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Chapter 1
Modelling of Power System
Components
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Contents
Basic Concepts
Single Phase
Three Phase Models Matrix Representation of Networks
Bus Admittance Matrix
Bus Impedance Matrix Network Reduction Techniques
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Basic Concepts
power system components Generation plant
Transformers
Transmission lines
FACTS devices Loads
HVDC converters
Phasor representation
Complex power supplied to a one port Conservation of complex power
Balanced Three phase
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Revise these concepts
Read Arthur from page 24 to end of chapter
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Example
For the following system, compute S13,S31,
S23,S32and SG3 using MATLAB
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Single Phase
Generator model
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Generator cross section
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Generator model contd
Open circuit voltage
Voltage due to field current
Assume ia=ib=ic=0
For a differential angle d
Taking over a whole Gaussian surface
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Generator modeling
For an N turn concentrated winding
Assuming uniform rate of rotation
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Generator modeling
Using circuit conventions
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Generator modeling
Armature reaction Air gap flux due to current in stator windings ia, ib
and ic
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Generator modeling
Over a small air gap
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Generator modeling
The spatial flux density distribution is
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Taking Fourier series of the flux density
With sinusoidal current input
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Taking the effect of other phases
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Terminal voltage
Obtained using superposition
Total air gap flux linkage
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Final generator model
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Generator model contd
Power delivered
Round rotor case
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Dynamic model and generation control
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Generator dynamic model
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Generator modeling
Basic relations
If speed of machine is
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Phase angle deviation
Speed deviation
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Relating it to torque
The relation between mechanical and
electrical powers is
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Similarly for torque
Using relation for power and torque
At steady state , electrical and mechanical
torque and power are equal
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Generator modeling contd
Using the relation between torque and speed
change
The change in power then
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Generatorload model
Most loads are motor loads
Where D is the change in load for a unit
change in power
It is based on a given base MVA
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Transmission line models
Per phase distributed parameter of TL
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TL modeling
Lumped and simplified models
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Complex power TX over TL
Consider two generators connected by TL
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Power circle diagram
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Transformer models
Equivalent pi-model is given by
Yoc=1/zoc, ysc=1/zsc where zoc is magnetizing
impedance and zsc is short circuit impedance
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Bus Admittance Matrix
Bus admittance matrix is a matrix which
relates the injected current to the voltage
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Rules for Y bus formation
Steps / rules
Can be applied to PS components or networks
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Example
Find the y bus for the transformer and TL
connection shown below
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Admittance matrix
For larger networks, the steps can be written
in a program using MATLAB
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Network solution
Finding V from given I values involves inverting Y
matrix
Gaussian elimination and triangular factorization
Where L and U are lower and upper triangular
factors of Y bus matrix
Splitting the equation
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Example in triangular factorization
Suppose we have a 3 by 3 matrix M
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LU factorization algorithm
Given an n by n Y matrix
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Bus Impedance Matrix
Inverse of Y bus is the impedance matrix
Where
Zkk is Thevenin equivalent of network at nodek
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Network Reduction Techniques
Bus with no generator or load
Has no current injection
Can be eliminated
Krone reduction
Reduction of size of Y matrix from n by n to n-k by n-k
where k is the number of buses with no current
injection
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Krone reduction
Consider a 3 by 3 Y matrix and nodal equation
Step 1- write V3 in terms of V1 and V2
Step 2- substitute into eq. 1 and eq. 2
Step 3- obtain the new Y matrix as 2 by 2
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Krone reduction contd
For a general n by n matrix
Assume node k has zero current injection
Where is ij element of the new admittance
matrix
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