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CHAPTER 6: ENERGY USE AND END-USE LOAD CHARACTERIZATION

TABLE OF CONTENTS

6.1 INTRODUCTION ..................................................... 6-2

6.1.1 Transformer Losses .............................................. 6-2

6.1.2 Transformer Loading ............................................. 6-3

6.2 HOURLY LOAD MODEL .............................................. 6-3

6.2.1 Overview of Hourly Load Model .................................... 6-3

6.2.2 Hourly Load Simulation Method .................................... 6-4

6.2.3 Hourly Load Simulation Inputs ..................................... 6-6

6.2.3.1 Utility Sample .......................................... 6-6

6.2.3.2 Customer Type ......................................... 6-10

6.2.3.3 Transformer Initial Peak Loading .......................... 6-10

6.2.3.4 Load Factor and Inverse Load Factor ........................ 6-11

6.2.3.5 Load Distribution Function ............................... 6-12

6.2.3.6 Hourly System Load ..................................... 6-13

6.3 MONTHLY LOAD MODEL ............................................ 6-14

6.3.1 Overview of Monthly Load Model ................................. 6-14

6.3.2 Monthly Load Estimation Steps .................................... 6-15

6.3.3 Monthly Load Model Inputs ...................................... 6-16

6.3.3.1 Customer Demand and Usage Data ......................... 6-16

6.3.3.2 Customer Sampling Weights .............................. 6-17

6.3.3.3 Initial Peak Load ....................................... 6-18

6.3.3.4 Transformer Load Factor ................................. 6-19

6.3.3.5 Root Mean Square-to-Average Load Ratio ................... 6-21

6.3.3.6 Transformer Peak Responsibility Factor ..................... 6-22

LIST OF TABLES

Table 6.2.1 Utility Sample for Hourly Load Simulation Data ....................... 6-7

LIST OF FIGURES

Figure 6.2.1 Histogram of the Peak Loading Distribution Generated in the Hourly 

Load Analysis .................................................. 6-11

Figure 6.2.2 Scatterplot of Inverse Load Factor versus Peak Load from 

Hourly ELCAP Data ............................................ 6-12

Figure 6.3.1 Histogram of Annual Load Factors Derived from Monthly 

CBECS 1995 Bill Data .......................................... 6-19

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Figure 6.3.2 Scatterplot of Annual Load Factor versus Annual Peak Demand from 

Monthly CBECS 1995 Bill Data ................................... 6-20

Figure 6.3.3 Scatterplot of Peak Responsibility Factor versus Load Factor ............ 6-23

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CHAPTER 6: ENERGY USE AND END-USE LOAD CHARACTERIZATION

6.1 INTRODUCTION

The energy use and end-use load characterization analysis produced energy use estimates

and end-use load shapes for distribution transformers. The energy use estimates enabled

evaluation of energy savings from the operation of distribution transformer equipment at various

efficiency levels, while the end-use load shapes allowed evaluation of the impact on electricity

monthly and peak demand from the operation of distribution transformers. The analysis

produced a distribution of results for a variety of building types and uses, covering a range of

climate locations in order to represent the diversity of use, and performance, of distribution

transformers.

The energy use by distribution transformers derives from no-load losses and load losses. 

No-load losses are constant over time and occur whenever a transformer is energized by power

lines. Load losses vary with the square of the load being served by the transformer. Because

liquid-immersed transformers are primarily owned by utilities which have marginal electricity

generation costs that can vary by the hour, the Department developed a statistical transformer

load simulation model to estimate the hourly characteristics of liquid-immersed transformer

loads. For dry-type transformers, the Department used empirical estimates of load characteristics

to estimate monthly average root mean square (RMS) loads and peak coincident loads for

distribution transformers owned by commercial and industrial (C&I) establishments. This

chapter first describes transformer losses, and then presents the details of the hourly and monthly

end-use load characterization models the Department developed.

