State of Health estimation for NCA-C Lithium-ion cells

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48-15 (2015) 376–382 StateIFAC-PapersOnLine of Health estimation for NCA-C State of Health estimation for NCA-C Lithium-ion cells State estimation for NCA-C Lithium-ion cells State of of Health Health estimation Lithium-ion cellsfor NCA-C cells ∗Lithium-ion ∗ ∗ ∗ Lithium-ion cells ∗ ∗ ∗ ∗ D. D. Di Di Domenico Domenico ∗ ,, P. P. Pognant-Gros Pognant-Gros ∗ ,, M. M. Petit Petit ∗ ,, Y. Y. Creff Creff ∗

D. Di Domenico ∗∗ , P. Pognant-Gros ∗∗ , M. Petit ∗∗ , Y. Creff ∗∗ ∗ , P. Pognant-Gros ∗ , M. Petit ∗ , Y. Creff ∗ D. Di Domenico D. Domenico , P. Rond-point Pognant-Gros , M. Petit Y. Creff IFP Energies nouvelles, de de BP IFPDi Energies nouvelles, Rond-point de l’´ l’´eechangeur changeur de ,Solaize Solaize BP 3, 3, IFP Energies nouvelles, Rond-point de l’´ e changeur de Solaize BP 3, 69360 Solaize, France; E-mail: {domenico.didomenico, 69360 Solaize, France; E-mail: {domenico.didomenico, IFP nouvelles, Rond-point l’´ changeur 69360 Solaize, France; E-mail:de IFP Energies Energies nouvelles, Rond-point de{domenico.didomenico, l’´eeyann.creff}@ifpen.fr changeur de de Solaize Solaize BP BP 3, 3, philippe.pognant-gros, martin.petit, philippe.pognant-gros, martin.petit, yann.creff}@ifpen.fr 69360 Solaize, France; E-mail: {domenico.didomenico, philippe.pognant-gros, yann.creff}@ifpen.fr 69360 Solaize, France;martin.petit, E-mail: {domenico.didomenico, philippe.pognant-gros, philippe.pognant-gros, martin.petit, martin.petit, yann.creff}@ifpen.fr yann.creff}@ifpen.fr Abstract: Abstract: The The Lithium-ion Lithium-ion batteries batteries suffer suffer from from several several aging aging phenomena, phenomena, which which imply imply aa Abstract: The Lithium-ion batteries suffer from several aging phenomena, which imply battery performance degradation, i.e. decreasing in energy storage capacity and power supply battery performance degradation, i.e. decreasing in energy storage capacity andwhich powerimply supplya Abstract: ThetheLithium-ion Lithium-ion batteries suffer from from several aging phenomena, capability. For electrified vehicles, monitoring the health of the battery is aa necessary battery performance degradation, i.e. decreasing in energy storage capacitypack andwhich power supplyaa Abstract: The batteries suffer several aging phenomena, imply capability. For the electrified vehicles, monitoring the health of the battery pack is necessary battery performance degradation, i.e. decreasing in capacity and power supply capability. For prevent the electrified vehicles, monitoring theenergy healthstorage of the battery pack a necessary battery degradation, i.e. decreasing energy storage capacity and power supply task can the defaults, alert for anomalies, and predict the battery endtask that thatperformance can prevent the system system defaults, alert in for anomalies, and predict theis battery endcapability. For the electrified vehicles, monitoring the health of the battery pack is a necessary task that can prevent the system defaults, alert for anomalies, and predict the battery endcapability. For the electrified vehicles, monitoring the health of the battery pack is a necessary of-life (EOL). This paper presents a two-phase cell state of health (SOH) estimation strategy, of-life (EOL). This paper presents a two-phase cell state of health (SOH) estimation strategy, task that can prevent the system defaults, alert for anomalies, and predict the battery enddepending on the cell current demand. The cell energy storage capacity is identified when the of-life (EOL). This paper presents a two-phase cell state of health (SOH) estimation strategy, task that can prevent the system defaults, alert for anomalies, and predict the battery enddepending on the cell current demand. The cell cell energy storage capacity is estimation identified when the of-life (EOL). This paper presents a two-phase two-phase statestorage of health (SOH) strategy, battery is at rest while the cell power supply capability is identified during the cell cycling. depending on the cell current demand. The cell energy capacity is identified when the of-life (EOL). This paper presents a cell state of health (SOH) estimation strategy, battery is at rest while the celldemand. power supply capability is identified during the cell cycling. depending on the cell current The cell energy storage capacity is identified when the battery is at the celldemand. power supply capability is identified during the cell cycling. depending onrest the while cell current The cell energy storage capacity is identified when the © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. Allcell rights reserved. battery is at rest while the cell power supply capability is identified during the cycling. battery is at rest while the cell power supply capability is identified during the cell cycling. 1. INTRODUCTION rameters identification or state estimation 1. INTRODUCTION rameters identification or state estimation [Gould [Gould et et al., al., 1. INTRODUCTION 2009, Ahmed Ahmed et al., al., 2015], 2015], and fuzzy fuzzy logic [Salkind [Salkind rameters identification or state estimation [Gould et 2009, et and logic et al., al., 1. INTRODUCTION rameters identification or state estimation [Gould et al., 1. INTRODUCTION 2009, Ahmed et al., 2015], andmodels fuzzy logic [Salkind rameters identification or state estimation [Gould et preal., 1999]. Electrochemical aging have been also 1999]. Electrochemical aging models have been also preLithium-ion 2009, Ahmed Ahmed et al., 2015], andmodels fuzzy logic [Salkind etpreal., Lithium-ion batteries batteries have have recently recently emerged emerged as as the the stanstan- 1999]. Electrochemical have beenand alsopower 2009, 2015], and fuzzy [Salkind et al., sented, which are able to predict both capacity sented, which et areal., able toaging predict bothlogic capacity and power dard electric electricbatteries power source for electrified such Lithium-ion have recently emergedvehicles, as the stan1999]. Electrochemical aging models have been also predard power source for electrified vehicles, such sented, which arethem ableas toaging predict both capacity and 1999]. Electrochemical models