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The gas transmission problem when the
merchant and the transport functions are
disconnected.
Bouchra Bakhouya,
Ieseg, Universit´e Catholique de Lille,
3 rue de la digue, 59.800 Lille, France,
e-mail: b.bakhouya@ieseg.fr
Daniel De Wolf,
Institut des Mers du Nord HEC Ecole de Gestion de l’ULG
Universit´eduLittoral Cˆote d’Opale Universit´edeLi`ege
49 Place d de Gaulle, BP 5529, Boulevard du Rectorat 7 (B31)
59 383 Dunkerque Cedex 1, France 4000 Li`ege, Belgium
e-mail: daniel.dewolf@univ-littoral.fr e-mail: daniel.dewolf@ulg.ac.be
January 8th, 2007
Abstract
In [7], De Wolf and Smeers consider the problem of the gas distri-
bution through a network of pipelines. The problem was formulated as
acost minimization subject to nonlinear flow-pressure relations, ma-
terial balances and pressure bounds. The practical application was the
Belgian gas transmission network.
This model does not reflect any more the current situation on the
gas market. Today, the transportation function and gas buying func-
tion are separated. This work considers the new situation for the trans-
portation company. The objective for the transportation company is to
determine the flows in the network that minimize the energy used for
1
the gas transport. This corresponds to the minimization of the power
used in the compressors.
1Introduction
In De Wolf and Smeers [7], the objective for the transportation company
is to satisfy the demand at several points of the network by buying gas at
minimal prize. Today, the transportation function and gas buying function
are separated. For example, on the Belgian gas market, the transport is
devoted to Fluxys company. On the other side, several actors are in charge
of gas supplying (such as Distrigaz, the former integrated company).
In order to reflect this new situation, a modelization of the compressors
is introduced in the model of De Wolf and Smeers. The introduction of the
compressor’s modelization change the nature of the optimization problem.
In fact, the problem considered by De Wolf and Smeers [7] was a non convex
but separable non linear problem. This problem was solved using successive
piecewise linear approximations of the problem.
In the present case, the relation between the pressure increase and the
flow in the compressor is non separable. See, for example, Jean Andr´eetal
[3], Babu et al [5] or Seugwon et al [10].
This non linear non convex problem is solved using a preliminary problem,
namely the problem suggested by Maugis [9], which is a convex problem easy
to solve. We show, on the example of the Belgian gas network that this
auxiliary problem gives a very good starting point to solve the highly non
linear complete problem.
Futures research are devoted to introduce this model in a dimension-
ing model such as [6]. Such a model will consider the trade-off between
the minimization of capital expenditures (as in [2]) and the minimization of
operational expenditures. In other terms, this model could balance any de-
crease in investment of pipelines with an increase of compressor power (and
conversely) regarding the costs.
The paper is organized as follows. The formulation of the problem is
presented in section 2, the solution method being discussed next. Section
4introduces a first test problem based on the Belgian transmission system.
Section 4 provides first results obtained by using the first problem to find a
good starting point. Conclusions terminate the paper.
2
2Formulation of the problem
The formulation of the problem presented in this section applies thus to a
situation where the gas merchant and transmission functions are separated.
The transportation company must decide the gas flow in each pipe and the
level of compression in each compressor to satisfy fixed demands distributed
over different nodes at some minimal guaranteed pressure, the income of gas
being also given. For the supply, we have preserved a flexibility close that
allows to take gas between 90 % and 110 % of the daily nominal quantity.
The network of a gas company consists of several supply points where the
gas is injected into the system, several demand points where gas flows out of
the system and other intermediate nodes where the gas is simply rerouted.
Pipelines or compressors are represented by arcs linking the nodes.
The following mathematical notation is used. The network is defined as
the pair (N,A) where N is the set of nodes and A ⊆ N × N is the set of arcs
connecting these nodes. The network can be represented as in Figure 1.
s
g
s
i
s
k
s
j
g
i
k
j
p
i
p
j
h
l
o
f
ij
d
h
= −s
h
d
l
= −s
l
d
o
= −s
o
Figure 1: Network representation
Since we have preserved the flexibility close, two variables are associated
to each node i of the network: p
i
represents the gas pressure at this node
and s
i
the net gas supply in node i.Apositive s
i
corresponds to a supply of
gas at node i.Anegative s
i
implies a gas demand d
i
= −s
i
at node i.
