Leak Detection in Gas Pipelines Using Accurate Hydraulic Models

Transcription

1 Artcle pubs.acs.org/iecr Leak Detecton n Gas Ppelnes Usng Accurate Hydraulc Models Mguel Bagajewcz*,, and Gary Valtnson Department of Chemcal Bologcal and Materals Engneerng, Unversty of Oklahoma, Norman, Oklahoma 73019, Unted States Ok-Solutons, Norman, Oklahoma 7307, Unted States ( ABSTRACT: In ths paper, we show the mplementaton of the Generalzed Lkelhood Rato (GLR) method to detect and also dentfy the sze and locaton of leaks n ppelnes. We ntroduce the use of accurate hydraulc models for hypothess testng and the use of economcs to determne the thresholds of detecton and dentfcaton. We compare the leak detecton power and costs to those of other smple leak detecton methods. The economc comparson ncludes computng the losses for not detectng the leaks (false negatves) and detectng leaks that do not exst (false postves). We also llustrate the mprovement n the power of our method by usng more-accurate nstrumentaton. INTRODUCTION Ppelnes are used to delver petroleum products, natural gas, lqud hydrocarbons, and water to consumers. They do so n a cost-effectve manner; however, safety and losses due to leaks are the prmary concern n ppelne operaton. Thus, the challenge s to use leak detecton methods capable of accurately detectng leaks and ther locaton n a tmely fashon. There are several hardware-based methods for leak detecton and locaton. They are generally senstve to small leak szes and qute accurate, wth regard to locaton of the leak. Typcally, nstrumentaton s run along the entre length of the ppelne, whch helps wth the detecton of both large and small leaks n a tmely fashon and allows for the detecton of a leak anywhere along the ppelne. Although sgnfcant nstrumentaton provdes many of the advantages assocated wth hardware leak detecton, t also provdes dsadvantages. The hgh level of nstrumentaton results n nstallaton and mantenance costs that are sgnfcant. Installaton s complex, requrng a consderable amount of work below the surface, snce many ppelnes are bured. These methods are based on several technologes: detecton of nose generated by leaks and the measurement of ultrasonc wave speed (acoustc methods), detecton of scattered lght due to leaks (fber optcs), detecton of vapors (typcally by runnng a small ppe wth an nert gas permeable to hydrocarbons wth an analyzer at certan dstances), among others. Software-based methods make use of exstng measurements of flow and pressure. The smplest one s the flow dscrepancy method. 1 It assumes steady state and t declares that there s a leak when two flow measurements separated by a certan dstance ndcate dfferent flow values. Ths technque only apples to sngle ppelnes wth flow movng n one drecton, so complex networks such as gas dstrbuton systems n urban areas do not apply. The volume balance does not help n detectng the locaton of leaks, and t also cannot dstngush between bases and leaks. If one can add the pressure measurements upstream and downstream, one can use the pressure drop equaton to look for dscrepances between flow and pressure drop. For lquds, one can use the gradent ntersecton method to addtonally detect the sze and locaton of the leak (see Fgure 1). ute clearly, wth some effort, ths can be extended to gases, where the pressure gradent s not lnear. Threshold values must be set n Fgure 1. Depcton of the gradent ntersecton method for lquds. the gradent ntersecton method, snce normal pressure drop fluctuatons occur. Many false alarms are the result of not settng these values, snce ths method s dependent on the tunng of the model, because measurement errors (along wth uncertanty) n flud propertes can cause dffcultes. Transent model-based methods attempt to dstngush the effects of a leak from all other phenomena n a ppelne. Whle the pressure analyss method cannot dstngush between a leak and anythng else that causes a pressure drop, transent models smulate transents n a system n real tme. Ths method s a numercal ntegraton of three dfferent equatons: the momentum, contnuty, and the energy equaton. Generally, an mplct matrx-based soluton s used wth all three of these equatons. One downsde to ths approach s that many parameters are needed for the method to work accurately. Specal Issue: Jame Cerda Festschrft Receved: Aprl 0, 014 Revsed: October 3, 014 Accepted: October 6, 014 Publshed: October 6, Amercan Chemcal Socety dx.do.org/10.101/e5013g Ind. Eng. Chem. Res. 014, 53,

