Chaotc Modelng and Smulaton (CMSIM) 1: 95-108 014 Stochastc modelng of hydaulc opeatng paametes n ppelne systems N.N. Novtsy 1 O.V. Vanteyeva 1 Enegy Systems Insttute Sbean Banch of the ussan Academy of Scences 130 Lemontov st. Iuts 664033 ussa (e-mal: ppenet@sem.se..u) Enegy Systems Insttute Sbean Banch of the ussan Academy of Scences 130 Lemontov st. Iuts 664033 ussa (e -mal: vanteeva@sem.se..u) Intoducton he poblems of calculatng hydaulc opeatng paametes ae the basc poblems n the analyss of opeatng condtons of ppelne systems when desgned opeated and contolled. hese poblems ae tadtonally solved usng models and methods whch howeve do not allow us to quanttatvely assess the satsfablty of opeatng condtons when consumpton s andom whch s typcal of many pactcal stuatons. hs s explaned by hgh complexty and dmensonalty of ppelne systems (heat - wate- gas supply systems etc.) as modelng obects excessve effots necessay to apply geneal methods of stochastc modelng (such as the Monte -Calo method) and dffcultes n obtanng ntal statstcal data. he pape pesents an appoach a set of mathematcal models and methods fo modelng the opeatng paametes of ppelne systems that wee developed n tems of stochastcs and dynamcs of consumpton pocesses and the establshed ules of the contol whch mae t possble to atonally combne the adequacy of modelng and ts hgh computatonal effots [1 ]. Poblem statement of the pobablstc calculaton of hydaulc opeatng paametes. Pobablstc descpton of defnte hydaulc opeatng paametes s educed to the pobablty densty functon whch s denoted hee by p( ) whee the value of a andom vecto of opeatng paametes (pessue flow ate etc.); dstbuton paametes. Most of the pactcal cases allow us to use the hypothess about nomal dstbuton of. hen { C} and the pobablstc descpton of hydaulc opeatng paametes can be educed to the specfcaton of values of mathematcal expectaton ( ) and covaance matx ( C ) fo value. Not evey combnaton of components s acceptable snce they eceved: 6 August 013 / Accepted: 30 Novembe 013 014 CMSIM ISSN 41-0503
96 Novtsy and Vanteyeva should satsfy the equatons of flow dstbuton model U () 0 (whee U non-lnea vecto functon). hese equatons esult fom geneal physcal consevaton laws and hence should be solved detemnstcally. he tadtonal detemnstc model of steady hydaulc opeatng paametes n a ppelne system as a hydaulc ccut wth lumped paametes can be epesented as [3] Ax Q U ()( )( U Y ) U x Q P 0. (1) A P f ( x) Hee the fst subsystem of equatons epesents the condtons of mateal balance at the nodes of hydaulc ccut (equatons of the fst Kchhoff law); the second subsystem the equatons of the second Kchhoff law; bounday condtons; Y unnown opeatng paametes; tansposton sgn; A m n - ncdence matx wth elements a 1( 1) f node s the ntal (end) node fo banch a 0 f banch s not ncdent to node ; m n numbe of nodes and banches of the hydaulc ccut; x n-dmensonal vecto of flow ate n banches Q P m-dmensonal vectos of nodal pessues and flow ates f ( x) n-dmensonal vecto-functon wth components f ( x ) eflectng the laws of hydaulc flow fo the banches; n -dmensonal vecto of paametes of these chaactestcs. Fo nstance f f ( x ) s x x H then { s H } whee x flow ate n the -th banch; s hydaulc esstance of the banch; H 0 ncease n pessue n the case of an actve banch (e.g. a banch epesentng a pumpng staton); H 0 n the case of a passve banch (e.g. a banch epesentng a ppelne secton). If n (1) all paametes s H 1 n ae set detemnstcally then ( ). x Q P hus the pobablstc model of steady flow dstbuton can be epesented as U () 0 ~( N) C whee N dmensonal nomal pobablty dstbuton; dmensonal of vecto. In the case of nomal dstbuton of f we neglect the non-lnea dstoton of dstbuton p[() Y Y ] () (whee Y () mplct functon gven by the flow dstbuton equatons) the poblem can be educed to the detemnaton of { C } wth the gven functon { C } and unde condton U ()( ) U 0 Y. Moeove the composton of should povde solvablty of equatons U ( ) Y 0 wth espect to Y.e. dm() Y dm() Uan( /) U Y whee U / Y Jacoban matx (of patal * devatves) unde fxed bounday condtons n the neghbohood of the * soluton pont Y dm() vecto dmensonal an() matx an.
