United States Department of Commerce

Tec nol Administration

National Institute of Standards and Technology

NIST Technical Note 1427

An Analysis of Efficiency Testing Under the Energy

Policy Conservation Act: A Case Study With

Application to Distribution Transformers '

IC L. Stricklett, angel, and 0. Petersons

EE-TP-98-550 COMMENT 41
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EE-TP-98-550 COMMENT 41
NIST Technical Me 1427

An Analysis of Efficiency Testing Under the Eneij-2^3f

Policy and Conservation Act: A Case Study With

Application to Distribution Transformers

K. L. Stridklett

Electricity Division

Electronics and Electrical Engineering Laboratory

National Institute of Standards and Technology

Gaithersburg, MD 20899-0001

M. Vangel

Statistical Engineering Division

Information Technology Laboratory

National Institute of Standards and Technology

Gaithersburg, MD 20899-0001

0. Petersons

Electricity Division

Electronics and Electrical Engineering Laboratory

National Institute of Standards and Technology

Gaithersburg, MD 20899-0001

April 1999

U.S. Department of Commerce

William M. Daley, Secretary

Technology Administration

Gary Bachula, Acting Under Secretary for Technology

National 1 of Standards and Technology

Raymond G. Kammer, Director

EE-TP-98-550 COMMENT 41
National Institute of Standards U.S. Government Printing Office For sale by the Superintendent of

and Technology Washington: 1999 Documents

Technical Note 1427 U.S. Government Printing Office

Natl. Inst. Stand. Technol. Washington, DC 20402

Tech. Note 1427

58 pages (April 1999)

CODEN: NTNOEF

EE-TP-98-550 COMMENT 41
Bibliographic Information

Abstract

The protocols for efficiency testing promulgated by 10 CFR Part 430 and established under the Energy Policy

and Conservation Act of 1975 as amended (EPCA) are discussed. The case of distribution transformers, which

are covered under the proposed new 10 CFR Part 432, is treated in detail. Model calculations are presented

that estimate the probability of demonstrating compliance and the average number of units tested under the 10

CFR Part 430 and Part 432 sampling plans. The results of model calculations for the sampling plan contained

in NEMA Standard TP 2 are also presented.

Keywords

distribution transformers; efficiency testing; energy conservation; energy policy; EPCA; operating characteristic;

sampling plan

ordering

Copies of this document are available from the National Technical Information Service, 5285 Port Royal Road,

Springfield, VA 22161, at (800) 553-6847 or (703) 487-4650.

EE-TP-98-550 COMMENT 41
Contents

1 Introduction 1

2 General guidelines 1

3 Industry practice 2

4 Methods of analysis 2

4.1 Loss representation ............................................ 4

5 Compliance testing 5

5.1 10 CFR Part 430 and the proposed 10 CFR Part 432 (Method I) ..............

.

....

5.1.1 10 CFR Part 430 ............................................

5.1.2 The proposed 10 CFR Part 432 .................................

5.1.3 Results ..............................................

5.2 NEMA Standard TP 2 (Method II and Method III)

5.2.1 Method II results .........................................

5.2.2 Method Ill results .........................................

5.3 Discussion .................................................

5

5

5

6

7

7

8

8

6 Enforcement testing 8

6.1 10 CFR Part 430 ......................... . .................... 9

6.2 Results .................................................... 10

7 Summary 10

8 Further information 10

Appendix A: The 10 CFR Part 430 and the proposed 10 CFR Part 432 sampling plans for

compliance testing 42

Appendix B: The NEMA Standard TP 2 sampling plan 43

Appendix C: A sampling plan for enforcement testing 47

Appendix D: Computational algorithms 49

iv

EE-TP-98-550 COMMENT 41
List of Figures

1. Operating characteristic of the 10 CFR Part 430 compliance criteria for a loss tolerance of 103 percent and a fixed sample of two. .... ........................ ..... 12

2. Operating characteristic of the 10 CFR Part 430 compliance criteria for a loss tolerance of 103 percent and a fixed sample of five. ..................................... 13

3. Operating characteristic of the 10 CFR Part 430 compliance criteria for a loss tolerance of 103 percent and a fixed sample of ten . ..................................... 14

4. Operating characteristic of the 10 CFR Part 430 compliance criteria for a loss tolerance of 103 percent and a fixed sample of twenty. ................................... 15

5. Operating characteristic of the 10 CFR Part 430 compliance criteria for a loss tolerance of 103 percent and a fixed sample of thirty . .................................... 16

6. Operating characteristic of the 10 CFR Part 430 compliance criteria for a 106 percent loss tolerance

and a fixed sample of two ......................................... 17

7. Operating characteristic of the 10 CFR Part 430 compliance criteria for a 106 percent loss tolerance

and a fixed sample of five.......................................... 18

8. Operating characteristic of the 10 CFR Part 430 compliance criteria for a 112 percent loss tolerance

and a fixed sample of two ......................................... 19

9. Operating characteristic of the 10 CFR Part 430 compliance criteria for a 112 percent loss tolerance

and a fixed sample of five . ........................................ 20

10. The operating characteristic of the 10 CFR Part 430 compliance criteria for a 103 percent loss

tolerance and variable sample size. ................................... 21

11. The testing burden of the 10 CFR Part 430 compliance criteria for a 103 percent loss tolerance

and variable sample size .......................................... 22

12. Operating characteristic of the Method II sampling plan for a fixed sample of five . ........ 23

13. Operating characteristic of the Method II sampling plan for a fixed sample of ten . ........ 24

14. Operating characteristic of the Method II sampling plan for a fixed sample of twenty ........ 25

15. Operating characteristic of the Method II sampling plan for a fixed sample of thirty . ....... 26

16. Operating characteristic of the Method II sampling plan with variable sample size . ........ 27

17. The testing burden under the Method II sampling plan . ....................... 28

18. Operating characteristic of the Method III sampling plan for a fixed sample of five .. ....... 29

19. Operating characteristic of the Method III sampling plan for a fixed sample of ten. ........ 30

20. Operating characteristic of the Method III sampling plan for a fixed sample of twenty . ...... 31

21. Operating characteristic of the Method III sampling plan for a fixed sample of thirty........ 32

22. Operating characteristic of the Method III sampling plan with variable sample size .. ....... 33

23. The testing burden under the Method III sampling plan ................. ....... 34

24. Operating characteristic of the sampling plan for enforcement testing with a loss tolerance of

103 percent ................................................. 35

25. Testing burden of the sampling plan for enforcement testing with a loss tolerance of 103 percent. 36

26. Operating characteristic of the sampling plan for enforcement testing with a loss tolerance' of

106 percent................................................. 37

27. Testing burden of the sampling plan for enforcement testing with a loss tolerance of 106 percent. 38

28. Operating characteristic of the sampling plan for enforcement testing with a loss tolerance of

112 percent ................................................. 39

29. Testing burden of the sampling plan for enforcement testing with a loss tolerance of 112 percent. 40

v

EE-TP-98-550 COMMENT 41
List of Tables

1. NEMA Class 1 efficiency levels for liquid-filled distribution transformers............... 3

2. NEMA Class 1 efficiency levels for dry-type distribution transformers . ............... 3

B1. NEMA Standard Publication TP 2 Table 5 .............................. 44

vi

EE-TP-98-550 COMMENT 41
An Analysis of Efficiency Testing Under

the Energy Policy and Conservation Act: A Case Study

with Application to Distribution Transformers

I Introduction

This report provides analysis of the protocols for efficiency testing promulgated by 10 CFR Part 430 [1] as

established under the Energy Policy and Conservation

Act of 1975 (EPCA) as amended [2], and proposed for

the new 10 CFR Part 432 [3]. Specifically, the performance of these testing protocols in establishing conformance with a rated efficiency or a rated energy use is

addressed. The case of distribution transformers, which

are covered under the proposed new 10 CFR Part 432,

is discussed in detail. This discussion includes analysis

of existing industry standards for transformer efficiencies. These results are presented, in part, to assess the

impact of EPCA rule making on the transformer industry.

In the case of distribution transformers, laboratory

measurements of efficiency are likely to be used for two

purposes: 1) for testing of compliance with a rated efficiency; and 2) for enforcement testing. The objectives

of testing for each of these purposes differ in significant

ways: Compliance testing is a one-time activity undertaken at the initiation of the program or upon the

introduction of a covered product, while enforcement

testing would be undertaken when the performance of

a specific product or products is contested . Enforcement testing is one of a series of requirements during

an enforcement action.

in NEMA Standard TP 2. A Sampling Plan for Enforcement Testing which is adapted form that provided

in 10 CFR Part 430 appears in appendix C. Finally,

the computational algorithms used to model each of

these sampling plans are presented in appendix D.

It should be emphasized that this report is not a statement of policy by the U.S. Department of Energy

(DOE) and that this report should ' be regarded only

as commentary on these testing protocols. DOE contacts for further information on the rule making for

distribution transformers and on EPCA legislation are

provided in Section 8.

2 General guidelines

In this section, we summarize briefly the general objectives of testing under EPCA. This material is included

to provide ground rules for evaluating these various

sampling plans.

A statement of purpose for the EPCA legislation is

given in 42 U.S.C. 6312(a):

It is the purpose of this Part to improve the

efficiency of electric motors and pumps and

certain other industrial equipment in order to

conserve the energy resources of the Nation.

The remainder of this report is organized as follows:

The general objectives and constraints for testing under EPCA are discussed in Section 2. Current industry practice regarding efficiency performance and efficiency testing is reviewed in Section 3. The methods and model assumptions used in the evaluation of

these testing protocols are presented in Section 4. A

discussion of the sampling plans for compliance testing provided in 10 CFR Part 430, the proposed new

10 CFR Part 432, and NEMA Standard TP 2 is presented in Section 5. The Sampling Plan for Enforcement Testing established by 10 CFR Part 430 is presented in Section 6. For the convenience of the reader,

each of these sampling plans are provided in appendices to this report: Appendix A contains an example

sampling plan from 10 CFR Part 430 and the proposed

10 CFR Part 432 sampling plan for compliance testing. Appendix B contains the sampling plan provided

To this end, EPCA establishes energy performance

standards that may specify acceptable levels of efficiency or energy use for each covered product. EPCA

further requires that any represented energy performance be accurate in 42 U.S.C. 6314(d)(1): .

no manufacturer , distributor retailer or private labeler may make any representation -

(A) in writing (including and representation

on a label), or

(B) in any broadcast advertisement,

respecting the energy consumption of such

equipment or cost of energy consumed by such

equipment, unless such equipment has been

tested in accordance with such test procedure

and such representation fairly discloses the results of such testing.

