(12h) Pamso : Parametric Autotuning Multi-Timescale Optimization Algorithm

In large systems consisting of different interrelated processes, decisions are taken at different time scales including high-level planning decisions as well as low-level scheduling and control decisions. In the standard decision hierarchy adopted by the process systems engineering community, decisions from high-level decision problems are passed on to low-level operating decision problems to obtain a complete execution (Shin & Lee, 2016). High-level decision problems include planning problems (e.g., supply chain design, capacity planning, production targets, and task selection) and provide a set of decisions on a longer time scale, i.e., on a yearly, monthly, or weekly basis. Low-level operating decision problems include scheduling and control decisions and provide a set of decisions on a shorter time scale, i.e., on an hourly scale or in seconds. These decision layers are interconnected and, a one-way top-down communication in the hierarchical structure, i.e., solving the high-level problems without input from the low-level problems and then fixing the high-level decisions in the low-level problems, can result in infeasible or suboptimal solutions. Therefore, it is necessary to integrate the different decision layers. To do this, multi-timescale decision problems can be solved which involve formulating a multi-timescale optimization model. Multi-timescale decision problems involve decision processes taking place at different levels of different timescales (Shin & Lee, 2019). Multi-timescale optimization models are significant in various domains, such as plant production and scheduling (Biondi et al., 2017) and electricity market and environmental management (Dowling et al., 2017).

When working with multi-timescale optimization models, a critical issue arises from their potential to involve an exceedingly high volume of variables and constraints, sometimes reaching into the billions. While the strategy of solving the entire model using full-space methods (Zantye et al., 2022) has been used for small multi-timescale optimization problems, it is difficult to use full-space methods to solve these problems as they become larger (in the order of millions of variables). To solve these problems, different methods such as decomposition methods (Barzanji et al., 2020; Erdirik-Dogan & Grossmann, 2008), metaheuristic methods (Lee & Ha, 2019), data-driven approaches involving surrogate models as well as reinforcement learning-based approaches (Dias & Ierapetritou, 2020; Li et al., 2016; Shin & Lee, 2019), as well as mathheuristics (Ramanujam et al., 2023) have been applied. Most of these methods are not very scalable to problems of extremely large sizes.

To address these scalability issues, this work presents an algorithm that we call the Parametric Autotuning Multi-Timescale Optimization algorithm (PAMSO) to solve multi-timescale optimization models. The algorithm is inspired by parametric cost function approximations (CFAs) (Perkins III & Powell, 2017). Parametric CFA is a method that has been used to solve sequential decision-making problems under uncertainty (Powell & Ghadimi, 2022). CFAs use deterministic models to achieve similar results to a multi-stage stochastic programming model at a significantly lower cost. The idea is to select some parameters in the deterministic model and tune them in a stochastic simulator through derivative-free or derivative-based stochastic search.

To the best of our knowledge, no work has been done to adapt CFA or similar ideas for solving multi-time scale optimization models. Adapting the ideas behind CFA, our algorithm involves tuning some selected parameters in a low-fidelity optimization model (single-scale model) so that it can achieve similar performance as a high-fidelity optimization model (multi-time scale model). The tunable parameters in the single-scale model reflect the mismatch between the single-scale model and the multi-scale model. The parameters can be tuned using derivative-free optimization methods. To further improve the scalability of our algorithm, we can select parameters in such a way we can transfer-learn the parameters i.e., we can pre-tune these parameters on one class of problems and fine-tune the trained parameters when it is applied to a slightly different problem. This helps reduce the computational costs significantly.

To illustrate the effectiveness of our algorithm, we implement the algorithm on a series of multi-time scale optimization models. One of the examples we will be focusing on is the application of multi-time scale optimization models in the decarbonization and electrification of chemical process systems. Electrification, a promising approach for decarbonization, entails powering chemical processes with electricity which can be sourced from renewable green energy rather than fossil fuels. However, these renewable resources have substantial spatial and temporal variations, and electricity prices also fluctuate over time. To fully exploit the economic possibilities of the entire system, it's crucial to account for these fluctuations in the design of electrified chemical plants and the renewable energy sources that fuel them. The optimal approach includes integrating the planning and scheduling of these plants and resources into a multi-timescale optimization model (Ramanujam et al., 2023).

