Explicit model predictive control through robust optimization

A strategy that calculates an explicit state feedback policy to regulate constrained uncertain discrete-time uncertain linear systems is presented. We consider uncertain processes, affected by box-bounded multiplicative uncertainty as well as bounded additive uncertainty with linear state and inputs constraints. The proposed method includes (i) the calculation of a terminal set constraint and (ii) the robust reformulation of state constraints in the prediction horizon. These features allow the derivation of the desired policy by solving a single multiparametric quadratic programming problem that guarantees feasible operation in the presence of uncertainty. Additionally, we employ variable and constraint elimination approaches to enhance the computational performance of the strategy. We demonstrate the steps and benefits of these developments with a numerical example and a chemical engineering case study.     

A data‐driven multistage adaptive robust optimization framework for planning and scheduling under uncertainty

A novel data‐driven approach for optimization under uncertainty based on multistage adaptive robust optimization (ARO) and nonparametric kernel density M‐estimation is proposed. Different from conventional robust optimization methods, the proposed framework incorporates distributional information to avoid over‐conservatism. Robust kernel density estimation with Hampel loss function is employed to extract probability distributions from uncertainty data via a kernelized iteratively reweighted least squares algorithm. A data‐driven uncertainty set is proposed, where bounds of uncertain parameters are defined by quantile functions, to organically integrate the multistage ARO framework with uncertainty data. Based on this uncertainty set, we further develop an exact robust counterpart in its general form for solving the resulting data‐driven multistage ARO problem. To illustrate the applicability of the proposed framework, two typical applications in process operations are presented: The first one is on strategic planning of process networks, and the other one on short‐term scheduling of multipurpose batch processes. The proposed approach returns 23.9% higher net present value and 31.5% more profits than the conventional robust optimization method in planning and scheduling applications, respectively. © 2017 American Institute of Chemical Engineers AIChE J, 63: 4343–4369, 2017     

Operational optimization of industrial steam systems under uncertainty using data-Driven adaptive robust optimization

This article addresses the operational optimization of industrial steam systems under device efficiency uncertainty using a data-driven adaptive robust optimization approach. A semiempirical model of steam turbine is first developed based on process mechanism and operational data. Uncertain parameters of the proposed steam turbine model are further derived from the historical process data. A robust kernel density estimation method is then used to construct the uncertainty sets for modeling these uncertain parameters. The data-driven uncertainty sets are incorporated into a two-stage adaptive robust mixed-integer linear programming (MILP) framework for operational optimization of steam systems to minimize the total operating cost. Integer variables are introduced to model the on/off decisions of the steam turbines and electrical motors, which are the major energy consumers of the steam system. By applying the affine decision rule, the proposed multilevel optimization model is transformed into its robust counterpart, which is a single-level MILP problem. The proposed framework is applied to the steam system of a real-world ethylene plant to demonstrate its applicability. © 2018 American Institute of Chemical Engineers AIChE J, 65: e16500 2019     

Multi‐stage adjustable robust optimization for process scheduling under uncertainty

Variations in parameters such as processing times, yields, and availability of materials and utilities can have a detrimental effect in the optimality and/or feasibility of an otherwise “optimal” production schedule. In this article, we propose a multi‐stage adjustable robust optimization approach to alleviate the risk from such operational uncertainties during scheduling decisions. We derive a novel robust counterpart of a deterministic scheduling model, and we show how to obey the observability and non‐anticipativity restrictions that are necessary for the resulting solution policy to be implementable in practice. We also develop decision‐dependent uncertainty sets to model the endogenous uncertainty that is inherently present in process scheduling applications. A computational study reveals that, given a chosen level of robustness, adjusting decisions to past parameter realizations leads to significant improvements, both in terms of worst‐case objective as well as objective in expectation, compared to the traditional robust scheduling approaches. © 2016 American Institute of Chemical Engineers AIChE J, 62: 1646–1667, 2016     

Constrained robust Bayesian optimization of expensive noisy black-box functions with guaranteed regret bounds

Many real-world design problems involve optimization of expensive black-box functions. Bayesian optimization (BO) is a promising approach for solving such challenging problems using probabilistic surrogate models to systematically tradeoff between exploitation and exploration of the design space. Although BO is often applied to unconstrained problems, it has recently been extended to the constrained setting. Current constrained BO methods, however, cannot identify solutions that are robust to unavoidable uncertainties. In this article, we propose a robust constrained BO method, constrained adversarially robust Bayesian optimization (CARBO), that addresses this challenge by jointly modeling the effect of the design variables and uncertainties on the unknown functions. Using exact penalty functions, we establish a bound on the number of CARBO iterations required to find a near-global robust solution and provide a rigorous proof of convergence. The advantages of CARBO are demonstrated on two case studies including a non-convex benchmark problem and a realistic bubble column reactor design problem.     