3
A gas flow f
ij
is associated with each arc (i, j) from i to j. There are two
typesofarcs: passive arcs (whose set is noted A
p
) correspond to pipelines
and active arcs correspond to compressors (whose set is noted A
a
).
The constraints of the model are as follows. At a supply node i, the gas
inflow s
i
must remain within take limitations specified in the contracts. A gas
contract specifies an average daily quantity to be taken by the transmission
company from the producer. Depending on the flexibility of the contract,
the transmission company has the possibility of lifting a quantity ranging
between a lower and an upper fraction (e.g. between 0.85 and 1.15) of the
average contracted quantity. Mathematically:
s
i
≤ s
i
≤ s
i
Atademand node, the gas outflow −s
i
must be greater or equal to d
i
, the
demand at this node.
The gas transmission company cannot take gas at a pressure higher than
the one insured by the supplier at the entry point. Conversely, at each exit
point, the demand must be satisfied at a minimal pressure guaranteed to the
industrial user or to the local distribution company. Mathematically:
p
i
≤ p
i
≤ p
i
The flow conservation equation at node i (see Figure 2 for the flow conser-
vation at a supply node) insures the gas balance at node i. Mathematically:
j|(i,j)∈A
f
ij
=
j|(j,i)∈A
f
ji
+ s
i
s
i
i
f
ij
f
ji
Figure 2: Supply node i
Now, consider the constraints on the arcs. We distinguish between the
passive and active arcs. For a passive arc, the relation between the flow
4
f
ij
in the arc (i, j) and the pressures p
i
and p
j
is of the following form (see
O’Neill and al.[11]):
sign(f
ij
)f
2
ij
= C
2
ij
(p
2
i
− p
2
j
), ∀(i, j) ∈ A
p
where C
ij
is a constant which depends on the length, the diameter and the
absolute rugosity of pipe and on the gas composition. Note that the f
ij
are
unrestricted in sign. If f
ij
< 0, the flow −f
ij
goes from node j to node i.
For an active arc corresponding to a compressor, the following expression
of the power used by the compressor can be found, for example, in Andr´eet
al [3], Babu et al [5] or Seugwon et al [10]):
W
ij
=
1
kη
ad
100
3600
P
0
T
0
T
i
Z
m
Z
0
γ
γ − 1
f
ij
p
j
p
i
γ−1
γ
− 1
(1)
with the following parameters:
P
0
= 1.01325 bar,
T
0
= 273.15 K,
γ = 1.309,
Z
0
=1,
k =0, 95 × 0, 97 × 0, 98,
Z
m
= mean compressibility factor of the gas,
η
ad
= the efficacity factor,
T
i
= the gas temperature at the entry of the compressor.
Note also that in (1), the gas flow is expressed in m
3
per hour, the dissi-
pated power in kW, the pressures p
i
and p
j
are in bar. Using mean values for
theses factors (Z
m
= 0,9, T
i
= 288,15 K, η
ad
= 0.75 for a turbo-compressor,
0.8 for a moto-compressor), the energy dissipated can be written as:
W
ij
= γ
1
f
ij
p
j
p
i
γ
2
− 1
(2)
with γ
1
= 0.167 for a turbo-compressor, 0.157 for a moto-compressor and γ
2
= 0.236.
The power used in the compressor must be lower than the maximal power,
noted
W
ij
:
W
ij
≤ W
ij
5
There is another constraint on the maximal pressure increase ratio :
p
j
p
i
≤ γ
3
The constant γ
3
= 1.6 in most of practical cases.
For active arcs, the direction of the flow is fixed:
f
ij
≥ 0, ∀(i, j) ∈ A
a
The objective function of the gas transmission company is to minimize
the energy used in the compressor. This can be written:
min z = α
(i,j)∈A
a
1
0, 9η
therm
W
ij
(3)
where α is the unitary energy price (in Keuro/kW) and η
therm
is the thermic
efficacity of the compression station.
