Research Journal of Applied Sciences, Engineering and Technology 3(2): 67-73, 2011
ISSN: 2040-7467
© Maxwell Scientific Organization, 2011
Received: September 22, 2010
Accepted: January 20, 2011
Published: February 20, 2011
Optimal Pipeline Connection for the West African Gas Pipeline Project
E.K. Donkoh, S.K. Amponsah and K.F. Darkwah
Department of Mathematics, Kwame Nkrumah University of Science
and Technology, Kumasi, Ghana
Abstract: Ghana and three other West African countries including Benin, Togo and Nigeria have installed
pipelines to establish the flow of natural gas. In this research paper, we combine Prim’s and Steiner Tree
algorithms with factor rating method to solve the single source shortest path offshore/onshore pipeline problem.
Data on the West African Gas Pipeline (WAGP) project was collected and analyzed. We used Prim’s algorithm
to find the minimum spanning tree of length 712.30 km. This is a reduction over the original 788.90 km WAGP
project design. Factor rating method was then used to find an alternative path of length 723.29 km. Steiner Tree
algorithm and geometry were used to obtain an optimal pipeline length of 707.75 km. This is 10.3% reduction
of the WAGP length. Our solution is shown to be topologically equivalent to the WAGP network and hence
optimal in pipeline distance and project cost.
Key words: Factor rating, graphs, networks, prim’s algorithm, steiner tree algorithm, trees
gas pipelines to establish the flow of natural gas to the
respective countries (WAGPCo, 2004a). The total
offshore distance is put at 678 km (WAGPCo, 2010)
while the onshore/offshore distance determined from grid
coordinates is 788.90 km (Nuamah, Accra, Ghana, Grid
coordinates data of WAGP in AutoCad). This research
study determines the optimal pipeline distance from the
single source, Alagbado ‘Tee’, Nigeria, to the final
destination, Aboadzi site of Takoradi, Ghana.
INTRODUCTION
Liquefied natural gas can be transported by means of
truck or ship. Due to its high cost of production and being
highly inflammable, it must be shipped at a very low
temperature. This makes shipping mode of transportation
prohibitively expensive. Thus the most efficient and
effective movement of large quantities of natural gas from
producing regions to consumption regions is through
pipeline transportation system especially for long
distance transportation (Wikipedia, 2008). Gas pipeline
transportation is unique in the transportation industry due
to its mode captivity, continuous transport system, time
of transport and as a commodity (Dott, 1997). Lurie
(2008) reviewed the mathematical models of fluid and gas
flow in the interior of a pipeline and discussed issues of
non-stationary gas flow in a pipeline and non-isothermal
gas flow in gas-pipelines. Hochbaum and Segev (1989)
observed that sending flow along a pipeline segment
involves fixed cost for using the segment and variable
cost for a unit of flow. The authors recommended the use
of Lagrange multiplier method in finding the minimum
cost of pipeline segments along which flow will be
directed in a network.
Li et al. (2008) established a topology optimization
technology for flow networks that finds the least cost
network topology while the seismic reliability between
the sources and each terminal satisfies prescribed
reliability constraints. The model is applied to a large city
in China and the solution is based on genetic algorithm.
Ghana and three other West African countries
including Benin, Togo and Nigeria have installed natural
LITERATURE REVIEW
The Minimum Spanning Tree (MST) problem is one
of the most typical and well known problems in
combinatorial optimization. Borçvka (1926) (cited in
Graham and Hell, 1985), used euclidean MST to find the
most economical construction of an electricity network in
Moravia. The author’s MST is applied to a network
with distinct edge distances (Wikipedia, 2010).
Schrijver (1996) cited Boruvka as the first to consider the
MST. The MST algorithms that are commonly used are
the Prim’s (1957) and Kruskal’s (1956) algorithms (cited
in Jayawant and Glavin, 2009). However, Kruskal
algorithm finds the minimum spanning forest if the
network is not connected (Agarwal, 2010).
