This document presents a dissertation defense on non-intrusive methods for mode estimation in power systems using synchrophasors, focusing on oscillations, mode identification principles, and optimal probing design. It discusses how to enhance mode estimation accuracy despite challenges such as non-white loads and ambient excitation, and outlines a prototype implementation for real-time monitoring. The findings suggest that well-designed probing signals can significantly improve estimation accuracy while minimizing costs and negative influences from non-white noise.

1. Non-intrusive Methods for Mode
Estimation in Power Systems using
Synchrophasors
Vedran S. Perić
March 15, 2016PhD Defense Presentation

2. Outline
1
• Oscillations – modes, ambient excitation
2
• Principles of mode estimation
3
• Optimal probing design and signal selection
4
• Mode estimation with non-white loads
5
• Prototype implementation
6
• Conclusions
2/28

3. Oscillations
 General phenomenon
 Excitation types
 Omnipresent random
excitation (ambient)
 Intrinsic property of
the system (modes)
3/28

4. Electromechanical oscillations in
power systems
”The largest and most complex machine
ever built by humankind”
4/28

5. 49.85
49.9
49.95
50
50.05
50.1
50.15
08:00:00 08:05:00 08:10:00 08:15:00
f[Hz]
20110219_0755-0825
Freq. Mettlen Freq. Brindisi Freq. Kassoe
Why do we care ?
PMU Data
Oscillations if lightly damped can lead to a system black-out
Occupy transmission capacities, increase losses, wear and tear
February 19th 2011 – North-South Inter-Area Oscillation
Continuously monitor frequency and damping
5/28

6. Application objectives
Avoid black-outs
Real-time monitoring
Decision support
tools
Control actions
6/28

7. Outline
1
• Oscillations – ambient excitation, modes
2
• Principles of mode estimation
3
• Optimal probing design and signal selection
4
• Mode estimation with non-white loads
5
• Prototype implementation
6
• Conclusions
7/28

8. Basic principles of mode estimation
Power system
dx/dt=Ax+Bu
y=Cx+Du
Inputs
(load noise)
Outputs
(PMUs)
Deterministic
signal
Exactly known excitation brings new information
that can be used for improved mode identification
Probing signals
FACTS devices
AVR
Turbine governors
H(θ,z)
G(θ,z)
e(t) y(t)
u(t) - designed
input signal
Aggregated
load noise
Single output model
8/28

9.  Model structure of the power system
o ARMAX
o Box Jenkins
Mathematical formulation
 Optimization problem:
min
𝜃
1
𝑁
𝑡=1
𝑁
𝜀 𝑡, 𝜃 2 𝜀(𝑡, 𝜃) = 𝑦(𝑡) − 𝑦
⌢
(𝑡|𝑡 − 1
Solution - identified model
( , ) ( , )
( ) ( ) ( )
( , ) ( , )
B z C z
y t u t e t
A z D z
 
 
  Contain information about
the critical modes/poles
9/28

10. Model order selection
 Too low model order => bias of the estimate
 Too high model order => large variance/inaccuracy
Data length Time resolutionvs
Methods:
 Residual analysis
 Singular value analysis
 Akaike Information Criterion
 Variance-Accounted-For
10/28

11. Outline
1
• Oscillations – ambient excitation, modes
2
• Principles of mode estimation
3
• Optimal probing design and signal selection
4
• Mode estimation with non-white loads
5
• Prototype implementation
6
• Conclusions
11/28

12. Optimal probing – problem formulation
 Objective: Identify the
critical damping ratio of G(z)
 

 
         




 
   
       
   
 
1 * *
0 0 0 02
1
( , ) ( , ) ( , ) ( , )
2 2
( )u euu e
N N
P F F d F F d
H(θ,z)
G(θ,z)
e(t) y(t)
u(t) - input signal
load measurement
How should the probing signal look like ?
12/28
Spectrum influences accuracy

1
P Good estimate

1
P Bad estimate
 ( )u
 ( )u
There is a limit how strong probing can be
Stronger probing provides better accuracy

13. Spectrum calculation of the probing signal
1) Control effort 2) System disturbance 3) Accuracy
Objective function
 
 
   
  
   
         
   
