Robust online control of cascading power grid blackoutsdano/talks/icsrc.pdf · 2009-01-07 ·...
Transcript of Robust online control of cascading power grid blackoutsdano/talks/icsrc.pdf · 2009-01-07 ·...
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Robust online control of cascading power gridblackouts
Daniel Bienstock
Columbia University, New York
ICS09
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 1 / 21
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Summary
Background. National-scale blackouts in North America andEurope since Summer/Fall 2003 due to cascading power gridfailures, specifically, cascading failures of transmission systems.Experts agree: more failures inevitable in the future; potentialeconomic and social impact enormous.
Goal. Develop an online robust control algorithm that can bedeployed in the event of a cascading failure, with the goal ofdiminishing or completely stopping the cascade.
Methodology. Adapt models of power grid cascades and employtechniques of modern robust and stochastic optimization.
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 2 / 21
![Page 3: Robust online control of cascading power grid blackoutsdano/talks/icsrc.pdf · 2009-01-07 · blackouts Daniel Bienstock Columbia University, New York ICS09 Daniel Bienstock ( Columbia](https://reader033.fdocuments.us/reader033/viewer/2022050418/5f8e398e1d45223b690ba566/html5/thumbnails/3.jpg)
Summary
Background. National-scale blackouts in North America andEurope since Summer/Fall 2003 due to cascading power gridfailures, specifically, cascading failures of transmission systems.Experts agree: more failures inevitable in the future; potentialeconomic and social impact enormous.
Goal. Develop an online robust control algorithm that can bedeployed in the event of a cascading failure, with the goal ofdiminishing or completely stopping the cascade.
Methodology. Adapt models of power grid cascades and employtechniques of modern robust and stochastic optimization.
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 2 / 21
![Page 4: Robust online control of cascading power grid blackoutsdano/talks/icsrc.pdf · 2009-01-07 · blackouts Daniel Bienstock Columbia University, New York ICS09 Daniel Bienstock ( Columbia](https://reader033.fdocuments.us/reader033/viewer/2022050418/5f8e398e1d45223b690ba566/html5/thumbnails/4.jpg)
Summary
Background. National-scale blackouts in North America andEurope since Summer/Fall 2003 due to cascading power gridfailures, specifically, cascading failures of transmission systems.Experts agree: more failures inevitable in the future; potentialeconomic and social impact enormous.
Goal. Develop an online robust control algorithm that can bedeployed in the event of a cascading failure, with the goal ofdiminishing or completely stopping the cascade.
Methodology. Adapt models of power grid cascades and employtechniques of modern robust and stochastic optimization.
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 2 / 21
![Page 5: Robust online control of cascading power grid blackoutsdano/talks/icsrc.pdf · 2009-01-07 · blackouts Daniel Bienstock Columbia University, New York ICS09 Daniel Bienstock ( Columbia](https://reader033.fdocuments.us/reader033/viewer/2022050418/5f8e398e1d45223b690ba566/html5/thumbnails/5.jpg)
A power grid has three components
TRANSMISSION
GENERATION
DISTRIBUTION
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 3 / 21
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Basic power flow model
A power flow satisfies flow conservation :∑ij f ij −
∑ij f ji = b i , for all i , where
0 ≤ b i ≤ SUi , for each i ∈ S (generation),
0 ≤ −b i ≤ Dmaxi for i ∈ D (demands),
and b i = 0, otherwise.
Flows further constraind by physics
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 4 / 21
![Page 7: Robust online control of cascading power grid blackoutsdano/talks/icsrc.pdf · 2009-01-07 · blackouts Daniel Bienstock Columbia University, New York ICS09 Daniel Bienstock ( Columbia](https://reader033.fdocuments.us/reader033/viewer/2022050418/5f8e398e1d45223b690ba566/html5/thumbnails/7.jpg)
Basic power flow model
A power flow satisfies flow conservation :∑ij f ij −
∑ij f ji = b i , for all i , where
0 ≤ b i ≤ SUi , for each i ∈ S (generation),
0 ≤ −b i ≤ Dmaxi for i ∈ D (demands),
and b i = 0, otherwise.
Flows further constraind by physics
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 4 / 21
![Page 8: Robust online control of cascading power grid blackoutsdano/talks/icsrc.pdf · 2009-01-07 · blackouts Daniel Bienstock Columbia University, New York ICS09 Daniel Bienstock ( Columbia](https://reader033.fdocuments.us/reader033/viewer/2022050418/5f8e398e1d45223b690ba566/html5/thumbnails/8.jpg)
Power flow constraints
Linearized flow model: f ij = x ij (θi − θj ),
x ij = “resistance” (parameter),θi = “phase angle” at node i (variable)
More accurate “active” power flow model “without losses”:f ij = x ij sin (θi − θj ),
|θi − θj | ≤ π/2
“Full” model including active and reactive flows:
Flows represented using complex numbers
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 5 / 21
![Page 9: Robust online control of cascading power grid blackoutsdano/talks/icsrc.pdf · 2009-01-07 · blackouts Daniel Bienstock Columbia University, New York ICS09 Daniel Bienstock ( Columbia](https://reader033.fdocuments.us/reader033/viewer/2022050418/5f8e398e1d45223b690ba566/html5/thumbnails/9.jpg)
Power flow constraints
Linearized flow model: f ij = x ij (θi − θj ),
x ij = “resistance” (parameter),θi = “phase angle” at node i (variable)
More accurate “active” power flow model “without losses”:f ij = x ij sin (θi − θj ),
|θi − θj | ≤ π/2
“Full” model including active and reactive flows:
Flows represented using complex numbers
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 5 / 21
![Page 10: Robust online control of cascading power grid blackoutsdano/talks/icsrc.pdf · 2009-01-07 · blackouts Daniel Bienstock Columbia University, New York ICS09 Daniel Bienstock ( Columbia](https://reader033.fdocuments.us/reader033/viewer/2022050418/5f8e398e1d45223b690ba566/html5/thumbnails/10.jpg)
Power flow constraints
Linearized flow model: f ij = x ij (θi − θj ),
x ij = “resistance” (parameter),θi = “phase angle” at node i (variable)
More accurate “active” power flow model “without losses”:f ij = x ij sin (θi − θj ),
|θi − θj | ≤ π/2
“Full” model including active and reactive flows:
Flows represented using complex numbers
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 5 / 21
![Page 11: Robust online control of cascading power grid blackoutsdano/talks/icsrc.pdf · 2009-01-07 · blackouts Daniel Bienstock Columbia University, New York ICS09 Daniel Bienstock ( Columbia](https://reader033.fdocuments.us/reader033/viewer/2022050418/5f8e398e1d45223b690ba566/html5/thumbnails/11.jpg)
Complexity issues:
How to compute a feasible power flow?
