3.1 Result of hybrid model construction
Through automatic model structure identification, the hybrid model is identified as Eq. (8a)-(8c) (all inactive terms are removed; active binary variables are not shown as they are equal to 1). Table 2 shows the parameter estimation result. Fig. 1 shows the modelling fitting result.
\begin{equation} \frac{dc_{X}}{\text{dt}}=u_{0}\bullet\frac{c_{N}}{c_{N}+K_{N}}\ \bullet\frac{I_{0}e^{-\tau\bullet c_{X}\bullet z}}{I_{0}e^{-\tau\bullet c_{X}\bullet z}+k_{s}}\ \bullet c_{X}+{a_{10}\bullet c}_{1}+{a_{14}\bullet c}_{1}{\bullet c}_{4}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(8a\right)\nonumber \\ \end{equation}\begin{equation} \frac{dc_{N}}{\text{dt}}=-Y_{N/X}\bullet u_{0}\bullet\frac{c_{N}}{c_{N}+K_{N}}\ \ \bullet\frac{I_{0}e^{-\tau\bullet c_{X}\bullet z}}{I_{0}e^{-\tau\bullet c_{X}\bullet z}+k_{s}}\bullet c_{X}+F_{\text{in}}\bullet c_{N,in}+{a_{20}\bullet c}_{2}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(8b\right)\nonumber \\ \end{equation}\begin{equation} \frac{dc_{L}}{\text{dt}}=Y_{L/X}\bullet u_{0}\bullet\frac{c_{N}}{c_{N}+K_{N}}\ \ \bullet\frac{I_{0}e^{-\tau\bullet c_{X}\bullet z}}{I_{0}e^{-\tau\bullet c_{X}\bullet z}+k_{\text{sL}}}\bullet c_{X}+{a_{31}\bullet c}_{3}{\bullet c}_{1}+{a_{33}\bullet c}_{3}{\bullet c}_{3}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(8c\right)\nonumber \\ \end{equation}
From the figure, it is seen that the hybrid model can accurately fit multiple datasets collected over a broad spectrum of operating conditions (e.g. substrate limiting to substrate excessive, low light intensity to high light intensity). In addition, the total number of binary variables assigned to the data-driven model is 15, whilst only 5 are active after solving the parameter estimation problem. The correct kinetic expression is also identified more efficiently than traditional model discrimination. Compared to the complex kinetic model shown in Eq. (1a)-(1d), the current hybrid model well fits all the datasets with a simpler model structure.
Table 2: Parameter values of the current hybrid model