# Impact of Model Depth on State of Charge Estimation in Second-Life Batteries

Lukas Böhning, Mathias Herget, Patrick Stock, Sven Fießer, Raphael Kress, Ulf Schwalbe

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Lukas Böhning, Mathias Herget, Patrick Stock, Sven Fießer, Raphael Kress, et al. Impact of Model Depth on State of Charge Estimation in Second-Life Batteries. The 26th European Conference on Power Electronics and Applications, GDR SEEDS France & EPE Association, Mar 2025, Paris, France. 10.34746/epe2025-0236. hal-05088951

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# Impact of Model Depth on State of Charge Estimation in Second-Life Batteries

Lukas Böhning, Mathias Herget, Patrick Stock, Sven Fieber, Raphael Kress, Ulf Schwalbe

Hochschule Fulda University of Applied Sciences

Department of Electrical Engineering and Information Technology

Leipziger Straße 123, 36037 Fulda, Deutschland

E-Mail: lukas.boehning@et.hs-fulda.de

URL: http://www.hs-fulda.de

*Index Terms*—Energy Storage, State of Charge, Energy Management System, Modelling

**Abstract**—This paper investigates the impact of different levels of modelling on the accuracy of state of charge estimation in battery storage systems. The aim is to compare simple and complex models to identify the optimal balance between accuracy and computational complexity. The results show that while deeper models improve estimation, the additional complexity provides only marginal benefits. A kinetic approach improves accuracy at high loads without significantly increasing the effort. Separating the inverter and battery models increases flexibility by allowing system changes without retooling. The results provide a basis for integrating temperature effects and optimising storage operation for economic efficiency.

## I. INTRODUCTION

The selection of the application, such as self-consumption optimisation, peak shaving or several grid services, is decisive for the economic efficiency of stationary battery storage systems (BSS) [1]. By utilizing stationary BSS for various applications at simultaneously, there is potential to increase economic gains [2]. This requires the development of algorithms that enable the operation of multiple use cases in parallel, as proposed in [3]. Therefore, a detailed model of a stationary BSS is essential for the accurate estimation of its behaviour [4, 5]. As illustrated in [6], a system is introduced which integrates multiple applications, resulting in increased returns. The applications are peak shaving and the use of flexible electricity prices as well as the reduction of grid connection power. A single battery string was isolated for this study, and the subsequent curves were derived from experimental assessments conducted

on this specific battery string. A concept of the system setup is shown in Fig. 1.

<figure><img src="image_2.png" /><figcaption>Fig. 1. Concept of second-life BSS and its interconnection with the inverter in a low-voltage distribution network.</figcaption></figure>

Before algorithms can be developed for a better energy management system or calculating the economic efficiency of these systems, it is necessary to develop a system model. One of the most important results of modelling a BSS is the estimation of the state of charge (SOC) depending on the input power [7]. Based on this, it can be estimated whether a certain load peak can be reduced or determine how the self-consumption of a photovoltaic system can be increased.

Modelling approaches can be categorized into whitebox, greybox, and blackbox models [8]. Due to the high complexity of various modelling approaches, this study focuses on utilizing published whitebox models from the literature. These models are adapted and trained using measured data. The objective is to develop a simple and robust model that can serve as a foundation for future research, such as optimization algorithms. Unlike many studies that emphasize detailed modelling of BSS, this work prioritizes an efficient and pragmatic approach to
ensure reliable results for economic feasibility assessments. The accuracy of SOC estimations is validated based on practical system data from [6], and the results are analyzed with respect to their applicability in further studies.

## II. PRACTICAL EXPERIMENTAL SETUP

The practical system consists of a second-life BSS and a fast charging station for electric vehicles, integrated to validate the developed models. The BSS consists of two battery strings, each comprising 192 cells connected in series. Each battery string has a capacity of 80 kWh in the configuration. These batteries are connected to a low voltage distribution grid, along with a fast charging station capable of delivering up to 150 kW of DC power. The system is designed to reduce the peak load of the grid by using the battery storage and recharging it during times without peak loads. This configuration allows the fast charging station to operate even in areas with limited grid capacity, temporarily increasing the available power [6].

The fast charging station is integrated with an energy management system (EMS) that controls the charging and discharging cycles of the battery storage. The EMS ensures that the grid connection is not overloaded by regulating the power flow between the grid, the battery, and the charging station. The system prioritizes maintaining stable power flow while ensuring that the battery operates within safe limits. The EMS also monitors the SOC of the BSS and adjusts the charging power based on real-time conditions, such as load demand and grid capacity [6].

