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Computing Power System Steady-state
Stability Using Synchrophasor Data
Karl Reinhard
ECE Power and Energy Group Colloquium
29 Oct 12
work with Peter Sauer and Alejandro Domínguez-García
Conjecture
• Collect synchrophasor measurements at 2 buses directly connected
by transmission Line
• Compute model parameters for a pair of Thevenin sources
• Connect Thevenin sources by the transmission line
2
Conjecture
• Collect synchrophasor measurements at 2 buses directly connected
by transmission Line
• Compute model parameters for a pair of Thevenin sources
• Connect Thevenin sources by the transmission line
• Resulting Thevenin source angle difference (AnglxSys) indicates
system stability stress
• Largest AnglxSys of all directly connected buses is proposed as an
indicator of the risk of losing system stability
3
Purpose
To report progress investigating this conjecture to date
Take Aways
• Synchrophasor data is not suitable for some Thevenin equivalent
formulations  equations poorly conditioned with field measurements
• A reduced Thevenin equivalent system meeting expected power
system physical constraints can be calculated
• Importance of verifying that computed values are consistent
with power system physics and model at each iteration
4
Stability Limits to Power System Operation
Stability
Stability – voltage collapse, steady-state
stability, transient stability, bifurcations
 margins to each critical point –
difficult to assess.
Voltage
Voltage – plus or minus 5% of nominal –
easy to assess from measurements.
Thermal
Thermal – short term and long term –
typically measured in Amps or Power
(MW or MVA) –
easy to assess from measurements.
5
Stability Limits to Power System Operation
300  400 
1.0 SIL = power delivered by a
“lossless” line to a load resistance equal
to the surge (characteristic) impedance

Flat voltage profile along entire line

Voltage and current are in phase along
entire line

VARS into line from shunt charging are exactly
equal to the total line VAR series losses
V2
SIL 
Rc
Voltage
Thermal

69 KV
138 KV
230 KV
345 KV
500 KV
765 KV






Stability
SIL 
L
C
12 MW
50 MW
140 MW
400 MW
1000 MW
2000 MW
6
Estimating Thevenin Equivalents w/ SPD
(SynchroPhasor Data)
E
E  I  ·( R  jX )  V 
Er  R I r  X I i  Vr
Ei  X I r  R I i  Vi
1

0

1
0

0  I r(1)
1
 I i(1)
0  I r( 2 )
1  I i( 2 )
I i(1) 
V
Er   r(1) 


 


 I r(1)  E
Vi(1) 
i
    
I i( 2 )   R  Vr( 2 ) 
   

 I r( 2 )  X  Vi( 2 )

 
7
Estimating Thevenin Equivalents w/ SPD
8
Estimating Thevenin Equivalents w/ SPD
1

0

1
0

0  I r(1)
1
 I i(1)
0  I r( 2)
1  I i( 2)
I i(1) 
V
Er   r(1) 


 
 I r(1)   E  Vi(1) 
i
    
I i( 2)   R  Vr( 2 ) 
   

 I r( 2)  X  Vi( 2)

 
9
Condition No. Analysis – Exact Soln
K ( A)  A
Condition Number
A
1

0
 
1
0

0  I r(1)
1  I i(1)
0  I r( 2 )
1  I i( 2 )
p
I i(1) 
1

0
 I r(1) 
  
I i( 2 ) 
1

0
 I r( 2 ) 
  b( a  c )  a  ( b  d )  ( a  c ) f 
b
a

1 
2
(a  c)
(b  d ) f ( a  c ) f

 a (a  c)  b  (b  d )  ( a  c) f 
a
b
1


2
(a  c) f
(b  d ) f ( a  c) f
A1  

(b  d )  ( a  c ) f
1

2
(a  c) f
(a  c) f


1
1

(a  c) f
(b  d ) f

where
f 
( a  c ) 2  (b  d ) 2
( a  c )(b  d )
A1
p
0 a
1 b
0 c
1 d
b
a 

d 

c 
b( a  c )  a  ( d  b)  ( a  c ) f 
(a  c)2 f
 a ( a  c )  b  ( a  c ) f  (b  d ) 
( a  c) 2 f
( a  c ) f  (b  d )
(a  c)2 f
1
(a  c) f

