ECONTECHMOD. AN INTERNATIONAL QUARTERLY JOURNAL 2015, Vol.4, No.3, MATHEMATICAL MODELLING OF DYNAMIC PROCESSES IN GAS TRANSMISSION

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1 ECONTECHMOD AN INTERNATIONAL QUARTERLY JOURNAL 5 Vol4 No MATHEMATICAL MODELLING OF DYNAMIC PROCESSES IN GAS TRANSMISSION N Prytula PJSC UKRTRANSGAZ Th Brach Rsarch Isttut of Gas Trasportato ; Ctr of Mathmatcal Modllg of IAPMM amd aftr YaSPdstryhach NAS of Ura; azarprytula@gmalcom Rcvd July 75: accptd August 85 Abstract Th problm of fdg th paramtrs of th gas trasmsso systm trms of ts oprato udr ustady opratg mods ar cosdrd W show mathmatcal modls of gas flows th basc faclts th systm wth a complcatd ppg dagram Th soluto s bult to solv a systm of partal dffrtal quatos wth th ft lmt mthod of larg dmso wth boudary codtos som of whch ar dsgd th procss of solvg th systm of quatos Ths s du to th provso of tchcal ad tchologcal costrats o th prssur of th cotrolld pots of th systm O th rsstac to fdg a soluto sgfcatly affcts a stp of th tm coordat Th proposd algorthm th dtg procss of flow dagrams provds a cras stablty of solvg systms of quatos ad sgfcatly rducs th tm for th smulato Ky words: trasmsso systm comprssor stato optmal opratg mod gas flow modl twor modl ppg dagram ft lmt mthod INTRODUCTION Th calculato of th paramtrs of th gas ppl systm s a complx mathmatcal problm Ma gas ppls wth comprssor statos ad othr faclts ar a sgl faclty all paramtrs whch ar hydraulcally trcoctd Chagg th mod of oprato of a dvdual obct s to chag th mod of th tr systm Th gas trasmsso systm (GTS) blogs to a class of olar systms wth dstrbutd paramtrs whch ar charactrzd by a twor layrd structur a larg spac ad tm dmso ad dstrbuto th prsc of cotuous ad dscrt cotrol actos a hgh lvl of ucrtaty of obctvs structur proprts ad stats as wll as flucs from th vromt Trasmsso systm whch obcts ar hydro thrmodyamc fltrato ad othr physcal procsss s th subct of cotrol Th physcal atur of th gas allows a wd rag to chag ts paramtrs - prssur tmpratur ad volum I ths rgard th gas-dyamc procsss that ta plac th faclts of th systm ar dscrbd by complx mathmatcal dpdcs Th olarty of th dyamc procsss that volv ovr larg spatal ad tm dmsos th wa prdctablty of put paramtrs a sgfcat fluc of xtral factors ad suffct mtrologcal support gratly complcat both th formulato ad soluto of problms of th aalyss optmzato ad fdg cotrol paramtrs of gas flows Not yt fully solvd th problm of cratg modllg softwar for optmzato of ustady opratg mods of gas trasmsso systms Th ffctvss solvg ustady flows of gas trasmsso wll mprov th opratv dspatch cotrol of gas flows Th wor for fdg solutos of th corrspodg olar systms of partal dffrtal quatos that dscrbs th procsss that ta plac th faclts of th GTS usd tratv approachs whch ar basd o th larzato of th orgal systm wth a furthr rfmt of th soluto by calculatg th rspctv dscrpacs THE ANALYSIS OF RECENT RESEARCHES AND PUBLICATIONS I gral th smulato of th trasmsso of gas through ppls s rducd to th aalyss of th complt systm of quatos of gas dyamcs whch cotas th quato of cotuty chag of momtum rgy ad th quato of gas stat Th complt systm of partal dffrtal quatos of gas dyamcs s grally a o-lar ad ts soluto s coctd wth cosdrabl dffcults At ths tm thay bult a sgfcat umbr of ways to solv problms of ths typ [-6] All of thm ca b dvdd to svral classs: aalytcal umrcal approxmat ad tratv Each has ts advatags ad dsadvatags As practc shows th choc of modls ad mthods dctatd maly problms that d to b solvd o th bass of th costructd mathmatcal modl Th o of ffctv soluto approachs hav larzato of th tal quatos wth th followg rfmt of solutos basd o th buldg of tratv procdurs Most modllg tchqus that hav practcal us lad to th cssty of solvg systms of dffrtal or algbrac quatos of hgh dmsoalty Ths rass ssus of rsstac ad th mmum tm for solvg systms wth guaratd accuracy

