Download Slide 1

Session 1b.
Electric Power Systems and How
They Work
1
What really makes power flow ?
• The amount of power flowing over the bulk
power system is proportional to the
difference between the power angles of
the machines
• Actually, proportional to the sine of the
angle
2
3
Power Swing Equation
P12
=
E1 E2 sine δ12
_____________
X12
Note: Above equation can be derived from Kirchhoff’s Laws
Note: Said derivation intended only for professional drivers
on a closed course – do not attempt this at home!
4
5
Stability
The property of a system by virtue of
which it will attain a new steady state
condition following any one of a list of
specified contingencies.
Not unlike an Alexander Calder mobile.
6
Stability (cont.)
• Simulation of the dynamic response of the
system, particularly the generators, to a sudden
contingency.
• Normally judged by examining plots of the power
angles of the various generators vs. time.
• In a stable case, the generator power angles will
oscillate, but settle at a new equilibrium. In an
unstable case, they will diverge, usually quite
dramatically.
7
8
9
10
Electric Systems of Pangaea
11
12
VARs
(Volt Amperes Reactive)
• VARs are based on “Imaginary Numbers”
• What’s an “Imaginary Number” ?
– The square root of 9 is 3
– The square root of 4 is 2
– The square root of 1 is 1
– What’s the square root of minus 1 ?
A number whose square is a minus
13
Blame it on the Italians
• 16th Century – Italian mathematicians
– Cubic & quadratic equations
– Solutions to some didn’t make sense
– Square root of minus 1
• Raffael Bombelli
– Developed first consistent theory of imaginary
and complex numbers c1550
– Published Algebra in 1572
14
Bombelli’s Great Insight
“Imaginary” numbers include the square root of
minus 1 – “real” numbers do not
Cannot combine “real” and “imaginary” numbers
• Ergo, must express them as “complex numbers”
with 2 distinct parts
• Form: a + bi (or, a + jb)
– where a is the “real” part and b is the “imaginary” part
– and i (or j) is the square root of minus 1
15
Bombelli’s Algebra
• What Newton’s Principia is to Physics …
• What the Thomas Aquinas’ Summa is to
Theology …
• Bombelli’s Algebra is to Mathematics
16
VARs (cont.)
WATTs / VARs
Real / Imaginary
Active / Reactive
VARs = VOLTAGE CONTROL
• Too few VARs, voltage goes down
• Too many VARs, voltage goes up
• Can’t live without ’em, can’t shoot ’em !
(apologies to Stephen Wright)
17
18
VARs (cont.)
•
•
•
•
Need VARs all along the way
Can’t move WATTs without VARs
VARs don’t travel well
Generally, they need to be produced
locally
• That’s because X >> R
• “You want VARs with that?”
19
20
Transmission Losses
Examples
New York “Central East”
• 1 – 765 kV line operated at 345 kV
• 1 – 345 kV line
• 2 – 230 kV lines, a number of 115 kV lines
At interface loading of about 3000 MW, losses are:
• 87.7 MW
• 861 MVAR (717 net after line charging)
Typical ECAR (Ohio Valley) to NPCC
• Incr. MW losses about 10% of incr. transfer
21
Where do VARs Come From?
(And Where do they Go to Die?)
VARs are produced by :
• generators & synchronous condensers
• capacitors (or capacitor banks)
• transmission lines (capacitive effect – “line charging”)
VARs are consumed by :
• the customers’ electrical load
• series reactors
• transmission lines (inductance – creates magnetic field)
• generators & synchronous condensers
French blackout – 12/7/78 – insufficient VARs
22
VAR Analogies
VARs are like your brother-in-law; they do no work.
But, unlike your brother-in-law, they’re absolutely
essential.
•
•
•
•
•
•
Steam locomotive
Dyeing fabric
Carrier wave
Telephone poles
Head on the beer (?)
Pushing car across inclined plane
23
VAR Analogies (cont.)
• You can think of power flow as consisting of two
separate and distinct kinds of STUFF -- which
can’t be mixed together, so you always have to
identify them separately.
• If you were ordering 6 hamburgers and 4 orders
of fries, you wouldn’t try to combine them and
say, “I want 10 (somethings)” !!
• NO! You’d say, “I want 6 hamburgers and 4
fries.”
24
Loehr’s First Law
• VARs roll down the voltage hill
– They flow in accordance with differences in
voltage magnitude
• WATTs roll down the angle hill
– They flow in accordance with differences in
voltage angle
25
Useful Approximation
Difference in voltage magnitude
equals
MVAR flow
times
inductive reactance (in per unit)
E2 - E1
=
Q
x
X12
26
Prof. Vladimir Karapetoff (Cornell Univ.)
upon death of Steinmetz:
“He was allowed to try to generate
electricity out of the square root of
minus one.”
27