have been alsopower prefade and to the effect of same microscopic Lithium-ion batteries havevehicles recently emergedvehicles, as the the stanstanfade and to link link them as the effect of the the same microscopic as plug-in plug-in hybrid-electric (PHEV), electric vehidard electric power source for electrified such fade Lithium-ion batteries have recently emerged as sented, which arethem able tothe predict both capacity and power as hybrid-electric vehicles (PHEV), electric vehiaging mechanisms [Prada et al.,both 2013, Li et al., power 2015, and to link as effect of2013, the same microscopic sented, which are able to predict capacity and dard electric power source for electrified vehicles, such aging mechanisms [Prada et al., Li et al., 2015, as plug-in hybrid-electric vehicles (PHEV), electric vehidard electric power for electrified vehicles, such fade cles (EV), and hybrid electric vehicles (HEV) [Chaturvedi fade and and to Delacourt, link them them as2011]. theet effect ofto the same microscopic cles (EV), and hybridsource electric vehicles (HEV) [Chaturvedi Due their mechanisms [Prada al., of 2013, Li mathematical etmicroscopic al., 2015, to link as the effect the same Safari and as al., plug-in hybrid-electric vehicles (PHEV), electric vehi- aging Safari and Delacourt, 2011]. Due to their mathematical cles (EV), and hybrid electric vehicles (HEV) [Chaturvedi as plug-in hybrid-electric vehicles (PHEV), electric vehiet 2010, Hu et al., 2011]. In this domain, to correctly aging mechanisms [Prada et al., 2013, Li et al., 2015, 2015, et al., 2010, Hu et al., 2011]. In this domain, to correctly Safari and Delacourt, 2011]. Due to their mathematical aging mechanisms [Prada et al., 2013, Li et al., complexity, they require heavy computational efforts preclesal., (EV), and hybrid electric vehicles (HEV) [Chaturvedi complexity, they require2011]. heavyDue computational efforts premanage the battery charge level and, as aa consequence, et 2010, Hu et al.,charge 2011].level In this domain, to correctly complexity, cles (EV), and hybrid electric vehicles (HEV) [Chaturvedi Safari and Delacourt, to their mathematical manage the battery and, as consequence, them on board aging estimation. In they require heavy computational efforts preSafari and Delacourt, 2011]. Due to their mathematical venting from using for et al., 2010, Hu et al., 2011]. In this domain, to correctly venting from using them for on board aging estimation. In manage the battery charge level and, as a consequence, et al., 2010, Hu et al., 2011]. In this domain, to correctly the vehicle range, the battery capacity, i.e. the maximal complexity, they require heavy computational efforts prethe vehicle range, thecharge battery capacity, i.e. the maximal venting order tofrom reduce the complexity of the problem, this paper paper using them for on board aging estimation. In complexity, they require heavy computational efforts premanage the battery level and, as a consequence, order to reduce the complexity of the problem, this the vehicle range, the battery capacity, i.e. the maximal manage the battery charge level and, as a consequence, amount of energy that the battery pack is able to store and venting from using them for on board aging estimation. In amount of energy that the battery pack is i.e. ablethe to store and order tofrom reduce thethem complexity of the problem, paper venting using for on board In presents an strategy that aims to characterthe vehicle vehicle range, the the battery capacity, maximal presents an identification identification strategy thataging aimsestimation. to this characteramount of energy that battery pack is2012, ablethe to store and presents the range, the battery capacity, i.e. maximal release, has to be known [Onori et al., Chaturvedi order tohealth reduce the complexity of variables, the problem, this paper release, has to be known [Onori et al., 2012, Chaturvedi independently an identification strategy that aims to characterorder to reduce the complexity of the problem, this paper ize the of a battery by two amount of energy that the [Onori battery pack is2012, able to to store and the health of a battery strategy by two variables, independently et al., 2010, 2010, Schweighofer et al., al., et 2003]. Moreover, forand an ize release, hasenergy to bethat known al.,is Chaturvedi amount of the battery pack able store presents ani.e. identification that aims aims to charactercharacteret al., Schweighofer et 2003]. Moreover, for an of health (E-SOH) and ize the health of a energy batterystate by two variables, independently presents an identification strategy that to identified, the release, has to bemanagement known [Onori [Onori et al., EV 2012, Chaturvedi identified, i.e. the energy state of health (E-SOH) and the the et al., 2010, Schweighofer et al., 2003]. Moreover, for an release, has to be known et al., 2012, Chaturvedi efficient and safe of PHEV, and HEV, the ize the the state health of a energy battery by two two variables, independently efficient and safe management of PHEV, EV and HEV, the power of health (P-SOH). In order to perform this identified, i.e. the state of health (E-SOH) and this the ize health of a battery by variables, independently et al., 2010, Schweighofer et al., 2003]. Moreover, for an power state of health (P-SOH). In order to perform efficient and safe management of PHEV, EV and HEV, the et al., 2010, Schweighofer et al., 2003]. Moreover, for an battery has to be permanently able to deliver the electric identified, i.e. the energy state of health (E-SOH) and the battery has bemanagement permanentlyofable to deliver the electric state health (P-SOH). order(E-SOH) perform identified, i.e.ofthe energy statefollow-up, of In health and this the task and ensure the battery at least two cell efficientdemand. and to safe PHEV, EV and and HEV, the power task and ensure the battery follow-up, atto least two cell battery has to bemanagement permanentlyof able to deliver the electric efficient and safe PHEV, EV HEV, the power powerand state of health health (P-SOH). In orderpaper toleast perform this power demand. task ensure the battery follow-up, at two cell power state of (P-SOH). In order to perform this parameters have to be estimated. In this the battery