电力系统分析英文课件OptimalDispatch.pdf

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1、ECE421/599ElectricEnergySystems7–OptimalDispatchofGenerationInstructor:KaiSunFall20141Background•Inapracticalpowersystem,thecostsofgeneratinganddeliveringelectricityfrompowerplantsaredifferent(duetofuelcostsanddistancestoloadcenters)•Undernormalconditions,thesystemgenerationcap

2、acityismorethanthetotalloaddemandandlosses.•Thus,thereisroomtoschedulegenerationwithincapacitylimits–Minimizingacostfunctionthatrepresents,e.g.•Operatingcosts•Transmissionlosses•Systemreliabilityimpacts•ThisiscalledOptimalPowerFlow(OPF)problem•AtypicalproblemistheEconomicDispat

3、ch(ED)ofrealpowergeneration2IntroductionofNonlinearFunctionOptimization•Unconstrainedparameteroptimization•Constrainedparameteroptimization–Equalityconstraints–Inequalityconstraints3UnconstrainedparameteroptimizationMinimizecostfunctionfxx(,,,)x12n1.Solvealllocalminimasatisfyi

4、ngtwoconditions(necessary&sufficient)Condition-1:Gradientvector∂∂ff∂fStationarypoint∇=f(,,,)=0∂∂xx∂x(wherefisflatinall12ndirections)Condition-2:HessianmatrixHispositivedefiniteLocalminimum(apuresourcein∇fvectorfield)2.Findtheglobalminimumfromalllocalminima4f(x,y)=−(cos2x+cos2y

5、)25•Minimizef(x,y)=x2+y2y∂∂ff∇=f(,)(2,2)0=xy=∂∂xy∇fxy=0,=00x22∂∂ff2∂x∂∂xy20H==∂2f02∂∂yx2∂y6ParameterOptimizationwithEqualityConstraintsMinimizefxx(,,,)x12nSubjecttogxx(,,,)0x=kK=1,2,,kn12•IntroduceLagrangeMultipliersλ~λ1KKLf=+∑λkkgi=1•Necessaryconditionsfor

6、thelocalminimaofL(alsonecessaryfortheoriginalproblem)K∂∂Lf∂gk=+=∑λk0∂∂xxiii=1∂xi∂L=g=0k∂λk7•Minimizef(x,y)=x2+y2Subjectto(x-8)2+(y-6)2=25→g(x,y)=(x-8)2+(y-6)2-25=0y2222∇fLxy=++−+−−λ[(x8)(y6)25]∂L=+−=2xxλ(216)0∂x6∂L=+−=2yyλ(212)0∂y08x∂L22=−+−−=(xy8)(6)250∂λSolutions(fromtheN-Rme

7、thod):λ=1,x=4andy=3(f=25)λ=3,x=12andy=9(f=225)8ParameterOptimizationwithInequalityConstraintsMinimizefxx(,,,)12xnSubjectto:gxx(,,,)0x=kK=1,2,,kn12uxx(,,,)0x≤jm=1,2,,jn12•IntroduceLagrangeMultipliersλ~λandµ~µ1K1mKmLf=++∑∑λµkkgjjukj=11=•Necessaryconditionsforthelocalminimaof

8、L∂∂LfKm∂g∂ukj=++=∑∑λµkj0in=1,,∂∂xxiiij=11∂xi=∂xi∂L=g=