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NIST Technical Note 1432

Test Procedures for Electric Motors Under 10 CFR

Part 431

K. L Stricklett and M. Vangel

National Institute of Standards and Technology • Technology Administration • U.S. Department of Commerce
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NIST Technical Note 1432

Test Procedures for Electric Motors Under 10 CFR

Part 431

Supersedes NIST Technical Note 1422

K.L. Stricklett

Electricity Division

Electronics and Electrical Engineering Laboratory

National Institute of Standards and Technology

Gaithersburg, MD 20899-811

3

M. Vangel

Statistical Engineering Division

Information Technology Laboratory

National Institute of Standards and Technology

Gaithersburg, MD 20899-8980

June 2000

«ft \ T .<f

U.S. Department of Commerce

William M. Daley, Secretary

Technology Administration

Dr. Cheryl L. Shavers, Under Secretary ofCommercefor Technology

National Institute of Standards and Technology

Raymond G. Kammer. Director
National Institute of Standards U.S. Government Printing Office For Sale by the Superintendent of and Technology Washington: 2000 Documents

Technical Note 1432 U.S. Government Printing Office

Natl. Inst. Stand. Technol. Washington, DC 20402-9325

Tech. Note 1432

20 pages (June 2000)

CODEN: NTNOEF
Bibliographic Information

Abstract

The procedures for electric motor efficiency testing promulgated by Part 431 of Chapter II of Title 10, Code

of Federal Regulations, are discussed. The operating characteristics of the sampling plans for certification and enforcement testing are presented together with examples of the application of the sampling plans. The

criteria for substantiation of an Alternative Energy Determination Method (AEDM) are also discussed.

Keywords

electric motor efficiency; electric motor testing; energy policy; operating characteristics; 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.

Ill

Contents

1 Introduction 1

2 General guidelines 1

3 Industry practice 2

4 Methods of analysis 2

5 Compliance certification 2

5.1 Examples of compliance testing 3

6 AEDM substantiation 4

7 Enforcement testing 5

7.1 Method 6

7.2 Enforcement testing example 8

8 Further information 9

Appendix A: Exerpts from 10 CFR Part 431 11

A.l Certification testing 11

A. 2 AEDM substantiation 11

A. 3 The Sampling Plan for Enforcement Testing 12
List of Figures

1. The operating characteristics of the sampling plan for certification of compliance 4

2. The operating characteristics of the Sampling Plan for Enforcement Testing 6

VI
List of Tables

1. The NEMA Nominal and Minimum Efficiencies 3

2. t coefficients for specified confidence 4

3. An abridged f-table 9

Vll

Test Procedures for Electric Motors Under 10 CFR Part 431

1 Introduction

This note provides analysis of the procedures for ef- ficiency testing of polyphase electric motors promulgated by Part 431 of the Code of Federal Regulations

[1]. It is intended to supplement the materials published in the Federal Register [1, 2], and to specifically

address issues regarding the performance of these

sampling plans in establishing conformance with the

minimum nominal efficiencies mandated by the Energy PoUcy and Conservation Act of 1975 (EPCA),

as amended [3]. Part 431 will be referred to herein

as the Final Rule. In the context of the Final Rule,

laboratory measurements of motor efficiency are used

for three purposes: 1) certification of efficiency per- formance; 2) substantiation of an Alternative Effi- ciency Determination Method (AEDM); and 3) en- forcement testing. This note discusses each of these

topics in turn and thus supersedes NIST Technical

Note 1422 [4], which dealt solely with enforcement

testing.

The remainder of this document is organized as fol- lows: Section 2 discusses the general objectives and

constraints for testing under EPCA; Section 3 dis- cusses the guidelines for motor efficiency labeling es- tabhshed by NEMA Standard MG 1-1993; Section 4

discusses the model assumptions used in calculat- ing the operating characteristics; Section 5 discusses

the operating characteristics of the sampling plan for certification testing; Section 6 discusses substantiation of an Alternate Energy Determination Method

(AEDM); Section 7 discusses the operating characteristics of the Sampling Plan for Enforcement Testing. For the convenience of the reader, relevant portions of the Final Rule are provided in Appendix A.

2 General guidelines

We begin with a brief summary of the general objectives and guidelines for testing under the EPCA

legislation. A general statement of purpose of the

EPCA legislation is provided 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.

To this end, EPCA establishes energy efficiency stan- dards, i.e., a set of minimum nominal efficiencies, for certain general purpose electric motors. The EPCA legislation establishes that a program of

systematic testing be used to demonstrate that en- ergy efficiency standards are met. The objectives and

limitations of testing under EPCA are stated in 42

U.S.C. 6314(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 burden- some to conduct.

