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ADE	AppendixA. SubmittedversionofPublication1: WDSLib: AWaterDistributionSystem SimulationTestBed Choice of scaling factors 779 The choice of the scaling factor, despite much research, is not well understood. In this 780 subsection, a choice for each scaling factor, based on the experience of the authors, is 781 recommended. There are two types of variables and parameters that need to be scaled: 782 invariants and variants. 783 Data sets that have very wide range of values can confound numerical accuracy. As 784 a result, it may be preferable to scale the data to a narrower range. The default scaling 785 factorforeachoftheinputdataischosentobeitsmaximumabsolutevalue. Forexample, 786 the scaling factor for demand is max(d), so that its values range from zero to one. 787 In contrast, it is more difficult to choose a scaling factor a priori for values that vary 788 between iterations (variants). This is because the range of variants can change as the 789 iteration progresses. As a result, the intermediate and the final results might not be 790 within the same range as the initial guesses. 791 There are two variants that need to be scaled: q, h. A good choice of the scaling 792 d factor for the flow rate is because the demand at each node must be satisfied by 793 n Pf the water sources in the WDS and it is a reasonable assumption that the all demands 794 are equally carried by each pipe that is directly connected to a water source and a good 795 choice of the scaling factor for nodal head is max(e ) because the maximum nodal head 796 l cannot exceed the maximum elevation head of the fixed nodes. 797 During the process of the computation, the matrices G and F are scaled because 798 their input variables are scaled. For the Darcy-Weisbach head loss model, the diagonal 799 elements of the matrix G are modelled by: 800 8 L j [G] = diag ( ) f q for j = 1,...,n 801 jj π2g D5 j \| j \| p (cid:26) j (cid:27) 157
ADE	AppendixB. SubmittedversionofPublication2: ABenchmarkingStudyofWater DistributionSystemSolutionMethods INTRODUCTION 25 Water Distribution Systems (WDSs) are frequently modeled by a system of nonlinear equa- 26 tions, the steady-state solutions of which, the flows and heads in the system, are used in WDS 27 design, management and operation. In a design setting, the solutions might be used as part of 28 an optimization problem to determine the best choices of some network parameters such as pipe 29 diameters. In a management setting, the solutions might be used for the calibration of network 30 parameterssuchasdemandpatterns. Inanoperationalenvironment,newsolutionsmightbeneeded 31 toadjustcontroldevicesettingswhenevernewsupervisorycontrolanddataacquisition(SCADA) 32 informationbecomesavailable. 33 ThemostwidelyusedWDSsimulationmethodincurrentuseistheGlobalGradientAlgorithm 34 (GGA) (Todini and Pilati 1988), which solves the non-linear system of equations representing 35 the WDS. The GGA and its implementations exhibit excellent convergence characteristics for a 36 wide range of starting values and a wide variety of WDS problems. However, some networks 37 have structural properties which can be exploited to further improve the efficiency of the solution 38 process. TheGGA,arangespacemethod,exploitstheblockstructureofthefullJacobianmatrixin 39 ordertoproduceasmallerkeymatrixinthelinearizationoftheNewtonmethod. Thereformulated 40 co-tree flows method (RCTM) (Elhay et al. 2014), a null-space method (Benzi et al. 2005), can 41 further exploit the block structure of the Jacobian matrix to produce, in realistic WDSs, an even 42 smallerkeymatrix. Thisisachievedbydealingseparatelywiththespanningtreeandtheco-treein 43 theNewtonmethodlinearization. 44 AnotheravenueforreducingcomputationcanbeexploitedbyusingtheForest-CorePartitioning 45 Algorithm (FCPA) (Simpson et al. 2012) to separate the problem into its linear and non-linear 46 components. TheobservationunderpinningtheFCPAisthatmostWDSshavetrees,thecollections 47 ofwhicharecalledforests. Thecomplementoftheforestinanetworkiscalledthecore. Theflows 48 inaforestcanbecomputeda-prioribyalinearprocess. Hence,thedimensionofthekeymatrices 49 inthesolutionprocesscanbesignificantlyreducedwhentheforestisalargepartofthenetwork. 50 With the development of different solution methods, WDS simulation package users are faced 51 171
ADE	AppendixB. SubmittedversionofPublication2: ABenchmarkingStudyofWater DistributionSystemSolutionMethods with a choice of which solution method or methods to apply. Previous publications performed 52 case studies comparing the performances of their respective methods to the GGA. However, these 53 comparisons were often done using different implementation languages, and different levels of 54 code optimization – which makes cross-comparison of methods difficult. Consequently, there is a 55 need for a study which reliably compares the relative performance of these methods using a fast, 56 carefully designed code implementation. To this end, this work presents a thorough benchmark 57 studytocomparetheperformanceofGGA,GGA-with-FCPA,RCTM,andRCTM-with-FCPAfor 58 a range of case study networks using a fast C++ implementation. The timings for these runs are 59 decomposed according to how often each solution component is executed in different simulation 60 settings. From these timings it is possible to accurately predict runtimes for long-run multiple 61 simulationsettings. Toconfirmtherelevanceoftheseresults,thetimingshavebeencomparedwith 62 the speed of the industrial and research standard toolkit of EPANET2 (Rossman 2000) and was 63 foundtobefasterinallcases. 64 Thispaperisorganizedasfollowed. Adetailedreviewofexistingsolutionmethodsisgivenin 65 thenextsection. Thesectionfollowingpresentsthemathematicalformulationofeachmethod. The 66 motivationforabenchmarkstudyisthengiven,followedbythemethodologyusedinthispaperto 67 carryoutabenchmarkstudy. Thedescriptionofthemodulecategorizationisthenpresented. This 68 is followed by a case study of the four solution methods applied to the eight case study networks. 69 Theresultsarediscussedinthenextsection. Thelastsectionofferssomeconclusions. 70 LITERATURE REVIEW 71 Thissectionprovidesareviewofthealgorithmsthataretestedinthispaper. Abriefdevelopment 72 history of WDSsolution algorithms is presentedin the first subsection. The nextsubsection gives 73 anoverviewoftheGGAanditsdevelopment,followedbyanoverviewofsolutionmethodswhich 74 usethenullspaceapproach(suchasco-treesflowmethod(CTM)andRCTM).Finally,areviewof 75 themethodsthatusegraphtheorytosimplifyproblemcomplexityarepresented. 76 172
ADE	AppendixB. SubmittedversionofPublication2: ABenchmarkingStudyofWater DistributionSystemSolutionMethods pipes exhibit negligible changes after a few iterations. As a result, there is no point in wasting 131 computer resources to re-compute the pipe head losses for the pipes that have little or no change 132 inflows. ThispartialupdatecanbeusedtoeconomizethecomputationalcomplexityoftheGGA, 133 theRCTMandtheirvariations. 134 Graphtheory 135 The forest-core partitioning algorithm (FCPA) (Simpson et al. 2012) speeds up the solution 136 process. This algorithm permutes the system equations to partition the linear component of the 137 problem, which is the forest of the WDS, from the non-linear component, which is the core of the 138 WDS. It can be viewed as a method that simplifies the problem by solving for the flows and the 139 heads in the forest just once instead of at every iteration. The FCPA reduces the number of pipes, 140 numberofjunctions,andthedimensionoftheJacobianmatrixinthecorebythenumberofforest 141 pipes(ornodes). 142 The graph matrix partitioning algorithm(GMPA) (Deuerlein et al. 2015) exploited the linear 143 relationshipsbetweenflowsoftheinternaltreesandtheflowsofthecorrespondingsuper-linksafter 144 theforestofthenetworkhasbeenremoved. Thiswasamajorbreakthrough. TheGMPApermutes 145 thenode-arcincidencematrixinsuchawaythatallofthenodeswithdegreetwointhecorecanbe 146 treatedasagroup. Bypartitioningthenetworkthisway,thenetworkcanbesolvedbyaglobalstep, 147 whichsolvesforthenodeswithdegreegreaterthantwo(supernodes)andthepipeswhichconnect 148 to them (path chords), and a local step, which solves for the nodes with degree two (interior path 149 nodes)andpipesconnectedtothem(path-treelinks). 150 In a recent paper by Elhay et al. (2018), they proposed a single framework for both the FCPA 151 and GMPA and extended the methods applicability to the pressure dependent problem. Although 152 the flows and heads in the forest component of a pressure driven WDS cannot be determined by a 153 linearprocess,theycanbesolvedbyasimilarlineariterativeprocessasthelocalstepintheGMPA. 154 MOTIVATION 155 Thus far, this paper has discussed the recent developments in the solution methods. Previous 156 workonWDSsimulationhasfocusedontworesearchareas: (i)hydraulicsolutionmethods(Nielsen 157 175
ADE	AppendixB. SubmittedversionofPublication2: ABenchmarkingStudyofWater DistributionSystemSolutionMethods networkhasasignificantforestcomponent. TheFCPAstartsbygeneratingapermutationmatrix 250 n n p j n S O f   n P O 251 P 1 = pc      (9) n O C  jc       n O T  f       S 252 , where   Rnp ×np is the square orthogonal permutation matrix for the pipes, S Rn f×np ∈ ∈ P     253 is the permutation matrix which identifies the pipes in the forest as distinct from those of the 254 core of the WDS, P Rnpc×np is the permutation matrix for the pipes in the core of the WDS, ∈ C 255   Rnj ×nj is the square orthogonal permutation matrix for the nodes, C Rnjc×nj is the ∈ ∈ T     256 permutation matrix for the nodes in the core of the WDS, T Rn f×nj is the permutation matrix ∈ whichidentifiesthenodesintheforestasdistinctfromthoseofthecoreoftheWDS. 257 Anewlemmaisproposedasfollows: 258 LEMMA1. Suppose 259 n m P 1 260 Q =  , m S 2      Q Rn n, is an orthogonal permutation matrix and that D = diag d ,d , ,d Rn n is 261 × 1 2 n × ∈ { ··· } ∈ diagonal. Then 262 PDST = 0 (10) 263 264 T T 265 Proof. P = e i 1,e i 2,...,e im1 forasetofindicesT = {i 1,i 2, ··· ,i m 1}andS = e j 1,e j 2,...,e jm2 (cid:18) (cid:19) (cid:18) (cid:19) for a set of indices V = j ,j , ,j . Note that T S = . Now for some 1 i m ,1 266 { 1 2 ··· m 2} ∩ ∅ ≤ ≤ 1 ≤ 180
ADE	AppendixB. SubmittedversionofPublication2: ABenchmarkingStudyofWater DistributionSystemSolutionMethods j m thereexisti , j suchthat 267 2 t s ≤ eTPDSTe = eTd e = 0 268 i j it it js fromwhich(10)follows. EndofLEMMA1(cid:3) 269 AfterapplyingtheFCPApermutation,thesystemequationsbecome 270 G A q A e 271 P 1  − 1  P 1T P 1   P 1  2 l  = O (11) × A T O × × × h − × d − 1                  andwiththispermutation,Eq. (3)leadsto 272 SGST O SA CT SA TT Sq SA e 1 1 2 l − −      O PGPT PA CT O Pq PA e 273     CA TST  CA TPT − O1  O       Ch   −    Cd2 l    = O (12) − 1 − 1               TA TST  O O  O   Th    Td   − 1               SA CT SA TT 1 1 274 where(i) − − ,whichistheoriginaltoprighttwo-by-twoblockinthefirst PA TCT PA TT − 1 − 1    matrix ofEq. (12), is the permutedA matrix, in which the (2,2) block, which is PA TT , 275 1 1 − becomes O because the pipes in the core do not connect to any nodes in the forest which are not 276 SGST SGPT 277 root nodes, and (ii)  , which is the original top left two-by-two matrix of Eq. PGST PGPT     (12), is the permuted G matrix, in which it is evident from the Lemma 1 that the (1,2) and (2,1) 278 blocks,whichareSGPT andPGST ,becomeO. 279 The top right block (the (1,2) block) of the permuted A matrix, SA TT , is invertible and 280 1 1 − canbepermutedtobelowertriangularformbecauseitrepresentstheunionofthetrees. Theflows 281 ofthepipesintheforest,Sq canbefounddirectlyfrom 282 181
ADE	AppendixB. SubmittedversionofPublication2: ABenchmarkingStudyofWater DistributionSystemSolutionMethods relationship between the co-tree flows and spanning tree flows. This is achieved by applying the 304 Schilders’ factorization to permute the A matrix into a lower triangular square block at the top, 305 1 representingaspanningtree,andarectangularblockbelow,representingthecorrespondingco-tree. 306 TheRCTMstartsbygeneratingapermutationmatrix: 307 n n p j n K O st 1   308 P 2 = n ctK 2 O  (17)     n  O R j       K 1 309 where   Rnp ×np is the square orthogonal permutation matrix for the pipes, in which ∈ K  2   K Rnst np is the permutation matrix that identifies the pipes in the spanning tree as distinct 310 1 × ∈ from those in the co-tree and K Rnct np is the permutation matrix for the pipes in the co-tree, 311 2 × ∈ R is the permutation matrix for the nodes to have the same sequence that are traversed by the 312 correspondingspanningtreepipes. 313 ThepermutedsystemequationoftheRCTMis: 314 G A q A e 315 P 2  − 1  P 2T P 2   P 2  2 l  = O (18) × × × × − × −Ab 1T Ob  h b  bd              b b b and(14)becomes: 316 K GK T O K A RT K q K A e 1 1 1 1 1 1 2 l −      317  bO K 2GK 2T −K 2Ab 1RT K 2q b−K 2Ab 2e l = O (19)            −RA 1TK 1T −RAb 1TK 2T Ob   Rh b    Rbd             b b b b inwhichthe(1,2)and(2,1)blocks,whichareK GK T andK GK T ,becomeOforthereasons 318 1 2 2 1 showninLemma1. 319 183
ADE	AppendixB. SubmittedversionofPublication2: ABenchmarkingStudyofWater DistributionSystemSolutionMethods METHODOLOGY 341 This section describes the methodology used to carry out a comparative study of the WDS 342 solution methods. The following describes the software platform used to run the benchmarking 343 simulations. Thisdescriptionisfollowedbytheproposedalgorithmevaluationmethod. 344 TheSoftwarePlatform 345 To run the benchmark tests required by this study a hydraulic simulation toolkit, WDSLib, 346 wascreated. Thistoolkit, writteninC++, incorporatedthesolutionmethodsstudiedinthispaper, 347 which include the GGA, the GGA with the FCPA, the RCTM, and the RCMT with the FCPA. In 348 order to provide a useful platform for comparison, the solution methods were implemented using 349 fast and modularized code. A focus of attention in this research has been the implementation 350 correctness,robustnessandefficiency. Thecorrectness∗ ofthetoolkithasbeenvalidatedagainsta 351 reference MATLAB implementation. The differences between all results (intermediate and final) 352 produced by the C++ toolkit and the MATLAB implementation were shown to be smaller than 353 10 10. In the interest of toolkit robustness, special attention has been paid to numerical processes 354 − to guard against avoidable failures, such as loss of significance through subtractive cancellation, 355 andnumericalerrors,suchasdivisionbyzero. Thedatastructuresandcodelibrariesinthetoolkit 356 aresharedandallsolutionmethodimplementationshavebeencarefullydesignedtoensurefairness 357 ofperformancecomparisonsbetweenalgorithms. 358 The following subsections describe the measures taken in the implementation the solution 359 methods to help ensure the validity of the timing experiments for the case study results. These 360 include measures to ensure accurate timing results, minimization of memory use, and numerical 361 robustness. 362 TimingConsiderations 363 C++waschosenastheimplementationlanguagebecausetimingsinMATLABareconfounded 364 by a variety of factors. The MATLAB programming language is a hybrid of interpreted language 365 and compiled language: some codes are interpreted by MATLAB with no compilation, some 366 ∗termsrecognizedinComputerSciencewillbedesignatedbyasterisksuperscript 185
ADE	AppendixB. SubmittedversionofPublication2: ABenchmarkingStudyofWater DistributionSystemSolutionMethods codes are partially compiled by a closed-source just-in-time (JIT) compiler, and some codes 367 are fully compiled. MATLAB may also perform additional work and bookkeeping between the 368 interpretationofonelineandthenext. 369 In contrast, C++ is a compiled language: the compiler translates the code into native machine 370 instructions which are later executed by the hardware. This faster and much simpler model of 371 executionovercomesmanyoftheproblemsassociatedwithMATLABtiming. Asaconsequence,a 372 C++implementationformsabetterbasisforafaircomparisonofdifferentWDSsolutionmethods. 373 When executing the timing experiments in this work, each code module reports the time 374 spent in it by sampling wall clock time at the start and end of its execution. Although the 375 overhead for sampling wall clock time is small, there are at least two special considerations 376 involved in the interpretation of these timings: (i) the operating system, at its own discretion, may 377 launch background processes (for example anti-virus software), which distort the timings and (ii) 378 extrapolatingthetimingformultiplesimulations(asmayoccur,forexample,inageneticalgorithm 379 or other evolutionary algorithm run) from a single analysis must be done with care because the 380 relationship between the different settings is not linear. More detail on these issues is given in a 381 latersectiondescribingtheproposedalgorithmevaluationmethod. 382 MemoryConsiderations 383 Memory management for each method was very carefully handled to advantage that method 384 in the interest of a fair comparison. To offer further assurance of the correctness of memory 385 management,Valgrind(NethercoteandSeward2007),aprogrammingtoolformemorydebugging, 386 memoryleakdetectionandprofilingtool,wasdeployedduringtestingtodetectanymemoryleaks, 387 memorycorruption,anddouble-freeing. 388 Numericalconsiderations 389 The calculations in this paper were performed in C++ under IEEE-standard double precision 390 floating point arithmetic with machine epsilon (cid:15) = 2.2204 16. The constants and parameters 391 mach − ineveryequationweregatheredandreplacedbyfull20-decimaldigitaccuracyvalues. Inaddition, 392 alldependentconstantsinmathematicalexpressionswereremoved. 393 186
ADE	AppendixB. SubmittedversionofPublication2: ABenchmarkingStudyofWater DistributionSystemSolutionMethods ModuleCategorization 420 Forthepurposesofmodelingexecutiontimes,codemodulesinamultiplesimulationruncanbe 421 dividedintothreecategories: (i)modulesthatrunonlyonceforeverymultiplesimulationarecalled 422 level-1modules(L ). Thelevelofamoduleisdeterminedbythenumberoftimesitwouldberun. 423 1 ExamplesofL modulesincludethemodulethatloadstheWDSnetworkconfigurationfileandthe 424 1 modulethatidentifiestheforestpipesinFCPA;(ii)modulesthatarerunonceforeverysimulation 425 arecalledlevel-2modules(L ). ExamplesofL modulesarethosethatinitialize,respectively,all 426 2 2 pipe flows in the GGA, core flows in FCPA and co-tree flows in RCTM; (iii) modules that are run 427 once for each iteration of every simulation are called level-3 modules (L ). An example of an L 428 3 3 module is the module computing the G and F matrices in any of the solution methods described 429 here. 430 In a once-off network simulation setting, for each trial, a given solver configuration is used to 431 solveaninputnetworkandthetimetocompletethesolutionismeasured. Inthissetting,theFCPA 432 andRCTMmodulesrequirecertaincomputationswhichonlyneedtobedoneonceeveryiterative 433 phase. The computation for these so-called invariants can be lifted* out of the iterative phase and 434 executed once per evaluation, thus saving computation time. The second setting considered here 435 is a multiple simulation run, such as one might find in a GA to optimize the design of a WDS, for 436 example. In this setting, a network with a fixed topology is solved multiple times for say different 437 pipediameters. Inthiscase,becauseofthefixedtopology,theFCPAandRCTMhavemodulesthat 438 need only be run once for each multiple simulation run. This again reduces the overall simulation 439 runtime. 440 CASE STUDIES 441 The implementation described above was used to evaluate the efficiency of the four solution 442 methods in two simulation settings: a once-off simulation setting, in which the steady-state heads 443 and flows are computed just once with the given WDS parameters, and a design setting, in which 444 thesteady-stateheadsandflowsneedtobecomputedmanytimesto,say,findtheleast-costdesign 445 by EA optimization. In the methodology section, the four solution methods, namely GGA, GGA 446 188
