Mathematical Problems in Engineering

Volume 2018, Article ID 7984079, 17 pages

https://doi.org/10.1155/2018/7984079

## Stationary Gas Networks with Compressor Control and Random Loads: Optimization with Probabilistic Constraints

Lehrstuhl Angewandte Mathematik 2, Department Mathematik, Friedrich-Alexander Universität Erlangen-Nürnberg (FAU), Cauerstr. 11, 91058 Erlangen, Germany

Correspondence should be addressed to Michael Schuster; ed.uaf@retsuhcs.ihcim

Received 27 April 2018; Revised 29 July 2018; Accepted 4 September 2018; Published 26 September 2018

Guest Editor: Gerhard-Wilhelm Weber

Copyright © 2018 Martin Gugat and Michael Schuster. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We introduce a stationary model for gas flow based on simplified isothermal Euler equations in a non-cycled pipeline network. Especially the problem of the feasibility of a random load vector is analyzed. Feasibility in this context means the existence of a flow vector meeting these loads, which satisfies the physical conservation laws with box constraints for the pressure. An important aspect of the model is the support of compressor stations, which counteract the pressure loss caused by friction in the pipes. The network is assumed to have only one influx node; all other nodes are efflux nodes. With these assumptions the set of feasible loads can be characterized analytically. In addition we show the existence of optimal solutions for some optimization problems with probabilistic constraints. A numerical example based on real data completes this paper.

#### 1. Introduction

In this paper, we present a way to model stationary gas networks with random loads. Natural gas as energy source has been popular for decades. Because of the nuclear power phase-out and aim to leave energy gained by coal behind, natural gas as energy source is more current than ever. So in this paper we study the problem of gas transport in a pipeline network mathematically. The aim of this paper is to get results in optimization problems with probabilistic constraints using the example of gas networks.

There are several studies about the mathematical problem of gas transport. A gas transport network in general can be modeled as a system of hyperbolic balance laws, like e.g., the isothermal Euler equations. In [1] one can find a great overview about existing models and their application areas. A network system is modeled as a graph with nodes end edges. It can be extended by different elements, like compressor stations, valves and resistors. In [2] the authors describes the functionality of these elements and how they can be included to the model.

We assume an active stationary state for our model that means we use compressor stations as controllable elements in a state of time-independent flows and pressures. Our model is based on the Weymouth-Equation (see [3]), a simplification of the isothermal Euler equation (see e.g., [4, 5]). Further [3] also studies the isothermal Euler equations for real gas. We assume our gas to be ideal. The optimal control problem in pipeline networks has been studied in, e.g., [4, 6]. In our work we assume the loads to be random, so this leads to optimization problems with probabilistic constraints (see [7]).

In the next section, we introduce our gas network model, which is based on the model in [8]. This is a passive one which we extend by compressor stations. So our model contains inner control variables which makes our model an active one. These controls will be one goal of the optimization in Section 4.

In Section 3 we characterize the set of feasible loads. Instead of searching a flow and a pressure vector (based on a given load vector), which fulfills the physical balance laws, we look for these vectors fulfilling a system of inequalities, which depend on bounds for the pressure. In addition, we assume the load vector to be random. So we adapt the system to the spheric-radial decomposition, which is our central tool to handle the uncertainty in the loads.

In Section 4, we consider some optimization problems with and without probabilistic constraints. The existence of optimal solutions is based on the analysis we did in Section 3. This should built a base for further works about optimal control problems with probabilistic constraints using the example of gas networks.

In Section 5 we present three numerical examples. The first is a minimal graph without inner control, which shows the idea of the spheric-radial decomposition. The second one is a minimal graph with inner control and the third one is based on real data, for which we present real values for a network.

In the last section we present a few ideas of extending this paper. One main aspect is the turnpike-theory.

#### 2. Mathematical Modeling

We start with the introduction of the model that is also used in [8]. This model does not include compressor stations. We extend this model to include compressor stations and we discuss the network analysis. Then the main tool, the so called spheric-radial decomposition, is introduced to handle the stochastic uncertainty.

##### 2.1. Model Description

We consider a connected, directed graph which represents a pipeline gas transport network. We assume that there are no cycles in the graph, so the network is a tree. We set and (). In our model, every node is either a influx node (gas enters the network) or an efflux node (gas leaves the network). An edge can either be a flux edge, so the pressure decreases along the edge, or a compressor edge, so the pressure increases along the edge. We define as the set of all flux edges and as the set of compressor edges, it follows with .

Let with denote the balanced load vector and assume for nodes with gas influx and for nodes with gas efflux (). The vector denotes the vector of all ones in the dimension . Furthermore, a vector describes the flows through the edges resp. through the pipes. The pressures at the nodes are defined in a vector . Let pressure bounds be given. For the pressure we consider the constraints . For further modeling we need the following definition:

*Definition 1. *Consider the graph :(i) denotes the head node of an edge and denotes the feet of an edge for all .(ii) denotes all edges which are connected to node .(iii)The matrix , with is called the incidence matrix of the graph .(iv)For is the (unique) directed path from the root to .(v)For is the (unique) directed path from to .

Note that definition (iv) only makes sense since we assumed the graph to be tree-structured, so there are no cycles and the union is finite. The model is based upon the conservation equations of mass and momentum. The mass equation for the nodes is formulated for every node by

This is equivalent to* Kirchhoff*’*s first law* (see [9]) and by using the definition of the incidence matrix, the equation for mass conservation for the whole graph is

As balance laws we use the isothermal Euler equations (see [3, 4, 10]) for a horizontal pipe, so the flow through every flux edge is modeled byHere, is the gas density, is the speed of sound, is a (constant) friction coefficient and is the diameter of the pipe.

We assume first, that the network is in a steady state, so the time derivatives are equal to zero. And second, we assume the gas flow to be slow, so the coefficient is negligible. These assumptions simplify the equations enormously. With the ideal gas equation (see [9]) we get a direct dependency between pressure and density. Solving this simplified equation leads us to the so called Weymouth-Equation (see [3]):Here, is the specific gas constant, is the temperature of the gas and is the length of the pipe . This equation shows the pressure drop along the pipes (see Figure 1). Some articles consider the isothermal Euler equations in gas transport without these simplifications, e.g., [5, 10].