L Vichard

A Ravey

P Venet

F Harel

S Pelissier

D Hissel

A method to estimate battery SOH indicators based on vehicle operating data only

Keywords: Battery SOH, Battery modeling, Vehicle operating data, Electric vehicle

Batteries are multi-physical systems and during actual operating conditions they are submitted to variable ambient operating conditions which can affect the dynamic behavior and the degradation. Therefore, a good understanding of the dynamic behavior and the degradation laws under actual operating conditions is the key to a durability improvement and to the development of better energy management strategies. The purpose of the proposed study is to use an experimental database issued from a three years monitoring of a ten postal vehicle fleet to model the batteries with respect to operating conditions. Based on an electrical circuit model, an optimization algorithm and a Kalman filter, the scientific contribution is to propose a simple but efficient method, using vehicle operating data only, to estimate on-board the state of charge and state of health indicators linked to internal resistance and available capacity. The proposed model presents a very good accuracy and state of health indicators estimations show promising results. In the future, the proposed method could be applied on-board to estimate and analyze the state of health during the entire battery lifetime in order to provide an accurate state of charge estimation and to contribute to a better understanding of the degradation laws.

Introduction

Nowadays, greenhouse gas emissions have to be significantly reduced. This is the reason why the popularity of low emission vehicles such as Electric Vehicles (EVs), Hybrid Electric Vehicles (HEVs) and Fuel Cell Hydrogen Electric Vehicles (FCHEVs) is highly growing [START_REF] Lipu | A review of state of health and remaining useful life estimation methods for lithium-ion battery in electric vehicles: Challenges and recommendations[END_REF][START_REF] Martinez | Energy Management in Plug-in Hybrid Electric Vehicles: Recent Progress and a Connected Vehicles Perspective[END_REF][START_REF] Han | Multisource coordination energy management strategy based on SOC consensus for a PEMFC-batterysupercapacitor hybrid tramway[END_REF]. Consequently, batteries used in transportation applications became a famous topic. However, batteries are frequently pointed out because of their high prices but also because of their low autonomy, low durability and their long time recharge which are limitations to a worldwide deployment [START_REF] Saw | Integration issues of lithium-ion battery into electric vehicles battery pack[END_REF].

Battery performances degrade with the repeatedly charging and discharging [START_REF] Dubarry | Identifying battery aging mechanisms in large format Li ion cells[END_REF] and because of calendar aging [START_REF] Ghossein | Degradation behavior of Lithium-Ion Capacitors during calendar aging[END_REF][START_REF] Mejdoubi | Experimental investigation of calendar aging of lithium-ion batteries for vehicular applications[END_REF]. Batteries are multiphysical systems and operating conditions can affect the dynamic behavior and the degradation in different manners [START_REF] Birkl | Degradation diagnostics for lithium ion cells[END_REF][START_REF] Barré | A review on lithium-ion battery ageing mechanisms and estimations for automotive applications[END_REF]. Moreover, as the batteries degrade overtime depending on the operating conditions, the battery maximum discharge capacity decreases during the long-term operation and affects the accuracy of the State Of Charge estimation (SOC). Therefore, a good understanding of the dynamic behavior and the degradation mechanisms occurring during actual operating conditions is of particular importance to improve the durability with the development of better materials and energy management strategies. To tackle this issue, a first step is to master a good online degradation quantification of the battery component.

The battery State of Health (SOH) is commonly used as a metric to quantify the aging level of a battery [START_REF] Berecibar | Critical review of state of health estimation methods of Liion batteries for real applications[END_REF][START_REF] Xiong | Towards a smarter battery management system A critical review on battery state of health monitoring methods[END_REF]. The SOH often refers to either the capacity fade [START_REF] Zheng | Incremental capacity analysis and differential voltage analysis based state of charge and capacity estimation for lithium-ion batteries[END_REF][START_REF] Guha | State of Health Estimation of Lithium-Ion Batteries Using Capacity Fade and Internal Resistance Growth Models[END_REF] or to the power fade [START_REF] Guha | State of Health Estimation of Lithium-Ion Batteries Using Capacity Fade and Internal Resistance Growth Models[END_REF]. The capacity fade refers to the loss in capacity (in Amps hours) of the component and the power fade refers to the internal impedance increment. On the one hand, accurate online estimation results are helpful to ensure the system running safely and to estimate precisely the SOC. On the other hand, SOH evolution can be analyzed with respect to operating conditions in order to contribute to a better understanding of the degradation laws.

A wide variety of studies have focused on SOH estimation [START_REF] Berecibar | Critical review of state of health estimation methods of Liion batteries for real applications[END_REF][START_REF] Xiong | Towards a smarter battery management system A critical review on battery state of health monitoring methods[END_REF]. The proposed approaches can be classified according to three differents categories: (1) direct measurement approaches, (2) data driven approaches and (3) model-based approaches. Each category has its own advantages and drawbacks in terms of accuracy, computational requirement, testing time or cost.

