11/20/14 Power System: Power Delivery and Stability
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
1 11/20/14 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”
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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?
2 11/20/14 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
3 11/20/14 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.
4 11/20/14 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?
5 11/20/14 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
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