One of the main challenges in using simple energy storage models is their inability to accurately represent the full operational characteristics of the system. This is particularly evident when considering the voltage profile over the SOC and the current limitations, such as the maximum charge and discharge currents, which depend on the minimum and maximum cell voltage. Simple models often assume a constant power output. However, in practice, the power output of the system is limited by the voltage-current characteristics of the batteries and inverters [6]. Additional influences, such as the temperature of the batteries and inverters, can also affect performance [9, 10]. These limitations can result in the BSS full power not being utilized, particularly at high or low SOC when the current capacity must be reduced to protect the battery cells. Therefore, detailed models are necessary to account for these nonlinear characteristics and to provide a more accurate estimation of available power and SOC. By integrating factors such

as inverter efficiency and the voltage-current profile of the battery system, these detailed models offer a more accurate representation of the system's true operational limits.

## III. MODELLING APROACHES FOR BATTERY STORAGE SYSTEMS

For the development of whitebox models of BSS, formal assumptions are made about the physical principles for the conversion and storage of electrical energy. The fundamental principle describes electrical energy storage, where losses occur during both the charging and discharging processes, influenced by various factors [11]. These models are crucial elements for the effective control and operation of BSS. Although there are different methods of the algorithmics and control of BSS, the main element of each approach is a model of the BSS [4].

The work by [4] identifies four different types of battery storage modelling. These include energy reservoir models (ERM), charge reservoir models (CRM), single-particle models, and pseudo two-dimensional models. The single-particle models and pseudo two-dimensional models are not applied in this paper.

ERM is a way of modelling the SOC using a relationship between electrical power in W and electrical energy storage capacity in Wh, shown in (1) [4]. These models are purely concerned with the ratio of power to SOC change.

$$ \frac{\delta \zeta}{\delta t} = \frac{p^{+} \cdot \eta^{+} + \frac{p^{-}}{\eta^{-}} + p_{sd}}{E_{N} \cdot SOH} \quad (1) $$

Where $\frac{\delta \zeta}{\delta t}$ represents the change in SOC, $p^+$ and $p^-$ represent the charging and discharge power, $\eta^+$ and $\eta^-$ represent the efficiency of charging and discharge, $p_{sd}$ represents the self-discharging power, SOH represents the state of health of the battery and $E_N$ for the maximum storable energy in the delivery state [4].

ERMs are categorised into three different types, including models without consideration of efficiency and self-discharge, models with linear self-discharge and efficiency, as shown in (1), and models with non-linear properties, where SOC-, load- or temperature-dependent efficiencies and self-discharge are taken into account [4].

For example, CRM model from [4] shown in (2) refer to cell capacity in Ah and battery current in A.

$$ \frac{\delta \zeta}{\delta t} = \frac{i_{bat}^{+} \cdot \eta^{+} + \frac{i_{bat}^{-}}{\eta^{-}} + i_{sd}}{C_{N} \cdot SOH} \quad (2) $$

Where $\frac{\delta \zeta}{\delta t}$ represents the change in SOC, $i_{bat}^+$ and $i_{bat}^-$ represent the charging and discharge power, $\eta^+$ and $\eta^-$
$\eta^-$ represent the charging and discharge efficiency, $i_{sd}$ for the self-continuous charge and $C_N$ for the maximum storage capacity in the delivery state. CRM are also subdivided into models with dynamic and non-dynamic voltage [4].

The influence of temperature in ERM and CRM can be taken into account, which can affect charging and discharge efficiency $\eta^+$ and $\eta^-$ [4]. This influence requires accurate modelling of the battery temperature as a function of load and ambient temperature. It is particularly important to mention that in this work the thermal influence is not modeled as a function of temperature due to continuous thermal conditioning to a fixed battery and system temperature in the case of the research project in [6].

According to [4], the current state of research on the modelling and optimal control of battery storage systems reveals several gaps. Most studies are simulation-based and do not extend their findings to practical applications, which limits their practical relevance. Although some experimental analyses exist, they are few and often lack complete integration of results into real operational environments.

A critical under-explored area is the role of nonlinear system dynamics and kinetic models in BSS. Although these dynamics are critical for accurate modelling and control, they are not adequately represented in existing algorithms and frameworks. For example, as highlighted in [4], the increasing complexity of advanced models often results in a significant number of tunable parameters, increasing the computational workload required to achieve an accurate representation of the battery. This is further complicated by the need to model the inverter separately from the power data, particularly for AC systems, to effectively simulate the entire BSS.

