# Modeling the steady state response of the solid phase diffusion process in Lithium-ion cells

Ghada Bakaraki, Houssam Rabab, Nicolas Damay, Khadija El Kadri Benkara, Christophe Forgez

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Ghada Bakaraki, Houssam Rabab, Nicolas Damay, Khadija El Kadri Benkara, Christophe Forgez. Modeling the steady state response of the solid phase diffusion process in Lithium-ion cells. The 26th European Conference on Power Electronics and Applications, GDR SEEDS France & EPE Association, Mar 2025, Paris, France. 10.34746/epe2025-0197 . hal-05101122

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# Modeling the steady state response of the solid phase diffusion process in Lithium-ion cells

Ghada Bakaraki<sup>(1)(2)</sup>, Houssam Rabab<sup>(1)(3)</sup>, Nicolas Damay<sup>*</sup><sup>(1)</sup>, Khadija El Kadri Benkara<sup>(1)</sup>, Christophe Forgez<sup>(1)</sup>.

E-Mail: nicolas.damay@utc.fr

URL: https://www.utc.fr

## Keywords

Battery, Lithium-ion, equivalent circuit model, solid-state diffusion, non-linearities.

## Abstract

Accurate battery models are crucial for the evaluation of Lithium-ion batteries states and performance. Physics based models can represent the battery's non-linear behavior but may not take into account the whole battery's phenomena given the high complexity of the equations. Despite their simplicity, traditional equivalent circuit models (ECM) may not represent non-linear behaviour especially in low state of charge (SoC) area. To take the best of the above modeling approaches, a physics-based ECM is presented. This model is built on separating solid and liquid diffusion effects in Li-ion cells whose diffusion is strongly non-linear with SoC. A detailed identification methodology of the solid diffusion's steady-state response was proposed. Thanks to this model, diffusion parameters present less non-linearities and are easier to be identified, while reducing the dimensions of lookup tables and characterization tests time.

## 1. Introduction

Nowadays, the whole world is facing major environmental challenges including pollution and energy shortage. To deal with these issues, new energy technologies have been emerging rapidly. Thanks to their high energy density, long life cycle, decreasing price and functional characteristics [1,2], lithium-ion batteries are becoming a vital element in many applications especially in the aeronautic and automotive fields [3]. Actually, electric aircrafts and vehicles are promptly emerging in the market since they

widely contribute in reducing the carbon footprint. However, a robust Battery Management System (BMS) is required to ensure safe, effective and reliable performance of lithium-ion batteries. The primary tasks of a BMS include: monitoring of the cell's characteristics (voltage, current and temperature) [4], real-time state estimation: state of charge (SoC), state of energy (SoE), state of power (SoP) and state of health (SoH) [5], ensuring energy management and life extension [6,7]. Thus, accurate state estimation is crucial for BMS to guarantee the safety and reliability of critical systems based on lithium-ion batteries [8].

In order to accurately predict the internal states of lithium-ion batteries, an accurate yet simple battery model is required. Therefore, Equivalent Circuit Models (ECM) have been widely utilized [9]. Nevertheless, these models are based on high-dimensional lookup tables. These tables present several drawbacks [10]: (1) they are difficult to be recalibrated as the battery ages, (2) the parameters values are usually found by optimization algorithms that may not always provide physically coherent values [11] and, most importantly, (3) they are only able to describe the non-linear phenomena of the batteries for discrete set points.

Actually, diffusion phenomena in Li-ion batteries are strongly non-linear especially in the low SoC region, and it was shown in [12,13] that a classic ECM is not able to represent the diffusion effects. Otherwise, very complex and time-consuming characterization tests are required.

In this paper, we present an approach to model and characterize the diffusion process in ECM. This approach is based on separating the lithium diffusion in the electrodes (called solid diffusion [14]) and the lithium ions diffusion in the

<footnote><sup>(1)</sup> Université de Technologie de Compiègne, Roberval (Mécanique, énergie et électricité), Centre de recherche Royallieu - CS 60319, 60203 Compiègne Cedex, France</footnote>

<footnote><sup>(2)</sup> Safran Electrical & Power 70-80 Rue Auguste Perret, 94000 Créteil, France</footnote>

