Seismic Reliability Analysis and Topology Optimization of Lifeline Networks

Transcription

1 The 4 th World Conference on Earthquake Engneerng October 2-7, 2008, Beng, Chna Semc Relablty Analy and Topology Optmzaton of Lfelne Network ABSTRACT: Je L and We Lu 2 Profeor, Dept. of Buldng Engneerng, Tong Unverty, Shangha. Chna 2 Lecturer, Dept. of Buldng Engneerng, Tong Unverty, Shangha. Chna Emal:le@mal.tong.edu.cn In th paper, n order to analyze emc relablty of large-cale lfelne network, the concept of tructural functon and complementary tructural functon are ntroduced. Then, a path-baed recurve decompoton algorthm and a cut-baed recurve decompoton algorthm are derved to calculate the emc relablty of lfelne network. Thee two algorthm can enumerate the dont mnmal path and the dont mnmal cut multaneouly. A the reult, a probabltc nequalty can be ued to gve the approxmate relablty. Baed on emc relablty analy of lfelne network, a topology optmzaton model etablhed. The goal of the model to fnd the leat-cot network whch atfe pecfed emc relablty. Combnng wth the path-baed recurve decompoton algorthm, three optmzaton algorthm, genetc algorthm, mulated annealng algorthm and mulated annealng genetc algorthm, are preented to obtan the optmal oluton. Above approache have been ued to a practcal ga network. KEYWORDS: Lfelne Network, Recurve Decompoton Algorthm, Topology Optmzaton. Introducton The lfelne ytem, ncludng water dtrbuton, ga upply and power network etc, are the arterae of modern cte. Wth the development of modern ocety, lfelne ytem play more and more mportant role n urban lfe (L, 2005). The nvetgaton of many prevou earthquake ndcated that the performance of lfelne ytem may have mportant effect on the property loe and caualte of cte durng the dater. In fact, almot all the lfelne ytem uffered erou damage durng many prevou trong earthquake. For lfelne network, ut lke buldng tructure, emc analy and emc optmzaton or degn are two mportant reearch feld. In th paper, the concept of network tructural functon and complementary tructural functon are ntroduced. By takng of the recurve decompoton of tructural functon and complementary tructural functon, a path-baed recurve decompoton algorthm and a cut-baed recurve decompoton algorthm are deduced to evaluate network emc relablty. A number of cae tude how that thee two algorthm form hgh effcent and accurate method to calculate the large-cale network emc relablty. Baed on thee technologe, the topology optmzaton model further explored. The goal of the model to fnd the leat-cot network whch atfe pecfed emc relablty. Three optmzaton algorthm, genetc algorthm(ga), mulated annealng algorthm(saa) and mulated annealng genetc algorthm(saga), are developed to get the optmal oluton. An actual ga network nvetgated to ndcate the valdty of the propoed approache. 2. Semc Relablty Evaluaton of Lfelne Network 2. Structural Functon and Complementary Structural Functon A network a graph wth a weght agned to each edge or node. In a network, the weght denote the ucce probablty (relablty) of correpondng edge or node. In th paper, only edge-weghted network,.e. edge are agned weght, are condered. However, for the node weghted network and general weghted network, the propoed algorthm alo avalable after makng mall change. For an edge weghted network, each edge can be n ether of two tate, operatve tate or faled tate. Therefore

