Matrix-based System Reliability Analysis of Urban Infrastructure Networks: A Case Study of MLGW Natural Gas Network

Transcription

1 Cover page Ttle: Matrx-based System Relablty Analyss of Urban Infrastructure Networks: A Case Study of MLGW Natural Gas Network The Ffth Chna-Japan-US Trlateral Symposum on Lfelne Earthquake Engneerng Authors: Mr. Lang Chang (Presenter) Department of Cvl and Envronmental Engneerng Unv. of Illnos at Urbana-Champagn Urbana, IL 680 USA Phone: E-mal: lchang9@uuc.edu Prof. Junho Song (Contact person) Department of Cvl and Envronmental Engneerng Unv. of Illnos at Urbana-Champagn Urbana, IL 680 USA Phone: Fax: E-mal: unho@uuc.edu

2 Ffth Chna-Japan-US Trlateral Symposum on Lfelne Earthquake Engneerng MATRIX-BASED SYSTEM RELIABILITY ANALYSIS OF URBAN INFRASTRUCTURE NETWORKS: A CASE STUDY OF MLGW NATURAL GAS NETWORK Lang Chang ) and Junho Song 2) ) Department of Cvl and Envronmental Engneerng, Unversty of Illnos at Urbana-Champagn Urbana, IL 680, U.S.A. lchang9@uuc.edu 2) Department of Cvl and Envronmental Engneerng, Unversty of Illnos at Urbana-Champagn Urbana, IL 680, U.S.A. unho@uuc.edu ABSTRACT Urban nfrastructure systems such as utlty networks for electrcty-, water-, sewage- and gasservces, transportaton systems and telecommuncaton networks are crtcal backbones of modern socetes. However, these systems are often susceptble to natural and man-made hazards. Structural damages of components n these nfrastructure networks not only dsrupt resdental and commercal actvtes, but also mpar post-dsaster response and recovery efforts, resultng n substantal socoeconomc losses. Therefore, estmatng the relablty of these nfrastructure systems s vtal to urban stakeholders such as utlty companes, urban planners, and polcy makers as well as to resdents and busness owners. Evaluaton of the performance and connectvty of such urban nfrastructure systems s complex n nature due to the large number of network components, complex network topology and component/system nterdependences. Due to ths complexty, network relablty analyss s often performed by repeated network analyses for random samples of hazard scenaros and component status, whch prevents rapd rsk assessment and near-real-tme rsk-nformed decson makng. In ths paper, an analytcal,.e. non-samplng-based network relablty analyss method s proposed for urban nfrastructure systems. Frst, a revew of prevous research on network relablty analyss s gven, followed by a bref summary of the recursve decomposton algorthm (RDA) and the matrx-based system relablty (MSR) method that consttute the proposed network relablty analyss methodology. As a case study, the proposed methodology s appled to a Memphs Lght, Gas and Water (MLGW) natural gas network of Shelby County of Tennessee. Based on sesmc hazard maps from the Md-Amerca Earthquake Center s rsk management software MAEvz and the topologcal characterstcs of an MLGW gas network, the relablty of the network components are computed by use of a geographc nformaton system, ArcGIS. All the dsont cut sets and lnk sets of the smplfed MLGW gas network are effcently dentfed by use of the RDA. The MSR method, whch can account for statstcal dependence between components and ncomplete nformaton, s employed to evaluate the connectvty relablty of the gas network n an effcent manner. The results of the system relablty analyss are presented for earthquake scenaros wth dfferent magntudes. By ntegratng the MSR method wth an advanced network analyss algorthm such as the RDA, we can perform network relablty analyss for a complex nfrastructure system wthout random samplngs. Ths approach wll enable us to perform rsk analyss and varous statstcal nferences rapdly. Based on the results of the case study, further research s n progress to () account for the statstcal dependence between sesmc ntenstes at adacent network components, (2) estmate the average downtme and soco-economc losses, (3) account for the nterdependency between dfferent nfrastructure systems, and (4) provde useful nformaton for decson makng on dsaster plannng, response, recovery, and mtgaton usng the proposed methodology.