The gas transmission problem can thus be formulated as follows:
min z(f, s, p, W )=
(i,j)∈A
a
W
ij
s.t.
j|(i,j)∈A
f
ij
=
j|(j,i)∈A
f
ji
+ s
i
∀i ∈ N (4.1)
sign(f
ij
)f
2
ij
= C
2
ij
(p
2
i
− p
2
j
) ∀(i, j) ∈ A
p
(4.2)
W
ij
= γ
1
f
ij
p
j
p
i
γ
2
− 1
∀(i, j) ∈ A
a
(4.3)
s
i
≤ s
i
≤ s
i
∀i ∈ N (4.4)
p
i
≤ p
i
≤ p
i
∀i ∈ N (4.5)
f
ij
≥ 0 ∀(i, j) ∈ A
a
(4.6)
p
j
p
i
≤ 1.6 ∀(i, j) ∈ A
a
(4.7)
W
ij
≤ W
ij
∀(i, j) ∈ A
a
(4.8)
(4)
It can be seen that the problem is separable for equation (4.2) but not
separable for equation (4.3).
6
3 Solution Procedure
We now consider the solution of the problem (4). As already seen in De Wolf
and Smeers [7], the formulation of the gas flow-pressure relation in pipes (4.2)
is clearly nonconvex. The problem without the compressors modelization
(4.3) was separable allowing to solve the problem by successive piecewise
linear approximations of the problem. Now the compressors modelization
(4.3) renders the problem non separable.
Some procedure is thus required in general to tackle the nonconvexity of
the problem, if only to find a local solution. The approach proposed here is
to proceed by successively solving two problems, the first one being expected
to produce a good initial point for the second one.
Convergence in nonlinear programming may indeed crucially depend on
agoodchoice of the starting point and this is especially true when the prob-
lem is nonconvex. Our first problem is obtained by relaxing the pressure
constraints and eliminating all compressors in the full model.
The solution of this problem is conjectured to provide a good starting
point for the second problem which is the complete model with pressure
bounds and compressors. The same procedure was already proposed for
solving the problem of an integrated transmission and merchant gas company
by De Wolf and Smeers [7]. But here, we shall show on examples that the
solution of first problem is a very good starting solution for the complete
problem.Infact, we shall prove that they have similar objective functions
(namely to minimize the energy dissipated in the network to transport the
gas).
3.1 First problem : find a good initial point.
Consider the following convex problem which only accounts for pressure losses
along the pipelines:
min h(f,s)=
(i,j)∈A
|f
ij
|f
2
ij
3C
2
ij
s.t.
j|(i,j)∈A
f
ij
−
j|(j,i)∈A
f
ji
= s
i
∀i ∈ N
s
i
≤ s
i
≤ s
i
∀i ∈ N
(5)
7
Since the problem is strictly convex in the flow variables, its optimal
solution is unique. The first constraints of (5) insure that the solution is also
unique in the supply variables. It is easy to prove the following results for
problem (5):
Proposition 1 The unique optimal solution of the problem (5) satisfies the
nonlinear flow pressure relation (4.2).
Proof: Let π
i
be the dual variable associated to the gas balance constraint
at node i. The Karush-Kuhn-Tucker necessary conditions (See Bertsekas [4,
page 284]) satisfied at the optimum solution of the problem (5) can be written
as:
sign(f
ij
)
f
2
ij
C
2
ij
= π
i
− π
j
∀(i, j) ∈ A
This is exactly the nonlinear flow pressure relation (4.2) if we define:
π
i
= p
2
i
, ∀i ∈ N
As remark in De Wolf and Smeers [7], there is no sign constraint on the
π variables since π
i
is the Lagrange multiplier associated to an equality con-
straint, namely the gas balance equation at node i.Thus directly replacing
π = p
2
i
is not allowed. Let π be the value of the lowest dual variable:
π
= min
i∈N
{π
i
}
Replace now
p
2
i
= π
i
− π.
We obtain exactly the flow pressure relations (4.2). 2
The optimal solution of (5) satisfies thus all the constraints of (4) except
the pressure bounds constraints (4.5) and the compressors modelization (4.3),
(4.7) and (4.8).
It can be shown (See De Wolf and Smeers [8]) that problem (5) has a
physical interpretation, in the sense that its objective function is the me-
chanical energy dissipated per unit of time in the pipes. This implies that
the point obtained by minimizing the mechanical energy dissipated in the
pipes should constitute a good starting point for the complete problem.
Definition 1 The mechanical energy is defined as the energy necessary for
compressing f
ij
from pressure p
i
to pressure p
j
.
8
Extending the work of Maugis for distribution network, we have the fol-
lowing proposition.
