Reliability Theory

This reference explains the mathematics behind power system reliability assessment — quantifying the risk of not meeting electricity demand.

What is Reliability?

Power system reliability has two aspects:

1. Adequacy: Are there enough resources (generation, transmission) to meet demand?
2. Security: Can the system withstand disturbances without cascading failures?

This reference focuses on adequacy assessment — the probability-based analysis of supply sufficiency.

Fundamental Concepts

Loss of Load

Loss of Load (LOL) occurs when available generation cannot meet demand:

LOL event: Available Capacity < Demand

This can happen due to:

• Generator outages (forced or planned)
• Transmission constraints
• Demand exceeding forecasts
• Renewable shortfalls

Capacity States

Each generator has two states:

• Available: Operating or ready to operate
• Unavailable: Failed or under maintenance

For a system with n generators, there are 2ⁿ possible capacity states.

Forced Outage Rate (FOR)

The probability a unit is unavailable due to unplanned failure:

FOR = (Forced Outage Hours) / (Service Hours + Forced Outage Hours)

Typical values:

• Coal/gas units: 2-10%
• Nuclear: 3-8%
• Hydro: 1-3%
• Wind/solar: Captured via capacity factor, not FOR

Reliability Indices

LOLE — Loss of Load Expectation

The expected number of hours (or days) per year when load exceeds available capacity:

$$\text{LOLE} = \sum_i p_i \cdot t_i$$

where:

• $p_i$ = probability of capacity state i
• $t_i$ = duration of load loss in state i (hours)

Planning standard: LOLE ≤ 0.1 days/year (2.4 hours/year) is common in North America.

Interpretation: On average, there will be 2.4 hours per year when some load cannot be served. This does NOT mean 2.4 hours of actual blackout — it's a probabilistic expectation.

LOLP — Loss of Load Probability

The probability that load exceeds capacity at any given time:

$$\text{LOLP} = \sum_i p_i \quad \text{(for all states where capacity < demand)}$$

Related to LOLE:

$$\text{LOLE} = \text{LOLP} \times 8760 \text{ hours/year}$$

EUE/ENS — Expected Unserved Energy

The expected energy (MWh) not delivered per year:

$$\text{EUE} = \sum_i p_i \cdot (D_i - C_i) \cdot t_i \quad \text{(for states where } C_i < D_i\text{)}$$

where:

• $D_i$ = demand in state i
• $C_i$ = available capacity in state i

Interpretation: Average annual energy shortfall. More meaningful than LOLE for economic analysis.

LOLH — Loss of Load Hours

Same as LOLE but explicitly in hours:

$$\text{LOLH} = \sum_i p_i \cdot (\text{hours in state } i \text{ with LOL})$$

Normalized Indices

For comparing systems of different sizes:

$$\text{LOLE/peak} = \frac{\text{LOLE}}{\text{Peak Demand}}$$

$$\text{EUE\%} = \frac{\text{EUE}}{\text{Annual Energy Demand}} \times 100\%$$

Analytical Methods

Capacity Outage Probability Table (COPT)

For small systems, enumerate all capacity states:

Example: Two 100 MW units, each with FOR = 0.05

Convolution Method

For larger systems, build COPT incrementally using convolution:

Starting with capacity probability distribution $p(C)$, add unit k with capacity $c_k$ and FOR $q_k$:

$$p_{\text{new}}(C) = (1-q_k) \cdot p_{\text{old}}(C-c_k) + q_k \cdot p_{\text{old}}(C)$$

This builds up the distribution one unit at a time without enumerating $2^n$ states.

Load Duration Curve

Load varies throughout the year. The Load Duration Curve (LDC) shows load ranked from highest to lowest:

     Load
     (MW)
       │     ╭──────╮
  Peak │─────╯       ╲
       │              ╲
       │               ╲
  Base │────────────────╲──────
       └──────────────────────── Hours
       0                    8760

LOLE Calculation with LDC

For each capacity state, find hours where demand exceeds capacity:

$$\text{LOLE} = \sum_i p_i \cdot H(C_i)$$

where $H(C)$ = hours on LDC where load > C.

EUE Calculation

$$\text{EUE} = \sum_i p_i \cdot E(C_i)$$

where $E(C)$ = area under LDC above capacity level C (MWh).