Moreover, a paucity of research has been identified that addresses the trade-off between the complexity of modelling, the computational effort required, and the accuracy of the results, particularly in the context of estimating the SOC. Although some studies have acknowledged this trade-off, there is a lack of systematic comparisons between different modelling depths under real conditions. The integration of measured data into these comparisons is essential to ensure the practicality and reliability of the models.

Therefore, this work aims to close these gaps by systematically comparing various modelling depths of BSS in the context of practical applications. The research will focus on balancing computational simplicity with sufficient accuracy for SOC estimation and system control. Using measured data, the study will evaluate

existing approaches to recommend an appropriate depth of the model that meets practical requirements while remaining computationally efficient. The findings will provide a foundation for better informed decisions in the design and operation of BSS.

## IV. MODELLING OF THE SYSTEM

This chapter first defines the requirements for simulation models. The evaluation criteria are then presented, based on which the models are compared in terms of accuracy and effort. In addition, initial battery models are introduced and extended based on the work of [4]. Furthermore, a model for the inverter is presented. The battery models are then fitted to a real system from [6], which has already been validated.

### A. Requirements

The goal of modelling a BSS is to estimate the SOC based on the data in the datasheet of the batteries shown in Tab. I, including the number and interconnection of cells, the cell capacity, and the input power (based on AC or DC, depending on the modelling level).

<table><caption>TABLE I
SPECIFICATIONS OF THE BATTERY STORAGE</caption><thead><tr><td>Parameter</td><td>Value</td></tr></thead><tbody><tr><td>Cell capacity</td><td>57.5 Ah</td></tr><tr><td>Number of cells in parallel</td><td>2</td></tr><tr><td>Number of cell pairs in series</td><td>192</td></tr><tr><td>Nominal voltage per cell</td><td>3.64 V</td></tr><tr><td>Total capacity</td><td>80.4 kWh</td></tr></tbody></table>

The experimentally measured open-circuit voltage curve in Fig. 2 is also used depending on the depth of the model.

<figure><img src="image_3.png" /><figcaption>Fig. 2. OCV curve as a function of the SOC of the battery cells.</figcaption></figure>

Real measurement curves from the prototype concept are used for modelling, which are shown in Fig. 3.

The main focus for modelling is to minimize the maximum deviation and the mean absolute deviation.
<figure><img src="image_4.png" /><figcaption>Fig. 3. AC input power at the inverter and battery storage SOC over a 16-hour period.</figcaption></figure>

The time range from zero to hour 16 is used for the fitting of the models. Later, a different time period will be used to evaluate the results. In a measured application, the models must be continuously synchronized with the actual SOC to address any deviations. Therefore, models are required to maintain accuracy only for a specified evaluation period. In this study, an evaluation period of 80 to 90 hours has been chosen. Similarly, the current SOH is considered for the calculation of the SOC, but the aging process itself is not determined.

## B. Metrics

The mean absolute error (MAE) and the root mean square error (RMSE) are used to evaluate the deviation over the period. The maximum error (ME), which evaluates the maximum deviation over the period, is also significant in times of high loads. It should be noted that the metrics presented here aim to estimate the SOC as accurately as possible. There are two reasons why it is not necessary to evaluate the deviation over significantly longer periods of time. Firstly, a calibration with the model occurs practically every time the battery is charged or discharged to 100% or 0%. Secondly, automatic calibration can be carried out, for example, at fixed intervals or under certain operating conditions (e. g. in idle mode, in which the storage system is not active). For other applications where the BSS is permanently in operation and no fully charged or discharged, it may be necessary to use other procedures.

## C. Battery Models

To model the overall system, the first step is to investigate which models are suitable methods for estimating the SOC. For this purpose, six different depth levels of the model (LVL) were derived from the work of [4], which differ in the properties of the adjusted parameters.

The SOH, which indicates the remaining capacity of the storage system, is an important factor, the charging

and discharging efficiency (CE and DE), and the self-discharge (SD) is also included. Kinetic models (KM) and the use of dynamic voltages (DV) also participate. An overview of the depths of the model is given in Tab. II.

