151 results found
Villanueva ME, Houska B, Chachuat B, 2014, Unified framework for the propagation of continuous-time enclosures for parametric nonlinear ODEs, Journal of Global Optimization, Vol: 62, Pages: 575-613, ISSN: 1573-2916
This paper presents a framework for constructing and analyzing enclosures ofthe reachable set of nonlinear ordinary differential equations using continuous-time setpropagation methods. The focus is on convex enclosures that can be characterized in terms of their support functions. A generalized differential inequality is introduced, whose solutionsdescribe 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 modelswith 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.
Houska B, Chachuat B, 2014, Branch-and-Lift Algorithm for Deterministic Global Optimization in Nonlinear Optimal Control, Journal of Optimization Theory and Applications, Vol: 162, Pages: 208-248, ISSN: 0022-3239
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. © 2013 Springer Science+Business Media New York.
Gros S, Chachuat B, 2014, Optimization-based load reduction during rapid shutdown of multi-megawatt wind turbine generators, Wind Energy, Vol: 17, Pages: 1055-1075, ISSN: 1095-4244
This paper describes an optimization-based approach to reducing extreme structural loads during rapid or emergency shutdown of multi-megawatt wind turbine generators. The load reduction problem is cast into an optimal control formulation, and a simple, low-order model is developed in order for this optimization problem to be tractable in reasonable time using state-of-the-art numerical methods. To handle the variations in wind speed and turbulence inherent to wind turbine operation as well as the presence of model mismatch, a real-time optimization strategy based on fast sensitivity updates is also considered, whose online computational burden is limited to the repeated solution of quadratic programs that are designed offline. The low-order model and both the open-loop and closed-loop optimal control strategies are validated against a high-fidelity model in the simulation environment Bladed™ for an industrial 3 MW wind turbine. Under favorable shutdown scenarios, i.e. when the wind turbine is operating properly and the actuators and sensors are not faulty, large reductions of the first compressive peak and subsequent compressive/tensile peaks of the tower load pattern are obtained at various above-rated wind speeds compared with normal pitch control shutdown. Extension to more challenging shutdown scenarios are also discussed.
Khor CS, Chachuat B, Shah N, 2014, Optimization of water network synthesis for single-site and continuous processes: milestones, challenges, and future directions, Industrial & Engineering Chemistry Research, Vol: 53, Pages: 10257-10275, ISSN: 1520-5045
Increasing water demand in the process and its allied industries coupled with global water stress and scarcity has underlined the importance of water as a crucial resource and elevated a need for widespread adoption of water reuse and recycle. This paper provides a state-of-the-art review of the area of water network synthesis focusing on single-site and continuous process problems since its inception in the 1980s. The survey centers around model-based optimization or mathematical programming methods for water network synthesis and covers key findings from the water pinch analysis technique, which are often essential in enhancing model formulations. Major modeling and computational challenges are discussed that explore the issues of nonconvexity, nonlinearity, and uncertainty inherent in water network synthesis problems. The review concludes by providing a perspective of future research directions to be tackled to address the challenges highlighted.
Khor CS, Chachuat B, Shah N, 2014, Fixed-flowrate total water network synthesis under uncertainty with risk management, Journal of Cleaner Production, Vol: 77, Pages: 79-93, ISSN: 0959-6526
This work addresses the problem of integrated water network synthesis under uncertainty with risk management. We consider a superstructure consisting of water sources, regenerators, and sinks that leads to a mixed-integer quadratically-constrained quadratic program (MIQCQP) for a fixed-flowrate total water network synthesis problem. Uncertainty in the problem is accounted for via a recourse-based two-stage stochastic programming formulation with discrete scenarios that gives rise to a multiscenario MIQCQP comprising network design in the first stage and its operation in the second stage acting as recourse. In addition, we extend the model to address risk management using the Conditional Value-at-Risk (CVaR) metric. Because a large number of scenarios is often required to capture the underlying uncertainty of the problem, causing the model to suffer from the curse of dimensionality, we propose a stepwise solution strategy to reduce the computational load. We illustrate this methodology on a case study inspired from the water network of a petroleum refinery in Malaysia. The presence of nonconvex bilinear terms necessitates the use of global optimization techniques for which we employ a new global MIQCQP solver, GAMS/GloMIQO and verify the solutions with BARON. Our computational results show that total water network synthesis under uncertainty with risk management problems can be solved to global optimality in reasonable time.
