State of Charge Estimation of Cells in Series Connection by Using only the Total Voltage Measurement

213 American Control Conference (ACC) Washington, DC, USA, June 17-19, 213 State of Charge Estimation of Cells in Series Connection by Using only the Total Voltage Measurement Xinfan Lin 1, Anna G. Stefanopoulou 1, Yonghua Li 2 an R. Dyche Anerson 2 Abstract The voltage of lithium ion batteries is usually monitore to prevent overcharge an overischarge. For battery packs consisting of hunres of cells, monitoring the voltage of every single cell as significant cost an complexity to the battery management system (BMS). Reucing voltage sensing by only measuring the total voltage of multiple cells in series connection is esirable if the state of charge () of iniviual cells can be correctly estimate. Such goal cannot be achieve by an extene Kalman filter, because the cell s are not observable in the linearize battery string moel. In this paper, an observer base on solving simultaneously multiple nonlinear equations along the trajectory of evolution is use for the estimation problem. Existence of the solution epens on the nonlinearity of the battery voltage- relationship. The observer is applie to a LiFePO 4 /graphite battery string with 2 cells, where the iniviual cell s are observable in low an high ranges. Experimental results show goo convergence of an voltage estimation, inicating that this new methoology can be applie to, at least, halve the voltage sensing in a battery pack. I. INTRODUCTION Automotive battery systems for hybri electric vehicles (HEV), plug-in hybri electric vehicles (PHEV), an battery electric vehicles (BEV) usually consist of hunres an even thousans of cells. The number of cells in a battery system is a function of the esire total voltage, power an energy, an the characteristics of the selecte cell chemistry. Battery packs in HEVs with Nickel Metal Hyrie (NiMH) batteries typically have 15-25 cells. Packs using lithium ion cells for the same type of vehicles have about 5-1 cells ue to the higher voltage inherent to the lithium ion chemistry. Prouction HEVs with NiMH batteries typically measure total voltage of moules with 5 to 16 cells in series. Due to the sensitivity of most lithium ion cells to overcharge an overischarge [1], [2], existing battery management systems (BMS) for lithium ion battery packs nee to monitor the state of charge () an voltage of every single cell. This single-cell monitoring strategy as significant cost to the lithium ion battery system incluing sensors, wiring an labor. Therefore, it is highly esirable to reuce the voltage sensing, e.g. by only measuring the total voltage of multiple cells, uner the conition that the an voltage of iniviual cells can still be monitore. *This work is supporte by the For Motor Company. 1 X. Lin an A. Stefanopoulou are with the Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 4819, USA. Email: xflin@umich.eu an annastef@umich.eu 2 Y. Li an R.D. Anerson are with the Vehicle an Battery Controls Department, Research an Avance Engineering, For Motor Company, Dearborn, MI 48121, USA. Email: yli19@for.com an raner34@for.com In this paper, the possibility of estimating the s of batteries in series connection by only using the total voltage measurement is investigate. First, we try to use the extene Kalman filter (EKF), which have been wiely use to estimate the battery. The EKF works well when the single cell voltage is measure for feeback [3], [4], [5]. However, it has foun that when only the total voltage of a battery string is measure, the EKF can only track the average of the string but not the iniviual cell s. The limitation arises from the lack of observability of the linearize battery string moel iscusse in Section III. In [6], the cell balancing circuit is use to augment the voltage measurement when only the total voltage is measure. In this paper, instea, it is shown that the iniviual cell s are observable in the sense of nonlinear observability for some lithium ion battery chemistry. For example, a 2-cell string with LiFePO 4 /graphite batteries, whose voltage- relationship is highly nonlinear, is shown to have observable iniviual cell s in low an high ranges. Such nonlinear observability provies the possibility of estimating iniviual cell s by using methos