<table><caption>TABLE II
SIMULATION OVERVIEW OF BATTERY MODELS</caption><thead><tr><th>LVL</th><th>SOH</th><th>CE</th><th>DE</th><th>SD</th><th>KM</th><th>DV</th></tr></thead><tbody><tr><td>0</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>1</td><td>✓</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>2</td><td>✓</td><td>✓</td><td>✓</td><td>-</td><td>-</td><td>-</td></tr><tr><td>3</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>-</td><td>-</td></tr><tr><td>4</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>-</td></tr><tr><td>5</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>-</td><td>✓</td></tr><tr><td>6</td><td>✓</td><td>SOC</td><td>SOC</td><td>✓</td><td>✓</td><td>✓</td></tr></tbody></table>

<table><tr><td>LVL</td><td>Modelling depth</td></tr><tr><td>SOH</td><td>State of health</td></tr><tr><td>SOC</td><td>State of charge-dependent losses</td></tr><tr><td>CE</td><td>Charging efficiency</td></tr><tr><td>DE</td><td>Discharging efficiency</td></tr><tr><td>SD</td><td>Self discharge</td></tr><tr><td>KM</td><td>Kinetic model</td></tr><tr><td>DV</td><td>Dynamic voltage</td></tr></table>

Models of level zero to four can be referred as ERM, while models five and six are CRM. The following boundaries apply to all models:

$$E_N = 80.4 \text{ kWh} \qquad (3)$$

Unlike the models of [4], a differentiation was introduced between efficiency during charging $\eta^+$ and positive power $p^+$ and the efficiency during discharging $\eta^-$ and the negative power $p^-$.

The characteristic curves of [6] are used to limit the charging power $p^+$ and the discharging power $p^-$ as a function of cell voltage $u_{cell}$. In (4) and (5), the input power $p$ is limited to the maximum power $p_{max}(u)$ and the minimum power $p_{min}(u)$ allowed for the given cell voltage.

$$p^+ = \max(0, \min(p, p_{max}(u_{cell}))) \qquad (4)$$

$$p^- = \min(0, \max(p, p_{min}(u_{cell}))) \qquad (5)$$

The determination of the SOC within the model level 0 is shown in (6). Model level one (7) includes the consideration of the SOH, model level two (8) additionally considers an efficiency for charging $\eta^+$ and discharging $\eta^-$. Model level three (9) is characterized by the additional self-discharge $p_{sd}$.
$$ \text{LVL 0:} \quad \frac{\delta \zeta}{\delta t} = \frac{p}{E_N} \qquad (6) $$

$$ \text{LVL 1:} \quad \frac{\delta \zeta}{\delta t} = \frac{p}{E_N * SOH} \qquad (7) $$

$$ \text{LVL 2:} \quad \frac{\delta \zeta}{\delta t} = \frac{p^+ \cdot \eta^+ + \frac{p^-}{\eta^-}}{E_N \cdot SOH} \qquad (8) $$

$$ \text{LVL 3:} \quad \frac{\delta \zeta}{\delta t} = \frac{p^+ \cdot \eta^+ + \frac{p^-}{\eta^-} + p_{sd}}{E_N \cdot SOH} \qquad (9) $$

Model level four in (10) and (11) takes into account a kinetic model for the SOC, as presented in [4].

$$ \text{LVL 4:} \quad \frac{\delta \zeta_1}{\delta t} = \frac{p^+ \cdot \eta^+ + \frac{p^-}{\eta^-} + p_{sd} + P_{bal}(\zeta_2 - \zeta_1)}{c_f \cdot E_N \cdot SOH} \qquad (10) $$

$$ \frac{\delta \zeta_2}{\delta t} = \frac{P_{bal}(\zeta_1 - \zeta_2)}{(1 - c_f) \cdot E_N \cdot SOH} \qquad (11) $$

Here $P_{bal}$ is a factor that describes the speed with which the two energy reservoirs are balanced. $\zeta_1$ is the SOC of the first energy reservoir, $\zeta_2$ is the SOC of the second energy reservoir. $c_f$ represents the factor to divide the capacity of the reservoirs.

For the CRM models, the in (12) shown calculations are assumed.

$$ C_N = CC \cdot CP = 115Ah \qquad (12) $$

$C_N$ represents the parallel connection capacity CP of two cells with a capacity of CC = 57.5 Ah. The charging current $i_{bat}^+$ and the discharge current $i_{bat}^-$ are also limited based on the characteristic curve of [6] as a function of the cell voltage $u_{cell}$ and the resulting maximum current $i_{max}$ and minimum current $i_{min}$. In contrast to the formulation [4], the current on the input side is also limited in the presented CRM and a distinction is made between the efficiencies for discharging and charging the battery. The battery current is calculated from the electrical power $p$ divided by the number of series connected cells CS and the open circuit voltage dependent on SOC $U(\zeta)$, as shown in (13) and (14).

$$ i_{bat}^{-} = \min(0, \max(\frac{p}{CS \cdot U(\zeta)}, i_{min}(u_{cell}))) \qquad (13) $$

$$ i_{bat}^{+} = \max(0, \min(\frac{p}{CS \cdot U(\zeta)}, i_{max}(u_{cell}))) \qquad (14) $$

