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Publication# Dynamic Optimization of Reaction Systems via Exact Parsimonious Input Parameterization

Abstract

This paper discusses the use of parsimonious input parameterization for the dynamic optimization of reaction systems. This parameterization is able to represent the optimal inputs with only a few parameters. In the context of batch, semibatch, and continuous reactors, the method takes advantage of the concept of extents to allow the analytical computation of adjoint-free optimal control laws. It is shown that this computation can be performed in a systematic way for all types of arcs in the solution, thereby resulting in a finite set of plausible arc sequences. For each arc sequence, the optimal values of the input parameters are computed via numerical optimization. The results are illustrated via simulated examples of reaction systems.

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Mathematical optimization

Mathematical optimization (alternatively spelled optimisation) or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternative

Reaction–diffusion system

Reaction–diffusion systems are mathematical models which correspond to several physical phenomena. The most common is the change in space and time of the concentration of one or more chemical substan

Parameter

A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element

The concept of reaction variants and invariants for lumped reaction systems has been known for several decades. Its applications encompass model identification, data reconciliation, state estimation and control using kinetic models. In this thesis, the concept of variants and invariants is extended to distributed reaction systems and used to develop new applications to estimation, control and optimization.
The thesis starts by reviewing the material and heat balances and the concept of variants and invariants for several lumped reaction systems. Different definitions of variants and invariants, in particular the vessel extents, are presented for the case of homogeneous reaction systems, and transformations to variants and invariants are obtained. The extension to systems with heat balance and mass transfer is also reviewed.
The concept of extents is generalized to distributed reaction systems, which include many processes involving reactions and described by partial differential equations. The concept of extents and the transformation to extents are detailed for various configurations of tubular reactors and reactive separation columns, as well as for a more generic framework that is independent of the configuration.
New developments of the extent-based incremental approach for model identification are presented. The approach, which compares experimental and modeled extents, results in maximum-likelihood parameter estimation if the experimental extents are uncorrelated and the modeled extents are unbiased. Furthermore, the identification problem can be reformulated as a convex optimization problem that is solved efficiently to global optimality.
The estimation of unknown rates without the knowledge or the identification of the rate models is described. This method exploits the fact that the variants computed from the available measurements allow isolating the different rates. Upon using a Savitzky-Golay filter for differentiation of variants, one can show that the resulting rate estimator is optimal and obtain the error and variance of the rate estimates.
The use of variants and invariants for reactor control is also considered. Firstly, offset-free control via feedback linearization is implemented using kinetic models. Then, it is shown how rate estimation can be used for control via feedback linearization without kinetic models. By designing an outer-loop feedback controller, the expected values of the controlled variables converge exponentially to their setpoints.
This thesis presents an approach to speed up steady-state optimization, which takes advantage of rate estimation without rate models to speed up the estimation of steady state for imperfectly known dynamic systems with fast and slow states. Since one can use feedback control to speed up convergence of the fast part, rate estimation allows estimating the steady state of the slow part during transient operation.
The application to dynamic optimization is also shown. Adjoint-free optimal control laws are computed for all the types of arcs in the solution. In the case of reactors, the concept of extents allows the symbolic computation of optimal control laws in a systematic way. A parsimonious input parameterization is presented, which approximates the optimal inputs well with few parameters. For each arc sequence, the optimal parameter values are computed via numerical optimization.
The theoretical results are illustrated by simulated examples of reaction systems.

