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Bus Impedance Matrix in Faults Calculations
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ELCT 751 Matrix Methods for Short Circuits 1
Lecture 8: Matrix Methods for Short-Circuit Studies
Bus Impedance MatrixNetwork Changes
Short-Circuit CalculationsConsideration of Large
Systems
ELCT 751 Matrix Methods for Short Circuits 2
Introduction For systems with more than a few
generators, use matrix methods forshort-circuit studiesNodal admittance matrix is easy to
calculateNodal impedance matrix is more
difficult to calculate but gives short-circuit currents easily
ELCT 751 Matrix Methods for Short Circuits 3
Example-j 4.0
-j 5.0 -j 3.0
-j 0.5
-j 0.1I1
I2
Admittances in per unit
1 2
3
ELCT 751 Matrix Methods for Short Circuits 4
ExampleNodal analysis:
I1 = -j 9.0 V1 + j 4.0 V2 + j 5.0 V3I2 = j 4.0 V1 j 7.1 V2 + j 3.0 V30 = j 5.0 V1 + j 3.0 V2 j 8.5 V3or I = Ybus Vwhere Ybus is the nodal (bus)
admittance matrix
ELCT 751 Matrix Methods for Short Circuits 5
Bus Impedance MatrixSolve for V in terms of I:
Then V = Ybus-1 I = Zbus Iwhere Zbus is the nodal impedance matrix or bus impedance matrix
The bus impedance matrix can be computed by inverting the bus admittance matrix or by direct formation
ELCT 751 Matrix Methods for Short Circuits 6
For example, line/transf data:
4
3
3
2
0.00 - j10.00.00 + j0.1034
0.00 - j2.500.00 + j0.4023
0.00 - j4.000.00 + j0.2512
0.00 - j5.000.00 + j0.2011
Admittance [per unit]
Impedance [per unit]
Bus Numbers
Branch Number
ELCT 751 Matrix Methods for Short Circuits 7
and shunt admittances:
0.00 j0.604
0.00 j0.003
0.00 j0.802
0.00 j0.801
Admittance [per unit]
Bus Number
ELCT 751 Matrix Methods for Short Circuits 8
Ybus =
-j 10.6j 10.0j 0.00j 0.00
j 10.0-j 16.5j 2.50j 4.00
j 0.00j 2.50-j 8.30j 5.00
j 0.00j 4.00j 5.00-j 9.80
ELCT 751 Matrix Methods for Short Circuits 9
Zbus =
j 0.5762j 0.5108j 0.4035j 0.4143
j 0.5108j 0.5415j 0.4277j 0.4392
j 0.4035j 0.4277j 0.5117j 0.4357
j 0.4143j 0.4392j 0.4357j 0.5036
ELCT 751 Matrix Methods for Short Circuits 10
Short-Circuit Calculations with Zbus
Fault
Rest of the system
1
2
3
4
ELCT 751 Matrix Methods for Short Circuits 11
Short-Circuit Calculations with Zbus
Fault simulation
Rest of thesystem
1
23
4
1.0
0
1.0
ELCT 751 Matrix Methods for Short Circuits 12
Z43
Z33
Z23
Z13
0Z34Z32Z31V30
-If
0
V4
-1
V1=
Z44Z42Z41
Z24Z22Z21
Z14Z12Z11
ELCT 751 Matrix Methods for Short Circuits 13
If = 1/Z22
V1 = - Z12 /Z22 V1 = 1 - Z12 /Z22
Vi is voltage of node i to 0 node,Vi = 1+Vi is voltage of node i to neutral
ELCT 751 Matrix Methods for Short Circuits 14
Previous Example
Zbus =j
0.57620.51080.40350.4143
0.51080.54150.42770.4392
0.40350.42770.51170.4357
0.41430.43920.43570.5036
ELCT 751 Matrix Methods for Short Circuits 15
ExampleFault at bus k
Current into fault (at k):
I k= 1/Zkk
Voltage at bus j:
Vj = 1 - Zjk /Zkk
ELCT 751 Matrix Methods for Short Circuits 16
0.11350.0
0.1643
0.1278
V3
0.00.0566
0.2116
0.1772
V4
0.29980.2809-j 1.73540.21020.1889-j 1.8473
0.00.1487-j 1.9542
0.13490.0-j 1.9861
V2V1Ikk
Fault at bus k is three-phase short circuit
ELCT 751 Matrix Methods for Short Circuits 17
Adding a Branch to Zbus To modify an existing Zbus to add a
new branch. Four cases:1) Radial branch connecting a new bus
to the reference node2) Radial branch connecting a new bus
to an existing system bus3) Branch from an existing system bus
to the reference node4) Branch between two existing buses
ELCT 751 Matrix Methods for Short Circuits 18
Case 1 is straightforwardAdd a new axis (row and column
n+1) to the Z matrix for the new bus having the branch impedance z in the diagonal and zeros off-diagonal
n+1n
1...n+11 ... n
Znew =z0 ... 0
0...0
Z
ELCT 751 Matrix Methods for Short Circuits 19
Case 2 a new branch from old bus p to new bus q (= n+1)A current Iq injected into q has the
same effect on the existing system as if it were injected at p
Copy axis p to a new axis q andZqq = z + Zpp
ELCT 751 Matrix Methods for Short Circuits 20
n+1
n1...
