(59j) A Parametric Cost Function Approximation Algorithm for Multiscale Decision-Making

In the last decade, great efforts have been made to shift energy systems from high carbon-emitting fossil fuels toward renewable energy sources. As reported by EIA, the share of U.S. power generation from renewables is projected to increase from 21% in 2021 to 44% in 2050, mainly due to the usage of wind and solar power [1]. However, renewable energy is known to be volatile and intermittent, which makes demand-side management (DSM) necessary to hedge against long-term seasonal fluctuations as well as short-term stochastic disturbances in its supply. In the chemical industry, energy-intensive production processes (e.g., industrial gas industry) could be benefitted from using renewable resources and DSM, as their energy cost can be further reduced by dynamically adjusting the production rates (or energy consumption) in response to changes in energy prices (e.g., electricity prices) [2]. However, this requires incorporating energy management into decision-making at all operational levels, ranging from supply chain management down to planning, scheduling, and control. In this sense, it is imperative to integrate different decision layers in a production system and collectively optimize the involved operations and energy use, such that the full economic potential of the entire system can be achieved while maintaining a high level of sustainability and a quick response to fast-changing market conditions [3].

Mathematical optimization is key to the integration of process operations, following the idea of enterprise-wide optimization (EWO) [4]. Many approaches have been developed to integrate the multiple decision layers, such as integrated planning and scheduling (iPS) [5, 6], integrated scheduling and control (iSC) [7, 8], and integrated planning, scheduling, and control (iPSC) [9, 10]. Against the traditional sequential method which may result in suboptimal or infeasible solutions, simultaneous models are generally preferred and developed by incorporating the detailed lower-level problem as additional constraints in the upper-level model. However, this direct integration approach usually become computationally intractable due to the introduction of a large number of variables and constraints. Although some approximation and decomposition techniques have also been proposed accordingly, the computational cost remains an issue when large time horizons and high-dimensional problems are considered [6, 8, 10].

To address the computational challenge, we propose to adapt parametric cost function approximations (CFAs) [11], a method developed by the reinforcement learning community (Powell and co-workers). CFA was originally aimed at solving single time-scale sequential decision-making problems under uncertainty. To the best of our knowledge, no work has been done to adapt CFA for solving multiscale problems arising from PSE applications. For illustration purposes, we consider the integration of two decision layers, an upper-level decision layer with low time resolution and a lower-level decision layer with more refined time resolution, e.g., the planning and the scheduling layers in the case of iPS. The most closely related work in the PSE literature to our proposed method is the works that train surrogate models representing lower-level decision layers to an upper-level decision layer [5, 12]. Our approach differs from existing surrogate approaches in several ways. First, instead of using machine-learning surrogates such as neural networks to incorporate in the upper-level decision layer which is not interpretable, we propose to select parameters in the upper-level decision layer as tunable parameters that have concrete physical meaning. The hypothesis is that, after proper tuning of these parameters, the single-scale model corresponding to the upper-level decision layer will achieve similar multiscale performance to the utopia multiscale model. The second novelty is in the algorithms to tune the parameters through an optimization-simulation-and-optimization framework. First, the parameterized upper-level problem is designed and solved to obtain a trial solution for the upper-level decisions. Second, with upper-level decisions fixed to the trial solution, we can easily “simulate” the lower-level problem to obtain the “true cost” of this fixed solution. Third, a stochastic search algorithm [11] can be utilized to optimize the selected parameters in the upper-level decision layer. We keep iterating between steps 1-3 until the stochastic search algorithm reaches some stability termination criterion. To illustrate the effectiveness of our proposed framework, a series of case studies on multiscale sustainability process systems is performed to show that high-quality solutions for integrated problems can be achieved with reduced computational time.

[1] U.S. Energy Information Administration, Annual energy outlook 2022 with projections to 2050. 2022.

[2] Zhang, Q., and Pinto, J. M., Energy-aware enterprise-wide optimization and clean energy in the industrial gas industry. Computers & Chemical Engineering 2022, 165, 107927.

[3] Andrés‐Martínez, O., and Ricardez‐Sandoval, L.A., Integration of planning, scheduling, and control: A review and new perspectives. The Canadian Journal of Chemical Engineering 2022, 100 (9), 2057–2070.

[4] Grossmann, I. E., Advances in mathematical programming models for enterprise-wide optimization. Computers & Chemical Engineering 2012, 47, 2–18.

[5] Maravelias, C. T., and Sung, C., Integration of production planning and scheduling: Overview, challenges and opportunities. Computers & Chemical Engineering 2009, 33(12), 1919–1930.

[6] Calfa, B. A., Agarwal, A., Grossmann, I. E., and Wassick, J. M., Hybrid bilevel-lagrangean decomposition scheme for the integration of planning and scheduling of a network of batch plants. Industrial & Engineering Chemistry Research 2013, 52(5), 2152–2167.

[7] Flores-Tlacuahuac, A., and Grossmann, I. E., Simultaneous cyclic scheduling and control of a multiproduct CSTR. Industrial & Engineering Chemistry Research 2006, 45(20), 6698–6712.

[8] Pattison, R. C., Touretzky, C. R., Johansson, T., Harjunkoski, I., and Baldea, M., Optimal process operations in fast-changing electricity markets: framework for scheduling with low-order dynamic models and an air separation application. Industrial & Engineering Chemistry Research 2016, 55(16), 4562–4584.

[9] Gutiérrez-Limón, M. A., Flores-Tlacuahuac, A., and Grossmann, I. E., MINLP formulation for simultaneous planning, scheduling, and control of short-period single-unit processing systems. Industrial & Engineering Chemistry Research 2014, 53(38), 14679–14694.

[10] Mora-Mariano, D., Gutiérrez-Limón, M. A., and Flores-Tlacuahuac, A., A Lagrangean decomposition optimization approach for long-term planning, scheduling and control. Computers & Chemical Engineering 2020, 135, 106713.

[11] Powell, W. B., and Ghadimi, S., The Parametric Cost Function Approximation: A new approach for multistage stochastic programming. ArXiv Preprint ArXiv:2201.00258, 2022

[12] Dias, L. S., and Ierapetritou, M. G., Integration of planning, scheduling and control problems using data-driven feasibility analysis and surrogate models. Computers & Chemical Engineering 2020, 134, 106714.