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

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IEEE Vehicle Power and Propulsion Conference (VPPC), September 3-5, 2008, Harbin, ChinaThe impedance spectra (Figure1) were obtained byapplying the “EISmeter” with no dc-current of theNi-MH battery at 70%SOC and temperature. Theequivalent elements included in the battery model can beidentified by evaluation of the impedance spectra [3] : Thespectra show an inductive behavior ( L ) at the highfrequencies; The intersection point with the real axis candetermine the battery pure ohmic resistance R ; Thesemicircle from the high frequencies to the lowerfrequencies can be modeled by a parallel connection of anon-linear resistor R ctand a capacitor Cdl; At thelower frequencies, the battery diffusion behavior of thebattery becomes apparent which is commonly modeledby a so-called Warburg impedance Zw. Hence, based onthe analysis of the impedance spectra, the basic Ni-MHbattery model structure can be developed as the figure2.However, according to the equivalent transform relations,different circuits may processes completely identicalimpedance spectroscopy. The basic model structurepresented in this paper is the combination of theconsideration of the simplicity and sufficient knowledgeinvolved in the impedance spectra.OCV( V)7.17.06.96.86.76.66.56.46.36.26.1Di s c har geChar ge6.00 10 20 30 40 50 60 70 80 90 100SOC( %)Figure 3. Measured Ni-MH battery Open-circuit voltage values underdifferent SOC, the lower curve measured after discharging steps (-1C)with rest periods (3600s) in between, and the upper curve measuredafter charging steps (+1C).Besides the test of completed charge/dischargeSOC-OCV relationship, we also gained the relationshipof OCV-SOC under partial charge/discharge as the figure4.LRC d lZ w7.17.06.9Di s c har geChar geFigure2. The basic equivalent electrical circuit structure of the Ni-MHbattery in frequency domain determined by the impedance spectrawithout modeling the equilibrium potential.R c tOCV( V)6.86.76.6B. Hysteresis Phenomenon in Ni-MH BatteryFor most electrochemical system the equilibriumpotential is unambiguously defined by the state of charge(SOC). However, the Ni-MH battery shows a significanthysteresis of the open circuit potential [4] , with thepotential on charge being higher than on discharge atevery SOC (Figure2). The more detailed explanation ofthe voltage hysteresis characteristics could refer to thepaper [4], [5].The figure 3 is the depiction of theSOC-OCV relationship of one certain Ni-MH batterieswhich is composed of five single cells in series. In thefigure 3, we can obviously recognize the stabledifference of the open-circuit potential undercharge/discharge condition. That means that the potentialdoes not depend solely on the state of charge but also onthe history of charging and discharging of the electrode.6.56.410 20 30 40 50 60 70 80 90 100SOC( %)Figure 4. Measured Ni-MH battery OCV-SOC under partialcharge/dischargeIII. SCHEMATIC PRESENTATION OF THEEQUIVALENT CIRCUIT MODELFinally, based on the impedance-based model structureand Ni-MH battery hysteresis phenomenon testbehaviors, one equivalent circuit model including thehysteresis voltage model is presented as the figure. In thefigure, one hysteresis voltage model Vhwas included inthe part of V oc.The inductive element L hasn’t beenincluded in the model since the Ni-MH batteries are lessused at so high frequencies. Besides, we used one linearresistor paralleled with the capacitor in the model and weneed the further investigation on the non-linear resistorin the future work. Any other else electric elements inthe model are the same as the explanations ofpre-provided impedance-based model.Authorized licensed use limited to: GOVERNMENT COLLEGE OF TECHNOLOGY. Downloaded on December 31, 2009 at 04:49 from IEEE Xplore. Restrictions apply.
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.
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