(1) Direct measurement approaches: these methods use raw measured data to estimate SOH without reproducing the dynamic behavior of the battery. For example, from the perspective of determining the battery capacity, Kong et al. [START_REF] Ng | Enhanced coulomb counting method for estimating state-of-charge and state-of-health of lithiumion batteries[END_REF] proposed a coulomb counting method to estimate the SOC and the SOH of a battery cell. Coulomb counting methods are very basics and do not consider operating conditions [START_REF] Berecibar | Critical review of state of health estimation methods of Liion batteries for real applications[END_REF]. Moreover, an accumulated error and sensor noise can lead to significant estimation errors. Some other studies proposed Open Circuit Voltage (OCV) based methods to determine the battery capacity by using the relationship between the OCV and the SOC [START_REF] Guo | State of health estimation for lithium ion batteries based on charging curves[END_REF][START_REF] Weng | A unified open-circuit-voltage model of lithium-ion batteries for state-of-charge estimation and state-of-health monitoring[END_REF]. OCV based methods can be performed in both online or offline states. Nonetheless, laboratory tests are necessary to determine the relationship between OCV and SOH. Electrochemical Impedance Spectroscopy (EIS) has also been considered as a tool to estimate battery SOH [START_REF] Andre | Characterization of high-power lithium-ion batteries by electrochemical impedance spectroscopy. II: Modelling[END_REF][START_REF] Galeotti | Performance analysis and SOH (state of health) evaluation of lithium polymer batteries through electrochemical impedance spectroscopy[END_REF][START_REF] Swierczynski | Electrothermal impedance spectroscopy as a cost efficient method for determining thermal parameters of lithium ion batteries: Prospects, measurement methods and the state of knowledge[END_REF][START_REF] Cui | State of health diagnosis model for lithium ion batteries based on real-time impedance and open circuit voltage parameters identification method[END_REF] and assure that EIS can be integrated in a vehicle by controlling the harmonic injection in the system to be characterized via the static converter [START_REF] Wang | A review of DC/DC converter-based electrochemical impedance spectroscopy for fuel cell electric vehicles[END_REF][START_REF] Wang | Online electrochemical impedance spectroscopy detection integrated with step-up converter for fuel cell electric vehicle[END_REF]. Incremental Capacity Analysis (ICA) and Differential Voltage Analysis (DVA) are both interesting methods that have been focused on. These methods provide information about the internal cell state using only the cell voltage and current measurements during a constant recharge or discharge [START_REF] Dubarry | Identify capacity fading mechanism in a commercial LiFePO 4 cell[END_REF][START_REF] Riviere | Innovative incremental capacity analysis implementation for C/LiFePO4 cell state-of-health estimation in electrical vehicles[END_REF][START_REF] Zheng | Incremental capacity analysis and differential voltage analysis based state of charge and capacity estimation for lithium-ion batteries[END_REF]. ICA refers to the ratio between a quantity of electric charges and a voltage variation and DVA is the mathematical inverse. Some studies used these methods during battery charging to analyze the dynamic behavior and quantify the SOH [START_REF] Dubarry | Identify capacity fading mechanism in a commercial LiFePO 4 cell[END_REF][START_REF] Riviere | Innovative incremental capacity analysis implementation for C/LiFePO4 cell state-of-health estimation in electrical vehicles[END_REF][START_REF] Zheng | Incremental capacity analysis and differential voltage analysis based state of charge and capacity estimation for lithium-ion batteries[END_REF][START_REF] Li | A quick on-line state of health estimation method for Li-ion battery with incremental capacity curves processed by Gaussian filter[END_REF]. Both methods are very promising and show very consistent results. However, as mentionned in Dubarry et al. [START_REF] Dubarry | Identify capacity fading mechanism in a commercial LiFePO 4 cell[END_REF], the consistency of the obtained information are directly linked to the current value: the lower the charging current the best the information, which can be very problematic in the case of vehicle applications.

(2) Data driven approaches: data driven methods have also been used to estimate battery SOH. This approach does not consider the physical laws of the battery system as the methods are purely based on data-driven tools such as artificial neural networks [START_REF] Li | Data-driven health estimation and lifetime prediction of lithium-ion batteries: A review[END_REF][START_REF] Wu | A novel state of health estimation method of Li-ion battery using group method of data handling[END_REF][START_REF] You | Real-time state-of-health estimation for electric vehicle batteries: A data-driven approach[END_REF], support vector machine [START_REF] Klass | A support vector machine-based stateof-health estimation method for lithium-ion batteries under electric vehi-cle operation[END_REF][START_REF] Deng | Feature parameter extraction and intelligent estimation of the State-of-Health of lithium-ion batteries[END_REF] or again deep learning [START_REF] Shen | A deep learning method for online capacity estimation of lithium-ion batteries[END_REF]. Although these black-box approaches can provide good estimation accuracy, they require a large amount of data for the training and high computational load. Because of these constraints, these methods may not be suitable for on-line real-time purpose as BMS often contain some microcontrollers or low computing performance systems.

(3) Model based approaches: a large number of studies have focused on the use of battery models to identify degradation and estimate the SOH [START_REF] Berecibar | Critical review of state of health estimation methods of Liion batteries for real applications[END_REF][START_REF] Tian | Online simultaneous identification of parameters and order of a fractional order battery model[END_REF][START_REF] Jafari | Deterministic models of Li-ion battery aging: It is a matter of scale[END_REF]. Models are used to reproduce the dynamic behavior of a battery: voltage response as a function of inputs such as current, temperature and SOC. Model parameters have certain relationships with the battery aging level. Thus, they can be used to estimate the SOH and their evolutions can be investigated overtime and correlated with operating conditions to analyze degradation laws. It is worth remembering that operating conditions such as SOC, temperature, current etc. affect battery degradation and dynamic behavior. Thus, many parameters are interdependent and empirical models can tend to be com-plex. This is the reason why electrical circuit models (ECM) and electrochemical models are commonly used. Electrochemical models are generally exploited to understand electrochemical phenomena. This kind of model is more complicated to implement as it needs many physico-chemical parameters as inputs and lead to important computational requirements [START_REF] Majdabadi | Simplified electrochemical multi-particle model for LiFePO4 cathodes in lithium-ion batteries[END_REF][START_REF] Li | An electrochemicalthermal model based on dynamic responses for lithium iron phosphate battery[END_REF][START_REF] Gu | On the suitability of electrochemical-based modeling for lithium-ion batteries[END_REF]. In contrast, ECM have been extensively studied in the literature for SOH estimation applications [START_REF] Bahramipanah | Enhanced Equivalent Electrical Circuit Model of Lithium-Based Batteries Accounting for Charge Redistribution, State-of-Health, and Temperature Effects[END_REF][START_REF] Shen | The co-estimation of state of charge, state of health, and state of function for lithium-ion batteries in electric vehicles[END_REF][START_REF] Chaoui | Online parameter and state estimation of lithiumion batteries under temperature effects[END_REF][START_REF] Ren | Design and implementation of a battery management system with active charge balance based on the SOC and SOH online estimation[END_REF][START_REF] Zhang | A novel approach of battery pack state of health estimation using artificial intelligence optimization algorithm[END_REF]. These approaches exhibit low complexity, low computational requirement and acceptable estimation accuracy. Ren et al. [START_REF] Ren | Design and implementation of a battery management system with active charge balance based on the SOC and SOH online estimation[END_REF] proposed a method based on an ECM and a Kalman filter to estimate SOC and SOH indicators of a battery pack. The method has been tested with Hybrid Pulse Power Characterization (HPPC) profiles and showed promising accuracy. Shen et al. [START_REF] Shen | The co-estimation of state of charge, state of health, and state of function for lithium-ion batteries in electric vehicles[END_REF] proposed a method to estimate concurrently the SOC, the SOH, and the state of function (SOF) for lithium-ion batteries in real-time applications. The battery capacity was estimated according to the accumulated charge between two specific moments with online OCV identification. Based on the capacity update from SOH estimation, the SOC could be estimated according to the battery aging. The tests were performed with HPPC profiles. Although the presented results show good accuracy, this method is based on OCV curves. Therefore, the method can be hard to be applied on other technologies such as LiFePO 4 which presents a relatively flat OCV curve for SOC between 80% and 20%. Given the previous cited studies, an ECM based method has been preferred in our study to estimate internal resistance and battery capacity as SOH indicators. It is worth noting that SOH estimation accuracy depends on the model accuracy. Consequently, the first step is to build an accurate and suitable model for real applications.