Villanueva ME, Houska B, Chachuat B, 2014, On the Stability of Set-Valued Integration for Parametric Nonlinear ODEs, 24TH EUROPEAN SYMPOSIUM ON COMPUTER AIDED PROCESS ENGINEERING, PTS A AND B, Vol: 33, Pages: 595-600, ISSN: 1570-7946
Houska B, Villanueva ME, Chachuat B, 2013, A validated Integration algorithm for nonlinear ODEs using Taylor models and ellipsoidal calculus, 52nd IEEE Annual Conference on Decision and Control (CDC), Publisher: IEEE, Pages: 484-489, ISSN: 0191-2216
his paper presents a novel algorithm for bounding the reachable set of parametric nonlinear differential equations. This algorithm is based on a first-discretize-then-bound approach to enclose the reachable set via propagation of a Taylor model with ellipsoidal remainder, and it accounts for truncation errors that are inherent to the discretization. In contrast to existing algorithms that proceed in two phases-an a priori enclosure phase, followed by a tightening phase-the proposed algorithm first predicts a continuous-time enclosure and then seeks a maximal step-size for which validity of the predicted enclosure can be established. It is shown that this reversed approach leads to a natural step-size control mechanism, which no longer relies on the availability of an a priori enclosure. Also described in the paper is an open-source implementation of the algorithm in ACADO Toolkit. A simple numerical case study is presented to illustrate the performance and stability of the algorithm.
Paulen R, Villanueva M, Fikar M, et al., 2013, Guaranteed parameter estimation in nonlinear dynamic systems using improved bounding techniques, 2013 European Control Conference (ECC), Publisher: IEEE, Pages: 4514-4519
This paper is concerned with guaranteed parameter estimation in nonlinear dynamic systems in a context of bounded measurement error. The problem consists of finding - or approximating as closely as possible - the set of all possible parameter values such that the predicted outputs match the corresponding measurements within prescribed error bounds. An exhaustive search procedure is applied, whereby the parameter set is successively partitioned into smaller boxes and exclusion tests are performed to eliminate some of these boxes, until a prespecified threshold on the approximation level is met. Exclusion tests rely on the ability to bound the solution set of the dynamic system for a given parameter subset and the tightness of these bounds is therefore paramount. Equally important is the time required to compute the bounds, thereby defining a trade-off. It is the objective of this paper to investigate this trade-off by comparing various bounding techniques based on interval arithmetic, Taylor model arithmetic and ellipsoidal calculus. When applied to a simple case study, ellipsoidal and Taylor model approaches are found to reduce the number of iterations significantly compared to interval analysis, yet the overall computational time is only reduced for tight approximation levels due to the computational overhead. © 2013 EUCA.
Hartmann P, Nikolaou A, Chachuat B, et al., 2013, A dynamic model coupling photoacclimation and photoinhibition in microalgae, ECC 2013, Publisher: IEEE, Pages: 4178-4183
Microalgae are often considered a promising alternative for production of renewable energy, particularly as a potential producer of biodiesel. In order to improve large-scale, industrial culturing systems, the development of mathematical models that are capable of predicting photosynthetic productivity under dynamic conditions is crucial. Especially important are the processes of growth inhibition due to excess light, known as photoinhibition, and of adjustment of the light harvesting capacity to photon flux, known as photoacclimation. In this paper, we develop a dynamic model that accounts for the processes of photoinhibition, photoacclimation and growth in microalgae, thereby spanning multiple time scales. The properties of the model are investigated under quasi steady-state conditions and the model is validated against several experimental data sets from the literature. We also discuss how the model can provide new insights into the mechanisms underlying photoacclimation. © 2013 EUCA.