batterydemand. has to to be be permanently permanently able able to to deliver deliver the the electric electric parameters have to bebattery estimated. In this paper thetwo battery power battery has task and and and ensure the follow-up, at least least cell capacity theto over-potential estimation arethe proposed, parameters have bebattery estimated. In this paper battery ensure the follow-up, at two cell It is that power demand. capacity and the over-potential estimation are proposed, It is known known that Lithium-ion Lithium-ion batteries batteries are are subject subject to to sevsev- task power demand. parameters have to be estimated. In this paper the battery related respectively to the E-SOH and the P-SOH. A twocapacity and the over-potential estimation are proposed, parameters have to be estimated. In this paper the battery It is aging knownphenomena, that Lithium-ion batteries are subject to sev- related respectively to the E-SOH and the P-SOH. A twoeral which a performance eral aging phenomena, which imply imply a battery battery performance capacity and theestimation over-potential estimation are proposed, proposed, It is is aging knownphenomena, that Lithium-ion batteries arecauses, subject to sevsevrespectively to the E-SOH and the P-SOH. A twocapacity and the over-potential estimation are degradation. TheLithium-ion cell aging aging has several that can related phase cell SOH strategy is eral which has imply a battery performance It known that batteries are subject to phase cell SOH estimation strategy is implemented: implemented: degradation. The cell several causes, that can related respectively to the E-SOH and the P-SOH. A twotwoeral aging phenomena, which imply a battery performance phase cell SOH estimation strategyand is implemented: related respectively to the E-SOH the P-SOH. A degradation. The cell aging has several causes, that can eral aging phenomena, which imply a battery performance be activated by different cell operating conditions [Vetter be activated by different cell has operating conditions [Vetter • When When the estimation cell is at rest, the estimation algorithm phase cell SOH strategy is implemented: degradation. The cell aging several causes, that can • the cell is at rest, the estimation algorithm cell SOH estimation strategy is implemented: 2005]. From aa macroscopic point of view, aging be activated cell has operating [Vetter degradation. Thedifferent cell aging several that can phase et al., et al., 2005]. by From macroscopic point conditions ofcauses, view, the the aging • When thethe cellresidual is at rest, estimation algorithm evaluates cell capacity to the be al., activated by different cell operating conditions [Vetter evaluates the residual cell the capacity to compute compute the results in decreasing the battery energy storage capacity et 2005]. From a macroscopic point of view, the aging be activated by different cell operating conditions [Vetter • When thethe cellresidual is at at rest, rest, the estimation algorithm results in decreasing the battery energy storage capacity evaluates cell capacity to compute the • When the cell is the estimation algorithm E-SOH et al., al.,the 2005]. From a macroscopic macroscopic point ofstorage view, the aging E-SOH and power supply capability [Spotnitz, 2003, Vetter results in decreasing the battery energy capacity et 2005]. From a point of view, the aging evaluates the the residual cell capacity torecursive computeleast the and theinpower supplythecapability [Spotnitz, 2003,capacity Vetter Conversely, during the cell cell capacity cycling, aato evaluates residual compute the results decreasing batterypack energy storage •• E-SOH Conversely, during the cell cycling, recursive least in and the supply [Spotnitz, 2003, Vetter results inpower decreasing thecapability battery energy storage capacity et al., 2005]. Even if the battery can be over-sized E-SOH algorithm et al., 2005]. Even if the battery pack can be over-sized in • Conversely, during the cell cycling, a recursive least E-SOH squares is used to estimate the cell overandal.,the the power supply capability [Spotnitz, 2003, Vetter squares algorithm isthe used estimate the cell overvehicle performance in of et 2005]. Even ifdesired thecapability battery pack can be 2003, over-sized in and power supply [Spotnitz, Vetter order to ensure the Conversely, during cellto cycling, recursive least order to ensure the desired vehicle performance in spite spite of algorithm is used tocycling, estimate the cell over•• squares Conversely, during the cell aa recursive least potential and then the P-SOH et al., 2005]. Even if the battery pack can be over-sized in potential and then the P-SOH the cells performance degradation, the monitoring of the order to ensure the desired vehicle performance in spite of et al., 2005]. Even if the battery pack can be over-sized in squares algorithm is used to estimate the cell overthe cells performance degradation, the monitoring of the potential and then is theused P-SOH squares algorithm to estimate the cell overorder to state ensure the desired vehicle performance in spite spite of A complete battery ofthe health (SOH) is aaperformance relevant task for the the cells performance degradation, the monitoring of the order to ensure desired vehicle in of battery health potential and then then thedescription P-SOH is A complete battery health description is then then achieved achieved and and battery state of health (SOH) is relevant task for potential and the P-SOH the cells performance degradation, the monitoring of the A complete battery health description is then achieved and battery state of health (SOH) is a relevant task for the the cells performance degradation, the monitoring of it can be used for both cell diagnosis and update of the electrified vehicle management. A regular pack diagnostic it can be used for both cell diagnosis and update of the electrified vehicle management. A regular pack diagnostic A complete complete battery healthcell description isand thenupdate achieved and battery state of health health (SOH) is aregular relevant task for the the it functions. can be used for both diagnosis of the electrified vehicle management. A pack diagnostic A battery health description is then