2 Industral & Engneerng Chemstry Research Artcle Fgure. Lnear ppelne wth n segments and compressors. Some of these ppelne parameters can be dffcult to obtan, such as the nsde ppe roughness, the current drft, and calbraton of the nstruments. Fnally, n order to perform calculatons n real tme, adaptve modelng must be used. Ths mples that certan parameters n the system wll be adjusted when compared to smulaton or measured values. A leak s detected f the dscrepancy between the actual data and the model data s greater than the determned lmts. If no leak s found, the dfferences between the measured and calculated values are used to adjust parameters, all ths just for one ppe segment. Bllmann and Isermann 3 showed that detectable leaks were larger than % for lqud and 10% for gas. One clearly needs to perform better leak detecton, especally for gases. In turn, frequency analyss methods use a steady oscllatory flow produced by perodcally openng and closng a valve. Pressure ampltude peaks are developed from ths oscllatory flow for a system wth leaks, and then the peaks are compared wth a system where no leaks are present. Ths process allows for dentfcaton of leaks, as well as ther locaton and magntude n a gven system, but t can be very complex, and normal ppelne operatons must be suspended for frequency analyss methods to be mplemented. 1 Fnally, the statstcally based generalzed lkelhood methods are based on hypothess testng and data reconclaton models. 4,5 Mukherjee and Narasmhan 5 were the frst to propose a leak detecton method based on the Generalzed Lkelhood Rato (GLR) method for networks (as opposed to just one ppe segment) that are also based on the use of hydraulc equatons. Ther soluton procedure s teratve and only for lquds. What prevented the use of ths method for gases n the past s the fact that the pressure drop correlatons are rather naccurate, and, therefore, the error they ntroduce would lead to a large number of false postves, as well as large errors n predcton when real leaks occur. We propose to apply the GLR method to gas ppelnes, usng accurate metamodels for the hydraulcs, as well as tunng the thresholds of hypothess testng usng economcs. The artcle s organzed as follows: We frst revew the GLR method, as appled to our partcular case. Then, we dscuss the power of the method (effectveness) and the economcs of leak detecton. Fnally, we present a smple llustraton. GENERALIZED LIKELIHOOD RATIO FOR PIPELINE LEAK DETECTION The Generalzed Lkelhood Rato (GLR) s a well-known hypothess testng statstcal method based on the comparson of the performance of two alternatve models and how they ft the data. One model, represented by the null hypothess, s that there s no leak, whle the alternatve model s that there s a leak. There s then a model (a data reconclaton problem for our case) for each hypothess. In the case of leaks n materal balancng systems, Narasmhan and Mah 4 proposed usng the flow measurements at dfferent ponts and constructng two data reconclaton models: one that assumed no leak and one that assumed a leak n one of the many segments of a ppelne (defned as ppelnes between flowmeters). The value of T has a central χ dstrbuton wth one degree of freedom under Ho. Snce the T values are not ndependent, the dstrbuton of T (the maxmum value of all T ) cannot be obtaned. However, Narasmhan and Mah 4 used arguments smlar to those used by Mah and Tamhane 6 and chose the upper 