Multi-machine Systems
Each synchronous machine is represented by a constant
voltage source behind the direct axis transient reactance
The input powers are assumed to remain constant
Using the pre-fault voltage, all loads are converted to
equivalent admitt
the nodal voltages for a given load and generation schedule
known real (P) and reactive (Q) power injections
known real (P) power injection and the voltage magnitude (V)
known voltage magnitude (V) and voltage angle ()
must have one generator as the slack
Network
Nodes represent substation bus bars
Branches represent transmission lines and transformers
Injected currents are the flows from generator and loads
Used to form the network model of an interconnected
p o w e r s y s te m
Iinj = Ybus Vnode
Ik Vk
Th
Overhead Conductor
Overhead Spacer Cable
Underground Cable
Three-Conductor Cable
Service Cables
Power Systems I
Transmission Lines
ACSR
Aluminum Conductor with
inner Steel Reinforced strands
ACAR
Aluminum Conductor with
inner Al allow Reinforced
strands
A
Power System Analysis
Fundamentals of Power Systems (EEL 3216)
basic models of power apparatus,
simple systems
transformers, synchronous machines, transmission lines
one feeder radial to single load
What more is there?
large interconnected systems
why hav
Multi-machine Systems
Each synchronous machine is represented by a constant voltage source behind the direct axis transient reactance The input powers are assumed to remain constant Using the pre-fault voltage, all loads are converted to equivalent admitt
Solving Non-linear ODE
Objective
Time domain solution of a system of differential equations Given a function or a system of functions: f(x) or F(x) Seek a time domain solution x(t) or x(t) which satisfy f(x) or F(x)
Integration of the differential equatio
Transient Stability
The ability of the power system to remain in synchronism when subject to large disturbances
Large power and voltage angle oscillations do not permit linearization of the generator swing equations
Lyapunov energy functions
simplified en
Steady State Stability
l
l
l
l
The ability of the power system to remain in synchronism when subject to small disturbances Stability is assured if the system returns to its original operating state (voltage magnitude and angle profile) The behavior can be
Stability
l
The ability of the power system to remain in synchronism and maintain the state of equilibrium following a disturbing force
u
Steady-state stability: analysis of small and slow disturbances
n
gradual power changes faults, outage of a line, sud
Common Unbalanced Network Faults
Single-line-to-ground faults Double-line-to-ground faults Line-to-line faults
March 2004
Power Systems I
1
Single Line to Ground Fault
Ia Va Vb Vc Ib Ic
Va = 0 I f = Ia Ib = Ic = 0 I a0 1 1 1 I a I = 1 1 a a 2 0 a1 3 I a2
Transformers
Equivalent Series Impedance:
Transformer bank of three single-phase transformers
Z 0 = Z1 = Z 2 = Z l
Three-phase transformer with a three-leg core
Z1 = Z 2 = Z l
Z0 > Zl
Wye-Delta Wound Transformers
Wiring connection will always cause a phas
Fault Analysis
Percentage of total faults
<5%
l
Fault types:
u
balanced faults
n
three-phase
u
unbalanced faults
n
n
n
single-line to ground double-line to ground line-to-line faults
60-75% 15-25% 5-15%
l
Unbalance fault analysis requires new tools
u
u
sy
The Bus Impedance Matrix
l
Definition
-1 Z bus = Ybus
l
Direct formation of the matrix
u
u
u
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, sp
The Power Flow Solution
Most common and important tool in power system
analysis
also known as the Load Flow solution
used for planning and controlling a system
assumptions: balanced condition and single phase analysis
Problem:
determine the voltage magnit
Example
Using the Newton-Raphson PF,
find the power flow solution
1
0.01 + j0.03
y12 = 10 j 20 pu
y13 = 10 j 30 pu
y23 = 16 j 32 pu
Power Systems I
0.0125 + j0.025
0.02 + j0.04
2
3
|V3| = 1.04
200 MW
400 + j 250
S =
= 4.0 j 2.5 pu
100
200
P3sch =
= 2.0 pu
Solving Non-linear ODE
Objective
Time domain solution of a system of differential equations
Given a function or a system of functions: f(x) or F(x)
Seek a time domain solution x(t) or x(t) which satisfy f(x) or F(x)
Integration of the differential equat
Transient Stability
The ability of the power system to remain in synchronism
when subject to large disturbances
Lyapunov energy functions
Large power and voltage angle oscillations do not permit
linearization of the generator swing equations
simplified en
Steady State Stability
l
l
l
l
The ability of the power system to remain in synchronism
when subject to small disturbances
Stability is assured if the system returns to its original
operating state (voltage magnitude and angle profile)
The behavior can be
Stability
l
The ability of the power system to remain in synchronism
and maintain the state of equilibrium following a
disturbing force
u
Steady-state stability: analysis of small and slow disturbances
n
u
gradual power changes
Transient stability: analys
Common Unbalanced Network Faults
Single-line-to-ground faults
Double-line-to-ground faults
Line-to-line faults
March 2004
Power Systems I
1
Single Line to Ground Fault
Va = 0
Ia
Va
I f = Ia
Ib
Vb
Ib = Ic = 0
Ic
Vc
I a0
1 1 1 I a
I = 1 1 a a 2 0
a1 3
Transformers
Equivalent Series Impedance:
Transformer bank of three single-phase transformers
Z 0 = Z1 = Z 2 = Z
Three-phase transformer with a three-leg core
Z1 = Z 2 = Z
Z0 > Z
Wye-Delta Wound Transformers
Wiring connection will always cause a phase s
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
Balanced 3-Phase Short Circuit
Consider a synchronous generator operating at 60 Hz
with constant excitation
Examine the impact on the stator currents when a threephase short circuit is applied to the generator terminals
The initial currents
Park
ia (0+ )
Chapter 8: Transient Analysis of
Synchronous Machines
FAMU-FSU College of Engineering
Synchronous Machines
Steady state modeling
rotor mmf and stator mmf are stationary with respect to each other
flux linkage with the rotor are invariant with time
no volt
i = 1, 2, k
u j ( x1 , x2 , xn ) 0
j = 1,2, , m
and the inequality constraints
gi ( x1 , x2 , xn ) = 0
subject to the equality constraints
)
The Lagrange multiplier is extended to include the
inequality constraints by introducing the m-dimensional
vector
there are many solution combinations for scheduling generation
in practice, power plants are not located at the same distance
from the load centers
power plants use different types of fuel, which vary in cost from
time to time
in the power flow analysis,
Fault Analysis
l
Analysis types
u
u
power flow - evaluate normal operating conditions fault analysis - evaluate abnormal operating conditions
l
Fault types:
u
balanced faults
n
three-phase
u
unbalanced faults
n
n
single-line to ground and double-line to g