The EPCA legislation, in 42 U.S.C. 6314(d)(1), fur- ther requires that the represented energy efficiency

be based on product testing:

... no manufacturer, distributor, retailer or private labeler may make any

representation

(A) in writing (including any 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.

To re-cap, the purposes of EPCA are met provided:

1) the average energy efficiency of each covered product is not less than the applicable EPCA efficiency

standard, and 2) the average energy efficiency of esuJi covered product is not less thaji the represented en- ergy efficiency. Compliance with EPCA energy effi- ciency standards and with the represented energy ef- ficiency is demonstrated by a program of systematic

testing. EPCA stipulates that testing should not be

unduly burdensome to conduct. Thus the two key cri- teria for the evaluation of a sampling plan established

under EPCA are: 1) the assurance provided by that
5 COMPLIANCE CERTIFICATION

plan that the average performance of that product

meets or exceeds the EPCA standard efficiency and

the represented efficiency, and 2) the burden placed on industry by testing under the plan.

3 Industry practice

Industry guidelines for efficiency labeling are estabUshed by NEMA Standard MG 1-1993 [5]. Table 12-8

of this standard establishes a set of "Nominal Effi- ciencies" that are to be used for purposes of labeling

and a "Minimum Efficiency" that is associated with

each Nominal Efficiency. The NEMA Nominal Effi- ciencies as well as the EPCA nominal efficiencies are

listed in Table 1. It should be noted that the EPCA

standard values were adapted from the NEMA stan- dard. Under §12.58.2 of the NEMA Standard, two

conditions must be satisfied for a motor to be labeled

with a given Nominal Efficiency:

1. " ... the Nominal Efficiency . . . shall be not

greater than the average efficiency of a large population of motors of the same design."

2. "The full-load efficiency . . . shall be not less than

the minimum value . . . associated with the nominal value. ..."

Under the Final Rule, measurement of motor effi- ciencies is based on two standards: 1) IEEE Stan- dard 112-1996 [6] and 2) CSA-390 [7]. The reader

is referred to these standards and to the Final Rule

for a discussion of efficiency measurements for electric motors. When under these standards as directed

by the Final Rule the determination of efficiency is based on two measured quantities: 1) a measurement

of the total losses, and 2) measurement of the output

power. The NEMA Standard establishes efficiency

levels that differ by increments of approximately 10%

of the full-load losses. The Minimum Efficiencies cor- respond to approximately 120% of the rated loss.

4 Methods of analysis

The sampling plans established by the Final Rule are examined here by means of model calculations to predict their operating characteristics, where the operating characteristics of a sampling are the estimated

probabilities of being found in compliance when test- ing a specified population of motors. Calculation of

the operating characteristics of these sampling plans

relies on numerical methods of approximation. A dis- cussion of these algorithms can be found in separate

reports [4, 8]. Detailed information regarding the dis- tribution of motor efficiencies is required to model the

operating characteristics. Following methods used

by the NEMA, Motors and Generators Section [9], we assume that motor efficiencies are normally dis- tributed. The distribution of efficiencies of units of a

basic model is thus characterized by two parameters:

the true mean, /x, and the standard deviation, a, of

the population.

The operating characteristics are given below in terms of the total losses. This method of presentation

has the advantage that the full range of the Nominal

Efficiencies can be displayed in a single plot. For this presentation, we define the Loss Fraction, LF, by the

ratio. LF

100 - RE

100 - TE

X 100, (1)

where RE is the rated efficiency and TE is the true

efficiency of the population. So stated, the loss frac- tion is a percentage of the rated full-load losses such

that 100% corresponds to the case where the true ef- ficiency of the population and the rated efficiency are

equal.

The open-ended nature of testing under the Final

Rule introduces a complication. Testing under the

Final Rule is open-ended in that the number of motors tested is not fixed from the outset of testing: In the case of certification testing, a manufacturer could

test as few as five motors, but may test any arbitrarily

large number of motors; likewise the Sampling Plan

for Enforcement Testing specifies an initial sample

of five but allows testing of up to 20 motors. This

scenario is difficult to characterize statistically, and we have chosen to treat testing in the approximation

that the sample size is fixed from the outset.

5 Compliance certification

The full text of the sampling plan for compliance certification may be found in Appendix A of this report.