6.1.1 Transformer Losses

The energy used by distribution transformers is characterized by two types of losses. The

first type are no-load losses that arise primarily from the switching of the magnetic field in the

transformer core material. No-load losses are roughly constant and exist whenever the

transformer is energized (i.e., connected to live power lines). The second type of losses are load

losses which are also known as resistance or I2R losses. Load losses vary with the load on the

transformer and at any point in time are proportional to the load squared plus a relatively small

(<15% for loads less than rated load) temperature correction. Considering both load and no-load

losses, the Department uses the following formula for estimating the energy used by a

distribution transformer:

EDT(t) = ENL + ELL × [Load(t)/(Lrated × PF)]2 Eq. 6.1

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where:

EDT(t) = the energy used by the distribution transformer at time t,

ENL = the no-load losses at rated load,

ELL = the load losses at rated load,

Load(t) = the load served by the transformer at time t,

Lrated = the rated load of the transformer, and

PF = the power factor of the load served by the transformer.

6.1.2 Transformer Loading

The key input for estimating distribution transformer energy use is the transformer load.

Because the application of distribution transformers varies significantly by type of transformer

(liquid-immersed or dry-type) and ownership (electric utilities own approximately 95% of liquidimmersed transformers, commercial/industrial entities use mainly dry-type), the Department

performed two separate end-use load analyses to evaluate distribution transformer efficiency. 

One analysis was designed for liquid-immersed transformers that are used mainly by electric

utilities, and the second analysis was designed for dry-type transformers that are used mainly by

C&I customers. Sections 6.2 and 6.3 describe these two separate load analyses.

6.2 HOURLY LOAD MODEL

This section describes the hourly load model the Department developed to estimate

transformer loads for use in analyzing potential efficiency standards for liquid-immersed

transformers.

6.2.1 Overview of Hourly Load Model

The Department estimated hourly loads on individual liquid-immersed transformers using

a statistical load simulation model. The statistical load simulation model takes several inputs and

performs a statistical simulation for the hourly load on a set of sample transformers. There are

five main inputs to the statistical load simulation model:

1. an estimate of the annual peak load; 

2. a relationship between peak load and load factor; 

3. a relationship between the load factor and the load distribution function;

4. the hourly system load; and 

5. the correlation between hourly system load and an individual transformer load.

The Department performed the hourly load simulation with a data-processing program

that generated load inputs for the life-cycle cost (LCC) spreadsheet. The Department estimated

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a

 If transformers are sized for moderate load growth on average (for example, 1 percent annual growth over 30

years), then the average peak loading should be somewhat below the rating stated on the nameplate. The Department

modeled the initial peak loading of liquid-immersed transformers in the hourly load model with a smooth distribution

with a minimum peak loading of 50 percent of nameplate, a median of 85 percent of nameplate, and a maximum 130

percent of nameplate. The median value selected by the Department provides some room for load growth while

utilizing the ability of the transformer to tolerate temporary loads greater than nameplate in the later years of the

transformer's lifetime. The Department assumed a fairly wide range of initial peak loadings for liquid-immersed

transformers and an average initial peak loading of 85 percent

b

 PJM coordinates the movement of electricity in all or parts of Delaware, Maryland, New Jersey, Ohio,

Pennsylvania, Virginia, West Virginia, and the District of Columbia.

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the annual peak load on the transformer from the rated size of the transformer and a distribution

of peak loads that ranged from 50 percent to 130 percent of rated load, with a mean of 85 percent

of the rated peak load of the transformer.a The Department derived a scaling relationship

between peak load and load factor from available hourly load monitoring data. The Department

determined a load distribution function with the estimated load factor and simulated the

transformer load consistent with the load distribution function. The Department obtained the

hourly system loads from the U.S. Federal Energy Regulatory Commission (FERC) and market

data from the Independent System Operators for the California, New England, and PJM

Interconnectionb for the year 1999. The Department determined the correlation between

transformer load and system load from distributions of correlation coefficients for commercial

and residential customers that it derived from available hourly load monitoring data. 

The economically important characteristics of transformer loads depend primarily on the

load factor, the relationship between peak load and transformer capacity, and the correlation

between transformer load and system load.

6.2.2 Hourly Load Simulation Method

The Department developed an hourly transformer load simulation model to capture the

varying dynamics and economics of transformer loads. The load simulation program takes utility

inputs that include the hourly system loads, hourly prices, or system lambdas (lambda is roughly

the marginal operating cost of generation), and then simulates a distribution of customer loads. 

The program provides summary data on customer loads, costs, and prices as output.

