Page - 211 - in Emerging Technologies for Electric and Hybrid Vehicles

Energies 2016,9, 594 Proof. Astheoptimalsolution isunique (theproblemisstrictlyconvex),findingoneoptimalsolution proves this theorem. It canbereadilyverified thatEquation (9) satisfiesbothEquations (7) and (8). Hence,byLemma2, the theoremisproven. Aconsequenceof this theoremis thatonly thenumberofcharging intervals Indetermines the changeδn inchargingpower. 5. Predictions Themethodspresented intheprevious twosectionsarebasedonproperpredictionsof thefill levelZn andthenumberofactivecharging intervals In foreachhousen. Tovalidate that thesevalues arerelativelyeasytopredict,weanalyzedthesevaluesusing90consecutivedaysofmeasurements fromaDutchhouse,andusedthistocalculatetheoptimalchargingprofileforeachofthedaysbetween 18:00and24:00, subdividedinto24 intervalsof15min. The results for different charging amountsCn (in kWh) are presented in Table 1. This table shows that thevalues forZn are in a small band for all 90days. See alsoFigure 3,where the four graphscorrespondto thedifferentCnvalues fromthis tablearegiven.Whenwetake themaximum encounteredZn asprediction Zˆn anduse thisprediction forall 90days, the last columnshowsthat highestrealizedcostsincreasebyatmost16%,althoughthemedianissignificantlylower(7%additional costs). Thecolumn √ Zˆ Z gives theboundof thecostsasestimatedbyTheorem1andshowsthat this theoremprovidesareasonableboundof thecosts. Table1.AnalysisofZn andCn for90days forchargingbetween18:00and24:00. Cn (kWh) Zn In Max.Power √ Zˆ Z C(Zˆ) C(Z) Min Med Max Min Med Max Min Med Max Min Med Max 6 1188 1463 1713 21 22 24 1118 1361 1584 1.20 1.00 1.07 1.16 12 2188 2492 2776 22 24 24 2118 2385 2643 1.13 1.00 1.05 1.11 18 3188 3492 3798 24 24 24 3118 3388 3643 1.09 1.00 1.04 1.09 24 4188 4492 4798 24 24 24 4118 4388 4643 1.07 1.00 1.03 1.07 0 10 20 30 40 50 60 70 80 90 0 1,188 1,713 2,188 2,776 3,188 3,798 4,188 4,798 Day Figure3.Fill levelsZn correspondingtoTable1. TheseresultssuggestanefficientwayofpredictingZn: usemeasurements fromthepreviousdays tocalculate theZn foreachof thesedays,andusethehighestencounteredvalueas Zˆn. Tocalculate 211