OPTIMAL LAY-OUT OF NATURAL GAS PIPELINE NETWORK

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23rd World Gas Coferece, Amsterdam 2006 OPTIMAL LAY-OUT OF NATURAL GAS PIPELINE NETWORK Ma author Tg-zhe, Ne CHINA

ABSTRACT I cha, there are lots of gas ppele etwork eeded to be desged ad costructed owadays. I order to optmze the lay-out of atural gas ppele etwork, the Hopfeld Neural Network optmal method was appled. Accordg to the models of atural gas etwork total cost optmal layout ad equvalet legth optmal layout, the eergy fucto model of optmal layout of atural gas etwork based o Hopfeld Neural etwork was establshed. The dyamc equato of state of eural was duced. Whe teratve tme creases, the eergy fucto decreased to a stablty value, the smallest amely. The etwork state gradually teds to stablty appearace. The mmum eergy fucto s respoded to the optmal layout whch to be wated. The results express that adoptg the legth shortest layout match the actual project. The satsfed result could be gotte through applyg Hopfeld Neural Network method gas etwork optmal layout.

TABLE OF CONTENTS Abstract Body of Paper. Preamble 2. The Aalyss of Gas Network Lay-Out 3. The Model of Optmal Lay-Out of Network 4. Descrpto of Hopfeld Neural Network 5. Gas Network Optmal Lay-Out Based O HNN 5. The Eergy Fuctos of Lay-Out 5.2 The State Equato of Neural 6. Example ad Aalyss 7. Cocluso Refereces Lst Tables Table : The testy desg coeffcet Table 2: The relatve coordates of ctes ad gas sources Table 3: The actual dstace betwee gas supply ctes (km) Table 4: The weghts of legth betwee gas supply ctes Table 5: The assocate matrx of layout of gas supply ctes Table 6: The parameters of several layout solutos Lst of Fgures Fgure : Locatos of gas supply ctes ad gas sources Fgure 2: The structure of eural Fgure 3: Feasble layout of ppele betwee te ctes Fgure 4: Optmal layout of actual dstace ppele betwee ctes Fgure 5: Optmal layout of ppele wth weghts betwee ctes

Paper. PREAMBLE Gas trasmsso etwork lay-out s a aspect of etwork optmzato. I the proceedgs of the 20 th world gas coferece, the C3 report dcated that the gas ppele etwork optmal s the ew problem [].The problem how to coect the all ode-pots to make the layout of gas trasmsso system reasoable should be solved after the basc data are obtaed for desg the gas dustry. Whe the ppele route goes through may ctes, the route s terra s rregular ad complcated, ad several programs ca be selected for the route, oly the program that s optmzed should be used. The ormal method to solve etwork layout clude mmum spag tree ad m cost crculato ad dyamc programmg etc. The mmum spag tree cosder gas etwork as udrected graph. The effcet method s Dkstra algorthm. The mmum spag tree s effcet for brach ppele layout but ot so effcet for crcle ppele layout [2].The m cost crculato method s a etwork algorthm used to solve the lear programmg problem [3].Dyamc programmg selects solutos accordg tme or phase. The operato amout creases dex wth the creased varables. There s the dmeso obstacle. It s dffcult to solve mult-varables optmal problem usg dyamc programmg. The atural gas trasmsso etwork layout problem smles the shortest path problem. The shortest path problem s the basc problem of combato optmzato- NP problem. The tellget optmal method recetly developed ca solve ths kd of problem effcetly [4,5]. Pedrycz W. etc. adopt ANNs modelg objectve ad parameter to desg atural gas ppele [6,7]. Neural etwork ca express the correlato clearly. Ths method s used for exstg solutos ot for detaled parameters determato. I 980s, Hopfeld (982) ad Tak (985) have used Aual eural etwork (ANN) to solve TSP ad gotte success. Neural etwork method gets atteto creasgly for ts strog self-search ablty. Ad t has bee appled trasportato ad electrc ad commucato etc. I ol ad gas feld the method beg to be appled ol feld collecto ad trasport system layout. The ma techque s adoptg rght eergy fucto accordg to the objectve fucto of problem to determe the weghts betwee the eural. Wth the chage of etwork state the eergy decreases to the mmum ad the state get balace. It coverges to a local optmal soluto. Sce the Hopfeld eural etwork (HNN) ca solve optmal problem such as TSP problem (Travelg Salesma Problem). The gas trasmsso optmal layout problem smlar to TSP. The Hopfeld eural etwork s proposed to solve the gas trasmsso system optmal layout. 2. THE ANALYSIS OF GAS NETWORK LAY-OUT The frst step of trasmsso costructo s survey ladform from gas sources to gas supply ctes. The trasport le s determed accordg to the survey formato. There are several solutos whe there are a lot of ctes ad the ladforms s complex ad maybe obstacles such as rvers or protect areas. It eeds to select the optmal le for the project. Geerally durg the early desg phase several solutos are gve for compare [8]. There are maly three form of trasmsso layout: truk, brach ad loop. The cost of the truk ad brach layout s lower. But the relablty of the loop layout s hgher tha the other two. Here maly aalyse the loop layout of gas trasmsso system [9].

