sample average approximation and cascading failures of ...dano/talks/graphex15.pdf · sample...
TRANSCRIPT
![Page 1: Sample average approximation and cascading failures of ...dano/talks/graphex15.pdf · Sample average approximation and cascading failures of power grids Daniel Bienstock, Guy Grebla,](https://reader035.vdocuments.site/reader035/viewer/2022071014/5fcc6dc5b76b0c5bcb1c62a7/html5/thumbnails/1.jpg)
Sample average approximation and cascadingfailures of power grids
Daniel Bienstock, Guy Grebla, Tingjun Chen, Columbia UniversityGraphEx ’15
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August 14 2003
Approximately 50 million people affected
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Other large-scale cascading failures
• Italy, 2003
• San Diego, 2011
• India, 2012
At fault:
unexpected event, cascading mechanism, noise and human error
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Cascade cartoon
Generator
Load (demand)
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Cascade cartoon
![Page 6: Sample average approximation and cascading failures of ...dano/talks/graphex15.pdf · Sample average approximation and cascading failures of power grids Daniel Bienstock, Guy Grebla,](https://reader035.vdocuments.site/reader035/viewer/2022071014/5fcc6dc5b76b0c5bcb1c62a7/html5/thumbnails/6.jpg)
Cascade cartoon
![Page 7: Sample average approximation and cascading failures of ...dano/talks/graphex15.pdf · Sample average approximation and cascading failures of power grids Daniel Bienstock, Guy Grebla,](https://reader035.vdocuments.site/reader035/viewer/2022071014/5fcc6dc5b76b0c5bcb1c62a7/html5/thumbnails/7.jpg)
Cascade cartoon
![Page 8: Sample average approximation and cascading failures of ...dano/talks/graphex15.pdf · Sample average approximation and cascading failures of power grids Daniel Bienstock, Guy Grebla,](https://reader035.vdocuments.site/reader035/viewer/2022071014/5fcc6dc5b76b0c5bcb1c62a7/html5/thumbnails/8.jpg)
Cascade cartoon
![Page 9: Sample average approximation and cascading failures of ...dano/talks/graphex15.pdf · Sample average approximation and cascading failures of power grids Daniel Bienstock, Guy Grebla,](https://reader035.vdocuments.site/reader035/viewer/2022071014/5fcc6dc5b76b0c5bcb1c62a7/html5/thumbnails/9.jpg)
Cascade cartoon
= lost demand
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Cascade cartoon
![Page 11: Sample average approximation and cascading failures of ...dano/talks/graphex15.pdf · Sample average approximation and cascading failures of power grids Daniel Bienstock, Guy Grebla,](https://reader035.vdocuments.site/reader035/viewer/2022071014/5fcc6dc5b76b0c5bcb1c62a7/html5/thumbnails/11.jpg)
Cascade cartoon
![Page 12: Sample average approximation and cascading failures of ...dano/talks/graphex15.pdf · Sample average approximation and cascading failures of power grids Daniel Bienstock, Guy Grebla,](https://reader035.vdocuments.site/reader035/viewer/2022071014/5fcc6dc5b76b0c5bcb1c62a7/html5/thumbnails/12.jpg)
A quote from:
Final Report on the August 14, 2003 Blackout in the UnitedStates and Canada: Causes and Recommendations,
(U.S.-Canada Power System Outage Task Force)
Cause 1 of the blackout was “inadequate system understanding”
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A quote from:
Final Report on the August 14, 2003 Blackout in the UnitedStates and Canada: Causes and Recommendations,
(U.S.-Canada Power System Outage Task Force)
Cause 1 of the blackout was “inadequate system understanding”
(stated at least twenty times)
![Page 14: Sample average approximation and cascading failures of ...dano/talks/graphex15.pdf · Sample average approximation and cascading failures of power grids Daniel Bienstock, Guy Grebla,](https://reader035.vdocuments.site/reader035/viewer/2022071014/5fcc6dc5b76b0c5bcb1c62a7/html5/thumbnails/14.jpg)
A quote from:
Final Report on the August 14, 2003 Blackout in the UnitedStates and Canada: Causes and Recommendations,
(U.S.-Canada Power System Outage Task Force)
Cause 1 of the blackout was “inadequate system understanding”
(stated at least twenty times)
Cause 2 of the blackout was “inadequate situational awareness”
![Page 15: Sample average approximation and cascading failures of ...dano/talks/graphex15.pdf · Sample average approximation and cascading failures of power grids Daniel Bienstock, Guy Grebla,](https://reader035.vdocuments.site/reader035/viewer/2022071014/5fcc6dc5b76b0c5bcb1c62a7/html5/thumbnails/15.jpg)
Very approximate cascade model
→ Initial fault event takes place (an “act of God”).