Linearized model: solve a linear system. Relatively fast:requires on the order of 0.001 − 0.01 seconds for a grid with103 arcs. Numerically challenging: LP solvers can (and do)produce significant roundoff errors due to the x ij parameters.
Full active/reactive flows: NP-complete (most likely).
Lossless model with active power flows: complexity unknown(maybe NP-hard, but possibly not too bad).
→ This talk: linearized model only
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 6 / 21
![Page 12: Robust online control of cascading power grid blackoutsdano/talks/icsrc.pdf · 2009-01-07 · blackouts Daniel Bienstock Columbia University, New York ICS09 Daniel Bienstock ( Columbia](https://reader033.fdocuments.us/reader033/viewer/2022050418/5f8e398e1d45223b690ba566/html5/thumbnails/12.jpg)
Complexity issues:
How to compute a feasible power flow?
Linearized model: solve a linear system. Relatively fast:requires on the order of 0.001 − 0.01 seconds for a grid with103 arcs. Numerically challenging: LP solvers can (and do)produce significant roundoff errors due to the x ij parameters.
Full active/reactive flows: NP-complete (most likely).
Lossless model with active power flows: complexity unknown(maybe NP-hard, but possibly not too bad).
→ This talk: linearized model only
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 6 / 21
![Page 13: Robust online control of cascading power grid blackoutsdano/talks/icsrc.pdf · 2009-01-07 · blackouts Daniel Bienstock Columbia University, New York ICS09 Daniel Bienstock ( Columbia](https://reader033.fdocuments.us/reader033/viewer/2022050418/5f8e398e1d45223b690ba566/html5/thumbnails/13.jpg)
Complexity issues:
How to compute a feasible power flow?
Linearized model: solve a linear system. Relatively fast:requires on the order of 0.001 − 0.01 seconds for a grid with103 arcs. Numerically challenging: LP solvers can (and do)produce significant roundoff errors due to the x ij parameters.
Full active/reactive flows: NP-complete (most likely).
Lossless model with active power flows: complexity unknown(maybe NP-hard, but possibly not too bad).
→ This talk: linearized model only
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 6 / 21
![Page 14: Robust online control of cascading power grid blackoutsdano/talks/icsrc.pdf · 2009-01-07 · blackouts Daniel Bienstock Columbia University, New York ICS09 Daniel Bienstock ( Columbia](https://reader033.fdocuments.us/reader033/viewer/2022050418/5f8e398e1d45223b690ba566/html5/thumbnails/14.jpg)
Complexity issues:
How to compute a feasible power flow?
Linearized model: solve a linear system. Relatively fast:requires on the order of 0.001 − 0.01 seconds for a grid with103 arcs. Numerically challenging: LP solvers can (and do)produce significant roundoff errors due to the x ij parameters.
Full active/reactive flows: NP-complete (most likely).
Lossless model with active power flows: complexity unknown(maybe NP-hard, but possibly not too bad).
→ This talk: linearized model only
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 6 / 21
![Page 15: Robust online control of cascading power grid blackoutsdano/talks/icsrc.pdf · 2009-01-07 · blackouts Daniel Bienstock Columbia University, New York ICS09 Daniel Bienstock ( Columbia](https://reader033.fdocuments.us/reader033/viewer/2022050418/5f8e398e1d45223b690ba566/html5/thumbnails/15.jpg)
Complexity issues:
How to compute a feasible power flow?
Linearized model: solve a linear system. Relatively fast:requires on the order of 0.001 − 0.01 seconds for a grid with103 arcs. Numerically challenging: LP solvers can (and do)produce significant roundoff errors due to the x ij parameters.
Full active/reactive flows: NP-complete (most likely).
Lossless model with active power flows: complexity unknown(maybe NP-hard, but possibly not too bad).
→ This talk: linearized model only
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 6 / 21
![Page 16: Robust online control of cascading power grid blackoutsdano/talks/icsrc.pdf · 2009-01-07 · blackouts Daniel Bienstock Columbia University, New York ICS09 Daniel Bienstock ( Columbia](https://reader033.fdocuments.us/reader033/viewer/2022050418/5f8e398e1d45223b690ba566/html5/thumbnails/16.jpg)
A critical detail
→ For a given level of supply an demand, power flows are uniquelydetermined by the physics – not subject to control.
BUT:
For each arc (i , j) there is a parameter u ij , the “rating” or capacity.
If |f ij | > u ij then thermal effects will destroy the arc. Alternatively,protective equiment will shut down the arc.
Typically, this takes minutes, or tens of minutes, rather thanseconds or less.
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 7 / 21
![Page 17: Robust online control of cascading power grid blackoutsdano/talks/icsrc.pdf · 2009-01-07 · blackouts Daniel Bienstock Columbia University, New York ICS09 Daniel Bienstock ( Columbia](https://reader033.fdocuments.us/reader033/viewer/2022050418/5f8e398e1d45223b690ba566/html5/thumbnails/17.jpg)
A critical detail
→ For a given level of supply an demand, power flows are uniquelydetermined by the physics – not subject to control.
BUT:
For each arc (i , j) there is a parameter u ij , the “rating” or capacity.
If |f ij | > u ij then thermal effects will destroy the arc. Alternatively,protective equiment will shut down the arc.
Typically, this takes minutes, or tens of minutes, rather thanseconds or less.
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 7 / 21
![Page 18: Robust online control of cascading power grid blackoutsdano/talks/icsrc.pdf · 2009-01-07 · blackouts Daniel Bienstock Columbia University, New York ICS09 Daniel Bienstock ( Columbia](https://reader033.fdocuments.us/reader033/viewer/2022050418/5f8e398e1d45223b690ba566/html5/thumbnails/18.jpg)
A critical detail
→ For a given level of supply an demand, power flows are uniquelydetermined by the physics – not subject to control.
BUT:
For each arc (i , j) there is a parameter u ij , the “rating” or capacity.
If |f ij | > u ij then thermal effects will destroy the arc. Alternatively,protective equiment will shut down the arc.
Typically, this takes minutes, or tens of minutes, rather thanseconds or less.
