Numerical solution approaches for robust nonlinear optimal control problems

Nonlinear equality and inequality constrained optimization problems with uncertain parameters can be addressed by a robust worst-case formulation that leads to a bi-level min–max optimization problem. We propose and investigate a numerical method to solve this min–max optimization problem exactly in the case that the underlying maximization problem always has its solution on the boundary of the uncertainty set. This is an adoption of the local reduction approach used to solve generalized semi-infinite programs. The approach formulates an equilibrium constraint employing first order derivatives of both the uncertainty set and the user defined constraints. We propose two different ways for computation of these derivatives, one similar to the forward mode, the other similar to the reverse mode of automatic differentiation. We show the equivalence of the proposed approach to a method based on geometric considerations that was recently developed by some of the authors. We show how to generalize the techniques to optimal control problems. The robust dynamic optimization of a batch distillation illustrates that both techniques are numerically efficient and able to overcome the inexactness of another recently proposed numerical approach to address uncertainty in optimal control problems.     

Integrating product and process design with optimal control: A case study of solvent recycling

This paper proposes the simultaneous integration of environmentally benign solvent selection (product design), solvent recycling (process design) and optimal control for the separation of azeotropic systems using batch distillation. The previous work performed by Kim et al. (2004. Entrainer selection and solvent recycling in complex batch distillation. Chemical Engineering Communications 191(12), 1606-1633) combines the chemical synthesis and process synthesis under uncertainty. For batch distillation, optimal operation is also important due to the unsteady state nature of the process and high operating costs. Optimal control allows us to optimize the column operating policy by selecting a trajectory for the reflux ratio. However, there are time-dependent uncertainties in thermodynamic models of batch distillation due to the assumption of constant relative volatility. In this paper, the uncertainties in relative volatility were modeled using Ito processes and the stochastic optimal control problem was solved by combined maximum principle and non-linear programming (NLP) techniques. Then the previous work of optimal solvent selection and recycling was coupled with optimal control. As a real world example for this integrated approach, a waste stream containing acetonitrile-water was studied. The optimal design parameters obtained by Kim et al. (2004. Entrainer selection and solvent recycling in complex batch distillation. Chemical Engineering Communications 191(12), 1606-1633), for this separation were used and the optimal control policy is computed first without considering uncertainties by variable transformation technique. The deterministic optimal control policy improves the product yield by 4.0% as compared to the base case, verified using a rigorous simulator for batch distillation. When the stochastic optimal control policy was computed representing the relative volatility as an Ito process, a similar recovery rate was obtained from simulations, but the batch time was reduced significantly, producing the most profitable operation.     

A bilevel NLP sensitivity‐based decomposition for dynamic optimization with moving finite elements

Optimal control has guided numerous applications in chemical engineering, and exact determination of optimal profiles is essential for operation of separation and reactive processes, and operating strategies and recipe generation for batch processes. Here, a simultaneous collocation formulation based on moving finite elements is developed for the solution of a class of optimal control problems. Novel features of the algorithm include the direct location of breakpoints for control profiles and a termination criterion based on a constant Hamiltonian profile. The algorithm is stabilized and performance is significantly improved by decomposing the overall nonlinear programming (NLP) formulation into an inner problem, which solves a fixed element simultaneous collocation problem, and an outer problem, which adjusts the finite elements based on several error criteria. This bilevel formulation is aided by a NLP solver (the interior point optimizer) for both problems as well as an NLP sensitivity component, which provides derivative information from the inner problem to the outer problem. This approach is demonstrated on 11 dynamic optimization problems drawn from the optimal control and chemical engineering literature. © 2014 American Institute of Chemical Engineers AIChE J, 60: 966–979, 2014     

A simultaneous approach for singular optimal control based on partial moving grid

Singular optimal control plays an important role in process engineering, including optimal operation of batch and semi-batch reactions. However, for many practical applications, accurate solution of singular optimal control profiles is still an open issue. In particular, numerical optimization must deal with an ill-conditioned problem that often leads to very slow convergence or failure. Starting from the nested approach in our previous work in 2016, this study develops a more efficient strategy for singular control through a heuristic approach for the outer problem. The approach includes three stages. Starting from a coarse distribution of finite elements, sufficiently many finite elements are inserted where control profiles are steep and fixed gridpoints are inserted on the basis of error estimation of state profiles. Then, moving gridpoints are inserted where the modified switching function is violated. Initial junctions are obtained by moving the latest inserted gridpoints. Moreover, further mesh refinements are considered based on switching point detection and a moving grid point update strategy, until modified switching conditions are satisfied over the whole-time span. A key feature of this approach is that only a subset of finite elements needs to move during optimization. Complexity of the optimization formulation is considerably decreased compared to our previous work. This approach is demonstrated on eight classical singular control problems with known solutions, as well as six complex singular control problems drawn from the chemical engineering literature.     