achieved and battery state of (SOH) is a relevant task for BMS estimation check can prevent the system defaults, alert for anomalies, BMS estimation functions. check can prevent the system defaults, alert for anomalies, it can canestimation be used used for for both cell cell diagnosis diagnosis and and update update of of the the electrified vehicle management. A regular pack diagnostic such as anprevent unexpected speedup in the aging process, and BMS functions. check can the system defaults, for anomalies, it be both electrified vehicle management. A pack diagnostic such as an unexpected speedup in regular the alert aging process, and The is as BMSpaper estimation functions. check can prevent the system system defaults, alert for anomalies, The paper is organized organized as follows. follows. In In section section 2 2 the the cell cell such as an unexpected speedup in the aging process, and BMS estimation functions. check can prevent the defaults, alert for anomalies, for the battery end-of-life (EOL). Furthermore, the battery for the battery end-of-life (EOL).in Furthermore, the battery is and paper is organized asparameters follows. Inidentification section 2 thestratcell model such as an unexpected speedup the aging(SOP), process, and The model is presented presented and the the parameters identification stratfor the battery end-of-life (EOL). Furthermore, the battery such as an unexpected speedup in the aging process, and state of charge (SOC) and state of power which Theispaper paper is organized organized asparameters follows. Inidentification sectionin 22sections thestratcell3 state of charge (SOC) and state of power (SOP), which egy introduced and briefly described. Then, model is presented andbriefly the The is as follows. In section the cell for the battery end-of-life (EOL). Furthermore, the battery egy is introduced and described. Then, in sections 3 state ofbattery charge (SOC) and state of power (SOP), which model is presented and the parameters identification stratfor theto end-of-life (EOL). Furthermore, themanagebattery need be known in order to ensure the power need to be known in order to ensure the power manageare egy is introduced and briefly described. Then, in sections 3 model is presented and the parameters identification stratand 4 the identification algorithms described in detail state of charge (SOC) and state of power (SOP), which and 4 the identification algorithms are described in detail ment of the vehicle powertrain [Plett, 2004], are impacted need to be known in order to ensure the power managestate charge (SOC) and state of power (SOP), which egy is introduced and briefly described. Then, in sections 3 ment of the vehicle powertrain [Plett, 2004], are impacted and 4 the identification algorithms are described in detail egy is introduced and briefly described. Then, in sections 3 for the cell capacity (E-SOH estimation) and the cell overneed to be known in order to ensure the power managethe cellidentification capacity (E-SOH estimation) and the cell overment ofbattery the vehicle powertrain [Plett, 2004], are impacted need be known order to theSanthanagopalan power manage- for by the aging [Moura et al., 2012, andthe 4 the the algorithms are described in detail detail by theto battery agingin [Moura et ensure al., 2012, Santhanagopalan For for cell capacity (E-SOH estimation) and the cell overand 4 identification algorithms are described in potential (P-SOH estimation). both the algorithms a ment of the vehicle powertrain [Plett, 2004], are impacted potential (P-SOH estimation). For both and the the algorithms a Knowledge of by the aging [Moura et al., 2012, Santhanagopalan ment ofbattery the 2010, vehicle powertrain [Plett, 2004], are impacted and White, Smith et 2008]. for the the cell cell(P-SOH capacity (E-SOH estimation) cell overoverand White, 2010, Smith et al., al., 2008]. Knowledge of SOH SOH potential validation is proposed on the basis of experimental data estimation). For both the the algorithms a for capacity (E-SOH estimation) and cell by the battery aging [Moura et al., 2012, Santhanagopalan validation is proposed on the basis of experimental data and White, 2010, Smith et al., 2008]. Knowledge of SOH by the battery aging [Moura et al., 2012, Santhanagopalan can thus also improve the precision of SOC and SOP potential (P-SOH estimation). For both the algorithms can thus also improve the precision of SOC and SOP validation is proposed on the basis of experimental data potential (P-SOH estimation). For both the algorithms aa collected on a Nickel Cobalt Aluminum Graphite (NCAand White, White, 2010, Smith ettheir al., 2008]. Knowledge Knowledge of SOH SOH onis aproposed Nickel Cobalt Aluminum - Graphite (NCAestimations, ensuring thatthe accuracy does notand decrease can thus also improve precision ofdoes SOC SOP collected and 2010, Smith et al., 2008]. of validation on the basis of experimental data estimations, ensuring that their accuracy not decrease Nickel Cobalt Aluminum - Graphite (NCAvalidation is aproposed on the basis of experimental data C) Li-ion cell. can thus thus also improve the precision ofdoes SOC and SOP collected C) Li-ion on cell. estimations, ensuring thatthe their accuracyof notand decrease can also improve precision SOC SOP with the aging. collected on a Nickel Nickel Cobalt Cobalt Aluminum Aluminum -- Graphite Graphite (NCA(NCAwith the battery battery aging. C) Li-ion on cell. collected a estimations, ensuring that their accuracy does not decrease with the battery aging. estimations, ensuring that their accuracy does not decrease C) Li-ion cell. Several methods have been proposed proposed for for SOH SOH estimation, estimation, C) Li-ion cell. with the themethods battery have aging.been Several with battery aging. Several methods have been proposed for SOH estimation, including empirical methods [Onori et al., 2012], cell including empirical methods [Onori etforal., 2012], cell papaSeveral methods methods have been proposed proposed SOH estimation, including empirical methods [Onori etfor al., 2012], cell paSeveral have been SOH estimation, including empirical empirical methods [Onori et al., 2012], cell paincluding Copyright © 2015 IFAC methods [Onori et al., 2012], cell pa-376 ∗ ∗ ∗ ∗ ∗ ∗