2 The 4 th World Conference on Earthquake Engneerng October 2-7, 2008, Beng, Chna the network alo own two tate, operatve tate or faled tate. By ntroducng Boolean algebra, the operatve tate and faled tate are repreented by and 0 repectvely. The termnal node are denoted a ource and nk. Then network tructural functon repreented by Φ ( G) defned a follow f network operate Φ ( G) = (2.) 0 f network fal When all edge of any path of the network are n operatve tate, the network operate. Vew network tructural functon and all edge n the network a boolean varant, the network tructural functon can be wrtten a ( G) where m the number of mnmal path (P) of G and m Φ = A (2.2) k = k A k a P of G. Correpondngly, a complementary tructural functon of network can be defned a follow f network fal Φ '( G) = Φ ( G) = (2.3) 0 f network operate Smlarly, the complementary tructural functon can be expreed a the unon of all mnmal cut (C) ( G) m' Φ ' = D (2.4) k = k where m are the number of C of G and D k repreent a C of G. 2.2 Path-baed Recurve Decompoton Algorthm In order to mplfy the calculaton, defne a hortet P from the ource to the nk a A = aa2 am, where a an edge of the network and m the number of edge n A. From A, accordng to Boolean law and De organ Law, the network tructural functon Φ ( G) become Φ ( G) = A + a Φ ( G ) + a a Φ ( G ) a a a a Φ ( G ) + + a a a Φ( G ) 2 2 m m (2.5) where G repreent a ubgraph after removng edge edge a ( =,2,... ) nto the ource n equence. a from G and mergng the node connectng wth the Above ubgraph can be clafed nto connected ubgraph and dconnected ubgraph. If takng the number of connected ubgraph a m p, then the number of dconnected ubgraph m c = m mp. eantme, f Φ ( ) form a dont mnmal cut (DC) of G. =, then complementary tructural functon can be wrtten a G a dconnected ubgraph, the coeffcent n front of G Let C aa2 a a m c (2.6) = Φ ( G) = C + Q 2

3 The 4 th World Conference on Earthquake Engneerng October 2-7, 2008, Beng, Chna where Q the remander term, denotng that C do not form whole DC et. A dconnected ubgraph ext no path, ther tructural functon equal to 0 and Φ ( G) can be rewrtten a m p Φ ( G) = A + B Φ( G ) (2.7) = B = a a a a and G denote the connected ubgraph. where 2 Decompoe above connected ubgraph tep by tep untl no connected ubgraph ext, then all dont mnmal path (DP) wll be enumerated. Fnally Φ ( G) can be expreed a Φ ( G) = L (2.8) = where L a DP and the number of all DP. At the ame tme, all DC are alo enumerated, that K Φ ( G) = C (2.9) = where C a DC and K the number of all DC. After enumeratng all DP and DC, the relablty and the falure probablty of the network can be deduced a R= P[ Φ ( G) = ] = P( L = ) K F = P[ Φ '( G) = ] = P( C = ) (2.0) = (2.) = For a large network, t mght be mpoble to enumerate all dont product. In thee cae, the upper bound and the lower bound of the network relablty can be calculated ung the probabltc nequalty (L, 2005) f PL ( = ) R PC ( = ) (2.2) = = K where and K f are the number of calculated DP and DC, and K f K. When the dfference between the upper bound and the lower bound maller than a pecfed error bound, the approxmate network relablty can be gven a K f R= 0.5 P( C = ) + P( L = ) = = (2.3) 3