EE-TP-98-550 COMMENT 41
NISTTN 1427 An Analysis of Efficiency Testing Under the Energy Policy and Conservation Act:

EPCA relies on a program of systematic testing to es- for Measuring the Energy Consumption of Distributablish that these performance levels are met. The tion Transformers " [5]. The efficiency requirements

objectives and limitations of testing under EPCA are for NEMA class-1 designation are established in secstated in 42 U.S.C. 6314(a)(2): tion 4 of the NEMA TP 1 standard, which applies to

Test procedures prescribed in ac=dance with

this section shall be reasonably designed to

produce test results which reflect energy efficiency, energy use, and estimated operating

costs of a type of industrial equipment (or

class thereof) during a representative average

use cycle (as determined by the Secretary),

and shall not be unduly burdensome to conduct.

This report assumes that the purpose of EPCA is satisfied if the average efficiency is not less than the efficiency standard established under EPCA and the rated

efficiency. In the case of an energy use standard, this

report assumes that the purpose of EPCA is satisfied

if the average energy use is not greater than the energy

use standard established under EPCA and the rated

energy use. -.

To re-cap, for purposes of this analysis, we assume

that the performance objectives are met provided that

the average energy efficiency (energy use) is not less

(greater) than the EPCA energy performance standard

and the rated value . Compliance with a rated energy

performance is established under EPCA by systematic

testing; and EPCA stipulates that testing should not

be unduly burdensome to conduct: For the purposes

of this analysis, two criteria are considered: 1) the assurance provided by a sampling plan that the average

performance of each covered product meets or exceeds

the rated performance, and 2) the burden placed on

industry by testing under that sampling plan.

3 Industry practice

This section includes a brief discussion of current industry standards for distribution transformers. In a rule

making, it may be useful to clarify current industry

practice and, where current industry practice is consistent with the purpose of EPCA, to harmonize any energy performance standards established under EPCA

with these practices.

The energy efficiency and energy consumption of distribution transformers is covered by two standards sanctioned by the National Electrical Manufacturers Association (NEMA): NEMA Standards Publication TP 1-

1996, "Guide for Determining Energy Efficiency for

Distribution Transformers" [4] and by NEMA Standards Publication TP 2-1996 "Standard Test Method

both liquid-filled and dry-type distribution transformers. The NEMA TP 1 Standard establishes "minimum

efficiencies" for liquid-filled and dry-type single- and

three-phase transformers, which are tabulated tabulated in Table 4-1 and Table 4-2, respectively. Dry

type transformers are further differentiated by their

voltage rating. The NEMA standard TP 1 tables are

reproduced in part here in Tables 1 and 2, respectively.

NEMA standard TP 1 states that these values are the

"minimum efficiencies" for transformers designated as

NEMA class 1. For purposes of this discussion, the

phrase "minimum efficiencies" will refer to the minimum average efficiency of a population. NEMA Standard TP 2 [5] section 7 provides a sampling plan designed to establish compliance with the TP 1 efficiencies. Section 7 of the TP 2 standard is included in

appendix B of this report and is discussed below in

section 5.2.

Two standards sponsored by the Institute of Electrical and Electronics Engineers (IEEE) are also relevant

to this discussion: ANSI Standard 57.12.00-1993 [6],

"General Requirements for Liquid-Immersed Distribution, Power and Regulating Transformers " and ANSI

Standard 57.12.01, "General Requirements for DryType Distribution and Power Transformers, Including Those with Solid Case and/or Resin-Encapsulated

Windings" [7]. These standards cover abroad range of

transformer requirements, including recommended tolerances on measured losses: under these standards, the

total losses of a single unit shall not exceed 106 percent

of the rated value, and the average loss for two or more

transformers shall not exceed the rated value. The second requirement, which is on the average loss, is being

debated within the sponsoring committee and may be

modified or deleted in future versions of these standards. Since the measured losses are directly related

to the efficiency, these standards, in effect, establish a

tolerance for the measured efficiency and thus provide

a level of quality assurance for efficiency.

4 Methods of analysis

Two figures-of-merit provide the basis for the evaluation of these testing protocols: the operating characteristic' and the testing burden. The operating characteristic of a sampling plan is the probability of demonstrating compliance when testing a specific distribution

of efficiencies under that sampling plan. This quantity

provides an estimate of the probability or risk that an

acceptable product could fail by chance or that an un2

EE-TP-98-550 COMMENT 41
A Case Study with Application to Distribution Transformers NISTT'N 1427

Table 1: NEMA Class 1 efficiency levels for liquid-filled distribution transformers. (Adapted from NEMA Standards

Publication TP 1, Table 4-1 (4j.)

kVA

Single Phase

Efficiency kVA

Three Phase

. Efficiency

10 98.3 15 98.0

15 98.5 30 98.3

25 98.7 45 98.5

. 37.5 98.8 75 98.7

50 98.9 112.5 98.8

75 99.0 150 98.9

100 99.0 225 99.0

167 99.1 300 99.0

250 99.2 500 99.1

333 99.2 750 99.2

500 99.3 1000 99.2

667 99.4 1500 99.3

833 99.4 2000 99.4

2500 99.4

Table 2: NEMA Class 1 efficiency levels for dry-type distribution transformers. (Adapted from NEMA Standards

Publication TP 1, Table 4-2 [41.)

Single Phase Three Phase

Efficiency Efficiency

kVA Low Voltage Medium Voltage kVA Low Voltage Medium Voltage

15 97.7 97.6 15 97.0 96.8

25 98.0 97.9 30 97.3 97.3

37.5 98.2 98.1 45 97.7 97.6

50 98.3 98.2 75 98.0 97.9

75 98.5 98.4 112.5 98.2 98.1

100 98.6 98.5 150 98.3 98.2

167 98.7 98.7 225 98.5 98.4

250 98.8 98.8 300 98.6 98.5

333 98.9 98.9 500 98.7 98.7

500 - 99.0 750 98.8 98.8

667 - 99.0 1000 98.9 98.9

833 - 99.1 1500 - 99.0

2000 - 99.0

2500 - 99.1

3

EE-TP-98-550 COMMENT 41
NISTTN 1427 An Analysis of Efficiency Testing Under the Energy Policy and Conservation Act:

acceptable product could pass by chance under that

sampling plan. The second figure-of-merit, the testing

burden, is the average number of units tested under

that sampling plan. The testing burden may be considered in estimating the average cost of testing and in

recommending minimum or maximum sample sizes.

These testing protocols are examined by means of

model calculations. Two approaches are taken in this

analysis: Firstly, where analytic expressions or numerical approximations for the desired figures-of-merit are

derived, these expressions are evaluated numerically.

A discussion of these algorithms is presented in appendix D and in [8, 9, 10]. Secondly, Monte Carlo [11]

simulations are used to estimate these figures-of-merit.

These approaches differ substantially, and each should

be considered an independent estimate of the figureof-merit. Comparison of the results obtained by numerical computation and Monte Carlo simulation may

thus lend support to the calculated value. Monte Carlo

methods have an added advantage for this analysis, in

that they may be adapted to evaluate sampling plans

where the sample size is not fixed, and may thus provide estimates of the operating characteristic and the

testing burden for such plans.

The nature of testing under the sampling plans given in

10 CFR Part 430, the. proposed new 10 CFR Part 432,

and NEMA Standard TP 2 introduces a computational

problem, in that the number of units tested may not

be fixed from the outset of testing: For most products

tested under the 10 CFR Part 430 sampling plan, a

manufacturer could test as few as two units, but may

test any larger number. One exception to this minimum

sample size of two is lamp ballasts for which not fewer

than four units must be tested. Under the proposed

new 10 CFR Part 432 for distribution transformers, a

manufacturer could test as few as five units of each basic model, but may test any larger number of units,

including all units. Under the TP 2 sampling plan a

manufacturer may test all units manufactured during

a period of 180 days, or may conduct monthly tests

over a period of 180 days of no fewer than five units.

The scenario in which the sample size is not fixed is

difficult to characterize by analytical methods, and we

have chosen to first treat these sampling plans in the

approximation that the sample size is fixed from the

outset. Results obtained under this approximation provide a lower bound on the probability of being found

in compliance . For example, while testing under the

10 CFR Part 430 sampling plan, a fixed sample of five

units would include some cases for which a manufacturer was found in compliance after testing two units.

and could have stopped testing at that point, but fails

after testing five units due to the final three test results.

The computed probabilities for a fixed sample size may

thus under-estimate the probability of being found in

compliance.

Detailed information regarding the distribution of efficiencies is required for these calculations; and we assume that the population of efficiency is normally distributed with mean, p, and standard deviation, a. We

further assume that the units tested are selected at random.

4.1 Loss representation

We have chosen, for this analysis, to represent transformer energy performance in terms of energy use, i.e.

losses. This representation is chosen because it provides

a direct comparison with industry standards. Further,

the loss representation has the advantage of being independent of efficiency: The tolerance on the measured

loss specified in the NEMA and IEEE standards is given

as a percentage of the rated loss and is independent of

efficiency. Thus the operating characteristic and testing

burden are independent of efficiency in the loss representation.

In the discussion that follows, the operating characteristic and testing burden of each sampling plan are

represented by contour plots in coordinates of the average loss and the standard deviation of loss, where both

are given as a percentage of the rated loss. In all cases,

the rated loss corresponds to 100 percent on the loss

axis.

Expressed as a percentage, transformer efficiency, E, is

given by the following equation:

E= PPL x100,

where P is the output power and L is the loss power.

In practice, the no-load loss is measured at the rated

voltage and the load loss at the rated current and a

determination of efficiency involves three critical measurements: power, voltage, and current. Of these, the

measurement of loss power is the most difficult resulting

in measurement uncertainties that are about an order

of magnitude greater than those of voltage and current

measurement.

In summary, the measurement uncertainty of the efficiency depends primarily on the uncertainty of the loss

measurement. This discussion is provided by way of

explanation for the specification of a tolerance on the

measured loss in the IEEE and NEMA standards. Since

the critical measurement contributing to the measurement uncertainty in the efficiency is the uncertainty in

the loss measurement, as a practical matter it is most

effective to specify a tolerance for the measurement of

loss.

4

EE-TP-98-550 COMMENT 41
A Case Study with Application to Distribution Transformers

5 Compliance testing

Three sampling plans are discussed: the sampling plan

put forward by 10 CFR Part 430 [ 1], the sampling plan

proposed for the new 10 CFR Part 432 [3], and the

sampling plan established by NEMA Standard TP 2 [5].

5.1 10 CFR. Part 430 and the proposed

10 CFR Part 432 (Method I) '.

In the discussion that follows, we refer to the sampling

plans provided in 10 CFR Part 430 and proposed for

the new 10 CFR Part 432 as Method I type plans.