Simulation of customer or building loads is a common method for estimating building

energy use when specific monitoring data are not available. A statistical simulation takes

information on the mathematical and statistical characteristics of building loads and performs a

Monte Carlo simulation to produce sets of building loads that have the same mathematical and

statistical characteristics as measured building data. A Monte Carlo simulation is a computer

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calculation with a built-in random process that estimates the probability of different results. 

With regard to the economics of energy use by transformers, the most important mathematical

characteristics of the building loads are their size relative to the transformer capacity, their

variation, and their correlation with system loads and prices. 

The Department determined the values used for customer load variability, and customer

load correlation with system loads, by analyzing available hourly building load data. In more

detail, the specific steps in the operation of the hourly load simulation program are as follows: 

1. The program selects a utility from a database of utilities that publicly report system load

and system lambda data. 

2. The program determines a sample weight for the utility, based on total kilowatt-hour

(kWh) sold. 

3. The program selects the capacity for the transformer, based on the design line. 

4. The program selects the transformer peak load relative to the transformer capacity, and

the corresponding weight from a transformer peak load distribution. 

5. The program selects the transformer load factor based on the peak load versus inverse

load factor distribution. The program calculates the characteristics of the load factor

distribution from End-Use Load and Consumer Assessment Program (ELCAP) data.1

6. The program selects the customer type served by the transformer and the appropriate

weight for that customer type. The program selects the relative customer weight from the

fraction of electricity sales to the customer type for the particular utility. 

7. From the transformer inverse load factor estimate, the program calculates the load

distribution function. The load distribution function in the program is an exponential

function with the selected inverse load factor.

8. Based on the customer type served, the program selects the (rank order) correlation with

system load, based on correlation coefficient distributions calculated from hourly load

data. 

9. The program then calculates the hourly system load rank orders for each hour of the year

by sorting the hourly system load values for the year and assigning a rank to each. 

10. The program calculates the hourly transformer load rank orders by adding the appropriate

random variation to the system rank order to produce transformer rank orders with the

selected rank order correlation. The program normalizes the rank orders of the

transformer load so that they are evenly distributed between 1 and 8760, inclusive. 

11. The program calculates the hourly transformer load from the simulated hourly

transformer rank order. This calculation uses the cumulative transformer load-probability

function, which is an integral of the load-distribution function. 

12. The program then calculates the coefficients of a Fourier representation of the transformer

load shape for each day type (i.e., weekday, weekend, and peak day). It then makes the

same calculation for the load-square weighted price curves which are a function of day

type, hour, and the month of the year. 

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13. The program provides output for use in the LCC spreadsheet in the form of tab-separated

values. These data contain a customer load identification (ID), transformer characteristic,

the utility ID, summary statistics on price and load, and load and price shape coefficients.

6.2.3 Hourly Load Simulation Inputs

The inputs used in the hourly load simulation are described in the following sections.

6.2.3.1 Utility Sample 

The utility sample is the subset of electric utilities in the Nation whose economic and load

data are represented in the hourly load analysis. In the hourly load analysis for liquid-immersed

transformer designs, the Department began with hourly system load and economic data from the

Nation's electric utilities. The Department compiled the full list of the Nation’s electric utilities

by combining all utilities that report retail data using DOE's Energy Information Administration's

(EIA) Form 861,2 with utilities that reported data in FERC Form 1,3 and utilities that are listed in

the FERC Form 7144 respondent list. The Department included a utility in its hourly load

simulation sample if the utility reported distribution transformer data in FERC Form 1 for the

year 2000, and if sufficient information was available to characterize the utility's load and hourly

marginal electricity costs, either from FERC Form 714 data or from the relevant electricity

market. Table 6.2.1 shows the list of 80 utilities that satisfy all of these data requirements. 

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Table 6.2.1 Utility Sample for Hourly Load Simulation Data

EIA Code* NERC Region** Name

296 SPP Alfalfa Electric Coop Inc.

733 ECAR Appalachian Power Co. 

963 MAAC Atlantic City Electric Co. 

1167 MAAC Baltimore Gas & Electric Co. 

1179 NPCC Bangor Hydro-Electric Co. 

1796 NPCC Blackstone Valley Electric Co.

1998 NPCC Boston Edison Co. 

2886 NPCC Cambridge Electric Light Co. 

3292 NPCC Central Vermont Pub Serv Corp. 

3266 NPCC Central Maine Power Co. 

3597 MAAC Citizens Electric Co. 

4062 ECAR Columbus Southern Power Co. 

4110 MAIN Commonwealth Edison Co. 

4089 NPCC Commonwealth Electric Co. 

4148 NPCC Concord Electric Co. 

4184 NPCC Connecticut Valley Elec. Co. Inc. 

4176 NPCC Connecticut Light & Power Co. 

4254 ECAR Consumers Energy Co. 

5027 MAAC Delmarva Power & Light Co. 

5109 ECAR Detroit Edison Co. 

5416 SERC Duke Energy Corp. 

5487 ECAR Duquesne Light Co. 

* The EIA Code is the unique identification number assigned by EIA to companies and entities operating in the

electric power industry.