For t = 1, 2, . . . ,
1. Reconfigure demands and generator output levels (if islanding occurs).
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Islanding
supply > demand
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The “swing” equation
• A second-order differential equation used to explain swings in a motor’sfrequency in response to a change of loads.
• To properly analyize islanding, we need to consider systems of swing equa-tions, plus physics of power flows.
•Which is a very difficult computational problem.
• Primary, secondary frequency response.
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Very approximate cascade model
→ Initial fault event takes place (an “act of God”).
For t = 1, 2, . . . ,
1. Reconfigure demands and generator output levels (if islanding occurs).
2. New power flows are instantiated.
![Page 19: Sample average approximation and cascading failures of ...dano/talks/graphex15.pdf · Sample average approximation and cascading failures of power grids Daniel Bienstock, Guy Grebla,](https://reader035.vdocuments.site/reader035/viewer/2022071014/5fcc6dc5b76b0c5bcb1c62a7/html5/thumbnails/19.jpg)
Power flow problem in rectangular coordinates, simplest form
Variables: Complex voltages ek + jfk, power flows Pkm, Qkm
Notation: For a bus k, δ(k) = set of lines incident with k;V = set of buses
∀km : Pkm = gkm(e2k + f 2k )− gkm(ekem + fkfm) + bkm(ekfm − fkem) (1a)
∀km : Qkm = −bkm(e2k + f 2k ) + bkm(ekem + fkfm) + gkm(ekfm − fkem) (1b)
∀km : |Pkm|2 + |Qkm|2 ≤ Ukm (1c)
∀k : Pmink ≤
∑km∈ δ(k)
Pkm ≤ Pmaxk (1d)
∀k : Qmink ≤
∑km∈ δ(k)
Qkm ≤ Qmaxk (1e)
∀k : V mink ≤ e2k + f 2k ≤ V max
k . (1f)
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Solving AC power flow problems
•When considering a grid in stable operation, Newton-Raphson or similarworks very well. Convergence in seconds
• But no theoretical foundation for this behavior exists.
•When studying a grid under distress, Newton-Raphson (or similar) doesnot work well. Non-convergence.
• Recently, renewed interest in semidefinite relaxations, and techniques fromreal algebraic geometry.
• These methodologies are much more accurate but also much slower.
• The mathematics is the same as that for systems of polynomial equations :Hilbert’s and Smale’s 17th problems.
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Very approximate cascade model
→ Initial fault event takes place (an “act of God”).
For t = 1, 2, . . . ,
1. Reconfigure demands and generator output levels (if islanding occurs).
2. New power flows are instantiated.
3. The next set of faults takes place.
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Line tripping
1. If a power line carries too much power, it will overheat.
2. At a critical temperature, the line will fail.
3. Before that point, the line will sag. A physical contact would lead toimmeadite tripping.
4. If a line is overloaded for too long, it will be protectively tripped.
5. Simplification: there is a line limit beyond which a line is consideredoverloaded.
5. IEEE Standard 738. An adaptation of the heat equation so as to takeinto account ...
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Line tripping
1. If a power line carries too much power, it will overheat.
2. At a critical temperature, the line will fail.
3. Before that point, the line will sag. A physical contact would lead toimmeadite tripping.
4. If a line is overloaded for too long, it will be protectively tripped.
5. Simplification: there is a line limit beyond which a line is consideredoverloaded.
5. IEEE Standard 738. An adaptation of the heat equation so as to takeinto account ... the state of the universe, pretty much
(current work: appropriate stochastic variants of the heat equation)
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Very approximate cascade model
→ Initial fault event takes place (an “act of God”).
For t = 1, 2, . . . ,
1. Reconfigure demands and generator output levels (if islanding occurs).
2. New power flows are instantiated.
3. The next set of faults takes place.
4. STOP if no more faults
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Simulation of 2011 San Diego Event
Joint work with A. Bernstein, D. Hay, G. Zussman, M. Uzunoglu(EE Dept. Columbia)
• Initiating event: human error
•We do not have complete or exact data
•Nevertheless, in our simulations a cascade does take place,with similar characteristics at the initial stages
• “inadequate system understanding”
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Beginning (partial picture: 13K buses total)
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More
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More
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More
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Cascade control
→ Initial fault event takes place (an “act of God”).
0. Compute control on the basis of initial data.
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Cascade control
→ Initial fault event takes place (an “act of God”).
