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     

Robust Optimal Operation of Two‐Chamber Microbial Fuel Cell System Under Uncertainty: A Stochastic Simulation Based Multi‐Objective Genetic Algorithm Approach

This investigation is performed to study the optimal operation decision of two‐chamber microbial fuel cell (MFC) system under uncertainty. To gain insight into the mechanism of uncertainty propagation, a Quasi‐Monte Carlo method‐based stochastic analysis is conducted not only to elucidate the effect of each uncertain parameter on the variability of power density output, but also to illustrate the interactive effects of the all uncertain parameters on the performance of MFC. Moreover, a systematic stochastic simulation‐based multi‐objective genetic algorithm framework is proposed to identify a set of Pareto‐optimal robust operation strategies, which is helpful to provide an imperative insight into the relationship between the mean and standard deviation of output power density. The results indicate that (1) the coefficient of variance (COV) value of output power density has a linear relationship with the COV value of each uncertainty parameter as well as all interactive parameters; and (2) a significant performance improvement with respect to both mean and standard deviation of power density is observed by implementing the multi‐objective robust optimization. These results thus validate that the proposed uncertainty analysis and robust optimization framework provide a promising tool for robust optimal design and operation of fuel cell systems under uncertainty.     

Characteristics‐Based Model Predictive Control of a Catalytic Flow Reversal Reactor

This paper describes the formulation and tuning of a model‐based controller for a catalytic flow reversal reactor (CFRR). A plug flow non‐linear pseudo‐homogeneous mathematical representation of the process is used to model the mass and energy transport phenomena for the model‐based controller. A combination of the method of characteristics and model predictive control (MPC) technology is used to formulate the controller (Shang et al., Ind. Eng. Chem. Res. 43 (9) 2140–2149 (2004)). Mass extraction from the midsection of the reactor is used as the manipulated variable. Numerical simulations are used to show the performance of the formulated controller. The performance of the controller is evaluated on a simulated catalytic flow reversal reactor unit for combustion of lean methane streams for reduction of greenhouse gases emissions.     

LMI-based robust model predictive control for a continuous MMA polymerization reactor

A linear matrix inequality (LMI)-based robust model predictive control (MPC) is applied to a continuous stirred-tank reactor for the polymerization of methyl methacrylate (MMA). The polytopic model is constructed to predict the responses to various control input sequences by using Jacobians of uncertain nonlinear model at several operating points and the controller design is characterized as the problem of minimizing an upper bound on the ‘worst-case’ infinite horizon objective function subject to constraints on the control input and plant output. Simulation studies under different conditions are conducted to validate the feasibility of the optimization problem and evaluate the applicability of such a control scheme. Simulation results show that, despite the model uncertainty, the LMI-based robust model predictive controller performs quite satisfactorily for the property control of the continuous polymerization reactor and guarantees the robust stability.     

Robust model predictive control and fault handling of batch processes

This work considers the control of batch processes subject to input constraints and model uncertainty with the objective of achieving a desired product quality. First, a computationally efficient nonlinear robust Model Predictive Control (MPC) is designed. The robust MPC scheme uses robust reverse‐time reachability regions (RTRRs), which we define as the set of process states that can be driven to a desired neighborhood of the target end‐point subject to input constraints and model uncertainty. A multilevel optimization‐based algorithm to generate robust RTRRs for specified uncertainty bounds is presented. We then consider the problem of uncertain batch processes subject to finite duration faults in the control actuators. Using the robust RTRR‐based MPC as the main tool, a robust safe‐steering framework is developed to address the problem of how to operate the functioning inputs during the fault repair period to ensure that the desired end‐point neighborhood can be reached upon recovery of the full control effort. The applicability of the proposed robust RTRR‐based controller and safe‐steering framework subject to limited availability of measurements and sensor noise are illustrated using a fed‐batch reactor system. © 2010 American Institute of Chemical Engineers AIChE J, 2011     

Optimal supply chain design and operations under multi‐scale uncertainties: Nested stochastic robust optimization modeling framework and solution algorithm

