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"«°s 03u - PUB^rIONS

Nisr United States Department of Commerce

Technology Administration

National Institute of Standards and Technology

NIST Technical Note 1422

Electric Motor Efficiency Testing Under

the New Part 431 of Chapter II of Title 10,

Code of Federal Regulations: Enforcement Testing

K. L. Stridden and M. Vangel

|753

K1422

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NIST Technical Note 1422

Electric Motor Efficiency Testing Under

the New Part 431 of Chapter II of Title 10,

Code of Federal Regulations: Enforcement Testing

K. L. Stricklett

M. Vangel

Electricity Division

Electronics and Electrical Engineering Laboratory

and

Statistical Engineering Division

Information Technology Laboratory

National Institute of Standards and Technology

Gaithersburg, MD 20899-0001

January 1997

^TES o* *

U.S. Department of Commerce

Michael Kantor, Secretary

Technology Administration

Mary L. Good, Under Secretary for Technology

National Institute of Standards and Technology

Arati Prabhakar, Director
National Institute of Standards U.S. Government Printing Office For sale by the Superintendent of and Technology Washington: 1997 Documents

Technical Note 1422 U.S. Government Printing Office

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

Tech. Note 1422

18 pages (Jan. 1997)

CODEN: NTNOEF
Bibliographic Information

Abstract

The provisions for electric motor efficiency testing under the proposed new Part 431 of Chapter II of Title 10, Code of Federal Regulations, as published for public comment in the Federal Register, Vol. 61, No. 230, Wednesday, November 27, 1996, pp. 60439-60475, are discussed. The criteria for demonstration of compliance with the energy efficiency requirements established by Energy Policy and

Conservation Act of 1975, as amended, are presented. The operating characteristics, i.e., the estimated

probability of demonstrating compliance based on the mean efficiency, standard deviation, and number

of units tested, of the Sampling Plan for Enforcement Testing recommended by the new Part 431 are evaluated by model calculations.

Keywords

electric motor efficiency; electric motor testing; Energy Policy Act; 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.

in

Electric Motor Efficiency Testing under

the Proposed Part 431 of Chapter II of Title 10,

Code of Federal Regulations: Enforcement Testing

Contents

1 Introduction

This document provides commentary on the rules

for efficiency testing of polyphase electric motors

proposed by the new Part 431 of the Code of

Federal Regulations (10 CFR Part 431) as published for public comment in the Federal Register [1]. The uses of actual testing, in the context of the new Part 431, are 1) for demonstration of compliance of a basic model with the levels of efficiency established by the Energy Policy

and Conservation Act of 1975, as amended; 2) for substantiation of an alternative efficiency deter- mination method (AEDM) as described in Subpart. B §431.24; and 3) for enforcement testing.

This document provides background information on the statistical methods recommended by the new Part 431 for enforcement testing, i.e., topic

3: Topics 1 and 2 will be discussed in a future

publication.

This report supplements the materials published

in the Federal Register and is intended for persons

having an interest in motor efficiency testing and

in the Energy Policy Act (EPAct) legislation, or involved in recommending motors for testing, in estimating the relative risks in choosing a specific motor for testing, in enforcement testing, or

in performing efficiency measurements in support

of the new Part 431. Actual testing under the

proposed rule is based on the IEEE 112 [2] and

CSA-390 [3] standards. The reader is referred to these documents for a discussion of the efficiency

measurements.

One note of caution: This report is not an official statement of policy by the Department of Energy

(DOE) regarding the proposed rule. This report

must be regarded only as extended commentary

on the recommendations contained in the pro- posed new Part 431. Readers are encouraged to contact the Department of Energy with questions

and/or comments on the proposed new Part 431.

Several points of contact are provided in Section 5 of this report.

The remainder of this document is organized by

sections: Section 2 discusses the criteria for compliance of a basic model with the mandated effi- ciencies; Section 3 discusses the objectives and

general guidelines considered in developing the

Sampling Plan for Enforcement Testing provided

in the new Part 431; Section 4 discusses of the operating characteristics of the Sampling Plan for Enforcement Testing; Section 5 provides points

of contact for further information, and Section 6 contains a list of references. Supplemental infor- mation provided to support those discussions is contained in: Appendix A: the full text of the

Sampling Plan for Enforcement Testing included

in the new Part 431 [1]; Appendix B: a listing of

the computer code used to model the operating

characteristics of the Sampling Plan for Enforcement Testing; Appendix C: an abridged table of

^-values for single-sided i-tests.

