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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.

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 [

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 [

In the next section, we introduce our gas network model, which is based on the model in [

In Section

In Section

In Section

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

We start with the introduction of the model that is also used in [

We consider a connected, directed graph

Let

Consider the graph

The matrix

is called the incidence matrix of the graph

For

For

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

As balance laws we use the isothermal Euler equations (see [

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

Pressure drop along a pipe for

Compressor stations counteract the pressure drop along the pipes. These stations are modeled as pipes without friction. We use the following model equation (see [

Compressor property for

Now, we define the set of feasible loads. We say that a vector

In reality, the future demand for gas can never be known exactly a priori. It depends on various physical effects like e.g., the temperature of the environment. Also personal reasons can be responsible for a higher or lower use of gas. Thus the load vector is considerd to be a random vector. Many papers study how the distribution functions of the load vector can be identified, e.g., [

In our case, we assume that the tree-structured graph has only one influx node, which gets the number zero, all remaining nodes are efflux nodes. Then the graph is numerated with breadth-first search or depth-first search and every edge

There are many studies for special sets

Let

The difference with our problem is, that our load vector is normal distributed with mean value

Let

Sample

Compute the one-dimensional sets

Set

The first step is quite easy. There are many possibilities to sample a set of uniformly distributed points on the sphere

The main challenge is to handle the one-dimensional sets of the second step. If the set

The third step is the approximation of the integral. The values

Distribution function of the Chi-square distribution for 3 (blue), 5 (red), 8 (green), and 12 (purple) degrees of freedom.

In this section we want to characterize the feasible set to use Algorithm

Network graph with two compressor edges (left) and new not connected network graph with three connected subgraphs after removing the compressor edges (right).

We set

The last equation in

Let

(i) Consider a vector

(ii) The other way around is quite similar. We consider a vector

Because our network graph is a tree, the mass conservation is fulfilled with a full rank incidence matrix

With this result, we can write the equation for momentum conservation depending on the incidence matrices for the parts of the tree. This makes us able to formulate the theorem for characterizing the feasible set

Analogously we define

With this notation and these values we define a sum for

Example graph for illustration of the used notation.

Graph of Example

If we want to compare the pressure bounds of node 11 and node 12, we need to know how the pressures change on the path from 2 to 11, resp., from 2 to 12. Node 11 is part of the subgraph

For given pressure bounds

A vector

Let us have a look at the inequalities. Inequalities (

For an implementation it is wise to define a set

Now we want to proof the theorem.

Because the columns of

“

So the set

With this theorem we have another characterization of the feasible set

Because we assumed that the graph has only one entry, we define the following set:

In this section we want to have a look at some optimization problems. We distinguish between problems with constant loads (without uncertainty on the demand) and problems with random loads. The latter leads us to optimization problems with so called probabilistic constraints or chance constraints (see [

There exists a unique solution for (

We have a look at the side constraint

For the optimization problem (

Consider pressure bounds

(a) If

Note that a vector

(a) For this part we use a proof by contradiction. Assume that

(b) As mentioned before, the vector

Note that the sufficient condition in Lemma

If the graph has only one compressor edge, i.e.,

This conclusion follows directly from the strict monotonicity of the objective function.

Let a sampling

For the proof, we fix a

In this section we show a few results of implementation. At first we show the idea of the spheric-radial decomposition by using an easy example. Next we show a easy example with one compressor edge and at last we use real data of the Greek gas network. The focus of the implementation is on the theorems proofed in Section

Our first example is an easy graph without inner control (see Figure

Assume that node 1 and 2 are gas consumers with mean demand

Feasible set of Example

Now consider the optimization problem:

Next we use the spheric-radial decomposition, especially Algorithm

Test results of easy example.

Test 1 | Test 2 | Test 3 | Test 4 | Test 5 | Test 6 | Test 7 | Test 8 | |
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The probability in all eight tests is nearly the same. The mean value is 0.3479 and the variance is

Last we want to solve the following optimization problem:

Test results of Example

Test 1 | Test 2 | Test 3 | Test 4 | Test 5 | Test 6 | Test 7 | Test 8 | |
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Now we add a compressor edge to the graph of Example

Graph of Example

Assume that node 1 and 3 are consumers, node 2 is a inner node. The mean demand

We assume the control to be switched off, so it holds

Separated graph of Example

Feasible set of the Example

Solving the problem

Here, one can see that the sufficient condition in Lemma

Now, we use again Algorithm

Test results of control example.

Test 1 | Test 2 | Test 3 | Test 4 | Test 5 | Test 6 | Test 7 | Test 8 | |
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Again we can see, that the probability is nearly the same in all eight cases. The mean probability is

Test results of control example.

Test 1 | Test 2 | Test 3 | Test 4 | Test 5 | Test 6 | Test 7 | Test 8 | |
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In this example, we use a part of a real network. The focus here is on realistic values for our model. We get these values from the GasLib (see [

“The goal of GasLib is to promote research on gas networks by providing a set of large and realistic benchmark instances.” (

We use a part of the Greek gas network (GasLib-134) (see Figure

Greek gas network from GasLib [

For using real values, we first need to know the constant

The length of the first pipe is given by

Values based on real data.

variable | | | | | | | |

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value | | | | | | | |

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unit | | | − | | | | |

Now we want to optimize the inner control in the network for a fix load vector. Therefor we choose two nominations:

Test results for nomination

Test 1 | Test 2 | Test 3 | Test 4 | Test 5 | Test 6 | Test 7 | Test 8 | |
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Test results for nomination

Test 1 | Test 2 | Test 3 | Test 4 | Test 5 | Test 6 | Test 7 | Test 8 | |
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Again one can see that the results are quite similar. The mean probability in Table

In this paper, we have used an existing stationary model for gas transport based on the isothermal Euler equations. We have extended this model by compressor stations and presented a characterization of the set of feasible loads. Especially Theorem

With the characterization of the feasible set, we have shown that the feasibility of a load depends significantly on the pressure bounds. These pressure bounds play a main role in our optimization problems. The other main parts are the inner controls modeled by compressor edges, which should lead in optimal control problems with probabilistic constraints. Here we have built a base for further works with this topic.

The idea of separating the graph and removing the compressor edges can be used to extend the model with other components, like, e.g., control valves and resistors. This would change the inequalities in Theorem

Finally the expansion to a dynamic model might be also possible. The idea is to use a space-mapping approach, because one can use a lot of the analysis we did here. Therefore upcoming papers should deal with these problems and present results of the turnpike-theory, which describes a relation between stationary and instationary solutions. In fact, turnpike-theory implies that the optimal dynamic state is close to the optimal static state in the interior of the time intervals for sufficiently long time horizons. A.J. Zaslavski explains the turnpike-phenomenon in [

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work was supported by DFG in the framework of the Collaborative Research Centre CRC/Transregio 154, Mathematical Modeling, Simulation and Optimization Using the Example of Gas Networks, project C03.