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