Copyright © 2015 IFAC 376 2405-8963 © 2015, IFAC (International Federation of Automatic Control) Copyright 2015 IFAC 376 Hosting by Elsevier Ltd. All rights reserved. Copyright 2015 IFAC 376 Peer review© of International Federation of Automatic Copyright ©under 2015 responsibility IFAC 376Control. 10.1016/j.ifacol.2015.10.054

IFAC E-COSM 2015 August 23-26, 2015. Columbus, OH, USA D. Di Domenico et al. / IFAC-PapersOnLine 48-15 (2015) 376–382

2. PROBLEM FORMULATION AND PROPOSED SOLUTION In order to model the Li-ion cell, several approaches have been proposed. The models can be based on an electrical equivalent circuit or on electrochemical laws, and they can exhibit different levels of complexity [Di Domenico et al., 2013]. According to the purpose of this paper, a cell model suitable for SOH estimation has to include several variables, which are (1) The cell current, I: the current drawn from, or inserted in, the cell. (2) The cell voltage, V : the voltage measured at cell terminals. (3) The cell capacity, C: the maximal amount of charge that the cell can store/deliver. It decreases with cell aging. (4) The discharged capacity, Q: the charge that is drawn from the cell. It is equal to zero when the cell is fully charged and it takes its maximal value, equal to the cell capacity, when the cell is completely discharged. (5) The electrochemical potentials of the positive and negative electrodes, Up and Un : they can be expressed as a function of the lithium quantity inserted in the electrode. The quantity of lithium that is inserted in each electrode can be expressed as a function of the cell discharged capacity Q, by the means of a relation depending on the cell aging. (6) The cell over-potential, η: includes the high, medium and low-frequency effects of cell cycling. It depends on discharged capacity and current, and increases with aging. Its dynamic evolution is not considered in this paper. (7) The maximal and minimal cell voltage, Vmax and Vmin : imposed by the manufacturers for safety reasons, they are constant and do not depend on the cell aging. The following mathematical model, relating these variables, is selected for the cell V = Up (Q) − Un (Q) + η(Q, I)

Q = Q0 −

t

I(τ )dτ,

Q ∈ [0, C]

(1)

(2)

t0

Up (0) − Un (0) = Vmax

(3)

Up (C) − Un (C) = Vmin

(4)

where Q0 is the discharged capacity at t = t0 . All the defined quantities also depend on the cell temperature. Temperature is neglected in this paper, but the proposed approach can be easily extended to take into account the cell thermal evolution. Equations (1)-(2) allow to predict the cell voltage, when the current demand is known. Voltage depends on the electrochemical potentials and over-potential, that, as said, are impacted by the cell aging. The cell capacity, appearing in equations (2) and (4), also 377

377

depends on the cell aging. The follow-up of the cell aging can be then indirectly performed by identifying the agingdependent parameters of the model. As shown in section 3, the aging evolution of cell capacity and electrochemical potentials are related, and they can be identified at the same time. As a consequence, in order to quantify the cell aging, the identification of two variables, i.e. the capacity and over-potential, is adequate. Depending on the cell operating conditions, the parameters are identified separately. When battery is at rest, i.e. current is zero and all the internal dynamics are at equilibrium, the battery capacity can be identified. Conversely, during the cell cycling, the voltage and current measurements are used to estimate the over-potential, i.e. the internal resistance, as discussed in section 4. 3. CELL CAPACITY IDENTIFICATION The proposed method for determining the cell capacity of a Li-ion cell has been presented in the patent [Creff et al., 2014]. It is based on the measurement of the Open Circuit Voltage (OCV), denoted with U0 and defined as the stabilized electrodes difference in potential when there is no load current. This condition requires the cell to be completely relaxed, i.e. current and over-potential have to be equal to zero. Imposing η = 0 and I = 0 in equations (1)-(4) gives U0 = Up (Q) − Un (Q) (5) U0 (0) = Vmax

(6)

U0 (C) = Vmin

(7)

As said, during cell life, several physical characteristics of the cell electrodes and electrolyte vary, causing a gradual reduction of the energy storage capacity [Vetter et al., 2005]. Loss of cyclable lithium (LCL) and loss of active material (LAM) are two examples of the main causes of the cell capacity decrease (see, for instance, [Ramadass et al., 2004, Spotnitz, 2003]). It is then possible to define a vector of n parameters ξE = (ξE1 , .....ξEn ), where each component ξEi represents an aging phenomenon. U0 is then a function of the discharged capacity Q as well as of ξE [Dubarry et al., 2012]. The number of aging phenomena that can be included in the model depends on the available experimental data from aging campaign and on the knowledge for modeling the aging effect on OCV. Such modeling is not detailed here, as it is out of the purpose of this paper. More details can be found in [Prada et al., 2013]. As an example, the next section shows how the OCV curve distortion can be numerically computed for n = 1 and for a given cell chemistry, after which the knowledge of U0 (Q, ξE ) is just considered as an input for the capacity identification algorithm. 3.1 Distortion of the Open Circuit Voltage curves with aging The aging phenomenon considered in this example is the LCL due to the growth of Solid Electrolyte Interphase (SEI). It is often considered as the main aging factor,

