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Unformatted text preview: inversion of the bus admittance matrix is a n3 effort
for small and medium size networks, direct building of the matrix
is less effort
for large size networks, sparse matrix programming with
gaussian elimination technique is preferred Direct formation of the matrix −1
Z bus = Ybus Definition Power Systems I The Bus Impedance Matrix 1 j 0.4 j0.2 3 j0.8 2
j0.4 j0.4 1 3 4
5 2 selected tree 3 1 0
1
3
extending
tree branch 2 1 3 4 0 5 2
2 loop
closing
cotree branch bus node = graph vertex line branch = edge Graph theory techniques helps explain the building
process Power Systems I Forming the Bus Impedance Matrix Z m
bus Partial
Network
0
Reference 1
2
i
j Vbus = Z bus I bus Basic construction of the network and the matrix Power Systems I Forming the Bus Impedance Matrix q 0
Reference Vq = V p + z qp I q Power Systems I Z m
bus Partial
Network 1
2
p
m Adding a Line Z q 0
Reference Vq = 0 + z q 0 I q m
bus Partial
Network 1
2
p
m Power Systems I Ⱥ V1 Ⱥ Ⱥ Z11
ȺV Ⱥ Ⱥ Z
Ⱥ 2 Ⱥ Ⱥ 21
Ⱥ Ⱥ Ⱥ
Ⱥ Ⱥ Ⱥ
ȺV p Ⱥ = Ⱥ Z p1
Ⱥ Ⱥ Ⱥ
Ⱥ Ⱥ Ⱥ
ȺVm Ⱥ Ⱥ Z m1
ȺV Ⱥ Ⱥ= Z
p1
Ⱥ q Ⱥ Ⱥ
Z m2
= Z p2 Z 21
Z 22
Z p2 Z1 p
Z2 p
Z pp
Z1m
Z 2m
Z pm
Z mp Z mm
= Z pp = Z pm
= Z1 p
= Z2 p
= Z pp Ⱥ Ⱥ I1 Ⱥ
Ⱥ Ⱥ I Ⱥ
Ⱥ Ⱥ 2 Ⱥ
Ⱥ Ⱥ Ⱥ
Ⱥ Ⱥ Ⱥ
Ⱥ Ⱥ I p Ⱥ
Ⱥ Ⱥ Ⱥ
Ⱥ Ⱥ Ⱥ
= Z mp Ⱥ Ⱥ I m Ⱥ
= Z pp + z pq Ⱥ Ⱥ I q Ⱥ
Ⱥ Ⱥ Ⱥ Adding a Line to an Existing Line Power Systems I Ⱥ V1 Ⱥ Ⱥ Z11
ȺV Ⱥ Ⱥ Z
Ⱥ 2 Ⱥ Ⱥ 21
Ⱥ Ⱥ Ⱥ
Ⱥ Ⱥ Ⱥ
ȺV p Ⱥ = Ⱥ Z p1
Ⱥ Ⱥ Ⱥ
Ⱥ Ⱥ Ⱥ
ȺVm Ⱥ Ⱥ Z m1
ȺV Ⱥ Ⱥ = 0
Ⱥ q Ⱥ Ⱥ Z1 p Z1m
Z 2 p Z 2m
=0 =0 =0 Z p 2 Z pp Z pm
Z m 2 Z mp Z mm Z 21
Z 22
Adding a Line from Reference
Ⱥ Ⱥ I1 Ⱥ
Ⱥ Ⱥ I Ⱥ
Ⱥ Ⱥ 2 Ⱥ
Ⱥ Ⱥ Ⱥ
Ⱥ Ⱥ Ⱥ
= 0 Ⱥ Ⱥ I p Ⱥ
Ⱥ Ⱥ Ⱥ
Ⱥ Ⱥ Ⱥ
= 0 Ⱥ Ⱥ I m Ⱥ
= z0 q Ⱥ Ⱥ I q Ⱥ
Ⱥ Ⱥ Ⱥ =0
=0
z pq I l + Vq − V p = 0 Power Systems I z p0 Il − Vp = 0 z pq I l = V p − Vq → 0
Reference Z m
bus Partial
Network 0
Reference 1
p
i
m z p0 Il = Vp − 0 → Z m
bus Partial
Network 1
p
q
m Closing a Loop T I [ n×1] bus [1×n ] Power Systems I Z ll I bus [1×n ] Ⱥ old ΔZΔZ T Ⱥ
= Ⱥ Z bus −
Ⱥ I bus
Z ll Ⱥ
Ⱥ ΔZ T [ n×1]
→ Il = −
I bus [1×n ]
Z ll ΔZ[1×n ]ΔZ T [ n×1] + Z ll [1×1] I l old