Chaotc Modelng and Smulaton (CMSIM) 1: 95-108 014 97 Methodologcal appoach. Let () be a andom devaton of possble ealzaton of bounday condtons fom ts mathematcal expectaton. Afte lneazng functon Y () n the neghbohood of we obtan Y Y ()( /) Y whee Y / s devatve matx at pont. Snce E() Y Y and E() 0 whee E s the opeaton of mathematcal expectaton then Y Y (). hus the mathematcal expectaton of unnown opeatng paametes ( Y ) s the functon of flow dstbuton equatons unde bounday condtons. Coespondngly () Y Y () and whee C CY C E C Y Y Y C Y Y Y Y Y CY E Y Y E C Y Y CY CY E() Y E C Y () Y Y. hus the geneal scheme fo solvng the poblem of pobablstc calculaton of hydaulc paametes s educed to the followng: 1) to obtan vecto Y by tadtonal methods fo calculatng the flow dstbuton wth the gven ; ) to detemne matx C whose ndvdual blocs ae detemned usng the nown matx C and devatve matx Y / at pont. Hee two man questons ase: 1) based on what do we set the dstbuton paametes of bounday condtons ( { C } ); ) what s the fnal fom of elatonshps fo the esultant covaance matces n dffeent vaants of the dvson of nto and Y snce n the tadtonal methods fo the flow dstbuton calculaton the devatves Y / ae not calculated n explct fom whch epesents a sepaate poblem. Pobablstc descpton of consume loads. A typcal example of ppelne systems opeatng unde the condtons of stochastc consume loads s wate supply systems. he appoach appled to the pobablstc descpton of these stochastc condtons s based on the use of the queung theoy methods and on esults of the studes [4 5 etc.] whch found the eflecton n the egulatoy documents [6]. Accodng to these esults the pobablty of usng plumbng unts ( p h ) can be descbed by Elang fomulas whch demonstate a dscete lmt dstbuton of used channels
98 Novtsy and Vanteyeva dependng on the chaactestcs of the flow of equests and the pefomance of the queung system. he suggested technque fo calculatng the mathematcal expectaton of consume flow ates ( q ) and the vaances ( ) consst n the h followng: 1. Knowng the numbe of plumbng unts at the consumpton node ( N ) and the pobablty of usng them p h [6] we can calculate m mh such that maxmum value ( p () max m ) acques the pobablty whee N Z N p 0! p() m h N p h m! m Z q h m 01... N (3) m s the numbe of smultaneously used plumbng unts; Nph s the usage ate.. We should detemne the aveage houly flow ate qh mh q0 h whee q0 h q0 h / 1000 houly wate flow ate by one devce m 3 /h; q h can be ntepeted as the mathematcal expectaton of flow ate at the consumpton node; q standadzed value l/h. 0h 3. When appoxmatng the dscete Elang dstbuton by the contnuous nomal dstbuton we should calculate the equvalent vaance by fomula m h 1/ () pmax m. 4. he vaance of the aveage houly flow ate wll be detemned as q. q h 0 h m h p h Fgue 1 pesents a dagam of functon (3) whee N=70 and =0.03. he dagam shows that the maxmum pobablty densty functon coesponds to m h whose aveage houly flow ate s q h. Geneal scheme of obtanng the covaance matx conssts of thee stages: 1) to lneaze system (1) at pont ; ) to educe lneazed U Y system 0 to Y ; 3) to obtan covaance matx of the vecto of unnown opeatng paametes C usng the opeaton E. Y Y
