Conference Agenda

Implementing Model Predictive Control (MPC) for large-scale nonlinear systems is often computationally challenging due to the intensive online optimization required. To address this, various reduced-order linearization techniques have been developed [1]. The Koopman operator linearizes a nonlinear system by mapping it into an infinite-dimensional space of observables [2], enabling the application of linear control strategies [3]. While Artificial Neural Networks (ANNs) can approximate the Koopman operator in a data-driven manner [4, 5], training these networks becomes computationally intensive for high-dimensional systems as the lifting into a higher-dimensional observable space significantly increases data size and complexity. In this work, we propose a technique, combining Proper Orthogonal Decomposition (POD) with an efficient modified ANN structure to reduce the training time of ANN for large order systems. By first applying POD, we obtain a low order projection of the system. Subsequently, we train the modified ANNs to approximate the Koopman operator, significantly decreasing training time without sacrificing accuracy. The methodology is demonstrated through an illustrative large-scale chemical engineering case study.

Keywords: model predictive control, data-driven methodology, artificial neural networks

References

1. Theodoropoulos, C. 2010. ‘Optimisation and Linear Control of Large Scale Nonlinear Systems: A Review and a Suite of Model Reduction-Based Techniques’. Coping with Complexity: Model Reduction and Data Analysis, 37–61.
2. Koopman, Bernard O. 1931. ‘Hamiltonian Systems and Transformation in Hilbert Space’. Proceedings of the National Academy of Sciences 17 (5): 315–18.
3. Korda, M, and I Mezić. 2018. ‘Linear Predictors for Nonlinear Dynamical Systems: Koopman Operator Meets Model Predictive Control’. Automatica 93:149–60.
4. Wang, M., X. Lou, W. Wu, and B. Cui. 2022. ‘Koopman-Based MPC with Learned Dynamics: Hierarchical Neural Network Approach’. IEEE Transactions on Neural Networks and Learning Systems.
5. Verma G, Heath W, Theodoropoulos C. Robust stability analysis of Koopman based MPC system. In Computer Aided Chemical Engineering 2024 Jan 1 (Vol. 53, pp. 1927-1932). Elsevier.

Recycling plastic waste is considered one of the most effective strategies to promote the circular economy, but needs to be further improved. Consumer engagement is one of the most influencing factors and as such is one of the objectives of government incentive measures to promote effective recycling of plastic waste. In this study, the Stackelberg Game Approach is used to investigate how government incentive mechanisms (leaders) can influence the recycling behaviour of consumers (followers). Various incentives are evaluated, including economic, policy and educational measures to ensure permanent consumer participation. Three scenarios are proposed based on different incentive combinations aligned with circular strategies, namely narrow (use less), slow (use longer) and close (reuse), and are evaluated with the aim of measuring the optimal benefit for both participants, i.e. minimising government costs, such as recycling subsidies, while maximising recycling rates and maximising consumer gains. The comparison of the scenarios highlights possible ways of combining different incentives to maximise consumer engagement. Future work could integrate Internet-of-Things technology to facilitate dynamic strategy optimisation.

Certifying feasibility in decision-making is critical in the process industries and can be framed as a constraint satisfaction problem (CSP). Feasible decisions are often identified through mathematical programming. In this case, a single feasible decision is returned which minimizes an objective posed. In this research, we consider algorithmic approaches to identify a set of potential feasible decisions. This has been well explored through domain reduction methods in mathematical programming [1], flexibility as proposed in [2], and design space identification as explored in widely within the pharmaceutical industry [3].

Specific focus is directed towards a CSP where the task is to find parameter values from a continuous domain that satisfy constraints defined on a directed acyclic graph of constituent functions. This problem setting may be used to describe the operation of a network of process unit operations without recycle streams. In this case, the parameters one would like to identify may represent set-points for the unit operations in the network. We additionally impose the condition that the identification of the feasible space must enable decentralised operation of each of the constituent units. In practice, this means that selection of any given unit set-point must be feasible for selection of any other unit’s set-point.

We assume that the constituent unit operations are described by general input-output functions that may not be available in closed form and could result from expensive simulations. The solution approach assumes one is only able to evaluate the input-output behaviour of the functions and the associated constraints. This lends itself to the use of sampling methods to gain an inner approximation of the feasible region. However, sampling faces challenges due to the curse of dimensionality. To address this, a decomposition approach is introduced, leveraging the network structure to break the problem into unit-wise subproblems of reduced dimension. A data-driven tuning is introduced to ensure maximum volume of the feasible region. The methodology is demonstrated through a two-unit batch reactor network. Future research will extend this approach to account for uncertain parameters in the constituent models robustly.