<footnote><sup>(3)</sup> TIAMAT, 72 rue des Jacobins, Amiens, 80000, France</footnote>
<table><thead><tr><th>Nomenclature</th><th></th></tr></thead><tbody><tr><td>R<sub>s</sub></td><td>Series resistance (Ω)</td></tr><tr><td>R<sub>surf</sub></td><td>Surface resistance (Ω)</td></tr><tr><td>τ<sub>surf</sub></td><td>Time constant of the surface phenomenon (s)</td></tr><tr><td>τ<sub>sd</sub></td><td>Time constant of the solid diffusion phenomenon (s)</td></tr><tr><td>Z<sub>diff</sub></td><td>Diffusion impedance</td></tr><tr><td>Z<sub>ld</sub></td><td>Liquid diffusion impedance</td></tr><tr><td>V<sub>diff</sub></td><td>Diffusion voltage (V)</td></tr><tr><td>V<sub>sd</sub></td><td>Solid diffusion voltage (V)</td></tr><tr><td>V<sub>ld</sub></td><td>Liquid diffusion voltage (V)</td></tr><tr><td>SEV</td><td>Surface equilibrium voltage (V)</td></tr><tr><td>ΔSoC</td><td>Horizontal shift in SoC attributed to the solid diffusion (Ah)</td></tr><tr><td>ΔSoC<sub>st</sub></td><td>Steady state response of the solid diffusion phenomenon (Ah)</td></tr><tr><td>K<sub>sd</sub></td><td>The effect factor of solid diffusion (h)</td></tr><tr><td>EaK<sub>sd</sub></td><td>Activation energy of Ksd (eV)</td></tr><tr><td>Eaτ<sub>sd</sub></td><td>Activation energy of τsd (eV)</td></tr><tr><td>I<sub>ref</sub></td><td>Current of the reference voltage curve used to measure ΔSoC (A)</td></tr><tr><td>T<sub>ref</sub></td><td>Reference temperature (K)</td></tr></tbody></table>

electrolyte (called liquid diffusion[15]).

The focus in this study is to develop a physical model of the non-linearities of the solid-state diffusion's steady state response. This enables us to reduce the ECM's non linearity yet simplifying its parameters identification.

Several Li-ion cells models based on modelling the solid diffusion phenomenon were found in the literature [16,17]. It was proved in [18] that ECM taking into account the solid diffusion effect outperforms the conventional ECM where the diffusion processes are not separated. Regarding the solid diffusion response, some authors consider that it follows a first-order system dynamics [1,3], while others represent it by a second and three order system response [19,20]. In this study, we will consider an order 3 response as a first approximation, and will be focusing on the identification of its steady state. Hence, modeling the solid diffusion dynamics as well as the liquid diffusion effects are beyond the scope of work and are being explored in an ongoing study.

The remainder of this paper is organized as follows: Section 2 presents a comparison between the classic ECM and the modified ECM based on the separation of the diffusion effects in Li-ion batteries. Section 3 introduces the solid diffusion model along with its temperature and current dependencies. The experimental protocol and parameters extraction methodology are detailed in Section 4. Parameters identification results are presented in Section 5. Section 6 concludes the paper.

## 2. Equivalent Circuit Models

Lithium-ion cells can be represented via several model structures. In fact, two main model categories can be found: physics-based models [16,21,22] and behavioral models [3]. On the one hand, physics-based models, like the “Single Particle Model” and the “Multiple Particle Model”, describe the internal electrochemical reactions of Li-ion cells by the means of partial differential equations. On the other hand, behavioral models, like the Equivalent Circuit Model (ECM), are composed of several circuits to represent the battery’s behavior. These models are relatively easier to implement in a BMS (Battery Management System) since they require less computational complexity than the physics-based ones.

A review on different modelling techniques for Li-ion cells was conducted in [23]. A comparison between physics-based models and equivalent circuit models is studied in [24].

Nevertheless, to identify the ECM parameters, lookup tables are needed to represent the non-linearities of the model. This declines the accuracy of the model especially in the low state of charge area. Therefore, we have modified the ECM in a way to simplify its parameter identification process.

In this section, we will present the classic and simplified ECMs.
<figure><img src="image_2.png" /><figcaption>Fig. 1: Classic equivalent circuit model</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Fig. 2: ECM-SP<sup>e</sup> [13]</figcaption></figure>

## 2.1 Classic Equivalent Circuit Model

As illustrated in Fig.1, the classic ECM [10] of a Li-ion cell is based on a modified Randles circuit and consists of the following components:

* OCV: a voltage source representing the open circuit voltage. This is the voltage of the cell in open circuit and at equilibrium.

* R<sub>s</sub>: a series resistance representing electrolyte, current connectors and contact resistances.

* R<sub>surf</sub>: a surface resistance related to the voltage drop between the surface of the active material and the electrolyte[10].

* τ<sub>surf</sub>=R<sub>surf</sub>×C<sub>surf</sub>: a time constant related to the surface phenomena, which are associated with charge transfer, double layers and SEI. It represents a quick dynamic (less than 1 second).

* Z<sub>diff</sub>: an impedance related to the lithium-ion diffusion phenomena in the electrolyte and lithium atoms in the electrodes. It represents slow dynamics (several minutes).

