power system state estimation under high penetration of
TRANSCRIPT
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Power System State Estimation under High Penetration of
Renewable Energy Sources:
Observability and Detectability Studies
Presented by
Dr. Ning Zhou
Associate Professor
Department of Electrical and Computer Engineering
Binghamton University
Vestal, NY 13902
02/23/2020
Online Webinar hosted by:
IEEE PES Binghamton, Mississippi Chapters, and Dynamic State and Parameter
Estimation Taskforce
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Collaborators and Sponsors of
Presented Work
Collaborators (sorted by their last names)
❑ Shahrokh Akhlaghi (Ulteig Engineers Inc.)
❑ Renke Huang, Zhenyu Huang, Shaobu Wang, et al. (PNNL)
❑ Da Meng, Shuai Lu (EnerMod)
❑ Greg Welch (University of Central Florida)
❑ Junbo Zhao (Mississippi State University)
Sponsors (sorted by their names)
❑ DOE Advanced Grid Modeling program
❑ NSF CAREER grant no. #1845523
1
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Outlines
• Background
• Dynamic State Estimation in Power Systems
• Observability and Detectability Analyses
• Conclusions and Future Work
2
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SCADA/EMS Systems
3Figure from Abur, Ali, and Antonio Gomez Exposito. Power system state estimation:
theory and implementation. CRC press, 2004.
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Critical role of state estimation in
power system operations
4 Figure from Wood, Allen J., Bruce F. Wollenberg, and Gerald B. Sheblé. Power
generation, operation, and control. John Wiley & Sons, 2013.
Inputs:
• Data
• Models
Outputs:
• Estimated states
• Improved models
Goal: Support well-informed decision making.
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Problem Statement
• State Estimator:
– Monitors operating conditions (variables)
of a power grid in a control center
– Supports real-time operations (e.g., ED,
OPF)
• Challenges:
– Noise and even gross errors in
measurements (Inaccuracy)
– Limited number of direct
measurements (limited scope)
5
?
?
Figure from Wood, Allen J., Bruce F. Wollenberg, and Gerald B. Sheblé. Power
generation, operation, and control. John Wiley & Sons, 2013.
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Classical Solutions: SSE
• Static State Estimation (SSE)
– Estimate 𝑥 , 𝑏𝑢𝑠 𝑣𝑜𝑙𝑡𝑎𝑔𝑒 𝑝ℎ𝑎𝑠𝑜𝑟𝑠
– By integrating
• SCADA/PMU measurements: z
• Power flow models
– To
• Filter out noise using spatial redundancy
• Estimate variables that are not measured
• Example Algorithms
– Weighted least squares (WLS)
– Least absolute values (LAV)6
𝑧 = ℎ 𝑥 + 𝑟
X12=0.2 pu
X13=0.4 pu
X23=0.25 pu
?
?
min𝑥
𝐽 𝑥 = 𝑖=1
𝑚 1
𝑅𝑖𝑖𝑧𝑖 − ℎ𝑖 𝑥
min𝑥
𝐽 𝑥 = 𝑧 − ℎ 𝑥 T𝑅−1 𝑧 − ℎ 𝑥
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Limitations of Conventional SSE
• Divergence caused by time-skewed measurements
• Lack of current and future visions
7
Significant frequency deviations from the nominal 60
Hz during the August 14, 2003 Northeast Blackout [1]
[1] Z. Huang, N. Zhou, R. Diao, S. Wang, S. Elbert, D. Meng and S. Lu, "Capturing
real-time power system dynamics: Opportunities and challenges," in 2015 IEEE
Power & Energy Society General Meeting, Denver, CO, USA, 2015.
