Highlights Simulation of Transients in Natural Gas Networks via A Semi-analytical Solution Approach Xin Xu Rui Yao Kai Sun Feng Qiu

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Highlights

Simulation of Transients in Natural Gas Networks via A Semi-analytical Solution Approach

Xin Xu, Rui Yao, Kai Sun, Feng Qiu

•Research highlight 1: We present a semi-analytical solu-

tion approach for the simulation of transients in natural gas

networks.

•Research highlight 2: To further reduce the computation

burden, the nonlinear terms in the model are simpliﬁed

which induces another SAS scheme that can greatly reduce

the time consumption and have minor impact on accuracy.

•Research highlight 3: We analyze the proposed approach

in terms of eﬃciency and accuracy, and compare it with

with the ﬁnite diﬀerence method.

arXiv:2210.03298v1 [math.DS] 7 Oct 2022

Simulation of Transients in Natural Gas Networks via A Semi-analytical Solution

Approach

Xin Xua,b,1, Rui Yaoa,∗, Kai Sunb, Feng Qiua

aEnergy System Division, Argonne National Laboratory, Lemont, 60439, IL, USA

bDept. of Electrical Engineering &Computer Science, the University of Tennessee, Knoxville, 37996, TN, USA

Abstract

Simulation and control of the transient ﬂow in natural gas networks involve solving partial diﬀerential equations (PDEs). This

paper proposes a semi-analytical solutions (SAS) approach for fast and accurate simulation of the natural gas transients. The

region of interest is divided into a grid, and an SAS is derived for each grid cell in the form of the multivariate polynomials, of

which the coeﬃcients are identiﬁed according to the initial value and boundary value conditions. The solutions are solved in a

“time-stepping” manner; that is, within one time step, the coeﬃcients of the SAS are identiﬁed and the initial value of the next

time step is evaluated. This approach achieves a much larger grid cell than the widely used ﬁnite diﬀerence method, and thus

enhances the computational eﬃciency signiﬁcantly. To further reduce the computation burden, the nonlinear terms in the model are

simpliﬁed, which induces another SAS scheme that can greatly reduce the time consumption and have minor impact on accuracy.

The simulation results on a single pipeline case and a 6-node network case validate the advantages of the proposed SAS approach

in accuracy and computational eﬃciency.

Keywords: Natural gas network, Semi-analytic approach, Simulation, Transient ﬂow

1. Introduction

Natural gas is an important energy source that can be used

for electricity production and heating supply [1, 2, 3]. As one

of the crucial infrastructures for gas delivery, the gas pipeline

networks have been investigated over decades on economic op-

eration, control and security assessment. Disturbances in oper-

ating natural gas network can lead to transient processes prop-

agating across the entire network, and potentially could cause

damage to the equipment and even more severe security threats.

Therefore, it is necessary to model and simulate the transient

processes of natural gas pipeline network under potential dis-

turbances. Simulations of the natural gas transient can directly

reﬂect the variation of the states like pressure within the gas

pipelines and provide important insight in the dynamics anal-

ysis and the system control design, which motivates numerous

relevant studies [4, 5, 6].

Natural gas pipeline transients can generally be formulated

as a boundary value problem (BVP) of partial diﬀerential equa-

tions (PDEs) which governs the dynamics within the pipelines

as well as network and terminal constraints. Many simulation

techniques have been developed and investigated to solve the

BVP. Two mainstream representatives are the method of charac-

teristics [4] and the ﬁnite diﬀerence method (FDM) [5, 6, 7, 8].

The method of characteristics is one of the early eﬀorts to solve

the BVP. Although an accurate result can be reached, it can be

∗Corresponding author

1This author is currently with Dominion Energy, Richmond, 23220, VA,

USA.

computationally intensive since tiny time steps are necessary to

keep the numerical stability [7]. On the other hand, the FDM

gains more attention in recent years since it is more eﬃcient and

easier to implement [5]. Other techniques are also investigated

like ﬁnite volume method (FVM) [9] and state-space method

[10], which are still under development and need more compre-

hensive investigation to justify their performance in terms of

accuracy and eﬃciency

Amongst all those methods, the FDM is one of the most

commonly used in both academia and industry. FDM is es-

sentially a numerical method that depends on the discretiza-

tion techniques, i.e. a grid should be selected to discretize the

spatio-temporal space. The grid is commonly a regular one with

square and rectangle cells. The basic idea of FDM is to approxi-

mate the partial derivatives in the PDEs by the values at the grid

nodes, and then the BVP is converted to a set of diﬀerence equa-

tions, whose solution is the approximate solution of the BVP.

The formulation of FDM can be either explicit or implicit in

time. The explicit formulation can be easier to implement, but

is usually limited to small time steps due to the stability conﬁne-

ments. The implicit formulation is more complex to implement

but it has better numerical stability and thus enables larger time

and spatial steps, which makes it the choice of many commer-

cial software tools [11]. Various studies have been conducted

based on the FDM in terms of modeling[12, 11], formulations

[5, 6], and accuracy and numerical stability [13, 14]. Note that

one bottleneck for improving the eﬃciency of FDM lies in the

grid cell size. Since all the constraints in the BVP are only en-

forced on the grid nodes, the grid cell size has to be limited in

order to ensure the accuracy, which in turn increases computa-

Preprint submitted to ArXiv October 10, 2022

tion burden and limits the improvement of eﬃciency.

