Kodjo S R Mawonou
email: kodjo.mawonou@centralesupelec.fr

Akram EDDAHECH b , Didier DUMUR a , Dominique Beauvois

Emmanuel Godoy

Improved state of charge estimation for Li-ion batteries using fractional order extended Kalman filter

Keywords: Li-ion battery, Fractional order model, Electrochemical impedance spectroscopy, EKF, SoC estimation, Recursive identification

An accurate state of charge (SoC) estimation by the battery management system (BMS) is crucial for efficient and nondestructive battery operation in automotive applications. The model identification of these batteries has consistently been the critical point to meet good accuracy. To that extent, a fractional order model (FOM) is derived, which provides a more meaningful insight into the battery physical phenomena without increasing the number of parameters as opposed to electrochemical models. This paper proposes FOM identification for Li-ion batteries in both frequency domain based on recorded impedance spectroscopy (EIS) data and time domain using a recursive least squares (RLS) algorithm. Fractional derivatives are overly sensitive to the value of their fractional order. A straightforward and efficient way to identify the fractional orders based on recorded EIS data is proposed in this paper. Furthermore, an extended Kalman filter (EKF) is also designed based on the derived model to estimate the SoC. The designed fractionasl order filter provides a higher accuracy level in comparison to the classical equivalent electric circuit (EEC).

Various results at several temperatures and driving profiles for both PHEV and EV batteries confirm that the FOM provides better accuracy and robustness compared to the classical integer order model.

Introduction

Electrified vehicles (EV, PHEV and HEV) are promising technologies to overcome CO2 emission issues. These green vehicles rely on a core component which is the battery. In order to make these electrified vehicles widespread in the global market, car users need to be comforted in terms of security and benefit such as autonomy and rechargeability.

Despite of Lithium-ion (Li-ion) cells are the most commonly used, due to their high energy, power densities and their longer lifespan [START_REF] Diouf | The potential of Li-ion batteries in ECOWAS solar home systems[END_REF], recently, deep researches are animated on battery technologies to ameliorate their global cells electrochemical performances compared to today NMC lithium-ion like recent studies on solid-state battery [START_REF] Suriyakumar | Metal organic framework laden poly(ethylene oxide) based composite electrolytes for all-solid-state Li-S and Li-metal polymer batteries[END_REF][START_REF] Takada | Progress in solid electrolytes toward realizing solid-state lithium batteries[END_REF][START_REF] Zhang | Poly(ethylene oxide) reinforced Li6ps5cl composite solid electrolyte for all-solid-state lithium battery: Enhanced electrochemical performance, mechanical property and interfacial stability[END_REF].

The fact that battery packs are made of numerous electrochemical cells makes the control of such complex system mandatory. The battery management system BMS [START_REF] Xiong | Towards a smarter battery management system: A critical review on battery state of health monitoring methods[END_REF] represents a key point to ensure exploring these new emerging technologies with optimized functional and safety aspects. The BMS is dedicated to ensuring not only diagnosis (fault detection from communication for example) and battery safety (prevent overvoltage, undervoltage, overcurrent and overtemperature) but also handle battery state estimators and balancing. This paper precisely considers chemistry algorithms part in the BMS software. Predominantly battery model improvement and SoC estimation are discussed.

Efficient and cautious use of Li-ion batteries typically requires the monitoring of numerous variables like the state of charge (SoC), the state of health (SoH), the acceptable/available power when charging/discharging, the charge capacity and the internal resistance [START_REF] Li | State-of-health estimation for Li-ion batteries by combing the incremental capacity analysis method with grey relational analysis[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][START_REF] Lucu | A critical review on self-adaptive Li-ion battery ageing models[END_REF]. Unfortunately, these variables are not directly measurable. The BMS has to estimate each of them, thanks to measured data such as current, voltages, temperatures and state observers [START_REF] Gregory | Extended Kalman filtering for battery management systems of LiPB-based HEV battery packs: Part 1. Background[END_REF]. Besides, these parameters change over time and charge-discharge cycles. These changes have to be taken into account to ensure a maximal level of performance, throughout the life of the battery [START_REF] Zheng | Investigating the error sources of the online state of charge estimation methods for lithium-ion batteries in electric vehicles[END_REF]. The lifetime of the battery is maximized by facilitating an ideal operating condition [START_REF] Eddahech | Performance comparison of four lithium-ion battery technologies under calendar aging[END_REF][START_REF] Vetter | Ageing mechanisms in lithium-ion batteries[END_REF].

The accuracy of estimations provided by the BMS depends on the chosen battery model. Given that batteries have complex electrochemical mechanisms and highly non-linear behaviour, it is challenging to accurately design easily implementable models. Several models are considered in the literature, such as equivalent electric circuit (EEC), electrochemical model (EM), fractional order model (FOM) and neural network models (NN). EEC models are made of a few resistor-capacitor (RC) networks in series with a voltage source corresponding to the open circuit voltage (OCV) of the battery. These models are the most commonly used and are easy to operate, but they produce an insufficient level of accuracy. A detailed framework of EEC models for Li-ion batteries SoC estimation is given in [START_REF] Meng | Low-complexity online estimation for LiFePO4 battery state of charge in electric vehicles[END_REF][START_REF] Gregory | Extended Kalman filtering for battery management systems of LiPB-based HEV battery packs: Part 1. Background[END_REF][START_REF] Gregory | Extended Kalman filtering for battery management systems of LiPB-based HEV battery packs: Part 2. Modeling and identification[END_REF]. Electrochemical models, however, are based on a set of partial differential equations (PDEs) that describe the electrochemical reactions that take place inside the battery [START_REF] Doyle | Design and Simulation of Lithium Rechargeable Batteries[END_REF]. Several parameters are to be identified involving sometimes destructive tests for the battery. In addition, the uncertainty from parameter identification makes these models hard to operate. These models are frequently utilized for design purposes and are unsuitable for online applications. There were several attempts to simplify these models leading to single particle models (SP) [START_REF] Zhao | Electrochemical-thermal modeling of lithium plating/stripping of Li(Ni0.6mn0.2co0.2)O2/Carbon lithium-ion batteries at subzero ambient temperatures[END_REF][START_REF] Bizeray | Advanced battery management systems using fast electrochemical modelling[END_REF]. Nevertheless to the best of our knowledge, there is no electrochemical operated BMS. Black box approaches such as neural network models [START_REF] Eddahech | Modeling and adaptive control for supercapacitor in automotive applications based on artificial neural networks[END_REF][START_REF] Chemali | State-of-charge estimation of Li-ion batteries using deep neural networks: A machine learning approach[END_REF] or adaptive neuro-fuzzy inference system (ANFIS) [START_REF] Zahid | State of charge estimation for electric vehicle power battery using advanced machine learning algorithm under diversified drive cycles[END_REF] produce satisfying results if enough training data are available. On several occasions this approach lacks robustness and demands high computational resources.

Latterly, it has been reasoned that solving SP models leads to fractional order equations [START_REF] Li | A physics-based fractional order model and state of energy estimation for lithium ion batteries. Part I: Model development and observability analysis[END_REF][START_REF] Sabatier | Lithium-ion batteries modeling: A simple fractional differentiation based model and its associated parameters estimation method[END_REF]. Also, the impedance frequency response of a Li-ion battery displays at low frequencies behaviours that are easily captured by constant phase or Warburg elements (CPE). The use of CPEs leads to fractional order model. Moreover, FOM may improve SoC estimation accuracy while involving fewer parameters than the classical n-RC equivalent circuit models. This observation has led to a growing number of attempts to build FOM for Li-ion battery monitoring [START_REF] Zou | A review of fractional-order techniques applied to lithium-ion batteries, lead-acid batteries, and supercapacitors[END_REF][START_REF] Jiang | Data-based fractional differential models for non-linear dynamic modeling of a lithium-ion battery[END_REF][START_REF] Wang | Fractional-order modeling and parameter identification for lithium-ion batteries[END_REF].

In [START_REF] Jiang | Data-based fractional differential models for non-linear dynamic modeling of a lithium-ion battery[END_REF] a continuous time state-variable filter was used to estimate parameters for a Li-ion battery. The author argued that using instrumental variable helped to improve the accuracy of the output of the least square algorithm used. The fractional order in his study was achieved by fitting the output voltage of his identified model to the measured voltage of the battery. In [START_REF] Hu | Lithium-ion battery modeling and parameter identification based on fractional theory[END_REF] a multi-swarm cooperative particle swarm optimization (MCPSO) is used to identify fractional order parameters for a Li-ion battery based on federal city driving schedule experimental data. In [START_REF] Zou | Nonlinear Fractional-Order Estimator with Guaranteed Robustness and Stability for Lithium-Ion Batteries[END_REF] a FOM was designed based on frequency domain identification using particle swarm optimization (PSO). A hybrid multi-swarm particle swarm optimization (HMPSO) was used in [START_REF] Wang | Fractional-order modeling and parameter identification for lithium-ion batteries[END_REF] to identify a FOM based on training data of eight Li-ion cells.

All these approaches are based on classical optimization used on time domain or frequency domain data. Fractional derivatives are overly sensitive to the value of their fractional order. Therefore, time domain identification is extremely sensitive to the initialization and noise level. The frequency domain approaches in the literature do not take advantage of the structure of the studied model. This paper presents a swift, straightforward and efficient way to identify the fractional orders based on recorded EIS data for Li-ion batteries.

The present paper aims to propose a complete framework for a high fidelity EKF SoC estimator based on identified fractional order model parameters in the frequency domain for Li-ion batteries. The remainder of this paper is organized as follows. The following section describes fractional order derivatives. Section 3 displays FOM models for Li-ion batteries with a presentation of a new frequency domain identification procedure, a time domain identification and their numerical approximation. A fractional order EKF based on the identified model is designed and discussed in Section 4. Ultimately, conclusions are presented in Section 5.

Fractional derivatives

Fractional-order calculus (FOC) is an extension of the classic integer-order derivative. The idea of fractional calculus emerged in the development of integer-order calculus. The first literature reference is associated with a letter, from Leibniz to l'Hospital in 1695 [START_REF] Petráš | An Effective Numerical Method and Its Utilization to Solution of Fractional Models Used in Bioengineering Applications[END_REF]. FOC are broadly used at present for diverse topics such as diffusive systems, electrical systems, control, etc [START_REF] Sabatier | Advances in Fractional Calculus[END_REF]. Several fractional derivatives and integrals exist [START_REF] Oliveira | A Review of Definitions for Fractional Derivatives and Integral[END_REF], but the most commonly used fractional derivatives in the literature are: the Riemann-Liouville (RL) definition, the Caputo (CP) definition, and the Grünwald -Letnikov (GL) definition.

