The Lifted Newton Method and its Use for Large Scale Dynamic Optimization and NMPC in Chemical Engineering

s of all of these talks are given below. Adaptive Shooting Methods for Dynamic Optimization – Concepts, Algorithms and Applications R. Hannemann, A. Hartwich, Wolfgang Marquardt, and Lynn Würth Process Systems Engineering Center for Computational Engineering Science RTWH Aachen University, D-52074 Aachen, Germany This talk will review some of the work on the development of adaptive shooting methods for the solution of optimal control problems carried out in our research group in recent years. The basic feature of the shooting method is an adaptive choice of the control vector parameterization in a shooting type of solution strategy. The control variables are adaptively discretized by multi-scale basis functions to resolve local detail with an appropriate number of parameters. Furthermore, the control switching structure of the solution is automatically detected during the refinement process of the adaptation of the control vector parameterization which gives insight into the solution features facilitating the interpretation of the result. The such detected structure is exploited to reprarameterize the single-stage into a multi-stage problem with a close to minimal number of control vector parameters. First and second order derivatives are computed by novel and highly efficient numerical algorithms exploiting forward as well as backward mode differentiation. Our numerical method conceptually links single-shooting and multiple type shooting on the one hand as well as direct and indirect methods on the other. The robustness and performance of the algorithms will be illustrated by different kinds of examples from chemical engineering of different complexity. The implementation has been proven to be very robust and highly efficient for large-scale optimal control problems with up to 15000 differential-algebraic equations with a number of control variables and many inequality path and endpoint constraints. Some extensions of the algorithm to cover real-time applications in nonlinear model-predictive and neighboring extremal control will be briefly discussed together with illustrating examples. The Lifted Newton Method and its Use for Large Scale Dynamic Optimization and NMPC in Chemical Engineering Jan Albersmeyer and Moritz Diehl We present a new full space exact Hessian SQP algorithm for large scale dynamic optimization that makes heavily use of two ingredients. We start by a process simulator (the DAE solver DAESOL-II (1)) that is able to generate adjoint sensitivites by the principle of internal numerical differentiation. Thus, a gradient computation is available at the cost of about five process simulations. Second, we work in the framework of Bock and Plitt’s direct multiple shooting method (2) by introducing intermediate but constrained ”node” variables into the optimization problem. It is a well known technique for reducing nonlinearity and increasing robustness of the optimization procedure, in particular for boundary value problems e.g. with end point quality constraints. By a combination of both ingredients, we are able to derive a full space exact hessian SQPmethod that iterates in the very large space of all node variables, yet needs to evaluate only thesame amount of derivatives as would be needed in a single shooting approach. This is similar toSchlöder’s Trick (3) which was however only applicable to least squares objective functions andnot yet combined with adjoint techniques for derivative generation. By a smart programming trick,the algorithm can easily be derived by ”lifting” a standard single shooting SQP method, thusavoiding the tedious programming work usually avoided with new variants of direct multipleshooting. The algorithm is advantageous in case of large process models with few degrees offreedom. We present also an extension to online optimization in nonlinear model predictivecontrol, and demonstrate its performance at examples from chemical engineering. REFERENCES[1] J. Albersmeyer and H.G. Bock: Efficient Derivative Generation in an adaptive BDFMethod.Proceedings of the HPSC 2006, Springer Verlag (submitted)[2] H.G. Bock and K.J. Plitt: A multiple shooting algorithm for direct solution of optimalcontrol problems, Proceedings 9th IFAC World Congress Budapest,1984,p. 243–247[3] J.P. Schlöder: Numerische Methoden zur Behandlung hochdimensionaler Aufgaben derParameteridentifizierung, Bonner Mathematische Schriften 187, 1988 Optimisation Problems in Advanced Operating Modes of ContinuousChromatography Malte KaspereitMax Planck Institute for Dynamics of Complex Technical Systems,Sandtorstr. 