Although strategic and operational uncertainties differ in their significance of impact, a “one‐size‐fits‐all” approach has been typically used to tackle all types of uncertainty in the optimal design and operations of supply chains. In this work, we propose a stochastic robust optimization model that handles multi‐scale uncertainties in a holistic framework, aiming to optimize the expected economic performance while ensuring the robustness of operations. Stochastic programming and robust optimization approaches are integrated in a nested manner to reflect the decision maker's different levels of conservativeness toward strategic and operational uncertainties. The resulting multi‐level mixed‐integer linear programming model is solved by a decomposition‐based column‐and‐constraint generation algorithm. To illustrate the application, a county‐level case study on optimal design and operations of a spatially‐explicit biofuel supply chain in Illinois is presented, which demonstrates the advantages and flexibility of the proposed modeling framework and efficiency of the solution algorithm. © 2016 American Institute of Chemical Engineers AIChE J, 62: 3041–3055, 2016     

Aqueous‐Phase Reforming in a Microreactor: The Role of Surface Bubbles

In heterogeneous catalysis, the creation of gaseous products as bubbles in a liquid phase on the catalytic surface is associated with slip phenomena. In a microreactor, the slip length at the gas‐liquid interface is in the same order of magnitude as the reactor dimensions, which can affect fluid dynamics and transport phenomena. Here, the interplay of momentum, heat and mass transfer in a microreactor, when bubbles form on the catalytic surface, was investigated using two‐dimensional simulations. The effect of bubbles on the endothermic process of aqueous‐phase reforming of a glycerol solution was evaluated in terms of conversion and conversion and temperature in the reactor. Altogether, this study highlights the impact of bubbles, not only on the transport phenomena but also on the reactor performance.     

Internal model control design for input‐constrained multivariable processes

Multivariable plants under input constraints such as actuator saturation are liable to performance deterioration due to control windup and directionality change. A two‐stage internal model control (IMC) antiwindup design for open loop stable plants is presented. The design is based on the solution of two low‐order quadratic programs at each time step, which addresses both transient and steady‐state behaviors of the system. For analyzing the robust stability of such systems against any infinity‐norm bounded uncertainty, stability test have also been developed. In particular, we note that the controller input‐output mappings satisfy certain integral quadratic constraints. Simulated examples show that the two‐stage IMC has superior performance when compared with other existing optimization‐based antiwindup methods. The stability test is illustrated for a plant with left matrix fraction uncertainty. A scenario where the proposed two‐stage IMC competes favorably with a long prediction horizon model predictive control is described. © 2011 American Institute of Chemical Engineers AIChE J, 2011     

Identification of uncertain MIMO Wiener and Hammerstein models

Several approaches can be found in the literature to perform the identification of block oriented models (BOMs). In this sense, an important improvement is to achieve robust identification to cope with the presence of uncertainty.In this work, two special and widely used BOMs are considered: Hammerstein and Wiener models. The models herein treated are assumed to be described by parametric representations. The approach introduced in this work for the identification of the multiple input-multiple output (MIMO) uncertain model is performed in a single step. The uncertainty is described as a set of parameters which is found through the solution of an optimization problem.A distillation column simulation model is presented to illustrate the robust identification approach. This process is an interesting benchmark due to its well-known nonlinear dynamics. Both Hammerstein and Wiener models are used to represent this plant in the presence of uncertainty. A comparative study between these models is established.     

Optimization of grade transitions in polyethylene solution polymerization process under uncertainty

Model-based dynamic optimization is an effective tool for control and optimization of chemical processes, especially during transitions in operation. This study considers the dynamic optimization of grade transitions for a solution polymerization process. Here, a detailed dynamic model comprises the entire flowsheet and includes a method-of-moments reactor model to determine product properties, a simple yet accurate vapor–liquid equilibrium (VLE) model derived from rigorous calculations, and a variable time delay model for recycle streams. To solve the grade transition problem, both single stage and multistage optimization formulations have been developed to deal with specification bands of product properties.This dynamic optimization framework demonstrates significant performance improvements for grade transition problems. However, performance can deteriorate in the presence of uncertainties, disturbances and model mismatch. To deal with these uncertainties, this study applies robust optimization formulations through the incorporation of back-off constraints within the optimization problem. With back-off terms calculated from Monte Carlo simulations, the resulting robust optimization formulation can be solved with the same effort as the nominal dynamic optimization problem, and the resulting solution is shown to be robust under various uncertainty levels with minimal performance loss. Additional case studies show that our optimization approach extends naturally to different regularizations and multiple sources of uncertainty.     

