Method for state of health and electrode state of health estimation of a lithium-ion battery cell

Abstract

Battery management systems (BMSs) are required to estimate many non-measurable values that describe the actual operating condition of batteries, such as the state of charge (SOC) or the state of health (SOH). In order to improve this evaluation, many physical states and parameters can be estimated using physics-based models (PBMs). These estimates could be used to improve the control and prognosis of batteries. The invention proposes a method to estimate the state of health (SOH) parameters of a lithium-ion battery cell, using interconnected sigma-point Kalman filters (SPKFs) and a single-particle model with electrolyte dynamics (SPMe). The observability of each eSOH parameter is assessed and the state / parameter estimation approach is described. The method is capable of estimating the eSOH parameters and all the degradation modes that can occur inside a lithium-ion battery cell using only cell voltage and current measurements.

Description

[0001] DESCRIPTIONMethod for state of health and electrode state of health estimation of a lithium-ion battery cellTECHNICAL FIELDThe present invention relates to a method for state of health (SOH) and electrode state ofhealth (eSOH) estimation of a lithium-ion battery cell. PRIOR ART The SOH of a battery cell is usually defined as: ^^^ =^ ^^^^(1)where Q is the actual useful capacity of the battery between the maximum and minimumvoltage ranges, and Qnom is the nominal capacity of the cell, measured with a fresh cell usingthe specified protocol by the manufacturer. This value represents the health of the cell at certain moment in terms of how its past usage has affected its capacity; however, it does not give enough information to determine how quickly the capacity of the battery will drop from this point on and when it is going to reach the end-of-life (EOL) condition.One of the main reasons for defining the SOH in this manner is the lack of information that thestate estimation algorithm has about the actual internal state of the battery. Nowadays, mostlyempirical and equivalent-circuit models are used for battery state-estimation and control [seereference 1]. These are phenomenological models that cannot represent any physical behaviorthat occurs inside the components of a battery. To change this and obtain more information,in recent years physics-based models (PBMs) have been considered and investigated forthese tasks [see references 1, 2, 3, 4, 5, 6, 7]. Due to the physical information that PBMs canprovide about the internal physical states of the battery, these models can offer superiorperformance when maintaining the battery out of highly degrading working conditions [seereferences 1, 4]. Furthermore, they can provide more information on battery aging than justthe traditional estimation of SOH based on cell capacity, providing information on differentdegradation mechanisms and modes [ see reference 8].In addition to the SOH value, the importance of estimating electrode-specific state of health(eSOH) parameters has been remarked in many works [see references 9, 10, 11, 12, 13]. Theerrors that can occur in the estimation of SOC and internal variables, if these parameters ofeSOH are not accurately estimated, were also discussed in the literature [see reference 14].The eSOH parameters define the utilization window (the concentration values that they reachwhen the cell is at maximum and minimum voltages) and the total capacity of each electrode.This information can be used to calculate the loss of active material (LAM) and the loss oflithium inventory (LLI) as in [see reference 10]. This eSOH parameter estimation approachgives much more information about the actual health of the battery than only the cell capacityvalue, and can be used to improve the prognosis and control of a battery.Lee et al. studied the identifiability of individual electrode capacity and utilization window,showing that phase transition data improved the identifiability [ see reference 9]. Mohtat et al.estimated electrode capacities and utilization windows using cell expansion measurementsand constrained Cramer-Rao bound formulation [see reference 10], showing that addingexpansion measurements to the voltage measurements increased the observability of theirsystem and helped estimating the eSOH parameters. Lee et al. used voltage fitting anddifferential voltage analysis to estimate the eSOH parameters [see reference 11]. Fan et al.performed optimizations on specific check-up profiles using a particle swarm optimizationalgorithm to obtain the values of eSOH parameters and other several degradation parameters[see reference 12]. Li et al. used a Cuckoo search algorithm to optimize the eSOH parametersand impedance-related parameters of their equivalent circuit model using low-dynamic andhigh-dynamic operation data [see reference 13]. These methods can be used to estimateeSOH parameters; however, they require constant-current or low-dynamic data sets, whichare generally not available in most applications, or additional sensors to the commonly usedcurrent and voltage sensors. Furthermore, these methods use optimization algorithms toobtain eSOH parameters, which is not an ideal solution for online estimation due to its computational requirements. In addition, these optimizations depend on the quality of the data, so if the application cannot obtain and store all the necessary high-quality data, these methodsmight not work properly. Therefore, it would be desirable to obtain a filter-based estimationmethod, such as is often used to estimate SOC, to estimate these eSOH parameters withouthaving to obtain constant-current data or using additional sensors. This would facilitate theimplementation of these algorithms in real-world BMSs.Some physics-based SOH estimation methods that do not require these specific data, havebeen reported in the literature [see references 6, 7, 15]. Allam and Onori [see reference 6]added a solid electrolyte interphase (SEI) layer growth model to the single-particle model(SPM) to estimate cell capacity, as well as SOC. The results showed that the system was ableto estimate these values adequately. However, despite showing a good capacity estimation intheir results, this approach is based on a SEI layer growth model, which is not the onlydegradation mechanism that can occur in a battery cell [see reference 8]. Furthermore, sincethe SEI layer only models the loss of lithium inventory, the other two degradation modes (theloss of active material in both electrodes) cannot be