## 2.2 Modified Equivalent Circuit Model

The proposed model is inspired of a “single-particle” model called SPM-SP<sup>e</sup> [13]. This approach is based on the separation of the diffusion phenomena that occur in Li-ion cells. In the subsection 2.2.1, we will detail the diffusion effects in Li-ion cells and how they can be represented in ECMs. The subsection 2.2.2 compares the classic and the modified ECM, and summarizes the modifications that we made for the classic model, so that it represents better the diffusion phenomena.

### 2.2.1 Diffusion effects in Li-ion batteries

Diffusion phenomenon in Li-ion cells is represented by Fick's law, which relates the diffusive flux to the concentration [2]. According to [2,13,25], diffusion occurs in both the electrodes and electrolyte.

On the one hand, electrode diffusion is due to the concentration gradient of lithium ions in the active material particles, referred to as “solid diffusion” [14]. On the other hand, electrolyte diffusion refers to the lithium ions concentration gradient in the electrolyte, referred to as “liquid diffusion” [15].

In this paper, we aim to model the diffusion phenomenon by separating the solid and liquid contributions. Conventional ECMs do not distinguish between solid and liquid diffusions, resulting in a low voltage simulation accuracy in the non-linear ranges, and could further cause low state estimation accuracy [1,3,13].

According to previous works [13,26], the SoC dependency of the diffusion in Li-ion cells is attributed to the solid diffusion. Consequently, liquid diffusion is supposed to be independent of the SoC. In addition, the solid and liquid diffusion are both current and temperature dependent [1,3,19,25,27]. Thus, the diffusion voltage, noted “V<sub>diff</sub>”, could be expressed as the sum of “V<sub>sd</sub>” and “V<sub>ld</sub>”, representing the solid and liquid diffusion voltages respectively (equation (1)).

$$V_{diff}(I, T, SoC) = V_{sd}(I, T, SoC) + V_{ld}(I, T) \quad (1)$$

Kremer et al. [28] used a single particle model for LTO cells. In the proposed model, the solid diffusion affects the lithium insertion rate at the electrode’s surface, influencing the cell’s equilibrium voltage. The latter refers to the insertion reaction equilibrium, and differs from the OCV in figure (1), which represents the cell’s global equilibrium: when all the chemical reactions reach the steady state [3,13].

Consequently, to consider the solid diffusion, the surface equilibrium voltage has to be taken into account in the model. According to [13], the surface equilibrium voltage is denoted as “SEV” (figure 2), and can be written as (equation (2)):

$$SEV(SoC) = OCV(SoC + \Delta SoC) \quad (2)$$
<figure><img src="image_4.png" /><figcaption>Fig. 3: Vertical and horizontal shifts</figcaption></figure>

Where $\Delta SoC$ represents the horizontal shift in SoC between the voltage of the cell and the OCV (figure 3).

The SEV represents the equilibrium voltage of the insertion reaction of Lithium at the surface of the electrodes whose rate is the ratio of the surface lithium-ion concentration to the maximum concentration active material particle of the cell [3,13]. Since we are working on the macroscopic scale, we do not have access to the insertion rate. In [29], a relation between the insertion rate and SoC has been proposed for LFP cells.

Actually, the SoC measured by experiment represents the average concentration of lithium ions in the electrodes, which differs from the SoC related to the surface insertion rate. The latter is denoted “$SoC_{surf}$”, and is equal to the sum of the average SoC and $\Delta SoC$, as represented in equation (3).

$$SoC_{surf} = SoC + \Delta SoC \quad (3)$$

According to [1,13,25,26,28], the horizontal shift, $\Delta SoC$, is attributed to the solid diffusion and varies with current and temperature.

In fact, when a cell discharges, we will notice not only vertical shift from the OCV caused by $V_s$, $V_{surf}$ and $V_{ld}$, but also horizontal shift in SoC. Hence, by using equation (2), the solid diffusion will be represented by $\Delta SoC$ and will be used in this paper.

In the next section, the ECM integrating diffusion effects will be summarized.

### 2.2.2 ECM integrating diffusion effects

The modified ECM, so called ECM-SP[13], depicted in figure 2, introduces a novel approach for modelling the diffusion effects. This approach represents the diffusion as two main phenomena: the solid diffusion, represented by SEV which is

function of $\Delta SoC$ (equation (2)), and the liquid diffusion represented by several RC circuits connected in series, that is equivalent to the liquid diffusion impedance, denoted “$Z_{ld}$”.

However, for the simulation results presented in Section 5, one RC circuit is used. Indeed, a higher number of RC circuits could be used, but this makes the implementation of the model more complex.

