美赛MCM A题M奖获奖论文Word格式.docx

美赛MCM A题M奖获奖论文Word格式.docx

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Forofficeuseonly

F1 F2 F3 F4

2017

MathematicalContestinModeling（MCM/ICM）SummarySheet

TheKaribaDamontheZambeziRiverisoneofthelargerdamsinAfrica.Itsconstructionwascontroversial,anda2015reportbytheInstituteofRiskManagementofSouthAfricaincludedawarningthatthedamisindireneedofmaintenance.AnumberofoptionsareavailabletotheZambeziRiverAuthority（ZRA）thatmightaddressthesituation.

Itisassumedthatthedesignparameters,siteselectionandconstructionofthedam（damgroup）afterrepair,reconstructionandreconstructionareinaccordancewiththerequirementsofwaterconservancyandhydropowerprojects.Thenumberandlocationofthedamgroupsareinlinewiththerequirementsofwaterconservancyandhydropowerprojects.Therefore,theresearchproblemsare:

a）Comparingthecomprehensivebenefitsofthedamsafterrestoration,reconstructionandreconstruction,establishingthemathematicalmodel,andexpoundingthecostsandbenefitsofeachcase;

b）Reconstructingthedams（rebuildingtheexistingdams10-20Alittledam）,inthedrought,heavyrain,moderaterain,lightrain,whenthewaterstorage,flooddischargetoensuretheprobabilityofdambreakthesmallest,thatis,damgroupriskdecisionoptimization.

Forthefirstproblem,thispaperchoosestheoptimalcostandbenefitplanforKalibaDamrestorationworkbyestablishingtheeconomicbenefitmodel,theeconomiclossmodel,thecomprehensivebenefitmodel,theabsolutereturnmodelafterrestorationandtheinvestmentpaybackperiodmodel.

Thecost-effectivenessofthethreeprogramswasevaluatedindetailaccordingtotheresultsofthemathematicalmodel.Option1,therestorationofthedamcanextendtheusefullifeof11years.Whiletheone-timecostofaccesstotheinvestmentrecoveryperiodof3.2years.Option2,100yearsafterthedesignofthereconstruction,theinvestmentrecoveryperiodof65.2years.Option3,afterrebuildingforthedamgroup,designedtouse100years,theinvestmentrecoveryperiodof47.8years.Threeprogramshavebetterbenefits,ofwhichthebestoverallbenefitsofthethreeprograms.

Fortheproblem1,thispaperusestheRlanguagesoftwaretodrawtheheatmaptovalidatetheconclusion.UsingtheMatlabsoftwaretobuildtheMSEmodel,theerroranalysisoftheaboveresultsisfurtherprovedtobethebest.

Forthesecondproblem,weuseMATLABandSPSSMODELERsoftwaretoestablishAuthorTopicModel,BPnetworkandLS-SVMmodeltovalidatetheLDAmodel,anduseMSEandABSmethodtoanalyzetheerrorofLDAmodel.TheresultsshowthattheLDAmodelisaccurateandreliable,andtheerrorissmall.Itfurther

showsthatthemathematicalmodelestablishedinthispaperhasacertaingeneralizationability.

Keywords:

Thezambeziriver ；

Karibadam;

Waterconservancy;

reconstruction

I.Introduction

InordertoindicatemanagingtheZambeziriver,thefollowingbackgroundisworthmentioning.

1.1ZambeziRiver

TheZambezi（alsospelledZambezeandZambesi）isthefourth-longestriverinAfrica,thelongesteast-flowingriverinAfricaandthelargestflowingintotheIndianOceanfromAfrica.Theareaofitsbasinis1,390,000squarekilometres（540,000sqmi）,[1][2]slightlylessthanhalfthatoftheNile.The2,574-kilometre-longriver（1,599mi）risesinZambiaandflowsthrougheasternAngola,alongtheeasternborderofNamibiaandthenorthernborderofBotswana,thenalongtheborderbetweenZambiaandZimbabwetoMozambique,whereitcrossesthecountrytoemptyintotheIndianOcean.

FromVictoriatotheFallsfortheupperreachesofVictoriaFalls,1287kmlong;

fromVictoriaFallstotheterritoryofMozambiqueKabulaBassareservoirforthemiddle,869kmlong;

CabraBassaisthefollowingdownstream,579kmlong.

Zambeziaverageslopeof0.4‰,runoffof223.2billioncubicmeters,secondonlytotheCongoRiver,rankedsecondinAfrica.

1.2KaribaDamConstruction

ThedoublecurvatureconcretearchdamwasdesignedbyCoyneetBellierandconstructedbetween1955and1959byImpresitofItaly[2]atacostof$135,000,000forthefirststagewithonlytheKaribaSouthpowercavern.FinalconstructionandtheadditionoftheKaribaNorthPowercavernbyMitchellConstruction[3]wasnotcompleteduntil1977duetolargelypoliticalproblemsforatotalcostof$480,000,000.Duringconstruction,86menlosttheirlives.[2][4]

ThegrasslandinthefloodplainsaroundtheZambeziRiverreliesonannualfloodsandseasonalrainfall,whilerainfallintheregionisricherthanintheverydrywest.FloodsandvegetationexpansionduringtherainyseasonislimitedinthenorthoftheZambeziRiver,indicatinganincreaseinthenorthernmarginofthefloodplain.Thesevegetationprovidefoodfornumerousfishandanimalspecies.

