电力系统分析英文课件OptimalDispatch

1Instructor:KaiSunFall2014ECE421/599ElectricEnergySystems7–OptimalDispatchofGeneration2Background•Inapracticalpowersystem,thecostsofgeneratinganddeliveringelectricityfrompowerplantsaredifferent(duetofuelcostsanddistancestoloadcenters)•Undernormalconditions,thesystemgenerationcapacityismorethanthetotalloaddemandandlosses.•Thus,thereisroomtoschedulegenerationwithincapacitylimits–Minimizingacostfunctionthatrepresents,e.g.•Operatingcosts•Transmissionlosses•Systemreliabilityimpacts•ThisiscalledOptimalPowerFlow(OPF)problem•AtypicalproblemistheEconomicDispatch(ED)ofrealpowergeneration3IntroductionofNonlinearFunctionOptimization•Unconstrainedparameteroptimization•Constrainedparameteroptimization–Equalityconstraints–Inequalityconstraints4UnconstrainedparameteroptimizationMinimizecostfunction1.Solvealllocalminimasatisfyingtwoconditions(necessary&sufficient)Condition-1:GradientvectorCondition-2:HessianmatrixHispositivedefinite2.Findtheglobalminimumfromalllocalminima12(,,,)nfxxx12(,,,)nffffxxx∂∂∂∇==∂∂∂0Stationarypoint(wherefisflatinalldirections)Localminimum(apuresourcein∇fvectorfield)5f(x,y)=−(cos2x+cos2y)26•Minimizef(x,y)=x2+y2(,)(2,2)0fffxyxy∂∂∇===∂∂222222002ffxxyHfyxy∂∂∂∂∂==∂∂∂∂0,0xy==xy0∇f7ParameterOptimizationwithEqualityConstraintsMinimizeSubjectto•IntroduceLagrangeMultipliersλ1~λK•NecessaryconditionsforthelocalminimaofL(alsonecessaryfortheoriginalproblem)12(,,,)nfxxx12(,,,)0kngxxx=1,2,,kK=1KkkiLfgλ==+∑10KkkiiiigLfxxxλ=∂∂∂=+=∂∂∂∑0kkLgλ∂==∂8∇f•Minimizef(x,y)=x2+y2Subjectto(x-8)2+(y-6)2=25Solutions(fromtheN-Rmethod):λ=1,x=4andy=3(f=25)λ=3,x=12andy=9(f=225)xy0862222[(8)(6)25]Lxyxyλ=++−+−−2(216)0Lxxxλ∂=+−=∂2(212)0Lyyyλ∂=+−=∂22(8)(6)250Lxyλ∂=−+−−=∂→g(x,y)=(x-8)2+(y-6)2-25=09ParameterOptimizationwithInequalityConstraintsMinimizeSubjectto:•IntroduceLagrangeMultipliersλ1~λKandµ1~µm•NecessaryconditionsforthelocalminimaofL12(,,,)nfxxx12(,,,)0kngxxx=1,2,,kK=0kkLgλ∂==∂12(,,,)0jnuxxx≤1,2,,jm=11KmkkjjkjLfguλµ===++∑∑1,,in=1,2,,kK=0jjLuµ∂=≤∂1,,jm=0jjuµ=0jµ≥Kuhn-Tucker(KKT)necessarycondition110KmjkkjijiiiiugLfxxxxλµ==∂∂∂∂=++=∂∂∂∂∑∑10•Minimizef(x,y)=x2+y2Subjectto(x-8)2+(y-6)2=25→g(x,y)=(x-8)2+(y-6)2-25=0Solutions:µ=0,λ=3,x=12andy=9(f=225)µ=5.6,λ=-0.2,x=5andy=2(f=29)µ=12,λ=-1.8,x=3andy=6(f=33)xy0862222[(8)(6)25](122)Lxyxyxyλµ=++−+−−+−−2(216)20Lxxxλµ∂=+−−=∂2(212)0Lyyyλµ∂=+−−=∂22(8)(6)250Lxyλ∂=−+−−=∂212xy+≥∇f(,)1220uxyxy=−−≤0,0,0jjjLuµµµ∂≤∂=≥12200or12200LxyLxyµµµµ∂=−−=∂∂=−−=∂→11OperatingCostofaThermalPlant•Fuel-costcurveofagenerator(representedbyaquadraticfunctionofrealpower)•Incrementalfuel-costcurve:2iiiiiiCPPαβγ=++2iiiiiidCPdPλγβ==+12ArealcaseGenIDPRIORFUELCO($/MBtu)PMAX(MW)PMIN(WM)HEMIN(MBtu/hr)X1(MW)Y1(Btu/kWh)X2(MW)Y2(Btu/kWh)X3(Btu/kWh)Y3(Btu/kWh)A101.9123065532658760176950726010072B200.5391065042550850175919810610341Btu/h=Xi×Yi×1000$/h=Btu/h×$/MBtu/1000,000$/MWh=Yi/1000×$/MBtu05010015020025030002468101214161820P(MW)Lambda($/MWh)0501001502002503000100020003000400050006000P(MW)Cost($/h)13EDNeglectingLossesandNoGeneratorLimitsIftransmissionlinelossesareneglected,minimizethetotalproductioncost:subjectto•ApplytheLagrangemultipliermethod(ng+1unknownstosolve):1gntiiCC==∑21niiiiiiPPαβγ==++∑1gniDiPP==∑1()gntDiiLCPPλ==+−∑00iLPLλ∂=∂∂=∂0tiCPλ∂−=∂2tiiiiiiCdCPPdPβγλ∂==+=∂1,,gin=1gniDiPP==∑12gniiiDPλβγ==−∑11212ggniDiiniiPβγλγ==+=∑∑2iiiPλβγ−=AllplantsmustoperateatequalincrementalcostSolvePi14Example7.4Thefuel-costfunctionsforthreethermalplantsareC1~C2in$/h.P1,P2andP3areinMW.PD=800MW.Neglectinglinelossesandgeneratorlimits,findtheoptimaldispatchandthetotalcostin$/h2111222223335005.30.0044005.50.0062005.80.009CPPCPPCPP=++=++=++11212ggniDiiniiPβγλγ==+=∑∑5.35.55.88000.0080.0120.0181110.0080.0120.018+++=++8001443.05558.5$/MWh263.8889+==1238.55.3400.00002(0.004)8.55.5250.00002(0.006)8.55.8150.00002(0.009)PPP−==−==−==11115.30.008dCPdPλ==+22225.50.012dCPdPλ==+33335.80.018dCPdPλ==+6682.5$/htC=2iiiPλβγ−=Equalincrementalcostλ15SolvingλbytheN-RMethod•Forageneralcase:112ggnniiDiiiPPλβγ==−==∑∑()()()()()()kkkDdffPdλλλλ+∆≈1()gniDifPPλ===∑()igPλ=()()()()1()()()()()()()()()gnkkkDikiDkkkiPPPfPdfdfdPdddλλλλλλλ=−−∆∆===∑∑(1)()()kkkλλλ+=+∆(1)()||kkλλε+−≤until1605010015020025030035040045055.566.577.588.599.51010.511P(MW)$/MWhApplytheR-NMethodinExample7.4(1)6.0λ=(1)16.05.387.50002(0.004)P−==()()2kkiiiPλβγ−=(1)2(1)36.05.541.66672(0.006)6.05.811.11112(0.009)PP−==−==(1)800(87.541.666711.1111)659.7222P∆=−++=(1)659.72221112(0.004)2(0.006)2(0.009)659.72222.5263.8888λ∆=++==(2)1(2)2(2)3(2)8.55.3400.00002(0.004)8.55.5250.00002(0.006)8.55.8150.00002(0.009)800(400250150)0.0PPPP−==−==−==∆=−++=()()()()1()2kkkkiiPPdPdλλγ∆∆∆==∑∑(2)6.02.58.5λ=+=6682.5$/htC=17EDNeglectingLossesbutIncludingGeneratorLimits•Consideringthemaximum(byrating)andminimum(forstability)generationlimits,MinimizeSubjectto(min)(max)1,2,,iiigPPPin≤≤=，1gntiiCC==∑21niiiiiiPPαβγ==++∑1gniDiPP==∑iiiiiiiidCdPdCdPdCdPλλµλλγλ==−≤=+≥(max)(min)11()[()()]ggnntDiiiiiiiiiLCPPPPPPλµγ===+−+−+−∑∑00iLPLλ∂=∂∂=∂0iiiiCPλµγ∂−+−=∂1gniDiPP==∑00iiLLµγ∂≤∂∂≤∂(min)(max)1,2,,iiigPPPin≤≤=,(max)(min)()0,0()