2 The criteria for demonstration of

compliance by actual testing

The intent of the EPAct legislation is to improve

the nation's efficient use of electric power by the

use of efficient electric motors. The intent of the

legislation is, therefore, satisfied if the mean full- load efficiency of the entire population of motors,

of each basic model covered by the legislation

equals or exceeds the statutory efficiency. Deter- mination of the accuracy of labeled motor efficiencies, e.g., establishing upper and lower bounds

for labeled motor efficiencies or estimating the true variability of the population of motors, is not

strictly required to demonstrate that the EPAct

objective has been met and is, therefore, beyond

the purview of this legislation.

The true mean full-load efficiency is an abstracl quantity that can be known only in principle since

its full determination requires that the efficiencies of all units of a basic model be measured without
Efficiency Testing under the New Part 431

error. Since this is not possible, any demonstration of compliance by actual testing must rely on

accepted statistical methods to estimate the confidence of achieving the mandated efficiency, i.e., to estimate the probability that the true mean

full-load efficiency is not less than the mandated

level. The choice of statistical confidence does

not affect the quality or average performance of

the motors being tested but rather indicates the

method used to establish a confidence interval about the mean of the sample: The choice of confidence level is governed largely by convention. For the purpose of demonstrating compli- ance with the mandated efficiency levels, the De- partment of Energy has determined that compliance is demonstrated provided the true mean

efficiency of a basic model is not less than the

mandated efficiency with confidence not less than

90 percent. It should be emphasized that the confidence specified is that for the estimate of the mean efficiency and not for the efficiency of a sin- gle unit. The criteria for demonstration of compliance pro- vided is Subpart B §431.24 Paragraph (b)(l)(iii)

of the new Part 431 follow:

For each basic model selected for test- ing, 1 a sample of units shall be se- lected at random and tested in accor- dance with §§431.23 and 431.25, and

appendix A, of this subpart. The sample shall be comprised of production

units of the basic model, or units that

are representative of such production

units, and shall be of sufficient size to ensure that any represented value of the nominal or average full load effi- ciency of the basic model is no greater

than the lesser of

(A) The average full load efficiency of

the sample, or

(B) The lower 90 percent confidence

limit of the average full load efficiency of the entire population divided by

tin- coefficient "K" applicable to the

represented value. The coefficients are

set forth in appendix B of this subpart.

'

',

, 1 1

1

1

.

. ,i,. ni ..I similai design may !>• subsl ituted without requiring additional testing if the represented measures of energy consumption continue to satisfy the applicable sampling provision.

3 Design of the Sampling Plan for

Enforcement Testing

A Sampling Plan for Enforcement Testing is pro- vided in the proposed new Part 431 to aid in the

interpretation of data obtained by actual testing

and to ensure uniform application of the rule. The objectives of the Sampling Plan are: 1) to obtain an estimate of the true mean full-load ef- ficiency, 2) to establish reasonable measurement

tolerances for motor efficiencies, and 3) to ensure

that results that are obtained by actual testing are significant within these tolerances.

In addition to these objectives, the Sampling Plan

for Enforcement Testing should conform to the

following general guidelines:

• The criteria for demonstration of compliance

specified for electric motor testing should be

consistent with those established for other in- dustries and/or products covered by the EPAct

legislation.

• Demonstration of compliance by actual testing

should, when feasible, rely on standards and

practices currently used by industry and should

not require new or unique standards or procedures.

• Compliance testing should fall within the realm

of normal testing for purposes of quality control, i.e., the development of unique facilities should not be required.

• The level of quality control and quality assur- ance required for demonstration of compliance

should be supported across the industry.

• The rules for compliance testing should apply

uniformly across the industry, i.e., the level of

quality control and quality assurance required

by the Sampling Plan should be consistent with

industry practice regarding motor size and type.

What statistical test is appropriate? The

legislation is supported by ensuring that the mean

efficiency of each basic model is not less than the

statutory full-load efficiency, SFE, i.e., the ef- ficiency levels established by the Energy Policy

and Conservation Act of 1975, as amended. This

objective may be satisfied by demonstrating that

the mean efficiency obtained by tests conducted

on a random sample of motors exceeds a lower
Design of the Sampling Plan

bound or lower control limit. The Sampling Plan

developed here seeks to estimate the true mean

full-load efficiency of the basic model and the confidence that this estimate exceeds a lower control

limit. The Sampling Plan must assume that the

true mean full-load efficiency, the standard deviation of the motor efficiencies, and, indeed, that

precise information on the distribution of motor

efficiencies are not known.