The major modification to the standard ECM lies in the OCV and the diffusion process. In fact, the classic ECM (Figure 1) uses “$Z_{diff}$” as the diffusion impedance. The latter is non-linear in terms of SoC because it represents all of the diffusion effects.

In our modeling approach, we integrate the SoC non-linearity of the diffusion in the OCV (equation 2). This reduces the model’s non-linearity since $Z_{ld}$ is independent of the SoC, unlike $Z_{diff}$ in the conventional ECM (figure 1). In the next section, the methodology used to model the current and temperature non-linearities of the diffusion will be presented. However, modeling the liquid diffusion effect is beyond the scope of this paper.

## 3. Solid phase diffusion model

According to [1,3,25], the solid-phase diffusion process, represented by $\Delta SoC$, follows a first-order system’s dynamic. However, it can be also modeled as second or third order system’s response [19,20]. Thus, $\Delta SoC$ can represented by the following expression (equation 4):

$$\Delta SoC(s) = \Delta SoC_{st} \prod_{m=1}^{n} \frac{1}{1 + \tau_{sdms}} \quad (4)$$

where "$\tau_{sdms}$" is the time constant of the solid-state diffusion process, "$\Delta SoC_{st}$" is the $\Delta SoC$ value when the solid diffusion reaches its steady state, "$n$" is the order of solid diffusion dynamic and "$s$" is the Laplace operator.

In this paper, we choose to model $\Delta SoC$ as a third order system. Hence, $n=3$. As mentioned in Section 2.2.1, $\Delta SoC$ depends on current and temperature. These dependencies will be detailed in the following subsections.

### 3.1 Current dependent model

With the increase of discharge rate, the cell’s discharge capacity decreases [25]. This increase of discharge current induces vertical and horizontal shifts of the voltage curves from the OCV, hereinafter the “reference curve”.
<figure><img src="image_5.png" /><figcaption>Fig. 4: Voltage and current profile of discharge tests</figcaption></figure>

Horizontal shifts are due to solid-phase diffusion effects, whereas vertical shifts are due to other effects like the liquid diffusion, the series resistance, the surface resistance and similar phenomena. As a result, ΔSoC varies with the current.

This current dependency is mainly depicted in ΔSoC<sub>st</sub> which can be expressed as a linear function of the discharge current [1,3,25,28]. Hence, ΔSoC<sub>st</sub> is represented as follows (equation 5):

$$ \Delta SoC_{st}(I) = K_{sd} \times (I - I_{ref}) \quad (5) $$

Where K<sub>sd</sub>, the slope, is the effect factor of the solid-phase diffusion, I is the discharge current and I<sub>ref</sub> is the current of the corresponding reference curve.

Meanwhile, little literature reported the variation of τ<sub>sdm</sub> with current.

## 3.2 Temperature dependent model

Temperature has a significant impact on the solid-phase diffusion [3,27]. In fact, as the temperature increases, the diffusion rate of lithium ions increases. Hence, the time taken by lithium ions to diffuse to the surface decreases. Consequently, ΔSoC is temperature-dependent and this dependency lies in ΔSoC<sub>st</sub>, and subsequently K<sub>sd</sub>, and τ<sub>sdm</sub>. These terms are supposed to decrease when increasing the temperature as reported in [3]. Thereby, Arrhenius law (equation 6 and 7) is used to simulate the temperature's effect on K<sub>sd</sub>, and τ<sub>sdm</sub>:

$$ K_{sd}(T) = K_{sd}(T_{ref}) \times \exp\left(\frac{Ea_{K_{sd}}}{Kb}\left(\frac{1}{T} - \frac{1}{T_{ref}}\right)\right) \quad (6) $$

$$ \tau_{sdm}(T) = \tau_{sdm}(T_{ref}) \times \exp\left(\frac{Ea_{sd}}{Kb}\left(\frac{1}{T} - \frac{1}{T_{ref}}\right)\right) \quad (7) $$

where Ea<sub>K<sub>sd</sub></sub>, Ea<sub>τ<sub>sd</sub></sub> are the activation energy of K<sub>sd</sub>, and τ<sub>sdm</sub> respectively, T<sub>ref</sub> is the reference temperature in Kelvin, and K<sub>b</sub> the Boltzmann's constant.

In this study our goal is to model ΔSoC<sub>st</sub>. The dynamics of ΔSoC (i.e. τ<sub>sdm</sub>) will be explored in a future work.

According to equations (5) and (6), modeling ΔSoC<sub>st</sub> comes down to identifying K<sub>sd</sub>, then K<sub>sd</sub>(T<sub>ref</sub>) and Ea<sub>K<sub>sd</sub></sub>. For that, we did several continuous discharge tests at different current rates and temperatures. Hence, we will present the experimental protocol used as well as the parameters identification methodology in the upcoming section.