RL derivation of a causal function f of time t at order α ∈ R * + definition is given as [START_REF] Chetoui | Identification de systèmes par modèle non entier à partir de signaux d'entrée sortie bruités[END_REF][START_REF] Petráš | An Effective Numerical Method and Its Utilization to Solution of Fractional Models Used in Bioengineering Applications[END_REF]:

D α RL f (t) = 1 Γ(n -α) d n dt n t 0 f (τ) (t -τ) α-n+1 dτ (1)
where n -1 < α < n, n ∈ N and Γ(.) is Euler's Gamma function defined in Eq. 2.

Γ(n) = t 0 t n-1 e -t dt (2)
CP derivation of a causal function f of time t at order α ∈ R * + definition is given as:

D α CP f (t) = 1 Γ(n -α) t 0 f (n) (τ) (t -τ) α-n+1 dτ (3)
where n -1 < α < n, n ∈ N, Γ(.) is the Gamma function and f (n) is the n-th integer order derivative of the temporal function f . GL derivation of a causal function f of time t at order α ∈ R * + definition is given as:

D α GL f (t) = lim T s →0 1 T α s t/T s j=0 (-1) j C j α f (t -jT s ) (4)
where T s is the sampling time and t/T s the integer part of t/T s , C j α the Newton binomial coefficient generalized to real numbers using Eq. 5.

C j α = Γ(α + 1) Γ( j + 1)Γ(α -j + 1) = α(α -1) . . . (α -n + 1) n! (5)
GL fractional derivative is the most used definition for numerical applications in the literature.

Frequency domain representation of a Fractional order transfer (FOT) function can be written as follows [START_REF] Vinagre | Some approximations of fractional order operators used in control theory and applications[END_REF][START_REF] Victor | Identification par modèle non entier pour la poursuite robuste de trajectoire par platitude[END_REF]:

H z (s) = b M s ν b M + b M-1 s ν b M-1 + . . . + b 0 s ν b 0 a N s ν a N + a N-1 s ν a N-1 + . . . + a 0 s ν a 0 (6)
where s ∈ C is the Laplace variable. One can then derive a fractional-order differential equation of the form [START_REF] Vinagre | Some approximations of fractional order operators used in control theory and applications[END_REF][START_REF] Victor | Identification par modèle non entier pour la poursuite robuste de trajectoire par platitude[END_REF]:

(7) a N D ν a N y(t) + a N-1 D ν a N-1 y(t) + . . . + a 0 D ν a 0 y(t) = b M D ν b M u(t) + b M-1 D ν b M-1 u(t) + . . . + b 0 D ν b 0 u(t)
where D ν denotes the RL, CP or GL fractional derivative depending on initial conditions and their physical meaning, and a 0 = 1 conventionally. A state-space representation of a fractional-order linear time-invariant system is given as follows:

         D α x(t) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) (8)
where x ∈ R n is the state vector, u ∈ R m the input variable, and y ∈ R p is the output vector of the system; A ∈ R n×n ,

B ∈ R n×m , C ∈ R p×n , D ∈ R p×m , and α = [α 1 , α 2 , . . . α n ] T are the fractional orders. If α 1 = α 2 = . . . = α n ≡ α,
the system in Eq. 8 is of commensurate-order, otherwise it is of an incommensurate-order [START_REF] Victor | Identification par modèle non entier pour la poursuite robuste de trajectoire par platitude[END_REF][START_REF] Petráš | An Effective Numerical Method and Its Utilization to Solution of Fractional Models Used in Bioengineering Applications[END_REF][START_REF] Chetoui | Identification de systèmes par modèle non entier à partir de signaux d'entrée sortie bruités[END_REF]. For fractional order system stability, one can refer to theories developed in [START_REF] Matignon | Stability properties for generalized fractional differential systems[END_REF][START_REF] Sabatier | Advances in Fractional Calculus[END_REF] for commensurate-order systems.

Fractional-order model for Li-ion batteries

The state of systems with diffusion at a given time depends on their configuration at previous times. Fractionalorder derivatives take into account this history in its definition as a convolution with a function of which the amplitude decays at earlier times as a power-law of time [START_REF] Petráš | An Effective Numerical Method and Its Utilization to Solution of Fractional Models Used in Bioengineering Applications[END_REF]. Fractional-order transfer functions (FOTs) allow the use of a more compact representation of a system where the classical integer transfer would demand a high order model [START_REF] Victor | Identification par modèle non entier pour la poursuite robuste de trajectoire par platitude[END_REF]. There is an extensive literature of FOM used for Li-ion batteries modeling. Two alternative approaches exist to justify the use of fractional-order elements.

To begin with, one can use the frequency domain representation of the impedance of the battery, based on impedance spectroscopy data, and derive a transfer function of its internal impedance [START_REF] Zhang | Fractional-order modeling and State-of-Charge estimation for ultracapacitors[END_REF][START_REF] Hao Mu | A novel fractional order model based state-of-charge estimation method for lithium-ion battery[END_REF][START_REF] Westerhoff | Analysis of Lithium-Ion Battery Models Based on Electrochemical Impedance Spectroscopy[END_REF]. The use of constant phase elements to model the frequency domain behaviour of Li-ion batteries leads to FOMs, given that CPEs are defined using Eq. 9:

CPE( jω) = 1
Cω α cos απ 2 + j sin απ 2 [START_REF] Gregory | Extended Kalman filtering for battery management systems of LiPB-based HEV battery packs: Part 1. Background[END_REF] where α ∈ [0, 1] is the fractional exponent and the constant-phase value is -απ 2 . In [START_REF] Eddahech | Behavior and state-of-health monitoring of Li-ion batteries using impedance spectroscopy and recurrent neural networks[END_REF] CPEs were used in combination with a recurrent neural network (RNN) to monitor the behaviour and state of health (SoH) of Li-ion batteries. In [START_REF] Wang | State-of-Charge Estimation for Lithium-Ion Batteries Based on a Nonlinear Fractional Model[END_REF] a FOM sliding mode observer for SoC estimation based on frequency domain identification was designed.

Next, one can equally utilize simplified electrochemical models as for instance in [START_REF] Li | A physics-based fractional order model and state of energy estimation for lithium ion batteries. Part I: Model development and observability analysis[END_REF][START_REF] Sabatier | Lithium-ion batteries modeling: A simple fractional differentiation based model and its associated parameters estimation method[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] where single particle models (SPMs) were used to design fractional-order transfer functions.This approach is more accurate than the previous one but requires identification of many parameters that may be destructive for the batteries. In this paper, the first approach was adopted leading to a second order FOM circuit displayed in Fig. 1.1. This is associated with the use of two CPEs to reproduce the frequency domain behaviour of the battery. The first CPE represents the charge transfer phenomenon at the electrodes whereas the second represents lithium diffusion.

R 0 R 1 I(t) V t (t) V z (t) V ocv (t) + - V CP E,1 V CP E,2 R 2 CP E 1 CP E 2
(1.1) 2 nd order equivalent circuit using two CPEs. V t is the measured terminal voltage and I is the measured input current.

-Im(Z) The transfer function corresponding to the displayed circuit in Fig. 1.1 is given by Eq. 10:

Re(Z) 0 Im min R 1 /2 ω α = 1 τ 1 β = 2 π atan( ∂Im(-Z) ∂Re(Z) ) ω → ∞ ω → 0 R 1 Charge transfer (H T ) Diffusion (H D ) (1.2) Frequency domain representation of H z illustration.
H z (s) = R 0 + R 1 1 + τ 1 s α + R 2 1 + τ 2 s β (10)
To identify the parameters of this model, both frequency and time domain identifications was conducted. The frequency domain identification results based on recorded EIS data at various SoCs are used as initial guess for the time domain identification to achieve rapid convergence.

Frequency domain identification

The frequency domain identification is conducted using recorded EIS data. The measurement was carried out in a galvanostatic mode using a Bio-Logic VMP300 with a 20 A booster and EC-Lab R software package. The EIS data were collected within the frequency range of 10 mHz to 10 kHz using 10 frequency points per decade. The amplitude of the applied current was 1 A (RMS), approximatively C/100 for EV cells and of course adapted for PHEV cells as well. The cells were introduced in a climatic chamber and well thermalized. The measured temperature is around

+/-2 • C of the target one.
There are frequency domain FOM identification tools in the literature such as: CRONE toolbox [START_REF] Oustaloup | The CRONE toolbox for Matlab[END_REF], or FOMCON toolbox [START_REF] Tepljakov | FOMCON: Fractional-Order Modeling and Control Toolbox[END_REF] based on Hartley [START_REF] Hartley | Fractional-order system identification based on continuous order-distributions[END_REF], Levy's or Vinagre [START_REF] Valério | Finding a fractional model from frequency and time responses[END_REF] algorithms for commensurate FOT. This paper takes advantage of the structure the studied model to propose a novel approach to identify its parameters without using any of the existing algorithms. One can model the frequency domain response of the transfer function H z using the representation in Fig 1 .2.

The ohmic resistance R 0 is easily retrieved, as one can set its value to the real part of the transfer function H z when its imaginary part is equal to zero. The impedance of the charge transfer part is described using Eq. 11:

H T (s) = R 1 1 + τ 1 s α = R 1 1 + τ 1 ω α cos απ 2 + jτ 1 ω α sin απ 2 (11)
When Im (H T ( jω)) reaches its maximum value, one can compute the charge transfer resistance R 1 and the charge transfer time constant τ 1 using Eq. 12:

τ 1 = 1 ω min α (12a) R 1 = 2Re (H T ( jω)) (12b)
where ω min is the corresponding measurement frequency. Noting the imaginary part for that frequency Im min = -Im(H T ) one can compute its fractional exponent using Eq. 13:

α = 2 π arccos            1 - 4Im 2 min R 2 1 1 + 4Im 2 min R 2 1            (13)
The impedance of the diffusion part is described using Eq. 15. One has to bear in mind that the separation between transfer functions H T and H D holds as long as τ 1 << τ 2 . At this stage, the remaining parameters (β, τ 2 and R 2 ) are identified using the frequential data in Eq. 14 :

ĤD (Ω D ) = H Z,mes (Ω D ) -ĤT (Ω D ) (14)
where Ω D = ω 1 , . . . , ω N D is a set of size N D containing the frequencies in the diffusion zone and H Z,mes are the measured EIS data.