2 58 N Prytula Publcatos o th subct of th wor carrd out by may groups [7-] Attto should b gv oly to thos wors that hav b tstd o ral data ad ca b usd for th dvlopmt of modr systms of dspatchg maagmt Rqurmts for such wors ar wll-ow - thy hav to wor stadfastly th tr rag of gas-dyamc procsss that produc rsults th modlg of gas-dyamc procsss systms wth complx ppg dagram dos ot rqur smplfcato or quvaltg parts of th faclts to b adaptabl to th dffrt charactrstcs of th actual oprato of gas trasmsso systms At ths tm th modls of gas flows for dvdual faclts ar suffctly dvlopd Th ma problms assocatd wth th dvlopmt of mthods ad algorthms for th mplmtato of modls of dvdual ad trrlatd systms of faclts Ths rqurs a mmum smplfd modls of systms provdg stablty of smulato procss th whol rag of flow paramtrs ad mt may of th tchcal ad tchologcal lmtatos to mmz th tm of obtag rsults ad maxmzg th automatd procss of formulatg ad solvg problms A crta part of th wors that rlatd to th dvlopmt of commrcal products thy ar typ of advrtsg ad advrts oly fuctoal part of th dvlopd systms rathr tha mathmatcal apparatus mplmtato Ths appls to softwar systms: Astra (Russa) SIMONE (dsgr SIMONE Rsarch Group sro) ad dvlopmts whch ar opratd by Schlumbrgr Th most formatv cofrc procdgs ar partcular cofrcs whch ar hld "Gazprom VNIIGAZ" [-4] MAIN RESULTS Ustady o-sothrmal modl of gas flow Ustady osothrmal gas flow ppls s dscrbd by th systm of quatos [569]: ( ρv) λv v dh + ( p + ρv ) = ρ + g t D dx ρ + ( ρv) = () t P 4 ( Tгр T ) dh ( ρe) + ρv E + = ρvg t ρ D dx whr: ρ - th dsty of th gas; p - gas prssur; υ - gas vlocty; λ - coffct of hydraulc rsstac; - coffct of hat trasfr from th ppl to th groud; T ad T - th gas tmpratur ad th sol гр tmpratur rspctvly; h - th dpth of th ppl; E - total rgy pr ut of gas mass; g - acclrato of gravty; D - th damtr of th ppl; x x [ l] - th coordat of th ppl; l - th lgth of th ppl To clos ths systm of quatos usg th quato of gas stat: p = ρ zrt () whr: z - comprssblty factor that charactrzs th dffrc btw a ral gas from th dal ad s dtrmd o th bass of mprcal rlatoshps bult [] R - gas costat Gas comprssblty factor s calculatd for th wllow formula [-3]: z = + f ( a bp) p (3) whr: a ad b - approxmato coffcts calculatd z for th ow procdurs - procdurs for calculatg gas comprssblty factor such as Hall-Yamburg f = 4 t C hr Rdlch-Kwog ad othrs ( 4 ) p( x ) - masurd atmosphrs Wh stady-stat codtos of gas flow th tmpratur dpds oly o th coordats ad calculatd by th formula: whr: T ( x) = T + T ax π D T = Tг T T = T Tг + T a = C M p = p p T g h = p D + al cpρ c p Th Ryolds umbr s calculatd by th formula: Dvp T + C 73 R = µ RT 73 + C T whr: M = ρq - th mass flow rat; T - tmpratur of th flow gas to th ppl; T gr - tmpratur of th sol; D h - Joul Thomso coffct; 3 p C p - hat trasfr coffct from th gas to th sol; h - hght dffrc btw th d ad th bgg of th ppl; p - th valu of th prssur at th bgg of th ppl; p = p( x) - th dstrbuto of th prssur alog th ppl; ρ - th gas dsty udr ormal codtos; x - rug coordat Gas costat ad comprssblty factor usually ta costat (3) Th larzd systm by vlocty occurs for horzotal ppls crta rags of vlocty chag alog th ppl Th larg-damtr pps wth sgfcat volums of gas trasmsso must ta to accout th chag ts tc rgy to whch lttl attto publcatos for th subct Larzato mthod whch allows you to spcfy th larzd modl ad tratv procss to buld solutos of th systm s as follows f v = v wll rplac th chord ad Th curv ( ) tagt Sc th trasmsso of gas vlocty chags from v to v th th quato of th chord ad th tagt that passs through th pot:

