NIST Technical Note 1456

Operating Characteristics of the Proposed Sampling

Plans for Testing Distribution Transformers

O. Petersons

K.L. Stricklett

C.R. Hagwood

National Institute of Standards and Technology  Technology Administration  U.S. Department of Commerce
NIST Technical Note 1456

Operating Characteristics of the Proposed Sampling

Plans for Testing Distribution Transformers

O. Petersons

K.L. Stricklett

Quantum Electrical Metrology Division

Electronics and Electrical Engineering Laboratory

National Institute of Standards and Technology

Gaithersburg, MD 20899-8172

C.R. Hagwood

Statistical Engineering Division

Information Technology Laboratory

National Institute of Standards and Technology

Gaithersburg, MD 20899-8980

May 2004

UNITED STATES OF AMERICA

DEPARTMENT OF COMMERCE

U.S. Department of Commerce

Donald L. Evans, Secretary

Technology Administration

Phillip J. Bond, Under Secretary for Technology

National Institute of Standards and Technology

Arden L. Bement, Jr., Director
Certain commercial entities, equipment, or materials may be identified in this document in order to describe an

experimental procedure or concept adequately. Such identification is not intended to imply recommendation or

endorsement by the National Institute of Standards and Technology, nor is it intended to imply that the entities, materials,

or equipment are necessarily the best available for the purpose.

National Institute of Standards and Technology Technical Note 1456

Natl. Inst. Stand. Technol. Tech. Note 1456, 31 pages (May 2004)

CODEN: NTNOEF

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 2004

For sale by the Superintendent of Documents, U.S. Government Printing Office

Internet: bookstore.gpo.gov—Phone: (202) 512-1800—Fax: (202) 512-2250

Mail: Stop SSOP, Washington, DC 20402-0001

ii
Bibliographic Information

Abstract

The sampling plans for efficiency testing contained in the Supplemental Notice of Proposed Rulemaking (SNOPR), Energy Conservation Program: Test Procedures for Distribution Transformers, are discussed. The proposed sampling plans test for compliance with

standards for average energy efficiency. The SNOPR includes: a sampling plan for demonstration of compliance with a represented energy efficiency; criteria for substantiation of

an alternative efficiency determination method; and a sampling plan for enforcement testing. Model calculations are presented that estimate the operating characteristics of the

sampling plans, that is the probability of demonstrating compliance when testing a specific

distribution of efficiencies.

Keywords

distribution transformers; efficiency testing; Energy Policy and Conservation Act; operating characteristics; sampling plan

iii
Contents

1 Introduction 1

2 EPCA guidelines 1

3 Industry practice 2

3.1 NEMA standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

3.2 IEEE standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

3.3 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

4 Methods of analysis 3

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

5 Compliance testing 5

5.1 Statistical background . ........................... 6

5.2 Operating characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

6 AEDM substantiation 7

7 Enforcement testing 8

7.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

7.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

8 Concluding remarks 10

8.1 Compliance testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

8.2 Enforcement testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

9 Further information 10

Appendix A: Compliance testing 21

Appendix B: Substantiation of an AEDM 22

Appendix C: Enforcement testing 23

iv
List of Figures

1. The operating characteristics of the sampling plan for compliance testing

for an expanded uncertainty (5=p

n) %, n = 1 . . . . . . . . . . . . . . . 12

2. The operating characteristics of the sampling plan for compliance testing

for an expanded uncertainty (5=p

n) %, n = 5 . . . . . . . . . . . . . . . 13

3. The operating characteristics of the sampling plan for compliance testing

for an expanded uncertainty (5=p

n) %, n = 10 . . . . . . . . . . . . . . 14

4. The operating characteristics of the sampling plan for compliance testing

for an expanded uncertainty (8=p

n) %, n = 1 . . . . . . . . . . . . . . . 15

5. The operating characteristics of the sampling plan for compliance testing

for an expanded uncertainty (8=p

n) %, n = 5 . . . . . . . . . . . . . . . 16

6. The operating characteristics of the sampling plan for compliance testing

for an expanded uncertainty (8=p

n) %, n = 10 . . . . . . . . . . . . . . 17

7. The operating characteristics of the Sampling Plan for Enforcement Testing for m = 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

8. The operating characteristics of the Sampling Plan for Enforcement Testing for m = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

9. The average number of units tested under the Sampling Plan for Enforcement Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

v
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. . . 4

vi
Operating Characteristics of the Proposed Sampling Plans

for Testing Distribution Transformers

1 Introduction

This Technical Note provides analysis of the sampling plans

for efficiency testing contained in the Supplemental Notice

of Proposed Rulemaking (SNOPR), Energy Conservation

Program: Test Procedures for Distribution Transformers

[1]. The performance of the proposed sampling plans in

demonstrating compliance with a represented efficiency is

specifically addressed. The SNOPR was issued to resolve

certain issues that arose during review of an earlier Notice

of Proposed Rulemaking (NOPR) [2]. One issue being the

properties of the proposed sampling plan for compliance

testing. The sampling plan included in the earlier NOPR,

which was modeled after the examples provided in 10 CFR

Part 430 [3], is discussed in an earlier report [4].

Under the provisions of the SNOPR, measurements of efficiency are contemplated for three purposes: 1) testing for

compliance with a represented efficiency; 2) qualification of

an Alternative Efficiency Determination Method (AEDM);

and 3) enforcement testing. The objectives of testing in each

of these circumstances differ in significant ways. Compliance testing is a one-time activity undertaken at the initiation of the program, upon enrollment into the program, or

for qualification of a new product. The sampling plan for

compliance testing is intended to ensure that the product in

question meets or exceeds the represented efficiency. The

SNOPR allows the use of an AEDM to determine the represented efficiency of distribution transformers, where an

AEDM is a predictive model that is based on analysis of

design and test data. An AEDM must, however, be substantiated before it may be used for this purpose. The criteria for

substantiation of an AEDM provide a broad intercomparison

of the performance predicted by the AEDM and test data.

Finally, an enforcement action may be undertaken when the

performance of a specific product or products is contested.

Enforcement testing is one of a series of requirements under

the Energy Policy and Conservation Act of 1975 (EPCA) as

amended [5] during an enforcement action. The Sampling

Plan for Enforcement Testing is designed to correctly determine whether a product is in compliance, while providing a

low risk that a compliant product could fail by chance during

an enforcement action.

The preparation of this report was undertaken at the request

of the U.S. Department of Energy (DOE). This report is intended to supplement the materials presented in the SNOPR.

While the authors believe the information presented here to

be accurate and factual, this report is not a statement of the

policies of the U.S. Department of Energy and must be regarded only as commentary on the proposed sampling plans.

The remainder of this report is organized as follows: The

general objectives of testing under EPCA are discussed in

Section 2. Industry practice regarding efficiency performance is briefly reviewed in Section 3. The methods and

model assumptions used in the evaluation of these sampling

plans are presented in Section 4. Section 5 discusses the

sampling plan for compliance testing; Section 6 discusses

the criteria for substantiation of an AEDM; and the Sampling Plan for Enforcement Testing is presented in Section 7.

Concluding remarks are provided in Section 8. Contact information for further information is provided in Section 9.

For the convenience of the reader, each of the sampling

plans are provided as appendices to this report: Appendix A

contains the proposed sampling plan for compliance testing, Appendix B contains the criteria for substantiation of

an AEDM, and the Sampling Plan for Enforcement Testing

appears in Appendix C.

