Planned and Random Outages

Contents

1. Random Outages
2. Repair Time Distribution
   1. Constant Distribution (the simplest case)
   2. Uniform Distribution
   3. Triangular Distribution
   4. Exponential Distribution
   5. Weibull Distribution
   6. Lognormal
   7. Smallest Extreme Value (SEV)
   8. Largest Extreme Value (LEV)
3. Planned (maintenance) Outages
4. References

1. Random Outages

The simulator can model random outages for Generator, Line, Gas Pipeline and Water Pipeline objects for use in a Monte Carlo simulation. Partial and full outages are supported. Outages occur at a frequency controlled by the user-defined forced outage rate which in combination with an expected outage duration implies a mean time between failures (MTBF).

NOTE: This discussion focuses on generator outages, but the same applies to outages on transmission lines and gas pipelines.

The expected number, timing, and severity (duration and size) of outages is determined by the Forced Outage Rate, the repair time distribution and the Outage Rating.

By default, Forced Outage Rate is the fraction of time that units at the generator are expected to be unavailable due to random failures. For example, an outage rate of 10% means that over the course of one year the units are out-of-service for 0.1 × 365 = 36.5 days. The input outage rate may vary over time e.g. annually, seasonally or monthly but not more frequently than the resolution of PASA as controlled by the Step Type setting.

Forced Outage Rate can, alternatively, be input as a percentage of Operating Hours rather than total hours. For example, a 10% outage rate would mean that in 10% of hours where a unit is either operating or intending to operate it out-of-service. The Generator setting Forced Outage Rate Denominator controls this forced outage rate interpretation. This has been developed with reference to the EFORd formula of IEEE Standard [1], in which EFORd stands for Equivalent Forced Outage Rate on demand and it is an industry standard index for evaluating generating unit performance in competitive markets.

Finally, forced outage rate can be linked to planned outages (defined by Units Out) with the setting EFOR Maintenance Adjust.

Note that it is possible to have different repair time distributions for forced and maintenance outages by putting the Forced Outage Rate and Maintenance Rate on different band numbers. See the Maintenance Rate property for examples.

Forced outage events are automatically created by the simulator for all generators with Forced Outage Rate defined. However if the repair time distribution is omitted then a warning will be issued by the simulator and no outages will be modelled for those objects i.e. you must define at a minimum the Mean Time to Repair.

The random number generator for each outage object (generator, line, etc) is initialized from the Model Random Number Seed, which itself is randomly generated if not defined. If you want finer control on the seeding of each outage object you can set the Random Number Seed for each object. The Diagnostic Random Number Seed outputs the seeds read or created to a text file.

Thus to repeat the sequence of random outages used in a simulation set the Model Random Number Seed property. Because the stream of random numbers used depends on the number of outage elements in the system you might get a different set of outage patterns if you add or remove some elements. To avoid this problem you may set the Generator Random Number Seed directly on each object causing them to seed independently.

The automatic generation of forced outages can be switched on/off using the setting Stochastic Outage Scope.

2. Repair Time Distribution

When an element goes out of service it takes some finite time to repair. The time taken usually follows a known distribution with you can derive from historical data. The simulator implements a wide range of distribution types for repair time:

2.1. Constant Distribution (The Simplest Case)

Repair times are constant (always the same duration).

Parameters required to model constant distribution:
T ~(μ)

where:
The location parameter μ is the constant repair time.

For the example in Table 1 the element is out of service on average 9% of the year and each outage event will be 8 hours long.

Table 1: Constant Distribution Example

Property	Value	Units	Band
Forced Outage Rate	9	%	1
Mean Time to Repair	8	hrs	1

2.2. Uniform Distribution

Repair times vary in length uniformly from a minimum and maximum value.

Figure 2.2.1: Uniform Distribution Probability Density Function

Figure 2.2.2: Uniform Distribution Cumulative Distribution Function

a ~ The minimum repair time
b ~ The maximum repair time

Parameters required to model uniform distribution:
T ~(μ, σ)
The location parameter, μ = a
The scale parameter, σ = b - a
For a < t < b

Probability density function:
f(t) = 1 / σ

Cumulative distribution function:
F(t) = (t - μ) / σ

Inverse cumulative distribution function:
tp = μ + σp

Expected value:
E(T) = μ + σ / 2

For the example in Table 2 the element is out of service on average 9% of the year and outage events vary in duration from 6 to 36 hours with uniform probability.