This paper proposes a semi-analytical solution (SAS) ap-

proach for the simulation of transients in natural gas pipeline

networks. The name “semi-analytical” comes from the algo-

rithm that ﬁrst sets a solution in a piece-wise analytical form

(multivariate polynomials) and then determines the coeﬃcients

in the polynomials, which approximates the solution of the

studied PDE. As will be shown later, SAS is more accurate and

more eﬃcient than FDM, because the constraints are enforced

on the whole grid instead of just the gird nodes and also a larger

grid cell can be achieved. In our previous works, SAS approach

has been veriﬁed to be more eﬃcient than many conventional

numerical methods in solving ordinary diﬀerential equations

(ODEs) [15], diﬀerential-algebraic equations (DAEs) [16, 17],

and partial diﬀerential equations [18]. The idea of SAS is also

applied for solving algebraic equations in [19, 20], and [21].

The rest of the paper is organized as follows. The BVP re-

garding the simulation of transient ﬂow is formulated in section

2. The algorithm of the SAS approach is introduced in sec-

tion 3. The overall simulation procedure is organized in sec-

tion 4. The simulation results on a single pipeline case and a

6-node network case are presented in section 5 and compared

with those of FDM. The conclusions and future works are in

section 6.

2. Boundary value problem of natural gas network tran-

sients

2.1. Natural gas network model

The natural gas network can be viewed as a graph with links

and nodes. The set of links Eincludes all the pipelines and the

set of nodes Vis the union of the following sets: (i) supply

nodes VPwhere the gas is supplied into the network and the

pressure pis usually controlled, (ii) demand nodes VQwhere

the gas is extracted out of the network and the mass ﬂow qis

determined, and (iii)VJjunction nodes that are not in VPor

VQand where pipelines are connected.

For the sake of convenience, when introducing the network

model, we use the superscripts (e) and (ν) to denote the asso-

ciated quantity/function of the pipeline e∈ E and node ν∈ V,

respectively.

The simpliﬁed hydraulic model of the pipeline is considered

which governs the transient ﬂow within the pipeline [6, 11].

Assume that pipelines are horizontal. For a segment of pipeline

e∈ E of the length L(e), the transients can be modeled by partial

diﬀerential equations (PDEs) (1). All the physical quantities

and the units are deﬁned in Table 1. Without loss of generality,

we choose the convention that the gas ﬂows from x=0 to

x=L(e).











∂tp(e)+v2

S(e)∂xq(e)=0

∂tq(e)+S(e)∂xp(e)+λ(e)v2q(e)|q(e)|

2d(e)S(e)p(e)=0(1)

The gas transients are also determined by the given initial

value and the constraints imposed at the nodes. The initial value

of the pipeline e∈ E is given at t=0 for the any location

Table 1: Physical Quantities of Gas Pipeline

Physical quantity Units

Pressure, pPa

Mass ﬂow, qkg/s

Sound speed, νm/s

Cross-section area of pipeline, Sm2

Diameter of pipeline, dm

Friction factor, λ-

Constant temperature, T0K

Speciﬁc gas constant, RJ/(kg ·K)

x∈[0,L(e)], as shown in (2) where P(e)

ini and Q(e)

ini deﬁnes the

initial value.











p(e)(x,0) =P(e)

ini (x)

q(e)(x,0) =Q(e)

ini (x)(2)

The constraints imposed at a node ν∈ V should consider

both the speciﬁc node and also the associated pipelines. For

a pipeline e∈ E whose inlet is connected to a supplying node

ν∈ VP, the controlled pfollows a Dirichlet boundary condition

as in (3), where P(ν)

Bdeﬁnes the boundary condition.

p(e)(0,t)=P(ν)

B(t) (3)

If the outlet of the pipeline e∈ E is connected to a demanding

node ν∈ VQ, the controlled qfollows a Dirichlet boundary

condition as in (4), where Q(ν)

Bdeﬁnes the boundary condition.

q(e)(L(e),t)=Q(ν)

B(t) (4)

If the pipelines ein,1,ein,2, ... have their inlet connected to a

junction node v∈ VJand the pipeline eout,1,eout,2, ... have their

outlet connected to the same junction node, (5) must be satis-

ﬁed, i.e. at the junction node, the pressure of all the pipelines

should be the same, and the mass ﬂow should be balanced. Q(ν)

determines the amount of gas extracted out of the network at

node v.











p(ein,1)(0,t)=p(ein,i)(0,t),i,1

p(ein,1)(0,t)=p(eout,j)(L(eout,j),t)

Pjq(eout,j)(L(eout,j),t)−Piq(ein,i)(0,t)=Q(ν)

J(t)

(5)

2.2. Grid selection and normalized BVP

The boundary value problem of natural gas network is to

identify the solution of pressure p(e)(x,t) and mass ﬂow q(e)(x,t)

regarding the temporal variable tand spatial variable x, by con-

sidering the constraints below:

•Transient ﬂow in the pipeline governed by the PDEs, (1).

•Given initial values, (2).

•Controlled pat the supplying nodes VP, (3).

•Controlled qat the demanding nodes VQ, (4).

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HighlightsSimulationofTransientsinNaturalGasNetworksviaASemi-analyticalSolutionApproachXinXu,RuiYao,KaiSun,FengQiuˆResearchhighlight1:Wepresentasemi-analyticalsolu-tionapproachforthesimulationoftransientsinnaturalgasnetworks.ˆResearchhighlight2:Tofurtherreducethecomputationburden,thenonlineartermsin...

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Highlights Simulation of Transients in Natural Gas Networks via A Semi-analytical Solution Approach Xin Xu Rui Yao Kai Sun Feng Qiu

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