(15) represents model level five. Equivalent to the ERM, the efficiency for charging $\eta^+$ and discharging $\eta^-$

the battery is multiplied by the charging current $i^+$ and the discharging current $i^-$. The battery is also discharged at a constant self-discharge rate $i_{sd}$.

$$ \text{LVL 5:} \quad \frac{\delta \zeta}{\delta t} = \frac{i^+ \cdot \eta^+ + \frac{i^-}{\eta^-} + i_{sd}}{C_N \cdot SOH} \qquad (15) $$

Model level six in (16) and (17) also takes into account two charge reservoirs whose SOC is represented by $\zeta_1$ and $\zeta_2$ in order to map the dynamic of the load on the SOC. $\eta(\zeta_1)^+$ and $\eta(\zeta_1)^-$ represent efficiency calculated as a function of SOC.

LVL 6:

$$ \frac{\delta \zeta_1}{\delta t} = \frac{i^+ \cdot \eta(\zeta_1)^+ + \frac{i^-}{\eta(\zeta_1)^-} + i_{sd} + i_{bal}(\zeta_2 - \zeta_1)}{c_f \cdot C_N \cdot SOH} \qquad (16) $$

$$ \frac{\delta \zeta_2}{\delta t} = \frac{i_{bal}(\zeta_1 - \zeta_2)}{(1 - c_f) \cdot C_N \cdot SOH} \qquad (17) $$

In order to fit the respective variables, boundaries and constraints must be defined. The MAE between the real SOC $\zeta_{real,i}$ and the calculated SOC $\zeta_{est,i}$ in (18) should be minimized in order to estimate the SOC with minimum deviation at any time:

$$ \min_x = \frac{1}{n} \sum_{i=1}^{n} |\zeta_{est,i} - \zeta_{real,i}| \qquad (18) $$

subject to 19 and 20:

$$ \begin{cases} 0 \le \eta^+ \le 1 & 0 \le \eta^- \le 1 \\ 0 \le c_f \le 1 & 0 \le SOH \le 1 \\ 0 \le i_{bal} & 0 \le p_{bal} \\ 0 \le \zeta \le 1 & 0 \le \zeta_1 \le 1 \\ 0 \le \zeta_2 \le 1 & \zeta_1 + \zeta_2 = 1 \end{cases} \qquad (19) $$

$$ \{\eta^+, \eta^-, c_f, SOH, i_{bal}, p_{bal}, \zeta, \zeta_1, \zeta_2\} \in \mathbb{R} \qquad (20) $$

Fitting the variables leads to the following results for the battery models shown in Table III.

<table><caption>TABLE III
RESULTS FOR THE PARAMETERS OF THE BATTERY MODEL
FITTING</caption><thead><tr><th>LVL</th><th>SOH [%]</th><th>η<sup>+</sup> [%]</th><th>η<sup>-</sup> [%]</th><th>p<sub>sd</sub> [W] / i<sub>sd</sub> [A]</th><th>p<sub>bal</sub> [W] / i<sub>bal</sub> [A]</th><th>c<sub>f</sub></th></tr></thead><tbody><tr><td>0</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>1</td><td>0.826</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>2</td><td>0.901</td><td>0.936</td><td>0.978</td><td>-</td><td>-</td><td>-</td></tr><tr><td>3</td><td>0.923</td><td>0.9875</td><td>0.9773</td><td>328.3</td><td>-</td><td>-</td></tr><tr><td>4</td><td>0.9538</td><td>0.9795</td><td>0.9563</td><td>98.3</td><td>0.5</td><td>1.4</td></tr><tr><td>5</td><td>0.8527</td><td>0.921</td><td>0.9914</td><td>0</td><td>-</td><td>-</td></tr><tr><td>6</td><td>0.9103</td><td>soc1</td><td>soc2</td><td>0.2</td><td>0.4</td><td>3.0</td></tr></tbody></table>
<table><tr><td>LVL</td><td>Model depth level</td></tr><tr><td>SOH</td><td>State of health</td></tr><tr><td>η<sup>+</sup></td><td>Charging efficiency</td></tr><tr><td>η<sup>-</sup></td><td>Discharging efficiency</td></tr><tr><td>p<sub>sd</sub>/i<sub>sd</sub></td><td>Self-discharge power/current</td></tr><tr><td>p<sub>bal</sub>/i<sub>bal</sub></td><td>Balancing power/current</td></tr><tr><td>c<sub>f</sub></td><td>Capacity factor</td></tr><tr><td>soc1 &amp; soc2</td><td>SOC-dependent efficiency</td></tr></table>

At the highest level of model depth level six, SOC-dependent efficiencies have been introduced to account for the variation in charging and discharging efficiencies as a function of SOC. These values are shown in Fig. 4.