NISTTN 1427

CFR Part 430, the statistical confidence specified in

(i)(B) and (ii)(B) ranges in value between 90 percent

and 99 percent. While the divisor ranges in value between 1.01 and 1.10 in (i)(13) and between 0.90 and 0.99

in (ii)(B).

Under this sampling plan a manufacturer is required

to test only as needed to demonstrate the specified statistical confidence. The rule language in 10 CFR Part

430 states that the sample should be of "sufficient size"

to ensure a statistical confidence that is not less than

971/2 percent, in the example cited. Assuming that

the statistical confidence is established from test data

alone, the minimum sample size under this plan is two.

The reason for this minimum value is that the sample

standard deviation is not defined for a sample of one.

5.1.1 10 CFR Part 430

10 CFR Part 430 establishes specific criteria for each

covered product when testing for compliance with a

rated energy performance. An example of these criteria, which applies to dishwashers, is provided in appendix A of this report. The general structure of this

sampling plan is followed for all other products covered under 10 CFR Part 430. To emphasize its salient

features, we paraphrase the Part 430 sampling plan as

follows:

5.1.2 The proposed 10 CFR Part 432

(i) Compliance with a rated energy consumption is

demonstrated provided;

(A) The average energy consumption of the

sample is not greater than the rated energy consumption, and

(B) The upper 971/2 percent confidence limit

of the average energy consumption of the

entire population divided by 1.05 is not

greater than the rated energy consumption.

(ii) Compliance with a rated efficiency is demonstrated

provided;

(A) The average efficiency of the sample is not

less than the rated efficiency, and

(B) The lower 971/2 Percent confidence limit

of the average efficiency of the entire population divided by 09 is not'less than the

rated efficiency.

A Method I type plan is also proposed for distribution

transformers in 10 CFR Part 432 [3]. The proposed

sampling plan is provided in appendix A of this report.

We paraphrase this sampling plan as follows:

A sample of not fewer than five units must be

tested. Compliance with a rated energy efficiency

is demonstrated provided:

(a) The mean efficiency of the sample is not

less than the rated efficiency, and

(b) The lower 95 percent confidence limit of

the average efficiency of the entire population divided by [1 - 0.03(1 - E,/100)],

where E, is the rated efficiency, is not less

than the rated efficiency.

Criteria for the demonstration of compliance with a

rated loss are not proposed in 10 CPR Part 432. However, the criteria proposed for compliance testing may

be stated in terms of loss; and such loss criteria are fully

equivalent to the sampling plan for compliance testing

proposed 10 CFR Part 432. These criteria are included

here for illustrative purposes and for consideration of

users who may wish to analyze intermediate test data

in terms of loss power. When stated in terms of loss,

the proposed sampling plan may be paraphrased as follows:

For a given population, the operating characteristic and

testing burden of these criteria depend on three factors:

1) the sample size, 2) the statistical confidence in (i)(B)

and (ii)(B) and 3) the value of the divisor specified in

(i)(B) and (ii)(B). The statistical confidence and the

divisors are underlined in the above text. Under 10

A sample'of not fewer than five units must be

tested. Compliance with a rated loss is demonstrated provided:

5

(a) The average loss power of the sample is

not greater than the rated loss power, and

EE-TP-98-550 COMMENT 41
NISTTN 1427 An Analysis of Efficiency Testing Under the Energy Policy and Conservation Act:

(b) The upper 95 percent confidence limit of

the loss power of the entire population divided by 1.03 is not greater than the rated

loss power.

It should be emphasized that the divisor in criterion

(b) above establishes a tolerance on the estimate of

the mean loss of the population. This tolerance on

the estimate of the mean loss should not be confused

with the considerably larger loss tolerance established

by ANSI Standards 57.12.00-1993 and 57.12.01 and

NEMA Standard TP2: The loss tolerance established

by these standards applies to the measured loss of a

single unit.

It may be noted that the divisors in criterion (b) differ

when the criteria for compliance testing are stated in

terms of efficiency or loss. This feature differs from the

1OCFR Part 430 sampling plans. In the example provided from 10 CFR Part 430, the divisors applied in the

cases of efficiency testing and energy consumption testing are 1+0.05 and 1-0.05, respectively. In effect, these

divisors establish a tolerance of 5 percent for both the

estimate of the mean efficiency and the estimate of the

mean energy consumption. The use of an equivalent

tolerance on the estimate of the mean efficiency and

mean loss may not apply for devices that are highly efficient such as distribution transformers. The relationship between the uncertainty in the measurement of

efficiency and of loss is given by m = -n(1- E,/100),

where m is the measurement uncertainty in the efficiency, n is the measurement uncertainty in the loss,

and E. is the rated value of efficiency for the product.

Clearly, in the limit that E. approaches 100 percent

the measurement uncertainty in the efficiency is much

less than in the loss.

5.1.3 Results

The results of model calculations for the Method I sampling plan are shown in Figs. 1-9, where data are presented for a statistical confidence of 95 percent and for

values of loss tolerance of 103 percent, 106 percent, and

112 percent. Sampling plans using both fixed samples

and variable sizes are treated. The results shown in

Figs. 2 to 5 are of particular interest to the transformer

case.

For samples of fixed size, a rated efficiency is demonstrated provided:

1. the sample average loss is less than 100 percent,

and

2. the upper 95 percent confidence limit of the mean

loss is not greater than the loss tolerance.

The condition on the loss tolerance LT is equivalent to

the use of a divisor in (i)(B). A sampling plan based on

a loss tolerance of 103 percent, for example, exhibits

the same behavior as the use of a 1.03 divisor.

Samples of fixed size are first examined . The operating characteristic of the Method I sampling plan for a

loss tolerance of 103 percent for samples of 2, 5, 10, 20,

and 30 units are shown in Figs. 1-5, respectively. The

dependence of the operating characteristic on the loss

tolerance is shown in Figs. 6-9, which are for the samples of two and five and for loss tolerances of 106 percent and 112 percent. Monte Carlo simulations were

also performed for each of these cases and results obtained by these methods are shown in half-tone in each

of these figures. Aside from the random fluctuations,

which are inherent to Monte Carlo methods, the two

methods appear to give equivalent results.

Examination of these data suggests the following interpretation of the Method I sampling plan: For samples

of fixed size, the Method I sampling plan provides assurance at the specified statistical confidence that the

average loss does not exceed the loss tolerance . In this

case, the statistical confidence is 95 percent and thus

the risk of exceeding the loss tolerance is not greater

than 5. percent. This behavior is indicated by the

asymptotic limit of the 0.05 contours, which in each

figure approach the loss tolerance at large standard deviation.

Two additional analyses, which may be modeled by

Monte Carlo simulations, are suggested by these results: 1) the operating characteristic and 2) the testing burden under this sampling plan when the sample

size is not fixed. In principle, the Monte Carlo simulation should provide a more realistic estimate of the

Method I operating characteristic and testing burden.

The conditions for a determination of compliance for

variable sample size are identical to those used for a

fixed sample; however, in this case, testing is continued up to a maximum sample of twenty or until the

following condition is satisfied:

>(LT

t5 1a

- 100)

where n is the sample size, LT is the loss tolerance, S is

the sample standard error , and t is the fifth percentile

of the t-distribution for n - 1 degrees of freedom.

The results of these simulations are presented in

Figs. 10 and 11, which respectively depict the operating characteristic and the testing burden at a loss

tolerance of 103 percent. These operating characteristics suggest that the assurance on loss performance

may be somewhat less than for fixed samples, as the

asymptotic limit of the 0 .05 contour appears to be on

the order of 106 percent. These data further suggest

6

EE-TP-98-550 COMMENT 41
A Case Study with Application to Distribution Transformers

that the marginal loss performance required to provide

a high probability of demonstrating compliance may be

appreciable under this sampling plan, where marginal

loss performance refers to the difference between the

rated loss and the true average loss. It should be emphasized, however, that the operating characteristics

shown in the figure are those for minimal testing' and

that, under the Method I sampling plan, a manufacturer may be able to reduce the marginal loss by increasing the sample size.

5.2 1EMA Standard TIS 2 (Method II

and Method III)

The NEMA TP 2 sampling plan for demonstrating the

efficiencies established by NEMA Standard TP 1 is next

presented. The NEMA TP 2 sampling plan is given in

sections 7.0 and 7.1 of the standard, which can be found

in appendix B of this report.

NEMA Standard TP 2 provides that the aggregate

measured input power of all transformers tested for a

period of 180 days be no greater than that allowed by

the rated efficiency. The proposed 10 CFR Part 432

contemplates measurement of the average efficiency of a

basic model, where a basic model may be distinguished

by the significant characteristics of that product such

as the kVA rating, voltage rating, and insulation type,

for example. The question of whether it is appropriate to aggregate a broad range of product types for

the purposes of determining an average energy performance, as is allowed under NEMA Standard TP 2, is

not addressed here. For purposes of this discussion, we

assume that the TP 2 sampling plan is applied to measure the average efficiency or the average energy use of

a basic model. When applied to a basic model, this

condition on the measured input power is equivalent

to an average measured efficiency that is not less than

the rated efficiency and an average measured loss that

is not greater than the rated loss. The TP 2 sampling

plan thus establishes a condition on the mean efficiency

or mean loss of a sample.

The TP 2 sampling plan provides further that no unit

may exceed 108 percent of the rated loss. This tolerance on the measured loss may be interpreted in two

ways: 1) The tolerance on the measured loss may establish a quality control limit for the NEMA Class I rating,

but this tolerance has no bearing on any demonstration

of compliance with a rated efficiency. This interpretation of the TP 2 sampling plan places a condition on

the mean of the sample. This scenario will be referred

to here as a Method II type sampling plan . 2) Another

interpretation of the TP 2 sampling plan places a condition on the mean of the sample and on the sample extremum, where the sample extremum is the maximum

NISTTN 1427

measured loss of any unit in the sample. This scenario

will be referred to here as a Method III type sampling

plan. Method III type criteria are used for quality assurance of various packaged goods [12, 13] and have

been proposed for use in testing whether electric motors comply with a rated efficiency [9]. An approximate

solution for the probability that such conditions on the

sample mean and on the sample extremum are satisfied

is discussed by Vangel [8].

Two scenarios for sampling are presented in TP 2:

all units manufactured during a 180 day period may

be tested, or units may be tested on a monthly. basis

for a period of 180 days. The standard provides that

no fewer than five units may be tested during these

monthly tests and that the sample size shall be sufficient to ensure a statistical confidence of not less that

95 percent. The statistical confidence is established by

the condition

n = (tSK)2,

where n is the sample size , S is the sample standard

deviation, the value of t is selected according to the

number of units tested as tabulated in TP 2, and the

coefficient K is given by the equation

_ 108 - 0.08SEL

K SEL(8 - 0.08SEL)'

where SEL is the standard efficiency. level in percent.