The large majority of SOH estimation methods published in the literature are based on laboratory tests or simplified driving cycles such as NEDC, FTP-75 or WLTP. Despite the greatest attention given to these studies, the generated electrical solicitations may not fully reflect the battery dynamic behavior operating in real applications. Therefore, the use of raw vehicle operating data directly issued from monitoring is essential to build more accurate models and more reliable SOH estimation methods.

In this framework, the main contribution of this study is to propose a simple and accurate model-based method to estimate both SOH indicators (capacity and internal resistance), of a LifePO 4 battery by using only physical values measured on-board during real driving cycles. In a first part the experimental context of the project and its application will be introduced. In a second part, the method to build the battery cell model considering the operating condition and to identify online the internal resistance will be detailed. In a third part a method based on an extended Kalman filter to estimate concurrently the SOC and the battery capacity will be exposed. Finally conclusion and perspectives will be drawn.

Experimental context

Vehicle fleet

Mobypost was a european project which aimed at developing a fleet of ten experimental fuel cell hydrogen electric vehicles (cf. Figure 1) dedicated to postal delivery applications [START_REF] Ravey | Energy management of fuel cell electric vehicle with hydrid tanks[END_REF], [START_REF] Vichard | Battery aging study using field use data[END_REF]. The vehicles were designed considering the ability to support an important number of starts and stops and also the possibility to operate with the fuel cell off. The power-train integrated two power sources : a LiFePO 4 battery pack and a fuel cell system.

The battery pack is composed of four modules connected in series. Each module presents a nominal voltage of 12.8V at 23°C. Nominal capacity given by the manufacturer is 110Ah at 23°C with a discharge current of C/5. Each module integrates four cell groups connected in series which are composed of LiFePO 4 3.2V cells connected in parallel to reach a capacity of 110Ah. The battery temperature values are measured inside each module.

A deep monitoring of the vehicles during the three years of experimentation led to create a rich database. More than one hundred physical variables from all the vehicle components were measured and saved on a server every second such as battery cell voltages, battery current, battery temperature, battery SOC, fuel cell system voltage and current, hydrogen tank SOC, vehicle speed, gps location to cite the main variables. The vehicles were parked in a garage where ambient temperature was controlled to be 12°C so the temperature in the vehicles has never been lower than 0°C during experimentation.

Database presentation

The fleet monitoring led to create a database that store more than 1500 real driving cycles. This database can be used to analyze the dynamic behavior with respect to the operating conditions and investigate their effects. In this study the data used to analyze the dynamic behavior are physical measurable variables : battery current (I), the battery cell voltage (Vcell) and the battery module temperature (T ). In addition, some characterizations have been performed on one battery pack at the beginning of the Mobypost project to measure OCV and to estimate battery capacity with constant charge/discharge profiles and internal resistance with HPPC profiles. Figure 2 draws an example of data recorded during a driving cycle with the vehicle in pure electric mode (fuel cell stopped). Negative current values refer to currents provided to the motors, and positive current values refer to energy recovery during breaking.

The Table 1 highlights the features of the fleet after three years of experimentation exposing: determined cycle number that the batteries suffered [START_REF] Vichard | Battery aging study using field use data[END_REF], number of post delivery that the vehicle realized, distance traveled by the vehicles and the total electric charge supplied by the batteries.

It is worth noting that battery packs of vehicles were over-sized and that all the vehicles were recharged after each postal delivery. In this context, no driving cycle reached a battery state of charge value below 50%. 3. ECM and parameters identification

ECM structure

Electrical circuit model are commonly used to reproduce the temporal battery voltage response. Several structures of equivalent circuits can be found in the litterature but a very common is highlighted in Figure 3 [40, 45, 46]. By integrating R//C circuits, this simple structure allows reproducing the dynamic behavior and the voltage response of a battery which is composed of transient periods of different durations. Therefore, the R//C circuits time constants (τ) will be different so that the model is able to reproduce different time order responses. R//C circuits are used to represent impedances linked to the charge transfer effects, double layer effects, diffusion and relaxation phenomena. The number of R//C integrated in the model refers to the desired accuracy and computing time limitations. Models can be found considering only one R//C circuit as in [START_REF] Guha | State of Health Estimation of Lithium-Ion Batteries Using Capacity Fade and Internal Resistance Growth Models[END_REF][START_REF] Li | Co-estimation of capacity and state-of-charge for lithium-ion batteries in electric vehicles[END_REF][START_REF] Yang | Battery states online estimation based on exponential decay particle swarm optimization and proportional-integral observer with a hybrid battery model[END_REF][START_REF] Dubarry | Development of a universal modeling tool for rechargeable lithium batteries[END_REF], or two R//C circuits [START_REF] Bahramipanah | Enhanced Equivalent Electrical Circuit Model of Lithium-Based Batteries Accounting for Charge Redistribution, State-of-Health, and Temperature Effects[END_REF][START_REF] Shen | The co-estimation of state of charge, state of health, and state of function for lithium-ion batteries in electric vehicles[END_REF][START_REF] Hu | A technique for dynamic battery model identification in automotive applications using linear parameter varying structures[END_REF][START_REF] Hu | Lithium-ion battery modeling and parameter identification based on fractional theory[END_REF], or with three R//C circuits [START_REF] Cao | Multi-timescale parametric electrical battery model for use in dynamic electric vehicle simulations[END_REF][START_REF] Kroeze | Electrical battery model for use in dynamic electric vehicle simulations[END_REF][START_REF] Hu | Electro-thermal battery model identification for automotive applications[END_REF]. In the proposed study the equivalent circuit integrates three R//C circuits in order to approach a good compromise between accuracy and computational requirements. Theoretically an infinite number of R//C circuits would be necessary to represent the complete voltage response but practically increasing the number of R//C circuits will proportionally increase the number of parameters of the model and consequently the computational requirements. In the proposed structure, three R//C circuits are used to represent the double layer effects, charge transfer phenomena, diffusion phenomena and relaxation phases according to the following distinction.

