9511

Optimization of Pipe Sizes for Distribution Gas Network

Design

Andrzej J. Osiadacz Warsaw University of Technology

and

Marcin Gbrecki Regional Gas Dispatching Center, Warsaw

1. Introduction

Historically, most interest in pipe networks has focused on the development of efficient algorithms

for the analysis of flow, and there are now very useful and efficient computer packages available

for simulation of existing and proposed new schemes. SimNet [6], for example, is widely used in

Poland for analysis and simulation of gas distribution systems.

In contrast, there has been comparatively little research interest into the development of

methodologies aimed at optimising the design of pipe networks. There are very few computer

packages available for commercial use that help the designer to produce truly optimal pipe

network designs. The current available methods for designing networks can be divided into three

groups:

- heuristic methods

- methods which assume a continuous range of diameters is available

- discrete optimisation methods

2. Heuristic methods

Although, these methods have developed through the years from an appreciation of what usually

constitutes a good and economic design, there is no guarantee that the designs produced will in

any sense be optimal. Even for a simple tree-like network there are a very large number of

possible designs that provide feasible solutions. It can be seen that the chances of hitting on the

best solution are very small.

In PILOT method [lo] the pipes are ordered for the tree definition according to the shortest

distance from the source. The program then proceeds in the usual way up to the point of balancing

the flows. The pipe resistances are evaluated, then recalculated using flows obtained by Hardy-

Cross loop balancing method and corresponding diameters are selected. Another Hardy-Cross to

balance the network is performed determining flows and pressures through the system. The

process of scaling resistances and performing Hardy-Cross is repeated until none of the pressures

falls below the minimum design pressure. Once the basic solution has been found the program

aims at further economic improvements.

In FP6 method [9], after all pipes have been set to minimum diameter the loops are defined and

initial flows assigned Hardy-Cross balancing flows. Pipes are then upgraded on maximum

-l-

flowpath from source outwards the criterion being the cost benefit. Several flowpaths are

upgraded simultaneously.

Areas of influence are assigned to each governor in CONGAS method [lo].

The tree system is designed for each area with minimum pressure at ends and even pressure drop

loop closing pipes are sized to a minimum and Hardy-Cross flow balancing performed. The

theoretical diameters are substituted by actual diameters and network is rebalanced.

3. Continuous methods

One way of solving the design problem is to initially suppose that any size of diameter is possible.

Doing this allows continuous optimization methods to be employed. The result of such

optimization will be a set of diameters which needs to be corrected to available diameter sizes.

SUMT is an algorithm for sequential unconstrained minimization technique and is a method

devised and programmed by G. Boyne in [2].

In this method the cost is assumed to be given by a function of the form :

C(Q)=c(a+bDp)L (1)

where:

D- is the vector of pipe diameters

L - is the pipe length

a, b, p - are cost parameters

The program minimizes a sequence of functions, where the minimum of one is used as the starting

point in the research for the minimum of the next. The functions to be minimized are:

F, = c(a+bD”)L+R,, c ’ nodes (Pi -Pimu)

(2)

where:

Pi, Pimin - are th e pressure and allowed pressure at node i and pi depends on D

R, - is a weight such that R,+I = R, /I 0

The sequential method works as follows.

First the diametersD,, minimising Fo are found. Then D, is used as a starting point in the search __ -

for the minimum of Fl. More generally, D. is used as a starting point in the minimization of Fi+l. 1

-2-

Each of these minimizations is carried out using the conjugate gradient method. The procedure

stops when Di+, is sufficiently close toD, . (Note that R, decreases rapidly with n so the last

solutions are effectively minimising cost).

This process yields a local optimum. The program now randomly generates a new set of diameters

and uses these to begin the whole procedure again and locates another local minimum. This

continues until a “sufficient” number of local optima are found and the cheapest of these is the

desired solution.

In addition this program uses the parametric technique to ,,round” up each local optimum.

Watanatada [l l] has used augmented Lagrangian methods to obtain optimised designs of water

distribution systems. The model he used is as follows:

Di - diameter of the i - th pipe i E [ 1, NP]

pj - pressure atj - th node j E [~,NN]

Qk - flow out of the k - th source k E [l,NS]

&tin i, pmirt i - the corresponding minimum allowed pressures

The flow equation, relating pressure drop and flow is used to eliminate pipe flows as independent

variables.

New variables are now introduced:

D; = Dmin, + X; i=l,NP

Pi = Pmini + ‘Z i=NP+l,NN+NP Qi = X” i=NN+NP+l,NN+NP+NS

- this last equation means that the total flow from each source must be non-negative

The problem can now be formulated as one of minimizing the cost C(X) subject to the total flow

at each node being zero -

T(X)= CQ-d, =0 i=I,NN (3) Jlows at node i

- di is the demand at each node, except for sources where it is the total flow Qi

The method used in [ 1 l] to solve this problem is to use the augmented Lagrangian function

-3-

This function is successively minimized with r = 0, I,2,. . . . Initially, EF) = 0, the value of .!,“I

is found from that of Ef) and the corresponding minimising x value, x”‘, by the relationship:

(5)

The value xtr) should then converge to a local minimum.

In [4] the use of the generalised reduced gradient method (GRG) in the design of expansion in

water distribution networks is described. In the optimization model the objective function includes

capital cost of pipelines and operating cost of pumping stations. The main constraints are a set of

non-linear hydraulic equations, upper and low bounds of diameters, and a minimum requirement

of pressure head at nodes. A modified Newton - Raphson technique is use in the hydraulic

simulation to accelerate calculation and the convergence and consistency of the algorithm is

investigated.