To emphasize the salient features of the sampling

plan, we paraphrase these criteria as follows: Compliance with a rated efficiency is demonstrated

provided:

(A) The average full-load efficiency of a sample of not fewer than five motors is not less than the

value given by the following expression,

100

1 + 1-05(^-1)'

where RE is the rated efficiency, and

(2)
5 COMPLIANCE CERTIFICATION

Table 1: The NEMA Nominal and Minimum Efficiencies. The EPCA nominal efficiencies, i.e., motor efficiencies that are specified explicitly as minimum nominal efficiencies, are shown in bold. This table is adapted from Table 12-8

of NEMA Standard MG 1-1993 [5].

Nominal Efficiency Minimum Efficiency Nominal Efficiency Minimum Efficiency

99.0 98.8 94.1 93.0

98.9 98.7 93.6 92.4

98.8 98.6 93.0 91.7

98.7 98.5 92.4 91.0

98.6 98.4 91.7 90.2

98.5 98.2 91.0 89.5

98.4 98.0 90.2 88.5

98.2 97.8 89.5 87.5

98.0 97.6 88.5 86.5

97.8 97.4 87.5 85.5

97.6 97.1 86.5 84.0

97.4 96.8 85.5 82.5

97.1 96.5 84.0 81.5

96.8 96.2 82.5 80.0

96.5 95.8 81.5 78.5

96.2 95.4 80.0 77.0

95.8 95.0 78.5 75.5 95.4 94.5 77.0 74.0

95.0 94.1 75.5 72.0

94.5 93.6

(B) No individual motor in the sample shall have

full-load efficiency less than the value given by

the following expression,

100

1 + 1.15(^-1)'

where RE is the rated efficiency.

(3)

The operating characteristics of this sampling plan

are shown in Figure 1. The data shown depict the

outcome when testing a sample of five motors. The

contours shown in the figure correspond to equal

probabilities of demonstrating compliance. For ex- ample, if the average efficiency of a basic model is equal to the rated efficiency, i.e., LF = 100%, and

the standard deviation of the total losses is approximately 6% of the total losses, then the probability of demonstrating compliance with the rated efficiency

is approximately 95%. The risk of a false determination of non-compliance is thus approximately 5%, in this case.

5.1 Examples of compliance testing

Several hypothetical cases of testing under this sampling plan are presented. Assume for this discussion

that a basic model is being tested to demonstrate

compliance with a rated efficiency of 89.5%. The condition on the sample mean is obtained by using Eq. 2, which, for a rated efficiency of 89.5, yields 89.0. The

condition on the sample minimum efficiency is obtained by using Eq. 3, which, for a rated efficiency of

89.5, yields 88.1.

A demonstration of compliance: Five motors

are selected at random from a representative population of motors and tested. The results of testing

yield efficiencies of 89.9, 89.2, 89.0, 89.3, and 89.4. The mean efficiency of the sample and the sample

minimum efficiency are thus 89.4 and 89.0, respectively. Since the sample mean is greater than 89.0

and the least efficient motor has an efficiency that is greater than 88.1, the criteria for a demonstration of

compliance are satisfied and the manufacturer may

represent the basic model to be 89.5% efficient.

Non-compliance due to a low mean: Five motors are selected at random from a representative

population of motors and tested. The test results yield efficiencies of 88.9, 88.8, 88.6, 89.0, and 89.1. The mean efficiency for the saxnple and the sample

minimum efficiency are thus 88.9 and 88.6, respectively. Since the sample mean is less than 89.0, the
6 AEDM SUBSTANTIATION

Increasing Efficiency

c

(DO

Q)Q.

C

o

'—

•

CO >

QQ

CO T3

c

CO CO

Loss Fraction (percent)

Figure 1: The operating chaxacteristics of the samphng plan for certification of compliance. The contours indicate the probability of demonstrating compliance, e.g., the 0.900 contour corresponds to a 90% likelihood of demonstrating

compliance while testing under the sampling plcin. The model calculations shown axe for a sample of five.

criteria for a demonstration of compliance are not

satisfied; the manufacturer may not represent the basic model to have a rated efficiency of 89.5. The manufacturer may elect to conduct further testing; however the initial test results must be included when

computing the new sample mean and sample minimum. Assume that the manufacturer elects to test two additional motors and that the new test results yield efficiencies of 89.3 and 89.5. Since the sample mean for the entire sample of 7 motors is equal to 89.0 and no unit in the sample tested below the minimum efficiency, the basic model is determined to be

in compliance with a rated efficiency of 89.5.

rated efficiency of 89.5 has not been demonstrated.