The character of loop layout s gas ca supply from two drectos to the users. The locato of gas sources ad statos are stated. The loop ppele coects the gas sources ad the trasmsso etwork ad supples gas to ctes. I order to determe the coecto of supples the sgle loop etwork s adopted. The stato pass oly oce ad the objectve of layout s the mmum total cost. The latest study cosdered the cost s the lear fucto of ppe legth. The objectve smulates to the shortest total legth [0]. But the area grades of pass ladform are dfferet. O the le the ladform s dfferet ad maybe meets the obstacle such as rvers or hlls. The actual cost of local area exceeds the same dstace le. For example the materal of the hgher grade area s grater. So t s ot smple to cosder the shortest le as the layout objectve. I order to make the layout calculato geeral ad smple here gve the weght legth ad weght. Frst assume the total cost per legth as the stadard cost o the base of talzed dameters ad mmum dameter ad oe grade ad geeral sol. The total cost brought by crease dameter ad obstacle dvde the stadard cost per legth obtas the covert legth. The sum of covert legth ad actual legth s the weght legth. Just as followg. l w S = la + S The l w s the weght legth (km). The l a s actual legth (km). The S s s the stadard cost per legth (0 4 /km). The S p s the crease cost (0 4 ). Defed the weght w, The, S p w = + S l l w = wl So o the codto of the same stadard cost per legth, the objectve adopt the shortest weght legth o the le. The weght legth s the sum of actual le s weght ad actual legth. The testy desg of ppele wall thckess s dfferece accordg to the ladform. Accordg to the accout of house ad dese of buldgs there are four area grades. The testy coeffcet F lsted followg table. Rego grade 0.72 2 0.6 3 0.5 4 0.4 p S S a a Itesty coeffcet F Table : The testy desg coeffcet Accordg to the F pass area determe the weght of dfferet dameter ad thckess. Ad the weght of rver ad hll etc. ca determe by cost. 3. THE MODEL OF OPTIMAL LAY-OUT OF NETWORK The gas trasmsso system supply atural gas to ctes. The locatos of gas sources ad ctes are gve. I order to supply users relablty t eeds to supply users from two drectos.

Fgure : Locatos of gas supply ctes ad gas sources There are statos. ad j s the supply ctes. The l (= ; j= ) s the ppe legth betwee ad j stato. The v s the coecto mode. Whe v equals to 0 ad j s t coectg. Whe v equals to ad j s t coectg. The v compose the assocate matrx. The ppele etwork s loop. The gas ca flow reverse drecto. The etwork s o drecto etwork. Ad the assocate matrx s symmetrcal matrx. The objectve fucto adopts the mmum total cost of etwork. At the same tme gve the costrats. The model s followg. m 2 j=, j =, j = j= = j= s. t. υ = 2 ( =,2..., ) υ = 2 ( j =, 2..., ) υ = 2 w l S υ s Where w s the weght of betwee stato ad j. The l s the actual legth betwee stato ad j. ut s km. The S s s the cost per legth (km). The v s the elemets of cojucto matrx for gas etwork. The s the umber of statos. I the optmal model, m (mmze) s the objectve fucto. The s.t. (subject to) s the costrats. The objectve fucto adopts total cost of ppele layout. The stadard cost of ppele per ut legth s same. The objectve equals to total Correspodg legth. The model s followg. m 2 j=, j =, j = j= = j= s. t. υ = 2 ( =, 2..., ) υ = 2 ( j =, 2..., ) υ = 2 w l υ 4. DESCRIPTION OF HOPFIELD NEURAL NETWORK