<figure><img src="image_5.png" /><figcaption>Fig. 4. SOC dependent efficiency of the battery model at level six.</figcaption></figure>

The accuracy of the different models is then compared using key metrics MAE, RMSE and ME. These metrics provide insight into the performance of the model in terms of precision and outlier sensitivity. The results of the different model depths are summarised in Fig. 5.

In model levels zero, the MAE is at a high value of 4%, as no efficiency losses are taken into account here. With the introduction of efficiencies in the models of model depths two and three, the accuracy increases significantly, due to the consideration of charging and discharging losses. The kinetic model, which is implemented at model depth four, reduces the error to an MAE of 1.16% but significantly affects the ME to 6.8%. Model level six achieves the best MAE, which is due to a large number of variables that have to be adjusted in these models, which significantly increases the complexity and computational effort. For higher accuracies, additional influencing factors must be identified and integrated into the models in order to further improve the estimation accuracy. However, based on the previously defined requirements, even more precise models are not necessary.

## D. Inverter Models

Three inverter models are defined to calculate the battery power. Model level zero has no influence on the

<figure><img src="image_6.png" /><figcaption>Fig. 5. Error metrics for different battery model levels. The MAE and ME represent the absolute error between the measured and calculated SOC, ranging from 0 to 1 (0 to 100%).</figcaption></figure>

DC power, model level one is characterized by charge and discharge-based efficiency and self-consumption. Model level two includes power-dependent efficiency and self-consumption. The battery power is calculated as the output. An overview is shown in Tab. IV.

<table><caption>TABLE IV
SIMULATION OVERVIEW OF INVERTER MODELS</caption><thead><tr><th></th><th>LVL</th><th>CE</th><th>DE</th><th>SC</th></tr></thead><tbody><tr><td>LVL</td><td>0</td><td>-</td><td>-</td><td>-</td></tr><tr><td>CE</td><td>1</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>DE</td><td>2</td><td>P</td><td>P</td><td>✓</td></tr></tbody></table><p>Modelling depth</p><p>Charging efficiency</p><p>Discharging efficiency</p><p>Self-consumption</p><p>Power-dependent efficiency</p>

Model level one is described in (21), model level two in (22) and model level three in (23):
$$ \text{LVL 0: } p_{\text{bat}} = p_{\text{ac}} \qquad (21) $$

$$ \text{LVL 1: } p_{\text{bat}} = p_{\text{ac}}^{+} \cdot \eta^{+} + \frac{p_{\text{ac}}^{-}}{\eta^{-}} + p_{\text{ic}} \qquad (22) $$

$$ \text{LVL 2: } p_{\text{bat}} = p_{\text{ac}}^{+} \cdot \eta^{+}(p_{\text{ac}}) + \frac{p_{\text{ac}}^{-}}{\eta^{-}(p_{\text{ac}})} + p_{\text{ic}} \qquad (23) $$

Where $p_{\text{bat}}$ represents the battery input power, $p_{\text{ac}}^{+}$, the AC input power of the inverter is multiplied by the efficiency $\eta^{+}$ or $\eta^{+}(p_{\text{ac}})$. The AC output power of the inverter $p_{\text{ac}}^{-}$ is also multiplied by the efficiency $\frac{p_{\text{ac}}^{-}}{\eta^{-}(p_{\text{ac}})}$ or $\frac{p_{\text{ac}}^{-}}{\eta^{-}(p_{\text{ac}})}$. The internal consumption of the inverter is represented by $p_{\text{ic}}$.

In order to fit the respective variables, boundaries and constraints must be defined. The MAE between the real battery power $P_{\text{bat,real},i}$ and the calculated battery power $P_{\text{bat,est},i}$ in (24) is to be minimized:

$$ \min_{x} = \frac{1}{n} \sum_{i=1}^{n} |P_{\text{bat,est},i} - P_{\text{bat,real},i}| \qquad (24) $$

subject to (25) and (26):

$$ \begin{array}{l} 0 \le \eta^{+} \le 1 \qquad 0 \le \eta^{-} \le 1 \\ 0 \le p_{\text{ic}} \qquad \{\eta^{+}, \eta^{-}, p_{\text{ic}}\} \in \mathbb{R} \end{array} \qquad (25) $$

$$ \{\eta^{+}, \eta^{-}, p_{\text{ic}}\} \in \mathbb{R} \qquad (26) $$

The results of the inverter model fitting are presented in Tab. V.