Table 2: t coefficients for specified confidence."

t Statistical Probability of

Confidence Exceeding E

3 99.7 0.003

2.56 99 0.010

2 95.5 0.045

1.96 95 0.050

1.64 90 0.100

"Adapted firom ASTM Std E 122-89

Non-compliance due to a low minimum: Five

motors are selected at random from a representative

population of motors and tested. The test results yield efficiencies of 89.9, 89.2, 88.0, 89.3, 89.4, and

89.9. The mean efficiency for the sample and the

sample minimum efficiency are thus 89.2 and 88.0,

respectively. Since the sample minimum is less than

88.1 the basic model is not in compliance; the manufacturer may not represent the basic model to have

a rated efficiency of 89.5. Since the 88.0 test result may not be excluded, further testing will serve no

purpose and testing is at an end. Compliance with a

6 AEDM substantiation

Per Subpart B §431.24(a)(3) of the Final Rule, laboratory efficiency measurements may also provide the

basis for substantiation of an Alternative Efficiency

Determination Method (AEDM); i.e., a mathematical model based on engineering or statistical analysis,

computer simulation or modeling, or other analytic

evaluation of performance data that may be used to

assign motor efficiency.
7 ENFORCEMENT TESTING

To emphasize the sahent features of the Final Rule we paraphrase §431. 24(a) (3) as follows: An AEDM is substantiated provided:

7 Enforcement testing

1. At least five different basic models are tested ac- cording to the procedures established for certifi- cation testing, and

2. The total power loss calculated by the AEDM

is within the interval of ±10% of the mean total power loss determined from the actual testing

for each of the basic models tested.

The scenario described, in which testing is required

to conform to a predetermined precision is addressed

by ASTM Standard E 122-89 [10]. This standard

is based on the t statistic and establishes the sample size required to determine a two-sided confidence

interval on the estimate of the mean. The follow- ing discussion provides an estimate of the number of

tests needed to support the ±10% precision required

by the Final Rule.

Following the ASTM standard, the sample size n

needed to support a tolerance E is given by the fol- lowing expression:

n = (4)

where a is the standard deviation, i is a coefficient that corresponds to the desired statistical confidence.

Values of t for commonly specified statistical confi- dences are presented in Table 2. Since a high statistical confidence is desired, we set the coefficient t to three. The error tolerance, E, is 10% of the total

loss, i.e.. E^ 0.10(100- RE). (5)

If we further assume, as discussed in section 3, that a 20% tolerance in total loss corresponds to three

standard deviations, i.e..

3a = 0.20(100 -RE),

we may arrive at the following result.

n =

0.20(100 - RE)

0.10(100 -HE)

(6)

(7)

BJid conclude that no fewer than four motors should

be tested. The Final Rule, in establishing a minimum

sample of five for compliance testing, appears to be

consistent with this result.

Enforcement testing is likely to be required only in the circumstance that all other means of resolving

conflicting interpretations of test results and/or la- beled motor efficiencies have been exhausted. Further, as it may be necessary to enter results obtained

by testing under the Sampling Plan for Enforcement

Testing into evidence, the Sampling Plan for Enforcement Testing is based on well-established statistical methods and follows clearly delineated procedures.

The Sampling Plan for Enforcement Testing is based on a procedure, which is due to C. Stein [11, 12], for obtaining a confidence interval on a mean.

Since the Sampling Plan for Enforcement Testing

may recommend that certain adverse actions be

taken against a manufacturer—e.g., relabeling, the

cessation of the sale and distribution of certain basic models, and the assessment of fines—the risk of a false determination of noncompliance should be

small. The sampling plan is based on a 97.5% statistical confidence, thus the risk to a manufacturer of a false determination of noncompliance is no greater

than 2.5%.

The legislation is supported by ensuring that the mean efficiency of each basic model is not less than

the EPCA nominal eflficiency and the rated efficiency.

This objective may be satisfied by demonstrating

that the mean efficiency obtained by tests conducted on a random sample of motors exceeds a lower control limit. The Sampling Plan for Enforcement Test- ing estimates the true mean full-load efficiency of the

basic model and the confidence that this estimate ex- ceeds a lower control limit. The Sampling Plan for Enforcement Testing assumes that the true mean full- load efficiency, the standard deviation of the motor

efficiencies, and the distribution of motor efficiencies are not known.

The Sampling Plan for Enforcement Testing has been

adapted from that provided for appliance testing under Part 430 [13]. This sampling plan is based on the

t-test. The t-test is well suited to this application as

it is well known to be insensitive to departures from

the assumption of normally distributed data: The

^-test is a test on a mean, i.e., an average of independent values obtained by a random sample. In general, sums of arbitrary, independent random values tend to have a distribution that is almost normal. Hence, the

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

the underlying distribution.
7 ENFORCEMENT TESTING

Increasing Efficiency

c

0)o

CDQ.