H D (s) = R 2 1 + τ 2 s β = R 2 1 + τ 2 ω β cos βπ 2 + jτ 2 ω β sin βπ 2 ( 15
)
6

By computing the derivative of the imaginary part of H D (s) with respect to its real part, one gets Eq. 16 from which the value of the fractional exponent β can be retrieved using Eq. 17:

∂Im(-H D ) ∂Re(H D ) = (τ 2 2 ω 2β -1) sin βπ 2 (τ 2 2 ω 2β + 1) cos βπ 2 + 2ω β τ 2 (16) lim ω→+∞ ∂Im(-H D ) ∂Re(H D ) = tan βπ 2 =⇒ β = 2 π arctan ∂Im(-H D ) ∂Re(H D ) (17)
The diffusion time constant τ 2 and impedance R 2 are obtained using the weighted linear least squares algorithm 1

Algorithm 1 Recursive weighted linear least squares.

Initialization:

Θ itr = 0 0 Θ = argmin τ 2 ,R 2 N D n=1           A 1 (ω n ) -1 A 2 (ω n ) 0                     τ 2 R 2           +           Re (H D (ω n )) Im (H D (ω n ))           2 while Θ itr -Θ ≤ T ol do Θ itr = Θ Θ = argmin τ 2 ,R 2 N D n=1           A 1 (ω n ) W(ω n ) -1 W(ω n ) A 2 (ω n ) W(ω n ) 0                     τ 2 R 2           +           Re(H D (ω n )) W(ω n ) Im(H D (ω n )) W(ω n )           2
end while [START_REF] Sanathanan | Transfer function synthesis as a ratio of two complex polynomials[END_REF], where coefficients A 1 (ω), A 2 (ω) and W(ω) are computed as follows:

                         A 1 (ω) = Re (H D ) ω β cos πβ 2 -Im (H D ) ω β sin πβ 2 A 2 (ω) = Im (H D ) ω β cos πβ 2 -Re (H D ) ω β sin πβ 2 W(ω) = 1 + τ 2 ω β cos πβ 2 2 + τ 2 ω β sin πβ 2 2 r Θ = τ 2 R 2 (18)
The parameter r ∈ [0 + ∞[ is a relaxation exponent. The increase of the value of r reduces the impact of high frequencies, which is useful given that the diffusion phenomenon occurs at low frequencies.

Using the frequency domain identification method beforehand designed designed on recorded spectroscopy data

FIT (%) = 100 × ω ∈ Ω mes H z,mes ( jω) -Ĥz ( jω) ω ∈ Ω mes H z,mes ( jω) (19)
The corresponding results are displayed in Table 1. At temperatures 0, 25 and 45 However, at -15 • C, the identification result displays a poor FIT. The frequency domain identified parameters are initial guesses for the time domain identification. Therefore, there is no urge to aspire to a 100% FIT at the current stage.

Numerical approximation

The simulation of a temporal domain response of a fractional order system requires all the previous output signal values. This necessity can become an issue as the memory length grows over time. For effective implementation, one can approximate the Grünwald -Letnikov definition in Eq. 4 using Eq. 20 if T s is chosen small enough:

D α x(t) = 1 T α s L m j=0 (-1) j C j α x(t -jT s ) ( 20
)
where L m is the "memory length" to be chosen [START_REF] Petrás | Fractional Derivatives, Fractional Integrals, and Fractional Differential Equations in Matlab[END_REF]. For |x(t)|≤ X max (the maximal amplitude), the memory length L m that satisfies an accuracy ξ for D α x(t) is given as [START_REF] Petrás | Fractional Derivatives, Fractional Integrals, and Fractional Differential Equations in Matlab[END_REF][START_REF] Dorcak | Numerical Models for the Simulation of the Fractional-Order Control Systems[END_REF]:

L m ≥ X max ξ|Γ(1 -α)| 1/α (21)
For EV applications T s is equal to 1 s, leading to T s > τ 1 with respect to the identified values (see Fig. 2.4-c).

For example, Fig. 3 Let's consider the fractional transfer function H(s) to be identified from data sampled at T s > τ 1 :

H z (s) = R ES R + R 2 1 + τ 2 s β (22)
where R ES R = R 0 + R 1 is the equivalent series resistance. The discrete transfer function corresponding to H z (s), associating z-transforms of I/O data can be written as following using GL derivatives and power series expansion:

H z,GL (z) = R ECR + R 2 T β s T β s + τ 2 L m k=0 (-1) k C k β z -k ( 23
)
where z is the Z-transform variable. Fig. 3.2 displays various frequency responses, the experimental one H z,mes , that of the identified continuous model H z (s), that of the approximate discrete model Ĥz,GL for different values of S oC ∈ {100%; 20%; 10%}. As expected, the numerical approximation Ĥz,GL reproduces well the diffusion phenomenon whereas the charge transfer is absent.

Time domain identification

The time domain identification requires the use of fractional differential Eq. 7, which one can rewrite for the studied system as follows:

V z (t) = - N n=1 a n D ν an V z (t) + M m=0 b m D ν bm I(t) (24)
where V z is the voltage of the internal impedance of the battery Z cell (R 0 , CPE 1 , CPE 2 ), I is the input current. Using a similar approach as in [START_REF] Miassa | Identification of a PEMFC fractional order model[END_REF][START_REF] Djouambi | Fractional system identification using recursive algorithms approach[END_REF][START_REF] Aoun | System identification using fractional hammerstein models[END_REF], one can write a linear form of the Eq. 24 as follows:

V z [k] = - N n=0 a n V * n [k] + M m=0 b m I * m [k] (25)
where:

a n = a n        N i=0 a i T ν an -ν a i s        -1 (26a) b m = b m        N i=0 a i T ν bm -ν a i s        -1 (26b) V * n [k] = k i=1 (-1) i C i ν an V z [k -i] (26c)
I * m [k] = k i=0 (-1) i C i ν bm I[k -i] (26d)
One can then notice that N n=0 a n = 1 using Eq. 26. Thus, the number of estimated parameters can be reduced by one, and the linear Eq. 25 becomes:

Vz [k, θr ] = -V * 0 [k] - N n=1 â n (V * n [k] -V * 0 [k]) + M m=0 b m I * m [k] ( 27
)
where θr

= [â 1 â 2 . . . â N b 0 b 1 . . . b M ] is the parameter vector. Noting the estimated output ŷ[k] = Vz [k, θr ] + V * 0 [k]
, parameters θr can be estimated using an instrumental variable recursive least square (IV-RLS) algorithm with a forgetting factor laid down in Eq. 28:

θr [k] = θr [k -1] + L[k] y[k] -θT r [k -1]φ[k] , (28a)
L[k] = P[k -1]φ IV [k] λ + φ T [k]P[k -1]φ IV [k] , (28b)
P[k] = 1 λ (P[k -1] -L[k]φ[k]P[k -1]) (28c)
where 0 << λ ≤ 1 is the forgetting factor, L[k] is the correction gain vector, P[k] the error covariance matrix at time index k, and the measurement vectors φ, φ IV are given in Eq. 29 [START_REF] Jiang | Data-based fractional differential models for non-linear dynamic modeling of a lithium-ion battery[END_REF]:

φ = -(V * 1 -V * 0 ) . . . -(V * N -V * 0 ) I * 0 . . . I * M T (29a) φ IV = -(V * 1 -V * 0 ) IV . . . -(V * N -V * 0 ) IV I * 0 . . . I * M T ( 29b
)
Φ IV is computed at each iteration using the output obtained by simulating the model with estimated parameters from the previous iteration. One can retrieve the initial parameters: a l and b m using Eqs. 30 and 31 [START_REF] Djouambi | Fractional system identification using recursive algorithms approach[END_REF][START_REF] Miassa | Identification of a PEMFC fractional order model[END_REF]:

                                     a 1 T ν a 0 s . . . a N T ν a 0 s T = M × a 1 . . . a N T M =                            (1-a 1 ) T νa 1 s -a 1 T -ν a 2 s . . . -a 1 T -ν a N s -a 1 T -ν a 1 s (1-a 2 ) T νa 2 s . . . -a 1 T -ν a N s . . . . . . . . . . . . -a N T -ν a 1 s -a N T -ν a 2 s . . . (1-a N ) T νa N s                            (30)
b m = b m N n=0 a n T ν bm -ν an s (31)
In the present study case, considering the transfer model in Eq. 22; the set of parameters and equations is given as following:

a = 1 τ 2 ν a = 0 β (32a) b = R 2 + R ES R τ 2 R ES R ν b = 0 β (32b) θ r = a 1 b 0 b 1 (32c) â1 = â 1 (1 -â 1 )T -β s (32d) b0 = b 0 (â 0 + â1 T β s ) b1 = b 1 (â 0 T β s + â1 ) (32e)
To asses the presented method, there are two types of time domain test that were run. First the current pulse excitations of the battery at various SoCs; and secondly, driving cycle current profile applied to the battery. Those tests are detailed in the following subsections.

Current pulse excitation

For this test, an EV Li-ion battery with a charge capacity Q n = 64 Ah is considered. Pulsed currents of 1C amplitude (64 A) for about 2 mins and an imposed relaxation of about 8 minutes were utilized as inputs. Knowing the SoC of this cell, one can easily compute its internal impedance voltage using Eq. 33. The transfer H z (s) between V z (s) and I(s) can be identified using an adaptive IV-RLS algorithm with a forgetting factor λ. To enable rapid convergence of the time domain identification, the initial guess of parameters τ 2 , R 2 and R ES R are derived from the frequency domain identification result using the model in Eq. 22:

V z [k] = V t [k] -OCV(S oC[k]) (33)
Fig. 4.1 displays a) the reference SoC, b) the pulsed current, and the internal impedance voltage obtained using Eq.

33.