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,

X

1 n

n *-^

i = \

(1)

where X{ 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 = 'E7=AXi-x)

(2)

is one measure of the variability of the motor ef- ficiencies. The standard error in the mean,

SE{X) (3)

provides an estimate of the standard deviation of

the mean efficiency as determined by tests con- ducted on samples of n units. If we assume that

the efficiencies of the entire population of motors are normally distributed about the true mean

full-load efficiency, fi, then the ratio

t H-X

SE(X) (4)

is distributed according to a probability density

function that is know in statistics literature as the ^-distribution. The values of t associated with

commonly specified percentiles are readily avail- able and are included in many references on statistics [4], for completeness, and for the convenience

of the reader, an abridged 2-table is provided in Appendix C of this report.

To be mathematically precise, much of the dis- cussion that follows is rigorously true if and only

if the motor efficiencies are normally distributed.

However, the t-test is well known to be insensitive to departures from the assumption of normally

distributed data: The t-test is a test on a mean,

i.e., an average of independent values obtained by a random sample; and, in general, sums of arbitrary, independent random values tend to have a

distribution that is almost normal. Hence, the i- test is not strongly influenced by the exact form

of the underlying distribution.

Establishing a lower control limit. The ratio t defined in eq (4) provides an expression for the mean of the sample:

X = n-tSE(X). (5) 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 statutory full-load efficiency.

If t is the 90th percentile of the ^-distribution appropriate for the sample size, then the probability

of obtaining a mean efficiency,

X > SFE-tSE(X), (6)

is not less than 90 percent, which recommends

the lower control limit,

LCL = SFE-tSE{X). (7)

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 lvalue is selected and the lower control

limit calculated. For example, by way of reference to the i-table provided in Appendix C, 90 per- cent confidence and a sample size of five units

yields a i-value of 1.533. Provided the mean ef- ficiency obtained from the random sample is not

less than the lower control limit, as defined by

eq (7), we may assert with a confidence not less than 90 percent that the true mean efficiency of

the entire population is not less than the mandated efficiency and thus that the basic model is in compliance.

In any statistical test there is some probability

of incorrectly concluding noncompliance. H\ il<- sign, the probability that the mean efficiency for a random sample drawn from this population oi
Efficiency Testing under the New Part 431

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 10 percent.

There is some probability that the estimate of the

standard deviation and, therefore, the standard

error in the mean is large and that the lower control limit may be set, by chance, to a value that

defeats the intent of the legislation. To avoid this circumstance, it is sufficient to establish a toler- ance 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.

put power, i.e., Pout = P'out , thus

* out l^min Pin + 0.20(-P; n — Pout

120 - 0.20/i

100

100, (10)

which is again expressed as a percentage. The

lower control limit must then satisfy two conditions: LCL = SFE-tSE{X) and (11)

SFE

£ 120- 0.20W00 - < 12>

The second condition is obtained from eq (10) by

setting the efficiency equal to the SFE.

The problem of defining a reasonable tol- erance for the standard error. The strat- egy recommended here is to establish acceptable

benchmarks for the lower control limit. One possible solution, is to base these tolerances on the

existing NEMA guidelines for motor efficiency la- beling [5]. These guidelines were developed by

consensus among motor manufacturers and they

are followed, on a voluntary basis, by a large

segment of the motor manufacturers. Although

the NEMA guidelines were developed primarily

to provide uniformity in motor efficiency labeling, they are used for purposes of quality control by many manufacturers and they may therefore provide a reasonable basis for estimating ef- ficiency tolerances among motors of different size and type. We argue that the NEMA 20 percent

loss tolerance is significant and meaningful and is likely to be supported across the industry.

By definition, the efficiency as a percentage can

be expressed as,

Pout , nn \.i = -^—100.