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 7 / 21
![Page 19: Robust online control of cascading power grid blackoutsdano/talks/icsrc.pdf · 2009-01-07 · blackouts Daniel Bienstock Columbia University, New York ICS09 Daniel Bienstock ( Columbia](https://reader033.fdocuments.us/reader033/viewer/2022050418/5f8e398e1d45223b690ba566/html5/thumbnails/19.jpg)
A critical detail
→ For a given level of supply an demand, power flows are uniquelydetermined by the physics – not subject to control.
BUT:
For each arc (i , j) there is a parameter u ij , the “rating” or capacity.
If |f ij | > u ij then thermal effects will destroy the arc. Alternatively,protective equiment will shut down the arc.
Typically, this takes minutes, or tens of minutes, rather thanseconds or less.
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 7 / 21
![Page 20: Robust online control of cascading power grid blackoutsdano/talks/icsrc.pdf · 2009-01-07 · blackouts Daniel Bienstock Columbia University, New York ICS09 Daniel Bienstock ( Columbia](https://reader033.fdocuments.us/reader033/viewer/2022050418/5f8e398e1d45223b690ba566/html5/thumbnails/20.jpg)
A model for cascading power grid failures
Adapted from Dobson, Carreras, Lynch, Newman (2003-2004)
(1) An initial, exogenous event (an “act of god”) takes place, resultingin the destruction of a (small) number of power lines.
(2) New power flows are instantiated (demand or output has notchanged)
(3) Under the new power flows, some arcs exceed their rating.
(4) After a certain period of time, some of those arcs are removedfrom the network. Go to 2 .
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 8 / 21
![Page 21: Robust online control of cascading power grid blackoutsdano/talks/icsrc.pdf · 2009-01-07 · blackouts Daniel Bienstock Columbia University, New York ICS09 Daniel Bienstock ( Columbia](https://reader033.fdocuments.us/reader033/viewer/2022050418/5f8e398e1d45223b690ba566/html5/thumbnails/21.jpg)
A model for cascading power grid failures
Adapted from Dobson, Carreras, Lynch, Newman (2003-2004)
(1) An initial, exogenous event (an “act of god”) takes place, resultingin the destruction of a (small) number of power lines.
(2) New power flows are instantiated (demand or output has notchanged)
(3) Under the new power flows, some arcs exceed their rating.
(4) After a certain period of time, some of those arcs are removedfrom the network. Go to 2 .
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 8 / 21
![Page 22: Robust online control of cascading power grid blackoutsdano/talks/icsrc.pdf · 2009-01-07 · blackouts Daniel Bienstock Columbia University, New York ICS09 Daniel Bienstock ( Columbia](https://reader033.fdocuments.us/reader033/viewer/2022050418/5f8e398e1d45223b690ba566/html5/thumbnails/22.jpg)
A model for cascading power grid failures
Adapted from Dobson, Carreras, Lynch, Newman (2003-2004)
(1) An initial, exogenous event (an “act of god”) takes place, resultingin the destruction of a (small) number of power lines.
(2) New power flows are instantiated (demand or output has notchanged)
(3) Under the new power flows, some arcs exceed their rating.
(4) After a certain period of time, some of those arcs are removedfrom the network. Go to 2 .
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 8 / 21
![Page 23: Robust online control of cascading power grid blackoutsdano/talks/icsrc.pdf · 2009-01-07 · blackouts Daniel Bienstock Columbia University, New York ICS09 Daniel Bienstock ( Columbia](https://reader033.fdocuments.us/reader033/viewer/2022050418/5f8e398e1d45223b690ba566/html5/thumbnails/23.jpg)
A model for cascading power grid failures
Adapted from Dobson, Carreras, Lynch, Newman (2003-2004)
(1) An initial, exogenous event (an “act of god”) takes place, resultingin the destruction of a (small) number of power lines.
(2) New power flows are instantiated (demand or output has notchanged)
(3) Under the new power flows, some arcs exceed their rating.
(4) After a certain period of time, some of those arcs are removedfrom the network. Go to 2 .
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 8 / 21
![Page 24: Robust online control of cascading power grid blackoutsdano/talks/icsrc.pdf · 2009-01-07 · blackouts Daniel Bienstock Columbia University, New York ICS09 Daniel Bienstock ( Columbia](https://reader033.fdocuments.us/reader033/viewer/2022050418/5f8e398e1d45223b690ba566/html5/thumbnails/24.jpg)
How does a cascade end?
Complete collapse – most or many of the arcs disabled, zero orvery little demand satisfied.
Spontaneous stop – cascade stops when no lines are over rating,some amount of demand lost (could be significant)
Induced blackout (“load shedding”) – power grid operators shutdown some amount of demand in order to stop or slow downcascade – US 2003.
Slow cascade – cascade does not stop, but goes on “for a longtime” with small amounts of demands lost. Controllable?
An important detail: we expect the pace of the cascade toaccelerate with time – slow changes at the start.
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 9 / 21
![Page 25: Robust online control of cascading power grid blackoutsdano/talks/icsrc.pdf · 2009-01-07 · blackouts Daniel Bienstock Columbia University, New York ICS09 Daniel Bienstock ( Columbia](https://reader033.fdocuments.us/reader033/viewer/2022050418/5f8e398e1d45223b690ba566/html5/thumbnails/25.jpg)
How does a cascade end?
Complete collapse – most or many of the arcs disabled, zero orvery little demand satisfied.
Spontaneous stop – cascade stops when no lines are over rating,some amount of demand lost (could be significant)
Induced blackout (“load shedding”) – power grid operators shutdown some amount of demand in order to stop or slow downcascade – US 2003.
Slow cascade – cascade does not stop, but goes on “for a longtime” with small amounts of demands lost. Controllable?
An important detail: we expect the pace of the cascade toaccelerate with time – slow changes at the start.
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 9 / 21
![Page 26: Robust online control of cascading power grid blackoutsdano/talks/icsrc.pdf · 2009-01-07 · blackouts Daniel Bienstock Columbia University, New York ICS09 Daniel Bienstock ( Columbia](https://reader033.fdocuments.us/reader033/viewer/2022050418/5f8e398e1d45223b690ba566/html5/thumbnails/26.jpg)
How does a cascade end?
Complete collapse – most or many of the arcs disabled, zero orvery little demand satisfied.
Spontaneous stop – cascade stops when no lines are over rating,some amount of demand lost (could be significant)
Induced blackout (“load shedding”) – power grid operators shutdown some amount of demand in order to stop or slow downcascade – US 2003.
Slow cascade – cascade does not stop, but goes on “for a longtime” with small amounts of demands lost. Controllable?
