3.2 Open-loop optimal control (offline optimisation)
To further assess advantages of this hybrid model, it is first used in offline optimisation to test its ability in long-term process predictive modelling and optimisation. The objective function is to maximise the final lutein production over a 5-day fed-batch operation system in which incident light intensity (100 μmol m-2s-1 to 800 μmol m-2s-1) and nitrate inflow rate (0 to 17.5 mL h-1 with a concentration of 0.1 M so that total added liquid volume is negligible) can change once per day. Once optimal control actions were estimated by the hybrid model, they were implemented and verified in the computational experiment (using the original complex kinetic model). In addition, the original model was also used to search for the theoretical maximum lutein production in this fed-batch process. This is used as a benchmark to evaluate the hybrid model’s predictive capability. Initial operating conditions of this process are listed in Table 1.
Fig. 2 and Fig. 3 shows the model prediction result and experimental verification results. From Fig. 2, it can be observed that the hybrid model can well predict the dynamic process behaviour over the entire operating time course before executing the process. This directly speaks of the predictive capability of the hybrid model. Furthermore, through (computational) experimental verification, the hybrid model is found to provide the same lutein production as the maximum theoretical value estimated by the original kinetic model, indicating its high performance in offline optimisation. The two processes (designed based on the hybrid model: Fig. 2(a), 2(e), 3(c); designed based on the original model: Fig. 2(b), 2(f), 3(d)) have the same trajectory for biomass growth, lutein production, and control actions of incident light intensity. However, culture nitrate concentration (Fig. 2(b) vs. Fig. 2(d)) and nitrate inflow rate (Fig. 3(a) vs. Fig. 3(b)) deviate substantially between the two processes. This indicates that this process is not sensitive to the change of nitrate concentration as long as it is sufficient (400 to 900 mg L-1). It also suggests that multiple optimal solutions exist for this offline optimisation problem.