For the expansion of an existing water distribution network, consisting of m branches and ~2 nodes

the objective function is as follows:

min COST= K,xLiDe +K2xQiHi (6) icp, ia

subject to:

g@&,B) = Q

(7) 0, 2 Dmini, i Ey,,

Hei,, -

where:

Q - vector of diameters, Q = [D, ,D, ,. . . D,,,]’

Q - vector ofloads, Q = [Q,,Q,...Q,]' - -

H - vector of pressure heads at nodes, H = [H, , H, . . . HnIT

Dmin - permissible minimum diameter

Hm, - permissible minimum nodal head

s(LL QZ) - vector of functions used in simulation - -

-4-

L - length of pipe

b, - the set of new branches

p,s - the set of source nodes with pumping station

Kl - a constant related to cost of pipe (weighting factor)

e - exponent related to pipe constant, e E [I,21

Kz - a constant dependent on the unit cost of energy

In the optimization model, the decision variables are the diameters of new pipes Di, i E bn, the

pressure head H will be solved in the simulation (eq.(7)) if the diameter Q and loads e are

known. H can be called state variable.

There are two problems associated with these methods. The first is how to correct from the ideal

diameter to available diameter sizes. The figures given in [2] suggest that the effects of such

,,rounding” can be made acceptably small. The second problem is that of local optima. A local

optimum is an acceptable network such that a small perturbation of the diameters either causes the

pressure to fall below the minimum or leads to an increased cost. Continuous design methods try

to find local optima because it is known that one of these will be the cheapest design.

4. Discrete methods

Rothfarb at al [8] give a simple and effective method for finding the cheapest design when the

network has no loops.

This method is based on the fact that when a diameter is assigned to a pipe then pressure drop

across that pipe and the cost of the pipe is determined because in a tree the flows are known. Pipes

may be joined in a tree network in one of the two ways - either in series or in a “V”.

(Fig. 1)

1 2

If two pipes are linked in series (Fig. 1) then the list of possible diameter assignments can be

reduced considerably simply by eliminating any pair of diameters of there is another pair which

produces a lower pressure drop at lower cost. This is because if the diameters with the higher

-5-

pressure drop are part of a feasible network then they can always be replaced by the pair with

lower pressure drop thereby reducing costs.

Pipes may also be joined in a “V” (see Fig.2).

(Fig.2)

<

This situation differs from the first only in the way the pressure drop is calculated. For any

diameter assignment to these pipes the pressure drop across the pair may be taken as a maximum

of the individual pressure drops. Therefore, provided these pipes are at the downstream end of a

network, any pair of diameters can be eliminated if there is another pair of diameters which, when

assigned to these pipes, produces a lower pressure drop at lower cost. If there are n possible

diameters for each pipe then this “V” elimination will produce at most 2n acceptable pairs. These

methods can be easily extended to networks of pipes. If we have a list of pressure drops and their

costs for arbitrary networks A and B which are joined at node 1 (Fig. 3) then the maximum

pressure drop across the combined system is the maximum of the pressure drop across A and the

pressure drop across B. Again any pressure drop across the combined system can be eliminated if

there is another pressure drop both lower and cheaper .

(Fig.3)

1

Likewise pipe C may be joined to network D (Fig. 4). Again, a list of the system pressures could

be drawn up consisting of pressures which are the obtained by summing one from the C list with

one from the D list. Then these pressures for which there is a lower and cheaper system pressure

-6-

would be removed from this list to give a list of pressures (and costs) for the C-D system. This

process ends when a list of acceptable pressures has been built up for the whole system. The

cheapest pressure on this list consistent with the design requirements is then chosen and the

associated diameters found.

(Fig.4)

c Beale [l] gives a method of optimization which removes non-linearities in the constraints by

introducing piecewise linear approximations. However, the problem as formulated by Beale can

give as solutions designs which are infeasible.

The trouble with his method is the assumption that in the optimization procedure the flow

equation :

can be replaced by the inequality

If, when the problem is solved, there is strict inequality in this expression then the values ofp and

e have no physical significance since they do not satis@ the flow equations. If the diameters we

allowed to be continuous then this problem would not arise because there would always be

equality (inequality implies that k(D) can be increased by a small amount while retaining the

values of e and y thereby giving a cheaper but still feasible solution). In the case of discrete

diameters strict inequality can occur, and as the example shows (see [l]) this produces a set of

pipes satisfying Beales conditions but in some cases does violate the true pressure requirements.

-7-

5. Proposed method

In order to optimize a network design it is first necessary to formulate a completely defined

mathematical model of the system. The system has to be optimised from a cost view point and thus

a relationship between the pipe cost and the system variable (i.e. diameter) must be specified. The

objective function which links pipeline cost to diameter for distribution gas network is:

f(Q) = 2.05.[4 L, K LmlT . Di (21) (8)

where:

L - length of the pipe,

D - diameter of the pipe,

m - number of pipes.

The network optimization problem involves minimizing the cost function expressed by above

equation subject to certain operating constraints.

For any network Kirchhoff s laws must apply.

I-st Kirchhoff s Law:

where:

A - nodal-branch incidence matrix, dim _A = (n x m):

aj 7 if branch j enters node i

aij = -aj , if branch j leaves node i

0 , if branch j is not connected to node i

(9)

d - vector of loads, dim d = (m x 1).

II-nd Kirchhoff s Law:

-8-

where:

I$ - loop-branch incidence matrix, dim B = (u x m), u - the number of independent

loops,

p j , if branch j has the same direction as loop i

bij = -pi , if branch j has opposite direction to loop i

0 > if branch j is not in loop i

y - flow equation exponent

In addition, the pressure at all nodes of the system must not fall below a given minimum working

pressure, i.e.:

Pi - Pimin ’ O (11) where:

pi - is the nodal pressure value

Another constraint was imposed on the gas velocity at each branch. It was assumed that

Vmin I v < v max (12)

for all branches i.

