Transcript Slide 1

ChE 391, Spring 2012
Power Systems Control
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© Alexis Kwasinski, 2012
Introduction
• Control variables in dc power systems
• Voltage
v(t )  V
• Control variables in ac power systems:
• Voltage amplitude
• Phase: (angular) frequency and angle
v(t )  V cos(t  V )
• Phasors
• Used to represent ac signals in single-frequency systems through
a fixed vector in the complex plane.
Imaginary
v(t )  Real(Ve jV e jt )
V
V  V V
2
V
© Alexis Kwasinski, 2012
Real
Introduction
• Power in ac systems
• Instantaneous power:
p(t )  v(t )i(t )  V cos(t  V ) I cos(t   I )
p(t )  v(t )i(t ) 
VI
cos(V  I )  cos(2t  V  I )
2
Constant part
• Real power: related with irreversible energy exchanges (work or
dissipated heat). That is, real power represents energy that leaves
or enters the electrical circuit under analysis per unit of time, so the
energy exchanges occur between the circuit and its environment.
P
VI
V I
cos(V  I ) 
cos(V  I )
2
2 2
P  VRMS I RMS cos( )
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© Alexis Kwasinski, 2012
V   I  
Introduction
• Power in ac systems
• Reactive power: related with reversible energy exchanges. That is,
reactive power represents energy that is exchanged between the
circuit and electric or magnetic fields in a cyclic way. During half of
the cycle energy from the sources are used to build electric fields
(charge capacitors) or magnetic fields (charge machines) and during
the other half cycle exactly the same energy is returned to the
source(s).
Q  VRMS I RMS sin( )
e.g. in an inductor:
P0
VI
I ( L) I LI 2 2
Q  sin(90) 

2
2
2 T
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© Alexis Kwasinski, 2012
Introduction
• Power in ac systems
• Complex power
• Notice that
1
1
1
P  Real(VI* )  Real VI V   I  Real VI  
2
2
2
• and that
1
1
1
Q  Imaginary(VI* )  Imaginary VI V   I  Imaginary VI  
2
2
2
• So a magnitude called complex power S is defined as



S  VI*  VRMS I RMS (cos   j sin  )
2
S  I RMS
( R  jX )

Q > 0 (inductive load)
Q = 0 (resistive load)
Q < 0 (capacitive load)
• Power factor (in power systems with one frequency) is defined as
P
p. f .  cos   
S
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P
P2  Q2
• It provides an idea of how efficient is the
process of using (and generating) electrical
power in ac circuits:
© Alexis Kwasinski, 2012
Introduction
• Synchronous generators
• Input:
• Mechanical power applied to the rotor shaft
• Field excitation to create a magnetic field constant in
magnitude and that rotates with the rotor.
• Output:
• P and Q (electric signal with a given frequency for v and i)
Field
Excitation
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Q
© Alexis Kwasinski, 2012
Introduction
• Synchronous generators
• Open circuit voltage:
e  NS
d
dt
ERMS  4.44Kd K p fNS 
E  N S 

1
NR IR
l
A
E
Magneto-motive force
(mmf)
IR
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© Alexis Kwasinski, 2012
Synchronous generators control
• Effect of varying field excitation in synchronous generators:
• When loaded there are two sources of excitation:
• ac current in armature (stator)
• dc current in field winding (rotor)
• If the field current is enough to generate the necessary mmf,
then no magnetizing current is necessary in the armature and
the generator operates at unity power factor (Q = 0).
• If the field current is not enough to generate the necessary
mmf, then the armature needs to provide the additional mmf
through a magnetizing current. Hence, it operates at an inductive
power factor and it is said to be underexcited.
• If the field current is more than enough to generate the
necessary mmf, then the armature needs to provide an opposing
mmf through a magnetizing current of opposing phase. Hence, it
operates at a capacitive power factor and it is said to be
overexcited.
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© Alexis Kwasinski, 2012
Synchronous generators control
• Relationship between reactive power and field excitation
http://baldevchaudhary.blogspot.co
m/2009/11/what-are-v-andinverted-v-curves.html
• The frequency depends on the rotor’s
speed. So frequency is controlled
through the mechanical power.
• Pmec is increased to increase f
• Pmec is decreased to decrease f
Field
Excitation
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© Alexis Kwasinski, 2012
Q
Voltage and frequency control
• The simplified equivalent circuit for a generator and its output equation
is:
Q, p E
LOAD
• Assumption: during short circuits or load changes E is
constant
• V is the output (terminal) voltage
pe 
E.V
E.V
sin  