ADE	AppendixB. SubmittedversionofPublication2: ABenchmarkingStudyofWater DistributionSystemSolutionMethods withFCPA,RCTM,andRCTMwithFCPA,weredecomposedintomodules. Thesemoduleswere 447 categorized into levels by using the method described in the previous section. Fig.1 shows the 448 module classifications and the level of repetition of different modules for the different solution 449 methods. The columns of the block diagram show different solution methods and the rows of the 450 block diagram show the levels of repetition of the steps as they would be executed in a multiple 451 simulation setting. In the body of the table, the different methods are separated by double vertical 452 lines where column(s) intersect a box, which means the modules that are represented by that box 453 are used by the method(s) that are presented by that column(s). For example, the modules for 454 RCTM that are required to be carried out once before a multiple simulation include: (i) load 455 the configuration file and read EPANET input file, (ii) find the Schilders’ spanning tree co-tree 456 factorizationand(iii)findandapplytheAMDbandwidthreduction. 457 Eightbenchmarknetworkswereusedtostudytheeffectivenessofeachmethodunderdifferent 458 designsettings. ThenetworksusedherewerederivedfromSimpsonetal.(2012)withsomeslight 459 modificationstoremovecontroldevices,patterns,curvesandpumps. Detailsofthesenetworksare 460 given in that paper. The basic network characteristics of the case study networks are summarized 461 in Table 2. All the case study networks are realistic. The ratio between the number of the forest 462 pipes and the total number of pipes ranges between 17% and 42%. The ratio between the number 463 of the co-tree pipes and the total number of the pipes ranges between 3% and 31%. Each of 464 thefour solutionmethodsand theGGAimplementation intheEPANET areappliedto theseeight 465 benchmarknetworks. IthasbeenpointedoutbyGuidolinetal.(2010)thatthecodeimplementation 466 in EPANET are highly optimized for its performance and not written to be readily decomposed 467 into modules for different tasks.. As a result, it is difficult, if not impossible, to apply the module 468 categorizationmethodproposedinthecurrentpapertoEPANET.ThetimestakenbybothENopen, 469 theEPANETfunctionforreadingtheinputfileandmemoryallocation,andENclose,theEPANET 470 functionformemorydeallocationarenotcountedinthefinalEPANETtiming. 471 Thenextsectionpresentsthetiminganalysisforthesecasestudynetworks. Ofcourse,thesame 472 benchmarktestsperformedonanothercomputingplatformwillproducequitedifferentresults,but 473 189
ADE	AppendixB. SubmittedversionofPublication2: ABenchmarkingStudyofWater DistributionSystemSolutionMethods theauthorsbelievethattherelativetimingswillremainthesame. 474 RESULTS AND DISCUSSION 475 The benchmark tests were performed on a Intel(R) Xeon(R) CPU E5-2698 v3 running at 2.30 476 GHzwith16coresand40MBL cacheonaHighPerformanceComputingmachinecalledPhoenix 477 3 attheUniversityofAdelaide. Thenumberofcoresallocatedtoeachtestwaslimitedtoone. Each 478 timingtestwasrepeated15timesoneachbenchmarknetwork. 479 Once-offSimulationSetting 480 The mean, minimum, maximum and median wall clock times for all modules were collected. 481 As an example, the detailed statistics of the time for each module of the GGA method applied to 482 N ,thefirstcasestudynetwork. Table3presentsthedetailedtimingresultsofallmodulesusedin 483 1 the toolkit implementation of the GGA without FCPA at the three levels of repetition: once every 484 multiple simulation (L ), once every simulation (L ) and once every solver iteration (L ). The 485 1 2 3 sub-total for each level is summarized after each level of repetition and the grand-total is shown 486 in the last row. The percentage inside the bracket shows the contribution of each of the modules 487 towards its level of repetition and the contribution of each of the levels towards the total runtime. 488 For example, the AMD permutation contributes 66.9% of the L time and the all L modules 489 1 1 contribute 14.8% towards the total runtime. The mean time for a once-off simulation of the N 490 1 networkis6.75ms. Ofthetotaltime,84.3%wasspentonL tasks. Thetwomosttime-consuming 491 3 tasks are the linear solver in the iterative process, which solves the linearization of the non-linear 492 problembyusingEq. (7),andgetGF-2,whichcomputesthederivativesofthehead-lossequations. 493 Table 4 shows the summary statistics of the 15 repetitions of each solution method applied to 494 the eight benchmark networks. Under a once-off simulation setting, the GGA implementation in 495 this paper has been able to achieve between a 26% and 73% speedup when compared with the 496 GGAimplementationinEPANETbyimplementingtheproposedmodulecategorization. Thebest 497 performingalgorithmcombinationforeachnetworkishighlightedinbold. BoththeGGAandthe 498 RCTM benefit from the use of the FCPA (between 19.3% and 37.2% of time saved for the GGA, 499 between7.6%and21.4%savedfortheRCTM). 500 190
ADE	AppendixB. SubmittedversionofPublication2: ABenchmarkingStudyofWater DistributionSystemSolutionMethods The number of non-zeros in the key matrices is commonly used as an indicator of the com- 501 putational complexity of the Cholesky factorization when sparse arithmetic is used. The numbers 502 of non-zeros in the key matrices of the four WDS solution methods are summarized in Table 5. 503 The number of non-zeros in the key matrix of the GGA is a topology-related constant whereas 504 the number of non-zeros in the key matrix of the RCTM is determined by the choice of spanning 505 tree. NetworkN istheonlycasewherethenumberofnon-zeroelementsinthekeymatrixofthe 506 8 RCTM is significantly greater than that of the GGA, therefore network N is the only case where 507 8 the per-iteration runtime of the RCTM linear solver is greater than that of the GGA (Table 6). 508 Using the FCPA with the GGA can reduce the number of non-zeros in its key matrix. Moreover, 509 the dimension of the non-linear problem reduces from n to n which reduces the per-iteration 510 p pc executiontime whencomputingthe headlossderivatives, secondphaseand thestoppingtest. Al- 511 thoughthenumberofnon-zerosinthekeymatrixoftheRCTMisindependentofwhetherornotthe 512 FCPA is used, using the FCPA does: (i) reduce the computation time of the matrix multiplication 513 in the linear solver, (ii) reduce the dimension of the search space which speeds up the process of 514 partitioningtheco-treepipesfromthespanningtreepipesintheRCTM,and(iii)reducethenumber 515 of pipes in the spanning tree. This can be seen by the per-iteration execution times for each of the 516 L modules,whichareshownintheTable6. 517 3 The number of iterations required for each of the four solution methods to satisfy the stopping 518 test for the eight case studies networks is shown in the Table 7. It is evident from Table 7 that the 519 GGA took exactly the same number of iterations to satisfy the stopping test with or without the 520 FCPA. The flows in the forest network satisfy a linear system, which does not change from one 521 iteration to the next. Therefore, the flows in the forest pipes reach their steady-state after the first 522 iteration. Similarly, the RCTM with or without FCPA takes the same number of iterations. In the 523 casesthatwereanalyzedinthisstudy,theRCTMrequiredagreaternumberofiterationstosatisfy 524 thestoppingtestcomparedtotheGGA.Thisisbecausedifferentmechanismsareusedtogenerate 525 asetofinitialflowsforthetwomethodsasdiscussedpreviously. 526 It is worth using the FCPA in conjunction with both the GGA and RCTM for a once-off 527 191
ADE	AppendixB. SubmittedversionofPublication2: ABenchmarkingStudyofWater DistributionSystemSolutionMethods simulation given that FCPA decreases the L per-iteration time without increasing the number of 528 3 iterations per module. Interestingly, a smaller per-iteration time is required by the L modules of 529 3 the RCTM except for network N . However, RCTM requires a greater number of iterations for all 530 8 thecasestudynetworks. ThissometimescausesagreatertimefortheRCTMtosatisfythestopping 531 test. 532 MultipleSimulationSetting 533 The performance of the four solution methods under the multiple simulation setting are com- 534 pared. Pipediametersfortheeightcasestudynetworkswererandomlygeneratedateachevaluation 535 tosimulateanevolutionaryalgorithmrun. Itisimportanttonotethattheuseofrandomlygenerated 536 pipe diameters gives an overestimate of the total runtime. This is because, as EA’s progress, the 537 pipe diameters in its population become increasingly realistic, which, on average, should reduce 538 thenumberofiterationsattheL level. 539 3 Table 8 and Table 9 show the detailed timing results of multiple simulations with number of 540 evaluations N = 100,000 for each of the four solution methods applied to the networks N and 541 E 1 N . Table 8 shows that exploiting the treed nature of network N gives the FCPA a 29% time 542 8 1 savingovertheGGAand15%timesavingovertheRCTM.Asmallersavingisachievedbytheuse 543 of the FCPA for network N : 14% for the GGA and 9% for the RCTM. In a multiple simulation 544 8 setting,theRCTMismoretiming-consumingthantheGGAwhenappliedtonetworkN because 545 8 ofthegreaternumberofnonzeroelementsinitskeymatrix(Table5). 546 Table 10 shows the actual multiple simulation runtime with 100,000 evaluations for each of 547 the four solution methods applied to the eight case study networks. Under a multi-run simulation 548 setting, the GGA implementation in this paper has been able to achieve between a 35% and 81% 549 speedupwhencomparedwiththeGGAimplementationinEPANETbyimplementingtheproposed 550 module categorization. Note that both the upper and lower range values of the speed-up achieved 551 by implementing the proposed module categorization in a multi-run simulation are higher than 552 thoseinaonce-offsimulation. Thisisbecausetheeffectivenessofproposedmodulecategorization 553 andthenumberofevaluationaredirectlyproportional. Thefastestsolutionmethodsforeachofthe 554 192
ADE	AppendixC. SubmittedversionofPublication3: ABridge-BlockPartitioningAlgorithm forSpeedingupAnalysisofWaterDistributionSystems operation,andmanagementofWDSsinresearchandindustry. Thesemodelshavebeenusedfor(1) 24 optimizing WDS network design parameters (such as pipe diameters), (2) for calibrating network 25 parameters (such as demand patterns), (3) conducting real-time monitoring and calibration of the 26 network elements in a supervisory control and data acquisition (SCADA) operational setting, and 27 (4)adjustingcontroldevices(suchasvalves). Inhydraulicsimulation,thesystemofequationscan 28 beformulatedasalargeandsparsenon-linearsaddle-pointproblem. Thereareseveralwell-known 29 iterative methods for solving the non-linear saddle-point problem. These include: range space 30 methods,nullspacemethods,andloop-basedmethods. 31 The most widely used WDS solution method is the Global Gradient Algorithm (Todini and 32 Pilati 1988). The GGA, a range space method, takes advantage of the block structure of the full 33 Jacobian matrix to achieve a smaller key matrix in the linearization of the Newton method. Since 34 thedevelopmentoftheGGA,numerousnewWDShydraulicsolutionmethodshavebeenproposed 35 andimprovementshavebeenmadetoexistingWDShydraulicsolutionmethods. Mostofthesenew 36 WDShydraulicsolutionmethodsemploygraphtheorytodecomposeorpartitiontheWDSnetwork 37 graphintosub-graphswhichresultsinasmallersystemofequations. Deuerlein(2008)introduced 38 a decomposition model for a WDS network graph, in which the one-connected components are 39 categorized as the forest component and the biconnected components are categorized as the core 40 component. After removing the forest component, the core component can be further partitioned 41 into blocks that are connected by bridge elements. After the partitioning processes, a loop flow 42 corrections method is then used. Simpson et al. (2012) proposed a matrix based identification 43 methodfortheforestcomponentandthecorecomponentandintroducedtheforest-corepartitioning 44 algorithm (FCPA). In the FCPA, flows and heads in the forest component can be solved for just 45 once. Theremainingsystemofequations,representingthecore–whichhasasmallerdimensionif 46 the network has a significant forest component – is then solved iteratively by the Newton method. 47 Deuerleinetal.(2015)proposedanothergraphpartitioningalgorithmwhichexploitstheproperties 48 ofnetworkcomponentsinserieswithinthecorecomponentofthenetwork. Thisalgorithmexploits 49 the fact that flows in the internal tree pipes are linearly dependent on the topological minor. This 50 209
ADE	AppendixC. SubmittedversionofPublication3: ABridge-BlockPartitioningAlgorithm forSpeedingupAnalysisofWaterDistributionSystems theseoftheotherblockcomponentsandthesolutionoftheflowsandheadsinabridgecomponent 96 isalinearprocess. TheconvergencerateforthesolutionofthecorecomponentofaWDS,without 97 the BBPA, is restricted to that of the worst block of the network. Solving each block separately 98 reducesthenumberofiterationsexecutedtothenumberofiterationsrequiredbythatblock. 99 There is a number of advantages to using the BBPA to identify the linear bridge components 100 andtheblockcomponentsofaWDSnetwork: 101 1. The number of iterations required by each block is bounded by that required by the unpar- 102 titioned system – solving the flows and heads in each block separately significantly reduces 103 theoverallcomputationaltimeforthenon-linearsolverinalmostallcases. 104 2. Itimprovesthenumericalreliabilityofthesolution. Thenumericalreliabilityofthesolution 105 canbedeterminedbytheconditionnumberoftheSchurcomplement. Theconditionnumber 106 of a matrix is the ratio of the largest to the smallest singular value of any square matrix. 107 A rough rule of thumb is: one digit of reliability in the solution is lost for every power of 108 ten in the condition number. If a square matrix is partitioned into block diagonal form by 109 orthogonalpermutations,theconditionnumbersofblockscanbenogreaterthanthatofthe 110 fullmatrix. Inmostcases,theconditionnumbersforalltheindividualblockswillbesmaller 111 than the condition number of the full matrix. This phenomenon is illustrated later in this 112 paper. 113 3. Itreducestheneedtoregularizeforthepresenceofzeroflows(ElhayandSimpson2011). It 114 hasbeenpointedoutbySimpsonetal.(2012)thatsolvingfortheflowsandheadsseparately 115 can avoid the numerical failure that occurs when there are nodes with zero demand present 116 in the forest. It is shown in this paper that there are blocks, in some networks, that have 117 zero accumulative demands. The solutions of these networks need a regularization method 118 to deal with the presence of the zero flows to avoid catastrophic numerical failure when the 119 Hazen-William head loss model is used. Using the BBPA avoids this failure which reduces 120 theneedforregularization. 121 212
ADE	AppendixC. SubmittedversionofPublication3: ABridge-BlockPartitioningAlgorithm forSpeedingupAnalysisofWaterDistributionSystems 4. Itreducesthecomputationaltimeinamanagementsettingbecausetheflowsintheblockswith 122 unchangednodaldemandsdonotneedtobesolvedagainandtheheadsinthecorresponding 123 blockonlyneedtobeadjustedaposteriori. 124 5. The solution of each block can be found in parallel in a demand-driven model because the 125 flows and heads in one block component can be found separately from those of the other 126 blockcomponents. 127 The main contributions of this paper are: (1) to extend the concept of using bridge and block 128 components in the loop flow correction method, proposed in Deuerlein (2008), to a generalized 129 graph partitioning algorithm that can be used with any demand-driven WDS solution method, (2) 130 toestablishthetheoreticaladvantagesofusingtheBBPAintermsofreducingcomputationalload 131 and improving numerical reliability, (3) to provide a detailed case study to demonstrate BBPA’s 132 usefulnessintermsofperformanceandaccuracy. 133 Thispaperisorganizedasfollows. Somedefinitionsandnotationsaregiveninthenextsection. 134 Thesectionfollowingprovidesthederivationofthemethodwithsomeexamples. Thealgorithmic 135 description of the BBPA is then given, followed by the a discussion of the relation of the BBPA 136 andothermethods. ThisisfollowedbyabenchmarkanalysisoftheBBPAappliedtotheeightcase 137 study networks that supports the claim about the advantages of using the BBPA. These results are 138 thendiscussedinthesectionthatfollows. Finally,thelastsectionsummarizestheoverallfindings. 139 GENERAL WDS DEMAND-DRIVEN STEADY-STATE PROBLEM 140 This section describes the general WDS demand-driven steady-state problem. The following 141 startswiththebasicdefinitionandnotations,followedbythesystemequations. Finally,theGlobal 142 GradientAlgorithm,whichisusedasthehydraulicsolvertoseparatelysolveeachblock,areshown. 143 DefinitionsandNotation 144 Consider a water distribution system that contains n pipes, n junctions, n fixed head nodes 145 p j r andn forestpipesandnodes. Thej thpipeofthenetworkcanbecharacterizedbyitsdiameter 146 f − 213
ADE	AppendixC. SubmittedversionofPublication3: ABridge-BlockPartitioningAlgorithm forSpeedingupAnalysisofWaterDistributionSystems D , length L , resistance factor r . The i th node of the network has two properties: its nodal 147 j j j − demandd anditselevationheadz . 148 i i T T Let q = q ,q ,....q denote the vector of unknown flows, h = h ,h ,....h denote 149 1 2 np 1 2 nj (cid:16) (cid:17) T (cid:16) (cid:17) the vector of unknown heads, r = r ,r ,....r denote the vector of resistance factors, d = 150 1 2 np T (cid:16) (cid:17) T d ,d ,.....d denote the vector of nodal demands, e = e ,e ....e denote the vector of 151 1 2 nj l l 1 l 2 lnr (cid:16) (cid:17) (cid:16) (cid:17) fixedheadelevations. 152 The head loss exponent n is assumed to be dependent only on the head loss model: n = 2 153 for the Darcy-Weisbach head loss model and n = 1.852 for Hazen-Williams head loss model. 154 The head loss within the pipe j, which connects the node i and the node k, is modelled by 155 h h = r q q n 1. Denote by G(q) Rnp np, a diagonal square matrix with elements 156 i k j j j − × − \| \| ∈ [G] = r q n 1 for j = 1,2,....n . Denote by F (q) Rnp np, a diagonal square matrix where 157 jj j \| j \| − p ∈ × the j-th element on its diagonal [F] = d [G] q . The matrix A is the full rank, unknown 158 jj dqj jj j 1 head,node-arcincidencematrix. ThematrixA isthefixed-headnode-arcincidencematrix. 159 2 SystemofEquations 160 The steady-state flows and heads in a WDS system are modeled by the demand-driven model 161 (DDM)continuityequations(1)andtheenergyconservationequations(2): 162 A Tq d = O (1) 163 1 − − G(q)q A h A e = O, (2) 164 1 2 l − − whichcanbeexpressedas 165 G(q) A q A e 1 2 l 166  −     = 0, (3) A T O h − d − 1               214
ADE	AppendixC. SubmittedversionofPublication3: ABridge-BlockPartitioningAlgorithm forSpeedingupAnalysisofWaterDistributionSystems as the generalized equations that can be applied when the head-loss is modeled by the Hazen- 184 William equation or the Darcy-Weisbach equation. The correct Jacobian matrix with the formula 185 forF,whenheadlossismodeledbyDarcy-Weisbachequation,canbefoundinSimpsonandElhay 186 (2010). They showed that the use of the correct Jacobian matrix restores the quadratic rate of 187 convergence. 188 DERIVATION OF THE BRIDGE-BLOCK PARTITIONING ALGORITHM 189 The following terminology will be used in this paper. Associated with a WDS is a graph 190 G=(V, E), where the elements of V are the nodes (vertices) of the graph G and elements of E are 191 the pipes (links or edges) of the graph G. Every WDS can be divided into two subgraphs: a treed 192 subgraph(forest)G = V ,E andaloopedsubgraph(core)G = (V ,E ),sothatE E = E, 193 f f f c c c f c ∪ (cid:16) (cid:17) E E = , V V = V. A cut-vertex is a node in a WDS graph, the removal of which will 194 f c f c ∩ ∅ ∪ increasethenumberofconnectedcomponents,andabridgeisapipeinaWDSgraph,theremoval 195 of which will separate its two end nodes. A block is a maximal connected subgraph without a 196 cut-vertex. AWDSgraphcanbedecomposedintoatreeofblocks,cut-vertices,andbridgescalled 197 ablock-cuttree(Diestel2005). Arootblockisablockwhichincludesoneormorewatersources. 198 Notethateverywatersourceisdefinedtobewithintherootblockofitsnetworkcomponent. That 199 is, all water sources are in the root block of their connected component of the network. The level 200 ofblockiinarootedblock-cuttreeisthelengthoftheuniquepath,composedofblocks,fromthe 201 rootblocktoblocki. Theparentofblockiistheblockconnectedtoblockionthepathtotheroot 202 block. If block i is the parent of block j, then block j is the child of block i. A block of a graph 203 G containing only one cut-vertex is called an end block of G. Note that any block except for the 204 rootblockhasauniqueparentblock,andanyblockexceptforanendblockcanhavemultiplechild 205 blocks. 206 A WDS graph can be divided into n subgraphs, G =(V ,E ), G =(V ,E ), ..., G = 207 b b 1 b 1 b 1 b 2 b 2 b 2 bnb (V ,E ). If two blocks, G =(E ,V ) and G =(E ,V ), are adjacent, then E E = and 208 bnb bnb bi bi bi bj bj bj bi ∩ bj ∅ V V = c where c is the cut-vertex that connects the parent block i and child block j. The 209 bi ∩ bj ij ij cut-vertex, c , in the parent block, b , is a cluster of the demands of this cut-vertex and all its 210 ij i 216
