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