1.3KaribaDam

KaribahydropowerstationislocatedintheAfricanZambiaandZimbabweatthejunctionoftheZambezi（Zambezi）RiverKaribaGorge,290kmawayfromSolihey,theprojectismainlyforpowergeneration.

ThedamsiteiscomposedofPrecambrianrocks.Thewholebed,theleftbankandtherightbankofthelowertwo-thirdsoftheheightbelowaremetamorphicbiotitegneiss,grayandwhitesurfacewithwhitestripes,foliationshowingabend.Rockoutcropsintheriverbedarehard,finefractures,thedepthofrockerosionbetweenthetwosidesof4.5~6m,susceptibletoweathering.Theupperpartoftherightbankishardbrownquartzite,withfracturesdeveloped,weatheredbiotitegneissintrusions,andgraniticgranitesappearingveins.Gneissisalmostentirelybiotitelensbody,duetothedistributionisnotwide,theundergroundprojectwillnothaveasignificantimpact.

OnNovember11,2013ItwasannouncedbyZimbabwe'

sFinanceMinister,PatrickChinamasathatcapacityattheZimbabwean（South）Karibahydropowerstationwouldbeincreasedby300megawatts.Thecostofupgradingthefacilityhasbeensupportedbya$319mloanfromChina.ThedealisaclearexampleofZimbabwe'

s"

LookEast"

policywhichwasadoptedafterfallingoutwithWesternpowers.[7]ConstructionontheKaribaSouthexpansionbeganinmid-2014andisexpecttobecompletein2019.[8]

ThenorthstationbelongingtoZambiahasbeeninoperationsince1976,andhasfourgeneratorsof150megawatts（200,000hp）eachforatotalof600megawatts（800,000hp）;

worktoexpandthiscapacitybyanadditional360megawatts（480,000hp）to960megawatts（1,290,000hp）wascompletedinDecember2013.Twoadditional180MWgeneratorswereadded.[9][10][11]

Figure1.1zambeziriverkaribadam

1.4KaribaDamRecentactivity

Onthe6thofFebruary2008,theBBCreportedthatheavyrainmightleadtoareleaseofwaterfromthedam,whichwouldforce50,000peopledownstreamtoevacuate.RisinglevelsledtotheopeningofthefloodgatesinMarch2010,requiringtheevacuationof130,000peoplewholivedinthefloodplain,andcausingconcernsthatfloodingmayspreadtonearbyareas.

InJanuary2016itwasreportedthatwaterlevelsatthedamhaddroppedto12%ofcapacity.Levelsfellby5.58metres（18.3ft）,whichisjust1.75metres（5ft9in）abovetheminimumoperatinglevelforhydropower.Lowrainfallsandoveruseofthewaterbythepowerplantshaveleftthereservoirnearempty,raisingtheprospectthatbothZimbabweandZambiawillfacewatershortages.

1.5Theremovalofthedam,restorationandreconstruction

TheKaribaDamontheZambeziRiverisoneofthelargerdamsinAfrica.Itsconstructionwascontroversial,anda2015reportbytheInstituteofRiskManagementofSouthAfricaincludedawarningthatthedamisindireneedofmaintenance.AnumberofoptionsareavailabletotheZambeziRiverAuthority（ZRA）thatmightaddressthesituation.ThreeoptionsinparticularareofinteresttoZRA:

（Option1）RepairingtheexistingKaribaDam,（Option2）Rebuildingthe

existingKaribaDam,or（Option3）RemovingtheKaribaDamandreplacingitwithaseriesoftentotwentysmallerdamsalongtheZambeziRiver.

II.TheDescriptionoftheProblem

2.1ModelAssumptions

1）Assumingthatthedesignparameters,siteselectionandconstructionofthedam（damgroup）afterrestoration,reconstructionandreconstructionareinaccordancewiththerequirementsofwaterconservancyandhydropowerprojects.Thenumberandlocationofthedamgroupsareinlinewiththerequirementsofwaterconservancyandhydropowerprojects.Therefore,theresearchproblemsare:

b）Reconstructingthedams（rebuildingtheexistingdams10-20Alittledam）,inthecaseofdrought,heavyrain,moderaterainandlightrain,whentheimpoundmentandflooddischargecanensuretheprobabilityofdambreak,theriskdecisionofdamgroupisoptimized.

2）Theriskofflooddischargeinemergenciesishigh,andthisstudydoesnotconsiderthesituationofemergencyflooddischarge.

3）Assumingtheecologicalrestorationofthewater-leveldrawdownareaafterthedemolitionofthedam,constructionwastetransportationandsedimentdepositionareideal.

4）Assumingthattheprobabilityofdambreakafterrainfalltoacertainwaterlevelisfixed.

5）Assumethattherainfallisacontinuousvariable,obeyingthenormaldistribution.

6）Assumingthatthedamgroupisaseriesmodel,donotconsidertheparallelandmixed

2.2Equirement 1.

Theannualeconomicbenefitsofthedamaremadeupofanumberofbenefits.Theseincludepowerg