Power System State Estimation under High Penetration of

Renewable Energy Sources:

Observability and Detectability Studies

Presented by

Dr. Ning Zhou

ningzhou@Binghamton.edu

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

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

Outlines

• Background

• Dynamic State Estimation in Power Systems

• Observability and Detectability Analyses

• Conclusions and Future Work

2

SCADA/EMS Systems

3Figure from Abur, Ali, and Antonio Gomez Exposito. Power system state estimation:

theory and implementation. CRC press, 2004.

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.

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.

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

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

Outlines

• Background

• Dynamic State Estimation in Power Systems

• Observability and Detectability Analyses

• Conclusions and Future Work

8

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

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

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

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)

Outlines

• Background

• Dynamic State Estimation in Power Systems

• Observability and Detectability Analyses

• Conclusions and Future Work

13

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

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)

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

Unobservable Systems and States

17

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

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

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

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

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)

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

(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

(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

24

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

(c): Marginally Observable & Unstable

25

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

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

Outlines

• Background

• Dynamic State Estimation in Power Systems

• Observability and Detectability Analyses

• Conclusions and Future Work

27

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).

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)..

Questions or Comments?

Ning Zhou

email: ningzhou@binghamto.edu

phone: (607)777-3195

30