other than the EKF. The Newton observer, which is base on solving simultaneously multiple nonlinear equations along the system trajectory, [7], [8], has been propose to aress the state estimation problem with high nonlinearity. In this paper, a similar observer base on the Levenberg-Marquart algorithm [9], [1] is esigne for iniviual cell an voltage estimation uner reuce voltage sensing. Experiments uner a constant charging current have been conucte to valiate the propose algorithm with a 2-cell battery string. It is emonstrate that goo convergence of an voltage estimation can be achieve in the high range, where monitoring overcharge is of critical importance. II. GENERALIZED BATTERY MODEL FORMULATION Some of the most commonly use control-oriente battery moels can be generalize in a state-space representation [11], as ẋ=ax+bi V = g(x,i), where the state matrix A epens on the transient ynamics associate with particle iffusion [3], B on the battery capacity, an g is the nonlinear voltage output function. The state(s) x relates to the energy storage level, the input I is the current an the output V is the voltage. Such moels inclue an are not limite to: Coulomb counting moel (CCM) [12], [13] (1) 978-1-4799-178-4/$31. 213 AACC 74

Equivalent circuit moel (ECM) [13], [14] Simplifie electrochemical moel featuring lithium ion transport an iffusion uring battery charge an ischarge (SECM) [3], [15]. Physical meaning of the variables an parameters in (1) for the above three moels are summarize in Table I. More etails on the moel generalization can be foun in [11]. It is note that there are more complicate partial ifferential equation (PDE) base battery moels, which capture etaile electrochemical processes uring the battery operation [16], [17]. These moels may not be written in the form of (1) with linear state equations an a nonlinear output equation. For a battery string connecte in series, the current input is the same for each cell, an the measure output, the total voltage, is the summation of all the single cell voltages. In this paper, we consier the situation that the cells may only iffer in their s, an all the cells have the same moel parameters. Such imbalance can be cause by factors such as ifference among cells in self-ischarge rate, which is typically relate to the temperature variability along the string. Inee, since the battery egraation is also temperatureepenent, this variability may also result in imbalance in capacity an resistance among cells. Such imbalance is more likely to be evelope over longer time-span an will be aresse in the future work. Base on the single cell moel in (1), a string with r cells in series can be moele as, where ẋ str =F(x str,i )=A str x str + B str I V str =G(x str,i)= r i=1 g(x i,i), x str = [ x 1 x r ] T, A str = iag(a,,a), B str = [ B T B T B T] T. (3) This general form will be use in the analysis in the subsequent sections. The state estimation of (1), where the single cell voltage is measure, is referre to as the estimation uner full voltage sensing, an the state estimation of (2), where only the total voltage is measure, is calle the estimation uner reuce voltage sensing. III. EXTENDED KALMAN FILTER AND OBSERVABILITY OF THE LINEARIZED BATTERY STRING MODEL The extene Kalman filter is one of the most commonly use methos for nonlinear state estimation problems, an has been wiely aopte to estimate the battery uner full voltage sensing [3], [4], [5]. However, it is shown here that the EKF cannot be use uner reuce sensing because the linearize battery string moel is not observable. The EKF for (1) takes the form ˆx=A ˆx+Bu+K(V ˆV) ˆV = g( ˆx,I), where ˆx an ˆV are the estimate state(s), e.g., an voltage, V is the measure voltage an K is the observer (2) (4) gain. The estimate state of the EKF will converge to the real state asymptotically, if the linearize system moel is observable along the system trajectory [18]. Furthermore, the linear observability practically guarantees that the errors in estimation by the EKF will be boune by some constant value even if the moel parameters are not precise [11]. Such observability conition is satisfie if the single cell voltage is measure, leaing to the popularity of EKF uner full voltage sensing. When only the total voltage of the string is measure, the linearize