Neural etwork method gets the tellget formato maages fucto by follow the bology dspose mode. Ad the method deals wth the mode formato laguage through complex jots ad the parallel dstrbuted mode. The basc characters of eural etwork are:. Parallel deal wth large scale. 2. Mstake tolerato. 3. Self adaptato ad self orgazato. The teracto betwee eutrals realze through the coecto weght. The structure of eural s followg. υ ω u υ j ω j I υ ω υ s Fgure 2: The structure of eural The u s the teral state of eural. The I s the threshold. The v j s the put sgal. The ω s the weght for the coecto betwee u ad u j. The S s the exteral put sgal. The put of eural s followg. u = ω υ + I s j j= The output of eural s the fucto of put. The output s v j =f (u ). The f s the fucto of eural. The ormal fucto has lear fucto ad sgmod fucto ad threshold fucto etc []. There are several eural coectg each other Hopfeld eural etwork. Gve the tal state of eural the etwork state wll reach a mmum that s the stable pot. Ad that s the stable state. The eergy of Hopfeld eural etwork ca be expressed by the state of all the eural of etwork. The eergy fucto of HNN s defed as followg [2]. υ E = ωυυ j Iυ + f ( ) d / R 2 υ υ 0 = j= = = Eergy fucto E s the fucto of tme. ω s the weght for the coecto betwee u ad u j. ω =ω j. The v s the exteral put sgal. The I s the threshold of eural. R s the postve costat. The frst tem of the rght of above equato meas the coecto weght betwee eural. The secod tem meas threshold of eural. The thrd tem meas the fucto betwee put ad output of eural. The etwork s composed by eural. The state vares by tmes ad reflects the varety of eergy of eural etwork. The eergy decreases by the varety of state. Just as followg. du dt E = υ I order to reflect the varety of state by tme adopts dyamc dfferetal equato. du dt u = + ω υ j + I R j = The u s the state varable of eural. The output of eural s the cotuous ad crease stmulat fucto of state varety. It ca adopt sgmod fucto. υ = f ( u ) = /( + e u ( T ) )

T s the parameter fluece the form of stmulat fucto. Hopfeld eural etwork s eergy fucto decreases by tme to the mmum. The problem cludes solvg the state of etwork whe the eergy fucto reaches the mmum that s the coecto stace betwee odes. 5. GAS NETWORK OPTIMAL LAY-OUT BASED ON HNN 5. The eergy fuctos of lay-out Accordg to Hopfeld eural etwork method, order to solute optmal problem, the erected model should base o the actual problem. The eergy fucto cludes objectve fucto ad costrat codtos. Just as followg. A B E = D w l + + 2 2 υ ( υ 2) ( υ 2) = j= 2 = j= 2 j= = j j C A + B + + 2 2 2 ( υ 2 ) υ ( υ ) = j= = j= j j The frst tem of the rght of above equato deotes the objectve fucto of gas etwork layout. Its meas the total legth s the shortest. The secod tem ad thrd tem meas that there are two elemets s ad the other s 0 every le ad every row ad every le cojucto matrx. The forth tem meas the maxmum of the sum of elemets s 2. The ffth tem esures the etwork output coverget to 0 or. Whe satsfy the costrats the last four tems equals to 0. The mmum of eergy fucto E respods to optmal solutos. Coeffcet A, B, C, D ad dstace l more tha 0. The eergy fucto E more tha 0. The followg judges the covergece of eergy fucto E. The tme partal dervatve of eergy fucto lst followg. E E υ du υ = = t υ t dt t j j du = ( ) j dt 2 υ u The output of eural s the cotuous ad mootoously creasg fucto of state varable. The Therefore υ j u j > 0 E t < 0 The eergy fucto s proved to be descedg by tme. It arrved to the mmum value. That s the stable state of the eural etwork. 5.2 The state equato of eural The dyamc equato of eural shows the chage process of state varable follow the tme. E υ Accordg to the eergy fucto expresso, get the dervatve ad the dyamc equato. = d u d t

