(558h) Safe Real-Time Optimization Using Multi-Fidelity Gaussian Processes

The advantages of real-time optimization (RTO) in industrial processes are well known; however, currently, most implementations rely on heuristics and trial and error to reach optimality. The fact that complex phenomena cannot be modelled precisely is the biggest challenge in real-time optimization; in this context, we refer to the 'plant' as the 'true' system and therefore refer to 'plant-model mismatch' as the error of our model with respect to the real system (the plant). Real-time optimization relies on the development of models utilized to conduct optimization [1]; these (imperfect) models are usually updated in real-time when measurements are available. The optimization is repeated with the updated models, and an iterative procedure between model refinement and optimization ensues.

This two-step procedure in RTO schemes is called the model-adaptation strategy.

Despite being the most widely used industry method, this model-adaptation strategy does not converge to the systems' optimal operating conditions. Modifier Adaptation [2] has been proposed to accommodate this issue inspired by the technique of Integrated System Optimization and Parameter Estimation [3]. Still, it differs in the fact that no parameter estimation is required. These RTO schemes can reach plant optimality upon convergence, despite the presence of a structural plant-model mismatch. However, this comes at the cost of having to estimate gradient terms from process measurements.

The use of derivative-free approaches can lift the issues with the estimation of gradients.

The natural choice of surrogates is Gaussian processes (GPs) within Bayesian optimization ideas [4]. These methods use the GPs as a discrepancy model that approximate the mismatch between the model and the plant. The prior model combined with the GP is then explicitly used in the optimization formulation; however, most industrially relevant models are not analytical. They are often complex black-box simulators and legacy code, e.g. systems of ordinary or partial differential equations or a set of `if/else if' rules.

This work's novelty lies on integrating derivative-free optimization schemes and multi-fidelity Gaussian processes [5] within a Bayesian optimization framework. The proposed method uses two Gaussian processes for the stochastic system; one emulates the (known) process model, and another, the true system through measurements. In this way, low fidelity samples can be obtained via a model, while high fidelity samples are obtained through the system's measurements. This framework captures the system's behaviour in a non-parametric fashion while driving exploration through acquisition functions. The benefit of using a Gaussian process to represent the system is the ability to perform uncertainty quantification in real-time and allow for chance constraints to be satisfied with high confidence. This results in a practical approach illustrated in numerical case studies, including a semi-batch photobioreactor Optimization problem. The results show that the use of the multi-fidelity GP and chance constraints significantly help both the convergence to the optimum and the system's feasibility.

References:

1. Chachuat B., Srinivasan B., Bonvin D. (2009) Adaptation strategies for real-time optimization. Comput. Chem. Eng. 33:1557-1567
2. Marchetti A.G., François G., Faulwasser T., Bonvin D. (2016) Modifier adaptation for real-time optimization—Methods and applications. Processes 4:55.
3. Roberts P.D., Williams T.W. (1981) On an algorithm for combined system optimisation and parameter estimation. Automatica 17:199-209.
4. del Rio Chanona A., Petsagkourakis P., Bradford E., Alves Graciano J. E.,Chachuat B. (2021) Real-time optimization meets Bayesian optimization and derivative-free optimization: A tale of modifier adaptation. Computers & Chemical Engineering. Volume 147.107249,
5. Perdikaris P., Raissi m., Damianou A., Lawrence N. D., and Karniadakis G. E. (2017) Nonlinear information fusion algorithms for data-efficient multi-fidelity modelling, Proceedings of the Royal Society A: Mathe-matical, Physical and Engineering Sciences, vol. 473, no. 2198