# 4. Experiments and parameter identification

For this study, Samsung 18650-25R cells (NCA+NMC/graphite+SiO) were used with a nominal capacity Q<sub>nom</sub> of 2.5 Ah, nominal voltage U<sub>nom</sub> of 3.6V, maximum current rate of 8C for continuous discharge and 1.6C for continuous charge. A Bio-Logic system was used for tests (BCS-815), offering 8 channels with a maximum current of 15A each. Additionally, the cells were placed in a climatic chamber to control their temperature.

## 4.1 Experimental protocol

Characterizing the current dependency of the solid diffusion's steady state response (equation 4), consists of constant current discharge tests at C/20, C/10, C/5, C/3, C/2, 1C, 2C and 3C. These tests were also conducted at three temperatures (10°C, 25°C and 35°C), which is a minimum
number required to validate Arrhenius law (equation 6 and 7).

The first step consists of a residual discharge at C/2 until reaching the cut-off voltage (2.5V). Then we recharge the cell at a constant current rate at C/2 until reaching 4.2V, and followed by charge at a constant voltage to the cut-off current rate of C/20 to fully charge the battery. The second step is to discharge the cell at C/20, succeeded by a charge identical to the one performed in the first step. The next step repeats the second one for the remaining current rates. We note that a rest time of 6 hours was added between each discharge/charge cycle to ensure that the cell's temperature and voltage are stable. However, this rest time could even be reduced to 4 hours for current rates lower than 2C.

In figure 4, the current and the corresponding voltage profile of the constant current discharge tests, at 25°C, are displayed. The 1C, 2C and 3C discharge pulses, as well as the intermediate charges and rest periods, were omitted to make the figure clearer.

## 4.2 Parameters extraction methodology

The solid diffusion's steady state parameters extraction consists of four main steps (figure 5):

* **Step 1** is to determine $\Delta SoC_{st}$ based on discharge tests described in section 4.1. For the cells used in this study, the effect of solid diffusion (i.e. the horizontal shift), is clearly manifested in the end of discharge, in the range 2.5V-2.8V, as shown in figure 6. Therefore, we will suppose that the $\Delta SoC_{st}$ values are associated to a discharge current $I_j$ and the corresponding end of discharge temperature $T'_j$, since the cell's temperature is no longer equal to the test temperature due to cell heating. Hence, we get $\Delta SoC_{st}(I_j, T'_j)$.

* **Step 2** is to perform temperature adjustment to $\Delta SoC_{st}(I_j, T'_j)$ points to bring them back to the actual test temperature $T_j$ by using equation 8 and 9:

$$ \Delta SoC_{st}(T'_j) = \Delta SoC_{st}(T_{ref}) \times \exp\left(\frac{E_{a,0}}{K_b}\left(\frac{1}{T_j} - \frac{1}{T_{ref}}\right)\right) \quad (8) $$

$$ \Delta SoC_{st}(T'_j) = \Delta SoC_{st}(T_j) \times \exp\left(\frac{E_{a,0}}{K_b}\left(\frac{1}{T'_j} - \frac{1}{T_j}\right)\right) \quad (9) $$

Firstly, we fit $\Delta SoC_{st}(T'_j)$ to equation 8 at each current $I_j$. Consequently, the optimization algorithm will identify $\Delta SoC_{st}(T_{ref})$ and $E_{a,0}$.

Secondly, we use $E_{a,0}$ and $\Delta SoC_{st}(T'_j)$ at each current $I_j$ along with equation 9. This results in $\Delta SoC_{st}(I_j, T_j)$ and $E_{a,0}$ will be used as an initial value for $E_{a,K_{sd}}$ in Step 4.

* **Step 3** is to fit, at each $T_j$, the $\Delta SoC_{st}(I_j)$ points to equation (5). Then, we find the slope $K_{sd}$ for $T_j=10^\circ C, 25^\circ C$ and $35^\circ C$. The obtained $K_{sd}$ value at $25^\circ C$ will be used as an initial value for $K_{sd}(T_{ref})$, with $T_{ref} = 298 K (25^\circ C)$, in Step 4. Steps 2 and 3 serve as a pre-identification and initialization step for the optimization algorithm in Step 4.

* **Step 4** is to fit $K_{sd}(T_j)$ to equation (6), to identify $K_{sd}(T_{ref})$ and $E_{a,K_{sd}}$.

# 5. Results and discussions

In this section, we present the results of each of the identification steps described in Section 4.2 along with some model simulation results.