 
2
1 2
(t)
min ( ) (s) ( )
2 2u uu
k k
J d G d
Constraint: var( ) T
i i i
e P e r
   r - tolerance
Input power Output power
(frequency deviation)
 Requirements :
Optimization problem in a form of LMI
The solution is the power spectrum of the probing signal
14/28

14. Time domain probing signal realization
 Spectrum calculation (solved)
 Time domain signal realization
LMI
Signal realization
max var(ζ)
Multisine
ACF (rd)
min(upeak
2
/urms
2
)
FIR filter
min(║ r-rd║2
)
white noise - e(t)
u(t)
u(t)
u(t)
Probing Φu(ω) calculation
13/28

15. A method for probing signal realization
 Power spectrum



  ( ) ( )
m
j r
u des
r m
ACF r e
1
1
( ) ( ) ( )
k
k
i
ACF u i u i
k 
 
 
 
 
2
( )
0
( ) ( )
M K
k des
u k
min ACF ACF

 



 Sample autocorrelation
 Optimization
 Efficient recursive algorithm
15/28
Aimed probing
autocorrelation
General expression
for autocorrelation
Time domain signal as
the decision variable

16. Optimal probing signal design results
0 0.5 1 1.5 2 2.5
0
5000
10000
15000
Frequency [Hz]
Powerspectrum
0 0.5 1 1.5 2 2.5
0
1
2
3
4
x 10
5
Frequency [Hz]
Powerspectrum
Input spectrum parameterization
White noise Multi-sine FIR filter
var{u(t)} 10410.0 1441.58 1933.55
var{y(t)} 1.6761 1.598 1.5515
var{uy(t)} 6881.10 2318.81 2518.24
The same accuracy obtained with the 5-7 times weaker
excitation
The same input power provides 4-5 times better accuracy
(0.25*10-5)
Damping variance < 10-5
Benefits of
the proposed
method
16/28
KTH Nordic 32
Minimized
input power
Minimized
disturbance2 critical modes
0.5Hz & 0.76Hz
Reactive power
probing

17. Optimal signal selection for mode
estimation
 Only a few signals are sufficient for accurate estimation
 Large number of signals introduce bias and extensive
computational effort
 Variance of the estimate describes estimation quality
Compute asymptotic variance for each measured signal
Rank the signals
Select the top ones
 

 
          
 

 
   
        
   
 
1 * *
0 0 0 02
1
( , ) ( , ) ( , ) ( , ) ( )
2 2e e u u u
N N
P F F d F F d



    



 
   
 

1 *
0 0
( , ) ( , )
2 e e
N
P F F d
17/28

18. 49 48 50 17 40 31 18 42 44 47
0.09
0.1
0.11
0.12
0.13
0.14
0.15
1 2 3 4 5 6 7 8 9 10
Voltage angles
37 26 36 32 34 39 38 43 22 21
0.15
0.16
0.17
0.18
0.19
0.2
0.21
0.22
0.23
1 2 3 4 5 6 7 8 9 10
Voltage magnitudes
Optimal signal selection results
18/28

19. Outline
1
• Oscillations – ambient excitation, modes
2
• Principles of mode estimation
3
• Optimal probing design and signal selection
4
• Mode estimation with non-white loads
5
• Prototype implementation
6
• Conclusions
19/28

20. Ambient mode estimation with
non-white loads - Approach
20
 Transmission/distribution
border points have PMUs
 Non-white loads are inputs
 Input-output identification
Source: EirGrid plc
Transmission
Distribution
20/28
Non-white noise excitation
corrupts results of the classical
mode estimators

21. Mode estimation considering spectral
characteristics of load
 Direct closed loop identification
 Only modes of the transmission
parts are identified
0 0.5 1 1.5 2
0.495
0.497
0.499
0.501
0.503
0.505

