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Machine learning-based model predictive control (ML-MPC) has been developed to control nonlinear processes with unknown first-principles models. While ML models can capture nonlinear dynamics of complex systems, the complexity of ML models leads to increased computation time for real-time implementation of ML-MPC. To address this issue, in this work, we propose an explicit ML-MPC framework for nonlinear processes using multi-parametric programming. Specifically, a self-adaptive approximation algorithm is first developed to obtain a piecewise linear affine function that approximates the behaviors of ML models. Then, multi-parametric quadratic programming (mpQP) problems are formulated to generate the solution map for states in discretized state–space. Furthermore, to accelerate the implementation of explicit ML-MPC, a neighbor-first search algorithm is developed. Finally, an example of a chemical reactor is used to demonstrate the effectiveness of the explicit ML-MPC. • An explicit MPC framework for MPCs using a general class of ML models. • A self-adaptive algorithm for approximating ML models with the desired accuracy. • Neighbor-first search algorithm for accelerating implementation of explicit ML-MPC. • Improvement of computational efficiency demonstrated in a chemical process example. [ABSTRACT FROM AUTHOR]

In this paper, we propose a graph classification approach for automatically determining whether to use a monolithic or a decomposition-based solution method. In this approach, an optimization problem is represented as a graph that captures the structural and functional coupling among the variables and constraints of the problem via an appropriate set of features. Given this representation, a graph classifier can be built to assist a solver in selecting the best solution strategy for a given problem with respect to some metric of choice. The proposed approach is used to develop a classifier that determines whether a convex Mixed Integer Nonlinear Programming problem should be solved using branch and bound or the outer approximation algorithm. Finally, it is shown how the learned classifier can be incorporated into existing mixed integer optimization solvers. • A deep learning framework is proposed to decide when to use a decomposition. • A feature-based graph representation of optimization problems is proposed. • The proposed approach is applied to Mixed Integer Nonlinear optimization problems. • Integration of the proposed approach with optimization solvers is discussed. [ABSTRACT FROM AUTHOR]

Highlights • A Gaussian process based model is constructed to simulate a complex biological system. • Model accuracy and predictive capability is compared against the neural network model. • Dynamic optimisation under uncertainty is conducted to maximise algal lutein production. • A framework to build bioprocess models with Gaussian Processes is presented. Abstract Dynamic modeling is an important tool to gain better understanding of complex bioprocesses and to determine optimal operating conditions for process control. Currently, two modeling methodologies have been applied to biosystems: kinetic modeling, which necessitates deep mechanistic knowledge, and artificial neural networks (ANN), which in most cases cannot incorporate process uncertainty. The goal of this study is to introduce an alternative modeling strategy, namely Gaussian processes (GP), which incorporates uncertainty but does not require complicated kinetic information. To test the performance of this strategy, GPs were applied to model microalgae growth and lutein production based on existing experimental datasets and compared against the results of previous ANNs. Furthermore, a dynamic optimization under uncertainty is performed, avoiding over-optimistic optimization outside of the model's validity. The results show that GPs possess comparable prediction capabilities to ANNs for long-term dynamic bioprocess modeling, while accounting for model uncertainty. This strongly suggests their potential applications in bioprocess systems engineering. [ABSTRACT FROM AUTHOR]

Highlights • Dynamic RTO of distributed MPC systems using closed-loop prediction. • The interaction between distributed MPCs and plant dynamics is fully considered. • Approximation techniques are applied to improve computational efficiency. • Significant reduction in computation times achieved with little loss in performance. • The two-level control architecture is readily applicable to industrial processes. Abstract A dynamic real-time optimization (DRTO) strategy has recently been proposed for coordination of distributed MPC systems. The scheme utilizes a prediction model that accounts for the impact of the distributed controllers on the plant response. Two techniques are presented for approximating the closed-loop prediction within the DRTO formulation - a hybrid closed-loop and an input clipping formulation. The hybrid formulation generates closed-loop predictions for a limited number of time intervals along the DRTO prediction horizon, followed by an open-loop optimal control formulation for the rest of the horizon. The input clipping formulation utilizes an unconstrained MPC optimization formulation for each distributed MPC, coupled with an input saturation mechanism. The performance of the approximation techniques is evaluated through application to case studies based on linear and nonlinear dynamic plant models respectively. The approximation techniques are demonstrated to be more computationally efficient than the rigorous counterpart without significant loss in performance. [ABSTRACT FROM AUTHOR]

