Page - 167 - in Emerging Technologies for Electric and Hybrid Vehicles

Energies 2017,10, 5 Inorder to further illustrate theaboveanalysis, a thirdorderRCnetworkcircuit is simulated in MATLAB;twoequivalent timeconstants (τ’short andτ’long)areestimatedfromthedifferentvalueofΔt. In thesimulation, theresistancesof the threeRCnetworksareall setas1mΩ, andthe timeconstants arepredeterminedasτ1=40s,τ2=200sandτ3=2000s (τ3>>τ2>τ1). Theappliedexcitationconsists ofa400-spulse-dischargingcurrentanda2-hrestperiod,andtheamplitudeof thecurrent is20A. Timeconstantsestimatedbydifferent lengthsof thevoltageresponsearegiven inTable2. It canbe clearlyseenfromTable2 thatbothτ’short andτ’longdecreasesimultaneouslywith thereducedvalueof Δt,which isconsistentwith thepreviousanalysis.Hence, toobtain theappropriatevaluesof the time constants,Δtshouldbepredeterminedproperly,which is illustrated indetailas follows. Table2.Equivalent timeconstantestimationresultswithdifferentvaluesofΔt. Δt (s) 7200 3600 1800 1400 1200 1000 900 850 800 τ’short (s) 88.67 67.18 48.53 45.10 43.74 42.59 42.08 41.83 41.63 τ’long (s) 971.0 484.3 284.4 256.7 245.3 235.3 230.9 228.8 226.8 k 1 4.049×10−12 4.395×10−5 0.1448 0.8759 2.154 5.299 8.311 10.41 13.03 1 k represents thedegreeof resistor-capacitor (RC)voltagevariability; thedetailedexpressioncanreferredto in Equation(13). DuringΔt, thederivativeofEquation(13)withrespect toτiduringtherestperiod isexpressedas:∣∣∣∣dVRC,idτi ∣∣∣∣= Δt|VRC,i(0)|τ2i e −Δtτi (11) whereVRC,i is thevoltageacross the i-thRCnetwork, i∈ {1,2,3, . . . , j},VRC,i(0) is thecorresponding initial voltage,Ri is the resistanceof the i-thRCnetworkandτi is the timeconstantof the i-thRC network,which issubject toτ1 <τ2 < . . . <τj. After thepulse-dischargingperiod,|VRC,i(0)|canbeexpressedas: |VRC,i(0)|= |I|Ri(1−e− D τi ) (12) whereDdenotes the lengthof thepulse-dischargingperiod. For the twowell-separatedtimeconstantsτiandτi+m (τi+m≥10τiand0<m<j− i), thevoltage across the shorter termRC networkVRC,i has a larger degree of variabilitywhen satisfying the followingrequirement: |dVRC,i/dτi| |dVRC,i+m/dτi+m|= k (13) where theconstantkdenotes thedegreeofvariability,andit is subject tok>1. SubstitutingEquations (11)and(12) intoEquation(13), thevalueofΔtcanbederivedas: Δt= ln ⎡ ⎣ Ri(1−e−Dτt)τ2i+m kRi+m(1−e− D τt+m)τ2i ⎤ ⎦ τiτi+m τi+m−τi (14) In Equation (14), since the values of Ri and Ri+m are nearly of the same order of magnitude[39,43,46], thevalueofRi/Ri+m canbeneglectedwhencomparedto thevalueofτ2i+m/τ 2 i ; thus,Δtcanbesimplifiedas: Δt= ln ⎡ ⎣ (1−e−Dτt)τ2i+m k(1−e− Dτt+m)τ2i ⎤ ⎦ τiτi+m τi+m−τi (15) 167