3 INTRODUCTION Urbana nfrastructure systems such as power, water, sewage, telecommuncaton and transportaton networks are crtcal backbones of modern socetes. In the past 20 years, the dsruptons of varous lfelne systems and equpments nduced by man-made and natural hazards have caused severe soco-economc consequences around the world (Abraham et al., 2004; Chang et al., 996; Wu et al., 2006). It also has been learned that lfelne systems play a crtcal role n dsaster plannng, mtgaton, response and recovery (Chang et al., 200). Wth the fast development of the world economy and hgher securty requrements, the safety of lfelne systems are of greater concern these days. Therefore, estmatng the relablty of these nfrastructure systems s vtal to urban stakeholders such as utlty companes, urban planners, and polcy makers as well as to resdents and busness owners. Evaluaton of the performance and connectvty of such urban nfrastructure systems s complex n nature due to the large number of network components, complex network topology, and component/system nterdependency. Due to ths complexty, network relablty analyss s often performed by use of random samplngs of hazard scenaros and component status, whch prevents rapd rsk assessment and decson makngs durng or after a hazardous event. Ths paper proposes a new non-samplng-based system relablty method for lfelne networks and demonstrates t through a case study of a Memphs Lght, Gas and Water (MLGW) natural gas network. After a bref summary of prevous research on network relablty analyss of cvl nfrastructure systems, ths paper proposes a new network relablty analyss methodology based on the matrx-based system relablty (MSR) analyss method (Kang et al., 2007; Song and Kang, 2007). It employs an advanced network algorthm such as the recursve decomposton algorthm (RDA; L and He, 2002). The results and observatons from the case study are summarzed, and the dentfed future research topcs are also presented. 2 NETWORK RELIABILITY ANALYSIS FOR CIVIL INFRASTRUCTURE SYSTEMS The uncertanty n the performance of lfelne networks arses not only from the components (e.g. ppelnes or substatons) and the confguraton of a network, but also from the envronment such as the magntude of an earthquake and the maxmum sustaned wnd speed. The relablty of a network can be measured n three perspectves (L, 2005): () structural component relablty; (2) connectvty relablty between node pars; and (3) performance relablty, e.g., mantanng mnmum water head/pressure. Ths paper focuses on analytcal,.e. non-samplng-based relablty analyss on the connectvty. The basc dea of analytcal network relablty analyss s to convert a complex network to the combnaton of smple sub-networks such as parallel or seres systems, and then compute the network relablty from the probabltes of the sub-systems. Kroft (967) frst used a shortest path algorthm to compute network relablty. Panousss (974) and Taleb-Agha (975 and 977) proposed to compute general network relablty by convertng complex lfelne networks to seres systems n parallel (SSP) networks. However, fndng the shortest paths s not always an easy task, especally for large-scale networks. Aggarwal and Msra (975) proposed dsont shortest path algorthm. Later, researchers (Dotson and Goben, 979; Yoo and Deo, 988; Torrer, 994; L and He, 2002) mproved ths algorthm to estmate the exact relablty of largescale complex networks. A full probablty analytc algorthm (Wu and Sha, 998) and ordered bnary decson dagram (OBDD) algorthm (Kuo et. al, 999) are also capable of fndng exact network relablty, but nether of them s able to handle large-scale networks. The RDA (L and He, 2002) s effcent n dentfyng cut sets or lnk sets for a wde class of networks, but t s not able to account for the statstcal dependence between the demands or capactes of network components. Recently, Kang et al. (2007) proposed a matrx-based system relablty (MSR) method, whch can estmate the relablty of general systems through effcent matrx-calculatons wth the statstcal dependence consdered. Ths paper proposes to use the MSR method n conuncton wth a network analyss algorthm such as the RDA.