du dt = D w l A ( υ 2) B ( υ 2) = j= = j= j= = j j C( υ 2 ) = j= j Accordg to the dyamc equato, the state ad eergy fucto are chaged. The Hopfeld eural etwork model was made cludg the eergy fucto ad eural dyamc equato. The soluto method cludes desgg electrc crcut to model dyamc equato ad computer modelg. The computer modelg arthmetc s smple. Here adopt computer modelg to solute the Hopfeld eural etwork model. 6. EXAMPLE AND ANALYSIS Accordg to the supplyg gas cty s locatos, the legth betwee statos s kow. The area grade alog the le ad the obstacle ad the ppe dameter are kow. The weght alog the le ad layout legth was determed. The optmal layout was gotte by the algorthm. The followg s the relatve coordates of ctes ad gas sources. Node x (km) y(km) 400.0 449.3 2 249.3 46.3 3 70.7 229.3 4 229.3 76.0 5 57. 94.4 6 873.2 653.6 7 687.8 52.9 8 848.8 360.9 9 668.3 253.6 0 69.5 263.4 Gas sources 00.0 800.0 Gas sources 2 900.0 600.0 Table 2: The relatve coordates of ctes ad gas sources The actual dstace betwee gas supply ctes ca be calculated by the relatve coordates. ode 2 3 4 5 6 7 8 9 0 0 338.4 37.8 355.4 505.8 55.4 296.8 457.4 332. 287.6 2 338.4 0 4.3 65.0 839.0 804. 577.4 636.8 432.5 388.3 3 37.8 4.3 0 534.9 79.9 820.7 594. 690.8 498.2 450. 4 355.4 65.0 534.9 0 339.7 652.8 57. 737.5 67.0 632.3 5 505.8 839.0 79.9 339.7 0 457.9 452.9 668.6 704.2 685.7 6 55.4 804. 820.7 652.8 457.9 0 227.4 293.7 449.4 465.4 7 296.8 577.4 594. 57. 452.9 227.4 0 227.7 269.0 267.4 8 457.4 636.8 690.8 737.5 668.6 293.7 227.7 0 20.0 249.2 9 332. 432.5 498.2 67.0 704.2 449.4 269.0 20.0 0 49.8

table 4. table 5. 0 287.6 388.3 450. 632.3 685.7 465.4 267.4 249.2 49.8 0 Table 3: The actual dstace betwee gas supply ctes (km) Accordg to testy desg coeffcet, the obstacle ad ppe dameter the weghts were gve Node 2 3 4 5 6 7 8 9 0 0.0.0.2.0.0.0.0.0.0 2.0 0.0.2.0.0.0.2.0. 3.0.0 0.0.0.0.0.0.0.2 4.2.2.0 0.0.0.0.0.0.0 5.0.0.0.0 0 5.0.0.0.0.0 6.0.0.0.0 5.0 0.0.0.0.0 7.0.0.0.0.0.0 0.0.0.0 8.0.2.0.0.0.0.0 0.0.0 9.0.0.0.0.0.0.0.0 0.0 0.0..2.0.0.0.0.0.0 0 Table 4: The weghts of legth betwee gas supply ctes The satsfed solutos were gotte through computer smulatg. The assocate matrx shows Node 2 3 4 5 6 7 8 9 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Table 5: The assocate matrx of layout of gas supply ctes Durg the solute process, the results show fgure 4 s the feasble soluto. Ths called feasble soluto. Fgure 5 s the satsfed soluto for ppele layout problem whose objectve s the mmum actual legth. Ths called actual satsfed soluto. Fgure 6 s the satsfed soluto for correspod legth. It called satsfed correspodg soluto whch s the optmal solutos. Fgure 3: Feasble Layout of ppele betwee te ctes

