# United States Department of Commerce Tec nol Administration National Institute of Standards and Technology



# NI ST Technical Note 1427 

# An Analysis of Efficiency Testing Under the Energy Pol i cy Conserva ti

on Act : A Case Study W

ith Appl i cat i on to Di str i but ion Transformers '

IC L. Stricklett, angel , and 0 . Petersonshe National Institute of Standards and Technology was established in 1988 by Congress to "assist industry in the development of technology

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# NIST Technical Me 1427 

# An Analysis of Efficiency Testing Under the Eneij-2^ 3f Policy and Conservation Act: A Case Study With Application to Distribution Transformers

K. L. Stridklett Electricity Division Electronics and Electrical Engineering Labo ratory National Institute of Standards and T echnology Gaithersburg, MD 20899-0001 

### M. Vangel Statistical Engineering Division Information Technology Laboratory National Institute of Standards and Technology Gaithersburg, MD 20899-0001 

### 0. Petersons Electricity Division Electronics and Electrical Engineering Laboratory National Institute of Standards and Technology Gaithersburg , MD 20899-0001 April 1999

U.S. Department of Commerce William M. Daley, Secretary Technology Administration Gary Bachula, Acting Under Secretary for Technology National 1 of Standards and T ec hnology Raymond G. Kammer, Director

### National Institute of Standards U.S. Government Printing Office For sale by the Superintendent of

and Technology Washington : 1999 Documents Technical Note 1427 U. S. Government Printing Office Natl. Inst

. Stand . Technol. Washington , DC 20402 Tech. Note 1427 58 pages (April 1999) CODEN: NTNOEF

# B ibliographic Information

Abstract The protocols for efficiency testing promulgated by 10 CFR Part 430 and established under the Energy Policy and Conservation Act of 1975 as amended (EPCA) are discussed . The case of distribution transformers, which are covered under the proposed new 10 CFR Part 432, is treated in detail

. Model calculations are presented that estimate the probability of demonstrating compliance and the average number of units tested under the 10 CFR Part 430 and Part 432 sampling plans . The results of model calculations for the sampling plan contained in NEMA Standard TP 2 are also presented. Keywords distribution transformers ; efficiency testing; energy conservation ; energy policy ; EPCA ; operating characteristic; sampling plan ordering Copies of this document are available from the National Technical Information Service, 5285 Port Royal Road, Springfield, VA 22161, at (800) 553-6847 or (703) 487-4650.

## Contents 1 Introduction 1 2 General guidelines 1 3 Industry practice 2 4 Methods of analysis 2 4.1 Loss representation ............................................ 4 5 Compliance testing 5 5.1 10 CFR Part 430 and the proposed 10 CFR Part 432 (Method I) ..............

. .... 

### 5.1.1 10 CFR Part 430 ............................................ 5.1.2 The proposed 10 CFR Part 432 ................................. 5.1.3 Results

.............................................. 

### 5.2 NEMA Standard TP 2 (Method II and Method III) 5.2.1 Method II results

......................................... 

### 5.2.2 Method Ill results

......................................... 5.3 Discussion ................................................. 5 5 5 6 7 7 8 8 6 Enforcement testing 8 

### 6.1 10 CFR Part 430 ......................... . .................... 9 6.2 Results .................................................... 10 7 Summary 10 8 Further i nformation 10 Appendix A: The 10 CFR Part 430 and the proposed 10 CFR Part 432 sampling plans for compliance testing 42 Appendix B: The NEMA Standard TP 2 sampling plan 43 Appendix C: A sampling plan for enforcement testing 47 Appendix D: Computational algorithms 49

iv

# List of Figures1. Operating characteristic of the 10 CFR Part 430 compliance criteria for a loss tolerance of 103 percent and a fixed sample of two

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

. ..................................... 133. Operating characteristic of the 10 CFR Part 430 compliance criteria for a loss tolerance of 103 percent and a fixed sample of ten . ..................................... 144. Operating characteristic of the 10 CFR Part 430 compliance criteria for a loss tolerance of 103 percent and a fixed sample of twenty

. ................................... 155. Operating characteristic of the 10 CFR Part 430 compliance criteria for a loss tolerance of 103 percent and a fixed sample of thirty . .................................... 16 6. Operating characteristic of the 10 CFR Part 430 compliance criteria for a 106 percent loss tolerance and a fixed sample of two ......................................... 17 7. Operating characteristic of the 10 CFR Part 430 compliance criteria for a 106 percent loss tolerance and a fixed sample of five

.......................................... 18 8. Operating characteristic of the 10 CFR Part 430 compliance criteria for a 112 percent loss tolerance and a fixed sample of two ......................................... 19 9. Operating characteristic of the 10 CFR Part 430 compliance criteria for a 112 percent loss tolerance and a fixed sample of five .

........................................ 20 10. The operating characteristic of the 10 CFR Part 430 compliance criteria for a 103 percent loss tolerance and variable sample size

. ................................... 21 11. The testing burden of the 10 CFR Part 430 compliance criteria for a 103 percent loss tolerance and variable sample size .......................................... 22 12. Operating characteristic of the Method II sampling plan for a fixed sample of five

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

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

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

. ....... 26 16. Operating characteristic of the Method II sampling plan with variable sample size . ........ 27 17. The testing burden under the Method II sampling plan . ....................... 28 18. Operating characteristic of the Method III sampling plan for a fixed sample of five

.. ....... 29 19. Operating characteristic of the Method III sampling plan for a fixed sample of ten. ........ 30 20. Operating characteristic of the Method III sampling plan for a fixed sample of twenty

. ...... 31 21. Operating characteristic of the Method III sampling plan for a fixed sample of thirty........ 32 22. Operating characteristic of the Method III sampling plan with variable sample size

.. ....... 33 23. The testing burden under the Method III sampling plan ................. ....... 34 24. Operating characteristic of the sampling plan for enforcement testing with a loss tolerance of 103 percent ................................................. 35 25. Testing burden of the sampling plan for enforcement testing with a loss tolerance of 103 percent. 36 26. Operating characteristic of the sampling plan for enforcement testing with a loss tolerance' of 106 percent ................................................. 37 27. Testing burden of the sampling plan for enforcement testing with a loss tolerance of 106 percent. 38 28. Operating characteristic of the sampling plan for enforcement testing with a loss tolerance of 112 percent ................................................. 39 29. Testing burden of the sampling plan for enforcement testing with a loss tolerance of 112 percent. 40

v

# List of Tables 1. NEMA Class 1 efficiency levels for liquid

-filled distribution transformers

............... 3 2. NEMA Class 1 efficiency levels for dry-type distribution transformers . ............... 3 B1. NEMA Standard Publication TP 2 Table 5 .............................. 44

vi

## An Analysis of Efficiency Testing Under the Energy Policy and Conservation Act: A Case Study with Application to Distribution Transformers

I Introduction
This report provides analysis of the protocols for efficiency testing promulgated by 10 CFR Part 430 [1] as established under the Energy Policy and Conservation Act of 1975 (EPCA) as amended [2], and proposed forthe new 10 CFR Part 432 [3]. Specifically, the performance of these testing protocols in establishing conformance with a rated efficiency or a rated energy use is addressed . The case of distribution transformers, which are covered under the proposed new 10 CFR Part 432, is discussed in detail . This discussion includes analysis of existing industry standards for transformer efficiencies

. These results are presented , in part, to assess the impact of EPCA rule making on the transformer industry. In the case of distribution transformers , laboratory measurements of efficiency are likely to be used for two purposes : 1) for testing of compliance with a rated efficiency; and 2 ) for enforcement testing . The objectives of testing for each of these purposes differ in significant ways : Compliance testing is a one-time activity undertaken at the initiation of the program or upon the introduction of a covered product, while enforcement testing would be undertaken when the performance of a specific product or products is contested

. Enforcement testing is one of a series of requirements during an enforcement action. in NEMA Standard TP 2. A Sampling Plan for Enforcement Testing which is adapted form that provided in 10 CFR Part 430 appears in appendix C. Finally, the computational algorithms used to model each of these sampling plans are presented in appendix D. It should be emphasized that this report is not a statement of policy by the U.S. Department of Energy (DOE) and that this report should ' be regarded only as commentary on these testing protocols. DOE contacts for further information on the rule making for distribution transformers and on EPCA legislation are provided in Section 8. 

## 2 General guidelines In this section, we summarize briefly the general objectives of testing under EPCA. This material is included to provide ground rules for evaluating these various sampling plans. A statement of purpose for the EPCA legislation is given in 42 U

. S.C. 6312(a): It is the purpose of this Part to improve the efficiency of electric motors and pumps and certain other industrial equipment in order to conserve the energy resources of the Nation. The remainder of this report is organized as follows: The general objectives and constraints for testing under EPCA are discussed in Section 2. Current industry practice regarding efficiency performance and efficiency testing is reviewed in Section 3. The methods and model assumptions used in the evaluation of these testing protocols are presented in Section 4. A discussion of the sampling plans for compliance testing provided in 10 CFR Part 430, the proposed new 10 CFR Part 432, and NEMA Standard TP 2 is presented in Section 5. The Sampling Plan for Enforcement Testing established by 10 CFR Part 430 is presented in Section 6. For the convenience of the reader, each of these sampling plans are provided in appendices to this report: Appendix A contains an example sampling plan from 10 CFR Part 430 and the proposed 10 CFR Part 432 sampling plan for compliance testing. Appendix B contains the sampling plan provided To this end , EPCA establishes energy performance standards that may specify acceptable levels of efficiency or energy use for each covered product. EPCA further requires that any represented energy performance be accurate in 42 U.S.C. 6314(d)(1): . no manufacturer , distributor retailer or private labeler may make any representation - (A) in writing (including and representation on a label), or (B) in any broadcast advertisement, respecting the energy consumption of such equipment or cost of energy consumed by such equipment, unless such equipment has been tested in accordance with such test procedure and such representation fairly discloses the results of such testing.