This condition on the sample size is equivalent to that

used under the Method I sampling plan and is that

recommended by ASTM Standard E 122-89114).

5.2.1 Method II results

The operating characteristics and testing burden of

Method II are presented in Figs. 12-17.

Under Method II, one condition must be satisfied to

demonstrate compliance with a rated performance:

1. the mean loss of the sample must be no greater

than 100 percent.

The operating characteristics for samples of fixed size

are shown in Figs. 12-15, which depict samples of 5,

10, 20, and 30 units. Monte Carlo simulations were

again performed for each of these cases and the results

of these simulations are shown in each of these figures

in half-tone. Numerical computation and Monte Carlo

simulation appear to yield consistent values.

The case of variable sample size is examined in Figs. 16

and 17, which depict the operating characteristics and

the testing burden of Method II, respectively. These

data apply to the case of monthly testing under TP 2

7

EE-TP-98-550 COMMENT 41
NISTTN 1427 An Analysis of Efficiency Testing Under the Energy Policy and Conservation Act:

where the the minimum units tested is thirty. The conditions for a demonstration of compliance for a variable

sample are identical to those for fixed sample size; however, in this case testing is continued until the condition

n>^ t5 2 LT -100)

is satisfied.

5.2.2 Method III results

The operating characteristics and testing burden of

Method III are presented in Figs. 18-23.

Under Method III, two conditions must be satisfied to

demonstrate compliance with a rated performance:

1. the mean loss of the sample must be no greater

than 100 percent, and

2. no single unit in the sample may exceed 108 percent of the rated loss.

It should emphasized that under Method III the entire population is not in compliance if any unit in the

sample exceeds 108 percent of the rated loss. The operating characteristics for samples of fixed size are shown

in Figs. 18-21, which depict samples of 5, 10, 20, and 30

units. Monte Carlo simulations were again performed

for each of these cases and the results of these simulations are shown in each of these figures in half-tone.

Numerical computation and Monte Carlo simulation

appear to yield consistent values.

The case of variable sample size is examined in Figs. 22

and 23, which depict the operating characteristics and

the testing burden of Method III, respectively. These

data apply to the case of monthly testing under TP 2

where the the minimum units tested is thirty. The conditions for a demonstration of compliance for a variable

sample are identical to those for fixed sample size; however, in this case testing is continued until the condition

> ( i00) 2

is satisfied or the sample exceeds twenty, whichever occurs first. It should be noted that a maximum sample

size is not stipulated in the TP 2 sampling plan: the

maximum sample size of twenty used here was included

to simplify the Monte Carlo simulation.

at small standard deviation. This is due to the condition placed on the sample mean loss under these sampling plans: Each of these sampling plans require that

the sample mean loss be no greater than the rated loss;

thus at small standard deviation, where the determination of-compliance is based primarily on the sample

mean loss, each of these sampling plans exhibit similar

operating characteristics.

It should be noted that the operating characteristic

of these sampling plans depends on the sample size.

However, these sampling plans exhibit very different

behavior with increased sample size. For example, the

marginal loss required under Method I to provide a high

probability of demonstrating compliance decreases with

sample size. Thus-the marginal loss may be reduced to

an acceptable level under the Method I sampling plan

by additional testing. The behavior of the Method III

sampling plan is much different. Under Method III, the

risk of failing to demonstrate compliance may actually

increase with increased sample size. The reason for this

behavior is that as the sample size increases the likelihood of sampling from the wing of the distribution of

loss increases.

The design performance of a transformer may depend

on both engineering and business factors. However, it

appears from this analysis that each of these sampling

plans favor loss performance below the rated value.

Since, if a transformer were designed to perform, and

indeed performed at the rated loss on average, the likelihood of demonstrating compliance with that rated performance is not greater than 50 percent under each of

these plans.

The Method I, Method U, and Method III sampling

plans all provide assurance that the mean performance

of a population meets or exceeds a rated value. The

Method I sampling plan appears to be perform well

in cases where large quantities are available for testing

and the cost of testing is low. However, one unique

character of distribution ' transformers is that the lot

size can be small. Indeed some transformers are manufactured in extremely limited numbers. In that the

marginal loss can be appreciable for small samples, the

Method I sampling plan may, in -effect, require that

certain transformers have average losses that are appreciably less than their rated loss. Alternately the

Method III sampling plan may place a high burden on

manufacturers when it is applied to large samples.

6 Enforcement testing

5.3 Discussion

The Sampling Plan for Enforcement Testing provided

For a fixed sample size, the operating characteristics of in 10 CFR Part 430 applies to all products covered unMethod I, Method II, and Method III are very similar der Part 430. A similar plan has been proposed in the

8

EE-TP-98-550 COMMENT 41
A Case Study With Application to Distribution Transformers NISTTN 1427

new 10 CFR Part 431 for electric motors. The proposed sample, i.e. the number of transformers tested, and 2)

10 CFR Part 432 [31 does not include a Sampling Plan the underlying variability in the entire population. The

for Enforcement Testing. Its development will be con- sample standard deviation,

sidered at a later date. While some details may change,

the general format of the plan is expected to remain

same if and when it is finalize for transformers.

The Sampling Plan for Enforcement Testing provided

in 10 CFR Part 430 establishes clearly delineated procedures to be followed under an enforcement action.

This sampling plan is based on a well established statistical method, which is due to C. Stein j15]; for obtaining a confidence interval on a mean. A discussion

of this procedure can also be found in Bickel and Doksum [16). The Sampling Plan for Enforcement Testing

provided in 10 CFR Part 430 covers both testing of

efficiency and for energy consumption and. it is very

general. The sampling plan presented in appendix C

has been adapted from 10 CFR Part 430; however, it

has been simplified somewhat to clarify the procedure

and it includes only the case for efficiency testing.

The sampling plan is based on a t-test. The t-test is

well suited to this application as it is known to be insensitive to departures from the assumption of normality:

The t-test is a test on a mean, which is an average

of independent values obtained by a random sample.

Since, sums of arbitrary, independent random values

tend to have a distribution that.is almost normal, the

t-test is not strongly influenced by the exact form of

the underlying distribution of efficiencies.

Since the test results obtained during enforcement testing may recommend that certain adverse actions be

taken against a manufacturer : relabeling of specific

products, the cessation of distribution and sale of certain basic models, and/or the assessment of fines, for

example; the risk to a manufacturer of a false determination of noncompliance during an enforcement action is set, by design, to a negligible level. The Sampling Plan for Enforcement Testing provided in 10 CFR

Part 430 is based on a 97.5 percent statistical confidence, thus the risk to a manufacturer of a chance false

determination of noncompliance is not greater than

2.5 percent.

6.1 10 CFR Part 430

An estimate of the true mean efficiency is first obtained

by a random sample,

1 "

JC=-EX;,

n

i=1

(1)

S= ^.(X{-X)2

n-1 ' (2)

provides an estimate of the population standard deviation a; and the standard error in the mean,

SE(X) _ =, (3)

provides an estimate of the standard deviation of the

mean for samples of n units. For a normal distribution

with mean p; the ratio,

f_ p-X

SE( ' (4)

is distributed according to a probability density function that is know in statistics literature as the tdistribution. Now eq. 4 may be rearranged to provide

an expression for the mean of the sample:

(5)

where the value oft is associated with a specific sample

size and statistical confidence. Values of t are readily

available and are included in many references on statistics [17].

In a test of compliance with a rated efficiency, RE,

we assume, by hypothesis, that the units to be tested

are drawn from a population of transformers for which

the mean efficiency is equal to or greater than the rated

efficiency. If t is the 97.5 percentile of the t-distribution

for n - 1 degrees of freedom, then the probability of

obtaining a mean efficiency,

X < RE + tSE(X ), (6)

is not less than 97.5 percent. This procedure recommends the upper control limit,

UCL - RE + tSE(X). (7)

To apply this method, a random sample is tested and

the mean and standard error in the mean are calculated. Based on the size of the sample and the

confidence desired, the appropriate t-value is selected

and the lower control limit calculated. For example,

97.5 percent confidence and a sample size of five units

yields a t-value of 3.18. Provided the mean efficiency

obtained from the random sample is not less than the

lower control limit, as defined by eq. 7, we may assert

:here X, is the measured efficiency of unit i, and n is with a confidence not less than 97.5 percent that the

the number of units tested. The uncertainty in this true mean energy consumption of the entire population

estimate depends on two factors: 1) the size of the is not greater than the rated efficiency.

9

EE-TP-98-550 COMMENT 41
NISTTN 1427 An Analysis of Efficlency Testing Under the Energy Policy and Conservation Act:

In any statistical test there is some probability of incorrectly concluding noncompliance. By design, the

probability that the mean efficiency for a random sample drawn from this population of transformers would

fail below the lower control limit, hence, the risk of

incorrectly concluding the basic model is in noncompliance, is not greater than 2.5 percent.

There is some probability that the sample standard deviation may be large and that the lower control limit

may be set, by chance, to an exceptionally low value.

This circumstance may be avoided by placing a tolerance on the standard error in the mean, SE(S). The

tolerance for the standard error should be chosen to

be appropriate for that product and to be supported

across the industry.

Choosing, for example, a loss tolerance of 108 percent,

P.41 - x 100 P;,, + 0.08(P{„ - P"t)

14 x 100, 108 - 0.0814 (8)

loss of the sample under each of these plans. Since under each of these plans the mean loss of the sample may

not exceed the rated loss, each of these plans provide

assurance that the mean efficiency of a population is

not less than the rated efficiency. When testing small

samples under Method I, the marginal loss performance

may be appreciable and may, in effect, require that certain transformers have average losses significantly less

than their rated values. However, under Method I, a

manufacturer may reduce the marginal loss by testing

additional units. Alternately the Method III sampling

plan appears to require an appreciable marginal loss

performance when it is applied to large samples and

may, in effect, require that certain transformers have

average losses significantly less than their rated values.

Enforcement testing: The Sampling Plan for Enforcement Testing included in 10 CFR Part 430 is based

on statistical methods that are widely used and well

documented. The sampling plan is robust, in that it

is a test on the mean and that it is not highly dependent on the form of the underlying distribution. The

sampling plan is designed to protect the interests of

the manufacturer, in that the risk of a false outcome

against a manufacturer may be limited to some acceptable level. Additionally, the risk of a false outcome is

independent of variance.

The analysis presented here is of value primarily as

a qualitative evaluation of the operating characteristic and the testing burden of these testing protocols

and secondarily as a quantitative estimate of the statistical confidence associated with testing under these

protocols. While the agreement noted between results

obtained by analytical methods and Monte Carlo simulations is encouraging, it should be noted, since these

results may depend on our model assumptions, that a

quantitative estimation of statistical confidence is more

tenuous than qualitative evaluation.

where P;,, and P..d are the input and output power.