An important detail: we expect the pace of the cascade toaccelerate with time – slow changes at the start.
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 9 / 21
![Page 27: Robust online control of cascading power grid blackoutsdano/talks/icsrc.pdf · 2009-01-07 · blackouts Daniel Bienstock Columbia University, New York ICS09 Daniel Bienstock ( Columbia](https://reader033.fdocuments.us/reader033/viewer/2022050418/5f8e398e1d45223b690ba566/html5/thumbnails/27.jpg)
How does a cascade end?
Complete collapse – most or many of the arcs disabled, zero orvery little demand satisfied.
Spontaneous stop – cascade stops when no lines are over rating,some amount of demand lost (could be significant)
Induced blackout (“load shedding”) – power grid operators shutdown some amount of demand in order to stop or slow downcascade – US 2003.
Slow cascade – cascade does not stop, but goes on “for a longtime” with small amounts of demands lost. Controllable?
An important detail: we expect the pace of the cascade toaccelerate with time – slow changes at the start.
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 9 / 21
![Page 28: Robust online control of cascading power grid blackoutsdano/talks/icsrc.pdf · 2009-01-07 · blackouts Daniel Bienstock Columbia University, New York ICS09 Daniel Bienstock ( Columbia](https://reader033.fdocuments.us/reader033/viewer/2022050418/5f8e398e1d45223b690ba566/html5/thumbnails/28.jpg)
How does a cascade end?
Complete collapse – most or many of the arcs disabled, zero orvery little demand satisfied.
Spontaneous stop – cascade stops when no lines are over rating,some amount of demand lost (could be significant)
Induced blackout (“load shedding”) – power grid operators shutdown some amount of demand in order to stop or slow downcascade – US 2003.
Slow cascade – cascade does not stop, but goes on “for a longtime” with small amounts of demands lost. Controllable?
An important detail: we expect the pace of the cascade toaccelerate with time – slow changes at the start.
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 9 / 21
![Page 29: Robust online control of cascading power grid blackoutsdano/talks/icsrc.pdf · 2009-01-07 · blackouts Daniel Bienstock Columbia University, New York ICS09 Daniel Bienstock ( Columbia](https://reader033.fdocuments.us/reader033/viewer/2022050418/5f8e398e1d45223b690ba566/html5/thumbnails/29.jpg)
Cascade model in more detail
(0) “Steady-state” power flows f (0)ij . “Act of God” happens.
Set t = 1.
(1.t) Stage t begins – power flows f (t )ij are realized.
(2.t) Compute the set of arcs to be removed at stage t .Arc (i , j ) is removed:
(strict rule) if |f (t )ij | > u ij
(random rule) if Pij (|f (t )ij |/u ij ) (̧Pij increasing, Dobson et al)
(“thermal” rule) if τ(t )ij > u ij .
τ(t )ij = αij |f (t )
ij | + (1− αij )τ(t−1)ij , 0 ≤ αij ≤ 1 (moving
average)
(3.t) Reset t ← t + 1 and go to 1.
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 10 / 21
![Page 30: Robust online control of cascading power grid blackoutsdano/talks/icsrc.pdf · 2009-01-07 · blackouts Daniel Bienstock Columbia University, New York ICS09 Daniel Bienstock ( Columbia](https://reader033.fdocuments.us/reader033/viewer/2022050418/5f8e398e1d45223b690ba566/html5/thumbnails/30.jpg)
Cascade model in more detail
(0) “Steady-state” power flows f (0)ij . “Act of God” happens.
Set t = 1.
(1.t) Stage t begins – power flows f (t )ij are realized.
(2.t) Compute the set of arcs to be removed at stage t .Arc (i , j ) is removed:
(strict rule) if |f (t )ij | > u ij
(random rule) if Pij (|f (t )ij |/u ij ) (̧Pij increasing, Dobson et al)
(“thermal” rule) if τ(t )ij > u ij .
τ(t )ij = αij |f (t )
ij | + (1− αij )τ(t−1)ij , 0 ≤ αij ≤ 1 (moving
average)
(3.t) Reset t ← t + 1 and go to 1.
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 10 / 21
![Page 31: Robust online control of cascading power grid blackoutsdano/talks/icsrc.pdf · 2009-01-07 · blackouts Daniel Bienstock Columbia University, New York ICS09 Daniel Bienstock ( Columbia](https://reader033.fdocuments.us/reader033/viewer/2022050418/5f8e398e1d45223b690ba566/html5/thumbnails/31.jpg)
Cascade model in more detail
(0) “Steady-state” power flows f (0)ij . “Act of God” happens.
Set t = 1.
(1.t) Stage t begins – power flows f (t )ij are realized.
(2.t) Compute the set of arcs to be removed at stage t .Arc (i , j ) is removed:
(strict rule) if |f (t )ij | > u ij
(random rule) if Pij (|f (t )ij |/u ij ) (̧Pij increasing, Dobson et al)
(“thermal” rule) if τ(t )ij > u ij .
τ(t )ij = αij |f (t )
ij | + (1− αij )τ(t−1)ij , 0 ≤ αij ≤ 1 (moving
average)
(3.t) Reset t ← t + 1 and go to 1.
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 10 / 21
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Cascade model in more detail
(0) “Steady-state” power flows f (0)ij . “Act of God” happens.
Set t = 1.
(1.t) Stage t begins – power flows f (t )ij are realized.
(2.t) Compute the set of arcs to be removed at stage t .Arc (i , j ) is removed:
(strict rule) if |f (t )ij | > u ij
(random rule) if Pij (|f (t )ij |/u ij ) (̧Pij increasing, Dobson et al)
(“thermal” rule) if τ(t )ij > u ij .
τ(t )ij = αij |f (t )
ij | + (1− αij )τ(t−1)ij , 0 ≤ αij ≤ 1 (moving
average)
(3.t) Reset t ← t + 1 and go to 1.
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 10 / 21
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Cascade model in more detail
(0) “Steady-state” power flows f (0)ij . “Act of God” happens.
Set t = 1.
(1.t) Stage t begins – power flows f (t )ij are realized.
(2.t) Compute the set of arcs to be removed at stage t .Arc (i , j ) is removed:
(strict rule) if |f (t )ij | > u ij
(random rule) if Pij (|f (t )ij |/u ij ) (̧Pij increasing, Dobson et al)
(“thermal” rule) if τ(t )ij > u ij .
τ(t )ij = αij |f (t )
ij | + (1− αij )τ(t−1)ij , 0 ≤ αij ≤ 1 (moving
average)
(3.t) Reset t ← t + 1 and go to 1.