### NISTTN 1427 An Analysis of Efficiency Testing Under the Energy Policy and Conservation Act:

EPCA relies on a program of systematic testing to es

- for Measuring the Energy Consumption of Distributablish that these performance levels are met. The tion Transformers " [5]. The efficiency requirements objectives and limitations of t esting under EPCA are for NEMA class-1 designation are established in secstated in 42 U.S.C. 6314(a)(2): tion 4 of the NEMA TP 1 standard, which applies to Test procedures prescribed in ac=dance with this section shall be reasonably designed toproduce test results which reflect energy efficiency, energy use , and estimated operating costs of a type of industrial equipment (or class thereof) during a representative average use cycle (as determined by the Secretary), and shall not be unduly burdensome to conduct. This report assumes that the purpose of EPCA is satisfied if the average efficiency is not less than the efficiency standard established under EPCA and the rated efficiency. In the case of an energy use standard, this report assumes that the purpose of EPCA is satisfied if the average energy use is not greater than the energy use standard established under EPCA and the rated energy use. -. To re- cap, for purposes of this analysis , we assume that the performance objectives are met provided that the average energy efficiency (energy use) is not less (greater) than the EPCA energy performance standard and the rated value

. Compliance with a rated energy performance is established under EPCA by systematic testing; and EPCA stipulates that testing should not be unduly burdensome to conduct : For the purposes of this analysis , two criteria are considered : 1) the assurance provided by a sampling plan that the average performance of each covered product meets or exceeds the rated performance , and 2) the burden placed on industry by testing under that sampling plan. 

# 3 Industry practice
This section includes a brief discussion of current industry standards for distribution transformers

. In a rule making , it may be useful to clarify current industry practice and , where current industry practice is consistent with the purpose of EPCA, to harmonize any energy performance standards established under EPCA with these practices. The energy efficiency and energy consumption of distribution transformers is covered by two standards sanctioned by the National Electrical Manufacturers Association (NEMA): NEMA Standards Publication TP 1- 1996 , "Guide for Determining Energy Efficiency for Distribution Transformers " [4] and by NEMA Standards Publication TP 2-1996 "Standard Test Method both liquid-filled and dry-type distribution transformers. The NEMA TP 1 Standard establishes "minimum efficiencies" for liquid-filled and dry-type single- and three-phase transformers , which are tabulated tabulated in Table 4-1 and Table 4-2, respectively. Dry type transformers are further differentiated by their voltage rating. The NEMA standard TP 1 tables are reproduced in part here in Tables 1 and 2, respectively. NEMA standard TP 1 states that these values are the "minimum efficiencies " for transformers designated as NEMA class 1 . For purposes of this discussion, the phrase "minimum efficiencies" will refer to the minimum average efficiency of a population. NEMA Standard TP 2 [5] section 7 provides a sampling plan designed to establish compliance with the TP 1 efficiencies . Section 7 of the TP 2 standard is included in appendix B of this report and is discussed below in section 5.2. Two standards sponsored by the Institute of Electrical and Electronics Engineers (IEEE) are also relevant to this discussion : ANSI Standard 57.12.00-1993 [6], "General Requirements for Liquid- Immersed Distribution, Power and Regulating Transformers " and ANSIStandard 57.12.01, "General Requirements for DryType Distribution and Power Transformers, Including Those with Solid Case and/or Resin

- Encapsulated Windings " [7]. These standards cover abroad range of transformer requirements , including recommended tolerances on measured losses : under these standards, the total losses of a single unit shall not exceed 106 percent of the rated value, and the average loss for two or more transformers shall not exceed the rated value. The second requirement , which is on the average loss, is being debated within the sponsoring committee and may bemodified or deleted in future versions of these standards. Since the measured losses are directly related to the efficiency, these standards, in effect , establish a tolerance for the measured efficiency and thus provide a level of quality assurance for efficiency. 

## 4 Methods of analysis Two figures

-of-merit provide the basis for the evaluation of these testing protocols: the operating characteristic' and the testing burden . The operating characteristic of a sampling plan is the probability of demonstrating compliance when testing a specific distribution of efficiencies under that sampling plan . This quantity provides an estimate of the probability or risk that anacceptable product could fail by chance or that an un2

### A Case Study with Application to Distribution Transformers NISTT'N 1427

Table 1 : NEMA Class 1 efficiency levels for liquid-filled distribution transformers . (Adapted from NEMA Standards Publication TP 1, Table 4-1 (4j.) kVA Single Phase Efficiency 

### kVA Three Phase

. Efficiency 10 98.3 15 98.0 

### 15 98.5 30 98.3 25 98.7 45 98.5 

### . 37.5 98.8 75 98.7 

### 50 98.9 112.5 98.8 

### 75 99.0 150 98.9 100 

### 99.0 225 99.0 

### 167 99.1 300 99.0 250 99.2 500 99.1 

### 333 99.2 750 99.2 

### 500 99.3 1000 99.2 667 99.4 1500 99.3 833 99.4 2000 99.4

2500 99.4

Table 2: NEMA Class 1 efficiency levels for dry -type distribution transformers . (Adapted from NEMA Standards Publication TP 1, Table 4-2 [41.) Single Phase Three Phase Efficiency Efficiency kVA Low Voltage Medium Voltage kVA Low Voltage Medium Voltage 

### 15 97.7 97.6 15 97.0 96.8 25 98.0 97.9 

### 30 97.3 97.3 37.5 98.2 98.1 45 97.7 97.6 

### 50 98.3 98.2 75 98.0 97.9 75 98.5 98.4 112.5 98.2 98.1 100 98.6 98.5 

### 150 98.3 98.2 

### 167 98.7 98.7 225 

### 98.5 98.4 250 98.8 98.8 300 98.6 98.5 333 98.9 98.9 500 98.7 98.7 

### 500 - 99.0 750 98.8 98.8 667
- 99.0 1000 98.9 98.9 

### 833
- 99.1 1500 

### - 99.0 

## 2000 - 99.0 2500
- 99.1

3

### NISTTN 1427 An Analysis of Efficiency Testing Under the Energy Policy and Conservation Act:

acceptable product could pass by chance under that sampling plan

. The second figure

- of-merit, the testing burden, is the average number of units tested underthat sampling plan. The testing burden may be considered in estimating the average cost of testing and in recommending minimum or m aximum s ample sizes . These testing protocols are examined by means of model calculations. Two approaches are taken in this analysis : Firstly, where analytic expressions or numerical approximations for the desired figures-of

-merit are derived, these expressions are evaluated numerically.A discussion of these algorithms is presented in appendix D and in [8, 9, 10]

. Secondly, Monte Carlo [11] simulations are used to estimate these figures-of

-merit. These approaches differ substantially, and each shouldbe considered an independent estimate of the figureof-merit. Comparison of the results obtained by numerical computation and Monte Carlo simulation may thus lend support to the calculated value. Monte Carlo methods have an added advantage for this analysis, in that they may be adapted to evaluate sampling plans where the sample size is not fixed, and may thus provide estimates of the operating characteristic and the testing burden for such plans. The nature of testing under the sampling plans given in 10 CFR Part 430, the. proposed new 10 CFR Part 432, and NEMA Standard TP 2 introduces a computational problem , in that the number of units tested may not be fixed from the outset of testing : For most products tested under the 10 CFR Part 430 sampling plan, a manufacturer could test as few as two units, but may test any larger number

. One exception to this minimum sample size of two is lamp ballasts for which not fewer than four units must be tested

. Under the proposed new 10 CFR Part 432 for distribution transformers, amanufacturer could test as few as five units of each basic model , but may test any larger number of units, including all units . Under the TP 2 sampling plan a manufacturer may test all units manufactured during a period of 180 days, or may conduct monthly tests over a period of 180 days of no fewer than five units. The scenario in which the sample size is not fixed is difficult to characterize by analytical methods, and we have chosen to first treat these sampling plans in the approximation that the sample size is fixed from the outset
. Results obtained under this approximation provide a lower bound on the probability of being found in compliance

. For example , while testing under the 10 CFR Part 430 sampling plan , a fixed sample of fiveunits would include some cases for which a manufacturer was found in compliance after testing two units. and could have stopped testing at that point , but fails after testing five units due to the final three test results. The computed probabilities for a fixed sample size may thus under-estimate the probability of being found in compliance.Detailed information regarding the distribution of efficiencies is required for these calculations; and we assume that the population of efficiency is normally distributed with mean, p, and standard deviation, a. We further assume that the units tested are selected at random. 

### 4.1 Loss representationWe have chosen, for this analysis, to represent transformer energy performance in terms of energy use, i.e. losses. This representation is chosen because it provides a direct comparison with industry standards. Further,the loss representation has the advantage of being independent of efficiency: The tolerance on the measured loss specified in the NEMA and IEEE standards is given as a percentage of the rated loss and is independent of efficiency. Thus the operating characteristic and testingburden are independent of efficiency in the loss representation.In the discussion that follows, the operating characteristic and testing burden of each sampling plan arerepresented by contour plots in coordinates of the average loss and the standard deviation of loss, where both are given as a percentage of the rated loss. In all cases, the rated loss corresponds to 100 percent on the loss axis. Expressed as a percentage , transformer efficiency, E, is given by the following equation:

E= 

# PPL x100, where P is the output power and L is the loss power. In practice , the no-load loss is measured at the rated voltage and the load loss at the rated current and adetermination of efficiency involves three critical measurements : power, voltage, and current

. Of these, the measurement of loss power is the most difficult resulting in measurement uncertainties that are about an order of magnitude greater than those of voltage and current measurement.In summary, the measurement uncertainty of the efficiency depends primarily on the uncertainty of the loss measurement

. This discussion is provided by way of explanation for the specification of a tolerance on the measured loss in the IEEE and NEMA standards. Sincethe critical measurement contributing to the measurement uncertainty in the efficiency is the uncertainty in the loss measurement , as a practical matter it is most effective to specify a tolerance for the measurement of loss.
4

### A Case Study with Application to Distribution Transformers



# 5 Compliance testing Three sampling plans are discussed : the sampling plan put forward by 10 CFR Part 430 [ 1], the sampling plan proposed for the new 10 CFR Part 432 [3], and the sampling plan established by NEMA Standard TP 2 [5]. 