Zachariasen (1998) noted that all known exact
algorithms for the Euclidean Steiner tree problem require
exponential time. The general consensus is to use
heuristics and approximation algorithms. However, one of
the first and easiest methods involves the use of minimal
spanning trees as approximation to the Steiner tree
algorithm. The Euclidean Steiner tree problem has its
Corresponding Author: E.K. Donkoh, Department of Mathematics, Kwame Nkrumah University of Science and Technology,
Kumasi, Ghana
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Res. J. Appl. Sci. Eng. Technol., 3(2): 67-73, 2011
Table 1: Grid and WGS-84 coordinates of the WAGP nodes
WAGP node points
---------------------------------------------------------------------------------------------------------------------------------------------WGS-84
WAGP grid co-ordinates
-------------------------------------------------------------------------------------------------------------------Node
Location
Latitude
Longitude
Easterns (Xm)
Northerns (Ym)
F
Takoradi R & M Station
4º58'30.846'N
1º39.39.422'W
241,893.442
550,381.671
A
Takoradi Subsea Tie-in
4º53'00.352'N
1º35'43.400"E
259,133.210
540,298.000
B
Tema offshore subsea Tie-in
5º32'33.322"N
0º07' 19.628"E
439,686.000
612,669.500
T
Tema R & M Station
5º40'04.378"N
0º02'17.112"E
440,493.093
626,529.525
C
Lome offshore subsea Tie-in
6º00'18.199"N
1º23'10.506"E
579.637.380
663,815.721
L
Lome R & M Station
6º09'20.808"N
1º18'16.843"E
570,289.540
680,467.020
Y
Lome C.E.B Power Plant
6º09'34.803"E
1º18'04.947E
570,223.429
680,896.350
D
Cotonou offshore subsea tie-in
6º03'17.694N
2º18'54.811"E
680,394.773
687,983.208
K
Cotonou R & M Station
6º23'12.938"N
2º15'26.839"E
675,945.754
706,248.423
M
C.E.B Facility(Maria Gleta)
6º25'33.060"N
2º18'18.828"E
681,217.340
710,569.390
E
Lagos Beach Comp. Station
6º25'02.210"N
2º59'18.213"E
756,810.938
709,913.811
V
Midline Isolation Valve
6º30'31.073"N
3º01'08.775"E
760,163.131
720,0.35.976
G
Alagbado Tee (Itoki)
6º43'24.000"N
3º56'53.400"E
862,848.602
744,382.001
In this research, we combined factor rating method
with Prim’s and Steiner tree algorithms and geometry to
obtain an optimal pipeline distance for the West African
Gas pipeline.
roots in Fermat problem whereby one finds in the plane a
point, the sum of whose distances from three given points
is minimal (Ivanov and Tuzhilin, 1994). The steiner ratio
compares the solution of the Steiner Minimum Tree
(SMT) problem to that of the MST. Moore in (Gilbert and
Pollack, 1968) put the lower bound of the Steiner ratio to
be 0.5 while Gilbert and Pollack (1968) conjectured the
upper bound to be 0.866. This means that the SMT can be
shorter than the MST by at most 13.4% (Bern and
Graham, 1989).
Dott (1997) reduced the 952 km Palliser pipeline
network to 832 km using the haMSTer program, which is
based on Prim algorithm. This was a 13% reduction over
the original length. The author conjectured that if the
SMT is used in addition to the haMSTer then 721 km will
be realized.
Brimberg et al. (2003), presented the optimal design
of an oil pipeline network for the South Gabon oil field.
The original design covered thirty-three (33) nodes with
hundred and twenty-nine (129) possible arcs having total
distance of 191.1 km. Using a variation of Prim’s
algorithm, this reduced the connection to 121.6 km, which
was a reduction of 36.4% of the total distance to be
covered.
Dott (1997) obtained an Optimal Design of Natural
Gas Pipeline of Amoco East Crossfield Gas pipeline
project, (Alberta, Canada). The pipeline, which covers a
distance of 66 km was reduced to 48.9 km with the use of
the haMSTer program software. Steiner tree algorithm
was later used to reduce the MST created by the haMSTer
to 48.84 km. This was 1% reduction over the MST.
Arogundade and Akinwale (2009) used Prim’s
algorithm to find the shortest distance between 88 villages
connected by 96 roads of Odeda local governement map,
Nigeria, and arrived at an MST of 388,270 m.
Nie et al. (2000) combined an algorithm of rectilinear
Steiner tree and the constraints and connectivity reliability
of road network to obtain a new method of rural road
network layout designing in a county area. The method is
applied to the rural road network layout designing of
Shayang County, China.