4 The 4 th World Conference on Earthquake Engneerng October 2-7, 2008, Beng, Chna 2.3 Cut-baed Recurve Decompoton Algorthm Smlar to the path-baed recurve decompoton algorthm, defne a C of G a organ Law and Boolean law, network complementary tructural functon become D = aa2 am. Ung De Φ '( G) = D + a Φ '( G ) + + a a a Φ'( G ) (2.4) 2 m m where G repreent a ubgraph after removng edge a ( =,2,... ) from G and mergng the node connectng wth the edge a nto the ource. Above ubgraph can be clafed nto aborbng ubgraph (the ubgraph that nk merged nto ource) and non-aborbng ubgraph. For each aborbng ubgraph, the coeffcent n front of t complementary tructural functon form a DP of network. For non-aborbng ubgraph, decompong them repeatedly untl no non-aborbng ubgraph ext, then all DC wll be enumerated. eantme, all DP are alo enumerated. Correpondngly, the probabltc nequalty can be ued to calculate the approxmate value. 3. Semc Relablty Optmzaton of Lfelne Network 3. Optmzaton odel Apparently, the emc relablty of the lfelne network ytem determned by t edge emc relablty and t topology. Uually, the tratege to mprove edge relablty nclude ung ductle ppelne materal and adoptng larger dameter ppelne and o on. But thee method may be not utable for extng lfelne network becaue ome ppelne have to be dcarded before they are out of ervce. So modfyng the network topology by addng everal edge to or removng everal edge from the network a feable way to mprove the network emc relablty. The network topology can be et a an optmzaton model. A t optmzaton obect actually to fnd the leat-cot network topology tructure whch atfy precrbed emc relablty contrant, the optmzaton model can be mathematcally formulated n the followng general form * mnmze CG ( ) = γ c ubect to Pk P0 k =,2,..., n (3.) * G a ubgraph of G where G repreent a network and uually generated emprcally, value of f edge ext n * G a oluton of the model, γ take * G and 0 nverely, P k repreent the emc relablty between ource and termnal k and can be calculated ung above path-baed recurve decompoton algorthm, P 0 repreent the relablty contrant and c repreent the cot of edge and can be evaluated n an actual lfelne network. Obvouly, above problem a typcal combnatoral optmzaton problem n whch γ the optmzaton varable. Conderng a network wth 60 edge and 30 node, the number of all potental network 60 7 C Aumng to ue a computer that can deal wth 00 network a econd, t wll take about = algorthm, uch a GA, SAA and SAGA, hould be ntroduced to olve th optmzaton model. 8 year to calculate all feable network. It mpoble n practce. Therefore, the modern optmal 3.2 Genetc algorthm GA wa poneered by J. H. Holland durng the md-970 n the feld of machne learnng (Holland, 975). It a method of earchng oluton n the oluton pace by mtatng the natural electon proce (Holland, 99). 4

5 The 4 th World Conference on Earthquake Engneerng October 2-7, 2008, Beng, Chna GA ha been wdely ued n varou optmzaton tak, ncludng numercal optmzaton, combnatoral optmzaton problem uch a knapack problem and arlne crew chedulng problem (Xuan et al, 2004). In order to ue GA n the topology optmal problem of lfelne network, a generaton ncludng many gene created ntally where each gene repreent a network. Then by ung electon, croover and mutaton operator, a new generaton evolved. The ftne of each gene determne whether t wll urvve or not. After a number of teraton or when ome crtera are met, a near-global optmal oluton could uually be found Repreentng graph a gene anpulatng a network wth GA requre that the network properly repreented. Note that any gene n GA a ubgraph of orgnal G. The mplet 0- bnary codng can be adopted here. An n bt array ued to repreent a graph and each bt repreent an edge of the graph G, where n the number of edge n G. A n the array mean that the gene cont of a correpondng edge of G whle a 0 mean not. For example, Fg. a brdge network. In th fgure, the orgnal G nclude all edge whle the ubgraph doen t nclude the edge 5(dah lne). Then the correpondng gene of the ubgraph can be wrtten a Fgure A brdge network Generatng ntal graph Becaue all the other generaton n the genetc algorthm evolve from the frt one, o t reaonable to hope the frt generaton contan a many dvere gene a poble. Therefore generatng a ere of random graph the man ob n th tep. To generate a random graph, a gene ntalzed to contan no edge. That mean the bt n the array of the gene are all 0. Then each bt n the array choen n turn and changed to at a precrbed probablty. In other word, each edge n G added to the graph at random. It mut be noted that not all graph generated from above method are feable n practce. Dconnected graph have no practcal meanng. Therefore one mut frt udge the connectvty of each graph n the ntal generaton and modfy the dconnected graph by addng everal edge at random. In order to keep the chema of the gene, only a mall number of edge can be added to a dconnected graph. If the graph tll a dconnected graph after modfyng, t dcarded and a new gene generated to replace t Evaluatng the graph At each generaton, the gene need to be evaluated to determne ther ftne. Snce the model to calculate the leat-cot ytem, evaluaton requre calculatng the cot of each edge. However, t mut be noted that not all gene n the generaton are feable oluton becaue ome of them may not atfy the relablty contrant. For thee gene, a penalty factor, a functon of the ytem emc relablty, appled to thoe unfeable oluton. The penalty factor can be defned a 4 5 t 0 P P QX ( ) = C C max( P ) P P < P mn 0 max mn mn mn mn 0 max( Pmn ) mn( Pmn ) (3.2) where P mn repreent the mnmum nodal emc relablty of gene, C max and C mn repreent the maxmum and mnmum cot of the gene n one generaton repectvely. The ftne of gene k can be wrtten a T( k) = S C( k) Q( k) (3.3) 5