The model of Figure 3 contains parameters corresponding to resistances and capacitors:

• A resistance R 0 representing the ohmic losses related to the physical nature of the electrodes and the electrolyte.

• A first R//C circuit linked to the effects of the double layer capacity and charge transfer phenomena. The time constant will be the smallest of the model.

• Two R//C circuits linked to the diffusion phenomenon in the electrolyte and the relaxation phases. Time constants of these circuits will be higher.

Parameter identification

A large number of methods has been presented during the last decades to identify battery model parameters. The recursive least square and the Kalman filter are the most common because of their low complexity. Zhou et al. [START_REF] Zhou | Online estimation of state of charge of Li-ion battery using an iterated extended Kalman particle filter[END_REF] proposed a method based on recursive least square algorithm to identify parameters of a one R//C circuit ECM. Results show satisfying accuracy but the method is tested with a simplified HPPC profile and variations of ambient conditions are not considered. Ren et al. [START_REF] Ren | Design and implementation of a battery management system with active charge balance based on the SOC and SOH online estimation[END_REF] used a Kalman filter to estimate the parameters of an ECM and to estimate the SOC. Based on the model and the SOC estimation the battery capacity can be quantified. Temperature variations are not considered and simulation and validation have been performed on simplified HPPC profiles which may not fully reflect actual operating conditions.

Zhang et al. [START_REF] Zhang | A novel approach of battery pack state of health estimation using artificial intelligence optimization algorithm[END_REF] assured that the accuracy of such adaptative methods is generally not satisfying. The authors preferred to use optimization algorithm in order to identify ECM parameters. In their study, they proposed an ECM based method to reproduce the dynamic behavior of a battery pack and estimate its capacity. Firstly the optimization algorithm was applied to identify ECM parameters, then a particle filter was used to estimate the SOC and finally a recursive least square algorithm was performed to quantify the battery capacity. Results showed satisfying accuracy. However, the method was tested on lab with simplified HPPC profile and temperature variations were not considered.

Considering the previous study [START_REF] Zhang | A novel approach of battery pack state of health estimation using artificial intelligence optimization algorithm[END_REF], the use of an optimization algorithms have been also preferred in our study as the feature of more effective in parameters identification. In a different approach, our contribution is here to build a reliable battery model able to reproduce the dynamic behavior during actual driving cycle with good accuracy by using data directly issued from a vehicle monitoring. Moreover, the model will be reliable regardless of varying operating conditions.

Model equations

The first step is to define the model equations and the mathematical relationships linking ECM parameters to operating conditions. From Figure 3, the equation governing the battery cell output voltage during discharge are given by:

V cell estimated = OCV -U 0 -U 1 -... -U i (1)
OCV being the Open Circuit Voltage.

dU i dt = 1 C i .I - 1 R i .C i .U i (2)
Solving the first order differential equation ( 2) leads to the following expression (3). It is worth noting that the acquisition frequency is 1Hz, so t is discretized with a sampling period of 1s to solve the equation.

U i (n) = U i (n -1).e -(t(n)-t(n-1)) R i .C i + R i .(1 -e -(t(n)-t(n-1)) R i .C i ).I(n) (3)
R i and C i being the resistive and capacitive parameters of the electrical equivalent circuit given in Figure 3.

In order to integrate thermal dependency to the model, resistive parameters will be dependent on the temperature as in (4). This equation form was prefered because of the temperature dependence following an Arrhenius law [START_REF] Liaw | Correlation of Arrhenius behaviors in power and capacity fades with cell impedance and heat generation in cylindrical lithium-ion cells[END_REF] and was demonstrated in [START_REF] Jaguemont | Characterization and modeling of a hybrid-electric-vehicle lithium-ion battery pack at low temperatures[END_REF], [START_REF] Kim | Parameter Estimation for a Lithium-Ion Battery from Chassis Dynamometer Tests[END_REF] or in [START_REF] Giordano | Model-based lithium-ion battery resistance estimation from electric vehicle operating data[END_REF] based on experiments.

R i (n) = a i .exp(b i .T (n)) + c i (4)
where a i , b i and c i are constants to be identified. T is the temperature in Celsius degree.

As presented in [START_REF] Wijewardana | Dynamic battery cell model and state of charge estimation[END_REF], battery internal resistance is also dependent on the SOC value. The relationship is exponential, the lower the stateof-charge, the higher the resistance. However, for a state of charge higher than 0.3, the internal resistance remains constant. In this study, driving profiles never reach SOC values lower than 0.3, consequently this relationship has not been considered in the proposed model.