[1] Puranik, Y., & Sahinidis, N. V. (2017). Domain reduction techniques for global NLP and MINLP optimization. Constraints, 22(3), 338-376.

[2] Swaney, R. E., & Grossmann, I. E. (1985). An index for operational flexibility in chemical process design. Part I: Formulation and theory. AIChE Journal, 31(4), 621-630.

[3] Kusumo, K. P., Gomoescu, L., Paulen, R., García Muñoz, S., Pantelides, C. C., Shah, N., & Chachuat, B. (2019). Bayesian approach to probabilistic design space characterization: A nested sampling strategy. Industrial & Engineering Chemistry Research, 59(6), 2396-2408.

Bayesian Optimization (BO) has demonstrated significant promise in enhancing data-driven optimization strategies across various fields. However, both the machine learning and process systems engineering communities face similar challenges when applying BO in safety-critical settings, where model discrepancies, noisy measurements, and stringent safety constraints are prevalent. This has led to the emergence of Safe BO, designed to operate effectively under these constraints. Despite these advancements, there still remains a limited comparative understanding on the effectiveness and applicability of these safe BO methods, particularly within process system engineering. Thus, this work provides a comprehensive examination of state-of-the-art safe BO methods, with our own enhancements, focusing on their performance in process systems.

While safe BO methods have been developed to address limitations in traditional BO methods, they still face significant challenges when applied to process systems. For instance, SafeOpt [1] and GoOSE [2], which have excellent performance in ML applications, rely on discretization of a system, suffer from heavy computational expense, and together with StableOpt [3,4], lack the capability to manage multiple safety constraints. To address these challenges, we introduced modifications that enable optimization under continuous system, handle multiple constraints and reduce computational costs, thereby enhancing their practical applicability.

Our study rigorously assessess these safe BO algorithms, including our own enhancements, with a reactory system from the process system engineering literature, focusing on convergence speed, mitigation of unknown constraints, practicality, and robustness against adversarial perturbations. We found that a performance of SafeOpt is often hindered by inefficiency due to excessive exploration, while GoOSE mitigates this inefficiency by incorporating an oracle to selectively expand the safe set, thus minimizing unnecessary evaluations. The integration of conventional Trust-Region with BO [5] demonstrates high performance, though its effectiveness is highly sensitive to the initial choices on trust-region parameters. StableOpt while guaranteeing feasible solutions under adversarial perturbations, often yields suboptimal solution due to its focus on the worst-case scenarios.

Overall, this study highlights the strengths and limitations of safe BO methods in process system engineering, advancing the field on data-driven approaches for decision-making in safety-critical processes while also identifies areas where further improvements are necessary.

References

[1] Yanan Sui, Alkis Gotovos, Joel Burdick, and Andreas Krause. Safe exploration for optimization with Gaussian processes. In International Conference on Machine Learning (ICML), pages 997–1005, 2015.
[2] Matteo Turchetta, Felix Berkenkamp, and A. Krause. Safe exploration for interactive machine learning. In NeurIPS, 2019.
[3] Ilija Bogunovic, Jonathan Scarlett, Stefanie Jegelka, and Volkan Cevher. Adversarially robust optimization with gaussian processes. In Advances in Neural Information Processing Systems, pages 5765–5775, 2018.
[4] Joel A. Paulson, Georgios Makrygiorgos, Ali Mesbah. Adversarially robust Bayesian optimization for efficient auto-tuning of generic control structures under uncertainty. AIChE Journal. p. e17591, 2022.
[5] E.A. del Rio Chanona, J.E. Alves Graciano, E. Bradford, B. Chachuat. Modifier-Adaptation Schemes Employing Gaussian Processes and Trust Regions for Real-Time Optimization. IFAC-PapersOnLine.

A critical part of process design and synthesis is ensuring that the designs generated are resilient against various uncertainties and disruptions that could affect the system during its lifetime. Various approaches have been proposed to address different aspects of this problem. Techniques such as flexibility analysis have been applied to tackle continuous uncertainties inherent in parameter values, such as cost or demand [1,2]. Reliability theory, which focuses on the likelihood that a system or component will perform or fail under specific conditions, has been utilized to design systems that can maintain a specific performance for a desired time duration [3]. To optimize operation under low probability-high impact disruptions, different scenarios are simulated on a given system design to minimize their impact through scheduling decisions [4,5]. While these techniques are well-established individually, a resilient system requires the integration of prevention, mitigation, and recovery actions under a general design framework that brings together consideration of continuous uncertainties and discrete disruption scenarios.