The above problem has been formulated as a nonlinear programming one with nonlinear

constraints in the form:

min f(z) (13)

subject to equality and inequality constraints:

ci(g)= 0 , i = 1,2,K ,m’ c&20 , i=m’+l,K ,m 1

(14)

and the functions f(z) and c;(z), i=1,2,... ,m are real and differentiable. To solve this problem, an

iterative method is used at each iteration which minimizes a quadratic approximation to the

Lagrangian function subject to sequentially linearized approximations to the constraints ([7]). To

begin the calculation a starting point 50 has to be chosen together with positive definite matrix ISo.

-9-

The iterations generate a sequence of points B (k=1,2,...) that usually converges to the required

vector of variables and also the generate a sequence of positive definite matrices & (k= 1,2,. . .).

At the beginning of the k-th iteration both a and & are known. The vector d=& is obtained by

minimizing the quadratic function:

subject to linear constraints:

ci(g,)+cf 4%,(x,) = 0 , i = 1,2,K ,m’

c~(~,)+~_~~VC~(X~)>O , i=m’+l,K ,m (16)

The vector a+1 has the form:

where:

&k+l =&,+akd, (17)

ak - is a positive step-length

The calculation of & is a quadratic programming problem having the property that, if all

constraints are linear and if f(z) is a quadratic function whose second derivative matrix is E& then

a+& is the required vector of variables. Given that & is the m-component vector of Lagrange

multipliers at the solution of the quadratic programming problem which defines 4, the definition

of &+I must depend on & in order to take account of any constraint curvature.

6. Results of investigations

To prove the correctness of the stated algorithm a non-trivial examples have been solved. We

will show the results for two networks.

The low pressure gas network shown in Fig.5 comprises 108 pipes, 83 nodes and 2 sources. List

of pipes and list of nodes are given in Table 1 and Table 2 respectively.

To calculate flow through each pipe, Pole’s equation [S] has been used:

Ap= 5.117~10-‘3 e L-Q

D5 (18)

where:

Ap - drop pressure along a pipe [Lb/in2],

Q - flow through pipe under standard conditions [cu.ft./h],

D - diameter of pipe [A]

-lO-

The results of simulation presented in Table 3 and Table 4 have shown that, network was badly

designed. The gas velocity in many pipes does not exceed lmls (heavy lines in Fig. 5).

The aim of optimization is to minimize objective function (eq.(8)) subject to constraints.

For this case:

Qi = a, . Di2 [CU$/h] (1%

where:

OT,i = 19.6354 X Vi,

Di [inch],

vi [is]

Api =P, .Q-’ [Lb/in21

where:

pi = 1.391X1O-6 X Li X Vi*,

Li [fit]

Vi [R/S]

(20)

y = I in equation (10)

It was assumed that:

16.4 ft / s I vi 2 32.8 ft / s

for each pipe,

pj 2 0.261 Lb / in2

for each node.

Results of optimization are given in Table 5.

The medium pressure gas network shown in Fig. 6 comprises 39 pipes, 36 nodes and 1 source.

List of pipes and list of nodes are given in Table 6 and Table 7 respectively.

For this case Renouard’s equation [6] has been used:

AP = p,! - p,? = 5.01 I39 * I o-” * Ps . L . Q’.”

4 82 s D.

[(Lb/in2)2J (21)

where:

-ll-

pS - density of gas (the subscript s refers to quantities at standard conditions

pS = 14.5 Lb/in2 and temperature T, = 32 “F).

The results of simulation are presented in Table 8 and Table 9.

Branches for which

vi 13.38 fth

or

vi 2 65.4 ft / s

are marked by heavy lines in Fig. 6.

For the purpose of optimization:

Q = aj -Q2 [cxjm]

where: -

CXi = 1.751 X p&s X Vi,

p&s - average absolute pressure in the pipe [psia],

Di [inch],

Vi [fi/S]

Api = pi . &‘.I8 [(Lb / in2)l]

where:

pi = 7.051 X 10m6 X Li X p:by X Vii’**,

Li [fil

y = 1.18 in equation (10)

It was assumed that:

32.8 ft / s s vi s45.8 ft / s

for each pipe,

pi 2 14.5 Lb / in2

for each node.

Results of optimization are given in Table 10.

(22)

(23)

-12-

7. Conclusions

In both cases, diameter of pipes were corrected to the closest available diameter sizes.

Investigations have shown, that developed algorithm works properly. Optimization of pipe

diameters for the first network (Fig. 5) gives total profit equal 44603.00 USD. Second case gives

less savings, only 2900.00 USD, because much smaller network was much better designed.

8. References

PI

PI

PI

PI

PI

PI

PI

PI

PI

Beale, E.M. Some Uses of Mathematical Programming Systems to Solve Problems that are

not Linear”. Operational Research Quarterly. 1975, Vol. 26,3. pp. 609-618

Boyne, G.G. ,,The Optimum Design of Fluid Distribution Networks with Particular

Reference to Low Pressure Gas Distribution Networks”. International Journal for Numerical

Methods in Engineering, Vol. 5, 253-270, 1972

Chamberlain, R.M., et al, ,,The Watchdog Technique for Forcing Convergence in Algorithms

for Constrained Optimization”, Report DATMP 80/NA9, University of Cambridge

Guoping Yu and Powell R. ,,A Generalized Reduced Gradient Approach to Expansion of

Water Distribution Networks”. International Conference on Pipeline Systems, 1992,

Manchester, U.K.

Osiadacz, A.J. Simulation and Analysis of Gas Networks”. E&FN Spon Ltd., London 1987

Osiadacz A.J., Zelman H. and Krawczynski T. ,,SimNet SSV - Package for Steady-state

Simulation of any Gas Network” (In Polish), GWiTS, Nr 6, 1995

Powell, M.J.D. ,,Extensions to subroutine VFO2AD”. In System Modelling and Optimization,

Lecture Notes in Control and Information Sciences 38, eds. R.F. Drenick and F. Kozin,

Springer Verlag, 1982

Rothfarb B., Frank M., Rosenbaum D.M., Steiglitz K. and Kleitman D.J. ,,Optimal Design of

Offshore Natural Gas Pipeline Systems”. Operational Research 1980, Vol. 18, Part 6

Stubbs C.W. and Thompson P.D. ,,Designing Gas Distribution Networks by Computer”.