estimated, leading to incorrect estimationsof the SOC and internal variables as it is shown in [see reference 14]. Thus, this SOHestimation approach is not able to obtain eSOH parameter values.Gao et al. [see reference 7] used a dual extended Kalman filter (EKF) to estimate the SOCand SOH of a cell. They used open-loop simulations of the positive electrode and theelectrolyte phase to improve the observability of the negative electrode solid-phaseconcentration. Consequently, if the positive electrode or electrolyte variables are incorrectlyinitialized, these estimates will be incorrect. The solid-phase concentration, the active materialvolume fraction and the resistance increase, all in the negative electrode, were estimated withthe SOC / SOH estimator. One of the inconveniences that result from this approach is that thestoichiometry limits of the negative electrode can only be estimated once the cell is fullycharged or discharged. Since 100 % and 0 % SOC stoichiometry values are not directlyobservable from the cell voltage, they propose to use the estimated concentration values atthe cell voltage limits. Therefore, if the cell does not reach one of these values, this informationwould be unknown. Furthermore, the approach only considers negative electrode degradation,which will generate an estimation error as the positive electrode ages. Summing up, thismethod can only give the negative electrodes’ total capacity value, and since the positiveelectrode aging is not taken into account, the utilization window cannot be well estimated, whatwill results in an erroneous SOC estimation. Besides, the utilization window values can onlybe obtained if and only if the cell is fully charged and discharged.Smiley et al. [see reference 15] used an interacting multiple-model Kalman filter to estimatethe SOH of a battery using a PBM. They precomputed different models that representeddifferent aging scenarios changing some parameters of their model that affected the cell SOH.Then, they used the filter to select the model that most accurately represented the currentaging state. One of the issues with this implementation is that loss of active material (LAM) inthe negative electrode was not considered. Therefore, it has the same SOC and internalvariable estimation issue when the positive electrode has aged as the works by Allam andOnori [see reference 6], and Gao et al. [see reference 7]. Another weakness of thisimplementation is that it loses accuracy if the aging case is not similar to one of theprecomputed models. Also, all the precomputed models have to be stored and a subset mustbe continuously executed by the IMM, which increases the memory requirements of theimplementation. DISCLOSURE OF THE INVENTIONThe object of the invention is to provide a method for state of health and electrode state ofhealth estimation of a lithium-ion battery cell, as defined in the claims.Compared to the previous cited methods, the estimation method proposed does not need anyconstant-current or low-dynamic data, nor additional sensors, and can compute all the eSOHparameters from online measurements of current and voltage values. The estimation methodis able to estimate all the eSOH parameters, as well as the SOC and the internal physicalvariables; obtain LLI and both electrodes LAM values, it can work with dynamic current profiles,and it does not need to fully charge or discharge the cell.These and other advantages and features of the invention will become apparent in view of the figures and the detailed description of the invention. DESCRIPTION OF THE DRAWINGSFigure 1 is a representation of a pseudo two-dimensional P2D model and single-particle modelwith electrolyte dynamics SPMe.Figure 2 shows the condition number of the observability matrix of the system for the volumefractions of the active materials for different input currents. Figure 3 shows a diagram of a triple interconnected SPKF for SOC estimation. Figure 4 shows charge-depleting UDDS current profile. Figure 5 shows estimation of states and parameters with 10 % LAM in the negative electrode and 5 % LLI. (a)-(b) Negative and positive electrode stoichiometries, respectively; (c) and (d) negative and positive electrode volume fractions of active materials, respectively; (e) SOC estimate; (f) SOH estimate; (g) 100 % and 0 % SOC stoichiometries; (h) degradation modes.Figure 6 shows state and parameter estimation results with 10% LAM in the negative electrode20% LAM in the positive electrode and 16% LLI. (a)-(b) Negative and positive electrodestoichiometries, respectively; (c) and (d) negative and positive electrode volume fractions ofactive materials, respectively; (e) SOC estimate; (f) SOH estimate; (g) 100 % and 0 % SOCstoichiometries; (h) degradation modes. DETAILED DISCLOSURE OF THE INVENTIONThe outline of the invention is the following; First, the PBM that has been used and the cellparameters are summarized. Later, the selection of degradation parameters and theirobservability are discussed. Afterwards, a state / parameter estimation approach is described.The results obtained with the proposed estimation method are then presented and discussed. Finally, conclusions are outlined. Physics-based battery modelMany physics-based battery models have been proposed in the literature, the pseudo two-dimensional (P2D) model [see references 16, 1] and the single-particle model (SPM) [seereference 17] being the most used for state estimation and control applications. The SPM is asimplification of the P2D model, which assumes that the solid-phase diffusion is the slowestprocess in the battery and, consequently, the most dominant, avoiding to describe the rest ofthe dynamics. In this way, the computational cost of the P2D model is significantly reduced,but the accuracy of the model decreases, especially at mid-high C-rates [see reference 18]. Inorder to improve the response of the SPM without increasing significantly the computationalcost, many authors proposed to extend the SPM by including the electrolyte dynamics. Thesemodels are called extended SPMs or single particle models with electrolyte dynamics (SPMe-s). Among the different proposed methods, Marquis et al. showed that their approach to derivethe SPMe outperformed other models presented in the literature. An illustration of the P2Dmodel and the SPMe is shown in figure 1.The SPMe derived by Marquis et al. [see reference 18] was used for the state estimatorproposed in the invention, due to its accuracy and reduced computational cost compared tothe P2D model. The equations of this SPMe are summarized in the following Table 1. Theorthogonal