(371y) A Systematic Approach for Characterizing Probabilistic Operating Space Under Hybrid Uncertainties

Reliable operations play a crucial role in ensuring the plantwide safety and product quality of complex process systems. In conventional practice, the tasks of making decisions and performing control measures mainly rely on operator experience. However, the presence of multiple uncertainties poses challenges in capturing system behavior, which may lead to operator confusion and increased risk of misoperation. To overcome this issue, a forecast is required to predict the outcome of each control setting under uncertain environment so that an informed decision can be made. Consequently, recent years have witnessed growing efforts in developing system modeling and model-based operational feasibility analysis techniques for operator assistance [1,2].

Uncertainty in process operations can originate from many aspects. Dias et al. proposed a classification scheme for uncertainty sources, highlighting process variability and model mismatch as the primary inherent uncertainties when investigating operational feasibility [3]. The uncertainty arising from process variability can be directly determined through analysis of historical data, whereas quantifying the uncertainty of model parameters is typically more complicated. Techniques for model parameter estimation vary from the classic maximum likelihood method [4] to Bayesian inference that accounts for prior knowledge of parameter uncertainty [5]. These uncertainties can then be propagated to predictions by model simulation, thereby providing insight into the influence of uncertain factors on critical system performance. Uncertainty propagation using traditional Monte Carlo method or its variants can be prohibitively expensive, since these techniques require excessive model evaluations. As surrogate modeling methods mature, the computational burden of handling uncertainties is significantly relieved. Among the various surrogates, Kriging and polynomial chaos expansion are most widely used for uncertainty analysis [6].

This study devotes to investigating operational feasibility of process systems under uncertainty by introducing a concept of probabilistic operating space (POS), which is defined by the envelope of operations that can provide assurance of process safety and product quality with desired reliability. A systematic approach illustrated as Figure 1 is proposed to efficiently characterize the POS, considering hybrid uncertainties of process variability and model mismatch. The step-by-step procedures include: (I) A first principle model is formulated to describe the system behavior and represent the hybrid uncertainties with bounded form and probability distribution. (II) Sobol indices are used as the global sensitivity measure to select the most influential parameters with an orthogonalization procedure. Potential issues with parameter estimability in estimation problem and curse of dimensionality in uncertainty propagation are addressed. (III) The probability-interval hybrid uncertainties are propagated by a polynomial chaos expansion surrogate with parameterized coefficients, which is constructed by a two-layer sampling strategy. Uncertainty in system response is characterized by a probability box and the computational costs of evaluating feasibility probabilities are greatly reduced. (IV) The nested sampling scheme is applied to allocate samples of operating points towards regions of increasing feasibility probability, further expediting the exploration of the operating space. (V) To facilitate operator assistance, the resulting POS is finally visualized as a probability map using geometric projection. In the proposed approach, computational efficiency is improved by novel techniques of uncertainty dimensionality reduction, hybrid uncertainty propagation and adaptive sampling.

The effectiveness of our proposed approach is demonstrated by a case study of the advanced modular high-temperature gas-cooled reactor (MHTGR) nuclear power plant. The MHTGR model formulation consists of nonlinear equations of neutron kinetics, thermal hydraulics, conservation laws and thermophysical property calculation [7], emphasizing the significance of our approach in computational efficiency improvement. The graphical POS results can aid nuclear operators in coordinating the flow rates of the primary and secondary circuits, thereby ensuring load matching and safe operation.

References:

[1] G. Giannakoudis, A. I. Papadopoulos, P. Seferlis, and S. Voutetakis, “Optimum design and operation under uncertainty of power systems using renewable energy sources and hydrogen storage,” Int J Hydrogen Energy, vol. 35, no. 3, pp. 872–891, Feb. 2010.

[2] F. Destro and M. Barolo, “A review on the modernization of pharmaceutical development and manufacturing – Trends, perspectives, and the role of mathematical modeling,” International Journal of Pharmaceutics, vol. 620. Elsevier B.V., May 25, 2022.

[3] L. S. Dias and M. G. Ierapetritou, “Integration of scheduling and control under uncertainties: Review and challenges,” Chemical Engineering Research and Design, vol. 116, pp. 98–113, Dec. 2016.

[4] W. Chen and L. T. Biegler, “Reduced Hessian based parameter selection and estimation with simultaneous collocation approach,” AIChE Journal, vol. 66, no. 7, Jul. 2020.

[5] S. Shin, O. S. Venturelli, and V. M. Zavala, “Scalable nonlinear programming framework for parameter estimation in dynamic biological system models,” PLoS Comput Biol, vol. 15, no. 3, Mar. 2019.

[6] G. Makrygiorgos, G. M. Maggioni, and A. Mesbah, “Surrogate modeling for fast uncertainty quantification: Application to 2D population balance models,” Comput Chem Eng, vol. 138, Jul. 2020.

[7] C. Yang, K. Wang, Z. Shao, and L. T. Biegler, “Integrated Parameter Mapping and Real-Time Optimization for Load Changes in High-Temperature Gas-Cooled Pebble Bed Reactors,” Ind Eng Chem Res, vol. 57, no. 28, pp. 9171–9184, Jul. 2018.