Manchester District Junior Gas Association, April 1979

[lo] Ward E. ,,Design of Gas Distribution Networks Using PILOT”, LRS T195, October 1974

[l l] Watanada T. ,,Least Cost Design of Water Distribution Systems”. Journal of Hydraulics

Division, ASCE, September 1973, HY9

-13-

TABLE 1.

LIST OF PIPES

Jo oi

y!?

2 3 4 5 6 7 8 9 IO 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38

LN. Gi

100 101 101 102 102 103 103 104 107 101 107 108 107 116 108 102 108 109 108 117 109 110 109 118 110 Ill 110 121 Ill 104 Ill 122 114 115 114 131 115 116 115 136 116 132 117 118 117 134 118 119 118 139 119 121 121 122 121 140 131 135 132 133 132 137 133 134 133 158 134 138 135 136 135 145 136 137 136 147

,ength

-El- 240 456 702 653 43

446 66 75

709 266 656 302 33

322 59

315 256 230 249 502 308 656 154 459 328 259 23

305 207 197 318 230 879 174 361 331 197 315

Diameter 1 jinch] 6.00 6.00 6.00 6.00 4.00 6.00 4.00 6.00 6.00 8.00 6.00 4.00 8.00 8.00 8.00 4.00 6.00 6.00 4.00 4.00 4.00 3.00 8.00 3.00 4.00 3.00 6.00 8.00 6.00 3.00 4.00 3.00 3.00 8.00 3.00 4.00 3.00 4.00

mm: 150 150 150 150 100 150 100 150 150 200 150 100 200 200 200 100 150 150 100 100 100 80

200 80 100 80 150 200 150 80 100 80 80

200 80 100 80 100

rlo of

3

40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76

137 148 138 139 138 149 139 140 139 150 140 141 141 144 144 154 145 146 145 147 147 148 148 157 149 150 149 160 149 161 150 151 150 152 152 153 153 154 154 155 156 157 157 158 159 160 162 161 48 49 48 64 49 50 50 51 51 52 52 53 52 54 54 55 55 98 64 65 64 69 65 66 66 67 66 83

,ength

-El- 302 732 390 728 341 52

367 118 295 240 220 312 827 499 400 295 492 230 289 276 322 220 184 164 184 52

308 817 922 102 394 62

886 554 72

256 525 722

Diameter 1 mm 100 100 200 100 100 150 150 150 80 100 100 100 150 300 150 100 150 150 150 150 80 80

300 100 300 150 300 300 300 400 300 300 200 150 150 125 150 100

inch’ 4.00 4.00 8.00 4.00 4.00 6.00 6.00 6.00 3.00 4.00 4.00 4.00 6.00 12.oc 6.00 4.00 6.00 6.00 6.00 6.00 3.00 3.00 12.0( 4.00 12.0( 6.00 12.0( 12.0( 12.0( 16.0( 12.0( 12.0( 8.00 6.00 6.00 5.00 6.00 4.00

TABLE 1 .CONT.

Vo of S.N, pipe

77 67 78 69 79 69 80 70 81 71 82 72 83 72 84 73 85 73 86 74 87 78 88 78 89 80 90 80 91 81 92 83

Gi

72 78 80 81 72 73 83 74 85 86 79 92 101 81 82 84

,ength

-PI.- 256 230 574 187 289 623 299 525 308 285 226 322 531 256 594 66

i- Diameter

:mm’ 300 150 100 100 100 100 300 100 100 100 80 150 100 100 100 300

[inch? 12.00 6.00 4.00 4.00 4.00 4.00 12.00 4.00 4.00 4.00 3.00 6.00 4.00 4.00 4.00 12.00

I No of

IagE

94 95 96 97 98 99 100 101 102 103 104 105 106 107 108

S.N. R.N

84 85 84 95 85 86 85 96 86 97 92 99 93 94 94 100 94 80 95 102 95 96 96 103 96 97 97 98 98 104 99 100

,ength

[rtl 614 269 535 295 285 302 200 361 256 276 689 269 551 46 230 295

Dia 171111: 100 300 80 100 100 150 100 100 100

80 100 80 80

200 150

reter jnch] 4.00 12.00 3.00 4.00 4.00 6.00 4.00 4.00 4.00 12.00 3.00 4.00 3.00 3.00 8.00 6.00

1

TABLE 2.

LIST OF NODES

Node no. ILoad 1 Node no. /Load

102 103 104 107 108 109 110 111 114 115 116 117 118 119 121 122 131 132 133 134 135 136 137 138 139 140 141 144 145 146 147 148 149 150 151 152 153 154 155 156

633.19 926.66 856.73 362.68 134.90 664.62 881.45 523.01 303.35

8.83 391.29 303.35

1140.66 1408.70

705.59 1250.49

246.14 0.00

431.55 974.69 616.59

2142.19 2532.77

536.08 1501.93 1611.41 1350.43

931.95 20.84

313.95 137.37 472.16 716.54

1961.73 2282.74

310.42 929.84

69.92 0.00

236.96 182.22

158 159 160 161 162 48 49 50 51 52 53 54 55 64 65 66 67 69 70 71 72 73 74 78 79 80 81 82 83 84 85 86

~ 92 93

~ 94 95 96 97 98 99

1124.77 0.00 0.00

588.70 41.32

253.91 174.81 229.90

98.88 233.43

0.00 233.43

35.31 125.37 163.51 713.71 666.04 260.27 171.98 228.49

1347.26 836.96 443.20 181.52 156.09 767.74 775.51 686.52

1082.40 1557.38 1867.09

883.22 284.28 165.27 455.21

1216.24 1261.09

821.77 35.31

300.88

rlo. of aipe

1 2 3 4 5 6 7 a 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

S.N.

