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A Power System Example

A Power System Example. Starrett Mini-Lecture #3. Power System Equations. Start with Newton again .... T = I a We want to describe the motion of the rotating masses of the generators in the system. The swing equation. 2H d 2 d = P acc w o dt 2 P = T w

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A Power System Example

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  1. A Power System Example Starrett Mini-Lecture #3

  2. Power System Equations Start with Newton again .... T = I a We want to describe the motion of the rotating masses of the generators in the system

  3. The swing equation • 2H d2d = Paccwo dt2 • P = T w • a = d2d/dt2, acceleration is the second derivative of angular displacement w.r.t. time • w = dd/dt, speed is the first derivative

  4. Accelerating Power, Pacc • Pacc = Pmech - Pelec • Steady State => No acceleration • Pacc = 0 => Pmech = Pelec

  5. Classical Generator Model • Generator connected to Infinite bus through 2 lossless transmission lines • E’ and xd’ are constants • d is governed by the swing equation

  6. Simplifying the system . . . • Combine xd’ & XL1 & XL2 • jXT = jxd’ + jXL1 || jXL2 • The simplified system . . .

  7. Recall the power-angle curve • Pelec = E’ |VR| sin( d ) XT

  8. Use power-angle curve • Determine steady state (SEP)

  9. Fault study • Pre-fault => system as given • Fault => Short circuit at infinite bus • Pelec = [E’(0)/ jXT]sin(d) = 0 • Post-Fault => Open one transmission line • XT2 = xd’ + XL2 > XT

  10. Power angle curves

  11. Graphical illustration of the fault study

  12. Equal Area Criterion • 2Hd2d = Paccwo dt2 • rearrange & multiply both sides by 2dd/dt • 2 ddd2d = wo Paccdd dt dt2 H dt => d {dd}2 = wo Pacc dd dt {dt } H dt

  13. Integrating, • {dd}2 = wo Pacc dd{dt} H dt • For the system to be stable, d must go through a maximum => dd/dt must go through zero. Thus . . .dm • wo Pacc dd = 0 = { dd }2 • H { dt } • do

  14. The equal area criterion . . . • For the total area to be zero, the positive part must equal the negative part. (A1 = A2) • Pacc dd = A1 <= “Positive” Area • Pacc dd = A2 <= “Negative” Area dcl do dm dcl

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

  16. In-class Exercise . . . • Draw a P-d 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?

  17. Clearing at 80 degrees

  18. Clearing at 120 degrees

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

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