4 3 PROPOSED METHODOLOGY The MSR method subdvdes the sample space of component events wth s dstnct states, n =,..., n, nto m = Π = s mutually exclusve and collectvely exhaustve (MECE) events. The probablty of any general system event s then descrbed by the nner product of two vectors: P( ) = c T p E sys () where c s the event vector whose element s f ts correspondng MECE event s ncluded n the target system E sys event, and 0 otherwse; and p s the probablty vector that contans the probabltes of all the MECE events. Effcent matrx-based procedures were proposed to effcently obtan these vectors by use of matrx computer language such as Matlab (Kang et al., 2007). The MSR method has the followng merts over exstng system relablty methods. Frst, the probablty of a system event s always calculated by a smple matrx multplcaton as n Equaton regardless of the complexty of the system event defnton. Second, the MSR method separates the tasks of dentfcaton of system event (c) and computaton of probablty calculatons (p), whch allows for an easy ntegraton wth other computaton modules, e.g. geographc nformaton system (GIS) or network analyss algorthms. Moreover, the matrx-based procedures proposed along wth the method help obtan c and p vectors effcently. Thrd, even f one has ncomplete nformaton on the component falure probabltes and/or ther statstcal dependence, the matrx-based framework stll enables us to obtan the narrowest possble bounds on any general system event (Song and Der Kureghan, 2003) based on the avalable nformaton. Fourth, once P( Esys) s obtaned, one can calculate the probabltes of other system events of nterest, condtonal probabltes and component mportance measures (Song and Der Kureghan, 2005) wthout addtonal probablty calculatons. A drawback of the MSR method s that the sze of the vectors ncreases exponentally wth the number of component events. Ths may be a crtcal ssue n case a network wth a large number of components s consdered. However, ths can be overcome by a mult-scale approach (Der Kureghan and Song 2007) or subdvde the system event nto multple dsont lnk sets or cut sets consttuted by smaller number of components. When dsont cut sets or lnk sets, S, =,..., Nset are dentfed, the system falure probablty or relablty s computed by summng up the results of the MSR analyses of ndvdual sets,.e. N sys S = set N set T P( E ) = P( ) = c p = (2) where c and p are the event and probablty vectors of the -th dsont cut set or lnk set. These dsont sets can be effcently dentfed by makng use of an advanced network algorthm such as RDA (L and He, 2002). RDA recursvely decomposes the network nto sub-graphs untl there exst no paths between the source and termnal nodes n all sub-graphs. When paths between source and termnal are found n sub-graphs, they are dsont lnk sets and thus contrbute to the network relablty; for those subgraphs contanng no paths, they are dsont cut sets that contrbute to the system falure probablty. RDA s applcable to all types of networks regardless of ther sze or topology. After the Boolean descrptons of the dsont cut sets and lnk sets are dentfed, the correspondng event vectors c s are obtaned by use of the followng matrx-based procedures (Kang et al., 2007): E E c = c E E E E c L 2 = c.* c.* L. * c c n E n E E2 = ( c ).* ( c ).* L.*( c E ULU En En ) (3)

5 where denotes a vector of s; and.* s the Matlab operator for element-by-element multplcaton. When component events are statstcally ndependent of each other, the probablty vectors p s are constructed based on the falure probabltes of the lfelne network elements by the followng recursve matrx-based procedure: p [] = [ P P ] T [ ] p [ ] P p = [ ] for = 2,3, K,n p ( P ) (4) where n s the number of the components n the -th cut set or lnk set, P s the falure probablty [ n ] or relablty of the -th component n the set; and p = p. The probablty vectors can be obtaned even n case the components have statstcal dependence (Kang et al., 2007). As seen n Equaton 4, the falure probabltes or relabltes of the network elements, P need to be obtaned. For lfelne systems subected to sesmc hazard, the falure probabltes of network elements are evaluated based on sesmc ntenstes measures such as peak ground acceleraton (PGA), peak ground velocty (PGV), or permanent ground deformaton (PGD). Accordng to the Gudelnes by Amercan Lfelne Allance (ALA, 200), the sesmc damages of gas ppelnes are affected by the earthquake ntensty, sol condton, ppelne materal (e.g. cast ron), type of connector, etc. A regonal rsk assessment software HAZUS-MH (FEMA, 2003) uses a PGV-based damage model for gas ppelnes n whch the rate of repars (per unt klometer) s gven as 2.25 λ PGV where PGV s gven n cm/s. In the case study of ths paper, we compute the repar rates of the ppelne elements usng ths PGV-based damage model n conuncton wth the PGV values by the Md-Amerca Earthquake Center s rsk management software MAEVz, whch generates PGV values at each locaton for a scenaro earthquake (Fernandez and Rx, 2006). The repar rate of each lnk s then calculated usng ArcGIS s Spatal Analyst extenson. In the case study of ths paper, the falures along a ppelne are modeled as a non-homogeneous Posson process. Further research efforts are currently beng made to account for the statstcal dependence between the falures at adacent locatons by use of theores of stochastc processes. The falure probablty of the -th lne element s gven as L N k λ ( l) dl λ ΔLk 0 k = P = e e (5) where λ ( ) and L are the repar rate functon and the total length of the -th ppelne, k respectvely; λ and ΔLk are the representatve repar rate and the length of the k-th segment; and N s the number of the dscretzed segments for the -th ppelne. The falure probabltes of the nodes are also obtaned. The PGA data of MAEvz are frst mported nto ArcGIS, and the falure probabltes of the nodes are obtaned by use of the fraglty curves of compressor staton by HAZUS-MH. Snce HAZUS-MH does not have the fraglty curves for gate statons, we assume the gate staton owns the same fraglty property as compressor statons n the case study. It s also assumed that a node loses ts connectvty when t s n ether extensve or complete damage state. The probablty of exceedng extensve damage state s gven by where ln a ln 0.77 P ( exceeds extensve PGA = a) = Φ (6) 0.65 Φ( ) s the cumulatve densty functon of the standard normal dstrbuton.