A unified framework for adjustable robust optimization with endogenous uncertainty

This work proposes a framework for multistage adjustable robust optimization that unifies the treatment of three different types of endogenous uncertainty, where decisions, respectively, (a) alter the uncertainty set, (b) affect the materialization of uncertain parameters, and (c) determine the time when the true values of uncertain parameters are observed. We provide a systematic analysis of the different types of endogenous uncertainty and highlight the connection between optimization under endogenous uncertainty and active learning. We consider decision-dependent polyhedral uncertainty sets and propose a decision rule approach that incorporates both continuous and binary recourse, including recourse decisions that affect the uncertainty set. The proposed method enables the modeling of decision-dependent nonanticipativity and results in a tractable reformulation of the problem. We demonstrate the effectiveness of the approach in computational experiments that cover a wide range of applications. The results show significant benefits from proper modeling of endogenous uncertainty and active learning.     

Soft-constrained model predictive control based on data-driven distributionally robust optimization

This article proposes a novel distributionally robust optimization (DRO)-based soft-constrained model predictive control (MPC) framework to explicitly hedge against unknown external input terms in a linear state-space system. Without a priori knowledge of the exact uncertainty distribution, this framework works with a lifted ambiguity set constructed using machine learning to incorporate the first-order moment information. By adopting a linear performance measure and considering input and state constraints robustly with respect to a lifted support set, the DRO-based MPC is reformulated as a robust optimization problem. The constraints are softened to ensure recursive feasibility. Theoretical results on optimality, feasibility, and stability are further discussed. Performance and computational efficiency of the proposed method are illustrated through motion control and building energy control systems, showing 18.3% less cost and 78.8% less constraint violations, respectively, while requiring one third of the CPU time compared to multi-stage scenario based stochastic MPC.     

Theoretical and computational comparison of continuous‐time process scheduling models for adjustable robust optimization

Coping with uncertainty in system parameters is a prominent hurdle when scheduling multi‐purpose batch plants. In this context, our previously introduced multi‐stage adjustable robust optimization (ARO) framework has been shown to obtain more profitable solutions, while maintaining the same level of immunity against risk, as compared to traditional robust optimization approaches. This paper investigates the amenability of existing deterministic continuous‐time scheduling models to serve as the basis of this ARO framework. A comprehensive computational study is conducted that compares the numerical tractability of various models across a suite of literature benchmark instances and a wide range of uncertainty sets. This study also provides, for the first time in the open literature, robust optimal solutions to process scheduling instances that involve uncertainty in production yields. © 2018 American Institute of Chemical Engineers AIChE J, 64: 3055–3070, 2018     

Adversarially robust Bayesian optimization for efficient auto-tuning of generic control structures under uncertainty

The performance of optimization- and learning-based controllers critically depends on the selection of several tuning parameters that can affect the closed-loop control performance and constraint satisfaction in highly nonlinear and nonconvex ways. Due to the black-box nature of the relationship between tuning parameters and general closed-loop performance measures, there has been a significant interest in automatic calibration (i.e., auto-tuning) of complex control structures using derivative-free optimization methods, including Bayesian optimization (BO) that can handle expensive unknown cost functions. Nevertheless, an open challenge when applying BO to auto-tuning is how to effectively deal with uncertainties in the closed-loop system that cannot be attributed to a lumped, small-scale noise term. This article addresses this challenge by developing an adversarially robust BO (ARBO) method that is particularly suited to auto-tuning problems with significant time-invariant uncertainties in an expensive system model used for closed-loop simulations. ARBO relies on a Gaussian process model that jointly describes the effect of the tuning parameters and uncertainties on the closed-loop performance. From this joint Gaussian process model, ARBO uses an alternating confidence-bound procedure to simultaneously select the next candidate tuning and uncertainty realizations, implying only one expensive closed-loop simulation is needed at each iteration. The advantages of ARBO are demonstrated on two case studies, including an illustrative problem and auto-tuning of a nonlinear model predictive controller using a benchmark bioreactor problem.     