•*in (8)

The input and output power are indicated by P{n and Pou t, respectively. So that the minimum ef- ficiency associated with a 20 percent tolerance in power loss is expressed as,

rP'out 100. (9)

Following the convention used by NEMA [5], the

minimum efficiency is calculated at constant outThe standard deviation allowed under the

Sampling Plan. An estimate of the width of

the distribution of motor efficiencies, i.e., the stan- dard deviation, that is allowed by the Sampling

Plan may be obtained by equating the right-hand

sides of eqs (11) and (12):

SFE

120-0.20SFE SFE-tSE{X), (13)

which, after substitution from eq (3), may be re- arranged to yield

y/a SFE(20-0.20SFE)

t 120 - 0.20SFE ' [

'

It should be noted that the NEMA minimum ef- ficiencies are determined by a table [5] while the new Part 431 relies on a formula. A comparison of the NEMA guidelines and the LCL rec- ommended by the new Part 431 is presented in Tab. 1. The column headed "NEMA 20 percent Loss Tolerance" lists 20 percent of the difference

SFE — Umin while the column headed "Part 431

20 percent Loss Tolerance" lists 20 percent of the

difference SFE — LCL. The maximum standard

deviation is calculated using eq (14) by assuming

a sample of five units.

Advantages of the Sampling Plan. By design, the tolerances for the motor efficiency specified by the rule are closely associated with the

NEMA guidelines for motor efficiency labeling,

and are thus likely to follow quality control practices used by industry. This has several potential
Operating characteristics

Table 1. Comparison of the NEMA 20 percent loss tolerance, the Part 431 20 percent loss tolerance, and

the maximum standard deviation allowed under the Sampling Plan.

Statutory Full-load

Efficiency

NEMA

Minimum

Efficiency

NEMA

20

percent

Loss

Tolerance

Part

431

LCL

-t—

c

CDO

t-H <D CD O oj

CD r—1 —

CO rO

+=> to

f-c CO

cd o

P-, hJ

Maximum

Standard

Deviation

75.5 72.0 3.5 72.0 3.5 5.1 80.0 77.0 3.0 77.0 3.1 4.5 82.5 80.0 2.5 80.1 2.8 4.1 84.0 81.5 2.5 81.4 2.6 3.8 85.5 82.5 3.0 83.1 2.4 3.5 86.5 84.0 2.5 84.2 2.3 3.4 87.5 85.5 2.0 85.4 2.1 3.1 88.5 86.5 2.0 86.5 2.0 2.9 89.5 87.5 2.0 87.7 1.8 2.6 90.2 88.5 1.7 88.4 1.8 2.6 91.0 89.5 1.5 89.4 1.6 2.2 91.7 90.2 1.5 90.2 1.5 2.2 92.4 91.0 1.4 91.0 1.5 2.2 93.0 91.7 1.3 91.7 1.3 1.9 93.6 92.4 1.2 92.4 1.2 1.8 94.1 93.0 1.1 93.0 1.1 1.6 94.5 93.6 0.9 93.5 1.0 1.5 95.0 94.1 0.9 94.1 0.9 1.3

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

tolerances recommended by the new Part 431 fol- low those currently 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, i.e., 1) the average full-load efficiency of the

entire population of motors exceeds the NEMA

nominal efficiency, and 2) no single unit falls be- low the NEMA minimum efficiency, then the probability of demonstrating compliance by actual test- ing is high.

Risks associated with adopting the Sampling Plan. We have assumed that the NEMA

guidelines reasonably represent the tolerance that

is maintained across the industry. There are. in- deed, very little data to test the validity of this assumption.

4 Operating characteristics of the

Sampling Plan for Enforcement

Testing

How the Sampling Plan will perform in practice

depends, to a large extent, on the type and magnitudes of the variabilities encountered in the ef- ficiency measurements. Unfortunately, there are limited data on which to base these judgments.

The Sampling Plan may be modeled, however,

and tested, in a sense, on worst-case or other sce- narios. A computer-based computation was de- veloped for this purpose and is included in Appendix B of this report. The program calculates the probability of demonstrating compliance

based on the true mean efficiency, the sample size, and standard deviation. These calculations as- sume that the full-load efficiencies are normally

distributed.

The results of several calculations are plotted in figures 1 and 2. The data shown are for standard

deviations equal to 20 percent, 10 percent, and

7 percent of the total loss, i.e., the 20 percent

loss tolerance corresponds to 1, 2, and 3 stan- dard deviations. These figures plot the probability of demonstrating compliance as a function of

the true mean efficiency. For example, by way of

reference to the lower panel in figure 1, the probability of demonstrating compliance for samples

of five motors drawn at random from a population having a true mean efficiency of 76 percent

is approximately 0.98, where a probability of 1.0 provides complete certainty of compliance. Thus

the risk to the manufacturer of a false outcome

would be 2 in 100. For these same conditions, the

probability that a basic model having a true mean

efficiency of 74.5 percent would demonstrate compliance is approximately 0.50.