### 5.1 10 CFR. Part 430 and the proposed 10 CFR Part 432 (Method I) '. In the discussion that follows, we refer to the sampling plans provided in 10 CFR Part 430 and proposed for the new 10 CFR Part 432 as Method I type plans. 

### NISTTN 1427 CFR Part 430, the statistical confidence specified in (i)(B) and (ii)(B) ranges in value between 90 percentand 99 percent. While the divisor ranges in value between 1.01 and 1.10 in (i)(13) and between 0.90 and 0.99 in (ii)(B). Under this sampling plan a manufacturer is requiredto test only as needed to demonstrate the specified statistical confidence

. The rule language in 10 CFR Part 430 states that the sample should be of "sufficient size" to ensure a statistical confidence that is not less than 971/2 percent, in the example cited

. Assuming that the statistical confidence is established from test data alone , the minimum sample size under this plan is two. The reason for this minimum value is that the sample standard deviation is not defined for a sample of one. 5.1.1 10 CFR Part 430 10 CFR Part 430 establishes specific criteria for each covered product when testing for compliance with a rated energy performance

. An example of these criteria, which applies to dishwashers , is provided in appendix A of this report

. The general structure of thissampling plan is followed for all other products covered under 10 CFR Part 430

. To emphasize its salient features , we paraphrase the Part 430 sampling plan as follows: 5.1.2 The proposed 10 CFR Part 432

(i) Compliance with a rated energy consumption is demonstrated provided; (A) The average energy consumption of the sample is not greater than the rated energy consumption, and (B) The upper 971/2 percent confidence limit of the average energy consumption of the entire population divided by 1.05 is notgreater than the rated energy consumption. (ii) Compliance with a rated efficiency is demonstrated provided; (A) The average efficiency of the sample is not less than the rated efficiency, and (B) The lower 971/2 Percent confidence limitof the average efficiency of the entire population divided by 09 is not'less than the rated efficiency. A Method I type plan is also proposed for distribution transformers in 10 CFR Part 432 [3]. The proposed sampling plan is provided in appendix A of this report. We paraphrase this sampling plan as follows: A sample of not fewer than five units must be tested

. Compliance with a rated energy efficiency is demonstrated provided: (a) The mean efficiency of the sample is not less than the rated efficiency, and (b) The lower 95 percent confidence limit ofthe average efficiency of the entire population divided by [1 - 0.03(1 - E,/100)], where E, is the rated efficiency, is not less than the rated efficiency.
Criteria for the demonstration of compliance with a rated loss are not proposed in 10 CPR Part 432. However , the criteria proposed for compliance testing may be stated in terms of loss; and such loss criteria are fully equivalent to the sampling plan for compliance testing proposed 10 CFR Part 432. These criteria are included here for illustrative purposes and for consideration of users who may wish to analyze intermediate test data in terms of loss power . When stated in terms of loss, the proposed sampling plan may be paraphrased as follows: For a given population , the operating characteristic and testing burden of these criteria depend on three factors: 1) the sample size, 2) the statistical confidence in (i)(B) and (ii)(B) and 3) the value of the divisor specified in (i)(B) and (ii)(B). The statistical confidence and the divisors are underlined in the above text. Under 10 A sample'of not fewer than five units must be tested

. Compliance with a rated loss is demonstrated provided:

5 (a) The average loss power of the sample is

not greater than the rated loss power, and

### NISTTN 1427 An Analysis of Efficiency T esting Under the Energy Policy and Conservation Act: (b) The upper 95 percent confidence limit of the loss power of the entire population divided by 1. 03 is not greater than the rated loss power. It should be emphasized that the divisor in criterion (b) above establishes a tolerance on the estimate of the mean loss of the population. This tolerance on the estimate of the mean loss should not be confused with the considerably larger loss tolerance established by ANSI Standards 57.12.00-1993 and 57.12.01 and NEMA Standard TP2: The loss tolerance established by these standards applies to the measured loss of a single unit. It may be noted that the divisors in criterion (b) differ when the criteria for compliance testing are stated in terms of efficiency or loss . This feature differs from the 1OCFR Part 430 sampling plans . In the example provided from 10 CFR Part 430, the divisors applied in the cases of efficiency testing and energy consumption testing are 1 +0.05 and 1-0.05, respectively. In effect, these divisors establish a tolerance of 5 percent for both the estimate of the mean efficiency and the estimate of the mean energy consumption . The use of an equivalent tolerance on the estimate of the mean efficiency and mean loss may not apply for devices that are highly efficient such as distribution transformers . The relationship between the uncertainty in the measurement of efficiency and of loss is given by m = -n(1- E,/100),where m is the measurement uncertainty in the efficiency, n is the measurement uncertainty in the loss, and E. is the rated value of efficiency for the product. Clearly, in the limit that E. approaches 100 percent the measurement uncertainty in the efficiency is much less than in the loss.

5.1.3 Results The results of model calculations for the Method I sampling plan are shown in Figs . 1-9, where data are presented for a statistical confidence of 95 percent and for values of loss tolerance of 103 percent, 106 percent, and 112 percent

. Sampling plans using both fixed samples and variable sizes are treated . The results shown in Figs. 2 to 5 are of particular interest to the transformer case. 

### For samples of fixed size, a rated efficiency is demonstrated provided: 1. the sample average loss is less than 100 percent, and 2. the upper 95 percent confidence limit of the mean loss is not greater than the loss tolerance. 

### The condition on the loss tolerance LT is equivalent to the use of a divisor in (i)(B). A sampling plan based on a loss tolerance of 103 percent, for example, exhibits the same behavior as the use of a 1.03 divisor. Samples of fixed size are first examined

. The operating characteristic of the Method I sampling plan for a loss tolerance of 103 percent for samples of 2, 5, 10, 20, and 30 units are shown in Figs

. 1-5, respectively. The dependence of the operating characteristic on the loss tolerance is shown in Figs

. 6-9, which are for the samples of two and five and for loss tolerances of 106 percent and 112 percent

. Monte Carlo simulations werealso performed for each of these cases and results obtained by these methods are shown in half

- tone in each of these figures. Aside from the random fluctuations, which are inherent to Monte Carlo methods , the two methods appear to give equivalent results.Examination of these data suggests the following interpretation of the Method I sampling plan: For samples of fixed size , the Method I sampling plan provides assurance at the specified statistical confidence that the average loss does not exceed the loss tolerance

. In this case, the statistical confidence is 95 percent and thus the risk of exceeding the loss tolerance is not greater than 5. percent. This behavior is indicated by the asymptotic limit of the 0

. 05 contours , which in eachfigure approach the loss tolerance at large standard deviation. Two additional analyses, which may be modeled by Monte Carlo simulations , are suggested by these results : 1) the operating characteristic and 2 ) the testing burden under this sampling plan when the sample size is not fixed

. In principle , the Monte Carlo simulation should provide a more realistic estimate of the Method I operating characteristic and testing burden. The conditions for a determination of compliance for variable sample size are identical to those used for a fixed sample ; however , in this case, testing is continued up to a maximum sample of twenty or until the following condition is satisfied:

>( LT t5 1a

- 100) where n is the sample size, LT is the loss tolerance, S is the sample standard error , and t is the fifth percentile of the t

- distribution for n - 1 degrees of freedom. The results of these simulations are presented inFigs. 10 and 11, which respectively depict the operating characteristic and the testing burden at a loss tolerance of 103 percent

. These operating characteristics suggest that the assurance on loss performance may be somewhat less than for fixed samples, as the asymptotic limit of the 0

. 05 contour appears to be on the order of 106 percent. These data further suggest 

### 6

### A Case Study with Application to Distribution Transformers that the marginal loss performance required to provide a high probability of demonstrating compliance may be appreciable under this sampling plan , where marginal loss performance refers to the difference between the rated loss and the true average loss

. It should be emphasized , however, that the operating characteristics shown in the figure are those for minimal testing' and that, under the Method I sampling plan, a manufacturer may be able to reduce the marginal loss by increasing the sample size. 

