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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 Offdiagonal 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: YBus 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 voltageregulating
transformers and by phaseshifting transformers The tapchanging 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 TapChanging 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 offnominal tap ratio is given as 1:a
the nominal turnsratio (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 TapChangers 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 TapChangers 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− * ȺVi + Ⱥ 2 ȺV j
Ⱥ a Ⱥ Ⱥ a Ⱥ Ⱥ Ⱥ Ⱥ yt Ⱥ I i = {yt } i + Ⱥ− ȺV j
V
Ⱥ a Ⱥ Matrix formation Power Systems I YBus Formation of TapChangers (a  1) yt / a i nontap side yt / a (1  a) yt / a2 j tap side the offdiagonal element represent the impedance across the two
buses
the remainder form the shunt element Valid for real values of a
Taking the ybus formation, break the diagonal elements
into two components Power Systems I PiCircuit Model of TapChangers ...
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 Fall '11
 THOMASBALDWIN

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