Paulen R, Villanueva M, Chachuat B, 2013, Optimization-based domain reduction in guaranteed parameter estimation of nonlinear dynamic systems, 9th IFAC Symposium on Nonlinear Control Systems, 2013, Publisher: International Federation of Automatic Control, Pages: 564-569, ISSN: 1474-6670
This paper is concerned with guaranteed parameter estimation in nonlinear dynamic systems in a context of bounded measurement error. The problem consists of finding-or approximating as closely as possible-the set of all possible parameter values such that the predicted outputs match the corresponding measurements within prescribed error bounds. An exhaustive search procedure is applied, whereby the parameter set is successively partitioned into smaller boxes and exclusion tests are performed to eliminate some of these boxes, until a prespecified threshold on the approximation level is met. In order to enhance the convergence of this procedure, we investigate the use of optimization-based domain reduction techniques for tightening the parameter boxes before partitioning. We construct such bound-reduction problems as linear programs from the polyhedral relaxation of Taylor models of the predicted outputs. When applied to a simple case study, the proposed approach is found to reduce the computational burden significantly, both in terms of CPU time and number of iterations. © IFAC.
Podmajersky M, Fikar M, Chachuat B, 2013, Measurement-based optimization of batch and repetitive processes using an integrated two-layer architecture, Journal of Process Control, Vol: 23, Pages: 943-955, ISSN: 1873-2771
This paper is concerned with optimal control of batch and repetitive processes in the presence of uncertainty. An integrated two-layer optimization strategy is proposed, whereby within-run corrections are performed using a neighboring-extremal update strategy and run-to-run corrections are based on a constraint-adaptation scheme. The latter is appealing since a feasible operating strategy is guaranteed upon convergence, and its combination with neighboring-extremal updates improves the reactivity and convergence speed. Moreover, these two layers are consistent in that they share the same objective function. The proposed optimization scheme is declined into two versions, namely an indirect version based on the Pontryagin maximum principle and a direct version that applies a control parameterization and nonlinear programming techniques. Although less rigorous, the latter approach can deal with singular extremals and path constraints as well as handle active-set changes more conveniently. Two case studies are considered. The indirect approach is demonstrated for a level-control problem in an experimental two-tank system, whereas the direct approach is illustrated in numerical simulation on a fed-batch reactor for acetoacetylation of pyrrole. The results confirm that faster adaptation is possible with the proposed integrated two-layer scheme compared to either constraint adaptation or neighboring-extremal update alone.
Villanueva M, Paulen R, Houska B, et al., 2013, Enclosing the Reachable Set of Parametric ODEs using Taylor Models and Ellipsoidal Calculus, 23 EUROPEAN SYMPOSIUM ON COMPUTER AIDED PROCESS ENGINEERING, Vol: 32, Pages: 979-984, ISSN: 1570-7946
One of the main bottlenecks of state of the art algorithm for global dynamic optimizationis the computation of enclosures of parametric differential equations (ODEs). Afterreviewing existing techniques based on Taylor model propagation and ellipsoidal calculusfor nonlinear dynamic processes, we introduce a novel algorithm for computing suchstate enclosures. Here, the bounding strategy employs a Taylor series with an ellipsoidalremainder bound. We analyze the convergence properties of the new ODE enclosure forsmall parameter intervals and provide conditions under which higher order convergencecan be proven. Moreover, we discuss implementation details and practical advantages byapplying the method to a numerical test example.
Nikolaou A, Hartmann P, Bernard O, et al., 2013, A dynamic model coupling photoacclimation and photoinhibition in microalgae, Pages: 351-352
Rajyaguru J, Chachuat B, 2013, Taylor models in deterministic global optimization for large-scale systems with few degrees of freedom, Vol: 32, Pages: 973-978, ISSN: 1570-7946
Many process systems applications comprise large sets of nonlinear model equations, whose participating variables can be split naturally into independent and dependent variable subsets. This structure can be exploited for deterministic global optimization based on a sequential approach, which performs the optimization in the reduced space of independent variables by considering the model as implicit equations. This paper presents a new method for constructing Taylor model estimators of the implicit equation solutions in order to generate tighter lower bounds on the reduced-space optimization problem. The convergence properties of these estimators are analyzed through numerical examples, and the global optimization approach is demonstrated on a numerical case study featuring a discretized PDE system. © 2013 Elsevier B.V.
