robust online control of cascading power grid blackoutsdano/talks/icsrc.pdf · 2009-01-07 ·...

Robust online control of cascading power gridblackouts

Daniel Bienstock

Columbia University, New York

Daniel Bienstock ( Columbia University, New York)Robust online control of cascading power grid blackouts ICS09 1 / 21

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.

Summary

A power grid has three components

TRANSMISSION

GENERATION

DISTRIBUTION

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

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

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

|θi − θj | ≤ π/2

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

Complexity issues:

A critical detail

→ For a given level of supply an demand, power flows are uniquelydetermined by the physics – not subject to control.

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.

A critical detail

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 .

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.

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.

Set t = 1.

average)

Set t = 1.

average)

Set t = 1.

average)

Set t = 1.

average)

Set t = 1.

average)

Online control

Set t = 1.

Compute control algorithm .

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

Online control

Set t = 1.

τ(t )ij = αij |g

(t )ij | + (1− αij )τ

(t−1)ij , 0 ≤ αij ≤ 1

Online control

Set t = 1.

τ(t )ij = αij |g

(t )ij | + (1− αij )τ

(t−1)ij , 0 ≤ αij ≤ 1

Online control

Set t = 1.

τ(t )ij = αij |g

(t )ij | + (1− αij )τ

(t−1)ij , 0 ≤ αij ≤ 1

Online control

Set t = 1.

τ(t )ij = αij |g

(t )ij | + (1− αij )τ

(t−1)ij , 0 ≤ αij ≤ 1

Online control

Set t = 1.

τ(t )ij = αij |g

(t )ij | + (1− αij )τ

(t−1)ij , 0 ≤ αij ≤ 1

Online control

Set t = 1.

τ(t )ij = αij |g

(t )ij | + (1− αij )τ

(t−1)ij , 0 ≤ αij ≤ 1

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

{f iju ij

}> 1, then

→ Algorithm = λ

{f iju ij

}> 1, then

→ Algorithm = λ

Application of component-wise control:

600 nodes, 1268 edges, 25 generators, 344 demands, 6 rounds

0 0.2 0.4 0.6 0.8 1

throughput

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.

Affine controls

{f iju ij

}, where

Affine controls

{f iju ij

}, where

Affine controls

{f iju ij

}, where

Affine controls

{f iju ij

}, where

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

Local optimum

Algorithm

3. Repeat.

Local optimum

Algorithm

3. Repeat.

Local optimum

Algorithm

3. Repeat.

Local optimum

Algorithm

3. Repeat.

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

4 CPUs

690 192 0.6408151479 434 0.7011231562 460 0.8451573055 898 0.8899154633 1599 0.9148655418 1983 0.916966

4 CPUs

690 192 0.6408151479 434 0.7011231562 460 0.8451573055 898 0.8899154633 1599 0.9148655418 1983 0.916966

0 50 100 150 200 250 300 350

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)

Robustness

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

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

τ(t )ij = αij |f

(t )ij | + (1− αij )τ

(t−1)ij , 0 ≤ αij ≤ 1

τ(t )ij = αij |f

(t )ij | + (1− αij )τ

(t−1)ij , 0 ≤ αij ≤ 1

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.

αij =

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