ADE	AppendixC. SubmittedversionofPublication3: ABridge-BlockPartitioningAlgorithm forSpeedingupAnalysisofWaterDistributionSystems descendant blocks. A block except for the end block can have multiple cut-vertices behaving as 211 clusters of demands because a parent block can have multiple child blocks. The cut-vertex, c , in 212 ij the child block, b , is considered as a pseudo-source. The head of the cut-vertex, c , that is found 213 j ij in the parent block, b , is used as the elevation head of the pseudo-source for the corresponding 214 i child block. With the exception of the root block, every block has a single cut-vertex that behaves 215 as a pseudo-source. The ancestors of a block are the blocks in the path from the root block to this 216 block, excluding the block itself and including the root block. The descendants of block i are all 217 theblocksthathaveblockiasanancestor. 218 The BBPA is now derived by generating two orthogonal permutation matrices and using them 219 tomanipulatethematrixA tofindn unknown-headnode-arcincidencematricesforeachblock, 220 1 b B , B , .....B , and n 1 fixed head node-arc incidence matrices, C , C , ..., C . 221 11 22 n bn b b − 1 2 n b−1 Notethatinthefollowing,B ,theblockinthei-thblockrowandthej-thblockcolumn,isusedto 222 ij denotethefixedheadnode-arcincidencematrices,wherethesubscriptsiandj areusedtoindicate 223 thelocationoftheblock,rowj andcolumni,andalsotoindicateadirectconnectionbetweenthe 224 blockiandblockj. 225 Recall that all blocks except for the root block have exactly one cut-vertex that behaves as a 226 pseudo-source. The terms involving these pseudo-sources are moved to the right-hand-side of the 227 system leaving the remaining node-arc incidence matrix full rank. This is because each of the 228 diagonalblockmatricesofA ,afullrankmatrix,isalsofullrank. Thepermutationmatrixthatis 229 1 usedtopermutethesystemequation,Eq.(3),is 230 n n p j n P O p 231 P 1 =  , (10) n O R j       T 232 where P = (cid:18)P e b1 P e b2 ...P enb(cid:19) ∈ Znp ×np is the square orthogonal permutation matrix 233 for the pipes in each block, in which P e bi ∈ Znp ×npbi, for i = 1,2,....n b, is the permutation 217
ADE	AppendixC. SubmittedversionofPublication3: ABridge-BlockPartitioningAlgorithm forSpeedingupAnalysisofWaterDistributionSystems matrix that identifies the pipes in the block i as distinct from the pipes in other blocks and 234 T 235 R = (cid:18)R v b1 R v b2 ...R vnb(cid:19) ∈ Znj ×nj is the square orthogonal permutation matrix for the 236 nodes in each block, in which R v bi ∈ Rnj ×nvbi, for i = 1,2,....n b, is the permutation matrix that identifiesthenodesintheblockiasdistinctfromthenodesinotherblocks. 237 ThepermutedsystemoftheBBPAequationsis: 238 G A q a 239 P 1 − 1 P 1TP 1  P 1  = O (11) A T O h − d − 1                  wherea = A e . Withthispermutation,Eq.(3)becomes: 240 2 l PGPT PA RT Pq Pa 1 241  −     = O (12) RA TPT O Rh − Rd − 1               where 242 B O ... O 11   B B ... O 243 PA 1RT =     . . .21 . . .22 ... . . .    ,       B B ... B   n b1 n b2 n bn b     in which all the block entries above the diagonal blocks become zero matrices because there is no 244 pipeinaparentblockthatconnectstoanynodeinanyofitschildblocks. Theblockentriesbelow 245 the diagonal blocks, B represent the connection between the nodes in the parent block, block j, 246 ij and the pipes in the child block, block i, which are O when block j and block i are not adjacent 247 blocks. It has been pointed out above that any block, except for the end block, can have multiple 248 childblocks. Furthermore,anyblock,exceptfortherootblock,canhaveonlyoneparentblock. As 249 aresult,eachblockcolumncanhavemorethantwonon-zeroblockentries(includingthediagonal 250 block in that block column) and each block row, except for the root block row, has exactly two 251 218
ADE	AppendixC. SubmittedversionofPublication3: ABridge-BlockPartitioningAlgorithm forSpeedingupAnalysisofWaterDistributionSystems blockisfound. Asstatedpreviously,eachblockrowofthematrixA hasonlyonenon-zeroblock 282 C entry below its block diagonal. The matrix B only has one column entry that is non-zero. This 283 ij column entry is the A matrix for that block, which is the node-arc incidence matrix representing 284 2 theconnectionbetweenthepseudo-sourceandthepipesinthechildblock. 285 Lemma 1. Suppose v Rnj 1 is a column vector of all ones A Rnp nj, is an unknown-head 286 × 1 × ∈ ∈ node-arc incidence matrix and A Rnp 1 is a fixed-head node-arc incidence matrix for one of 287 2 × ∈ theWDS’sblocksthatisnottherootblock. Then 288 A v = A (18) 289 1 2 − 290 Proof. Denotebyp ,asetofindicesforthepipesthatarenotconnectedtoawatersource;byp ,aset 291 1 2 T 292 ofindicesforthepipesthatareconnectedtoawatersource. LetA 1 = a 1T a 2T ... a npT . (cid:18) (cid:19) The i-th row of the matrix A has two non-zero entries, 1 and -1, and the i-th row of the matrix 293 1 A is zero if i p . It is evident that the inner product of a and v becomes 0. The j-th row of 294 2 1 i ∈ the matrix A has only one entry, -1, and the j-th row of the matrix A has only one entry, 1, if 295 1 2 j p . Itisevidentthattheinnerproductofa andvT is-1. EndofLEMMA1. (cid:3) 296 2 j ∈ The relationship shown in Eq. (18) can be used to calculate term A h in Eq. (16). The 297 c B relationship between the unknown head node-arc incidence matrix, B , and the fixed head node- 298 ii arcincidencematrix,B ,is 299 ij B = B v, (19) 300 ij ii − thetransposeofwhichisB T = vTB T andmultiplyingbothsidesbyq ,theflowsinblock 301 ij − ii B j j ,wegetB Tq = vTB Tq . Therefore, 302 ij B j − ii B j B Tq = vTd , (20) 303 ij B j − B j which is in fact the sum of the demands in the child block to the cut-vertex in the parent block. 304 221
ADE	AppendixC. SubmittedversionofPublication3: ABridge-BlockPartitioningAlgorithm forSpeedingupAnalysisofWaterDistributionSystems Eq. (20) is used repeatedly from the end block to the root block until the A Tq in Eq. (16) has 305 c B beenreplaced. Thisprocessisperformedonlyoncebeforetheiterativephase. 306 MultiplyingbothsidesoftheEq.(19)bytheunknownheadatcut-vertexc ,h ,weget 307 j cj B h = B vh , (21) 308 ij cj − ii cj which is used to move A h from the left-hand-side of the equation to the right-hand-side of the 309 c b equation so that each block can be solved in parallel. The heads need only be computed just once 310 aftertheiterationsforallblockshavebeencompleted. 311 Thepropertiesofthesystemofequationsafterbridge-blockpartitioning 312 IntheBBPA,afullWDSnetworkispartitionedinton smallerindependentnon-linearsystems 313 b by permuting the original full system of equations using two orthogonal permutations P and R. 314 One of the main contributions of this paper is to show that the use of the BBPA can significantly 315 reducethecomputationalloadsandimprovethenumericalreliabilityoftheresults. 316 TheBBPAcanbeusedtoimprovethereliabilityofsolutionoftheloopedcomponentinthefinal 317 WDS solution. This is because the condition number, the ratio between the largest to the smallest 318 singular value of a matrix, can be used to estimate the loss of reliable digits in solving a linear 319 systemwiththatmatrix. TheorthogonalpermutationsoftheBBPAshufflethen singularvaluesof 320 j the Schur Complement into their corresponding blocks. This is because pre-and-post-multiplying 321 a matrix by orthogonal matrices preserves the singular values. The upper bound of the largest 322 singular value of all blocks is the largest singular value of the full system and the lower bound for 323 thesmallestsingularvalueofallblocksisthesmallestsingularvalueforthefullsystem. Therefore, 324 the condition number of each block at the solution is bounded above by the condition number of 325 the full system of equations but in most cases will be smaller. Moreover, the only occasions when 326 one of the blocks has the same condition number as the full system is where both the highest and 327 lowest singular values are present in the same block. Even in this particular case the other blocks 328 inthesystemwillhavelowerconditionnumbersthanthefullsystem. 329 222
ADE	AppendixC. SubmittedversionofPublication3: ABridge-BlockPartitioningAlgorithm forSpeedingupAnalysisofWaterDistributionSystems Furthermore, the use of the BBPA can minimize the need to use regularization methods for 330 handling zero-flows. In the FCPA paper (Simpson et al. 2012), the authors pointed out that it is 331 commonforzeroflowstooccurattheendsoftreeswithzerodemands. Similarly,itisalsopossible 332 for all nodes in the end blocks to have zero demands. The GGA fails catastrophically at these 333 blocks when the head loss is modelled by the Hazen-William head loss model. One side-effect of 334 identifying these end blocks with zero nodal demands is zero flows can be assigned to all pipes in 335 these blocks and the head of pseudo-source can be assigned to all nodes in these blocks. When 336 zero flows occur in other blocks, regularization is needed only for the blocks with the presence of 337 zeroflowsinsteadofthefullsystem. 338 Inadditiontotheimprovementofthenumericalreliabilityofthefinalresult,theuseoftheBBPA 339 can significantly reduce computational loads. This reduction in computational loads is achieved 340 through: (1)thebridgecomponentbeingsolvedbyalinearprocess,theremovalofwhichreduces 341 thenumberofnon-zeroesinSchurcomponent,(2)theprobablereductioninthenumberiterations 342 required by each block as shown in the Appendix, and (3) the non-linear system of equations for 343 eachblockisindependentofotherblockswhichallowseachblocktobesolvedinparallel. 344 BRIDGE-BLOCK PARTITIONING ALGORITHM 345 The steps of the BBPA are now described. The BBPA starts with a forest search algorithm 346 to identify the forest component as distinct from the core. This is followed by identifying all the 347 blocks and bridges in the core, and updating the demands for the cut-vertices by using Stage 1 as 348 given below, a variation of the algorithm detailed by Hopcroft and Tarjan (1973). Note that this 349 algorithmisbasedonthedepth-firstsearchandrunsinlineartime. Therearetwowaystosolvethe 350 coreofthenetwork: inparallelorserially. 351 Parallel: Itcanbemoreefficienttosolvealltheblocksinparallelwhenthesolutionoftheentire 352 system is needed, such as in a design setting. After the network has been permuted, each block is 353 then individually solved by using Stage 2 in no particular order. Once the solutions for all blocks 354 are found, the heads for the core nodes are recovered by using Stage 3 from the root block to the 355 223
ADE	AppendixC. SubmittedversionofPublication3: ABridge-BlockPartitioningAlgorithm forSpeedingupAnalysisofWaterDistributionSystems 1 0 0 0 0 0 1 −     1 0 0 0 0 1 0  −             1 1 0 0 0 0  0  −             0 1 1 0 0 0  0 A 1 =    −   ,A 2 =      .  0 0 1 1 0 0  0  −             0 1 0 1 0 0  0  −             0 1 0 0 1 0  0  −             0 0 0 0 1 1 0  −            Thesystemofpipeheadlossandnodalcontinuityequationsfortheexamplenetworkis 374 G 1 1 0 0 0 0 0 q 1 e l7        G 2 −1 0 0 0 0 1   q 2   0           G 3 −1 1 0 0 0 0    q 3     0           G 4 0 −1 1 0 0 0    q 4     0           G 5 0 0 1 −1 0 0    q 5     0           G 6 0 −1 0 1 0 0    q 6     0        375    G 7 0 −1 0 0 1 0    q 7  =  0  . (22)         G 8 0 0 0 0 −1 1    q 8     0          1 −1 −1 0 0 0 0 0     h 1     d 1          0 0 1 −1 0 −1 −1 0     h 2     d 2         0 0 0 1 1 0 0 0   h 3   d 3              0 0 0 0 −1 1 0 0     h 4     d 4          0 0 0 0 0 0 1 −1     h 5     d 5         0 1 0 0 0 0 0 1   h 6   d 6            Bypermutingtherows(pipes)intheorderinggivenbyp = 1;2;3;7;8;4;5;6 andthecolumns 376 { } (nodes) in the ordering given by v = 1;6;2;5;3;4 , the system of equations in Eq. (22) can be 377 { } rearrangedintothefollowingblockstructure: 378 227
ADE	AppendixC. SubmittedversionofPublication3: ABridge-BlockPartitioningAlgorithm forSpeedingupAnalysisofWaterDistributionSystems Pipes Nodes Block B 1 B 2 B 3 B 1 B 2 B 3 B 1 G 1 1 0 0 0 0 0 q 1 el7       379                           B B2 3                            G 2 G 3 G 7 G 8 G 4 G 5 G 6 − − 0 0 0 0 01 1 1 0 0 1 0 0 0 − − −1 1 10 1 0 0 −0 0 1 0 0 01 0 0 0 0 1 1 0 −10 0 0 0 0 1                                                    q q q q q q q2 3 7 8 4 5 6                          =                          0 0 0 0 0 0 0                           (23)  B 1    1 −1 −1 0 0 0 0 0       h 1       d 1                      B B2 3                    00 0 0 0 01 0 0 0 10 0 0 0 −0 1 0 01 −01 0 01 −0 0 1 01 −00 0 1 1 −0 0 0 11                                       h h h h h6 2 5 3 4                                       d d d d d6 2 5 3 4                    380 *theboldnumbersinthematrixrepresentthecut-vertices Eq. (23) has three graph blocks as shown in Fig. 2 include Block 1 (a bridge), Block 2, and 381 Block 3. Note that, for cross-referencing purposes, this equation has been labeled with the block 382 numbers(affiliatedwithpipesandnodes)correspondingtoeachentityintheexamplenetwork. The 383 cut-vertices (cv and cv in Fig. 2) are highlighted in bold in their corresponding matrix blocks. 384 1 2 In the equation, it is evident that the permuted A matrix is a block three by three, lower block 385 1 triangularmatrixwhichrepresentsaWDSwiththethreegraphblocks(B ,B ,andB ). 386 1 2 3 Theendblock(B inFig.2)isasub-networkconsistingofthreepipes{4;5;6},twonodes{3; 387 3 4}, and a pseudo-source at node {2}. The nodal demands of this block do not need to be updated 388 because this is the end block. The head of the node 2 (cv ) , which is the cut-vertex behaving as 389 2 thepseudo-sourceforthisblock,canbemovedtotheright-hand-sideofsystemofequationsusing 390 Eq. (16). The solution of block B can be found separately after the head of the pseudo-source at 391 3 node{2}isfound. 392 The second block diagonal row (B in Fig. 2) is a sub-network consisting of four pipes {2; 393 2 3; 7; 8}, three nodes {2; 5; 6}, and one pseudo-source at node {1}. This is an intermediate 394 228 sepiP sedoN
ADE	AppendixC. SubmittedversionofPublication3: ABridge-BlockPartitioningAlgorithm forSpeedingupAnalysisofWaterDistributionSystems block so that the demand at the node 2 (cv ), a cut-vertex that is not a pseudo-source, needs to be 395 2 updated by increasing its demand by the sum of demands at all nodes of its child block (B ) as 396 3 follows: d = d +d +d using Eq. (20). Node 1 (cv ), which is the cut-vertex behaving as the 397 2 2 3 4 1 pseudo-soburceforthisblock,B ,canbemovedtotheright-hand-sideofsystemofequationsusing 398 2 Eq. (16). The solution of block B can be found separately after the head of the pseudo-source at 399 2 node{1}isfound. 400 Finally, the root block (B in Fig. 2) is a sub-network consisting of pipe {1}, node {1}, and 401 1 source {7}. Block B is a bridge component. The bridge component can be solved by using a 402 1 linearprocess. Thedemandforthenode1inFig.2(cv ),acutvertexintherootblock,isupdated 403 1 by increasing its demand by the sum of demands at all nodes of its child block (B ) as follows: 404 2 d = d +d +d +d +d +d andtheelevationheadforthesourcestaysthesame. Afterupdating 405 1 1 2 3 4 5 6 tbhedemandsandheads,thesystemofequationsinEq.(23)becomes: 406 Pipes Nodes Block B1 B2 B3 B1 B2 B3 B1 G1 1 0 0 0 0 0 q1 el7 407                                               B B B B B2 3 1 2 3                                                    1 00 0 0 0 −G 01 0 0 02 1 −G 10 0 0 03 1 −G 0 0 1 0 07 1 −G 00 1 0 018 −G 0 0 0 1 04 1 −G 0 00 0 1 15 −G 0 0 0 0 16 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 −0 1 0 0 0 01 −10 0 1 0 0 0 0 0 0 0 1 1 0 −0 0 0 0 0 11                                                                                                      h h h h h hq q q q q q q2 3 7 8 4 5 6 1 6 2 5 3 4                                                   =                                                   d1+d2 d+ 2d +3h h h h d d d d d+0 0 0 6 3 5 3 41 1 2 2 d +4 d+ 4d5+d6                                                    (24) NotethatthesystemofequationsobtainedinEq.(24)isequivalenttoperformingblockGauss- 408 Jordan elimination on Eq. (23). Solving the system of equations in this way requires solving each 409 blockinaparticularsequence,fromtherootblock(B )totheendblock(B ). Thesequencethat 410 1 3 229 sepiP sedoN
ADE	AppendixC. SubmittedversionofPublication3: ABridge-BlockPartitioningAlgorithm forSpeedingupAnalysisofWaterDistributionSystems isrequiredintheexamplenetworkinFig.2is: (1)tofindthesolutionofblockB ,therootblock; 411 1 (2) to find the solution of block B using the head of the node one, cv , in block B ; and (3) to 412 2 1 1 findthesolutionofblockB ,theendblock,usingtheheadofthenodetwo,cv ,inblockB . 413 3 2 2 Furthermore, the second pipe head-loss block equation or the second block equation (B ) in 414 2 Eq.(24)is: 415 G q B h = B h , 416 b 2 b 2 − 22 b 2 21 b 1 whichexpandsto: 417 G q 1 0 0 h 2 2 1      h   6 G q 0 1 0   h  3  3    1 418       +   h 2 =    , (25)  G q  0 1 1   0  7  7  −         h         5    G q  1 0 1  0  8 8  −                   theright-hand-sideofwhichcanberewrittenas: 419 B h = B [v h ], (26) 420 21 b 1 − 22 3 1 whichexpandsto: 421 h 1 0 0 1     h 1 h 0 1 0    1   422     =    h 1 0 0 1 1      −      h      1 0 1 0 1     −           usingEq.(21). SubstitutingitbackintoEq.(25),weget: 423 G q B h = B [v h ], 424 b 2 b 2 − 22 b 2 − 22 3 1 230
ADE	AppendixC. SubmittedversionofPublication3: ABridge-BlockPartitioningAlgorithm forSpeedingupAnalysisofWaterDistributionSystems whichexpandsto: 425 G q 1 0 0 1 0 0 2 2      h   h 6 1 G q 0 1 0   0 1 0    3  3     426       +   h 2 =    h 1,  G q  0 1 1   0 1 1    7  7  −    −       h   h       5   1  G q  1 0 1  1 0 1   8 8  −    −                 whichcanfurthersimplifiedinto: 427 G q B [h +v h ] = O, 428 b 2 b 2 − 22 b 2 3 1 whichexpandsto: 429 G q 1 0 0 2 2      h h 6 1 − G q 0 1 0    3  3   430       +   h 2 −h 1 = O.  G q  0 1 1    7  7  −       h h       5 − 1  G q  1 0 1   8 8  −             Thethirdpipehead-lossblockequationorthethirdblockequation(B )inEq.(24)is: 431 3 G q B h = B h , 432 b 3 b 3 − 33 b 3 32 b 2 whichexpandsto: 433 G q 1 0 h 4 4 2      h   3 434  G 5 q 5+ 1 −1   = 0 . (27)      h        4    G q  0 1   h   6 6     2               231
ADE	AppendixC. SubmittedversionofPublication3: ABridge-BlockPartitioningAlgorithm forSpeedingupAnalysisofWaterDistributionSystems Eq.(27)canbefurthersimplifiedto 435 G q 1 0 4 4      h h 3 2 436  G 5 q 5+ 1 −1  −  = O      h h      4 − 2  G q  0 1    6 6              usingasimilarmanipulationasforBlock2above. Finally,thesystemofequationsinEq.(24)may 437 berewrittenas: 438 Pipes Nodes Block B1 B2 B3 B1 B2 B3 B1 G1 1 0 0 0 0 0 q1 el7 439                            B B2 3                             G2 G3 G7 G8 G4 G5 G6 0 0 0 0 0 0 0 1 0 0 1 0 0 0 −10 1 0 0 0 0 −0 0 1 0 0 01 0 0 0 0 1 1 0 −0 0 0 0 0 11                                                       q q q q q q q2 3 7 8 4 5 6                            =                            0 0 0 0 0 0 0                             (28)  B1    1 −1 −1 0 0 0 0 0        h1        d1+d2+d3+d4+d5+d6                      B B2 3                    00 0 0 0 01 0 0 0 10 0 0 0 −0 1 0 01 −01 0 01 −0 0 1 01 −00 0 1 1 −0 0 0 11                                       h h h h h6 2 5 3 4− − − − −h h h h h1 1 1 2 2                                        d2+ ddd d d536 3 4+d4                     Solvingtheexamplenetwork 440 Consider the network shown in Fig. 2 and its permuted system of equations, Eq. (28). Each 441 blockbecomesanindependentsystemandcanbesolvedsequentiallyfromtherootblocktotheend 442 block. The system of equations for the root block, B (Block 1 in Fig. 2), which also represents a 443 1 bridge,is: 444 G 1 q e 445  1  1  =  l 7 , (29) 1 0 h d +d +d +d +d +d   1  1 2 3 4 5 6           232 sepiP sedoN
ADE	AppendixC. SubmittedversionofPublication3: ABridge-BlockPartitioningAlgorithm forSpeedingupAnalysisofWaterDistributionSystems thesolutionofwhichcanbeusedtosolveitschildblock,blockB (Block2inFig.2)byusing: 446 2 G 1 0 0 q 0 2 8      G 0 1 0 q 0  3  7               G 0 1 1  q   0   7 −  3              447   G 8 1 0 −1   q 2   =   0  , (30)            1 0 0 1 h h   d    6 − 1  6             0 1 1 0 h h  d +d +d   −  2 − 1  2 3 4            0 0 1 1 h h   d   −  5 − 1  5            andfinally,theendblock,blockB (Block3inFig.2)canbesolvedbyusing: 448 3 G 1 0 q 0 4 6      G 1 1 q 0  5 −  5              449   G 6 0 1    q 3   =  0 . (31)            1 1 0 h h  d    3 − 2  3            0 1 1 h h  d   −  4 − 2  4           Thesystemsofequationsforeachofthethreeblockscanalsobesolvedinparallel. 450 Notethat,whenusingBBPA,iftheheadlossoftheexamplenetworkshowninFig.2ismodeled 451 by the Hazen-William formula and the nodal demands at nodes three and four are zero, this does 452 not cause a failure of the method due to singularity of the Schur complement, unlike the GGA 453 andRCTMonthesamenetwork(ElhayandSimpson2011). Inaddition,theblockwithzerototal 454 demandcanbesolved(1)priortotheiterativephasebyassigningzeroflowstoallapplicablepipes 455 and(2)byassigningtheheadsofthesourcetoallnodesinthisblockaftertheiterativephase. 456 RELATION OF BBPA TO OTHER SOLUTION METHODS 457 TheBBPAcanbedescribedasapre-and-post-processingmethodforthefollowingreasons: (1) 458 it finds the blocks and bridges of a WDS, (2) the bridges can be solved by using a linear process 459 similar to the forest component, and (3) then uses any WDS solution method, for example GGA, 460 233
