Seite - 208 - in Emerging Technologies for Electric and Hybrid Vehicles

Energies 2016,9, 594 Theorem1. WhenZ≤ Zˆ, the ratiobetween theobjectivevalueof theoptimal solutionandthatbasedonthe prediction Zˆ isgivenby C(Zˆ) C(Z)≤ √ Zˆ Z . Proof. Note thatwhen zˆm> Zˆ foran intervalm,wehave zˆm= zmbecause Zˆ≥Z, i.e., bothsolutions are thesamefor such intervalsm. Asaconsequence, if the instancehasan intervalmwith zˆm> Zˆ, replacing this instance by an instancewith pm−qm = Zˆ leads to an increase of the ratio C(Zˆ)C(Z) as C(Zˆ)≥C(Z), andboth zˆm and zmdecreasebythesameamount; i.e.,wemayassumefor thisproof w.l.o.g. that zˆm≤ Zˆ forall intervalsm. Similarly,wemayassumew.l.o.g. thatxm< x¯, sinceotherwise Z≤ zm≤ Zˆandtheratioonly improves. Becauseof this latterassumption, the instancemeets the requirements forLemma1. Let ZΣ = ∑Mm=1zm. First, we show that C(Zˆ) ≤ √ ZˆZΣ. For this, consider a solution of Algorithm1,which is characterized by zˆm = xˆm−qm+ pm. Since the obtained result is feasible, wehave M ∑ m=1 zˆm=ZΣ. Now,wehave C(Zˆ)= √√√√ M∑ m=1 zˆ2m ≤ √√√√ M∑ m=1 Zˆzˆm = √√√√Zˆ M∑ m=1 zˆm= √ ZˆZΣ. Combiningthis inequalitywithC(Z)≥√ZZΣ (duetoLemma1),weget: C(Zˆ) C(Z)≤ √ ZˆZΣ√ ZZΣ = √ Zˆ Z , whichproves the theorem. This theoremgives aboundon the relativedeviationof theobjectivevalueof the solutionof Algorithm1comparedtotheobjectivevalueof theoptimalsolution.Note that thisbounddepends onlyontherelativedeviationof theusedestimate for thefill levelcomparedto theoptimalfill level, andthat thisdependencyonthis relativedeviation isdampenedbythesquareroot function. 4. FleetPlanning Thissectionextends theresults fromtheprevioussection tochargingmultipleEVs. Inadomestic situation, cars are typically chargedwhen their owners arrive athome,which commonly is in the eveningandcoincideswithadomestic consumptionpeak. EspeciallywhenmultipleEVsarecharging simultaneouslywithin aneighborhood, there is a risk of highpeaks and therefore of overloading the transformer. Asweargued in theprevious sections,predictinga loadprofile isdifficult,while controlling an individual EVwith only fewpredictions canbedone. Weextend this approach to multiple EVs by adding anothermethod that shaves loadpeaks at the neighborhood level at the moment theyarenoticed(e.g., inanonlinesetting). For this,wepropose the followingsolution: 1. The charging of EVs is planned locally within the houses such that the total household consumptionpowerprofile (includingtheEV)becomesasflataspossible. 208