1 β quantle of the χ dstrbuton (χ 1,1 β ) as the crtcal value. For a gven level of sgnfcance α (usually 95%), β 1 (1 α) 1/p, where p s the number of gross errors hypotheszed (we use a value of 1). Ths would ensure that the probablty of a Type I error s less than or equal to α. For the case of only flow measurements, Narasmhan and Mah 4 developed analytcal expressons. A smlar approach was used by Mukherjee and Narasmhan 5 for lqud ppelnes. In ths latter case, as n ours, they ncorporated pressure measurements. They actually used some approxmatons for the data reconclaton models. Our Generalzed Lkelhood Rato Approach. Our proposed method uses pressure measurements at the begnnng and end of each ppelne secton, n addton to flow measurements. Wthout any loss of generalty, consder a ppelne wth n segments (see Fgure ). The lkelhood of the null hypothess (no leak) s represented by the followng data reconclaton model: ( ) ( P P ) ( P P ) O n n ns, ns, ne, ne, mn σ σ σ n N n Pns, Pne, (1) subject to A ( P P ) B Δz n n, s n, e n n n Ln n N... 1 N (3) In these equatons, ñ, P ñ,s, and P ñ,e are the measured values of flow rates and pressures for each ppe segment n, whle n, P n,s, and P n,e are the estmators. Fnally, σ n, σ Pn,s and σ Pn,e are the varances of the respectve flow and pressure measurements. Note that, for the hydraulc equaton, whch represents the relatonshp between flow and pressures (eq ), we use a smplfed metamodel 7 that we developed and has a much smaller error than that of exstng proposed equatons (Panhandle, Weymouth, etc.). We now buld the alternatve model (a leak hypotheszed n segment ). In the case where there s a leak of magntude l at some dstance x n the ppelne, the pressure drop equaton for that segment must be wrtten for each nterval before and after the leak (see Fgure 3). Thus, we wrte AP ( s Pl ) BΔz x () (4) dx.do.org/10.101/e5013g Ind. Eng. Chem. Res. 014, 53,

3 Industral & Engneerng Chemstry Research Fgure 3. Schematc depcton of a segment that has a leak. AP ( l Pe ) BΔz ( l) L x When a leak s assumed n ppe segment, we solve the expresson O mn subject to n N (5) ( ) ( P P ) ( P P ) n n ns, ns, ne, ne, + + σ σ σ n Pns, Pne, (6) A ( P P ) B Δz n n, s n, e n n n Ln, ( l) n N; n (7) A( P s P, l ) BΔz( x) x (8),, A( P l P e ) BΔz( L x) L x... 1 (10) + 1 l (11) N (1) Here, P,l s the pressure at the locaton of the leak. In turn, Δz (x ) s the change n heght between the begnnng of the ppe and the locaton of the leak and Δz (L x ) s the change n heght downstream from the leak. Equatons 8 and 9 present the problem that Δz (x ) and Δz (L x ) are functons and not parameters as n the other equatons. These equatons can be smplfed as follows: We multply 8 by x and 9 by L x and add them together to obtan s, e, x + ( l) ( L x) A( P P ) BΔz( L) (9) (13) where we used the fact that Δz (L )Δz (x )+Δz (L x ). Substtutng eqs 8 and 9 by eq 13 elmnates the unmeasured varable P,l, whch s an observable quantty that can be calculated after the problem s solved (although ths s rarely needed). The leak detecton procedure s as follows: (1) Hypothesze leak n every branch and solve data reconclaton problems () Obtan GLR test statstcs for each branch T log(o 0 ) log(o ) (3) Determne the maxmum test statstc T (4) Followng the GLR method, we compare the max test statstc wth the chosen threshold value (T*). If the statstcs s larger than ths threshold (.e., T > T*), the leak s sad to have been dentfed and located n the branch correspondng to the maxmum test statstc ute clearly, the method can be extended to multple branches, by just modfyng eq 10 accordngly. Artcle POWER OF THE METHOD The overall power of the method s the rato of the number of leaks correctly detected or dentfed to the number of leaks smulated. One can add qualfers to such power. For example, one can defne correct dentfcaton f the ppelne segment s correctly dentfed, or f the estmator of the leak s close to the real value wthn a