ADE	AppendixC. SubmittedversionofPublication3: ABridge-BlockPartitioningAlgorithm forSpeedingupAnalysisofWaterDistributionSystems RCTM,orGMPA,to,independently,solveeachblock. 461 TheBBPAcanalsobeusedtoidentifytheforestcomponentofthenetwork. However,theuse 462 oftheFCPArequireslessoverheadthantheBBPA. 463 The same topological properties exploited by FCPA and BBPA are partly responsible for 464 the savings achieved by partial-update (Abraham and Stoianov 2015). The forest and bridge 465 components - being linear - converge after just one iteration of application of a non linear solver. 466 Thepartialupdateschemeisabletoexploitthisbycheckingforconvergenceeveryiteration. Once 467 the convergence test for a pipe has been met, the head-loss of the converged component does 468 not need to be re-computed, whereas the linear solver for the full system is required until the 469 convergence tests for all pipes have been met. In contrast, FCPA and BBPA have the advantage 470 of identifying these components in advance and removing them from non-linear solution process. 471 BBPA also has the additional advantage of being able to exploit earlier convergence of different 472 blocks in the core network and removing them from the problem once they have converged. As a 473 result, the authors recommend that it is inefficient to implement the partial update for a full WDS 474 system before applying the FCPA and the BBPA. The usefulness of applying the partial update to 475 eachblockrequiresfurtherinvestigation. 476 CASE STUDIES 477 AcomparisonoftheGGAwithorwithoutBBPAoneightcasestudynetworkshasbeencarried 478 outinordertosupporttheabovediscussion. NotethatthefirststepeachmethodistouseFCPAto 479 removetheforestcomponentfromthecasestudynetworks,toensureafaircomparison. 480 TheefficiencyandreliabilityoftheBBPAinaonce-offsimulationsetting,inwhichthesteady- 481 state heads and flows are computed just once with the given WDS parameters, was benchmarked 482 against an efficient GGA implementation. As a baseline, timings of the solution process for the 483 benchmarknetworksusingEPANET2werealsorecorded. Thebenchmarktestswereperformedon 484 aIntel(R)Core(TM)CPUi5-4590runningat3.30GHzwith4coresinC++underIEEE-standard 485 doubleprecisionfloatingpointarithmeticwithmachineepsilon(cid:15) = 2.22 10 16. Thenumber 486 mach − × ofcoresallocatedtoeachtestwaslimitedtoone. Eachtimingtest,measuringwall-clocktime,was 487 234
ADE	AppendixC. SubmittedversionofPublication3: ABridge-BlockPartitioningAlgorithm forSpeedingupAnalysisofWaterDistributionSystems TABLE 1. Benchmark networks summary, their core network size, the number of blocks and the numberofbridges FullNetwork Corenetwork BBPA Network n n n n n The The p j s jc pc numberof numberof blocks bridges N 934 848 8 573 487 33(1) 118 1 ∗ N 1118 1039 2 797 718 10(2) 45 2 N 1975 1770 4 1152 947 7 6 3 N 2465 1890 3 2036 1461 47(3) 62 4 N 2509 2443 2 1087 1741 8(1) 45 5 N 8585 8392 2 6735 6542 7(2) 58 6 N 14830 12523 7 11898 9591 487(19) 895 7 N 19647 17971 15 15232 13557 17(2) 59 8 numbersinthebracketsreferstothenumberofblockswithnonodaldemands ∗ repeated15timesoneachbenchmarknetwork. 488 It is shown that the use of an efficiently implemented BBPA can provide a significant runtime 489 reduction and improvement in the reliability of the solution. The BBPA with the GGA and the 490 standalone GGA were each applied to eight case studies with between 932 and 19,651 pipes and 491 between848and17,977nodeswithnopumpsandnovalves. 492 RESULTS AND DISCUSSION 493 The basic details of the case study networks considered in this study are described in columns 494 2to4inTable1andmoreinformationcanbefoundinSimpsonetal.(2012). Thesizeofthecore 495 componentforeachoftheeightcasestudiesisshowninthecolumns5and6,thenumberofblocks 496 in column 7, with the number of blocks with no nodal demands in the brackets, and the number 497 of bridges in column 8. Table 2 shows the detailed profile of the size of each block in each of the 498 eightcasestudynetworks. Thesizeofthelargestblock,smallestblock,andmedianblockandthe 499 number in brackets is the percentage of the corresponding block size as a proportion of the core 500 componentofthenetwork 501 Table3showsthesummarystatisticsofthe15repetitionsofeachsolutionmethodappliedtothe 502 eightbenchmarknetworks. TheGGAbenefitsfromtheuseoftheBBPAbybetween33%and70%. 503 235
ADE	Abstract Abstract This PhD research has developed new measurement strategies and analysis techniques to enable hydraulic transient-based condition assessment of targeted pipe sections in complex pipe systems. The conventional practice of hydraulic transient-based pipeline condition assessment involves analysis of signals from a single pressure sensor located at each measurement site. Although multiple measurement sites can be used, they are typically far apart from each other since the access points (e.g. air valves or fire hydrants) are usually sparsely located. The pressure measurement obtained from a single sensor is a superposition of reflections coming from both upstream and downstream of the sensor. This superposition makes the measured wave reflections often too complex to analyse, especially in complex pipe systems where multiple features (e.g. deteriorated sections, branches and cross-connections, and other unknown features) often exist in the pipe section of interest. The research presented in this thesis has proposed a dual-sensor measurement strategy that uses two closely placed pressure sensors at a measurement site, and has developed a wave separation algorithm that enables the extraction of the two directional pressure waves travelling upstream and downstream. The wave separation can significantly simplify the signal to be analysed, and the unprecedented directional information enables advanced condition assessment techniques to be developed. Numerical and experimental verification has been conducted, with an application to pipe wall condition assessment. I
ADE	Abstract In the experimental verification, conventional flush-mounted pressure transducers have been used by connecting through closely located tapping points on the pipe wall. In addition, a customised in-pipe fibre optic pressure sensor array has been developed and tested in the laboratory, as a step towards real-world implementation. The sensor array cable can be inserted into a pipe through a single access point, avoiding the use of multiple tapping points. Complexities introduced by the in-pipe cable have been investigated, and accordingly, adjustments to the wave separation and wall condition assessment techniques have been made. The wave separation technique has been further developed by using a two- source-four-sensor transient testing configuration to enable the virtual isolation of a targeted pipe section in complex systems. Two dual-sensor units (i.e. two pairs of pressure sensors) are used to bracket the targeted pipe section, with the two sensors in each pair being located in close proximity. Two transient pressure wave generators are used, which bracket the four sensors and the “virtually” isolated pipe section. This measurement strategy enables the extraction of the transfer matrix of the “virtually” isolated pipe section, which is a full representation of the characteristics of this section independent from any complexities outside the section bounded by the sensors. A novel leak detection technique has been developed based on the analysis of the extracted transfer matrix, and has been validated by numerical simulation. The technique determines the leak location and impedance (related to the leak size), and it is applicable to the detection of multiple leaks. II
ADE	Statement of Originality Statement of Originality I certify that this work contains no material which has been accepted for the award of any other degree or diploma in my name, in any university or other tertiary institution and, to the best of my knowledge and belief, contains no material previously published or written by another person, except where due reference has been made in the text. In addition, I certify that no part of this work will, in the future, be used in a submission in my name, for any other degree or diploma in any university or other tertiary institution without the prior approval of the University of Adelaide and where applicable, any partner institution responsible for the joint-award of this degree. I acknowledge that copyright of published works contained within this thesis resides with the copyright holder(s) of those works. I also give permission for the digital version of my thesis to be made available on the web, via the University’s digital research repository, the Library Search and also through web search engines, unless permission has been granted by the University to restrict access for a period of time. I acknowledge the support I have received for my research through the provision of an Australian Government Research Training Program Scholarship. Signed: ………… …………………………Date: ………………….. III
ADE	Chapter 1 Chapter 1 Introduction 1.1 Structural deterioration of water supply systems and associated challenges Water distribution systems (WDS) are fundamental to modern civilisation; however, the sustainable management of large scale WDSs is a global challenge. In Australia, despite the fact that that water authorities spend about AU$4 billion in capital expenditure every year, an estimated 19,000 breaks in water mains occur annually, resulting in the loss of more than 265 GL of potable water (Bureau of Meteorology 2016). Almost all developed countries face the same problem due to the ageing of their water infrastructure. For example, in the US, it is estimated that more than US$1 trillion will be required between 2011 to 2035 to replace ageing water mains and address projected growth (American Water Works Association 2012). The majority of a water distribution system (WDS) infrastructure consists of pipelines that form complex networks. During construction, the structural integrity of pipeline systems can be compromised due to improper handling and poor workmanship (Gould et al. 2016). After commissioning, pipelines suffer from structural deterioration due to various sources such as traffic loading 1
ADE	Chapter 1 (Rakitin and Xu 2015), ground movement (Tucker 2010), corrosion (Świetlik et al. 2012), biological activity (Beech and Sunner 2004), and excessive hydraulic transient activity (Rezaei et al. 2015). Leakage in WDSs is a global issue, and the leakage rate ranges from about 10% in well-maintained WDSs (Beuken et al. 2006) to above 50% in poorly managed systems (Mutikanga et al. 2009). The annual potable water loss in Australia (265 GL) is equivalent to the annual consumption of 1.5 million homes and represents a value over $700 million. Leaking water pipes also impose risks to public health, since polluted water with harmful bacteria may enter into the system through the leak openings during low pressure events (Mora-Rodríguez et al. 2014). Structurally deteriorated pipe sections will also result in pipe bursts, which damage properties and interrupt traffic. The economic cost of pipe bursts is staggering. As shown by an investigation in the US, the average cost of a single large diameter (greater than 500 mm) water main failure is about US$1.7 million (Gaewski and Blaha 2007). The deterioration of water pipelines is not uniform, and faults are difficult to detect due to the sheer scale of the pipe network and the fact that most pipes are buried underground. Due to a lack of information on the actual condition of pipes, current asset management practice is often reactive and on the basis of standard economic life: typically remedial actions are taken only after pipe bursts or service interruptions have occurred; and pipeline replacement programs are often guided by indicative surrogate factors, such the age of the pipe and the number of historical pipe bursts. The current practice is not sustainable. For example, Water Corporation of Western Australia predicts that 2
ADE	Chapter 1 the potential cost for replacing water pipes in the metropolitan area of Perth alone will reach almost AU$1 billion in 2050-59, which is an increase of more than a factor of 12 compared to the cost of AU$76.5 million in 2010-19 (Water Corporation WA 2014). Better and more sustainable strategies for pipe asset management are urgently needed, which has to be guided by the actual pipe condition and the risk of pipe failure, such that high-risk pipes are replaced in time while the useful life of pipes in reasonable condition is extended. Cost-effective pipeline condition assessment is essential to obtain the critical information of pipe condition and failure risk. However, current technologies all have limitations, and more advanced pipeline condition assessment technologies need to be developed. 1.2 Conventional techniques for pipeline condition assessment There are several pipe leak detection and wall condition assessment techniques available in the market; however, none of them can achieve cost-effective condition assessment for long distance pipe systems or pipe networks. The conventional techniques can be categorised into three groups: acoustic and ultrasonic methods, electromagnetic methods and optical methods. 1.2.1 Acoustic and ultrasonic methods Acoustic methods are traditionally used for leak detection in pipelines. Two acoustic sensors (e.g. hydrophones or accelerometers) placed in different locations are used to measure leak-induced acoustic signals, then a software algorithm is used to calculate the cross-correlation function of the two leak 3
ADE	Chapter 1 signals to determine the location of the leak (Fuchs and Riehle 1991). Leak detection using acoustic correlators can only cover a limited distance in one test (typically less than 100 m), and is ineffective in plastic pipes, where acoustic signals attenuate much more quickly than in metallic pipes (Muggleton and Brennan 2004). Acoustic measurement and correlation analysis have also been used for pipeline condition assessment through wave speed analysis. The acoustic wave speed analysis uses a correlation method to calculate the average wave speed in a pipe section bounded by two acoustic sensors, from which the average pipe wall thickness is then calculated (Bracken et al. 2010). However, the average wave speed can be misleading if the section of pipe includes unregistered reaches with a much lower wave speed. For example, a polyvinyl chloride (PVC) pipe has a much lower wave speed than that of a metal or asbestos cement (AC) pipe, and the existence of an undocumented PVC replacement would result in a low average wave speed even though the metal or AC parts are in good condition. Ultrasonic-based pipeline condition assessment methods involve generating ultrasonic waves and measuring the wave reflections. For localised pipe wall thickness detection, an ultrasonic transmitter sends an ultrasonic ping and the signals reflected from the external and internal surfaces of the pipe wall are measured. The time between the two reflections is used to compute the thickness of the pipe wall (Liu and Kleiner 2012). For extended detection, guided wave ultrasound methods, in which the ultrasonic waves propagate along the axial direction of a pipe and the propagation and reflection are guided by the pipe wall, have been developed for detecting cracks along the pipe wall 4
ADE	Chapter 1 (Lowe et al. 1998; Demma et al. 2004). However, the range of detection is typically very limited due to fast signal dissipation and the complexities in the wave reflection, especially for pipes buried underground (Liu and Kleiner 2013). 1.2.2 Electromagnetic methods Electromagnetic pipeline condition assessment methods include techniques using magnetic flux leakage (MFL), remote field eddy current (RFEC), broadband electromagnetic (BEM) and ground penetrating radar (GPR) (Liu and Kleiner 2012). The MFL method uses strong magnets to induce a saturated magnetic field around a short section of ferrous pipe wall. If the pipe section contains damaged areas, a magnetic sensor detects the flux leakage from the air. The RFEC and BEM both use eddy current based techniques. A transmitter coil creates a current to the pipe surface, which generates a magnetic field. Flux lines from the magnetic field pass through the metallic pipe wall, and generate a voltage across it. The voltage produces eddy currents in the pipe wall, which induce a secondary magnetic field. Wall thickness is indirectly estimated by measuring signal attenuation and phase delay of the secondary magnetic field. RFEC methods use relatively low frequencies for testing, and the BEM techniques transmit a signal that covers a broad frequency spectrum. The MFL, RFEC and BEM can only be used for fault detection in ferrous pipes. They require excavation and pipe wall cleaning, and each test can only cover a few metres of pipe. For practical applications, the limited spatial extent of these methods means that only a few spatial points along a pipeline can be tested. As 5
ADE	Chapter 1 a result, highly approximate statistical inference methods are used to estimate the condition of a pipe based on these few points. GPR methods (Costello et al. 2007; Donazzolo and Yelf 2010) use electromagnetic wave pulses and their reflections to identify the interface between different material layers. The GRP technique can locate water pipe of all types of materials, but for buried pipes the resolution is not enough for pipe wall condition assessment. 1.2.3 Optical methods Closed-circuit television (CCTV) (Jo et al. 2010) inspection and the laser scanning (Duran et al. 2003) technique are two well-adopted optical methods for the inspection of a pipe’s inner surface. These methods introduce a carrier with the CCTV camera or laser sensors into the pipe via an access point. The moving velocity and sampling rate of the carrier determine the resolution and affect the accuracy of the scanning. The inspection is complicated by the roughness as well as the colour of the pipe surface. Currently, available optical inspection systems are intrusive, costly and only used in de-watered pipes (Tur and Garthwaite 2010). 1.3 Hydraulic transient-based techniques for pipeline condition assessment Research in the past three decades has demonstrated that controlled hydraulic transient pressure waves, also known as water hammer waves, can be used as a tool for pipeline condition assessment. This process is similar to the use of sonar waves to detect remote objects within marine environments. As a pressure wave propagates along a pressurised water pipeline at a high speed (typically 1000 to 6
ADE	Chapter 1 1200 m/s in water-filled metallic pipes, and 800 to 1000 m/s in asbestos cement pipes), part of the wave energy is reflected at pipe sections where the structural properties of the pipe cross-section changes (e.g. due to leaks, spalling of cement-mortar lining, or internal and/or external corrosion). This results in the creation of wave reflections that can be observed by appropriately placed pressure sensors, and then analysed by appropriate computer algorithms. Measurement and analysis of these reflections enables a diagnosis of the pipeline condition. A number of hydraulic transient based techniques have been developed and those for leak detection and pipe wall condition assessment are reviewed in the following sub-sections. 1.3.1 Transient-based techniques for pipe leak detection Transient based pipeline leak detection methods can be generally divided into three categories: time-domain-reflectometry (TDR) methods, inverse transient analysis (ITA) methods and frequency domain methods. Leak detection using time-domain-reflectometry TDR-based leak detection techniques analyse leak-induced wave reflections directly in their raw form, or analyse the transformed impulse response function (IRF). The direct wave reflection analysis uses the magnitude of a pressure wave reflection to determine the leak size and uses the arrival time to estimate its location (Brunone 1999; Lee et al. 2007a). However, the leak-induced reflections can be difficult to identify when the pressure response is complex due to background noise or the existence of multiple features. 7