Proposition 2 The objective function of problem (5) corresponds up to a
multiplicative constant to the mechanical energy dissipated per unit of time
in the pipes
Proof : At node i, the power W
i
given by a volumetric outflow of Q
i
units
of gas per second at pressure p
i
can be calculated in the following manner.
The total energy released by the gas when changing pressure from p
i
to the
reference pressure p
o
is:
W
i
=
p
i
p
0
Q(p)dp
By using the perfect gas state relation (p
0
Q
0
= pQ), we can write:
W
i
=
p
i
p
0
p
0
Q
0
dp
p
= p
0
Q
0
log(
p
i
p
0
)
Denote the volumetric flow going though arc (i, j) under standard condi-
tions by Q
0
ij
and the pressures at the two ends of the arc by p
i
and p
j
. The
power lost in arc (i, j) can be calculated by:
W
ij
= W
i
− W
j
= Q
0
ij
p
0
log(
p
i
p
j
)
This power loss can be expressed through the head discharge variable H
ij
=
p
2
i
− p
2
j
as
W
ij
= Q
0
ij
p
0
2
log(
p
2
i
p
2
j
)
Introducing the mean of square of pressure p
M
defined by
p
2
M
=
p
2
i
+ p
2
j
2
,
the power discharge can be rewritten as
W
ij
= Q
0
ij
p
0
2
log(
p
2
M
+
H
ij
2
p
2
M
−
H
ij
2
)
= Q
0
ij
p
0
2
log(1 +
H
ij
2p
2
M
) − log(1 −
H
ij
2p
2
M
)
9
Note that since H
ij
is small with respect to p
2
M
,weobtain as a first order
approximation
W
ij
≈ Q
0
ij
p
0
H
ij
p
2
M
= Q
0
ij
p
0
p
2
i
− p
2
j
p
2
M
The power discharge through the whole network is thus (we suppose that p
M
is similar for each arc (i, j) and can be factored out in the following sum):
W ≈
p
0
p
2
M
(i,j)∈A
Q
0
ij
(p
2
i
− p
2
j
)=
p
0
p
2
M
(i,j)∈A
(Q
0
ij
)
3
C
2
ij
sign(Q
0
ij
)
which corresponds, up to a multiplicative constant, to the first term of the
objective of problem (5). We can thus conclude that the function h cor-
responds to the mechanical energy dissipated per unit time in the
network due to the flow of gas in the pipes up to a multiplicative
constant. 2
This proposition was suggested to us by Mr. Zarea, direction de la
recherche of Gaz de France.
Problem (5), being a convex separable problem, can be solved by any
convex non linear optimization method. We have solve this problem using
GAMS/CONOPT. The resolution with GAMS leads to a solution (f
∗
,s
∗
)
with associated Lagrange multipliers π
∗
which are used as starting point for
the complete problem.
3.2 Second problem : the complete problem.
The solution to the first problem satisfies all the constraints of problem (4)
except for the pressure bounds and the compressors modelization equations.
We now return to problem (4) and solve it using also GAMS/CONOPT. And
as we shall see in the following section, the flow variables founded in the first
problem are the (local) optimal values of the complete problem. Thus the
first problem give a very good starting point for the second problem.
This is not a surprizing since the objective function of problem (5) is
the mechanical energy dissipated in all the network since the objective of the
complete problem (4) is to minimize the energy dissipated in the compressors.
The only task for CONOPT for solving the complete problem (4) from the
solution of problem (5) is to restore the bounds on pressure.
10
4 Application to the Belgium gas network
To illustrate the utility of the first problem (5), we apply our suggested
solution procedure to the schematic description of the Belgium gas network
given in De Wolf and Smeers [7] (See figure 3).
Algeriangas
From storage
Norvegian gas
To France
To Luxemburg
Zeebrugge
Dudzele
Brugge
Zomergem
Gent
Antwerpen
Loenhout
Poppel
Lowgas
Highgas
Hasselt
Brussel
Mons
Blaregnies
P
eronnes
Anderlues
Warnand-Dreye
Namur
Wanze
Sinsin
Arlon
Berneau
Li`ege
From storage
Dutchgas
Node
Compressor
Voeren
From storage
Figure 3: Schematic Belgium gas network
Recall that Belgium has no domestic gas resources and imports all its nat-
ural gas from the Netherlands, from Algeria and from Norway. The Belgium
gas transmission network carries two types of gas and is therefore divided
in two parts. The high calorific gas (10 000 kilocalories per cubic meter),
comes from Algeria and Norway. The gas from Algeria comes in LNG form
at the Zeebrugge terminal and the gas from Norway is piped through the
Netherlands and crosses the Belgian border at Voeren (See Figure 3). The
gas coming from the Netherlands is a low calorific gas (8 400 kilocalories per
cubic meter). We consider only to the high calorific network.