100 101 101 102 102 103 103 104 107 101 107 108 107 116 108 102 108 109 108 117 109 110 109 118 110 111 110 121 111 104 111 122 114 115 114 131 115 116 115 136 116 132 117 118 117 134 118 119 118 139 119 121 121 122 121 140 131 135 132 133 132 137 133 134 133 158 134 138 135 136

TABLE 3.

RESULTS OF SIMULATION

3.N. ,ength PI

240 456 702 653

43 446

66 75

709 266 656 302

33 322

59 315 256 230 249 502 308 656 154 459 328 259

23 305 207 197 318 230 879 174 361

Diameter [mm] 1 [inch]

150 6.00 150 6.00 150 6.00 150 6.00 100 4.00 150 6.00 100 4.00 150 6.00 150 6.00 200 8.00 150 6.00 100 4.00 200 8.00 200 8.00 200 8.00 100 4.00 150 6.00 150 6.00 100 4.00 100 4.00 100 4.00 80 3.00

200 8.00 80 3.00

100 4.00 80 3.00

150 6.00 200 8.00 150 6.00

80 3.00 100 4.00

80 3.00 80 3.00

200 8.00 801 3.00

:low $u.ft./h]

3128.53 -1013.53 -1184.81 -3182.21 -3814.34 -3932.64 7613.85 4774.90

-1453.55 -7917.56 -2310.64

-23.66 -5736.52 2905.34

-6652.58 612.36

-2931.12 2923.00

-4630.82 1309.47 2679.33

401.88 -9458.34

14.48 -1043.20

-690.05 -367.27 1330.66 2923.00 -950.67 3199.16

-3235.18 1311.23

-13309.76 270.16

Jeloc. IfUs]

4.53 1.48 1.71 4.59

12.40 5.68

24.77 6.89 2.10 6.43 3.35 0.07 4.66 2.36 5.41 2.00 4.23 4.23

15.09 4.27 8.73 2.03 7.68 0.07 3.38 3.51 0.52 1.08 4.23 4.82

10.43 16.44

6.69 10.83

1.38

37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73

136 136 137 138 138 139 139 140 141 144 145 145 147 148 149 149 149 150 150 152 153 154 156 157 159 162 48 48 49 50 51 52 52 54 55 64 64

145 137 147 148 139 149 140 150 141 144 154 146 147 148 157 150 160 161 151 152 153 154 155 157 158 160 161 49 64 50 51 52 53 54 55 98 65 69

TABLE 3. CONT.

j

197 80 315 100 302 100 732 100 390 200 728 100 341 100

52 150 367 150 118 150 295 80 240 100 220 100 312 100 827 150 499 300 400 150 295 100 492 150 230 150 289 150 276 150 322 80 220 80 184 300 164 100 184 300

52 150 308 300 817 300 922 300 102 400 394 300

62 300 886 200 554 150

3.00 4.00 4.00 4.00 8.00 4.00 4.00 6.00 6.00 6.00 3.00 4.00 4.00 4.00 6.00

12.00 6.00 4.00 6.00 6.00 6.00 6.00 3.00 3.00

12.00 4.00

12.00 6.00

12.00 12.00 12.00 16.00 12.00 12.00 8.00 6.00

ISOl 6.00

:low :cu.ft./h ]

513.48 -1107.47

157.15 1555.61 1363.50

-16173.43 -59.33

-1230.36 -0.35

-1008.94 -1029.78

137.37 62.15

-252.85 586.58

6087.90 -24849.89

630.01 310.42

2266.50 1337.37 1267.44 236.96

-182.22 -186.46

24850.60 -41.32

-13143.78 12892.34

-13320.00 -13548.84 -13648.07 -27452.24 13573.91 13337.66 13305.17 4843.06 7924.27

Jeloc. :ft/s]

1.67 5.64 0.52 5.09 4.43

13.12 0.20 4.00 0.00 1.44 1.48 0.69 0.20 0.82 1.90 8.79 8.96 0.92 1.02 3.28 I .94 1.84 0.33 0.92 0.95 8.96 0.13 4.76

18.60 4.79 4.89 4.92 5.58 4.89 4.82

10.79 6.99

11.45

TABLE 3. CONT.

74 65 66 256 125 75 66 67 525 150 76 66 83 722 100 77 67 72 256 300 78 69 78 230 150 79 69 80 574 100 80 70 81 187 100 81 71 72 289 100 82 72 73 623 100 83 72 83 299 300 84 73 74 525 100 85 73 85 308 100 86 74 86 285 100 87 78 79 226 80 88 78 92 322 150 89 80 101 531 100 90 80 81 256 100 91 81 82 594 100 92 83 84 66 300 93 84 85 614 100 94 84 95 269 300 95 85 86 535 80 96 85 96 295 100 97 86 97 285 100 98 92 99 302 150 99 93 94 200 100

100 94 100 361 100 101 94 80 256 100 102 95 102 276 300 103 95 96 689 80 104 96 103 269 100 105 96 97 551 80 106 97 98 46 80 107 98 104 230 200 108 99 100 295 150

R.N. ILength 1 Diameter r f 1 [ft] [[mm1 iinch]

5.00 6.00 4.00

12.00 6.00 4.00 4.00 4.00 4.00

12.00 4.00 4.00 4.00 3.00 6.00 4.00 4.00 4.00

12.00 4.00

12.00 3.00 4.00 4.00 6.00 4.00 4.00 4.00

12.00 3.00 4.00 3.00 3.00 8.00 6.00

=fow :cu.ft./h]

4679.90 3026.47

940.78 2361.49 5168.31 2496.40 -171.98 -228.49 789.64

0.35 -79.81 34.61

-523.01 156.09

4831.05 305.83

1634.01 686.52

-0.71 796.35

-2494.98 -223.90 -810.83

-1629.42 4547.12 -165.27 -828.48 209.77

-4019.52 310.06

-1141.02 -619.42

-3070.61 10198.53 4246.59

r/eloc. ifffs]

9.74 4.36 3.05 0.85 7.45 8.10 0.56 0.75 2.56 0.00 0.26 0.10 1.71 0.79 6.99 0.98 5.31 2.23 0.00 2.59 0.89 1.15 2.62 5.31 6.56 0.52 2.69 0.69 1.44 1.57 3.71 3.15

15.58 8.30 6.14

lode no.