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 10 / 21
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Cascade model in more detail
(0) “Steady-state” power flows f (0)ij . “Act of God” happens.
Set t = 1.
(1.t) Stage t begins – power flows f (t )ij are realized.
(2.t) Compute the set of arcs to be removed at stage t .Arc (i , j ) is removed:
(strict rule) if |f (t )ij | > u ij
(random rule) if Pij (|f (t )ij |/u ij ) (̧Pij increasing, Dobson et al)
(“thermal” rule) if τ(t )ij > u ij .
τ(t )ij = αij |f (t )
ij | + (1− αij )τ(t−1)ij , 0 ≤ αij ≤ 1 (moving
average)
(3.t) Reset t ← t + 1 and go to 1.
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 10 / 21
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Online control
(0) “Steady-state” power flows f (0)ij . “Act of God” happens.
Set t = 1.
Compute control algorithm .
(1.t) Stage t begins – power flows f (t )ij are realized.
1 Apply control .2 Let g (t )
ij be the new flows post-control.
(2.t) Arc (i , j ) is removed if τ(t )ij > u ij .
τ(t )ij = αij |g
(t )ij | + (1− αij )τ
(t−1)ij , 0 ≤ αij ≤ 1
(3.t) Reset t ← t + 1 and go to 1.
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 11 / 21
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Online control
(0) “Steady-state” power flows f (0)ij . “Act of God” happens.
Set t = 1.
Compute control algorithm .
(1.t) Stage t begins – power flows f (t )ij are realized.
1 Apply control .2 Let g (t )
ij be the new flows post-control.
(2.t) Arc (i , j ) is removed if τ(t )ij > u ij .
τ(t )ij = αij |g
(t )ij | + (1− αij )τ
(t−1)ij , 0 ≤ αij ≤ 1
(3.t) Reset t ← t + 1 and go to 1.
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 11 / 21
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Online control
(0) “Steady-state” power flows f (0)ij . “Act of God” happens.
Set t = 1.
Compute control algorithm .
(1.t) Stage t begins – power flows f (t )ij are realized.
1 Apply control .2 Let g (t )
ij be the new flows post-control.
(2.t) Arc (i , j ) is removed if τ(t )ij > u ij .
τ(t )ij = αij |g
(t )ij | + (1− αij )τ
(t−1)ij , 0 ≤ αij ≤ 1
(3.t) Reset t ← t + 1 and go to 1.
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 11 / 21
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Online control
(0) “Steady-state” power flows f (0)ij . “Act of God” happens.
Set t = 1.
Compute control algorithm .
(1.t) Stage t begins – power flows f (t )ij are realized.
1 Apply control .2 Let g (t )
ij be the new flows post-control.
(2.t) Arc (i , j ) is removed if τ(t )ij > u ij .
τ(t )ij = αij |g
(t )ij | + (1− αij )τ
(t−1)ij , 0 ≤ αij ≤ 1
(3.t) Reset t ← t + 1 and go to 1.
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 11 / 21
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Online control
(0) “Steady-state” power flows f (0)ij . “Act of God” happens.
Set t = 1.
Compute control algorithm .
(1.t) Stage t begins – power flows f (t )ij are realized.
1 Apply control .2 Let g (t )
ij be the new flows post-control.
(2.t) Arc (i , j ) is removed if τ(t )ij > u ij .
τ(t )ij = αij |g
(t )ij | + (1− αij )τ
(t−1)ij , 0 ≤ αij ≤ 1
(3.t) Reset t ← t + 1 and go to 1.
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 11 / 21
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Online control
(0) “Steady-state” power flows f (0)ij . “Act of God” happens.
Set t = 1.
Compute control algorithm .
(1.t) Stage t begins – power flows f (t )ij are realized.
1 Apply control .2 Let g (t )
ij be the new flows post-control.
(2.t) Arc (i , j ) is removed if τ(t )ij > u ij .
τ(t )ij = αij |g
(t )ij | + (1− αij )τ
(t−1)ij , 0 ≤ αij ≤ 1
(3.t) Reset t ← t + 1 and go to 1.
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 11 / 21
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Online control
(0) “Steady-state” power flows f (0)ij . “Act of God” happens.
Set t = 1.
Compute control algorithm .
(1.t) Stage t begins – power flows f (t )ij are realized.
1 Apply control .2 Let g (t )
ij be the new flows post-control.
(2.t) Arc (i , j ) is removed if τ(t )ij > u ij .
τ(t )ij = αij |g
(t )ij | + (1− αij )τ
(t−1)ij , 0 ≤ αij ≤ 1
(3.t) Reset t ← t + 1 and go to 1.
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 11 / 21
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A simple control algorithm
“Adaptive load shedding”: let 0 < λ < 1 be a parameter.
At time t , if max ij
{f iju ij
}> 1, then
Scale all demands by a factor of λ
“Component-wise” version: apply the rule above to eachconnected component separately
→ Algorithm = λ
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A simple control algorithm
“Adaptive load shedding”: let 0 < λ < 1 be a parameter.
At time t , if max ij
{f iju ij
}> 1, then
Scale all demands by a factor of λ
“Component-wise” version: apply the rule above to eachconnected component separately
→ Algorithm = λ
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 12 / 21
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A simple control algorithm
“Adaptive load shedding”: let 0 < λ < 1 be a parameter.
At time t , if max ij
{f iju ij
}> 1, then
Scale all demands by a factor of λ
“Component-wise” version: apply the rule above to eachconnected component separately
→ Algorithm = λ
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 12 / 21
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Application of component-wise control:
600 nodes, 1268 edges, 25 generators, 344 demands, 6 rounds
0
1000
2000
3000
4000
5000
6000
7000
0 0.2 0.4 0.6 0.8 1
throughput
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 13 / 21
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Affine controls
For each demand node k , let sk , bk , be parameters
At time t , letκk = max (i ,j )∈C(k )
{f iju ij
}, where
C(k ) = component containing node k .
If κk > 1, we scale the demand at k by a factor of sk κk + bk .
→ Algorithm: compute the parameters sk , bk for every k .
→ NP-hard already for the one-round problem.
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Affine controls
For each demand node k , let sk , bk , be parameters
At time t , letκk = max (i ,j )∈C(k )
{f iju ij
}, where
C(k ) = component containing node k .
If κk > 1, we scale the demand at k by a factor of sk κk + bk .
→ Algorithm: compute the parameters sk , bk for every k .
→ NP-hard already for the one-round problem.
