## APPENDIX 8-C. LIFETIME DISTRIBUTIONS

TABLE OF CONTENTS

8-C.1 INTRODUCTION ................................................................................................. 8-C-1

8-C.2 DERIVATION OF WEIBULL DISTRIBUTION PARAMETERS ..................... 8-C-1

8-C.3 WATER HEATER LIFETIME DISTRIBUTIONS .............................................. 8-C-2

8-C.4 GAS-FIRED POOL HEATERS LIFETIME DISTRIBUTIONS.......................... 8-C-7 8-C.5 DIRECT HEATING EQUIPMENT LIFETIME DISTRIBUTIONS.................... 8-C-8

LIST OF TABLES

Table 8-C.3.1 Water Heaters............................................................................................. 8-C-2

Table 8-C.4.1 Gas-Fired Pool Heaters.............................................................................. 8-C-7 Table 8-C.5.1 Direct Heating Equipment ......................................................................... 8-C-8

LIST OF FIGURES

Figure 8-C.3.1 Fraction of the Gas-Fired Storage Water Heaters Failing ......................... 8-C-3 Figure 8-C.3.2 Cumulative Lifetime Length of Gas-Fired Storage Water Heaters ........... 8-C-3 Figure 8-C.3.3 Fraction of the Electric Water Heaters Failing .......................................... 8-C-4 Figure 8-C.3.4 Cumulative Lifetime Length of Electric Water Heaters ............................ 8-C-4 Figure 8-C.3.5 Fraction of the Oil-fired Storage Water Heaters Failing ........................... 8-C-5 Figure 8-C.3.6 Cumulative Lifetime Length of Oil-fired Storage Water Heaters ............. 8-C-5 Figure 8-C.3.7 Fraction of the Gas-Fired Instantaneous Water Heaters Failing................ 8-C-6 Figure 8-C.3.8 Cumulative Lifetime Length of Gas-Fired Instantaneous Water
Heaters ....................................................................................................... 8-C-6 Figure 8-C.4.2 Fraction of the Gas-Fired Pool Heater Failing........................................... 8-C-7 Figure 8-C.4.3 Cumulative Lifetime Length of Gas-Fired Pool Heaters........................... 8-C-8 Figure 8-C.5.2 Fraction of the Direct Heating Equipment Failing .................................... 8-C-9 Figure 8-C.5.3 Cumulative Lifetime Length of Direct Heating Equipment ...................... 8-C-9

8-C-i

## APPENDIX 8-C. LIFETIME DISTRIBUTIONS

8-C.1 INTRODUCTION

For each product class, DOE characterized the product lifetime using a Weibull probability distribution that ranged from the minimum to maximum lifetime estimates described in Ch. 8. The Weibull distribution is recommended for applying lifetime data because it can be shaped to match low, most likely or average, and high values, yet still allow some probability of exceeding the high value. The Weibull distribution has a long tail, which the analyst can use to specify a distribution with some chance of exceeding the high value. 1, 2



## 8-C.2 DERIVATION OF WEIBULL DISTRIBUTION PARAMETERS

DOE utilized an approach for calculating the Weibull distribution using product lifetime data to determine low, average, high value, and percentile of high value. A similar approach is

3 

## described by Crystal Ball Technical Note, which uses most likely value instead of average. Available data is used to assign low, average, and high values to a random variable with unknown distribution parameters.

The Weibull distribution can be defined as:

β ⎛ ⎜ ⎝ L ⎞ ⎟ ⎠ − β

− 1 x 

## ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ β

− 

## x

− L

α 

## f ( x ) = 

## Eq. 1 exp α α

Where:



## L = location



## α = scale 

## β = shape

Therefore, the cumulative distribution is

β ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ x

− L

− α 

## F ( x ) = 1

− e xp 

## Eq. 2

Based on available data, Weibull distribution parameters are specified in the following

manner:

1. The output deviates must be greater than the expert opinion for low value,

2. The average X avg 

## , must be equal to the average value from the available data,

3. The high value, xb, must correspond to some particular percentile point (e. g., 95%, or 90%)

8-C-1

## The values for the parameters in Eq.1 and Eq. 2 are determined using the approach

3 

## outlined in Crystal Ball’s Technote.

This solution can be checked using Crystal Ball by specifying a Weibull distribution with the calculated parameters (location, scale, and shape) in an assumption cell and generate a forecast that equals the assumption. Forecast histogram and statistics verify that the Weibull distribution matches the desired shape.