6 The 4 th World Conference on Earthquake Engneerng October 2-7, 2008, Beng, Chna where S a large value and C(k) repreent the cot of gene k GA operator There are three operator, electon operator, croover operator and mutaton operator, to be ued on the gene n current generaton to produce the gene n the next generaton. The electon operator a core operator of GA. It elect the uperor gene, ndvdual wth hgh ftne, at a hgh probablty and nferor gene, ndvdual wth low ftne, at a low probablty and pae them to next generaton. Although many electon operator are avalable, roulette wheel electon operator and eltt electon operator (Chen et al, 996) are adopted here. The croover operator an operator whch guarantee the global earchng capablty of GA. Th operator take two gene at a precrbed probablty and produce two offprng. In th paper, one-pont croover operator (Chen et al, 996) adopted for producng new gene. The mutaton operator ued to guarantee the local earchng capablty of GA by perturbng the gene generated from croover operator. The proce of th operator very mple, electng a gene and changng each bt of gene nto 0 f t wa and f t wa 0 at a precrbed mutaton probablty Stoppng crtera In th paper, the algorthm top when the number of teraton reache to maxmum teraton number. 3.3 Smulated annealng algorthm SAA wa frt ntroduced by Krkpatrck et al (983) and ndependently by Cerny (985) a a problem-ndependent combnatoral optmzaton technque. SAA ha been appled to a wde range of dffcult combnatoral optmzaton problem, uch a travelng aleman problem (Aart et al, 998), large-cale ntegraton computer-aded degn (Wong et al, 988), computer communcaton network degn (Samuel et al, 995) and o on. SAA a earch procedure n whch the current oluton contnually compared to oluton whch are obtaned by carryng out a perturbaton. The perturbaton reult accepted at a probablty decrbed a followng: P ( ) = exp [ ]/ ( f () f ( ) t) ( ) ( ) ( ) > ( ) f f f f (3.4) where f () energy functon of oluton determned by t cot and emc relablty and t current temperature, a control parameter whch decreae a the proce of SAA goe on and approache 0 at lat. Apparently f the perturbaton reult an mproved oluton, t accepted and the current oluton updated accordngly. Otherwe, t can alo be accepted at a probablty decrbed n Eq.(3.4). By acceptng a worenng oluton, SAA avod beng trapped too early n a local optmal oluton. On the other hand, the probablty of acceptng a worenng perturbaton oluton decreae becaue t decreae a the proce of SAA goe on, whch guarantee the algorthm wll eventually converge and be le lkely to move away from a global optmal oluton after havng approached t. For the network topology optmzaton problem, a oluton repreented a the ame a the gene n GA and the proce of SAA can be decrbed a followng. Produce an ntal oluton ung the ame method a producng the gene n ntal generaton of GA; 2Determne current temperature t baed on the ntal temperature T and coolng chedule. If the current 6