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 14 / 21
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Affine controls
For each demand node k , let sk , bk , be parameters
At time t , letκk = max (i ,j )∈C(k )
{f iju ij
}, where
C(k ) = component containing node k .
If κk > 1, we scale the demand at k by a factor of sk κk + bk .
→ Algorithm: compute the parameters sk , bk for every k .
→ NP-hard already for the one-round problem.
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 14 / 21
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Affine controls
For each demand node k , let sk , bk , be parameters
At time t , letκk = max (i ,j )∈C(k )
{f iju ij
}, where
C(k ) = component containing node k .
If κk > 1, we scale the demand at k by a factor of sk κk + bk .
→ Algorithm: compute the parameters sk , bk for every k .
→ NP-hard already for the one-round problem.
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 14 / 21
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Affine controls
For each demand node k , let sk , bk , be parameters
At time t , letκk = max (i ,j )∈C(k )
{f iju ij
}, where
C(k ) = component containing node k .
If κk > 1, we scale the demand at k by a factor of sk κk + bk .
→ Algorithm: compute the parameters sk , bk for every k .
→ NP-hard already for the one-round problem.
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 14 / 21
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Local optimum
Notation: let F (b, s) = throughput obtained by applying the affinecontrol (b, s).We want to choose (b, s) so as to maximize F (b, s)
Algorithm
1. Given (b, s), estimate the gradient ∇b,sF
2. Step: (b, s) ← (b, s) + ε∇b,sF (line search for ε)
3. Repeat.
→ Each step 1 and 2 requires multiple cascade simulations
→ But parallelizable
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Local optimum
Notation: let F (b, s) = throughput obtained by applying the affinecontrol (b, s).We want to choose (b, s) so as to maximize F (b, s)
Algorithm
1. Given (b, s), estimate the gradient ∇b,sF
2. Step: (b, s) ← (b, s) + ε∇b,sF (line search for ε)
3. Repeat.
→ Each step 1 and 2 requires multiple cascade simulations
→ But parallelizable
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 15 / 21
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Local optimum
Notation: let F (b, s) = throughput obtained by applying the affinecontrol (b, s).We want to choose (b, s) so as to maximize F (b, s)
Algorithm
1. Given (b, s), estimate the gradient ∇b,sF
2. Step: (b, s) ← (b, s) + ε∇b,sF (line search for ε)
3. Repeat.
→ Each step 1 and 2 requires multiple cascade simulations
→ But parallelizable
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 15 / 21
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Local optimum
Notation: let F (b, s) = throughput obtained by applying the affinecontrol (b, s).We want to choose (b, s) so as to maximize F (b, s)
Algorithm
1. Given (b, s), estimate the gradient ∇b,sF
2. Step: (b, s) ← (b, s) + ε∇b,sF (line search for ε)
3. Repeat.
→ Each step 1 and 2 requires multiple cascade simulations
→ But parallelizable
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 15 / 21
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Local optimum
Notation: let F (b, s) = throughput obtained by applying the affinecontrol (b, s).We want to choose (b, s) so as to maximize F (b, s)
Algorithm
1. Given (b, s), estimate the gradient ∇b,sF
2. Step: (b, s) ← (b, s) + ε∇b,sF (line search for ε)
3. Repeat.
→ Each step 1 and 2 requires multiple cascade simulations
→ But parallelizable
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Example: 600 nodes, 990 arcs, 344 demand nodes, 98 generators
Starting with (bk , sk ) = (0.80, 0) for all k , yield = 0.63997
4 CPUs
Run Wall-clock Yieldtime (sec.) (fraction)
690 192 0.6408151479 434 0.7011231562 460 0.8451573055 898 0.8899154633 1599 0.9148655418 1983 0.916966
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Example: 600 nodes, 990 arcs, 344 demand nodes, 98 generators
Starting with (bk , sk ) = (0.80, 0) for all k , yield = 0.63997
4 CPUs
Run Wall-clock Yieldtime (sec.) (fraction)
690 192 0.6408151479 434 0.7011231562 460 0.8451573055 898 0.8899154633 1599 0.9148655418 1983 0.916966
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Example: 600 nodes, 990 arcs, 344 demand nodes, 98 generators
Starting with (bk , sk ) = (0.80, 0) for all k , yield = 0.63997
4 CPUs
Run Wall-clock Yieldtime (sec.) (fraction)
690 192 0.6408151479 434 0.7011231562 460 0.8451573055 898 0.8899154633 1599 0.9148655418 1983 0.916966
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0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
0 50 100 150 200 250 300 350
yield
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Robustness
Catastrophic cascades are very rare
During a cascade we will face a very noisy environment
Difficult to formulate a precise mathematical model
Need to “train” a control algorithm, by “exposing” it to noise
Cannot expect to obtain an exact optimization tool – it’s a meansto an end (robustness)
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Robustness
Catastrophic cascades are very rare
During a cascade we will face a very noisy environment
Difficult to formulate a precise mathematical model
Need to “train” a control algorithm, by “exposing” it to noise
Cannot expect to obtain an exact optimization tool – it’s a meansto an end (robustness)
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 18 / 21
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Robustness
Catastrophic cascades are very rare
During a cascade we will face a very noisy environment
Difficult to formulate a precise mathematical model
Need to “train” a control algorithm, by “exposing” it to noise
Cannot expect to obtain an exact optimization tool – it’s a meansto an end (robustness)
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 18 / 21
![Page 63: Robust online control of cascading power grid blackoutsdano/talks/icsrc.pdf · 2009-01-07 · blackouts Daniel Bienstock Columbia University, New York ICS09 Daniel Bienstock ( Columbia](https://reader033.fdocuments.us/reader033/viewer/2022050418/5f8e398e1d45223b690ba566/html5/thumbnails/63.jpg)
Robustness
Catastrophic cascades are very rare
During a cascade we will face a very noisy environment
Difficult to formulate a precise mathematical model
Need to “train” a control algorithm, by “exposing” it to noise
Cannot expect to obtain an exact optimization tool – it’s a meansto an end (robustness)
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 18 / 21
![Page 64: Robust online control of cascading power grid blackoutsdano/talks/icsrc.pdf · 2009-01-07 · blackouts Daniel Bienstock Columbia University, New York ICS09 Daniel Bienstock ( Columbia](https://reader033.fdocuments.us/reader033/viewer/2022050418/5f8e398e1d45223b690ba566/html5/thumbnails/64.jpg)
Robustness
Catastrophic cascades are very rare
During a cascade we will face a very noisy environment
Difficult to formulate a precise mathematical model
Need to “train” a control algorithm, by “exposing” it to noise
Cannot expect to obtain an exact optimization tool – it’s a meansto an end (robustness)
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 18 / 21
![Page 65: Robust online control of cascading power grid blackoutsdano/talks/icsrc.pdf · 2009-01-07 · blackouts Daniel Bienstock Columbia University, New York ICS09 Daniel Bienstock ( Columbia](https://reader033.fdocuments.us/reader033/viewer/2022050418/5f8e398e1d45223b690ba566/html5/thumbnails/65.jpg)
Robustness
Catastrophic cascades are very rare
During a cascade we will face a very noisy environment
Difficult to formulate a precise mathematical model
Need to “train” a control algorithm, by “exposing” it to noise
Cannot expect to obtain an exact optimization tool – it’s a meansto an end (robustness)
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 18 / 21
![Page 66: Robust online control of cascading power grid blackoutsdano/talks/icsrc.pdf · 2009-01-07 · blackouts Daniel Bienstock Columbia University, New York ICS09 Daniel Bienstock ( Columbia](https://reader033.fdocuments.us/reader033/viewer/2022050418/5f8e398e1d45223b690ba566/html5/thumbnails/66.jpg)
Basic methodology for the cascade
At time t , arc (i , j ) is removed if τ(t )ij > u ij . Here,
τ(t )ij = αij |f
(t )ij | + (1− αij )τ
(t−1)ij , 0 ≤ αij ≤ 1
where
f (t)ij = flow on (i , j)
τ(t)ij = moving average of flow on (i , j)
What is αij ? Does it actually exist?