The lower control limit must then satisfy two conditions:

UCL = RE - tSE(.X) (9)

and

UCL > 108 -

o SRE x 100, (10)

where,the second condition is obtained from eq. (8) by

setting the efficiency equal to the RE.

6.2 Results

The operating characteristic and the testing burden of

the Sampling Plan for Enforcement Testing are shown

in Figs. 24-29. Factors that influence the operating

characteristic and testing burden include : 1) the size

of the initial sample, 2) the statistical confidence, and

3) the tolerance set on the standard error.

It may be noted that the 97.5 percent contour lies along

100 percent loss in each of these figures and that the risk

of a false outcome is therefore independent of variability

under this sampling plan.

7 Summary

Compliance * test : The operating characteristic

of the Method I, Method II, and Method III sampling

plans are equivalent in the limit of small standard deviation. This is due to the condition placed on the mean

8 Further information

For information on the rule making for distribution

transformers and on EPCA legislation contact:

Kathi Epping

U.S. Department of Energy

Energy Efficiency and Renewable Energy

Mail Station EE-43

1000 Independence Avenue, SW

Washington, DC 20585-0121

202-586-7425

kathi.epping®ee.doe.gov

10

EE-TP-98-550 COMMENT 41
A Case Study with Application to Distribution Transformers

Edward Levy, Esq.

U.S. Department of Energy

Office of General Counsel

Mail Station GC-72

1000 Independence Avenue, SW

Washington, DC 20585-0103

202-586-9507

edward.levy©hq.doe.gov

For information on this report contact:

Ken Stricklett

National Institute of Standards and Technology

100 Bureau Dr. Stop 8113

Gaithersburg, MD 20899-8113

301-975-3955

ken.stricklettC9nist.gov

Oskars Petersons

National Institute of Standards and Technology

100 Bureau Dr. Stop 8113

Gaithersburg, MD 20899-8113

301-975-2417

oskars.petersons©nist.gov

Mark Vangel

National Institute of Standards and Technology

100 Bureau Dr. Stop 8980

Gaithersburg, MD 20899-8980

301-975-3208

mark.vangel®nist.gov

Acknowledgments

This report was completed with support from the U.S.

Department of Energy and the Technology Administration of the U.S. Department of the Commerce.

NISTfN 1427

11

EE-TP-98-550 COMMENT 41
Increasing Efficiency

12

0

115 110 105 100 95

Loss (percent)

90 85

Figure 1: Operating characteristic of the Method I compliance criteria for a fixed sample size of two . The statistical confidence is 95 percent and the loss tolerance

is 103 percent. The contours indicate the probability of demonstrating compliance , e.g., the 0.90 contour corresponds to a 90 percent likelihood of demonstrating

compliance. The results of Monte Carlo simulations are shown in half-tone.

EE-TP-98-550 COMMENT 41
Increasing Efficiency

12

0

3

115

0

110 105 100 95

Loss (percent)

90 85

Figure 2: Operating characteristic of the Method I compliance criteria for a fixed sample size of five . The statistical confidence is 95 percent and the loss tolerance

is 103 percent. The contours indicate the probability of demonstrating compliance , e.g., the 0.90 contour corresponds to a 90 percent likelihood of demonstrating

compliance. The results of Monte Carlo simulations are shown in half-tone.

EE-TP-98-550 COMMENT 41
Increasing Efficiency

110 105 100 95

Loss (percent)

90 85

Figure 3: Operating characteristic of the Method I compliance criteria for a fixed sample size of ten . The statistical confidence is 95 percent and the loss tolerance

is 103 percent. The contours indicate the probability of demonstrating compliance , e.g., the 0.90 contour corresponds to a 90 percent likelihood of demonstrating

compliance. The results of Monte Carlo simulations are shown in half-tone.

EE-TP-98-550 COMMENT 41
Increasing Efficiency

110 105 100 95

Loss (percent)

90 85

Figure 4: Operating characteristic of the Method I compliance criteria for a fixed sample size of twenty . The statistical confidence is 95 percent and the loss tolerance

is 103 percent. The contours indicate the probability of demonstrating compliance , e.g., the 0.90 contour corresponds to a 90 percent likelihood of demonstrating

compliance . The results of Monte Carlo simulations are shown in half-tone.

EE-TP-98-550 COMMENT 41
Increasing Efficiency

105 100 95

Loss (percent)

Figure 5: Operating characteristic of the Method I compliance criteria for a fixed sample size of thirty . The statistical confidence is 95 percent and the loss tolerance

is 103 percent. The contours indicate the probability of demonstrating compliance , e.g., the 0.90 contour corresponds to a 90 percent likelihood of demonstrating

compliance. The results of Monte Carlo simulations are shown in half-tone.

EE-TP-98-550 COMMENT 41
Increasing Efficiency

110 105 100 95

Loss (percent)

90 85

Figure 6: Operating characteristic of the Method I compliance criteria for a fixed sample two. The statistical confidence is 95 percent and the loss tolerance is

106 percent. The contours indicate the probability of demonstrating compliance , e.g., the 0.90 contour corresponds to a 90 percent likelihood of demonstrating

compliance.

EE-TP-98-550 COMMENT 41
. Increasing Efficiency

12

0

3

115

0

n= E11

110

O®

105 100 95

Loss (percent)

90 85

Figure 7: Operating characteristic of the Method I compliance criteria for a fixed sample five. The statistical confidence is 95 percent and the loss toleranceis

106 percent. The contours indicate the probability of demonstrating compliance , e.g., the 0.90 contour corresponds to a 90 percent likelihood of demonstrating

compliance.

EE-TP-98-550 COMMENT 41
Increasing Efficiency

12

U)

0

115 110 105 100 95

Loss (percent)

90 85

Figure 8: Operating characteristic of the Method. I compliance criteria for fixed sample of two. The statistical confidence is 95 percent and the loss tolerance

is 112 percent. The contours indicate the probability of demonstrating compliance , e.g., the 0.90 contour corresponds to a 90 percent likelihood of demonstrating

compliance.

EE-TP-98-550 COMMENT 41
Increasing Efficiency

12

C

a) is2

0

115

V0

n®

110

0

105 ' 100 95

Loss (percent)

90

Figure 9: Operating characteristic of the Method I compliance criteria for fixed sample of five. The statistical confidence is 95 percent and the loss tolerance

is 112 percent. The contours indicate the probability of demonstrating compliance, e.g., the 0.90 contour corresponds to a 90 percent likelihood of demonstrating

compliance.

EE-TP-98-550 COMMENT 41
lncreasin^ Efficiency

12

C

v

L

0

® ° N ° A a

!o.

115 110 105 100 95

Loss (percent)

90

Figure 10: The operating characteristic of Method I. The results of Monte Carlo simulations for an initial sample size of two and a maximum sample of twenty are

shown. The statistical confidence is 95 percent and the loss tolerance is 103 percent . The operating characteristic of the Method I compliance criteria . The contours

indicate the probability of demonstrating compliance , e.g., the 0.90 contour corresponds to a 90 percent likelihood of demonstrating compliance.

EE-TP-98-550 COMMENT 41
Increasing Efficiency

12

3

115

0

110 105 100 95

Loss (percent)

90

Cb

ro

2

85

Figure 11; The testing burden of Method I. The results of Monte Carlo simulations for an initial sample size of two and a maximum sample of twenty are shown.

The statistical confidence is 95 percent and the loss tolerance is 103 percent . The contours indicate the average sample size at the end of testing.

b

EE-TP-98-550 COMMENT 41
Increasing Efficiency

12

0

3

115

0

"0

0

110 105 100 95

Loss (percent)

90 85

Figure 12: Operating characteristic of the Method II sampling plan for fixed sample of five . The contours indicate the probability of demonstrating compliance, e.g.,

the 0.90 contour corresponds to a 90 percent likelihood of demonstrating compliance . The results of Monte Carlo simulations are shown in half-tone.

EE-TP-98-550 COMMENT 41
Increasing Efficiency

12

0

115

0

110

is

105 100 95

Loss (percent)

90 85

Figure 13: Operating characteristic of the Method II sampling plan for a fixed sample of ten. The contours indicate the probability of demonstrating compliance,

e.g., the 0.90 contour corresponds to a 90 percent likelihood of demonstrating compliance . The results of Monte Carlo simulations are shown in half-tone.

EE-TP-98-550 COMMENT 41
ncreasing Efficiency

12

3

0

115

n= 0

110 105 100 95

Loss (percent)

90 85

Figure 14: Operating characteristic of the Method II sampling plan for a fixed sample of twenty. The calculations are based on an extremism of 108 percent of

the rated loss. The contours indicate the probability of demonstrating compliance, e.g., the 0.90 contour corresponds to a 90 percent likelihood of demonstrating

compliance. The results of Monte Carlo simulations are shown in half-tone.

EE-TP-98-550 COMMENT 41
Increasing Efficiency

12

3

0

115 110 105 100 95

Loss (percent)

90 85

Figure 15: Operating characteristic of the Method II sampling plan for a fixed sample of thirty . The contours indicate the probability of demonstrating compliance,

e.g., the 0.90 contour corresponds to a 90 percent likelihood of demonstrating compliance. The results of Monte Carlo simulations are shown in half-tone.

EE-TP-98-550 COMMENT 41
Increasing Efficiency

12

110 105 100 95 90 85

Loss (percent)

Figure 16: Operating characteristic of Method II sampling plan with variable sample size. Monte Carlo simulation for an initial sample size of five we shown. As in

the previous figures, the contours indicate the probability of demonstrating compliance.

EE-TP-98-550 COMMENT 41
I ncreasing Efficiency

12

40

35

0

3

0

115 110 105 100 95

Loss (percent)

90 85

Figure 17: The testing burden of the Method II sampling plan. The results of Monte Carlo simulations are shown. Under the TP 2 sampling plan, compliance with

the rated efficiency may be established by monthly tests conducted over a period of 180 days . Each monthly test would consist of no fewer , than 5 units. The contours

indicate the average sample size at the end of testing, e.g., the 30 contour corresponds to thirty units tested on average.

EE-TP-98-550 COMMENT 41
Increasing Efficiency

12

U)

0

0

115 110 105 100 95

Loss (percent)

90 85

Figure 18: Operating characteristic of the Method III sampling plan for fixed sample of five . The calculations are based on an extremum of 108 percent of the rated

loss. The contours indicate the probability of demonstrating compliance , e.g., the 0.90 contour corresponds to a 90 percent likelihood of demonstrating compliance.

The results of Monte Carlo simulations are shown in half-tone.