Data: WGS 84 AutoCad map (of the pipeline route and
the grid coordinates) was obtained from the West African
Gas Pipeline Company (WAGPCo) (P. Nnamah, Accra,
Ghana. Grid coordinates data of WAGP in AutoCad ).
Figure 1 shows, in AutoCAd map, the thirteen (13)
labeled WAGP nodes consisting of nine (9) R & M
stations and four (4) subsea tie-ins, in addition to four (4)
shore crossing points (which are not labeled) and pipeline
segments.
Table 1 shows the grid and the WGS-84 coordinates
of the pipeline nodes. The location listing is movement
from left to right along the pipeline shown in Fig. 1 and
from Takoradi R & M station to Alagbado Tee.
Table 2 depicts the internodal distances and pipeline
diameters of the pipeline segments. Column 1 shows that
the pipeline consists of offshore and onshore segments as
well as offshore and onshore laterals which respectively
are segments from offshore and onshore to shore
crossings. Link lines are auxiliary onshore segments.
Columns 2 and 3 show the end-point locations of each
pipeline segment. The total length of the pipeline is
788.90 km.
From Table 2, the maximum diameter of onshore
pipeline segment is 76.20 cm (7.62x10G4 km) and that of
offshore is 50.80 cm (5.08x10G4 km). The respective per
length unit curved surface areas of pipeline are
2.3939x10G3 and 1.5959x10G3 /km.
Table 3 is the WAGP grid coordinates of the nodes in
Table 1 converted to kilometers. Node G is the connection
source representing the gas distribution source.
COMPUTATIONAL EXPERIENCE
The distances between all pairs of nodes were
computed using the Euclidean distance formula. The
results are shown in Table 4. Prim’s algorithm was then
used to find the internodal distances that form a minimum
68
Res. J. Appl. Sci. Eng. Technol., 3(2): 67-73, 2011
Table 2: Pipeline nodes, length of segments and pipeline diameters of WAGP
Segment type
From
To
Onshore mainline (Nigeria)
Alagbado ‘Tee’ (G)
Midline Isolation Valve (V)
Midline Isolation Valve (V)
Lagos Beach Compr. Station (E)
Offshore mainline
Lagos Beach Compr. Station(E)
Cotonou Offshore Subsea Tie-in (D)
Subtotal
Offshore laterals
Subtotal
Offshore total
Onshore laterals
Subtotal
Link lines
Length (km)
115.32
12.18
86.93
Cotonou Offshore Subsea Tie-in (D)
Lome` Offshore Subsea Tie-in (C)
106.60
Lome` Offshore Subsea Tie-in (C)
Tema Offshore Subsea Tie-in (B)
153.40
Tema Offshore Subsea Tie-in (B)
Takoradi Subsea Tie-in (A)
Cotonou Offshore Subsea Tie-in(D)
Lome` Offshore Subsea Tie-in(C)
Cotonou Shore Crossing (D!)
Lome R & M Station (L)
221.00
695.43
15.06
19.80
Tema Offshore Subsea Tie-in (B)
Takoradi Subsea Tie-in(A)
Tema Shore Crossing (B!)
Takoradi Shore Crossing (A!)
16.48
19.06
Cotonou Shore Crossing(D!)
Lome Shore Crossing(C!)
Cotonou R & M Station (K)
Lome R & M Station (L)
70.40
765.83
5.10
0.15
Tema Shore Crossing(B!)
Takoradi Shore Crossing(A!)
Tema R & M Station (T)
Takoradi R & M Station (F)
0.52
0.90
Cotonou R & M Station (K)
Site of Future CEB Facility
at Maria Gleta (M)
CEB Power Plant (Y)
Lom R & M Station (L)
Link lines total
Grand total
The shore crossing points are indicated with (! )
6.67
9.50
6.90
Diameter
30 in (76.2 cm)
30 in (76.2 cm)
20 in (50.8 cm) base case or
18 in (45.7 cm)
20 in (50.8 cm) base case or
18 in (45.7 cm)
20 in (50.8 cm) base case or
18 in (45.7 cm)
20 in (50.8 cm)
8 in (20.3 cm)
10 in (25.4 cm) base case or
18 in (45.7 cm)
18 in (45.7 cm)
20 in (50.8 cm) base case or
18 in (45.7 cm)
8 in (20.3 cm)
10 in (25.4 cm) base case or
18 in (45.7 cm)
18 in (45.7 cm)
20 in (50.8 cm) base case or
18 in (45.7 cm)
10 in (25.4 cm) base case or
18 in (20.3)
8 in (20.3 cm) base case or
10 in (25.4 cm)
16.40
788.90
Fig. 1: WGS 84 AutoCAD map of the WAGP, R & M stations are represented by
points by
spanning tree. The internodal distances of the MST are
shown Table 4 as bold and in brackets.
