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  • 1. Optimization of Pipe Sizes for Distribution Gas Network Design Andrzej J. Osiadacz Warsaw University of Technology and Marcin Gbrecki Regional Gas Dispatching Center, Warsaw
  • 2. 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-
  • 3. 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 asfollows. 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-
  • 4. 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 asfollows: 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, pmirti - 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) Jlowsat nodei - di is the demand at each node, except for sources where it is the total flow Qi The method used in [ 1l] to solve this problem is to use the augmented Lagrangian function -3-
  • 5. 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 ~2nodes the objective function is asfollows: 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-
  • 6. 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 Ebn, 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-
  • 7. 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-
  • 8. 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 casesdoes violate the true pressure requirements. -7-
  • 9. 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 entersnode i aij = -aj , if branch j leavesnodei 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-
  • 10. where: I$ - loop-branch incidence matrix, dim B = (u x m), u - the number of independent loops, p j , if branch j has the samedirection as loop i bij = -pi , if branch j hasopposite 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 VminI v < vmax (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 ,m1 (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-
  • 11. 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 eL-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-
  • 12. 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,?= 5.01I39 *I o-” *Ps .L .Q’.” 482s D. [(Lb/in2)2J (21) where: -ll-
  • 13. 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 65.4ft / 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 10m6X 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-
  • 14. 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-
  • 15. 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 1jinch] 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
  • 16. 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 INo 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
  • 17. 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
  • 18. 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
  • 19. 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
  • 20. 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 rf 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
  • 21. 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. RESULTS OF SIMULATION 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 ILb/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
  • 22. 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
  • 23. 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
  • 24. 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 [$] I44.603.57
  • 25. 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 IT 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
  • 26. 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. RESULTS OF SIMULATION 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 1a.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
  • 27. TABLE 8. RESULTS OF SIMULATION 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
  • 28. 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 lesult 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
  • 29. J32 -1337 ,,134 ,138 139 146, . 156 157 159 d 160 p=5000 Pa FIGURE 5 : low pressure gas network
  • 30. 3L 29-- 7+ 30 !26 P 28 7 y 6 8 FIGURE 6: medium pressure gas network 36 1 p=360 kPa