(251d) On the Calculation of Operability Sets of Nonlinear High-Dimensional Processes

The calculation of operability sets for nonlinear high-dimensional processes is a difficult task because of the immense number of calculations required. If the available input is a 3D parallelepiped, then there are six 2-D faces to map and if we define a 2-D grid of N^2 points per face, then the total calculations are 6N^2. For N=10-100 the number of calculation points is 600-60,000. This number increases substantial when dealing with plant-wide problems in which the number of input and outputs that we need to consider will be in the hundreds or thousands. Furthermore, the image of the bounding surface of the input set (AIS) is not sufficient in defining the bounding surface of the output set (AOS), in the case of a process with input multiplicity.

Here we propose a new approach, called Design of Selective Calculations (DoSC), motivated by the well established design of experiments procedure. It enables the judicious selection of points in the input set for which the corresponding points in the output set are calculated using the detailed steady state model of the continuous process. Designing lever-3 calculation points, in a similar fashion to the design of level-3 experiments, enables the collection of enough data for the estimation of the parameters of a quadratic response surface model (RSM) of the process that is much simpler than the detailed model. The DoSC approach can very efficiently calculate the achievable output set (AOS) from the available input sets (AIS). The developed method is illustrated with two motivating examples and a plant-wide industrial process, the Tennessee Eastman challenge problem. For all the examples considered, it is shown that the AOS obtained from the RSM model differs minimally, if at all, from the one obtained from the detailed model.

A particular interest is whether the quadratic response surface models are able to accurately describe the phenomenon of input multiplicity, which causes the output map of the boundary of the AIS to be different from the boundary of the AOS. This is a substantial limitation of previous methods that aimed to map just boundary points. It is shown here that the RSM is able to very accurately represent such challenging cases with just quadratic terms. The calculation of the bounding surface of the AOS in the cases with input multiplicity is easily achieved using the RSM rather than the detailed model. Two calculation methods are proposed