Fgure 4: Optmal layout of actual dstace ppele Fgure 5: Optmal layout of ppele wth weghts The followg table s comparso amog above solutos. Iclude terato umber, eergy fucto, ad actual legth ad correspod legth. Layout solutos Iterato umber Eergy fucto Layout legth(km) Correspod legth(km) feasble soluto 05 0.00002 2787.7 469.3 Actual legth satsfed soluto 74 0.00005 2698.7 459.9 Correspod satsfed soluto 57 0.0000 2749.3 2859.2 Table 6: The parameters of several layout solutos Accordg to the above comparso, whe cosder the weghts, correspod satsfed soluto s better tha actual legth soluto. The correspodg legth ca reflect the egeerg cost of ppele etwork. From the optmal result, the eergy fucto decrease alog wth the creasg terato umber. The eergy fucto gets the stable value whch s the mmum. The etwork state gets to the stable state. The soluto of layout problem s t the optmst soluto but the satsfed soluto. It s dffcult to solve complex etwork adoptg Hopfeld eural etwork. Here the layout s sgle le to trasmt the gas. The sample s smple, but t show the self adapt ad self search ablty. I order to crease the relablty of future trasmsso system, the crcle etwork s better ad the complex atural gas etwork layout soluto s mportat. The Hopfeld eural etwork optmal method s feasble for atural gas trasmsso system layout. Ad adoptg correspod legth reflect actual egeerg. 7. CONCLUSION The atural gas trasmsso system cludes several kds of elemet. Ad the optmal desg area s ew area eeded to be mproved. There are several aspects to be studed farther. The ecoomcal parameter data eeded to be statc ad aalyzed. Hopfeld eural etwork method ca be used to solve the atural gas trasmsso etwork layout optmal problem. It s eeded to make the total

cost model ad the correspodg legth model. Accordg to the model, get the eergy fucto ad eural dyamc equato. The method smulates the layout of ppele etwork. Hopfeld eural etwork method used to solve optmal layout of gas trasmsso system solve the dmeso obstacle dyamc programmg. But oly ppele legth s t eough. There are detal thgs such as partcular ladform ad spa etc. The cost s t lear wth le legth. There are localzatos before. The mprovemet s adoptg cost replace legth ad cosders other factor. Parameter optmzato ad layout are combed. The complexty creases correspodgly. The result shows correspod legth result reflect actual egeerg better ad the soluto s the optmal layout. REFERENCES. Iteratoal gas uo (998).The proceedgs of the 20th world gas coferece. 2. Hu Yuqua (998). Operatoal research ad applcato. Ht publsh. 3. Federgrue, Aw. Groeevelt, Hery (986),.Optmal Flows Networks wth Multple Sources ad Sks, wth Applcatos to Ol ad Gas lease Ivestmet programs. Operatos Research. 34(2):28~225 4. Nrwa Asar Edw Hou. (L Ju ad Ba Zhaoq) (999). Computatoal Itellgece for Optmzato. 5. Wag Shtog (998). Fuzzy system fuzzy eural etwork ad applcato. 6. W. Pedrycz, J. Davdso, I. Goulter (992). Neural Network Based Decso Model Used for Desg of Rural Natural Gas Systems. IEEE Iteratoal Coferece o Fuzzy Systems.29~226 7. J. Davdso, I. Goulter (99). Rule-based Desg of Layout of Rural Natural Gas Networks. Joural of Computg Cvl Egeerg. 5(3):300~34 8. Ne Tgzhe (2004). Fuzzy optmzato desg ad relablty desg of atural gas ppele. PhD Dssertato. 9. L Chagju (2000). Natural gas trasmsso. Petroleum publshes. 0. Ne Tgzhe ad Dua Chaggu (2005). Layout Optmzato of Gas Trasmsso System by Hopfeld Neural Network. NATUR. GAS IND. 2:55-57.. Zhuag Zhequa, Wag Xufa etc (992). Neural etwork ad eural computer. 2. Wag Lg (200). Itellget optmzato algorthms wth applcatos. Tup, Sprger publsh.

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