predetermned error and/or f the locaton s dentfed wthn a certan predetermned error. The key operatve word n these defntons s the word correctly, whch needs to be defned. More specfcally, we dstngush these dfferent types of power: (1) Detecton Power: Ths s related to the frequency at whch the presence of a leak s detected, regardless of ts sze and locaton. The outcome of ths test s ether there s a leak or there s no leak. Some methods can only do ths. () Identf caton Power: We call dentfcaton the ablty to detect the leak and also gve addtonal nformaton. Of these, there are two varetes: (a) Leak Sze Identfcaton Power: In ths case, the addtonal nformaton s the sze of the leak. (b) Leak Sze and Locaton Identfcaton Power: In ths case, the addtonal nformaton s the sze of the leak, as well as ts locaton. To obtan the power, several leak szes were tred at specfc locatons, samplng 100 dfferent sets of smulated measurements normally dstrbuted around the actual flows and pressures. Our analyss focuses on the ablty of our method (ALINA) to detect a leak wthn a partcular segment and locate the leak. As n all GLR-based methods, one must tune the threshold, T*, usng some procedure. We use economcs to determne ths threshold. We compare ALINA wth other methods: The use of GLR based on flow measurements only (as mplemented by Narasmhan and Mah 4 ). Ths method uses a threshold (T*) smlar to that used n ALINA, and ts data reconclaton model s gven by eqs 6, 10, 11, and 1. Ths method does not provde locaton, as t does not regard the hydraulcs. Flow dscrepancy: In ths case, the dfference of two measurements ( ĩ ĩ+1 ) s compared to the sum of the two standard devatons (σ + σ +1 ). A leak s declared f the aforementoned dfference s postve and larger than the sum of the standard devatons multpled by a threshold, T σ. Because ths method does not nvolve the use of reconclaton, one has several choces. Smlar to GLR based on flow measurements, the flow dscrepancy method does not localze the leak. Hydraulc dscrepancy: Ths method nvolves acceptng the pressure measurements as true and obtanng a representatve flow as follows: # A( P, s P, e ) BΔz L (14) Ths flow s now compared to the flow measurement correspondng to the begnnng of the same ppe segment. The dfference between the measured flow at the begnnng of the ppe ĩ and the flow predcted by hydraulcs determnes the presence of a leak based on a threshold, T λ. That s, f ĩ # > T λ, then a leak has been dentfed. To obtan the locaton, two pressure profles are constructed. They obey eqs 8 and 9. As dscussed above, these two equatons dx.do.org/10.101/e5013g Ind. Eng. Chem. Res. 014, 53,

4 Industral & Engneerng Chemstry Research can be rewrtten, elmnatng the pressure at the leak locaton to obtan eq 13. Thus, the locaton of the leak can be obtaned manpulatng 13 to obtan the followng: x A Δ ( P, s P, e ) B z( L) L (15) Economcs. To calculate the cost of each leak of sze l each tme t occurs (LC), we add the cost of a detected leak (locaton cost + repar cost + cost of the loss), and the cost of an undetected leak. The cost of a real leak s gven n the followng expresson: LC( ) RCD [ + CD+ C( D+ D)] + (1 P ) l l l r r f l l r l Cf 365 (16) The frst term s the cost of the detected leak. Ths s obtaned by multplyng the expected number of successful dentfcatons of a leak R (ncdents per year of the leak of sze l ) wth three terms: The cost of physcally locatng the leak: Ths s gven by the cost of locatng the leak (C l ), mutlpled by the number of days requred to locate the leak (D l ). It s assumed that once a leak has been detected, the task of sendng crews to physcally dentfy the exact locaton takes a few days. The cost of reparng the leak: Ths s gven by the cost per day (C r ) multpled by the number of days needed to repar the leak (D r ). The fnancal loss due to the loss of flud, durng the days needed to locate and repar the leak: Ths s gven by the prce of flud (C f ) multpled by the