Communication and
computation delay
Time skew
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Outlines
• Background
• Dynamic State Estimation in Power Systems
• Observability and Detectability Analyses
• Conclusions and Future Work
8
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Our Visions on
Dynamic State Estimation (DSE)
• An integrated dynamic state estimator (iDSE):
– Spatial and temporal correlations
• Features– Future visions
– Accurate estimates
– Fast responses
9
Future
Current Time
Forecasting
Resolution
Forecasting Horizon
Historical Record Forecasted States
Past
Power System
States
Confidence
Intervals
𝑧𝑘 = ℎ 𝑥𝑘 + 𝑣𝑘
𝑥𝑘+1 = 𝑓 (𝑥𝑘 , 𝑢𝑘) + 𝑤𝑘 𝑄 = 𝐸 𝑤𝑘𝑤𝑘𝑇
𝑅 = 𝐸 𝑣𝑘𝑣𝑘𝑇
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Benefits of DSE
not Available to SSE
10
Communication and
computation delay
Time skew
Forecasting
Horizon
▪ SSE provides a vision of past
▪ SSE may suffer from the time
skew problem
❑ iDSE can align the measurements
❑ iDSE can forecast into the future
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Formulation of Dynamic State
Estimation (DSE) Approaches
11
• Two-Step Procedure:– Prediction through dynamic simulation
– Correction to include new measurements
• DSE Algorithms:– EKF (Extended Kalman filter)
– EnKF (Ensemble Kalman filter)
– UKF (Unscented Kalman filter)
– PF (Particle filter)
Prediction
through dynamic simulation
Correction
includes new measurementInitialization
x0, k=1
𝑥𝑘− = 𝑓 (𝑥𝑘−1
+ , 𝑢𝑘−1)
Measurements: zk
𝑥𝑘+ = 𝑥𝑘
− + 𝑧𝑘 − ℎ(𝑥𝑘−)
k=k+1
𝑥𝑘−
𝑥𝑘+
𝑥𝑘+𝑥𝑘−1
+
Q R
Ref: N. Zhou, D. Meng, Z. Huang and G. Welch, "Dynamic State Estimation of a Synchronous Machine Using PMU Data:
A Comparative Study," in IEEE Transactions on Smart Grid, vol. 6, no. 1, pp. 450-460, Jan. 2015
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Practical Imperfections and Solutions
• System’s Non-linearity : interpolation[1]
• Unknown and varying noise covariance: adaptive tuning[2]
12
Residual Analysis:
Prediction
through dynamic simulation
Correction
includes new measurementInitialization
x0, k=1
𝑥𝑘− = 𝑓 (𝑥𝑘−1
+ , 𝑢𝑘−1)
zk
𝑥𝑘+ = 𝑥𝑘
− + 𝑧𝑘 − ℎ(𝑥𝑘−)
k=k+1
𝑥𝑘−
𝑥𝑘+
𝑥𝑘+𝑥𝑘−1
+
Q R
Adaptive Tuning Interpolation
𝑅𝑒𝑠 𝑢𝑎𝑙 = 𝑧𝑘 − ℎ(𝑥𝑘+)
𝑧𝑘 = ℎ 𝑥𝑘 + 𝑣𝑘
𝑥𝑘+1 = 𝑓 (𝑥𝑘, 𝑢𝑘) + 𝑤𝑘
𝑄 = 𝐸 𝑤𝑘𝑤𝑘𝑇
𝑅 = 𝐸 𝑣𝑘𝑣𝑘𝑇
[1] S. Akhlaghi, N. Zhou and Z. Huang, "A Multi-Step Adaptive Interpolation Approach to Mitigating the Impact of Nonlinearity on Dynamic
State Estimation," in IEEE Transactions on Smart Grid, vol. 9, no. 4, pp. 3102-3111, July 2018
[2] S. Akhlaghi, N. Zhou and Z. Huang, "Adaptive adjustment of noise covariance in Kalman filter for dynamic state estimation," 2017 IEEE
Power & Energy Society General Meeting, Chicago, IL, 2017 (Best Conference Paper on Power System Modeling and Analysis)
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Outlines
• Background
• Dynamic State Estimation in Power Systems
• Observability and Detectability Analyses
• Conclusions and Future Work
13
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Motivation of Observability and
Detectability Analysis for DSE
Objectives of a DSE observer:
• Estimate states (observability & detectability)
• Minimize cost
Approach:
• Measurement placement
• Model setup (inputs/outputs)
14
𝑧𝑘 = ℎ 𝑥𝑘 + 𝑣𝑘
𝒙𝒌+𝟏 = 𝑓 (𝒙𝒌, 𝑢𝑘) + 𝑤𝑘
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Problem Formulation for
Observability and Detectability Analyses
Given zk, uk, h and f
❑ Observability (initial-state estimation):
Can x0 be uniquely determined?
❑ Detectability (asymptotic estimation):
Can xk be uniquely determined as 𝑘 → ∞ ?
