Assessment and Future Directions of Nonlinear Model Predictive ...

Close-Loop Stochastic Dynamic Optimization 307The developed model considers both the reactor and the cooling jacket energybalance. Thus, the dynamic performance between the cooling medium flow rateas manipulated variable and the controlled reactor temperature is also includedin the model equations. The open-loop optimal control is solved first for thesuccessive optimization with moving horizons involved in NMPC. The objectivefunction is to maximize the production of B at the end of the batch CB f whileminimizing the total batch time t f with β =1/70:min (−CB f + β · t f ) (5)∆t, ˙V cool , feedsubject to the equality constraints (process model equations (2) – (4)) as wellas path and end point constraints. First, a limited available amount of A to beconverted by the final time is fixed to ∫ t ft n 0=0 A(t)dt = 500mol.Furthermore,soasto consider the shut-down operation, the reactor temperature at the final batchtime must not exceed a limit (T (t f ) ≤ 303 K). There are also path constraintsfor the maximal reactor temperature and the adiabatic end temperature T ad .The latter is used to determine the temperature after failure. This is a safetyrestriction to ensure that even in the exterme case of a total cooling failureno runaway will occur (T (t) ≤ 356 K; T ad (t) ≤ 500 K) [1]. Additionally, thecooling flow rate changes from interval to interval are restricted to an upperbound: ∥ ˙V cool (t +1)− ˙V cool (t) ∥ ≤ 0.05. The decision variables are the feed flowrate into the reactor, the cooling flow rate, and the length of the different timeintervals. A multiple time-scale strategy based on the orthogonal collocationmethod on finite elements is applied for both discretization and implementationof the optimal policies according to the controller’s discrete time intervals (6 –12 s; 600 – 700 intervals). The resulting trajectories of the reactor temperatureand the adiabatic end temperature (safety constraint) for which constraints havebeen formulated are depicted in Fig. 1. It can be observed that during a largepart of the batch time both states variables evolve along their upper limits i.e.the constraints are active. The safety constraint (adiabatic end temperature),in particular, is an active constraint over a large time period (Fig. 1 right).Although operation at this nominal optimum is desired, it typically cannot beahieved with simultaneous satisfaction of all contraints due to the influence ofuncertainties and/or external disturbances. However, the safety hard-constraintsshould not be violated at any time point.3 Dynamik Adaptive Back–Off StrategyBased on the open-loop optimal control trajectories of the critical state variables,in this section, a deterministic NMPC scheme for the online optimizationof the fed-batch process is proposed. Furthermore, the momentary criteria on therestricted controller horizon with regard to the entire batch operation is howeverinsufficient. Thus, the original objective of the nominal open-loop optimization