket problems and have been realized within the framework of an adaptive interactive decision making system..H\:RUGV Load Management, Power and Energy Shortage Allocation, Resource Allocation, MulticriteriaOptimization, Fuzzy Set Theory, Aggregation Operators, Adaptive Interactive Decision Making System.,QWURGXFWLRQDifferent conceptions of load management (forexample, considered in [1,2]) may be united by thefollowing: elaboration of control actions isperformed on the two-stage bases. On the level ofenergy control centers, optimization of allocatingpower and/or energy shortages (natural or associatedwith the economic feasibility of load management) iscarried out for different levels of planning andcontrol hierarchy. This allows one to draw up tasksfor consumers. On their level, control actions arerealized in accordance with these tasks.Thus, the questions of power and energy shortageallocation play a large role in a family of loadmanagement problems. These questions are complexand should be considered from the economical andtechnological as well as from the social andecological points of view. Besides, when resolvingthese questions, it is necessary to account forconsiderations of creating incentive influences forconsumers.Considering this, it should be noted that methodsof power and energy shortage allocation based onfundamental principles of allocating resources havedrawbacks. Their overcoming is possible on the basisof formulating and solving the problems within theframework of multicriteria models. This allows oneto consider and to minimize diverse consequences ofpower and energy shortage allocation and to createincentive influences for consumers.The application of the multicriteria approach toload management permits one to give a new look atproblems generated by processes of the electricityindustry deregulation and restructuring [3] to filltheir statement by new, most likely, more realisticcontent. In particular, market participants aspire tomaximize their benefits (including economical andalso technological, social, political, etc. factors).Thus, a criterion for many energy market problemscannot be presented as a unified function. The goalsof market participants, as a rule, come in conflict,which may be resolved by search for a compromise.Its objective is to create mutually advantageous andharmonious relations between market participants.Taking the above into account, the present paperis dedicated to posing and solving problems of powerand energy shortage allocation within the frameworkof multicriteria optimization models. Their analysisis based on using the Bellman-Zadeh approach todecision making in a fuzzy environment. The resultsof the paper have been realized as an adaptiveinteractive decision making system (AIDMS).The use of the results of the paper permits one toimprove the validity and efficiency in allocatingpower and energy shortages. They can serve as amethodological, informational and computational

¦§¤£¦¤©¡¤+%)$"(¤!*#/.40=40"3?!332221#111!#73:3?3>62922111100¢µ ¥~$ = {,...,[,µ ( [)},[ ∈ /,S = 1 T (8)where~( [)is a membership function of $ [8].~A fuzzy solution ' with setting up the fuzzy sets(8) is turned out as a result of their intersection~ ~' = I $ with the membership functionµ=1( ) = ∧ µ( ) = min µ[ [¨= 1= 1,...,¨ ©([),[∈/. (9)Using (9), it is possible to obtain the solutionproviding the maximum degreemax ( ) max µ = min µ ( [)(10)[ ∈ = 1,...,~of belonging to ',and the problem (5) is reduced to= arg max min0[∈ = 1,..., µ( [). (11)There are theoretical basis (for example, [9]) ofthe validity of applying PLQ operator in (9)-(11).However, there exist many families of aggregationoperators [10] that may be used in place of PLQoperator. Considering this, it is possible togeneralize (9) as follows:µ ( [)= agg ( µ ( [),µ ( [),...,µ ( [)),[ ∈ /. (12)1 2Despite that some properties of the aggregationoperators have been established, there is no clear andintuitive interpretation of these properties, norunifying interpretation of the operators themselves[10]. It is possible to state the following question:among many types of aggregation operators, how isone selected, which is adequate for a particularproblem, and how is the selection justified?Although some selection criteria are suggested in[8], the majority of them deals with empirical fit.Thus, it is possible to assert that the selection of theoperators, in large measure, is based on experience.Considering this, the present paper includes resultsof experimental comparing the validity of