To numerically reproduce this phenomenon, firstly the dependence of the OCV on the discharged capacity is derived from the mass balance of lithium inserted in both electrodes for a cell at beginning-of-life (BOL). Then, the effect of the LCL on the OCV is introduced. The law of the conservation of the mass allows to assert that even if the number of lithium moles inserted in the positive and negative electrodes, nLi+ and nLi− , varies at each cell equilibrium point, the total number of moles of cyclable lithium, ntot , is a constant for a given cell aging: ntot = nLi+ + nLi− (8) Moreover, given the manufacturers constraints, at Q = 0, i.e. when the cell is completed charged, it is nLi+ = nLi+,min (9) nLi− = nLi−,max = ntot − nLi+,min (10) and (11) Up (nLi+,min ) − Un (nLi−,max ) = Vmax while at Q = C, i.e. when the cell is completely discharged, nLi− = nLi−,min (12) nLi+ = nLi+,max = ntot − nLi−,min (13) and (14) Up (nLi+,max ) − Un (nLi−,min ) = Vmin where nLi+,max , nLi+,min , nLi−,max , nLi−,min are, respectively, the maximal and minimal number of lithium moles that can be inserted in the positive electrode and the maximal and minimal number of lithium moles that can be inserted in the negative electrode. They define the stoichiometric operational window for the positive and negative electrodes. Assuming that the electrochemical potentials of the electrodes as a function of nLi+ and nLi− are known (see for instance [Smith et al., 2007]) and that ntot is a known quantity for a BOL cell, the stoichiometric operational window can be found by solving the non-linear system composed of equations (10), (11), (13) and (14). Based on the obtained solution for nLi+,max , nLi+,min , nLi−,max and nLi−,min , the cell capacity is then C = (nLi+,max − nLi+,min )F = (nLi−,max − nLi−,min )F (15) where F is the Faraday constant, and the discharged capacity is Q = (nLi+ − nLi+,min )F = (nLi−,max− − nLi− )F (16) The equations 16 allow to obtain nLi+ and nLi− as a function of Q 378

Negative Electrode Electrode potential [V] Potential [V] Circuit Voltage[V] OpenOpen Circuit Voltage [V]

in particular for the calendar aging [Vetter et al., 2005, Safari and Delacourt, 2011]. It is indeed known that during battery life, a parasitic reaction occurs at the negative electrode surface, during which lithium from the negative electrode is desinserted and then captured in the SEI. This induces a shift of the negative electrode insertion rate with respect to the positive electrode as illustrated by Kassem thanks to postmortem analysis of Li-ion cells [Kassem and Delacourt, 2013]. In order to maintain the operational voltage constraints Vmin and Vmax given by the manufacturer, the stoichiometric operational window, i.e. the extreme values of both electrode insertion rates, is reduced and the nominal capacity of the cell is reduced accordingly. As a consequence the resulting OCV curve is modified, as can be seen in Figure 1.

Positive Electrode Electrode potential [V] Potential [V]

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4 3.8 3.6 nLi+,min

nLi+,max

1 LCL 0.5

0 4

nLi-,max

nLi-,min

Q=0

Q=C

Vmax

3.5 3

Vmin

Fig. 1. Change of stoichiometric window dues to LCL for a NCA/C cell resulting in a capacity fade at two instants of the cell life Q − nLi+,min nLi+ = (17) F and Q (18) nLi− = nLi−,max − F Finally, OCV can be expressed as a function of Q, as U0 = Up (nLi+ ) − Un (nLi− ) = (19) Q Q = Up ( − nLi+,min ) − Un (nLi−,max − ) F F As said, because of the SEI formation, the stoichiometric window and the quantity of cyclable lithium are not fixed and vary with the cell aging. Let introduce an aging parameter ξE , varying between 0 and 1, et let assume that the number of moles of cyclable lithium ntot decreases by a factor (1−ξE ). At a given aging, the mass balance becomes nLi+ + nLi− = ntot (1 − ξE ) (20) Accordingly, the equations 10 and 13, become nLi−,max = ntot (1 − ξE ) − nLi+,min (21) and nLi+,max = ntot (1 − ξE ) − nLi−,min (22) that, together with the two voltage constrains (23) Up (nLi+,min ) − Un (nLi−,max ) = Vmax (24) Up (nLi+,max ) − Un (nLi−,min ) = Vmin result in a non-linear system of 4 equations in 4 unknowns. The solution can be find numerically and allows to obtain the stoichiometric window for the given cell aging. Based on this values, the cell capacity, the residual discharge capacity and the OCV can be computed accordingly from the equations 15, 16 and 19. When repeated for several value of ξE , this procedure allows to generate a collection of OCV curves, each corresponding to a given level of aging. Similar procedures apply to other aging mechanisms, provided that their impact on the OCV is well established and modeled. Introducing more aging mechanisms results in increasing the dimension of the vector ξE . It is to highlight that the value of the residual capacity obtained with this procedure is a theoretical value, that

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Open Circuit Voltage [V]

Then ξE can be identified by solving the nonlinear leastsquares minimization problem

C=26.5 Ah C=23.9 Ah C=21.4 Ah C=19.2 Ah C=17.2 Ah C=14.7 Ah

4 3.8 3.6

min

{Q0,k },{∆Qk },ξE

3.2 10 20 Discharged Capacity [Ah]

30

Up [V]