Concerning the OCV, it is known that the latter is a measure of the battery electromotive force which have a monotonic relationship with the battery SOC [START_REF] Jaguemont | Characterization and modeling of a hybrid-electric-vehicle lithium-ion battery pack at low temperatures[END_REF][START_REF] Pattipati | Open circuit voltage characterization of lithium-ion batteries[END_REF]. Therefore, accurate knowledge of this nonlinear relationship is required for adaptive SOC estimation throughout the use of the battery. Equally, relationship between the OCV and the temperature of the battery cells has been proven [START_REF] Xing | State of charge estimation of lithium-ion batteries using the open-circuit voltage at various ambient temperatures[END_REF][START_REF] Damay | Contribution à la modélisation thermique de packs batteries LiFePO4 pour véhicules décarbonés[END_REF]. The consequence is a variation of the usable battery capacity [START_REF] Jaguemont | Lithium-Ion Battery Aging Experiments at Subzero Temperatures and Model Development for Capacity Fade Estimation[END_REF]. Therefore, to better reproduce the battery dynamic behavior, both relationships have to be considered in the proposed model.

For this purpose, OCV characterizations have been performed in order to translate mathematically its evolution with respect to the temperature and the SOC. It is worth noting that, as highlighted on the Figure 2, when the vehicle is stopped (speed = 0) ancillaries require a constant current of 2A. To fit with this application, the choice was made to characterized pseudo OCV by measuring discharge curves at very low current (C/22). Figure 4 highlights three pseudo OCV evolutions measured at three different temperatures. As shown in Figure 4, a decomposition of each part of the curves allowed to identify mathematical expressions able to reproduce the path of the curves. For example, it can be noticed that the OCV value decreases exponentially when the temperature decreases. Mathematically, this phenomenon is governed by the expression α.exp(β.T (t)). The deduced empiric mathematical function ( 5) can be used to reproduce OCV evolution according to the temperature and SOC. Parameters a, b, c, d, e, f, g, h and i are constants to be identified. Based on the measured operating data, the parameter identification allows to adjust the OCV evolution path specific to each LiFePO 4 cell. Consequently, OCV of a LiFePO 4 cell can be estimated online without further characterizations.

OCV(t) = a + b.(20 -T (t)) S OC(t) + c sqrt(S OC(t)) + d.S OC(t)
+ e.ln(S OC(t)) + f.ln(1.001 -S OC(t))

+ g.ln(1.01 -S OC(t)) + h.exp(i.T (t)) (5)
T is the temperature in Celsius degree.

State of charge reference

As the battery SOC is an input of the proposed model, a reference value is required to identify ECM parameters. Coulomb counting is a common method to estimate the SOC easily [START_REF] Berecibar | Critical review of state of health estimation methods of Liion batteries for real applications[END_REF]. The SOC could be computed with equation [START_REF] Ghossein | Degradation behavior of Lithium-Ion Capacitors during calendar aging[END_REF].

S OC(t) = S OC(0) - 1 Q batt × 3600 t 0 I.dt (6)
where Q batt represents the battery capacity in Ah. However, this method can be inaccurate as the battery capacity varies with respect to the temperature, as exposed in Figure 4, the aging and the current [START_REF] Xiong | Towards a smarter battery management system A critical review on battery state of health monitoring methods[END_REF]. To determine a good SOC reference, we need a relationship between the capacity and the temperature. This would enable to adjust the battery capacity in [START_REF] Ghossein | Degradation behavior of Lithium-Ion Capacitors during calendar aging[END_REF]. Based on characterizations which aimed at measuring the battery capacity under three different temperatures (2°C, 20°C, 30°C), the empirical relationship (7) was determined with a polynomial regression.

Q batt = -0.025 × T 2 + 1.6 × T + 86.76 ( 7
)
where T is the temperature in Celsius degree. Figure 5 illustrates this empiric relationship. The nominal capacity is 110Ah at 23°C for a discharge current of C/5.

Optimization algorithm

Optimization methods can be summarized as identifying a set of input parameters (x n ) minimizing a mathematical function f(x n ). The nature is a great inspiration for proposing optimization methods such as Genetic Algorithms, Ant Colony System, Simulated Annealing, Particle Swarm Optimization or again big-bang big-crunch [START_REF] Koza | Genetic Programming[END_REF][START_REF] Kakas | Ant Colony Optimization[END_REF][START_REF] Kirkpatrick | Optimization by simulated annealing[END_REF]. Optimization algorithms have been already used to identify parameters of a battery model [START_REF] Zhang | A novel approach of battery pack state of health estimation using artificial intelligence optimization algorithm[END_REF][START_REF] Yang | A novel method on estimating the degradation and state of charge of lithium-ion batteries used for electrical vehicles[END_REF]. The purpose is to identify the set of input parameters model minimizing the objective function. In this case the objective function to minimize is the Normalized Root Mean Square Error (NRMSE) [START_REF] Birkl | Degradation diagnostics for lithium ion cells[END_REF].

NRMS E = RMS E max(Vcell real ) -min(Vcell real ) ( 8
)
where RMSE is the Root Mean Square Error, max(Vcell real is the real cell voltage maximum value, min(Vcell real is the real cell voltage minimum value.

In this work the Big-Bang Big-Crunch (BB-BC) algorithm has been prefered as it presents good results and performances in [START_REF] Morando | Reservoir Computing optimisation for PEM fuel cell fault diagnostic[END_REF]. It is worth noting that in this study the purpose is not to demonstrate which optimization algorithm is the best. The proposed BB-BC algorithm refers to one of the theories of the universe evolution [START_REF] Morando | Reservoir Computing optimisation for PEM fuel cell fault diagnostic[END_REF][START_REF] Erol | A new optimization method: Big BangBig Crunch[END_REF]. This theory, explains that the universe expansion phase due to the big bang will end and will be replaced by a universe contraction phase named big crunch.

The BB-BC algorithm is constituted by the following steps:

• Step 1: create an initial population of N pop candidates randomly, respecting the search space limitations.

• Step 2: evaluate performance of each candidate by applying the objective function.

• Step 3: identifying the best candidate (named center of mass). Candidate leading to the lowest error.

• Step 4: create a new population around the center of mass and reduce the search space. However an exploration probability is preserved to get out of a possible local minimum: some candidates are created in the entire search space looking for a global optimum.

• Step 5: return to step 2 while stopping criteria is not met.