Knowing that the SoC is near 20%, the initial guess transfer function was obtained using the frequency domain result at S oC = 20% from section 3.1. A lower bound of the "memory length" can be computed using Eq. 21. For example, if V z,max = 400 mV and β = 0.7; to achieve a 10 mV accuracy : L m must be superior to 40. For simplicity the memory length is chosen as L m = T/T s ; where T is a time constant to be chosen at least superior to the response time of the system. The accuracy of the numerical approach will improve as the value of T increases. Unfortunately, the memory length required for the computation also increases, consequently increasing the computational time. In both of these figures, one can clearly confirm that FOM results are more accurate than first order equivalent circuit model. Also the increase of the value of T improves the accuracy as displayed in Table 2. However, to limit the computational burden, the value of T will not be increased too considerably.

Driving cycle excitation

For this test, recorded driving cycle data were used as inputs of the proposed model. Knowing the S oC the same scheme and Eq. 33 were utilized to retrieve voltage V z of internal impedance of the battery displayed on Fig. 5.1.

The transfer function H z (s) between V z (s) and I(s) can be identified implementing an adaptive RLS algorithm with a As observed for current pulse excitations, when using the FOM, the estimation errors are reduced in comparison to first order EEC model. Also the increase of the value of T improves the accuracy of the identification. The error comparison is displayed in table 3. It is essential to notice that in this case, there is no need to increase significantly the value of L m . This is cause by the fact that the input current does not excite low frequencies very much.

C (T s = 1 s, L m = T/T s ).

Fractional-order-EKF for Li-ion batteries

The aim in this section is to design and test an extended Kalman observer based on FOM for Li-ion batteries.

First, a generalized filter for a second order FOM will be designed. Following that, the filter will be adapted to the presented study case.

Filter design

Differential equation describing the evolution of the voltages of the constant phase elements (CPEs) displayed in Fig. 1.1 is given as:

D α i V CPE,i (t) = - 1 τ i V CPE,i (t) + R i τ i I(t) (34)
where α i ∈ {α, β}. Eq. 34 leads to the discrete time Eq. 35, using numerical approximation of the Grünwald-Letnikov fractional derivative displayed in Eq. 20:

(35) D α i V CPE,i [k + 1] = 1 T α i s {V CPE,i [k + 1] + L m j=1 (-1) j C j α i V CPE,i [k + 1 -j]}
This allows the computation of the CPE voltage at time index k, as linear combination of its previous values and the input current value I[k]:

(36) V CPE,i [k + 1] = α i - T α i s τ i V CPE,i [k] - L m j=2 (-1) j C j α i V CPE,i [k + 1 -j] + R i T α i s τ i I[k]
For each CPE, one can consider the state vector to be:

(37) x i [k] = V CPE,i [k], V CPE,i [k -1], . . . , V CPE,i [k + 1 -L m ] T
The state equation of one CPE is computed as follows:

x i [k + 1] = A i i [k] + B i I[k] (38)
where

                                                   A i =                                    α i -T α i s τ i W 2 W 3 . . . W L m 1 0 0 . . . 0 0 1 0 . . . . . . . . . . . . . . . . . . 0 0 . . . 0 1 0                                    B i = R i T α i s τ i 0 . . . 0 T W j = (-1) j+1 C j α i (39)
Considering the S oC definition in Eq. 40, one writes the global state equation of the system using Eq. 41:

S oC[k + 1] = S oC[k] + I[k]T s Q n × 3600 (40)
x

[k + 1] = Ax[k] + BI[k] (41a) V t [k] = 2 i=1 V CPE,i [k] + OCV(S oC[k]) + R 0 I[k] (41b)
where:

                                   A =                  A 1 0 L m ×L m 0 L m ×1 0 L m ×L m A 2 0 L m ×1 0 1×L m 0 1×L m 1                  B = B 1 B 2 T s /(3600 × Q n ) T x[k] = x 1 [k] x 2 [k] S oC[k] T (42)
A ∈ R (2 * L m +1)×(2 * L m +1) , B ∈ R (2 * L m +1)×1 , B ∈ R 1×(2 * L m +1
) and V t is the terminal voltage of the battery. The function OCV(S oC) is an optimized lookup table provided by the manufacturer of the Li-ion batteries used in this paper. Similar solutions were adopted in the literature. For example, in [START_REF] Li | A physics-based fractional order model and state of energy estimation for lithium ion batteries. Part II: Parameter identification and state of energy estimation for LiFePO 4 battery[END_REF] a fractional Kalman filter was derived based on a simplified electrochemical model. Whereas, in [START_REF] Hao Mu | A novel fractional order model based state-of-charge estimation method for lithium-ion battery[END_REF] an unscented Kalman filter (UKF) was designed for SoC estimation while the parameters were estimated using a genetic algorithm. In [START_REF] Zou | Nonlinear Fractional-Order Estimator with Guaranteed Robustness and Stability for Lithium-Ion Batteries[END_REF] a sliding mode observer was designed based on frequency domain identification using particle swarm optimization (PSO). A multi-swarm cooperative particle swarm optimization (MCPSO) is used to identify fractional order parameters while the SoC was estimated using an EKF in [START_REF] Hu | Lithium-ion battery modeling and parameter identification based on fractional theory[END_REF]. This paper proposes a simple EKF based SoC estimation with a FOM derived from a new frequency domain data identification for Li-ion batteries. The EKF algorithm steps are given below:

Prediction: x-[k] = A x[k -1] + BI[k -1], (43a)
P - x [k] = AP + x [k -1]A T + Q x . (43b)
Update:

L x [k] = P - x [k]C T CP - x [k]C T + R -1 , (43c)
x+ [k] = x-[k] + L x [k] V t [k] -C x-[k] -DI[k] , (43d)
P + x [k] = P + x [k -1] -L x CP - x [k]. (43e)
where P - x and P + x are the predicted and updated state error covariance matrix, Q x and R the state and measurement noises covariance matrix, L x the correction gain matrix; C and D are given in the following equation:

           C = 1 0 . . . 1 0 . . . ∂OCV ∂S oC | S oC[k-1] D = R 0 (44)
Using Eq. 41 jointly with the parameter estimator described in Section 3.3, one can estimate the SoC of the Li-ion cell as displayed Fig. 6.1. It is a dual estimator scheme [START_REF] Gregory | Extended Kalman filtering for battery management systems of LiPB-based HEV battery packs: Part 1. Background[END_REF] with an RLS component that estimates the FOM model parameters using measured currents I[k] I[k -1] and the estimated state of charge Ŝ oC[k -1]. A convergence test is run to retrieve converged parameters R 0,conv , (τ 1 , R 1 ) conv and (τ 2 , R 2 ) conv . These converged parameters are then used by the EKF observer to build matrix A, B, C and D when estimating the state vector:

x[k] = x 1 [k] x 2 [k] S oC[k] T , of size 2 × L m + 1. EKF RLS R 0,conv Convergence Test - + (FOM) V t [k] I[k] I[k -1] V t [k] I[k] I[k -1] SoC[k] SoC[k -1] (τ 1 , R 1 ) conv (τ 2 , R 2 ) conv R 0 [k] τ 1 [k], R 1 [k] τ 2 [k], R 2 [k] (FOM) V CP E,i [k -1] V CP E,i [k -L m ] V CP E,i [k] V CP E,i [k + 1 -L m ] OCV (SoC[k]) (6
         V t,mes = V t + b V I mes = I + b I (45)
         Q x = σ 2 b I × BB T R = σ 2 b I × R 2 ES R + σ 2 b V (46)
where σ 2 b I = 10 -4 A 2 and σ 2 b V = 10 -7 V 2 are the current and voltage measurement noise variances. Those noises are considered to be additive and of zero mean and white. In table 4 several driving profiles were computed, at temperatures 0 • C and 25 • C using FOM and EEC on not only EV but also a PHEV battery. The SoC estimation errors comparison is displayed in Tables 5 and6. For both EV and PHEV profiles one can notice that all RMS and maximum errors are improved when using FOM models. In addition, the maximum error improvement at low temperature is more significant. The averaged improvement of the SoC maximum error for all PHEV profiles at 25 In this paper, the proposed method (FOM) is compared to the existing method (EEC) using single cells. Out of the 12 tests performed (8 PHEV tests and 4 EV tests), the execution time ration between FOM and EEC approach is about 1.68 with a standard deviation of 0.066. EVs and PHEVs frequently feature several (up to N = 100) cells organized in modules (or stacks). Increasing the computational time by 68% can therefore be challenging.

Globally, a compromise should be considered between increasing estimator complexity leading to increased time and resources consumption, and on the other hand, improving battery state accuracy benefits. Electrified vehicles are continuously challenged, that's why more accuracy could be important in extending autonomy range, charging time optimization and even more precise understanding of battery degradation. That is why the suggested approach must be considered in a broader framework. In fact, in [START_REF] Senou | Li-ion battery pack soc estimation for electric vehicles[END_REF] the "Bar-Delta" approach was proposed to reduce the CPU load for battery pack state estimation. It was argued that in the battery pack only the limiting cells (most charged during charge and least charged during discharge) are relevant. This CPU reduction strategy can be combined with the proposed FOM for the limiting cells. In doing so, the overall CPU load is reduced in comparison with the direct approach consisting of N different EEC based EKFs. On top of that, the accuracy is improved for the limiting cells thanks to the FOM approach, as suggested in the present paper.

Conclusion

A method to estimate a second order fractional model parameters for Li-ion batteries using recorded EIS data for an EV and PHEV Li-ion batteries was presented in this paper. The proposed identification approach is original. The studied FOM parameters are simply and efficiently identified by taking advantage of the structure of the proposed model. Frequency domain estimation results were used to initialize the time domain estimation for pulsed and driving cycle current inputs, which enabled fast convergence and good estimation results. The study suggests that FOM is more suitable than the classical integer order systems for Li-ion batteries SoC estimation. To that extent, a joint estimator for FOM parameters and the state of charge S oC was then presented. The parameters are estimated using an IV-RLS while the battery SoC is estimated using a FOM-EKF. The designed fractional-order filter provides a higher accuracy level in comparison to the classical one based on the integer order model, therefore fulfilling the requirement of SoC accuracy improvement. The improved model enables a better behaviour understanding and could be helpful for ageing mechanism investigation. For example, using the identified model parameter, all along the life, ageing law and SoH could be predicted.