using PLQoperator and SURGXFWoperator. The last operator hasfound wide applications in decision makingproblems. Its use reduces (12) toµ ( [)= µ ( [)(13)∏= 1,...,and permits one to construct the problemmax µ ( ) = max µ (& [)(14)to find∏[ '∈= 1,...,∏= arg max µ ( ) . (15)0,[ - [∈= 1,...,To obtain the solution (11) or (15), it is necessaryto construct the membership functions ( [),S 1,...,= T that reflect a degree of achieving “own”optimums by( [),[ ∈ 1/,S = T)". This demandis satisfied by using the membership functions⎡ ( ) − min ( ) ⎤µ⎣⎦for objective functions that must be maximized orthe membership functionsµ) [) [∈( [)= ⎢⎥ (16)⎢max) ( [)− min ) ( [)⎥∈∈λ⎡ max ) ( [)− ) ( [)⎤∈( [)= ⎢⎥ . (17)⎢max) ( [)− min ) ( [)⎥∈∈⎣for objective functions that must be minimized.In (16) and (17), λ , ,...,S = 1 T are importance5factors for the corresponding objective functions.The construction of (16) or (17) demands to solvethe following problems:( [)→ min , (18))8); [∈( ) → max . (19)Thus, the solution of the problem (5) demandsanalysis of 2 T + 1 monocriteria problems (18), (19)and (10) or (14), respectively.0Since the solution [ must belong to Ω ⊂ / , ifwe solve, for example, the problem (10), it isnecessary to construct the membership function( ) & µ = ∧ µ ( [ ' ) ∧ µ[

¢£¢¤¤ ¨§ " ¦ ¥©#¥conditions that are difficult to formalize. Theconsideration of conditions defined by the linguisticvariable does not change the solution technology:the availability of P additional conditions leads toS = 1 ,..., T + P in (9)-(11) and (13)-(15).Furthermore, the AIDMS includes procedures forbuilding and correcting the vector λ = λ ,..., λ ) of( 1the importance factors. The group procedures arerelated to ordering goals and to giving marks forthem. The individual procedure is based on the Saatyapproach [12] associated with processing of pairedqualitative comparisons of the importance of goals.Following the procedure, DM has to indicate whichamong two goals is more important with estimatingthe perception of distinction using a rank scale. Thecomparisons allow one to build a " ¡matrix % = [ E ],, W = 1 T . The eigenvector corresponding to itsS ,...,maximum eigennumber serves as λ = λ ,..., λ )( 1The use of this procedure allows for testing thequality of comparisons from the standpoint of theirtransitiveness. If the maximum eigennumber of % isclose to T , then λ , ,...,S = 1 T are acceptable;"otherwise the comparisons should be reconsidered.As a simple example of multicriteria powershortage allocation on the basis of decision makingin a fuzzy environment with using PLQand SURGXFWoperators as well as the traditional scalarizationmethod [6] (this method has found the mostwidespread use in power engineering problems), wecan consider the following problems.It is necessary to allocate power shortages123$ =20000 kW, $ =30000 kW, $ =40000 kW,4$ =50000 kW and5$ =60000 kW among 6consumers with considering the goals 1, 12, 15 and16 listed above. These goals are described by thelinear objective functions (1) and we have:6( ; ) = ∑ F [ ,£ S = 1,15 ,16 (21)) ¢= 1that must be minimized and∑¤) ( (22)= 6 ;12) F [12,= 1that must be maximized.Table 1 includes initial information to solve theproblems. The results obtained on the basis of theBellman-Zadeh approach ( 0[ ) , use of SURGXFW00000operator ( [ ) and the traditional method ( [ ) for15$ =20000 kW and $ =60000 kW are presented inTable 2 and Table 3.To reflect the quality of solutions obtained on thebasis of different approaches Table 4 includes themean magnitudes of absolute values of deviations ofmembership function levels, for example, ( 0[ µ¦ )ˆ0from their mean levels µ ( [ ) calculated asfollows:LL1∆( ) = µ ( ) − µ ˆ ( ) . (23), F 1,monetaryunits/kWh∑© [ [000[ T = 1Table 1:Initial information12 F 15,F 16 ,,F ,hours$ , kW1 1.50 5.40 0.63 15.30 140002 4.10 6.20 0.33 17.20 60003 1.40 5.80 0.28 21.10 40004 2.20 5.30 0.21 18.50 70005 1.20 4.20 0.26 17.40 190006 2.13 4.70 0.36 19.60 140001,0[Table 2: Power shortage allocation1,00[1,000[5,0[5,00[5,000[1 5398 5804 0 13020 14000 140002 2515 1104 0 5076 5731 60003 2399 870 0 3986 4000 40004 950 6898 1000 6223 7000 70005 6738 5324 19000 19000 19000 190006 0 0 0 12695 10269 10000Table 3: Levels of membership functionsS1 2 3 4µ1,0( [ ) 0.604 0.605 0.605 0.6061,00µ ( [ ) 0.615 0.590 0.633 0.6301,000µ ( [ ) 0.974 0.020 0.951 0.5965,0µ ( [ ) 0.428 0.431 0.428 0.4285,00µ ( [ ) 0.366 0.700 0.353 0.7145,000µ ( [ ) 0.321 0.750 0.357 0.741Table 4 covers the cases reflected in Table 2 and23Table 3 as well as $ =30000 kW, $ =40000 kW4and $ =50000 kW. The data of Table 4 bring out000000clearly that ( ) ( µ f µ [ ) f µ ( [ ) . The[ 0high quality of the solutions [ is also confirmed by000inequalities min ( ) µ > min µ ( [ ) and[ 0000min µ ( ) > min µ ( [ ) observed for all cases.[ 00000The inequalities min ( ) ! "# µ > min µ ( [ ) also[ !take place for all cases. This permits one to assertthat decision making in a fuzzy environment evenwith using SURGXFW operator provides us with