collocation method on Chebyshev polynomials [see reference 19] was used tosolve the model accurately and with reduced computational cost. For the state estimationalgorithm, the model was represented in state-space form and discretized in time using theforward Euler method. A detailed description of this model can be found in [see reference 14].The total number of collocation points used for the state-space SPMe was of 29 points: 10 ineach active material particle (positive and negative), 3 for the negative electrode thickness, 3for the separator and 3 for the positive electrode. Table 1: Equations of the SPMe. A high-fidelity P2D model was used as a reference to obtain the true values of the simulations.The P2D model was also solved using the orthogonal collocation method, in the same way asexplained in [see reference 19]. To obtain an accurate solution, more collocation points thanfor the state-space SPMe were used: 15 points in each active material particle of bothelectrodes, 6 for the positive electrode, 3 for the separator and another 6 for the negativeelectrode; 195 collocation points in total. The parameters used for the simulations wereacquired from Chen et al. [see reference 20] and are shown in the following Table 2.Table 2: LG M50 cell parameters [see reference 20] Parameter Description Negative Separator Positive^^(m2s-1) Solid-phase diffusivity 3.3 × 10^^^ 4 × 10^^^^ (S m-1) Electric conductivity 215 0.18^^,^^^(kmol m-3) Maximum solid-phase concentration 33.133 63.104^^,^(kmol m-3) Initial electrolyte concentration 1 1 1^ (m2) Area 0.1027 0.1027 0.1027^ (m) Thickness 8.52 × 10^^ 1.2 × 10^^ 7.56 × 10^^^^ (m) Particle radius 5.86 × 10^^ 5.22 × 10^^^ Charge transfer coefficient 0.5 0.5^^ (Ω) Film resistance 0.02 0^^ Active material volume fraction 0.75 0.665^^ Porosity 0.25 0.47 0.335^^^^ Bruggeman’s exponent 1.5 1.5 1.5^^ % Stoichiometry at 0 % SOC 0.027 0.8536^^^^ % Stoichiometry at 100 % SOC 0.9014 0.27^^,^^^^(mol m-2 s-1) Normalized reaction rate 7.04 × 10^^ 7.07 × 10^^^^^ Transference number 0.2594 0.2594 0.2594The functions for the negative and positive open circuit potentials are given by: 0.2482 − 0.0909 × tanh^29.8538(^ − 0.1234)^ −0.04478 × tanh^14.9159(^ − 0.2769)^ − 0.0205 × tanh^30.4444(^ − 0.6103)^ (2)and ^^^^^ (^) = −0.809^ + 4.4875 − 0.0428 × tanh^18.5138(^ − 0.5542)^ − 17.7326 ×tanh^15.789(^ − 0.3117)^ + 17.5842 × tanh^15.9308(^ − 0.312)^ (3), respectively. The ionic conductivity of the electrolyte is given by ^(^ ) = 1.297 × 10^^^^ − ^^ ^.^ ^^^ ^^^ 7.94 × 10 ^^ + 3.329 × 10 ^^ , (4)and the electrolyte diffusivity De(Ce) by ^(^ ) = 8.794 × 10^^^^ − 3. ^^^ ^^^^ ^^^ 972 × 10 ^^ + 4.862 × 10 . (5)Aging parameter selection and observability analysis As batteries age due to usage and parasitic effects that accumulate over time, their internalcharacteristics change, making cell voltage predictions of the battery model inaccurate (if themodel is not adjusted to account for this aging), and results in incorrect state estimates. Withthe aim of maintaining the accuracy of the state estimates shown in [see reference 14]throughout the lifetime of the battery, we would like to adapt certain parameters of the modelto consider the effects of aging in the battery. To represent battery aging as faithfully aspossible, it is important to select the appropriate parameters to estimate with the parameterestimation algorithm proposed. These parameters will define how accurate battery cell voltageprediction is compared to the real cell voltage value, which will affect the accuracy of the estimation algorithm.Two of the main contributors to changes in the cell voltage value are the OCP values (opencircuit potential values) of both electrodes. Hence, it is very important that these two curvesremain accurate during the entire lifetime of the battery. The estimates of these curves can be used to calculate the amount of lithium that has been lost in the cell, as well as the amount of active material that has been lost from each electrode. Many degradation mechanisms can occur in a battery cell [see reference 8], and it is extremely difficult, if not impossible, to discern which mechanisms are the ones that are taking place in the battery and at which rate they are degrading its elements. However, these mechanisms can be grouped in three degradation modes that can be estimated from the open circuit voltage (OCV) and OCP curves [seereference 21]: LLI, LAMn and LAMp. Since these degradation modes are observable from the cell OCV value, its estimation is more tractable than the estimation of each degradation mechanism and could give very useful information to improve the control and prognosis of the battery. Therefore, it has been determined that the parameters that must be updated to maintain accurate state estimates are the eSOH parameters. As mentioned above, the eSOH parameters define the total capacities of the electrodes, as well as their operating windows between some fixed cell voltage limits. When these parametersare estimated, all degradation modes can be calculated. In order to adapt the proposed modelto the new utilization windows, we would like to actualize the parameters ^^^^ % , ^^ %, ^^^^^ %and ^ ^ ^^^ % of the proposed SPMe, which denote the stoichiometry values of the electrodes whenthe cell is fully charged or discharged, and therefore they define the operating windows of the electrodes. For the change in capacity of each electrode, known as LAMnand LAMp, we also want to adapt the volume fractions of the active material ^^^and ^^^. These two parameters define the amount of active material per volume unit that can be used for lithium intercalation and deintercalation. Thus, by estimating ^^^and ^ ^ ^, and knowing the electrode dimensions andmaximum lithium concentrations, the amount of capacity that each electrode has, in Ah, can be calculated as ^^ = ^^^^^^,^^^^^^^^ / 3600 (6)where superscript r denotes the negative or positive electrode.In order to estimate these six parameters using a filter, we first have to ensure that they are observable from the output equation. On the one hand, examining the equations of the SPMe, it can be seen that the active material volume fractions affect the output voltage equation because they are related to ^^^and ^^^by the relation ^^^ = 3^^^ / ^^^(where superscript ^ denotes the negative or positive electrode). These two parameters contribute to the calculation of the reaction flux j, resulting in a change in the cell voltage value. On the other hand, it can be observed that the stoichiometric limits are not directly related to the output voltage equation. The