Nonlinear robust optimization for process design

A novel robust optimization framework is proposed to address general nonlinear problems in process design. Local linearization is taken with respect to the uncertain parameters around multiple realizations of the uncertainty, and an iterative algorithm is implemented to solve the problem. Furthermore, the proposed methodology can handle different categories of problems according to the complexity of the problems. First, inequality‐only constrained optimization problem as studied in most existing robust optimization methods can be addressed. Second, the proposed framework can deal with problems with equality constraint associated with uncertain parameters. In the final case, we investigate problems with operation variables which can be adjusted according to the realizations of uncertainty. A local affinely adjustable decision rule is adopted for the operation variables (i.e., an affine function of the uncertain parameter). Different applications corresponding to different classes of problems are used to demonstrate the effectiveness of the proposed nonlinear robust optimization framework. © 2017 American Institute of Chemical Engineers AIChE J, 64: 481–494, 2018     

A generalized cutting-set approach for nonlinear robust optimization in process systems engineering

We propose a novel computational framework for the robust optimization of highly nonlinear, non-convex models that possess uncertainty in their parameter data. The proposed method is a generalization of the robust cutting-set algorithm that can handle models containing irremovable equality constraints, as is often the case with models in the process systems engineering domain. Additionally, we accommodate general forms of decision rules to facilitate recourse in second-stage (control) variables. In particular, we compare and contrast the use of various types of decision rules, including quadratic ones, which we show in certain examples to be able to decrease the overall price of robustness. Our proposed approach is demonstrated on three process flow sheet models, including a relatively complex model for amine-based CO₂ capture. We thus verify that the generalization of the robust cutting-set algorithm allows for the facile identification of robust feasible designs for process systems of practical relevance.     

Distributional uncertainty analysis and robust optimization in spatially heterogeneous multiscale process systems

Multiscale models have been developed to simulate the behavior of spatially‐heterogeneous porous catalytic flow reactors, i.e., multiscale reactors whose concentrations are spatially‐dependent. While such a model provides an adequate representation of the catalytic reactor, model‐plant mismatch can significantly affect the reactor's performance in control and optimization applications. In this work, power series expansion (PSE) is applied to efficiently propagate parametric uncertainty throughout the spatial domain of a heterogeneous multiscale catalytic reactor model. The PSE‐based uncertainty analysis is used to evaluate and compare the effects of uncertainty in kinetic parameters on the chemical species concentrations throughout the length of the reactor. These analyses reveal that uncertainty in the kinetic parameters and in the catalyst pore radius have a substantial effect on the reactor performance. The application of the uncertainty quantification methodology is illustrated through a robust optimization formulation that aims to maximize productivity in the presence of uncertainty in the parameters. © 2016 American Institute of Chemical Engineers AIChE J, 62: 2374–2390, 2016     

Data-driven distributionally robust optimization of shale gas supply chains under uncertainty

This article aims to leverage the big data in shale gas industry for better decision making in optimal design and operations of shale gas supply chains under uncertainty. We propose a two-stage distributionally robust optimization model, where uncertainties associated with both the upstream shale well estimated ultimate recovery and downstream market demand are simultaneously considered. In this model, decisions are classified into first-stage design decisions, which are related to drilling schedule, pipeline installment, and processing plant construction, as well as second-stage operational decisions associated with shale gas production, processing, transportation, and distribution. A data-driven approach is applied to construct the ambiguity set based on principal component analysis and first-order deviation functions. By taking advantage of affine decision rules, a tractable mixed-integer linear programming formulation can be obtained. The applicability of the proposed modeling framework is demonstrated through a small-scale illustrative example and a case study of Marcellus shale gas supply chain. Comparisons with alternative optimization models, including the deterministic and stochastic programming counterparts, are investigated as well. © 2018 American Institute of Chemical Engineers AIChE J, 65: 947–963, 2019     

Resilient supply chain design and operations with decision-dependent uncertainty using a data-driven robust optimization approach

To addresses the design and operations of resilient supply chains under uncertain disruptions, a general framework is proposed for resilient supply chain optimization, including a quantitative measure of resilience and a holistic biobjective two-stage adaptive robust fractional programming model with decision-dependent uncertainty set for simultaneously optimizing both the economic objective and the resilience objective of supply chains. The decision-dependent uncertainty set ensures that the uncertain parameters (e.g., the remaining production capacities of facilities after disruptions) are dependent on first-stage decisions, including facility location decisions and production capacity decisions. A data-driven method is used to construct the uncertainty set to fully extract information from historical data. Moreover, the proposed model takes the time delay between disruptions and recovery into consideration. To tackle the computational challenge of solving the resulting multilevel optimization problem, two solution strategies are proposed. The applicability of the proposed approach is illustrated through applications on a location-transportation problem and on a spatially-explicit biofuel supply chain optimization problem. © 2018 American Institute of Chemical Engineers AIChE J, 65: 1006–1021, 2019     

Steady‐state process optimization with guaranteed robust stability under parametric uncertainty

Interest in chemical processes that perform well in dynamic environments has led to the development of design methodologies that account for operational aspects of processes, including flexibility, operability, and controllability. In this article, we address the problem of identifying process designs that optimize an economic objective function and are guaranteed to be stable under parametric uncertainties. The underlying mathematical problem is difficult to solve as it involves infinitely many constraints, nonconvexities and multiple local optima. We develop a methodology that embeds robust stability constraints to steady‐state process optimization formulations without any a priori bifurcation analysis. We propose a successive row and column generation algorithm to solve the resulting generalized semi‐infinite programming problem to global optimality. The proposed methodology allows modeling different levels of robustness, handles uncertainty regions without overestimating them, and works for both unique and multiple steady states. We apply the proposed approach to a number of steady‐state optimization problems and obtain the least conservative solutions that guarantee robust stability. © 2011 American Institute of Chemical Engineers AIChE J, 2011     