2 EPCA guidelines

The purpose of the EPCA legislation is stated in 42 U.S.C.

x6312(a) [6]:

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.

To this end, EPCA authorizes the establishment of energy

performance standards that may specify energy efficiency

or energy use for each covered product.

EPCA relies on a program of systematic testing to establish that energy performance standards are met. The objectives and limitations of testing under EPCA are stated in

42 U.S.C. x6314(a)(2):

Test procedures prescribed in accordance 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.

1
NIST TN 1456 Operating Characteristics of the Proposed Sampling Plans

EPCA also addresses the represented energy performance of

covered equipment in 42 U.S.C. x6314(d)(1):

...no manufacturer, distributor, retailer, or private

labeler may make any representation—

(A) in writing (including a 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.

In the case of energy efficiency, the purpose of EPCA is

met provided the average energy efficiency of each type of

covered equipment 1) is not less than the EPCA energy efficiency standard for that product, and 2) is not less than

the represented energy efficiency of that product. Energy

efficiency must be demonstrated under EPCA through use

of DOE test procedures. EPCA stipulates that such testing

should not be unduly burdensome to conduct. Thus the two

key criteria for evaluation of sampling plans for efficiency

testing are: 1) the assurance provided that the average efficiency of that product meets or exceeds the represented

efficiency, and 2) the burden placed on a manufacturer by

testing under the plan.

3 Industry practice

Voluntary industry standards are typically developed by consensus among manufacturers, customers, and other stakeholders and are thus likely to describe technical capabilities

that are broadly supported by that industry. A brief summary of references to the energy performance of distribution

transformers in voluntary standards follows.

3.1 NEMA standards

Criteria for the energy performance of distribution transformers are included in two standards sanctioned by the

National Electrical Manufacturers Association (NEMA):

NEMA Standards Publication TP 1-2002, “Guide for Determining Energy Efficiency for Distribution Transformers”

[7], and NEMA Standards Publication TP 2-1998 “Standard

Test Method for Measuring the Energy Consumption of Distribution Transformers” [8].

NEMA TP 1: Section 4 of the NEMA TP 1 standard establishes energy efficiency levels for the NEMA class-1 designation. The minimum efficiencies for liquid-filled singleand three-phase transformers are tabulated in Table 4-1 of

the TP 1 standard, and efficiency values for dry-type singleand three-phase transformers are tabulated in Table 4-2.

Dry-type transformers are further delineated by voltage rating and basic insulation level (BIL): the standard provides

efficiencies for low-voltage transformers and for mediumvoltage transformers having basic insulation levels less than

or equal to 60 kV and for basic insulation levels greater than

60 kV. The NEMA standard TP 1 tables are reproduced below in Tables 1 and 2. NEMA standard TP 1 states that the

tabulated values are the “minimum efficiencies” for NEMA

class 1 designation, which we interpret as “minimum [average] efficiencies.”

NEMA TP 2: NEMA standard TP 2 stipulates that no

individual transformer may exceed 108 % of the rated

loss under specific load conditions: The load condition

for liquid-filled transformers and dry-type, medium-voltage

transformers is 50 % of the rated load; and the load condition for dry-type, low-voltage transformers is 35 % of the

rated load. Section 7 of the TP 2 standard provides a sampling plan for establishing compliance with the TP 1 efficiencies. Conformance with the TP 1 efficiencies is based

on a weighted average of transformers manufactured in a

period of 180 d.

3.2 IEEE standards

Two standards sponsored by the Institute of Electrical

and Electronics Engineers (IEEE) are also relevant to

this discussion: IEEE Standard C57.12.00-2000, “General Requirements for Liquid-Immersed Distribution, Power

and Regulating Transformers” [9]; and IEEE Standard

C57.12.01-1998, “General Requirements for Dry-Type Distribution and Power Transformers, including those with

Solid Case and/or Resin-Encapsulated Windings” [10].

The IEEE standards prescribe maximum loss limits for an

individual transformer of 110 % of the rated value for the

no-load loss and 106 % of the rated value for the total losses.

Individual units having losses above these limits can be rejected.

3.3 Comments

Energy performance criteria for distribution transformers

are provided in voluntary standards sponsored by NEMA

and by IEEE. The NEMA standards establish minimum average efficiencies for distribution transformers and also set

2
for Testing Distribution Transformers NIST TN 1456

Table 1: NEMA Class 1 efficiency levels for liquid-filled distribution transformers [7].

Reference Condition Temperature % of Nameplate Load

Load Loss 55 C 50 %

No Load Loss 20 

C 50 %

Single Phase Three Phase

kVA Efficiency kVA Efficiency

10 98.4 15 98.1

15 98.6 30 98.4

25 98.7 45 98.6

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

limits on the maximum allowable loss power. The condition

on the maximum loss power applies to individual transformers tested under specific load conditions. The IEEE standards also prescribe limits on the maximum loss power of

an individual transformer. Since the measured loss power

is directly related to the energy efficiency, these standards

establish a tolerance on the measured efficiency of a single

unit.

The minimum average efficiencies established in NEMA

Standard TP 1 are stated as a percentage and are given to

three significant figures, which implies that the resolution

of energy efficiency measurements is at least 0.1 %. The

limits on loss power are stated as a percentage of the rated

loss power in both the NEMA and IEEE standards. The loss

power is given to the nearest percent, which implies that the

resolution of measured loss power is at least 1 % of the rated

loss.

The NEMA standard TP 2 imposes a maximum total loss

limit of 108 % of the rated value for liquid-filled transformers and for medium-voltage, dry-type transformers operated

at 50 % of the rated load. The same limit of 108 % applies to low-voltage transformers operated at 35 % of the

rated load. IEEE standards limit the measured loss power to

106 % of the rated loss power at full load and 110 % of the

rated value for no-load conditions. Under full load conditions the loss power is approximately four times larger than

the no-load loss. Due to the reduced loading under NEMA

standard TP 2 the no-load loss and the load loss are approximately equal, the maximum loss limits in both the IEEE and

the NEMA standards are consistent.

4 Methods of analysis

Two figures-of-merit are considered and provide the basis

for the evaluation of the sampling plans: 1) the operating

characteristic, and 2) the testing burden. The operating

characteristic of a sampling plan is the probability of demonstrating compliance when testing a specific distribution of

efficiencies. This quantity provides an estimate of the probability or risk that an acceptable product could fail by chance

or that an unacceptable product could pass by chance under

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

burden, is the number of units tested. The minimum number

of units to be tested is specified in the SNOPR for each sampling plan. However there are several exceptions to the recommended minimum sample size: the proposed sampling

plans for compliance testing and enforcement testing both

contain provisions for testing basic models that are produced

in limited number; the Sampling Plan for Enforcement Testing tests whether the number of units tested is sufficient to

support the conclusion and may require that additional units

be tested; finally a manufacturer may elect to test units over

and above the minimum required.

The sampling plans can be examined by means of model

calculations. The model calculations presented assume that

energy efficiency, and therefore loss power, is normally distributed with mean, , and standard deviation, . The normal distribution is well known and may be evaluated numerically with high accuracy. Monte Carlo methods [11] can be

adapted to evaluate sampling plans where the sample size is

not fixed and are used in this document to estimate the testing burden of the Sampling Plan for Enforcement Testing.

