Battery ParameterEstimation
Battery Parameter Estimation
ImportanceofBatteryparameterEstimationThe model parameters changes with operatingconditions and battery aging. If they are notupdated, the model is no longer accurate. ThenSoc estimation will lose accuracy.The capacity Q reduces due to battery agingEstimation of Q is the main task for SOHestimationThe parameters reflect battery conditions andare critical indicators for battery diagnosis
Importance of Battery parameter Estimation • The model parameters changes with operating conditions and battery aging. If they are not updated, the model is no longer accurate. Then SOC estimation will lose accuracy. • The capacity Q reduces due to battery aging. Estimation of Q is the main task for SOH estimation. • The parameters reflect battery conditions and are critical indicators for battery diagnosis
Parameter Estimation of RC-BranchBatteryModelsRpOCIState equation is linearROutput equation is nonlinearv(t) = vocv + Ri(t)+v,(t) = f(s(t)+v,(t)+ Ri(t)Theparametersto beestimated:Q,R, Rp,Cp,VocvThe nonlinearfunction f(s)
State equation is linear 1 1 Q s(t) = 1 i(t) vp (t) =− v p (t) + i(t) RpCp Cp Rp R v i vocv vp + - + - Cp v(t) = vocv + Ri(t) + vp (t) = f (s(t)) + vp (t) + Ri(t) Output equation is nonlinear The parameters to be estimated: Q, R, Rp, Cp, Vocv The nonlinear function f(s) Parameter Estimation of RC-Branch Battery Models
APractical Method forObtaining R,Vp,Rp,VocvinRC-BranchModels byOff-line Experimental MethodRVocv
Rp R v i vocv vp + - + - Cp A Practical Method for Obtaining R, Vp, Rp, Vocv in RC-Branch Models by Off-line Experimental Method
Switchoffatt=oBatteryDigital storageoscilloscopeMeasureboththeterminal voltage(V)andcurrent (l)atatime ofdischarge
Battery Measure both the terminal voltage (V) and current (I) at a time of discharge Switch off at t = 0
ObservetheTerminalVoltage(Data Collection)VoltageUsethis curvetoidentifyRpand CpSlowfinal riseV2toOcV,Immediaterise involtage,V,VoUsethisjumptoidentifyRTimeTimeofcurrentinterruptVocychangesslowlyandisassumedtobeaconstant.The RC branch is at the steady state, so that V,is a constant before switching
Observe the Terminal Voltage (Data Collection) Use this curve to identify Rp andCp Use this jump to identifyR V1 V2 V0 • Vocv changes slowly and is assumed to be a constant. • The RC branch is at the steady state, so that Vp is a constant before switching
CalculationofParameters(1) Voc,= Vo +Vi + V2VR:(2) V。 +Vi = Vocv - I(0-)R=)I(0-)V2R=(3) V2 = V,(O) = R,I(0-) =)I(0-)(4) V =V,(0), V,(t) = V,(0)e R.c, Use any data point on the curve to calculate C
Calculation of Parameters t RpCp V1 − (1) Vocv = V0 +V1+V2 (2) V0 +V1 =Vocv − I(0−)R R = I(0−) V2 (3) V2 = Vp (0) = Rp I(0−) R = I(0−) (4) V2 =Vp (0), Vp (t) =Vp (0)e Use any data point on the curve to calculate Cp
ToEstimate Q,we dolocal linearization of f(s) firstAt an SOC point So, the vocy equation can be linearized locallyvo = f(so),af(s)~ +SSy.+c0oCoCVoCVasLinearizedFunctionIs=Soneartheoperatingpoints=(s -so)v= v(t) -vo=cs+v (t) +Ri(t)1oCVp30[ci,1]+ Ri(t)= Cx + Ri(t)2VocvVp.na5uedoC =[ci,1],x =Vs1Supposethatis C,known.010.200.3oTsState ofCharge (SOC)0andRhavealreadybeencalculated.SoVocv
To Estimate Q, we do local linearization of f(s) first 0 1 ocv ocv ocv At an SOC point s0 , the vocv equation can be linearized locally v 0 s=s0 = f (s ), v v 0 + f (s) (s − s ) = v 0 0 ocv + c s, s s = (s − s0 ) v = v(t) − v 0 = c s + v (t) +Ri(t) ocv 1 p s = [c1 ,1] + Ri(t) = Cx + Ri(t) vp s C = [c1 ,1], x = v p s0 0 V ocv Linearized Function near the operatingpoint • Suppose that is C1 known. • 0 V ocv and R have already been calculated
Estimation of OSupposethat C,isknown.0andRhavealreadybeencalculatedVocvaFrom the operating SOC point s and vodefine0OCVs(t) = s(t)-s, v(t) = v(t)-vo=cs+v (t)+Ri(t)1OCVps(t) = li(t)QLet a= I that is to be estimated.9(tRLet the sampling interval be t.Define the sampledvaluesSk= s(kt),Vp,k=V,(kt),i =i(kt),Vh=v(kt)Sk+I=S,+atik'p.k+I=Vpp.kRNote: yp, can be calculated from yp.o step by step (iteratively) since all parameters are known
Estimation of Q From the operating SOC point s and v 0 , define 0 ocv s(t) = s(t) − s , v(t) = v(t)−v 0 = c s + v (t) +Ri(t) 0 ocv 1 p • Suppose that C1 is known. • 0 V ocv and R have already been calculated. 1 i(t) 1 RpCp 1 Cp Let = that is to be estimated. Q p,k +1 s(t) = 1 i(t) Q vp (t) =− vp (t) + Let the sampling interval be . Define the sampled values sk = s(k ),vp,k = vp (k ),ik = i(k ),vk = v(k ) sk +1 = sk + ik p,k p,k R p C p C p Note: vp,k can be calculated from vp,0 step by step (iteratively) since all parameters are known. v = v + − 1 v + 1 i k
Vk+1=C,Sk+I+Vp.k+I+Rik+1+dk =CiS++Ac,Tik +Vp,k+I+Rik+1+dkd, is the measurement noise or error.Vk+1-Vp.k+1-Rik+= CSt +Ac,Ti+dk=CiSk-I +Ac,Tik-I +Ac,Tit+dk...= c,So+ a(c,tik-1 +Ctio+Cti)+dDefine yk = Vk+I-Vp,k+1-Ri+1, u = CTik-I +CTio +CTikk=1,2....,Nyh=Aus+dk,Now, we can use the least-squares estimation method[d uyi..Yn = ::D.IdI[u ]LyNYN=UNa+DNN =(UTUN)"UTYN
dk is the measurement noise or error. Define yk vk +1 = c1 sk +1 + vp,k +1 + Rik +1 + dk = c1 sk+ c1ik +vp,k +1 + Rik +1 + dk vk +1 − vp,k +1 − Rik +1 = c1 sk + c1ik + dk = c1 sk −1 + c1ik −1 + c1ik + dk = = c1 s0 + (c1ik −1 + c1i0 + c1ik ) + dk YN N N N N N N N ˆ ) −1UT Y = vk +1 − vp,k +1 −Rik +1 , uk = c1ik −1 + c1i0 + c1ik yk = uk + dk , k = 1,2, ,N Now, we can use the least-squares estimation method: y1 u1 d1 , U = , D = = yN uN dN YN = UN + DN = (U TU