*2.2. Equivalent Circuit Model*

Equivalent circuit modelling is a common approach to model lithium-ion batteries. ECMs describe battery dynamics with only a few states and parameters. Due to their simplicity, they are often used to predict the SOC or the SOH of batteries [33,34]. There is no agreement in the literature about the type of equivalent circuit to be used for lithium-ion batteries [2]: Simple empirically oriented versions of ECMs model battery dynamics with a voltage source, a serial resistor and one or more RC elements [33,35–40]. Electrochemically oriented models will typically include a Warburg diffusion element (either in series with the RC element or within the RC element). A more detailed analysis, particularly in the context of the present combination with NODEs, is out of the scope of the present study.

One can take into account that the circuit parameters may depend on SOC, temperature, the battery current, and the cycle number [36,40].

As in Ref. [17], we used a simple ECM as a basis for battery modelling. The chosen ECM is shown in Figure 2. It is composed of an SOC-dependent voltage source, a serial resistor, and one RC circuit. The open-circuit voltage of phase-change active materials such as LFP is known to exhibit a path dependency [21]: The measured voltage is different after discharge with a subsequent rest phase or after charge with a subsequent rest phase at the same SOC. To describe this effect with our model, we included a hysteresis voltage drop representing the particular feature of the studied LFP cell.

The following equation system describes the chosen ECM including parameter dependencies on battery current and SOC:

$$\frac{\text{dSOC}}{\text{dt}} = -\frac{1}{\text{C}\_{\text{bat}}} i\_{\text{bat}} \tag{4}$$

$$\frac{\mathrm{d}\upsilon\_{\mathrm{RC}1}}{\mathrm{d}t} = \frac{1}{C\_1} \cdot \left(\dot{\imath}\_{\mathrm{bat}} - \frac{1}{R\_1(\mathrm{SOC}, \dot{\imath}\_{\mathrm{bat}})} \cdot \upsilon\_{\mathrm{RC}1}\right) \tag{5}$$

$$\upsilon\_{\rm bat} = \upsilon\_{\rm CC}(\rm SOC) - \upsilon\_{\rm hys} \cdot \text{sgn}(i\_{\rm bat}) - R\_{\rm S} \cdot i\_{\rm bat} - \upsilon\_{\rm RC1},\tag{6}$$

where *C*bat is the battery capacity, *R*<sup>S</sup> the serial resistance, *R*1(SOC, *i*bat) the charge-transfer resistance in the RC circuit depending on SOC and battery current, and *C*<sup>1</sup> the doublelayer capacitance (which, in our case, may include other physical contributions to voltage dynamics, for example, solid-state diffusion). It should be noted that considering a nonconstant *C*<sup>1</sup> could improve the approximation capability of the model. However, we

decided to use a constant double-layer capacitance at the present stage because we wanted to focus on the most important effects which we expect from the charge-transfer resistance and its dependency on the battery current and the SOC. The SOC-dependent open-circuit voltage (OCV) is labelled *v*OC(SOC) and the hysteresis voltage drop is given by *v*hys times the signum function of the battery current sgn(*i*bat). The hysteresis voltage drop could have also been modelled by the current- and SOC-dependent resistance *R*1. However, we did not include the voltage hysteresis into *R*<sup>1</sup> to maintain the physical characteristics of both *v*hys and *R*1. The battery voltage *v*bat is the output of the dynamic system and the battery current *i*bat is the external variable. We define the current positive for battery discharge and negative for battery charge. Note that Equations (4) and (5) represent 'standard', physics-derived ordinary differential equations (ODEs).

**Figure 2.** ECM of a battery consisting of an SOC-dependent voltage source, a hysteresis voltage drop, a series resistor, and an RC circuit.