## 8-C.3 WATER HEATER LIFETIME DISTRIBUTIONS

Table 8-C.3.1 shows the average, minimum, maximum lifetime, and maximum percentile values used to determine the Weibull distribution parameters alpha and beta. The analysis utilized a maximum percentile of 99%.



## Table 8-C.3.1 Water Heaters

Product Class Expert Opinion Values Weibull

Parameters

Minimum

(years) Average (years) Maximum (years) Maximum percentile (%) Alpha (scale) Beta (shape)
Gas-Fired/LPG Storage 7 12 20 99 5.6258 1.8233
Electric Storage 6 14 20 99 8.9079 3.3778
Oil-Fired Storage 5 9 13 99 4.5001 2.6548

Gas-Fired Instantaneous 8 20 30 99 13.4203 3.0897



## Figure 8-C.3.2 to Figure 8-C.3.9 show the Weibull distribution as well as the cumulative Weibull distribution for each water heater product class.

8-C-20% 2% 4 % 6 % 8 % 10 % 1 2 % 14 %

0 2 4 6 8 10 12 14 16 18 20 22 24

Ag e of Equipment (Years)Probability of Failure (%)



## Figure 8-C.3.2 Fraction of the Gas-Fired Storage Water Heaters Failing

0% 10% 2 0 % 3 0 % 4 0 % 5 0 % 6 0 % 7 0 % 8 0 % 9 0 % 1 0 0%

0 2 4 6 8 10 1 2 14 16 18 20 22 24

Ag e of Equipment (Years)Probability (%)



## Figure 8-C.3.3 Cumulative Lifetime Length of Gas-Fired Storage

Water Heaters

8-C-30% 2% 4 % 6 % 8 % 1 0 % 1 2 % 1 4 %

0 2 4 6 8 10 1 2 14 16 18 20 22 24

Ag e of Equipment (Years)Probability of Failure (%)



## Figure 8-C.3.4 Fraction of the Electric Water Heaters Failing

0% 10% 2 0 % 3 0 % 4 0 % 5 0 % 6 0 % 7 0 % 8 0 % 9 0 % 1 0 0%

0 2 4 6 8 10 1 2 14 16 18 20 22 24

Ag e of Equipment (Years)Probability (%)



## Figure 8-C.3.5 Cumulative Lifetime Length of Electric Water

Heaters

8-C-40% 5% 1 0 % 1 5 % 2 0 % 2 5 %

0 2 4 6 8 10 1 2 14 16

Ag e of Equipment (Years)Probability of Failure (%)



## Figure 8-C.3.6 Fraction of the Oil-fired Storage Water Heaters Failing

0% 10% 2 0 % 3 0 % 4 0 % 5 0 % 6 0 % 7 0 % 8 0 % 9 0 % 1 0 0%

0 2 4 6 8 10 1 2 14 16

Ag e of Equipment (Years)Probability (%)



## Figure 8-C.3.7 Cumulative Lifetime Length of Oil-fired Storage

Water Heaters

8-C-50% 2% 4 % 6 % 8 % 1 0 %

0 2 4 6 8 10 1 2 14 16 18 20 22 24 26 28 30 32 34

Ag e of Equipment (Years)Probability of Failure (%)



## Figure 8-C.3.8 Fraction of the Gas-Fired Instantaneous Water Heaters Failing

0% 10% 2 0 % 3 0 % 4 0 % 5 0 % 6 0 % 7 0 % 8 0 % 9 0 % 1 0 0%

0 2 4 6 8 10 1 2 14 16 18 20 22 24 26 28 30 32 34

Ag e of Equipment (Years)Probability (%)



## Figure 8-C.3.9 Cumulative Lifetime Length of Gas-Fired

Instantaneous Water Heaters

8-C-6

## 8-C.4 GAS-FIRED POOL HEATERS LIFETIME DISTRIBUTIONS

Table 8-C.4.1 shows the average, minimum, maximum lifetime, and maximum percentile values used to determine the Weibull distribution parameters alpha and beta. The analysis utilized a maximum percentile of 99.99%.