### 5.2 1EMA Standard TIS 2 (Method II and Method III)
The NEMA TP 2 sampling plan for demonstrating the efficiencies established by NEMA Standard TP 1 is next presented . The NEMA TP 2 sampling plan is given in sections 7.0 and 7

. 1 of the standard, which can be found in appendix B of this report. NEMA Standard TP 2 provides that the aggregate measured input power of all transformers tested for a period of 180 days be no greater than that allowed by the rated efficiency. The proposed 10 CFR Part 432 contemplates measurement of the average efficiency of a basic model , where a basic model may be distinguished by the significant characteristics of that product such as the kVA rating , voltage rating, and insulation type,for example. The question of whether it is appropriate to aggregate a broad range of product types forthe purposes of determining an average energy performance , as is allowed under NEMA Standard TP 2, is not addressed here

. For purposes of this discussion, we assume that the TP 2 sampling plan is applied to measure the average efficiency or the average energy use of a basic model

. When applied to a basic model, this condition on the measured input power is equivalent to an average measured efficiency that is not less than the rated efficiency and an average measured loss that is not greater than the rated loss. The TP 2 sampling plan thus establishes a condition on the mean efficiency or mean loss of a sample. The TP 2 sampling plan provides further that no unitmay exceed 108 percent of the rated loss. This tolerance on the measured loss may be interpreted in two ways : 1) The tolerance on the measured loss may establish a quality control limit for the NEMA Class I rating, but this tolerance has no bearing on any demonstration of compliance with a rated efficiency

. This interpretation of the TP 2 sampling plan places a condition on the mean of the sample

. This scenario will be referred to here as a Method II type sampling plan

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

### NISTTN 1427 measured loss of any unit in the sample

. This scenario will be referred to here as a Method III type sampling plan. Method III type criteria are used for quality assurance of various packaged goods [12, 13] and havebeen proposed for use in testing whether electric motors comply with a rated efficiency [9]. An approximate solution for the probability that such conditions on the sample mean and on the sample extremum are satisfied is discussed by Vangel [8]. Two scenarios for sampling are presented in TP 2: all units manufactured during a 180 day period may be tested , or units may be tested on a monthly. basis for a period of 180 days. The standard provides that no fewer than five units may be tested during thesemonthly tests and that the sample size shall be sufficient to ensure a statistical confidence of not less that 95 percent. The statistical confidence is established by the condition n = (tSK)2, where n is the sample size , S is the sample standard deviation , the value of t is selected according to the number of units tested as tabulated in TP 2 , and the coefficient K is given by the equation

_ 108 - 0.08SEL 

### K SEL(8 - 0.08SEL)' where SEL is the standard efficiency

. level in percent. This condition on the sample size is equivalent to that used under the Method I sampling plan and is that recommended by ASTM Standard E 122

-89114).


### 5.2.1 Method II results The operating characteristics and testing burden of Method II are presented in Figs . 12-17. Under Method II, one condition must be satisfied to demonstrate compliance with a rated performance: 1. the mean loss of the sample must be no greater than 100 percent. The operating characteristics for samples of fixed size are shown in Figs. 12-15, which depict samples of 5, 10, 20, and 30 units. Monte Carlo simulations were again performed for each of these cases and the results of these simulations are shown in each of these figures in half-tone. Numerical computation and Monte Carlo simulation appear to yield consistent values. The case of variable sample size is examined in Figs. 16 and 17, which depict the operating characteristics and the testing burden of Method II, respectively. These data apply to the case of monthly testing under TP 2 7

### NISTTN 1427 An Analysis of Efficiency Testing Under the Energy Policy and Conservation Act:

where the the minimum units tested is thirty. The conditions for a demonstration of compliance for a variablesample are identical to those for fixed sample size; however, in this case testing is continued until the condition



### n>^ t5 2 LT -100 ) is satisfied.
5.2.2 Method III results The operating characteristics and testing burden of Method III are presented in Figs . 18-23. 

### Under Method III, two conditions must be satisfied to demonstrate compliance with a rated performance: 1. the mean loss of the sample must be no greater than 100 percent, and2. no single unit in the sample may exceed 108 percent of the rated loss. It should emphasized that under Method III the entire population is not in compliance if any unit in the sample exceeds 108 percent of the rated loss. The operating characteristics for samples of fixed size are shown in Figs . 18-21, which depict samples of 5, 10, 20, and 30 units . Monte Carlo simulations were again performed for each of these cases and the results of these simulations are shown in each of these figures in half-tone. Numerical computation and Monte Carlo simulation appear to yield consistent values. The case of variable sample size is examined in Figs. 22 and 23, which depict the operating characteristics and the testing burden of Method III, respectively. These data apply to the case of monthly testing under TP 2 where the the minimum units tested is thi r ty. The conditions for a demonstration of compliance for a variable sample are identical to those for fixed sample size; however , in this case testing is continued until the condition

> 00 

# ( i ) 2 

### is satisfied or the sample exceeds twenty, whichever oc

### curs first. It should be noted that a maximum sample size is not stipulated in the TP 2 sampling plan: the maximum sample size of twenty used here was included 

### to simplify the Monte Carlo simulation. at small standard deviation

. This is due to the condition placed on the sample mean loss under these sampling plans : Each of these sampling plans require that the sample mean loss be no greater than the rated loss;thus at small standard deviation, where the determination of-compliance is based primarily on the sample mean loss, each of these sampling plans exhibit similar operating characteristics. It should be noted that the operating characteristic of these sampling plans depends on the sample size. However, these sampling plans exhibit very different behavior with increased sample size . For example, the marginal loss required under Method I to provide a high probability of demonstrating compliance decreases with sample size. Thus-the marginal loss may be reduced to an acceptable level under the Method I sampling plan by additional testing. The behavior of the Method III sampling plan is much different . Under Method III, the risk of failing to demonstrate compliance may actually increase with increased sample size . The reason for this

behavior is that as the sample size increases the likeli hood of sampling from the wing of the distribution of loss increases. The design performance of a transformer may depend on both engineering and business factors. However, it appears from this analysis that each of these sampling plans favor loss performance below the rated value. Since, if a transformer were designed to perform, and indeed performed at the rated loss on average, the likelihood of demonstrating compliance with that rated performance is not greater than 50 percent under each of these plans. The Method I, Method U, and Method III sampling plans all provide assurance that the mean performance of a population meets or exceeds a rated value. The Method I sampling plan appears to be perform well in cases where large quantities are available for testing and the cost of testing is low . However, one unique character of distribution ' transformers is that the lot size can be small

. Indeed some transformers are manufactured in extremely limited numbers. In that the marginal loss can be appreciable for small samples, the Method I sampling plan may, in -effect, require thatcertain transformers have average losses that are appreciably less than their rated loss. Alternately the Method III sampling plan may place a high burden on manufacturers when it is applied to large samples. 

## 6 Enforcement testing 5.3 Discussion The Sampling Plan for Enforcement Testing provided For a fixed sample size , the operating characteristics of in 10 CFR Part 430 applies to all products covered unMethod I , Method II, and Method III are very similar der Part 430. A similar plan has been proposed in the 

### 8

### A Case Study With Application to Distribution Transformers NISTTN 1427

new 10 CFR Part 431 for electric motors. The proposed sample, i.e. the number of transformers tested, and 2) 10 CFR Part 432 [31 does not include a Sampling Plan the underlying variability in the entire population. The for Enforcement Testing

. Its development will be con

- sample standard deviation, sidered at a later date. While some details may change, the general format of the plan is expected to remain same if and when it is finalize for transformers. The Sampling Plan for Enforcement Testing providedin 10 CFR Part 430 establishes clearly delineated procedures to be followed under an enforcement action.This sampling plan is based on a well established statistical method, which is due to C. Stein j15]; for obtaining a confidence interval on a mean . A discussion of this procedure can also be found in Bickel and Doksum [16). The Sampling Plan for Enforcement Testing provided in 10 CFR Part 430 covers both testing of efficiency and for energy consumption and. it is very general

. The sampling plan presented in appendix C has been adapted from 10 CFR Part 430; however, it has been simplified somewhat to clarify the procedure and it includes only the case for efficiency testing. The sampling plan is based on a t-test . The t-test iswell suited to this application as it is known to be insensitive to departures from the assumption of normality: The t-test is a test on a mean , which is an average of independent values obtained by a random sample. Since, sums of arbitrary, independent random values tend to have a distribution that. is almost normal, the t-test is not strongly influenced by the exact form of the underlying distribution of efficiencies.Since the test results obtained during enforcement testing may recommend that certain adverse actions be taken against a manufacturer : relabeling of specific products , the cessation of distribution and sale of certain basic models , and/or the assessment of fines, for example ; the risk to a manufacturer of a false determination of noncompliance during an enforcement action is set, by design, to a negligible level. The Sampling Plan for Enforcement Testing provided in 10 CFR Part 430 is based on a 97

. 5 percent statistical confidence , thus the risk to a manufacturer of a chance false determination of noncompliance is not greater than 2.5 percent.


### 6.1 10 CFR Part 430
An estimate of the true mean efficiency is first obtained by a random sample, 1 "

JC=-EX;, n i=1 

### (1) 

## S= ^.(X{-X)2 n-1 ' 

### (2)provides an estimate of the population standard deviation a; and the standard error in the mean,

SE(X) _ =, (3) provides an estimate of the standard deviation of the mean for samples of n units. For a normal distribution with mean p; the ratio,



### f_ p-X SE( ' (4)is distributed according to a probability density function that is know in statistics literature as the tdistribution. Now eq. 4 may be rearranged to provide an expression for the mean of the sample: (5) 

### where the value oft is associated with a specific sample size and statistical confidence . Values of t are readilyavailable and are included in many references on statistics [17]. In a test of compliance with a rated efficiency, RE, we assume , by hypothesis, that the units to be tested are drawn from a population of transformers for which the mean efficiency is equal to or greater than the rated efficiency

. If t is the 97.5 percentile of the t-distribution for n

- 1 degrees of freedom , then the probability of obtaining a mean efficiency,



### X < RE + tSE(X ), (6) is not less than 97.5 percent

. This procedure recommends the upper control limit,



### UCL - RE + tSE(X). (7) To apply this method , a random sample is tested andthe mean and standard error in the mean are calculated. Based on the size of the sample and the confidence desired , the appropriate t-value is selected and the lower control limit calculated

. For example, 97.5 percent confidence and a sample size of five units yields a t-value of 3

. 18. Provided the mean efficiency obtained from the random sample is not less than the lower control limit, as defined by eq. 7, we may assert 

### :here X, is the measured efficiency of unit i , and n is with a confidence not less than 97.5 percent that the the number of units tested. The uncertainty in this true mean energy consumption of the entire population estimate depends on two factors: 1) the size of the is not greater than the rated efficiency. 9

### NISTTN 1427 An Analysis of Efficlency Testing Under the Energy Policy and Conservation Act:

In any statistical test there is some probability of incorrectly concluding noncompliance. By design, theprobability that the mean efficiency for a random sample drawn from this population of transformers would fail below the lower control limit, hence, the risk ofincorrectly concluding the basic model is in noncompliance, is not greater than 2.5 percent.There is some probability that the sample standard deviation may be large and that the lower control limit may be set, by chance, to an exceptionally low value.This circumstance may be avoided by placing a tolerance on the standard error in the mean , SE(S). The tolerance for the standard error should be chosen to be appropriate for that product and to be supported across the industry. 