battery string moel can be obtaine by linearizing the output voltage function g of each single cell, as g(x i,i)= C i x+d i I. (5) As liste in Table I, C is mainly the slope of the open circuit voltage (OCV) curve an D is the ohmic resistance. Since g is a nonlinear function of the state, the values of C an D can be ifferent for cells with ifferent s. Hence, the linearize battery string moel will be ẋ str = A str x+b str I V str = C str x+d str I C str = [ ] C 1 C 2 C r (6) D str = r i=1 D i, with C i C j an D i D j if i j. The moel is observable if an only if its observability matrix C str C 1 C 2 C r U = C str A str = C 1 A C 2 A C r A C str A n 1 str C 1 A n 1 C 2 A n 1 C r A n 1 (7) }{{} U 1 }{{} U 2 }{{} U r is of full rank [18], [19], where n is the imension of the A str matrix. Accoring to Tab.I, for the coulomb counting an the simplifie electrochemical moels, C i s are proportional to each other since they only have one non-zero element, C j = η i, j C i, (8) where η i, j = α j α i for CCM, an η i, j = α j+β j α i +β j for SECM. Hence the U i s in (7) is linearly epenent on each other, C j η i, j C i C i U j = C j A = η i, j C i A = η C i A i, j (9) C j A n 1 η i, j C i A n 1 C i A n 1 = η i, j U i. As a result, the observability matrix will be short of full rank, inicating the the linearize battery string moel is unobservable. The same conclusion can be rawn for the equivalent circuit moel upon similar analysis. Simulation has been conucte to show the inaequacy of the EKF for the estimation problem uner reuce voltage sensing. A coulomb counting moel of the LiFePO 4 /graphite 75

TABLE I MODEL GENERALIZATION Coulomb counting moel (CCM) Equivalent circuit moel (ECM) Simplifie electrochemical moel (SECM) State x, spatially istribute lithium concentrations RC voltages: V c,1v c,m c s,1c s,q Input u current I current I current I Output y voltage V voltage V voltage V 2 2 1 1 3 A R c,1 C 2 2 2 c,1 D s 2 r... 1 R q 1 c,mc c,m q q 1 q R c,i : equivalent resistance D s : iffusion coefficient C c,i : equivalent capacitance r : spatial iscretization parameter B Q 1 Q 1 (A e δfa s r ) 1 [,,, q+1 q ]T Q: battery capacity C 1 c,i A e :electroe area, δ: electroe thickness, F: Faraay constant, a s : active particle surface g OCV()+IR, OCV(c s,q )+OP(c s,q,i)+ir f OCV : open circuit voltage OCV() Σ m i V c,i+ IR OP: over potential R: internal resistance [ ] [ R f : ohmic resistance ] C α α 1 1 1 α+ β α: local slope of OCV w.r.t. β: local slope of OP w.r.t. D R R R f + γ γ: local slope of OP w.r.t. I battery is use to emulate a battery string with 2 cells. The output voltage function of the LiFePO 4 /graphite battery uner a constant current is shown in Fig. 1. In simulation, 3.2 3 2.8 2.6.2.4.6.8 1 1 SÔC 1.8 1.6 SÔC 2 2.4 5 1 15 2 3.7 3.5 ˆ ˆ 3.3 5 1 15 2 t (s) Fig. 1. Voltage- relationship of LiFePO 4 batteries uner a constant charging current Fig. 2. Estimation of single cell an string voltage by the EKF one cell is initialize to 5%, an the other to 4%. The two cells are charge with the constant current until the cell with higher reaches 1% mark. The total voltage of the two cells is fe back to the extene Kalman filter. Simulation results are shown in Fig. 2. It can be seen that as the two batteries are charge up, their s increase linearly with time an the ifference between them stays at 1%. The two estimate s, however, instea of tracing 1 an 2 respectively, both converge to the intermeiate values that woul make the estimate voltage match the average voltage of the string. IV. OBSERVABILITY ANALYSIS OF THE NONLINEAR BATTERY STRING MODEL Although the unobservable linearize battery string moel prevents the EKF from estimating the iniviual cell s uner reuce voltage sensing, it oes not rule out the possibility of solving the estimation problem with a nonlinear estimation technique. The existence of the solution epens on the observability of the nonlinear battery moel, which is to be investigate in this section. A nonlinear system is observable at a certain state if the graient (matrix) of the Lie erivative vector of the output function is of full rank [2]. For the string moel in (2), the ith orer Lie erivative of the output function G is L i F(G)= V (i) str = G str F(x str,i), (1) where the current I is fixe at a constant value [2]. The Lie erivative vector is efine as LF (G) l(x str )=, (11) (G) L n 1 F 76