A discussion of the algorithms used to calculate the figuresof-merit can be found in [12–14].

3
NIST TN 1456 Operating Characteristics of the Proposed Sampling Plans

Table 2: NEMA Class 1 efficiency levels for dry-type distribution transformers [7].

Reference Condition Temperature % of Nameplate Load

Low Voltage 75 C 35 %

Medium Voltage 75 

C 50 %

Single Phase Efficiency Three Phase Efficiency

kVA Low Voltage Medium Voltage kVA Low Voltage Medium Voltage

 60 kV BIL > 60 kV BIL  60 kV BIL > 60 kV BIL

15 97.7 97.6 97.6 15 97.0 96.8 96.8

25 98.0 97.9 97.9 30 97.5 97.3 97.3

37.5 98.2 98.1 98.1 45 97.7 97.6 97.6

50 98.3 98.2 98.2 75 98.0 97.9 97.9

75 98.5 98.4 98.4 112.5 98.2 98.1 98.1

100 98.6 98.5 98.5 150 98.3 98.2 98.2

167 98.7 98.8 98.7 225 98.5 98.4 98.4

250 98.8 98.9 98.8 300 98.6 98.6 98.5

333 98.9 99.0 98.9 500 98.7 98.8 98.7

500 — 99.1 99.0 750 98.8 98.9 98.8

667 — 99.2 99.0 1000 98.9 99.0 98.9

833 — 99.2 99.1 1500 — 99.1 99.0

2000 — 99.2 99.0

2500 — 99.2 99.1

4.1 Loss representation

Transformer energy efficiency, , is determined by two

quantities: the output power, Po, and the loss power, Pl .

Expressed as a percentage, transformer efficiency is given

by the following equation:

 =

Po Po + Pl

100: (1)

The practice of the transformer industry is to measure transformer loss power and to perform intermediate computations in terms of loss power. Transformer energy efficiency

is typically calculated for purposes of final validation and is

based on the measured loss power.

The energy performance of distribution transformers is

stated in the SNOPR in terms of energy efficiency. We have

chosen, for this analysis however, to represent transformer

energy performance in terms of energy use or loss power.

The loss representation of energy performance has the advantage of being independent of efficiency; figures-of-merit

can thus be summarized globally rather than on a case-bycase basis for each energy efficiency. This representation

also provides direct comparison with the tolerance on the

measured losses specified in the NEMA and IEEE standards.

In the discussion that follows, the figures-of-merit are represented by contour plots, see Fig. 1, for example. The coordinates for these plots are the normalized average loss and

the normalized standard deviation of the population. The

average loss and standard deviation are normalized to the

represented loss and are given as a percentage of the represented loss. In all cases, the represented loss corresponds to

100 % on the average loss axis.

The NEMA and IEEE standards both place conditions on the

maximum allowable loss power of a single unit. It may be

inferred then that there is some risk that an individual transformer may have exceptionally high loss power and may be

rejected on that basis. Rejecting a distribution transformer

during final validation is costly and a manufacture must limit

such rejections. Statements made during the public hearing on the earlier NOPR [15] suggest that the maximum

loss limits in the NEMA and IEEE standards correspond

to approximately three standard deviations for at least some

transformers and manufacturers, which implies that the rate

of rejection is on the order of one per thousand. Assuming

that distribution transformers are designed to meet the represented loss power on average yields a standard deviation of

approximately 2.7 % when using the figure of 108 % from

NEMA standard TP 2. A higher value of 4.0 % for the standard deviation was mentioned during the public hearing. A

subsequent letter from NEMA [16] also cites the 4.0 % figure for the standard deviation. For the larger standard deviation, the design point would have to be set below the represented loss in order to maintain a low rate of rejection. The

distribution of loss power would likely differ between manufacturers and basic models. Indeed a statement made at the

public hearing indicated that the data on product variability

maintained by manufacturers are proprietary. We conclude

that the population standard deviation of the loss power is

likely to range in value between approximately 2.7–4.0 %.

The sampling plans are tailored to this range of values.

4
for Testing Distribution Transformers NIST TN 1456

5 Compliance testing

The sampling plan for compliance testing is contained in

x432.12(b)(2) of the proposed rule. For the convenience of

the reader it has been included as Appendix A to this report.

The proposed rule groups distribution transformers into basic models for compliance testing, where basic model is

fully defined in x432.10. For the purposes of this discussion, basic model will refer to distribution transformers having energy efficiencies that are nominally equivalent.

The sampling plans for compliance testing contained in

10 CFR Part 430 and Part 431 are tailored to the characteristics of the specific covered products [4, 12, 13]. Distribution

transformers as a product have characteristics that differ substantially from the products covered under 10 CFR Part 430

and Part 431: in particular, some basic models are produced

in very limited number, indeed, a basic model as defined in

x432.10 may describe a single transformer.

The sampling plans for compliance testing provided in

10 CFR Part 430 are based on the t-test in statistics. As

discussed below, the t-test presents difficulties when testing

small samples: For small sample sizes the value of the t

statistic is large, resulting in broad confidence intervals and

low precision in the determination of the population mean

from the sample. The t-test cannot be used to evaluate the

results of a single test.

The z-test allows testing of small samples, including samples of one. Test methods based on the z-test can be employed if the population standard deviation is known. Since

a manufacturer will likely know the range of standard deviations from test results obtained on similar transformers, the

z-test may be appropriately applied to compliance testing.

EPCA energy performance standards may place a lower

bound on the average efficiency or an upper bound on the

average energy use of a covered product. The proposed rule

places a condition on the mean efficiency of the sample in a

demonstration of compliance with a represented efficiency.

To emphasize the salient features of the proposed sampling

plan for compliance testing, we paraphrase the sampling

plan as follows:

A sample of not fewer than five units shall be

tested whenever possible. However, if fewer than

five units are produced in a period of 180 d, then

each unit produced in a period of 180 d shall be

tested. Compliance with a represented average efficiency is demonstrated provided:

The average efficiency of the sample is not

less than the represented efficiency minus an

expanded uncertainty [17].

The criterion for a demonstration on compliance can be

stated as a condition on the average efficiency of the sample

or as a condition on the average energy use of the sample. A

compliance demonstration can be performed on the basis of

energu use by comparing the measured average loss power

of a sample with the represented loss calculated from the

represented efficiency.

Criterion for efficiency testing: Stated in terms of efficiency, the condition for compliance demonstration is

X 

100

1 + 1 + 0

p:05n  

100

RE  1

 ; (2)

where X is the mean efficiency of a sample in percent, RE

is the represented efficiency of the basic model in percent,

and n is the number of units tested. Equation (2) is provided

in the proposed rule as the condition on the mean efficiency

of the sample required for a demonstration of compliance

with a represented efficiency.

In the process of determining compliance, the transformer

no-load and load losses are measured, the measurement data

are adjusted to reference conditions and loading levels, the

efficiency of each transformer and the average efficiency of

the sample are calculated, all according to the test procedure

proposed in the SNOPR. The resulting sample average efficiency is compared with the represented efficiency of the

standard using eq. (2) to determine the compliance status

of the basic model. Efficiency is used as the measure of

energy consumption of distribution transformers because it

combines the effects of both the no-load and load losses.