C

o

>?

0) Q

CT3 U

c

o5

Loss Fraction (percent)

Figure 2: The operating characteristics of the SampHng Plan for Enforcement Testing. Model calculations are for samples of five and a maximum of 20. The contours indicate the probability of demonstrating compliance, e.g., the

0.90 contour corresponds to a 90% likelihood of demonstrating compliance.

7.1 Method

The best estimate of the true mean efficiency that

may be obtained by tests conducted on a random

sample is the mean efficiency of that sample,

1 "

(8) i=l

where Xi is the measured full-load efficiency of unit i, and n is the number of units tested. The uncertainty

of this estimate depends on two factors: 1) the size of the sample, i.e., the number of motors tested, and

2) the underlying variability in the entire population

of motors. The sample standard deviation.

S = n-l (9)

is one measure of the variability of the motor efficiencies. The standard error in the mean,

SE{X) =

\/n'

(10)

provides an estimate of the standard deviation of the mean efficiency as determined by tests conducted on

samples of n units. If we assume that the efficiencies of the entire population of motors are normally dis- tributed about the true mean full-load efficiency, fi, then the ratio.

t =

A* X

SE{X)' (11)

is distributed according to a probability density function that is known in statistics literature as the t- distribution. The values of t associated with com- monly specified percentiles are readily available and

are included in many references on statistics [14].

Establishing a lower control limit. Equation (11) provides an expression for the mean of the

sample:

X ^li-tSEiX). (12)

We may assume, by hypothesis, that the units to be tested are drawn from a population of motors

for which the mean full-load efficiency is equal to

or greater than the rated efficiency RE. If t is the

97.5th percentile of the ^-distribution appropriate to the sample size, then the probability of obtaining a

6
7 ENFORCEMENT TESTING

mean efficiency,

X>RE-tSE{X),

control limit must then satisfy two conditions:

LCL ^ RE- tSE{X) and (13)LUL = HE - tSE[X) and (17)

RE

is not less than 97.5%, which recommends the lower — 120 — 0.20i?£^

'

control limit, rj.^^ second condition is obtained from Eq. (16) by

LCL = RE- tSE{X). (14)

setting the efficiency equal to the RE.

To apply this method, a random sample of motors

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 i- value is selected and the lower control limit calculated. For example, 97.5% confidence and a sample size of five units recommends a t-walue of 2.776. Provided the mean efficiency obtained from the random sample is not less than the lower control limit, as defined by

Eq. (14), we may assert with a confidence not less than 97.5% that the true mean efficiency of the entire

population is not less than the rated efficiency and

thus that the basic model is in compliance.

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

probability that the mean efficiency for a random

sample drawn from this population of motors would

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

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

There is some probability that the estimate of the

standard deviation and, therefore, that the standard

error in the mean is large and that the lower control

limit may be set, by chance, to an exceptionally large

value. To avoid this circumstance, it is sufficient to

establish a tolerance for the standard error in the

mean, SE{X). The tolerance for the standard error should be chosen to be appropriate for the size and

type of motor being tested and to be supported across the industry.

By definition, the efficiency as a percentage can be

expressed as.

A* - -pr- ^ 100, (15)

where Pin and Pout are the input and output power,

respectively. Following the convention used by

NEMA [5], the minimum efficiency is calculated at constant output power, thus

fJ-r out Pin + 0.20{Pin - Pout)

P

X 100

X 100, (16)

120 - 0.20/x

which is again expressed as a percentage. The lower

Discussion. By design, the tolerances for the motor efficiency specified by the Final Rule are closely

associated with the NEMA guidelines for motor effi- ciency labeling, and are thus likely to follow quality

control practices used by industry. This has several

potential advantages: 1) industry should be better

able to estimate the risk involved with the selection

of a basic model for testing and thus better manage

their financial risk, and 2) the investment required for personnel training should be reduced since the tol- erances recommended by Part 431 follow those cur- rently used by industry.

The Sampling Plan has an additional advantage: If a manufacturer is in compliance with the voluntary

NEMA guidelines for motor efficiency labeling, the

probability of demonstrating compliance by actual

testing is high. As discussed, the probability of fail- ure during enforcement testing due to a low mean

value is not greater than 2.5%. A motor may also fail during enforcement testing due to high variability.