** The North American Electric Reliability Council (NERC) regions, in the order they appear in this utility sample,

are: Southwest Power Pool (SPP), East Central Area Reliability Coordination Agreement (ECAR), Mid-Atlantic

Area Council (MAAC), Northeast Power Coordinating Council (NPCC), Mid-America Interconnected Network,

Inc. (MAIN), Southeastern Electric Reliability Council (SERC), Mid-Continent Area Power Pool (MAPP), Western

Electricity Coordinating Council (WECC), Electric Reliability Council of Texas, Inc. (ERCOT), and Florida

Reliability Coordinating Council (FRCC). FERC provides load and lambda data in files organized by NERC

region.

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Table 6.2.1 Utility Sample for Hourly Load Simulation Data (continued)

EIA Code* NERC Region** Name

5389 NPCC Eastern Edison Co. 

5659 ECAR Edison Sault Electric Co. 

5860 SPP Empire District Electric Co. 

13478 SPP Entergy New Orleans Inc. 

814 SPP Entergy Arkansas Inc. 

7806 SPP Entergy Gulf States Inc. 

11241 SPP Entergy Louisiana Inc. 

12685 SPP Entergy Mississippi Inc. 

6077 NPCC Exeter & Hampton Electric Co. 

6342 SPP First Electric Coop Corp. 

6374 NPCC Fitchburg Gas & Elec. Light Co. 

26510 NPCC Granite State Electric C.o 

7601 NPCC Green Mountain Power Corp. 

9162 MAPP IES Utilities Inc. 

9324 ECAR Indiana Michigan Power Co. 

9392 MAPP Interstate Power Co. 

9726 MAAC Jersey Central Power & Light Co. 

22053 ECAR Kentucky Power Co. 

10331 ECAR Kingsport Power Co. 

11118 SERC Lockhart Power Co. 

* The EIA Code is the unique identification number assigned by EIA to companies and entities operating in the

electric power industry.

** The North American Electric Reliability Council (NERC) regions, in the order they appear in this utility sample,

are: Southwest Power Pool (SPP), East Central Area Reliability Coordination Agreement (ECAR), Mid-Atlantic

Area Council (MAAC), Northeast Power Coordinating Council (NPCC), Mid-America Interconnected Network, Inc.

(MAIN), Southeastern Electric Reliability Council (SERC), Mid-Continent Area Power Pool (MAPP), Western

Electricity Coordinating Council (WECC), Electric Reliability Council of Texas, Inc. (ERCOT), and Florida

Reliability Coordinating Council (FRCC). FERC provides load and lambda data in files organized by NERC region.

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Table 6.2.1 Utility Sample for Hourly Load Simulation Data (continued)

EIA Code* NERC Region** Name

11804 NPCC Massachusetts Electric Co. 

12390 MAAC Metropolitan Edison Co. 

12341 MAPP MidAmerican Energy Co. 

12796 ECAR Monongahela Power Co. 

13214 NPCC Narragansett Electric Co. 

13407 WECC Nevada Power Co. 

13441 NPCC New Hampshire Elec Coop Inc. 

13549 NPCC Newport Electric Corp. 

13781 MAPP Northern States Power Co. 

13780 MAPP Northern States Power Co. 

14006 ECAR Ohio Power Co. 

14063 SPP Oklahoma Gas & Electric Co. 

14328 WECC Pacific Gas & Electric Co. 

14940 MAAC PECO Energy Co. 

14711 MAAC Pennsylvania Electric Co. 

15270 MAAC Potomac Electric Power Co. 

15263 ECAR Potomac Edison Co. 

15466 WECC Public Service Co. of Colorado 

15473 WECC Public Service Co. of NM 

15472 NPCC Public Service Co. of NH 

* The EIA Code is the unique identification number assigned by EIA to companies and entities operating in the

electric power industry.

