Page - 205 - in Emerging Technologies for Electric and Hybrid Vehicles

Energies 2016,9, 594 Note that theaimof theobjective is tobring thevaluesofzn,m ascloseaspossible tozero,where large valuesareaddressedfirst inorder tominimize theEuclideandistance (i.e.,due toquadratic terms). Toease thediscussionandnotation further,weomit the indexn (weconsidera singlehouse), andwemay assumew.l.o.g. that a := an = 1 and d := dn = M. This leads to the following optimizationproblem: Problem1. min x √√√√ M∑ m=1 z2m, (1) subject to M ∑ m=1 xm=C, (2) zm= pm+xm−qm, form∈{1,. . . ,M}, (3) 0≤ xm≤ x¯, form∈{1,. . . ,M}. (4) Onecommonlyconsideredobjective is tocreateapowerprofilethat isasflataspossible, i.e., toset all elementsof thevector q toaconstant (e.g., theaverageof the loads). Awell-knownpropertyof Problem1is thatall constantvectors q=(q,q, . . . ,q) leadto thesameoptimal result x, hence,wemay use q= 0 (andthus z= p+ x,which is the total load) toease thenotation. Severalpapers studyalgorithms that cansolve thisproblemoptimally. An intuitiveapproach is theso-calledvalleyfillingapproach [16]. Thisapproach isdemonstratedgraphically inFigure1, whereinanEV(arrivingat18:00, tobe fullychargedat07:00) is chargedwith thegoal toobtainaflat profile (i.e., q= 0 asmentionedbefore). Theoptimal solution xflattens the total houseprofile by charging insuchawaythat the“valleys”arefilleduptoacertainfill levelZ. Thisoptimalfill levelZ dependsontheshapesof the“valleys”andtheamountC tobecharged. InFigure1,C corresponds to thegrayarea.Note thatZ canalsobe interpretedas theoptimalminimalmismatchbetweenthe obtained anddesiredprofile thatwehave to accept. Ifwehave a lowermismatch in someother interval (i.e., zm<Z), thisneeds tobecompensated insomeother interval byamismatcheven greater thanZ (i.e.,z >Z),whichis, in termsofProblem1, lessflatandthussuboptimal.Chargingup to the levelZ (i.e., zm≥Z)makes sure that theEVbattery is fully chargedexactlyat thedeadline without introducingpeaks. 18:00 07:00 200 400 600 Z Time NoEV( p) House+EV( p+ x) Figure1.OptimalEVplanning. In [16], a line search tofindthevalueZ isproposed. Adifferentapproach is takenbyvander Klauwetal. [4]. Theydeterminethe intervals inwhichnochargingshouldtakeplace,afterwhichthey canstraightforwardlycalculateZ. Theiralgorithmfindstheoptimalplanning inO(M logM) time. TheEVchargingproblemfalls inamoregeneralclassofproblemsthatarereferredtoas“resource allocationproblems” [18]. HochbaumandHong [19] study several resource allocationproblems, andtheypresentanalgorithmforageneralizedversionofProblem1,whichalsocantake intoaccount 205