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What do you think about this system response? Time Rotor Angle.

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Presentation on theme: "What do you think about this system response? Time Rotor Angle."— Presentation transcript:

1 What do you think about this system response? Time Rotor Angle

2 How about this response? Time Rotor Angle

3 Compare these two responses Time Rotor Angle

4 What about these responses? Time Rotor Angle

5 Compare these instabilities Time Rotor Angle

6 Steady-state = stable equilibrium l things are not changing l concerned with whether the system variables are within the correct limits

7 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

8 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?

9 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"

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

11 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

12 Primary components to be modeled l Synchronous generators

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

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

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

16 May be used to help control transient stability problems

17 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

18 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

19 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

20 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

21 Mass-Spring Analogy l Mass-Spring System

22 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

23 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

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

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

26 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 ’

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

28 Transient Stability? l Does x tend to become unbounded? l Do any of the system variables violate limits in the transition?

29 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

30 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

31 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

32 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

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

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

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

36 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

37 Power angle curves

38 Graphical illustration of the fault study

39 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

40 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

41 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

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

43 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?

44 Clearing at 80 degrees

45 Clearing at 120 degrees

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


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