The reduction of the search space is represented by a function inversely proportional to the number of generation. The equation ( 9) illustrates the research space division (RS div ) according to the number of generations (gen):

RS div (gen) = 1 gen (9)
This function radically reduces the search space for the first generations which can lead to progress to a local minimum instead of the desired global minimum. In order to tackle this issue, another way to reduce the search space, proposed in [START_REF] Morando | Reservoir Computing optimisation for PEM fuel cell fault diagnostic[END_REF], has been used in this study. The proposed method divides more slowly the search space during first generations and then accelerates to find the results more efficiently. For this purpose, an exponential function replaces the linear function in the proposed version of the algorithm.

Training data

To improve robustness of the model and avoid extrapolations, identifying model parameters according to the entire operating range is of particular importance. For that purpose, four driving cycles (cf. Figure 7) have been chosen as they have been realized at four different temperatures as detailed below. The entire operating temperature range of the vehicle is therefore considered (from 1°C to 25°C).

• Cycle 1 : average temperature 3°C

• Cycle 2 : average temperature 7°C

• Cycle 3 : average temperature 18°C

• Cycle 4 : average temperature 24°C

Modeling results

As a result of the parameters identification, the Figure 6 draws a comparison between the estimated and the measured voltage during the driving cycle 3 used for model building. It is worth remembering that this voltage is the result of a parallel association of single LFP cells which reach a capacity of 110Ah. Results highlight the good accuracy of the model. The obtained error is NRMS E = 0.016. The model is able to provide an accurate cell voltage estimation with respect to the temperature and the SOC. Tables 2 and3 present the values of the identified model parameters. Three different time constants are obtained: 2 seconds to reproduce the end of charge transfer and double layer effects and the beginning of diffusion phenomena, 40 seconds and 400 seconds to reproduce both diffusion phenomena and relaxation phases. The efficiency could be improved by adding others R//C circuits with higher time constants up to several hours in order to reproduce relaxation phenomena at best [START_REF] Li | Fast characterization method for modeling battery relaxation voltage[END_REF]. Figure 8 illustrates the evolution of the internal resistance Rin with respect to the temperature. In this study Rin is defined as the sum of the four resistive parameters. Figure 9 compares OCV estimation based on operating data and OCV measures from characterization.

Model parameter

Value Battery temperature (°C) Then, the model must be validated by verifying its accuracy for an input profile (current and temperature) different from one used for parameters identification. Thereby, Figure 10 compares both estimated and real voltage responses for a driving cycle selected randomly from the database. The battery temperature range is from 13 • C to 17 • C. The error is NRMSE = 0.02 which proves the validity of the proposed model. In this section, a method has been proposed to model a battery with respect to temperature and SOC dependencies. This method allows to estimate the cell voltage response with a good accuracy whatever the operating conditions.

τ 1 2 s τ 2 40 s τ 3 409 s C 1 1266F C 2 47772F C 3 454800F a 3.566 V b -1.013 × 10 -4 V • C -1 c -0.0700 V d 0.0735 V e -0.0191 V f -0.0744 V g 0.0657 V h -0.023 V i -0.00201 • C -1

On board estimation of SOH indicators

The previous parameter identification could be applied during the vehicle lifetime to follow model parameters evolution according to the aging. In particular, the internal resistance Rin evolution can be dissociated from the temperature influence. Consequently, SOH referring to power fade, could be quantified on-board all along the vehicle operation. However, the proposed modeling method needs to know the value of the SOC (cf. equation ( 5)) and SOC is directly depending on battery capacity. Therefore monitoring this capacity is a key for on board estimation of SOH indicators (internal resistance and capacity).

As mentioned before, the capacity varies with respect to the temperature, the current and the aging. The coulomb counting is a simple method but not sufficient for a good estimation in embedded applications. This study proposes a method to estimate simultaneously the battery SOC and the battery capacity based on the previous presented model.

The proposed approach to estimate the usable battery capacity is based on SOC variations (∆S OC) and electric charge variations ∆Q (in Ah). The battery capacity is defined by the expression [START_REF] Berecibar | Critical review of state of health estimation methods of Liion batteries for real applications[END_REF].

Qbatt estimated = ∆Q ∆S OC (10)
In this case, as ∆S OC is at the denominator of the expression (10), a small ∆S OC will easily lead to oscillations. This is why it is advisable to consider sufficiently large SOC variations to minimize estimation errors. In the proposed method, ∆S OC is defined as the difference between the SOC at time k and the initial SOC, so the estimation will be more accurate through the time steps.

The proposed approach is governed by the following algorithm 1. All along a driving cycle, measures are acquired at a sampling period ∆t. At each time step, the sum of supplied current is computed and stored in variable int. With this sum, the quantity of electric charges ∆Q (in Ah) delivered by the battery can be determined. Finally, estimated capacity is determined by the ratio between ∆Q and ∆S OC.

Algorithm 1 Capacity estimation

while k < end do int = int + I(k) ∆Q(k) = int.∆t/3600 ∆S OC(k) = S OC(0) -S OC(k) Qbatt estimated (k) = ∆Q(k) ∆S OC(k) k = k + 1 end while
Then, the first step is to propose an SOC estimation. Several methods have been exposed in the literature to estimate battery SOC such as artificial neural networks [START_REF] Chen | State-of-charge estimation of lithium-ion battery using an improved neural network model and extended Kalman filter[END_REF][START_REF] Yang | State-of-charge estimation of lithiumion batteries based on gated recurrent neural network[END_REF], Kalman filters, particle filter, recursive least square or again fuzzy logic. Considering the literature, Kalman filtering is certainly the most common method to estimate SOC due to its good accuracy, low computational requirement and easy implementation [START_REF] Li | Co-estimation of capacity and state-of-charge for lithium-ion batteries in electric vehicles[END_REF][START_REF] Plett | Battery Management System Volume II Equivalent-circuit Methods[END_REF][START_REF] Zhu | A state of charge estimation method for lithium-ion batteries based on fractional order adaptive extended kalman filter[END_REF][START_REF] Guo | A multi-scale parameter adaptive method for state of charge and parameter estimation of lithium-ion batteries using dual Kalman filters[END_REF]. For theses reasons a Kalman filter based method has been preferred in this study.