## Table 8-C.4.1 Gas-Fired Pool Heaters

Product Class Expert Opinion Values Weibull

Parameters

Minimum

(years) Average (years) Maximum (years) Maximum percentile (%) Alpha (scale) Beta (shape)

Gas-Fired Pool Heaters 3 6 20 99.99 3.2703 1.3470



## Figure 8-C.4.1 and Figure 8-C.4.2 show the Weibull distribution as well as the cumulative Weibull distribution for each water heater product class.

0% 5% 1 0 % 1 5 % 2 0 % 2 5 %

0 2 4 6 8 10 1 2 14 16 18 20

Ag e of Equipment (Years)Probability of Failure (%)



## Figure 8-C.4.1 Fraction of the Gas-Fired Pool Heater Failing

8-C-70% 10% 2 0 % 3 0 % 4 0 % 5 0 % 6 0 % 7 0 % 8 0 % 9 0 % 1 0 0%

0 2 4 6 8 10 1 2 14 16 18 20

Ag e of Equipment (Years)Probability (%)



## Figure 8-C.4.2	 Cumulative Lifetime Length of Gas-Fired Pool

Heaters

8-C.5 DIRECT HEATING EQUIPMENT LIFETIME DISTRIBUTIONS

Table 8-C.5.1 shows the average, minimum, maximum lifetime, and maximum percentile values used to determine the Weibull distribution parameters alpha and beta. The analysis utilized a maximum percentile of 99%.



## Table 8-C.5.1	 Direct Heating Equipment

Product Class Expert Opinion Values Weibull

Parameters

Minimum

(years) Average (years) Maximum (years) Maximum percentile (%) Alpha (scale) Beta (shape)
All DHE Equipment 10 15 20 99 5.6257 2.6548



## Figure 8-C.5.1 to Figure 8-C.5.2 show the Weibull distribution as well as the cumulative Weibull distribution for each water heater product class.

8-C-80% 2% 4 % 6 % 8 % 1 0 % 1 2 % 1 4 % 1 6 % 1 8 % 2 0 %

0 2 4 6 8 10 1 2 14 16 18 20 22

Ag e of Equipment (Years)Probability of Failure (%)



## Figure 8-C.5.1 Fraction of the Direct Heating Equipment Failing

0% 10% 2 0 % 3 0 % 4 0 % 5 0 % 6 0 % 7 0 % 8 0 % 9 0 % 1 0 0%

0 2 4 6 8 10 1 2 14 16 18 20 22 24

Ag e of Equipment (Years)Probability (%)



## Figure 8-C.5.2 Cumulative Lifetime Length of Direct Heating Equipment

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## REFERENCES

1. Barnes, P. R., J. W. Van Dyke, B. W. McConnell, S. M. Cohn, and S. L. Purucker, The Feasibility of Replacing or Upgrading Utility Distribution Transformers During Routine

Maintenance , 1995. Oak Ridge National Laboratory. Oak Ridge, TN. Report No. Report

No. ORNL-6804/R1. < http://www.ornl.gov/~webworks/cpr/v823/rpt/78562.pdf >

2. Karr, T., Making the Most of Life Test Data . Appliance Magazine , 2003.

< http://www.appliancemagazine.com/print.php?article=197&zone=1&first=1 >

3. Crystal Ball, Technote: Derivation of Weibull Distribution Parameters When the Minimum, Most Likely Value, and a Percentile are Known, 2007.

< http://www.crystalball.com/support/simulation/cbl_gen_021A.html >

4. U.S. Department of Energy - Energy Information Administration, Residential Energy Consumption Survey: 2001 Public Use Data Files, 2001.

< http://www.eia.doe.gov/emeu/recs/recs2001/publicuse2001.html >

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