Highlights • An enhanced surrogate assisted framework for constrained global optimization is proposed. • Maximizing probability of improvement approach is used for selecting infill points. • Kriging meta-models of objective and constraints functions are updated in every iteration. • Meta-model of objective function is local optimized when it's sufficient mature. • Numerical results indicate that the framework is suitable for use in solving computationally expensive and constrained black-box optimization. Abstract An enhanced surrogate assisted framework, based on Probability of Improvement (PI) method, is proposed in this paper. We made some modifications to the original PI approach to enhance the performance of the modeling and optimization framework, leading to fewer rigorous simulations to find the optimal solution without loss of accuracy. We also extended the algorithm for handling general constraints using a fully probabilistic approach. The behavior of the proposed framework was investigated through a set of 9 Unconstrained Test Functions (UTF), 7 Constrained Optimization Problems (COP) and 3 Chemical Engineering Problems (CEP). The numerical results indicate that a lower number of rigorous model simulations were needed for optimizing UTF compared to the classic PI method and that the proposed framework was capable of achieving sustained near optimal solutions for COP and CEP. These results indicate that the proposed framework is suitable for solving computationally expensive constrained black-box optimization problems. [ABSTRACT FROM AUTHOR]

Highlights • Simulation study to economics optimizing control and robust multi-stage optimizing control. • Discussion of numerical aspects of optimizing control as well as the software realization using high-fidelity high-dimensional process models of complex chemical processes. • Control problem formulation for the dynamic product changeover in a multi-product process. • Robustification of economics optimizing control by multi-stage nonlinear model predictive control in case of the presence of model uncertainty. Abstract By the use of online optimization, the profitability of chemical processes can be enhanced while meeting process-related, environmental and ecological constraints. Two well-known techniques to achieve an economically optimal operation of chemical processes are repetitive stationary optimization (RTO) combined with advanced control and direct economics optimizing control on a receding horizon (also called one-layer approach or dynamic RTO). Direct economics optimizing control can react faster to disturbances and is better suited for the optimization of transients between different products. On the other hand it is computationally demanding and requires sufficiently accurate dynamic models. In this paper, we discuss economics optimizing control of a very complex process, a multi-product transesterification reaction that is realized in a reactive distillation process. The process model, the specification of the economics optimizing controller, the implementation of the dynamic optimization, and the robustification of the controller by a multi-stage approach are discussed using simulation studies. [ABSTRACT FROM AUTHOR]

Highlights • A novel piecewise state feedback controller is introduced. • The chemical process optimal control problem is formulated as an optimal control problem of switched dynamical systems. • A gradient-based parallel optimization algorithm is developed for solving the optimal control. • Convergence results indicate that the gradient-based parallel optimization algorithm is global convergent. • Three chemical process optimal control problems are given to illustrate the effectiveness of the proposed algorithm. Abstract This paper considers an optimal control problem of chemical processes with a novel piecewise state-feedback controller. Firstly, the chemical process optimal control problem is formulated as a switched dynamical system optimal control problem, which can be transformed into a parameter optimization problem. Next, to achieve rapid convergence from remote starting points, we propose a novel gradient-based optimization algorithm, which is suitable to a parallel implementation because the step is improved by updating its direction as well as its length simultaneously before moving to the next iteration, and the step computation involves only the inner products of vectors. Then, the convergent properties of the parameter optimization problem to the original optimal control problem are discussed. Finally, the numerical simulation results show that the gradient-based parallel optimization algorithm is an effective alternative method for solving the chemical process optimal control problems. [ABSTRACT FROM AUTHOR]