Extended Kalman Filter equations

Kalman Filters (KF) can be used to estimate a system state which cannot be directly measured, in this case the battery state of charge. The Extended Kalman Filter (EKF) is a variant of the basic Kalman Filter which is usable only if the system is linear. EKF is one of the commonly used methods for battery parameter and state estimation [START_REF] Li | Co-estimation of capacity and state-of-charge for lithium-ion batteries in electric vehicles[END_REF][START_REF] Guo | A multi-scale parameter adaptive method for state of charge and parameter estimation of lithium-ion batteries using dual Kalman filters[END_REF]. This filter is a recursive algorithm that combines a model which is used to estimate a measurable value and a measurement data set to seek for an optimal estimation of the internal state of the system. Once a new measurement is available, the prediction error is used to correct the state prediction. A good detailed introduction on the EKF and its applications for SOC estimation has been provided in [START_REF] Plett | Battery Management System Volume II Equivalent-circuit Methods[END_REF]. EKF relies on two functions : a state equation [START_REF] Xiong | Towards a smarter battery management system A critical review on battery state of health monitoring methods[END_REF] and a measurement equation [START_REF] Zheng | Incremental capacity analysis and differential voltage analysis based state of charge and capacity estimation for lithium-ion batteries[END_REF]. These two equations are governed by the nonlinear vector functions f and h, where x k is the state vector at time index k, u k is the system input, w k is the process noise, y k is the output of the system and v k is the measure noise that affects the measurement of the output.

x k = f (x k-1 , u k-1 ) + w k (11)
y k = h(x k , u k ) + v k (12)
The function f can be used to compute the predicted state from the previous estimate and the function h can be used to compute the predicted output from the predicted state. As KF cannot work in a nonlinear model, functions f and h cannot be applied directly so a linearized process method is required for EKF. State equation [START_REF] Xiong | Towards a smarter battery management system A critical review on battery state of health monitoring methods[END_REF] and measurement equation [START_REF] Zheng | Incremental capacity analysis and differential voltage analysis based state of charge and capacity estimation for lithium-ion batteries[END_REF] are linearized iteratively by Taylor series expansion. The Jacobian matrices of the first order Taylor expansion are computed as in [START_REF] Guha | State of Health Estimation of Lithium-Ion Batteries Using Capacity Fade and Internal Resistance Growth Models[END_REF] and [START_REF] Ng | Enhanced coulomb counting method for estimating state-of-charge and state-of-health of lithiumion batteries[END_REF]. F k and H k are the first partial derivative matrices of functions f and h with respect to x k .

F = ∂ f (x k , u k ) ∂x k (13)
H = ∂h(x k , u k ) ∂x k (14)
The filter consists of two phases : one for predicting the measured value and one for correcting. The prediction phase is governed by the two following equations ( 15) and ( 16) :

Prediction of the state variable at time k :

x - k = f ( x k-1 , u k-1 ) (15)
Prediction of covariance error :

P - k = F k .P k-1 .F T k + Q k ( 16
)
And the measurements update equations are governed by ( 17)-( 20): Update of innovation covariance :

S k = H k .P - k .H T k + R k (17)
Update of Kalmain gain :

K k = P - k .H T k .S -1 k ( 18
)
Optimal estimation of system state at time k :

x k = x -+ K k .(y k -y k ) (19)
Update of system state error covariance matrix :

P k = (I -K k .H k ).P - k (20)
Q k and R k are the covariances of the process and measurement noises. P k is the a posteriori covariance matrix which is a measure of the estimated accuracy of the state estimation.

To apply EKF, the first step is to transform the SOC expression [START_REF] Ghossein | Degradation behavior of Lithium-Ion Capacitors during calendar aging[END_REF] to a discrete form [START_REF] Wang | A review of DC/DC converter-based electrochemical impedance spectroscopy for fuel cell electric vehicles[END_REF]:

S OC(k) = S OC(k -1) -I(k).
∆t Qbatt estimated × 3600 [START_REF] Wang | A review of DC/DC converter-based electrochemical impedance spectroscopy for fuel cell electric vehicles[END_REF] With Qbatt estimated the usable battery capacity estimation obtained through the Algorithm 1.

The Kalman filter state equation f (x k , u k ) [START_REF] Xiong | Towards a smarter battery management system A critical review on battery state of health monitoring methods[END_REF] and measurement equation h(x k , u k ) ( 12) can now be expressed respectively as in [START_REF] Wang | Online electrochemical impedance spectroscopy detection integrated with step-up converter for fuel cell electric vehicle[END_REF] and [START_REF] Dubarry | Identify capacity fading mechanism in a commercial LiFePO 4 cell[END_REF].

    S OC k+1 U 1 k+1 U 2 k+1 U 3 k+1     =         S OC k -I(k). ∆ t Qbatt U 1 k .e -∆t R 1 .C 1 + R 1 .(1 -e -∆t R 1 .C 1 ).I k U 2 k .e -∆t R 2 .C 2 + R 2 .(1 -e -∆t R 2 .C 2 ).I k U 3 k .e -∆t R 1 .C 1 + R 3 .(1 -e -∆t R 3 .C 3 ).I k         (22) h(x k , u k ) = OCV k -U 1 k -U 2 k -U 3 k -R 0 k .I k (23)
with the battery current defined as the input signal u k = I k . Jacobian matrices F k and H k can be expressed respectively as in ( 24) and ( 25) :

F k = ∂ f (x k , u k ) ∂x k =       1 0 0 0 0 e -∆t R 1 .C 1 0 0 0 0 e -∆t R 2 .C 2 0 0 0 0 e -∆t R 3 .C 3       (24)
H k = ∂h(x k , u k ) ∂x k =       ∂OCV ∂S OC -1 -1 -1       T (25)