ADE	Chapter 1 The use of IRFs is an improvement over the direct wave reflection analysis. The pipe system’s IRF is independent of the waveform of the input signal, and leak- induced reflections are represented by spikes in the IRF response (Liou 1998; Vítkovský et al. 2003; Kim 2005; Lee et al. 2007b). This help to enhance the accuracy in localisation. Recent work has shown that the use of pseudo random sequences and advanced signal processing techniques can enhance the robustness and accuracy of pipeline IRF extraction (Nguyen et al. 2018). However, the leak-induced spikes are typically rather small in the IRF, and the method may still encounter challenges when applied to real pipelines with multiple features and complex wave responses. Leak detection using inverse transient analysis Pudar and Liggett (1992) first proposed that leaks may be detected by solving an inverse problem to match the measurement of steady-state pressure and flow at multiple locations. Liggett and Chen (1994) extended the steady-state work to transient measurement and analysis, and the technique is known as the inverse transient analysis (ITA) method. ITA methods inversely calibrate a numerical pipeline model by minimising the calculated and measured transient pressure responses, and the pipe numerical model providing the best match is considered as the most likely representation of the real pipe system (Vítkovský et al. 2000; Kapelan et al. 2004; Jung and Karney 2008; Covas and Ramos 2010). ITA based pipeline leak detection has been extended to simple pipe networks (Shamloo and Haghighi 2010). The implementation of ITA can be very time-consuming, because it iteratively calibrates the pipeline parameters by comparing the measured transient pressure 8
ADE	Chapter 1 trace with the numerical results from the pipeline in the forward model. The forward modelling is typically conducted in the time domain and by using the Method of Characteristics (MOC) (Wylie and Streeter 1993; Chaudhry 2014). Recently development of frequency-domain inverse analysis has the potential to enhance the computational efficiency, where the forward modelling is conducted in the frequency domain using the impedance method (Kim 2014) or the admittance matrix method (Capponi et al. 2017). The successful calibration of a pipeline system relies on accurate forward simulation. However, varying boundary conditions, parameter uncertainties in real pipelines, and the difficulty in accurately simulating transient behaviour make errors in the forward modelling almost inevitable (Vítkovský et al. 2007). In addition, when the parameters to be calibrated are significant in number the results may be non- unique (Vítkovský et al. 2007; Zhang et al. 2018a). Leak detection using frequency-domain analysis Frequency domain leak detection techniques have been studied extensively (Colombo et al. 2009). One innovation in this area has been that steady oscillatory flow can be adopted to extract a system’s response to signals of different frequencies, which is known as the system’s frequency response function (FRF) or transfer function and can be used for leak detection. In this approach, both the head and flow are assumed to be composed of the steady- state average and oscillatory components. Impedance or transfer matrix methods (Wylie and Streeter 1993; Chaudhry 2014) are commonly used to solve the frequency response of a pipeline system. 9
ADE	Chapter 1 Jönsson and Larson (1992) first proposed that the spectral analysis of a measured pressure trace could be used for leak detection. (Mpesha et al. 2001) proposed that leaks would introduce extra resonant peaks in the frequency response diagram (FRD) of a system. However, research by Ferrante and Brunone (2003) demonstrated that extra peaks would not be observed unless leak size is larger than a critical value. Covas et al. (2005) proposed a standing wave difference method, which uses the spectral analysis of an FRD to determine the leak location. Lee et al. (2005b) observed that a leak in a single pipeline would introduce a sinusoidal pattern on the resonant responses, and the location and size of the leak can be determined from the period and amplitude of this pattern. Sattar and Chaudhry (2008) found that the leak-induced sinusoidal pattern could be observed on the anti-resonant responses in some situations. The factors that decide whether the leak-induced sinusoidal pattern would appear at the resonant or at the anti-resonant frequencies have been explained by Gong et al. (2014a). Gong et al. (2013a) developed a leak detection technique that only uses the first three resonant peaks, which significantly reduces the requirement on the bandwidth of the excitation signal. A customised solenoid valve that generates pseudorandom binary sequences (PRBS) was developed and numerical and laboratory experiments confirmed its usefulness in pipeline FRD extraction (Gong et al. 2013b; Gong et al. 2016b). Frequency response-based leak detection methods have much better computational efficiency compared to the time-domain ITA. However, the successful implementation of these techniques replies on accurate measurement of the FRF of a pipeline system, which is difficult in the field due to the 10
ADE	Chapter 1 complexities of the pipeline configuration and the limitation in the bandwidth of the transient input signal (Gong et al. 2013b; Lee et al. 2013). 1.4 Transient-based technique for pipe wall condition assessment Research of transient-based pipe wall condition assessment has been focused on the detection of thinner-walled pipe sections (e.g. sections with extended internal/external corrosion or the spalling of cement-mortar lining), and on sections with extended blockages (e.g. sections with extended tuberculation). 1.4.1 Detection of thinner-walled pipe sections Stephens et al. (2008; 2013) were the first to investigate transient analysis applied to the detection of changes in pipe wall thicknesses. Stephens et al. (2013) studied a mild steel cement-mortar lined (MSCL) water transmission main in the field, and calibrated the wave speed along the pipe for the detection of sections with extended spalling of cement-mortar lining and/or internal corrosion by using time-domain ITA. The calibrated remaining pipe wall thicknesses (as derived from the wave speed) were consistent with those from ultrasonic pipe wall thickness inspection. However, due to the structural complexity and parametric uncertainties of real pipelines, the efficiency and accuracy of the ITA-based pipe wall condition assessment techniques need to be improved. Zhang et al. (2018b) proposed a head-based MOC technique with flexible computational grids to speed up the forward modelling part of an ITA analysis. Associated research by Zhang et al. (2018a) confirmed that the ITA- based pipe wall condition assessment technique suffers from the problem of 11
ADE	Chapter 1 multiple solutions, since different combinations of the pipe wave speed distributions can result in very similar transient pressure responses at some locations. Hachem and Schleiss (2012) proposed a technique for detecting a structurally weak section in single pipelines using a steep transient pressure wave and TDR analysis. The wave speed in the deteriorated section was estimated by comparing the measured wave speed with that of an intact pipe. This technique would have difficulties in determining the wave speeds should multiple deteriorated sections exist. Gong et al. (2013c) developed a technique for detecting deteriorated pipe sections based on direct analysis of the magnitude and the TDR principle. The magnitude of the wave reflection induced by a section with wall thickness changes is indicative of the impedance of that section, which can then be directly used to calculate the wall thickness and wave speed in that section. The technique is efficient and effective for single pipelines with only a few deteriorated pipe sections, and it has been verified in field trials on a MSCL water main (Gong et al. 2015) and an asbestos cement (AC) water main (Gong et al. 2016c) in regional Australia. A further development based on the direct reflection analysis-based condition assessment technique is the reconstructive method of characteristic (RMOC) technique (Gong et al. 2014b). The RMOC technique enables the reconstruction of the impedance continuously along a pipe by using a measured transient response trance and by calculating along the characteristic lines of MOC backward in time. However, the original RMOC technique is only applicable to reservoir-pipeline-valve (R-P-V) systems, where the valve closure is used to 12
ADE	Chapter 1 generate a step transient wave and pressure response is measured at the upstream face of the valve. Zeng et al. (2018b) has developed a technique to reconstruct the pipeline impedance of an R-P-V system using a modified layer- peeling method, which analyses the IRF instead of the response from a step incident wave. Zhang et al. (2019) has generalised the RMOC technique to be independent of the boundary conditions of the pipeline system. This is achieved by using two pressure transducers in close proximity – an inspiration from the sensor array measurement strategy used in the PhD research presented here. In the frequency domain, Zecchin et al. (2009) developed a mathematical framework for transient simulation in arbitrary pipe networks using the admittance matrix method. The framework was used in general calibration of pipeline parameters in a network environment (Zecchin et al. 2014a). The principle is to find an optimal pipe model whose response matches the measured response, which is similar to that of conventional ITA but in the frequency domain. However, the work so far has been limited to numerical studies, and applications to real pipeline networks will be challenging. 1.4.2 Detection of extended blockages Duan et al. (2012) proposed a technique to detect extended blockages (pipe sections with larger wall thicknesses but the same external diameter as the intact part) using the FRD of single pipe systems. The principle is that extended blockages could cause the resonant frequencies of a single pipeline system to shift, and the frequency shift can be used to determine the properties of the extended blockage. Although the concept of the technique has been validated in the laboratory (Duan et al. 2013); however, many challenges are expected in 13
ADE	Chapter 1 real applications (Duan 2016b). As acknowledged by the same authors, the shifts of the resonant peaks due to extended blockages are typically insignificant (Duan et al. 2011) therefore difficult to determine accurately. Zeng et al. (2018a) developed a technique for extended blockage detection using a modified layer-peeling method. It was found through numerical simulations that the wave speed and internal diameter can be reconstructed even for non-uniform extended blockages. The technique requires an R-P-V system configuration with the generation and measurement points at the valve end. 1.5 Key challenges to address A common limitation to all the conventional transient-based pipeline defect detection and wall condition assessment techniques is that they are difficult to be extended to applications involving complex pipe systems or pipe networks. This is because convenional single pipe transient based methods make explicit assumptions about boundary conditions that are incompatable with network junction interactions. That is, the bounary conditions imposed on a single pipe section from the surrounding network lead to measured transient responses that can be too complex to analyse with conventional methods. The most common measurement strategy used in the field is illustrated in Figure 1.1. Multiple single pressure transducers are sparsely placed along a pipeline at existing access points, such as air valves or fire hydrants. Pressure waves travel along a pressurised pipeline in two directions – towards both the upstream and the downstream directions. For a single pressure transducer, the measurement is always a superimposed signal of both the waves travelling in the two directions. When multiple deteriorated sections exist on both sides of a transducer, which 14
ADE	Chapter 1 is the most common scenario in real pipeline systems, reflections are complex in waveform due to the wave superposition. In water distribution systems, where pipe branches are significant in number and size (compared to the size of the main pipe), the measured pressure signal can be very complex and difficult to interpret. Pressure Transient generator G and Pressure sensor T 1 pressure sensor T 2 sensor T 3 Partial blockages Spalling of lining and External internal corrosion corrosion Upstream Directional pressure waves Downstream Figure 1.1 Schematic diagram showing the current transient pressure measurement strategy used in the field. To avoid the complexity, many conventional transient-based techniques use a single pressure transducer at a dead-end, such that pressure waves can only come from one direction. Additionally, the FRD-based techniques also require the whole pipe system to be simple in configuration (e.g. reservoir-pipeline- valve or reservoir-pipeline-reservoir). However, these requirements are difficult to achieve in field pipelines, which are typically buried underground with limited access and can also be embedded in a complex network environment. 15
ADE	Chapter 1 Gong et al. (2012b) proposed that the use of the transient pressure measurements from two pressure transducers installed at different locations along a pipe could separate the pressure waves travelling upstream and downstream. In a following study, Gong et al. (2012a) found that arranging the two pressure transducers in close proximity can facilitate the wave separation. Zecchin et al. (2014b) found that the use of a pair of pressure transducers can enable the determination of the system IRF. While these preliminary numerical studies have proven the concept that the directional pressure waves can be separated to facilitate pipeline condition assessment, it is envisaged that many challenges exist in real applications due to the uncertainties in the pressure measurement and pipeline parameters. Similar applications can be found in acoustic research, where two or more microphones are used to separate the travelling acoustic waves in ducts filled with air (De Sanctis and Van Walstijn 2009; Kemp et al. 2013). However, these acoustic techniques cannot be directly applied to transient pressure analysis in water pipelines due to the difference in the dominant physical processes, and associated modelling paradigms, for acoustic waves versus hydraulic transient waves. 1.6 Research objectives The overall objective of this PhD research is to address the complexity in the interpretation and analysis of transient pressure data associated with the supposition of the two directional waves underlying any pressure measurement. The research here proposes to use paired pressure sensors (two pressure transducers in close proximity) to measure the pipeline transient pressure response at each station, instead of just using a single pressure sensor, as is the 16
ADE	Chapter 1 conventional approach. New and practical techniques are developed to extract the directional information of travelling pressure waves, which then enables the development of advanced pipeline leak detection and wall condition assessment techniques for targeted pipe sections in complex pipeline systems. The specific aims of this research are as follows: Aim 1: To develop a robust wave separation algorithm that can extract the two directional travelling pressure waves in a pipeline from the pressure traces measured by a pair of pressure transducers located in close proximity (a dual- sensor unit). This separation of the measurement of pressure into its respective wave components allows for the directional attribution of observed pressure fluctuations. That is, the wave reflections induced by anomalies located on either side of the dual-sensor unit are able to be separated, through the reconstruction of the directional travelling waves. This enables a significant reduction in the complexity of the wave form, and the wave reflections will be attributable to their source. Aim 2: This twofold aim is to develop: (i) a system identification algorithm that can determine the system transfer matrix of a section of pipe bounded by two dual-sensor units; and (ii) an associated technqiue for detecting leaks in the pipe section to utilise this system transfer matrix. The two dual-sensor units provide information about the pressure waves travelling into, and out of, the section of interest, even if the section is part of a complex network. Given a linear systems framework, this complete acquisition of the input/output signals enables the determination of the associated system transfer matrix, which is a full representation of the physical characteristics of the section of interest. As an 17
ADE	Chapter 1 outcome of this aim, this specific section of pipeline can be isolated from a complex pipeline network for independent analysis (e.g. leak detection). Aim 3: To develop and assess the utility of a fibre optic dual-sensor array. The in-pipe fibre optic pressure sensor will enable distributed measurement of transient pressure through a single access point on the pipe. This restriction to a single access point is a critical limitation for the practical implementation of dual-sensor approaches in the field. Laboratory experiments will be conducted to validate the approach and explore its utility for condition assessment.. 1.7 Organisation and overview of the thesis This PhD thesis contains five chapters. Chapter 1 (this current chapter) is an introduction of the research project, with a literature review, a summary of the research challenges, a statement of the research aims, and an outline of the organisation of the thesis. The main body of this thesis – Chapters 2 to 4 – constitutes the three journal manuscripts arising from the research. The final chapter contains the conclusions and recommendations for future work. Chapter 2 presents an advanced wave separation algorithm for transient analysis in pipelines. The technique enables the extraction of directional travelling pressure waves by using two closely placed pressure sensors at one measurement site (referred to as a dual-sensor). The dynamic relationship between the two pressure transducers can be calibrated in-situ to enhance the robustness of the wave separation. In addition to numerical simulations, the research has also conducted the first experimental verification of transient wave separation on a copper pipeline in the laboratory, where a step wave generated 18
ADE	Chapter 1 by a side-discharge valve was used as the excitation and two adjacent pressure sensors flush mounted on the experimental pipe were used for transient measurements. Comparison of the wave separation results with their numerically predicted counterparts has shown that the wave separation algorithm is successful. The results have also shown that the proposed wave separation technique facilitates transient-based pipeline condition assessment by reducing the complexity of the wave form. The research findings have been published in the Journal of Hydroinformatics (DOI: 10.2166/hydro.2017.146). Chapter 3 presents an innovative approach for leak detection in a targeted pipe section using hydraulic transient waves and the dual-sensor measurement strategy. The new concept is to utilize a special transient pressure generation and sensing configuration, combined with custom developed signal processing algorithms, to “virtually” break any complex pipeline system down to its simplest form – a single pipe section – for independent condition diagnosis. The virtual isolation of a pipe section is achieved by a two-source-four-sensor transient testing strategy: two dual-sensor units are used to bracket the targeted pipe section (with the two sensors in each unit being in close proximity); and two transient pressure wave generators are used, which bracket the four sensors and the targeted pipe section. This testing strategy enables the extraction of the transfer matrix of the in-bracket pipe section, independent from any hydraulic components outside of the two transient generators. A transfer matrix of a pipe section is a full representation of the wave propagation characteristics (in the format of frequency response functions) as governed by the physical properties of the pipe section. Given this, the extracted 19
ADE	Chapter 1 transfer matrix can be used for leak detection. It has been found that a linear combination of two elements in the extracted transfer matrix is sensitive to leaks, where a leak will introduce a sinusoidal with the period and the magnitude of the pattern related to the location and impedance of the leak, respectively. Multiple leaks introduce multiple sinusoidal patterns. An algorithm has been developed to extract the leak information from the extracted transfer matrix of the “virtually” isolated pipe section, and the technique has been validated by numerical simulations. Note that the technique and the findings are different from that of conventional FRD-based leak detection techniques, in which the boundary conditions of the pipe system need to be known and the entire system needs to be simple (e.g. reservoir-pipe-reservoir or reservoir-pipe-valve). The work presented in this thesis is the first to utilise the two-source-four-sensor transient generation and measurement strategy for leak detection in targeted pipe sections embedded in complex systems. This concept is also useful for other applications such as blockage detection and pipe wall condition assessment. The research findings have been submitted to the Journal of Hydraulic Engineering for peer review. Chapter 4 presents a customised in-pipe fibre optic pressure sensor array and its application to transient wave separation and pipe wall condition assessment in the laboratory. The sensor array consists of five fibre Bragg grating (FBG)- based pressure sensors in close proximity (∼0.5 m apart). The cable that protects the optical fibre is made from a plastic material, and has a diameter of approximately 4 mm. At each FBG pressure sensor, a 10 mm window is open in the protective cable, and a flexible elastomeric sleeve is used to cover the FBG. This fibre optic sensor array represents a second generation of 20