Fig. 2. Distorted OCV resulting from simulation of LCL from the beginning-of-life (BOL)

m − U0 (Q0,k , ξE ))2 )+ {αk (Vi,k

k=1

where αk , βk and γk are real positive numbers introduced in order to weight the different contributions and the notation {xk } indicates a vector of p components xk , one for every available set of three measurements. The problem convexity is obviously not guaranteed in general, as it depends on the non-linear function U0 . As each material of the electrodes has a different nonlinear electrochemical potential, in the last analysis the convexity depends on the cell chemistry. It is indeed not possible to analyze if the problem is well-posed in general. Nevertheless, for every particular case, several solutions can be proposed to partially elude the possible problem of local minima, such as imposing that aging increases with respect to time. Once ξE is identified, the corresponding cell capacity C(ξE ) can be computed by imposing that

4

U0 (C(ξE ), ξE ) = Vmin

3.5 0 Un [V]

p 

m 2 + βk (Vf,k − U0 (Q0,k + ∆Qk , ξE ))2 + γk (∆Qm k − ∆Qk ) } (25)

3.4

3 0

379

10

(26)

Finally, assuming a linear variation of E-SOH with the cell capacity and imposing that E-SOH lies between 0 and 1, it can be defined as

20

0.5 E-SOH =

0 0

C(ξE ) − CEOL ∈ [0, 1] CBOL − CEOL

(27)

where CBOL is the BOL cell capacity and CEOL is the EOL cell capacity, usually defined as equal to 75% of the nominal capacity.

10 20 Discharged Capacity [Ah]

Fig. 3. Positive and negative electrochemical potentials for a NCA/C cell

3.3 Validation

corresponds to an infinitely slow charge or discharge rate. In reality the over-potential reduces the effective capacity, as the maximal or minimal voltage constrains are reached when the cell is not completely relaxed. Figure 2 shows the OCV curves distortion due to the LCL for a 22 Ah NCA-C Li-ion cell. Note that the measured BOL cell capacity is larger than the nominal value given by the manufacturer.

An exhaustive validation of the algorithm is still ongoing. Nevertheless, in this section, the result of an application case is presented. The proposed strategy has been tested by using the current and voltage measurements collected during an aging campaign test at IFPEN battery facilities. The pack was composed of a series of 8 modules, each of them composed of series of 6 pairs of cells. Every pair is composed of 2 cells connected in parallel, for a total of 96 NCA-C Li-ion cells. The nominal capacity of each cell is 22 Ah. The voltage measurement of every pair of cells was recorded. Figure 4 shows the pack current demand Ipack (t) which is a part of one of the regular pack check-ups of the campaign. The current profile was obtained by simulating the current demand for a pack of a typical PHEV during a New European Driving Cycle (NEDC) repeated for 5 consecutive times. During the test the pack temperature is regulated at 298 K.

3.2 Estimation Algorithm Given a collection of OCV curves such as shown in Figure 2, an aging state of the cell, that is described by a vector of n parameters ξE = ξE1 , ....., ξEn , is associated to each curve. The patent [Creff et al., 2014] presents a method to identify the cell aging parameter ξE . This algorithm is based on a collection of p ≥ n sets of three measurements, each of them including: (1) An OCV measurement at time ti , indicated as Vim (2) An OCV measurement at time tf , indicated as Vfm (3) The variation of discharged capacity, measured bet tween ti and tf , ∆Qm = tif I(τ )dτ 379

During the check-up, the capacity of the pack was also measured. In order to test the capacity estimation algorithm, two open circuit voltage measurements for a cell of the pack, V1m and V2m , were used, corresponding at the instants t1 and t2 highlighted in Figure 4.

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of the cells reaches the maximal or minimal voltage limit, the capacity pack value is imposed by the lowest value between the capacities of the cells in the pack. The estimation algorithm result is then in agreement with experimental measurement, as s C(ξE ) > Cmin (ξE ). Further tests are ongoing in order to support this preliminary result.

100

Cell current [A]

50 0 −50

4. INTERNAL RESISTANCE IDENTIFICATION

−100

In the following, an identification method is described and used to evaluate the over-potential η(Q, I) of a cell, and then to obtain a P-SOH indicator. In the equivalent circuit modeling framework, the over-potential dynamics are usually modeled as a series of resistance-capacitance circuits [Diard et al., 1996, Di Domenico et al., 2013], each of then representing a different time-scale. Three main terms are usually considered:

−150 −200 0

1

2

3 Time [h]

4

5

6

Fig. 4. Battery pack check-up current profile. The equilibrium point measurements used for the algorithm validation are taken at the two instants highlighted by the markers in the plot.

identified OCV BOL COV EOL OCV

4

Cell OCV [V]

3.8

(29)

where ηhf is the high frequency term, representing the ohmic resistance of the cell, ηct is the charge transfer term, representing medium frequency over-potential due to the chemical reaction and ηdif f is the diffusion over-potential, representing the slow phenomena. In this paper the low frequencies term is neglected and two time-scales, at 10s and 30s, are alternatively considered. The over-potential η can be then approximated as ηhf (Q, I) = R0 (Q)I

3.6

(30)

where R0 is the cell internal resistance, that can be considered as the real component of the impedance at the given time-scale. In discrete time and by following the over-potential approximation, the cell model (1)-(4) becomes, at each step k

3.4 3.2 3 0

η(Q, I) = ηhf (Q, I) + ηct (Q, I) + ηdif f (Q, I)

V (k) = U0 (k) + R0 (k)I(k)

(31)

where also (5) was used.

