Modeling and Simulation of Vehicular Power Systems - webfiles its ...

C = C I I( 5-2)( n−1)1 2(2/1)Where• C = capacity of the battery• Subscripts 1 and 2 refer different discharge-rate statesFrom this relationship the state of charge (SOC) at a constant discharge rate is:SOC = 1 − ( I × t)/ C( 5-3)For non-constant discharge rates the above equation must be modified and evaluated insmall time steps:ΔSOC = I( 5-4)( n−1)2Δt/ 3600 / C1(I2/ I1)In the equation 5-4, it is assumed that a given combination of current and discharge rate(C 1 and I 1 ) is known. Given the current at the present time step (I 2 ), the correspondingdischarge rate is calculated using 5-2 for C 2 and plugged into an incremental form ofequation for SOC.5.2 Shepherd Model EquationThe Shepherd model is perhaps the best known and most often used battery model forHybrid vehicle analysis. The model describes the electrochemical behavior of the batterydirectly in terms of voltage and current. It is often used in conjunction with the Peukertequation to obtain battery voltage and state of charge given power draw variations [23]:Et= E − R I − K ( 1−q)( 5-5)oiiWhere• E t = battery terminal voltage [volts]• E o = open circuit voltage of a battery cell when fully charged [volts]• R i = internal (ohmic) resistance of the battery [ohms]• K i = polarization resistance [ohms]• C = battery capacity [ampere-hour]• I = instantaneous current [amps]• q = accumulated ampere-hours divided by full battery capacity.The fractional state of charge is then found via Peukert's equation.5.3 Unnewehr Universal ModelThe Shepherd model is based on constant current discharges at low current levels. TheShepherd equation tries to find the cut-off point beyond which the terminal voltage26