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 l things are not changing l concerned with whether the system variables are within the correct limits

Transient Stability l "Transient" means changing l The state of the system is changing l We are concerned with the transition from one equilibrium to another l The change is a result of a "large" disturbance

Primary Questions l 1. Does the system reach a new steady state that is acceptable? l 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 l Instability => at least one rotor angle becomes unbounded with respect tothe rest of the system l Also referred to as "going out of step" or "slipping a pole"

Additional Concerns: limits on other system variables l Transient Voltage Dips l Short-term current & power limits

Time Frame l Typical time frame of concern y seconds l Model system components that are "active" in this time scale l Faster changes -> assume instantaneous l Slower changes -> assume constants

Primary components to be modeled l Synchronous generators

Traditional control options l Generation based control yexciters, speed governors, voltage regulators, power system stabilizers

Traditional Transmission Control Devices l Slow changes l modeled as a constant value

FACTS Devices l May respond in the 1-30 second time frame l modeled as active devices

May be used to help control transient stability problems

Kundur's classification of methods for improving T.S. l Minimization of disturbance severity and duration l Increase in forces restoring synchronism l Reduction of accelerating torque by reducing input mechanical power l Reduction of accelerating torque by applying artificial load

Commonly used methods of improving transient stability l 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! l Deregulation & separation of transmission & generation functions of a utility l FACTS devices can help to control transient problems from the transmission system

3 Minute In-Class Activity l 1. Pick a partner l 2. Person wearing the most blue = scribe Other person = speaker l 3. Write a one-sentence definition of "TRANSIENT STABILITY” l 4. Share with the class

Mass-Spring Analogy l Mass-Spring System

Equations of motion l Newton => F = Ma = Mx’’ l Steady-state = Stable equilibrium = Pre-fault  F = -K x - D x’ + w = M ball x’’ = 0 l Can solve for x

Fault-on system l New equation of motion  F = -K x - D x’ + (M ball + M bird )g = (M ball + M bird ) x’’ l Initial Conditions? l x = x ss x’ = 0

How do we determine x(t)? l Solve directly l Numerical methods y(Euler, Runge-Kutta, etc.) l Energy methods

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

Post-fault system l "New" equation of motion  F = -K x - D x’ + w = M ball x’’ l Initial Conditions? l x = x c x’ = x c ’

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

Transient Stability? l Does x tend to become unbounded? l 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 d 2  = P acc  o dt 2 P = T   = d 2  /dt 2, acceleration is the second derivative of angular displacement w.r.t. time  = d  /dt, speed is the first derivative

l Accelerating Power, P acc l P acc = P mech - P elec l Steady State => No acceleration l P acc = 0 => P mech = P elec

Classical Generator Model l Generator connected to Infinite bus through 2 lossless transmission lines l E’ and x d ’ are constants  is governed by the swing equation

Simplifying the system... l Combine x d ’ & X L1 & X L2 l jX T = jx d ’ + jX L1 || jX L2 l The simplified system...

Recall the power-angle curve P elec = E’ |V R | sin(  ) X T

Use power-angle curve l Determine steady state (SEP)

Fault study l Pre-fault => system as given l Fault => Short circuit at infinite bus  P elec = [E’(0)/ jX T ]sin(  ) = 0 l Post-Fault => Open one transmission line yX T2 = x d ’ + X L2 > X T

Power angle curves

Graphical illustration of the fault study

Equal Area Criterion 2H d 2  = P acc  o dt 2 rearrange & multiply both sides by 2d  /dt 2 d  d 2  =  o P acc d  dt dt 2 H dt => d {d  } 2 =  o P acc d  dt {dt } H dt

Integrating, {d  } 2 =  o P acc d  {dt} H dt For the system to be stable,  must go through a maximum => d  /dt must go through zero. Thus...  m  o P acc d  = 0 = { d  2 l H { dt }  o

The equal area criterion... l For the total area to be zero, the positive part must equal the negative part. (A1 = A2) P acc d  = A1 <= “Positive” Area P acc d  = A2 <= “Negative” Area  cl oo mm

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 l For a clearing angle of 80 degrees yis the system stable? ywhat is the maximum angle? l For a clearing angle of 120 degrees yis the system stable? ywhat is the maximum angle?

Clearing at 80 degrees

Clearing at 120 degrees

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