Table 2: Uniform Distribution Example

Property	Value	Units	Band
Forced Outage Rate	9	%	1
Min Time to Repair	6	hrs	1
Max Time to Repair	36	hrs	1

2.3. Triangular Distribution

Repair time happens from a minimum to a maximum value with the mode value known.

Figure 2.3.1: Triangular Distribution Probability Density Function Figure 2.3.2: Triangular Distribution Cumulative Function

a ~ The minimum repair time
b ~ The maximum repair time
c ~ The mode repair time

Parameters required to model triangular distribution:
tp = μ + σp
The location parameter, μ = a
The scale parameter, σ = b - a
The shape parameter, κ = (c - a) / (b - a)

For a < t < b

Probability density function:
f(t) = 2(t-μ) / σ2κ for t ≤ c
f(t) = 2(μ+σ-t) / σ2(1-κ) for t ≥ c

Cumulative distribution function:
F(t) = (t - μ)2 / σ2κ for t ≤ c
F(t) = 1 - (μ+σ-t)2 / σ2(1-κ) for t ≥ c

Inverse cumulative distribution function:
tp = μ + σ√(κp) for p ≤ κ
tp = μ + σ(1-√[(1-κ)(1-p)]) for p ≥ κ

Expected value:
E(T) = μ + σ(1+κ) / 3

  For the example in Table 3 the element is out of service on average 9% of the year and outage events vary in duration from 6 to 36 hours with 12 hours being the highest frequency.

Table 3: Triangular Distribution Example

Property	Value	Units	Band
Forced Outage Rate	9	%	1
Min Time to Repair	6	hrs	1
Mean Time to Repair	12	hrs	1
Max Time to Repair	36	hrs	1

2.4. Exponential Distribution

Repair times happen at a constant average rate, λ. That is, it does not matter how long the time since the last failure, the repair time will be the same. It is commonly used for high quality electronic circuits, or for components that exhibit wearout only after the expected technological life of the component.

Figure 2.4.1: Exponential Distribution Probability Density Function

Figure 2.4.2: Exponential Distribution Cumulative Function

a ~ The minimum repair time
λ ~ The rate parameter

Parameters required by PLEXOS to model exponential distribution:
T ~(μ, σ)
The location parameter, μ = a
The scale parameter, σ = 1 / λ

For t > μ

Probability density function:
f(t) = exp(-(t-μ)/σ) / σ

Cumulative distribution function:
F(t) = 1 - exp(-(t-μ)/σ)

Inverse cumulative distribution function:
tp = μ - σ.ln(1-p)

Expected value:
E(T) = μ + σ

In the following example the element is out of service on average 9% of the year and outage events with minimum duration of 6 hours and the rate of 1.

Table 4: Exponential Distribution

Exponential Distribution 

Property	Value	Units	Band
Forced Outage Rate	9	%	1
Min Time to Repair	6	hrs	1
Repair Time Scalar	1	-	1

2.5. Weibull Distribution

Figure 2.5.1: Weibull distribution probability density function

Figure 2.5.2: Weibull Distribution Cumulative Function

Parameters required to model Weibull distribution:
T ~(μ, σ, κ)
The minimum repair time, μ
The scale parameter, σ
The shape parameter, κ

For t > μ

Probability density function:
f(t) = (κ/σ)[(t-μ)/σ]κ-1exp(-[(t-μ)/σ]κ)

Cumulative distribution function:
F(t) = 1 - exp(-[(t-μ)/σ]κ)

Inverse cumulative distribution function:
tp = μ + σ.(-ln(1-p))1/κ

Expected value:
E(T) = μ + σ.Γ(1+1/κ)

where gamma function, Γ(z) = ∫tz-1.e-tdt from 0 to ∞

Example

In the following example the element is out of service on average 9% of the year and outage events with minimum duration of 6 hours, the shape parameter of 2 and the scale parameter of 1.