Highlights • A superstructure of methanation unit is proposed, and the topology of methanation unit is optimized for the coal-to-SNG process. • Simultaneous optimization and heat integration are performed along with the area targeting. • A sequential method is used to provide initial values for the simultaneous model. • Compared with industrial data, the exergy efficiency is improved by 0.77% and TAC can reduce by 10.36 MM$·a−1 for the coal-to-SNG process. Abstract The area of the heat recovery network has a significant impact on the economic performance of the coal-to-SNG process. This paper proposes a superstructure of the methanation unit, and the number of the methanator, recycle gas extraction position (RGEXP), as well as the operating conditions, are optimized. Simultaneous optimization and heat integration are performed along with the area targeting to weigh against the operating cost and capital cost of the coal-to-SNG process. Area cost is evaluated by assuming vertical heat transfer between cold and hot composite curves. A sequential method is used to provide initial values for the simultaneous model. Results show that Case7 (2) has the best economic performance, which has 7 methanators and recycled gas is extracted from the 2nd methanator. The total annual cost of Case7 (2) can be reduced by 10.36 MM$·a−1, and exergy efficiency can be also improved by 0.77% compared with an industrial plant. [ABSTRACT FROM AUTHOR]

Highlights • A novel Kriging-based adaptive sampling framework is proposed for the optimization of stochastically constrained black-box systems. • With an innovative feasibility-enhanced infill criterion, the proposed algorithm is more robust in maintaining feasibility while returning a near-optimal solution. • The efficacy of this algorithm is demonstrated with low-dimensional benchmark problems and a pharmaceutical case study. Abstract Stochastically constrained simulation optimization problems are challenging because the inherent noise terms to a black-box system lead to the need of considering the uncertainty at both the objective and the constraint function. To address this problem, we propose a Kriging-based optimization framework, which uses stochastic Kriging models to approximate the objective and the constraint functions and an adaptive sampling approach to sequentially search for the next point that is promising to have a better objective. The main contribution of this work is to incorporate a feasibility-enhanced term to the infill sampling criterion, which improves the Kriging-based algorithm's capabilities of returning a truly feasible near-optimal solution for stochastic systems. The efficacy of the Kriging-based algorithm is demonstrated with eight benchmark problems and a case study from the pharmaceutical manufacturing domain. [ABSTRACT FROM AUTHOR]

Highlights • Heuristics with performance guarantees for the minimum number of matches problem. • Algorithm design classes used: relaxation rounding, water filling, greedy packing. • Improved heuristic numerical results with efficient running times are obtained. • Efficient heat exchanges and tighter big-M parameters for MILP models are derived. • New MILP formulation and approximation algorithm in one temperature interval. Abstract Heat exchanger network synthesis exploits excess heat by integrating process hot and cold streams and improves energy efficiency by reducing utility usage. Determining provably good solutions to the minimum number of matches is a bottleneck of designing a heat recovery network using the sequential method. This subproblem is an NP -hard mixed-integer linear program exhibiting combinatorial explosion in the possible hot and cold stream configurations. We explore this challenging optimization problem from a graph theoretic perspective and correlate it with other special optimization problems such as cost flow network and packing problems. In the case of a single temperature interval, we develop a new optimization formulation without problematic big-M parameters. We develop heuristic methods with performance guarantees using three approaches: (i) relaxation rounding, (ii) water filling, and (iii) greedy packing. Numerical results from a collection of 51 instances substantiate the strength of the methods. [ABSTRACT FROM AUTHOR]

Heat exchanger network synthesis exploits excess heat by integrating process hot and cold streams and improves energy efficiency by reducing utility usage. Determining provably good solutions to the minimum number of matches is a bottleneck of designing a heat recovery network using the sequential method. This subproblem is an NP -hard mixed-integer linear program exhibiting combinatorial explosion in the possible hot and cold stream configurations. We explore this challenging optimization problem from a graph theoretic perspective and correlate it with other special optimization problems such as cost flow network and packing problems. In the case of a single temperature interval, we develop a new optimization formulation without problematic big-M parameters. We develop heuristic methods with performance guarantees using three approaches: (i) relaxation rounding, (ii) water filling, and (iii) greedy packing. Numerical results from a collection of 51 instances substantiate the strength of the methods. [ABSTRACT FROM AUTHOR]