an the graient of l(x str ) takes the form l(x str )= l = str L F (G) str LF n 1 (G) str. (12) Consier the coulomb counting moel as an example to analyze the moel observability, whose l(x str ) matrix is g g x1 xr 2 g x1 I l(x str )= 2 Q 2 g xr I 2 Q r g x1 r ( Q I )n 1 r, (13) g xr r ( Q I )n 1 with x i being the of the ith cell in the string. It can be seen that the rank of (13) epens on the graients of g with respect to the. Clearly, if g is linear, all its graients higher than 2n orer will be zero, an the rank of (13) will be 1. The system will thus be unobservable uner reuce voltage sensing. Nevertheless, it is possible to have a full-rank matrix in (13) if the voltage- relationship g is nonlinear. For example, the g(x) function of the LiFePO 4 /graphite battery, is shown in Fig. 1 an its graients in Fig. 3. In the range between 1% an 9%, g(x) is 2 V 2 V 15 1 5.2.4.6.8 1 1 1 2 3 4.2.4.6.8 1 Fig. 3. Graients of g(x) for LiFePO 4 /graphite batteries uner a constant charging current close to linear, making the battery string moel unobservable. However, in the low ( 1%) an high (9 1%) ranges, as shown in Fig. 1 an Fig. 3, g(x) is highly nonlinear with non-zero 1st an 2n (an even higher) orer graients. It can be checke numerically that for battery strings with two cells, the l matrix is of full rank for any combination of iniviual cell s along the evolution trajectory. Therefore, it is possible to istinguish the iniviual cell s when they are in those ranges. In general cases, for strings with r cells, up to (r 1)th orer erivatives nee to be calculate to etermine the rank of (13) an hence the observability. Base on the above conclusion, a nonlinear observer will be esigne to estimate iniviual cell s in a 2-cell battery string, while the methoology is applicable to strings with more cells. V. NONLINEAR OBSERVER BASED ON THE LEVENBERG-MARQUARDT ALGORITHM Apart from extening the linear estimation theory, e.g. EKF, nonlinear observers can also be constructe base on solving multiple nonlinear equations along the system evolution trajectory, such as the newton observer [7], [8]. Such algorithms, instea of hanling the output ata one point at a time, process the output trajectory over a certain time span simultaneously. As a result, compare with methos extene from the linear estimation theory base on moel linearization, they retain the nonlinearity of the system an have a wier range of application. Since the nonlinear observer is intene for onboar BMS, it will be esigne in iscrete-time omain to accommoate sample measurements. In iscrete-time omain, the moel in (2) can be written as x str,k+1 = F (x str,k,i k ) V str,k = G(x str,k,i k ), where k is the time step. The function F equations in iscrete-time omain, as F (x str,k,i k )=A str, x str,k + B str, I k A str, = e A strt, B str, = T e A strτ B str τ, (14) is the state (15) where T is the sampling perio. At each estimation step, a set of N consecutive measurements an inputs, V str,[k N+1,k] = V str,k N+1,I [k N+1,k] = V str,k are use for estimation. Let us efine I k N+1 I k, (16) F I k+1 F I k (x ( ) str,k) := F F (x str,k,i k ),I k+1, (17) as stans for function composition. It can be erive from (14) that G I k N+1(x str,k N+1 ) V str,[k N+1,k] = G I k N+2 F I k N+1 (x str,k N+1 ) (18) G I k F I k 1 F I k N+1 (x str,k N+1 ) = H(x str,k N+1,I [k N+1,k] ). The problem is then reuce to solving (18) for x str,k N+1 given V str,[k N+1,k] an I str,[k N+1,k]. Evolution of the state can then be etermine base on (14). The Newton observer has been propose in [7] an [8] to solve (18). At each estimation step, x str,k N+1 is calculate through multiple iterations base on the Newton Raphson algorithm, [ x j+1 str,k N+1 = x j str,k N+1 + H ( x j str,k N+1,I [k N+1,k]) str,k N+1 ( Vstr,[k N+1,k] G( x j str,k N+1,I [k N+1,k]) ). (19) ] 1 77