<table><caption>TABLE V
RESULTS OF INVERTER MODEL FITTING</caption><thead><tr><th>LVL</th><th>η<sup>+</sup> [%]</th><th>η<sup>-</sup> [%]</th><th>p<sub>ic</sub> [W]</th></tr></thead><tbody><tr><td>0</td><td>-</td><td>-</td><td>-</td></tr><tr><td>1</td><td>97.5</td><td>99.31</td><td>381.1</td></tr><tr><td>2</td><td>p1</td><td>p2</td><td>384.9</td></tr></tbody></table>

<table><tr><td>LVL</td><td>Model depth level</td></tr><tr><td>η<sup>+</sup></td><td>Charging efficiency of the inverter</td></tr><tr><td>η<sup>-</sup></td><td>Discharging efficiency of the inverter</td></tr><tr><td>p<sub>ic</sub></td><td>Self-consumption of the inverter</td></tr><tr><td>p1 & p2</td><td>Power-dependent efficiency</td></tr></table>

Fig. 6 shows the load-dependent efficiency levels of the inverter as a function of the percentage power.

<figure><img src="image_7.png" /><figcaption>Fig. 6. Power-dependent efficiency of the inverter.</figcaption></figure>

The different model depths based on the metrics MAE, RMSE and ME are then shown in Fig. 7.

<figure><img src="image_8.png" /><figcaption>Fig. 7. Results of modelling different inverter model depths.</figcaption></figure>

The inverter model level three with a load-dependent efficiency characteristic curve achieves a lower error. The effort for fitting the data is low compared to battery models with more complex properties and more parameters.

### E. Battery storage system model

The BSS model is now combined from the battery and inverter models. As shown in Fig. 8, the inverter model is connected in series with the BSS model. AC power is transferred to the inverter as an input variable, the result of the battery storage model is the SOC.

<figure><img src="image_9.png" /><figcaption>Fig. 8. Block diagram of a BSS, consisting of the inverter and the battery model.</figcaption></figure>

Four combinations of different battery and inverter models were created based on the configurations and
fittings considered. Level zero comprises the battery and inverter model zero. Level one consists of the inverter model level one and the battery model level three. Level two is a combination of the inverter model level one and the battery model level four. The deepest models presented, from battery models level six and inverter model level two, are selected as the last model level three. An overview is given in Table VI.

<table><caption>TABLE VI
SIMULATION OVERVIEW OF BSS MODELS</caption><thead><tr><th>LVL</th><th>SOH</th><th>η<sup>+</sup><sub>B</sub>, η<sup>-</sup><sub>B</sub></th><th>p<sub>sd</sub>, i<sub>sd</sub></th><th>KM</th><th>DV</th><th>η<sup>+</sup><sub>I</sub>, η<sup>-</sup><sub>I</sub></th></tr></thead><tbody><tr><td>0</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>1</td><td>✓</td><td>✓</td><td>✓</td><td>-</td><td>-</td><td>-</td></tr><tr><td>2</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>-</td><td>-</td></tr><tr><td>3</td><td>✓</td><td>SOC</td><td>✓</td><td>✓</td><td>✓</td><td>P</td></tr></tbody></table>

<table><tr><td>LVL</td><td>Model depth level</td></tr><tr><td>SOH</td><td>State of Health</td></tr><tr><td>η<sup>+</sup><sub>B</sub>, η<sup>-</sup><sub>B</sub></td><td>Charging and discharging efficiency of the battery</td></tr><tr><td>p<sub>sd</sub>, i<sub>sd</sub></td><td>Self-discharge power/current</td></tr><tr><td>KM</td><td>Kinetic model for energy balancing</td></tr><tr><td>DV</td><td>Dynamic voltage consideration</td></tr><tr><td>η<sup>+</sup><sub>I</sub>, η<sup>-</sup><sub>I</sub></td><td>Charging and discharging efficiency of the inverter</td></tr><tr><td>P</td><td>Power-dependent efficiency</td></tr><tr><td>SOC</td><td>State of charge-dependent losses</td></tr></table>

V. EXPERIMENTAL VALIDATION

Based on the model fitting, the results of an experimental validation are presented. New power and SOC curves, derived from an additional dataset from a different period, are used to make a comparative assessment of the results. For example, Fig. 9 compares the estimated SOC curves from the level zero and level three models with the measured SOC curves over a period of 170 hours.

As can be seen, the influence of the model depth is particularly large at low discharge depths and high loads. In order to evaluate the differences between the model depths, the error of the models from level zero to three is shown in the following Fig. 10.

