(371t) Two-Stage Chance-Constrained Programming for Refinery Optimization Under Uncertainty

The two-stage chance-constrained program (CCP) is developed for the crude oil procurement and refining operations under uncertainty. In stage-I, the refinery decision-makers determine the type and quantity of crude oil purchase under operational uncertainties to maximize the expected profit under all possibilities. In stage-II, process unit flowrates are adjusted based on the realized uncertainties and available crude oil, while introducing probabilistic constraints to manage the off-spec risk. To solve such a two-stage optimization problem, we propose a novel approach using Gaussian mixture model (GMM) to characterize uncertainties, and piecewise linear decision rule to design stage-II operations. Comparing to the conventional scenario-based mixed-integer linear program (MILP), our new approach offers three advantages. First, it leverages a well-developed global optimization scheme for joint CCP to avoid unseen scenario incurred bias. Second, the data driven GMM enables CCP to handle uncertainties with general distributions. Third, the stage-II variables are parameterized via Gaussian component induced piecewise linear decision rule to achieve a balance between optimality and computational time.

The new contributions of this work lie in the integration of GMM and decision-rule with stochastic programming formula. We employ GMM to model uncertainty distributions due to the following merits:

• GMM is a parametric approach to approximate complex distributions with arbitrary shapes by combining multiple Gaussian components.
• As a clustering method, GMM not only approximates the true distribution of uncertainty, but also enables cluster-dependent decision rule for two-stage optimization.
• GMM can be easily built from data or scenario through the well-developed expectation-maximization (EM) algorithm. Compared with the scenario tree approach, GMM is more efficient to characterize uncertainties.

The piecewise linear decision rule is employed due to the following advantages:

• Piecewise function can approximate the optimal decision, which is nonlinear in nature.
• Local function is still kept as linear to facilitate relaxation.

The GMM and piecewise linear decision rule integrated formula for two-stage CCP can be optimized through the adaptive outer approximation, second-order cone relaxation, branch-and-bound, and bound tightening techniques. Both stage-I and stage-II variables can be determined simultaneously without relying on decomposition techniques. A simplified refinery plant, consisting of distillation, cracker, reformer, isomerization, and desulfurization units, is studied to demonstrate the superiority of the proposed optimization method in solution time, probabilistic feasibility, and optimality over the large-scale sample average approximation (SAA). Our results show that the proposed method converges with relative gap less than 1% using 3000-6000 seconds, whereas the SAA takes more than 14400 seconds to achieve relative gap 12%-25%. In addition, the proposed method finds better stage-I solutions than SAA in all tested cases [1].

Reference

[1] Y. Yang, "Two-stage chance-constrained programming based on Gaussian mixture model and piecewise linear decision rule for refinery optimization," Computers & Chemical Engineering, vol. 184, pp. 108632, 2024.