Vedran Peric's PhD Defense Presentation: Non-intrusive Methods for Mode Estimation in Power Systems using Synchrophasors
Thesis available at:
http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-182134
Abstract [en]
Real-time monitoring of electromechanical oscillations is of great significance for power system operators; to this aim, software solutions (algorithms) that use synchrophasor measurements have been developed for this purpose. This thesis investigates different approaches for improving mode estimation process by offering new methods and deepening the understanding of different stages in the mode estimation process.
One of the problems tackled in this thesis is the selection of synchrophasor signals used as the input for mode estimation. The proposed selection is performed using a quantitative criterion that is based on the variance of the critical mode estimate. The proposed criterion and associated selection method, offer a systematic and quantitative approach for PMU signal selection. The thesis also analyzes methods for model order selection used in mode estimation. Further, negative effects of forced oscillations and non-white noise load random changes on mode estimation results have been addressed by exploiting the intrinsic power system property that the characteristics of electromechanical modes are predominately determined by the power generation and transmission network.
An improved accuracy of the mode estimation process can be obtained by intentionally injecting a probing disturbance. The thesis presents an optimization method that finds the optimal spectrum of the probing signals. In addition, the probing signal with the optimal spectrum is generated considering arbitrary time domain signal constraints that can be imposed by various probing signal generating devices.
Finally, the thesis provides a comprehensive description of a practical implementation of a real-time mode estimation tool. This includes description of the hardware, software architecture, graphical user interface, as well as details of the most important components such as the Statnett’s SDK that allows easy access to synchrophasor data streams.
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
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
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