Nomenclature

Figure 1 : 2

 12 Figure 1: 2 nd order fractional model representation of a Li-ion battery.

at 25 •

 25 C and 0 o C, a second order FOM parameters were estimated. Figs. 2.1 and 2.2 are display identified transfer functions (dotted lines) versus measured spectroscopy data (plain lines) at S oC ∈ {100%, 20%, 10%} and temperatures 25 • C and 0 • C. The fit is quite satisfying, and this approach provides more insight and understanding of the identified parameters values as the obtained transfer functions are mathematically fitted transfer functions. We have established a global trend of various parameters of the model in Eq. 10 for SoC values ranging from 0% to 100% and temperature T ∈ {-15, 0, 25, 45 • C}. Fig. 2.3 displays a) ohmic resistance R 0 , b) fractional exponent α, c) fractional exponent β. One can notice that the ohmic resistance R 0 values increase as the S oC and the temperature(2.1) Frequency domain identification of H z at low SoCs at 25 • C. Identified transfer functions (dotted lines --H--) versus measured spectroscopy data (plain lines -#-).(2.2) Frequency domain identification of H z at low SoCs at 0 • C. Identified transfer functions (dotted lines--H--) versus measured spectroscopy data (plain lines -#-).(2.3) Frequency domain identification results at different temperatures -15, 0, 25 and 45 • C: a) ohmic resistance R 0 , b) fractional exponent α, c) fractional exponent β.

( 2 . 4 )

 24 Frequency domain identification results at different temperatures -15, 0, 25 and 45 • C: a) charge transfer resistance R 1 , b) diffusion resistance R 2 , c) charge transfer time constant τ 1 , d) Diffusion time constant τ 2 .

Figure 2 :Fig. 2 .

 22 Figure 2: Frequency domain identification results

  .1 displays a) a one hour urban driving cycle input current I(t) and b) its single-sided amplitude spectrum |I( f )|. The relevant excited frequencies are inferior to 2Hz. One can recall that the estimated values of τ 1 displayed in Fig. 2.4 are less than 0.5 s. As a consequence, one can neglect the charge transfer time response.

( 3 . 1 )( 3 . 2 )

 3132 Spectrum study of an urban driving cycle input current sampled at T s = 0.1 s,at temperature 25 • C : a) Imput current I(t), b) Single-sided amplitude spectrum of the current. Numerical approximation of the identified FOM transfer function with T s = 1 s, at 25 • C and L m = 40. Measured spectroscopy data (plain lines -#-), Identified transfers (dotted lines--H--) and approximation using GL (plain lines -I-).

Figure 3 :

 3 Figure 3: GL numerical approximation based on input current spectrum.

Fig. 4 .

 4 2 displays the results for two distinctive values of the time constant T ∈ {120 s, 600 s}. Similarly at 0 • C around S oC = 30% the same test was conducted leading to the result displayed in Fig. 4.3. (S oC = 20%, 25

( 4 . 1 )

 41 Input data for a Q n = 64 Ah cell at 25 • C a) State of Charge b) (left) Applied pulsed current and (right) internal impedance voltage obtained using Eq. 33 (T s = 1 s).

( 4 .

 4 2) a) Estimated Voltage b) Estimation error for a Q n = 64 Ah charge capacity cell, around 20% SoC at 25 • C (T s = 1 s, L m = T/T s ).

( 4 .

 4 3) a) Estimated Voltage b) Estimation error for a Q n = 64 Ah charge capacity cell, around 30% SoC at 0 • C (T s = 1 s, L m = T/T s ).

Figure 4 :

 4 Figure 4: Parameters identification for a pulsed current excitation.

( 5 . 1 )

 51 Input data for a Q n = 64 Ah cell at 25 • C for highway driving cycle a) State of Charge b) (left) Applied pulsed current and (right) internal impedance voltage obtained using Eq. 33 (T s = 1 s).

( 5 .

 5 2) a) Estimated Voltage b) Estimation error for a Q n = 64 Ah charge capacity cell, for SoC ranging from 100% to 87% at 25 • C (T s = 1 s, L m = T/T s ).

Figure 5 :

 5 Figure 5: Parameters identification for a driving cycle excitation.

. 1 )

 1 Dual filter diagram for SoC and parameters estimation.

( 6 . 2 )( 6 . 3 )

 6263 Highway driving cycle input current at 25 • C (Experimental data). Highway driving cycle output voltage at 25 • C. (Experimental data).

Figure 6 :

 6 Figure 6: GL numerical approximation based on input current spectrum.

Fig. 7 .( 7 . 1 )( 7 . 2 )

 77172 Fig. 7.1 displays a) the reference state of charge S oC re f (blue plain line) and the estimated state of charge Ŝ oC (dotted blue line), b) the estimation error S oC (red plain line). Fig. 7.2 displays a) the estimated diffusion time constant τ 2 , b) the estimated diffusion resistance R 2 (right) and the Equivalent series resistance R ES R (left). One can notice the increase of τ 2 and R 2 values as the state of charge decreases.

Figure 7 :

 7 Figure 7: Estimated State of charge and parameters for a highway driving cycle input current at 25 • C.

20

 20

( 8 . 1 )( 8 . 3 )

 8183 Spectrum study of a Highway driving cycle input current sampled at T s = 0.1 s,at temperature 25 • C : a) input current I(t), b) Single-sided amplitude spectrum of the current. (8.2) Spectrum study of a countryside road driving cycle input current sampled at T s = 0.1 s,at temperature 25 • C : a) input current I(t), b) Single-sided amplitude spectrum of the current. Spectrum study of an NEDC driving cycle input current sampled at T s = 0.1 s,at temperature 25 • C : a) input current I(t), b) Single-sided amplitude spectrum of the current.

Figure 8 :

 8 Figure 8: Driving cycle input current used in this paper.

  Abbreviations

Table 1 :

 1 Frequency domain identification FIT values ([%]) at different temperatures.

	S oC	at -15 • C	at 0 • C	at 25 • C	at 45 • C
	100	79.3	93.4	97.8	97.07
	97	78.3	93.28	97.8	96.66
	95	82.4	93.23	97.6	96.76
	92	82.2	93.10	97.7	97.17
	90	82	93.11	97.6	97.16
	80	83.7	92.77	97.7	97.18
	70	83	92.8	97.7	96.81
	60	82.6	93.36	97.2	96.53
	50	82.1	93.16	97.7	97.41
	40	81.3	93.16	97.8	96.86
	30	82.2	91.15	97.5	97.25
	20	78.7	92.14	97.4	96.26
	10	81.8	89.6	97.4	96.70
	7	78.8	90.82	96.4	95.86
	5	80.8	89.9	94.8	93.52

• C, the FIT values are satisfactory.

Table 2 :

 2 Voltage estimation error for pulsed current inputs at 25 • C and 0 • C

	• C)	7.01	3.11	10.02
	Max, at (S oC = 20%, 25 • C)	11	10.98	14.66
	RMS , at (S oC = 30%, 0 • C)	15.82	8.8	22.02
	Max, at (S oC = 30%, 0 • C)	24.51	12.71	30.95

Table 3 :

 3 Voltage estimation error comparison for a highway driving cycle and SoC ranging from 100% to 87% at 25•

	Errors	FOM: L m = 30	FOM: L m = 300	EEC
	RMS (mV)	8.94	7.59	12.49
	Max (mV)	37.99	37.49	39.01

Table 4 :

 4 are displayed SoC estimation errors using FOM and first-order EEC. As expected, the use of the FOM model leads to lower SoC estimation error values, justifying the use of FOM for Li-ion batteries. To go even further, SoC estimation error comparison for a highway driving cycle input current at 25 o C

	Errors	1//RC (EEC Model)	1R//CPE (FOM)
	RMS (%)	0.58	0.41
	Max(%)	1.86	1.18

Table 5 :

 5 • C is about 1% compared to 2.5% at 0 • C. This can be explained by a more pronounced non-linear behaviour of Li-ion batteries at low temperatures. All these improvements justify the use of the FOM for SoC estimation. The several driving cycles used as input for the tests in Tables5 and 6are displayed in Figs. 8.1, 8.2 SoC estimation error comparison for several driving profiles for an EV battery of charge capacity Q n = 64 Ah, L m = 40.

				25 • C					0 • C
	Profiles	FOM		EEC		SoC range	FOM		EEC		SoC range
		Max (%) RMS (%) Max (%) RMS (%)		Max (%) RMS (%) Max (%) RMS (%)
	Urban	1.15	1.12	1.35	1.9	90 -0	0.8	0.43	1.41	0.68	93 -6
	Highway	1.18	0.41	1.86	0.58	94 -3.5	0.94	0.47	1.82	1.15	90 -1
				25 • C					0 • C
	Profiles	FOM		EEC		SoC range	FOM		EEC		SoC range
		Max (%) RMS (%) Max (%) RMS (%)		Max (%) RMS (%) Max (%) RMS (%)
	NEDC	0.34	0.18	1.44	1	94 -6	1.8	1.3	4.54	3	94 -12
	Urban	2.1	1.42	3.61	2.27	100 -40	6.58	3.65	7.27	3.9	92 -9
	Road	1.44	0.71	2.25	1.18	91 -6	1.92	0.84	5.3	3.2	91 -7
	Highway	0.82	0.397	0.82	0.68	91 -40	1.07	0.45	3.5	2.27	92 -4.6

Table 6 :

 6 SoC estimation error comparison for several driving profiles for a PHEV battery, L m = 40.

and 8.3 for the convenience of the reader.


