Benoît Chachuat

Adaptation Strategies for Real-time Optimization

Challenges in real-time process optimization mainly arise from the inability to build and adapt accurate models for complex physico-chemical processes. This paper surveys different ways of using measurements to compensate for model uncertainty in the context of process optimization. Three approaches can be distinguished according to the quantities that are adapted: model-parameter adaptation updates the parameters of the process model and repeats the optimization, modifier adaptation modifies the constraints and gradients of the optimization problem and repeats the optimization, while direct input adaptation turns the optimization problem into a feedback control problem and implements optimality via tracking of appropriate controlled variables. This paper argues in favor of modifier adaptation, since it uses a model parameterization and an update criterion that are well tailored to meeting the KKT conditions of optimality. These considerations are illustrated with the real-time optimization of a semi-batch reactor system.

Convex/concave relaxations of parametric ODEs using Taylor models

This paper presents a discretize-then-relax method to construct convex/concave bounds for the solutions of a wide class of parametric nonlinear ODEs. The algorithm builds upon Taylor model methods recently developed for verified solution of parametric ODEs. To enable the propagation of convex/concave state bounds, a new type of Taylor model is introduced, in which convex/concave bounds for the remainder term are computed in addition to the usual interval bounds. At each time step, a two-phase procedure is applied: a priori convex/concave bounds that are valid over the entire time step are calculated in the first phase; then, pointwise-in-time convex/concave bounds at the end of the time step are obtained in the second phase. This algorithm is implemented in an object-oriented manner using templates and operator overloading. It is demonstrated and compared to other available approaches on a selection of problems from the literature.

Dynamic optimisation of small size wastewater treatment plants including nitrification and denitrification processes

In this paper, dynamic optimisation of small size wastewater treatment plants is studied. The problem is stated as a hybrid dynamic optimisation problem which is solved using a gradient-based method. The aeration policy which minimises the energy consumption and satisfies discharge requirements under specified constraints (process and physical constraints) is then determined. The comparison between usual rule-based control policies and optimised aeration strategies showed that the optimised aeration profiles lead, to reductions of energy consumption of at least 30%.

A superstructure optimization approach for water network synthesis with membrane separation-based regenerators

Abstract This work addresses the problem of water network synthesis. We propose a superstructure with fixed topology for a water network that consists of three layers, similar to a pooling problem: sources for reuse/recycle; regenerators for contaminants removal; and sinks for acceptance of water for reuse/recycle. The superstructure encompasses multiple freshwater sources, membrane separation-based partitioning regenerators of the industrially favored ultrafiltration and reverse osmosis, and sinks for incineration and deep ocean discharge. A mixed-integer nonlinear program is formulated based on this superstructure to determine the optimal interconnections in terms of total flowrates and contaminant concentrations. The main decisions include determining the split fractions of the source flowrates, extents of regeneration, and mixing ratios of the sources and regenerated streams subject to compliance with the maximum allowable inlet contaminant concentration limits of the sinks and discharge regulations. We also develop linear models for the membrane regenerators that admit a more general expression for the retentate stream concentration based on liquid-phase recovery factors and removal ratios. Computational studies are performed using GAMS/BARON on an industrially significant case study of a petroleum refinery water system. We incorporate linear logical constraints using 0–1 variables that enforce certain design and structural specifications to tighten the model formulation and enhance solution convergence. A globally optimal water network topology is attained that promotes a 27% savings equivalent to about

Unified framework for the propagation of continuous-time enclosures for parametric nonlinear ODEs

218,000/year reduction in freshwater use.

Modelling of Microalgae Culture Systems with Applications to Control and Optimization

This paper presents a framework for constructing and analyzing enclosures of the reachable set of nonlinear ordinary differential equations using continuous-time set-propagation methods. The focus is on convex enclosures that can be characterized in terms of their support functions. A generalized differential inequality is introduced, whose solutions describe such support functions for a convex enclosure of the reachable set under mild conditions. It is shown that existing continuous-time bounding methods that are based on standard differential inequalities or ellipsoidal set propagation techniques can be recovered as special cases of this generalized differential inequality. A way of extending this approach for the construction of nonconvex enclosures is also described, which relies on Taylor models with convex remainder bounds. This unifying framework provides a means for analyzing the convergence properties of continuous-time enclosure methods. The enclosure techniques and convergence results are illustrated with numerical case studies throughout the paper, including a six-state dynamic model of anaerobic digestion.

Towards global bilevel dynamic optimization

Mathematical modeling is becoming ever more important to assess the potential, guide the design, and enable the efficient operation and control of industrial-scale microalgae culture systems (MCS). The development of overall, inherently multiphysics, models involves coupling separate submodels of (i) the intrinsic biological properties, including growth, decay, and biosynthesis as well as the effect of light and temperature on these processes, and (ii) the physical properties, such as the hydrodynamics, light attenuation, and temperature in the culture medium. When considering high-density microalgae culture, in particular, the coupling between biology and physics becomes critical. This chapter reviews existing models, with a particular focus on the Droop model, which is a precursor model, and it highlights the structure common to many microalgae growth models. It summarizes the main developments and difficulties towards multiphysics models of MCS as well as applications of these models for monitoring, control, and optimization purposes.

Local optimization of dynamic programs with guaranteed satisfaction of path constraints

The global solution of bilevel dynamic optimization problems is discussed. An overview of a deterministic algorithm for bilevel programs with nonconvex functions participating is given, followed by a summary of deterministic algorithms for the global solution of optimization problems with nonlinear ordinary differential equations embedded. Improved formulations for scenario-integrated optimization are proposed as bilevel dynamic optimization problems. Solution procedures for some of the problems are given, while for others open challenges are discussed. Illustrative examples are given.

Branch-and-Lift Algorithm for Deterministic Global Optimization in Nonlinear Optimal Control

An algorithm is proposed for locating a feasible point satisfying the KKT conditions to a specified tolerance of feasible inequality-path-constrained dynamic programs (PCDP) within a finite number of iterations. The algorithm is based on iteratively approximating the PCDP by restricting the right-hand side of the path constraints and enforcing the path constraints at finitely many time points. The main contribution of this article is an adaptation of the semi-infinite program (SIP) algorithm proposed in Mitsos (2011) to PCDP. It is proved that the algorithm terminates finitely with a guaranteed feasible point which satisfies the first-order KKT conditions of the PCDP to a specified tolerance. The main assumptions are: (i) availability of a nonlinear program (NLP) local solver that generates a KKT point of the constructed approximation to PCDP at each iteration if this problem is indeed feasible; (ii) existence of a Slater point of the PCDP that also satisfies the first-order KKT conditions of the PCDP to a specified tolerance; (iii) all KKT multipliers are nonnegative and uniformly bounded with respect to all iterations. The performance of the algorithm is analyzed through two numerical case studies.

A model of chlorophyll fluorescence in microalgae integrating photoproduction, photoinhibition and photoregulation

This paper presents a branch-and-lift algorithm for solving optimal control problems with smooth nonlinear dynamics and potentially nonconvex objective and constraint functionals to guaranteed global optimality. This algorithm features a direct sequential method and builds upon a generic, spatial branch-and-bound algorithm. A new operation, called lifting, is introduced, which refines the control parameterization via a Gram–Schmidt orthogonalization process, while simultaneously eliminating control subregions that are either infeasible or that provably cannot contain any global optima. Conditions are given under which the image of the control parameterization error in the state space contracts exponentially as the parameterization order is increased, thereby making the lifting operation efficient. A computational technique based on ellipsoidal calculus is also developed that satisfies these conditions. The practical applicability of branch-and-lift is illustrated in a numerical example.