solid-phase concentration changes the OCPs of the electrodes, but the fact that the 100 % and 0 % SOC stoichiometric limits are higher or lower does not directly affect the output. Hence, some other method to estimate these four parameters should be found. To define how observable are the volume fractions of active material, and to confirm that thestoichiometry limits do not contribute directly in the output voltage equation, the same methodthat was presented in [see reference 14] was used. First, we reformulate the model in a generalnonlinear form as,^̇ = ^(^, ^) (7)^ = ℎ(^, ^),where ^ denotes the eSOH parameters in this case. The observability of a system can beevaluated by calculating the condition number of its observability matrix, where the conditionnumber is defined as the ratio of the maximum to the minimum singular value. In the case ofnonlinear systems, the observability matrix can be obtained using the Lie derivatives of ℎ [seereference 22], which are calculated as where ^ ^^ℎ(^) = ℎ (^, ^) . The observability matrix is then defined with the Jacobian of the Liederivatives as, where n is the state vector dimension. ^ is an ^ × ^ matrix that must be full rank for the systemto be observable.Using this method, the observability of the active material volume fractions was analyzed. Todo so, the observability matrix of the system was obtained with respect to these twoparameters. By analyzing its condition number for different parameter values, we saw that inall cases, except when the current was 0, the matrix was full rank and had a relatively lowcondition number, meaning that these parameters are observable from the output equation.This can be observed in figure 2, where the condition number of the observability matrix isshown for different currents. When there is no current in the cell, the observability matrix is notfull rank, and the condition number grows to infinity.After analyzing the observability matrix of the stoichiometry limits, we found that the conditionnumber of the system is not full rank and, thus, that the parameters cannot be calculateddirectly using the output voltage equation. Summing up, it was determined that the activematerial volume fractions are observable from the cell voltage equation unless there is nocurrent and thus can be estimated using a filter. However, the analysis of the stoichiometrylimits has shown that these parameters are not direct contributors to the cell voltage equation,meaning that they are not representative to the present voltage value, unless the cell is at 100% or 0 % SOC of course. Therefore, another method must be designed to estimate theseparameters. Estimator design Based on the observability analysis presented above, an estimation approach to obtainupdated values of the six eSOH parameters was designed. The method is divided into twoparts. The first part consists of estimating the active material volume fractions of bothelectrodes, which are observable from the output voltage equation, meaning that they can beestimated using a Sigma Point Kalman filter SPKF, for example. The second part is related tothe four stoichiometry limits. Since it was determined that these parameters cannot beestimated with a filter using the present voltage value, another method was developed.Active material volume fraction estimation εsAs explained above, the active material volume fractions ^^ are observable from the cellvoltage equation, so they were added to the SPKF-based estimation approach presented in[see reference 14]. A representation of the state estimator is shown in figure 3. This way,together with the internal states of the battery, the active material volume fractions areestimated periodically by the filter. To do so, the state estimates, ^, were interconnected withthe parameter estimates, ^, as in a dual state and parameter estimator [see reference 23].The state equations for the internal physical states are given by: ^^[^] = ^^^ ^^[^ − 1] + ^^^ ^[^ − 1]^ ^^[^] = ^^ ^^[^ − 1] + ^^^ ^[^ − 1] (10)^^[^] = ^^^ ^^[^ − 1] + ^^^^[^ − 1]where ^^, ^^, and ^^ are the negative electrode solid-phase lithium concentration, positiveelectrode solid-phase lithium concentration and electrolyte concentration estimatesrespectively. ^^^^^^, ^^^^, ^^ , ^^^^and ^^ are the state-space form discrete-time ^ and ^ matricespresented in [see reference 14], and ^ denotes the input of the system, which is the cell current.For the eSOH parameter estimation, two additional estimators were included. These additionalestimators are used to estimate the active material volume fraction, ^^, of each electrode. Thestate equations for the parameter estimators are given by: where ^^and ^^are the negative electrode and positive electrode active material volume fraction estimates respectively. These parameters are assumed to change very slowly in timecompared to the states, so the ^^^^and ^^^^matrices are equal to 1, while ^^^^ and ^^^^ are equal to 0. The interconnected state estimator has been updated to estimate the two active material volume fractions as follows:Step 1a: State prediction time update. The prediction of the two parameters is added to therest of the predictions as: Step 1b: Error covariance time update. The error covariance matrices for the parameters aregiven by Step 1c: Output prediction. To predict the output, the sigma points of the active material volumefractions have to be calculated together with the state variable sigma points. The set of sigmapoints for the two parameter estimators are calculated as:where ℎ is a tuning square root of the error covariance matrices, which have been computed using a Choleskydecomposition. The sigma points (the vectors of the sets ^) are then used to compute theoutput equation and to obtain the output sigma points. Since the active material volumefractions affect in the output equation, the output sigma points of the state estimators, ^, haveto calculated considering the changes in these parameters.