Bompadre A, Mitsos A, Chachuat B, 2012, Convergence analysis of Taylor models and McCormick-Taylor models, Journal of Global Optimization, Vol: 57, Pages: 75-114, ISSN: 1573-2916
This article presents an analysis of the convergence order of Taylor models and McCormick-Taylor models, namely Taylor models with McCormick relaxations as the remainder bounder, for factorable functions. Building upon the analysis of McCormick relaxations by Bompadre and Mitsos (J Glob Optim 52(1):1–28, 2012), convergence bounds are established for the addition, multiplication and composition operations. It is proved that the convergence orders of both qth-order Taylor models and qth-order McCormick-Taylor models are at least q + 1, under relatively mild assumptions. Moreover, it is verified through simple numerical examples that these bounds are sharp. A consequence of this analysis is that, unlike McCormick relaxations over natural interval extensions, McCormick-Taylor models do not result in increased order of convergence over Taylor models in general. As demonstrated by the numerical case studies however, McCormick-Taylor models can provide tighter bounds or even result in a higher convergence rate.
Radivojevic A, Chachuat B, Bonvin D, et al., 2012, Exploration of trade-offs between steady-state and dynamic properties in signaling cycles, PHYSICAL BIOLOGY, Vol: 9, ISSN: 1478-3967
Deshpande SA, Bonvin D, Chachuat B, 2012, DIRECTIONAL INPUT ADAPTATION IN PARAMETRIC OPTIMAL CONTROL PROBLEMS, SIAM Journal on Control and Optimization, Vol: 50, Pages: 1995-2024, ISSN: 1095-7138
This paper deals with input adaptation in dynamic processes in order to guarantee feasible and optimal operation despite the presence of uncertainty. For those optimal control problems having terminal and mixed control-state path constraints, two orthogonal sets of adaptation directions can be distinguished in the input space: the sensitivity-seeking directions, along which a small variation from an optimal nominal solution will not affect the respective active constraints, and the complementary constraint-seeking directions, along which a variation will affect the respective constraints. It follows that selective input adaptation strategies can be defined, namely, adaptation in the sensitivity- and constraint-seeking directions. This paper proves the important result that, for small parametric perturbations, the cost variation resulting from adaptation in the sensitivity-seeking directions (over no input adaptation) is typically smaller than the cost variation due to adaptation in the constraint-seeking directions. It is also established that no selective input adaptation along a sensitivity-seeking direction can reduce the dominant, first-order term in the optimality gap; adaptation along a constraint-seeking direction is necessary to cancel it out. These results are illustrated with two numerical case studies.
Khor CS, Chachuat B, Shah N, 2012, A superstructure optimization approach for water network synthesis with membrane separation-based regenerators, European Symposium of Computer Aided Process Engineering - 21, Publisher: Elsevier, Pages: 48-63, ISSN: 0098-1354
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 $218,000/year reduction in freshwater use.
Mairet F, Bernard O, Cameron E, et al., 2012, Three-reaction model for the anaerobic digestion of microalgae, Biotechnology and Bioengineering, Vol: 109, Pages: 415-425, ISSN: 1097-0290
Coupling an anaerobic digester to a microalgal culture has received increasing attention as an alternative process for combined bioenergy production and depollution. In this article, a dynamic model for anaerobic digestion of microalgae is developed with the aim of improving the management of such a coupled system. This model describes the dynamics of inorganic nitrogen and volatile fatty acids since both can lead to inhibition and therefore process instability. Three reactions are considered: Two hydrolysis–acidogenesis steps in parallel for sugars/lipids and for proteins, followed by a methanogenesis step. The proposed model accurately reproduces experimental data for anaerobic digestion of the freshwater microalgae Chlorella vulgaris with an organic loading rate of 1 gCOD L−1 d−1. In particular, the three-reaction pathway allows to adequately represent the observed decoupling between biogas production and nitrogen release. The reduced complexity of this model makes it suitable for developing advanced, model-based control and monitoring strategies.
