Coal- and biomass-based polygeneration with multiple energy products is a promising technology to supplement the current oil-based energy processes. A certain degree of operational flexibility can be introduced into a polygeneration process by over-sizing equipment so that one can alter the production rates of individual products in response to changing market conditions. The resulting flexible polygeneration plant achieves potentially higher profit and lower economic risk than the static plant [1].

In the design of flexible polygeneration systems, the optimal trade-off between operational flexibility and equipment capacity (or capital cost) needs to be determined. This problem can be formulated as a multi-period optimization problem, which is a potentially large-scale nonconvex mixed-integer nonlinear program (MINLP) and cannot be solved to global optimality by state-of-the-art global optimization solvers, such as BARON [2], within a reasonable time. A duality-based decomposition method, recently developed for the design and operation of natural gas production networks [3,4], can exploit the special structure of this mathematical programming problem and enable faster solution. In this method, the nonconvex MINLP is relaxed into a convex lower-bounding MINLP which can be further reformulated into a relaxed master problem according to the principles of projection, dualization and relaxation. The relaxed master problem yields an increasing sequence of lower bounds for the original problem, but these lower bounds may converge to a global optimum slowly due to the high nonconvexity of the problem.

In this work, an enhanced decomposition method with an improved relaxed master problem is developed by introducing additional dual information into the relaxed master problem, which requires solving additional nonconvex NLPs. The solutions of these nonconvex NLPs lead to stronger dual cuts that can significantly improve the convergence rate of the algorithm. The enhanced decomposition algorithm guarantees to find an ε-optimal solution in a finite number of iterations.

The computational advantages of the enhanced decomposition method are demonstrated via several case studies on the optimal design and operation of a flexible energy polygeneration system using coal and biomass as feedstocks and co-producing electricity, liquid fuels (naphtha and diesel) and chemicals (methanol). The computational results show that the enhanced decomposition algorithm achieves much faster convergence than both BARON and the original decomposition algorithm, and it solved the large-scale nonconvex MINLPs to ε-optimality in practical times (e.g., a problem with 70 binary variables and 14712 continuous variables was solved within 5 hours).

**Reference**

[1] Chen Y, Adams TA, Barton PI. Optimal design and operation of flexible energy polygeneration systems. *Industrial & Engineering Chemistry Research*. 2011;50(8):4553-4566.

[2] Tawarmalani M, Sahinidis NV. Global optimization of mixed-integer nonlinear programs: A theoretical and computational study. *Mathematical Programming*. 2004;99(3):563-591.

[3] Li X, Armagan E, Tomasgard A, Barton PI. Stochastic pooling problem for natural gas production network design and operation under uncertainty. *AIChE Journal*. DOI: 10.1002/aic.12419.

[4] Li X, Tomasgard A, Barton PI. Decomposition strategy for natural gas production network design under uncertainty. *Proceedings of the 49*^{th}* **IEEE Conference on Decision and Control*, 188-193 (2010).

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