Optimal processing network design under uncertainty for producing fuels and value‐added bioproducts from microalgae: Two‐stage adaptive robust mixed integer fractional programming model and computationally efficient solution algorithm

Fractional metrics, such as return on investment (ROI), are widely used for performance evaluation, but uncertainty in the real market may unfortunately diminish the results that are based on nominal parameters. This article addresses the optimal design of a large‐scale processing network for producing a variety of algae‐based fuels and value‐added bioproducts under uncertainty. We develop by far the most comprehensive processing network with 46,704 alternative processing pathways. Based on the superstructure, a two‐stage adaptive robust mixed integer fractional programming model is proposed to tackle the uncertainty and select the robust optimal processing pathway with the highest ROI. Since the proposed problem cannot be solved directly by any off‐the‐shelf solver, we develop an efficient tailored solution method that integrates a parametric algorithm with a column‐and‐constraint generation algorithm. The resulting robust optimal processing pathway selects biodiesel and poly‐3‐hydroxybutyrate as the final fuel and bioproduct, respectively. © 2016 American Institute of Chemical Engineers AIChE J, 63: 582–600, 2017     

On the relation between flexibility analysis and robust optimization for linear systems

Flexibility analysis and robust optimization are two approaches to solving optimization problems under uncertainty that share some fundamental concepts, such as the use of polyhedral uncertainty sets and the worst‐case approach to guarantee feasibility. The connection between these two approaches has not been sufficiently acknowledged and examined in the literature. In this context, the contributions of this work are fourfold: (1) a comparison between flexibility analysis and robust optimization from a historical perspective is presented; (2) for linear systems, new formulations for the three classical flexibility analysis problems—flexibility test, flexibility index, and design under uncertainty—based on duality theory and the affinely adjustable robust optimization (AARO) approach are proposed; (3) the AARO approach is shown to be generally more restrictive such that it may lead to overly conservative solutions; (4) numerical examples show the improved computational performance from the proposed formulations compared to the traditional flexibility analysis models. © 2016 American Institute of Chemical Engineers AIChE J, 62: 3109–3123, 2016     

Integration of Design and Control: A Robust Control Approach Using MPC

This paper presents a new method to integrate process control with process design. The process design is based on steady‐state costs, .i.e., capital and operating costs. Control is incorporated into the design in terms of a variability cost. This term is calculated based on the non‐linear process model, represented here as a nominal linear model supplemented with model parameter uncertainty. Robust control tools are then used within the approach to assess closed‐loop robust stability and to calculate closed‐loop variability. The integrated method results in a non‐linear constrained optimization problem with an objective function that consists of the sum of the steady costs and the variability cost. Optimization using the traditional sequential approach and the new integrated method was applied to design a multi‐component distillation column using a Model Predictive Control (MPC) algorithm. The optimization results show that the integrated method can lead to significant cost savings when compared to the traditional sequential approach. In addition, an RGA analysis was performed to study the effects of process interactions on the optimization results.     

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     

Input/output linearization with a two-degree-of-freedom scheme for uncertain nonlinear processes

This work presents a new method to control processes with unmeasured input disturbance and random noise parametric uncertainty. The developed method takes advantage of a two-degree-of-freedom control structure in which setpoint regulation and load disturbance rejection are integrated in the controller synthesis. Input/output linearization is selected to provide the setpoint tracking ability. For disturbance rejection, the high-gain technique is used to compensate for the effect of the uncertainty. The control performance of the method is evaluated through numerical simulation of continuous stirred tank reactors with uncertainty. The simulation results show that both unmeasured disturbance and parametric uncertainty can be effectively compensated for by the proposed control method.     

Robust control of a reverse-flow reactor

This paper is focused on the design of a robust controller for a catalytic fixed-bed reactor with periodical inversion of the flow direction (reverse-flow reactor, RFR). The analogy between the RFR operated at infinite switching frequency and the countercurrent reactor is the basis of the simplified mathematical model of the reactor.The control system uses dilution and internal electric heating to ensure complete conversion of the reactants and to prevent overheating of the catalyst. As the state of the system is not fully available, apart from some temperature measurements, an observer is designed and used in the control algorithm. This is a typical case of nonlinear system with uncertainties. Following the procedure described in detail by Fissore [2008. Robust control in presence of parametric uncertainties: observer-based feedback controller design. Chemical Engineering Science, in press, doi:10.1016/j.ces.2007.12.019.], the extended model for the process is setup, thus taking into account all the simplifications of the model and linking performance and robustness to the control law, which is a simple state feedback. Simulations with randomly varying feeding concentration have been carried out in order to demonstrate the effectiveness of the proposed control system.     