❑ Observability Detectability
𝑥0
𝑥𝑘+1=𝒇 (𝑥𝑘,𝒖𝒌)𝑥𝑘
15
𝒛𝒌 = 𝒉 𝑥𝑘
𝑥𝑘+1 = 𝒇 (𝑥𝑘 , 𝒖𝒌)
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Observability Analysis
❑Observability analysis method– Observability matrix (linearization of the analytical models)
– Empirical observability Gramian matrix (simulation models)
– Lie-derivative method (accurate, but high computation cost)
16
𝑦(𝑡) = ℎ 𝑥 𝑡 , 𝑢(𝑡)
ሶ𝑥(𝑡) = 𝑓 (𝑥 𝑡 , 𝑢(𝑡))
𝑦 𝑡 = ҧ𝐶𝑥 𝑡 + ഥ𝐷𝑢(𝑡)
ሶ𝑥(𝑡) = ҧ𝐴𝑥(𝑡) + ത𝐵𝑢(𝑡)෨𝑂 =
ҧ𝐶ҧ𝐶 ҧ𝐴⋮
ҧ𝐶 ҧ𝐴𝑛−1
𝑆𝑆𝑉 = 𝑠𝑣 ෨𝑂SSV: smallest singular values
J. Qi, K. Sun and W. Kang, "Optimal PMU placement for power system dynamic state estimation by using empirical observability Gramian," IEEE Transactions on Power Systems, vol. 30, no. 4, pp. 2041-2054, 2015.
K. Sun, J. Qi and W. Kang, "Power system observability and dynamic state estimation for stability monitoring using synchrophasor measurements," Control Engineering Practice, vol. 53, pp. 160-172, 2016.
A. Rouhani and A. Abur, "Observability analysis for dynamic state estimation of synchronous machines," IEEE Transactions on Power Systems, vol. 32, no. 4, pp. 3168-3175, 2017.
ቐ𝑆𝑆𝑉 = 0 𝑈𝑛𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑏𝑙𝑒
0 < 𝑆𝑆𝑉 ≤ 𝜖 𝑀𝑎𝑟𝑔 𝑛𝑎𝑙𝑙𝑦 𝑂𝑏𝑠𝑒𝑟𝑣𝑎𝑏𝑙𝑒𝑆𝑆𝑉 > 𝜖 𝑂𝑏𝑠𝑒𝑟𝑣𝑎𝑏𝑙𝑒
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Unobservable Systems and States
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ҧ𝑥 𝑡 = 𝑈𝑇𝑥 𝑡
𝒚 𝒕 = ഥ𝑪 ҧ𝑥 𝑡 + ഥ𝐷𝑢(𝑡)
ሶ ҧ𝑥(𝑡) = ҧ𝐴 ҧ𝑥 (𝑡) + ത𝐵𝑢(𝑡)Kalman decomposition
ሶ𝑥𝑛𝑜(𝑡)ሶ𝑥𝑜(𝑡)
=𝐴𝑛𝑜 𝐴12
0 𝐴𝑜
𝑥𝑛𝑜 𝑡
𝑥𝑜 𝑡+
𝐵𝑛𝑜
𝐵𝑜𝑢 𝑡
𝑦 𝑡 = 0 𝐶𝑜𝑥𝑛𝑜 𝑡
𝑥𝑜 𝑡+ 𝐷 𝑢 𝑡
U ֚ kernel ෨𝑂 and its complementary basis
𝑥𝑜 𝑡 ∈ 𝑅𝑟 represents the observable states
𝑥𝑛𝑜 𝑡 ∈ 𝑅𝑛−𝑟 represents the unobservable states
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DSE Observers of the
Unobservable States (xno)• System:
ሶ𝑥𝑛𝑜(𝑡)ሶ𝑥𝑜(𝑡)
=𝐴𝑛𝑜 𝐴12
0 𝐴𝑜
𝑥𝑛𝑜 𝑡
𝑥𝑜 𝑡+
𝐵𝑛𝑜
𝐵𝑜𝑢 𝑡
• Its observer:ሶො𝑥𝑛𝑜(𝑡)ሶො𝑥𝑜(𝑡)
=𝐴𝑛𝑜 𝐴12
0 𝐴𝑜
ො𝑥𝑛𝑜 𝑡
ො𝑥𝑜 𝑡+
𝐵𝑛𝑜
𝐵𝑜𝑢 𝑡