In this work, we propose a two-stage design under uncertainty and disruption framework for flexibility and resilience considerations. In the first stage, a multi-parametric programming reformulation of the flexibility analysis is solved to capture trade-offs among investment cost, design decisions, and flexibility for a desired range of uncertain parameters. In particular, to represent the probability distribution of the uncertain parameters at this stage, the stochastic flexibility method is applied. In the second stage, disruption scenarios are simulated on designs generated in the previous stage with known cost and flexibility index. The resilience performance of each design against disruption scenarios is assessed, providing design strategies that feature both the flexibility to handle parameter fluctuations and the resilience to manage discrete disruptions. The proposed methodology offers a path to exploring trade-offs among cost, flexibility, and resilience at the process design stage through multi-parametric programming. An illustrative case study is presented on an energy system under threat of internal failures as well as fluctuating market conditions.

References

1. Di Pretoro, Alessandro, et al. "Flexibility assessment of a biorefinery distillation train: Optimal design under uncertain conditions." Computers & Chemical Engineering 138 (2020): 106831.
2. Tian, Huayu, et al. "Feasibility/Flexibility-based optimization for process design and operations." Computers & Chemical Engineering 180 (2024): 108461.
3. Ade, Nilesh, et al. "Investigating the effect of inherent safety principles on system reliability in process design." Process Safety and Environmental Protection 117 (2018): 100-110.
4. Badejo, Oluwadare, and Marianthi Ierapetritou. "Enhancing Pharmaceutical Supply Chain Resilience: A Multi-Objective Study with Disruption Management." Computers & Chemical Engineering (2024): 108769.
5. Gong, Jian, and Fengqi You. "Resilient design and operations of process systems: Nonlinear adaptive robust optimization model and algorithm for resilience analysis and enhancement." Computers & chemical engineering 116 (2018): 231-252.

Spherical Tokamak for Energy Production (STEP) is a project by the UK Atomic Energy Authority to build and demonstrate the viability of a fusion power plant for generating clean electricity. As a heat source, fusion tokamaks present challenges for the design of the power plant, such as multiple heat sources of different qualities and pulsed operation leading to pulsed heat supplies. Pulsed operation means that dynamic modelling is necessary to simulate and evaluate designs. Thermal energy storage is chosen as a solution to mitigate the impact of consequent large thermal power transients on electricity generation, as already used in commercial solar thermal power plants.

This work presents a dynamic model of a power conversion system suitable for use in optimization-based design. The model is implemented in Modelica using OpenModelica. The power conversion system uses three different alternative heat sources as energy inputs. These energy inputs vary in time due to the pulsed nature of the fusion plant operation. A two-tank molten salt sensible heat storage system is used to provide heat during a tokamak dwell to a steam Rankine cycle and to buffer sensitive components, like the turbine, from thermal fluctuations. The variable nature of the operation necessitates the incorporation of a control system to regulate process flows due to the changing tokamak modes. Lumped-parameter models are favoured due to their computational efficiency, the robustness against simulation failures relative to more complex models, and the simplicity of parameterisation.

Multi-objective optimization is used to explore the design space, with each design evaluation involving a full dynamic simulation. Two objectives are considered simultaneously: robustness in the presence of pulsed operation and equipment sizing, such as the size of the molten salt tanks, as a proxy for the economics. Design variables include equipment sizing and controller tuning parameters. The latter are considered as different designs may require different controller actions. Integration of an external simulation program into an optimization algorithm poses some challenges, such as handling simulation failures with minimal information on the cause.

Due to the computational expense involved in dynamic simulation, a meta-heuristic optimization approach is used for design. Specifically, a population-based plant propagation meta-heuristic algorithm is used [1]. Populations can consist of both feasible and infeasible solutions. This facilitates handling failed simulations without losing potentially valuable information about the search space. A recently developed multi-agent system [2] is used to support multiple instances of the optimization method working together to more efficiently explore the design space.

A case study shows that the methodology can generate a diverse set of non-dominated designs in reasonable time despite using relatively computationally expensive simulations in the objective function. The resulting set of non-dominated designs is shown to capture the trade-off between economics and most efficient use of the heat storage system.

[1] A. Salhi, E. S. Fraga, 2011, Nature-inspired optimisation approaches and the new plant propagation algorithm , Proceedings of ICeMATH 2011

[2] E. S. Fraga, V. Udomvorakulchai, L. G. Papageorgiou, 2024, A multi-agent system for hybrid optimization, Computer Aided Chemical Engineering, 53