Chapter ­ ­ GG G G G. G G. LG L G. L. ­ In any lumped network, for any of its loops and at any time, 2 6 1 the algebraic sum of the branch voltages around the loop is zero. 5 7 8 G 4 3 ¥ nodes b\ n 1 2 3 1 –1 0 0 2 0 –1 0 3 0 0 0 4 0 0 –1 5 0 1 –1 6 1 –1 0 7 0 1 0 4 0 0 –1 1 0 0 1 ­ 3 2 1 4 4 2 1 3 0 ¥ ¥ nodes b\ n 1 2 3 4 1 –1 0 0 0 2 0 –1 0 0 3 0 0 0 –1 4 0 0 –1 1 1 2 - 1 2 3 4 5 6 7 - - - - 1 - - 2 3 4 = = = = = = = ...(.) = if the bus to reference bus and is orientedinthesamedirection. ¥ ¥ t\n 1 2 3 4 nodes 1 2 3 –1 0 0 0 –1 0 0 0 –1 0 0 –1 4 0 0 –1 0 - - - - - - - - = - ­ ¥ 7 L3 1 6 L2 4 5 2 4 3 L1 2 1 3 basicloopandisorientedinthesamedirection. = ,ifthe basicloopandisorientedintheopposite direction. = 0,ifthe basicloop. ThematrixfortheexampleofFig..0isgivenby Loops L1 L 2 1 0 1 2 1 –1 3 –1 0 4 –1 0 5 1 0 6 0 1 7 0 0 b\ l L3 0 1 –1 0 0 0 1 ¥ ¥ ­ Loop ...(. ) - -- - ¥ b/ l 1 2 3 4 5 6 7 Open loops Basic loops L 4 L5 L6 L7 L1 L2 L3 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 1 –1 1 0 –1 0 –1 1 –1 0 0 0 1 0 0 0 0 1 0 0 0 0 1 V1 + i1 – L1 M L2 + i2 V2 – + ­ + COMPUTER TECHNIQUES AND MODELS IN POWER SYSTEMS andaretheselfinductancesofthetwocoilsand =+ =+ ...(.6) Itisimportanttonotethat(.6)doesnotdependontheconnectivitybetweenthetwocoils andiswrittentakingintoaccountonlytheindividualelementcurrentsandvoltages.Wecancall thetwocoilsasprimitiveelementsandtogethertheyformaprimitivenetwork. jsk p ik + – vsk q E + Vk – Geer prmtve eemet. ­ s 3 NETWORK TOPOLOGY ik – E + ek vk + – Impedce form of the prmtve eemet. + 06 0 0 3 .7 ­ ¥ COMPUTER TECHNIQUES AND MODELS IN POWER SYSTEMS jk p E ik q yk vk + – Admttce form of the prmtve eemet. ­ - ­ ­ 2.6 ­ - - SINGULAR TRANSFORMATIONS 2.6.1 Bus Admittance and Bus Impedance Matrix = = ­ ­ ¥ 3 3 4 2 5 1 2 1 4 ­ ­ ­ ­ ­ - - - - - - - - - - - - - - - ­ 1 2 3 4 1 y10 y20 2 Z12 Z24 Z13 Z23 Z34 3 4 y30 y40 + + + + + + + + + + + + + + + ­ ­ ­ ­ 3 4 1 2 Xg1 Xg2 ­ ­ + ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ 3 3 4 CS3 2 1 1 CS1 4 Tree branch Link 5 CS2 2 - - - - - - - - ­ ­ ­­ ­ - - - - - 3 2 5 L2 1 L1 4 - - ­ - - ­ ­ 4 5 4 2 2 1 1 3 3 7 6 0 ¥ ¥ ¥ ¥ 4 3 1 2 - - - - - - - - ­ - - ­ jk ik ik v= Fictitious node Link 0 Fictitious tree branch ­ ­ + = + + + + = + ­ = 1 0 = = 1 1 = ­ = = ­ 3 2 4 = ­= ­= ­= ­= = ­ = = ­ ­ ­ ­ = Thus, = ­ 1= l] (.6)isanon-singulartransformationof[­ ...(.65) ­ ­ = = = ­ ­ ­ ­ 2.7.6 Power Invariance of Singular Transformations ­ Branch Admittance Matrix \ ­ ­ ­ ­ ­ ­ 2.8.2 Singular Transformations 2.8.3 Non-singular Transformations 2.8.4 Current and Voltage Relationships - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -1 - = -1 -1 - -1 - = - - - - -1 - -1 -1 -1 ­ ­ 1 ­ ­ oo(looimedancematrix) andoo(looadmittancematrix),whichcanbeformedfromtherimitivematricesbysingular transformation.Allthecoefficientmatricescanalsobeobtainedfromnon-singulartransformationsbyusingaugmentedloo-incidencematrixandaugmentedcutsetincidencematrix. 1 T
