A Power System Example

Starrett Mini-Lecture #3

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?