二元提馏式间歇精馏的优化操作与最小汽化总量

理想操作条件下二元提馏式间歇精馏优化操作的汽化总量与最小汽化总量的计算是约束函数优化问题。本文采用罚函数法,将此约束函数优化转变为无约束函数优化,并采用固定双步长因子梯度法数值求解该函数的极值。计算表明:固定双步长因子梯度法具有良好的收敛性,同时,降低分段数较多时,数值截断误差积累对计算结果的影响。二元提馏式间歇精馏优化操作较恒残液组成操作的能耗低的原因如下:在理论板数相对较少(接近二元提馏式间歇精馏恒残液组成操作所需的最少理论板)时,优化操作通过控制再沸比提高了能耗效率;在理论板数相对较多时,优化操作通过控制再沸比,在保证过程的能耗效率较高的同时,可尽可能快地将物料移出系统,减少了精馏过程中塔顶贮槽内液体的混合熵产。通过对计算结果的归纳与外推,得到了理想操作条件下理论板数为无穷多时二元提馏式间歇精馏优化操作再沸比的变化方式以及最小汽化总量的计算公式。     

Recovery of lactic acid by batch reactive distillation

Lactic acid, being virtually a non‐boiling compound, is difficult to separate from its aqueous solution by conventional methods such as distillation. It is necessary to convert it to the relatively volatile ester and the separation of the ester, followed by hydrolysis, is recommended as an appropriate method of recovery. In the present work, we explore and investigate a novel reactive distillation strategy to perform esterification, distillation and hydrolysis in a single unit. The experiments were performed in a batch reactive distillation set‐up and the results have been explained with the help of an appropriate model. An unsteady state mathematical model based on an equilibrium stage concept was developed for batch reactive distillation. A pseudo‐homogeneous model was used for the determination of reaction kinetics. The effect of operating parameters such as feed concentration, mole ratio, catalyst loading, boil‐up rate, etc. on the recovery of lactic acid was studied with the help of simulation and experimental results. The feasibility issue of reactive distillation has been discussed based on the results obtained. Copyright © 2006 Society of Chemical Industry     

Design of multipurpose production facilities: A RTN decomposition-based algorithm

A general mathematical formulation for the design of multipurpose facilities has recently been presented by Barbosa-Póvoa and Pantelides (1997). The model proposed permits a detailed consideration of the design problem taking account of trade-offs between capital costs, revenues and operational flexibility. The optimal solution involves the selection of the required processing and storage equipment items and the required levels of provision of other production resources such as utilities, manpower, cleaning and transportation equipment.In order to guarantee solution optimality, the above design formulation has to consider a large number of equipment items, out of which it will select the ones that will actually be incorporated in the plant. This may result in large mixed-integer linear programming (MILP) problems that are expensive to solve.This paper presents a decomposition approach for the solution of large batch process design problems. The approach involves the iterative solution of a master problem (representing a relaxation of the original design problem) and a design sub-problem (in which several of the design decisions are already fixed).An example illustrating the effectiveness of the proposed decomposition approach is presented.     

A highway in state space for reactors with minimum entropy production

Thousands of numerical solutions of an optimal control problem for plug flow reactors were found to give, what we call a “highway in the reactors’ state space”. The problem was to find the heat transfer strategy which minimise the entropy production in reactors with fixed chemical conversion. The control variable was always the temperature of the heating/cooling medium along the reactor. The highway represents the most energy efficient way to travel far in state space. Such highways were studied for five reactor systems, endothermic and exothermic ones. Numerical analysis showed that the reactor highway is characterised by approximately constant thermodynamic driving forces/local entropy production for reasonable process intensities. Each solution represents a compromise between the entropy production of reactions, heat transfer and frictional flow (pressure drop). The solutions enter and leave the highway at different positions depending on how far from the highway their initial and final destinations are. Knowledge about the nature of the highway, e.g. when the reactor operates in a reaction mode or a heat transfer mode, may be important for energy efficient reactor design. The theoretical formulation of the optimisation problem is valid for plug flow as well as batch reactors. We showed that important results in literature like the Spirkl-Ries quantity, the theorems of equipartition of entropy production and equipartition of forces are contained in our general formulation. The numerical results showed that the analytical results are good approximations to the optimum also in problems where they do not apply in a strictly mathematical sense.     