^ ^^,^ = ℎ(^^,^ [^], ^^^[^], ^^^ [^], ^^^̂[^], ^^^̂[^], ^[^]^^,^ = ℎ(^^^[^], ^ ^^,^ [^], ^^^ [^], ^^^̂[^], ^^^̂[^], ^[^]^^,^ = ℎ(^^^[^], ^^^[^], ^ ^^,^ [^], ^^^̂[^], ^^^̂[^], ^[^] (15)^ ^^^,^ = ℎ(^^[^], ^^^[^], ^^^ [^], ^ ^^^[^], ^^^̂[^], ^[^]^ ^ ^ ^^^,^ = ℎ(^^[^], ^^[^], ^^ [^], ^^^̂[^], ^ ^^^[^], ^[^]The five cell voltage predictions are calculated as the weighted mean of these sigma points: Where^(^) ^,^ , ^(^) ^,^ , ^(^) ^^,^and ^ (^) ^^,^are the constants used to calculate the weightedStep 2a: Estimator gain matrix. The covariance matrices for the parameter estimators arecalculated in the same way as for the state estimator: where ^ (^)^^,^ and ^ (^)^^,^ are the constants used to calculate the weighted covariance. Once thesematrices are computed we calculate the gains of the estimators: Step 2b: State estimate measurement update. The parameter estimates are then calculatedas: Step 2c: Error covariance measurement update. Lastly, the error covariance matrices areupdated as With this combined state / parameter estimator, solid-phase and electrolyte concentrations, aswell as active material volume fractions, are estimated for both electrodes. The following statevariables and parameters are obtained from these five SPKF estimators’ interconnection: It is not mandatory to interconnect one estimator per state variable and per parameter. As has been shown in the observability analysis, the parameters could be estimated using a uniqueestimator. However, the interconnection makes the tuning of the filters easier, more stable androbust. Stoichiometric window estimation As mentioned above, as the stoichiometry limits are not direct contributors to the voltageequation in the SPMe, another method to estimate these variables is designed. For that, asystem of four equations is defined, which can be solved to obtain the rest of the eSOHparameters (the 100 % and 0 % SOC stoichiometries). The procedure to obtain each equationis explained below.The available capacity, in Ah, of an electrode between some defined voltage limits is given by where superscript ^ denotes the negative, ^, or positive, ^, electrode. Since the lost capacity over a full charging or discharging process is relatively small compared to the capacities of the electrodes, it can be assumed that the usable capacities of both electrodes are equivalent: ^^^^ ≈ ^^ ≈ ^^ . (24)The same occurs in the charging process: ^^ ^^^ ≈ ^ ≈ ^ . (25)Therefore, This equation shows the six eSOH parameters that we want to estimate: ^^^^^ %, ^^^^ % , ^^^^ %, ^^^% ,^^^and ^^^. If the useful capacities are equivalent, the remaining capacities until full discharge or charge must also be the same. The full discharge equivalency is given by: and for a full charge: where ^^ and ^^ are the current lithiation states of the electrodes at any given moment. Notethat we use ^^ and ^^ instead of ^^ and ^^. This is because the stoichiometry limits are definedusing OCP values, which are obtained when the surface concentration is equal to or almostequal to the state of lithiation of the active material particle. When operating the battery withhigher C-rates, the intercalation process in both electrodes can be different, as the surfaceconcentration may change faster or slower depending on the diffusion coefficients of theelectrodes. This can generate a difference between the available capacities of both electrodes.Therefore, it is more accurate to define the available electrode capacities using the state oflithiation rather than the stoichiometry. In this way, any difference caused by the differentdiffusion rates of the positive and negative electrodes is avoided. Since we also know our fixed operation voltage limits, we know that the 100 % and 0 % positive and negative stoichiometries must fulfill and ^^^^^ (^^ %) − ^ ^^^^ (^^ %) = ^^^^ (30) We assume that the OCPs are fixed functions of active materials that do not change withbattery aging, nor do the values of ^^, ^ ^^,^^^ , ^^ and ^. Therefore, the only unknowns of thisfour-equation system are the six eSOH parameters and the state of lithiation of both electrodes^^and ^^. Nonetheless, ^^and ^^are estimated by the SOC estimator with the interconnectedSPKFs. ^^^and ^ ^ ^are also obtained from the two additional interconnected SPKFs that wehave defined above. Therefore, the resulting system of equations has four equations to solvefor four variables. By solving this system of equations, the remaining four eSOH parameterscan be obtained, completing the full estimation of eSOH parameters.As mentioned above, the stoichiometry limits are not estimated every time step by theinterconnected SPKF filter, instead, for the validation process, the equation system was solvednumerically using the MATLAB® vpasolve solver. In the simulations, the system was solvedat the end of the charging process, however, it can be performed at any other moment.. Thisis less demanding computationally, and since the eSOH parameters will not change its valueoften, it is not necessary to change their values in every time step. The equation system wassolved for several values of ^^ and ^^ of the last discharge / charge cycle for each end-of-charge event. These estimates were taken every 1000 s from the last cycle. Later, the ^^^% , ^^^^^ % , ^^ ^ ^%and ^^^^ % values obtained from all the solutions are filtered (using the MATLAB®function rmoutliers) and averaged. This is the approach that we decided to follow to obtain theresults; however, the equation system can be solved in just one point if it is preferable. Thereason for doing so is that, in this way, an overall estimate value was obtained from the differentestimates of the entire discharge / charge cycle.Degradation mode and SOH estimationOnce the eSOH parameters are estimated, the three degradation modes LLI, LAMn andLAMpcan be calculated. The LAM of each electrode, in %, can be defined as 100, (31) Where r denotes the positive or negative electrode, ^^^is the total capacity of the aged electrode r calculated as ^^^ = ^^^ ^^,^^^^^^^^,^ / 3600 (32)where ^^^,^is the estimated active material volume fraction. ^^^is the total capacity of a fresh electrode r, calculated as ^^^ = ^^^ ^^,^^^^^^^^,^ / 3600 (33) where ^^^,^ is the initial active material volume fraction of the fresh cell. Therefore, since theonly variable that changes is ^^^, the LAM can be defined as 100, (34) The total amount of the intercalated lithium in the cell at any SOC can be calculated as Any ^^and ^^values can be chosen from the entire SOC range (both from the same SOC). In our case, we have chosen to take the 0 % SOC values for example. Then, the LLI can be calculated as 100. (36) where ^^^^ is the aged lithium inventory, and ^ ^^^ is the fresh lithium amount in the cell.Since the total electrode capacities, ^^ and ^^, are calculated with the ^^ estimates, and thelithiation states were also estimated, the calculation of LLI is straightforward. In addition, anSOH value can be calculated by comparing the fresh cell capacity with the aged cell capacity.The available capacity of the aged cell can be obtained by substituting the estimated eSOHparameter values in the electrode capacity equation 23. Then, the SOH of the cell can becalculated as where ^^is the aged cell capacity calculated with the