Protecting against national-scale power blackouts

Daniel Bienstock, Columbia University

Collaboration with:

Sara Mattia, Universitá di Roma, Italy

Thomas Gouzènes, Réseau de Transport d’Electricité, France

• August 2003: North America. 50 million people affected during two days; New York City loses power

• September 2003: Switzerland-France-Italy. 57 million people affected during one day; Italy loses power

• Other major incidents in recent years in Europe and Brazil

• The potential economic and human consequences of a prolongued national-scale blackout are significant

Recent major incidents

Were the blackouts due to insufficient generation capacity?

No: they were due to inadequately protected transmission networks

U.S.-Canada task force: The leading cause of the blackout was

Inadequate System Understanding

A power grid has 3 components

The transmission network is the key ingredient in modern grids

Modern transmission networks are “lean” and, as a result, “brittle”

An inconvenient fact

The power flows in a grid are controlled by the laws of physics

When analyzing a hypothetical change in a network, the behavior of the power flows must be computed -- it cannot be dictated

Two popular methodologies:

• AC power flow models

• DC (linearized) flow models

Summary

“AC” models for computing power flows

• Account for both “active” and “reactive” power flows• Fairly accurate• Non-convex system of nonlinear equations• Computationally intensive, Newton-like methods• Solution methods tend to require a good initial guess • Heavy data requirements

“DC” models• Linearized approximations of AC models• Much faster• Usually preferred by the industry for large-scale analysis

How does a blackout develop?

Individual power lines fail due to:

• External effects: fires, lightning strikes, tree contacts, malicious agents (?)

• Thermal effects: an overloaded line will melt -- usually requires several minutes(protection equipment will shut it down first)

The physics and engineering underlying line failures are well understood

Individual line failures system collapse

A model for system collapse

Initial set of externally caused faults:

Several lines are disabled

The network is altered – new power flows ensue

flows in some of the lines exceed the line ratings

Further line shutoffs

New network: new power flows

Cascade ! (sometimes)

Simulation

Round No. of shut-off lines

No. of connected components (“islands”)

Demand served (%)

1 2 1 100.0

2 8 3 100.0

3 17 8 87.66

4 20 16 82.72

What we are doing

• Proactive planning: how to economically engineer a network so as to ride-out potential failure scenarios

Each “scenario” is an “interesting” combination of externally caused faults. Example from industry: “N – k” modeling

• Reaction planning: what to do if a significant event materializes

• From a theoretical standpoint, very intractable

• Multiple time scales

The adversarial model

Proactive model

We can upgrade a network in a number of ways. Examples:

Upgrade individual lines

Add new lines:

Join/split nodes:

Integer programming approach

• 0/1 vector x: each entry represents whether a certain action is taken, or not

• x has an entry for each line of the network

• example: a line parallel to a certain line is added, or not

• total cost = cT x, for a certain cost vector cProblem: find x feasible, of minimum cost

What is feasible?

In each scenario (of a certain list), the networkaugmented as per vector x survives the cascade

Solution approach: game against an adversaryMaintain a “working model” M,

which describes conditions that a protection plan x must satisfy

This model may be incomplete

Solve the problem

FIND x OF MINIMUM COST THAT SATISFIES THE CONDITIONS

STIPULATED BY M,

with solution x*

Is x* adequate in all scenarios?

YES - DONENO

In some scenario, x* does not suffice. State this fact algebraically

Add this algebraic statement to M

Solution approach: Bender’s decompositionMaintain a “working formulation”

Ax b

of inequalities valid for feasible x

Solve the problem

Minimize cTx

subject to: Ax b, x 0/1

With solution x*

x* feasible?

YES - DONENO

Find a valid inequalityTx

with Tx* <

Add Tx

to Ax b

Simple example:

• we “protect” power lines – a 0/1 variable x per each line

• the grid survives a cascade if 70% of demand is met

• if the grid survives two rounds then it survives

First round after initial event:• lines 1 – 7 shut off• 5 islands, 80% of demand is met

Second round:• lines 8 – 13 shut off, 15 islands• 61% < 70% of demand met, collapse

x1 + x2 + x3 + x6 + x11 + x12 + x13 1

Experiments, and lessons

• Algorithm converges in few iterations, even with thousands of scenarios

• But each iteration is expensive because of the need to simulate scenarios to test if a

certain network is survivable – in the worst case, all scenarios must be simulated

And where do the scenarios come from?

A model for system collapse, revisited

Initial set of externally caused faults:

Several lines are disabled

The network is altered – new power flows ensue

flows in some of the lines exceed the line ratings

Further line shutoffs

New network: new power flows

Research topic: can this process be efficiently approximated?

How are scenarios generated?

Today: “N-1” analysis

• It can prove too slow on large networks

• Many of the scenarios are uninteresting

• The generalization: “N – k” analysis is prohibitively expensive

A different technique

Stochastic simulation: assign a fault probability to each network component, and simulate the entire system

• We are dealing with extremely low probability events

• The interesting scenarios have very low probability, which will likely be incorrectly estimated

• And in any case we will generate many unimportant scenarios

Ongoing work: adversarial problem

ProblemProblem: find a smallest initial set of faults, following which a cascade occurs

Enumerating all k-subsets for k 5 is computationally infeasible for large grids

Approach we are using: combination of

approximate dynamic programmingand integer programming

One approach

• Adversary enumerates sets of k (small) lines at a time

• Adversary chooses the best set according to an appropriate merit function

• Examples: number of overloaded lines, nonlinear function of overloads (e.g. exponential), cost of flow under nonlinear cost function

A difficulty: problem is not monotone

“Braess’ Paradox”

Example:

if we cut lines a, b, and c the system cascades but if we cut a, b, c, and d it does not

Solution approach: game against an adversaryMaintain a “working model” M,

which describes conditions that a protection plan x must satisfy

This model may be incomplete

Solve the problem

FIND x OF MINIMUM COST THAT SATISFIES THE CONDITIONS

STIPULATED BY M,

with solution x*

Can the adversary collapse the system protected by plan x*?

YES - DONENO

In some scenario, x* does not suffice. State this fact algebraically

Add this algebraic statement to M