6 4 CASE STUDY: MLGW NATURAL GAS NETWORK Earthquake mpacts on urban natural gas system may cause not only outage of gas servces, but also post-dsaster fres, whch may result n serous consequences, e.g. the 923 Great Kanto Earthquake. The area of ths case study, Shelby County of Tennessee, ncludng the cty of Memphs, s a maor urban area wth a populaton more than 9,438 (US Census 2006 data). Memphs Lght, Gas, and Water Dvson (MLGW) provdes utlty servces to more than 420,000 customers (MLGW, 2007) n ths area. The natural gas facltes operated by MLGW cover 750,000 square mle servce terrtory (Benson, 992) through more than 3,500 mles of mans and 2,200 mles of servce lnes (Chang et al., 996). For the purpose of demonstratng the proposed methodology wthout undue complexty, we use a smplfed gas transmsson network model shown n Fgure (Chang et al., 996), whch conssts of 6 nodes (gas statons and regulator statons) and 7 lnks (transmsson lnes). The gate statons, where MLGW receves the purchased gas are consdered supply facltes or source nodes for the gven gas network. The gas transmsson mans are consdered dstrbutng elements whle regulator statons and other servce faclty consdered demand nodes. Although the network used n ths research only contans 33 elements, a mult-scale approach (Der Kureghan and Song, 2007) can be used to assess the system relablty by aggregatng or dsaggregatng servce areas nto equvalent servce nodes. Smplfed MLGW Gas Network (37-node) Gate Staton Regulator Staton Other Nodes Lne Nodes Subunctve Source Node Fgure. MLGW gas network n Shelby County, TN (modfed from Chang et al., 996). Fgure 2. Drected graph model of MLGW gas network. To apply RDA to the MLGW gas network, we frst represent the network as a drected graph model shown n Fgure 2. The arrows ndcate the drectvty of the gas flow, from source to customers, and from ppelnes wth hgh pressure to those wth low pressure (Bowker, 2007). A subunctve source node, Node 37 s added to the graph to facltate fndng the mult-source network s node-par connectvty (L and He, 2002) and the ppelnes are replaced wth vrtual lnk nodes n the graph. As a result, the smplfed network n Fgure s represented by a 37-node and 40-arcs network n Fgure 2. For example, f we want to evaluate the connectvty between sources and the servce area represented by Node 6 (source node = 37 and termnal node = 6), RDA dentfes four dsont lnk sets: {(37, 34,, 9, 6), ( 34, 2, 6, 20, 35, 37), (, 2, 6, 20, 34, 35, 37), ( 9,, 2, 6, 20, 34, 35, 37)} n whch the numbers wth and wthout an upper bar ndcate the falures and survvals of the correspondng elements, respectvely. Based on the sesmc hazard maps from MAEVz software as shown n Fgure 3, we calculate the repar rates of lnks (Fgure 4), the falure probabltes of lnks (Equaton 5) and the falure probabltes of nodes (Equaton 6), all by use of ArcGIS. In ths case study, we use a set of scenaro earthquakes wth dfferent magntudes wth the epcenter N35.535º W-90.43º at Blythevlle, AR.