Vbus [1×n ] = Z bus [ n×n ] I bus [1×n ] − 0 = ΔZ Vbus [1×n ] ΔZ[1×n ] Ⱥ Ⱥ I bus [1×n ] Ⱥ
Ⱥ
Z ll [1×1] Ⱥ Ⱥ I l Ⱥ
Ⱥ
Ⱥ Ⱥ
old
= Z bus [ n×n ] I bus [1×n ] + ΔZ[1×n ] I l old
ȺVbus [1×n ] Ⱥ Ⱥ Z bus [ n×n ]
Ⱥ 0 Ⱥ = ȺΔZ T
[ n×1]
Ⱥ
Ⱥ Ⱥ
Ⱥ Eliminating a node from the system Kron Reduction
Z pp
Z qp
Z mp
Power Systems I
Z mm Z qm
Z pm Z1m Z qq −Z pq Z qm −Z pm Z mq Z qq
Z pq Z1q Z ll = z pq + Z pp + Z qq − 2 Z pq Z qp −Z pp Z1 p Then execute Kron reduction on Zll Ⱥ V1 Ⱥ Ⱥ Z11
Ⱥ Ⱥ Ⱥ
Ⱥ Ⱥ Ⱥ
ȺV p Ⱥ Ⱥ Z p1
Ⱥ Ⱥ Ⱥ
ȺVq Ⱥ = Ⱥ Z q1
Ⱥ Ⱥ Ⱥ
Ⱥ Ⱥ Ⱥ
ȺVm Ⱥ Ⱥ Z m1
Ⱥ 0 Ⱥ Ⱥ Z q1−Z p1
Ⱥ Ⱥ Ⱥ Adding a Line between two Lines Z1q −Z1 p Ⱥ Ⱥ I1 Ⱥ
Ⱥ Ⱥ Ⱥ
Ⱥ Ⱥ Ⱥ
Z pq −Z pp Ⱥ Ⱥ I p Ⱥ
Ⱥ Ⱥ Ⱥ
Z qq −Z qp Ⱥ Ⱥ I q Ⱥ
Ⱥ Ⱥ Ⱥ
Ⱥ Ⱥ Ⱥ
Z mq −Z mp Ⱥ Ⱥ I m Ⱥ
Z ll Ⱥ Ⱥ I l Ⱥ
Ⱥ Ⱥ Ⱥ Z1 p
Z pp
Z ip
Z mp
− Z pp
Power Systems I Z1m
Z pm
Z im
Z mi Z mm
− Z pi − Z pm Z1i
Z pi
Z ii Then execute Kron reduction on Zll Z ll = z p 0 + Z pp Ⱥ V1 Ⱥ Ⱥ Z11
Ⱥ Ⱥ Ⱥ
Ⱥ Ⱥ Ⱥ
ȺV p Ⱥ Ⱥ Z p1
Ⱥ Ⱥ Ⱥ
Ⱥ Vi Ⱥ = Ⱥ Z i1
Ⱥ Ⱥ Ⱥ
Ⱥ Ⱥ Ⱥ
ȺVm Ⱥ Ⱥ Z m1
Ⱥ 0 Ⱥ Ⱥ− Z p1
Ⱥ Ⱥ Ⱥ − Z1 p Ⱥ Ⱥ I1 Ⱥ
Ⱥ Ⱥ Ⱥ
Ⱥ Ⱥ Ⱥ
−Z pp Ⱥ Ⱥ I p Ⱥ
Ⱥ Ⱥ Ⱥ
− Z ip Ⱥ Ⱥ I q Ⱥ
Ⱥ Ⱥ Ⱥ
Ⱥ Ⱥ Ⱥ
− Z mp Ⱥ Ⱥ I m Ⱥ
Z ll Ⱥ Ⱥ I l Ⱥ
Ⱥ Ⱥ Ⱥ Adding a Line from a Line to Reference Ⱥ V1 Ⱥ Ⱥ Z11
Ⱥ Ⱥ Ⱥ
Ⱥ Ⱥ Ⱥ
ȺV p Ⱥ Ⱥ Z p1
Ⱥ Ⱥ Ⱥ
Ⱥ Vi Ⱥ = Ⱥ Z i1
Ⱥ Ⱥ Ⱥ
Ⱥ Ⱥ Ⱥ
ȺVm Ⱥ Ⱥ Z m1
Ⱥ 0 Ⱥ Ⱥ Z l1
Ⱥ Ⱥ Ⱥ
Z new
jk =Z
Z mp
Z lp Z pm
Z im Z1m old
jk − Z ll Z jl Z lk
Z mi Z mm
Z li Z lm Z ii
Z pi
Z pp
Z ip Z1i Z1 p Z1l Ⱥ Ⱥ I1 Ⱥ
Ⱥ Ⱥ Ⱥ
Ⱥ Ⱥ Ⱥ
Z pl Ⱥ Ⱥ I p Ⱥ
Ⱥ Ⱥ Ⱥ
Z il Ⱥ Ⱥ I q Ⱥ
Ⱥ Ⱥ Ⱥ
Ⱥ Ⱥ Ⱥ
Z ml Ⱥ Ⱥ I m Ⱥ
Z ll Ⱥ Ⱥ I l Ⱥ
Ⱥ Ⱥ Ⱥ Kron reduction removes an axis (row & column) from a
matrix while retaining the axis’s numerical influence Power Systems I Kron Reduction zq0 the new offdiagonal row and column filled with (0)
the diagonal element (m+1),(m+1) filled with the element impedance start with existing network matrix [m × m]
create a new network matrix [(m+1) × (m+1)] with Rule 1: Addition of a branch to the reference Power Systems I ZBus Building Rules the new offdiagonal row and column filled with a copy of row p and
column p
the diagonal element (m+1),(m+1) filled with the element impedance