flow rate of the leak ( l ), multpled by the number of days the flud s beng lost (D l +D r ); one can also assume that the leak s stopped once t s located, so the number of days of loss reduces to D l. We assume that the cost of physcal locaton and repar costs are ndependent of the sze of the leak. The second term s the cost of the undetected leak, whch s obtaned by multplyng the frequency at whch the method does not detect the leak (1 R) by the sze of the leak ( l ), multpled by the cost of the flud (C f ), multpled by 365 days per year. There s also a cost of false postves (detectng a leak when there s none); such a cost s gven by the number of false postves FP per year multpled by the daly cost of locatng a leak (C l ), multpled by the tme t takes to gve up the search for the nonexstent leak (D FP ). Thus, the false postve cost (FPC) s FPC FPC l D FP (17) We consder the value D FP to be economcally equvalent to the tme needed to locate a leak wthout a localzng method. Therefore, FPC s a fxed cost per year. In turn, the number of false postves (FP) s related to the value of the threshold used. The smaller the threshold, compared to the standard devaton of the estmators through data reconclaton, the larger the number of postves. Two ways of ameloratng the mpact of these false postves s to ncrease the threshold, whch, n turn, may ncrease the false negatves, or force the use of more-accurate nstrumentaton, whch wll reduce the standard devaton of the estmators. Thus, ths s an economc ssue. Somethng else that can be consdered when estmatng the cost of a false postve s the use of a threshold related to the leak sze. If a leak of sze 1% s detected 90% of the tme, whle leak szes of <1% are detected sgnfcantly less frequently, t may be justfable to gnore leaks below a sze of 1%. Ths, n turn, reduces the number of false postves. We do not explore ths added threshold n the present artcle. To evaluate any method, one must compute the expected annual cost. False postves have a fxed cost, so the expected value s already gven by FPC. For false negatves, the expected cost can be calculated by ntegratng the product of the annual cost for each leak sze by the probablty of such leak sze: ELC P ( l ) LC ( l )d l 0 (18) Fgure 4 provdes the probablty of a leak per year for a gven sze of leak. Ths was constructed usng the sgnfcant ncdents Fgure 4. Probablty of leak for a gven sze. data from the Ppelne and Hazardous Materal Safety Admnstraton. 8 As should be expected, large leaks are not as lkely as smaller leaks. Optmum Threshold. The optmum threshold (T*) can be obtaned by calculatng the expected cost for each threshold value and selectng the one that provdes the smaller annual cost. Ths apples to three other thresholds as well: T* for GLR wthout flow, T σ for flow dscrepances, and T λ for hydraulc dscrepances. The mnmum expected cost s the locaton of the optmum threshold. Ths pont corresponds to the pont where false postves have as much economc mpact as the false negatves. ILLUSTRATION Our example conssts of a four-segment ppelne, wth the gas beng recompressed at the begnnng of each segment. Measurements of nlet as well as outlet pressures and flow rates for each segment are made. Usng Pro II (SIMSCI, Invensys), we smulated natural gas through a sngle ppelne 400 Table 1. Economc Parameters Artcle parameter symbol value annual gas prce a,b C f $5.00/MMBTU cost of physcally locatng a leak a C l $5,000/day cost to repar a leak a C r $10,000/day tme to repar a leak D r 5 days tme to physcally locate a leak all methods except ALINA D l 3 days ALINA D l 1 day tme before gvng up search for nonexstent leak (all methods) D FP 3 days a Prces and costs gven n terms of U.S. dollars (USD). b Data taken from ref dx.do.org/10.101/e5013g Ind. Eng. Chem. Res. 014, 53,