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

### P.41 - x 100 P;,, + 0.08(P{„ - P"t) 14 x 100, 108 - 0.0814 (8) loss of the sample under each of these plans. Since under each of these plans the mean loss of the sample may not exceed the rated loss, each of these plans provide assurance that the mean efficiency of a population is not less than the rated efficiency. When testing small samples under Method I, the marginal loss performancemay be appreciable and may, in effect, require that certain transformers have average losses significantly less than their rated values . However, under Method I, a manufacturer may reduce the marginal loss by testing additional units . Alternately the Method III sampling plan appears to require an appreciable marginal loss performance when it is applied to large samples and may, in effect, require that certain transformers have average losses significantly less than their rated values. Enforcement testing : The Sampling Plan for Enforcement Testing included in 10 CFR Part 430 is based on statistical methods that are widely used and well documented. The sampling plan is robust, in that itis a test on the mean and that it is not highly dependent on the form of the underlying distribution. The sampling plan is designed to protect the interests of the manufacturer, in that the risk of a false outcome against a manufacturer may be limited to some acceptable level. Additionally, the risk of a false outcome is independent of variance. The analysis presented here is of value primarily asa qualitative evaluation of the operating characteristic and the testing burden of these testing protocolsand secondarily as a quantitative estimate of the statistical confidence associated with testing under these protocols. While the agreement noted between resultsobtained by analytical methods and Monte Carlo simulations is encouraging, it should be noted, since these results may depend on our model assumptions, that a quantitative estimation of statistical confidence is more tenuous than qualitative evaluation. where P;,, and P..d are the input and output power.The lower control limit must then satisfy two conditions: 

### UCL = RE - tSE(. X) (9) and UCL > 108 - o SRE x 100, (10) where ,the second condition is obtained from eq. (8) by setting the efficiency equal to the RE.


### 6.2 Res ults The operating characteristic and the testing burden of the Sampling Plan for Enforcement Testing are shown in Figs

. 24-29

. Factors that influence the operating characteristic and testing burden include : 1) the size of the initial sample , 2) the statistical confidence, and 3) the tolerance set on the standard error. It may be noted that the 97.5 percent contour lies along 100 percent loss in each of these figures and that the risk of a false outcome is therefore independent of variability under this sampling plan. 

# 7 Summary Compliance *test : The operating characteristic of the Method I, Method II, and Method III samplingplans are equivalent in the limit of small standard deviation

. This is due to the condition placed on the mean 

## 8 Further information For information on the rule making for distribution transformers and on EPCA legislation contact: 

### Kathi Epping U.S. Department of Energy Energy Efficiency and Renewable Energy Mail Station EE-43 

### 1000 Independence Avenue, SW Washington, DC 20585-0121 202-586-7425 kathi.epping®ee.doe.gov

10

### A Case Study with Application to Distribution T ransformers Edward Levy, Esq. U.S. Department of Energy Office of General Counsel Mail Station GC-72 1000 Independence Avenue, SW Washington, DC 20585-0103 202-586-9507 edward.levy©hq.doe.gov
For information on this report contact:
Ken Stricklett National Institute of Standards and Technology 100 Bureau Dr. Stop 8113 Gaithersburg , MD 20899-8113 301-975-3955 ken.stricklettC9nist.gov Oskars Petersons National Institute of Standards and Technology 100 Bureau Dr. Stop 8113 Gaithersburg, MD 20899-8113 301-975-2417 oskars

. petersons©nist.gov Mark Vangel National Institute of Standards and Technology 100 Bureau Dr. Stop 8980 Gaithersburg , MD 20899-8980 301-975-3208 mark

. vangel ®nist.gov 

## Acknowledgments This report was completed with support from the U.S.Department of Energy and the Technology Administration of the U

. S. Department of the Commerce. 

### NISTfN 1427

11

# Increasing Eff iciency 12



# 0 115 110 105 100 95



# Loss ( 

# perce nt) 90 

# 85

Figure 1 : Operating characteristic of the Method I compliance criteria for a fixed sample size of two

. The statistical confidence is 95 percent and the loss tolerance is 103 percent

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

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

# Increasing Efficiency 12

0

3



# 115 0 110 

# 105 100 95



# Loss ( percent) 90 85

Figure 2: Operating characteristic of the Method I compliance criteria for a fixed sample size of five

. The statistical confidence is 95 percent and the loss tolerance is 103 percent

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

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

# Increasing Efficiency

110 105 100 95



# Loss (percent) 90 8 5

Figure 3: Operating characteristic of the Method I compliance criteria for a fixed sample size of ten

. The statistical confidence is 95 percent and the loss tolerance is 103 percent

. The contours indicate the probability of demonstrating compliance , e.g., the 0

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

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

# Increasing Efficiency

110 105 100 95



# Loss ( 

# percent) 90 85

Figure 4 : Operating characteristic of the Method I compliance criteria for a fixed sample size of twenty
. The statistical confidence is 95 percent and the loss tolerance is 103 percent

. The contours indicate the probability of demonstrating compliance , e.g., the 0

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

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

# Increasi ng Efficiency

105 100 95 

# Loss 

# (percent)
Figure 5 : Operating characteristic of the Method I compliance criteria for a fixed sample size of thirty

. The statistical confidence is 95 percent and the loss tolerance is 103 percent

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

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

# Increasing Efficiency

110 105 100 95



# Loss (perce 

# nt) 9 

# 0 85

Figure 6: Operating characteristic of the Method I compliance criteria for a fixed sample two. The statistical confidence is 95 percent and the loss tolerance is 106 percent

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

# . I 

# ncre

# as i ng Eff ici en 

# c

# y 12



# 0

3

115 

# 0 

# n= E11

110 

# O®

105 100 95 

# Loss ( pe rce nt) 90 85

Figure 7: Operating characteristic of the Method I compliance criteria for a fixed sample five. The statistical confidence is 95 percent and the loss toleranceis 106 percent

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

# Increasing Efficiency 12



# U) 0 115 110 105 100 95



# Loss (percent) 90 85

Figure 8: Operating characteristic of the Method

. I compliance criteria for fixed sample of two

. The statistical confidence is 95 percent and the loss tolerance is 112 percent

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

# Increasing Efficiency 12

C 

# a) is 2



# 0115 V

0

n®
110 

# 0

105 ' 100 95 

# Loss (percent) 90

Figure 9: Operating characteristic of the Method I compliance criteria for fixed sample of five

. The statistical confidence is 95 percent and the loss tolerance is 112 percent. The contours indicate the probability of demonstrating compliance, e.g., the 0.90 contour corresponds to a 90 percent likelihood of demonstrating compliance.

# lncreasin^ Eff ici ency 12

C 

# v L



# 0 ® ° N ° A a !o.



# 115 110 105 100 95



# Loss (pe rcent) 90

Figure 10 : The operating characteristic of Method I. The results of Monte Carlo simulations for an initial sample size of two and a maximum sample of twenty are shown . The statistical confidence is 95 percent and the loss tolerance is 103 percent

. The operating characteristic of the Method I compliance criteria . The contours indicate the probability of demonstrating compliance , e.g., the 0.90 contour corresponds to a 90 percent likelihood of demonstrating compliance.

# I n creasi ng Effici ency 12

3

115 0 110 105 100 95



# Loss 

# (p

# ercent) 90 

# Cb

ro

2



# 85

Figure 11; The testing burden of Method I. The results of Monte Carlo simulations for an initial sample size of two and a maximum sample of twenty are shown. The statistical confidence is 95 percent and the loss tolerance is 103 percent . The contours indicate the average sample size at the end of testing.

b

# I n creasi ng Eff ici ency 12

0



# 3

115 0 "0 0



# 110 105 100 95



# Loss (perc ent) 90 85

Figure 12 : Operating characteristic of the Method II sampling plan for fixed sample of five

. The contours indicate the probability of demonstrating compliance, e.g., the 0

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

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

# Increasing Eff iciency 12

0 115 

# 0

110 

# i s



# 105 100 95



# L

# oss (percent) 90 8 5

Figure 13: Operating characteristic of the Method II sampling plan for a fixed sample of ten

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

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

# ncreasing Efficiency 12



# 3

0 

# 115 

# n= 0

110 105 100 95



# L

# oss (pe rcent) 90 85

Figure 14 : Operating characteristic of the Method II sampling plan for a fixed sample of twenty
. The calculations are based on an extremism of 108 percent of the rated loss. The contours indicate the probability of demonstrating compliance , e.g., the 0.90 contour corresponds to a 90 percent likelihood of demonstrating compliance

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

# Increasing Eff iciency 12

3

0 115 110 105 100 95



# Loss (percent) 90 85

Figure 15: Operating characteristic of the Method II sampling plan for a fixed sample of thirty

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

# I ncreasi ng Effici e ncy 12

110 105 100 95 90 85



# Loss 

# (pe r

# ce nt)
Figure 16: Operating characteristic of Method II sampling plan with variable sample size . Monte Carlo simulation for an initial sample size of five we shown. As in the previous figures, the contours indicate the probability of demonstrating compliance.

