**2. General Modeling Approach**

The general electro-thermal model framework is shown in Figure 1. It consist of three submodels: an electrical model, a heat generation model and a thermal model. The goal of the total model is to estimate the core temperature of the battery cell. Therefore, a commonly used electro-thermal co-simulation is performed to capture the temperature dependency of several model parameters [21,24,25,27]. The core temperature as and output of the thermal submodel is fed back to the other two submodels, which are described in the following sections. For the integration of the model in BEV applications, an observer is implemented that is described in detail in Section 3.4. The current *I*, which is determined via measurement in a realistic BEV system, is the input to the total model structure, as well as the observer. For the investigation in this work, two thermal models are implemented and compared that both fit in the same framework. The thermal models are described in detail in Section 3.

**Figure 1.** Schematic representation of the modeling framework of an electro-thermal battery cell model, consisting of an electrical, a heat generation, a thermal submodel, and an observer. Two different thermal models are implemented for comparison.

### *2.1. Electrical Model*

An Equivalent Circuit Model (ECM) is used as an electrical model, as frequently seen in thermal modeling of battery cells [27,51]. In this work, the model consists of a voltage source, representing the Open Circuit Voltage (*OCV*) of the cell, a series resistance *R*<sup>0</sup> corresponding to the cell's ohmic resistance and two RC-elements, which stand, for the voltage drop due to overvoltages, e.g., charge transfer, diffusion, phase change overvoltages and other losses. The cell terminal voltage *U* can be calculated from Equation (1) below.

$$
\mathcal{U}I = \mathcal{O}\mathcal{C}V + \mathcal{R}\_0 \cdot I + \mathcal{U}\_{\mathcal{R}\mathcal{C}\_1} + \mathcal{U}\_{\mathcal{R}\mathcal{C}\_2} \tag{1}
$$

All the components of the ECM, e.g., resistances, capacitances and the *OCV* are dependent on the cell core temperature and the SOC. The temperature is fed back from the thermal submodel and the SOC is calculated inside the electrical submodel using coulomb counting. The resistances and capacitances used for the model parametrization were determined analytically for the 25 Ah cell and published in a previous publication [52].

#### *2.2. Heat Generation Model*

The heat generation model comprises two parts. It separately calculates the heat generation in the cell and the electronics components. For the heat generation in the jelly roll, Equation (2) is used based on the simplified energy balance in electrochemical systems by Bernardi et al. [53]:

$$
\dot{Q}\_{\text{gen}} = \dot{Q}\_{\text{irr}} + \dot{Q}\_{\text{rev}} = I^2 \cdot R\_0 + \frac{\mathcal{U}\_{\text{RC}\_1}^2}{R\_1} + \frac{\mathcal{U}\_{\text{RC}\_2}^2}{R\_2} + I \cdot T \cdot \frac{\text{dOCV}}{\text{d}T} \tag{2}
$$

In Equation (2), the first three terms stand for the irreversible heat generation resulting from the voltage drop at the electrical resistances shown in the electrical submodel. It is always positive and therefore, leads to the heating of the cell. The last term in Equation (2) is reversible heat generation, which results from the entropy change during intercalation and the deintercalation of the lithium ions. It may be positive or negative, depending on the SOC and the direction of the current and, therefore, may heat or cool the cell, respectively. The entropy coefficient d*OCV*/d*T* is dependent on the temperature and was also determined experimentally in [52].

The second part of the heat generation model calculates the heat generation in the electronics based on joule heating by

$$\dot{Q}\_{\text{elec}} = \sum\_{i=1}^{n} I\_i^2 \cdot R\_{\text{elec},i} \tag{3}$$

*R*elec,*<sup>i</sup>* is the ohmic resistance of the electronics' pieces, respectively, and is strongly dependent on each components' temperature. Thereby, n is the number of all different electronics segments. A detailed description of the electronics and the corresponding model is given in Section 3.1 .

#### **3. Thermal Models of an Intelligent Cell**

The purpose of the thermal models is to estimate the jelly roll temperature for the given heat input by the heat generation model. In order to compare a physics-based to a data-driven model, a TECM and an ANN are implemented for the example of a prismatic cell prototype for intelligent batteries published in [19]. Thereby, the geometrical as well as boundary conditions for this prototype are defined by the reference system and taken into account for the modeling. Since the focus of this work is on the differences of the two approaches, they are implemented in the same electro-thermal framework to compare accuracy, computational effort and time and applicability to an in-use temperature estimation in a BTMS.
