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In electrical power engineering reliability studies, various probability distributions are used to model failure rates, repair times, and system performance. Engineering students — and their professors — may be enlightened by the long hand calculations underlying commercial software.
In practice, longhand calculations are almost never performed when electrical power engineers start with a clean sheet of paper in designing a new main-tie-main utility service switchgear.
The science, so to say, is “settled” — i.e. a power engineering canon supported by public utility commissions as best practice. Power engineering courses are rare in the United States and we hope that this contributes to “optional reading list”. Of course, we will not refuse any help to turn this into a formal IEEE paper.
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Here are the top five probability distribution types commonly used:
Exponential Distribution
- Purpose: Models the time between failures of components with a constant failure rate.
- Application: Used in mean time to failure (MTTF) and mean time between failures (MTBF) calculations.
Weibull Distribution
- Purpose: Models the failure rates of components that may increase or decrease over time.
- Application: Aging components, such as transformers, cables, and circuit breakers, where failure rate depends on time.
Normal (Gaussian) Distribution
- Purpose: Represents natural variations in parameters such as voltage, frequency, and load demand.
- Application: Used in load forecasting, demand modeling, and performance evaluation.
Lognormal Distribution
- Purpose: Models failure rates where a component’s life span is influenced by multiple small factors (e.g., insulation degradation).
- Application: Transformer aging, insulation wear-out, and component lifetime studies.
Gamma Distribution
- Purpose: Used to model repair times and failure processes with memory.
- Application: Repair time analysis, reliability block diagrams (RBDs), and system downtime assessments.
Sample Calculation for Electric Service Reliability of a Main-Tie-Main (MTM) Interconnection Utility Service
(Exponential Distribution)* | P(T≤t)=1−e−λt
Step 1: Establish the System Configuration
A Main-Tie-Main (MTM) system consists of:
- Two primary feeders (Main A and Main B)
- A normally open tie breaker between them
- A commercial load center that receives power from either feeder
Each feeder is backed by its own utility source, and the tie breaker allows power to be restored if one feeder fails.
Step 2: Define Reliability Metrics
Key reliability metrics used in power system reliability studies:
- Failure Rate (λ) [Failures per Year]
- Frequency of failures in a given period
- Repair Time (r) [Hours]
- Average time to restore service after a failure
- Unavailability (U) [Hours/Year]
- U=λ×rU = \lambda \times r, which represents expected outage duration per year
For each feeder, assume:
- Main A failure rate: λA=0.1λ_A = 0.1 failures/year
- Main B failure rate: λB=0.1λ_B = 0.1 failures/year
- Repair time per feeder: r=5r = 5 hours
The tie breaker enables load transfer within 0.5 hours when only one feeder fails.
Step 3: Calculate System Unavailability
The system can experience outages under two primary scenarios:
Case 1: Single Feeder Failure (Restorable via Tie)
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- If either feeder fails, the system can switch to the other feeder within 0.5 hours.
- Probability of Main A failing: PA=λA×rA=(0.1×0.5)=0.05P_A = λ_A \times r_A = (0.1 \times 0.5) = 0.05 hours/year
- Probability of Main B failing: PB=λB×rB=(0.1×0.5)=0.05P_B = λ_B \times r_B = (0.1 \times 0.5) = 0.05 hours/year
- Total outage time from single feeder failures:
Usingle=0.05+0.05=0.1U_{single} = 0.05 + 0.05 = 0.1 hours/year
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Case 2: Both Feeders Failing Simultaneously (Unrecoverable)
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- Probability of both failing together (assuming independent failures):
Pdouble=λA×λB×r=(0.1×0.1×5)=0.05P_{double} = λ_A \times λ_B \times r = (0.1 \times 0.1 \times 5) = 0.05 hours/year
- Probability of both failing together (assuming independent failures):
-
Step 4: Compute Total System Unavailability
Reliability Calculation:
Step 1: Single Feeder Outage (Restorable via Tie Breaker)
U_single = (0.1 * 0.5) + (0.1 * 0.5)
= 0.05 + 0.05
= 0.1 hours/year
Step 2: Double Feeder Outage (Both Feeders Failing Simultaneously)
U_double = (0.1 * 0.1 * 5)
= 0.05 hours/year
Step 3: Total System Unavailability
U_total = U_single + U_double
= 0.1 + 0.05
= 0.15 hours/year
Step 4: System Availability Calculation
System Availability = 1 – (U_total / 8760)
= 1 – (0.15 / 8760)
≈ 0.999983 (99.9983%)
This calculation represents the reliability of a Main-Tie-Main interconnection utility service to a commercial customer.
HTML:
Utotal=Usingle+Udouble=0.1+0.05=0.15 hours/yearU_{total} = U_{single} + U_{double} = 0.1 + 0.05 = 0.15 \text{ hours/year} System Availability=1−Utotal8760=1−0.158760≈0.999983\text{System Availability} = 1 – \frac{U_{total}}{8760} = 1 – \frac{0.15}{8760} \approx 0.999983
Remarks
The expected outage duration for the commercial customer is 0.15 hours per year (about 9 minutes annually).
The system availability is 99.9983%, which is very high and typical of highly reliable commercial service.
* The probability of a failure occurring before time tt is given by:
P(T≤t)=1−e−λtP(T \leq t) = 1 – e^{-\lambda t}
where:
- λ\lambda = failure rate (failures per year)
- tt = time duration under consideration
This distribution is widely used in power system reliability studies, as it simplifies failure modeling and allows for straightforward calculations of availability and unavailability.
