## 2 Kalman Filter Design for Real-Time Estimation of Electric Power Characteristic Of Fuel Cells
### Fundamental Current-Voltage and Current-Power Characteristics of SOFCs
In this case, the parameter estimates listed in Table 1 are obtained, where the coefficients \(a_{4}\), \(a_{5}\), and \(a_{6}\) can be set to zero without any loss of information as long as \(\dot{m}_{\mathrm{H_{2},in,}k}\) is constant. Additive offset terms are then fully included in the coefficient \(a_{0}\). This offline identification has been repeated in Figure 6 for larger, temporally slowly varying hydrogen mass flows \(\dot{m}_{\mathrm{H_{2},in,}k}\) at the anode inlet. Again, the same analytic expressions, Equations (2) and (3), were used to compute reasonably accurate approximations. The corresponding numerical parameter values are summarized in Table 2. For both scenarios, they have been computed by a linear least-squares parameter identification similar to the one used in [22] for the dynamics of the electrochemical fuel cell behavior. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline \(a_{0}\) & \(-23.3553\) \\ \(a_{1}\) & \(135.3495\) \\ \(a_{2}\) & \(-10.0203\) \\ \(a_{3}\) & \(-229.1167\) \\ \(a_{4}\) & \(0\) \\ \(a_{5}\) & \(0\) \\ \(a_{6}\) & \(0\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Offline identified parameters of the current–voltage and current–power characteristics as a function of the stack current \(l_{k}\) for constant hydrogen mass flow \(\dot{m}_{\mathrm{H_{2},in,}k}\approx 5.1\cdot 10^{-6}\ \frac{\mathrm{kg}}{\mathrm{s}}\) in Figure 5d,e. Figure 6: Offline identification of the current–voltage and current–power characteristic for larger, temporally varying hydrogen mass flows \(\dot{m}_{\mathrm{H_{2},in}}\). (**a**) Fuel cell voltage depending on the terminal current and hydrogen mass flow. (**b**) Fuel cell power depending on the terminal current and hydrogen mass flow. For that purpose, Equation (2) has been re-written in the following matrix-vector form  
\[\mathbf{\mathcal{U}}_{K}=\mathbf{\mathcal{D}}_{K}\cdot\mathbf{a}\enspace, \tag{4}\]  
where the vector  
\[\mathbf{\mathcal{U}}_{K}= \begin{bmatrix}U_{1}&\ldots&U_{K}\end{bmatrix} ^{T} \tag{5}\]  
consists of all voltage measurements over a time window \(k\in\{1,\ldots,K\}\), where \(K\) denotes a sufficiently large number of (typically a few hundred) measurement points. In addition, current and mass flow measurements are included in the data matrix  
\[\mathbf{\mathcal{D}}_{K}= \begin{bmatrix}1&I_{1}&I_{1}^{2}&\ln\left(I_{1}\right)& \dot{m}_{\mathrm{H}_{\mathrm{H}_{2}},\mathrm{in},1}&I_{1}\cdot\dot{m}_{ \mathrm{H}_{\mathrm{H}_{2}},\mathrm{in},1}&I_{1}\cdot\dot{m}_{\mathrm{H}_{ \mathrm{H}_{2}},\mathrm{in},1}^{2}\\ \vdots&&\vdots\\ 1&I_{K}&I_{K}^{2}&\ln\left(I_{K}\right)&\dot{m}_{\mathrm{H}_{\mathrm{H}_{2}}, \mathrm{in},K}&I_{K}\cdot\dot{m}_{\mathrm{H}_{\mathrm{H}_{2}},\mathrm{in},K}&I _{K}\cdot\dot{m}_{\mathrm{H}_{\mathrm{H}_{2}},\mathrm{in},K}^{2}\end{bmatrix} \enspace, \tag{6}\]  
while  
\[\mathbf{a}= \begin{bmatrix}a_{0}&a_{1}&a_{2}&a_{3}&a_{4}&a_{5}&a_{6}\end{bmatrix}  ^{T} \tag{7}\]  
denotes the parameter vector to be identified. This identification is performed by the classical least-squares solution expressed in terms of  
\[\mathbf{a}=\mathbf{\mathcal{D}}_{K}^{+}\cdot\mathbf{\mathcal{U}}_{K}\enspace, \tag{8}\]  
where \(\mathbf{\mathcal{D}}_{K}^{+}\) is the left pseudo inverse of the data matrix \(\mathbf{\mathcal{D}}_{K}\) according to  
\[\mathbf{\mathcal{D}}_{K}^{+}=\left(\mathbf{\mathcal{D}}_{K}^{T}\cdot\mathbf{\mathcal{D}}_ {K}\right)^{-1}\cdot\mathbf{\mathcal{D}}_{K}^{T}\enspace. \tag{9}\]  
Note, both experimental datasets and the presented analytic representations for the current-power characteristic are used for validating the filter-based identification and maximum power point tracking procedure in the remainder of this paper. For this maximum power point tracking, it is essential to estimate variations of the current-voltage as well as current-power characteristics in real time, as soon as new measured data become available. This task can be solved partially by classical recursive formulations of least-squares estimation schemes.