5 Industral & Engneerng Chemstry Research Artcle Fgure 5. Power of leak detecton n the correct ppe for varous thresholds (T*) for ALINA. Fgure 6. Frequency of ALINA to detect false postves at varous values of T*. Fgure 7. Annual cost of leak for a threshold of T* 1 usng ALINA. km long, wth a total of four compressors. The nomnal ppe dameter s 0 n.; the nlet temperature and pressure to each segment are 45 C and 100 ps, respectvely. The volumetrc flow rate was mllon standard cubc feet (SCF) per day. Once we obtaned the pressure nto and out of the compressors, we then appled random error to these values to produce 100 ponts of data. The economc parameters are provded n Table 1. We assumed t takes 5 work days (8 h per day) to complete the repar. Snce ALINA can detect the locaton of the leak, we used 3 days to locate the leak when usng GLR on flows only and 1 day wth Fgure 8. Cost of leak for a threshold of T* 1 usng ALINA. ALINA. Fnally, we assumed that the flow measurements have 1% error, whle the pressure measurements have 0.1% error. We provde ALINA s power curves to detect a leak n the correct segment for varous thresholds n Fgure 5. As Fgure 5 ndcates, a leak sze of 1% s detected 88% of the tme for T* 0.1 and 1. Below ths leak sze of 1%, the number of leaks detected decreases by more than 0%, because the leak s more dffcult to detect (t may have the same effect n the data reconclaton as nose) wthout decreasng the threshold, T*. However, decreasng T* ncreases the number of false postves. Fgure 6 provdes the frequency of false postves, compared to the threshold, T*. As can be seen, ncreasng the threshold decreases the number of false postves. Indeed, at T* and T* 3, there s no leak detected when there s no leak (no false postves). When T* 1, a leak s detected 3% of the tme when there s no leak, whle for T* 0.1, ALINA detects a false postve 63% of the tme. Now that we have the power curves at varous thresholds for ALINA, we can assess the economcs and optmze the method. Frst, we calculate the annualzed cost for each threshold, T*. Fgure 7 shows the annualzed cost from ALINA usng a threshold of T* 1. At the sze of a leak of 0%, the cost s fnte due to false postves. At the begnnng of the curve, the cost ncreases because of false negatves. In ths case, the false postves have an annual cost of approxmately $ per year. Below an exstng leak sze of %, the prmary cost assocated wth the annual cost s strctly false negatves. Eventually, the leak s detected the majorty of the tme (% n ths case), causng the dx.do.org/10.101/e5013g Ind. Eng. Chem. Res. 014, 53,

6 Industral & Engneerng Chemstry Research Artcle Fgure 9. Optmal threshold (T*) determnaton for leak detecton usng ALINA based on detecton of a leak n the correct ppe. Fgure 10. Power to detect a leak n a partcular segment for dfferent methods. Table. False Postves for Each Method at Optmal Thresholds ALINA GLR flow dscrepancy hydraulc dscrepancy threshold T* 1 T* 1 T σ 1.5 T λ 6.5 false postves 3% 1% 15% 11% cost to be prmarly the prce of locatng t physcally, reparng the leak, and the product lost before locaton and whle reparng. Once the annual cost s determned, we can then determne the expected cost. Fgure 8 shows the cost of a leak versus the leak sze as a percentage of total flow. It s the product of the annualzed cost and the probablty (see Fgure 4). As the probablty decreases wth leak sze, so does the mpact of that partcular leak sze. As wth the annual cost, we used the curve for a threshold of T* 1 usng ALINA. The expected cost s obtaned by computng the area under the curve. Fgure 9 shows a curve based on the T* values from ALINA. The mnmum of ths curve s the optmum threshold where the false negatves and false postves balance each other economcally. As prevously ndcated, the smaller the threshold, the less Fgure 11. Power of ALINA to detect and locate a leak wthn a dstance from the orgn of the leak at the optmal economc threshold, T* 1. false negatves occur; however, the number of false postves wll ncrease. Ths optmal pont corresponds to the pont where the false postves and negatves are balanced n the economcs. As Fgure 9 ndcates, the optmum threshold for ALINA s 1 wth an expected annual cost of $1.09 mllon per year. It s clear dx.do.org/10.101/e5013g Ind. Eng. Chem. Res. 014, 53,