b
a

(b  d ) f ( a  c ) f 


a
b


(b  d ) f ( a  c) f 

1

(a  c) f


1

(b  d ) f


10
Condition No. Analysis
K ( A)  A
Condition Number
A
A 1 ~ A  ~1
A
1
1
~ A
1

~ 10
3
A 
1
1

0
 
1
0

0  I r(1)
1
 I i(1)
0  I r( 2 )
1  I i( 2 )
p
A1
pp
~ 10 3
I i(1) 
1 0


 I r(1) 
0 1
  
I i( 2 ) 
1 0

 I r( 2 ) 
0 1
101 101 

101 101 
101 101 

101 101 
 b(a  c)  a  (b  d )  (a  c) f 
b
a

1 
2
(

)
(

)
(

a
c
b
d
f
a
c) f

 a (a  c)  b  (b  d )  (a  c) f 
a
b
1


(a  c) 2 f
(b  d ) f (a  c) f


(b  d )  (a  c) f
1

2

a
c) f
a
c
f
(

)
(


1
1

(a  c) f
(b  d ) f

I
I

  (b  d ) ~ 10
r (1)
 I r ( 2 )  ( a  c ) ~ 103
i (1)
 Ii (2)
b ( a  c )  a  ( d  b)  ( a  c ) f 
(a  c) 2 f
a (a  c)  b  (a  c) f  (b  d )
(a  c) 2 f
(a  c) f  (b  d )
(a  c) 2 f
1
(a  c) f

b
a


(b  d ) f ( a  c) f 

a
b


(b  d ) f ( a  c) f 

1

( a  c) f


1

(b  d ) f


3
( a  c ) 2  (b  d ) 2
(103 ) 2  (103 ) 2
f 
~
~ 1
( a  c ) (b  d )
(103 ) (103 )
11
Condition No. Analysis
Condition Number
K ( A)  A
p
A1
p
~ 10 3
12
Condition No. Analysis – Least Squares Estimate
(m x n matrix)
1  1   2   2
K ( A) 
1
,
n
1     i     n
13
Estimating Reduced Thevenin Equivalent w/ SPD
Fix: | E1 |  | E2 | to 1 p.u.
R1  R 2  0
11  I  · ( j X 1 )  V11
sin1 
V2  2  I  · ( j X 2 )  1 2
1  cos2 1
I 2 X 12  2 V1 I sin( ) X 1  (V12  (1) 2 )  0
1
I
1

I
X1 
  V sin(   ) 
1
 1
X2
  V sin(   ) 
2
 2
(1) 2  V12 cos 2 (  1 ) 

(1) 2  V22 cos 2 (   2 ) 

14
Estimating Reduced Thevenin Equivalent w/ SPD
Fix: | E1 |  | E2 | to 1 p.u.
R1  R 2  0
11· I    j ( X 1  X l  X 2 ) I 2  1 2 · I  
*
Real Power Eqn
*
1
I
1

I
X1 
  V sin(   ) 
1
 1
X2
  V sin(   ) 
2
 2
(1) 2  V12 cos 2 (  1 ) 

(1) 2  V22 cos 2 (   2 ) 

cos(1   )  cos( 2   )  1   2
cos ( )  cos ( )
  2  2  1   
 ( X1  X l  X 2 ) I 

2


 ( X  Xl  X 2 )I 
 2    sin 1  1

2


1   2
2
1    sin 1 
15
Estimating Reduced Thevenin Equivalent w/ SPD
1
I
1

I
X1 
  V sin(   ) 
1
 1
X2
  V sin(   ) 
2
 2
(1) 2  V12 cos 2 (  1 ) 

(1) 2  V22 cos 2 (   2 ) 

 ( X1  X l  X 2 ) I 

2


 ( X  Xl  X 2 )I 
 2    sin 1  1

2


1    sin 1 
16
Estimating Reduced Thevenin Equivalent w/ SPD
1
I
1

I
X1 
  V sin(   ) 
1
 1
X2
  V sin(   ) 
2
 2
(1) 2  V12 cos 2 (  1 ) 

(1) 2  V22 cos 2 (   2 ) 

 ( X1  X l  X 2 ) I 

2


 ( X  Xl  X 2 )I 
 2    sin 1  1

2


1    sin 1 
1  1   2   2
17
Estimating Reduced Thevenin Equivalent w/ SPD
1  1   2   2
18
RECAP
Purpose: To report progress investigating the conjecture that a
Thevenin Equivalent from Synchrophasor data indicates system stress
Take Aways
• Synchrophasor data is not suitable for some Thevenin equivalent
formulations  equations poorly conditioned with field measurements
• A reduced Thevenin equivalent system meeting expected power
system physical constraints can be calculated
• Importance of verifying that computed values are consistent
with power system physics and model at each iteration
• Next – Does AnglxSys indicate system stress??
• Simulation using MATLAB and Power World
19
QUESTIONS?
Karl Reinhard
reinhrd2@illinois.edu
20
Phasor Measurement Unit – System Model
21
GPS Timing Data
2 Satellite Signals
Intersection – Circle
3 Satellite Signals
Intersection – 2 Points
4th Satellite – Timing
Signal Correction
Satellite 3
•
•
Due to Clock errors, Unlikely 4th satellite’s sphere will intersect
either of 2 intersection points…
Distance from the valid GPS receiver position estimate to the 4th
satellite sphere surface enables timing error determination /
correction:
da
e( t ) 
c
da
Satellite 4
Receiver
22
GPS Timing Data
23
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