Chaotc Modelng and Smulaton (CMSIM) 1: 95-108 014 99 р 018 016 014 01 01 008 006 004 00 0 m pcs. 1 3 4 5 6 7 8 9 101111314151617 m h Fg. 1. Contnuous appoxmaton of Elang dstbuton fo the pobablty of smultaneously used devces fo the case whee N =70 and x hus fo the case whee Q Y P 1 1 U A 0 x () f x A M f A ; 1 Q ; M P CQ CQx C QP C CxQ Cx CxP CPQ CPx C P Pm const const : 1 1 1 CQ CQM A() f x CQM 1 1 1 1 1 1 1 1 1 ()()()() f x A M CQ f x A M CQM A f x f x A M CQM 1 1 1 1 1 1 M CQ M CQM A() f x M CQM whee f x dagonal matx wth elements f ( x ) / x ; CQ nown covaance matx of nodal flow ate; C P Cx covaance matx of nodal pessue and covaance matx of flow ate n banches; C C covaance matx of nodal flow ate and flow ate n banches; covaance matx of nodal pessue and flow ate; C Px C p h =0.03. xp C Qx PQ xq C QP covaance matx of nodal pessue and flow ate n banches. hus nowng C CQ we can calculate C. No specal equements ae mposed on matx C Q howeve n pactce t s usually taen as a dagonal matx fom consdeatons
100 Novtsy and Vanteyeva of statstcal ndependence of consume loads. hs means that cov( Q ) Q fo t and cov( Q ) Q 0 fo t. t Q Covaance matx fo the geneal case of settng bounday condtons ( Q P ) whee at each node we can set ethe the flow ate o the pessue and each banch s chaactezed by n -dmensonal vecto (e.g. { s H } n ) of hydaulc paametes whch s specfed n the pobablstc fom n full o patally [1 ]. Dvde the set of nodes n the desgn scheme nto subsets of nodes wth the gven flow ate ( J ) and pessue ( J ) and the set of banches nto subsets Q of banches wth hydaulc paametes gven n the pobablstc ( detemnstc ( I D P t I V ) and ) foms. We omt the concluson and gve the fnte expessons fo the covaance matx of unnown opeatng paametes: 1) Covaance matx of unnown nodal pessue PY xv xv PY CPY PY PY AQV CV AQV Q V V Q PY PY PY PY CQ CP Q Q P P ) Covaance matx of flow ate n the banches wth detemnstcally specfed chaactestcs xd xd xd xd x D x D x D PY P PY PY P P C C C 3) Covaance matx of flow ate n the banches wth pobablstcally specfed chaactestcs xv xv CxV xv x V CPY PY PY ; xv xv xv xv CP CV P P V V 4) Covaance matx of unnown nodal flow ates CQY QY QY APDC xd APD APV CxV APV whee A QD () m Q n D -dmensonal ncdence matx wth elements a J Q I D ; A QV () m Q n V - dmensonal ncdence matx wth elements a JQ IV ; A PD () m P n D -dmensonal ncdence matx wth elements a J P I D ; A PV () m P n V -dmensonal ncdence matx ; ;
Chaotc Modelng and Smulaton (CMSIM) 1: 95-108 014 101 a) b) Fequency % 80 60 me h 40 0 3 0 1 56 9 16 1 14 86 76 70 80 3 Flow ate m 3 /h 67 6 wth elements 1 a J P IV ; x P 1 D Y f x xd D 1 QD A x P D f x xv f xv xv f xv xv f xv f xv AQV APV matces of PY xv P xv V xv V patal devatves of the coespondng combnatons of paametes whch f xd f xv f xv mplctly depend on thee matces only: and whose x x stuctue s detemned by the type of banch chaactestcs. Moeove the fst two of them ae dagonal and theefoe easly nvetble. hus based on the gven elatons we can sequentally calculate the covaance matces of all the opeatng paametes f we now the covaance matces of nodal flow ate set n the pobablstc fom ( C ) nodal pessue ( C P ) and hydaulc chaactestcs of banches ( C V ). Pobablstc calculaton of dynamcs of hydaulc opeatng paametes. Stochastc bounday condtons ntate the change n hydaulc opeatng paametes wth tme. As a esult we face the poblem of pobablstc modelng and analyss of opeatng paamete dynamcs () t 0 t as a andom pocess fo the calculaton peod. Fgue pesents the gaphs of ealzaton-fequency dstbuton of two hydaulc opeatng paametes (the nodal flow ate and the nodal pessue). he fst paamete can be consdeed as a dstubance the second as a esponse. Fgue a shows the gaph of wate flow ate fequences fo an D V 1 Q xd V D 1 A PD me h Fequency % 100 80 60 40 0 0 36 19 Pessue w.c.m Fg.. Daly change n the fequency dstbuton of hydaulc opeatng paametes а) Fo the nodal flow ate b) Fo the nodal pessue
