Contingency Analysis and N-1 Security

The power grid must keep the lights on even when things break. Contingency analysis tests whether the system can survive equipment failures, and the N-1 criterion is the fundamental security standard.

Why Contingencies Matter

Equipment fails. Transmission lines get struck by lightning. Generators trip offline. Transformers overheat. A secure power system must handle these events without:

Cascading outages
Widespread blackouts
Equipment damage

The August 2003 Northeast blackout — 55 million people without power — demonstrated what happens when contingencies cascade out of control.

The N-1 Criterion

N-1 security means the system must survive the loss of any single component:

After any single credible contingency, the system must remain stable with no thermal overloads and voltages within limits.

If you have $N$ components (lines, generators, transformers), you must be able to lose any one and still operate safely.

What Counts as a "Credible Contingency"?

Typically:

Loss of any single transmission line
Loss of any single generator
Loss of any single transformer
Loss of any single bus (rare, but included for critical substations)

Not usually included:

Multiple simultaneous failures (covered by N-2)
Extreme events (hurricanes, earthquakes)
Common-mode failures (multiple lines on same tower)

The Mathematical Statement

For all single contingencies $c \in C$:

Before contingency (base case):

$$P_{\text{line}} \leq P_{\max} \quad \text{(all lines)}$$ $$V_{\min} \leq V \leq V_{\max} \quad \text{(all buses)}$$

After contingency $c$:

$$P_{\text{line}}^{(c)} \leq P_{\max} \quad \text{(remaining lines)}$$ $$V_{\min}^{(c)} \leq V^{(c)} \leq V_{\max}^{(c)} \quad \text{(all buses)}$$

How Contingency Analysis Works

Step 1: Define Contingency List

Create a list of all contingencies to test:

All transmission line outages
All generator outages
Critical transformer outages

A 1000-bus system might have 1500+ contingencies.

Step 2: Solve Base Case

Run power flow for the intact system. Verify it's feasible (no violations).

Step 3: Test Each Contingency

For each contingency:

Remove the component from the network model
Re-solve power flow (or use sensitivity factors)
Check for violations:
- Line flows > thermal limit
- Voltages outside [0.95, 1.05] p.u.
- Generator reactive limits exceeded
Record severity of any violations

Step 4: Report Results

List contingencies that cause violations, ranked by severity:

Worst thermal overloads (% above limit)
Worst voltage violations
Number of cascading issues

Screening with Sensitivity Factors

Running full AC power flow for 1500 contingencies is slow. Sensitivity factors provide quick approximations:

PTDF (Power Transfer Distribution Factor)

How does power redistribute when a generator-load pattern changes?

$$\Delta P_{\text{line}} = \text{PTDF} \times \Delta P_{\text{injection}}$$

LODF (Line Outage Distribution Factor)

If line $k$ trips, how does flow redistribute to other lines?

$$P_{\ell}^{\text{post}} = P_{\ell}^{\text{pre}} + \text{LODF}_{k \to \ell} \times P_k^{\text{pre}}$$

LODF screening workflow:

Compute LODFs once (offline)
For each contingency: multiply base flows by LODFs
Flag contingencies with potential violations
Run full AC power flow only for flagged cases

This reduces computation from 1500 power flows to perhaps 50.

N-2 and Beyond

N-2 security requires surviving any two simultaneous failures:

Two transmission lines
One line + one generator
Two generators

N-2 is required for:

Extra-high voltage (EHV) lines
Critical generation interconnections
Major load centers

N-k security generalizes to $k$ simultaneous failures, but becomes combinatorially explosive:

N-1: ~1500 contingencies
N-2: ~1,000,000 contingencies
N-3: ~500,000,000 contingencies

Practical N-2 analysis screens for "credible" N-2 events (related failures, common modes).

Security-Constrained OPF

Standard OPF minimizes cost subject to base case constraints only. Security-constrained OPF (SCOPF) adds contingency constraints:

$$\min \sum_g c_g(P_g)$$

Subject to:

$$\text{Base case power flow constraints}$$ $$P_{\text{line}} \leq P_{\max} \quad \text{(base case)}$$ $$P_{\text{line}}^{(c)} \leq P_{\max} \quad \text{(all contingencies } c)$$

This ensures the dispatch is secure even if a contingency occurs.

The Cost of Security

SCOPF solutions cost more than standard OPF because:

Some cheap generation must be backed off to create margin
More expensive units may need to run for post-contingency support
Transmission constraints bind more tightly

The difference represents the cost of security — what we pay to avoid blackouts.

Corrective vs. Preventive Actions

Two philosophies for handling contingencies:

Preventive (Pre-contingency)

Operate so that no action is needed after a contingency — the system automatically stays within limits.

Pros: Simple, safe Cons: More conservative, higher cost

Corrective (Post-contingency)

Operate closer to limits, but have automatic actions ready:

Generator runback
Load shedding
Line switching

Pros: More efficient base case operation Cons: Requires fast automation, higher risk

Modern systems use a mix: preventive for severe contingencies, corrective for less critical ones.

Real-Time Contingency Analysis

Utilities run contingency analysis continuously:

State estimator provides current system state
Contingency analysis tests all N-1 events
Operator displays show any violations
Alarms trigger if security margin is low

Cycle time: every 2-5 minutes.

If a contingency shows violations, operators take action:

Redispatch generation
Adjust voltage setpoints
Call for emergency procedures

Example: Line Outage

Consider a three-bus system with two parallel paths:

    Gen ──[Line 1]──┬── Load
                    │
         ──[Line 2]──┘

Base case: Each line carries 50% of the load (100 MW each), well within 150 MW limits.

Contingency (Line 1 trips): All 200 MW must flow through Line 2.

Result: Line 2 overloads at 200 MW vs. 150 MW limit. N-1 violation!

Solution: Either:

Reduce load to 150 MW (load shedding)
Build a third parallel path (expansion)
Install flow control (phase shifter)
Accept the risk (if contingency is rare)

Cascading Failures

The real danger is when one failure triggers another:

Initial contingency: Line A trips
Overload: Lines B and C now exceed limits
Protection operates: Line B trips on overcurrent
Further overload: Line C now at 200% limit
Cascade: Line C trips, island separates
Frequency collapse: Island has generation-load mismatch

N-1 security is designed to prevent step 2 — no overloads after the first failure.

GAT Contingency Analysis

GAT's gat-algo crate includes contingency analysis:

gat contingency network.arrow --n1

This:

Identifies all single contingencies
Computes LODFs for screening
Runs AC power flow for critical contingencies
Reports violations and severity

Options:

--n2: Test double contingencies
--thermal-limit 100: Override default limits (%)
--voltage-limits 0.95,1.05: Voltage bounds

Key Takeaways

N-1 criterion: Survive any single equipment failure
Contingency analysis tests all credible outages
LODFs enable fast screening without full power flow
Security-constrained OPF embeds contingency constraints in dispatch
Cascading failures are why we need security margins