solutions more harmonious than on the basis of thetraditional method.∆Table 4: Mean magnitudes ofdeviation absolute values1$2$3$4$5$∆ ( 0 [ ) 0 0.003 0.052 0.060 0.001∆ ( 00 [ ) 0.015 0.010 0.100 0.192 0.174∆ ( 000 [ ) 0.327 0.327 0.290 0.194 0.203&RQFOXVLRQThe problems of power and energy shortage (naturalor associated with the advisability of loadmanagement) allocation have been formulated withinthe framework of multicriteria optimization models toconsider and to minimize diverse consequences oflimiting consumers as well as to create incentiveinfluences for them. The general list of goals has beendeveloped. This list is of a universal character and canserve for building models at different levels of loadmanagement hierarchy. It has been shown the utilityof applying the Bellman-Zadeh approach to analyzingmulticriteria models. The advantages and capabilitiesopened byits use have been demonstrated. The resultsof the paper have been realized as the adaptiveinteractive decision making system. Its applicationpermits one to improve the validity and factualefficiency in allocating power and energy shortages.The results of the paper may directly be extendedto energy market problems, whose solution is relatedto functions of the Independent System Operator orother independent institutions [3]. It may demand:1. Additional substantial analysis of energy marketproblems for adapting the general list of goals. Thelist is to serve for building models at different levelsof territorial, temporal and situational hierarchy ofplanning and operation, considering their differentstructures [3];2. Construction of diverse types of sensitivityindices that are necessary to solve energy marketproblems. This construction is expedient to realize onthe basis of experimental design techniques [13];3. Development of aggregation and decompositionprocedures to solve the electricity market problemsfor different territorial levels. This development isalso related to posing and solving problems ofplanning and operation of multi-zone markets [3].$FNQRZOHGJHPHQWVThis research is supported by the National Council forScientific and Technological Development of Brazil andFoundation for Superior Level Personnel Improvement ofthe Ministry of Education of Brazil.5HIHUHQFHV[1] S. Talukdar and C. Gellings, /RDG0DQDJHPHQW, IEEE Press, 1987.[2] A. Prakhovnik, P. Ekel and A. Bondarenko,0RGHOV DQG 0HWKRGV RI 2SWLPL]LQJ DQG&RQWUROOLQJ 0RGHV RI 2SHUDWLRQ RI (OHFWULF3RZHU 6XSSO\ 6\VWHPV, ISDO, 1994, inUkrainian.[3] M. Ilic, F. Galiana and L. Fink, 3RZHU 6\VWHPV5HVWUXFWXULQJ (QJLQHHULQJ DQG (FRQRPLFV,Kluwer, 1998.[4] V. Burkov and V. Kondrat’ev, 0HFKDQLVPV RI)XQFWLRQLQJ 2UJDQL]DWLRQDO 6\VWHPV, Nauka,1981, in Russian.[5] P. Ekel, L. Terra, M. Junges, A. Prakhovnik andO. Razumovsky, Multicriteria load managementin power systems, 3URFHHGLQJV RI WKH,QWHUQDWLRQDO &RQIHUHQFH RQ (OHFWULF 8WLOLW\'HUHJXODWLRQ DQG 5HVWUXFWXULQJ DQG 3RZHU7HFKQRORJLHV, London, 2000, pp. 167-172.[6] S. Rao, (QJLQHHULQJ2SWLPL]DWLRQ, Wiley, 1996.[7] P. Ekel, Methods of decision making in fuzzyenvironment and their applications, 1RQOLQHDU$QDO\VLV, Vol. 47, No. 5, 2001, pp. 979-990.[8] H.-J. Zimmermann, )X]]\ 6HW 7KHRU\ DQG ,WV$SSOLFDWLRQV, Kluwer, 1990.[9] R. Bellman and M. Giertz, On the analyticformalism of the theory of fuzzy sets,,QIRUPDWLRQ 6FLHQFHV, Vol. 5, No. 2, 1974, pp.149-157.[10] G. Beliakov and J. Warren, Appropriate choiceof aggregation operators in fuzzy decisionsupport systems, ,((( 7UDQVDFWLRQV RQ )X]]\6\VWHPV, Vol. 9, No. 6, 2001, pp. 773-784.[11] P. Ekel, Fuzzy sets and models of decisionmaking, ,QWHUQDWLRQDO -RXUQDO RI &RPSXWHUVDQG0DWKHPDWLFVZLWK$SSOLFDWLRQV, to appear.[12] T. Saaty, A scaling method for priorities inhierarchical structures, 0DWKHPDWLFDO3V\FKRORJ\, Vol. 15, No. 3, 1977, pp. 234-281.[13] P. Ekel, M. Junges, J. Morra and F. Paletta,Fuzzy logic based approach to voltage andreactive power control in power systems,,QWHUQDWLRQDO -RXUQDO RI &RPSXWHU 5HVHDUFK,Vol. 11, No. 2, 2002, pp. 159-170.

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