3 of vw ( v) a v b MATHEMATICAL MODELLING OF DYNAMIC PROCESSES 59 ( v f ( v )) ( v f ( v )) ϕ = whr: = ar for chord ad = - a tagt Ad whr: a = a = v + v b vv b = 5 v + v Gv th abov th systm () a sothrmal mod of flow ca b wrtt as: or ( ) = та ( ) ρ v p λ a λ b λ ρ + + ρ v ρ = v ϕ ( v) t x D D D ( ρ v) p = x c t whr dcatd: ω p λ ρ + + a ω b p = v ϕ ( v) t x D (4) p ω + = c t x b λ b = = v DgzRT ω λ a a = D To dtrm th hydraulc rsstac of th gas flow thr s mprcal formula: α4 α α3 λ = α ш + R D ρ (5) whr: α = 4 - ar som ow costats ш - roughss of ppls ( ш ad D mm) µ - th dyamc vscosty udr ormal codtos C - th Suthrlad s costat Formula (5) gvs th gratst rror udr ustady opratg mods (lamar to turbult flow ad vc vrsa) Prst systm () s a y systm of quatos of ustady modl of o-sothrmal gas flow th ppl tag to accout th profl of th ppl Ustady flow modl of gas trasmsso twors [9-] Mathmatcal modl of th gas trasmsso systm s basd o ts ppg dagram Structural proprts of th ppg dagram affct both th dmso of th systm (systm modl) ad th complxty of ts solutos Coductd umrcal xprmts showd that som smplfcato of th ppg dagram provds gratr stablty ad rduc th tm for solvg th corrspodg systms Th basc graph opratos such s th uo of dgs ad cotracto of th dgs to th vrtx Importat s th squc of opratos Buldg ad dtg ppg dagrams GTS ppg G = V E whr V - th dagram prstd as a graph ( ) st of vrtcs (ods) E - a st of dgs Edgs rprstd obcts that hav xtso spac vrtcs - all othr obcts I th cas of mathmatcal modlg of procsss that ta plac th dgs ach of whch s dvdd to a crta amout (whch dpds o th lgth of a dg) sgmts Th d rsult s th calculato of ustady flow whch ta plac th GTS s rducd to solvg th systm of quatos A mportat prformac crtro s th choosg of th mmum umbr of sgmts of dgs to rcv th smallr dmso of th systm of quatos ad thraftr t wll b solvd O th othr had - for hghr accuracy smulato of ustady procsss d mor of ths sgmts It should thrfor b a balacd approach to rduc th dmsoalty of th systm of quatos Th obvous fact s that th graph must ot cota zro lgth dgs or damtrs thrfor ths dgs ar dtfd wth o of th vrtcs (ths cluds op valvs bypass valvs ad tc) It s also approprat to cosdr a squc of dgs that hav th sam damtr as o dg That s f crta adact dgs = ( ) ( v v ) v v + D = hav th sam damtr + + D D < ε whr ε D - th tolrac valu for th damtr of dgs ad + ad assg a valu to th lgth of a w dg qual to th sum of th two combd L * = L + L + ad D * = D Aothr paramtr that allows you to smplfy th systm of quatos s to st a mmum lgth of dgs th graph L sh If a dg s shortr tha L sh t s dtfd wth a vrtx Ths rducs th umbr of dgs ad thrfor th umbr of quatos By choosg th valu L sh cssary to ta car gv that th volum of th gomtrc dgs of th modfd graph was ot sgfcatly dffrt from th orgal graph of th GTS as wll as th topology was ot chagd Cosdrg t s also cotmplatd ot to coduct shrag of th dg dspt th fact that L < Lsh It should also b otd that crta paramtrs of th vrtx v of ay cdt to th ot dgs (prssur or flow or outflow of gas) whch s abst th modfd graph should b cosdrd at th vrtx of th start v поч or d of th rsultg dg R Th algorthm s mplmtd choc of th vrtx wth rgard to th dstac corrspodg to th vrtcs that s f L v v L v v th th chag wll occur to th ( ) ( ) поч пкін vrtx v поч Ntwor modl of ustady flow [4-5] Sc wh ustady gas flow Krchhoff's scod law s ot fulflld th th dsg of th corrspodg mathmatcal modl should b carrd out o othr prcpls - cosrvato quatos For as dscrpto cosdr a systm wth a sgl vrtx that cotas M ppl sctos Suppos that

