Page - 210 - in Emerging Technologies for Electric and Hybrid Vehicles

Energies 2016,9, 594 maybeestimatedtobeconstant, andtherefore the last termof theequationabovedoesnothave tobe consideredforoptimization. Thismeans thatwecanapproximate Cˆn(δ)by: Cˆn(δ) :=(Zn+δ)2+ In ( Zn− δIn )2 , (6) where thefirst termgives thecostsof changingthefirst intervalandthesecondtermexpresses the costs of theother active charging intervals. To study the influenceof δn on the costs,weneed the derivativeof Cˆn(δn), givenby Cˆn′(δ)=2(Zn+δ)−2 ( Zn− δIn ) =2δ ( 1+ 1 In ) . Thefollowinglemmaprovidesasufficientconditionofoptimality forProblem2,which is later usedtofindanoptimalsolution. Lemma2. Let Cˆn(δn)begivenby (6). Ifwehavea solution δ=(δ1, . . . ,δN) toProblem2that satisfies δi ( 1+ 1 Ii ) = δj ( 1+ 1 Ij ) , for all i, j, (7) and N ∑ n=1 δn=Δ, (8) this solution is optimal. Proof. UsingthemethodofLagrangemultipliers (i.e.,∇f+λ∇g=0),weobtainasetofequations thatgivesufficientconditions foroptimality: Cˆ1′(δ1)+λ=2δ1 ( 1+ 1 I1 ) +λ=0 ... CˆN′(δN)+λ=2δN ( 1+ 1 IN ) +λ=0 N ∑ n=1 δn=Δ, withλbeingtheLagrangemultiplier. It canbereadilycheckedthatEquations (7)and(8) solve this set ofequations. The intuition behind this lemma is thatwhenC′i(δi)< C ′ j(δj), the costs can bedecreased by slightly increasingδi anddecreasingδjbythesameamount. Usingthis lemma,wecanderive the followingtheoremthatprovides theoptimalvaluesδn. Theorem2. Let Cˆn(δ)begivenby (6). Then, theoptimal solution toProblem2 is forall n: δn= Δ S(2+2/In), (9) where S := N ∑ i=1 1 2+2/Ii . 210