100 101 102 103 104 107 108 109 110 111 114 115 116 117 118 119 121 122 131 132 133 134 132 13E 137 13e 13s 14c 141 144 14: 14E 14i 14E 14s 1% 151 15; 15: 15L 15f 15c

TABLE 4.


Vessure Lb/in21

0.699 0.698 0.698 0.699 0.702 0.696 0.699 0.700 0.702 0.702 0.659 0.661 0.681 0.701 0.700 0.700 0.701 0.701 0.658 0.673 0.675 0.703 0.658 0.657 0.660 0.706 0.701 0.701 0.701 0.701 0.657 0.657 0.657 0.657 0.71s 0.70: 0.70: 0.702 0.702 0.701 0.701 0.657

lode no.

157 158 159 160 161 162

48 49 50 51 52 53 54 55 64 65 66 67 69 70 71 72 73 74 78 79 80 81 82 83 84 85 8E 92 92 94 95 9c 97 9E 95

Vessure I Lb/in2 0.657 0.657 0.725 0.723 0.718 0.718 0.718 0.719 0.720 0.722 0.725 0.725 0.724 0.724 0.714 0.708 0.701 0.698 0.712 0.696 0.698 0.698 0.697 0.697 0.709 0.709 0.698 0.696 0.694 0.698 0.698 0.697 0.697 0.705 0.698 0.698 0.698 0.697 0.7oc 0.702 0.702

TABLE5

RESULTSOFOPTIMIZATION

40. of Length

Pipe ml

1 240 2 456 3 702 4 653 5 43 6 446 7 66 8 75 9 709 10 266 11 656 12 302 13 33 14 322 15 59 16 315 17 256 18 230 19 249 20 502 21 308 22 656 23 154 24 459 25 328 26 259 27 23 28 305 29 207 30 197 31 318 32 230 33 879 34 174 35 361 36 331 37 197

old result of new AD= diameter optimization diameter old-new

[inch] [inch] [inch] [inch] 6.00 3.42333 4.00 2.00 6.00 4.02147 5.00 1 .oo 6.00 1.79539 2.00 4.00 6.00 4.92069 5.00 1.00 4.00 0.05000 0.50 3.50 6.00 5.59558 6.00 0.00 4.00 5.55803 6.00 -2.00 6.00 7.06769 8.00 -2.00 6.00 2.33288 2.50 3.50 8.00 3.33702 4.00 4.00 6.00 0.81741 1 .oo 5.00 4.00 1.83676 2.00 2.00 8.00 2.19912 2.50 5.50 8.00 1.59502 1.50 6.50 8.00 2.87766 3.00 5.00 4.00 1.58207 1.50 2.50 6.00 4.07753 5.00 1 .oo 6.00 4.07416 5.00 1 .oo 4.00 4.10231 5.00 -1 .oo 4.00 1.59862 1.50 2.50 4.00 3.09081 3.00 1 .oo 3.00 1.68912 2.00 1 .oo 8.00 2.17738 2.50 5.50 3.00 1.46055 1.50 1.50 4.00 0.53192 0.50 3.50 3.00 0.23999 0.50 2.50 6.00 1.31860 1.50 4.50 8.00 0.58544 0.50 7.50 6.00 4.07416 5.00 1.00 3.00 2.05621 2.50 0.50 4.00 1.99633 2.00 2.00 3.00 1.68122 2.00 1 .oo 3.00 2.00701 2.00 1 .oo 8.00 0.05000 0.50 7.50 3.00 2.50377 2.50 0.50 4.00 1.91797 2.00 2.00 3.00 1.36608 1.50 1.50

AF

[zt] 629.69 617.58

3,425.51 884.16 150.75

172.52 219.62

3,090.60 1,472.14 3,800.95

679.07 238.57

2,658.81 396.95 859.79 346.56 311.01 317.90

1,370.29 364.53 700.64

lJ21.28 710.94

1,159.63 609.75 123.07

2,768.70 279.91 108.27 715.97 245.22 938.86

1,577.86 198.49 745.50 304.69

TABLESCONT.

rlo. of pipe

Length WI

38 315 39 302 40 732 41 390 42 728 43 341 44 52 45 367 46 118 47 295 48 240 49 220 50 312 51 827 52 499 53 400 54 295 55 492 56 230 57 289 58 276 59 322 60 220 61 184 62 164 63 184 64 52 65 308 66 817 67 922 68 102 69 394 70 62 71 886 72 554 73 72 74 256 75 525 76 722 77 256