5 Further information

For information regarding the status of the pro posed new Part 431 and for details regarding pub-
Efficiency Testing under the New Part 431

lie review of the new Part 431 contact: 6 References

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

Ed.Levy@hq.doe.gov

For information regarding this report contact:

[1] "Energy Conservation Program for Certain

Commercial and Industrial Equipment: Test

Procedures, Labeling, and Certification Requirements of Electric Motors; Proposed

Rule," Federal Register, Vol. 61, No. 230,

Wednesday, November 27, 1996, pp. 60439-60475.

[2] IEEE Standard Test Procedure for Polyphase

Induction Motors and Generators, IEEE

Std 112-1991, 1991, The Institute of Electrical and Electronics Engineers, Inc., New York, NY,

USA.

[3] Energy Efficiency Test Methods for Three-Phase Induction Motors, C390-93, 1993,

Canadian Standards Association, Rexdale,

Ontario, Canada.

[4] See, for example, R.S. Schulman, Statistics in Plain English with Computer Applications,Vax\

Norstand Reinhold, New York, NY, USA, 1992.

[5] Motors and Generators, MG 1-1993, 1993,

National Electrical Manufacturers Association,

Washington, DC, USA.

Ken Stricklett National Institute of Standards and Technology

Bldg 220 Rm B344

Gaithersburg, MD 20899

301-975-3955

stricklett@nist.gov

Mark Vangel

National Institute of Standards and Technology

NIST North 353

Gaithersburg, MD 20899

301-975-3208

vangel@nist.gov

The FORTRAN code used for the model calculations presented here and documented in Appendix B is available on request.

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.

The authors wish to thank Jim Raba, DOE; and

Dr. Roger Daugherty, Reliance Electric, for their

i omments.
References

LU

o

LL Q.

O ^

>- °

E °

d CD

CD Z

< F

CD S

O C£

DC hQ_ CO

LU

Q

60 65 70 75 80

TRUE MEAN FULL-LOAD EFFICIENCY

Figure 1. The probability of demonstrating compliance for sample size, n, and standard deviation , S. I In statutory full-load efficiency, SFE, is 75.5 percent. The motor efficiencies are assumed to be normally dis- tributed with the standard deviation expressed as a percentage of the total losses, i.e., (100 — ji). where /i is the true mean full-load efficiency.
Efficiency Testing under the New Part 431

1.0

0.8 -

0.6 -

0.4 -

0.2

LU

o 0.0

-z. <

_J 1.0

U_ Q_

O ^ > o 0.8

h- o

d CD 0.6

CO ~Z.

OBA RATI 0.4

DC hQ_ CO

IZ 0.2

O

^

LU 0.0

Q

1.0

0.8 -

0.6

0.4

0.2

0.0

SFE = 95%

S = 20% TOTAL LOSS

n = 2 -^-^^_

n = 3 v,.v.-,,.v,.-,,.v,,. n = 5 ..„.

J I L J I I L

t

1 r

SFE = 95% S= 10% TOTAL LOSS

n = 2 ^^^^^—

n = 3

n = 5

n = 20

' ' _l |_

i 1 r

J I I L

t

1 1 1 1 1 r

SFE = 95% S= 7% TOTAL LOSS

n = 2 -^^^—

n = 3 n = 5 n = 20

I I I I

86 88 90 92 94 96 98

TRUE MEAN FULL-LOAD EFFICIENCY

100

Figure 2. The probability of demonstrating compliance for sample size, n, and standard deviation, S. The

statutory full-load efficiency, SFE, is 95 percent. The motor efficiencies are assumed to be normally distributed with the standard deviation expressed as a percentage of the total losses, i.e., (100 — /i), where /j is the true mean full-load efficiency.
Appendix A: Sampling Plan for Enforcement Testing

Appendix A

Sampling Plan for Enforcement Testing

The Sampling Plan for enforcement as is published in the Federal Register [1] follows:

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 n x units in the

first sample as follows:

i — \ where Xi is the measured full-load efficiency of unit. i. Step 3. Compute the sample standard deviation (5"i) of the measured full-load efficiency of

the ni units in the first sample as follows:

Step 4. Compute the standard error (SE(Xi)) of the mean full-load efficiency of the first sample as follows: SE(X1 ) = -?L. (3)

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

applicable statutory full-load efficiency (SFE) as the desired mean as follows:

LCLi =SFE-tSE{X1 ). (4)

Here t is the 10th percentile of a i-distribution for a sample size of n\ and yields a

90 percent confidence level for a one-tailed 2-test. Step 6. Compare the mean of the first sample {X\) with the lower control limit (LCLi) 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 deter- mination of compliance or noncompliance can be made; proceed to Step 7.