Criterion for energy use testing: The mean loss power of

the sample must not exceed the represented loss plus an expanded uncertainty. The expanded uncertainty proposed is

5 % of the represented loss scaled by a factor of 1=p

n where

n is the number of units in the sample. The condition on

the sample mean efficiency stipulated in the proposed rule

is equivalent to the following condition on the mean loss

power of the sample:

X

l  RL 

1 + 0:05

p

n 

; (3)

where X

l is the mean loss power of the sample in watts and

RL is the represented loss power of the basic model in watts.

Condition (3), expressed as a percentage of the represented

loss, is given by

X

l 



100 +

5

p

n

; (4)

where X

l is the normalized mean loss power of the sample

in percent.

5
NIST TN 1456 Operating Characteristics of the Proposed Sampling Plans

5.1 Statistical background

Finding a confidence interval on a mean and testing the

mean of a population from a sample are well known problems in statistical engineering [18].

The sampling plan for compliance testing is a test on a mean;

and an estimate of the mean loss power of the population is

obtained by a random sample

X

l =

1

n Xn

i=1

Xi ; (5)

where Xi is the measured loss power of unit i, and n is the

number of units tested. The uncertainty in this estimate depends on three factors: 1) the size of the sample, that is, the

number of transformers tested, 2) the underlying variability

in the population, and 3) the measurement uncertainty.

The condition for compliance with a represented loss is,

l  RL; (6)

where l is the average loss power of the population and RL

is the represented loss power. In a test of compliance with

a represented loss power, it is assumed by hypothesis that

the mean loss power of the population is not greater than the

represented loss power. This hypothesis is not rejected, provided the mean loss power of a random sample is not greater

than the represented loss plus an expanded uncertainty. In a

statistical test there is some probability of concluding falsely

that an acceptable product is not compliant or that an unacceptable product is compliant. The form of the expanded

uncertainty influences the statistical confidence, that is, the

probability that the mean loss power of the population actually satisfies condition (6).

Two cases are considered: 1) the variance of the population, l

2

, is unknown and must be estimated from the sample; and

2) the variance is known.

Unknown variance: When the variance is not known, the

population standard deviation, l , must be estimated from

the sample data. The sample standard deviation,

S = sPn

i=1(Xi  X

l)2

n  1

; (7)

provides an estimate of the population standard deviation;

and the standard error of the mean,

SE(X

l) = S

p

n

; (8)

provides an estimate of the standard deviation of the mean

for a sample of n units. For a normal distribution with mean

l ; the ratio,

t = X

l  l

SE(X

l)

; (9)

is distributed according to a probability density function that

is known in statistics literature as the t-distribution. t may

range in value from minus infinity to infinity, and the density

function, f (t), is defined such that

Z 1

1

f (t)dt = 1: (10)

The a quantile, ta, is defined such that the integral of density

from minus infinity to ta equals a, Z ta 1

f (t)dt = a (11) = P r[t  ta];

where P r[t  ta] is the probability that t will assume values

less than or equal to ta. (1  a) is then the probability that t

will assume values greater than ta, P r[t>ta] = (1  a).

Equation (9) may be rearranged to provide the following expression for the mean of the sample:

X

l = l + tSE(Xl); (12)

which suggests the condition,

X

l  RL + taSE(Xl); (13)

on the mean of the sample in a demonstration of compliance

with a represented loss power. The value of ta is associated with a specific sample size and statistical confidence a.

Values of ta are readily available and are included in many

references on statistics [19].

Known variance: The standard deviation of the sample

mean for a sample of n units is l=p

n, where l is the population standard deviation. The standardized random variable, z, is realized as the ratio,

z = X

l  l l=p

n

; (14)

where z may assume values from minus infinity to infinity

and is normally distributed with mean 0. The standard normal density function, (z), is defined such that

Z 1

1

(z)dz = 1: (15)

The a quantile, za, is defined such that the integral of standard normal density from minus infinity to za equals a, Z za 1

(z)dz = a (16) = P r[z  za];

6
for Testing Distribution Transformers NIST TN 1456

where P r[z  za] is the probability that z will assume values less than or equal to za. (1  a) is then the probability

that z will assume values greater than za, P r[z > za] =

(1  a).

Equation (14) may be rearranged to provide an expression

for the mean of the sample

X

l = l + z l

p

n

; (17)

which suggests the condition,

X

l  RL + za l

p

n

; (18)

on the mean of the sample for compliance with a represented

loss with confidence a.

Conditions (13) and (18) for the t-test and the z-test, respectively, are similar. Indeed, for n greater than 30, the two tests

yield essentially identical results. In a z-test the standard deviation of the population, l , is known from previous tests,

whereas in a t-test the standard deviation of the population

is estimated from the sample data. Because the variability is

estimated from the sample in the t-test, the confidence limits are broader. It should be noted that the sample standard

deviation (7) is not defined for n = 1. The value of t is

thus not defined, and the t-test cannot be used for samples

of one. The z-test satisfies a key criterion for testing distribution transformers: The z-test performs well when testing

small samples.

5.2 Operating characteristic

The operating characteristic is a function of the mean, l ,

and the standard deviation, l , of the loss power. Using

eq. (3), the operating characteristic is given by

p(l ; l) = P r X

l  RL(1 + 0:05

p

n

)

= P r X

l  l l=p

n



RL  l l=p

n

+

0:05RL

l 

=  RL  l l=p

n

+

0:05RL

l 

; (19)

where (z) is the cumulative distribution function of z.

The level curves of the operating characteristic function,

a = p(l ; l), 0 <a< 1, determine what sampling design

parameters l and l produce compliance with the stated

probability a.

The operating characteristics of the proposed sampling

plan for compliance testing are summarized graphically in

Figs. 1–3. The data shown in the figures are for an expanded

uncertainty of (5=p

n) % and for sample sizes of n = 1, 5

and 10, respectively. Level curves are plotted in the figures

for a = 0:999, 0.99, 0.90, 0.75, 0.50, 0.25, 0.10, 0.01, and

0.001, for l ranging from 95–110 % and l ranging from

0–6 %.

The level curves turn out to be straight lines, as can be seen

by noting that a = p(l ; l) implies

a =  RL  l l=p

n

+

0:05RL

l 

: (20)

The points (l ; l) that satisfy this equation are

za =

RL  l l=p

n

+

0:05RL

l

, or  =

RL  l

za=pn

+

0:05RL

za

: (21)

The represented loss power is normalized to 100 % in the

figures. Thus the region of compliance with a represented

loss power is for l  100 % and the region of noncompliance is for l > 100 %. Equation (3) implies that

a basic model manufactured with losses 5=p

n % above the

represented value, that is, l = 100 + 5=p

n %, will have

0.5 probability of demonstrating compliance for all standard

deviations. Indeed substituting l = RL(1+ 0:05=p

n) into

eq. (19) gives (0) = 0:50. Hence, the 0:50 contour is a vertical line intersecting the horizontal axis at 100 + 5=p

n %.

The other level curves converge to the same point on the horizontal axis, and the slope of each level curve is (p

n=za).

The value of the expanded uncertainty influences the operating characteristics. The operating characteristics for an expanded uncertainty of (8=p

n) % are shown in Figs. 4–6 for

comparison. The data shown are again for samples of n = 1,

5 and 10.