ADE	Chapter 1 development and is especially designed for high-speed pressure measurement under relatively large pressure conditions (2 bar to 10 bar). The sensors have a wider pressure applicable range and a better linearity compared to the first generation of the fibre optic pressure sensors tested in 2014 (which is reported in a conference paper itemised in the List of Publications, but this work is not included in this PhD thesis). Extensive laboratory experiments have been conducted in the Robin Hydraulics Laboratory at the University of Adelaide on this fibre optic pressure sensor array. The sensor array was inserted into the pipeline through a single entrance point. Pressure response data were successfully collected from the fibre optic sensor array with a sampling rate up to 20 kHz. The previously developed wave separation algorithm was adapted to analyse the transient pressure measurement from the FBG sensors. The resultant directional pressure waves were then used to detect pipe sections with a thinner wall thickness. A challenge is the influence of the in-pipe fibre optic sensing cable on the transient pressure measurement. The impact was analysed and adjustments to the pipeline condition assessment algorithm were undertaken to resolve the issue. The successful experimental application has provided a verification of the usefulness of the in-pipe fibre optic sensor array, which can facilitate transient-based pipeline condition assessment for buried water pipes with limited access points. The results of the research have been published in the Journal of Hydroinformatics (DOI: 10.2166/hydro.2019.051). 21
ADE	Chapter 2 2.1 Introduction The aging of water distribution systems worldwide has brought many issues, ranging from significant water and energy losses (Colombo and Karney 2002) to risks to public health due to possible pathogen intrusion (Karim et al. 2003). Over the past two decades, hydraulic transients (water hammer waves) have been identified as a useful tool for non-invasive pipeline leak detection (Mpesha et al. 2002; Ferrante and Brunone 2003; Covas et al. 2005; Lee et al. 2005a; Soares et al. 2010; Ferrante et al. 2012; Duan 2016a), blockage detection (Sattar et al. 2008; Meniconi et al. 2013; Massari et al. 2014), wall condition assessment (Gong et al. 2013c; Stephens et al. 2013) and general system parameter identification (Zecchin et al. 2013; Zecchin et al. 2014a). When undertaking a hydraulic transient analysis of a pipeline system, a transient disturbance (a pulse or a step pressure wave) is typically introduced by abruptly operating a valve. Then the transient pressure wave propagates along the pipe in both upstream and downstream directions. Any physical changes or anomalies in a pipeline will affect the propagation of transient pressure waves, resulting in specific reflections. These reflections can be analysed in either the time or frequency domain, in order to diagnose the anomalies in the pipeline system. Most existing studies are based on the analysis of the transient pressure measured by a single sensor, or by multiple sensors usually separated by a significant distance along pipes. However, there are often simultaneous waves travelling in opposite directions. The hydraulic pressure at any single point in a 29
ADE	Chapter 2 pipeline can be expressed as the sum of a travelling pressure wave coming from upstream of the measurement point and a travelling pressure wave coming from downstream. As a consequence, for a single pressure sensor, the measured signal is always a superimposed signal of waveforms propagating upstream and downstream. One measurement strategy to avoid a superposition problem is by placing the measurement point at a dead end to ensure the reflection comes only from one direction (Gong et al. 2014b). However, when investigating transmission mains, which may be tens of kilometres long, it is not always practical to find an ideal measurement point at a dead end. Moreover, when multiple anomalies exist on both sides of a sensor, which is the common case in most pipelines, the measured pressure signal can be very complex and difficult to interpret, even when multiple measurement sites are used (Gong et al. 2016c). To investigate and extract the directional information of travelling transient pressure waves, Gong et al. (2012b) proposed a technique that uses two pressure sensors (100 m spaced in a pipe with an internal diameter 600 mm) to separate the pressure waves travelling downstream from those travelling upstream along a pipeline. In a subsequent study by Gong et al. (2012a), a new measurement strategy, which involved the use of two pressure sensors in close proximity (1 m spaced in a pipe with an internal diameter 600 mm), was proposed to facilitate the wave separation. However, these preliminary studies were limited to numerical simulations with ideal conditions where the pipeline between two sensors is assumed to be lossless and the incident wave is a sharp step signal. 30
ADE	Chapter 2 Zecchin et al. (2014b) proposed a technique for extracting the impulse response of a single pipeline using a pair of sensors (10 m spaced in a pipe with an internal diameter of 200 mm) for measurement, and using hydraulic noise as the excitation. The hydraulic noise is in the form of wide-sense stationary pressure signals (the mean function and correlation function do not change over time). In that study, a theoretical propagation loss between two sensors was considered. However, the directional travelling waves were not extracted from the measurements. The wide-sense stationary pressure signals are difficult to achieve in practice, and only a numerical case study was conducted in that paper. It should be noted that the use of two pressure sensors in close proximity (referred to as a “dual-sensor” in the following) for wave separation has been studied in the acoustics research area, where acoustic waves in ducts measured by two (or more) microphones are analysed (Chung and Blaser 1980). However, hydraulic transient waves in water-filled pipes have many differences from acoustic waves propagating in the air, namely, they have a different signal bandwidth, wave magnitude and wave propagation properties where wall friction plays a much more significant role. Moreover, the research in acoustic ducts focuses on calculating the reflection and transmission coefficients, rather than splitting the directional travelling waves explicitly, which is the focus of the wave separation method developed in the current paper. In the field of pipeline transient analysis, the use of two pressure sensors in close proximity has been used for unsteady flow measurement (Washio et al. 1996a; Kashima et al. 2013) However, except for the preliminary numerical work reported in Gong et al. (2012b; 2012a) and Zecchin et al. (2014b), to the knowledge of the 31
ADE	Chapter 2 authors, there is no study on the separation of hydraulic transient waves using a dual-sensor in pressurised pipelines. The research reported in the current paper develops a systematic wave separation algorithm that can extract the two directional travelling hydraulic transient waves from pressure traces as measured by two closely spaced pressure sensors. Compared to the preliminary numerical work in Gong et al. (2012b; 2012a) and Zecchin et al. (2014b), the new developments include: (1) an experimental data-driven approach to estimate the transfer function between the two sensors, which enables wave separation without the knowledge of the specific pipe parameters (e.g. flow rate, friction factor, and diameter of the pipe); (2) the extraction and removal of the incident waves, making the algorithm applicable to real incident waves with curved wave fronts rather than the theoretical sharp incident waves used in previous studies; and (3) the first experimental verification of the wave separation technique. To validate the wave separation algorithm, a pulse incident wave is used in a numerical study and a step wave is considered in a laboratory study. In both studies, as presented in this paper, the comparison between the extracted directional reflection trace with its counterpart, which has a reflection from one direction only, shows the wave separation algorithm is successful. The wave separation algorithm provides the directional information of travelling transient pressure waves in pipelines and simplifies the interpretation of the signals. The directional waves, as obtained in the laboratory study, are then used to determine the properties of two deteriorated pipe sections in the experimental system (simulated by short pipe sections with thinner wall thicknesses). The 32
ADE	Chapter 2 results demonstrate that the proposed wave separation technique can adequately facilitate transient-based pipeline condition assessment. Limitations of the technique and practical challenges are discussed before drawing the conclusions. 2.2 Wave separation algorithm using a dual- sensor 2.2.1 Hydraulic wave propagation theory The transient behaviour of pressurised fluid within a closed conduit pipeline system is governed by the so called water hammer equations, which are a series of two one-dimensional (1-D) quasi-linear hyperbolic differential equations describing mass and momentum conservation (Wylie and Streeter 1993; Chaudhry 2014). The solution of the water hammer equations can be expressed in terms of pressure waves travelling upstream and downstream (Wylie and Streeter 1993; Chaudhry 2014). This is a consequence of the mathematical properties of the hyperbolic equations, but also reflects the physics of the fluid, that is, the fluid pressure at any single point in a pipeline can be expressed as the sum of a pressure wave travelling in the positive direction and a pressure wave travelling in the negative direction, i.e.: p(x,t) p(x,t) p(x,t) (2.1) 33
ADE	Chapter 2 Experimental representation As an alternative to the analytic expression for H(s), the properties of the transfer function can also be determined experimentally. In hydraulic transient analysis, a pipeline system is typically excited by abruptly opening or closing a valve, which results in a discrete wave with a short duration as an incident pressure wave (e.g. a sharp step or pulse wave). Under these conditions, an assumption can be made that during the time of the incident wave entering into the system at T then exiting the system at T , there are no transient waves 1 2 entering the system from T (i.e. p (t)0 in Figure 2.1). This assumption is 2 2 often valid when the incident wave is short, and the system is excited from a steady state condition. Therefore, the LTI system in Figure 1(a) can be temporarily treated as a single-input and single-output system for this short time period. The input is the incident wave p(t) p (t) at T , and the output is 1 1i 1 the incident wave p (t) p (t) at T . The incident waves p (t) and p (t) 2 2i 2 1i 2i p (t) p (t) can be extracted from the original measured pressure trace and by 1 2 applying a rectangular time window (i.e. truncating the short signal section that includes the wave front out of the whole pressure trace). The transfer function H(s) is the linear mapping from an input to an output in the Laplace domain, and can be given by: P (s) H(s) 2i (2.14) P (s) 1i 39
ADE	Chapter 2 where P (s) and P (s) are the Laplace transforms of the incident waves at T 1i 2i 1 and T , respectively. Note that the experimental approach does not require any 2 flow rate information except that there is no wave (or relatively very small wave) in one direction, which is an advantage over the analytical approach. 2.2.3 The wave separation algorithm In Figure 2.1(a), when an incident pressure wave is generated at G and arrives at sensor T , the positive travelling pressure wave at T contains the incident 1 1 wave and the reflected wave coming from upstream of T . The reflected wave 1 is the focus because it carries the pipeline information that can be used for pipeline condition assessment. However, compared to the incident wave, the reflections due to wall deterioration are usually small (Gong et al. 2015). Given this, removing the dependence of the incident wave from the wave separation results leads to clearer separated directional travelling waves, and the method is described below. In Figure 2.1(a), the positive travelling waves can be written as the sum of the incident wave and the reflected wave coming from upstream: p(t) p (t) p (t) (2.15) 1 1i 1r p (t) p (t) p (t) (2.16) 2 2i 2r where p (t) and p (t) are the reflected waves coming from upstream of the 1r 2r measurement points T and T respectively. The negative travelling waves 1 2 40
ADE	Chapter 2 P (s)P (s)H(s) P (s) 1r 2r (2.17) 1r 1H2(s) P (s)H(s)P (s)H2(s) P (s) 2r 1r (2.18) 1r 1H2(s) The inverse Laplace transforms of Equations (2.21) and (2.22) will give the positive travelling reflected waves and the negative travelling reflected waves in the time domain. For analysis of real pipeline systems where the pressure signals measured by sensors are used, the value of the Laplace variable is restricted to the imaginary axis, i.e. s = i , where i is the imaginary unit, and  is the radial frequency. Consequently, the Fourier transform can be used instead of the Laplace transform. To apply the wave separation algorithm to pressure traces [ p (t) and p (t)] 1 2 measured by a dual-sensor as in Figure 2.1(a), the following steps will be performed: 1. Time-windowing to separate incident waves and reflections in the original pressure traces measured by a dual-sensor using Equations (2.19) and (2.20). 2. Transfer incident waves and reflections into the frequency domain by using the Fourier transform. 3. Determine the transfer function between two sensors using the analytic Equation (2.11) or using the experimental Equation (2.14). 42
ADE	Chapter 2 4. Extract the directional reflection waves in the frequency domain using Equations (2.21) and (2.22). 5. Transfer wave separation results into the time domain using an Inverse Fourier Transform, or using the results directly in the frequency domain for further analysis, e.g. determine the frequency response of the pipe section. It should be noted that, when other incident waves are used in place of discrete waves with a short duration, step 1 can be ignored and Equations (2.9) and (2.10) in step 4 used instead. Before the wave separation algorithm is applied to real data, pre-processing may be needed, including determining the effective frequency range to minimise the impact of high frequency noise on the time domain reconstruction of the separated reflected waves. Because the analysis is built on linear systems theory, the incident waves should be small perturbations to limit the effect of linearization (Lee and Vitkovsky 2010). 2.3 Numerical verification To verify the dual-sensor wave separation algorithm, numerical simulations have been conducted. A single pulse hydraulic pressure wave is used as the incident wave in the numerical investigations since it has never been studied previously for wave separation. 2.3.1 System layout and procedure For the numerical study, a metallic pipeline system with two short deteriorated sections and one relatively long section with a change of pipe class is considered. The layout of the numerical pipeline system is given in Figure 2.2. The physical 43
ADE	Chapter 2 details of the pipe sections are summarised in Table 2.1. The length of each reach is carefully designed to satisfy the Courant condition for MOC simulations (with a time step of 0.05 ms). The system is a reservoir-pipeline- reservoir (R-P-R) system. Reservoir 1 has a constant head of 60 m, and the constant head for Reservoir 2 is 57 m. The total length of the pipeline is 1 km. The steady-state flow is calculated as 0.264 m3/s, corresponding to a velocity of 1.34 m/s. For the normal pipe sections, the internal diameter is 500 mm, the wall thickness is 8 mm, the Reynolds number is 4.75105 (indicating turbulent flow) and the wave speed is 1154 m/s. Two pipe sections L and L which have 2 9 thinner wall thicknesses (6 and 5 mm), larger internal pipe diameters (504 and 506 mm) and smaller wave speeds (1083 and 1036 m/s) are placed in the system to simulate the deteriorated sections (e.g. extended internal corrosion). Pipe section L with a length of approximately 150 m, the same internal diameter as 7 the majority of the pipe, but a thinner wall thickness (7 mm) and thus a lower wave speed (1123 m/s), is placed in the system to simulate a section of a lesser pipe class. A significantly higher Darcy-Weisbach friction factor (0.03) has been assigned to sections L and L to represent the effect of a much higher wall 2 9 surface roughness as would result from a pipe that has experienced corrosion. The dual sensor (with a sensor spacing of 0.9809 m) is placed in the middle of the pipeline system at T and T , respectively. A side-discharge valve which is 1 2 located at 0.9809 m upstream from T is used as the transient generator. The 1 steady-state discharge through the side-discharge valve is set as 0.01 m3/s. The length of each pipe section has been selected to satisfy the Courant condition for the time domain method of characteristics (MOC) simulations so that no interpolation scheme is required (Chaudhry 2014). 44
ADE	Chapter 2 previously. The reflected waves p (t) and p (t) are shown in Figure 2.3(b). 1r 2r It can be seen from Figure 2.3(b) that the pressure reflections as recorded by the dual-sensor possess a complex form of pressure wave fluctuations, although only three uniform sections with lower wave speeds are considered. This complexity is due to the superimposition of the reflections from the three thinner-walled sections. Figure 2.4(a) shows the reflections from upstream of T and Figure 2.4(b) gives 1 the reflections from downstream. The pressure waves p (t) and p (t) 1r_A 1r_A are obtained by using the analytic expression of the transfer function between two sensors according to Equations (2.11) and (2.12), while p (t) and 1r_E p (t) are calculated from the experimental transfer function which is 1r_E estimated by using the extracted incident waves according to Equation (2.14). The analytically and experimentally determined transfer functions are consistent within the bandwidth of the incident wave. For a comprehensive comparison, the wave separation results are compared with predicted results as computed directly from the MOC. The predicted results for upstream reflections ( p (t) as shown in Figure 2.4(a)) are 1r_P obtained from MOC modelling the system similar to that depicted in Figure 2.2, but only with one deteriorated section L existing on the upstream side of the 2 sensors. On the downstream side of the sensors, there are just uniform intact pipes (i.e. L and L are set as the same as the intact sections). Hence, the 7 9 simulated reflections only come from upstream and are a result of section L . 2 48
ADE	Chapter 2 Similarly, the predicted results for downstream reflections ( p (t) as shown 1r_P in Figure 2.4(b)) are acquired by MOC modelling with no defective sections existing on the upstream side of the sensors. So that reflections only happen on the downstream side of the sensors and include reflections from sections L and 7 L . 9 It can be seen in Figure 2.4 that the reflections from the three thinner-walled pipe sections are separated and clearly shown in the directional waves p(t) 1r and p(t) respectively. The separation results from two different transfer 1r function calculation methods (analytical and experimental) are almost identical, and both of them have an excellent match with the MOC predictions. It should be noted that the separated results of directional waves include multiple reflections while the predicted results do not. The multiple reflections are due to secondary reflections between anomalous sections on the two sides of sensors. For example, when all three thinner-walled sections are considered in the simulation, the major wave reflections from sections L and L (as shown 7 9 in Figure 2.4(b)) will propagate from downstream to upstream, pass the dual- sensor and then be reflected by section L as secondary reflections. These 2 secondary reflections will then propagate downstream as part of p(t) . 1r Nevertheless, the numerical simulation demonstrates that the proposed wave separation approach is valid for pipelines excited by single pulse incident pressure waves. 49