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

tSi(12Q-0.2SFE)'1 (5) SFE(20 - 0.2SFE)

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

120 - 0.2SFE

SFE{2Q - Q.2SFE)

is based on a 20 percent tolerance in the total power loss at full load and tixed output power.

Given the value of n, determine one of the following:
10 Efficiency Testing under the New Part 431

(i) If the value of n is less than or equal to r%\ and if the mean energy efficiency of

the first sample (X\) is equal to or greater than the lower control limit (LCL\),

the basic model is in compliance and testing is at an end.

(ii) If the value of n is greater than nj, 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 — n.\. If the value of 77.2 so calculated is greater

than 20 — 77 1, set 772 equal to 20 — n\.

Step 8. Compute the combined mean (X2) of the measured energy performance of the n\

and n-i units of the combined first and second samples as follows:

1 «i+n 2

Step 9. Compute the standard error (SE(X2)) of the mean full-load efficiency of the ni

and 7i2 units in the combined first and second samples as follows:

SE(X2 ) =

Sl

M

. (7)

(Note that S\ is the value obtained above in Step 3.) Step 10. Set the lower control limit (LCL2) to, LCL2 =SFE-tSE{X2 ), (8) 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 non-compliance is made in Steps 6, 7 or 11, above, the manufacturer may request that additional testing be conducted, in accordance with the following

procedures.

Step A. The manufacturer requests that an additional number, 77.3, of units be tested, with

773 chosen such that 711 + 772 + 77,3 does not exceed 20. 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 n\ + 712 + 773 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.
Appendix B: Determining the probability of compliance 11 Appendix B

The probability of compliance

The Algorithm

If one makes the model assumptions that the population of motor efficiencies is normally distributed

with mean // and standard deviation <r, then the probability of compliance can be calculated using

straightforward numerical integration.

In the following expressions, n\ > 5 denotes the minimum sample size in the sampling plan, and

n2 = 20 is the maximum sample size 5 and 20, respectively). In order to simplify the equations, n : — 1

is represented by u, and t is the 95th percentile of the t distribution for a sample size of ni, which

appears in the sampling plan.

The probability of compliance is

(SFB- n)yfi-tx

p

r„/2

2"/2-'3 1 p

f-le -v*/*>*dXmt

where the limits of integration are kUi =0,

SFE(20-.2SFE)

Ki = - —y/i — 1 for i — n\ + 1, . . ., ri2, '120-.2SFE)< ' ' '

and k„ 2+ i = oo. The function $(u) is defined to be the probability that a normally distributed random variable is greater than u, that is ru

e -x 2 /2

<E>(u) = / —-=dx.

J-oo v27T

There are many public-domain routines available for numerically calculating this integral efficiently and

accurately. The complete gamma function with parameter i//2, Tu / 2 i can al so De calculated without

writing special software.
12 Efficiency Testing under the New Part 431

The FORTRAN Code

A partial listing of the computer source code used for the model calculations.

implicit double precision (a-h, o-z)

C

C Test problem for probability of compliance

C calculation. The correct output is:

C

C Prob. of compliance = 0.4163048163619565

C dimension x(500), w(500)

nquad = 500

nl =5

xmu = 88

sfe = 90

sig = dfloat(100-xmu)/3

x(l) = -1

call comply (nl, xmu, sig, sfe, prob, x, w, nquad)

write (*,*) ' Prob. of compliance = ' ,prob

stop

C end

SUBROUTINE COMPLY

$ (nl, xmu, sig, sfe, prob, xvec, wvec, nquad)

C***BEGIN PROLOGUE COMPLY

C***DATE WRITTEN 950530 (YYMMDD)

C***REVISI0N DATE 000000 (YYMMDD)

C***CATEG0RY NO.

C***KEYW0RDS

C***AUTH0R VANGEL, M.

C*** (NATIONAL INSTITUTE OF STANDARD AND TECHNOLOGY)

C***PURP0SE Calculates the probability of compliance for motors tested

according to the proposed test plan.