6 AEDM substantiation

The SNOPR contemplates the use of an Alternative Efficiency Determination Method (AEDM) to determine the

represented efficiency of a basic model. The SNOPR stipulates, however, that a manufacturer must substantiate the

accuracy and reliability of an AEDM before it can be used

as the basis for the represented efficiency. The criteria for

substantiation of an AEDM are provided in x432.12(a)(3) of

the proposed 10 CFR Part 432. The text of the proposal is

provided in Appendix B to this report.

An AEDM is substantiated by comparison of the predicted

loss power and the measured loss power. The SNOPR places

several conditions on this comparison: The predicted loss

power must be compared with the measured loss power for

not fewer than five basic models; not fewer than five units

7
NIST TN 1456 Operating Characteristics of the Proposed Sampling Plans

must be tested for each basic model that is selected for testing; and the predicted loss power must be within plus or minus 5 % of the mean loss power of the sample for each basic

model tested. Specific criteria are provided in x432.12(b)(1)

for the selection of basic models for testing that are intended

to provide a broad sample of the basic models manufactured.

The SNOPR also places a condition on the global behavior

of the AEDM: The predicted loss power expressed as a percentage of the mean loss power of the sample is calculated

for each basic model tested. The predicted loss power averaged over all basic models tested must not be less than 97 %

nor greater than 103 %.

7 Enforcement testing

The Sampling Plan for Enforcement Testing proposed for

distribution transformers is provided in Appendix B to Subpart B of Part 432. For the convenience of the reader, the text

of the proposed sampling plan is included in Appendix C.

The proposed sampling plan is similar to that provided in

10 CFR Part 430 and in Part 431. Part 430, however, contains standards for both efficiency and energy consumption

and the Sampling Plan for Enforcement Testing included

there is very general. Since, in the case of transformers, only

standards for efficiency are contemplated, the proposed sampling plan is simplified somewhat from the Part 430 plan to

include only efficiency testing.

The proposed Sampling Plan for Enforcement Testing is

based on a well established statistical method, which is due

to C. Stein [20], for obtaining a confidence interval on a

mean. A discussion of this procedure can be found in Bickel

and Doksum [18], for example. The procedure 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.

Since test results obtained during an enforcement action

may recommend that adverse actions be taken against a

manufacturer—such actions may include, for example, relabeling of specific products, the cessation of distribution

and sale of certain basic models, and/or the assessment of

fines—the risk to a manufacturer of a false determination

of non-compliance during enforcement testing is set, by design, to a negligible level. The proposed Sampling Plan for

Enforcement Testing is based on a 97.5 % statistical confidence, and thus the risk of a chance false determination of

non-compliance is not greater than 2.5 %.

7.1 Method

The discussion of the proposed Sampling Plan for Enforcement Testing is again presented in terms of loss power rather

than efficiency. The conditions provided here on loss power

are equivalent to the conditions on efficiency provided in the

proposed sampling plan.

An estimate of the mean loss power of the population, l , is

obtained by a random sample,

X

l =

1

n Xn

i=1

Xi ; (22)

where Xi is the measured loss power of unit i, and n is the

number of tests conducted.

The sample standard deviation,

S = sPn

i=1(Xi  X

l)2

n  1

; (23)

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

SE(X ) = S

p

n

; (24)

provides an estimate of the standard deviation of the mean

for samples of n tests. For a normal distribution with mean

l ; the ratio,

t = X

l  l

SE(X

l)

; (25)

is distributed according to a probability density function that

is known in the statistics literature as the t-distribution.

Equation (25) may be rearranged to provide an expression

for the mean of the sample as follows:

X

l = l + taSE(Xl); (26)

where the value of ta is associated with a specific sample

size and statistical confidence a. Values of ta are readily

available and are included in many references on statistics

[19].

In a test of compliance with a represented loss power, RL,

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

drawn from a population of transformers for which the mean

loss power is not greater than the represented loss power. If

ta is the 97.5 percentile of the t-distribution for n1 degrees

of freedom, then the probability of obtaining a mean sample

loss power, X

l , such that

X

l  RL + taSE(Xl) (27)

8
for Testing Distribution Transformers NIST TN 1456

is not less than 97.5 %. This procedure recommends an upper control limit,

UCL = RL + taSE(X

l): (28)

To apply this method, a random sample is tested and the

mean and standard error of the mean are calculated. Based

on the number of tests conducted and the desired confidence

interval, the appropriate t-value is selected and the upper

control limit is calculated. For example, 97.5 % confidence

and a sample size of five tests yields a t-value of 2.78 for four

degrees of freedom. Provided the mean loss power obtained

from the random sample is not greater than the upper control

limit, as defined by eq. (28), we may assert with confidence

of at least 97.5 % that the mean loss power of the population

is not greater than the represented value.

In a statistical test there is some probability of incorrectly

concluding that the product is not compliant. By design, the

probability that the mean loss power for a random sample

drawn from a compliant population of transformers would

fall above the upper control limit, hence, the risk of incorrectly concluding that the basic model is not compliant, is

not greater than 2.5 %.

The standard deviation of the population is estimated from a

random sample, and there is some probability that the sample standard deviation may be exceptionally large and that

the upper control limit may be set, by chance, to a high

value. This circumstance may be avoided by placing a tolerance on the standard error of the mean, SE(X

l). The proposed sampling plan imposes the following condition on the

standard error of the mean: The 97.5 % confidence interval

on the mean loss power must not exceed 5 %, that is,

taSE(X

l)  5; (29)

where ta is selected for a 97.5 % confidence interval and an

initial sample of n1 tests. An expression for the number of

units to be tested, n, is obtained from eqs. (24) and (29):

n 



taS

5 2

: (30)

The proposed sampling plan requires further testing if the

number of tests conducted initially, n1, is less than the minimum value of n from eq. (30).

As indicated earlier, some basic models of distribution transformers may be produced in limited numbers. Two features

of the proposed Sampling Plan for Enforcement Testing are

provided to allow for tests of small samples: 1) a sample

size discount is included in the calculation of the upper control limit, and 2) a minimum number of tests rather than a

minimum number of units tested is specified.

The sample size discount, SSD, in the proposed Sampling

Plan for Enforcement Testing is modeled after the sampling

plan for compliance testing. The sample size discount is

given by:

SSD = 100 + 5

pm

; (31)

where m is the number of units selected for testing. The

sampling plan recommends that 20 units be selected for testing, however, it allows testing when fewer than 20 units are

selected for testing. Indeed, the value of m may range between 1 and 20. The sample size discount is included in the

calculation of the upper control limit,

UCL = SSD + taSE(X

l): (32)

It should be noted that the sample size discount proposed

for purposes of enforcement testing is consistent with the

control limit (4) proposed for compliance testing.

The proposed sampling plan differentiates between the number of units selected for testing, m, and the number of tests

conducted, n. Criteria for establishing the number of tests

initially conducted are provided in x432.13(a)(5) through

(8). These criteria provide that a single unit would be tested

four times, two units would be tested twice each, three units

would be tested twice each, and four or more units would

be tested once each. The value of t selected is based on the

number of tests initially conducted rather than the number

of units initially tested.

7.2 Results

The operating characteristics of the proposed Sampling Plan

for Enforcement Testing are shown in Figs. 7 and 8. The

data shown in Fig. 7 are for a 97.5 % statistical confidence,

20 units selected for testing and five initial tests. Level

curves are plotted in the figures for a = 0:999, 0.99, 0.975,

0.90, 0.75, 0.50, 0.25, 0.10, 0.01, and 0.001, for l ranging

from 95–110 % and l ranging from 0–6 %. The results of

two computational methods are presented: numerical computations and Monte Carlo simulations [4]. The continuous

level lines indicate the numerical results and the results of

Monte Carlo calculations are indicated by the irregular lines.