Dr Arnaud Viarouge
email: arnaud.viarouge@parisdescartes.fr

Olivier Houdé

Grégoire Borst

Evidence for the role of inhibition in numerical comparison: A negative priming study in 7-to 8-year-olds and adults

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Research on numerical cognition has identified basic systems of numerical processing [START_REF] Feigenson | Core systems of number[END_REF]. One of them, the approximate number system (ANS) [START_REF] Halberda | Developmental change in the acuity of the 'Number Sense': The Approximate Number System in 3-, 4-, 5-, and 6-year-olds and adults[END_REF], allows us to estimate the number of objects in a set, providing us with a sense of discrete quantity [START_REF] Dehaene | The number sense: How the mind creates mathematics[END_REF]. Many studies have investigated the characteristics of numerical estimation and its development [START_REF] Dehaene | The neural basis of the Weber-Fechner law: a logarithmic mental number line[END_REF][START_REF] Cantlon | Functional Imaging of Numerical Processing in Adults and 4-y-Old Children[END_REF][START_REF] Halberda | Developmental change in the acuity of the 'Number Sense': The Approximate Number System in 3-, 4-, 5-, and 6-year-olds and adults[END_REF]. In general, these studies use the non-symbolic numerical comparison task, whereby participants perform numerical comparative judgments between two rapidly presented dot arrays. In such a comparison task, performance depends on the ratio between the two presented numerosities (i.e., the quantity represented by a set of objects), the ANS following Weber's law [START_REF] Piazza | Tuning curves for approximate numerosity in the human intraparietal sulcus[END_REF]. One difficulty researchers face when using this task resides in the continuous, non-numerical dimensions of magnitude (size, total surface area, density, etc.) that one can perceive in the stimuli. These continuous dimensions can affect numerical judgments throughout development [START_REF] Rousselle | Magnitude comparison in preschoolers: what counts? Influence of perceptual variables[END_REF]Szücs, Nobes, Devine, Gabriel, & Gebuis, 2013;[START_REF] Tokita | Effects of perceptual variables on numerosity comparison in 5-6-year-olds and adults[END_REF] and adulthood (Gebuis & Reynvoet, 2012a[START_REF] Gebuis | The interplay between nonsymbolic number and its continuous visual properties[END_REF][START_REF] Gebuis | Sensory-integration system rather than approximate number system underlies numerosity processing: A critical review[END_REF].

Inhibiting the irrelevant dimensions to focus on number

One of these continuous dimensions is spatial extent. Since Piaget's seminal work on the influence of length on numerical abilities in children [START_REF] Piaget | The child's conception of number[END_REF], behavioral and brain imaging studies have shown evidence of interactions between number and spatial extent that are present very early in infancy [START_REF] De Hevia | Representations of space, time, and number in neonates[END_REF]Pinel, Piazza, Le Bihan, & Dehaene, 2004; see [START_REF] Hubbard | Interactions between number and space in parietal cortex[END_REF], for a review). These interactions have been reported in various studies in adults. For instance, perceiving Arabic digits can bias estimation of sizes [START_REF] De Hevia | The representational space of numerical magnitude: illusions of length[END_REF][START_REF] Viarouge | The role of numerical magnitude and order in the illusory perception of size and brightness[END_REF] and relative conceptual size (e.g., pictures of an elephant vs. a dog) can affect parity judgments [START_REF] Gabay | Size before numbers: Conceptual size primes numerical value[END_REF].

Several studies have investigated the impact of size on symbolic numbers using the size congruity (or numerical Stroop) task [START_REF] Arend | Choosing the larger versus choosing the smaller: Asymmetries in the size congruity effect[END_REF][START_REF] Henik | Is three greater than five: The relation between physical and semantic size in comparison tasks[END_REF][START_REF] Leibovich | The importance of being relevant: modulation of magnitude representations[END_REF][START_REF] Pavese | Further evidence on the effects of symbolic distance on Stroop-like interference[END_REF][START_REF] Schwarz | On the relative speed account of number-size interference in comparative judgments of numerals[END_REF], in which the participants' performances decrease when a numerical comparison is made between two Arabic digits with incongruent sizes (e.g., 6 vs 9).

The number/size congruency effect has also been found in non-symbolic numerical comparison tasks [START_REF] Nys | Judgement of discrete and continuous quantity in adults: number counts![END_REF][START_REF] Hurewitz | Sometimes area counts more than number[END_REF]. For instance, Hurewitz and collaborators (2006) reported lower performances in adults when the sizes of the presented items were incongruent with their number (i.e, the set with larger dots was less numerous), than when they were congruent or neutral. Similar congruency effects have been observed in the context of non-symbolic numerical comparison tasks in preschoolers and school-aged children [START_REF] Defever | Task-and age-dependent effects of visual stimulus properties on children's explicit numerosity judgments[END_REF][START_REF] Fuhs | ANS acuity and mathematics ability in preschoolers from low-income homes: contributions of inhibitory control[END_REF][START_REF] Fuhs | The Role of Non-Numerical Stimulus Features in Approximate Number System Training in Preschoolers from Low-Income Homes[END_REF][START_REF] Gilmore | Congruency effects in dot comparison tasks: convex hull is more important than dot area[END_REF]Szücs, Nobes, Devine, Gabriel, & Gebuis, 2013). While most studies in preschoolers report congruency effects on accuracy rates [START_REF] Fuhs | ANS acuity and mathematics ability in preschoolers from low-income homes: contributions of inhibitory control[END_REF][START_REF] Fuhs | The Role of Non-Numerical Stimulus Features in Approximate Number System Training in Preschoolers from Low-Income Homes[END_REF], studies in early primary school years report both less accurate and slower responses on incongruent trials whereby dot area and/or total surface area of the dots were among the manipulated nonnumerical dimensions [START_REF] Gilmore | Individual differences in inhibitory control, not non-verbal number acuity, correlate with mathematics achievement[END_REF]Szücs et al., 2013).

Several studies have investigated the development of these congruency effects with age, showing that age influences the amplitude of the difference in performance between congruent and incongruent trials in non-symbolic numerical comparison tasks [START_REF] Defever | Task-and age-dependent effects of visual stimulus properties on children's explicit numerosity judgments[END_REF][START_REF] Gebuis | The development of automated access to symbolic and non-symbolic number knowledge in children: an ERP study[END_REF][START_REF] Soltész | Relationships between magnitude representation, counting and memory in 4-to 7-year-old children: A developmental study[END_REF]Szücs et al., 2013). In particular, studies have reported larger congruency effects between school-aged children and adults. For instance Szücs and colleagues (2013) found a larger congruency effect in 7-year-olds compared to adult participants, in a non-symbolic numerical comparison task whereby numerosity could be either congruent or incongruent with several non-numerical dimensions including dots' average size and total surface area.

Consistent with studies suggesting that congruency effects could reflect the role of inhibition in numerical comparison tasks [START_REF] Fuhs | ANS acuity and mathematics ability in preschoolers from low-income homes: contributions of inhibitory control[END_REF][START_REF] Fuhs | The Role of Non-Numerical Stimulus Features in Approximate Number System Training in Preschoolers from Low-Income Homes[END_REF][START_REF] Rousselle | The development of automatic numerosity processing in preschoolers: Evidence for numerosity-perceptual interference[END_REF], Szücs and colleagues propose that larger congruency effect in children could be due to lower inhibitory control abilities (see also [START_REF] Defever | Task-and age-dependent effects of visual stimulus properties on children's explicit numerosity judgments[END_REF] for a similar account). This is consistent with the protracted development of inhibitory control from childhood to late adolescence [START_REF] Luna | What has fMRI told us about the development of cognitive control through adolescence?[END_REF].

Recently, several authors have gone so far as proposing that inhibitory control is necessary for numerical development [START_REF] Clayton | Inhibition in Dot Comparison Tasks[END_REF][START_REF] Leibovich | From 'sense of number' to 'sense of magnitude': The role of continuous magnitudes in numerical cognition[END_REF], Szücs et al., 2013), either to develop numerical representations from a general sense of magnitude [START_REF] Leibovich | From 'sense of number' to 'sense of magnitude': The role of continuous magnitudes in numerical cognition[END_REF] or to handle the interference between an innate sense of number and the continuous dimensions of magnitude at all ages [START_REF] Clayton | Inhibition in Dot Comparison Tasks[END_REF] see [START_REF] Cappelletti | Number skills are maintained in healthy ageing[END_REF] for a study in aging). However, a few studies have failed to observe congruency effects, casting doubt on the critical causal role of inhibitory control in non-symbolic numerical comparison. For instance, [START_REF] Odic | Developmental change in the acuity of approximate number and area representations[END_REF][START_REF] Odic | Developmental change in the acuity of approximate number and area representations[END_REF][START_REF] Odic | Hysteresis affects approximate number discrimination in young children[END_REF][START_REF] Odic | Hysteresis affects approximate number discrimination in young children[END_REF] failed to observe lower performance in trials in which number was incongruent with total cumulative area, than in trials in which it was congruent, in children or in adults.

In order to clarify the role of inhibition in numerical development, it is crucial to precisely evaluate the impact of inhibitory control processes in numerical comparison tasks. So far, the efficiency to block irrelevant dimensions of magnitude in the stimuli has been assessed either through a posteriori analyses of performances in incongruent trials [START_REF] Fuhs | ANS acuity and mathematics ability in preschoolers from low-income homes: contributions of inhibitory control[END_REF][START_REF] Gilmore | Individual differences in inhibitory control, not non-verbal number acuity, correlate with mathematics achievement[END_REF] or via correlational analyses using seminal inhibitory control tasks (e.g., the inhibition subtest of the NEPSY-II, [START_REF] Keller | Inhibitory control may not explain the link between approximation and math abilities in kindergarteners from middle class families[END_REF].

The current study

In the current study, we took a different approach to provide evidence for or against the role of inhibitory control in numerical comparison tasks in 7-8-year-olds and adults, by devising a negative priming paradigm. The logic of the negative priming paradigm is as follows: if one needs to inhibit irrelevant information or distractors on a given incongruent item (i.e., the prime), the processing of that information (or distractor) on a subsequent congruent item (i.e., the probe) should be impaired (i.e., longer response times or higher error rates) as compared to a control condition in which the information to process on the probe is not related to the one to process (or to inhibit) on the prime (see [START_REF] Tipper | Does negative priming reflect inhibitory mechanisms? A review and integration of conflicting views[END_REF]. This paradigm, initially used by [START_REF] Tipper | The negative priming effect: inhibitory priming by ignored objects[END_REF] to study inhibitory control processes at play in selective attention, was later adapted to demonstrate the role of inhibition in various tasks, from classic color-word Stroop tasks [START_REF] Dalrymple-Alford | Examination of some aspects of the Stroop Color-Word Test[END_REF][START_REF] Pritchard | Avoiding the potential pitfalls of using negative priming tasks in developmental studies: assessing inhibitory control in children, adolescents, and adults[END_REF] to logicomathematical reasoning tasks such as Piaget's number-conservation task [START_REF] Borst | Negative priming in logicomathematical reasoning: The cost of blocking your intuition[END_REF][START_REF] Houdé | Negative priming effect after inhibition of number/length interference in a Piaget-like task[END_REF]. To provide evidence that a non-numerical dimension of magnitude (i.e., size) needs to be inhibited to process numerosity in both 7-8year-olds and adults, we adapted a numerical comparison task to a negative priming paradigm. Previous studies have shown that, in experimental contexts with such explicit congruence/incongruence between number and non-numerical dimensions of magnitude, preschoolers' performance could be very low on incongruent trials [START_REF] Soltész | Relationships between magnitude representation, counting and memory in 4-to 7-year-old children: A developmental study[END_REF].