0. Compute control on the basis of initial data.
For t = 1, 2, . . . , T
1. Reconfigure demands and generator output levels (if islanding occurs).
2. New power flows are instantiated.
![Page 32: Sample average approximation and cascading failures of ...dano/talks/graphex15.pdf · Sample average approximation and cascading failures of power grids Daniel Bienstock, Guy Grebla,](https://reader035.vdocuments.site/reader035/viewer/2022071014/5fcc6dc5b76b0c5bcb1c62a7/html5/thumbnails/32.jpg)
Cascade control
→ Initial fault event takes place (an “act of God”).
0. Compute control on the basis of initial data.
For t = 1, 2, . . . , T
1. Reconfigure demands and generator output levels (if islanding occurs).
2. New power flows are instantiated.
3a.Obtain measurements.
![Page 33: Sample average approximation and cascading failures of ...dano/talks/graphex15.pdf · Sample average approximation and cascading failures of power grids Daniel Bienstock, Guy Grebla,](https://reader035.vdocuments.site/reader035/viewer/2022071014/5fcc6dc5b76b0c5bcb1c62a7/html5/thumbnails/33.jpg)
Cascade control
→ Initial fault event takes place (an “act of God”).
0. Compute control on the basis of initial data.
For t = 1, 2, . . . , T
1. Reconfigure demands and generator output levels (if islanding occurs).
2. New power flows are instantiated.
3a.Obtain measurements.
3b.Apply control.
![Page 34: Sample average approximation and cascading failures of ...dano/talks/graphex15.pdf · Sample average approximation and cascading failures of power grids Daniel Bienstock, Guy Grebla,](https://reader035.vdocuments.site/reader035/viewer/2022071014/5fcc6dc5b76b0c5bcb1c62a7/html5/thumbnails/34.jpg)
Cascade control
→ Initial fault event takes place (an “act of God”).
0. Compute control on the basis of initial data.
For t = 1, 2, . . . , T
1. Reconfigure demands and generator output levels (if islanding occurs).
2. New power flows are instantiated.
3a.Obtain measurements.
3b.Apply control.
3. The next set of faults takes place.
![Page 35: Sample average approximation and cascading failures of ...dano/talks/graphex15.pdf · Sample average approximation and cascading failures of power grids Daniel Bienstock, Guy Grebla,](https://reader035.vdocuments.site/reader035/viewer/2022071014/5fcc6dc5b76b0c5bcb1c62a7/html5/thumbnails/35.jpg)
Cascade control
→ Initial fault event takes place (an “act of God”).
0. Compute control on the basis of initial data.
For t = 1, 2, . . . , T
1. Reconfigure demands and generator output levels (if islanding occurs).
2. New power flows are instantiated.
3a.Obtain measurements.
3b.Apply control.
3. The next set of faults takes place.
4. If t = T proportionally shed enough load to stop cascade.
![Page 36: Sample average approximation and cascading failures of ...dano/talks/graphex15.pdf · Sample average approximation and cascading failures of power grids Daniel Bienstock, Guy Grebla,](https://reader035.vdocuments.site/reader035/viewer/2022071014/5fcc6dc5b76b0c5bcb1c62a7/html5/thumbnails/36.jpg)
A basic form of affine control
• Control consists of nonnegative parameters u1, u2, . . . , uT .
• At time t, on an island C with max line overload κC, scale demandsby
1 + ut (1 − κC)
• κC > 1 implies loss of demand
• Easy to apply?
![Page 37: Sample average approximation and cascading failures of ...dano/talks/graphex15.pdf · Sample average approximation and cascading failures of power grids Daniel Bienstock, Guy Grebla,](https://reader035.vdocuments.site/reader035/viewer/2022071014/5fcc6dc5b76b0c5bcb1c62a7/html5/thumbnails/37.jpg)
Clairvoyant control
• For any island C that exists at time t, a parameter ut,C.
• At time t, on island C with max line overload κC, scale demands by
1 + ut,C (1 − κC)
• κC > 1 implies loss of demand
• Easy to apply? Does it even make sense?
![Page 38: Sample average approximation and cascading failures of ...dano/talks/graphex15.pdf · Sample average approximation and cascading failures of power grids Daniel Bienstock, Guy Grebla,](https://reader035.vdocuments.site/reader035/viewer/2022071014/5fcc6dc5b76b0c5bcb1c62a7/html5/thumbnails/38.jpg)
A basic form of affine control
• Control consists of nonnegative parameters u1, u2, . . . , uT .
• At time t, on an island C with max line overload κC, scale demandsby
1 + ut (1 − κC)
• κC > 1 implies loss of demand
• Easy to apply?
![Page 39: Sample average approximation and cascading failures of ...dano/talks/graphex15.pdf · Sample average approximation and cascading failures of power grids Daniel Bienstock, Guy Grebla,](https://reader035.vdocuments.site/reader035/viewer/2022071014/5fcc6dc5b76b0c5bcb1c62a7/html5/thumbnails/39.jpg)
An even simpler form of affine control
• Control consists of nonnegative parameters λ1, λ2, . . . , λT .