X
X
Electric power provided to the load
XQ
E V 
E
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© Alexis Kwasinski, 2012
Voltage and frequency control
• It can be found that
d
  (t )  syn
dt
• Generator’s angular frequency
• Grid’s angular frequency
• Ideally, the electrical power equals the mechanical input power.
The generator’s frequency depends dynamically on δ which, in
turn, depends on the electrical power (=input mechanical power).
So by changing the mechanical power, we can dynamically change
the frequency.
• Likewise, the reactive power controls the output voltage of the
generator. When the reactive power increases the output voltage
decreases.
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© Alexis Kwasinski, 2012
Voltage and frequency control
• Droop control
• It is an autonomous approach for controlling frequency and voltage
amplitude of the generator and, eventually, the grid.
• It takes advantage that real power controls frequency and that
reactive power controls voltage
f  f0  kP ( P  P0 )
V  V0  kQ (Q  Q0 )
V
f
f0
V0
P0
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P
© Alexis Kwasinski, 2012
Q0
Q
Voltage and frequency control
• Droop control
•Then a simple (e.g. PI) controller can be implemented. It considers
a reference voltage and a reference frequency:
•If the output voltage is different, the field excitation is changed
(and, thus, changes Q and then V).
•If the frequency is different, the prime mover torque is
changed (and thus, changes P and then f).
V  V0  kQ (Q  Q0 )
f  f0  kP ( P  P0 )
V
f
f0
V0
P0
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P
© Alexis Kwasinski, 2012
Q0
Q
Voltage and frequency control
• Operation of a generator connected to a large grid
• A large grid is seen as an infinite power bus. That is, it is like a
generator in which
• Changes in real power do not cause changes in frequency
• Changes in reactive power do not originate changes in voltage
• Its droop control curves are horizontal lines
V
f
P
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© Alexis Kwasinski, 2012
Q
Voltage and frequency control
• Operator of a generator connected to a large grid
• When connected to the grid, the voltage amplitude and frequency
is set by the grid.
• In order to synchronize the oncoming generator, its frequency
needs to be slightly higher than that of the grid, but all other
variables need to be the same.
V
f
f gen
VG
fG
P
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© Alexis Kwasinski, 2012
Q
Voltage and frequency control
• Operator of a generator connected to a large grid
• After the generator is paralleled to the grid then its output
frequency and voltage will remain fixed and equal to the grid’s
frequency and voltage, respectively.
• Output power is controlled by attempting a change in frequency by
controlling the prime mover’s torque. By “commanding” a decrease
in frequency, the output power will increase.
• A similar approach is followed with reactive power control, by
controlling field excitation in an attempt to change output voltage.
Higher commanded
frequencies
f
Higher power output
Operating frequency
No load droop line
P1
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P2
© Alexis Kwasinski, 2012
P
Stability
• From mechanics
Moment of inertia
d 2 m
J
 Tm (t )  Te (t )
2
dt
angular
acceleration
mechanical
torque
electrical
torque
• If a synchronous reference frame is considered then
m (t )  m,synt   m (t )
Mechanical equivalent of its
electrical homologous
variable
Synchronous speed
# poles
x
xm
2
• Swing equation:
d 2 (t )
 p.u (t )
 pm, p.u . (t )  pe, p.u (t )
2
syn
dt
2H
where “p.u.” indicates per unit and
H
• So if pe  pm
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0.5 J m2 ,syn
Srated
,
m,syn
 p.u (t ) 
m (t )
 (t ) decreases and if pe  pm  (t ) increases
© Alexis Kwasinski, 2012
Stability
• Equal area criterion: Assume that the mechanical power suddenly
increases.
4) Because of rotor inertia
increases up to here
2) The rotor
accelerates
3) The rotor decelerates
pe  pm
pe  pm
5) After oscillating, δ(t)
comes to a rest at
 (t )  1
1) Initial condition
pe 
E.V
sin 
X
• The equal area criterion says that A1 = A2
• If  2   3 the generator looses stability because pe  pm and the
generator continues to accelerate.
• Sudden changes in pm are not common. But changes in pe do happen.
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© Alexis Kwasinski, 2012
Stability
• Equal area criterion during faults a most critical case: a fault.
Both areas are
equal
pe  pm