7 Industral & Engneerng Chemstry Research Artcle Fgure 1. Comparson of power to detect the locaton of a leak between ALINA and basc hydraulcs at optmal thresholds of T* 1 for ALINA and T λ 6.5 for the hydraulc method. Table 3. Expected Cost Comparson between ALINA and Varous Leak Detecton Methods method ALINA GLR wth only flows flow dscrepancy hydraulc dscrepancy threshold T* 1 T* 1 T σ 1.5 T λ 6.5 expected cost $1.09 $5.5 $5.63 $6.96 (M USD/yr) dfference from ALINA (M USD/yr) $0.00 $4.43 $4.54 $5.87 from ths fgure that ths optmum threshold determnaton s crucal, because cost changes rather rapdly. Knowng the optmal threshold, we can now compare ALINA to other methods. To compare the other methods, we had to perform the same assessment to determne ther optmal thresholds. We obtaned the optmal thresholds for the other methods (GLR, flow dscrepancy, and hydraulc dscrepancy) and Fgure 14. Frequency of ALINA to detect false postves at varous values of T* wth 0.1% error for flow and pressure. compare ther power curves to detect a leak n Fgure 10. Leak detecton through reconclaton of only flows uses the GLR method and thus requres a threshold (T*) smlar to that for ALINA. Fgure 10 ndcates that ALINA has the best performance, achevng 88% detecton for a leak sze of 1%. We provde the percentage of tme that a false postve wll occur n Table. Fgure 11 depcts the power of ALINA to locate the leak wthn a specfc dstance from the true locaton of the leak. Once agan, ths was usng the optmal threshold found for ALINA, n ths case, T* 1. As the graph ndcates, ALINA wll have an easer tme detectng a leak wthn 10 km than wthn.5 km. That sad, beng wthn 10 km s stll sgnfcantly better than lookng over the full 100-km ppe segment. We then make a comparson between the ablty of ALINA to locate a leak and locatng a leak wth the hydraulc dscrepancy method. Fgure 1 shows a comparson between ALINA at ts optmal threshold of T* 1 and the hydraulc dscrepancy method wth ts optmal threshold at T λ 6.5. The other methods Fgure 13. Power of ALINA to dentfy a leak n the correct ppe at varous thresholds, T*, wth 0.1% error for flow and pressure dx.do.org/10.101/e5013g Ind. Eng. Chem. Res. 014, 53,

8 Industral & Engneerng Chemstry Research Artcle Fgure 15. Power to locate a leak wthn a dstance from the orgn of the leak wth nstrument error at 0.1% for both flow and pressure nstruments. The threshold s T* 1. do not provde an estmator for locaton, because pressure measurements are requred to locate a leak. ALINA locates small leaks of 0.5% of the total flow 5% 15% of the tme more often than the hydraulc dscrepancy method. On the hgh end, ALINA acheves >50% localzng wthn.5 km and >90% localzng wthn 7.5 km for a 5% leak. Meanwhle, the hydraulc dscrepancy method remans at <50% localzng for both.5 km and 7.5 km for a leak sze of 5% of the total flow. We compled the optmal thresholds and expected costs for the three other leak detecton methods gven n Table 3. Ths ncludes the use of GLR wth flows only, flow measurement dscrepancy, and hydraulc dscrepancy. As the table ndcates, ALINA outperforms these other methods, wth $4.43 mllon USD n savngs at the low end and $5.87 mllon USD n savngs at the hgh end. We then looked at a case where both nstruments have hgh accuracy (flow measurements wth 0.1% error, pressure measurements wth 0.1% error). Ths would be a more-extreme (and more-expensve) case where the accuracy of all nstruments s the best possble. Fgure 13 provdes the power curves for multple thresholds to dentfy a leak n the correct ppe segment under these condtons. The frequency of false postves usng ALINA for each threshold s provded n Fgure 14. As these fgures ndcate, wth a threshold of T* 1, ALINA can detect a leak equvalent to 