We next estimate the likelihood that a motor labeled

in accordance with the MG 1 guidelines would fail during enforcement testing due to insufficient sample

size. Step 7 of the NOPR Sampling Plan for Enforcement Testing sets a condition on the sample size. To

demonstrate compliance, the initial sample size rii must satisfy the following condition:

Til >

tSiil20 - 0.2RE)

(19) RE{20 - 0.2RE)

where RE is the rated efficiency and 5i is the stan- dard deviation of the sample. This equation may be

rearranged to yield a condition on the value of t: f^RE{2Q-Q.2RE)' *-

[ Si{l20-0.2RE) J-

^^°^

Following our earlier discussion, we assume that the

difference between the NEMA Nominal and Minimum efficiencies corresponds to three stcmdeird deviations, and use the following approximation: a 0.20(100 - RE)

Si

Sy/rn

Upon substitution into Eq. 20, the following condition on t is obtained:

RE{20 - 0.20RE)

t < Ziii 0.20(100 - RE){\20 - Q.2RE)

'
7 ENFORCEMENT TESTING

For an initiaJ sample of five, t must exceed ten for the sample to fail due to insufficient sample size. The

probability that t would exceed 10 by chance is less than 1 in 1000, for a sample of five. We conclude that

it is highly unlikely that a product that is labeled in accordance with the MG 1 guidelines would require

testing beyond the initial sample of five. The operating characteristics of the Sampling Plan

for Enforcement Testing are shown in Figure 2, which

present data for an initial sample of five and testing as many as 20. It may be noted that the 97.5% contour

is independent of the standard deviation.

7.2 Enforcement testing example

A specific example of the use of the Samphng Plan

for Enforcement Testing follows. We assume that a

basic model is tested for a rated efficiency of 89.5. The manufacturer may select at random no fewer

than five units and no more than 20 units from a

representative population of motors.

Step 1. The first sample size (ni) must be five or more units. In this example, we assume that the manufacturer elects to test an initial sample of five motors, ni — 5.

Step 2. Compute the mean (Xi) of the measured en- ergy performance of the ni units in the first sample as follows:

follows:

1 "1 x, = ^Tx, (21)

where Xt is the measured full-load efficiency

of unit i. Assume, as in the earlier example of compli- ance testing, that the results of testing yield

efficiencies of 89.9, 89.2, 89.0, 89.3, 89.4. The mean of the sample is thus 89.4.

Step 3. Compute the sample standard deviation (Si)

of the measured full-load efficiency of the ni

units in the first sample as follows:

5i =

'Erii(^.-^i)'

ni — 1 (22)

The sample standard deviation is 0.3.

Step 4. Compute the standard error {SE{X\)) of the mean full-load efficiency of the first sample as

SE{Xi) =

Si

(23)

The standard error of the mean is 0.2.

Step 5. Compute the lower control limit (LCLi) for the mean of the first sample using RE as the

desired mean as follows:

LCLi =RE- tSE{Xi) (24)

where RE is the applicable EPCA nominal

full-load efficiency when the test is to deter- mine compliance to the applicable statutory

standard, or is the labeled nominal full-load

efficiency when the test is to determine compliance with the labeled efficiency value, and

t is the 2.5th percentile of a t-distribution

for a sample size of rii, which yields a 97.5%

confidence level for a one-tailed i-test. Next select the t-coefficient for a sample

of five units and a statistical confidence of

97.5%. The value of the t-coefficient may

he obtained from standard mathematical tables. A table of t-coefficients is provided here

in Table 3. It should be noted that the t- coefficient is often tabulated according to the

"degrees of freedom, " which is (ni — 1)

.

For a sample size of five the degrees of free- dom is 4, thus the t-coefficient is 2.776 and

the lower control limit is 89.1.

Step 6. Compare the mean of the first sample {Xi)

with the lower control limit {LCL\) 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.

Since the mean of the initial sample is greater than the lower control limit, proceed

to Step 7.

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

tSi{l2Q-0.2RE)

RE{20 - 0.2RE)

^ 2 (25)
8 FURTHER INFORMATION

where Si and t have the values used in Steps

4 and 5, respectively. The factor

120 - 0.2RE

RE{20 - 0.2RE)

is based on a 20% tolerance in the total power

loss at full load and fixed output power.

The recommended sample size is two. Given the value of n, determine one of the

following:

(i) If the value of n is less than or equal

to Til and if the mean energy efficiency

of the first sample {Xi) is equal to or greater than the lower control limit (LCLi), the basic model is in compli- ance and testing is at an end.

(ii) If the value of n is greater than ni , 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 — rii. If the value of ^2 so calculated is greater

than 20 — ni , set n2 equal to 20 — ni

.

Since the initial sample size is greater than

the recommended sample size, the basic model is in compliance and testing is at an

end.