EE-TP-98-550 COMMENT 41
Increasing Efficiency

8

12

0

115 110 -105 100 95

Loss (percent)

90 85

Figure 19: Operating characteristic of the Method III sampling plan for a fixed sample of ten . The calculations are based on an extremum of 108 percent of the rated

loss. The contours indicate the probability of demonstrating compliance, e.g., the 0.90 contour corresponds to a 90 percent likelihood of demonstr ating compliance.

The results of Monte Carlo simulations are shown in half-tone.

EE-TP-98-550 COMMENT 41
Increasing Efficiency

co

12

0

3

0

If 20

115 110 105 100 95

Loss (percent)

90

0

85

Figure 20: Operating characteristic of the Method III sampling plan for a fixed sample of twenty. The calculations are based on an extremum of 108 percent of

the rated loss. The contours indicate the probability of demonstrating compliance, e.g., the 0 .90 contour corresponds to a 90 percent likelihood of demonstrating

compliance. The results of Monte Carlo simulations are shown in half-tone.

EE-TP-98-550 COMMENT 41
Increasing Efficiency

115 110 105 100 95

Loss (percent)

12

0

3

0

n=

90 85

Figure 21: Operating characteristic of the Method 1II sampling plan for a fixed sample of thirty. The calculations are based on an extremum of 108 percent of

the rated loss. The contours indicate the probability of demonstrating compliance , e.g., the 0.90 contour corresponds to a 90 percent likelihood of demonstrating

compliance. The results of Monte Carlo simulations are shown in half-tone.

EE-TP-98-550 COMMENT 41
Increasing Efficiency

12

.3

115

0

0 0/

110 105 100 95

Loss (percent)

o

oyJ

90 85

Figure 22: Operating characteristic of Method III sampling plan with variable sample size. Monte Carlo simulation for an initial sample size of five are shown. Based

on an extremum of 108 percent of the rated loss. As in the previous figures, the contours indicate the probability of demonstrating compliance.

EE-TP-98-550 COMMENT 41
Increasing Efficiency

12

45

9

35

30

3

0

115 110 105 100 95

Loss (percent)

90 85

Figure 23 : The testing burden of the Method III sampling plan. The results of Monte Carlo simulations are shown . The statistical confidence is 95 percent and the

loss tolerance is 108 percent . Under the TP 2 sampling plan, compliance with the rated efficiency may be established by monthly tests conducted over a period of

180 days. Each monthly test would consist of no fewer than 5 units . The contours indicate the average sample size at the end of testing, e.g., the 30 contour corresponds

to thirty units tested on average.

EE-TP-98-550 COMMENT 41
Increasing Efficiency

115 110 105 100 95 .90 85

Loss (percent)

Figure 24: Operating characteristic of the Sampling Plan for Enforcement Testing. Model calculations for an initial sample of four and maximum sample of twenty

are shown. The statistical confidence is 97.5 percent and the loss tolerance is 103 percent. The contours indicate the probability of demonstrating compliance, e.g.,

the 0.90 contour corresponds to a 90 percent likelihood of demonstrating compliance.

EE-TP-98-550 COMMENT 41
Increasing Efficiency

12

3

115

0

110 105 100 95

Loss (percent)

16

14

12

90

14

fl

85

Figure 25: Testing burden of the Sampling Plan for Enforcement Testing . Model calculations for an initial sample of four and maximum sample of twenty are shown.

The statistical confidence is 97.5 percent and the loss tolerance is 103 percent. Contours indicate the average sample size at the completion of testing.

EE-TP-98-550 COMMENT 41
Increasi ng Efficiency

W

0

115 110 105 100 95

Loss (percent)

90 85

Figure 26: Operating characteristic of the Sampling Plan for Enforcement Testing . Model calculations for an initial sample of four and maximum sample of twenty

are shown. The statistical confidence is 97.5 percent and the loss tolerance is 106 percent. The contours indicate the probability of demonstrating compliance, e.g.,

the 0.90 contour corresponds to a 90 percent likelihood of demonstrating compliance.

EE-TP-98-550 COMMENT 41
Increasing Efficiency

0

115 110 105 100 95

Loss (percent)

90 85

Figure 27: Testing burden of the Sampling Plan for Enforcement Testing. Model calculations for an initial sample of four and maximum sample of twenty are shown.

The statistical confidence is 97 .5 percent and the loss tolerance is 106 percent . Contours indicate the average sample size at the completion of testing.

EE-TP-98-550 COMMENT 41
Increasing Efficiency

% 115 - 110 105 100 95

Loss (percent)

90 85

Figure 28: Operating characteristic of the Sampling Plan for Enforcement Testing. Model calculations for an initial sample of four and maximum sample of twenty

are shown. The statistical confidence is 97.5 percent and the loss tolerance is 112 percent. The contours indicate the probability of demonstrating compliance, e.g.,

the 0.90 contour corresponds to a 90 percent likelihood of demonstrating compliance.

EE-TP-98-550 COMMENT 41
Increasing Efficiency

12

® C 0

0

0

115 110 105 100 95

Loss (percent)

90

4

85

Figure 29: Testing burden of the Sampling Plan for Enforcement Testing . Model calculations for an initial sample of four and maximum sample of twenty are shown.

The statistical confidence is 97 .5 percent and the loss tolerance is 112 percent . Contours indicate the average sample size at the completion of testing.

EE-TP-98-550 COMMENT 41
A Case Study with Application to Distribution Transformers

References

[1] Energy conservation program for consumer products. Code of Federal Regulations, Title II, Chapter 10, Part 430, 1-1-97 edition, pages 93-263,

1997. Office of the Federal Register, National

Archives and Records Administration, Washington, DC.

[2] Energy policy and conservation act of 1975, pubic law 94-163; national energy conservation policy

act of 1978, public law 95-619; national appliance

energy conservation act of 1987, public law 100-

12; national appliance energy conservation amendments of 1988, public law 100-357; and section

122(d), energy policy act of 1992, public law 102-

486, october 24, 1992.

[3] 10 cfr part 432 energy conservation program:

[11]

NISTTN 1427

chapter II of title 10, Code of Federal Regulations:

Enforcement testing. NIST Technical Note 1422,

National Institute of Standards and Technology,

Technology Administration, U.S. Department of

Commerce, December 1996.

P.R. Bevington and D .K. Robinson. Data Reduction and Error Analysis for the Physical Sciences,

chapter 5. McGraw-Hill, 1992.

[12] C.E. Brickenkamp, S. Hasko, and M.G. Natrella.

[13]

Test procedures for distribution transformers; pro- [14]

posed rule. Federal Register, 63(218):63360-63372,

Thursday, November 12 1998.

[4] NEMA Standards Publication TP 1-1996, Guide [15]

for determining energy efficiency for distribution

transformers. National Electrical Manufacturers

Association, Rosslyn, VA, 1996.

[5] NEMA Standards Publication TP 2-1998, Standard test method for measuring the energy comsumption of distribution transformers. National

Electrical Manufacturers Association, Rosslyn,

VA, 1998.

[6] ANSI C57.12.00-1993 IEEE Standard, General

requirements for liquid-immersed distribution,

power and regulating transformers. American National Standards Institute, New York, 1993.

[7] ANSI C57.12.01-1989 IEEE Standard, General requirements for dry-type distribution and power

transformers, including those with solid case

and/or resin-encapsulated windings. American

National Standards Institute, New York, 1989.

[8] M.G. Vangel.. The joint distribution of a normal

[9]

mean and extremum with applications to quality

control. Manuscript in prepar ation.

K.L. Stricklett and M.G. Vangel. Analysis of proposals for compliance and enforcement testing under the new part .431; title 10, Code of Federal

Regulations. NISTIR 6092, National Institute of

Standards and Technology, Technology Administration, U.S. Department of Commerce, January

1998.

[10] K.L. Stricklett and M.G. Vangel. Electric motor efficiency testing under the new part 431 of

[16]

[17]

[18]

41

Checking the net contents of packaged goods.

Nbs handbook 133, National Bureau of Standards,

Gaithersburg, MD, 1988.

M.C. Croarkin and G.L. Yang. Acceptance probabilities for a sampling procedure based on the

mean and an order statistic . J. Res. Nat. Bur.

Standards, 87:485-511, 1987.

ASTM Standard E 122-89, Standard practice for

choice of sample size to estimate a measure of quality for a lot or process, 1989.

C. Stein. A two-sample test for a linear hypothesis

whose power is independent of variance . Annals of

Mathematical Statistics, 16:243-258, 1945.

P.J. Bickel and K.A. Doksum. Mathematical

Statistics, pages 158-159. Holden Day, Oakland,

CA, 1977.

R.S. Schulman. ' Statistics in Plain English with

Computer Applications. Van Norstand Reinhold,

New York, 1992.

Energy conservation program for certain commercial and industrial equipment: Test procedures, labeling, and certification requirements of

electric motors; proposed rule. Federal Register, 61(230):60439-60475, Wednesday, November

271996.

EE-TP-98-550 COMMENT 41
Appendix A

The 10 CFR 430 and the proposed 10 CFR Part 432 sampling

plans for compliance testing

The following test criteria are presented as an example of compliance testing under 10 CFR Part 430. Similar

criteria have been proposed for distribution transformers in the new 10 CFR Part 432 [3] and are also presented.

The 10 CFR Part 430 criteria presented below apply specifically to dishwashers, which is one of the products

covered under 10 CFR Part 430 . Similar sampling plans are provided in 10 CFR Part 430 §430.24 for all

other covered products. The statistical confidence and divisor used in criteria (i)(B) and (ffi)(B) differ between

products and are specific to each product. The values of these coefficients are chosen to be consistent with

current industry practice for that product.

§430.24 Units to be tested.

When testing of a covered product is required to comply with section 323 (c) of the Act, or to comply

with rules prescribed under sections 324 or 325 of the Act, a sample shall be selected and tested

composed of units which are production units, or are representative of production units of the basic

model being tested, and shall meet the following applicable criteria.

(c)(1) For each basic model of dishwashers, a sample of sufficient size shall be tested to insure that-

(i) Any represented value of estimated annual operating cost, energy consumption or other measure of energy consumption of a basic model for which consumers would favor lower values

shall be no less that the higher of (A) the mean of the sample or (B) the upper 971/2 percent

confidence limit of the true mean divided by 1.05, and

(ii) Any represented value of the energy factor of other measure of energy consumption of a basic

model for which consumers would favor higher values shall be no greater than the lower of

(A) the mean of the sample or (B) the lower 971/2 percent confidence limit of the true mean

divided by 0.95.

The criteria given below are proposed for distribution transformers in the new 10 CFR Part 432 . The values of

statistical confidence and the formulation of the divisor specified in (b) are chosen to be consistent with current

industry practice.

§432.24 Units to be tested.