7 The 4 th World Conference on Earthquake Engneerng October 2-7, 2008, Beng, Chna temperature lower than the termnal temperature, top. 3Perturb current oluton and generate a new oluton. Calculate t energy functon and determne the acceptng probablty of the new oluton. 4Generate a number vared from 0 to at random and compare t wth above acceptng probablty. If the random number maller than the acceptng probablty, the new oluton accepted and the current oluton updated. Otherwe, the new oluton dcarded and current oluton preerved. 5Judge whether the number of perturbaton ha reached precrbed value or not at current temperature, f ye, go to tep 2, or go to tep Smulated annealng genetc algorthm Although GA a very ueful algorthm for combnatoral optmzaton problem, t ha a maor lmtaton that premature convergence wll occur when the genetc algorthm cannot fnd the optmal oluton due to lo of ome mportant character. The reaon that GA depend heavly on croover operator, and the mutaton probablty generally too mall to move the earch nto another pace. To overcome th problem, many reearcher (Ilkwon et al, 996; Yu et al, 2000) notce that SAA good at wdenng the earchng pace and t operator very mlar wth the mutaton operator n GA. So a hybrd algorthm whch replace the mutaton operator n GA wth the perturbaton n SAA at the ame temperature developed. In other word, each gene generated by croover operator wll be perturbed to produce everal new oluton and be updated accordng to the acceptng probablty n Eq.(3.4). Heren the ntal temperature ued for the frt generaton and the temperature doen t change n one generaton but decreae n next generaton. 4. Cae Study A ga network locatedn a Cty of Chna, whch cont of 463 node and 977 ppelne, nvetgated n th paper. Ung path-baed recurve decompoton algorthm, the emc relablty of network calculated and the reult hown n Fg.2. Jut for uch a complexty network, path-baed recurve decompoton algorthm can gve the reult accurately and quckly. Alo, let P0 equal to 0.8, GA and SAGA have been ued to olve the emc topology optmzaton problem of the ga network and the reult hown n table 4.. Alo the reult calculated by GA hown n Fg.3. Apparently, after optmzaton, the network emc relablty of the network ncreae much more, all node are n the tate of relable (relablty between 0.9 and ) or lghtly damaged (relablty between 0.7 and 0.9). Fgure 2 Semc relablty of the network Fgure 3 Semc relablty of the network after optmzaton Table 4. Comparon of GA and SAGA for the actual network Algorthm Cot GA 573,03,000 ($74,47,272) SAGA 570,820,000 ($74,32,467) 7

8 The 4 th World Conference on Earthquake Engneerng October 2-7, 2008, Beng, Chna 5. Concluon Semc relablty analy and optmal degn of lfelne ytem mportant for modern cte that located on earthquake area. Baed on tructural functon and complementary tructural functon, everal ytematcal recurve decompoton algorthm were developed to gve the emc relablty of lfelne network. Combnng wth thee algorthm, dfferent optmal algorthm, genetc algorthm, mulated annealng algorthm and mulated annealng genetc algorthm, are explored to olve the topology optmzaton problem of lfelne ytem. Cae tudy ndcate that above approache provde an effcent route on emc relablty analy and optmzaton of lfelne network. Reference Aart EHL, Kort, JH, van Laarhoven PJ (988). Solvng travelng aleman problem by mulated annealng. Journal of Stattcal Phyc; 50: Cerny V (985). Thermo dynamcal approach to the travelng aleman problem: an effcent mulaton algorthm. Journal of Optmzaton Theory and Applcaton; 45:4-5 Chen Guolang, Wang Xufa et.al (996), Genetc algorthm and t applcaton(n Chnee), Beng: Pot & Telecom Pre Holland JH (975), Adaptaton n neural and artfcal ytem, Ann Arbor: Unverty of chgan Pre Holland JH(99). Genetc algorthm. Scentfc Amercan; 265: Ilkwon Jeong, Juang Lee(996). Adaptve Smulated Annealng Genetc Algorthm for Sytem Identfcaton, Engneerng Applcaton of Artfcal Intellgence, 9(5): Krkpatrck S, Gelatt Jr C D, Vecch.P(983). Optmzaton by mulated annealng. Scence, 220: 67~680. L Je(2005). Lfelne earthquake Engneerng-Bac ethod and Applcaton. Scence Pre, Beng, Chna. Samuel Perre, chel-ange Hyppolte, Jean-mare Bourolly, Oumar Doume(995), Topologcal degn of computer communcaton network ung mulated annealng. Engneerng Applcaton and Artfcal Intellgence, 8():6-69 Wong DF, Leong HW, Lu CL(988). Smulated annealng for VLSI degn. Boton, A; Kluwer Academc Publher. Xuan Guangnan, Ceng Runwe et.al(2004), Genetc algorthm and engneerng optmzaton(n Chnee), Beng: Tnghua Unverty Pre Yu Hongme, Hapeng Fang, Pngng Yao, et al(2000), A combned genetc algorthm/mulated annealng algorthm for large cale ytem energy ntegraton, Computer and Chemcal Engneerng, 24: