Probabilistic constrained optimization on flow networks

Michael Schuster; Elisa Strauch; Martin Gugat; Jens Lang

doi:10.1007/s11081-021-09619-x

What is it about?

In optimization, robustness is important, since the data are ususally not known exactly. Of course, robustess of a solution comes with a prize tag: If you want to be very conservative in the sense that the decision is the best worst-case decision, even if this worst case will not occur almost surely, then you should use the robustness approach of worst-case optimization. If however, there is some insight into the probability distribution of the data, then it makes sense to use probabilistic robustness where you accept to use decisions that will only be feasible with a prescribed probability, for example 0.99. Of course this comes with a risk, but there is no such thing as a free lunch!

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Why is it important?

In companies, it is important not to be over-conservative in order to optimize the operation. In the application of gas networks, the precise customer demand is uncertain and the operation has to be able to react to variations of this demand. In the operation of gas flow, the pressure is increased at certain compressor stations and the evolution of the pressure along the pipe is influenced by the flow rate. The system is governed by a hyperbolic system of balance laws, which has to be taken into account in order to determine the probability distribution of the pressure at the outflow nodes as a function of the customer demand.

Perspectives

Probabilistic constrains are an important modelling tool that allow to take into account information about the probability distribution of the problem data in rational decisions that are the solution of well-defined optimization problems. In the operation of engineering systems often well-established models that are governed by partial differential equations are available. In order to combine the advantages of both tools, both should be incorporated into problems of optimal control.
Martin Gugat
Friedrich-Alexander-Universitat Erlangen-Nurnberg

This page is a summary of: Probabilistic constrained optimization on flow networks, Optimization and Engineering, April 2021, Springer Science + Business Media,
DOI: 10.1007/s11081-021-09619-x.
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Probabilistic constrained optimization on flow networks
Uncertainty often plays an important role in dynamic fow problems. In this paper, we consider both, a stationary and a dynamic fow model with uncertain boundary data on networks. We introduce two diferent ways how to compute the probability for random boundary data to be feasible, discussing their advantages and disadvantages. In this context, feasible means, that the fow corresponding to the random boundary data meets some box constraints at the network junctions. The frst method is the spheric radial decomposition and the second method is a kernel density estimation. In both settings, we consider certain optimization problems and we compute derivatives of the probabilistic constraint using the kernel density estimator. Moreover, we derive necessary optimality conditions for an approximated problem for the stationary and the dynamic case. Throughout the paper, we use numerical examples to illustrate our results by comparing them with a classical Monte Carlo approach to compute the desired probability.

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Martin Gugat
Friedrich-Alexander-Universitat Erlangen-Nurnberg

Operation of gas pipeline networks: How to satisfy the customer demand in 99% of all cases

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