ModeFrequency[Hz]
0 0.5 1 1.5 2 2.5 3
0.495
0.497
0.499
0.501
0.503
0.505
0 0.5 1 1.5 2
1
1.8
2.6
3.4
4.2
5
0 0.5 1 1.5 2 2.5 3
1
1.8
2.6
3.4
4.2
5
Kpf
ModeDamping[%]
Mode frequency vs 
Mode frequency vs Kpf
Mode damping vs 
Mode damping vs Kpf
𝑃𝐿 = 𝑃0
𝑉
𝑉0
𝛼
(1 + 𝐾 𝑝𝑓 𝛥𝑓)
Transmission
Load Power State (V,θ)
Loads
Independent disturbance
(white noise)
21/28
 Load effects are compensated
afterwards

22. Results – Presence of forced oscillations
0 20 40 60 80 100
0
0.2
0.4
0.6
0.8
1
1.2
Damping ratio [%]
Frequency[Hz]
Estimated modes
Real system modes
0 20 40 60 80 100
0
0.2
0.4
0.6
0.8
1
1.2
Damping ratio [%]
Frequency[Hz]
Estimated modes
Real system modes
Yule-Walker method
Forced oscillation (0.45 Hz)
0 20 40 60 80 100
0
0.2
0.4
0.6
0.8
1
1.2
Damping ratio [%]
Frequency[Hz]
Estimated modes
Real system modes
N4SID method
 KTH Nordic32 system with forced oscillation at 0.45 Hz
 Monte Carlo simulations (large number of mode estimations)
 Classical methods:
– Forced oscillation estimated as a critical mode
– Real system mode masked by the forced oscillation
Classical methods Proposed method
22/28
Proposed method discerns the forced oscillation

23. Results – white noise input
 When all assumptions used in
classical mode estimators are
fully satisfied
 All methods provide similar
results
0 20 40 60 80 100
0
0.2
0.4
0.6
0.8
1
1.2
Damping ratio [%]
Frequency[Hz]
Estimated modes
Real system modes
0 20 40 60 80 100
0
0.2
0.4
0.6
0.8
1
1.2
Damping ratio [%]
Frequency[Hz]
Estimated modes
Real system modes
0 20 40 60 80 100
0
0.2
0.4
0.6
0.8
1
1.2
Damping ratio [%]
Frequency[Hz]
Estimated modes
Real system modes
Classical methods
Yule-Walker method N4SID method
Parameters
Yule-
Walker
N4SID
Proposed
Method
Mode1
Mean {f} [Hz] 0.4972 0.4978 0.4966
Mean {} [%] 3.3081 4.1213 3.8104
Var {f} 8.8670e-6 1.6260e-5 1.6592e-5
Var {} 0.2728 0.7942 0.8561
Mode2
Mean {f} [Hz] 0.7334 0.7309 0.7386
Mean {} [%] 3.8184 4.7095 3.5497
Var {f} 1.4072e-5 6.2711e-5 2.3151e-5
Var{} 0.3563 1.7925 0.5965
Proposed method
23/28

24. Outline
1
• Oscillations – ambient excitation, modes
2
• Principles of mode estimation
3
• Optimal probing design and signal selection
4
• Mode estimation with non-white loads
5
• Prototype implementation
6
• Conclusions
24/28

25.  Power system or real-time simulator
 Phasor measurement units (PMUs).
 Phasor data concentrator (PDC)
SDK
PMU 1
PMU 2
PMU n
PDC
Comm.
Network
IEEE C37.118 Protocol
KTH SmarTS Lab
Wide Area Measurements System
Real
system
Real time
simulator
User
application
Source Measurements Comm. Infrastructure Decoder
 Communication Network
 Software Development Kit (SDK)
 User applications
25/28

26. Prototype mode estimation application
 LabVIEW platform
 Real-life data
 The critical mode 0.39 Hz
damping ratio 9 %
(average)
 Other modes are
observable at 0.2 Hz, 1 Hz
and 1.4 Hz
 0.5 Hz mode sporadically
appears as a poorly
damped mode
26/28

27. Conclusions
 A comprehensive way of dealing with mode estimation
uncertainty.
 Probing when carefully designed provides a good mode
estimation accuracy at low cost.
 Some signals provide better mode estimates than others.
Quality signal criterion is proposed.
 A method that eliminates negative effects of non-white ambient
excitation is proposed.
 Testing of the mode estimation tools needs to include other
components of the system.
27/28

28. Thank you!
Questions?
Vedran Perić
email:vperic@kth.se
28/28