+ 𝑛𝑜
𝑜𝑦 𝑡 − 0 𝐶𝑜
ො𝑥𝑛𝑜 𝑡
ො𝑥𝑜 𝑡− 𝐷𝑢 𝑡
• Estimation error: Δ𝑥 𝑡 = ො𝑥 𝑡 − 𝑥 𝑡
Δ ሶ𝑥𝑛𝑜(𝑡)Δ ሶ𝑥𝑜(𝑡)
=𝐴𝑛𝑜 𝐴12 − 𝑛𝑜𝐶𝑜
0 𝐴𝑜 − 𝑜𝐶𝑜
Δ𝑥𝑛𝑜 𝑡
Δ𝑥𝑜 𝑡
18
𝑦 𝑡 = 0 𝐶𝑜𝑥𝑛𝑜 𝑡
𝑥𝑜 𝑡+ 𝐷 𝑢 𝑡
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Detectability Analysis• For estimation errors
• DSE observers converge (i.e., detectable) if
– Δ ሶ𝑥𝑛𝑜 𝑡 → 0
– Δ ሶ𝑥𝑜 𝑡 → 0𝑎𝑠 𝑡 → ∞
• Approach:
– Select model/measurement to make eig(𝐴𝑛𝑜) stable
– Select K0 to make eig(𝐴𝑜 − 𝑜𝐶𝑜) stable
19
Δ ሶ𝑥𝑛𝑜(𝑡)Δ ሶ𝑥𝑜(𝑡)
=𝐴𝑛𝑜 𝐴12 − 𝑛𝑜𝐶𝑜
0 𝐴𝑜 − 𝑜𝐶𝑜
Δ𝑥𝑛𝑜 𝑡
Δ𝑥𝑜 𝑡
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Application in the Selection of
Measurements and Models for DSE
DSE observer converges
(1) if the system is observable;
(2) or if the eigenvalues of
unobservable states are stable.
Take-away point:
Using voltage phasor (𝑉∠𝜃𝑣 ) of
terminal bus as input
→ stability of the model
→ convergence of the DSE
20
5. Identify the unobservable and
marginally observable states (xno) using (8)
2. Linearize the model
around operation points
4. Is the model
observable according to (7)?
6. Are the eigenvalues
of Ano in (8) stable?
Identified a convergent
model for DSE
Candidate models with different
measurement setups for DSE
Yes
No
No
A divergent
model for DSE
3. Is the model stable?
No
Yes
Yes
1. Select one model
𝑉∠𝜃𝑣
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Case Studies
21
30
2
1
G10
G1
39
G8
37
25
35
22
38
29
36
23
33
19
34
20
32
3110
6
9 8
5
7
4
3
11
12
13
1
4
1
5
18 17
27
26
16
28
21
24
G9
G2
G3
G5
G6
G7
G4
One-line diagram of IEEE 10-machine 39-bus system
PMU
Generator’s ODEs:
𝛿
𝑡= 𝜔 − 𝜔𝑠 , (A1)
𝜔
𝑡=
𝜔𝑠
2𝐻𝑇𝑀 − 𝑃′𝑒 − 𝐷 𝜔 − 𝜔𝑠 , (A2)
𝐸𝑞′
𝑡=
1
𝑇𝑑0′ −𝐸𝑞
′ − 𝑋𝑑 − 𝑋𝑑′ 𝐼𝑑 + 𝐸𝑓𝑑 , (A3)
𝐸𝑑′
𝑡=
1
𝑇𝑞0′ −𝐸𝑑
′ − 𝑋𝑞 − 𝑋𝑞′ 𝐼𝑞 . (A4)
Exciter’s ODEs:
𝐸𝑓𝑑
𝑡=
1
𝑇𝐸− 𝐸 + 𝑆𝐸 𝐸𝑓𝑑 𝐸𝑓𝑑 + 𝑉𝑅 , (A5)
𝑉𝐹
𝑡=
1
𝑇𝐹−𝑉𝐹 +
𝐹
𝑇𝐸𝑉𝑅 −
𝐹
𝑇𝐸 𝐸 + 𝑆𝐸 𝐸𝑓𝑑 𝐸𝑓𝑑 , (A6)
𝑉𝑅
𝑡=
1
𝑇𝐴−𝑉𝑅 + 𝐴 𝑉𝑟𝑒𝑓 − 𝑉𝐹 − 𝑉 . (A7)
Turbine-governor’s ODEs:
𝑇𝑀
𝑡=
1
𝑇𝐶𝐻−𝑇𝑀 + 𝑃𝑆𝑉 , (A8)