10 20 Cell Capacity [Ah]

30

Fig. 5. Identification of OCV curve during the pack checkup compared to the BOL OCV curve and EOL OCV curve The set of three measurements was then completed by computing the discharged capacity as

∆Q =

�t2

Ipack (τ ) dτ 2

(28)

t1

The minimization problem was then applied by using the set of OCV curves obtained in the example for n = 1 and shown in Figure 2. The method used for solving the minimization problem is the Levenberg-Marquadt algorithm. Figure 5 shows the identified OCV curve. The corresponding cell capacity is C(ξE ) = 19.3 Ah. This value can be compared to the measurement of the capacity of the pack, that gives Cmin (ξE ) = 17.3 Ah. It is to highlight that for a pack, the capacity test consists in integrating the pack current during a complete charge or discharge. As the pack charge or discharge stops when one 380

It can be equivalently written as a prediction model � � R0 (k) = φ(k)θ(k) (32) Vˆ (k) = [ I(k) 1 ] U0 (k) � � R0 (k) . Both U0 (k) where φ(k) = [ I(k) 1 ] and θ(k) = U0 (k) and R0 (k) are considered as parameters to be identified. In order to decouple each parameter convergence, two separated forgetting factors λ1 and λ2 can be introduced by using the following minimization criterion   k k � � 1 Jk =  λk−j e2 (j)2  λk−j e1 (j)2 + (33) 2 2 j=1 1 j=1 The prediction errors e1 and e2 are defined by � � θ (j) e1 (j) = V (j) − φ(j) 1∗ θ2 (j)

and e2 (j) = V (j) − φ(j)

�

θ1∗ (j) θ2 (j)

�

(34)

(35)

where it is supposed that θ2 parameter has converged to its final value θ2∗ , and the same for θ1 . From [Vahidi et al., 2005], a recursive form can be obtained:

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381

 ǫ(k) = V (k) − φ(k)θ(k − 1) expressed as tλ = Ts / ln(λ), with a sample time Ts . λ2     has been set in order to fit experimental measurements of   P (k − 1)φ (k)    1 1 R , basically internal resistances measured after t = 10 s  0   1 λ1    and t = 30 s. L(k) =   P2 (k − 1)φ2 (k) 1 + P1 (k − 1)φ1 (k)2 + P2 (k − 1)φ2 (k)2   The present approach has been tested on experimental   λ2 measurements collected during Power Pulse Current tests θ(k) = θ(k − 1) + L(k)ǫ(k) of the pack aging campaign. The persistent excitation   � �   condition for the proposed model has been also tested 2  1 P (k − 1)φ (k) 1 1   P1 (k) = P (k − 1) P1 (k − 1) − 1 for the considered input profile. This analysis confirmed   λ1 λ1 + P1 (k − 1)φ1 (k)2   that persistent excitation condition is satisfied during the  � �   1 P2 (k − 1)φ2 (k)2  current pulses. The comparison is presented in Figure 6. X P2 (k) = P2 (k − 1) P2 (k − 1) −  λ2 λ2 + P2 (k − 1)φ2 (k)2 axis represents the number of the experiments during the (36) battery aging. Y-axis represents the resistance in Ohms. For tλ2 = 5 s and 20 s, the estimated internal resistance values are closed to R10s , less than 10% of error. R30s is well estimated by a tuning of tλ2 = 300 s. In Figure 7, P-SOH defined by (38) with a = 2, is plotted for two modules (9 and 10), composed of 6 cells. X-axis still represents the number of the experiment during the battery aging protocol.

Fig. 6. Comparison between estimated and measured resistances for 10s and 30s The prediction error is then separated and implies two scalars P1 (k) and P2 (k) instead of a full matrix in the standard identification approaches. The main benefit of this algorithm is that it does not assume that the parameters vary with similar rates, allowing to avoid that the errors due to two parameters are lumped into a single scalar term. For more details on this approach see [Vahidi et al., 2005]. The application of the algorithm leads to evaluate at each step k the internal resistance θ1 (k) = R0 (k). At a step time k, the value of R0 (k) can be the consequence of the cell capacity and/or temperature variation. In order to separate the effect of aging on the internal resistance, an average of R0 (k) is calculated along the duration of the estimation test, defined by a current profile, as N 1 � R0est = R0 (k) (37) N k=1

Thus, an indicator for P-SOH can be introduced as aR0init − R0est (38) P -SOH = 100 init aR0 − R0init where a > 1, is a real number that is chosen by fixing the maximal allowable value of the internal resistance, corresponding to P-SOH= 0. In the following, the forgetting factor λ1 is set to 1 meaning that R0 is seen as a constant parameter to identify. On the contrary, θ2 = U0 is considered as a slowly time varying parameter, and implies a tuning for λ2 . With a forgetting factor λ < 1, the associated time constant tλ can be 381

Fig. 7. Comparison between measured P-SOH (1st column) and estimated P-SOH (2nd column) The left plots are considered with measurements of R30s . The right plots use R0 estimation using tλ2 = 300 s. Even if an error of 5% exists, the proposed algorithm allows a good estimation of the resistance evolution with respect to the aging. The relative values of the resistances of the selected cells are also well estimated. 5. CONCLUSION A two-phase strategy for Li-ion battery SOH estimation has been presented. When the cell is at rest and the persistent excitation condition is not satisfied, the estimation algorithm of patent [Creff et al., 2014] is used to evaluate the E-SOH, that allows to compute the residual cell energy storage capacity. Conversely, during the cell cycling, a recursive least squares algorithm is used to estimate the P-SOH, and as a consequence, to evaluate the cell power

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