old result of new AD= diameter optimization diameter old-new

[inch] [inch] [inch] [inch] 4.00 1.68126 2.00 2.00 4.00 0.67477 0.75 3.25 4.00 2.68523 3.00 1 .oo 8.00 3.12227 3.00 5.00 4.00 1.99334 2.00 2.00 4.00 1.43064 1.50 2.50 6.00 0.35160 0.50 5.50 6.00 1.66458 2.00 4.00 6.00 1.68390 2.00 4.00 3.00 0.65310 0.75 2.25 4.00 1.50908 1.50 2.50 4.00 1.90735 2.00 2.00 4.00 1.36666 1.50 2.50 6.00 4.02766 5.00 1 .oo 12.00 5.46139 6.00 6.00 6.00 1.39587 1.50 4.50 4.00 0.98174 1.00 3.00 6.00 2.58368 2.50 3.50 6.00 1.94637 2.00 4.00 6.00 1.88977 2.00 4.00 6.00 0.85776 1.00 5.00 3.00 0.75219 1.50 1.50 3.00 0.73196 0.75 2.25 12.00 5.46583 6.00 6.00 4.00 0.34730 0.50 3.50 12.00 8.26838 10.00 2.00 6.00 8.22056 10.00 -4.00 12.00 8.30113 10.00 2.00 12.00 8.34402 10.00 2.00 12.00 8.36239 10.00 2.00 16.00 10.05071 12.00 4.00 12.00 7.26529 8.00 4.00 12.00 7.25219 8.00 4.00 8.00 7.25189 8.00 0.00 6.00 5.92402 6.00 0.00 6.00 5.62595 6.00 0.00 5.00 5.88101 6.00 -1 .oo 6.00 4.83836 5.00 I .oo 4.00 2.99342 3.00 1.00 12.00 4.61971 5.00 7.00

AF

[a 708.59

1,013.70 864.79

2,624.26 1,638.62

931.44 323.56

1,792.79 576.25 642.65 653.80 494.54 850.83

lJ19.65 4,679.80 2J45.01

934.10 2J46.25 1,120.50 1,408.62 1,596.40

497.66 478.41

1,724.14 579.82 612.64 317.57

1,028.36 2,724.07 3,074.15

728.86 2,548.73

403.55

346.56 710.89 853.16

2.748.03

TABLE 5. cont.

No. of

pipe

Length

PI

78 230 79 574 80 187 81 289 82 623 83 299 84 525 85 308 86 285 87 226 88 322 89 531 90 256 91 594 92 66 93 614 94 269 95 535 96 295 97 285 98 302 99 200 100 361 101 256 102 276 103 689 104 269 105 551 106 46 107 230 108 295

old result of new AD=

diameter )ptimizatior diameter old-new [inch] [inch] [inch] [inch] 6.00 4.12945 5.00 1 .oo 4.00 3.77279 4.00 0.00 4.00 0.73075 0.75 3.25 4.00 0.84228 0.75 3.25 4.00 1.22750 1.25 2.75 12.00 3.86555 4.00 8.00 4.00 0.89517 1 .oo 3.00 4.00 0.53908 0.50 3.50 4.00 1.47561 1.50 2.50 3.00 0.69617 0.75 2.25 6.00 4.00051 5.00 1 .oo 4.00 2.53359 2.50 1.50 4.00 2.25244 2.50 1.50 4.00 1.45999 1.50 2.50 12.00 4.53236 5.00 7.00 4.00 3.93353 4.00 0.00 12.00 5.58385 6.00 6.00 3.00 2.57409 2.50 0.50 4.00 3.86451 4.00 0.00 4.00 3.39790 4.00 0.00 6.00 3.88863 4.00 2.00 4.00 0.71635 0.75 3.25 4.00 1.25257 1.25 2.75 4.00 0.59800 0.50 3.50 12.00 5.78219 6.00 6.00 3.00 2.45573 2.50 0.50 4.00 4.28132 5.00 -1 .oo 3.00 2.55947 2.50 0.50 3.00 3.97317 4.00 -1 .oo 8.00 5.79830 6.00 2.00 6.00 3.76659 4.00 2.00

TOTAL PROFIT [zl] 105,264.42

AF

WI 311.01

628.05 969.63

1,840.90 3,586.68 1,660.62 1,090.05

779.18 492.70 435.42 920.55 443.23

1,621.06 704.62

2,524.63 294.12

793.58 672.13

1,065.79 904.51

2,586.20 378.93 342.99 303.15

54.29 668.41 776.33

TOTAL PROFIT [$] I 44.603.57

TABLE 6.

LIST OF PIPES

lo. of pipe S.N R.N

1 1 26 2 1 32 3 2 5 4 2 35 5 3 5 6 3 15 7 4 19 8 5 14 9 6 34 10 7 10 II 7 33 12 8 6 13 8 9 14 8 36 15 9 22 16 10 2 17 10 11 18 11 3 19 11 12 20 12 4 21 12 9 22 14 1 23 14 27 24 15 4 25 15 16 26 16 17 27 16 18 28 18 14 29 18 25 30 19 20 31 19 24 32 20 21 33 21 23 34 26 28 35 26 29 36 29 30 37 29 31 38 33 1 39 35 13

Length

-IElI- 1706 525 66 66 197 131 213 295 492 197 482 1247 230 7546 148 591 197 541 279 525 213 180 1378 148 197 197 197 197 394 164 180 197 66

427 295 164 98

469 164

I T Diameter

(mm) (inch) 50 2.00 65 2.50 32 1.25 32 1.25 32 1.25 40 1.50 40 1.50 32 1.25

200 8.00 32 1.25 50 2.00 200 8.00 65 2.50 200 8.00 40 1.50 32 1.25 32 1.25 32 1.25 32 1.25 40 1.50 40 1.50 65 2.50 50 2.00 40 1.50 32 1.25 40 1.50 32 1.25 65 2.50 65 2.50 40 1.50 25 1.00 32 1.25 50 2.00 40 1.50 40 1.50 40 1.50 40 1.50 50 2.00 32 1.25

TABLE 7.

LIST OF NODES

Node No. Load [cu.ft./h]

1 1425.65 2 722.89 3 962.33 4 904.41 5 593.64 6 0.00 7 753.97

~ a 0.00 9 294.88

10 962.33 11 962.33 12 844.73 13 194.23 14 1508.29 15 294.88

~ 16 294.88 17 453.79 la 785.05 19 593.64 20 525.13 21 0.00 22 194.23 23 194.23 24 194.23 25 1400.58 26 814.36 27 919.59 28 194.23 29 525.13 30 198.47 31 198.47 32 5297.21 33 0.00 34 183636.44 35 0.00 36 0.00

TABLE 9.


qode No.

