Overview • Simple generator connected to power system model o Infinite bus = rest of power system
• The Power Angle o Simple generator model o Power – Angle graph
• Real power transfer – simple expression • Power-angle graph
EGR 325 November 19, 2014
Recap: Example • A (3-phase) synchronous generator is connected to an infinite bus. o The terminal voltage of the generator is 5 kV o The equivalent field voltage is 4.8 kV. o The synchronous reactance of the generator is 10 Ω.
• Compute the maximum power the generator can deliver before it will be pulled out of synchronism
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Maximum Power Delivery Xs
Ia
P=
Vt
Ef
Vt E f Xs
sin δ
P Pmax P
δl
90o
δ
4
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Power Angle
Power Angle à Springs Analogy • First example: Springs
Xs Ef
Ia
Ia Xs
δ
Vt
Ef
Ia
Vt
θ
o Twisting a stiff spring vs. a weak spring and notice the relative angular position of both ends o Restoring force returns it to its resting position o If you twist too far, it cannot return à “loss of synchronism”
• Second example: Hand generators o Relative angular position of shaft o Before and after a disturbance
• A deceleration or an acceleration • An imbalance of PM and PE
E f = Vt + I a X s
• Power angle
Two Important Angles: • θ = θ__ – θ__ = power factor angle
• δ = δ__ – δ__ = “power angle” 5
Power Angle • Generators rotate with angular velocity ωm o This is the angular velocity of the ‘rotor’
• δm = rotor angular position with respect to a synchronously rotating reference • δm is the power angle • This is also the phase angle of the voltage phasor • The mechanical angle is the electrical angle o Coupling of the electro-mechanical system
o Angular position of the generator rotors, o (Relative to a rotating synchronous position) o Angular position of rotor (mechanical angle) = phase angle of voltage phasor (electrical angle) = power angle
Power Delivered Across a Line P=
V1 V2 sin δ X
• This mechanical angle is the electrical phase Ø Coupling of the electro-mechanical system
• What is the role of this “power angle?” o We know Z (X) is a fixed parameter. o Goal of good system operations is to keep|Vi|, voltage magnitude, nearly constant
• This means that δ is what we change in order to change real power flow o How does an operator change δ?
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Connection Through a Transmission Line… Xs V t I a
Pm
• Assume a step change to PM changing mechanical power from pm0 to pm1
Vo
Xl
G
Graphically: PM & PE vs. δ
P Pmax
P= 90o
δl
Vt V0 Xl
sin δ
δ
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Interpreting Dynamics
GSO reading
Interpreting Dynamics
• In steady-state pe = pm = pm0 andδ = δ0
• But pm(0+) = pm1
• A step change in pm from pm0 to pm1 occurs at time t = 0. • Due to rotor inertia, the rotor position cannot change instantaneously
• This means that pm(0+) > pe(0+)
o pe0 = pm0 = pmax sin(δ0)
o δ(0+) =δ(0-) = δ0
• This means that electrical power output remains unchanged o pe(0+)
=
pe(0-)
o Mechanical power (energy) has changed o i.e., supply > demand
• So, there is a positive, momentary, acceleration of the rotor • The rotor accelerates and δ increases o Recall δ is the angular difference between the rotor positions at either end of the lines (as well as the difference in their voltage phase angles) o Until pe = pm1 at point δ= δ1
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Assume a sudden change to PM changing mechanical power from Pm0 to Pm1
To Increase Power Delivery
Pm
Xs V t I a
Vo
Xl
G
Pm > Pe = area above curve
Pm < Pe = area below curve
Vt V0
P=
Xl
sin δ
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HW Question 1
Dynamics • Steady-state, point a o Pm0 = Pe0 = Pmax sin(δ0)
• Suddenly, Pm increases!
o Perhaps a steam valve was opened o This causes the rotor speed to increase
• Accelerating power, Pa = Pm1 – Pe
o This causes the rotor speed and rotor angle to increase, momentarily
• With an increase in δ, the power delivery also increases P=
Vt V0 Xl
sin δ
• A synchronous generator is connected to an infinite bus through a transmission line. The infinite bus voltage is 15kV and the equivalent field voltage of the machine is 14kV. The transmission line inductive reactance is 4Ω, and the synchronous reactance of the machine is 5Ω. o Compute the (power) transfer capability of the system. o If a 2Ω capacitor is connected in series with the transmission line, compute the new capacity of the system.
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HW Question 2 • A 100 MVA synchronous generator is connected to a 25kV infinite bus through two parallel transmission lines. • The synchronous reactance of the generator is 2.5Ω, and the inductive reactance of each transmission line is 2Ω. • The generator delivers 100 MVA to the infinite bus at 0.8 power factor lagging. • Suppose a lightning strike causes one of the transmission lines to open. Assume that the mechanical power and excitation of the generator are unchanged. • Can the generator still deliver the same amount of power to the infinite bus?
Renewables and Power Balance Mechanical Turbine Electrical Power Load/Demand
• The net load variability with wind and solar variability directly affects the grid frequency, and can harm load motors
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A disturbance can also be loss of a Tx line – changing ‘Xeq’ – changing Pmax – changing the curve itself
Restarting a System • New Jersey and New York remained in electrical blackout longer than expected. • Once all the transmission lines are reconnected, what do power system engineers need to do to be able to restart the system? o They cannot simply start all the generators up separately and say they are done – why not?
• What can the power transfer equation tell us? • What can knowledge of reactive power, Q, tell us?
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Summary
Ancillary Services
• Power delivery into a power system o Role of the “power angle”
• Power system dynamics o Spring analogy o Coupling of mechanical and electrical elements via the power angle o Loss of a transmission line o Adding series compensation o Variability of wind and solar power – maintaining system energy balance o Restarting a system after a blackout
• To have generators that can provide specific services, not only low cost o Already discussed the need for ramping
• What services are required to support the transmission of energy from generators to customers?
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Ancillary Services (CAISO) • Regulation (frequency stays at 60 Hz) • Spinning Reserve • Non-Spinning Reserve • Voltage Support • Black Start and others as identified… • Load following • Ramping service
