power system components

7/22/2019 Power System Components

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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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