1 2 3 4 5 6 7 a 9

10 11 12 13 14 15 16 17 la 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

I

I

,

,

,

I

I

/

/

/

I

I

/

Pressure [Lb/in21

1 a.329 21.052 22.785 24.827 21.047 42.781 19.753 43.993 43.116 21.539 24.789 33.294 21.046 la.421 22.983 20.552 20.539 18.457 24.719 24.697 24.692 43.115 24.692 24.703 18.436 17.571 18.266 17.564 17.484 17.481 17.483 la.003 19.045 42.292 21.050 52.214

TABLE 8.


No. of pipe

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39

S.N. R.N.

1 26 1 32 2 5 2 35 3 5 3 15 4 19 5 14 6 34 7 10 7 33 8 6 8 9 8 36 9 22 10 2 10 11 11 3 11 12 12 4 12 9 14 1 14 27 15 4 15 16 16 17 16 18 18 14 18 25 19 20 19 24 20 21 21 23 26 28 26 29 29 30 29 31 33 1 35 13

Length

WI 1706 525 66 66 197 131 213 295 492 197 482 1247 230 7546 148 591 197 541 279 525 213 180

1378 148 197 197 197 197 394 164 180 197 66

427 295 164 98

469 164

r Diameter

[mm1 50 65 32 32 32 40 40 32

200 32 50

200 65

200 40 32 32 32 32 40 40 65 50 40 32 40 32 65 65 40 25 32 50 40 40 40 40 50 32

[inch] 2.00 2.50 1.25 1.25 1.25 1.50 1.50 1.25 8.00 1.25 2.00 8.00 2.50 8.00 1.50 1.25 1.25 1.25 1.25 1.50 1.50 2.50 2.00 1.50 1.25 1.50 1.25 2.50 2.50 1.50 1 .oo 1.25 2.00 1.50 1.50 1.50 1.50 2.00 1.25

Flow Jelocity Jcu.ft./h] [ftk]

1930.65 11.42 5297.21 18.44 346.44 4.56 194.23 2.56

4721.93 60.83 -2620.70 21.06 1507.23 11.52 4474.73 61.29

183636.44 38.75 -4638.94 61.91 3884.97 22.01

183636.44 38.16 23207.76 45.54 206843.50 39.76

194.23 1.02 1263.56 16.57 -6864.82 85.50 3063.20 37.53

-10890.35 117.42 10983.58 75.75 -22718.65 129.59 4768.54 16.47 919.59 5.38

-8571.94 67.03 5656.00 73.16 453.79 3.87

4907.33 67.65 2721.70 9.38 1400.58 4.82 719.36 5.51 194.23 3.81 194.23 2.33 194.23 0.95 194.23 1.80 922.07 8.63 198.47 1.87 198.47 1.87

3884.97 22.47 194.23 2.56

TABLE IO.

RESULTS OF OPTIMIZATION

uo. of 3ipe

1 2 3 4 5 6 7 8 9

IO 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39

,ength

:ft] 1706 525

66 66

197 131 213 295 492 197 482

1247 230

7546 148 591 197 541 279 525 213 180

1378 148 197 197 197 197 394 164 180 197

66 427 295 164 98

469 164

old diameter [inch]

2.00 2.50 1.25 1.25 1.25 1.50 1.50 1.25 8.00 1.25 2.00 8.00 2.50 8.00 1.50 1.25 1.25 1.25 1.25 1.50 1.50 2.50 2.00 1.50 1.25 1.50 1.25 2.50 2.50 1.50 1 .oo 1.25 2.00 1.50 1.50 1.50 1.50 2.00 1.25

l esult of optimization ‘inch]

0.80193 1.31959 0.83177 0.25144 0.98843 0.82543 0.69913 1.17346 7.64524 1.03909 1.05826 7.62555 2.71055 7.97146 0.24858 0.99321 1.40752 0.9117

1.67957 1.5627

2.33017 1.30756 0.5522

1.36832 1.1382 0.3852

1.05762 0.88437 0.67699 0.48467 0.25162 0.25236 0.25334 0.25684 0.55905 0.26006 0.26001 1.06147

0.2516

new diameter

jnch] 1.25 1.50 1 .oo 0.50 1 .oo 0.75 0.76 1.25 8.00 1.25 1.25 8.00 3.00 8.00 0.50 1 .oo 1.50 1 .oo 2.00 1.50 2.50 1.25 0.75 1.50 1.25 0.50 1.25 1 .oo 0.75 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50

A D= old-new

‘inch] 0.75 1 .oo 0.25 0.75 0.25 0.75 0.74 0.00 0.00 0.00 0.75 0.00

-0.50 - 0.00 1 .oo 0.25

-0.25 - 0.25

-0.75 - 0.00

-1.00 - 1.25 1.25 0.00 0.00 1.00 0.00 1.50 1.75 1 .oo 0.50 0.75 1.50 1 .oo 1 .oo 1 .oo 1.00 1.50 0.75

AF

[zll 1,200.04

523.79 13.80 38.15 41.39 82.49

132.46

339.24

126.31

118.81 124.18

43.97 113.84 196.16

212.79 220.36

1,527.67

158.41

281.79 640.33 132.01 66.96

114.44 84.30

343.22 237.62 132.01

79.21 602.76

95.37

TOTAL PROFIT [zt] 6J65.44 TOTAL PROFIT [$] 2,909.08

J32 -133 7 ,, 134

,138 139 \

146, . 156 157

159 d 160 p=5000 Pa

FIGURE 5 : low pressure gas network

3L 29 -- 7+ 30

! 26

P 28

7

y 6 8

FIGURE 6: medium pressure gas network

36 1

p=360 kPa