11
The reader is referred to De Wolf and Smeers [7] for a more detailed de-
scription of the Belgium network. We summarize all the relevant information
concerning the nodes in Table 1.
node town s
i
s
i
p
i
p
i
10
6
scm 10
6
scm bar bar
1 Zeebrugge 8.870 11.594 0.0 77.0
2 Dudzele 0.0 8.4 0.0 77.0
3 Brugge -∞ -3.918 30.0 80.0
4 Zomergem 0.0 0.0 0.0 80.0
5Loenhout 0.0 4.8 0.0 77.0
6Antwerpen -∞ -4.034 30.0 80.0
7 Gent -∞ -5.256 30.0 80.0
8Voeren 20.344 22.012 50.0 50.0
9 Berneau 0.0 0.0 0.0 66.2
10 Li`ege -∞ -6.365 30.0 66.2
11 Warnand 0.0 0.0 0.0 66.2
12 Namur -∞ -2.120 0.0 66.2
13 Anderlues 0.0 1.2 0.0 66.2
14 P´eronnes 0.0 0.96 0.0 66.2
15 Mons -∞ -6.848 0.0 66.2
16 Blaregnies -∞ -15.616 50.0 66.2
17 Wanze 0.0 0.0 0.0 66.2
18 Sinsin 0.0 0.0 0.0 63.0
19 Arlon -∞ -0.222 0.0 66.2
20 P´etange -∞ -1.919 25.0 66.2
Table 1: Nodes description
For each compressor, the maximal power is given in in Table 2. Note that
Localization W
ij
(kW)
Berneau 20 888
Sinsin 3 356
Table 2: Compressors description
the compressor in Berneau corresponds to a large compression station.
12
For each pipeline in the network, the length, interior diameter and corre-
sponding C
2
ij
are given in Table 3.
arc from to diameter length C
2
ij
[mm] [km]
1 Zeebrugge Dudzele 890.0 4.0 9.07027
2 Zeebrugge Dudzele 890.0 4.0 9.07027
3 Dudzele Brugge 890.0 6.0 6.04685
4 Dudzele Brugge 890.0 6.0 6.04685
5 Brugge Zomergem 890.0 26.0 1.39543
6Loenhout Antwerpen 590.1 43.0 0.100256
7Antwerpen Gent 590.1 29.0 0.148655
8 Gent Zomergem 590.1 19.0 0.226895
9 Zomergem P´eronnes 890.0 55.0 0.659656
10 Voeren Berneau 890.0 5.0 7.25622
11 Voeren Berneau 395.5 5.0 0.108033
12 Berneau Li`ege 890.0 20.0 1.81405
13 Berneau Li`ege 395.5 20.0 0.0270084
14 Li`ege Warnand 890.0 25.0 1.45124
15 Li`ege Warnand 395.5 25.0 0.0216067
16 Warnand Namur 890.0 42.0 0.863836
17 Namur Anderlues 890.0 40.0 0.907027
18 Anderlues P´eronnes 890.0 5.0 7.25622
19 P´eronnes Mons 890.0 10.0 3.62811
20 Mons Blaregnies 890.0 25.0 1.45124
21 Warnand Wanze 395.5 10.5 0.0514445
22 Wanze Sinsin 315.5 26.0 0.00641977
23 Sinsin Arlon 315.5 98.0 0.00170320
24 Arlon P´etange 315.5 6.0 0.0278190
Table 3: Pipe-lines description
4.1 Optimal solution for PLP
The energy used is the compressors founded by GAMS/CONOPT is:
z
∗
=6393.825 kW
13
The optimal flow pattern is given in table 4. Note that we have converted
each couple of two parallel arcs in one equivalent greater pipe (See appendix
A).