## 5.1 Solid diffusion identification results

For **Step 1**, we assumed $I_{ref}= C/5$. According to the manufacturer's specifications for the cells used in this study, the discharge capacity is calculated using a C/5 rate. Additionally, the use of higher C-rates reduces the errors resulting from the current integration while computing the SoC. Therefore, we suggest using the C/5 curve as the "reference curve" and with that, $I_{ref}$ is equal to 0.5 A. Then, we calculated the SoC difference between the C/5 curve and the ones at higher rates noted "$\Delta SoC$".

<figure><img src="image_6.png" /><figcaption>Fig. 5: Flowchart of parameters extraction</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>Fig. 6: Discharge voltage profiles</figcaption></figure>

Figure 7 illustrates, for each tested temperature, the ΔSoC data points, measured at the end of discharge, at each current. It is obvious that for current rates greater than C/2 (1.25A), data points are no more aligned with the previous ones. This is induced by the dynamic behavior of the solid diffusion as well as the significant cell heating at higher rates at the end of discharge. As a result, we can divide the ΔSoC curve into two regions noted “1” and “2” respectively (figure 7). In region 1, where ΔSoC is linear, we only have the temperature effect. While in region 2, where ΔSoC is non-linear, both ΔSoC dynamics and temperature effects are combined. Here, we would like to note that the dynamic behavior of the solid diffusion is present regardless of the discharge current. However, we supposed that we could neglect the solid diffusion dynamics effects at low discharge currents (region 1).

As our goal is to model the solid diffusion's steady state response, we need to consider the part of the ΔSoC curve with no dynamic effects. Hence, we have chosen the first three points (at C/5, C/3 and C/2 respectively) called “ΔSoC<sub>st</sub> points”, where we supposed that there is no dynamic effect. These points will be used in the next steps. The rest of the data points were discarded, but will be used in future works to identify the time constant τ<sub>sdm</sub>.

Figure 8, shows ΔSoC<sub>st</sub> points after temperature adjustment (Step 2), with E<sub>a,0</sub> = 0.54 eV. In fact, this step affects the points at 10°C the most, because lower temperatures result in higher cell heating. Besides, we notice a linear behavior of the adjusted data points.

Consequently, **Step 3** is applied and we obtain three linear curves, as shown in figure 9.

The linear fit results (Step3) are summarized in Table 1. It is clear that K<sub>sd</sub> decreases as the temperature T increases, which in accordance with [1,3,27].

<table><caption>Table I: K<sub>sd</sub> values at each tested temperature</caption><tr><td></td><td>T=10°C</td><td>T=25°C</td><td>T=35°C</td></tr><tr><td>K<sub>sd</sub></td><td>0.0797</td><td>0.0247</td><td>0.0132</td></tr></table>

<figure><img src="image_8.png" /><figcaption>Fig. 7: Evolution of ΔSoC with I-I<sub>ref</sub></figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 8: Temperature adjustment of ΔSoC<sub>st</sub></figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 9: Linear fit of ΔSoC<sub>st</sub></figcaption></figure>
<figure><img src="image_11.png" /><figcaption>Fig. 10: Simulation results at 25°C</figcaption></figure>

After fitting these points to equation (6), considering $E_{a,0}$ and with $K_{sd}$ at 25°C resulting from Steps 2 and 3, we obtain $K_{sd}(T_{ref}) = 0.0252$ hours and $Ea_{K_{sd}} = 0.54$ eV, where $T_{ref} = 298$ K (25°C) in **Step 4**. We notice that the value of $K_{sd}$ (25°C) is very close to the one we got, at the same temperature, in Step 3 (Table 1). Hence, the results of figure 9 demonstrate that the linear model of $\Delta SoC_{st}$ is close to the measurements for the studied temperatures.

## 5.2 Model simulation results

Despite being beyond the scope of this work, we identified the solid diffusion time constant $\tau_{sdm}$ as well as the liquid diffusion represented by $Z_{ld}$ (figure 2). Regarding the rest of the model's parameters, they were identified following the same methodology represented in [10]. Then, we tested our model at 25°C for a constant discharge at C/2, 1C, 2C and 3C. The simulation results are represented in figure 10. We notice that the simulated voltage is very close to the measured voltage even in the low SoC region. This is important to accurately estimate the state of the battery in the low SoC region.

However, vertical shifts appear between the simulated and measured voltage, mainly in two SoC intervals. The first, ranges between 0.35 Ah and 1.37 Ah, and the second one ranges between 1.96 Ah and 2.37 Ah. This might be due to the OCV curve used in the model, or the values of the model's resistances and time constants. These parameters were identified using optimization

algorithms, whose results might not always be accurate. Also, the smearing effects that are often observed during discharges at higher C-rates, possibly due to a heterogeneous behavior, are not represented in our model [30]. This may explain the difference at 3C around SoC 0.5 Ah. Moreover, the proposed model neglects the diffusion at the positive electrode. This can lead to voltage estimation errors especially in the medium-high SoC regions.

