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Statistical Early Warning Signs of Instability in Synchrophasor Data

IEEE GMS/PES Synchrophasor Meeting October 12, 2016

Goodarz Ghanavati, Taras Lakoba, Paul Hines*

*To whom all blame is dueFunding gratefully acknowledged: NSF Awards ECCS-1254549, DGE-1144388, DOE Award DE-OE0000447 ,

US Northeast and CanadaAugust 14, 200350 million people

Hines, 25 Jan 2013

California, Arizona, MexicoSeptember 8, 2011

5 million people

Hines, 25 Jan 2013

Photo: Bikas Das/AP PhotoIEEE Spectrum, Oct. 2012

Northern IndiaJuly 30, 2012: 350 million peopleJuly 31, 2012: 700 million people

Hines, 25 Jan 2013

5

Bangledesh. 1 November 2014

6washingtonpost.com

Washington DC, April 7, 2015

http://washingtonpost.com

Situational Awareness

7

Sure enough…statistics can be useful indicators

9

Sure enough…statistics can be useful indicators

9

Cotilla-Sanchez, Hines, Danforth, IEEE Trans Smart Grid, 2012. See also: DeMarco and Berge, IEEE Trans on Ckt & Sys, 1987.Dhople, Chen, DeVille, Domínguez-García, IEEE Trans on Ckt Sys, 2013Podolsky and Turitsyn, arXiv:1307.4318, Jul. 2013.Susuki and Mezic, IEEE Trans. Power Syst., 2012 (and others)

How can we find the useful* statistical early warning signs?

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*Useful: A sign that shows up early enough that we might actually be able to do something about it, even if there is

measurement noise

First let’s define our SDEs

11


11

x = f

�x, y

�

0 = g

�x, y, u

�

Differential equations. (swing eqs., governors, exciters, etc.)

Algebraic equations


11

x = f

�x, y

�

0 = g

�x, y, u

�


Algebraic equationsr.v. for stochastic load perturbations


11

x = f

�x, y

�

0 = g

�x, y, u

�



u = �Eu+ C⇠ Loads modeled as Ornstein– Uhlenbeck process


11

x = f

�x, y

�

0 = g

�x, y, u

�




Encodes corr. time of load fluctuations


11

x = f

�x, y

�

0 = g

�x, y, u

�




Encodes corr. time of load fluctuations

Ind. Gaussian r.v.s, 1% std. dev.

Choose an operating point, and linearize around that point

12

�y =⇥�g

�1y

g

x

�g

�1y

g

u

⇤ �x

�u

�

�x

�u

�=

f

x

� f

y

g

�1y

g

x

�f

y

g

�1y

g

u

0 �E

� �x

�u

�+

0C

�⇠


12

�y =⇥�g

�1y

g

x

�g

�1y

g

u

⇤ �x

�u

�

�x

�u

�=

f

x

� f

y

g

�1y

g

x

�f

y

g

�1y

g

u

0 �E

� �x

�u

�+

0C

�⇠

Jacobian matrix: df/dx


12

�y =⇥�g

�1y

g

x

�g

�1y

g

u

⇤ �x

�u

�

�x

�u

�=

f

x

� f

y

g

�1y

g

x

�f

y

g

�1y

g

u

0 �E

� �x

�u

�+

0C

�⇠


12

�y =⇥�g

�1y

g

x

�g

�1y

g

u

⇤ �x

�u

�

�x

�u

�=

f

x

� f

y

g

�1y

g

x

�f

y

g

�1y

g

u

0 �E

� �x

�u

�+

0C

�⇠

Which gives us a system of SDEs in Ornstein–Uhlenbeck form:


12

�y =⇥�g

�1y

g

x

�g

�1y

g

u

⇤ �x

�u

�

�x

�u

�=

f

x

� f

y

g

�1y

g

x

�f

y

g

�1y

g

u

0 �E

� �x

�u

�+

0C

�⇠

z = Az +B⇠

Which gives us a system of SDEs in Ornstein–Uhlenbeck form:

Now solve the SDEs

13

Now solve the SDEs

13

I’d like to tell you that we came up with new, elegant mathematics to solve. In reality…

Now solve the SDEs

13


A�z + �zAT= �BBT

E

⇥z (t) zT (s)

⇤= exp [�A|t� s|]�z

Now solve the SDEs

13

Lyapanov eq.



E

⇥z (t) zT (s)

⇤= exp [�A|t� s|]�z

Now solve the SDEs

13

Lyapanov eq.


And then reverse the Kron reduction to compute the variance and autocorrelation of voltage and current magnitudes.


E

⇥z (t) zT (s)

⇤= exp [�A|t� s|]�z

and choose a time delay for autocorrelation measurements

14

Check to make sure that the analytical and numerical line up

15

And add measurement noise

16

And add measurement noise

16

Which we can subsequently filter to largely regain our original signal,

with the interesting side-effect that some of the variance now appears as autocorrelation.

At key locations, we can see clear signs of instability in Autocorrelation and Variance

17

At key locations, we can see clear signs of instability in Autocorrelation and Variance

17

How do we measure“detectability”to distinguish

useful statistical signals from non-useful ones?

Which statistics provide useful (detectable)

early warning?

18

Variance of voltages

19

Not useful

Useful

Det

ecta

bilit

y

Why is variance in voltage useful?

20

Autocorrelation of currents

21

Not useful

Useful

Det

ecta

bilit

y

0

0.1

0.2

0.3

Can we find the location/source of a problem given the statistics?

22

Can we find trends that would not show up in mean values?

23

Why not just monitor critical modes/eigenvalues?

24

In summary

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In summary• Autocorrelation and variance are, sometimes, useful

indicators of proximity to instability.

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• Variances of voltages near loads are consistently good indicators of proximity to voltage collapse, even when voltage magnitudes are not.

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• Autocorrelations of currents near generators (particularly smaller ones) are generally good indicators of system-wide stability issues (e.g., inter-area oscillations—Hopf bifurcation)

25




• Autocorrelations of currents near generators (particularly smaller ones) are generally good indicators of system-wide stability issues (e.g., inter-area oscillations—Hopf bifurcation)

• Frequently, fluctuations can identify the locations of emerging problems in the network

25

Statistical Early Warning Signs of Instability in Synchrophasor Data

IEEE GMS/PES Synchrophasor Meeting October 12, 2016

Goodarz Ghanavati, Taras Lakoba, Paul Hines*

*To whom all blame is dueFunding gratefully acknowledged: NSF Awards ECCS-1254549, DGE-1144388, DOE Award DE-OE0000447 ,

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