Given that negative priming effects can only be observed if incongruent prime trials are performed correctly, we chose to test primary school children. This also ensured that we would be able to base our analyses of negative priming on both accuracy levels and reliable reaction times, as is usually the case in the negative priming literature [START_REF] Tipper | Does negative priming reflect inhibitory mechanisms? A review and integration of conflicting views[END_REF][START_REF] Neill | Selective attention and Inhibitory control of cognition, in Interference and Inhibition in Cognition[END_REF][START_REF] Borst | Inhibitory control efficiency in a Piaget-like class-inclusion task in school-age children and adults: A developmental negative priming study[END_REF]. If both 7-8 year-olds and adults need to inhibit the irrelevant dimension of size when comparing numerosities in incongruent items (i.e., dot array with more dots is smaller than the dot array with less dots and thus size interferes with number), they should require more time and commit more errors when congruent items (i.e., dot array with more dots is larger than the dot array with less dots) are preceded by incongruent items than by neutral items (i.e., the two dot arrays have identical size).

Given the protracted development of inhibitory control until late childhood and adolescence [START_REF] Luna | What has fMRI told us about the development of cognitive control through adolescence?[END_REF], and following our hypothesis of a causal link between inhibitory control and numerical comparison abilities, we should observe differences in the amplitude of the negative priming effects in the context of the non-symbolic numerical comparison task between our two age groups. Previous studies have shown that the amplitude of the negative priming effect could reveal the efficiency of the inhibitory control processes at play in a given context, by comparing primary school children and adults (e.g., [START_REF] Aïte | Taking a third-person perspective requires inhibitory control: Evidence from a developmental negative priming study[END_REF][START_REF] Borst | Inhibitory control efficiency in a Piaget-like class-inclusion task in school-age children and adults: A developmental negative priming study[END_REF]. In other words, if numerical comparison abilities are rooted in part on the ability to inhibit an irrelevant non-numerical dimension of size, then the amplitude of the negative priming effect should be smaller in 7-8-year-olds than in adults, due to lower inhibitory control abilities in children than adults.

Method

Participants

Fourty-seven primary school children (23 girls and 24 boys, mean age = 7.92 years, SD = .87) and thirty-two adults (21 females and 11 males, mean age = 27.86 years, SD = 6.13) with normal or corrected-to normal vision, participated in this study. The proportion of males to females did not differ between the two groups, χ²(1) = 2.15, p = .21. Children were recruited in primary schools, and participated voluntarily after their parents gave informed consent.

Adult participants were recruited through a mailing list. They gave informed consent and received 5 euros for their participation to the study. All participants were tested in accordance with national and international norms that govern the use of human participants.

Materials and Procedure Numerical Comparison Task Adapted to a Negative Priming Paradigm

The task stimuli were pairs of dot arrays, generated using Matlab and presented on the left and right side of the computer screen. The participants were seated approximately 50 cm from the screen. Each array of black dots was presented on a white circle (diameter visual angle of 8°) at a visual angle of 5.3° from a central-fixation cross. The sets of dots were generated using the Number, Size and Spacing dimensions as defined by DeWind and colleagues [START_REF] Dewind | Modeling the approximate number system to quantify the contribution of visual stimulus features[END_REF]. This allowed us to manipulate the congruency between Number and Size while keeping the Spacing dimension constant. Size in both arrays were either identical (neutral item), congruent (congruent items) or incongruent with the number of dots (incongruent items). We used a 1:2 numerical ratio (8 vs. 16 or 9 vs. 18 dots) in prime items (i.e., neutral and incongruent items) to ensure a high level of performance which is needed to run the analyses of the negative priming effects on the probe items. We presented 4:5 (12 vs. 15 dots) and 5:6 (10 vs. 12 dots) numerical ratios on the probe items (i.e., congruent items). These ratios were chosen so that the difficulty level would vary across the trials while keeping the total duration of the experiment acceptable for children (~5 min). For the incongruent and congruent items, following Salti, Katzin, Katzin, Leibovich, & Henik (2017), we controlled the saliency of the non-numerical dimension relative to the numerical dimension, i.e. the level of participants' reliance on non-numerical dimensions across trials, by equating the non-numerical ratio between two sets of dots to their numerical ratio. Specifically, the Size ratio was always in a 1:3 ratio with the numerical ratio (e.g. when the sets of dots are in a 1:2 numerical ratio, the Sizes in each set are in a 1:6 ratio, and when the numerical ratio is 4:5, the Size ratio is 4:15). For neutral prime items, Size was approximately halfway between the Size of the less numerous and more numerous arrays across the congruent and incongruent items. Across all the items, the dot array pairs were generated so that the percent difference in the occupied field area (defined by the minimum bounding circle) between the two arrays was less than 10%. Each trial consisted in a primeprobe item pair. Incongruent prime items followed by a congruent probe item will be further referred to as the test trials, whereas neutral prime items followed by a congruent probe item will be referred to as the control trials. Each trial type (test and control) was presented 24 times (12 per numerical ratio). We added 8 filler trials consisting of the following combinations of prime and probe items (one for each of the two numerical ratios used on the probe items: congruent /incongruent, incongruent /incongruent, incongruent /neutral and neutral/incongruent) to prevent the participants from noticing the pattern of incongruent, neutral and congruent items between the prime and probe. Thus, participants performed 56 experimental trials (112 items) presented in a random order.

The participants were instructed to press one of two keyboard keys ("D" or "K") on the side of the more numerous array as quickly and accurately as possible. Each trial started with a gray screen (500 ms), followed by a fixation cross and two white circles (500 ms). The arrays of dots from the prime item were then presented in the white circles until the participant gave a response, for a maximum duration of 1000 ms, at which time they were replaced by a gray screen. As soon as the participant gave their response, the same series of events occurred for the probe item. A mask was presented for 2000 ms as soon as the participants provided a response on the probe item to prevent transfer of processes between the probe and the prime of the next trial (Figure 1). The total duration of the task was approximately 5 minutes. The experiment started with 4 training trials (2 test and 2 control trials), using 1:2 numerical ratios for all of the items presented and a feedback on the accuracy of the participants' response was provided.

Results

The data for one child was removed due to technical failure. We first removed individual trials with RTs differing by two standard deviations from the average individual RT on either the prime or probe items of the control or test trials, which corresponded to 6.7% of the total number of trials across all participants in the children group, and 7.9% of the total number of trials across all participants in the adults group. For each participant, we averaged the proportion of correct responses and the response times (RTs) on items correctly performed, separately on the prime and the probes of the control and the test trials. The data from three additional children and two adults were removed due to overall accuracy rates (ARs) more than two standard deviations below their age group's average, leading to a final analyzed sample of 43 children and 30 adults. We verified that all the analyses described below remained identical when including these participants back into the sample. We then combined RTs and ARs by computing an inverse efficiency score (IES, [START_REF] Townsend | Methods of modeling capacity in simple processing systems[END_REF][START_REF] Townsend | Stochastic modeling of elementary psychological processes[END_REF], i.e. by dividing participants' average RTs by their average ARs on the prime and the probe items of the control and test trials. Note that in accordance with the logic of the Negative priming paradigm, we only analyzed ARs, RTs and IES on the probe items preceded by prime items performed correctly.

Congruency Effects on the Prime items

In order to test for the presence of a congruency effect in the prime items, we first ran a 2 age group (Children vs. Adults) x 2 congruency (Incongruent vs. Neutral) repeated measures ANOVA on the IES (see Table 1 for descriptive statistics on the prime items). ). The analysis also revealed a main effect of congruency, F(1, 71) = 17.48, p < .0001, η 2 p = 0.2, with lower IES for the neutral items (M = 607, SD = 147) than for the incongruent items (M = 708, SD = 292). We also observed a significant interaction between congruency and age group, F(1, 71) = 11.17, p = .001, η 2 p = 0.14.

We then ran paired t-tests between the incongruent and neutral trials on both age groups separately. IES was significantly higher for the incongruent condition than for the neutral

Negative Priming Effects: Analyses of the Probe Items

We conducted 2 age group (children vs. adults) x 2 trial type (test vs. control) repeated measures ANOVA on the probe IES (see Table 2 for descriptive statistics on the probe items).

Discussion

Using a negative priming paradigm, we provided additional evidence for the role of inhibitory control processes in non-symbolic numerical comparison tasks. First, consistent with previous studies [START_REF] Hurewitz | Sometimes area counts more than number[END_REF]Szücs et al., 2013), we observed a congruency effect on the prime items. Both 7-8-year-olds and adults performed significantly worse when comparing two dot arrays in a 1:2 numerical ratio when Size was incongruent with numerosity than when Size was constant across the two arrays.

Second, we observed significant negative priming effects both in 7-8 year-olds and in adults:

When comparing two arrays of dots in which numerosity was congruent with Size, the performances were lower when the preceding item showed a conflict between number and Size than when Size was identical in the two arrays of the preceding item. These negative priming effects provide evidence that 7-8-year-olds and adults inhibited Size to process numerosity in the incongruent prime items of the non-symbolic numerical comparison task.