The resulting x str,k N+1 at the final iteration step, is taken as the estimation for x str,k N+1, as ˆx str,k N+1 = x str,k N+1. (2) Estimates of the subsequent states, ˆx str,k N+2,..., ˆx str,k, can then be etermine by ˆx str,k i = F I k i... F I k N+1 ( ˆx str,k N+1 ). (21) At the next estimation step, V str,[k N+1,k] an I [k N+1,k] will be upate by the newly acquire measurements, an the initial guess of the state estimation is etermine base on the previous estimation. Although the Newton s metho usually converges fast, it might not be robust uner some circumstances. For example, when the graient of g(x) is small, H in (19) becomes illconitione, making it ifficult to calculate its inverse. One convenient remey is to replace the Newton iteration in (19) with the Levenberg-Marquart iteration [9], [1], as [ x j+1 str,k N+1 = x j str,k N+1 + H ( x j str,k N+1,I [k N+1,k])+ str,k N+1 λi] 1( V str,[k N+1,k] G( x j str,k N+1,I [k N+1,k]) ), (22) where λ is a scalar use to lower the conition number of the inverte matrix when g is small, an I is the ientity matrix. VI. EXPERIMENTAL VALIDATION Experiments have been conucte to valiate the esigne nonlinear observer. Two A123 2665 LiFePO 4 /graphite batteries are use for testing. Before the experiment, the two cells were first initialize with ifferent s, aroun 5% an % respectively. The two cells were then connecte in series an charge up by a single current source with a constant current of 2 A. The voltage of each single cell was monitore to prevent overcharge. The actual cell s were calculate base on current integration for valiation. The current was cut off when the voltage of the cell with higher hit V. The actual s an voltages are shown in Fig. 4. The nonlinear observer in (22) is use to estimate the iniviual cell s an voltages by only using the total voltage measurement. The coulomb counting moel is use for the battery string moel, which is aequate uner the constant current charging scenario. Estimation starts at the 2th secon, an the initial guess of the is taken as the same for both cells by inverting the average measure voltage. At each estimation step, 15 secons length of ata are use, corresponing to an span of 4.15%. The time interval between ata points is 1 s an a total of 15 sample ata are processe at each step. The estimation results are plotte in Fig. 5 an Fig. 6. The plotte value correspons to the last point at each estimation step. The final values are shown in Tab.(II). It can be seen from Fig. 5 that the iniviual cell s are not istinguishable before the 12th estimation step, when 1.8.6.4.2 1 2 5 1 15 2 25 3 t (s) 3.2 3 2.8 5 1 15 2 25 3 t (s) 1.95.9.85.8.75 Fig. 4. Actual s an voltages of iniviual cells SÔC 1 1 SÔC 2 2.7 1 2 3 4 5 6 Estimation step 5 3.55 3.5 5 Fig. 5. estimation of iniviual cells 1 2 SÔC 2 SÔC 1 3.35 1 2 3 4 5 6 Estimation step Fig. 6. ˆ ˆ ˆ Voltage estimation of iniviual cells ˆ 78

TABLE II FINAL ESTIMATION RESULTS Estimation Measurement error (%) 1 95.99% 94.36% 1.73 2 99.81% 1%.19 6 V 4 V.58 3.59 V V.28 both of them are below 85%. This has been preicte by the nonlinear observability analysis, since the moel is not observable in that range ue to almost linear voltage- relationship, as shown in Fig. 1. When the cell s get above 9%, the estimates start to converge graually to respective actual values as the moel becomes observable. It can be observe that the an voltage estimation of cell 1, which is the cell with the higher, is more accurate than that of cell 2. The reason can be attribute to the graients of g(x) at the s of the two cells. The change in total voltage can be linearize locally as δ( + )= g δ 1 + g 1 δ 2. (23) 2 Since cell 1 has higher, g 1 is larger than g 2 accoring to Fig. 3. Consequently, the change in total voltage will be more sensitive to the change in 1. This will lea to more accurate estimation of 1, which is important when the cell is nearly fully charge. VII. CONCLUSIONS This paper investigates the estimation of the iniviual cell s with only the total voltage measurement for cells in series connection. It is pointe out that the existence of the solution relies on the observability of the nonlinear battery string moel. For battery chemistry with linear voltage- relationship, the iniviual cell s are not observable uner reuce voltage sensing. Nevertheless, for some battery chemistry, such as LiFePO 4, the nonlinearity of the voltage- relationship reners moel observability in certain ranges. A nonlinear observer base on the Levenberg-Marquart algorithm is then esigne to estimate the iniviual cell s an voltages. The algorithm has been implemente to a LiFePO 4 /graphite battery string with 2 cells. 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