Optimal Inflow Control Penalizing Undersupply in Transport Systems with Uncertain Demands

Abstract

In this paper, we address the task of setting up an optimal production plan taking into account an uncertain demand. The energy system is represented by a system of hyperbolic partial differential equations (PDEs) and the uncertain demand stream is captured by an Ornstein-Uhlenbeck process. We determine the optimal inflow depending on the producer's risk preferences. The resulting output is intended to optimally match the stochastic demand for the given risk criteria. We use uncertainty quantification for an adaptation to different levels of risk aversion. More precisely, we use two types of chance constraints to formulate the requirement of demand satisfaction at a prescribed probability level. In a numerical analysis, we analyze the chance-constrained optimization problem for the Telegrapher's equation and a real-world coupled gas-to-power network.

Optimal inflow control in transport systems with uncertain demands ‐ A comparison of undersupply penalties

“…With only a slight modification of the reformulation of equation 5in [5], we again have a deterministic reformulation of the SOC problem (3). We use the numerical procedure set up in [5] adapted to the alternative cost function OF penI to test our hypothesis (H1) and (H2) numerically. We take 10 3 Monte Carlo repetitions for the following parameter setting: In Figure 1, we see that the number of undersupply cases is higher for OF penI .…”

“…We will compare different types of undersupply penalties in terms of different cost functions OF (Y s , t 0 , y t0 , y(s)). The first one is taken from [5]:…”

“…The multiple α gives the possibilty to control the intensity of penalization. For the effect of different intensities on the optimal supply, we refer the reader to [5]. We formulate an alternative penalization of undersupply as follows:…”

Control strategies for transport networks under demand uncertainty

“…The Jacobi process admits for the Markov property and belongs to the wide class of Pearson diffusion processes consisting of a deterministic drift term and a stochastic diffusion term. Other members of this class, as for instance the Ornstein-Uhlenbeck process or the CIR-Process, have been used to model uncertain demands in various applications [9,17,18,25]. The Ornstein-Uhlenbeck process is given by the solution to the SDE…”

Chance‐constrained optimal inflow control in hyperbolic supply systems with uncertain demand

“…One possibility to overcome problems of infeasibility and to reduce the average undersupply is to introduce an undersupply penalty term in the cost function. The effect of an undersupply penalty on the optimal output has been analyzed in Reference 38. A comparison of different types of undersupply can be found in Reference 39.…”