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A Hybrid Interval Constraint Propagation and Mathematical Programming Method for Energy Systems Synthesis

Mahmoud Bahy Noureldin and Majid M Al-Gwaiz. Consulting Services Department, Saudi Aramco, Dhahran, Saudi Arabia

Worldwide industrial community is ravenous to producing more-with-less through cost effective but benign new products and processes. In the last three decades, for instance, simultaneous synthesis of processes and utilities systems has become very important field in chemical engineering applications. Systematic techniques emerged in the late seventies of the last century for applications of such kind has resulted in more and more emphasis on research areas linked to large systems optimization. These systems are difficult to solve and a lot of simplifying assumptions are still needed to keep these problems mathematically tractable in industrial applications.

The good news is that while simultaneous process and utilities synthesis problems embed huge space of possibilities; it is in the same time, due to safety health and environmental concerns as well as capital limitations, highly constrained.

Constraint Programming field, known to be little younger than mathematical programming, is getting every day ground in reducing the computational effort needed to solve combinatorial optimization problems. A crucial component in Constraint Programming approach is the Constraint Propagation.

Constraint Propagation (CP) is a very efficient method for reducing the search space of combinatorial search and optimization problems and has become more important in the last two decades. The basic idea of constraint propagation methods is to detect and remove inconsistent variable(s) assignment(s) that can not participate in any feasible solution through the repeated analysis and evaluation of the variables, domains and constraints describing a specific problem instance.

Grossmann and co-workers in 2004 have combined the power of both Constraint Programming and MILP on the algorithmic level to solve continuous time scheduling of multipurpose batch plants. They used MILP to optimize the high level decisions and Constraint Programming to obtain feasible detailed schedule. Hooker and co-workers in 2002 specified two ways to integrate Constraint Programming and mathematical programming. One by combining them on the modeling level and try to find the algorithm that can solve such hybrid model. The second by decomposing the original problem into two sub-problems; one is MILP and the other is Constraint Programming. Each model is solved separately and information obtained from one help solving the other.

In this paper we introduce new method of integrating CP with mathematical programming for solving large energy systems synthesis. In our method, CP with interval labels is used first after decomposing the large problem to several small stand alone sub-problems, of-course with some overlapping/complicating constraints among these sub-problems, to check feasibility of all sub-problems that depict the large problem. Mathematical programming formulation enjoying the new insights obtained from the CP sub-problems solutions and the decision maker analysis is then used to find the optimal solution details.

The paper will review current state-of-art techniques, introduce new method and conclude with several case studies.

Case studies from energy systems application literatures are used to illustrate the application of the new method. One example is the HEN synthesis problem under varying driving force distribution and under all possible combinations of process parameters modifications. This lareg energy synthesis problem is decomposed into two bi-level programming problems, where each one is solved separately using CP. The cost minimization/ minimum number of units HEN objective will be solved subject to minimum energy utility cost. Mathematical programming is finally used to find details of the HEN problem solution.

Keywords: Interval Constraints Propagation; Mathematical Programming, Process Synthesis, Constraint Satis