## 2 Fuel Cell Model
### Model Description  
[PERSON] et al. [10] developed the model, and it is an adapted version of the [PERSON] et al. [8] model. The difference is that the adapted model has the property of analytical differentiability, which is the reason for its selection. This property is beneficial for various control methodologies [20; 21], and parameter sensitivity analyses (Section 3.2). The model is a zero-dimensional physical PEMFC model, and Figure 1 gives a schematic overview. It utilizes, amongst others, mass balances, linear nozzle equations, diffusion equations, electrochemical equations, and the ideal gas law [22; 23; 24]. However, it does not consider energy balances and thus cannot model the FC temperature over time. The model treats the measured temperature as an input. This temperature is assumed to be the uniform temperature for the whole FC. The model consists of a cathode, anode, membrane, and electrochemical submodel, which are interconnected. The model equations and their derivations are not the focus of this work. More information in this regard is available in [10]. The cathode submodel has four mass states (oxygen mass \(m_{\text{ca,O}_{2}}\), nitrogen mass \(m_{\text{ca,N}_{2}}\), vapor mass \(m_{\text{ca,vap}}\), and liquid water mass \(m_{\text{ca,liq}}\)) and two pressure states (supply manifold pressure \(p_{\text{ca,sm}}\), and exit manifold pressure \(p_{\text{ca,en}}\)). The anode submodel also has four mass states (hydrogen mass \(m_{\text{an,H}_{2}}\), nitrogen mass \(m_{\text{an,N}_{2}}\), vapor mass \(m_{\text{an,vap}}\), and liquid water mass \(m_{\text{an,liq}}\)) but only one pressure state (exit manifold pressure \(p_{\text{an,em}}\)). The membrane submodel only has the membrane water activity state \(a_{\text{m}}\). The model inputs are the air mass flow \(m_{\text{ca,in}}\), the hydrogen mass flow \(m_{\text{an,in}}\), the anode supply manifold pressure \(p_{\text{an,sm}}\), the relative humidity of the air mass flow \(\varphi_{\text{ca,in}}\), the purging signal \(\alpha_{\text{purge}}\), the FC temperature \(T\), the atmospheric pressure \(p_{\text{atm}}\), and the current \(I\). The outputs are the cathode supply manifold pressure \(p_{\text{ca,sm}}\), the cathode exit manifold pressure \(p_{\text{ca,em}}\), the anode exit manifold pressure \(p_{\text{an,em}}\), and the voltage \(U\). Hence, the following equations describe the nonlinear FC state-space model:  
\[\dot{\mathbf{x}}_{\text{nr}} =\mathbf{f}_{\text{nr}}(\mathbf{x}_{\text{nr}},\mathbf{u},\theta) \tag{1}\] \[\mathbf{y} =\mathbf{g}_{\text{nr}}(\mathbf{x}_{\text{nr}},\mathbf{u},\theta) \tag{2}\]  
Here, \(\mathbf{x}_{\text{nr}}=\mathbf{x}_{\text{nr}}(t)\) denotes the non-reduced state vector, \(\mathbf{u}=\mathbf{u}(t)\) the input vector, \(\mathbf{y}=\mathbf{y}(t)\) the output vector, \(\theta\in\mathbb{R}^{25}\) the parameter vector, \(\mathbf{f}_{\text{nr}}\) the non-reduced system function, \(\mathbf{g}_{\text{nr}}\) the non-reduced output function, and \(t\) the time [25]. The respective vectors are structured as follows:  
\[\mathbf{x}_{\text{nr}}= \begin{bmatrix}p_{\text{ca,sm}}\\ m_{\text{ca,O2}}\\ m_{\text{ca,N2}}\\ m_{\text{ca,vap}}\\ m_{\text{ca,in}}\\ m_{\text{ca,in}}\\ m_{\text{an,n,N2}}\\ m_{\text{an,vap}}\\ m_{\text{an,in}}\\ p_{\text{an,em}}\\ q_{\text{m}}\end{bmatrix} ,\ \mathbf{u}= \begin{bmatrix}\dot{m}_{\text{ca,in}}\\ \dot{m}_{\text{an,in}}\\ \dot{m}_{\text{an,in}}\\ \dot{p}_{\text{an,sm}}\\ \dot{q}_{\text{ca,in}}\\ \dot{q}_{\text{purge}}\\ T\\ p_{\text{am}}\\ I\end{bmatrix} ,\ \mathbf{y}= \begin{bmatrix}p_{\text{ca,sm}}\\ p_{\text{ca,em}}\\ p_{\text{pa,em}}\\ \dot{p}_{\text{an,em}}\\ \dot{m}_{\text{an,em}}\\ I\end{bmatrix}