A battery model including hysteresis for State-of-Charge estimation ...

IEEE Vehicle Power and Propulsion Conference (VPPC), September 3-5, 2008, Harbin, ChinaIU 0=V ocC 0RV h Z wFigure 5. Equivalent circuit model for the Ni-MH battery.According to some former works on the Ni-MHbattery hysteresis phenomenon and the reference [6], thehysteresis voltage model can be described as thefollowing equation:Charge:t−RC h0 h,maxU = U * + U ∗(1 − RC h ) (1)hDischarge:et−RC h0 h,maxU = U * −U∗(1 − RC h ) (2)heAnd the implementation of the Warburg impedanceZ in the Matlab/Simulink will refer to the paper [7].wIV.V tTHE Ni-MH BATTERY SOC ESTIMATIONUSING EKF BASED ON THE PRESENTED MODELIn order to use Kalman-based methods for the Ni-MHbattery SOC estimation, we must first have a cell modelin a discrete-time state-space form. Specifically, weassume the form:x = + 1f( x , u ) + w(3)k k k ky = gx ( , u)+ v(4)k k k kEq.(3) is called the “state equation” or “processequation”, and Eq.(4) is the “output equation”. Then, weconvert the proposed battery model to the state equation.The battery SOC is constrained to be a member of thestate vector xk; the vector ukcontains theinstantaneous cell current ik.the cell loaded terminalvoltage is considered to the output yk. Finally, theoverall state-space equations for the Ni-MH batterymodel are:⎡ 0 sign ( ik) F( ik)⎤⎡Uhk ( + 1) ⎤ ⎡1 −Fi(k) 0⎤⎡Uhk ( ) ⎤ ⎢ ikη t⎥⎡ ⎤⎢ ⎥= ⎢Z 0 1⎥⎢ ⎥+ Δ0⎢U⎥(5)k+1Z ⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ k ⎦ h,max⎢ C⎣ ⎦⎣ n ⎥ ⎦U = U ( z ) + i R+ U + U(6)eek ocv k k ct ( k ) h( k )The battery SOC and hysteresis voltage are denotedZU hkask( )( )and respectively. Where andZkare the two members of the state vectorxk . In theUEq.(), Eq.(),k is the battery loaded terminal voltage,−C dl−R ctttU hkikis the discharge/charge current, R is the cell internalUocvand Uctare the open-circuit voltageR C in parallel,resistance,and the voltage of the ct dlUh,maxisthe max value of the hysteresis voltage.Finally the EKF basic principle can be applied toestimate the state of the equation. The main EKFalgorithm equations are summarized in the TableI:TABLE I.SUMMARY OF THE NON-LINEAR EXTENDKALMAN FILTER (EKF) [9]Non-linear state-pace modelxk + 1= f ( xk , uk) + wkyk= g( xk, uk)+ vkDefinitionsˆ ∂ f ( xk, uk) ,Ak − 1=ˆ ∂ g ( xk, uk)Ck=∂ xk+ ∂ xx k = xˆk −k−1x k = xˆkInitializationˆ + + + Tx0 = E ( x0), P ˆ ˆ0= E[( x0 − x0 )( x0 − x0) ]ComputationFor k=1,2,3,…compute− +State estimation time update: xˆ( ˆk= f xk−1, uk−1)−Error covariance time update: P Tk= Ak− 1Pk−1Ak−1+Qk−1− T − T −1Kalman gain matrix: Kk = P C ( )k kCkP Ck k+ RkState estimation measurement update:+ − −xˆ ˆ [ ( ˆk= xk + Kkyk − g xk , uk)]Error covariance measurement update:+ −Pk = ( E − KkCk)PkThen, we should summarize some model function, andaccording to the basic principle of the EKF, someunknown parameters and function in the EKF algorithmcould also be derived from the presented model equation(5),(6), which are summarized in the Table IITABLE II.SUMMARY OF THE PARAMETERS IN THE MODELAND EKF ALGORITHMF ( i ) = exp( −βi Δt)kk⎡sign( ik) F ( ik)Uh,max⎤⎡1 − F( ik) 0⎤ f ( x , u ) = x⎢η t⎥⎢ u0 1⎥ + Δ⎣⎦⎢ ⎥⎢⎣Cn ⎥⎦g ( x , u ) = U ( x [1]) + x [2] + U + RuAˆk − 1k k k kk k ocv k k ct( k )k∂ f ( x , )1 (k) 0ku− F ik⎡⎤= =∂ x⎢k0 1 ⎥⎣ ⎦+x k = xˆk − 1ˆ ∂ g ( x , u ) dUCk= = ( ,1)dSOCk k OCV∂ xk x −k = x ˆ kThe parameter β could be derived from the hysteresisAuthorized licensed use limited to: GOVERNMENT COLLEGE OF TECHNOLOGY. Downloaded on December 31, 2009 at 04:49 from IEEE Xplore. Restrictions apply.