Arc from to Flow
(10
6
SCM)
1+2 Zeebrugge Dudzele 11.594
3+4 Dudzele Brugge 19.994
5 Brugge Zomergem 16.076
6Loenhout Antwerpen 4.800
7Antwerpen Gent 0.766
8 Gent Zomergem -4.490
9 Zomergem P´eronnes 11.586
10 +11 Voeren Berneau 19.344
12 +13 Berneau Li`ege 19.344
14+15 Li`ege Warnand 12.979
16 Warnand Namur 10.838
17 Namur Anderlues 8.718
18 Anderlues P´eronnes 9.918
19 P´eronnes Mons 22.464
20 Mons Blaregnies 15.616
21 Warnand Wanze 2.141
22 Wanze Sinsin 2.141
23 Sinsin Arlon 2.141
24 Arlon P´etange 1.919
Table 4: Optimal flows
The corresponding pressure and supply patterns are given in table 5.
Note that in Voeren, there is one unit of gas that is taken and which remains
at ’Voeren. This is possible because there is no price associated to the gas
taken at the source nodes.
The two compressors located at Berneau and Sinsin are in use. Table 6
give the used powers and the compression ratio p
j
/p
i
which must be lower
than 1.6.
14
Node Town Supply Demand Pressure
(10
6
SCM) (10
6
SCM) (Bars)
1 Zeebugge 11.594 56.710
2 Dudzele 8.400 56.678
3 Brugge 3.918 56.532
4 Zomergem 54.869
5Loenhout 4.800 56.174
6Antwerpen 4.034 54.090
7 Gent 5.256 54.053
8Voeren 20.344 1. 50.000
9in Berneau 49.579
9 out Berneau 57.820
10 Li`ege 6.365 56.126
11 Warnand 55.140
12 Namur 2.120 53.893
13 Anderlues 1.200 53.110
14 P´eronnes 0.960 52.982
15 Mons 6.848 51.653
16 Blaregnies 15.616 50.000
17 Wanze 54.326
18 in Sinsin 47.300
18 out Sinsin 58.726
19 Arlon 0.222 27.520
20 P´etange 1.919 25.000
Table 5: Optimal supplies and pressures
Localization W
ij
(kW) W
ij
(kW) p
j
/p
i
Berneau 4 973.991 20 888 1.166
Sinsin 780.452 3 356 1.242
Table 6: Compressors description
15
4.2 Role of the first problem
Consider now the utility by resorting to the first problem. First of all, we
have asked GAMS/CONOPT to solve directly the complete problem (4) from
scratch. Due to the nonconvexity of the problem, CONOPT can not find a
feasible solution to the problem. This already justifies the utility of the first
problem.
But if we consider the solution, in term of flows, of the first problem and
of the complete problem, we can see that the optimal solution of problem
(5) is the optimal solution of problem (4)!
Note that there is some work for CONOPT to solve the complete problem
(4) starting from the solution of the first problem (5). In fact, if we consider
the pressure computed from the dual variables of the first problem, we can
see that they are not feasible (See Table 7). As can be seen, the work of
GAMS/CONOPT is to satisfy the upper bound on pressure variables using
the compressors at a minimum level.
Perhaps this phenomena is due to the arborescent structure of the Belgian
gas network.Wehave try to add arcs on the Belgian gas network to have
a cycle (See figure 4). We have added two arcs allowing to supply the city
of Arlon directly from the entry point Voeren (distance between Voeren and
Arlon is 162 km divided in two equal parts of 81 km with a diameter of 395.5
mm (giving a c
2
ij
= 0.00666873148 for each part). In the middle point, we
have place a compressor with the same maximal power than in Sinsin, namely
3 356 kW. To force the use of this new canalization, we have increased all
the offers and demands by 10% . And the same phenomena appears: the
solution, in term of flows of the first problem is the optimal solution of the
complete problem.
This is not so surprizing if we recall the interpretation of the first problem,
namely the minimization of the energy dissipated in the network.
5 Conclusions
In this paper we have updated the gas transmission model of De Wolf and
Smeers [7] to the new situation in several european countries. Namely the
fact that the merchant and the transportation function are now separated.
