(700f) Design of Flare Systems Under Uncertainty: A Chance-Constrained Nonlinear Programming Approach

Tovar-Facio, J., Universidad Michoacana de San Nicolás de Hidalgo
Ponce, J. M., Universidad Michoacana de San Nicolás de Hidalgo
Cao, Y., University of Wisconsin-Madison
Zavala, V. M., University of Wisconsin-Madison
Flare systems are used as safety (relief) devices to manage abnormal situations in infrastructure

systems (natural gas and oil processing plants, and pipelines), manufacturing facilities (chemical

plants, offshore rings), and power generation facilities. Flare design is influenced by several uncertain

factors such as the amount and composition of the waste stream and the ambient conditions.

Here, we propose to use a chance-constrained nonlinear programming (CC-NLP) formulation to find

designs that minimize cost and that mitigate the risk that the combustion radiation exceeds a certain

safety threshold.

CCs are a powerful modeling paradigm that helps decision-makers control risk. Unfortunately,

CCs cannot be handled directly by off-the-shelf optimization solvers. CCs can reformulated into

a standard NLP when the quantile function of the constraint can be expressed in algebraic form.

This property has been exploited extensively in the special case in which the constraint mapping

is linear in both arguments and the random data vector is Gaussian [1]. Under these assumptions,

the constraint mapping is Gaussian and its quantile can be expressed as a weighted sum of its two

moments (the expectation and the variance). For more general settings it is possible to derive exact

reformulations using integer variables, as originally proposed in [3]. Unfortunately, in the context of

CC-NLP, integer reformulations would lead to large-scale and nonconvex MINLPs.

We review different strategies to tackle large-scale CC-NLPs arising in flare system design. In particular,

we consider the special case in which the algebraic structure of the moments and of the quantile

function of the constraint mapping is known or approximately known and propose to use moment

matching (MM) to compute its parameters and to derive its quantile function. We argue that this procedure

can be applied to a wide range of distributions that go beyond Gaussian such as uniform,

log-normal, and generalized extreme value (Weibull, Frechet, Gumbel). The reason is that many

random phenomena observed in science and engineering can be explained using asymptotic results

such as the central limit and the extreme value (Fisher-Tippett) theorem or because basic transformations

and fundamental relationships between distributions can be exploited. While this approach

cannot be applied to general settings, we believe that there is value in studying the structure of the

problem at hand to explore if suitable algebraic approximations emerge. To handle more general set

tings, we consider the use of a recently proposed sigmoidal approximation approach (SigVaR), which

can be used to solve large-scale NLPs with chance constraints by using powerful serial and parallel

solvers such as Ipopt and PIPS-NLP [2, 4]. We demonstrate that this approach outperforms existing

conservative approximations such as the conditional value-at-risk (CVaR) approximation that

not offer a mechanism to enforce convergence to a solution of CC-P, and as the almost-surely (AS)

approximation which enforces the constraint constraint with probability one (almost surely). This is

equivalent to enforce the constraint for all possible realizations of the uncertainty. Our flare design

study confirms that MM and SigVaR provide nearly exact solutions of the CC-NLP while AS and

CVaR solutions exhibit extreme conservatism.


[1] D. Bienstock, M. Chertkov, and S. Harnett. Chance-constrained optimal power flow: Risk-aware

network control under uncertainty. SIAM Review, 56(3):461–495, 2014.

[2] N.-Y. Chiang and V. M. Zavala. An inertia-free filter line-search algorithm for large-scale nonlinear

programming. Computational Optimization and Applications, 64(2):327–354, 2016.

[3] J. Luedtke, S. Ahmed, and G. L. Nemhauser. An integer programming approach for linear programs

with probabilistic constraints. Mathematical Programming, 122(2):247–272, 2010.

[4] A. W¨achter and L. T. Biegler. On the implementation of a primal-dual interior point filter line

search algorithm for large-scale nonlinear programming. Mathematical Programming, 106:25–57,



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