The two methods yield consistent results. The data shown

in Fig. 8 are for the case where one unit has beeb selected

for testing, that is, m = 1. It may be noted that the 97.5 %

contour lies along the (100 + 5=pm) % loss line in each of

the figures, and that the risk of a false outcome is therefore

independent of the variance for (100 + 5=pm) % average

loss.

Data indicating the estimated testing burden, that is, the

average number of units tested, are shown in Fig. 9. The

data shown are obtained by Monte Carlo calculations for a

97.5 % statistical confidence, 20 units selected for testing,

and five initial tests. Level curves are plotted in the figure

9
NIST TN 1456 Operating Characteristics of the Proposed Sampling Plans

for n = 5, 6, 7, 8, and 9, for l ranging from 95–110 % and l ranging from 0–6 %. These data suggest that six tests

would likely be required during an enforcement action.

8 Concluding remarks

8.1 Compliance testing

The sampling plan for compliance testing contained in the

earlier NOPR [2] was modeled after examples provided in

10 CFR Part 430. That sampling plan required the average

efficiency of a sample to be not less than the represented

value and also placed a condition on the lower confidence

limit of the efficiency. The probability of demonstrating

compliance under such a plan is thus not greater than 0.5

when testing a product having a population mean efficiency

that is equal to the rated value. The sampling plan contained

in the earlier NOPR provides very high assurance that the

product meets or exceeds the represented efficiency: under

that plan, in order to provide a reasonable likelihood of succeeding during a test of compliance with a represented efficiency the energy performance of the product must exceed

the represented value. Comments were made during the

public hearing [15] to suggest that distribution transformers are manufactured to meet the rated efficiency on average and that significant over design is inconsistent with industry practice. The sampling plan for compliance testing

contained the SNOPR is designed to provide a higher probability of demonstrating compliance than afforded under the

earlier NOPR when a product with energy performance at or

near the represented value is tested.

The operating characteristics of the proposed sampling plan

depend on the numerical value and the functional form of

the expanded uncertainty. The specific numerical value of

(5=pn) % for the expanded uncertainty provides a reasonable likelihood that a product with average energy performance at the represented value would succeed during a test

of compliance and yet not afford a significant probability of

success when the product is not compliant. Although the

sampling plan for compliance testing contained the SNOPR

provides a higher probability of demonstrating compliance

than afforded under the earlier NOPR, the proposed sampling plan still encourages average performance over and

above the represented efficiency.

The proposed sampling plan allows testing of large and

small samples, including a sample of one unit. The standard

deviation of the mean of a sample scales as 1=p

n, where n

is the size of the sample. The scaling factor 1=pn included

in the expanded uncertainty thus removes any bias due to

sample size. For products manufactured at the represented

value, the probability of demonstrating compliance is independent of sample size.

8.2 Enforcement testing

The Sampling Plan for Enforcement Testing proposed for

10 CFR Part 432 is based on statistical methods that are

widely used and well documented. The factors that influence the operating characteristics include: 1) the size of the

initial sample, 2) the statistical confidence, and 3) the tolerance set on the standard error. The sampling plan is robust, in that it is a test on the mean and that it is not highly

dependent on the exact 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 an acceptable level.

9 Further information

For information on the test procedure rulemaking for distribution transformers or on EPCA legislation contact:

Cyrus Nasseri

U.S. Department of Energy

Energy Efficiency and Renewable Energy

Building Technologies

Mail Station EE-2J

1000 Independence Avenue, SW

Washington, DC 20585-0121

202-586-9138

Cyrus.Nasseri@ee.doe.gov

For information on this report contact:

Oskars Petersons

National Institute of Standards and Technology

100 Bureau Drive, MS-8172

Gaithersburg, MD 20899-8172

oskars.petersons@nist.gov

Ken Stricklett

National Institute of Standards and Technology

100 Bureau Drive, MS-8172

Gaithersburg, MD 20899-8172

301-975-3955

ken.stricklett@nist.gov

Charles Hagwood

National Institute of Standards and Technology

100 Bureau Drive, MS-8980

Gaithersburg, MD 20899-8980

301-975-2846

charles.hagwood@nist.gov

10
for Testing Distribution Transformers NIST TN 1456

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.

References

[1] Energy conservation program: Test procedures for distribution transformers. Federal Register, in press, 2004.

[2] Energy conservation program: Test procedures for distribution transformers. Federal Register, 63(218):63359–63372,

Thursday, November 12, 1998.

[3] Energy conservation program for consumer products. 10

Code of Federal Regulations Part 430, Chapter II, Office of

the Federal Register, National Archives and Records Administration, Washington, DC, 1997. pp 93–263.

[4] K. L. Stricklett, M. Vangel, and O. Petersons. An analysis of

efficiency testing under the Energy Policy and Conservation

Act: A case study with application to distribution transformers. NIST Technial Note 1427, National Institute of Standards and Technology, Technology Administration, U.S. Department of Commerce, April 1999.

[5] Energy Policy and Conservation Act of 1975, Public Law

94-163, as amended by 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.

[6] United States Code, Title 42—The Public Health and Welfare, Chapter 77—Energy Conservation, Subchapter III—

Improving Energy Efficiency, Part A-1—Certain Industrial

Equipment, 2000 Edition. United States Government Printing Office, Washington.

[7] NEMA Standards Publication TP 1-2002, Guide for determining energy efficiency for distribution transformers. National Electrical Manufacturers Association, Rosslyn, VA,

2002.

[8] NEMA Standards Publication TP 2-1998, Standard test

method for measuring the energy comsumption of distribution transformers. National Electrical Manufacturers Association, Rosslyn, VA, 1998.

[9] C57.12.00-2000 IEEE Standard General requirements for

liquid-immersed distribution, power and regulating transformers. Institute of Electrical and Electronics Engineers,

New York, 2000.

[10] C57.12.01-1998 IEEE Standard General requirements for

dry-type distribution and power transformers, including those

with solid case and/or resin-encapsulated windings. Institute

of Electrical and Electronics Engineers, New York, 1998.

[11] P. R. Bevington and D. K. Robinson. Data Reduction and

Error Analysis for the Physical Sciences, chapter 5. McGrawHill, 1992.

[12] K. L. Stricklett and M. G. Vangel. Electric motor efficiency

testing under the new part 431 of 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.

[13] 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.

[14] M. G. Vangel. The joint distribution of a normal mean and

extremum with applications to quality control. In Proceedings of the Section on Physical and Engineering Sciences.

American Statistical Association, 1998.

[15] Public hearing on energy efficiency test procedures distribution transformers. U.S. Department of Energy, Wednesday,

January 6, 1999. Transcript of the Public Hearing.

[16] Anthony Balducci. Comments on the Proposed Rule for Test

Procedures for Distribution Transformers. U.S. Department

of Energy, Docket No. EE-TP98-550, July 23, 1999. Letter

submitted by the Transformer Section of the National Electrical Manufacturers Association (NEMA).

[17] Barry N. Taylor and Chris E. Kuyatt. Guidelines for evaluating and expressing the uncertainty of NIST measurement

results. NIST Technial Note 1297, National Institute of Standards and Technology, Technology Administration, U.S. Department of Commerce, January 1993.