The energy transition in Switzerland specifies reduction of the country’s primary energy consumption and greenhouse gas emissions. In the building sector, one way to achieve this is through adoption of efficient and sustainable technologies. With this aim, the building energy systems are gaining complexity. Modelling tools are becoming increasingly important in the design and evaluation of such systems. Two modelling methods are commonly used, simulation and optimization, which substantially differ in their approach and purpose. Simulation is a descriptive tool used to virtually represent systems’ behaviour under given conditions and operation strategies. Optimization approaches explore the possible scenarios under given system limits; they determine the best solution by optimizing an objective function described in mathematical form. In this project, the interactions and complementary use of these methods are investigated through evaluation of the newly installed energy systems of multi-familiy houses built in Zurich. The systems include renewable sources from solar and geothermal energy. The main components are a heat pump coupled with a borehole heat exchanger (BHE) and solar photovoltaic panels (PV) or hybrid photovoltaic panels (PV/T) for electricity production. Two system variants including different rates of borehole regeneration resulting from free cooling of the houses or injection of heat produced by PV/T are evaluated. The systems are divided into interlinked blocks representing the main components of the system: the borehole heat exchanger and surrounding ground, the heat pump, the storage tanks, the piping systems and the photovoltaic installations. The blocks are simulated with an hourly time scale. Energy consumption, self-consumption of on site produced electricity as well as performance degradation due to the long term operation are some of the results obtained from simulation. Monitoring data are used for calibration and validation of the simulation model. Separate optimization models of the evaluated energy systems are then developed. They are improved based on system characteristics obtained from the simulation model. In this way, optimal operation strategies which take into account specific operational limits are identified. Different levels of precision of parameters integration from the simulation results (constant over the year and hourly defined) are implemented and the influences on the optimization results are investigated. The results of the optimization are subsequently implemented in the simulation model. The results of the simulation found that the boreholes are sustainably exploited due the conservative design of the system according the simulated operation conditions. The influence of the regeneration is noticeable, however the simulation of the long term operation of a hypothetical variant without regeneration induces only a limited efficiency decrease and can also be considered as sustainable. The optimization results showed that the self consumption of electricity produced on site can be significantly improved by adapting the heat pump production profile with the electricity production; this was achieved through judicious management of the storage tanks. The analysis of the different level of parameter integration in the optimization model revealed that the optimization results are sensitive to the level of precision of the parameters. The iterative process allowed to combine the strengths of both modelling methods: the simulation was used to precisely describe the systems behaviour. Operational profiles from optimization were subsequently used in the simulation. The outcome of this process helped to better understand the system, identify optimal operating strategies while considering system limitations, as well as presenting

2017Julien Léo Billeter, Dominique Bonvin, Diogo Filipe Mateus Rodrigues

The identification of reaction kinetics represents the main challenge in building models for reaction systems. The identification task can be performed via either simultaneous model identification (SMI) or incremental model identification (IMI), the latter using either the differential (rate-based) or the integral (extent-based) method of parameter estimation. This contribution presents an extension of extent-based IMI that guarantees convergence to globally optimal parameters. In SMI, a rate law must be postulated for each reaction, and the model concentrations are obtained by integration of the balance equations. The procedure must be repeated for all combinations of rate candidates. This approach is computationally costly when there are several candidates for each reaction, and convergence problems may arise due to the large number of parameters. In IMI, the identification task is decomposed into several sub-problems, one for each reaction [1]. Since IMI deals with one reaction at a time, only the rate candidates for that reaction need to be compared. In addition, convergence is facilitated by the fact that only the parameters of a single reaction rate are estimated. In rate-based IMI, the parameters are estimated by fitting the simulated rates to the experimental rates obtained by differentiation of measured concentrations. In extent-based IMI, the simulated rates are integrated to yield extents, and the parameters are estimated by fitting the simulated extents to experimental extents obtained by transformation of measured concentrations [2]. The simulated rates are functions of concentrations. Hence, since each reaction is simulated individually, the simulated rates must be computed from measured concentrations. Most parameter estimation methods converge to local optimality, which may result in an incorrect model. It turns out that extent-based IMI is particularly suited to global optimization since each estimation sub-problem (i) involves only a small set of parameters, and (ii) can be rearranged as an algebraic problem, where the objective function is polynomial in the parameters with coefficients computed only once prior to optimization using a Taylor expansion. These features facilitate the task of finding a global optimum for each reaction. Instead of the classical branch-and-bound approach, this technique relies on reformulating the estimation problem as a convex optimization problem, taking advantage of the equivalence of nonnegative polynomials and conical combination of sum-of-squares polynomials on a compact set to solve the problem as a semidefinite program [3]. A simulated example of an identification problem with several local optima shows that extent-based IMI can be used to converge quickly to globally optimal parameters. References: [1] Bhatt et al., Chem. Eng. Sci., 2012, 83, p. 24 [2] Rodrigues et al., Comput. Chem. Eng., 2015, 73, p. 23 [3] Lasserre, SIAM J. Optim., 2001, 11(3), p. 796

2017