## 3 Real-time Capable Robust Maximum Power Point Tracking  
### Algorithm for Combined Parameter Identification and Maximum Power Point Tracking  
The proposed algorithm for the real-time capable robust maximum power point tracking summarized in Figure 7 consists of two phases, where only scalar and low-dimensional matrix-vector operations are involved. This leads to computing times that are much smaller than the typical sampling rates of any SOFC controller. In the _first phase_, the online identification of the current-power characteristic using the Kalman filter summarized in Section 2 is initialized and executed. For that purpose, a sufficiently rich excitation signal for the electric current is required. In the scenario under investigation, this input signal is given by a piecewise constant current \(I_{k}\) that increases and decreases successively, typically chosen from the desired operating region of Ohmic polarization. In each temporal discretization step, the Kalman filter with the prediction and innovation steps in Equations (19) and (20) as well as Equations (21) and (22), respectively, is evaluated. This open-loop identification is continued up to the point, where the computed standard deviation \(\sigma_{P_{\text{EL}},k}\) falls below the threshold value \(\sigma_{P_{\text{EL}},\min}\) for the first time. The same has to hold for the matrix trace \(\tau_{k}\) defined in Equation (25) to ensure sufficiently small uncertainty in the estimated coefficients of the approximated current-power characteristic in Equation (12). As soon as this point has been reached, the _second phase_ of the algorithm is initialized. Instead of using a predefined current variation \(I_{k}\), changes in the current now result form an online maximizationof the power characteristic. In an uncertainty-free setting, the current \(I_{k}^{*}\) at the maximum of the power curve in Equation (12) would be given by the necessary optimality condition  
\[\frac{\partial P_{\text{EL},k}}{\partial I_{k}}= \begin{bmatrix}0&1&2I_{k}^{*} \end{bmatrix} \cdot\mathbf{x}_{k}=0 \tag{32}\]  
leading to the optimal current value  
\[I_{k}^{*}=-\frac{x_{2,k}}{2x_{3,k}}\enspace. \tag{33}\]  
Figure 7: Structure diagram of the Kalman filter-based parameter identification and maximum power point tracking. However, uncertainty in the estimated parameters \(\mathbf{x}_{k}\) for the locally approximated current-power characteristic may lead to an undesirable overshoot over the maximum, corresponding to currents in areas in which the power curve has a negative slope. To prevent this phenomenon, the variance estimates for the second and third element of \(\mathbf{x}_{k}\) are taken into account in the current update according to  
\[I_{k}^{*}=\frac{-\mu_{x,2,k}^{\mathrm{e}}+\eta\cdot\sqrt{\mathbf{e}_{2}^{T} \cdot\mathbf{C}_{x,k}^{\mathrm{e}}\cdot\mathbf{e}_{2}}}{2\cdot\left(\mu_{x,3,k }^{\mathrm{e}}-\eta\cdot\sqrt{\mathbf{e}_{3}^{T}\cdot\mathbf{C}_{x,k}^{\mathrm{ e}}\cdot\mathbf{e}_{3}}\right)}\enspace, \tag{34}\]  
where \(\eta>0\) represents a user-defined safety margin and \(\mathbf{e}_{2}= \begin{bmatrix}0&1&0\end{bmatrix} ^{T}\) and \(\mathbf{e}_{3}= \begin{bmatrix}0&0&1\end{bmatrix} ^{T}\). To further smoothen the variation of the current update (avoiding the excitation of non-modeled high-frequency dynamics in the power characteristic violating the assumption of the integrator disturbance model for the system dynamics in Equation (10)), the actual current for the subsequent time step is chosen as the low-pass filtered system input  
\[I_{k+1}= \begin{cases}\bar{I}_{k+1}&\text{for}\quad\bar{I}_{k+1}\leq I_{k}^{*} \\ I_{k}^{*}-|\bar{I}_{k+1}-I_{k}^{*}|&\text{else}\end{cases}  \tag{35}\]  
with  
\[\bar{I}_{k+1}=I_{k}+(I_{k}^{*}-I_{k})\cdot\sqrt{\frac{\mathbf{e}_{2}^{T}\cdot \mathbf{C}_{x,k}^{\mathrm{e}}\cdot\mathbf{e}_{2}}{\mathrm{trace}\left\{ \mathbf{C}_{x,k}^{\mathrm{e}}\right\}}}\enspace. \tag{36}\]  
**Remark 3**.