lecture3

lecture3 - Network ￿ ￿ ￿ Nodes represent substation...

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Unformatted text preview: Network ￿ ￿ ￿ Nodes represent substation bus bars Branches represent transmission lines and transformers Injected currents are the flows from generator and loads Used to form the network model of an interconnected p o w e r s y s te m Iinj = Ybus ⋅ Vnode Ik Vk The matrix equation for relating the nodal voltages to the currents that flow into and out of a network using the admittance values of circuit branches Power Systems I ￿ ￿ The Bus Admittance Matrix ￿ ￿ 1 1 yij = = zij rij + j xij impedances are converted to admittances I k −inj = yk 0 Vk + yk 1 (Vk − V1 ) + yk 2 (Vk − V2 ) + ￿ + ykn (Vk − Vn ) form the nodal solution based upon Kirchhoff’s current law Constructing the Bus Admittance Matrix (or the Y bus matrix) Power Systems I ￿ The Bus Admittance Matrix line 23 z = j0.2 line 34 z = j0.08 line 12 z = j0.4 Power Systems I 2 generator 2 z = j0.8 4 Network Diagram 3 line 13 z = j0.2 1 generator 1 z = j1.0 j0.4 j0.08 V2 j0.2 4 Impedance Diagram 3 j0.2 2 j1.0 1 j0.8 V1 Matrix Formation Example y34 = -j12.5 4 3 y12 = -j2.5 y13= -j5 y23= -j5 y20= -j1.25 y10= -j1.0 Power Systems I Admittance Diagram 1 I1 KCL Equations 0 = y43 (V4 − V3 ) 0 = y31 (V3 − V1 ) + y32 (V3 − V2 ) + y34 (V3 − V4 ) I 2 = y20V2 + y21 (V2 − V1 ) + y23 (V2 − V3 ) I1 = y10V1 + y12 (V1 − V2 ) + y13 (V1 − V3 ) 2 I2 Matrix Formation Example Power Systems I Ⱥ I1 Ⱥ Ⱥ( y10 + y12 + y13 ) Ⱥ I Ⱥ Ⱥ − y21 Ⱥ 2 Ⱥ = Ⱥ − y31 Ⱥ 0 Ⱥ Ⱥ Ⱥ Ⱥ Ⱥ 0 Ⱥ 0 Ⱥ Ⱥ − y13 − y23 − y32 0 − y43 ( y31 + y32 + y34 ) − y12 ( y20 + y21 + y23 ) Matrix Formation of the Equations 0 = − y43V3 + y43V4 0 = − y31V1 − y32V2 + ( y31 + y32 + y34 )V3 − y34V4 I 2 = − y21V1 + ( y20 + y21 + y23 )V2 − y23V3 Rearranging the KCL Equations I1 = ( y10 + y12 + y13 )V1 − y12V2 − y13V3 Matrix Formation Example 0 Ⱥ ȺV1 Ⱥ 0 Ⱥ ȺV2 Ⱥ Ⱥ ⋅ Ⱥ Ⱥ − y34 Ⱥ ȺV3 Ⱥ Ⱥ Ⱥ Ⱥ y43 Ⱥ ȺV4 Ⱥ Power Systems I 0 Ⱥ ȺV1 Ⱥ j5.00 Ⱥ I1 Ⱥ Ⱥ− j8.50 j 2.50 Ⱥ I Ⱥ Ⱥ j 2.50 − j8.75 0 Ⱥ ȺV2 Ⱥ j5.00 Ⱥ 2 Ⱥ = Ⱥ Ⱥ ⋅ Ⱥ Ⱥ j5.00 − j 22.50 j12.50 Ⱥ ȺV3 Ⱥ Ⱥ 0 Ⱥ Ⱥ j5.00 Ⱥ Ⱥ Ⱥ Ⱥ Ⱥ Ⱥ 0 0 Ⱥ Ⱥ 0 j12.50 − j12.50Ⱥ ȺV4 Ⱥ Ⱥ Completed Matrix Equation Y11 = ( y10 + y12 + y13 ) = − j8.50 Y23 = Y32 = − y23 = j5.00 Y12 = Y21 = − y12 = j 2.50 Y33 = ( y31 + y32 + y34 ) = − j 22.50 Y13 = Y31 = − y13 = j5.00 Y34 = Y43 = − y34 = j12.50 Y22 = ( y20 + y21 + y23 ) = − j8.75 Y44 = y34 = − j12.50 Matrix Formation Example Power Systems I Matrix is symmetrical along the leading diagonal ￿ Yij = Y ji = − yij j≠i Off-diagonal elements: j =0 Yii = ∑ yij Square matrix with dimensions equal to the number of b us e s Convert all network impedances into admittances n Diagonal elements: ￿ ￿ ￿ ￿ Y-Bus Matrix Building Rules Power Systems I L in e g1 g2 L1 L2 L3 L4 L5 L6 System Data Start End X value 1 0 1 .0 0 5 0 1 .2 5 1 2 0 .4 0 1 3 0 .5 0 2 3 0 .2 5 2 5 0 .2 0 3 4 0 .1 2 5 4 5 0 .5 0 Example ￿ ￿ ￿ bus i 1 :a bus j complex number a can be a The flow of real power along a network branch is controlled by the angular difference of the terminal voltages The flow of reactive power along a network branch is controlled by the magnitude difference of the terminal voltages Real and reactive powers can be adjusted by voltage-regulating transformers and by phase-shifting transformers The tap-changing transform gives some control of the power network by changing the voltages and current m a g n i tu d e s a n d a n g l e s b y s m a l l a m o u n t s Power Systems I ￿ Tap-Changing Transformers Ii yt Vx = 1 V j a Vx I i = −a* ⋅ I j basic circuit equations: Vi I i = yt (Vi − Vx ) 1 :a Ij Vj the off-nominal tap ratio is given as 1:a the nominal turns-ratio (N1/N2) was addressed with the conversion of the network to per unit the transformer is modeled as two elements joined together at a fictitious bus x Power Systems I ￿ ￿ ￿ ￿ Modeling of Tap-Changers I i = yt (Vi − Vx ) yt yt yt 1 I j = − * (Vi − a V j ) = − * Vi + 2 V j a a a I j = − a1* I i I i = −a* ⋅ I j I i = yt (Vi − 1 V j ) a Vx = 1 V j a M a k i n g s u b s ti t u ti o n s Power Systems I ￿ Modeling of Tap-Changers Ⱥ I i Ⱥ Ⱥ yt Ⱥ I Ⱥ = Ⱥ− y a * Ⱥ j Ⱥ Ⱥ t − yt a Ⱥ ȺVi Ⱥ 2 Ⱥ ⋅ Ⱥ yt a Ⱥ ȺV j Ⱥ Ⱥ Ⱥ y Ⱥ Ⱥ t Ⱥ Ⱥ yt Ⱥ I j = Ⱥ− * Ⱥ Vi + Ⱥ 2 Ⱥ V j Ⱥ a Ⱥ Ⱥ a Ⱥ Ⱥ Ⱥ Ⱥ yt Ⱥ I i = {yt } i + Ⱥ− Ⱥ V j V Ⱥ a Ⱥ Matrix formation Power Systems I ￿ YBus Formation of Tap-Changers ￿ ￿ (a - 1) yt / a i non-tap side yt / a (1 - a) yt / a2 j tap side the off-diagonal element represent the impedance across the two buses the remainder form the shunt element Valid for real values of a Taking the y-bus formation, break the diagonal elements into two components Power Systems I ￿ ￿ Pi-Circuit Model of Tap-Changers ...
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