For each basic model of distribution transformers, a random sample of sufficient size, but no fewer

than five productions units, shall be tested to insure that any represented value of efficiency shall be

no greater than the lower of the:

(a) Mean of the sample; or

(b) The lower 95% confidence limit of the estimated true mean divided by a number equal to [1-

0.03(1- E,/100)], where E. is the represented value of efficiency claimed for that particular basic

model.

42

EE-TP-98-550 COMMENT 41
Appendix

The ICE A. Standard TP 2 sampling plan

Sections 7.0 and 7.1 of NEMA Standard TP 2 follow. The reader may note minor differences between this

appendix and the NEMA standard TP 1, e.g., the pagination and the reference to the included table in this

appendix differ from those in the NEMA standard TP 1.

SECTION 7

Demonstration of Compliance

7.0 General

This section provides a methodology for proving compliance in achieving the specified efficiency levels. It specifies

monthly sampling over a 180 day period for the cases where 100% of the units are not tested . This standard

requires that no individual transformer shall be considered acceptable if its measured losses exceed the allowance

by more than 8%.

For a transformer population of a specific kVA rating, compliance with the energy efficiency standards as defined

in Section 4 of NEMA TP 1 shall be demonstrated through measurements of No-Load and Load losses according

to the procedures described in this standard. A transformer model is defined as all transformers of the same

kVA rating and type as described in the efficiency tables of NEMA TP 1.

According to the IEEE Standards C57.12 .00 and C57. 12.01 the loss tolerance for an individual unit as related

to guarantee is defined as follows:

Limit Beyond Guarantee

No Load Loss 10% max.

Total Loss at 100% Load 6% max.

At 100% load, the Load Loss is normally four times the No Load loss, suggesting that a Load Loss variability

of approximately 5% is allowed. The sum of this allowance and the 10% variability for the No Load loss yields

the Total Loss variability of 6%.

Since TP-1 tables reflect loss measurements at 35% or 50% of rated load where No Load Loss is equal to Load

Losses, the 6% loss tolerance cannot be used and therefore a new total loss tolerance of 8% shall be applicable

at these measurement points. This is consistent with the IEEE Standards C57.12.00 and C57.12.01.

7.1 Number of Units to be Tested

NEMA TP-1 requires that the overall efficiency of the entire population of transformers must meet the specified efficiency standards. This intent is satisfied if the mean efficiency of the entire population satisfies this

requirement. The compliance of a group of transformers shall be demonstrated by testing all or randomly drawn

samples of these transformers.

7.1.1 Compliance Demonstration Through test on all Transformers:

Manufacturers may choose to test all units of various kVA ratings manufactured during a production period of

180 days to demonstrate compliance with the efficiency standard NEMA TP 1. The intent of this standard is

satisfied if the Total Measure kVA Input (TMI) of this batch of transformers is equal to or less than the Total

Allowed kVA Input (TAI) calculated based on the measured and specified efficiency levels specified in TP 1.

Each individual unit from this production batch must meet or exceed the minimum acceptable efficiency level

calculated as follows:

Minimum Acceptable Efficiency level = 108 0.08SEL X 100

Where

SEL = the standard percent efficiency level from NEMA Standard TP I.

Note: The Minimum Acceptable Efficiency level calculation is based on an 8% tolerance on total loss at the load

levels considered for the Efficiency levels specified in NEMA TP-1; i.e. no individual unit shall be considered

acceptable if its measured losses exceed the allowance by more than 8%.

43

EE-TP-98-550 COMMENT 41
A Case Study with Application to Distribution Transformers NISTTN 1427

Sample size n t at 95% confidencelevel Sample size n t at 95% confidence level

2 6.314 11 1.812

3 2.920 12 1.796

4 2.353 13 1.782

5 2.132 14 1.771

6 2.015 15 1.761

7 1.943 16 1.753

8 1.895 17 1.746

9 1.860 18 1.740

[LO_ 1.833 19 1.734

20 1.729

To demonstrate compliance with the Efficiency standard, proceed as follows:

Step I. Calculate the Total Allowed KVA input:

Where .

i = 1,2,3,4,...

kVA; = kVA ratings of various transformers included in a production batch

manufactured in 180 days.

Ili = Specified Efficiency Level in TP 1 for transformer rating kVA1

L; = Per unit load at which the efficiency is specified per TP 1

Step 2. Calculate the Total Measured kVA input:

TMI = L{ kVA,

itma

Where

i = 1,2,3,4,...

i1mi = Measured Efficiency Level for the transformer.

L{ = Per unit load at which the efficiency is specified per TP 1

Step 3. IfTMI is equal to or less than TAI; the compliance of the production batch has been demonstrated.

7.1.2 Compliance Demonstration Through Tests on a Statistically Valid Sample:

The manufacturer may choose to demonstrate the compliance of a plurality of units by a random sampling of

the units of each kVA rating produced in a period of 180 days. Statistically valid numbers of units but not less

than 5 shall be drawn on a monthly basis from the units of each kVA rating produced during this period for

testing. This will assure the randomness of the samples (30 units minimum).

All the units drawn in 180 day period shall be tested for computing the mean efficiency of each kVA rating.

None of individual units in a sample shall be considered acceptable if its measured losses exceed the allowance

by more than 8%.

It is the responsibility of the manufacturer to assure through adequate quality control procedures and/or random

testing that the conformance of various kVA rating transformers is maintained.

For a random sample to be statistically valid, a minimum number of units n in the sample must be tested to

assure that the standard deviation of the test results is no more than the standard deviation S of the population

with 95% confidence. The minimum sample size shall be determined as follows:

n = (tSK)2

b

44

EE-TP-98-550 COMMENT 41
A Case Study with Application to Distribution Transformers NISTTN 1427

where

_ 108 - 0.08SEL

K SEL(8 - 0.08SEL)

and t Statistic is determined from Table B1 corresponding to sample size of n at 95% confidence level.

To demonstrate compliance with this standard , proceed as follows:

Step 1. Choose a sample size of ni units (5 min.)

Step 2. Compute the mean A'i and standard deviation S1 as follows:

Xi= 1 EXa, ni

S1 =

E(X, - Xi)2

ni-1

Where

X1 = The average efficiency of the first sample.

Xi = The efficiency of the unit i

n1 = The number of units in the first set of samples (the subscript refers

to the sample number)

Si = The computed sample standard deviation of the first sample

Step 3. Calculate the minimum sample size n as follows:

n = (t1S1K)2

Where ti Statistic is chosen from Table BI corresponding to the sample size ni.

Step 4. If n < n1, the sample size is adequate to yield the acceptable standard deviation . Proceed with

Step 6. Otherwise, test additional units and increase the sample size to n2 such that:

n2 > n, where n2 is the total number of units tested.

Step 5. Repeat Steps 2 and 3 until each sample size for each kVA rating produced is larger than the

minimum sample size n.

Step 6. Calculate the average efficiency %j of all the samples tested in 180 days for each rating kVA;

manufactured within this period.

Step 7. Calculate the Total Allowed kVA input:

TAI = E Ni Li kVAi

IN

Where

i = 1,2,3,4,...

kVA; = kVA ratings of various transformers included in a production batch

manufactured in 180 days

n; = Specified Efficiency Level in TP 1 for transformer rating kVA;

L{ = Per unit load at which the efficiency is specified per TP 1

Ni = Total number of units produced with rating kVA;

Step 8. Calculate the Total Measured kVA input:

TMI=EN1L1kVA®

45

EE-TP-98-550 COMMENT 41
A Case Study with Application to Distribution Tmnsf0 rs

ere

i

kVAc

1,2,3,4,...

kVA ratings of various transformers included in a production batch

manufactured in 180 days

Measured efficiency Level for transformers rated kVA,

Per unit load at which the efficiency is specified per TP 1

Total number of units produced with rating kVA

NIST N 1427

Step 9. IfTMI is equal to or less than TAM; the compliance of the production batch has been demonstrated.

46

EE-TP-98-550 COMMENT 41
Appendix C

A sampling plan for enforcement testing

The following sampling plan is adapted from the Sampling Plan for Enforcement Testing contained 10 CFR.

Part 430. The plan is similar to that proposed for electric motors [18].

Step 1. The first sample size (ni ) must be four or more units.

Step 2. Compute the mean (XI) of the measured energy performance of the nj units in the first sample as

follows:

n,

XI = Xtl (Cl) . nj e-1

where Xi is the measured efficiency of unit i.

Step 3. Compute the sample standard deviation (SI) of the measured efficiency of the nj units in the first

sample as follows:

S+1=

I (X'- x1)2 ni-1

Step 4. Compute the standard error (SE(RI)) of the mean efficiency of the first sample as follows:

SE(X1) = sal .

(C2)

(C3)

Step 5. Compute the lower control limit (UCLI) for the mean of the first sample using the applicable rated

efficiency (RE) as the desired mean as follows:

UCLI = RE - tSE(XI ). (C4)

Here t is the 2.5 percentile of a t-distribution for a sample size of nj and yields a 97.5 percent

confidence level for a one-tailed t-test.

Step 6. Compare the mean of the first sample (XI) with the lower control limit (UCLI) to determine one

of the following:

(i) If the mean of the first sample is below the lower control limit , then the basic model is in

noncompliance and testing is at an end.

(ii) If the mean is equal to or greater than the lower control limit, no final determination of

compliance or noncompliance can be made; proceed to Step 7.

Step 7. Determine the recommended sample size (n) as follows:

n

tSi(108-0.08RE) 2 (C5)

[. RE(8 - 0.08RE) j

where S1 and t have the values used in Steps 4 and 5, respectively. The factor

108 - 0.08RE

RE(8 - 0.08RE)

is based on a 8 percent tolerance in'the total power loss at and fixed output power.

Given the value of n, determine one of the 'following:

(i) If the value of n is less than or equal to nj and if the mean energy efficiency of the first sample

(XI) is equal to or greater than the lower control limit (UCLI), the basic model is in compliance

and testing is at an end.

47

EE-TP-98-550 COMMENT 41
A Case Study with Application to Distribution Transformers NISTTN 1427

(ii) If the value of n is greater than n1 , the basic model is in noncompliance. The size of a second

sample n2 is determined to be the smallest integer equal to or greater than the difference n-nt.

If the value of n2 so calculated is greater than 20 - nt , set n2 equal to 20 - nt.

Step S. Compute the combined mean (X2) of the measured energy performance of the nj and n2 units of

the combined first and second samples as follows:

nl+n2

X2 = 1 Xt.

Tit + net

(C6)

Step 9. Compute the standard error (SE(X2)) of the mean efficiency of the nj and n2 units in the combined

first and second samples as follows:

St SE(X2) =

nl-t52

(Note that Si is.the value obtained above in Step 3.)

Step 10. Set the lower control limit (UCL2) to,

(C7)

UCL2 = RE - tSE(X2), (C8)

where t has the value obtained in Step 5, and compare the combined sample mean (X2) to the lower

control limit (UCL2) to find one of the following:

(i) If the mean of the combined sample (X2) is less than the lower control limit (UCL2), the basic

model is in noncompliance and testing is at an end.