Rotating corrugated photoreactor design: Experimental and computational analysis of TiO2‐based photocatalysis

A TiO₂‐coated rotating corrugated reactor is experimentally characterized and simulated for ultraviolet (UV)‐based photo degradation of methylene blue (MB). Using degradation kinetics derived from flat‐plate experiments and finite‐element discrete‐ordinate models simulating spatially and temporally varying radiation patterns from a UV‐lamp array, the simulated reactor performance is experimentally validated and extended to explore the impact of geometry variations. Mass‐transfer limitations between the corrugated surface and the reservoir bulk are identified as a determining factor in the degradation rate of MB. Optimization studies suggest minimizing the corrugation half‐angle to improve reactor performance for a given area, increasing the number of corrugations rather than the corrugation depth subject to practical limitations arising from undesirable bubble formation at small aperture widths. Given the relevance of mass transfer to this system, future work is needed to elucidate the experimental impact of reservoir volume and geometry while validating the use of flat‐plate correlations for mass transfer. © 2012 American Institute of Chemical Engineers AIChE J, 59: 560–570, 2013     

Heterogeneous Catalytic Reactor Design with Non‐Uniform Catalyst Considering Shell‐Progressive Poisoning Behavior

A novel methodology has been developed to design an optimum heterogeneous catalytic reactor, by considering non‐uniform catalyst pellet under shell‐progressive catalyst deactivation. Various types of non‐uniform catalyst pellets are modelled in combination with reactor design. For example, typical non‐uniform catalyst pellets such as egg‐yolk, egg‐shell and middle‐peak distribution are developed as well as step‐type distribution. A progressive poisoning behavior is included to the model to produce correct effectiveness factor from non‐uniform catalyst pellet. As opposed to numerical experiment with limited type of kinetic application to the model in the past, this paper shows a new methodology to include any types of kinetic reactions for the modeling of the reactor with non‐uniform catalyst pellet and shell‐progressive poisoning. For an optimum reactor design, reactor and catalyst variables are considered at the same time. For example, active layer thickness and location inside pellet are optimised together with reactor temperature for the maximisation of the reactor performance. Furthermore, the temperature control strategy over the reactor operation period is added to the optimization, which extends the model to three dimensions. A computational burden has been a major concern for the optimization, and innovative methodology is adopted. Application of profile based synthesis with the combination of SA (Simulated Annealing) and SQP (Successive Quadratic Programming) allows more efficient computation not only at steady state but also in dynamic status over the catalyst lifetime. A Benzene hydrogenation reaction in an industry scale fixed‐bed reactor is used as a case study for illustration.     

Model‐Aided Design of a Three‐Phase Gas‐Lift Reactor for Oxidation Accompanied by Catalyst Reversible Deactivation

A mathematical model of a three‐phase gas‐lift reactor (GLR) is developed to aid the design of a target reactor for simultaneous substrate catalytic oxidation in riser and a deactivated reactivation catalyst in the downcomer section of the multifunctional reactor. In the GLR model, the hydrodynamics of a real GLR and the kinetics of glucose oxidation by air over a palladium catalyst are incorporated. The GLR model searches for the optimal geometry of the target reactor. With regard to the GLR optimal geometry, the reactor productivity is maximal for given input operational conditions. An algorithm of the GLR model is presented together with simulation results of the target GLR and with insight into the parametric sensitivity of the model. Effects of the reaction components concentrations and the gas‐phase superficial velocity on the location of the target reactor optimal geometry and on the reactor productivity are discussed.     

A Comparative Study of Fuzzy Techniques to Handle Uncertainty: An Industrial Grinding Process

Fuzzy‐based approaches like fuzzy chance constrained programming (FCCP) and fuzzy expected value model (FEVM) have been applied to a multi‐objective optimization problem of the industrial grinding process to carry out the uncertainty analysis. Results are compared with respect to the power of risk averseness adopted in the approaches used. The extent of constraint satisfaction due to the presence of uncertain parameters can be accommodated assuming credibility of constraint satisfaction under the FCCP framework whereas the robust set of parameters in the FEVM approach is determined by considering the expectation terms for objectives and constraints. Nonlinear relation of uncertain parameters has been handled by adopting simulation‐based approaches while computing the credibility. These approaches are very generic and can be applied for the study of parametric sensitivity for any process model in a novel manner.