7 Fgure 3. PGA map of Shelby County, TN by MAEvz. Fgure 4. Repar rates of MLGW gas network computed by ArcGIS. The event (c) and probablty (p) vectors are obtaned for each lnk set or cut set by use of matrx-based procedures n Equatons 3 and 4. The probablty of outage at each dstrbuton node,.e. ts dsconnecton from all the sources s computed by Equaton 2. Fgure 5 shows the relablty and falure probabltes of network nodes as a result of the system relablty analyss. Fgure 6 shows the probablty of outage at Node 6 wth earthquake magntude n the earthquake scenaros vared. The results are then mported nto ArcGIS for dsplayng spatal dstrbuton of the relabltes/falure probabltes. Fgure 7 shows the system connectvty falure probabltes of the nodes (bar graphs) and the lnks (colors) for the gven earthquake scenaro wth magntude M = 7.2. MLGW Gas Network Connectvty Relablty Under a M7.2 Earthquake 0.6 MLGW Gas Network System Falure Probablty at Node 6 Network Connectvty Node No. System Falure Probablty System Relablty System Falure Probablty Earthquake Magntude Fgure 5. Connectvty relabltes and falure probabltes of network nodes. Fgure 6. Falure probablty at Node 6 versus earthquake magntude. 5 CONCLUSIONS A new approach for network relablty analyss s proposed based on a matrx-based system relablty method. The outage or avalablty of dstrbuton nodes are dentfed as dsont cut sets or lnk sets by use of an advanced network analyss algorthm such as the recursve decomposton algorthm. The probablty of each dsont lnk set or cutest s computed by use of matrx-based procedures of the matrx-based system relablty method. The proposed methodology s demonstrated through a case study on the Memphs Lght, Gas and Water (MLGW) s natural gas network n Shelby County, TN, but t can be appled to general lfelne systems for evaluaton of ther connectvty relabltes. Ths approach, n essence, reduces the sze and complexty of the large-scale urban nfrastructure systems by decomposng the complex system nto sub-systems whose relabltes are easy to evaluate. Wth ths approach, we are able to handle large-scale nfrastructure systems and obtan valuable statstcal nferences. Based on the results of the case

8 study, further research s n progress to () account for the statstcal dependence between sesmc ntenstes at adacent network components, (2) estmate the average downtme and soco-economc losses, (3) account for the nterdependency between dfferent nfrastructure systems, and (4) provde useful nformaton for decson makng on dsaster plannng, response, recovery, and mtgaton usng the proposed methodology. Fgure 7. Connectvty falure probabltes of MLGW gas network (ArcGIS presentaton) ACKNOWLEDGEMENTS Ths paper s based on research supported by the Md-Amerca Earthquake Center under a grant from the Earthquake Engneerng Research Center s program of the Natonal Scence Foundaton (Award No. EEC ). The authors wsh to thank Mr. Rchard Bowker (MLGW) for hs valuable comments and suggestons. Dscussons wth Dr. Yang Zhang (Unversty of Illnos at Sprngfeld), Dr. Zhenghong Tang (Wake County Plannng Department, NC), Dr. YoungSuk Km, Mr. Won-Hee Kang and Mr. Young Joo Lee (Unversty of Illnos at Urbana-Champagn) are also greatly apprecated. REFERENCES Abraham, S., R.J. Efford et al., 2004, Fnal report on the August 4, 2003 Blackout n the Unted States and Canada: Causes and Recommendatons, U.S.-Canada Power System Outage Task Force. Aggarwal, K.K., and K.B. Msra, 975, A fast algorthm for relablty evaluaton, IEEE Trans. on Relablty, 24(): Amercan Lfelnes Allance (ALA), 200, Sesmc fraglty formulatons for water systems, Reston, VA: Amercan Socety of Cvl Engneers, Aprl. Benson, J., 992, MLGW keepng pace wth RTU evoluton, Ppelne and Gas Journal, 29(9). Bowker, R., 2007, Personal communcaton, Manager, Informaton Tech. Dept., MLGW, Memphs, TN. Chang, S.E., A.Z. Rose, M. Shnozuka, and K.J. Terney, 200, Modelng earthquake mpact on urban lfelne systems: Advances and Integraton n Loss Estmaton, Earthquake Engneerng Fronters n the New Mllennum, B.F. Spencer and Y. Hu, eds. Swets & Zetlnger.

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