Page - 249 - in Emerging Technologies for Electric and Hybrid Vehicles

Energies 2017,10, 616 Whenthevoltagevector isat itsmaximumusmax,wecanget thechangerateofaxisq’s current fromEquation(13): diq dt = √ u2smax−(ωrLqiq)2−ωrψf−Rsiq Lq . (13) InEquation(13),becauseof themechanical inertiabeinggreater thantheelectronic inertia,wecan assumethatωr is constantduringtorque integration time tint, butduringthis time iq increases from0 tothemaximumofstatorcurrentvectoramplitude, ismax. Inorder tosimplify the torquefollowing process into a linearprocess, diq/dtmustbea constant. Bymaximizing iq to ismax,we canget the minimumofdiq/dt foran increasedintegrationtimeandthusahigher torquesaturationthreshold. Ifweneglect statorelectric resistanceRs, thenwegetdiq/dtas: diq dt = √ u2smax− ( ωrLqismax ) 2−ωrψf Lq . (14) It isobvious thatwithmotorrotaryvelocityωr increasing, thepartialvoltageofelectromotive force increases, voltageof stator gets almost saturated, and thevoltage allowanceused to control motorcuttentdecreases,which in turnresults indiminishingdiq/dt.Whenωr approaches theright boundaryof thenon-flux-weakeningregion’sexternal characteristics incurrentwindingmode,diq/dt slips to0, implying that thedrivesystemis losingcontrolof themotor’s current. Inorder toguarantee acertainamountofvoltagemarginusedtocontrol themotor’scurrent,we, inaccordancewithbase speedratio, canget themotorangularvelocityωrsusedtocalculatediq/dt ineachwindingmodeby Equation(15): ωrs= ksωrb. (15) InEquation(15),ωrb is therotator’sbaseangularvelocityof themotor incurrentwindingmode. WhenignoringRs in id=0control strategy,wecanget: ωrb= usmax ψs = usmax√ ψ2f+(Lqismax) 2 . (16) Because theapproximationofdiq/dt is constant, the integrationtime tint of thisprocess is: tint= Δiq diq/dt = Lqismax√ u2smax−(ωrsLqismax)2−ωrsψf = Lqismax√ u2smax−(ksωrbLqismax)2−ksωrbψf . (17) InEquation (17), ks is the rotator speed sensitivity coefficient and ks ∈ (0,1); thebigger ks is, thecloser themotor’sangularvelocity ksωrbused tocalculate integration time tint is to thecurrent windingmode’sbaseangularvelocity,andthesmallervoltagemarginanddiq/dtare,which leads to bigger tint andviceversa. ks reflects toleranceof torquefollowingspeed.Thebiggerks is, the lower therequirements for torque followingspeed,which leads toaslowerresponseof the torquesaturation judgmentalalgorithm. In theabove twoexpressions, ismax is themaximumof thestatorcurrentvector in thepresentwindingmodeandusmax is themaximumofanyanglevoltagevector in thepresent windingmode. Because theapproximationof diq/dt is constant, the torque followingprocess is simplified to a linearprocess. The integrationof torqueerrorof thisprocess is: Ilim= ∫ tint 0 |ΔTdt|= ∫ tint 0 |Tl−T∗e |dt= 1 2 Temaxtint. (18) 249