• At time t, all demands scaled by λt
• Easy to apply.
• But conservative?
![Page 40: Sample average approximation and cascading failures of ...dano/talks/graphex15.pdf · Sample average approximation and cascading failures of power grids Daniel Bienstock, Guy Grebla,](https://reader035.vdocuments.site/reader035/viewer/2022071014/5fcc6dc5b76b0c5bcb1c62a7/html5/thumbnails/40.jpg)
Theorem
On a network with m arcs, an optimal control of the aboveform can be computed in time
O(
mT
(T−1)!
)
![Page 41: Sample average approximation and cascading failures of ...dano/talks/graphex15.pdf · Sample average approximation and cascading failures of power grids Daniel Bienstock, Guy Grebla,](https://reader035.vdocuments.site/reader035/viewer/2022071014/5fcc6dc5b76b0c5bcb1c62a7/html5/thumbnails/41.jpg)
Theorem
On a network with m arcs, an optimal control of the aboveform can be computed in time
O(
mT
(T−1)!
)
→ What is the mathematics to explain “games” of this sort?
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One-parameter control: s
• At time t, on an island C with max line overload κC, scale demandsby 1 + s (1 − κC).
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One-parameter control: s
• At time t, on an island C with max line overload κC, scale demandsby 1 + s (1 − κC).
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
yie
ld (
%)
s
T = 5, Eastern Interconnect (23 K lines)
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One-parameter control: stochastic line failures
• At time t, on an island C with max line overload κC, scale demandsby 1 + s (1 − κC).
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
yie
ld (
%)
s
average- 1 dev+ 1 dev
T = 5, Eastern Interconnect (23 K lines)
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Robustness through noise: the sample average approach
• Let L̂j be the limit of line j, j = 1, . . . ,m (= number of lines).
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Robustness through noise: the sample average approach
• Let L̂j be the limit of line j, j = 1, . . . ,m (= number of lines).
• For s = 1, . . . , S, sample values Lt,sj , 1 ≤ t ≤ T .
• Here, ELt,sj = L̂j for all t and s.
• Interpretation: Lt,sj is a noisy estimate for L̂j.
![Page 47: Sample average approximation and cascading failures of ...dano/talks/graphex15.pdf · Sample average approximation and cascading failures of power grids Daniel Bienstock, Guy Grebla,](https://reader035.vdocuments.site/reader035/viewer/2022071014/5fcc6dc5b76b0c5bcb1c62a7/html5/thumbnails/47.jpg)
Robustness through noise: the sample average approach
• Let L̂j be the limit of line j, j = 1, . . . ,m (= number of lines).
• For s = 1, . . . , S, sample values Lt,sj , 1 ≤ t ≤ T .
• Here, ELt,sj = L̂j for all t and s.
• Interpretation: Lt,sj is a noisy estimate for L̂j.
• For a given s, the values {Lt,sj } are called a sample of the line limits.
![Page 48: Sample average approximation and cascading failures of ...dano/talks/graphex15.pdf · Sample average approximation and cascading failures of power grids Daniel Bienstock, Guy Grebla,](https://reader035.vdocuments.site/reader035/viewer/2022071014/5fcc6dc5b76b0c5bcb1c62a7/html5/thumbnails/48.jpg)
Robustness through noise: the sample average approach
• Let L̂j be the limit of line j, j = 1, . . . ,m (= number of lines).
• For s = 1, . . . , S, sample values Lt,sj , 1 ≤ t ≤ T .
• Here, ELt,sj = L̂j for all t and s.
• Interpretation: Lt,sj is a noisy estimate for L̂j.
• For a given s, the values {Lt,sj } are called a sample of the line limits.
Optimization problem
Compute a control that maximizes the yield averaged across all samples.
Theorem. Can be done in O(S) power flow computations.
![Page 49: Sample average approximation and cascading failures of ...dano/talks/graphex15.pdf · Sample average approximation and cascading failures of power grids Daniel Bienstock, Guy Grebla,](https://reader035.vdocuments.site/reader035/viewer/2022071014/5fcc6dc5b76b0c5bcb1c62a7/html5/thumbnails/49.jpg)
Experiment: robust vs. non-robust solutions(Table shows yield)
T 2 3 4 5
Non-robust solution 65.46% 65.46% 74.44% 86.84%and non-robust model
Non-robust solution 31.92% 30.46% 47.75% 23.07%and robust model
robust solution 62.19% 62.19% 70.73% 78.36%and robust model