The consequence of this new situation is the necessity of precisely modelize
16
Node Town Pressure end Pressure end Maximum
first problem problem (4) pressure
1 Zeebugge 86.664 56.710 77.0
2 Dudzele 86.634 56.678 77.0
3 Brugge 86.500 56.532 80.0
4 Zomergem 84.975 54.869 80.0
5Loenhout 86.171 56.174 77.0
6Antwerpen 84.263 54.090 80.0
7 Gent 84.230 54.053 80.0
8Voeren 88.019 50.000 50.0
9in Berneau 87.686 49.579 66.2
9 out Berneau 87.686 57.820 66.2
10 Li`ege 86.127 56.126 66.2
11 Warnand 85.223 55.140 66.2
12 Namur 84.084 53.893 66.2
13 Anderlues 83.370 53.110 66.2
14 P´eronnes 83.254 52.982 66.2
15 Mons 82.048 51.653 66.2
16 Blaregnies 80.555 50.000 66.2
17 Wanze 84.479 54.326 66.2
18 in Sinsin 78.139 47.300 63.0
18 out Sinsin 78.139 58.726 66.2
19 Arlon 36.505 27.520 66.2
20 P´etange 25.000 25.000 66.2
Table 7: Optimal pressures for first and complete problems
17
Algeriangas
From storage
Norvegian gas
To France
To Luxemburg
Zeebrugge
Dudzele
Brugge
Zomergem
Gent
Antwerpen
Loenhout
Node
Compressor
Mons
Blaregnies
P
eronnes
Anderlues
Warnand-Dreye
Namur
Wanze
Sinsin
Arlon
Berneau
Li`ege
Voeren
From storage
From storage
Figure 4: Schematic Belgium gas network with a cycle
18
the compressors. This is the first objective of the present paper. The second
is to present a procedure to solve this non linear non convex non separable
problem. We have seen, on the example of the Belgium Gas Network that
the preprocessing trough the Maugis problem [9] is very efficient in this case.
Namely, on the presented examples, the solution in term of flow variables
of the preliminary problem is the optimal solution of the gas transmission
problem. We have given a physical interpretation to the objective function
of the Maugis problem, namely the power used in the network due to the
flows in the pipes. This constitutes a justification to this phenomena.
Futures research are devoted to introduce this model in a dimension-
ing model such as [6]. Such a model will consider the trade-off between
the minimization of capital expenditures (as in [2]) and the minimization of
operational expenditures. In other terms, this model could balance any de-
crease in investment of pipelines with an increase of compressor power (and
conversely) regarding the costs.
Acknowledgment
Thanks are due to Mr Jean Andr´e, of “Direction de la Recherche de Gaz
de France” which provided most of the funding for the modelization of the
compressors. Our gratitude also extends to Mr Zarea of Gaz de France for
helping us to find a physical interpretation to the first problem.
19
AFormula for two parallel pipes-lines
Consider two parallel pipes-lines (indiced by a and b)between nodes i and j.
We note respectively f
a
ij
and f
b
ij
the flow in the two parallel arcs. The (4.2)
formula can be written:
sign(f
a
ij
)(f
a
ij
)
2
=(C
a
ij
)
2
(p
2
i
− p
2
j
)
sign(f
b
ij
)(f
b
ij
)
2
=(C
b
ij
)
2
(p
2
i
− p
2
j
)
Since, at the optimum, the flows in the two parallel arcs must be in the
same direction, we can choose the direction of arc (i, j)sothat the two sign
functions are equal to one:
(f
a
ij
)
2
=(C
a
ij
)
2
(p
2
i
− p
2
j
)
(f
b
ij
)
2
=(C
b
ij
)
2
(p
2
i
− p
2
j
)
If we replace the two parallel arcs by one equivalent greater diameter with
corresponding parameter C
ij
, the corresponding flow f
ij
is the sum of the
two flows:
f
ij
= f
a
ij
+ f
b
ij
We compute now the relation between, on one side, C
ij
and, on the other
side, C
a
ij
and C
b
ij
:
f
2
ij
=(f
a
ij
+ f
b
ij
)
2
=(f
a
ij
)
2
+2f
a
ij
f
b
ij
+(f
b
ij
)
2
=(C
a
ij
)
2
(p
2
i
− p
2
j
)+2C
a
ij
(p
2
i
− p
2
j
)C
b
ij
(p
2
i
− p
2
j
)+(C
b
ij
)
2
(p
2
i
− p
2
j
)
=[(C
a
ij
)
2
+2C
a
ij
C
b
ij
+(C
b
ij
)
2
](p
2
i
− p
2
j
)
The (4.2) formula can be written:
f
2
ij
= C
2
ij
[p
2
i
− p
2
j
]
By identification of the two last equations we deduced the desired formula:
C
2
ij
=(C
a
ij
)
2
+2C
a
ij
C
b
ij
+(C
b
ij
)
2
(6)
20
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22