Figure 2 depicts the graphical representation of the
MST in matlab figure window. The total length of the
MST was obtained to be 712.3 km. Segments AB, CD are
offshore. Segments TL, KM, ME, EV and VG are
onshore.
From Fig. 2 segment BC (149.02 km) is alternative to
segment TL (140.65 km). Similarly, segment DE (97.50
km) is alternative to ME (75.59 km). WAGPCo (2004b)
reports on factors that impact offshore and onshore
subsea tie-ins by
and shore crossing
pipeline segments. We used cost-benefit analysis to
compare offshore and onshore pipeline segments. This
comparison is relevant when an onshore pipeline segment
in the MST had an alternative. Table 5 shows the relative
impact of these factors.
These factors are evaluated for offshore and onshore
segments using factor rating method as a tool for locating
alternatives to the MST pipeline segments TL and ME.
These factors have been assigned weights from 0 to 1 and
their weighted score calculated. The “halo effect” in
rating scales has been considered when scores were being
69
Res. J. Appl. Sci. Eng. Technol., 3(2): 67-73, 2011
750
700
y(km)
V
KM
Table 3: Grid coordinates of nodes of the WAGP
Node
x(km)
F
241.89
A
259.13
B
439.67
T
440.39
C
579.64
L
570.29
Y
570.22
D
680.39
K
675.95
M
681.22
E
756.81
V
760.16
G
862.85
G
E
D
650
C
T
B
600
F
550
A
500
200
300
500
x(km)
400
600
800
700
900
Fig. 2: Calculated MST representation of West African ga
pipeline network
750
WAGP
<120°
650
G
E
B
600
550
assigned to these factors (Gay, 1999). From Table 6, the
rated pipeline segment for the offshore is 7.21 and is less
than that of the onshore which is 7.35. The offshore
segment has the least negative impact and hence is the
preferred path.
From the pipeline rating and the fact that BC and DE
are respective alternatives to TL and ME, the MST is
adjusted with such replacements leading to adjusted MST
total of 723.29 km. Figure 3 is a matlab plot of the
adjusted MST of the WAGP with BC replacing TL and
DE replacing ME.
We modified the adjusted MST using Steiner tree
algorithm. Every T-junction and L-junction with angle
less than 120º was converted to a Steiner point by using
Simpson’s method (Herring, 2004). Converting the Tjunctions or L-junctions to Steiner points we obtained the
Steiner Spanning Tree combined with the adjusted MST
points as shown in Fig. 4.
The total length of the Steiner Spanning Tree is
738.19 km. The geographical and the grid coordinates of
the Steiner points are shown in Table 7.
The offshore nodes A, B, C and D are not demand
centres nor production points and so they are removed to
allow for direct flow from S4 toS3 ,S3 to S2 andS2 to S1 .
The geometric property of direct edges being shorter than
sum of two edges in a triangle is used (Stewart, 2008).
Figure 5 depicts the resulting matlab plot.
D
C
T
V
M
K
YL
700
F
A
500
200
300
400
500
600
x(km)
700
800
900
Fig. 3: Adjusted MST of WAGP, arrow head points to angle less
than 120º
750
WAGP
Y
y(km)
700
650
T S2
B
600
550
F
L
S3
C
K
G
V
M
S4 E
D
S1
A
500
200
300
600
500
x(km)
400
700
800
900
Fig. 4: Steiner Spanning Tree for the WAGP are the Steiner
points and are offshore
750
WAGP
M
y(km)
700
Y
650
500
200
L
S2
S4
G
V
Topological equivalence: Two systems are topologically
equivalent if there exist homeomorphic transformation
from one system to the other (Clooman, 1989). Our SMT
of Fig. 5 and WAGP tree system of Fig. 1 are
topologically equivalent when the nodes A, B, C and D of
Fig. 1 are, respectively replaced by S1,S2,S3, and S4 of
Fig. 5. The nodes of the two systems are topologically
equivalent when the adjacency list of the two systems is
matched. This is illustrated in columns 1 and 6 of
Table 8. The nodes in columns 2 and 7 are adjacent nodes.