Hence, our results demonstrate, first, that inhibitory control processes are required to block irrelevant dimensions of magnitude during numerical comparison, impacting both the performance on the current and the following item, in children at the beginning of primary school and even in adults with a precise numerical acuity (adults can accurately perceive the difference between two numerosities in a numerical ratio up to 9:10). Second, we observe that, for a given level of numerical difficulty, the amplitude of the negative priming effect decreases between 7-8-year-olds and adults, supporting the idea that the ability to compare numerosities could be rooted at least in part on the growing ability to inhibit an irrelevant non-numerical dimension of Size, in line with a more general protracted development of inhibitory control [START_REF] Dempster | Interference and Inhibition in Cognition and Behavior: Comments on the Commentaries[END_REF][START_REF] Harnishfeger | A developmental perspective on individual differences in inhibition[END_REF]. This

decrease in the amplitude of the negative priming effect is consistent with the one reported in previous developmental negative priming studies including negative priming adaptation of logico-mathematical tasks [START_REF] Borst | Negative priming in logicomathematical reasoning: The cost of blocking your intuition[END_REF][START_REF] Borst | Inhibitory control in number-conservation and class-inclusion tasks: A neo-Piagetian inter-task priming study[END_REF], 2013;[START_REF] Houdé | Negative priming effect after inhibition of number/length interference in a Piaget-like task[END_REF]. This is, however, in contrast with other studies having found either no difference in negative priming between children and adults [START_REF] Pritchard | Avoiding the potential pitfalls of using negative priming tasks in developmental studies: assessing inhibitory control in children, adolescents, and adults[END_REF], or smaller if not an absence of negative priming in children compared to adult participants (Tipper et al., 1989). These discrepant findings could be accounted for by the different inhibitory control processes involved in different experimental contexts. For instance, Lechuga and colleagues (2006) have argued that a distinction should be drawn between intentional and automatic inhibitory control. In our study, it is possible that the inhibitory control required to block the irrelevant, non-numerical dimensions of magnitude in the numerical comparison task, is an automated process in adults, but not yet automatized in 7-8year-olds, consistent with the idea of a development of numerical representation relying on a progressive ability to abstract number from non-numerical dimensions [START_REF] Piazza | Learning to focus on number[END_REF]. In a study investigating the development of negative priming effects between children, adolescents and adults, [START_REF] Pritchard | Avoiding the potential pitfalls of using negative priming tasks in developmental studies: assessing inhibitory control in children, adolescents, and adults[END_REF] recommend to correct reaction time data for age-related differences in overall response speed. We did so in the present study by performing our analyses on a percentage change in IES, rather than on absolute difference. Thus, our observed differences in the amplitude of the negative priming effect cannot be accounted for by overall differences in reactions times between children and adults. However, further studies investigating multiple age ranges between the ones reported in the current study need to be conducted in order to fully describe the developmental trajectory of the role played by inhibitory control in non-symbolic numerical comparison judgements.

One limit of using negative priming paradigm is its applicability to very young population.

First, negative priming effects are usually reported on the basis of analyses of reaction times, which are less easily collectable in preschoolers. Most studies investigating numerical comparison abilities in preschool children report analyses on accuracy levels, rather than reaction times, which are often derived from button presses performed by the experimenters themselves. Second, the logic of the negative priming paradigm requires to restrict the analyses of performance on probe items preceded by correctly performed incongruent prime items. Low accuracy scores on incongruent trials in preschoolers could have precluded the analysis of a negative priming effect in this population. In the current study, we were able to report negative priming effects calculated on the basis of IES in 7-8 year old children and in adults having performed the same numerical comparison task, both in terms of presentation times and numerical ratios.

One could argue that participants are less efficient to compare numerosities in congruent items preceded by incongruent than neutral ones (i.e., negative priming effect) not because they need to inhibit Size in incongruent items but because neutral items are easier to perform than incongruent ones which could in turn affect the level of internal confidence of the participant [START_REF] Odic | Hysteresis affects approximate number discrimination in young children[END_REF]. However, these hysteresis effects have been observed after prolonged exposure to difficult trials, and not from one trial to the next, as in our task.

Moreover, our participants were not given any feedback on their performance, which has been shown to be crucial to evoke such hysteresis effect during numerical judgment tasks [START_REF] Odic | Hysteresis affects approximate number discrimination in young children[END_REF]Experiment 3). Additionally, negative priming effects have been observed

initially in the Color-Word Stroop task [START_REF] Dalrymple-Alford | Examination of some aspects of the Stroop Color-Word Test[END_REF][START_REF] Pritchard | Avoiding the potential pitfalls of using negative priming tasks in developmental studies: assessing inhibitory control in children, adolescents, and adults[END_REF] in which test and control trials were equivalent in terms of the difficulty of the primes (both items were incongruent), and differed solely by the color to be activated in the following probe items (either identical or different than the color inhibited in the prime).

Although the negative priming effects reported in the present study suggest that item size must be inhibited in incongruent items to process numerosities in children and adults, further studies are needed to determine whether (a) other irrelevant dimensions must be inhibited too and (b) whether the amplitude of the negative priming effect would also decrease with age especially for irrelevant dimensions that remain highly salient throughout development.

Convex hull might be a good candidate to test this hypothesis given that its impact remains relatively stable throughout development, unlike the impact of item size which decreases with age [START_REF] Gilmore | Congruency effects in dot comparison tasks: convex hull is more important than dot area[END_REF].

Conclusion

Our study reports the first evidence of negative priming effects in numerical estimation tasks in 7-8-year-olds and adults, and thus provides an additional piece of evidence for the role of inhibitory control processes involved in basic numerical intuitions. Together with the observed developmental effect, our findings support the role of executive functions, and in particular of inhibitory control, in numerical development, by using a different methodological approach. This is in line with recent accounts of numerical development [START_REF] Gebuis | Sensory-integration system rather than approximate number system underlies numerosity processing: A critical review[END_REF], Leibovich et al., 2017), as well as with recent evidence showing that the development of numerical skills relies on the ability to filter irrelevant dimensions of quantity [START_REF] Piazza | Learning to focus on number[END_REF].

Consistent with these accounts of numerical development, some studies have reported that the association between numerical estimation and math could be explained by interindividual differences in inhibitory control efficiency [START_REF] Fuhs | ANS acuity and mathematics ability in preschoolers from low-income homes: contributions of inhibitory control[END_REF][START_REF] Gilmore | Individual differences in inhibitory control, not non-verbal number acuity, correlate with mathematics achievement[END_REF][START_REF] Merkley | Of Huge Mice and Tiny Elephants: Exploring the Relationship Between Inhibitory Processes and Preschool Math Skills[END_REF]; but see [START_REF] Keller | Inhibitory control may not explain the link between approximation and math abilities in kindergarteners from middle class families[END_REF], for contradictory results). Adapting numerical estimation tasks to a negative priming paradigm such as described in our study allows measuring the ability to inhibit a specific irrelevant dimension of magnitude (i.e. the dimension which is incongruent with numerosity in the prime items).

Adapting such method in preschool children (in particular, by ensuring high accuracy levels on the incongruent prime items) could provide an alternative way of measuring children's inhibitory control abilities for specific irrelevant dimensions, allowing to subsequently determine their respective predictive value for the development of more general mathematics skills. Our findings thus have important implications for the current debates regarding the predictive value of numerical intuitions vs. executive functions for math achievement, by providing tentative evidence for the latter.

Figure 1 :

 1 Figure 1: Adaptation of the non-symbolic comparison task to a negative priming paradigm. Sequences of events in a test and control trial of the numerical comparison task.

  condition both for children (incongruent: M = 918, SD = 368; neutral: M = 736, SD = 181), t(42) = 4.55, p < .0001, d = 0.934, and adults (incongruent: M = 498, SD = 76.2; neutral: M = 477, SD = 63), t(29) = 2.64, p = .01, d = 0.523. Finally, we computed a percentage change in IES between neutral and incongruent items to compare the congruency effect between children and adults while controlling for overall difference in IES. A t-test confirmed that the congruency effect was larger in children (M = 23.8%, SD = 28.1%) than in adults (M = 4.25%, SD = 8.03%), t(71) = 3.7, p < .001, d = 0.89 (Figure2).

Figure 2 :

 2 Figure 2: Congruency Effect in children and adults. IES percentage change between the neutral and incongruent prime items of the non-symbolic numerical comparison task in children and adults. Error bars represent ± one standard error of the mean.

  (M = 822, SD = 535) showing overall lower IES than children (M = 1406, SD = 535), F(1, 71) = 20.99, p < .0001, η 2 p = 0.23. Importantly, we observed a significant main effect of the type of trial (test vs. control) with a smaller IES in the control (M = 924, SD = 330) than in the test (M = 1305, SD = 857) trials, F(1, 71) = 21.09, p < .0001, η 2 p = 0.23. There was a significant interaction between age group and trial type, F(1, 71) = 12.45, p = .001, η 2 p = 0.15. When performing paired t-tests looking at both age groups separately, we observed a significant negative priming effect on IES for both children (test: M = 1743, SD = 1057, control: M = 1069, SD = 355), t(42) = 4.91, p < .0001, d = 1.039, and adults (test: M = 867, SD = 352, control: M = 778, SD = 273), t(29) = 3.4, p = .002, d = 0.748. . In addition, we found that the amplitude of the negative priming effect computed as a percent change of IES between the control and the test trials was larger in children (M = 61.8%, SD = 75.7%) than in adults (M = 10.8%, SD = 15.4%), t(71) = 3.63, p = .001, d = 0.87 (Figure 3).

Figure 3 :

 3 Figure 3: Negative Priming Effect in children and adults. IES percentage change on the probe between the control and test trials of the non-symbolic numerical comparison task in children and adults. Error bars represent ± one standard error of the mean.

Table 1

 1 Average and range of ARs and RTs (ms) for the incongruent and neutral prime items of the

	non-symbolic numerical comparison task
		7-8-year-olds		Adults
		M (SD)	Range	M (SD)	Range
	Incongruent prime items
	Ars	0.88 (0.12) 0.54 -1	0.99 (0.04)	0.77 -1

Table 2

 2 Average and range ofARs and RTs (ms) for the congruent test and control probe items of the non-symbolic numerical comparison task.

		7-8-year-olds		Adults
		M (SD)	Range	M (SD)	Range
	Test probe items
	Ars	0.62 (0.23) 0.14 -1	0.82 (0.12) 0.55 -1
	RTs	907 (260)	470 -1871	681 (201)	435 -1340
	Control probe items
	Ars	0.79 (0.13) 0.5 -1	0,85 (0.13) 0.52 -1
	RTs	819 (228)	412 -1514	635 (173)	413 -1224

The analysis revealed a significant effect of age group, with adults