EKF estimation of SOC and capacity

Figures 11, 12 and 13 highlight SOC estimation with the proposed EKF. The figures compare the EKF estimation and a basic coulomb counting during three driving cycles realized at three different temperatures : 20°C, 5°C and 25°C. These cycles have been selected in the database in order to represent the full temperature range in which the driving cycles were realized. The Figure 11 highlights two SOC estimations compared to a coulomb counting for two driving cycles realized at 20°C. It can be seen that both estimations achieve same SOC final value than the coulomb counting regardless of the final state of charge value. For a temperature of 5°C (cf. Figure 12), the estimation differs from the coulomb counting as the usable battery capacity changed due to temperature variation. In the third case of 25°C (cf. Figure 13), the difference is less visible. Kalman filter estimation remains close to the coulomb counting as at higher temperatures the variation of usable capacity is less significant (cf. Figure 5). This shows that the proposed estimation is able to consider battery capacity variations. To verify the proposed method, SOC estimations in Figures 12 and13 have been compared with coulomb counting reference performed with the characterized battery capacity at both temperatures. This way, it can be seen that estimations from the Kalman filter consider battery capacity decreasing due to the low temperature and small capacity increasing due to higher temperature which is faithful to characterizations. Results above show that the Kalman filter is able to concurrently estimate the state of charge and the battery usable capacity. The estimation of the battery capacity is an important challenge as it is directly linked to the vehicle autonomy and is not constant during the battery lifetime. That is why the method for SOC estimation needs to be robust regarding the knowledge of the real value of capacity. This is thus mandatory to verify that SOC estimation remains reliable whatever the knowledge of the battery capacity. For that purpose the previous presented method has been performed with two different capacity values initialized in the algorithm. The Figure 14 highlights the proposed SOC estimation at 20°C and compare it with a basic coulomb counting initialized with two different initial capacity value. As expected, the coulomb counting estimations differ as the capacity values in the SOC expression (cf. equation ( 6)) are different. In contrary, the value of the initial capacity Qbatt in the Kalman filter state equation [START_REF] Wang | A review of DC/DC converter-based electrochemical impedance spectroscopy for fuel cell electric vehicles[END_REF] does not affect the SOC estimation. Indeed, even if the initial capacity value is different (210 Ah instead of 110 Ah), the SOC estimation converges to the same value. The proposed method corrects the battery capacity value and does not require to further measure the real value to provide a reliable SOC estimation. The proposed approach can be applied on-board all along the vehicle lifetime coupled with ECM parameters identifications to estimate the battery state of charge and capacity, which directly refers to SOH indicator, depending on the temperature and the battery aging. Contrary to the initial estimation (cf. section 3.4) this EKF estimation is based on battery model. Therefore, if model parameters vary with the aging, the battery SOC and usable capacity can be concurrently estimated based on these new parameters. In this way aging will be considered.

Conclusion

Based on vehicle operating only, this paper proposed a method to identify state of health indicators of a LiFePO 4 battery cell: available capacity and internal resistance referring to the available power. The method is based on an electrical circuit model to reproduce and simulate battery dynamic behavior according to operating conditions. The proposed model is able to reproduce the battery dynamic behavior: cell voltage with respect to thermal and state of charge dependencies, with a very promising accuracy. Model parameters can be used to estimate the internal resistance which is a first state of health indicator. Based on a Kalman filter, the model can then be used to estimate concurrently the state of charge and the available capacity, which is a second state of health indicator, considering the temperature variations and the aging. The proposed method could be integrated on board and applied during the entire vehicle lifetime to provide a reliable state of charge estimation and to follow state of health in order to contribute to a better understanding of the degradation laws and to develop better energy management strategies.

Otherwise, future aging data will be generated to extend the database in order to validate the proposed method, to analyze battery state of health evolution and to study effects of actual operating conditions on the battery degradation.

Acknowledgement

This research work was funded by the region of Bourgogne Franche-Comté, France.

Figure 1 :

 1 Figure 1: Three vehicles of the vehicle fleet

Figure 2 :

 2 Figure 2: Data measured during a postal delivering

Figure 4 :

 4 Figure 4: OCV mathematical function decomposition

Figure 5 :

 5 Figure 5: Characterized battery capacity versus temperature

Figure 8 :

 8 Figure 8: Internal resistance evolution versus temperature

Figure 9 :

 9 Figure 9: Comparison between estimated and measured OCV at three different temperatures

Figure 7 :Figure 10 :

 710 Figure 7: Driving cycles used for parameter identification

Figure 12 :

 12 Figure 12: State of charge estimation for battery temperature at 5°C

Figure 13 :

 13 Figure 13: State of charge estimation for battery temperature at 25°C

  with initial Q batt =210Ah Coulomb counting with initial Q batt =210Ah Kalman filter estimation with initial Q batt =110Ah Coulomb counting with initial Q batt =110Ah Initial value of estimated SOC

Figure 14 :

 14 Figure 14: State of charge estimation with two different initial capacities

Table 1 :

 1 fleet features

					Electric
		Battery	Postal	Distance	charge
	Vehicle	cycle	delivery	traveled	supplied
	number	number	number	(km)	(Ah)
	1	74	160	2600	7913
	2	83	203	2950	8816
	3	69	154	2320	7640
	4	89	243	3200	9600
	5	76	145	1980	8015
	6	39	110	1500	4184
	7	62	149	2140	6579
	8	83	188	3016	9354
	9	60	159	1925	6133
	10	69	171	2176	6860

Table 2 :

 2 Model parameters for T = 20 • C

		3.34
		3.32
		3.3							20
		3.28
	Vcell (V)	3.24 3.26							T(°C) 18
		3.22							16
		3.2					Estimated Vcell
							Real Vcell
		3.18					T OCV		14
		2.12	2.13	2.14	2.15	2.16	2.17	2.18	2.19
					Time (s)			10 4
	Figure 6: Cell voltages comparison with a driving cycle used for parameters
	identification

Table 3 :

 3 Model resistive parameters

	Model
	element	a i (mΩ)	b i ( • C -1 )	c i (mΩ)
	R 0	0.407	-0.067	0.145
	R 1	0.678	-0.058	0.305
	R 2	0.889	-0.04	0.186
	R 3	1.1	-0.14	0.427