[18] P. J. Bickel and K. A. Doksum. Mathematical Statistics.

Holden Day, Oakland, CA, 1977.

[19] R. S. Schulman. Statistics in Plain English with Computer

Applications. Van Norstand Reinhold, New York, 1992.

[20] C. Stein. A two-sample test for a linear hypothesis whose

power is independent of variance. Annals of Mathematical

Statistics, 16:243–258, 1945.

11
NIST TN 1456 Operating Characteristics of the Proposed Sampling Plans

110 108 106 104 102 100 98 96

1

5

3

0

2

4

6

Standard Deviation

(Percent of Represented Loss)

Loss Power (Percent of Represented)

Increasing Efficiency

.001

.01

.05

.10

.25

.50 .75 .90

.95

.99

.999

n=1

Figure 1: The operating characteristics of the sampling plan for compliance testing for an expanded uncertainty (5=pn) % and a sample

of one. The normalized loss of the basic model is indicated on the horizontal axis and the normalized standard deviation of the losses is

indicated on the vertical axis. The contours indicate the probability of demonstrating compliance, e.g., the 0.90 contour corresponds to

a 90 % likelihood of demonstrating compliance. The represented loss is indicated by the vertical broken line at 100 %. The horizontal

band from 2.7–4.0 % indicates the range of standard deviations likely to be encountered for distribution transformers.

12
for Testing Distribution Transformers NIST TN 1456

0

2

4

6

110 108 106 104 102 100 98 96

1

5

3

Standard Deviation

(Percent of Represented Loss)

Loss Power (Percent of Represented)

Increasing Efficiency

.001

.01

.05

.10

.25 .50 .75 .90

.95

.99

.999

n=5

Figure 2: The operating characteristics of the sampling plan for compliance testing for an expanded uncertainty (5=pn) % for a sample

of n = 5 units. The normalized loss of the basic model is indicated on the horizontal axis and the normalized standard deviation of

the losses is indicated on the vertical axis. The contours indicate the probability of demonstrating compliance, e.g., the 0.90 contour

corresponds to a 90 % likelihood of demonstrating compliance. The represented loss is indicated by the vertical broken line at 100 %.

The horizontal band from 2.7–4.0 % indicates the range of standard deviations likely to be encountered for distribution transformers.

13
NIST TN 1456 Operating Characteristics of the Proposed Sampling Plans

0

2

4

6

110 108 106 104 102 100 98 96

1

5

3

Standard Deviation

(Percent of Represented Loss)

Loss Power (Percent of Represented)

Increasing Efficiency

.001

.01

.05 .10 .25 .50 .75 .90 .95 .99

.999

n = 10

Figure 3: The operating characteristics of the sampling plan for compliance testing for an expanded uncertainty (5=pn) % for a sample

of n = 10 units. The normalized loss of the basic model is indicated on the horizontal axis and the normalized standard deviation of

the losses is indicated on the vertical axis. The contours indicate the probability of demonstrating compliance, e.g., the 0.90 contour

corresponds to a 90 % likelihood of demonstrating compliance. The represented loss is indicated by the vertical broken line at 100 %.

The horizontal band from 2.7–4.0 % indicates the range of standard deviations likely to be encountered for distribution transformers.

14
for Testing Distribution Transformers NIST TN 1456

0

2

4

6

110 108 106 104 102 100 98 96

1

3

5

Standard Deviation

(Percent of Represented Loss)

Loss Power (Percent of Represented)

Increasing Efficiency

.001

.01

.05

.10

.25

.50 .75 .90

.95

.99

.999

n=1

Figure 4: The operating characteristics of the sampling plan for compliance testing for an expanded uncertainty (8=pn) % for a sample

of n = 1 units. The normalized loss of the basic model is indicated on the horizontal axis and the normalized standard deviation of

the losses is indicated on the vertical axis. The contours indicate the probability of demonstrating compliance, e.g., the 0.90 contour

corresponds to a 90 % likelihood of demonstrating compliance. The represented loss is indicated by the vertical broken line at 100 %.

The horizontal band from 2.7–4.0 % indicates the range of standard deviations likely to be encountered for distribution transformers.

15
NIST TN 1456 Operating Characteristics of the Proposed Sampling Plans

0

2

4

6

110 108 106 104 102 100 98 96

1

5

3

Standard Deviation

(Percent of Represented Loss)

Loss Power (Percent of Represented)

Increasing Efficiency

.001

.01

.05

.10 .25 .50 .75 .90

.95

.99

.999

n=5

Figure 5: The operating characteristics of the sampling plan for compliance testing for an expanded uncertainty (8=pn) % for a sample

of n = 5 units. The normalized loss of the basic model is indicated on the horizontal axis and the normalized standard deviation of

the losses is indicated on the vertical axis. The contours indicate the probability of demonstrating compliance, e.g., the 0.90 contour

corresponds to a 90 % likelihood of demonstrating compliance. The represented loss is indicated by the vertical broken line at 100 %.

The horizontal band from 2.7–4.0 % indicates the range of standard deviations likely to be encountered for distribution transformers.

16
for Testing Distribution Transformers NIST TN 1456

0

2

4

6

110 108 106 104 102 100 98 96

1

5

3

Standard Deviation

(Percent of Represented Loss)

Loss Power (Percent of Represented)

Increasing Efficiency

.001

.01

.05 .10 .25 .50 .75 .90 .95 .99

.999

n = 10

Figure 6: The operating characteristics of the sampling plan for compliance testing for an expanded uncertainty (8=pn) % for a sample

of n = 10 units. The normalized loss of the basic model is indicated on the horizontal axis and the normalized standard deviation of

the losses is indicated on the vertical axis. The contours indicate the probability of demonstrating compliance, e.g., the 0.90 contour

corresponds to a 90 % likelihood of demonstrating compliance. The represented loss is indicated by the vertical broken line at 100 %.

The horizontal band from 2.7–4.0 % indicates the range of standard deviations likely to be encountered for distribution transformers.

17
NIST TN 1456 Operating Characteristics of the Proposed Sampling Plans

0

2

4

6

110 108 106 104 102 100 98 96

1

5

3

Standard Deviation

(Percent of Represented Loss)

Loss Power (Percent of Represented)

Increasing Efficiency

.001

.01

.05

.10

.25

.50

.75

.90 .95 .99

.999

m = 20

.975

Figure 7: The operating characteristics of the Sampling Plan for Enforcement Testing for m = 20. The normalized loss is indicated

on the horizontal axis and the normalized standard deviation of the losses is indicated on the vertical axis. The contours indicate the

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

represented loss is indicated by the vertical broken line at 100 %. The horizontal band from 2.7–4.0 % indicates the range of standard

deviations likely to be encountered for distribution transformers.

18
for Testing Distribution Transformers NIST TN 1456

0

2

4

6

110 108 106 104 102 100 98 96

1

5

3

Standard Deviation

(Percent of Represented Loss)

Loss Power (Percent of Represented)

Increasing Efficiency

.75

.90

.95 .99

.999

m=1

.975

Figure 8: The operating characteristics of the Sampling Plan for Enforcement Testing for m = 1. The normalized loss is indicated

on the horizontal axis and the normalized standard deviation of the losses is indicated on the vertical axis. The contours indicate the

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

represented loss is indicated by the vertical broken line at 100 %. The horizontal band from 2.7–4.0 % indicates the range of standard

deviations likely to be encountered for distribution transformers.