# Increasing Effici ency 12

40

35



# 0 3

0 115 110 105 100 95



# Loss 

# (perce nt) 90 85

Figure 17 : The testing burden of the Method II sampling plan. The results of Monte Carlo simulations are shown

. Under the TP 2 sampling plan, compliance with the rated efficiency may be established by monthly tests conducted over a period of 180 days

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

# Increasing Efficiency 12

U) 0



# 0 115 110 105 100 95 

# Loss 

# (p erc 

# e nt) 90 85

Figure 18 : Operating characteristic of the Method III sampling plan for fixed sample of five
. The calculations are based on an extremum of 108 percent of the rated loss. The contours indicate the probability of demonstrating compliance , e.g., the 0.90 contour corresponds to a 90 percent likelihood of demonstrating compliance. The results of Monte Carlo simulations are shown in half-tone.

# Increasing Efficiency

8 

# 12

0115 110

- 105 100 95



# Loss (percent) 90 85

Figure 19 : Operating characteristic of the Method III sampling plan for a fixed sample of ten

. The calculations are based on an extremum of 108 percent of the rated loss. The contours indicate the probability of demonstrating compliance, e.g., the 0.90 contour corresponds to a 90 percent likelihood of demonstr ating compliance. The results of Monte Carlo simulations are shown in half-tone.

# Increasing Efficiency

co 12

0



# 3



# 0 

# If 20

115 110 105 100 95



# Loss (percent) 90 

# 0

8 5

Figure 20 : Operating characteristic of the Method III sampling plan for a fixed sample of twenty. The calculations are based on an extremum of 108 percent of the rated loss. The contours indicate the probability of demonstrating compliance, e.g., the 0
. 90 contour corresponds to a 90 percent likelihood of demonstrating compliance

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

# Increasing Effici ency

115 110 105 100 95



# Loss ( 

# perc 

# e

# nt ) 12

0

3

0 

# n=
85 90

Figure 21 : Operating characteristic of the Method 1II sampling plan for a fixed sample of thirty. The calculations are based on an extremum of 108 percent of the rated loss. The contours indicate the probability of demonstrating compliance , e.g., the 0

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

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

# Increasing Effici ency 12

.3

115 0 0 0/



# 110 105 100 95



# Loss (percent) o oyJ



# 90 85

Figure 22: Operating characteristic of Method III sampling plan with variable sample size. Monte Carlo simulation for an initial sample size of five are shown. Based on an extremum of 108 percent of the rated loss. As in the previous figures, the contours indicate the probability of demonstrating compliance.

# Increasing Efficiency 12 45



# 9 35

30 

# 3



# 0 115 110 105 100 95



# Loss 

# (percent) 90 85

Figure 23 : The testing burden of the Method III sampling plan. The results of Monte Carlo simulations are shown

. The statistical confidence is 95 percent and the loss tolerance is 108 percent

. Under the TP 2 sampling plan, compliance with the rated efficiency may be established by monthly tests conducted over a period of 180 days

. Each monthly test would consist of no fewer than 5 units

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

# Increasing Efficiency

115 .90 85

110 105 100 95



# Loss ( 

# perc

# e n

# t

# )

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

. The statistical confidence is 97.5 percent and the loss tolerance is 103 percent
. The contours indicate the probability of demonstrating compliance, e.g., the 0

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

# Increasing Efficiency 12

3

115 0 110 105 100 95



# Loss (percent) 16 14 12



# 90 14 

### fl



# 85

Figure 25: Testing burden of the Sampling Plan for Enforcement Testing

. Model calculations for an initial sample of four and maximum sample of twenty are shown. The statistical confidence is 97

. 5 percent and the loss tolerance is 103 percent

. Contours indicate the average sample size at the completion of testing.

# Increasi ng Efficiency

W

0 115 110 105 100 95 

# Loss ( 

# pe rcent) 90 85

Figure 26 : Operating characteristic of the Sampling Plan for Enforcement Testing

. Model calculations for an initial sample of four and maximum sample of twenty are shown

. The statistical confidence is 97

. 5 percent and the loss tolerance is 106 percent

. The contours indicate the probability of demonstrating compliance, e.g., the 0

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

# Increasing Efficiency



# 0 115 110 105 100 95



# Loss (percent) 90 85

Figure 27: Testing burden of the Sampling Plan for Enforcement Testing. Model calculations for an initial sample of four and maximum sample of twenty are shown. The statistical confidence is 97

. 5 percent and the loss tolerance is 106 percent

. Contours indicate the average sample size at the completion of testing.

# Increasing Efficiency

% 115 - 110 105 100 95 

# Loss (percent) 90 85

Figure 28 : Operating characteristic of the Sampling Plan for Enforcement Testing . Model calculations for an initial sample of four and maximum sample of twenty are shown
. The statistical confidence is 97

. 5 percent and the loss tolerance is 112 percent

. The contours indicate the probability of demonstrating compliance, e.g., the 0

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

# Increasin g Effi ci en cy 12 ® C 0

0

0

# 115 110 105 100 95



# Loss ( 

# pe rce nt) 9 

# 0 4



# 8 

# 5

Figure 29: Testing burden of the Sampling Plan for Enforcement Testing

. Model calculations for an initial sample of four and maximum sample of twenty are shown. The statistical confidence is 97

. 5 percent and the loss tolerance is 112 percent

. Contours indicate the average sample size at the completion of testing.

### A Case Study with Application to Distribution Transformers



## References[1] Energy conservation program for consumer products. Code of Federal Regulations, Title II, Chapter 10, Part 430, 1-1-97 edition, pages 93-263, 1997. Office of the Federal Register, NationalArchives and Records Administration, Washington, DC.[2] Energy policy and conservation act of 1975, pubic law 94-163 ; national energy conservation policy act of 1978 , public law 95

- 619; national appliance energy conservation act of 1987 , public law 100-12; national appliance energy conservation amendments of 1988 , public law 100

- 357; and section 122(d), energy policy act of 1992, public law 102- 486, october 24, 1992. [3] 10 cfr part 432 energy conservation program: 

### [11] NISTTN 1427 chapter II of title 10, Code of Federal Regulations: Enforcement testing . NIST Technical Note 1422, National Institute of Standards and Technology, Technology Administration, U.S. Department of Commerce, December 1996. P.R. Bevington and D

. K. Robinson

. Data Reduction and Error Analysis for the Physical Sciences, chapter 5. McGraw-Hill, 1992. 

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

[13]

Test procedures for distribution transformers; pro- [14] 

### posed rule

. Federal Register, 63(218 ): 63360

-63372, Thursday, November 12 1998. [4] NEMA Standards Publication TP 1-1996, Guide [15] for determining energy efficiency for distribution transformers

. National Electrical Manufacturers Association, Rosslyn, VA, 1996.[5] NEMA Standards Publication TP 2-1998, Standard test method for measuring the energy comsumption of distribution transformers

. National Electrical Manufacturers Association, Rosslyn, VA, 1998. [6] ANSI C57.12.00-1993 IEEE Standard, General requirements for liquid- immersed distribution, power and regulating transformers . American National Standards Institute , New York, 1993. [7] ANSI C57.12. 01-1989 IEEE Standard , General requirements for dry-type distribution and power transformers , including those with solid case and/or resin-encapsulated windings . American National Standards Institute, New York, 1989. 

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

# [9] mean and extremum with applications to quality control

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

. NISTIR 6092 , National Institute of Standards and Technology, Technology Administration , U.S. Department of Commerce , January 1998.[10] K.L. Stricklett and M.G. Vangel. Electric motor efficiency testing under the new part 431 of [16]


### [17]
[18]

41 Checking the net contents of packaged goods. Nbs handbook 133, National Bureau of Standards, Gaithersburg, MD, 1988.M.C. Croarkin and G.L. Yang. Acceptance probabilities for a sampling procedure based on the mean and an order statistic

. J. Res

. Nat. Bur. Standards, 87:485-511, 1987. ASTM Standard E 122-89, Standard practice forchoice of sample size to estimate a measure of quality for a lot or process, 1989. C. Stein. A two-sample test for a linear hypothesis whose power is independent of variance

. Annals of Mathematical Statistics , 16:243

-258, 1945. P.J. Bickel and K.A. Doksum. Mathematical Statistics, pages 158-159

. Holden Day, Oakland, CA, 1977. R.S. Schulman

. ' Statistics in Plain English with Computer Applications. Van Norstand Reinhold, New York, 1992. Energy conservation program for certain commercial and industrial equipment : Test procedures, labeling , and certification requirements of electric motors ; proposed rule. Federal Register, 61(230):60439 -60475, Wednesday, November 271996.

## Appendix A

The 10 CFR 43 0 and the proposed 10 CFR Part 432 s ampling plans for compli ance testing
The following test criteria are presented as an example of compliance testing under 10 CFR Part 430. Similar criteria have been proposed for distribution transformers in the new 10 CFR Part 432 [3] and are also presented. The 10 CFR Part 430 criteria presented below apply specifically to dishwashers , which is one of the products covered under 10 CFR Part 430

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

. The statistical confidence and divisor used in criteria (i)(B) and (ffi)(B) differ between products and are specific to each product

. The values of these coefficients are chosen to be consistent with current industry practice for that product. 

### § 430.24 Units to be tested. When testing of a covered product is required to comply with section 323 (c) of the Act, or to comply with rules prescribed under sections 324 or 325 of the Act, a sample shall be selected and tested composed of units which are production units, or are representative of production units of the basic model being tested , and shall meet the following applicable criteria. (c)(1) For each basic model of dishwashers , a sample of sufficient size shall be tested to insure that-(i) Any represented value of estimated annual operating cost, energy consumption or other measure of energy consumption of a basic model for which consumers would favor lower values shall be no less that the higher of (A) the mean of the sample or (B) the upper 971 /2 percent confidence limit of the true mean divided by 1.05, and (ii) Any represented value of the energy factor of other measure of energy consumption of a basic model for which consumers would favor higher values shall be no greater than the lower of (A) the mean of the sample or (B) the lower 971/2 percent confidence limit of the true mean divided by 0.95. The criteria given below are proposed for distribution transformers in the new 10 CFR Part 432

. The values of statistical confidence and the formulation of the divisor specified in (b) are chosen to be consistent with current industry practice. 