10 Novtsy and Vanteyeva ndvdual esdental buldng n the wate supply system that s constucted based on the expemental data. Fgue b shows the gaph of pessue fequences at the connecton node of the esevo n the wate supply system n one of the Iuts dstcts that s obtaned by pocessng the data of the dspatchng depatment fo 490 days. Analyss of both pocesses n Fg. ndcates that: 1) the fequency dstbuton at any coss-secton of both pocesses s appoxmated by the nomal (Gaussan) dstbuton satsfactoly enough; ) the vaance of evey pocess () s pactcally nvaable. he oot-mean-squae devaton () fo daly wate flow ate changes neglgbly.e. wthn 10 pe cent (able 1) fo pessue wthn 7 pe cent; 3) the mathematcal expectaton fo both pocesses changes dung a day (Fg. 3a); 4) the autocoelaton functon stablzes at the zeo value (fo the nodal value n Fg. 3b) fast enough. P() t m 59 a) () t 1 b) 09 08 58 07 06 57 05 04 03 56 0 01 55 1 3 4 5 6 7 8 9 10 11 1 13 14 15 16 17 18 19 0 1 3 4 t h 0 t h 1 3 4 5 6 Fg. 3. Statstcal chaactestcs of change n the nodal pessue as a andom pocess а) Dynamcs of mathematcal expectaton; b) Gaph of the autocoelaton functon of pessue n the esevo he hydaulc opeatng paametes vay n tme n esponse to thee man dstubng actons (bounday condtons): 1) andom actons of egula chaacte (consume loads); ) detemnstc actons of egula chaacte (contol actons); 3) andom actons of egula chaacte (fes accdents). he second type of dstubances s taen nto account algothmcally on the bass of the specfed contol ules. Analyss of the consequences of elatvely ae dstubances of the thd type s the subect of the elablty theoy of ppelne systems and s not caed out hee.
Chaotc Modelng and Smulaton (CMSIM) 1: 95-108 014 103 able 1. Values of mathematcal expectatons and oot-mean-squae devatons of the nodal flow ate dung day hous fo the condtons n Fg. а. Day hou Q m 3 /h 100% 1 4.60.14 3.11.3 1.94 6.5 3 1.88 1.99 4.1 4 1.66 1.94 6.5 5 1.87 1.84 7.97 6 3.8. 6.00 7 7.88.0.67 8 10.80.08 0. 9 10.88 1.96 5.56 10 1.40.6 8.89 11 1.48.0.67 1 1.13.8 9.86 Day hou Q m 3 /h 100% 13 11.97.5 841. 14 11.77. 6.00 15 11.8.15 3.59 16 11.16.07 0.6 17 11.53.0 3.63 18 1.3.09 0.70 19 1.35.17 4.56 0 13.34.05 1. 1 13.68.04 1.71 14.34.0.67 3 1.51 1.85 10.86 4 9.10.18 5.04 Dynamcs of hydaulc opeatng paametes () t 0 t may be consdeed as a andom pocess wth the dscete tme (a quasdynamc appoach). At each tme nstant of the pocess the opeatng paametes obey the nomal dstbuton. Vaaton of the opeatng paametes at the adacent nstants may be consdeed as nsgnfcant and the flow dstbuton as steady. hus the poblem of pobablstc calculaton of hydaulc opeatng paamete dynamcs s educed to the detemnaton of [(0) (1)...() ] and C based on the specfed paametes E [(0) (1)...() ] A()()() t P t y t y()()() t f x t t C and the condtons A()()( t x t) Q t P t 0.... In ths case the suggested analytcal pobablstc models and the calculaton methods can be appled to each calculaton nstant whch wll shaply decease computatonal effots. he computng expements n able have shown the decease n unnng tme by tens of tmes.