ADE	Chapter 2 bounded by two pressurised tanks. The pressurised tanks can be isolated by an in-line valve to make the system a reservoir-pipeline-valve (R-P-V) configuration. A step incident pressure wave is used to avoid repetition from the numerical study and it better represents the incident waves used in the field. 2.4.1 System layout and procedure The layout of the pipeline system used in the experiments is shown in Figure 2.5 and the physical details are given in Table 2.2. The wave speeds are calculated using the theoretical wave speed formula (Wylie and Streeter 1993). The following parameter values are used: Young’s modulus of a copper pipe E 124.1 GPa, restraint factor for thick-walled copper pipe anchored throughout c 1.006 , bulk modulus of elasticity of water at 15 ºC is 1 K 2.149 GPa, and density of water at 15 ºC is 999.1 kg/m3. The restraint factor is a dimensionless parameter that depends on the elastic properties and the constraint condition of the pipe (Wylie and Streeter 1993). Figure 2.5 System layout of the experimental pipeline system. 51
ADE	Chapter 2 Table 2.2 Physical details of the pipeline system used in the laboratory experiments. Internal diameter Wall thickness Wave speed Pipe (symbol = value (symbol = value (symbol = value class (mm)) (mm)) (m/s)) A D = 22.14 e =1.63 a = 1319 0 0 0 B D = 22.96 e =1.22 a = 1273 1 1 1 C D = 23.58 e = 0.91 a = 1217 2 2 2 The majority of the pipeline is in Class A. Two short pipe sections of Class B and C, respectively, which have thinner wall thicknesses than that of Class A, are placed in the system to simulate pipe sections with wall deterioration. The head in the pressurised tank was controlled at approximately 31 m during the experiments. The in-line valve at the other end was kept closed during the experiments. A solenoid side-discharge valve was used as the transient generator (G) and placed at the same location as pressure sensor T . The other pressure sensor (T ) 1 2 was located upstream (on the left) of the transient generator separated by a distance of 0.99 m. A step pressure wave was generated by abruptly closing the solenoid valve in approximately 3 ms. The pressure responses were measured by the two sensors with a sampling frequency of 20 kHz. 52
ADE	Chapter 2 to avoid effects of noise in the high frequency range and also to cover the effective bandwidth of the reflected waves. 250 p (t) 1r p (t) 200 2r e d u 150 t ni g a M 100 50 0 0 200 400 600 800 Frequency (Hz) Figure 2.7 Amplitude spectrum of the reflected waves. The positive and negative travelling pressure reflection waves p(t) 1r (propagating towards the closed in-line valve) and p(t)(propagating towards 1r the tank) are determined by Equations (2.21) and (2.22) for frequencies up to 600 Hz in Step 4. The results are given in Figure 2.8 and compared with the predicted results generated by MOC simulations (the procedure is the same as that used in the numerical study, i.e. only deterioration on one side is considered when generating the predicted results). The steady-state pressure in the MOC model is set equal to the measured steady-state pressure in the laboratory. The step incident wave in the MOC model is designed according to the measured incident step wave with a rise time of 3 ms and a pressure head magnitude of 6.60 m. The shape of a cosine function changing from  to 2 is adopted to simulate the curved wave front. 55
ADE	Chapter 2 size of a wave reflection and the relative change in the wall thickness is derived as: (K /)(1e )  K Ea2c 2 rc  12e 0 1   K /e a2 rc K /a2   p  rc 0 0 (2.23) n (K /)(1e )  K Ea2c 2 rc  12e 0 1   K /e a2 rc K /a2   rc 0 0 where p represents the normalized head perturbation of the reflected wave n and is defined as p p  p  p , where p and p are the sizes of reflected n r i i r i wave and incident wave respectively; e is the relative change in wall thickness rc and is defined as e e e  e , where e and e represent the wall rc d 0 0 0 d thickness in the intact and deteriorated section respectively; a is the wave 0 speed in the intact pipe. Note that Equation (2.23) is derived under an assumption for lossless elastic pipelines. Figure 2.10 Relationship between the normalized wave reflection ( p ) and the n relative change in the wall thickness (e ) for the experimental pipeline. rc 58
ADE	Chapter 2 The plot of Equation (2.23) is given in Figure 2.10. The theoretical wave speed in the intact (Class A) pipe is considered, which is a 1319 m/s as in Table 0 2.2. The range of e used is from e = – 0.5 to e = 0, which represents wall rc rc rc thickness variation from half the original wall thickness to the original wall thickness. The plot can serve as a look-up chart for condition assessment for pipes with internal changes in wall thickness. It is obvious that the original pressure measurements as shown in Figure 2.9 cannot be directly used for condition assessment because the reflections from the Class B and the Class C sections are superimposed. In contrast, the separated directional wave reflections as shown in Figure 2.8 show the wave reflections from the two sections separately and clearly, and they can easily be used for further analysis. The values of the relative wave reflections ( p  p ) from the Class C and Class r i B sections are determined from the minima of p(t) and p(t) respectively 1r 1r as shown in Figure 2.8, for which the results are p  p = –0.64 m and 1r i p  p = –0.40 m respectively. The magnitude of the incident step wave is 1r i determined as p = 6.60 m from the measured trace shown in Figure 2.6(b). As i a result, the normalized reflections for the Class C and Class B sections are p n = –0.097 and p = –0.061, respectively. Referring to the look-up chart in n Figure 2.10, the relative change in wall thickness corresponding to these two wave reflections are e = –0.44 and e = –0.29, respectively. Finally, using the rc rc wall thickness of the Class A pipe of e = 1.63 mm, the wall thicknesses in the 0 59
ADE	Chapter 2 Class C and Class B sections are determined by the reflection analysis as e = 0.91 mm and e = 1.16 mm, respectively. Compared with the wall thicknesses as given by the manufacturer (e = 0.92 mm and e = 1.22 mm as shown in 2 1 Table 2.2), the wall thicknesses are estimated with relatively high accuracy. These results have demonstrated that the wave separation algorithm as developed in this research can significantly facilitate pipeline condition assessment by resolving the complexity due to wave superposition. 2.5 Discussion Some practical issues related to real applications of transient-based pipeline condition assessment are discussed in this section. Recommended future work is also presented. 2.5.1 Detection resolution The spatial resolution of detection is limited by the effective bandwidth of the pressure waves, which is itself related to the sharpness of the wave front. For a ramp incident wave, theoretically one can accurately diagnose deteriorated sections only with a length longer thanTa /2, where T is the rise time of the i d i ramp wave front and a is the wave speed in the deteriorated pipe section (Gong d et al. 2013c). Sections shorter than that may still be detectable but will not give a full-sized reflection, therefore the change in wall thickness will be underestimated. In the experimental study, the rise time of the incident step was about 3 ms. Using a wave speed of 1,300 m/s, the threshold is calculated as approximately 2 m. 60
ADE	Chapter 2 2.5.2 Detection range The length of pipe that can be assessed reliably mainly depends on the signal- to-noise ratio (SNR). It is expected that measurements in the field can have stronger noise than in the laboratory (e.g. due to pump operations). The frequency range to include in the analysis should be selected carefully to balance the SNR and detection resolution (as discussed above). Usually low frequency waves have better SNR than high frequency components, because the latter typically have less initial energy and suffer higher damping rates. A spectrum analysis for the wave reflections (as described in the Experimental verification section) will help to determine the useful bandwidth. Nevertheless, field trials by the authors confirmed that a step transient pressure wave can travel many kilometres with insignificant attenuation in water transmission mains (diameter 600 mm) (Stephens et al. 2013; Gong et al. 2015; Gong et al. 2016c). However, the sharpness of the wavefront decreases over the distance of propagation. 2.5.3 Non-uniform deterioration In real pipelines, deteriorated sections most likely have non-uniform wall thickness variations. As a result, the wave reflections may not have sharp edges as shown in the laboratory study. In such cases, the extrema of the reflections should be used to calculate the normalized head perturbation. The determined wall thickness represents the general condition of the deteriorated section. 61
ADE	Chapter 2 2.5.4 Other sources of reflections In addition to deteriorated pipe sections, wave reflections can be induced by other sources, which typically include changes in pipe material and class, leaks, blocks, branches and air pockets. Priori information of pipeline systems (e.g. as constructed drawings) will be helpful in identifying the source of wave reflections. The characteristics of the wave form can also facilitate the categorisation (e.g. discrete blocks introduce extended positive reflections while leaks introduce extended negative reflections). Note that pipe joints typically do not introduce noticeable reflections, since the dimension of joints is much smaller than the effective wavelength. 2.5.5 Accuracy of transfer function A topic for future work is to enhance the accuracy in the determination of the transfer function between the two pressure sensors. Error in the transfer function will affect the wave separation and therefore the condition assessment. It can be induced by background noise and the inconsistency among pressure transducers (i.e. for the same pressure condition, different sensors may give slightly different readings). The use of sensor arrays to provide redundant information may be helpful in enhancing the accuracy. 2.6 Conclusions A wave separation algorithm has been developed for extracting the directional hydraulic transient pressure waves that travel along a pipeline in the downstream and upstream directions, respectively. Discrete incident transient waves, such as a single pulse or a step wave which are commonly used in 62
ADE	Chapter 2 transient-based pipeline fault detection, are used to excite a pipeline system and induce reflections from deteriorated pipe sections. The wave separation is achieved by analysing the pressure responses of the pipeline as measured by a proximity dual-sensor setup. The wave separation resolves the complexity of the superposition of travelling pressure waves in a pipeline, providing directional information of wave reflections and simplifying the wave forms. The key contributions of the research include: (1) the development of an experimental technique for estimating the transfer function between two sensors that is more practical than analytical estimation for real pipelines with parameter uncertainties; (2) the further development of the wave separation algorithm to enhance the accuracy for the separation of the relatively small wave reflections by removing the dependence of the relatively large incident wave; and (3) the verification of the wave separation technique by numerical and laboratory experiments. In the numerical simulations, a discrete pulse pressure wave is considered as the incident wave, which has not been studied previously for hydraulic transient wave separation in pipelines. Three thinner-walled pipe sections are placed in the numerical pipeline system, with two of them simulating deteriorated sections due to internal corrosion and one simulating a section with a lower pipe class. The wave separation algorithm has been successfully implemented, with the resultant directional reflection waves consistent with the predicted results. Experimental verification of the hydraulic transient wave separation algorithm has been conducted. A step transient pressure wave generated by a fast closure of a side-discharge valve is considered as the incident wave. The original 63
ADE	Chapter 3 Abstract Leak detection in complex pipeline systems is challenging due to complex wave reflections. This research proposes a new technique for leak detection in targeted pipe sections within complex water supply pipe systems using controlled hydraulic transient pressure waves and a two-source-four-sensor transient testing configuration. To “virtually” isolate a targeted pipe section for independent analysis, a two-source-four-sensor transient testing configuration is used to extract the transfer matrix of the targeted pipe section. Two pairs of pressure sensors are used to bracket the targeted pipe section by “virtually” isolating it, with the two sensors in each pair being in close proximity. Two transient pressure wave generators are used, which bracket the four sensors and the “virtually” isolated pipe section. It is found that the imaginary part of the difference between two elements in the transfer matrix is sensitive to leaks. The result should be zero if no leak is present, while a leak will introduce a sinusoidal pattern. The period and the magnitude of the pattern are related to the location and impedance of the leak, respectively. An algorithm is developed to extract the leak information, which is applicable to multiple leaks. Two numerical case studies are conducted to validate the new leak detection technique. Case 1 is on a single pipe system with two leaks and deteriorated pipe sections, and pulse pressure waves are used as the excitation. Case 2 is on a simple pipe network with one leak and pseudo-random binary signals are used as the excitation. The successful determination of the leak location and impedance proves the concept. Challenges in field applications are also discussed. 69
ADE	Chapter 3 3.1 Introduction Pressurized pipeline systems are used globally to transmit and distribute all types of fluids, such as water, gas and oil. Leakages in pipeline systems can cause economic loss and sometimes environmental hazards. Leakage in water distribution systems (WDSs) is a global issue, and the leakage rate ranges from about 10% in well-maintained WDSs (Beuken et al. 2006) to above 50% in poorly managed systems (Mutikanga et al. 2009). In Australia, every year an estimated 19,000 breaks in water transmission mains occur, resulting in the loss of 265 GL of potable water (Bureau of Meteorology 2016). This water loss is equivalent to the annual consumption of 1.5 million homes and represents a value over $700m. Leak detection in WDSs, however, is challenging due to the sheer size of the pipe network and the fact that most pipes are buried under ground. Acoustic correlation analysis is the most commonly used technique for leak detection in water pipelines at present (Li et al. 2015). Two acoustic sensors are attached to two separate fittings on a pipeline and record the vibration on the pipe fittings (using accelerometers) or the acoustic pressure in water (using hydrophones). Cross-correlation of the two measured signals can indicate whether there is a common acoustic source (a leak) in the pipe, and also the time difference for the acoustic wave to travel from the source to the two sensors (Muggleton et al. 2006). The time difference, together with the know distance and wave speed between the two sensors, can be used to calculate the leak location. The acoustic correlation-based leak detection techniques are relatively easy to implement since only passive listening is required. However, 71
ADE	Chapter 3 leak-induced acoustic waves are prone to interference from water network background noise and environmental noise, also the propagation is sensitive to the pipe material (Butterfield et al. 2018). An alternative is the hydraulic transient-based leak detection approach (Puust et al. 2010). Controlled hydraulic transient pressure waves can be generated in pipelines by transient wave generators. Usable devices include valves (Meniconi et al. 2011b; Shucksmith et al. 2012; Gong et al. 2016b), portable pressure tanks (Brunone et al. 2008), and spark plugs (Gong et al. 2018a). The incident wave typically has a magnitude of a few meters of pressure head, and propagates along the pipe under test at high speed (around 1200 m/s in metallic pipes). Wave reflections occur at physical discontinuities (e.g. a leak), and can be measured by pressure transducers. Over the past two decades, a number of transient-based leak detection techniques have been developed, and they can be generally allocated into the following categories: (1) techniques that analyse wave reflections (either from the raw data or pre-processed data) using principles of time-domain reflectometry (TDR) (Shucksmith et al. 2012; Nguyen et al. 2018); (2) techniques that analyse the frequency response function (FRF) of a pipe system (Covas et al. 2005; Lee et al. 2005b; Gong et al. 2013a); (3) techniques that focus on the damping of transient pressure responses in a pipeline system (Wang et al. 2002); and (4) inverse transient analysis (ITA)-based techniques that search for an optimal numerical pipe model whose response matches the pressure measurements (Kapelan et al. 2003; Covas and Ramos 2010; Capponi et al. 2017). The transient-based techniques are attractive because a single test can cover up to kilometres of pipe length, 72
ADE	Chapter 3 and the active testing approach can reveal other information such as blockages (Meniconi et al. 2013) and pipe wall condition (Gong et al. 2016c). Despite that many transient-based leak detection techniques have been proposed, applications in real water pipeline systems are limited. A significant challenge to all the transient-based techniques is the complexity of real water pipeline systems. For the TDR-based techniques, leak-induced reflections can be difficult to distinguish from other reflections, such as those from cross- connections and unknown wall thickness changes. The FRF of a single pipe system is more sensitive to leaks than extended wall thickness changes (Duan et al. 2011), therefore the FRF-based techniques are advantageous over the TDR-based techniques in detecting small leaks. However, most FRF-based techniques are only applicable to reservoir-pipeline-reservoir (R-P-R) or reservoir-pipeline-valve (R-P-V) systems. Duan (2016a) has recently extended the FRF-based leak detection to simple pipe systems with a branch or a loop. The conventional FRF-based approach is difficult to be further extended to more complex pipe systems, because the FRF considered in all previous studies is a representation of the overall system, and complex systems will produce FRFs that too complex to analyse. The transient-damping-based technique is also difficult to apply to complex pipe systems, in which the damping can be related to many factors (Nixon and Ghidaoui 2006). The ITA-based techniques require iterative parameter calibration using optimization algorithms. The process is computationally costly and not robust if the number of parameters to calibrate is large (Vítkovský et al. 2007). 73
ADE	Chapter 3 The current research presented here proposes a new frequency-domain technique for leak detection in targeted pipe sections. A key innovation of the new technique is the concept of utilizing a special transient pressure generation and sensing configuration, combined with custom developed signal processing algorithms, to “virtually” break any complex pipeline systems down to its simplest form – a single pipe section – for independent condition diagnosis. To the authors’ knowledge, this work is the first to utilize this approach for leak detection in targeted pipe sections embedded in complex systems. The proposed approach is opposite to the conventional research idea of gradually adapting the transient-based leak detection techniques developed for simple pipeline systems (e.g. reservoir-pipeline-valve or reservoir-pipeline-reservoir systems) to more complex pipe systems and networks (Ghazali et al. 2012; Duan 2016a; Capponi et al. 2017). The virtual isolation of a pipe section is achieved by a two-source-four-sensor transient testing strategy, which enables the extraction of the transfer matrix of a selected pipe section out of any complex pipe system. A transfer matrix of a pipe section is a full representation of the wave propagation characteristics as defined by the physical properties of the section (Wylie and Streeter 1993; Chaudhry 2014). This testing strategy was originally developed and used in the field of acoustic analysis of ducts (Munjal and Doige 1990; Salissou and Panneton 2010), and recently it was validated using a short water pipeline in the laboratory by Yamamoto et al. (2015) for studying the transfer matrix of resistance (orifices) and compliance (trapped air). Note that the focus of Yamamoto et al. (2015) was purely on the individual components, and not on long pipe sub-systems. The current research adapts this technique to the transfer 74