Model based control of a liquid swelling constrained batch reactor subject to recipe uncertainties

This work presents the application of nonlinear model predictive control (NMPC) to a simulated industrial batch reactor subject to safety constraint due to reactor level swelling, which can occur with relatively fast dynamics. Uncertainties in the implementation of recipes in batch process operation are of significant industrial relevance. The paper describes a novel control-relevant formulation of the excessive liquid rise problem for a two-phase batch reactor subject to recipe uncertainties. The control simulations are carried out using a dedicated NMPC and optimization software toolbox OptCon which implements efficient numerical algorithms. The open-loop optimal control problem is computed using the multiple-shooting technique and the arising nonlinear programming problem is solved using a sequential quadratic programming (SQP) algorithm tailored for large-scale problems, based on the freeware optimization environment HQP. The fast response of the NMPC controller is guaranteed by the initial value embedding and real-time iteration technologies. It is concluded that the OptCon implementation allows small sampling times and the controller is able to maintain safe and optimal operation conditions, with good control performance despite significant uncertainties in the implementation of the batch recipe.     

Stochastic maximum principle for optimal control under uncertainty

Optimal control problems involve the difficult task of determining time-varying profiles through dynamic optimization. Such problems become even more complex in practical situations where handling time dependent uncertainties becomes an important issue. Approaches to stochastic optimal control problems have been reported in the finance literature and are based on real option theory, combining Ito’s Lemma and the dynamic programming formulation. This paper describes a new approach to stochastic optimal control problems in which the stochastic dynamic programming formulation is converted into a stochastic maximum principle formulation. An application of such method has been reported by Rico-Ramirez et al. (Computers and Chemical Engineering, 2003, 27, 1867) but no details of the derivation were provided. The main significance of this approach is that the solution to the partial differential equations involved in the dynamic programming formulation is avoided. The classical isoperimetric problem illustrates this approach.     

Integrated design and control of semicontinuous distillation systems utilizing mixed integer dynamic optimization

Semicontinuous distillation systems are notoriously difficult to design and optimize because the structural parameters, operational parameters, and control system must all be determined simultaneously. In the past 15 years of research into semicontinuous systems, studies of the optimal design of these systems have all been limited in scope to small subsets of the parameters, which yields suboptimal and often unsatisfactory results. In this work, for the first time, the problem of integrated design and control of semicontinuous distillation processes is studied by using a mixed integer dynamic optimization (MIDO) problem formulation to optimize both the structural and control tuning parameters of the system. The public model library (PML) of gPROMS is used to simulate the process and the built-in optimization package of gPROMS is used to solve the MIDO via the deterministic outer approximation method. The optimization results are then compared to the heuristic particle swarm optimization (PSO) method.     

OPTIMAL OPERATION OF BATCH DISTILLATION

The maximum-profit problem in the operation of a batch distillation column separating a binary mixture is studied. From the objective function made in this paper, such problem can be transformed under special condition, into either maximum-distillate or minimum-time problem. Reflux ratio is chosen as control variable. Digital method is worked out by using optimal control theory to find the condition of optimal operation. In deriving present mathematic model, the liquid hold-up in plates and in condenser as well as the plate efficiency are taken into account. The proposed optimal operation is compared theoretically with the conventional constant reflux ratio and constant overhead composition policies and the results show that the proposed optimal operation can lead to/nuch higher profits. To demonstrate the practical usefulness of the proposed optimal policy, experimental work is carried out in a 4 sieve trays 0.17M diameter batch distillation column with ethanolwater system. Agreement is found between the experimental data and the computed results from present model.     