## 6. Conclusion

In this paper, a physics-based ECM for NMC+NCA/graphite cells was presented. This model is based on the separation between both solid and liquid diffusion phenomena that occur in lithium-ion cells. The effect of the solid diffusion is mostly attributed to the negative electrode in this work and is represented as a state of charge shift ($\Delta SoC$) between the cell's voltage and the OCV. Liquid diffusion is assumed to be independent of SoC and can be modeled by several parallel RC circuits connected in series. As a first study, we opted for a third-order system dynamics to represent $\Delta SoC$ and our focus was on characterizing and modeling the non-linearities of its steady state response ($\Delta SoC_{st}$).

An experimental protocol along with $\Delta SoC_{st}$ identification methodology were proposed. $\Delta SoC_{st}$ varies linearly with the discharge current and its temperature dependency follows Arrhenius law. This model can be a promising solution to reduce the model's SoC dependencies and consequently the characterization time required to identify them compared to the use of multidimensional lookup tables.

The non-linear model proposed in this study was tested for several discharges after identifying $\tau_{sdm}$ and $Z_{ld}$. As a preliminary result, the output of the model was close to the measured voltage especially in low SoC region. The model has also good performance even in the low SoC region, where smearing effects occur and that are not that easy to model. The proposed physics-based non-linearities model has the potential to provide physical insights into solid-state diffusion, simplifies the characterization of the diffusion phenomena and enhances the model's performance in the low SoC region. Hence, the accuracy of the state estimators is more likely to improve. As perspectives of this work, the liquid diffusion phenomenon may be identified. Additionally, a state of health dependency may be added to the model.
References

[1]. X. Xu, S. Tang, H. Ren, X. Han, Y. Wu, L. Lu, X. Feng, C. Yu, J. Xie, M. Ouyang, W. Liu, Y. Yan, Joint state estimation of lithium-ion batteries combining improved equivalent circuit model with electrochemical mechanism and diffusion process, Journal of Energy Storage, Volume 56, 2022.

[2]. A. Berrueta, A. Urtasun, A. Ursúa, P. Sanchis, A comprehensive model for lithium-ion batteries: From the physical principles to an electrical model, Energy, Volume 144, 2018.

[3]. C. Chen, Y. Huang, X. Yu, M. Zhang, X. Han, Y. Sun, Y. Zheng, Improving the accuracy of voltage estimation in the low charge state range at low temperature: An equivalent circuit model considering the influence of temperature on solid phase diffusion process, Journal of Energy Storage, Volume 88, 2024.

[4]. L. Lu, X. Han, J. Li, J. Hua, M. Ouyang, A review on the key issues for lithium-ion battery management in electric vehicles, Journal of Power Sources, Volume 226, 2013.

[5]. L. Su, M. Wu, Z. Li, J. Zhang, Cycle life prediction of lithium-ion batteries based on data-driven methods, eTransportation, Volume 10, 2021.

[6]. X. Xu, C. Yu, S. Tang, X. Sun, X. Si, and L. Wu, Remaining Useful Life Prediction of Lithium-Ion Batteries Based on Wiener Processes with Considering the Relaxation Effect, Energies 12, no. 9: 1685, 2019.

[7]. S. Tang, C. Yu, X. Wang, X. Guo, and X. Si, Remaining Useful Life Prediction of Lithium-Ion Batteries Based on the Wiener Process with Measurement Error, Energies 7, no. 2: 520-547, 2014.

[8]. Y. Wang, L. Wang, M. Li, Z. Chen, A review of key issues for control and management in battery and ultra-capacitor hybrid energy storage systems, eTransportation, Volume 4, 2020.

[9]. Y. Zheng, M. Ouyang, X. Han, L. Lu, J. Li, Investigating the error sources of the online state of charge estimation methods for lithium-ion batteries in electric vehicles, Journal of Power Sources, Volume 377, 2018.

[10]. Fernanda Vendrame, Nicolas Damay, Houssam Rabab, Christophe Forgez, Marie Sayegh, Non-invasive diagnosis of solid electrolyte interphase and charge transfers resistances for degraded lithium-ion batteries, Journal of Power Sources, Volume 595, 2024.

[11]. L. Mihet-Popa, O. M. F. Camacho, P. B. Norgard. Charging and discharging tests for

obtaining an accurate dynamic electro-thermal model of high power lithium-ion pack system for hybrid and EV applications, 2013 IEEE Grenoble Conference, 2011.