104 Novtsy and Vanteyeva able. me equed fo pobablstc calculaton by the Monte Calo and analytcal methods [7] Numbe of scheme me fo method tm C nodes and banches Analytcal Monte Calo t 6 nodes and 8 banches 3. s 3 mn 56.5 1 nodes and 19 banches 4.8 s 8.5 mn 356.5 1 nodes and 9 banches 16. s 1.5 h 77.77 - Analyt. In some cases such as avalablty of esevos t s mpotant to tae account of the laggng facto of ntenal esponses of ppelne systems when the successve opeatng condton depends on the pehstoy of condtons. Avalablty of esevos can be taen nto account by usng the addtonal dynamc elaton P P 1 g( t /) F Q whee t duaton of the - th condton; F lqud suface aea n the esevo; ndex of the node wth a esevo; g gavtatonal acceleaton; lqud densty. he esevo opeaton can be modeled by nseton of a dummy banch connected to a dummy node wth zeo (o a) pessue. he hydaulc chaactestc of such a banch has the fom: y s x H whee H P 1 s / g t F. Let f H be a vecto of dummy pessue ses n the banches f that epesent all the esevos. he covaance matx of vecto H that s * used at the -th calculaton step wll have the fom: CH ()() t CPY t 1 whee * CP Y () t 1 bloc of covaance matx PY f C that was calculated at the pevous step and s attbuted to the pessues at the nodes wth esevos. Calculaton of pobablstc opeatng paametes of ppelne systems. he suggested appoach to the calculaton of statstcal paametes of ppelne system opeaton offes an oppotunty to obtan pobablstc estmates of vtually any opeatng paametes of ppelne systems dependng on the opeatng condtons by the nown fomulas of the pobablty theoy. Fo example the pobablty that any nondegeneate subset of opeatng paametes belongs to a gven ange at the tme t wll be detemned by the fomula v v 1 1 p d d n 1 1... exp C (4) 1 n n () C vn v 1 whee n-dmensonal vecto (subvecto) of opeatng paamete values at the tme nstant t ; n-dmensonal vecto of mathematcal expectaton ; C ( n n )-dmensonal covaance matx fo ; p pobablty
Chaotc Modelng and Smulaton (CMSIM) 1: 95-108 014 105 that belongs to a specfed ange [ v v ]; v [ v1... v n ] and v [ v1... v n ] vectos of uppe and lowe boundaes of the studed ange whose components can tae nfnte values to tae account of one-sded ntevals o the absence. he assessment of pobablty that belongs to a specfed ange [ v v ] dung peod Т wll be detemned by the fomula whee K K K K (5) p p t t p t 1 1 1 the numbe of calculated peods ove peod K t ; t 1 duaton of the -th condton. Equatons (4) and (5) can be appled to estmate the opeaton of ppelne system ts fagments o ndvdual components n a defnte opeatng condton o ove the peod of tme fo example n tems of the extent to whch they ae loaded consume demand s satsfed o pocess constants ae met etc. Numecal example Let us consde a numecal example of calculatng the stochastcs of the hydaulc opeatng paametes fo the netwo pesented n Fg.4. he netwo conssts of 7 nodes and 11 banches of whch: one node has a fxed pessue; two nodes have lumped loads; two nodes ae nonfxed loads dependng on pessue; one banch epesents a pumpng staton wth an nceasng head Н 0 =1 m; one dummy banch smulates a esevo (wate level n the esevo Н f =16.4 m); two dummy banches smulate nonfxed loads the esstances ae andom values. hus ths example llustates the possbltes of the suggested appoach n tems of the andom composton of bounday condtons. he nput nfomaton specfed n the pobablstc fom s: ( Q P )( Q4 Q) 5 P7 s9 s10 = (5. 1.8 0 0.30359 1.407); C a dagonal matx wth nonzeo elements (1.065 0.3969 0.0001 0059 0.51564). esstances n the dummy banches 9 and 10 that smulate nonfxed flow ates at consumes ae detemned by the fomula [1 ] s P /() Q and vaances 4() P /() Q 6 s Q whee P Q desgn (equed) pessues and flow ates fo ths consume ndex of the ntal node of the -th banch. Coespondngly n the example Q 7.7 9.61 P 18 Q3 7.11 3 0.81 P3 1. esstances n the banches that wee specfed detemnstcally ae: s1 0.0057 s 0.8996 s3 0.00408 s4 0.095 s5 0.67 s6 0.067 s7 0.0957 s8 0.00646 s11 0.014. he calculaton esults fo nodes ae pesented n able 3 and fo banches n able 4.