4 6 N Prytula th systm cossts of th M put ad M out output sctos that ar dxd th approprat ordr Dot x th pot of cocto Dot th lgth ad th J damtr of th - scto through L ad rspctvly ad ( = M ) O Each st w choos a pot { xj } S whch s farly clos to x J (th "+" or "-" th dx dpds o th drcto of gas flow) Th for scto systm () tas th form: Smulato of comprssor statos [6-8] Th modl of a comprssor stato (CS) s basd o th modl of th structur ad th modls of ts faclts Th structur modl s rprstd as a graph whch th obcts that hav xtso rprstd as dgs ad all othrs as vrtcs Th ma obct s th gas comprssor ut (GCU) ad cosst of th g ad th ctrfugal comprssor (CC) It s ow [6] that th paramtrs of th gas flow ad outflow of CC ar assocatd by a st of mprcal quatos: ω( t x) p( t x) + + Cωω( t x) + CP p( t x) = t p( t x) ω( t x) + c = = M t whr: c - th spd of soud th gas ω = ρv λ c C v g dh ω = CP = S zrt dx v - avrag spd c { } x wh M or xj { } x x L J + wh M > (6) ε ϕ = [ q] p H p 3 = N γ ϕ 3 вих вих вх вх ( ) p η ϕ [ q] пол ([ q] ) p = η пол T z = T z ε p н t t pa N N K = N Kt t Othr opratg paramtrs of th GCU folow th quatos: Gv th qualty of prssurs for all sctos at th pot of cocto ad th Krchhoff's frst law w wll hav modl of th gas flow th vcty of th cocto vrtx: ω ( t x) p( t x) + + Cω ω( t x) + CP p( t x) = = M t p ( t x) ω( t x) + c = t { p( t xj )} = p( t xj )} = M M M { Sω( t xj )} { Sω( t xj )} = = = M + whr: x { } { } wh M or xj x x L J + wh M > (7) Systm (7) wll dscrb th gas flow th M sctos whch ar coctd at o pot x J Ths approach ca b asly gralzd to th cas of mor complx twor structur (wth lots of vrtcs) To do ths w dsg a drctd graph whos dgs corrspod to sctos of th ppls Th for vry dg ad vrtx (vrtcs dgr whch s gratr tha ) w ca wor out th st of quatos (6) ad (7) rspctvly To fd a umrcal approxmato of th soluto (6) (7) t s advsabl to prform a spatal ad tmporal samplg of th modl O ach tm stp w ta th umbr of tratos to solv th larzd systm of quatos wth suffct accuracy ad hav solutos of olar quatos () N t + 73 p a pg = pg t ( ) N t q q K q pg = 86N 3 η Q N : ( ) = N ηmk N whr: - spd of th CC q - th gas flow rat through th CC η пол - polytropc ffccy q pg - omal flow rat of ful gas ε - prssur rato N - ratd powr of th th gas turb; K - coffct of tchcal stat of N th gas turb; K t - coffct whch tas to accout th ffct of ar tmpraturr; t - tmpratur of th ar at th lt of th gas turb; t - omal tmpratur of th ar at th lt of th gas turb; p a - th absolut prssur of th ar dpdg o th hght abov sa lvl H; Q - th omal lowr hatg valu; η - ratd polytropc ffccy; η m - mchacal ffccy tchcal codto accordg to powr K N - Th dvlopd algorthm of CS oprato for a st of put data ( ρ c P P T q{ M }) (th dsty of gas at stadard codtos th gas prssur at th lt outlt gas prssur gas flow rat cout of GCU ach worshop) calculats th opratg mod of th CS ( T s ε q N ) (th outlt tmpratur; schm of pg GCU coctos; - stag umbr; - umbr of GCU th stag; - th typ of GCU; CC spd; prssur rato; th amout of ful gas ad gas turb powr) Idcs ( ) { N N } ( N - st of stags of CC; N - st of CC at - stag)