estimated eSOH parameters, and is the fresh cell capacity. State / parameter estimation validationTo validate the proposed estimator, the accuracy of the state and parameter estimates wasanalyzed. For that, the estimates were compared to the high-fidelity P2D model mentionedabove, which was used to obtain the true values of the states and parameters. A charge-depleting UDDS current profile, scaled to cell level for a 1C maximum current, was used asdynamic input for the simulations. The profile is shown in figure 4.Two different aging scenarios were studied to assess the performance of the estimator. Theresponse of the algorithm was evaluated under incorrect state and parameter initializations inboth cases. First, the same SOC and internal variable estimation with incorrect eSOHparameters shown in [see reference 14] was performed with the state / parameter estimator tosee if the estimator was capable of correcting the SOC and internal variable estimates. Later,larger eSOH parameter differences were used, including significant LAM values in bothelectrodes, to discuss the estimation performance in harsher conditions.In the first case, a 10% LAM value was inserted in the negative electrode, and 5% LLI in thecell. In [see reference 14], it was concluded that without estimating these eSOH parameters,the SOC and internal variable estimation was inaccurate. The SOC estimate root mean square(RMS) error was 3.63 %, the electrolyte concentration estimate was very poor, and the solid-phase concentration estimates were significantly worse than in the cases with no eSOHparameter uncertainties. In this case, the eSOH parameters are estimated and updated duringthe simulation as defined above. The results of this simulation are shown in figure 5.To analyze all the dynamics of the parameters properly, 15 charge / discharge cycles weresimulated. In each cycle, the cell was discharged using consecutive charge-depleting UDDSprofiles, as the one shown in figure 4, until the minimum cell voltage was reached. Later, thecell was charged at C / 2 constant current until the maximum voltage value was reached. TheSOC was initialized with 20% error and the active material volume fractions were initialized at80% of their beginning-of-life values.The differences in eSOH parameters with respect to the fresh cell are shown in table 3.Table 3: Variations of the eSOH parameters for the first aging case. Parameter Fresh Agedεns0.75 0.675 εps0.665 0.665 zn0% 0.027 0.0256 zp0% 0.8536 0.8148 zn100% 0.9014 0.9350 zp100% 0.27 0.2661Looking at figure 5, it can be observed that the initial state and parameter estimates areinaccurate. However, as the volume fractions of the active materials are slowly corrected, thestate estimates improve. This can be seen in the SOC estimate of figure 5 (e). The SOC erroris notorious at the beginning, as the stoichiometry limits shown in figure 5 (g) are inaccurate.Little by little, estimates of active material volume fractions of figures 5 (c) and (d) are correctedand, electrode stoichiometry estimates shown in figures 5 (a) and (b) become more accurate.The RMS error of the SOC in the full simulation is 1.4 %, while the RMS error in the last cycleis 0.29 %. This shows how the parameter estimator improves the SOC estimate by updatingthe aged active material volume fractions and the stoichiometry limits. The same happens inthe negative and positive stoichiometries, which have 1.72 % and 0.73 % RMS errors,respectively, for the entire simulation, while in the last cycle both stoichiometry estimates have0.11 % RMS error. Regarding the parameters, the negative and positive volume fractions ofthe active materials have 2.94 % and 6.2 % RMS error respectively (normalized with the freshcell value), while they have 0.59 % and 0.85 % RMS error in the last cycle.As these four estimates converge to the true values, the equation system (27-30) to solveθn100%, θp100%, θn0% and θp0% provides better estimates, improving the SOC estimate.Normalized to the beginning-of-life values, θn100%, θp100%, θn0% and θp0% have 5.14 %, 0.9 %,1.94 % and 1.94 % error respectively in the entire simulation, while in the last cycle the RMSerrors are 0.22 %, 0.85 %, 1.34 % and 0.05 %.As the eSOH parameters are estimated, the degradation modes can be calculated usingequations 34-36. The results of these estimates can be seen in figure 5 (h).As happens withthe rest of the estimates, the results are accurate once the eSOH parameters are corrected.Additionally, an SOH estimate was obtained by computing equation 37. As can be observed,the estimate converges accurately to the true value.The results for a different aging scenario is shown in figure 6. In this case, in contrast to theprevious case, both electrodes suffer from LAM, not just the negative.The LAM for the positive electrode is of 20%, 10% for the negative electrode, and an LLI of16%. The eSOH parameters of this case are shown in table 4. Table 4: Variations of the eSOH parameters for the second aging case. Parameter Fresh Agedεns 0.75 0.675 εps 0.665 0.532 zn0% 0.027 0.0278 zp0% 0.8536 0.891 zn100% 0.9014 0.8566 zp100% 0.27 0.2657Figures 6 (a) and (b) show the stoichiometries of the negative and positive electrodes,respectively. The estimates of both variables are inaccurate in the first cycles, especially thepositive electrode stoichiometry, due to the incorrect active material volume fractions, shownin figure 6 (c) and (d), which affect the SPMe predictions of these variables due to the changein the reaction flux, j. After the first cycle, the positive electrode stoichiometry is corrected asthe positive electrode active material volume fraction converges to its true value. The RMSerror of the negative and positive electrode stoichiometries are 1.04% and 0.67%, respectively,for the entire simulation, and 0.11 % and 0.06 % in the last cycle. The negative electrode activematerial volume fraction estimate takes longer than the positive. This is probably due to thehigher sensitivity of the positive electrode stoichiometry in the cell voltage. For the end of thesimulation, the active material volume fraction of the negative electrode converges well to thetrue value, improving the negative electrode stoichiometry estimate. The RMS errors of thenegative and positive active material volume fractions are 2.72% and 4.14% respectively, whilein the last cycle are 0.11% and 0.29%.In the case of the SOC estimation, shown in figure 6 (e), a higher error can be seen in the firstcycles compared to the last cycles. This error comes from two sides; First, the stoichiometryestimates are not accurate until the volume