0.5% of the total flow rate 98% of the tme, wth only 4% of the leaks detected beng false postves. Fgure 15 provdes an overvew of ALINA s ablty to locate the leak wth these mproved flow nstruments. Comparng Fgure 11 and Fgure 15, we see that ALINA experences a 5% 10% ncrease n ts ablty to locate a leak. Thus, as s to be expected, hgh accuracy equates to better detecton and localzng of a leak wth ALINA and wll have an mpact on economcs. CONCLUSIONS We have studed the economcs of the mplementaton of the Generalzed Lkelhood Rato (GLR) to leak detecton n ppelnes. Our method, called ALINA, used both flow and pressure wth GLR to detect a leak and locate a leak. As we demonstrated, the threshold has a drect effect on ALINA s ablty to detect a leak and locate a leak, as well as n the economcs of leak detecton. We compared our method to the use of leak detecton by GLR wth only flow measurements, leak detecton va flow dscrepancy, and leak detecton through hydraulc dscrepances. We found that ALINA outperforms all of these methods, by 4.4 mllon USD or more (Table 3) for our partcular small example. We expect ths number to ncrease for larger-szed ppelnes wth more compresson statons. We also provded an assessment of ALINA s capabltes f nstrumentaton s even more accurate (partcularly wth flow measurements beng at 0.1% accurate, versus 1%). Ths showed a jump n the detecton rate for leaks 0.5% n sze to 98%. The accuracy of ALINA to locate a leak s also ncreased by 5% 10%. Thus, ALINA shows a clear capablty to detect and locate leaks of at least % of the total flow and moderately well below ths sze wth the example consdered n ths paper. In future work, we wll address the mpact of the uncertanty assocated wth the values of the parameters A n and B n, a problem that all model-based leak detecton formulatons share. Other work to be consdered s the ssue of bases (the method presented n ths paper assumes well-calbrated, bas-free nstruments), the addton of new nstruments, the economcs tradeoffs between nvestment and leak cost reducton, and consderng other thresholds (such as sze of the leak to be accepted as such) n tandem wth the thresholds used n ths paper. AUTHOR INFORMATION Correspondng Author *Tel.: E-mal: bagajewcz@ou.edu. Notes The authors declare no competng fnancal nterest dx.do.org/10.101/e5013g Ind. Eng. Chem. Res. 014, 53,

9 ACKNOWLEDGMENTS The followng undergraduate students worked on portons of the paper and deserve recognton: Davd Mannel, Andrew S. Negley, Rachel N. Weber, Erc Penner, Josh Stephens, Eljah Odusna, and James Akngbola. Graduate students Mesude Ozturk and Lutfye Hacoglu also helped at varous ponts. REFERENCES (1) Whaley, R. S.; Ncholas, R. E.; Van Reet, J. D. Tutoral on Software Based Leak Detecton Technques. In Proceedngs of the 4th PSIG Annual Meetng, Corpus Chrst, TX, Oct. 3, 199. () Reddy, H. R. Leak Detecton n Gas Ppelne Networks Usng Transfer Functon Based Dynamc Smulaton Model. Thess, Indan Insttute of Technology, Madras Chenna, Inda, 006. (3) Bllman, L.; Iserman, R. Leak detecton methods for ppelnes. Automatca 1987, 3 (3), (4) Narasmhan, S.; Mah, R. H. S. Generalzed Lkelhood Rato Method for Gross Error Identfcaton. AIChE J. 1987, 33 (9), (5) Mukherjee, J.; Narsmhan, S. Leak detecton networks of ppelnes by the Generalzed Lkelhood Rato Method. Ind. Eng. Chem. Res. 1996, 35 (6), (6) Mah, R. H. S.; Tamhane, A. C. Detecton of gross errors n process data. AIChE J. 198, 8 (5), (7) Bagajewcz, M.; Valtnson, G. On the computaton of natural gas ppelne hydraulcs. Ind. Eng. Chem. Res. 014, 53 (6), (8) U.S. Department of Transportaton. Ppelne and Hazardous Materals Safety Admnstraton. March 13, 014. Avalable va the Internet at (9) Petroleum & other lquds. U.S. Energy Informaton Admnstraton: Washngton, DC. Avalable va the Internet at petroleum/data.cfm#prces, accessed March 6, 014. Industral & Engneerng Chemstry Research Artcle 1697 dx.do.org/10.101/e5013g Ind. Eng. Chem. Res. 014, 53,