Optimization under uncertainty (OUU) allows for uncertain parameters in the process model to be systematically accounted for during optimization, which reduces the risk of poor performance far from the expected values (e.g., average) of the uncertain parameters. This is especially important for design and analysis of carbon capture systems, which must be designed using models with uncertain parameters (as many systems are novel and industrial scale validation data are not available yet) and to effectively operate in a variety of scenarios due to load following. A popular approach for process design under uncertainty involves formulation of a two-stage stochastic (e.g., bi-level optimization) problem, where design variables (e.g., flowsheet topology, size and number of units, etc.) are determined in the first step and operating variables (e.g., flowrates, stream pressures, etc.) are determined in the second step, after the uncertain parameters are realized. This allows for recourse, or reaction in response to additional information regarding uncertain parameters. Traditionally, stochastic programs for process design (or planning/scheduling) are typically solved in algebraic modeling environments such as GAMS or AMPL, which automatically calculate exact derivations and interface with powerful, large-scale optimization algorithms. However, for process development, commercial flowsheet simulation software packages (e.g., AspenPlus, gPROMs, etc.) are preferred. This paper focuses on computationally tractable formulations for process optimization under uncertainty with rigorous models and process simulators. The methods are demonstrated in a solid sorbent based post combustion CO

_{2}capture system design case study.

A novel approach for OUU is developed, where the OUU problem is treated as a bi-level optimization problem, as described above. In an outer loop, the design variables are determined using a derivative free optimization algorithm (e.g., BOBYQA), and in the inner loop, the optimal operating variables are determined for each scenario using the Sequential Quadratic Programming (SQP) algorithms built into Aspen Customer Modeler (ACM) or gPROMs. Scenarios represent the discrete joint probability distribution for the unknown parameters. For example, assume there are two unknown parameters for a CO_{2} capture system design problem: flue gas flowrate *f* (related to net power output) and a CO_{2} adsorption kinetic parameter *k.* Each scenario would consist of a value for *f* and *k*along with the probability of this scenario occurring. The bi-level structure allows for the inner problems to be decoupled and solved in parallel. Optionally, the approximation of the joint probability distribution for continuous uncertain parameters may be enhanced using response surface models, which reduces the number of required scenarios and decreases computational demands. The methods described in this paper are part of the Framework for Optimization and Quantification of Uncertainty (FOQUS) software package developed as part of the Carbon Capture Simulation Initiative (CCSI), which supports solution of up to 1,000 inner optimization problems in parallel using Turbine (another software package in the CCSI toolset). Interfaces with standard chemical engineering process simulation software (e.g., AspenPlus, ACM, gPROMs and Excel) are established with the SimSinter software package (also in the CCSI toolset).

The methods presented in this paper are demonstrated in a retrofit design case study of a solid sorbent-based post combustion CO_{2} capture system for a 650 MW pulverized coal supercritical power plant. Several kinetic parameters, power output/flue gas flowrate, and capture level are considered as uncertain parameters. The capture system model and the inner optimization problem are solved in ACM. The ACM model consists of approximately 150,000 equations which result from discretization of partial differential algebraic equation adsorber and regenerator bed models. The inner optimization problem has six operating decision variables (cooling water flowrates, sorbent recirculation rate, steam injection rate, and CO_{2} recycle fraction), and requires approximately 3 CPU-minutes to solve on average. The outer problem considers of 17 design variables, including the dimensions/size of equipment, number of parallel units, and heat exchange areas. The problem formulation seeks to minimize the cost of electricity while satisfying CO_{2} capture/recovery specifications. These results are compared against deterministic optimization of the nominal (e.g., expected value) case, and the benefits of optimization under uncertainty are discussed. Finally, the graphical interface to these methods through FOQUS will be presented.

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