fractions are corrected, and second, thestoichiometry limits shown in figure 6 (g) have a considerable error until the volume fractionsand the solid-phase concentrations are corrected. After some cycles, the SOC estimateimproves considerably, and in the last cycles becomes very accurate as it is shown in figure 6(e). The SOC RMS error of the full simulation is 1.73%, while in the last cycle the errordecreases to 0.33%. In the case of θn100%, θp100%, θn0% and θp0%, the RMS errors are 2.49%,0.75%, 2.78% and 2.28% respectively, while in the last cycle the errors are 0.14%, 0.68%,1.64% and 0.26%.Regarding the estimates of SOH and degradation modes shown in figures 6 (f) and (h),respectively, the SOH estimate improves considerably for the second cycle and remains veryaccurate until the end of the simulation. LAMp and LLI estimates are very accurate from thesecond cycle on. Since the LAMn value is obtained from the estimate of the active materialvolume fraction of the negative electrode, it takes more time to correct its value, but itconverges accurately. DiscussionThe proposed parameter estimation method can be used to estimate all the eSOH parameters,and therefore all the degradation modes, allowing the acquisition of very valuable informationfor prognosis and control. Additionally, other parameters could be added to the estimationapproach to obtain better voltage predictions from the model. Nevertheless, many parametersof the SPMe are bulk properties of the materials, and therefore, if the composition of theelectrode and electrolyte materials does not change, it should not be necessary to updatethem.In addition to the eSOH parameters, we would like to highlight the need to add parameters thatrepresent power fade in the battery to this estimation approach, such as the electric resistancegrowth or the porosity decrease. These updates in the estimator would improve state of powerand internal variable estimates, helping control algorithms.The equation system has to be solved numerically to obtain the stoichiometry values at 100%and 0% SOH. Even, it is not desirable for a BMS to require of numerical computations to obtainthese parameter values, however, since the eSOH parameters change very slowly over time,it is not necessary to do these calculations online. The stoichiometry limits can be obtainedapart from the online state / parameter estimation procedure, ideally after the estimates becomestable. For example, if the eSOH parameters of an electric vehicle (EV) battery have to beobtained, the numerical computations can be done when the EV arrives at a charging station,or wherever it has a connection to a more powerful computing source.One of the benefits of this estimation method is the estimation of the degradation modes.These estimates would certainly aid in the prediction of battery degradation “knee” trajectories[see reference 24], and several battery applications could benefit from this added knowledge.For example, second-life batteries could be more reliable if the degradation modes were wellknown and if there were a way to know the probability that a battery would be near adegradation “knee”. Moreover, electric vehicle or stationary applications could improve batterycontrol algorithms based on these degradation modes and internal variable knowledge.Accurate physics-based state estimation algorithms could prevent accelerated batterydegradation and safety issues. Besides, accurate stoichiometry estimates as the ones shownin this invention could be used to define dynamic safe operating areas in future BMSs,increasing the energy and power capabilities, while providing longer battery lifespans.Conclusion A novel eSOH parameter estimation method was presented which is capable of estimating all degradation modes from current and voltage measurements in normal battery operation; with dynamic input current profiles and without the need of any additional experiment. The method was validated in simulation for two different aging scenarios; one of them with 10 % LAM in the negative electrode and 5% LLI, and the other with 20% LAM in the positive electrode, 10% in the negative electrode and 16% LLI in the battery. The results showed very good agreement with the true values. The internal variable estimates were accurately corrected by the filter and improved while the eSOH parameter estimates converged. Besides, degradation modes and SOH were estimated with high accuracy in both studied cases. References: [1] Gregory L. Plett. Battery Management Systems, Volume 1: Battery Modeling, volume 1. Artech House, 2015. [2] Kirk D. Stetzel, Lukas L. Aldrich, M. Scott Trimboli, and Gregory L. Plett. Electrochemical state and internal variables estimation using a reduced-order physics-based model of a lithium- ion cell and an extended Kalman filter. J. Power Sources, 278:490–505, 2015. [3] E Miguel, Gregory L Plett, M Scott Trimboli, I Lopetegi, L Oca, U Iraola, and E Bekaert. Electrochemical Model and Sigma Point Kalman Filter Based Online Oriented Battery Model. IEEE Access, 9:98072–98090, 2021. [4] Gustavo Florentino and M. Scott Trimboli. Lithium-ion Battery Management Using Physicsbased Model Predictive Control and DC-DC Converters.2018 IEEE Transp. Electrif. Conf. Expo, ITEC 2018, pages 1046–1053, 2018. [5] Anirudh Allam and Simona Onori. An Interconnected Observer for Concurrent Estimation of Bulk and Surface Concentration in the Cathode and Anode of a Lithium-ion Battery. IEEE Trans. Ind. Electron., 65(9):7311–7321, sep 2018. [6] Anirudh Allam and Simona Onori. Online Capacity Estimation for Lithium-Ion Battery Cells via an Electrochemical Model-Based Adaptive Interconnected Observer. IEEE Trans. Control Syst. Technol., 29(4):1636–1651, sep 2020. [7] Yizhao Gao, Kailong Liu, Chong Zhu, Xi Zhang, and Dong Zhang. Co-Estimation of State-of-Charge and State-of- Health for Lithium-Ion Batteries Using an Enhanced ElectrochemicalModel. IEEE Trans. Ind. Electron., 69(3):2684–2696, 2022. [8] Jacqueline S. Edge, Simon O’Kane, Ryan Prosser, Niall D. Kirkaldy, Anisha N. Patel,Alastair Hales, Abir Ghosh, Weilong Ai, Jingyi Chen, Jiang Yang, Shen Li, Mei-Chin Pang,Laura Bravo Diaz, Anna Tomaszewska, M. Waseem Marzook, Karthik N. Radhakrishnan,Huizhi Wang, Yatish Patel, Billy Wu, and Gregory J. Offer. Lithium ion battery degradation:what you need to know. Phys. Chem. Chem. Phys., 23(14):8200–8221, 2021.[9] Suhak Lee, Peyman Mohtat, Jason B. Siegel, and Anna G. Stefanopoulou. BeyondEstimating Battery State of Health: Identifiability of Individual Electrode Capacity andUtilization. Proc. Am. Control Conf., 2018-June:2288–2293, 2018.