5 MATHEMATICAL MODELLING OF DYNAMIC PROCESSES 6 To calculat opratg mod paramtrs at tm ( ) t t = t + t s cssary to solv a systm of + + quatos for th uows q (volum flow rat) ad P (gas prssur) I ths systm of quatos amog othr codtos ar mt parg If w cosdr th CS as th dg of th GTS t s cssary to st th comprsso rato ε whch rachd by CC worg for a gv powr W = ( W W ) whr - th cout of worshops of th CS ad ths dg wll provd th fulfll of th quato P = P ε To calculat ε w hav ralzd fucto that calculats th opratg mod of th CS ( T s ε qpg N ) usg data ( ρ c P P T q{ M }) Trasto of th CS to th o opratg mod occurs usg comprsso rato ε = Turg o CS s prformd wth N stps of approchg comprsso O rato ε from ε m to ε dfault Th thr s th opratg mod of th CS th stuato that has occurd ad w fx th powr of th CS Chagg powr of th CS (cras or dcras) s proportoal to all th opratg worshops by a crta prctag κ I ths cas κ dfd as th prctag dvato of th motord paramtrs from th optmum valu Th boudary codtos At th puts ad outputs of th twor of gas ppls ar st boudary codtos o th prssur ad flow rat Som boudary codtos ar calculatd durg th smulato of gasdyamc procsss O calculato of th boudary codtos hav a sgfcat mpact tchcal ad tchologcal rstrctos: th wor pots o th charactrstcs of CC surg ara; CC maxmum volum flow rat; CC shaft spd m max ; maxmum powr of th gas turb; CC maxmum tal prssur whch s dtrmd by th strgth of ppls at th lt of CC; th maxmum tmpratur at th outlt of CC s dfd by th sulato coatg ppls; mmum valu of prssur at th outlt of ach CC; codtos of th cosstcy of cocto schm wth lt ad outlt ppls Th umrcal xprmt Th umrcal xprmt was coductd o a ral gas trasmsso systm whch blogs to o of th dpatmts of th PJSC "Urtrasgaz" (s Fg) Th ma obctv was to aalyz th mpact of topology chags o th dyamcs of chags th flow paramtrs th dstrbuto ovr tm Th chag of topology s th opg of thr taps at a spcfd tm Bfor th smulato was carrd out to dtfy modls of gas flows th faclts of th systm Opg taps lastd 3 muts Stp-by-tm varabl was qual t = хв Aftr smulatg paramtrs of th gas flows w got valus for ach faclty typ vrtx - prssur ad tmpratur ad typ of dg volum flow rat ovr a smulato tm Th smulato rsults as a xampl ar show for o of th valvs Fg Th currt systm of fdg of th boudary codtos ad smulato of ustady mods wth varabl stp provds smulato ad cotrol valvs - prssur rducrs ad valv systms for hydraulc cotrol fuctos ad protcto (cotrol valvs MOKVELD) Th rato of smulato tm wth a complx ppg dagram to th ral-tm trasmsso of gas dyamc procsss s th rag : 5 - : whch s compltly accptabl for practcal us Th ma factor of fluc o th stablty of th smulato - chag stats of a fw taps smultaously I ths cas stablty of th mthod s shurd by a spd of modl gas dyamc procsss th vcty of taps as wll as th rducto of th tm stp th umrcal aalyss Fg Ppg dagram of gas trasmsso systm of ppl oprator "Lvvtrasgaz"

6 6 N Prytula Fg Th graph of th gas volum flow rat (abov) ad th graph of th gas lt prssur of tap (bottom) CONCLUSIONS Th proposd approach to th calculato of th paramtrs of gas trasmsso systms wth complx ppg dagram udr th codtos of ts oprato ustady codtos has b tstd o ral data ad dmostratd a hgh lvl of stablty Stablty of th mthod for solvg systms of quatos of larg dmso s surd by dtg optos of ppg dagram ad adaptv mthods for passag spd of gas-dyamc procsss 3 It s rmas th op problm of optmal cotrol trasto opratg mods from th currt to a crta optmal