(400e) A Novel Volumetric Flexibility Index Calculation Method Using Physics-Informed Neural Networks

Chemical process systems are inherently subject to uncertainties, which can stem from a variety of sources, such as exogenous disturbances or internal parameter variations that cannot be precisely defined [1]. The ability of a chemical system to maintain feasible operation despite these uncertain deviations from nominal conditions is known as operational flexibility [2]. Operational flexibility is an important indicator of system performance, as it reflects the system's ability to respond and adapt to changing conditions.

Traditional process design approaches that rely solely on nominal values of process parameters may not provide realistic solutions for chemical systems operating under uncertain conditions. Various flexibility analysis methods have been proposed to address this issue. Among those methods, the volumetric flexibility index [3] represents the ratio of the hypervolume of the feasible region to that of a hypercube bounded by the expected upper and lower limits of uncertain process parameters. Compared to other flexibility indices, such as the traditional flexibility index [4,5] and the stochastic flexibility index [6], the volumetric flexibility index provides a more precise and convenient evaluation of flexibility. Several approaches have been developed to compute the volumetric flexibility index, including the use of auxiliary vectors [3] and the triangulation strategy [7]. However, these methods may not always be efficient or accurate when solving general problems with non-convex, non-simply connected, or disconnected regions.

In the proposed approach, we proposed a physics-informed neural network learning approach to address the volumetric flexibility index calculation problem. First, we proposed a learning procedure to eliminate the nonlinear equality constraints with a feedforward neural network. Specifically, we incorporate the process model (equality constraints) into the loss function and train a neural network to predict the system state x based on the control input z and the uncertain parameter θ. We then construct another neural network to map all feasible parameters from the input space to an output space. In this output space, feasible parameters are mapped to the interior of a hypersphere, while infeasible parameters are pushed outside, depending on whether they satisfy the inequality constraints or not. This approach enables the mapping of parameters into a higher-dimensional space, where feasible and infeasible parameters can be distinguished. The distance between the hypersphere center and mapped boundary points is the threshold radius. A parameter is identified as feasible if the distance between its mapped points and the hypersphere center is less than the threshold radius. Finally, we use Monte Carlo simulation to calculate the ratio of feasible parameters, which allows us to obtain the volumetric flexibility index.

We have tested our proposed approach on several cases and demonstrated its good performance, particularly its universality in solving problems with nonlinear process model and non-convex inequality constraints. Our approach offers a more efficient and accurate method for calculating the volumetric flexibility index, which can aid in the design and optimization of chemical systems operating under uncertain conditions.

References:

[1] Lima, Fernando V., and Christos Georgakis. "Design of output constraints for model-based non-square controllers using interval operability." Journal of Process Control 18.6 (2008): 610-620.

[2] Halemane, Keshava Prasad, and Ignacio E. Grossmann. "Optimal process design under uncertainty." AIChE Journal 29.3 (1983): 425-433.

[3] Lai, Sau M., and Chi-Wai Hui. "Process flexibility for multivariable systems." Industrial & engineering chemistry research 47.12 (2008): 4170-4183.

[4] Grossmann, Ignacio E., Keshava Prasad Halemane, and Ross E. Swaney. "Optimization strategies for flexible chemical processes." Computers & Chemical Engineering 7.4 (1983): 439-462.

[5] Zhang, Q., Grossmann, I. E., & Lima, R. M. (2016). On the relation between flexibility analysis and robust optimization for linear systems. AIChE Journal, 62(9), 3109-3123.

[6] Pistikopoulos, E. N., and T. A. Mazzuchi. "A novel flexibility analysis approach for processes with stochastic parameters." Computers & Chemical Engineering 14.9 (1990): 991-1000.

[7] Adi, Vincentius Surya Kurnia, Rosalia Laxmidewi, and Chuei-Tin Chang. "An effective computation strategy for assessing operational flexibility of high-dimensional systems with complicated feasible regions." Chemical Engineering Science 147 (2016): 137-149.