A new method is proposed for the steady-state optimization of biochemical systems described by Generalized Mass Action (GMA) models. In this method, a bi-level programming with a two-layer nested structure is established. In this bi-level problem, the upper-level objective is to maximize a flux or a metabolite concentration, and the lower-level objective is to minimize the total sum of metabolite concentrations of biochemical systems. The biological significance of the presented bi-level programming problem is to maximize the production rate or concentration of the desired product under a minimal metabolic cost to the biochemical system. To efficiently solve the above NP-hard, non-convex and nonlinear bi-level programming problem, we reformulate it into a single-level optimization problem by using appropriate transformation strategies. The proposed framework is applied to four case studies and has shown the tractability and effectiveness of the method. A comparison of our proposed method and other methods is also given. [ABSTRACT FROM AUTHOR]

Linear model predictive control is extended to both control and optimize a product grade schedule. The proposed methods are time-scaling of the linear dynamics based on throughput rates and grade-based objectives for product scheduling based on a mathematical program with complementarity constraints. The linear model is adjusted with a residence time approximation to time-scale the dynamics based on throughput. Although nonlinear models directly account for changing dynamics, the model form is restricted to linear differential equations to enable fast online cycle times for large-scale and real-time systems. This method of extending a linear time-invariant model for scheduling is designed for many advanced control applications that currently use linear models. Simultaneous product switching and grade target management is demonstrated on a reactor benchmark application. The objective is a continuous form of discrete ranges for product targets and economic terms that maximize overall profitability. [ABSTRACT FROM AUTHOR]

Flux balance analysis (FBA) is a linear programming-based framework widely used to predict the behavior, in terms of the resulting flux distribution, of cellular organisms in different media. FBA models are constructed using only stoichiometric information, and for this reason they sometimes fail in predicting fluxes precisely. In this work, we formally define the concept of hybrid FBA/kinetic models, in which kinetic information of key processes is used to tighten the search space of standalone FBA formulations, thereby enhancing their predictive capabilities. This approach leads to non-linear non-convex models that may exhibit multiple local optima. To solve them to global optimality, we use a customized outer-approximation algorithm that exploits the structure of the kinetic equations. Numerical results show that our method enhances the quality of standalone FBA models, providing more accurate predictions. [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel algorithm for the rigorous design of distillation columns that integrates a process simulator in a generalized disjunctive programming formulation. The optimal distillation column, or column sequence, is obtained by selecting, for each column section, among a set of column sections with different number of theoretical trays. The selection of thermodynamic models, properties estimation, etc. is all in the simulation environment. All the numerical issues related to the convergence of distillation columns (or column sections) are also maintained in the simulation environment. The model is formulated as a Generalized Disjunctive Programming (GDP) problem and solved using the logic based outer approximation algorithm without MINLP reformulation. Some examples involving from a single column to thermally coupled sequence or extractive distillation shows the performance of the new algorithm. [ABSTRACT FROM AUTHOR]

Highlights: [•] A new modeling framework that exploits the synergistic combination of commercial process simulators and GDP models. [•] Our methodology allows to include easily logical relationships among alternatives. [•] The proposed tool uses a logic based Outer Approximation algorithm. [•] The methodology is applied to the synthesis of a methanol plant where different alternatives. [Copyright &y& Elsevier]