Table 8 shows that the two systems are diagraphs and thus
satisfy the indegree and outdegree theorem (Lipschutz and
Lipson, 2007), and for each system, the sum of indegree
E
T
S2
600
550
K
F
S1
300
400
500
600
x(km)
700
800
y(km)
550.38
540.30
612.67
626.53
663.82
680.47
680.90
687.98
706.25
710.57
709.91
720.04
744.38
900
Fig. 5: Steiner Minimum Tree (SMT) of a total length 707.75
km accounting for minimum cost and distance
70
Res. J. Appl. Sci. Eng. Technol., 3(2): 67-73, 2011
Table 4: Matrix of internodal distances showing edges of the MST
Nodee F
A
B
T
C
L
F
4
19.97
207.38
203.30
356.29
353.23
4
194.50
200.73
341.49
341.27
A
19.97
B
207.38 194.50 4
16.68
149.02
147.23
4
144.16
140.65
T
203.30 20073 16.6
C
356.29 341.49 149.02
144.16
4
19.10
L
353.23 341.27 147.23
140.65
19.10
4
Y
353.32 341.39 147.30
140.75
19.51
0.44
D
459.58 146.40 252.22
247.74
103.61
110.36
K
461.20 448.64 254.14
248.68
105.24
108.76
M
467.62 455.14 260.64
255.07
111.82
114.944
E
539.07 525.79 331.71
327.22
183.07
188.83
V
545.33 532.29 337.99
333.16
189.07
193.93
G
650.566 637.28 443.20
438.59
294.44
299.46
Y
353.32
341.39
147.30
140.75
19.511
0.44
4
110.40
108.73
114.90
188.831
193.93
299.44
D
459.58
146.40
252.22
247.74
103.61
110.36
110.40
4
18.81
22.61
79.50
85.97
190.98
Table 5: Comparative advantage of potential impact on both offshore and onshore segments
Factors
Offshore
Environmental
Fast laying process with little disturbances.
Crosses less sensitive areas.
Technological
Higher pressure to delivery points.
Greater outputs.
Security
Less risks due to difficult accessibility.
Lower presence of humans in the proximity.
Economical
Higher investments
Lower operational costs.
No fuel and gas needed
WAGPCo (2004b)
Table 6: Ratings of the factors which impact on offshore and onshore pipeline segments
Factor score
----------------------------------------------Factor
Offshore
Onshore
Laying process with little disturbances
10
6
Crosses sensitive areas.
7
11
Pressure to delivery points
9
6
Greater outputs of gas.
6
5
Risks due to accessibility
2
7
Presence of humans in the proximity
5
8
Investments
9
11
Operational costs
5
7
Repair and maintenance
7
4
Total
Table 7: Grid and geographical coordinates of S1 , S2 , S3 and S4
Grid coordinates (x , y)
Latitude
S1 (249 750.1799, 543 774.4451)
4º 54' 56.7178" N
S2 (440 658.9333, 617 432.8707)
5º 35' 08.4733" N
S3 (581 177.6222, 668 291.0011)
6º 02' 43.8061" N
S4 (685 068.6184, 695 779.3449)
6º 17' 31.2037" N
Table 8: WAGP tree table of indegree and outdegree vertex with edges
Actual design of WAGP
-----------------------------------------------------------------------------------Vertex
Adjacencysts Outdegree
Indegree
Edges
F
0
1
FA
A
F
1
1
AB
B
T, A
2
1
BT
T
0
1
BC
C
B, L
2
1
CL
L
Y
Y
1
1
Y
0
1
CD
D
C, K
2
1
DK
K
M
1
1
KM
M
0
1
DE
E
D
1
1
EV
V
E
1
1
VG
G
V
1
0
Sum
12
12
12
K
461.20
448.64
254.14
248.68
105.24
108.76
108.73
18.81
4
6.81
80.94
85.33
190.75
M
467.62
455.14
260.64
255.07
111.82
114.94
114.90
22.61
6.81
4
75.59
79.51
184.75
E
539.07
525.79
331.71
327.22
183.07
188.83
188.83
79.50
80.94
75.59
4
10.67
111.50
V
545.33
532.29
337.99
333.16
189.07
193.95
193.93
85.97
85.33
79.51
10.67
4
105.54
G
650.56
637.28
443.20
438.59
294.444
299.46
299.44
190.98
190.75
184.75
111.50
105.54
4
Onshore
Slow laying process with much disturbances
Crosses much sensitive areas.