Scott JK, Chachuat B, Barton PI, 2012, Nonlinear convex and concave relaxations for the solutions of parametric ODEs, Optimal Control Applications & Methods, Vol: 34, Pages: 145-163, ISSN: 1099-1514
Convex and concave relaxations for the parametric solutions of ordinary differential equations (ODEs) are central to deterministic global optimization methods for nonconvex dynamic optimization and open-loop optimal control problems with control parametrization. Given a general system of ODEs with parameter dependence in the initial conditions and right-hand sides, this work derives sufficient conditions under which an auxiliary system of ODEs describes convex and concave relaxations of the parametric solutions, pointwise in the independent variable. Convergence results for these relaxations are also established. A fully automatable procedure for constructing an appropriate auxiliary system has been developed previously by the authors. Thus, the developments here lead to an efficient, automatic method for computing convex and concave relaxations for the parametric solutions of a very general class of nonlinear ODEs. The proposed method is presented in detail for a simple example problem.
Khor CS, Chachuat B, Shah N, 2012, Optimal water network synthesis with membrane separation-based regenerators, 22nd European Symposium on Computer Aided Process Engineering (ESCAPE), Publisher: ELSEVIER SCIENCE BV, Pages: 36-40, ISSN: 1570-7946
Khor CS, Chachuat B, Shah N, 2012, Optimal water network synthesis with detailed membrane-based regenerator models, 11th International Symposium on Process Systems Engineering (PSE), Publisher: ELSEVIER SCIENCE BV, Pages: 1457-1461, ISSN: 1570-7946
Chachuat B, Villanueva M, 2012, Bounding the Solutions of Parametric ODEs: When Taylor Models Meet Differential Inequalities, 22nd European Symposium on Computer Aided Process Engineering (ESCAPE), Publisher: ELSEVIER SCIENCE BV, Pages: 1307-1311, ISSN: 1570-7946
Gros S, Chachuat B, 2012, Methodology for Emergency Shut-Down of Multi-Megawatt Wind Turbine Generators, 11th International Symposium on Process Systems Engineering (PSE), Publisher: ELSEVIER SCIENCE BV, Pages: 1251-1255, ISSN: 1570-7946
Seong Khor C, Chachuat B, Shah N, 2012, Optimal water network synthesis with membrane separation-based regenerators, Vol: 30, Pages: 36-40, ISSN: 1570-7946
This article is concerned with the water network synthesis problem. We propose a superstructure 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 direct or regeneration-reuse/recycle. The superstructure accounts for membrane separation-based regenerators such as ultrafiltration and reverse osmosis. Linear models are developed for the membrane regenerators. A MINLP is then formulated based on the superstructure to determine the optimal interconnections in terms of total flowrates and contaminant concentrations. Computational experiments are performed using GAMS/BARON on an industrial case study of a petroleum refinery water system. We include model-tightening linear logical constraints to enhance solution convergence, resulting in a globally optimal water network that promotes 27% savings in freshwater use. © 2012 Elsevier B.V.
Marchetti A, Gopalakrishnan A, Chachuat B, et al., 2011, Robust Real-Time Optimization of a Solid Oxide Fuel Cell Stack, JOURNAL OF FUEL CELL SCIENCE AND TECHNOLOGY, Vol: 8, ISSN: 1550-624X
Sahlodin AM, Chachuat B, 2011, Discretize-then-relax approach for convex/concave relaxations of the solutions of parametric ODEs, APPLIED NUMERICAL MATHEMATICS, Vol: 61, Pages: 803-820, ISSN: 0168-9274
Sahlodin AM, Chachuat B, 2011, Convex/concave relaxations of parametric ODEs using Taylor models, COMPUTERS & CHEMICAL ENGINEERING, Vol: 35, Pages: 844-857, ISSN: 0098-1354
Sahlodin AM, Chachuat B, 2011, Tight Convex and Concave Relaxations via Taylor Models for Global Dynamic Optimization, 21st European Symposium on Computer Aided Process Engineering (ESCAPE-21), Publisher: ELSEVIER SCIENCE BV, Pages: 537-541, ISSN: 1570-7946
Khor CS, Giarola S, Chachuat B, et al., 2011, An optimization-based framework for process planning under uncertainty with risk management, Pages: 449-450
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