4) Because of
rotor inertia  (t )
increases up to
here
pe 
1) Initial condition
E.V
sin 
X
pe  pm
3) Fault is
cleared here
2) During the
fault pe = 0
• After reaching  2 δ(t) will oscillate until losses and the load damp
oscillations and  (t )   0
• If  2   3 the generator looses stability because pe  pm and the
generator continues to accelerate.
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© Alexis Kwasinski, 2012
A brief summary
• In ac systems, large machine inertia helps to maintain
stability.
• Since frequency needs to be regulated at a precise value,
imbalances between electric and mechanical power may
make the frequency to change. In order to avoid this issue,
mechanical power applied to the generator rotor must follow
load changes. If the mechanical power cannot follow the
load alone (e.g. due to machine’s inertia), energy storage
must be used to compensate for the difference. This is a
situation often found in microgrids.
•
Reactive power is used to regulate voltage.
•
Droop control is an effective autonomous controller.
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© Alexis Kwasinski, 2012
Some additional comments
• Large machine’s inertia contributes to system stability but makes it
difficult to follow fast changing loads.
• At every time instant the goal is power generation = load + losses.
• Hence a combination of generation technologies are used to achieve
good stability performance while still be able to follow the load.
• A dispatch center solves the power flow equations and commands the
generation units so generation = load + losses in an optimally
economical and technically feasible way (economic dispatch problem).
Summer day
Winter day
1.1
1.1
Peaking plants (gas
turbines and some
diesel
1.0
1.0
0.9
0.9
0.8
0.8
Load following plants
(gas turbines and some
hydro and nuclear)
0.7
0.7
0.6
0.6
0.5
0.5
0
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Base load plants (coalfired and most nuclear
plants)
3
6
9
12
15
18
21
24
0
3
6
© Alexis Kwasinski, 2012
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12
15
18
21
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Some additional comments
• Although not explicitly mentioned before, the analysis considered
some implicit assumptions:
• single phase equivalent for the circuits
• No harmonics
• Linear loads and components (e.g. no saturation in machines)
• Ideal lumped components
• The fast average frequency is the same everywhere in the grid.
• The voltage changes along the grid. Hence, then voltage is
compensated everywhere along the grid with capacitors and voltage
compensators (autotransformers) and other means (e.g. static VAR
compensators).
• Although it seems that control of a dc grid is simpler, the need for
power electronic interfaces create nonlinear instantaneous constantpower loads that have a de-stabilizing effect on the dc grid. Moreover,
autonomous controllers are more difficult to implement because the only
static existing variable is bus voltages.
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© Alexis Kwasinski, 2012
One grid or many grids?
• US: The same nominal frequency but 3 main grids
• Japan: Two different
nominal frequencies and
3 grids
Tie
Source: NPR
Source: Tosaka
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© Alexis Kwasinski, 2012
Examples
Slow Frequency Variation
Wednesday, November 7, 2007, 4:15 PM
Texas
0.1Hz
8 minutes
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© Alexis Kwasinski, 2012
Examples
Large Generator Trip
Tuesday, November 13, 2007, 4:19 PM
Texas
0.16Hz
8 minutes
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© Alexis Kwasinski, 2012
Examples
Unusual Wind-Related Event?
Sunday, May 13, 2007, 3:11am
8 minutes
• Intermittent (non-dispatchable) generation sources, such as wind
generators or PV modules) may have a severe negative effect on
grid’s stability if they are not properly controlled.
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© Alexis Kwasinski, 2012
Examples
Multiple Generator Trips
Saturday, August 25, 2007, 3:32 AM
California
0.10Hz
8 minutes
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© Alexis Kwasinski, 2012
Examples
Onset of Rotating Blackout
Monday, April 17, 2006, 4pm
(Unusually hot day, and many generators out for maintenance)
Texas
0.2Hz
Insufficient
Spinning
Reserve
Generator
Trip
Generator
Trip
Voluntary load
shedding begins
8 minutes
Stage 1 of automatic load shedding (5%) kicks in at 59.7Hz
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© Alexis Kwasinski, 2012