Monte Carlo Simulation

For complex systems with dependencies, analytical methods become intractable. Monte Carlo simulation samples random scenarios.

Sequential Monte Carlo

Simulates system operation chronologically:

1. Initialize: All units available, time = 0
2. Sample next event: Unit failure or repair (exponential distribution)
3. Update state: Mark unit up/down
4. Check adequacy: If capacity < demand, record LOL
5. Advance time: Move to next event or hour
6. Repeat: Until end of year
7. Average: Over many year replications

Advantages:

• Captures chronological effects (ramp limits, storage)
• Models dependent failures
• Handles maintenance schedules

Disadvantages:

• Computationally expensive
• Requires many samples for convergence

Non-Sequential Monte Carlo

Samples independent hourly snapshots:

1. For each hour in the year:
   • Sample each unit state (Bernoulli with FOR)
   • Sum available capacity
   • Compare to demand
   • Record LOL if capacity < demand
2. Repeat: Many times (1000+ replications)
3. Average: Compute expected values

Advantages:

• Much faster than sequential
• Easily parallelizable
• Sufficient for adequacy assessment

Disadvantages:

• Ignores chronological dependencies
• Can't model storage or ramps

Convergence

Standard error decreases as $1/\sqrt{N}$:

$$\text{SE}(\text{LOLE}) \approx \frac{\sigma}{\sqrt{N}}$$

For 1% relative error with LOLE ≈ 2.4 hours:

• Need ~10,000 samples
• Or use variance reduction techniques

GAT Implementation

use gat_algo::reliability::{MonteCarlo, ReliabilityMetrics};

let mc = MonteCarlo::new(num_scenarios);
let metrics: ReliabilityMetrics = mc.compute_reliability(&network, seed)?;

println!("LOLE: {:.2} hours/year", metrics.lole);
println!("EUE: {:.0} MWh/year", metrics.eue);

Multi-Area Reliability

Real power systems span multiple interconnected areas.

Corridor Constraints

Areas are connected by tie-lines (corridors) with limited transfer capacity:

Area A ═══════╦═══════ Area B
              ║
         Transfer limit
           (e.g., 500 MW)

Power can flow between areas, but only up to corridor limits.

Multi-Area LOLE

Each area has its own LOLE, affected by:

• Local generation adequacy
• Import capability from neighbors
• Neighbor's adequacy (can they export?)

$$\text{LOLE}_A = f(\text{local capacity, import limit, availability of imports})$$

Coordinated Assessment

The multi-area problem considers:

1. Sample capacity states in all areas
2. Compute optimal power transfers (respecting limits)
3. Determine LOL in each area after transfers
4. Aggregate across scenarios

GAT Multi-Area Implementation

use gat_algo::{MultiAreaSystem, MultiAreaMonteCarlo};

let mut system = MultiAreaSystem::new();
system.add_area(AreaId(0), network_a)?;
system.add_area(AreaId(1), network_b)?;
system.add_corridor(Corridor::new(0, AreaId(0), AreaId(1), 500.0))?;

let mc = MultiAreaMonteCarlo::new(1000);
let metrics = mc.compute_multiarea_reliability(&system)?;

println!("Area A LOLE: {:.2}", metrics.area_lole[&AreaId(0)]);
println!("Area B LOLE: {:.2}", metrics.area_lole[&AreaId(1)]);

ELCC — Effective Load Carrying Capability

ELCC measures the capacity value of variable resources.

Definition

ELCC = Additional load the system can serve at the same reliability level when a resource is added.

$$\text{ELCC}(\text{resource}) = \text{Load}_{\text{with resource}} - \text{Load}_{\text{without resource}} \quad \text{(at constant LOLE)}$$

Calculation Method

1. Compute base case LOLE with existing system
2. Add new resource (e.g., 100 MW wind farm)
3. Increase load until LOLE returns to base level
4. ELCC = load increase amount

Capacity Credit

$$\text{Capacity Credit} = \frac{\text{ELCC}}{\text{Nameplate Capacity}} \times 100%$$

Typical values:

• Thermal units: 90-95%
• Wind: 10-30%
• Solar: 30-70% (depends on peak timing)
• Storage: Varies with duration

Why ELCC < Nameplate for Renewables

Variable resources aren't always available when needed:

• Wind may be calm during peak demand
• Solar unavailable at night (evening peaks)
• Correlation with system stress matters

Distribution Reliability Indices

For distribution systems serving end customers:

SAIDI — System Average Interruption Duration Index

Average outage duration per customer:

$$\text{SAIDI} = \frac{\sum(\text{Customer Interruption Durations})}{\text{Total Customers}}$$

Units: minutes or hours per customer per year.

