(273f) A Mixed-Integer Conic Programming Formulation for Computing the Flexibility Index Under Multivariate Gaussian Random Variables

Quantifying the flexibility of complex systems such as chemical plants and power distribution networks is drawing increased interest. Flexibility seeks to quantify the ability of a system to counteract disturbances and maintain feasible operation [2]. Over the past 35 years, Grossmann and co-workers have proposed several strategies to quantify flexibility [8, 9]. This can be done by solving the so-called flexibility index problem, which seeks to find the largest uncertainty set under which the system maintains feasible operation [3]. This problem is conceptually attractive, but it is computationally challenging to solve due to the presence of nested optimization problems [4]. Floudas and co-workers showed that, for hyperbox uncertainty sets and convex constraints, the flexibility index problem can be reformulated as a mixed-integer linear program (MILP). This approach is scalable, but hyperbox uncertainty sets do not capture correlation between random variables. As a result, hyperbox representations can provide highly conservative solutions. Motivated by this limitation, Straub and Grossmann proposed the stochastic flexibility index which represents the probability that the system will exhibit feasible operation [6, 7]. This index can be computed by using a quadrature scheme, but the approach suffers from scalability issues [5, 7].

In this work, we propose a scalable approach to compute the flexibility index for multivariate Gaussian variables. The Gaussian representation enables us to capture correlations. We show that the problem can be cast as a mixed-integer conic program (MICP) that seeks to find the largest radius of a hyperellipsoidal uncertainty set under which the system maintains feasible operation. We also prove that the radius can be directly related to the probability of feasible operation, which is precisely the definition of the stochastic flexibility index. Consequently, both indexes are equivalent in this case. The proposed approach can leverage the recent emergence of efficient MICP solvers [1].

References

1. Berthold, S. Heinz, and S. Vigerske. Extending a cip framework to solve miqcp s. In Mixed integer nonlinear programming, pages 427–444. Springer, 2012.
2. Deffuant and N. Gilbert. Viability and resilience of complex systems: concepts, methods and case studies from ecology and society. Springer Science & Business Media, 2011.
3. E. Grossmann, B. A. Calfa, and P. Garcia-Herreros. Evolution of concepts and models for quantifying resiliency and flexibility of chemical processes. Computers & Chemical Engineering, 70:22–34, 2014.
4. E. Grossmann, K. P. Halemane, and R. E. Swaney. Optimization strategies for flexible chemical processes. Computers & Chemical Engineering, 7(4):439–462, 1983.
5. Ma, V. Rokhlin, and S. Wandzura. Generalized gaussian quadrature rules for systems of arbitrary functions. SIAM Journal on Numerical Analysis, 33(3):971–996, 1996.
6. Pistikopoulos and T. Mazzuchi. A novel flexibility analysis approach for processes with stochastic parameters. Computers & Chemical Engineering, 14(9):991–1000, 1990.
7. A. Straub and I. E. Grossmann. Integrated stochastic metric of flexibility for systems with discrete state and continuous parameter uncertainties. Computers & Chemical Engineering, 14(9):967– 985, 1990.
8. E. Swaney and I. E. Grossmann. An index for operational flexibility in chemical process design. part i: Formulation and theory. AIChE Journal, 31(4):621–630, 1985.
9. E. Swaney and I. E. Grossmann. An index for operational flexibility in chemical process design. part ii: Computational algorithms. AIChE Journal, 31(4):631–641, 1985.