(382n) Improving the Subset Selection Regression of Helmholtz Energy Equations

Authors: 
Engle, M., Carnegie Mellon University
Sahinidis, N. V., Carnegie Mellon University
Accurate thermodynamic properties are important for the development of new energy system processes and technologies. Equations of state typically used, such as Peng Robinson and Soave-Redlich Kwong, are inaccurate in critical regions of new technologies. A recent development in equations of state uses first principles to determine one unifying equation explicit in Helmholtz energy to overcome these inaccuracies. These equations have been developed for over 100 pure substances and have been expanded to mixtures [1, 2, 3, 4].

A challenge in the development of Helmholtz energy equations is the multi-objective regression of varying datasets containing correlated data to the main unifying equation [3]. The current fitting procedures depend on cycling between linear and nonlinear regression techniques that are restricted to equality constraints and result in local solutions requiring multiple starts or an experienced user manually selecting an initial starting point [4]. We have developed a model-building procedure to address this challenge, using a global deterministic solver that has the capabilities to enforce inequality constraints on the resulting models to control the thermodynamic behavior, improve extrapolation behavior, and maintain thermodynamically feasible limitations on the regressed equation while simultaneously fitting the data. We use best subset selection to choose a set of functional terms to best represent the different phase regions and thermodynamic behavior. In order to avoid overfitting, our procedure systematically selects a subset of these terms to optimally fit the multiple thermodynamic property data sets.

We rely on a deterministic global optimization solver [5] to find an optimal solution that optimizes the fit as well as the number of terms in the model according to an information criterion. Due to the size of the bank of functional terms and the thermodynamic dataset, this regression with a global deterministic model can become intractable. To enable the regression for the formulation, we apply additional bounds based on theoretical limitations of the thermodynamic properties to facilitate the bounding procedures by the global solver. Additionally, the sensitivity and robustness of the model are evaluated for use in optimization routines [6, 7].

References cited

[1] Lemmon, E. W.; Huber, M. L.; McLinden, M. O. NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties-REFPROP, Version 9.1, National Institute of Standards and Technology. 2013; https://www.nist.gov/srd/refprop.

[2] Span, R. Multiparameter equations of state: An accurate source of thermodynamic property data; Springer-Verlag, 2000.

[3] Span, R.; Wagner, W.; Lemmon, E. W.; Jacobsen, R. T. Multiparameter equations of state - recent trends and future challenges. Fluid Phase Equilibria 2001, 183-184, 1-20.

[4] Lemmon, E; Tillner-Roth, R. A Helmholtz energy equation of state for calculating the thermodynamic properties of fluid mixtures, Fluid Phase Equilibria 1999, 165, 1-21.

[5] Tawarmalani, M.;Sahinidis, N. V. Global optimization of mixed-integer nonlinear programs: A theoretical and computational study, Mathematical Programming, 2004, 99, 563-591.

[6]Thierry, D. and Biegler, L. T., Dynamic real‐time optimization for a CO2 capture process. AIChE J. 2019, doi:10.1002/aic.16511

[7] Glass, M.; Djelassi, H.; and Mitsos, A. (2018), Parameter estimation for cubic equations of state models subject to sufficient criteria for thermodynamic stability. Chem. Eng. Sci. 2018, 192, 981-992.