ADE	Chapter 3 matrix extraction of long sections in complex water pipe systems, with significantly more complex wave interaction phenomena. Different from the sine-sweep approach for system excitation as used in Yamamoto et al. (2015), the current research numerically tests pulse pressure waves that are easy to generate and pseudo-random binary signals that are tolerant to interference. A major contribution of the current research is the development of a new leak detection algorithm based on the analysis of the transfer matrix of a “virtually” isolated pipe section. This transfer matrix is related to the “virtually” isolated pipe section only, and is independent from any complexities of the rest of the pipe system (e.g. boundary conditions and other network connectivity). As a result, the extracted transfer matrix is much simpler than the transfer matrix of the overall pipe system, and the analysis is more straightforward. In contrast, conventional FRF-based leak detection algorithms use the transfer matrix of the overall pipe system to derive the frequency-domain pressure response at particular locations, which is system specific and can be difficult to analyse for complex pipe systems. In this research, it has been found that the imaginary part of the difference between two elements in the transfer matrix is sensitive to leaks. The result should be zero if no leak is present, while a leak will introduce a sinusoidal pattern. The period and the magnitude of the pattern are related to the location and impedance of the leak, respectively. Multiple leaks will introduce multiple sinusoidal patterns. An algorithm is developed to extract the leak information, including the number of leaks as well as their locations and impedance (which relates to the size of the leak). 75
ADE	Chapter 3 In the following, the technique for extracting the transfer matrix of a targeted pipe section and the new algorithm for leak detection of a “virtually” isolated pipe section are described. Two numerical case studies (a simple pipeline system and a simple pipe network) are conducted to validate the transfer matrix extraction technique and the proposed leak detection algorithm. Challenges in real world applications are also discussed. 3.2 Transfer matrix extraction for a targeted pipe section 3.2.1 Transfer matrix of a uniform pipe section For a uniform single pipe section, the relation between the two sets of pressure and flow as observed at the two ends of the section can be written as (Wylie and Streeter 1993; Chaudhry 2014)  1  Q  cosh(L)  sinh(L) Q    Z   (3.1) H  P  H D Z sinh(L) cosh(L)  U P where H and Q are complex pressure head and flow in the frequency domain; the footnotes D and U represent the downstream and the upstream boundary of the pipe section respectively; L is the length of the pipe; Z is the P characteristic impedance of the pipe section; and  is the propagation factor. The propagation factor is described by (Wylie and Streeter 1993; Chaudhry 2014) 76
ADE	Chapter 3 2  jgAR  (3.2) a where  is the angular frequency; j  1 is the imaginary unit; g is the gravitational acceleration; A is the cross sectional area of the pipe; a is the wave speed; and R is the frictional resistance term. For turbulent and laminar flows, R fQ / gDA2 and R32/ gD2A respectively, in which f is 0 the Darcy-Weisbach friction factor; Q is the steady-state flow rate; D is the 0 diameter of the pipe; and  is the kinematic viscosity of the fluid. The characteristic impedance is (Wylie and Streeter 1993; Chaudhry 2014) a2 Z  (3.3) P jgA 3.2.2 Two-source-four-sensor testing strategy for water pipes The proposed configuration for extracting the transfer matrix of a targeted pipeline section using the two-source-four-sensor strategy and hydraulic transient testing is illustrated in Figure 3.1. Two pairs of pressure transducers (T , T and T , T ) bracket the section of pipe under investigation. The distance A B C D between the two transducers in each pair, L for the distance between T and AB A T , and L for that between T and T , are recommended to be short B CD C D (recommended to be 2 m or less) in real pipelines, such that the transfer function of the short pipe reach can be calibrated or theoretically determined (Shi et al. 2017). Two transient pressure wave generators are used, with one on each side of the pipe section of interest. 77
ADE	Chapter 3 Transient Pressure transducer Transient generator G1 generator G2 e tdC e h t o t d e t c e n n o Ch t f o t r a p m a e r t s p u m e t s y s e p i p Ups𝑝 trT 𝐵− eA a mT B 𝑝 𝐵+ Seg PLm 1 i1 p L e ee n sa t ek c 1 ti o… n… o fL inea tS ek re L eNN g sNm + t+ 1 1e nt 𝑝 D𝐶−T o wC nsT t 𝑝 rD e𝐶+ a m metsys epip eh fo trap maertsnwo eht ot detcenno Figure 3.1 Test configuration for extracting the transfer matrix of a targeted pipe section with N leaks. The pipe section between transducer T and T can be considered as a linear- B C time-invariant (LTI) system. The directional travelling waves p  and p  B C which are travelling into the pipe section are considered as the input to the LTI system; while the waves p  and p  that are travelling out of the section are B C taken as the output. A pair of directional travelling pressure waves (e.g. p  B and p ) can be determined from the pressure waves (pressure perturbations) B as measured by two transducers in close proximity (e.g. p and p as the A B pressure perturbations measured by T and T ) and using a wave separation A B technique (Shi et al. 2017). Once the directional pressure waves at the two boundaries of a pipe section are obtained, the pipe section can be regarded as an independent system since the boundary conditions are entirely specified. As a result, two pairs of transducers enable the analysis of a specific section of pipe independently from the complexities of the rest of the pipeline system. 78
ADE	Chapter 3 3.2.3 Determination of the transfer matrix using pressure measurements For a pipe section with unknown conditions, the transfer matrix have four elements (U , U , U and U ) to be determined. Two independent 11 12 21 22 transient tests are needed to establish four equations to solve these four unknowns. This can be achieved by generating transient excitation from the two sides of the pipe section one at a time, and measure the pressure responses by the four transducers in each test. Based on Equation (3.1), the flowing matrix can be established Q Q  U U Q Q   C1 C2    11 12  B1 B2  (3.4) H H U U H H      C1 C2 21 22 B1 B2 where H , Q , H and Q are the complex pressure head and flow in the B1 B1 C1 C1 first transient test (using transient generator 1), andH , Q , H and Q B2 B2 C2 C2 are the parameters in the second test (using transient generator 2). The complex head parameters at the location of T (H and H ) and those at the location B B1 B2 of T (H and H ) can be readily obtained by transforming the measured C C1 C2 time-domain pressure perturbations ( p and p ) into the frequency domain. B C The complex flow parameters ( Q , Q , Q and Q ) are not directly B1 B2 C1 C2 measured but can be obtained from the directional pressure waves (Yamamoto et al. 2015). 79
ADE	Chapter 3 Note that more information on pipe transient flow determination using multiple pressure measurements can be found in the literature (Washio et al. 1996b; Kashima et al. 2013). Once the head and flow are all known, elements in the transfer matrix can be obtained by solving the matrix in Equation (3.4). 3.3 Leak detection for a targeted pipe section using transfer matrix 3.3.1 Transfer matrix for a pipe section with leaks For a uniform pipe section with N leaks, as depicted in Figure 3.1, the relationship between the two sets of pressure and flow as observed at the two boundaries can be can be written as Q Q   U   (3.10) H N H D U where U is the overall transfer matrix for the pipe section with N leaks. N Considering the effect of pipe wall friction is small for large diameter water pipelines and to highlight the leak-induced effect, the effect of friction is neglected in the following derivation but discussed later. The field matrix F i for a frictionless and uniform pipe segment i is given as (Chaudhry 2014)  L  j L  cos i  sin i       a  Z  a  F  c  (3.11) i  L  L   jZ sin  i  cos  i    c  a   a   81
ADE	Chapter 3 where Z a/gA and it is the characteristic impedance of the frictionless pipe; c and L is the length of the ith pipe segment. i The point matrixP for the ith leak is given as (Lee et al. 2005b; Gong et al. i 2013a)  1  1    P  Z (3.12) i  Li 0 1  where Z 2H /Q and it is the impedance of the ith leak, H is the steady- Li Li Li Li state head at the leak and Q is the steady-state discharge out of the leak. Li The overall transfer matrix U for the pipe section with N leaks can be N expressed by orderly multiplying the field matrices and point matrices from downstream to upstream and written as U U  U   11.N 12.N F P ...F PF (3.13) N U U N1 N 2 1 1   21.N 22.N where the footnote N denotes the number of leaks in the pipe section. Now considering a uniform pipe section with one leak, the overall transfer matrix U is 1 U U  U   11.1 12.1 FPF (3.14) 1 U U 2 1 1   21.1 22.1 82
ADE	Chapter 3 After substituting Equations (3.11) and (3.12) into Equation (3.14) and performing appropriate matrix operations, the analytical expressions of the transfer matrix elements are given as L jZ L jZ 12x L U cos   c sin   c sin L1  (3.15) 11.1  a  2Z  a  2Z  a  L1 L1 j L 1 L 1 12x L (3.16) U  sin   cos   cos L1  12.1 Z  a  2Z  a  2Z  a  c L1 L1 L Z 2 L Z 2 12x L (3.17) U jZ sin   c cos   c cos L1  21.1 c  a  2Z  a  2Z  a  L1 L1 L jZ L jZ 12x L U cos   c sin   c sin L1  (3.18) 22.1  a  2Z  a  2Z  a  L1 L1 where x is the dimensionless leak location, which is defined as the ratio of L1 the distance from the leak to the upstream end of the pipe to the total length of the pipe L . For the ith leak, x L L ...L /L. The transfer matrix Li 1 2 i U for a pipe section with N leaks can be derived following the same N procedure. 3.3.2 Extraction of the leak-induced feature The impact of a leak on the transfer matrix can be seen through comparing the transfer matrix of the pipe section with one leak [Equations (3.15) to (3.18)] with that of an intact pipe [Equation (3.11)]. In this research, one of the leak- induced features, the imaginary part of U U , is selected to determine the 22 11 83
ADE	Chapter 3 leak location and size. When there is no leak, U U is null since the two 22 11 elements should be identical according to Equations (3.1) and (3.11). For a pipe section with only one leak, the imaginary part of the difference between U [Equation (3.18)] and U [Equation (3.15)] is defined as T 22.1 11.1 1 and given as Z (12x )L T ImU U  c sin L1 (3.19) 1 22.1 11.1 Z   a   L1 where Im  gives the imaginary part of the parameter in the bracket. It can be seen from Equation (3.19) that T is a sinusoidal function that is related 1 to the leak impedance (which relates to the leak size) and the leak location (except for a leak at a normalized location of 0.5). The leak locations defines the period of the sinusoidal pattern and the leak impedance defines the amplitude of the pattern. This finding is similar to that observed from the pressure response of a reservoir-pipeline-valve (R-P-V) system with a leak (Lee et al. 2005b), however this sinusoidal function is different from the one observed in the previous work. The expression in Equation (3.19) is much simpler and independent from any boundary conditions. Using the same approach as outlined above, the leak-induced effects for a pipe system with two leaks, T , can be derived as 2 84
ADE	Chapter 3 Z (12x )L Z (12x )L T ImU U  c sin L1  c sin L2 (3.20 2 22.2 11.2 Z   a   Z   a   L1 L2 ) Equation (3.20) indicates that two leaks will introduce two sinusoidal patterns with different periods. For a pipe system with three leaks, the analytical expression of T is derived as 3 Z (12x )L T ImU U  c sin L1 3 22.3 11.3 Z   a   L1 (3.21) Z (12x )L Z (12x )L  c sin L2  c sin L3 T Z   a   Z   a   h L2 L3 where T is a higher order term h sin12x L/a   L1    sin12x L/a  Z Z Z   L2   T  c c c   (3.22) h 4Z L1Z L2Z L3 sin 12x L3L/a     sin 12x L12x L2 2x L3L/a  The ratio of the characteristic impedance of pipe and the impedance of the ith leak can be described as Z a C A c  d L (3.23) Z 2gH A Li L where C A / A is the normalized leak size. For small leaks (which are difficult d L to detect by conventional techniques and are the focus of this research), the impedance of the leak is much larger than the characteristic impedance of the 85
ADE	Chapter 3 pipe (i.e. the value of Z Z is much smaller than 1). Consequently, the value c Li of the higher order term T will be significantly smaller than the values of the h first three items in Equation (3.21) and negligible. For a pipe section with more than three leaks, the higher order term will be even smaller. As a result, the leak-induced effect on the transfer matrix of a pipe section with n leaks can be described as N Z 12x L T ImU U  c sin Li  (3.24) N 22.N 11.N Z a i1 Li   3.3.3 Determination of the leak location and size T in Equation (3.24) is a frequency domain signal with the x-axis being the N frequency and in the unit of Hz. If assuming the x-axis to be a time axis, the leak-induced signal T has a wave form equivalent to a superposition of N N sinusoidal waves. The period/frequency of each sinusoidal wave corresponds to the location of a leak, and the amplitude is related to the leak impedance. In other words, the frequency and amplitude of each sinusoidal wave in the T N signal can be used to determine the location and impedance (size) of a leak. The frequency and amplitude information of the N sinusoidal waves can be extracted by applying the Fourier transform to the T signal (i.e. treat it like a N time-domain signal) and analysing the resultant signal T . Since the leak- N induced signals in T are sinusoidal waves, based on the theory of the discrete N Fourier transform (Oppenheim et al. 1997), each leak will be represented by a 86
ADE	Chapter 3 spike in the imaginary part of T . If the normalized leak location is in the range N of (0, 0.5), the corresponding spike in the imaginary part of T will be negative N in value; if the normalized leak location is in the range of (0.5, 1), the corresponding spike will be positive in value. As a result, the location of the ith leak is determined by 1   F a x Li  2Sgn Im T N(F Pi)   2P Li   (3.25) where F is the “frequency” that corresponds to the ith peak in the imaginary Pi part of T , T (F ) is the complex value at the peak frequency, and Sgn  N N Pi assesses the sign of the parameter in the bracket. The ratio of the pipe characteristic impedance to the impedance of the ith leak is determined by Z   Zc 2Abs T N(F Pi) (3.26) Li where Abs  gives the absolute value of the parameter in the bracket. The effective leak size can be determined by substituting Equation (3.26) into Equation (3.23) and performing appropriate mathematical operations, with the final expression being A 2gH   C A 2Abs T (F ) L (3.27) d L N Pi a 87
ADE	Chapter 3 3.4 Numerical simulations Two numerical case studies are conducted to validate the proposed targeted leak detection technique. The system in Case 1 is a transmission main and that in Case 2 is a water distribution network. 3.4.1 Case 1: A single pipe with two leaks System information The layout of the pipeline system studied in Case 1 is given in Figure 3.2. The system is an R-P-V system with two leaks and two deteriorated pipe sections (e.g. sections with extended corrosion). The pipe deterioration is represented by a reduction in wave speed. The pipe section of interest (the targeted pipe section) is the section between T and T . The length information is given in Figure 3.2 B C and other system parameters are summarized in Table 3.1. The normalized leak locations are x = 0.2 and x = 0.7, respectively. The ratios of the leak L1 L2 impedance to the characteristic impedance of pipe are Z Z = 0.00415 and c L1 Z Z = 0.0116, respectively. c L2 Reservoir Deteriorated Deteriorated section 1 G 1 T A T B Leak 1 Leak 2 T C T D G 2 section 2 Valve Length 1800 115 120 1.21 .2 120 300 180 1.21 .2 180 22 1800 (m) L 1 L 2 L Figure 3.2 Layout of the single pipeline system in Case 1. 88
ADE	Chapter 3 Table 3.1 System information for Case 1. Parameter Value Reservoir head, H 60 m r Pipe internal diameter, D 500 mm Effective opening area of Leak 1, C A 22 mm2 d1 L1 Effective opening area of Leak 2, C A 63 mm2 d2 L2 Steady-state flow through valve, Q 0.2 m3/s 0 Steady-state flow through Leak 1, Q 0.75 L/s L1 Steady-state flow through Leak 2, Q 2.08 L/s L1 Wave speed in intact pipe, a 1200 m/s 0 Wave speed in deteriorated section 1, a 1150 m/s 1 Wave speed in deteriorated section 2, a 1100 m/s 2 Darcy-Weisbach friction factor, f 0.015 Normalised location of Leak 1, x 0.2 L1 Normalised location of Leak 2, x 0.7 L2 Impedance ratio of pipe to Leak 1, Z /Z 0.00415 c L1 Impedance ratio of pipe to Leak 2, Z /Z 0.0116 c L2 Pressure response The method of characteristics (MOC) (Wylie and Streeter 1993; Chaudhry 2014) is used to simulate the transient response of the pipeline system. Steady friction is considered to evaluate its impact on the leak detection. The time step used is 0.0001 s. Two transient tests are simulated: in the first test, a pulse pressure 89
ADE	Chapter 3 wave with a duration of 10 ms and a peak size about 6 m is generated at G (by 1 opening and then closing a side-discharge valve); and in the second test, a pulse pressure wave with the same characteristics is generated at G . The pressure 2 traces at T and T as obtained from the first test are shown in Figure 3.3. The A D standing pressure at T is lower than that at T because of the effect of steady D A friction. The two large pulses in the T and T traces are the incident pulse A D wave, arriving at T and T in sequence. A number of small pulses can be seen A D in both traces, and they are reflections from the two leaks and the two deteriorated pipe sections. Due to the complexity introduced by the deteriorated pipe sections, it is difficult to identify the leaks from the pressure responses even if the reflections are clear. Reflections from leaks and deteriorated sections Incident pulse wave Figure 3.3 Pressure responses at T and T as obtained from transient test 1 A D (using generator G1) in Case 1. Transfer matrix extraction The pressure measurements at T to T are transformed to the frequency A D domain by the Fourier transform after the steady-state head being offset from 90
ADE	Chapter 3 the original measurement. The calculations outlined in previous sections are then conducted to obtain the transfer matrix for the pipe section between T and B T . The imaginary part of the numerically obtained transfer matrix element U C 22 is shown in Figure 3.4, together with the theoretical counterpart for the same pipe section with two leaks and that for the same pipe section without any leak (only the results up to 30 Hz are shown for clarity). The theoretical results are calculated using Equation (3.13) with the friction effect neglected. It can be seen from Figure 3.4 that the numerically determined ImU  (solid 22 line) is highly consistent with the theoretical result for the same pipe section with two leaks (dotted line), except for the small error close to the zero frequency. In contrast, the theoretical ImU  for the same pipe section but 22 with no leaks is quite different because it only has one sinusoidal component that is related to the fundamental frequency of the pipe section [refer to Equation (3.11)]. 91
ADE	Chapter 3 Figure 3.5 Imaginary part of (U – U ) as obtained from numerical 22 11 simulations and the transfer matrix extraction technique for the pipe section with two leaks in Case 1 (solid line), and the theoretical result for the same pipe section with two leaks (dotted line). Leak detection Leak detection is conducted by analysing the numerically obtained ImU U  using the technique outline in Equations (3.25) and (3.26), and 22 11 the results are shown in Figure 6. The two distinctive pikes indicate that there are two leaks in the pipe section of interest. The normalized locations are determined as x = 0.20 and x = 0.70, respectively, as shown by the x-axis, L1 L2 and the values of the impedance ratio are Z Z = 0.00427 and Z Z = c L1 c L2 0.0119, respectively, according to the size of the two spikes. The results are highly consistent with the theoretical values as shown in Table 3.1. The successful detection has validated the effectiveness of the proposed targeted leak detection technique. 93

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