• Novel two-stage stochastic programming GDP (Generalized Disjunctive Programming) model with reliability constraints to deal with both the exogenous and endogenous uncertainties in process synthesis. • The reliability model is incorporated into the flowsheet superstructure optimization, considering the impact of selecting parallel units for improving plant availability. • Improved LOA algorithm to solve the hybrid GDP model with implicit nested disjunctions, obtaining optimal solutions by avoiding zero-flow numerical difficulties. • Quantification of the value of stochastic solution (VSS) and value of reliable solution (VRS) are used as the key measures for assessing the benefits of stochastic programming and reliability-based design optimization. • Simultaneous optimization of reliability and exogenous uncertainty in process design provides potential improvement for operational flexibility and economic performance. Plant availability and operating uncertainties are critical considerations for the design and operation of chemical processes as they directly impact service level and economic performance. This paper proposes a two-stage stochastic programming GDP (Generalized Disjunctive Programming) model with reliability constraints to deal with both the exogenous and endogenous uncertainties in process synthesis, where the reliability model is incorporated into the flowsheet superstructure optimization. The proposed stochastic programming model anticipates the market uncertainties through scenarios for selecting the optimal flowsheet topology, equipment sizes and operating conditions, while considering the impact of selecting parallel units for improving plant availability. An improved logic-based outer approximation algorithm is applied to solve the resulting hybrid GDP model, which effectively avoids numerical difficulties with zero flows and provides high quality design solutions. The applicability of the proposed modeling framework and the efficiency of solution strategy are illustrated with two well-known conceptual design case studies: methanol synthesis process and toluene hydrodealkylation process. The model, which integrates reliability (endogenous uncertainty) and exogenous uncertainty, shows the best economic performance with the increasing operational flexibility and plant availability. [ABSTRACT FROM AUTHOR]

Highlights: [•] MINLP model for simultaneous well placement and production planning in an oil reservoir with water injection. [•] To our knowledge, the first mathematical programming model to embed rigorous subsurface multiphase flow dynamics. [•] Our model gives new/infill producer wells, their locations, and production/injection rates of all wells. [•] Enhanced outer-approximation algorithm (Grossmann and co-workers) for this complex, dynamic, and nonconvex MINLP. [Copyright &y& Elsevier]

Abstract: Optimization methods play a central role in systems biology studies as they can help in identifying key processes that can be experimentally changed so that specific biological goals can be attained. Standard optimization methods used in this field rely on simplified linear models that may fail in capturing the underlying complexity of the target metabolic network. Within this general context, we present a novel approach to globally optimize metabolic networks. The approach presented relies on (1) adopting a general modeling framework for metabolic networks: the Generalized Mass Action (GMA) representation; (2) posing the optimization task as a non-convex nonlinear programming (NLP) problem; and (3) devising an efficient solution method for globally optimizing the resulting NLP that embeds a GMA model of the metabolic network. The capabilities of our method are illustrated through two case studies: the anaerobic fermentation pathway in Saccharomyces cerevisiae and the citric acid production using Aspergillus niger. Numerical results show that the method presented provides near optimal solutions in low CPU times even in cases where the commercial global optimization package BARON fails to close the optimality gap. [ABSTRACT FROM AUTHOR]

Abstract: In this work we address the problem of solving multiscenario optimization models that are deterministic equivalents of two-stage stochastic programs. We present a heuristic approximation strategy where we reduce the number of scenarios and obtain an approximation of the original multiscenario optimization problem. In this strategy, a subset of the given set of scenarios is selected based on a proposed criterion, and probabilities are assigned to the occurrence of scenarios in the reduced set. The original stochastic programming model is converted into a deterministic equivalent using the reduced set of scenarios. A mixed-integer linear program (MILP) is proposed for the reduced scenario selection. We apply this practical heuristic strategy to four numerical examples and show that reformulating and solving the stochastic program with the reduced set of scenarios yields an objective value close to the optimum of the original multiscenario problem. [Copyright &y& Elsevier]

• Novel iterative procedure to generate hybrid models within an optimization framework to solve design problems. • Hybrid models based on first principle and surrogate models (SMs) and represent potential plant process units embedded within a superstructure representation • Iterative procedure: generation of initial SMs with simple algebraic regression models and refinement with adding Gaussian Radial Basis Functions • Three-step refinement: initial SM refinement, domain exploration, and, after solution of the optimal design problem, further exploitation of the domain region • The superstructure optimization problem modeled as a Generalized Disjunctive Programming problem and solved with the Logic-based Outer Approximation algorithm. • Two case studies: methanol synthesis and propylene production plant design via olefin metathesis. • Compared to the optimal design determined with rigorous models, the proposed hybrid models give the same optimal configuration and objective functions with relative differences less than 1.1 %. We propose a novel iterative procedure to generate hybrid models (HMs) within an optimization framework to solve design problems. HMs are based on first principle and surrogate models (SMs) and they may represent potential plant units embedded within a superstructure. We generate initial SMs with simple algebraic regression models and refine them by adding Gaussian Radial Basis Functions in three steps: initial SM refinement, domain exploration, and, after solving the optimal design problem, further domain exploitation, until the convergence criterion is fulfilled. The superstructure optimization problem is formulated with Generalized Disjunctive Programming and solved with the Logic-based Outer Approximation algorithm. We addressed methanol synthesis and propylene plant design problems. Compared to rigorous model-based optimal design, the proposed HMs gave the same configuration, objective function and decision variables with maximum relative differences of 1 and 7 %, respectively. A sensitivity analysis shows that the proposed strategy reduced CPU time by 33 %. [ABSTRACT FROM AUTHOR]

A large number of problems in production planning and scheduling, location, transportation, finance, and engineering design require that decisions be made in the presence of uncertainty. Uncertainty, for instance, governs the prices of fuels, the availability of electricity, and the demand for chemicals. A key difficulty in optimization under uncertainty is in dealing with an uncertainty space that is huge and frequently leads to very large-scale optimization models. Decision-making under uncertainty is often further complicated by the presence of integer decision variables to model logical and other discrete decisions in a multi-period or multi-stage setting.This paper reviews theory and methodology that have been developed to cope with the complexity of optimization problems under uncertainty. We discuss and contrast the classical recourse-based stochastic programming, robust stochastic programming, probabilistic (chance-constraint) programming, fuzzy programming, and stochastic dynamic programming. The advantages and shortcomings of these models are reviewed and illustrated through examples. Applications and the state-of-the-art in computations are also reviewed.Finally, we discuss several main areas for future development in this field. These include development of polynomial-time approximation schemes for multi-stage stochastic programs and the application of global optimization algorithms to two-stage and chance-constraint formulations. [Copyright &y& Elsevier]

• A novel difference-of-convex approach to piecewise linear approximation of nonconvex multivariate functions. • An iterative approach that guarantees convergence to a piecewise linear approximation with an arbitrary tolerance. • Subdomains of the piecewise linear function can be any convex polyhedral sets and are determined automatically. • Case study results demonstrate the advantages of the proposed approach over a start-of-the-art method. We address the problem of finding a continuous piecewise-linear (CPWL) approximation of a nonlinear function that satisfies predefined error-tolerances, and keeps the number of polytopes low. We introduce the difference-of-convex (DC) CPWL representation that represents any CPWL function as the difference of two convex CPWL functions. Any convex CPWL function can be represented as the maximum of a set of affine functions, so the polytopes defining a DC CPWL function can be implicitly defined by the affine functions. By simply searching the parameters of affine functions, CPWL approximations can be produced which contain polytopes of any type. We use the DC CPWL representation to develop a CPWL approximation algorithm and prove its finite convergence. We compare the quality of CPWL approximations produced from the proposed algorithm to the best-known existing method. We find that the DC CPWL approximation consistently requires fewer polytopes to meet the same error-tolerance. [ABSTRACT FROM AUTHOR]

• There are many possible connections from process systems engineering to theoretical computer science. • We show how approximation algorithms can address several process systems engineering applications. • We pose challenges that would enable major breakthroughs in process systems engineering. Designing and analyzing algorithms with provable performance guarantees enables efficient optimization problem solving in different application domains, e.g. communication networks, transportation, economics, and manufacturing. Despite the significant contributions of approximation algorithms in engineering, only limited and isolated works contribute from this perspective in process systems engineering. The current paper discusses three representative, NP -hard problems in process systems engineering: (i) pooling, (ii) process scheduling, and (iii) heat exchanger network synthesis. We survey relevant results and raise major open questions. Further, we present approximation algorithms applications which are relevant to process systems engineering: (i) better mathematical modeling, (ii) problem classification, (iii) designing solution methods, and (iv) dealing with uncertainty. This paper aims to motivate further research at the intersection of approximation algorithms and process systems engineering. [ABSTRACT FROM AUTHOR]