→ Robustify the model by allowing αij , randomly or adversarially
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 19 / 21
![Page 67: Robust online control of cascading power grid blackoutsdano/talks/icsrc.pdf · 2009-01-07 · blackouts Daniel Bienstock Columbia University, New York ICS09 Daniel Bienstock ( Columbia](https://reader033.fdocuments.us/reader033/viewer/2022050418/5f8e398e1d45223b690ba566/html5/thumbnails/67.jpg)
Basic methodology for the cascade
At time t , arc (i , j ) is removed if τ(t )ij > u ij . Here,
τ(t )ij = αij |f
(t )ij | + (1− αij )τ
(t−1)ij , 0 ≤ αij ≤ 1
where
f (t)ij = flow on (i , j)
τ(t)ij = moving average of flow on (i , j)
What is αij ? Does it actually exist?
→ Robustify the model by allowing αij , randomly or adversarially
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 19 / 21
![Page 68: Robust online control of cascading power grid blackoutsdano/talks/icsrc.pdf · 2009-01-07 · blackouts Daniel Bienstock Columbia University, New York ICS09 Daniel Bienstock ( Columbia](https://reader033.fdocuments.us/reader033/viewer/2022050418/5f8e398e1d45223b690ba566/html5/thumbnails/68.jpg)
Basic methodology for the cascade
At time t , arc (i , j ) is removed if τ(t )ij > u ij . Here,
τ(t )ij = αij |f
(t )ij | + (1− αij )τ
(t−1)ij , 0 ≤ αij ≤ 1
where
f (t)ij = flow on (i , j)
τ(t)ij = moving average of flow on (i , j)
What is αij ? Does it actually exist?
→ Robustify the model by allowing αij , randomly or adversarially
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 19 / 21
![Page 69: Robust online control of cascading power grid blackoutsdano/talks/icsrc.pdf · 2009-01-07 · blackouts Daniel Bienstock Columbia University, New York ICS09 Daniel Bienstock ( Columbia](https://reader033.fdocuments.us/reader033/viewer/2022050418/5f8e398e1d45223b690ba566/html5/thumbnails/69.jpg)
Embedded Markov chain model(s)
There are K possible values for α: α(1) < α(2) < . . . < α(K )
Assuming that at time t , αij = α(k ), thenat time t + 1
αij =
α(k +1), with probability πk ,k +1
α(k ), with probability πk ,k
α(k −1), with probability πk ,k −1
These probabilities are known , πk ,k −1 + πk ,k + πk ,k +1 = 1 andπ1,0 = πK ,K +1 = 0.
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 20 / 21
![Page 70: Robust online control of cascading power grid blackoutsdano/talks/icsrc.pdf · 2009-01-07 · blackouts Daniel Bienstock Columbia University, New York ICS09 Daniel Bienstock ( Columbia](https://reader033.fdocuments.us/reader033/viewer/2022050418/5f8e398e1d45223b690ba566/html5/thumbnails/70.jpg)
Embedded Markov chain model(s)
There are K possible values for α: α(1) < α(2) < . . . < α(K )
Assuming that at time t , αij = α(k ), thenat time t + 1
αij =
α(k +1), with probability πk ,k +1
α(k ), with probability πk ,k
α(k −1), with probability πk ,k −1
These probabilities are known , πk ,k −1 + πk ,k + πk ,k +1 = 1 andπ1,0 = πK ,K +1 = 0.
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 20 / 21
![Page 71: Robust online control of cascading power grid blackoutsdano/talks/icsrc.pdf · 2009-01-07 · blackouts Daniel Bienstock Columbia University, New York ICS09 Daniel Bienstock ( Columbia](https://reader033.fdocuments.us/reader033/viewer/2022050418/5f8e398e1d45223b690ba566/html5/thumbnails/71.jpg)
Embedded Markov chain model(s)
There are K possible values for α: α(1) < α(2) < . . . < α(K )
Assuming that at time t , αij = α(k ), thenat time t + 1
αij =
α(k +1), with probability πk ,k +1
α(k ), with probability πk ,k
α(k −1), with probability πk ,k −1
These probabilities are known , πk ,k −1 + πk ,k + πk ,k +1 = 1 andπ1,0 = πK ,K +1 = 0.
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 20 / 21
![Page 72: Robust online control of cascading power grid blackoutsdano/talks/icsrc.pdf · 2009-01-07 · blackouts Daniel Bienstock Columbia University, New York ICS09 Daniel Bienstock ( Columbia](https://reader033.fdocuments.us/reader033/viewer/2022050418/5f8e398e1d45223b690ba566/html5/thumbnails/72.jpg)
Embedded Markov chain model(s)
There are K possible values for α: α(1) < α(2) < . . . < α(K )
Assuming that at time t , αij = α(k ), thenat time t + 1
αij =
α(k +1), with probability πk ,k +1
α(k ), with probability πk ,k
α(k −1), with probability πk ,k −1
These probabilities are known , πk ,k −1 + πk ,k + πk ,k +1 = 1 andπ1,0 = πK ,K +1 = 0.
