Hines early warning - IEEEsites.ieee.org/gms-pes/files/2016/10/Statistical-Early-Warning... · In summary • Autocorrelation and variance are, sometimes, useful indicators of proximity

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

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Bangledesh. 1 November 2014

6washingtonpost.com

Washington DC, April 7, 2015

http://washingtonpost.com

Situational Awareness

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Sure enough…statistics can be useful indicators

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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

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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


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x = f

�x, y

�

0 = g

�x, y, u

�



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


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x = f

�x, y

�

0 = g

�x, y, u

�




Encodes corr. time of load fluctuations


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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

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�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

�⇠


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�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


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�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

�⇠


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�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:


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�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⇠



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�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⇠


Now solve the SDEs

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Now solve the SDEs

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

Now solve the SDEs

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Now solve the SDEs

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A�z + �zAT= �BBT

E

⇥z (t) zT (s)

⇤= exp [�A|t� s|]�z

Now solve the SDEs

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Lyapanov eq.



E

⇥z (t) zT (s)

⇤= exp [�A|t� s|]�z

Now solve the SDEs

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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

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Check to make sure that the analytical and numerical line up

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And add measurement noise

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And add measurement noise

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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

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How do we measure“detectability”to distinguish

useful statistical signals from non-useful ones?


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How do we measure“detectability”to distinguish

useful statistical signals from non-useful ones?

Which statistics provide useful (detectable)

early warning?

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Variance of voltages

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Not useful

Useful

Det

ecta

bilit

y

Why is variance in voltage useful?

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Autocorrelation of currents

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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?

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

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Why not just monitor critical modes/eigenvalues?

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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)

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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)

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

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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 ,

Hines early warning - IEEEsites.ieee.org/gms-pes/files/2016/10/Statistical-Early-Warning... · In summary • Autocorrelation and variance are, sometimes, useful indicators of proximity

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