Table 5: Weibull Distribution

Property	Value	Units	Band
Forced Outage Rate	9	%	1
Min time to repair	6	%	1
Repair Time Shape	2	-	1
Repair Time Scale	1	-	1

2.6. Lognormal

It is useful for modelling components with a decreasing repair time (due to a small proportion of defects in the population). It is used to describe time to failure for certain degradation processes.

Figure 2.6.1: Lognormal distribution probability density function

Figure 2.6.2: Lognormal distribution cumulative function

Parameters required by PLEXOS to model Lognormal distribution:
The minimum repair time, μ
The scale parameter, σ
The shape parameter, κ

For t > μ

Probability density function:
f(t) = (1/κ(t-μ)).φ_nor([ln(t-μ)-σ]/κ)

Cumulative distribution function:
F(t) = Φ_nor([ln(t-μ)-σ]/κ)

Inverse cumulative distribution function:
tp = μ + exp(σ+κ.Φ_nor^-1(p))

Expected value:
E(T) = μ + exp(σ+κ2/2)

where φ_nor and Φ_nor are the pdf and cdf for standardized normal, and Φ_nor^-1(p) is the p quantile for a standardized normal.

Example

In the following example the element is out of service on average 9% of the year and outage events with minimum duration of 6 hours, the shape parameter of 2 and the scale parameter of 1.

Table 6: Lognormal

Property	Value	Units	Band
Forced Outage Rate	9	%	1
Min Time to Repair	6	hrs	1
Repair Time Shape	2	-	1
Repair Time Scale	1	-	1

2.7. Smallest Extreme Value (SEV)

It can be used to model components with an increasing repair time.

Figure 2.7.1: Smallest extreme value distribution probability density function

Figure 2.7.2: Smallest Extreme value distribution cumulative function

Parameters required by PLEXOS to model SEV distribution:
T ~(μ, σ)
The location parameter, μ
The scale parameter, σ

For -∞ < t < ∞

Probability density function:
f(t) = (1/σ).exp((t-μ)/σ - exp((t-μ)/σ))

Cumulative distribution function:
F(t) = 1 - exp(-exp((t-μ)/σ))

Inverse cumulative distribution function:
tp = μ + σ.ln(-ln(1-p))

Expected value:
E(T) = μ - 0.5772.σ

Example

In the following example the element is out of service on average 9% of the year and outage events with the location parameter of 6 hours and the scale parameter of 1.

Table 7: Smallest Extreme Value(SEV)

Property	Value	Units	Band
Forced Outage Rate	9	%	1
Min Time to Repair	6	hrs	1
Repair Time Scale	1	-	1

2.8. Largest Extreme Value (LEV)

It can be used to model components with an increasing repair time which remains constant after a period of time.

Figure 2.8.1: Largest extreme value distribution probability density function

Figure 2.8.2: Extreme value distribution cumulative function>

Parameters required by PLEXOS to model LEV distribution:
T ~(μ, σ)
The location parameter, μ
The scale parameter, σ

For -∞ < t < ∞

Probability density function:
f(t) = (1/σ).exp(-(t-μ)/σ - exp(-(t-μ)/σ))

Cumulative distribution function:
F(t) = exp(-exp(-(t-μ)/σ))

Inverse cumulative distribution function:
tp = μ - σ.ln(-ln(p))

Expected value:
E(T) = μ + 0.5772.σ

Example

In the following example the element is out of service on average 9% of the year and outage events with the location parameter of 6 hours and the scale parameter of 1.

Table 8: Largest Extreme Value (LEV)

Property	Value	Units	Band
Forced Outage Rate	9	%	1
Min Time to Repair	6	hrs	1
Repair Time Scale	1	-	1

3. Planned (Maintenance) Outages

Generator, transmission line and gas pipeline planned outages can be input with the properties:

• Generator Units Out
• Line Units Out
• Gas Pipeline Units Out

However, a complete forecast schedule of maintenance outages is rarely available for all plant. The PASA simulation phase can automatically schedule additional outages and time those outages and time those outages appropriately at times of maximum reserve.

Further the maintenance class can be used to perform value-based reliability analysis by optimally timing outages accounting for all system costs and constraints.

4. References

[1] IEEE Std 762, "IEEE Standard Definitions for Use in Reporting Electric Generating Unit Reliability, Availability, and Productivity", 2008