1, D-39106 Magdeburg, Germany Chemical processes most generally consist of a reaction step (synthesis) and a subsequentresolution of reactants and products (separation). The separation step can be very difficult (andexpensive), in particular in the production of valuable pharmaceuticals and fine chemicals. In thiscontext, chromatographic processes are an important option. Chromatography is performed usingcolumns containing a stationary phase. The figure below shows two general process options –discontinuous batch chromatography (single column, periodic injection of small feed amounts;left), and Simulated Moving Bed (SMB) chromatography (periodic switching of multiplecolumns, continuous feed; right). Due to its superior economic performance, the SMB process isreceiving more and more attention.However, the design of chromatographic processes cannot be performed on the basis of intuition,but requires optimisation schemes to determine suitable parameters. Mathematical processmodels are computationally expensive, since they typically involve a set of non-linear PDEscoupled by the involved thermodynamic equilibria. Occuring phenomena like shock fronts oftennecessitate a fine spatial discretisation (up to several thousand grid points per column).Furthermore, the process includes discrete events.Due to the computational efforts related this, optimisations of SMB processes could be performedonly recently. Different strategies have been proposed; for example, genetic algorithms [1],sequential quadratic programming (SQP) [2], a two-level approach [3], and the use of feedbackcontrol [4]. Besides standard SMB configurations, currently different advanced operating modesare subject to investigations. These allow to further increase the economic performance of SMBprocesses and to broaden the range of applications. Examples for advanced operating conceptsinclude:• combinations of SMB and complementary separation processes,• additional periodic variation of parameters (e.g., column configuration, flowrates, feed concentration),• introduction of gradients with respect to solvent strength or temperature,• implementation of chemical reactions within SMB processes.However, optimisation of such processes remains a challenge; only few results have beenpublished. Mainly, processes with periodic variations (see above) have been considered usinggenetic algorithms [5]. More recently, non-linear optimization with full discretisation [6] wasapplied successfully to several of the above process options. In the presentation, an overview willbe given on optimisation problems related to (continuous) chromatographic processes. After anintroductory review of fundamental principles and modelling approaches, recent developmentswith respect to advanced operating modes for SMB processes and optimisation problems will beexplained using concrete examples. [1] Z. Zhang, K. Hidajat, A. Ray, M. Morbidelli. AIChE J 48 (2002) 2800.[2] M. Kaspereit, K. Gedicke, V. Zahn, A. W. Mahoney, A. Seidel-Morgenstern.J Chromatogr A 1092 (2005) 43.[3] M. Minceva, A. E. Rodrigues. Comp Chem Engng 29 (2005) 2215.[4] H. Schramm, S. Grüner, A. Kienle. J Chromatogr A 1006 (2003) 3.[5] Z. Zhang, M. Mazzotti, M. Morbidelli. Korean J Chem Eng 21 (2004) 454.[6] Y. Kawajiri, L. T. Biegler. J Chromatogr A 1133 (2006) 226. On the Optimality of Superstructures for Simulated Moving Beds Sebastian Sager, Yoshiaki Kawajiri and Lorenz T. Biegler The past decade has seen a variety of operating modifications for Simulated Moving Bed (SMB)processes, including Three-Zone, VARICOL, and PowerFeed. In recent studies, we have shownthat these can all be embedded within a superstructure optimization problem with time-variantflow rates. Moreover, the resulting dynamic optimization problem has yielded a number ofinteresting and useful insights on novel SMB operations. In many cases these solutions havea ’bang-bang’ character, i.e., the feed, desorbent, extract and raffinate streams appear in only onelocation. Remarkably, this occurs even though the optimization is performed without anintroduction of binary decision variables. In this study we analyze and present conditions where‘bang-bang’ solutions are optimal. We also demonstrate cases where these conditions do not holdand where ’bang-bang’ solutions are suboptimal. To show that these properties are independent ofthe column model and solution strategy, we demonstrate both ‘bang-bang’ and ‘non-bang-bang’cases for two different column models and two optimization environments.