106 Novtsy and Vanteyeva Н 0 =1 8 1 1 9 3 7 10 3 5 4 4 6 Н f =164 11 7 5 6 Fg.4. Example of the calculated scheme of the ppelne system fo the geneal case of bounday condtons eal secton; dummy banch smulatng nonfxed consume loads; dummy banch smulatng esevo; dummy banch smulatng pumpng staton; 4 node wth the specfed nodal loads; 7 node wth the specfed pessue. able 3. Calculaton esults fo nodes able 4. Calculaton esults fo banches Paametes Р Mwc P Q m 3 /h Q 1 18. 0.89 17.11 1.5 9.19 4.03 3 14.96 1.1 6.48 1.08 4 16.70 1.5 5 16.01 0.83 6 16.37 0.0 7.67 9.07 Paametes x m 3 /h x 1 0.75 3.9 1.54 0.0 3 10.01 0.03 4-3.3 0.59 5-1.61 0.0 6 3.0 0.98 7-1.9.1 8 0.75 3.9 9 9.19 4.03 10 6.48 1.08 11 1.8 0.66 Fgue 5 pesents a gaphcal ntepetaton of the calculated pobablty of povdng consumes wth a equed flow ate. Fo example fo the consume at the second node p(0) Q Q 0.344 o p() Q Q 0.64446 and at the thd node p(0) Q3 Q3 0.71914 o p() Q 3 Q3 0.8083 whee Q s the equed flow ate.
Chaotc Modelng and Smulaton (CMSIM) 1: 95-108 014 107 p a) p b) Q Q Q Q Fgue 5. Illustaton to the calculaton of pobablty of povdng consume wth a equed flow ate: a) at node b) at node 3. Q Calculated value of mathematcal expectaton of consume flow ate consdeng ts dependence on nodal pessue Q equed value of consume flow ate. Conclusons 1. he pape pesents: - a technque fo apo calculaton of statstcal chaactestcs of a pobablstc pocess of the tanspoted medum consumpton as a queung pocess; - a geneal scheme fo pobablstc calculaton of ppelne system hydaulc opeatng paametes. he calculaton suggests detemnng statstcal chaactestcs of the opeatng paametes by specfed chaactestcs of bounday condtons and flow dstbuton model. It s shown that such a calculaton s educed to solvng a tadtonal poblem of flow dstbuton at the pont of mathematcal expectaton of bounday condtons n combnaton wth an addtonal pocedue fo calculatng covaance matces of opeatng paametes; - a technque fo obtanng the analytcal expessons fo covaance matces of opeatng paametes as well as the expessons fo the geneal case of specfyng bounday condtons; - a technque fo pobablstc modelng of changes n the hydaulc opeatng paametes on the bass of developed analytcal pobablstc flow dstbuton models. hs technque povdes a consdeable educton n computatonal effots aganst the nown methods of smulaton modelng.. he suggested technque fo modelng ppelne systems povdes the possblty of obtanng pobablstc estmates of pactcally any ppelne system opeatng paametes that depend on opeatng condtons.
108 Novtsy and Vanteyeva 3. A numecal example of pobablstc calculaton of the steady flow dstbuton n the ppelne system s gven fo the geneal case of bounday condtons. he example llustates the suggested pobablstc appoach. efeences 1. N.N. Novtsy O.V. Vanteyeva. Poblems and methods of pobablstc modelng of hydaulc condtons of ppelne systems // Scentfc techncal bulletn of StPSU. 008. No.1. P. 68-75.. N.N. Novtsy O.V. Vanteyeva. Modelng of stochastc flow dstbuton n hydaulc ccuts // Izv AN. Enegeta. 011. No.. P.145-154. 3. A.P. Meenov V.Y. Khaslev. heoy of hydaulc ccuts. M. Naua 1985. 80 p. 4. N.N. Abamov. elablty of wate supply systems M. Stozdat 1979. p.31. 5. L.A. Shopensy. Study on the opeatng condtons of wate ppelnes n esdental buldngs: PhD dssetaton thess. M. 1968. 6. Intenal santaton of buldngs (Constucton Noms and ules.04.01-85 *) M.: CIP Gossto oss 004. 58 с. 7. O.V. Vanteyeva. Pobablstc models and methods fo analyss of ppelne system opeatng condtons. PhD dssetaton. Iuts 011. 139 p.