Chapter 1

Chapter 1Modelling of Power System ComponentsContents Basic Concepts Single Phase Three Phase Models Matrix Representation of NetworksBus Admittance MatrixBus Impedance MatrixNetwork Reduction TechniquesBasic Concepts power system componentsGeneration plant Transformers Transmission lines FACTS devices Loads HVDC convertersPhasor representation Complex power supplied to a one port Conservation of complex power Balanced Three phase Revise these concepts Read Arthur from page 24 to end of chapter ExampleFor the following system, compute S13,S31, S23,S32 and SG3 using MATLAB

Single PhaseGenerator model

Generator cross section

Generator model contdOpen circuit voltageVoltage due to field current Assume ia=ib=ic=0For a differential angle d

Taking over a whole Gaussian surface

Generator modeling For an N turn concentrated winding

Assuming uniform rate of rotation

Generator modeling Using circuit conventions

Generator modeling Armature reactionAir gap flux due to current in stator windings ia, ib and ic

Generator modeling Over a small air gap

Generator modeling The spatial flux density distribution is

Taking Fourier series of the flux density

With sinusoidal current input

Taking the effect of other phases

Terminal voltageObtained using superpositionTotal air gap flux linkage

Final generator model

Generator model contdPower deliveredRound rotor case

Dynamic model and generation control

Generator dynamic model

Generator modeling Basic relations

If speed of machine is

Phase angle deviation

Speed deviation

Relating it to torque

The relation between mechanical and electrical powers is

Similarly for torque

Using relation for power and torque

At steady state , electrical and mechanical torque and power are equal

Generator modeling contdUsing the relation between torque and speed change

The change in power then

Generator load 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

Transmission line models Per phase distributed parameter of TL

TL modeling Lumped and simplified models

Complex power TX over TLConsider two generators connected by TL

Power circle diagram

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

Bus Admittance MatrixBus admittance matrix is a matrix which relates the injected current to the voltage

Rules for Y bus formationSteps / rules

Can be applied to PS components or networks

Example Find the y bus for the transformer and TL connection shown below

Admittance matrix For larger networks, the steps can be written in a program using MATLAB

Network solutionFinding 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

Example in triangular factorization Suppose we have a 3 by 3 matrix M

LU factorization algorithmGiven an n by n Y matrix

Bus Impedance MatrixInverse of Y bus is the impedance matrix

Where

Zkk is Thevenin equivalent of network at node k

Network Reduction TechniquesBus with no generator or loadHas 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 Krone reduction Consider a 3 by 3 Y matrix and nodal equation

Step 1- write V3 in terms of V1 and V2Step 2- substitute into eq. 1 and eq. 2 Step 3- obtain the new Y matrix as 2 by 2

Krone reduction contdFor a general n by n matrix Assume node k has zero current injection

Where is ij element of the new admittance matrix