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 20 / 21
![Page 73: Robust online control of cascading power grid blackoutsdano/talks/icsrc.pdf · 2009-01-07 · blackouts Daniel Bienstock Columbia University, New York ICS09 Daniel Bienstock ( Columbia](https://reader033.fdocuments.us/reader033/viewer/2022050418/5f8e398e1d45223b690ba566/html5/thumbnails/73.jpg)
Embedded Markov chain model(s)
There are K possible values for α: α(1) < α(2) < . . . < α(K )
Assuming that at time t , αij = α(k ), thenat time t + 1
αij =
α(k +1), with probability πk ,k +1
α(k ), with probability πk ,k
α(k −1), with probability πk ,k −1
These probabilities are known , πk ,k −1 + πk ,k + πk ,k +1 = 1 andπ1,0 = πK ,K +1 = 0.
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 20 / 21
![Page 74: Robust online control of cascading power grid blackoutsdano/talks/icsrc.pdf · 2009-01-07 · blackouts Daniel Bienstock Columbia University, New York ICS09 Daniel Bienstock ( Columbia](https://reader033.fdocuments.us/reader033/viewer/2022050418/5f8e398e1d45223b690ba566/html5/thumbnails/74.jpg)
Ongoing work: stochastic gradients method
Repeat:
Compute a sample path for each of the parametersαij : α1,ij , α2,ij , . . . , αT ,ij .
Compute the gradient ∇b,sF assuming the sampled αij
Step: (b, s) = (b, s) + ε∇b,sF .
→ Can be proved to converge to a (local) optimumunder appropriate assumptions (modifications)
→ Highly parallelizable
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 21 / 21
![Page 75: Robust online control of cascading power grid blackoutsdano/talks/icsrc.pdf · 2009-01-07 · blackouts Daniel Bienstock Columbia University, New York ICS09 Daniel Bienstock ( Columbia](https://reader033.fdocuments.us/reader033/viewer/2022050418/5f8e398e1d45223b690ba566/html5/thumbnails/75.jpg)
Ongoing work: stochastic gradients method
Repeat:
Compute a sample path for each of the parametersαij : α1,ij , α2,ij , . . . , αT ,ij .
Compute the gradient ∇b,sF assuming the sampled αij
Step: (b, s) = (b, s) + ε∇b,sF .
→ Can be proved to converge to a (local) optimumunder appropriate assumptions (modifications)
→ Highly parallelizable
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 21 / 21
![Page 76: Robust online control of cascading power grid blackoutsdano/talks/icsrc.pdf · 2009-01-07 · blackouts Daniel Bienstock Columbia University, New York ICS09 Daniel Bienstock ( Columbia](https://reader033.fdocuments.us/reader033/viewer/2022050418/5f8e398e1d45223b690ba566/html5/thumbnails/76.jpg)
Ongoing work: stochastic gradients method
Repeat:
Compute a sample path for each of the parametersαij : α1,ij , α2,ij , . . . , αT ,ij .
Compute the gradient ∇b,sF assuming the sampled αij
Step: (b, s) = (b, s) + ε∇b,sF .
→ Can be proved to converge to a (local) optimumunder appropriate assumptions (modifications)
→ Highly parallelizable
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 21 / 21
![Page 77: Robust online control of cascading power grid blackoutsdano/talks/icsrc.pdf · 2009-01-07 · blackouts Daniel Bienstock Columbia University, New York ICS09 Daniel Bienstock ( Columbia](https://reader033.fdocuments.us/reader033/viewer/2022050418/5f8e398e1d45223b690ba566/html5/thumbnails/77.jpg)
Ongoing work: stochastic gradients method
Repeat:
Compute a sample path for each of the parametersαij : α1,ij , α2,ij , . . . , αT ,ij .
Compute the gradient ∇b,sF assuming the sampled αij
Step: (b, s) = (b, s) + ε∇b,sF .
→ Can be proved to converge to a (local) optimumunder appropriate assumptions (modifications)
→ Highly parallelizable
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 21 / 21
![Page 78: Robust online control of cascading power grid blackoutsdano/talks/icsrc.pdf · 2009-01-07 · blackouts Daniel Bienstock Columbia University, New York ICS09 Daniel Bienstock ( Columbia](https://reader033.fdocuments.us/reader033/viewer/2022050418/5f8e398e1d45223b690ba566/html5/thumbnails/78.jpg)
Ongoing work: stochastic gradients method
Repeat:
Compute a sample path for each of the parametersαij : α1,ij , α2,ij , . . . , αT ,ij .
Compute the gradient ∇b,sF assuming the sampled αij
Step: (b, s) = (b, s) + ε∇b,sF .
→ Can be proved to converge to a (local) optimumunder appropriate assumptions (modifications)
→ Highly parallelizable
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 21 / 21
![Page 79: Robust online control of cascading power grid blackoutsdano/talks/icsrc.pdf · 2009-01-07 · blackouts Daniel Bienstock Columbia University, New York ICS09 Daniel Bienstock ( Columbia](https://reader033.fdocuments.us/reader033/viewer/2022050418/5f8e398e1d45223b690ba566/html5/thumbnails/79.jpg)
Ongoing work: stochastic gradients method
Repeat:
Compute a sample path for each of the parametersαij : α1,ij , α2,ij , . . . , αT ,ij .
Compute the gradient ∇b,sF assuming the sampled αij
Step: (b, s) = (b, s) + ε∇b,sF .
→ Can be proved to converge to a (local) optimumunder appropriate assumptions (modifications)
→ Highly parallelizable
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 21 / 21
![Page 80: Robust online control of cascading power grid blackoutsdano/talks/icsrc.pdf · 2009-01-07 · blackouts Daniel Bienstock Columbia University, New York ICS09 Daniel Bienstock ( Columbia](https://reader033.fdocuments.us/reader033/viewer/2022050418/5f8e398e1d45223b690ba566/html5/thumbnails/80.jpg)
Ongoing work: stochastic gradients method
Repeat:
Compute a sample path for each of the parametersαij : α1,ij , α2,ij , . . . , αT ,ij .
Compute the gradient ∇b,sF assuming the sampled αij
Step: (b, s) = (b, s) + ε∇b,sF .
→ Can be proved to converge to a (local) optimumunder appropriate assumptions (modifications)
→ Highly parallelizable
Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 21 / 21