[0010] Peyman Mohtat, Suhak Lee, Jason B. Siegel, and Anna G. Stefanopoulou. Towards better estimability of electrode-specific state of health: Decoding the cell expansion. J. Power Sources, 427(January):101–111, 2019.

[0011] Suhak Lee, Jason B. Siegel, Anna G. Stefanopoulou, Jang-Woo Lee, and Tae-Kyung Lee. Electrode State of Health Estimation for Lithium Ion Batteries Considering Half-cell Potential Change Due to Aging. J. Electrochem. Soc., 167(9):090531, 2020.

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[0014] Iker Lopetegi, Gregory L. Plett, M. Scott Trimboli, Aloisio Kawakita de Souza, Laura Oca, Eduardo Miguel 4 and Unai Iraola. A new battery SOC / SOH / eSOH estimation method using an electrochemical model and interconnected SPKFs: Part 1. SOC estimation. [Manuscript submitted for publication]. Journal of The Electrochemical Society.

[0015] Adam Smiley and Gregory L Plett. An adaptive physics-based reduced-order model of anaged lithium-ion cell, selected using an interacting multiple-model Kalman fi lter. J. EnergyStorage, 19(July):120–134, 2018.

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Claims

CLAIMS1. Method for state of health and electrode state of health estimation of a lithium-ionbattery having a negative electrode and a positive electrode, wherein the state of healthSOH of the battery is:-where Q is the capacity of the battery at any given moment, working betweena maximum voltage Vmax and a minimum voltage Vmin, and- Qnom is the nominal capacity of the battery when the battery is fresh,and the capacity Q, in amperes hour Ah, of each electrode between the definedmaximum voltage Vmax and minimum voltage Vmin, is:Qr = Ar F crs,max Lr εrs ( θr100% - θr0% ) / 3600 ,- where r denotes the negative electrode n or the positive electrode p of thebattery, -Ar is the area of the electrode,- F is the Faraday constant,- crs,max is the maximum solid phase concentration of the electrode,- Lr is the thickness of the electrode,- εrs is the active material volume fraction of the electrode, and- θr100% and θr0% are the 100% and 0% state of charge SOC stoichiometry valuesof the electrode, the method comprising assuming that the capacity of the negative electrode Qn and thecapacity of the positive electrode Qp between the maximum voltage Vmax and theminimum voltage Vmin are equivalent, Qn ≈ Qp, because the lost capacity of the batteryover a full charging or discharging process is relatively small compared to the capacities of the electrodes Qn and Qp, therefore the following equation is obtained:An F cns,max Ln εns ( θn100% - θn0% ) = Ap F cps,max Lp εps ( θp100% - θp0% )and, if the capacities of both electrodes Qn and Qp are equivalent, the remaining usefulcapacity of the electrodes Qn and Qp at any given moment until full charge or fulldischarge of the battery are also equivalent,the method further comprising: ^obtaining a first equation for the full discharge of the battery:An F cns,max Ln εns ( zn - θn0% ) = Ap F cps,max Lp εps ( θp0% - zp )^ obtaining a second equation for full charge of the battery:An F cns,max Ln εns ( θn100% - zn ) = Ap F cps,max Lp εps ( zp - θp100% )- where zn and zp are the current lithiation states of the electrodes at any givenmoment, ^obtaining a third equation of the maximum voltage Vmax of the battery:Upocp (θp100%) – Unocp (θn100%) = Vmax^ obtaining a four equation of the minimum voltage Vmin of the battery:Upocp (θp0%) – Unocp (θn0%) = Vmin- where Upocp and Unocp are the open circuit potentials of the electrodes,^ establishing a system of equations having the four aforementioned equations, whereparameters zp , zn , εps and εns are known estimated parameters, and the parametersof Upocp and Unocp , Ar , F, crs,max and Lr are known fixed parameters that do not changewith battery aging, thus the only unknown parameters of the system of fourequations are θn100%, θn0% , θp100% and θp0%, and^ solving the system of four equations to obtain the parameters θn100%, θn0% , θp100% andθp0%.

2. Method according to claim 1, wherein the loss of active material LAM of each electrodeis obtained according to the following equation:100,-where r denotes the negative electrode n or the positive electrode p of thebattery, -Qra is the total capacity of the positive or negative electrode when the batteryis aged, and -Qrf is the total capacity of the positive or negative electrode when the batteryis fresh,and wherein Qra is calculated as:Qra= ArF crs,maxLrεrs,a / 3600 , Qrf is calculated as:Qra = ArF crs,max Lrεrs,f / 3600 , -where εrs,a is the known estimated parameter of active material volume fractionof the electrode when the battery is aged and εrs,f is the active material volumefraction parameter of the electrode when the battery is fresh,therefore, since the only variable that change is εrs, the loss of active material LAM ofeach electrode is obtained according to the following equation:100,3. Method according to any of the preceding claims, wherein the loss of lithium inventoryLLI is obtained according to the following equation:-where naLi is the amount of lithium in the battery when the battery is aged andnfLi is the amount of lithium in the battery when the battery is fresh.

4. Method according to the preceding claim, wherein the amount of lithium ^^^ in thebattery is obtained according to the following equation: 3600 ^^^=( ^ ^ ^ ^)^^ ^ + ^ ^ .- where zn and zp are the current lithiation states of the electrodes at any givenmoment and Qn and Qp are the capacities of the electrodes.

5. Method according to any of the preceding claims, wherein the known estimatedparameters zp, zn, εps and εns are estimated using an estimator with interconnectedsigma-point Kalman filters SPKFs.

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