Lower pressure to delivery points
Lower outputs
Higher risks due to easy access
Easy repair and maintenance.
Higher investments
Higher operational costs
Fuel and gas needed.
Weighted score
--------------------------------------------------------------------Rating ratio
Offshore
Onshore
0.18
1.80
1.08
0.12
0.84
1.32
0.15
1.35
0.90
0.07
0.42
0.35
0.08
0.16
0.56
0.10
0.50
0.80
0.12
1.08
1.32
0.10
0.50
0.70
0.08
0.56
0.32
1.0
7.21
7.35
Longitude
1º 35' 23.760.6" W
0º 07' 51.1102" E
1º 24' 00.8006" E
2º 20' 22.5955" E
Authors SMT design of the WAGP network
-------------------------------------------------------------------------------------Vertex
Adjacency lists Outdegree
Indegree
Edges
F
0
1
FS
S1
F
1
1
S 1 S2
S2
T, S1
2
1
S2 T
T
0
1
S3 S 2
S3
S2,L
2
1
LS3
LY
L
Y
1
YL
Y
0
1
S 3 S4
S4
S3,K
2
1
S3 K
K
M
1
1
KM
M
0
1
S4 E
E
S4
1
1
EV
V
E
1
1
VG
G
V
1
Sum
12
12
12
71
Res. J. Appl. Sci. Eng. Technol., 3(2): 67-73, 2011
addition both pipeline systems have the same number of
edges, indegreees and out degrees and hence can be
compared in terms of distance and cost.
is 12 and the sum of outdegree is also 12. Hence the two
systems are different but represent the same information.
DISCUSSION
ACKNOWLEDGEMENT
Prim’s algorithm for MST solution of the WAGP
gathering system resulted in a pipeline distance of 712.30
km, which is a reduction over the WAGP total distance of
788.90 km.
We introduced cost-benefit analysis by using factor
rating to find an alternative cheaper path. This resulted in
onshore segment TL and ME being replaced by BC and
DE, respectively producing an adjusted MST pipeline
distance of 723.29 km. Using the per length unit pipeline
curved surface areas, the curved areas for TL (0.33670
km2) and ME (0.18095 km2) are respectively higher than
BC (0.23825 km2) and DE (0.15560 km2). Hence BC and
DE have least material cost. This confirms the cost aspect
of the factor rating method.
Steiner tree algorithm was used on the adjusted MST
to obtain Steiner points S1, S2, S3 an S4, which gave
Steiner spanning tree of total length 738.19 km when the
MST points are added. Following Bern and
Graham (1989), nodes A, B, C and D were replaced by S1,
S2, S3 and S4, respectively. The geometric property of
direct edges being shorter than sum of two edges in a
triangle is used (Stewart, 2008). This resulted in an SMT
with shorter distance of 707.75 km.
The Steiner ratio of the SMT to the MST is 0.9936.
That of the SMT to the adjusted MST is 0.9785. These
values could not be used to judge Gilbert and
Pollack (1968) conjectured upper bound of 0.866 since
our Steiner points were obtained from the adjusted MST.
However, the optimal distance of 707.75 km is 10.3%
reduction over the WAGP distance and from
Clooman (1989), the topological equivalence achieved
means our solution is both of shorter distance and the
least cost.
The authors would like to acknowledge the onshore
engineer Mr. Rubson Mawunyo, the Chief Lead Surveyor
Mr. Paul Nnamah, formerly of the WAGP company for
providing the data for this paper and Prof. I.A Adetunde
of University of Mines and Technology, Tarkwa, Ghana
for his valuable suggestions and seeing through the
publication of this study.
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