### §432.24 Units to be tested. For each basic model of distribution transformers, a random sample of sufficient size , but no fewer than five productions units, shall be tested to insure that any represented value of efficiency shall be no greater than the lower of the: (a) Mean of the sample; or (b) The lower 95% confidence limit of the estimated true mean divided by a number equal to [1- 0.03(1- E,/100)], where E. is the represented value of efficiency claimed for that particular basic model.

42

## Appendix The ICE A. Standard TP 2 sam pling pl an
Sections 7.0 and 7.1 of NEMA Standard TP 2 follow

. The reader may note minor differences between this appendix and the NEMA standard TP 1, e.g., the pagination and the reference to the included table in this appendix differ from those in the NEMA standard TP 1. SECTION 7 Demonstration of Compliance 

### 7.0 General This section provides a methodology for proving compliance in achieving the specified efficiency levels

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

. This standard requires that no individual transformer shall be considered acceptable if its measured losses exceed the allowance by more than 8%. For a transformer population of a specific kVA rating, compliance with the energy efficiency standards as defined in Section 4 of NEMA TP 1 shall be demonstrated through measurements of No-Load and Load losses according to the procedures described in this standard

. A transformer model is defined as all transformers of the same kVA rating and type as described in the efficiency tables of NEMA TP 1. According to the IEEE Standards C57.12

. 00 and C57

. 12.01 the loss tolerance for an individual unit as related to guarantee is defined as follows: Limit Beyond Guarantee No Load Loss 10% max. Total Loss at 100% Load 6% max. At 100% load, the Load Loss is normally four times the No Load loss, suggesting that a Load Loss variability of approximately 5% is allowed. The sum of this allowance and the 10% variability for the No Load loss yields the Total Loss variability of 6%. Since TP-1 tables reflect loss measurements at 35% or 50% of rated load where No Load Loss is equal to Load Losses, the 6% loss tolerance cannot be used and therefore a new total loss tolerance of 8% shall be applicable at these measurement points. This is consistent with the IEEE Standards C57.12.00 and C57.12.01. 7.1 Number of Units to be Tested NEMA TP-1 requires that the overall efficiency of the entire population of transformers must meet the specified efficiency standards . This intent is satisfied if the mean efficiency of the entire population satisfies this requirement

. The compliance of a group of transformers shall be demonstrated by testing all or randomly drawn samples of these transformers. 7.1.1 Compliance Demonstration Through test on all Transformers: Manufacturers may choose to test all units of various kVA ratings manufactured during a production period of 180 days to demonstrate compliance with the efficiency standard NEMA TP 1. The intent of this standard is satisfied if the Total Measure kVA Input (TMI) of this batch of transformers is equal to or less than the Total Allowed kVA Input (TAI) calculated based on the measured and specified efficiency levels specified in TP 1. Each individual unit from this production batch must meet or exceed the minimum acceptable efficiency level calculated as follows: 

### Minimum Acceptable Efficiency level = 108 0

. 08SEL X 100 Where SEL = the standard percent efficiency level from NEMA Standard TP I. Note : The Minimum Acceptable Efficiency level calculation is based on an 8% tolerance on total loss at the load levels considered for the Efficiency levels specified in NEMA TP

- 1; i.e. no individual unit shall be considered acceptable if its measured losses exceed the allowance by more than 8%. 

### 43

### A Case Study with Application to Distribution Transformers NISTTN 1427

Sample size n 

### t at 95% confidencele vel ple size n Sam t at 95% confidence level 2 6.314 11 1.812 3 2.920 12 1.796 4 2.353 13 1.782 5 2.132 14 1.771 6 2.015 15 1.761 7 1.943 16 1.753 

### 8 1.895 17 1.746 9 1.860 18 1.740 

### LO 1.833 19 1.734 [

_ 20 1.729
To demonstrate compliance with the Efficiency standard, proceed as follows: 

### Step I. Calculate the Total Allowed KVA input:

Where . i = 1,2,3,4,... kVA; = kVA ratings of various transformers included in a production batch

manufactured in 180 days. Ili = Specified Efficiency Level in TP 1 for transformer rating kVA1 L; = Per unit load at which the efficiency is specified per TP 1 Step 2 . Calculate the Total Measured kVA input: 

### TMI = L{ kVA,

itma Where i = 1,2,3,4,... i1mi = Measured Efficiency Level for the transformer. L{ = Per unit load at which the efficiency is specified per TP 1 

### Step 3 . If TMI is equal to or less than TAI; the compliance of the production batch has been demonstrated. 7.1.2 Compliance Demonstration Through Tests on a Statistically Valid Sample: The manufacturer may choose to demonstrate the compliance of a plurality of units by a random sampling of the units of each kVA rating produced in a period of 180 days

. Statistically valid numbers of units but not less than 5 shall be drawn on a monthly basis from the units of each kVA rating produced during this period for testing

. This will assure the randomness of the samples (30 units minimum). All the units drawn in 180 day period shall be tested for computing the mean efficiency of each kVA rating. None of individual units in a sample shall be considered acceptable if its measured losses exceed the allowance by more than 8%. 

### It is the responsibility of the manufacturer to assure through adequate quality control procedures and/or random testing that the conformance of various kVA rating transformers is maintained. For a random sample to be statistically valid, a minimum number of units n in the sample must be tested to assure that the standard deviation of the test results is no more than the standard deviation S of the population with 95% confidence

. The minimum sample size shall be determined as follows: 

### n = (tSK)2 b

44

### A Case Study with Application to Distribution Transformers NISTTN 1427

where 

### _ 108 - 0.08SEL K SEL (8 - 0.08SEL) and t Statistic is determined from Table B1 corresponding to sample size of n at 95 % confidence level. To demonstrate compliance with this standard , proceed as follows: Step 1. Choose a sample size of ni units (5 min.) Step 2 . Compute the mean A'i and standard deviation S1 as follows: Xi= 1 EXa, ni

S1 = 

### E(X, - Xi)2 

# ni-1
Where X1 = The average efficiency of the first sample. Xi = The efficiency of the unit i n1 = The number of units in the first set of samples (the subscript refers

to the sample number) Si = The computed sample standard deviation of the first sample Step 3 . Calculate the minimum sample size n as follows: n = (t1S1K)2 Where ti Statistic is chosen from Table BI corresponding to the sample size ni. Step 4

. If n < n1, the sample size is adequate to yield the acceptable standard deviation
. Proceed with Step 6

. Otherwise , test additional units and increase the sample size to n2 such that: n2 > n, where n2 is the total number of units tested. Step 5. Repeat Steps 2 and 3 until each sample size for each kVA rating produced is larger than the minimum sample size n. Step 6

. Calculate the average efficiency %j of all the samples tested in 180 days for each rating kVA; manufactured within this period. Step 7. Calculate the Total Allowed kVA input: 

### TAI = E Ni Li kVAi

IN Where i = 1,2,3,4,... kVA; = kVA ratings of various transformers included in a production batch

manufactured in 180 days n; = Specified Efficiency Level in TP 1 for transformer rating kVA; L{ = Per unit load at which the efficiency is specified per TP 1 Ni = Total number of units produced with rating kVA; Step 8. Calculate the Total Measured kVA input: 

### TMI=EN

1L1kVA®

45

### A Case Study with Application to Distribution Tmnsf0 rs

ere i 

### kVAc 1,2,3,4,...

kVA ratings of various transformers included in a production batch manufactured in 180 days Measured efficiency Level for tr ansformers rated kVA, Per unit load at which the efficiency is specified per TP 1 Total number of units produced with rating kVA 

### NIST N 1427

Step 9

. If TMI is equal to or less than TAM; the compliance of the production batch has been demonstrated.

46

## Appendix C A s ampling plan for enforcem ent testing The following s ampling plan is adapted from the Sampling Plan for Enforcement Testing contained 10 CFR. Part 430. The plan is similar to that proposed for electric motors [18]. Step 1. The first sample size (ni ) must be four or more units. Step 2. Compute the mean (XI) of the measured energy performance of the nj units in the first sample as follows: n, XI = Xtl (Cl) . nj e-1 where Xi is the measured efficiency of unit i. 

### Step 3. Compute the sample standard deviation (SI) of the measured efficiency of the nj units in the first sample as follows:
S+1= 

# I (X'- 

# x 1)2 ni-1 

### Step 4. Compute the standard error (SE(RI)) of the mean efficiency of the first sample as follows: SE(X1) = sal . (C2)

(C3) Step 5. Compute the lower control limit (UCLI) for the mean of the first sample using the applicable rated efficiency (RE) as the desired mean as follows: UCLI = RE - tSE (XI ). (C4) Here t is the 2

. 5 percentile of a t

-distribution for a sample size of nj and yields a 97.5 percent confidence level for a one

-tailed t-test. 

### Step 6

. Compare the mean of the first sample (XI) with the lower control limit (UCLI) to determine one of the following: (i) If the mean of the first sample is below the lower control limit , then the basic model is in noncompliance and testing is at an end. (ii) If the mean is equal to or greater than the lower control limit, no final determination of compliance or noncompliance can be made ; proceed to Step 7. 