[12]. M. Juston, N. Damay, C. Forgez, Extracting the diffusion resistance and dynamic of a battery using pulse tests, Journal of Energy Storage, Volume 57, 2023.

[13]. H. Rabab, “Modeling of sodium-ion batteries for on-board diagnosis of their states of charge and health,” PhD Thesis, University of Technology of Compiegne, France, 6 2024.

[14]. D. M. Bernardi, J.-Y. Go. Analysis of pulse and relaxation behavior in lithium-ion batteries. Journal of Power Sources, 196(1), 2011.

[15]. X. Zhou, J. Huang, Z. Pan, M. Ouyang. Impedance characterization of lithium-ion batteries aging under high-temperature cycling: Importance of electrolyte-phase diffusion. Journal of Power Sources, 2019.

[16]. P. Rodríguez-Iturriaga, D. Anseán, S. Rodríguez-Bolívar, V. Manuel García, M. González, J. Antonio López-Villanueva, Modeling current-rate effects in lithium-ion batteries based on a distributed, multi-particle equivalent circuit model, Applied Energy, Volume 353, Part A.

[17]. D. Guo, G. Yang, X. Feng, X. Han, L. Lu, M. Ouyang, Physics-based fractional-order model with simplified solid phase diffusion of lithium-ion battery, Journal of Energy Storage, Volume 30, 2020.

[18]. C. Li, N. Cui, Z. Cui, C. Wang, C. Zhang, Novel equivalent circuit model for high-energy lithium-ion batteries considering the effect of nonlinear solid-phase diffusion, Journal of Power Sources, Volume 523, 2022.

[19]. X. Han, M. Ouyang, L. Lu, J. Li, Simplification of physics-based electrochemical model for lithium ion battery on electric vehicle. Part I: Diffusion simplification and single particle model, Journal of Power Sources, Volume 278, 2015.

[20]. M. Ouyang, G. Liu, L. Lu, J. Li, X. Han, Enhancing the estimation accuracy in low state-of-charge area: A novel onboard battery model through surface state of charge determination, Journal of Power Sources, Volume 270, 2014.

[21]. P. Rodríguez-Iturriaga, D. Anseán, S. Rodríguez-Bolívar, M. González, J. C. Viera, J. Antonio López-Villanueva, A physics-based fractional-order equivalent circuit model for time and frequency-domain applications in lithium-ion batteries, Journal of Energy Storage, Volume 64, 2023.
[22]. C. Brivio, V. Musolino, M. Merlo and C. Ballif, "A Physically-Based Electrical Model for Lithium-Ion Cells," in IEEE Transactions on Energy Conversion, vol. 34, no. 2, pp. 594-603, June 2019.

[23]. A. E. Jayasinghe, N. Fernando, S. Kumarawadu and L. Wang, "Review on Li-ion Battery Parameter Extraction Methods," in IEEE Access, vol. 11, pp. 73180-73197, 2023.

[24]. M. Lagnoni, C. Scarpelli, G. Lutzemberger, A. Bertei, Critical comparison of equivalent circuit and physics-based models for lithium-ion batteries: A graphite/lithium-iron-phosphate case study, Journal of Energy Storage, Volume 94, 2024.

[25]. Y. Zheng, W. Gao, X. Han, M. Ouyang, L. Lu, D. Guo, An accurate parameters extraction method for a novel on-board battery model considering electrochemical properties, Journal of Energy Storage, Volume 24, 2019.

[26]. H. Rabab, N. Damay, C. Forgez, A. E. Mejdoubi and A. Quelin, "Equivalent Circuit Model For Sodium-Ion Batteries With Physical-Based Representations Of Their Non-Linearities," 2023 IEEE Vehicle Power and Propulsion Conference (VPPC), Milan, Italy, 2023.

[27]. R. Xiong, J. Huang, Y. Duan, W. Shen, Enhanced Lithium-ion battery model considering critical surface charge behavior, Applied Energy, Volume 314, 2022.

[28]. F. Kremer, S. Rael, M. Urbain, 1D electrochemical model of lithium-ion battery for a sizing methodology of thermal power plant integrated storage system, AIMS Energy 8, 2020.

[29]. K. M. Mbeya, N. Damay, G. Friedrich, C. Forgez, M. Juston, Off-line method to determine the electrode balancing of li-ion batteries, Mathematics and Computers in Simulation 183, 2021.

[30]. M. Farkhondeh, M. Pritzker, M. Fowler, C. Delacourt. Mesoscopic Modeling of a LiFePO4 Electrode: Experimental Validation under Continuous and Intermittent Operating Conditions. Journal of The Electrochemical Society, 2017.