OPTIMAL DESIGN AND OPERATION OF BATCH PROCESSES

The economical design of continuous chemical processes to produce commodity products has reached an advanced state of development. Modern computer tools are used routinely to simulate and optimize these processes. This is not the case, however, for the manufacture of speciality products which must be made in batch operations. The continuing shift towards the production of higher value-added specialty products by the CPI has stimulated efforts aimed at developing good computer assisted design strategies for batch processes.

This paper discusses the formulation of the problem for the optimal design and operation of batch processes. The batch problem differs from the continuous one in a number of important ways. First, batch plants do not operate at steady state. There are important trade-offs between the processing time and the severity (intensity) of processing in single units. Cycle time and performance trade-offs also exist among the various units in the process. Second, batch plants produce multiple products in many cases. There is often a competition for shared resources (labor, utilities, and equipment) among the various products. This paper presents a hierarchical solution approach for the design and optimization of a batch process. The approach is demonstrated by solving an example problem which illustrates the fundamental economic trade-offs.     

Handling multi‐rate and missing data in variable duration economic model predictive control of batch processes

In the present work, we consider the problem of variable duration economic model predictive control of batch processes subject to multi‐rate and missing data. To this end, we first generalize a recently developed subspace‐based model identification approach for batch processes to handle multi‐rate and missing data by utilizing the incremental singular value decomposition technique. Exploiting the fact that the proposed identification approach is capable of handling inconsistent batch lengths, the resulting dynamic model is integrated into a tiered EMPC formulation that optimizes process economics (including batch duration). Simulation case studies involving application to the energy intensive electric arc furnace process demonstrate the efficacy of the proposed approach compared to a traditional trajectory tracking approach subject to limited availability of process measurements, missing data, measurement noise, and constraints. © 2017 American Institute of Chemical Engineers AIChE J, 63: 2705–2718, 2017     

A modeling strategy for integrated batch process development based on mixed-logic dynamic optimization

This paper introduces an optimization-based approach for the simultaneous solution of batch process synthesis and plant allocation, with decisions like the selection of chemicals, process stages, task-unit assignments, operating modes, and optimal control profiles, among others. The modeling strategy is based on the representation of structural alternatives in a state-equipment network (SEN) and its formulation as a mixed-logic dynamic optimization (MLDO) problem. Particularly, the disjunctive multistage modeling strategy by Oldenburg and Marquardt (2008) is extended to combine and organize single-stage and multistage models for representing the sequence of continuous and batch units in each structural alternative and for synchronizing dynamic profiles in input and output operations with material transference. Two numerical examples illustrate the application of the proposed methodology, showing the enhancement of the adaptability potential of batch plants and the improvement of global process performance thanks to the quantification of interactions between process synthesis and plant allocation decisions.     

Simultaneous synthesis of distillation trains and heat exchanger networks

A new mathematical formulation for the optimal synthesis of heat-integrated distillation systems that accounts for constrained heat matches between columns and the rest of the process is presented. The definition of temperature intervals commonly used by heat exchanger network synthesis procedures is embedded in the proposed formulation that can be regarded as a generalization of the network flow model of Cerdá and Westerberg (1983, Chem. Engng Sci.38, 1723) under conditions of variable stream temperatures. The minimization of the total flow-sheet utility cost has been chosen as the design target.The problem has been represented through a mixed-integer linear program. By performing the search with a standard branch-and-bound technique the global optimality of the solution found is guaranteed. Three case studies involving the separation of a ternary mixture by distillation have been solved. The consideration of the rest of the process produces in two of them new distillation train designs costing less overall.     

Inferential Control of Reactive Destillation Columns – An Algorithmic Approach

Decentralized control system design comprises the selection of a suitable control structure and controller parameters. Here, mixed integer optimization is used to determine the optimal control structure and the optimal controller parameters simultaneously. The process dynamics is included explicitly into the constraints using a rigorous nonlinear dynamic process model. Depending on the objective function, which is used for the evaluation of competing control systems, two different formulations are proposed which lead to mixed‐integer dynamic optimization (MIDO) problems. A MIDO solution strategy based on the sequential approach is adopted in the present paper. Here, the MIDO problem is decomposed into a series of nonlinear programming (NLP) subproblems (dynamic optimization) where the binary variables are fixed, and mixed‐integer linear programming (MILP) master problems which determine a new binary configuration for the next NLP subproblem. The proposed methodology is applied to inferential control of reactive distillation columns as a challenging benchmark problem for chemical process control.