19
NIST TN 1456 Operating Characteristics of the Proposed Sampling Plans

0

2

4

6

110 108 106 104 102 100 98 96

1

5

3

Standard Deviation

(Percent of Represented Loss)

Loss Power (Percent of Represented)

Increasing Efficiency

7

5

6

8

9

Figure 9: The average number of units tested under the Sampling Plan for Enforcement Testing. The normalized loss is indicated

on the horizontal axis and the normalized standard deviation of the losses is indicated on the vertical axis. The contours indicate the

average number of units tested, e.g., the 5 contour corresponds to 5 units tested. The represented loss is indicated by the vertical broken

line at 100 %. The horizontal band from 2.7–4.0 % indicates the range of standard deviations likely to be encountered for distribution

transformers.

20
Appendix A

Compliance testing

The sampling plan for compliance testing appears in x432.12(b)(2) of the proposed 10 CFR Part 432:

(2) Selection of units for testing within a basic model. For each basic model a manufacturer selects for testing, it shall select a sample of units at random and test them. The sample shall be comprised of production units of the basic model, or units that are representative of such production units. The sample

size shall be not fewer than five units, except that when the manufacturer would produce fewer than five

units of a basic model over a reasonable period of time (approximately 180 days), then it must test each

unit. However, a manufacturer may not use a basic model with a sample size of fewer than five units

to substantiate or verify an AEDM pursuant to paragraphs (a)(3) or (a)(4) of this section. In a test of

compliance with a represented efficiency:

The average efficiency of the sample, X , which is defined by

X =

1

n Xn

i=1

Xi

where Xi is the measured efficiency of unit i and n is the number of units tested, shall satisfy the

condition:

X 

100

1 + 1 + 0

p:05n  

100

RE  1



where RE is the represented efficiency.

21
Appendix B

Substantiation of an AEDM

The criteria for substantiation of an Alternative Efficiency Determination Method (AEDM) are provided in x432.12(a)(3)

of the proposed 10 CFR Part 432:

(3) Substantiation of an alternative efficiency determination method.Before using an AEDM, the manufacturer must substantiate the AEDM’s accuracy and reliability as follows:

(i) Apply the AEDM to at least five of the manufacturer’s basic models that have been selected

for testing in accordance with paragraph (b)(1) of this section, and calculate the power loss for

each of these basic models;

(ii) Test at least five units of each of these basic models in accordance with the applicable test

procedure and paragraph (b)(2) of this section, and determine the power loss for each of these

basic models;

(iii) The predicted total power loss for each of these basic models, calculated by applying the AEDM

pursuant to paragraph (a)(3)(i) of this section, must be within plus or minus five percent of the

mean total power loss determined from the testing of that basic model pursuant to paragraph

(a)(3)(ii) of this section; and

(iv) Calculate for each of these basic models the percentage that its power loss calculated pursuant to

paragraph (a)(3)(i) is of its power loss determined from testing pursuant to paragraph (a)(3)(ii),

compute the average of these percentages, and that calculated average power loss, expressed as

a percentage of the average power loss determined from testing, must be no less than 97 percent

and no greater than 103 percent.

22
Appendix C

Enforcement testing

Appendix B to Subpart B of Part 432—Sampling Plan for Enforcement Testing

Step 1. The number of units in the sample (m1) shall be in accordance with xx432.13(a)(4), 432.13(a)(5), 432.13(a)(6)

and 432.13(a)(7) and shall not be greater than twenty. The number of tests in the first sample (n1) shall be in

accordance with x432.13(a)(8) and shall be not fewer than four.

Step 2. Compute the mean (X

1) of the measured energy performance of the n1 tests in the first sample by using

equation 1 as follows:

X

1 =

1

n1 Xn1

i=1

Xi ; (1)

where Xi is the measured efficiency of test i.

Step 3. Compute the sample standard deviation (S1) of the measured efficiency of the n1 tests in the first sample by

using equation 2 as follows:

S1 = sPn1

i=1(Xi  X

1)2

n1  1

: (2)

Step 4. Compute the standard error (SE(X

1)) of the mean efficiency of the first sample by using equation 3 as

follows:

SE(X

1) = S1

p

n1

: (3)

Step 5. Compute the sample size discount (SSD(m1)) by using equation 4 as follows:

SSD(m1) = 100

1 + 1 + p:05m1 



100

RE  1



(4)

where m1 is the number of units in the sample, and RE is the applicable EPCA efficiency when the test is

to determine compliance with the applicable statutory standard, or is the labeled efficiency when the test is to

determine compliance with the labeled efficiency value.

Step 6. Compute the lower control limit (LCL1) for the mean of the first sample by using equation 5 as follows:

LCL1 = SSD(m1)  tSE(X

1) (5)

where t is the 2.5 percentile of a t-distribution for a sample size of m1, which yields a 97.5 percent confidence

level for a one-tailed t-test.

Step 7. Compare the mean of the first sample (X

1) with the lower control limit (LCL1) 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 non-compliance

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

non-compliance can be made; proceed to Step 8.

23
NIST TN 1456 Operating Characteristics of the Proposed Sampling Plans

Step 8. Determine the recommended sample size (n) by using equation 6 as follows:

n =



tS1(105  0:05RE)

RE(5  0:05RE) 2

(6)

where S1 and t have the values used in Steps 3 and 6, respectively. The factor

105  0:05RE

RE(5  0:05RE)

is based on a 5 percent tolerance in the total power loss.

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

(i) If the value of n is less than or equal to n1 and if the mean energy efficiency of the first sample (X

1) is

equal to or greater than the lower control limit (LCL1), the basic model is in compliance and testing is

at an end.

(ii) If the value of n is greater than n1, and no additional units are available for testing, testing is at an end

and the basic model is in non-compliance. If the value of n is greater than n1, and additional units are

available for testing, select a second sample n2. The size of a second sample n2 is determined to be the

smallest integer equal to or greater than the difference n  n1. If the value of n2 so calculated is greater

than 20  n1, set n2 equal to 20  n1.

Step 9. After testing the n2 sample, compute the combined mean (X

2) of the measured energy performance of the n1

and n2 units of the combined first and second samples by using equation 7 as follows:

X

2 =

1

n1 + n2 nX1+n2

i=1

Xi : (7)

Step 10. Compute the standard error (SE(X

2)) of the mean efficiency of the n1 and n2 tests in the combined first and

second samples by using equation 8 as follows:

SE(X

2) = S1

p

n1 + n2

: (8)

(Note that S1 is the value obtained above in Step 3.)

Step 11. Set the lower control limit (LCL2) to,

LCL2 = SSD(m1)  tSE(X

2); (9)

where t has the value obtained in Step 5 and SSD(m1) is sample size discount from Step 5. Compare the

combined sample mean (X

2) to the lower control limit (LCL2) to find one of the following:

(i) If the mean of the combined sample (X

2) is less than the lower control limit (LCL2), the basic model is

in non-compliance and testing is at an end.

(ii) If the mean of the combined sample (X

2) is equal to or greater than the lower control limit (LCL2), 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.

24
for Testing Distribution Transformers NIST TN 1456

Step A. The manufacturer requests that an additional number, n3, of units be tested, with n3 chosen such that n1 +

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 (LCL2) 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 n1 + 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 non-compliance.

25