stady mod REFERENCES Colquho I Mmdz A ad Dovco R 998 Mtod ylds safty for -l spcto data Ol &Gas J Vol 96 Nr Roach P J 976 Computatoal Flud Damcs Rvsd Prtg Hrmosa Publshrs Albuqurqu NM 66 3 Harlow F H 964 Th Partcl--Cll Computg Mthod for Flud Dyamcs Mthods Computatoal Physcs Acadmc Prss Nw Yor Vol 3 4 Amst AA 966 Th Partcl--Gll Mthod for th Calculato of th Dyamcs of Comprssbl Fluds Los Alamos Sctfc Laborator Rport LA Los Alamos 69 5 Sardaashvly SA 5 Calculato mthods ad algorthms Moscow: Publshg Hous of th "Ol ad Gas" 577 ( Russa) 6 Slzv VE Alsh VV ad Klsh GS Mthods ad tchqus of umrcal smulato of ppl systms Moscow: Edtoral URSS 448 ( Russa) 7 Evdomov AG Dubrovsy VV ad Tvyashv AD 99 Modllg ad optmzato of flow grg twors Moscow: Stroyzdat 365 ( Russa) 8 Novtsy NN Suharv MG Tvyashv AD ad t al Ergy Ppl systms: mathmatcal modlg ad optmzato Naua Novosbrs 49 ( Russa) 9 Tvyashv AD Smrov VS 8 Mathmatcal modlg of ustady flow of osothrmal gas ppl scto Elctrocs ad Computr Scc -7 ( Russa) Kul N Lptsy IK Lshcho ICh Uslotsv AB Mtg th challgs of opratoal maagmt mod of gas trasmsso systms basd o ustady modls Problms of gral rgtcs 8-5 ( Russa) Lscho ІCh 7 Optmzato of oprato of comprssor worshops as a mas of mprovg th rgy ffccy of gas trasmsso Problms of gral rgy ( Uraa) Prytula N Paylo Ya ad Prytula M 6 Calculato of stady gas flow th ma ppls Bullt of th Natoal Uvrsty "Lvv Polytchc" Computr Scc ad Iformato Tchology ( Uraa) 3 Prytula N 7 Calculato of th flow dstrbuto th gas trasmsso systm (stady cas) Physcal ad mathmatcal modlg ad formato tchologs ssu ( Uraa) 4 Sіchu Yu Prytula N Prytula M Smulato of taps o gas ppls Bullt of th Natoal Uvrsty "Lvv Polytchc Computr Scc ad Iformato Tchology ( Uraa) 5 Sіchu Yu Prytula N ad Prytula M Smulato of ustady mods of gas twors Bullt of th Natoal Uvrsty "Lvv Polytchc Computr Scc ad Iformato Tchology ( Uraa) 6 Boyar I Borovoy R Grіv O ad t al Optmzato of th multshop comprssor statos wth a multtypd gas comprssor uts Bullt of th Natoal Uvrsty "Lvv Polytchc" Computr Scc ad Iformato Tchology ( Uraa) 7 Prytula N Problms of flow cotrol gas trasmsso systm Bullt of th Natoal Uvrsty "Lvv Polytchc" Computr Scc ad Iformato Tchology ( Uraa) 8 Prytula N Ad Gryv O 3 Modlg of comprssor statos wth adact ppls Bullt of th Natoal Uvrsty "Lvv Polytchc" Computr Scc ad Iformato Tchology 75 ( Uraa) 9 Prytula N 3 Adaptv algorthm of fdg a mathmatcal modl of th gas trasmsso systm wth varabl structur Bullt of th Natoal Uvrsty

7 MATHEMATICAL MODELLING OF DYNAMIC PROCESSES 63 "Lvv Polytchc" Computr Scc ad Iformato Tchology ( Uraa) Dols A Dragaov B Kozrs V Noqulbrum Stat of Egrg systms ECONTECHMOD A tratoal quartrly oural Vol Wróbl K Styła S ad Sumor A Us of GIS Systms th Costructo of Hydraulc Modl of Ntwors Ecotchmod A Itratoal Quartrly Joural Vol Pyaylo YaD Prytula NM Prytula MG ad Lopuh NB 4 Modls of mass trasfr gas trasmsso systms Mathmatcal modlg ad computg Vol Nr Datsyu AV Frolov V Gladu SV ad t al 7 Computr tchologs for support dcso mag th maagmt of gas trasmsso ad producg systms 3rd It sctfc ad grg Cof "Computr tchology support dcso mag th cotrol ad maagmt of gas trasport ad producg systms": Moscow 7 ( Russa) 4 Prytula N Prytula M Gryv O ad t al Th calculato of th optmal paramtrs of flow cotrol gas trasmsso systm 5th It sctfc ad tchcal cof "Computr tchology support dcso mag th cotrol ad maagmt of gas trasport ad producg systms": abstracts Moscow - 63 ( Russa)

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