zpq plus the diagonal impedance Zpp connecting to existing bus p
start with existing network matrix [m × m]
create a new network matrix [(m+1) × (m+1)] with Rule 2: Addition of a branch to an existing bus Power Systems I ZBus Building Rules the new offdiagonal row and column filled with a copy of row q
minus row p and column q minus column p
the diagonal element (m+1),(m+1) filled with zpq + Zpp + Zqq  2 Zpq
perform Kron reduction on the m+1 row and column connecting to existing buses p and q
start with existing network matrix [m × m]
create a new network matrix [(m+1) × (m+1)] with Rule 3: Addition of a linking branch Power Systems I ZBus Building Rules the new offdiagonal row and column filled with a copy of the
negative of row p and the negative of column p
the diagonal element (m+1),(m+1) filled with
zp0 + Zpp perform Kron reduction on the m+1 row and column connecting to existing bus p and reference
start with existing network matrix [m × m]
create a new network matrix [(m+1) × (m+1)] with Rule 4: Addition of a linking branch Power Systems I ZBus Building Rules Network j0.8
j0.4 j0.4 2 1
3 4 2 3
Graph 1 0 Line adding order: 10, 20, 13, 12, then 23 3 Power Systems I 1 j 0.4 j0.2 Example 5 2 [ j 0.2] 0
j 0. 4
0 0 Ⱥ
j 0.4Ⱥ
Ⱥ Power Systems I Ⱥ j 0.2
2. Ⱥ
Ⱥ 0
Ⱥ j 0.2
3. Ⱥ 0
Ⱥ
Ⱥ j 0.2
Ⱥ 0.
1. Example j 0.2Ⱥ
0 Ⱥ
Ⱥ
j 0.6Ⱥ
Ⱥ Ⱥ j 0.171
Ⱥ j 0.057
Ⱥ
Ⱥ j 0.171
Ⱥ 0.285
j 0.057 j 0.057 j 0.2 j 0.171Ⱥ
j 0.057 Ⱥ
Ⱥ
j 0.571Ⱥ
Ⱥ j 0.2 Ⱥ
− j 0.4Ⱥ
0
Ⱥ
j 0.6 j 0.2 Ⱥ
Ⱥ
j 0.2
j1.4 Ⱥ
( j 0.2)( j 0.2) = j 0.17
Z11 = j 0.2 −
j1.4 0
Ⱥ j 0.2
Ⱥ 0
j 0.4
Ⱥ
4.
Ⱥ j 0.2
0
Ⱥ
Ⱥ j 0.2 − j 0.4 Power Systems I Ⱥ j 0.171
Ⱥ j 0.057
5. Ⱥ
Ⱥ j 0.171
Ⱥ
Ⱥ j 0.114
Ⱥ j 0.16
Ⱥ j 0.08
Ⱥ
Ⱥ j 0.12
Ⱥ Example
j 0.171 j 0.114 Ⱥ
j 0.285 j 0.057 − j 0.228Ⱥ
Ⱥ
j 0.057
j 0.571 j 0.514 Ⱥ
Ⱥ
j1.14 Ⱥ
− j 0.228 j 0.514
j 0.08 j 0.12Ⱥ
j 0.24 j 0.16Ⱥ
Ⱥ
j 0.16 j 0.34Ⱥ
Ⱥ j 0.057 ...
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This note was uploaded on 02/01/2012 for the course EEL 4213 taught by Professor Thomasbaldwin during the Fall '11 term at FSU.
 Fall '11
 THOMASBALDWIN

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