APPENDIX 8

- C. LIFETIME DISTRIBUTIO NS

TABLE OF CO NTENTS

8

- C.1 INTRODUCTION

................................ ................................ ................................ . 8

- C

- 1 8

- C.2 DERIVATION OF WEIBUL L DISTRIBUTION PARAM ETERS

..................... 8

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- 1

LIST OF TABLES

Table 8

- C.2.1 Residential Dehumidifier Lifetime Parameters

................................ ......... 8

- C

- 2

LIST OF FIGURE S

Figure 8

- C.2.1 Fraction of Portable Unit Dehumidifier Failing

................................ ......... 8

- C

- 3 Figure 8

- C.2.2 Cumulative Lifetime Length of Portable Unit Dehumidifiers

................... 8

- C

- 3 Figure 8

- C.2.3 Fraction of Whole

- home Dehumidifier Failing

................................ .......... 8

- C

- 4 Figure 8

- C.2.4 Cumulative Lifetime Length of Whole

- home Dehumidifiers

.................... 8

- C

- 4

8

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- iAPPENDIX 8

- C. LIF ETIME DISTRIBU TIONS

8

- C.1 IN TRODUCTION
The U.S. Department of Energy ( DOE ) characterized the lifetime of both product types of dehumidifiers (i.e. portable unit dehumidifiers, whole

- home dehumidifiers) being considered for new energy efficiency standards. DOE characterized residential dehumidifier l ifetimes using a Weibull probability distribution that ranged from the minimum to maximum lifetime estimates

, as described in chapter 8

, section 8.2. 2

. The We ibull distribution is recommended for evaluating lifetime data

, because it can be shaped to match low, average, and high values . The probability of exceeding the high value is contained in the long tail of the Weibull distribution

. 1

, 2

8

- C.2 DERIVATION OF WEIBULL DISTRIBUTION PARAMETERS

Weibull distributions utilize available data to assign low, average, and high values to a random variable that has unknown distribution parameters. DOE appli ed Weibull distribution s to product lifetime data to de riv e low, average, and high lifetime value s , along with a percentile containing a high value. A similar approach is described in a technical note to the software Crystal Ball , which uses a most likely value in place of an average value. 3 The Weibull distribut ion can be defined as:

W here :

L = location

, α (alpha) = scale , and β (beta) = shape

.

T he cumulative distribution therefore is :

Based on available data, Weibull distribution parameters are specified as follow s.

1. The output deviates must be greater than the expert opinion of 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 percent or 90 percent )

.

8

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- 1The values for the parameters in the equations we re determined using the approach outlined in Crystal Ball’s technical note

. 3

Crystal Ball can be used to check a solution by (1) specifying a Weibull distribution that has the calculated para meters (location, scale, and shape) in an assumption cell , then (2) generat ing a forecast that equals th at assumption. The f orecast histogram and statistics will confirm whether the Weibull distribution matches the desired shape.

This solution can be chec ked 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 dist ribution matches the desired shape.

Table 8C. 2 .1 shows the average, minimum, maximum lifetime, and maximum percentile values used to determine the Weibull distribution parameters alpha and beta. For portable unit dehumidifiers, DOE used lifetime estimate s from the pre vious DOE rulemaking for dehumidifiers. 4 DOE estimated that portable unit dehumidifier lifetimes d o not vary by capacity size

. DOE assumed whole

- home dehumidifiers have the same life span as residential room air conditioner. DOE therefore applied the lifetime parameters derived for room air conditioners to whole

- home dehumidifiers. 5 DOE estimated that the maximum lifetime percentile for both product type s was 99 percent.

Table 8

- C.2.1 Residential Dehumidifier Lifetime Parameters

Product A pplication Expert Opinion Values Weibull Parameters

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

Portable unit Dehumidifiers -- 11. 0

-- 99 11.00 4.20

Whole

- home Dehumidifiers 3.0 10.5 20.00 99 20.30 2.50

Figures 8C.2.1 to 8C.2

. 4 show the Weibull distribution as well as the cumulative Weibull distribution for each product type of residential dehumidifiers

.

8

- C

- 2Figure 8

- C.2

. 1 Fraction of P ortable Unit Dehumidifier Failing

Figure 8

- C.2

. 2 Cumulative Lifetime Length of Portable Unit Dehumidifiers 0% 2% 4% 6% 8% 10% 12% 14%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25Porbability of Failure (%)

Age of Equipment (Years)

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20Cumulative Porbability (%)

Age of Equipment (Years)

8

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- 3Figure 8

- C.2

. 3 Fraction of Whole

- home Dehumidifier Failing

Figure 8

- C.2

. 4 Cumulative Lifetime Length of Whole

- home Dehumidifiers 0% 1% 1% 2% 2% 3% 3% 4% 4% 5% 5%

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45Porbability of Failure (%)

Age of Equipment (Years)

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45Cumulative Porbability (%)

Age of Equipment (Years)

8

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- 4REFERENCES

1 B arnes, 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.

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

. Appliance Magazine

. October 2003. (Last accessed March 23, 2010.)

< www.appliancemagazine.com/ae/editorial.php?article=197&zone=215&first=1 >

3 Crystal Ball. T echnote: Derivation of Weibull Distribution Parameters When the Minimum, Most Likely Value, and a Percentile are Known . 2007. (Last accessed December 4, 2008.) Available from: www.crystalball.com/support/simulation/cbl_gen_021A.html

.

4 DOE

– Energy Efficiency and Renewable Energy (EERE). Energy Conservation Program for Consumer Products: Technical Support Document: Energy Efficiency Program for Consumer Products and Commercial and Industrial Equipment, Residential Dishwashers, Dehumidifiers, and Cooking Products, and Commercial Clothes Washers. 2009. Washington, DC. (Last accessed October 10, 2013.)

< http://www1.eere.energy.gov/buildings/appliance_standards/residential/cooking_products_f inal_rule_tsd.html >

5 DOE

– EERE. Energy Conservation Program for Consumer Products. Technical Support Documen t: Energy Efficiency Program for Consumer Products and Commercial and Industrial Equipment, Residential Clothes Dryers and Room Air Conditioners. 2011. Washington, D.C. (Last accessed October 25, 2013.)

< http://www.regulations.gov/#!documentDetail;D=EERE

- 2007

- BT

- STD

- 0010

- 0053 >

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