n+11 ... n
Znew =Zpp+ zZp1 ... Zpn
Z1p...Znp
Z
where 1 p n
ELCT 751 Matrix Methods for Short Circuits 21
Case 3 a new branch from old bus p to the reference node (loop-closing)Add a new axis by the algorithm of
Case 2, creating a fictitious new busShort the new bus to reference (it now
has zero voltage)Use Kron reduction to eliminate the
new axis of the matrix
ELCT 751 Matrix Methods for Short Circuits 22
Case 4 a new branch z between existing buses p and q (loop closing)Create a new axis (row and column)
to represent the new loop createdDiagonal:
Zn+1,n+1 = Zpp+Zqq-Zpq-Zqp+z
Off diagonal: Zn+1, j = Zpj-ZqjZi, n+1 = Zip-Ziq
ELCT 751 Matrix Methods for Short Circuits 23
Case 4 (continued):Use Kron reduction to eliminate the
new axis representing the loopZijnew = Zij (Zi, n+1 Zn+1,j )/ Z n+1,n+1
Notice that this will change almost every element in the existing part of the matrix
ELCT 751 Matrix Methods for Short Circuits 24
Zbus Building Method The Zbus modification outline
above can also be used to create Zbus, by starting with any Case 1 branch to get the 1 1 matrix for the starting point:Z = z
Then proceed to add other branches in any convenient order
ELCT 751 Matrix Methods for Short Circuits 25
Removing a Branch
Any branch (of impedance z) may be removed during a study by adding a new branch in parallel with impedance of zZeq = -z2/(z-z) = -z/0 an open
circuit
ELCT 751 Matrix Methods for Short Circuits 26
Axis Discarding Method Large-scale systems have full Zbus
matrices that can be difficult to store and use Instead we can store the sparse LU
factors (as discussed previously)Or we can discard unneeded axes by
simply striking them off (if their current injections are zero)
ELCT 751 Matrix Methods for Short Circuits 27
Axis Discarding
If I1= 0 and we dont need V1 then:V1 = Z11 I1 + Z12 I2 + Z13 I3V2 = Z21 I1 + Z22 I2 + Z23 I3V3 = Z31 I1 + Z32 I2 + Z33 I3
becomes:V2 = Z22 I2 + Z23 I3V3 = Z32 I2 + Z33 I3
ELCT 751 Matrix Methods for Short Circuits 28
Axis discarding for short-circuit calculation: Divide system into study system and
external systemStart building Z matrix in external
systemAfter all branches are added to external
node, its axis is discardedAll study system nodes are kept
Axis Discarding
ELCT 751 Matrix Methods for Short Circuits 290.0379076
0.2000084
0.0599083
0.0414073
0.0820053
0.0420043
0.1160052
0.1737071
0.1763031
0.1983021
0.0185060
0.0175010
XRji
Branch Z [pu]Bus Numbers
Example: Set up Z matrix, discarding all axes except 1, 2 and 5
ELCT 751 Matrix Methods for Short Circuits 30
Solution: Add 0-6: Z = 6j 0.0185
Add 6-7: Z = 7j 0.0564j 0.0185
j 0.0185 6j 0.0185
Discard 6: Z = 7j 0.0564
Add 7-3: Z = 3j 0.0978j 0.0564
j 0.0564 7j 0.0564
ELCT 751 Matrix Methods for Short Circuits 31
Add 7-1:
Z = j 0.0564j 0.0978j 0.0564
3j 0.0564j 0.05641j 0.2301j 0.0564
j 0.0564 7j 0.0564
Discard 7: Z = 1j 0.2301j 0.0564
j 0.0564 3j 0.0978
ELCT 751 Matrix Methods for Short Circuits 32
Add 3-1:Z =
-j 0.1737j 0. 2301j 0.0564
1-j 0.1737j 0.0564Lj 0.3914j 0.0414
j 0.0414 3j 0.0978
Kron reduction on L:
1j 0.15301j 0.07477j 0.07477 3j 0.09342
Continue this process, adding 3-4, 3-5, 3-8
Z =
ELCT 751 Matrix Methods for Short Circuits 33
0.093420.093420.135420.07477
40.093420.093420.07477
0.093420.17542
0.07477
50.093420.0747780.153320.07477
0.07477 10.15301
Discard 3:
Z= j
ELCT 751 Matrix Methods for Short Circuits 34
Add 4-8 creating fictitious loop axis L. Note that 4 and 8 are to be immediately discarded, so there is no need to compute those elements in the Kron reduction
Result: Z = j50.175420.07477
0.07477 10.15301
ELCT 751 Matrix Methods for Short Circuits 35
Add 1-0 perform Kron Reduction. Then add 2-5 and 1-2 and perform another Kron Reduction:
0.068560.102800.01004
50.068560.0100420.120900.01208
0.01208 10.01556Z = j
ELCT 751 Matrix Methods for Short Circuits 36
0.068560.102800.01004
50.068560.0100420.120900.01208
0.01208 10.01556Z = j
Fault at bus 2:If = 1/j0.1209 = -j 8.27 puV5 = 1 - j0.06856/j0.1209 = 0.433 puI52 = (V5-V2)/z52 = (0.433-0.00)/j0.116
= -j 3.73 pu
ELCT 751 Matrix Methods for Short Circuits 37
Bus-Tie Circuit Breakers A normally-open circuit breaker may
connect two buses when closedStudy might require closing and/or
opening of bus-tie circuit breakerNeed to represent a zero-impedance
branch that can be openedAdd z in series with z between buses
ELCT 751 Matrix Methods for Short Circuits 38
Bus-Tie Circuit Breakers1
2
NC
NC
NO
0 2
1Zg1
Zg2
z-z N+1
One-linediagram
Circuitrepresentation
Fictitious new bus