Robust dynamic optimization for nonlinear chemical processes under measurable and unmeasurable uncertainties

We formulate an integrated framework for the robust dynamic optimization of nonlinear chemical processes under measurable and unmeasurable uncertainties. An affine decision rule is proposed to approximate the causal dependence of the wait-and-see decision variables on the gradually revealed measurable uncertainties. To overcome the computational intractability of the proposed model, a linearization technique based on the first-order Taylor expansion is introduced around the nominal values of uncertainties to derive the robust dynamic counterpart, which can be discretized to a large-scale nonlinear programming (NLP) formulation. Effects of first discretizing the dynamic models or introducing the affine decision rule are investigated. The proposed framework is also compared with the state-of-the-art re-optimization and traditional robust optimization approaches. An illustrative example and an industrial semi-batch 2-mercaptobenzothiazole production case are involved to demonstrate the advantages and applicability of the proposed framework.     

Multistage distributionally robust optimization for integrated production and maintenance scheduling

In chemical manufacturing processes, equipment degradation can have a significant impact on process performance or cause unit failures that result in considerable downtime. Hence, maintenance planning is an important consideration, and there have been increased efforts in scheduling production and maintenance operations jointly. In this context, one major challenge is the inherent uncertainty in predictive equipment health models. In particular, the probability distribution associated with the stochasticity in such models is often difficult to estimate and hence not known exactly. In this work, we apply a distributionally robust optimization (DRO) approach to address this problem. Specifically, the proposed formulation optimizes the worst-case expected outcome with respect to a Wasserstein ambiguity set, and we apply a decision rule approach that allows multistage mixed-integer recourse. Computational experiments, including a real-world industrial case study, are conducted, where the results demonstrate the significant benefits from binary recourse and DRO in terms of solution quality.     

A method for flexibility analysis of continuous processing plants

A methodology has been developed for the analysis of operational flexibility of a continuous processing plant. An application has been demonstrated for a multipurpose plant with uncertain operating conditions. A practical flexibility analysis procedure using a direct search optimization algorithm is proposed to solve the problem. The method provides a heuristic screening technique that makes it possible to avoid the exhaustive enumeration of every constraint vertex, while the direct search optimization offers a simple and effective means to identify bottlenecks. A refinery multiperiod plant problem has been investigated in order to highlight the procedures which lead to the construction of flexibility indices for each operating mode.     

Distributionally robust chance-constrained optimization with Sinkhorn ambiguity set

A novel distributionally robust chance-constrained optimization (DRCCP) method is proposed in this work based on the Sinkhorn ambiguity set. The Sinkhorn ambiguity set is constructed based on the Sinkhorn distance, which is a variant of the Wasserstein distance with the entropic regularization. The proposed method can hedge against more general families of uncertainty distributions than the Wasserstein ambiguity set-based methods. The presented approach is formulated as a tractable conic model based on the Conditional value-at-risk (CVaR) approximation and the discretized kernel distribution relaxation. This model is compatible with more general constraints that are subject to uncertainty than the Wasserstein-based methods. Accordingly, the presented Sinkhorn DRCCP is a more practical approach that overcomes the limitations of the traditional Wasserstein DRCCP approaches. A numerical example and a nonlinear chemical process optimization case are studied to demonstrate the efficacy of the Sinkhorn DRCCP and its advantages over the Wasserstein DRCCP.     

A mathematical programming formulation for temporal flexibility analysis

Realistic chemical processes are often operated in the presence of complex and uncertain dynamics. While an ill designed system may become inoperable due to variations in some process parameters at certain instances, the cumulative effects of temporary disturbances in finite time intervals can also result in serious consequences. The latter issue is studied in the present study on the basis of a novel concept – temporal flexibility. Specifically, the mathematical program used for evaluating the corresponding performance measure is built with a dynamic system model, which usually consists of a set of differential-algebraic equations (DAEs). The numerical technique of differential quadrature (DQ) is adopted to approximate these DAEs with equality constraints. As a result, any solution strategy for the conventional steady-state flexibility analysis is applicable. Two examples, a simple liquid storage tank and a solar thermal driven membrane distillation desalination process, are adopted to demonstrate the usefulness of temporal flexibility analysis. All results obtained in case studies show that the proposed approach is convenient and effective for assessing realistic issues in operating complex dynamic chemical processes.