𝑃𝑆𝑉
𝑡=
1
𝑇𝑆𝑉−𝑃𝑆𝑉 + 𝑃𝐶 −
1
𝑅𝐷
𝜔
𝜔𝑠− 1 . (A9)
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Testing Models
22
Model Case (a) (b) (c) (d)
Inputs Voltage phasor (𝓥) Voltage phasor (𝓥) Current phasor (𝓘) Current phasor (𝓘)
Outputs Pe & Qe None Pe & Qe None
UKF Converged Converged Converged Diverged
Smallest Singular Value 7.03 ± 3.90 × 10−9 0 ± 0 9.56 ± 12.90 × 10−9 0 ± 0
System observability Marginally Observable Unobservable Marginally Observable Unobservable
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(b): Unobservable and Stable
23
Model Case Case (a) Case (b) Case (c) Case (d)
Inputs Voltage phasor (𝓥) Voltage phasor (𝓥) Current phasor (𝓘) Current phasor (𝓘)
Outputs Pe & Qe None Pe & Qe None
UKF Converged Converged Converged Diverged
Observability
Gramian
SSV 7.03 ± 3.90 × 10−9 0 ± 0 9.56 ± 12.90 × 10−9 0 ± 0
Relative SSV 1.39 ± 0.77 × 10−11 (N/A) 1.45 ± 1.95 × 10−12 (N/A)
Observability
matrix
SSV 0.0959 ± 0.0007 0 ± 0 0.0357 ± 0.00003 0 ± 0
Relative SSV 8.32 ± 0.04 × 10−9 (N/A) 1.70 ± 0.002 × 10−8 (N/A)
Proposed
method
𝝁𝑹 0.0138 ± 0.0001 0 ± 0 0.0135 ± 0.00001 0 ± 0
𝝁′𝑹 1.37 ± 0.01 × 10−4 (N/A) 9.37 ± 0.01 × 10−5 (N/A)
Eigenvalue of the marginally
observable state−0.4878 (N/A) −0.4820 (N/A)
Rightmost eigenvalue of the system −0.0606 ± 0.0002 −0.0606 ± 0.0002 1.7641 ± 0.0013 1.7641 ± 0.0013
System observability Marginally Observable Unobservable Marginally Observable Unobservable
System stability Stable Stable Not Stable Not Stable
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(b) Stability Results in the Convergence of DSE
T’d0𝝀𝒓𝒊𝒈𝒉𝒕𝒎𝒐𝒔𝒕
MSE of
E’d
MSE of
E’q
3.3 -0.213 4.8410-6 1.1410-5
33.0 -0.061 7.5110-6 1.1710-5
330.0 -0.007 23.1010-6 6.2810-5
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StatesDominant Participation Factor of
𝝀𝒓𝒊𝒈𝒉𝒕𝒎𝒐𝒔𝒕 = −0.0606 ± 0.0002Differential Equations
𝐸𝑑′ 0.07584 ± 0.0003
𝐸𝑑′
𝑡=
1
𝑇𝑞0′ −𝐸𝑑
′ − 𝑋𝑞 − 𝑋𝑞′ 𝐼𝑞 .