(ii) If the mean of the combined sample (X2) is equal to or greater than the lower control limit

(UCL2), the basic model is in compliance and testing is at an end.

MANUFACTURER OPTION TESTING

If a determination of non-compliance is made in Steps 6 , 7 or 11, above, the manufacturer may request that

additional testing be conducted, in accordance with the following procedures.

Step A. The manufacturer requests that an additional number , n3, of units be tested, with n3 chosen such

that nj + n2 + n3 does not exceed 20.

Step B. Compute the mean efficiency, standard error, and lower control limit of the new combined sample

in accordance with the procedures prescribed in Steps 8 , 9, and 10, above.

Step C. Compare the mean performance of the new combined sample to the lower control limit (UCL2) to

determine one of the following:

(a) If the new combined sample mean is equal to or greater than the lower control limit, the basic

model is in compliance and testing is at an end.

(b) If the new combined sample mean,is less than the lower control limit and the value of nj+n2+n3

is less than 20, the manufacturer may request that additional units be tested . The total of all

units tested may not exceed 20. Steps A , B, and C are then repeated.

(c) Otherwise, the basic model is determined to be in noncompliance.

48

EE-TP-98-550 COMMENT 41
Appendix D

Computational algorithms

Compliance testing under 10 CFR. Part 430 (Method 1)

Let x1 i x2, ... , xn be measured losses, with sample mean i and sample standard deviation s. In a test of

compliance, we require that

and

i + is/-,Fn < (1 + 0)k, (D2)

where t denotes the appropriate percentile of a Student-t distribution with v = n -1 degrees of freedom, and 0

is a small positive number (typically, 0 = .03). Assume that the data {xj }{1 are an independent, identicallydistributed sample from a normal (Gaussian) distribution with population mean p and population standard

deviation a.

The probability of compliance is given by the expression

p(a, b) = 2n Ia. L' e_n12/sX2 [t2 (a + b - y)2] dy ,. (D3)

where

a = ku and b = ck. (D4)

In this expression, x!(u) is the probability that a X2 random variable V,, having v = n - 1 degrees of freedom

is less than u; that is

Pr(V„ < u) = X2.(u) = r(Y)2-

j zv/2-1e z/2dz, (D5)

which is a function easily expressed and calculated in terms of an incomplete r-function . Function (D5) can be

evaluated numerically using a wide range of public domain software. Gauss-Legendre quadrature is adequate

for the calculation of (D3), and it is sufficient to use p - 5a/ ffor the lower bound of the numerical integral.

The limits of p(a, b) as or goes to zero or infinity are of particular interest; we consider these in turn

Limit of small a: As a -► 0, b -i oo. If a is allowed to go to infinity as well , then p(a, b) -► 1, which isn't

useful. So we require that b -> oo and that a remains finite; that is

-oo < lim k - p . = q < oo. (D6)

a-► 0 a

If this condition holds, then

r_n 9 _nya/2'

l m p(a, b) j e dy

1 J-v -z2/2

727

where di(z) denotes the probability that a standard normal random variable is less than z, and 0 < a < 1 is a

compliance probability which identifies a compliance probability contour, in the (p, a) plane, whose asymptotic

behavior we are investigating. Hence, if z. denotes the 100a percentile of a standard normal random variable

(that is,'(xa) = a), then

q=za// . (D7)

49

EE-TP-98-550 COMMENT 41
A Case Study with Application to Distribution Transformers NISTTN 1427

Asymptotically, as a -> 0, the a contour of compliance probability is linear; to be specific,

p - k - xv,a/ f . (D8)

We could have guessed this result by reasoning that as a -+ 0, a -+ 0, and condition (D2) must always hold. So

we need only be concerned with condition (D1), and we require that

Pr(2 < k) = a #-

Pr

x/v"

n- - k/s =a *

(a a n)

0 (/f)_a

k=μ+xaa/-,fn-:

Limit of large a: Asa -- oo, b -4 0. If a -} 0, then p(a, b) -4 0 if k > v, p(a, b) -+ 1 if k < p, and

p(a, b) -+ 1/2 if k = p. This is not particularly interesting or realistic, so we require that

(D9)

for finite v. It follows that

lim p(a, b) = Fn j e "V2/2X, [ tz (v - U)a] dy.

Let h = v - y, and note that

lim p(a, b) = Fn e-" X2 (

ea

hs) dh a, +00 10 00

_ t

e

- [ -v]'X

2

(

/ a rx = a. ) dx

2^rv 0

(D 10)

For the limiting behavior of the 100a percent contour in the (p, a) plane, we must solve the transcendental

equation

2^j

t a°° e [*r"J2x (x) (D11)

for v. The contour will be asymptotically linear, i.e., as a -► oo

p -k - va. (D12)

This asymptotic result could also be derived by reasoning that, in the limit of large or, s will almost certainty be

large and 2 < k will hold automatically, provided that condition (D2) is satisfied. So the compliance probability

is determined in this limit in terms of the second condition alone, which leads to the result developed in this

subsection.

Derivation of p(a, b): Under a normal distribution model, the sample mean and standard deviation, 2

and s, are statistically independent. The sample mean has a normal distribution with mean p and standard

deviation a/f. The sample standard deviation is proportional to the square root of a Xv random variable,

with proportionality constant a/V5.

It's easy to see that the probability of compliance can be expressed as

Pr(-* < k and 2 + ts/ f < (1 + 9)k) = Pr I a < k and 8 < (1 + B)k - 2^ (D13)

\ t/f

50

EE-TP-98-550 COMMENT 41
A Case Study With Application to Distribution Transformers NISTTN 1427

Because of independence (which holds only under a normal model), s has the same probability distribution

whatever the value of a. So we condition on x taking on all possible values -oo < x < oo, and integrate:

Pr(a<kands < (1+8)k-a)

t/,Fn l

f 00 ( (1+®)k-xl _

00

Pr k > x ands < t/^ J Pr(x - x)dx

1

00

Pr(k > x) Pr (s < (1 O)k xl Pr(x = x)dx. t/V

Note that we can write the joint robability distribution as a product because of independence . Substituting

expressions for the normal and X2 distributions, into the above expressions and simplifying leads to p(a, b) as

given in (D3).

Testing under TTY 2

Assume that the random variables Y, are normal with mean μ and standard deviation a. Since we will assume

that p and v can be regarded as approximately known from previous data, we can , without loss of generality,

employ the standardized sample X; _ (Y,• - pp)/a, having order statistics X(,) = (Y;) - μ)/a.

Then the probability of being found in compliance under the TP 2 sampling plan can be shown to be

p = 1- [Fxu) (t) + Fg (t) - Fx(,),X (ti, t2)] • (D14)

The function FX(,) (t) denotes the CDF of X(I), and Pr(t) denotes the CDF of X; that is

FX(,) (t) = Pr(Xil) < t) = 1 - (1- 4► (t))",

and

Fg (t) = Pr(. < t) = lb (,/nt) ,

where s1(•) is the standard normal CDF, which is defined to be the probability that a normally distributed

random variable is less than u, that is

-I'M f

U e-e

This integral may be calculated efficiently and accurately using public-domain routines.

A saddle-point approximation to the bivariate CDF

Fx(,),x(tl,t2 ) = Pr(X(I) < tI and 1 5 t2)

is derived by Vangel [8]. Let qi(t) denote the normal density, and let

h(t) _ O(t)

- t(t)

be the normal hazard function. The saddle-point approximation Fx(1),g(tl, t2), for t1 < t2, is

Fx(,,,g (tI, t2

) = f%. t(ft2)A(t)dt + ftoo t J VG- [tI + "nI (h(t) - t)] } A(t)dt

. f . A(t)dt

A(t) = h-("-I)(t) exp l (" 2n1)2 [h(t) -t]2 + (n - 1)t[h(t) - t]} 1- h2(t) + th(t),

where to is the (unique) solution to the equation

n-1 [h(ta)- t4]=t2-tin

(D15)

(D16)

51

EE-TP-98-550 COMMENT 41
A Case Shady with Application to ®isteibution Teansformses NISTTN 1427

The probability of demonstrating compliance by testing under the Sampling Plan for Enforcement Testing can

be calculated using straightforward numerical integration.

In the following expressions, ni > 4 denotes the minimum sample size specified in the sampling plan, and

rat = 20 is the maximum sample. In order to simplify the equations, ni - 1 is represented by v, and t is the

97.5th percentile of the Student-t distribution for the sample size nj.

The probability of compliance is

& = 2g-,r l2) d.-m1 l+ae 6

where the limits of integration are u,,, = 0,

r.j `LT -10U a-2fori=rai+1,...,n2, t

and rc„a+i = oo, and the loss tolerance, LT, is a percentage.

52

EE-TP-98-550 COMMENT 41
Technical Publications

Periodical

Journal of Research of the National Institute of Standards and Technology-Reports NISI research

and development in those disciplines of the physical and engineering sciences in which the Institute is

active. These include physics, chemistry, engineering, mathematics, and computer sciences. Papers cover a

broad range of subjects, with major emphasis on measurement methodology and the basic technology

underlying standardization. Also included from time to time are survey articles on topics closely related to

the Institute's technical and scientific programs. Issued six times a year.

onperiodicals

Monographs-Major contributions to the technical literature on various subjects related to the

Institute's scientific and technical activities.

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Special Publications-Include proceedings of conferences sponsored by NIST, NIST annual reports, and

other special publications appropriate to this grouping such as wall charts, pocket cards, and bibliographies.

National Standard Reference Data Series--Provides quantitative data on the physical and chemical

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performance criteria related to the structural and environmental functions and the durability and safety

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Technical Notes-Studies or reports which are complete in themselves but restrictive in their treatment of

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Voluntary Product Standards-Developed under procedures published by the Department of Commerce

in Part 10, Title 15, of the Code of Federal Regulations. The standards establish nationally recognized

requirements for products, and provide all concerned interests with a basis for common understanding of

the characteristics of the products. MST administers this program in support of the efforts of private-sector

standardizing organizations.

Order the following NIST publications--PIPS and NISTIRs from the National Technical Information

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Federal Information Processing Standards Publications (FIPS PUB)-Publications in this series

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official source of information in the Federal Government regarding standards issued by NIST pursuant to

the Federal Property and Administrative Services Act of 1949 as amended, Public Law 89-306 (79 Stat.

1127), and as implemented by Executive Order 11717 (38 FR 12315, dated May 11, 1973) and Part 6 of

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published subsequently in more comprehensive form.

EE-TP-98-550 COMMENT 41
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and Thchnology

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Official Business

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EE-TP-98-550 COMMENT 41