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

n tSi(108-0.08RE) 2 (C5) [. RE(8

- 0.08RE) j where S1 and t have the values used in Steps 4 and 5 , respectively

. The factor 108 - 0.08RE RE(8 - 0.08RE) is based on a 8 percent tolerance in 'the total power loss at and fixed output power. Given the value of n, determine one of the 'following: (i) If the value of n is less than or equal to nj and if the mean energy efficiency of the first sample (XI) is equal to or greater than the lower control limit (UCLI), the basic model is in compliance and testing is at an end. 

### 47

### A Case Study with Application to Distribution Transformers NISTTN 1427 (ii) If the value of n is greater than n1 , the basic model is in noncompliance. The size of a second sample n2 is determined to be the smallest integer equal to or greater than the difference n-nt. If the value of n2 so calculated is greater than 20 - nt , set n2 equal to 20 - nt. Step S. Compute the combined mean (X2) of the measured energy performance of the nj and n2 units of the combined first and second samples as follows: nl+n2 X2 = 1 Xt. Tit + net 

### (C6)
Step 9. Compute the standard error (SE(X2 )) of the mean efficiency of the nj and n2 units in the combined first and second samples as follows: 

### St SE(X2) = nl

-

t52 (Note that Si is

. the value obtained above in Step 3.) Step 10

. Set the lower control limit (UCL2) to, 

### (C7)

UCL2 = RE - tSE(X2), (C8) where t has the value obtained in Step 5, and compare the combined sample mean (X2) to the lower control limit (UCL2) to find one of the following: (i) If the mean of the combined sample (X2) is less than the lower control limit (UCL2), the basic model is in noncompliance and testing is at an end. (ii) If the mean of the combined sample (X2) is equal to or greater than the lower control limit (UCL2), the basic model is in compliance and testing is at an end.



### MANUFACTURER OPTION TESTING
If a determination of non

- compliance is made in Steps 6 , 7 or 11, above , the manufacturer may request that additional testing be conducted , in accordance with the following procedures. Step A

. The manufacturer requests that an additional number , n3, of units be tested , with n3 chosen such that nj + n2 + n3 does not exceed 20. Step B. Compute the mean efficiency , standard error , and lower control limit of the new combined sample in accordance with the procedures prescribed in Steps 8 , 9, and 10 , above. 

### Step C. Compare the mean performance of the new combined sample to the lower control limit (UCL2) to determine one of the following: (a) If the new combined sample mean is equal to or greater than the lower control limit , the basic model is in compliance and testing is at an end. (b) If the new combined sample mean , is less than the lower control limit and the value of nj+n2+n3 is less than 20, the manufacturer may request that additional units be tested

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

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

48

## Appendix D Computational algorithms
Compliance testing under 10 CFR. Part 430 (Method 1)
Let x1 i x2, ... , xn be measured losses, with sample mean i and sample standard deviation s. In a test of compliance, we require that

and i + is/-,Fn < ( 1 + 0)k, (D2) where t denotes the appropriate percentile of a Student

-t distribution with v = n -1 degrees of freedom, and 0 is a small positive number (typically, 0 = .03). Assume that the data {xj }{ 1 are an independent , identicallydistributed sample from a normal (Gaussian) distribution with population mean p and population standard deviation a. The probability of compliance is given by the expression 

### p(a, b) = 2n 

# I

a. L' 

### e_n12/sX2 [ t2 

### (a + b
- y)2] 

### dy,. (D3)
where a = ku and b = ck. (D4) In this expression , x ! (u) is the probability that a X2 random variable V ,, having v = n - 1 degrees of freedom is less than u; that is Pr(V„ < u) = X2 .(u) = r (Y)2- j zv/2-1e z/2dz, (D5) which is a function easily expressed and calculated in terms of an incomplete r-function

. Function (D5) can be evaluated numerically using a wide range of public domain software

. Gauss-Legendre quadrature is adequate for the calculation of (D3), and it is sufficient to use p - 5a / f for the lower bound of the numerical integral. The limits of p(a , b) as or goes to zero or infinity are of particular interest; we consider these in turn
Limit of small a: As a - ► 0, b -i oo

. If a is allowed to go to infinity as well , then p (a, b) - ► 1, which isn't useful. So we require that b -> oo and that a remains finite ; that is -oo < lim k - p . = q < oo. (D6) a- ► 0 a If this condition holds, then

r_ n 9 _nya/2' l m p(a, b) j e dy

1 J-v -z2/2 

### 727 where di(z) denotes the probability that a standard normal random variable is less than z, and 0 < a < 1 is a compliance probability which identifies a compliance probability contour, in the (p, a) plane, whose asymptotic behavior we are investigating. Hence, if z. denotes the 100a percentile of a standard normal random variable (that is,'(xa) = a), then 

### q=za// . (D7) 49

### A Case Study with Application to Distribution Transformers NISTTN 1427 Asymptotically, as a -> 0, the a contour of compliance probability is linear; to be specific, p - k - xv

,a/ f . (D8) We could have gu essed this result by reasoning that as a -+ 0, a -+ 0, and condition (D2) must always hold. So we need only be concerned with condition (D1), and we require that Pr(2 < k) = a #- Pr x/v"

n- - k/s =a * (a a n) 0 (/f)_a k=μ+xaa/-,fn-:

Limit of large a : Asa -- oo, b -4 0

. If a -} 0 , then p (a, b) -4 0 if k > v, p(a, b) -+ 1 if k < p, and p(a, b) -+ 1 /2 if k = p . This is not particularly interesting or realistic, so we require that 

### (D9) for finite v. It follows that lim p(a, b ) = Fn j e "V2/2X, [ tz (v - U)a] dy. Let h = v - y, and note that 

### lim p(a, b) = Fn 

### e-" X2 

### ( 

### ea hs) dh

1 0 00 

### a, +00

_ t e - [ -v]' X 2 ( / a ) dx rx = a. 2^rv 0 (D 10)

For the limiting behavior of the 100a percent contour in the (p, a) plane , we must solve the transcendental equation t a 2^j °° e [*r"J2x (x) (D11) for v. The contour will be asymptotically linear , i.e., as a - ► oo p -k - va. (D12) This asymptotic result could also be derived by reasoning that, in the limit of large or, s will almost certainty be large and 2 < k will hold automatically, provided that condition (D2) is satisfied. So the compliance probability is determined in this limit in terms of the second condition alone, which leads to the result developed in this subsection. Derivation of p(a , b): Under a normal distribution model , the sample mean and standard deviation, 2 and s , are statistically independent

. The sample mean has a normal distribution with mean p and standard deviation a/f. The sample standard deviation is proportional to the square root of a Xv random variable, with proportionality constant a/V5. It's easy to see that the probability of compliance can be expressed as



### Pr(-* < k and 2 + ts/ f < (1 + 9)k) = Pr I a < k and 8 < ( 1 + B)k - 2^ (D13) \ t/f 50

### A Case Study With Application to Distribution Transformers NISTTN 1427

Because of independence (which holds only under a normal model), s has the same probability distribution whatever the value of a. So we condition on x taking on all possible values

- oo < x < oo, and integrate: Pr(a<kands < ( 1+8)k-a) t/,Fn l 

### f 00 ( (1+®)k-xl _ 00 Pr k > x ands < t/^ J Pr( x - x)dx 

# 1 00 Pr(k > x) Pr (s < (1 O )k xl Pr( x = 

### x)dx. t/V Note that we can write the joint robability distribution as a product because of independence

. Substituting expressions for the normal and X2 distributions , into the above expressions and simplifying leads to p(a, b) as given in (D3). 

### Testing under TTY 2 Assume that the random variables Y, are normal with mean μ and standard deviation a. Since we will assume that p and v can be regarded as approximately known from previous data, we can , without loss of generality, employ the standardized sample X; _ (Y,• - pp)/a, having order statistics X(,) = (Y;) - μ)/a. Then the probability of being found in compliance under the TP 2 sampling plan can be shown to be p = 1- [Fxu) (t) + Fg (t) - Fx(,),X (ti, t2)] • (D14) The function FX(,) (t) denotes the CDF of X(I), and P r(t) denotes the CDF of X; that is 

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

### Fg (t) = Pr(. < t) = lb 

# (,/nt) , where s1(•) is the standard normal CDF, which is defined to be the probability that a normally distributed random variable is less than u, that is 

### -I'M f U e-e This integral may be calculated efficiently and accurately using public-domain routines. A saddle

-point approximation to the bivariate CDF Fx(,),x(tl,t2 ) = Pr(X(I ) < tI and 1 5 t2) is derived by Vangel [8]. Let qi (t) denote the normal density , and let 

### h(t) _ O(t) - t(t) be the normal hazard function

. The saddle

-point approximation Fx(1),g(tl, t2), for t1 < t2, is

Fx(,,,g (tI, t2 ) = f%. t(ft2)A(t)dt + ftoo t J VG- [tI + " nI (h(t) - t)] } A(t)dt . f . A(t)dt 

## A(t) = h-("-I)(t) exp 

## l (" 2n1)2 

## [h(t)
- t]2 

## + (n 

## - 1)t 

## [h(t) - t]} 1- h2(t) + th( 

### t 

## ), where to 

### is the (unique) solution to the equation n-1 [h(ta)- t4]=t2-tin (D15)

(D16)


### 51

### A Case Shady with Application to ® isteibution Teansformses NIS TTN 1427

The probability of demonstrating compliance by testing under the Sampling Plan for Enforcement Testing can be calculated using straightforward numerical integration. In the following expressions , ni > 4 denotes the minimum sample size specified in the sampling plan, and rat = 20 is the maximum sample

. In order to simplify the equations , ni - 1 is represented by v, and t is the 97.5th percentile of the Student-t distribution for the sample size nj. The probability of compliance is

& = 2g-,r l2) d.-m1 l+ae 6 where the limits of integration are u,,, = 0, 

### r.j `LT -10U a-2fori=rai+1,...,n2, t and rc„a+i = oo, and the loss tolerance, LT, is a percentage.

52

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