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Published byJacqueline Billingsley Modified about 1 year ago

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

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How about this response? Time Rotor Angle

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Compare these two responses Time Rotor Angle

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What about these responses? Time Rotor Angle

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Compare these instabilities Time Rotor Angle

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

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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

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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?

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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"

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

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Time Frame l Typical time frame of concern y1 - 30 seconds l Model system components that are "active" in this time scale l Faster changes -> assume instantaneous l Slower changes -> assume constants

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Primary components to be modeled l Synchronous generators

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

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

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

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May be used to help control transient stability problems

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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

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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

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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

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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

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Mass-Spring Analogy l Mass-Spring System

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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

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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

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

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

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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 ’

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

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

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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

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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

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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

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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

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

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

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

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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

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Power angle curves

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Graphical illustration of the fault study

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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

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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

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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 oo mm

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

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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?

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Clearing at 80 degrees

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Clearing at 120 degrees

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

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