𝐸𝑞′ 0.9102 ± 0.0003
𝐸𝑞′
𝑡=
1
𝑇𝑑0′ −𝐸𝑞
′ − 𝑋𝑑 − 𝑋𝑑′ 𝐼𝑑 + 𝐸𝑓𝑑
Δ ሶ𝑥𝑛𝑜(𝑡)Δ ሶ𝑥𝑜(𝑡)
=𝐴𝑛𝑜 𝐴12 − 𝑛𝑜𝐶𝑜
0 𝐴𝑜 − 𝑜𝐶𝑜
Δ𝑥𝑛𝑜 𝑡
Δ𝑥𝑜 𝑡
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(c): Marginally Observable & Unstable
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Model Case Case (a) Case (b) Case (c) Case (d)
Inputs Voltage phasor (𝓥) Voltage phasor (𝓥) Current phasor (𝓘) Current phasor (𝓘)
Outputs Pe & Qe None Pe & Qe None
UKF Converged Converged Converged Diverged
Observability
Gramian
SSV 7.03 ± 3.90 × 10−9 0 ± 0 9.56 ± 12.90 × 10−9 0 ± 0
Relative SSV 1.39 ± 0.77 × 10−11 (N/A) 1.45 ± 1.95 × 10−12 (N/A)
Observability
matrix
SSV 0.0959 ± 0.0007 0 ± 0 0.0357 ± 0.00003 0 ± 0
Relative SSV 8.32 ± 0.04 × 10−9 (N/A) 1.70 ± 0.002 × 10−8 (N/A)
Proposed
method
𝝁𝑹 0.0138 ± 0.0001 0 ± 0 0.0135 ± 0.00001 0 ± 0
𝝁′𝑹 1.37 ± 0.01 × 10−4 (N/A) 9.37 ± 0.01 × 10−5 (N/A)
Eigenvalue of the marginally
observable state−0.4878 (N/A) −0.4820 (N/A)
Rightmost eigenvalue of the system −0.0606 ± 0.0002 −0.0606 ± 0.0002 1.7641 ± 0.0013 1.7641 ± 0.0013
System observability Marginally Observable Unobservable Marginally Observable Unobservable
System stability Stable Stable Not Stable Not Stable
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Case (c) Detectability Analysis
26
Cases 𝝀𝒎𝒂𝒓𝒈𝒊𝒏𝒂𝒍𝒍𝒚 𝒐𝒃𝒔𝒆𝒓𝒗𝒂𝒃𝒍𝒆 𝝀𝒐𝒃𝒔𝒆𝒓𝒗𝒂𝒃𝒍𝒆
(𝑪) −0.482 −5.276 ± 7.920𝑗, −3.111, −1.885, −0.377, −0.193 ± 0.159𝑗, 1.785
StatesDominant Participation Factor of
𝝀𝒎𝒂𝒓𝒈𝒊𝒏𝒂𝒍𝒍𝒚 𝒐𝒃𝒔𝒆𝒓𝒗𝒂𝒃𝒍𝒆 = −0. 𝟒𝟖𝟐2Differential Equations
𝐸𝑑′ 1.241
𝐸𝑑′
𝑡=
1
𝑇𝑞0′ −𝐸𝑑
′ − 𝑋𝑞 − 𝑋𝑞′ 𝐼𝑞 .
𝐸𝑞′ −0.285
𝐸𝑞′
𝑡=
1
𝑇𝑑0′ −𝐸𝑞
′ − 𝑋𝑑 − 𝑋𝑑′ 𝐼𝑑 + 𝐸𝑓𝑑
Case𝑻𝒒𝟎
′
(𝒔)eig(Amo)
𝑴𝑺𝑬 𝑬𝒅′
(× 𝟏𝟎−𝟕)
(c.1) 5.5 -0.482 0.724
(c.2) 55.0 -0.025 53.1
(c.3) -55.0 0.025 inf
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Outlines
• Background
• Dynamic State Estimation in Power Systems
• Observability and Detectability Analyses
• Conclusions and Future Work
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Conclusions (Observability & Detectability)
• DSE observers exist if
– Either the system is observable,
– Or the unobservable (or marginally observable) states have
stable eigenvalues.
• For a subsystem in the power grid,
– Models with terminal voltage as their inputs are good
candidates for DSE because nearly all of them are required to be
stable when connected to an infinite bus.
28
[3] N. Zhou, S. Wang, J. Zhao and Z. Huang, "Application of Detectability Analysis for Power System Dynamic State Estimation," in
IEEE Transactions on Power Systems, vol. 35, no. 4, pp. 3274-3277.
[4] N. Zhou, S. Wang, J. Zhao, Z. Huang, R. Huang, "Observability and Detectability Analyses for Dynamic State Estimation of the
Marginally Observable Model of a Synchronous Machine," (submitted).
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Conclusions (DSE)
• Capturing dynamics is necessary because the power
grid is trending to be more dynamic
• Dynamic state estimation:– Synchronize measurement data (overcome the time-skew problem)
– Forward-Looking vision
• Interpolation can be used to mitigate the negative impact
of non-linearity on estimation accuracy[1]
• Adaptive Q &R estimation can improve estimation
accuracy by balance the model noises and
measurement noises[2]
29
[1] Akhlaghi, Zhou, Huang, “A Multi-Step Adaptive Interpolation Approach to Mitigating the Impact of Nonlinearity on Dynamic State
Estimation”, IEEE Transactions on Smart Grid, 2018.
[2] Akhlaghi, Zhou, Huang, "Adaptive Adjustment of Noise Covariance in Kalman Filter for Dynamic State Estimation." 2017 IEEE
Power and Energy Society General Meeting (Best Conference Paper)..