A lower model depth, such as model zero and one, is found to have weaknesses in the area of low discharge depths and high discharge powers, in which the deviation between measured and calculated SOC is particularly high. Model levels two and three, on the other hand, have the advantage of taking a kinetic approach into account, which takes better consideration of the influence of the load on the SOC calculation, resulting in a more accurate estimation. The metrics MAE, ME and RMSE are now used for the final evaluation of the SOC curves from

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 9. Comparison of estimated and real SOC curves over a 170-hour period.</figcaption></figure>

<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 10. Comparison of the absolute error between the calculated SOC of the models level zero to three and the measured SOC over a period of 170 hours.</figcaption></figure>

the models in comparison to the measured curves, as illustrated in Fig. 11.

The BSS model at level zero has an MAE of 3.1% and a significant ME of 20.9%. The RMSE is 0.174. Model level one can reduce the MAE to 1.7% and the
<figure><img src="image_14.png" /><figcaption>Fig. 11. Results of different model depths</figcaption></figure>

ME to 15.9%, with the RMSE reduced to 0.109. A substantial enhancement is observed in model level two, where the MAE is reduced to 1.33% and the ME to 10.6%, with an RMSE of 0.063. Conversely, model level three fails to demonstrate any significant improvements, with the MAE remaining at 1.33%, the ME at 11.2%, and the RMSE at 0.066.

## VI. DISCUSSION

The validation demonstrates that the depth of the model has a substantial impact on the accuracy of the SOC estimation. Simple models with low complexity (e. g. level zero) exhibit greater deviations, while models with increasing depth (e. g. level two or three) enable a more precise SOC calculation. However, an increase in model complexity does not necessarily result in a significant improvement in estimation accuracy. The discrepancy in accuracy between level two and level three is negligible, while the level of computational effort and the number of parameters to be optimised increase considerably.

Energy reservoir models have been shown to provide adequate results in the absence of a kinetic approach. However, when the kinetic approach is incorporated, the estimation is shown to improve, particularly in the context of load changes and instances of high short-term loads. This observation assumes particular significance in the context of second-life batteries, as the kinetic approach facilitates the more accurate modelling of dynamic effects. Conversely, simpler models that do not consider kinetic effects exhibit deficiencies at high load

peaks, as these peaks result in greater deviations from the SOC.

The division of inverters and battery models has proven to be a flexible approach, particularly in the context of second-life batteries. This division facilitates the replacement of inverters or the development of new systems with different inverters without the need for refitting the battery model. This approach significantly improves the flexibility of future battery storage models. In instances where practical measured values for an inverter are not available, the manufacturer's characteristic curves can be used as a substitute to increase the precision of the simulation results. This approach can also be employed to evaluate alternative concepts, such as those that use an AC-DC converter for battery storage system charging, while a DC-DC converter is used for the fast-charging station.

Each new battery model requires refitting, as its parameters are empirically determined and tailored to the specific fitted state. The future aging state of the battery is not taken directly into account. However, the model could be further adapted and continuously re-fitted during operation to ensure that the system is modelled in an age-adequate way.

Another aspect that was not taken into account in this approach is temperature influences. These were excluded due to the constant air conditioning of the cells, as the cell temperature is always kept stable in the real application. Nevertheless, this could be a starting point for future extensions, especially if a deliberate variation of the operating temperature is considered as an optimisation strategy.

## VII. CONCLUSION

In this work, the influence of modelling depth on the accuracy of state of charge estimation in battery storage systems was investigated. The aim was to find a balance between modelling complexity and accuracy. The analysis showed that simple models with a kinetic approach already provide good results and are sufficient for many applications. In particular, models based on the energy reservoir approach with two storage units enable a sufficiently accurate estimation of the state of charge with low computational effort.

The models deliver robust results, especially when the kinetic approach is included. The separation of the inverter and battery models increases flexibility and enables simple adaptation to different systems. The models can also be used for second-life batteries and provide reliable state of charge estimations.
A new battery model requires a new fitting, as the parameters were determined empirically, the ageing process of the battery is not taken into account directly, but only via the current ageing value. Temperature-dependent effects were not included, as the air conditioning in the real application remains constant. The inclusion of a mandatory manufacturer specification of parameters, such as the efficiency rate during charging and discharging, as well as self-discharge, on the data sheet of the battery storage system, has the potential to enhance transparency and facilitate more rapid and precise integration into applications during the planning phase of stationary battery storage systems.

In future work, the model will be extended to include temperature dependencies by specifically varying the target temperature of the air conditioning. This could provide new insights into optimized operational management. In addition, the models developed should serve as a basis for optimization algorithms that improve the economic operation of the storage system and open up new applications.

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