Such a multi-timescale optimization model can have around a million constraints and variables with a significant number of the variables being integers even for a moderate-size problem. For example, one of the case studies we will focus on has around 10 million constraints, 2 million continuous variables, and 300,000 integer variables. Commercial solvers and existing decomposition methods cannot give a good solution even in one day. Implementing the PAMSO algorithm in this case study using transfer learning gives a solution that has an optimality gap of 2.8% compared with the LP relaxation within 2 hours. Thus, we can get a good solution with significantly reduced computational costs.

Another example is the integrated design and scheduling for the production of chemicals in a resource task network (RTN). Design decisions that involve the sizing of units need to consider the demand for chemicals as well as the scheduling dynamics of the process. Therefore, to maximize the economic potential, we need to integrate the design and scheduling decisions for a plant. The best way would involve formulating a multi-timescale optimization model. Since sizing of vessels in chemical processes can indeed involve economies of scale, i.e., larger vessels often have a lower cost per unit volume compared to smaller vessels, the optimization model would be a Mixed Integer Non-linear programming (MINLP) model, intractable by the state-of-the-art algorithms. We obtained good feasible solutions to this multi-timescale model with tens of thousands of variables within 24 hours.

Barzanji, R., Naderi, B., & Begen, M. A. (2020). Decomposition algorithms for the integrated process planning and scheduling problem. Omega, 93, 102025. https://doi.org/https://doi.org/10.1016/j.omega.2019.01.003

Biondi, M., Sand, G., & Harjunkoski, I. (2017). Optimization of multipurpose process plant operations: A multi-time-scale maintenance and production scheduling approach. Computers & Chemical Engineering, 99, 325–339. https://doi.org/https://doi.org/10.1016/j.compchemeng.2017.01.007

Dias, L. S., & Ierapetritou, M. G. (2020). Integration of planning, scheduling and control problems using data-driven feasibility analysis and surrogate models. Computers & Chemical Engineering, 134, 106714. https://doi.org/https://doi.org/10.1016/j.compchemeng.2019.106714

Dowling, A. W., Kumar, R., & Zavala, V. M. (2017). A multi-scale optimization framework for electricity market participation. Applied Energy, 190, 147–164. https://doi.org/https://doi.org/10.1016/j.apenergy.2016.12.081

Erdirik-Dogan, M., & Grossmann, I. E. (2008). Simultaneous planning and scheduling of single-stage multi-product continuous plants with parallel lines. Computers & Chemical Engineering, 32(11), 2664–2683. https://doi.org/https://doi.org/10.1016/j.compchemeng.2007.07.010

Lee, H. C., & Ha, C. (2019). Sustainable Integrated Process Planning and Scheduling Optimization Using a Genetic Algorithm with an Integrated Chromosome Representation. Sustainability, 11(2). https://doi.org/10.3390/su11020502

Li, J., Xiao, X., Boukouvala, F., Floudas, C. A., Zhao, B., Du, G., Su, X., & Liu, H. (2016). Data-driven mathematical modeling and global optimization framework for entire petrochemical planning operations. AIChE Journal, 62(9), 3020–3040. https://doi.org/https://doi.org/10.1002/aic.15220

Perkins III, R. T., & Powell, W. B. (2017). Stochastic optimization with parametric cost function approximations. ArXiv Preprint ArXiv:1703.04644.

Powell, W. B., & Ghadimi, S. (2022). The Parametric Cost Function Approximation: A new approach for multistage stochastic programming.

Ramanujam, A., Constante-Flores, G. E., & Li, C. (2023). Distributed manufacturing for electrified chemical processes in a microgrid. AIChE Journal, n/a(n/a), e18265. https://doi.org/https://doi.org/10.1002/aic.18265

Shin, J., & Lee, J. H. (2016). Multi-time scale procurement planning considering multiple suppliers and uncertainty in supply and demand. Computers & Chemical Engineering, 91, 114–126. https://doi.org/https://doi.org/10.1016/j.compchemeng.2016.03.024

Shin, J., & Lee, J. H. (2019). Multi-timescale, multi-period decision-making model development by combining reinforcement learning and mathematical programming. Computers & Chemical Engineering, 121, 556–573. https://doi.org/https://doi.org/10.1016/j.compchemeng.2018.11.020

Zantye, M. S., Gandhi, A., Wang, Y., Vudata, S. P., Bhattacharyya, D., & Hasan, M. M. F. (2022). Optimal design and integration of decentralized electrochemical energy storage with renewables and fossil plants. Energy Environ. Sci., 15(10), 4119–4136. https://doi.org/10.1039/D2EE00771A