What do you think about this system response?

Time

Rotor Angle

How about this response?

Time

Rotor Angle

Compare these two responses

Time

Rotor Angle

What about these responses?

Time

Rotor Angle

Compare these instabilities

Time

Rotor Angle

Steady-state = stable equilibrium

things are not changing concerned with whether the system

variables are within the correct limits

Transient Stability

"Transient" means changing The state of the system is changing We are concerned with the

transition from one equilibrium to another

The change is a result of a "large" disturbance

Primary Questions

1. Does the system reach a new steady state that is acceptable?

2. Do the variables of the system remain within safe limits as the system moves from one state to the next?

Main Concern: synchronism of system synchronous machines

Instability => at least one rotor angle becomes unbounded with respect tothe rest of the system

Also referred to as "going out of step" or "slipping a pole"

Additional Concerns: limits on other system variables

Transient Voltage Dips Short-term current & power limits

Time Frame

Typical time frame of concern 1 - 30 seconds

Model system components that are "active" in this time scale

Faster changes -> assume instantaneous

Slower changes -> assume constants

Primary components to be modeled

Synchronous generators

Traditional control options

Generation based control exciters, speed governors, voltage

regulators, power system stabilizers

Traditional Transmission Control Devices

Slow changes modeled as a constant value

FACTS Devices

May respond in the 1-30 second time frame

modeled as active devices

May be used to help control transient stability problems

Kundur's classification of methods for improving T.S.

Minimization of disturbance severity and duration

Increase in forces restoring synchronism

Reduction of accelerating torque by reducing input mechanical power

Reduction of accelerating torque by applying artificial load

Commonly used methods of improving transient stability

High-speed fault clearing, reduction of transmission system impedance, shunt compensation, dynamic braking, reactor switching, independent and single-pole switching, fast-valving of steam systems, generator tripping, controlled separation, high-speed excitation systems, discontinuous excitation control, and control of HVDC links

FACTS devices = Exciting control opportunities!

Deregulation & separation of transmission & generation functions of a utility

FACTS devices can help to control transient problems from the transmission system

3 Minute In-Class Activity

1. Pick a partner 2. Person wearing the most blue =

scribe Other person = speaker 3. Write a one-sentence definition

of "TRANSIENT STABILITY” 4. Share with the class

Mass-Spring Analogy

Mass-Spring System

Equations of motion

Newton => F = Ma = Mx’’ Steady-state = Stable equilibrium

= Pre-fault

F = -K x - D x’ + w = Mball x’’ = 0 Can solve for x

Fault-on system

New equation of motion

F = -K x - D x’ + (Mball + Mbird)g = (Mball + Mbird) x’’

Initial Conditions? x = xss x’ = 0

How do we determine x(t)?

Solve directly Numerical methods

(Euler, Runge-Kutta, etc.) Energy methods

Simulation of the Pre-fault & Fault-on system responses

Post-fault system

"New" equation of motion

F = -K x - D x’ + w = Mball x’’ Initial Conditions? x = xc x’ = xc’

Simulation of the Pre-fault, Fault-on, and Post-fault system responses

Transient Stability?

Does x tend to become unbounded? Do any of the system variables

violate limits in the transition?

Power System Equations

Start with Newton again ....T = I

We want to describe the motion of the rotating masses of the generators in the system

The swing equation

2H d2 = Pacc

o dt2

P = T = d2/dt2, acceleration is the second

derivative of angular displacement w.r.t. time

= d/dt, speed is the first derivative

Accelerating Power, Pacc

Pacc = Pmech - Pelec

Steady State => No acceleration Pacc = 0 => Pmech = Pelec

Classical Generator Model

Generator connected to Infinite bus through 2 lossless transmission lines

E’ and xd’ are constants is governed by the swing equation

Simplifying the system . . .

Combine xd’ & XL1 & XL2

jXT = jxd’ + jXL1 || jXL2

The simplified system . . .

Recall the power-angle curve

Pelec = E’ |VR| sin( ) XT

Use power-angle curve

Determine steady state (SEP)

Fault study

Pre-fault => system as given Fault => Short circuit at infinite bus

Pelec = [E’(0)/ jXT]sin() = 0

Post-Fault => Open one transmission line XT2 = xd’ + XL2 > XT

Power angle curves

Graphical illustration of the fault study

Equal Area Criterion

2H d2 = Pacc

o dt2

rearrange & multiply both sides by 2d/dt

2 d d2 = o Pacc d dt dt2 H dt

=>d {d}2 = o Pacc ddt {dt } H dt

Integrating,

{d}2 = o Pacc d{dt} H dt

For the system to be stable, must go through a maximum => d/dt must go through zero. Thus . . . m

o Pacc d = 0 = { d2

H { dt } o

The equal area criterion . . .

For the total area to be zero, the positive part must equal the negative part. (A1 = A2)

Pacc d = A1 <= “Positive” Area

Pacc d = A2 <= “Negative” Area

cl

o

m

cl

For the system to be stable for a given clearing angle , there must be sufficient area under the curve for A2 to “cover” A1.

In-class Exercise . . .

Draw a P- curve

For a clearing angle of 80 degrees is the system stable? what is the maximum angle?

For a clearing angle of 120 degrees is the system stable? what is the maximum angle?

Clearing at 80 degrees

Clearing at 120 degrees

What would plots of vs. t look like for these 2 cases?