SAIFI — System Average Interruption Frequency Index

Average number of outages per customer:

$$\text{SAIFI} = \frac{\sum(\text{Customer Interruptions})}{\text{Total Customers}}$$

Units: interruptions per customer per year.

CAIDI — Customer Average Interruption Duration Index

Average duration of an interruption:

$$\text{CAIDI} = \frac{\text{SAIDI}}{\text{SAIFI}} = \frac{\sum(\text{Durations})}{\sum(\text{Interruptions})}$$

Units: minutes or hours per interruption.

CAIFI — Customer Average Interruption Frequency Index

Average interruption frequency for affected customers:

$$\text{CAIFI} = \frac{\sum(\text{Interruptions})}{\text{Customers Affected}}$$

N-1 and N-2 Criteria

Deterministic security standards complement probabilistic reliability.

N-1 Criterion

The system must survive any single contingency:

• Loss of one generator
• Loss of one transmission line
• Loss of one transformer

Without:

• Overloads on remaining elements
• Voltage violations
• Loss of load

N-2 Criterion

For critical facilities, survive two simultaneous failures:

• Double-circuit tower collapse
• Substation busbar fault
• Common-mode failures

Relationship to LOLE

N-1/N-2 are deterministic (pass/fail). LOLE is probabilistic (expected value).

Both are needed:

• N-1 ensures immediate security
• LOLE ensures long-term adequacy

Practical Considerations

Data Requirements

Reliability assessment requires:

• Generator capacities and FOR values
• Load forecast (hourly for 8760 hours)
• Transmission limits (for multi-area)
• Maintenance schedules (planned outages)
• Renewable profiles (for ELCC)

Sensitivity Analysis

Key sensitivities to test:

• FOR uncertainty (±20% typical)
• Load forecast error
• Renewable correlation assumptions
• Transmission limit changes

Computational Efficiency

For large systems:

• Use variance reduction (importance sampling, control variates)
• Parallel Monte Carlo across scenarios
• Smart sampling (focus on high-impact states)

GAT uses non-sequential Monte Carlo with parallel scenario evaluation.

Mathematical Appendix

Exponential Failure Model

Time to failure follows exponential distribution:

$$P(\text{failure before time } t) = 1 - e^{-\lambda t}$$

where $\lambda$ = failure rate (failures/hour).

Mean time to failure:

$$\text{MTTF} = \frac{1}{\lambda}$$

FOR relationship:

$$\text{FOR} = \frac{\text{MTTR}}{\text{MTTF} + \text{MTTR}}$$

where MTTR = mean time to repair.

Markov Model for Two-State Unit

States: Up (1), Down (0)

Transition rates:

• $\lambda$: failure rate (Up → Down)
• $\mu$: repair rate (Down → Up)

Steady-state probabilities:

$$P(\text{Up}) = \frac{\mu}{\lambda + \mu}$$

$$P(\text{Down}) = \frac{\lambda}{\lambda + \mu} = \text{FOR}$$

Convolution Formula Derivation

If $C_1$ has distribution $p_1(c)$ and $C_2$ has distribution $p_2(c)$, the sum $C = C_1 + C_2$ has:

$$p(c) = \sum_x p_1(x) \cdot p_2(c - x)$$

This is the discrete convolution of $p_1$ and $p_2$.

References

Textbooks

• Billinton & Allan, Reliability Evaluation of Power Systems — The standard reference
• Billinton & Allan, Reliability Evaluation of Engineering Systems — General reliability theory
• Endrenyi, Reliability Modeling in Electric Power Systems — Advanced topics

Standards

• IEEE 762: Standard Definitions for Use in Reporting Electric Generating Unit Reliability
• NERC TPL: Transmission Planning Standards
• NERC MOD: Modeling Standards

GAT Documentation

• Reliability Guide — Practical usage
• Analytics Reference — Reliability metrics in GAT
• Glossary — Term definitions