Table 3: An abridged t-table. t coefficients based on a

statistic confidence of 97.5%, sample size n, and degrees

of freedom v. n ly t 5 4 2.776

6 5 2.571

7 6 2.447

8 7 2.365

9 8 2.306

10 9 2.262

11 10 2.228

12 11 2.201

13 12 2.179

14 13 2.160

15 14 2.145

16 15 2.131

17 16 2.120

18 17 2.110

19 18 2.101

20 19 2.093

8 Further information

For further information regarding Final Rule contact:

James Raba

U.S. Department of Energy

Energy Efficiency and Renewable Energy

Mail Station EE-43

1000 Independence Avenue, SW

Washington, DC 20585-0121

202-586-8654

Jim.Raba@hq.doe.gov

Edward Levy, Esq.

U.S. Department of Energy

Office of General Counsel

Mail Station GC-72

1000 Independence Avenue, SW

Washington, DC 20585-0103

202-586-2928

edward.levy@hq.doe.gov

For information regarding this report contact:

Ken Stricklett

National Institute of Standards and Technology

Bldg 220 Rm B344

100 Bureau Drive Stop 8113

Gaithersburg, MD 20899-8113

301-975-3955

ken.stricklett@nist.gov

Mark Vangel

National Institute of Standards and Technology

NIST North 353

100 Bureau Drive Stop 8980

Gaithersburg, MD 20899-8980

301-975-3208

mark.vangel@nist.gov

Acknowledgments

This report was completed with the support of the

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

References

[1] Energy efficiency program for certain commercial and industrial equipment: Test procedures,

labeling, and certification requirements for electric motors. Federal Register, 64(192) :541 14-

54172, Tuesday, October 5, 1999.

[2] 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 27,

1996.

[3] 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 Appli- ance 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.

[4] 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.

[5] National Electrical Manufacturers Association

Standards Publication MGl-1993, Motors and

Generators, and Revisions 1, 2, 3 and 4. National

Electrical Manufacturers Association, Rosslyn,

VA, 22209.

[6] Institute of Electrical and Electronics Engineers,

Inc., Standard test procedure for polyphase in- duction motors and generators, IEEE Std 112- 1996, and the correction to the calculation at item (28) in section 10.2 Form B-Test Method

B issued by IEEE on January 20, 1998. Insti- tute of Electrical and Elecronics Engineers, Inc., Piscataway, NJ, 08855.

[7] CSA International (or Canadian Standards Association) Standard C390-93, Energy Efficiency Test Methods for Three-Phase Induction Motors. CSA International, Etobicoke (Toronto),

Ontario, Canada, M9W 1R3.

[8] 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.

[9] Statistical methods for motor efficiency data.

Motor and Generator Section Publication, National Electrical Manufacturers Association,

Washington, DC 20037, May 1978.

[10] ASTM Standard E 122-89, Standard practice for choice of sample size to estimate a measure of

quality for a lot or process, 1989.

[11] C. Stein. A two-sample test for a linear hypothesis whose power is independent of variance.

Annals of Mathematical Statistics, 16:243-258,

1945.

[12] P. J. Bickel and K. A. Doksum. Mathematical

Statistics. Holden Day, Oakland, CA, 1977.

[13] Energy conservation program for consumer

products. Code of Federal Regulations, Chapter II, Vol. 10, Part 430, pp. 93-263, Office of the

Federal Register, National Archives and Records

Administration, Washington, DC, 1-1-97 1997.

[14] R. S. Schulman. Statistics in Plain English with

Computer Applications. Van Norstand Reinhold, New York, 1992.

10
Appendix A

Exerpts from the Final Rule

This appendix contains exerpts from 10 CFR Part 431, "Energy Efficiency Program for Certain Commercial

and Industrial Equipment: Test Procedures, Labeling, and Certification Requirements for Electric Motors,"

Federal Register, Vol. 64, No. 192, Tuesday, October 5, 1999, pp. 54114-54172. The criteria for establishing

compliance with the EPCA standard efficiencies and for substantiation and subsequent verification of an AEDM are prescribed in §431.24, pp. 54152-54154. The Sampling Plan for Enforcement Testing is published

in the Appendix B to Subpart G, pp. 54171 and 54172.

