(523d) Data-Based Construction of Convex Region Surrogate Models for Multiscale Optimization
Mathematical models of complex processes typically involve large numbers of variables and constraints, are often nonlinear and nonconvex, and may include discrete events that are difficult to model as a mathematical programming problem. This makes optimization problems involving such processes extremely hard to solve. However, in multiscale optimization, often only a small subset of the process variables is needed at the higher level of decision-making. For example, on a plant scheduling level, usually the only variables of interest are the production rates of the final products, yet not necessarily all temperatures, pressures and intermediate flowrates (Maravelias & Sung, 2009). This has led to the idea of reduced order modeling, which is to substitute a complex process model with a surrogate model that is of lower dimensionality and has a simpler representation of the process, which consequently reduces the computational burden when solving the optimization problem (Lang et al., 2009).
In this work, we develop a general data-based approach to construct a surrogate model that approximates the feasible region in the space of selected variables by disjoint convex regions that can be described by mixed integer linear constraints. This kind of surrogate model is especially useful in multiperiod optimization problems that are already formulated as mixed integer linear programs (MILPs). Thus, by keeping the process model linear, one can make use of highly efficient MILP solvers such as CPLEX and GUROBI.
Ierapetritou (2001) applies the concept of flexibility analysis (Swaney & Grossmann, 1985) and finds a convex hull to approximate the feasible region by using the Quickhull algorithm. The constructed convex hull is guaranteed to be inscribed in the actual feasible region if the feasible region is convex. Sung & Maravelias (2007) propose an attainable region approach that determines the attainable production levels of the underlying MILP production scheduling model and an underestimation of the production cost in an integrated planning and scheduling framework. In a subsequent work (Maravelias & Sung, 2009), the same authors develop an extension of the attainable region approach that takes nonconvexities into account by combining multiple polytopes.
The approaches reviewed above all require the full mathematical formulation of the model to be approximated. However, often an algebraic process model is not available or is too complex to be included in those frameworks. For example, in the works of Karwan & Keblis (2007) and Mitra et al. (2012), only feasible operating points of the process are available. Using those data points, a convex hull in the product space is then constructed as an approximation of the feasible region and a linear correlation is used for the optimal power consumption.
In this work, we propose a surrogate modeling approach that only takes data points drawn from the process as input. We apply an algorithm that assigns the given data points into subsets such that disjoint convex regions can be formed for each subset. The subsets are chosen in a way such that the union of the resulting convex regions provides an approximation of the feasible region that takes nonlinearities and nonconvexities into account. This surrogate model, which we call a Convex Region Surrogate (CRS) model, can be incorporated in an optimization framework by means of disjunctive programming, which introduces binary variables that define which convex region to choose. We present the proposed algorithm with which a CRS model can be constructed and apply it to an illustrative example as well as to an industrial application related to an air separation plant.
Ierapetritou, M. G., 2001. New approach for quantifying process feasibility: Convex and 1-D quasi-convex regions. AIChE Journal, 47(6), pp. 1407-1417.
Karwan, M. H. & Keblis, M. F., 2007. Operations planning with real time pricing of a primary input. Computers & Operations Research, 34(3), pp. 848-867.
Lang, Y.-D. et al., 2009. Reduced Order Model Based on Principal Component Analysis for Process Simulation and Optimization. Energy & Fuels, Volume 23, pp. 1695-1706.
Maravelias, C. T. & Sung, C., 2009. Integration of production planning and scheduling: Overview, challenges and opportunities. Computers & Chemical Engineering, 33(12), pp. 1919-1930.
Mitra, S., Grossmann, I. E., Pinto, J. M. & Arora, N., 2012. Optimal production planning under time-sensitive electricity prices for continuous power-intensive processes. Computers & Chemical Engineering, Volume 38, pp. 171-184.
Sung, C. & Maravelias, C. T., 2007. An Attainable Region Approach for Production Planning of Multiproduct Processes. AIChE Journal, 53(5), pp. 1298-1315.
Sung, C. & Maravelias, C. T., 2009. A Projection-Based Method for Production Planning of Multiproduct Facilities. 55(10), pp. 2614-2630.
Swaney, R. E. & Grossmann, I. E., 1985. An Index for Operational Flexibility in Chemical Process Design - Part I: Formulation and Theory. AIChE Journal, 31(4), pp. 621-630.