C***DESCRIPTI0N

C

C Abstract

C

C This subroutine calculates the probability of compliance using

C Gauss-Legendre quadrature. The first time you call the

subroutine, set xvec(l) to some negative value. The quadrature

C abscissas will be calculated and returned in 'xvec', and the

C quadrature weights will be returned in 'wvec'. In subsequent

C calls, these weights and abscissas will not be recalculated unless

C x(l) is set to a value LE 0. C

C Description of Parameters

CC —Input

nl - Minimum no. of motors tested (integer, input)

C xmu - Mean efficiency of population (in '/.) (double precision, input)
Appendix B: Determining the probability of compliance 13

C sig - Standard deviation of population (in '/.) (double precision, input)

C sfe - Statutory full-load efficiency (in '/,) (double precision, input)

C prob - Probability of compliance (double precision, output)

C xvec - Vector of length 'nquad' of abscissas (double precision, input/output)

C wvec - Vector of length 'nquad' of weights (double precision, input/output)

C nquad - Number of Gauss-Legendre quadrature points (integer, input)

C (nquad should be at least 50. To be safe, set nquad = 100

C or even nquad = 500.)

C

C —Output

C (NONE)

CC***REFERENCES NONE

C ***R0UTINES CALLED GAULEG, TPPF, N0RMP

C***END PROLOGUE COMPLY

implicit double precision (a-h, o-z)

real preal , t dimension xvec(l), wvec(l), crit(21)

data zero, one/0. dO, l.dO/

C

C — If xvec(l) LE 0, then calculate the quadrature abscissas and weights, otherwise

C assume that the these values have already been calculated.

CC***FIRST EXECUTABLE STATEMENT COMPLY

if (xvec(l) .le. 0) then

call gauleg(zero, one, xvec, wvec, nquad)

end if

C

C — Get the appropriate t-quartile

preal = . 90E0

call tppf (preal, nl-1, t)

C

C — Calculate the points at which n changes

con = sfe*(20-.2d0*sfe)/((120-.2d0*sfe)*t)

crit(nl-l) = do 10 i=nl, 19

crit(i) = sqrt (df loat (i) ) *con

10 CONTINUE

C

C — Pick a big number to 'approximate infinity' for upper limit of final integral.

crit(20) - 1000 *sig/sqrt(df loat (nl)

)

C

C — Do the integrals

prob = do 20 i=nl-l, 19

dlt = crit(i+l) -crit(i)

dO = crit(i)

do 30 j=l, nquad

C

C — Weights and abscissas were calculated for integration on [0,1]. They must be

C transformed for integration on [crit(i), crit(i+l)].

x - dO +dlt*xvec(j)

w = dlt*wvec(j)
14 Efficiency Testing under the New Part 431

C

C — Argument for standard normal distribution

arg = (sqrt (df loat(i+l) )*(sf e-xmu) -t*x)/sig

CC — Upper tail probability for normal is q

call normp (arg, p, q, pdf)

CC — Here the argument of the integrand is calculated,

arg = log(q) +df loat(nl-l)/2 * $ log((nl-l)/sig**2) +

$ (nl-2)*log(x) -(nl-l)*x**2/(2*sig**2) -

$ (dfloat(nl-l)/2-l)*log(dfloat(2)) -

$ dlngam(dfloat(nl-l)/2)

arg = exp(arg)

C

C — The numerical integrations are performed by accumulating a weighted sum

C of the values of the integrands,

prob - prob +arg *w

30 CONTINUE

20 CONTINUE

C RETURN

END
Appendix C: An abridged t-table 15

Appendix C

An abridged i-table

The t values covering the range of sample sizes and confidences for the Sampling Plan are presented

below.

Table Cl. An abridged table of t percentiles

Sample

Size n

Confidence

90% 95% 99%

2 3.078 6.314 31.82

3 1.886 2.920 6.965

4 1.638 2.353 4.541

5 1.533 2.132 3.747

6 1.476 2.015 3.365

7 1.440 1.943 3.143

8 1.415 1.895 2.998

9 1.397 1.860 2.896

10 1.383 1.833 2.821

11 1.372 1.812 2.764

12 1.363 1.796 2.718

13 1.356 1.782 2.681

14 1.350 1.771 2.650

15 1.345 1.761 2.624

16 1.341 1.753 2.602

17 1.337 1.746 2.583

18 1.333 1.740 2.567

1!) 1.330 1.734 2.552

20 1.328 1.729 2.539

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