A.l Certification testing

431.24(b)(2) Selection of units for testing. For each basic model selected for testing, a sample of units shall be selected at random and tested. 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 fewer than five units of a basic model would be produced over a reasonable period of time

(approximately 180 days), then each unit shall be tested. In a test of compliance with a represented average

of nominal efficiency:

(i) The average full-load efficiency of the sample X which is defined by

1 " n ^-^

where Xi is the measured full-load efficiency of unit % and n is the number of units tested, shall satisfy

the condition:

1 + 1.05 ()M _ 1)

where RE is the represented nominal full-load efficiency, and

(ii) The lowest full-load efficiency in the sample Xmin-, which is defined by

Xmin = min(Xi)

shall satisfy the condition

^min ^

100

1 + 1.15(^-1)

A. 2 AEDM substantiation

§431. 24(a)(3) Substantiation of an alternative efficiency determination method. Before an AEDM is used,

its accuracy and reliability must be substantiated as follows:

(i) The AEDM must be applied to at least five basic models that have been tested in accordance with

§431.23 of this subpart, and

(ii) The predicted total power loss for each basic model, calculated by applying the AEDM, must be within

plus or minus 10% of the mean total power loss determined from testing of that basic model.

(4) Subsequent verification of an AEDM.

11
A EXERPTS FROM 10 CFR PART 431

(i) Each manufacturer shall periodically select basic models representative of those to which it has applied an AEDM, and for each basic model selected shall either:

(A) Subject a sample of units to testing in accordance with §§431.23 and 431.24(b)(2) by an accredited

laboratory that meets the requirements of §431.25,

(B) Have a certification body recognized under §431.27 certify its nominal full load efficiency, or

(C) Have an independent state-registered professional engineer, who is qualified to perform an evaluation of electric motor efficiency in a highly competent manner and who is not an employee of the

manufacturer, review the manufacturer's representations and certify that the results of the AEDM

accurately represent the total power loss and nominal full load efficiency of the basic model.

A. 3 The Sampling Plan for Enforcement Testing

Appendix B to Subpart G of Part 431—Sampling Plan for Enforcement Testing

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

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

follows:

1 "'

2=1

(26)

where Xt is the measured full-load efficiency of unit i.

Step 3. Compute the sample standard deviation (5i) of the measured full-load efficiency of the rii units in the first sample as follows:

S, = jSil^i^Zl)!. (27) V

ni - 1

Step 4. Compute the standard error {SE{Xi)) of the mean full-load efficiency of the first sample as follows:

SE{X,) = -|L. (28)

Step 5. Compute the lower control limit [LCLi) for the mean of the first sample using RE as the desired mean as follows: LCLi^RE-tSE{Xi) (29)

where RE is the applicable EPCA nominal full-load efficiency when the test is to determine compliance to the applicable statutory standard, or is the labeled nominal full-load efficiency when the

test is to determine compliance with the labeled efficiency value, and t is the 2.5th percentile of a

i-distribution for a sample size of rii, which yields a 97.5% confidence level for a one-tailed i-test.

Step 6. Compare the mean of the first sample (Xi) with the lower control limit {LCL\) 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.

12
A EXERPTS FROM 10 CFR PART 431

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

(30)

tSiil20-0.2REy'^

RE{20 - 0.2RE)

where Si and t have the values used in Steps 4 and 5, respectively. The factor 120-0.2RE

RE{20-0.2RE)

is based on a 20% tolerance in the total power loss at full load 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 rii and if the mean energy efficiency of the first sample

(Xi) is equal to or greater than the lower control limit {LCLi), the basic model is in compliance

and testing is at an end.

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

sample 712 is determined to be the smallest integer equal to or greater than the difference n — ni.

If the value of n2 so calculated is greater than 20 — ni, set 77,2 equal to 20 — ni.

Step 8. Compute the combined mean (X2) of the measured energy performance of the ni and 712 units of

the combined first and second samples as follows:

^ n\-\-n2 X2 = —^

Y. ^- (31)

"-1 + 712 ^

Step 9. Compute the standard error {SE{X2)) of the mean full-load efficiency of the iii and 7^2 units in the

combined first and second samples as follows:

SE{X.2) = -^==. (32)

Vni + 7i2

(Note that 5i is the value obtained above in Step 3.)

Step 10. Set the lower control limit {LCL2) toLCL2 = RE- tSE{X2), (33)

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

control limit {LCL2) to find one of the following:

(i) If the mean of the combined sample (X2) is less than the lower control limit {LCL2), 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 {LCL2), the basic model is in compliance and testing is at an end.

MANUFACTURER-OPTION TESTING

If a determination of noncompliance 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, n-s, of units be tested, with ;i3 chosen such

that ni + no + 7x3 does not exceed 20.

13
A EXERPTS FROM 10 CFR PART 431

Step B. Compute the mean full-load 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 rii +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.

14


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