** The North American Electric Reliability Council (NERC) regions, in the order they appear in this utility sample,

are: Southwest Power Pool (SPP), East Central Area Reliability Coordination Agreement (ECAR), Mid-Atlantic

Area Council (MAAC), Northeast Power Coordinating Council (NPCC), Mid-America Interconnected Network, Inc.

(MAIN), Southeastern Electric Reliability Council (SERC), Mid-Continent Area Power Pool (MAPP), Western

Electricity Coordinating Council (WECC), Electric Reliability Council of Texas, Inc. (ERCOT), and Florida

Reliability Coordinating Council (FRCC). FERC provides load and lambda data in files organized by NERC region.

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Table 6.2.1 Utility Sample for Hourly Load Simulation Data (continued)

EIA Code* NERC Region** Name

15477 MAAC Public Service Electric & Gas Co. 

8901 ERCOT Reliant Energy HL&P 

17535 MAPP South Beloit Water Gas & Elec. Co. 

17539 SERC South Carolina Electric & Gas Co. 

17609 WECC Southern California Edison Co. 

17633 ECAR Southern Indiana Gas & Elec. Co. 

17718 SPP Southwestern Public Service Co. 

18454 FRCC Tampa Electric Co. 

24211 WECC Tucson Electric Power Co. 

19497 NPCC United Illuminating Co. 

19840 WECC Valley Electric Assn. Inc. 

19876 SERC Virginia Electric & Power Co. 

20387 ECAR West Penn Power Co. 

20455 NPCC Western Massachusetts Elec. Co. 

20521 ECAR Wheeling Power Co. 

20856 MAPP Wisconsin Power & Light Co. 

20847 MAIN Wisconsin Electric Power Co. 

20860 MAIN Wisconsin Public Service Corp. 

* The EIA Code is the unique identification number assigned by EIA to companies and entities operating in the

electric power industry.

** The North American Electric Reliability Council (NERC) regions, in the order they appear in this utility sample,

are: Southwest Power Pool (SPP), East Central Area Reliability Coordination Agreement (ECAR), Mid-Atlantic

Area Council (MAAC), Northeast Power Coordinating Council (NPCC), Mid-America Interconnected Network, Inc.

(MAIN), Southeastern Electric Reliability Council (SERC), Mid-Continent Area Power Pool (MAPP), Western

Electricity Coordinating Council (WECC), Electric Reliability Council of Texas, Inc. (ERCOT), and Florida

Reliability Coordinating Council (FRCC). FERC provides load and lambda data in files organized by NERC region.

The hourly load model uses data from each utility as a basis for the hourly simulation and

valuation of transformer loads in that utility. The Department statistically simulated a set of

transformer loads for each utility that were correlated with the utility’s hourly system load. The

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Department also calculated average hourly price profiles using hourly system lambdas (or market

prices). The Department weighted the prices by the square of the statistically simulated

transformer load to calculate average prices (since load losses are proportional to the square of

the load on the transformer).

6.2.3.2 Customer Type

The Department used the data on electricity sales by customer classes reported in EIA

Form 861 to characterize the different types of customer loads as residential, commercial, or

industrial. The customer type for a transformer is the class of customers whose loads are served

by the transformer. 

The data simulation model for hourly loads is a Monte Carlo simulation. This means that

the Department sampled transformer and customer characteristics from probability distributions

using a random number generator. The results of the simulation are output with weights that are

proportional to the probability of the computed result. In the hourly load simulation, the

Department selected the customer type for the transformer after it had selected the sample utility. 

The Department simulated an equal number of sample transformers for each customer type and

each utility. The Department then assigned a weight to each sample transformer in the

simulation that represented the population of transformers represented by that sample. The

assignment of customer-type weights assumed that the population of transformers for each

customer type is roughly proportional to the electricity sales to that customer type. 

6.2.3.3 Transformer Initial Peak Loading

The Department used a distribution of initial peak loading values to characterize the

annual peak load served by each transformer in its simulation. The transformer initial peak

loading is the ratio of the transformer peak load in the first year of operation to the rated load of

the transformer. The Department selected a distribution of initial peak loadings with a median of

85 percent, a minimum of 50 percent and a maximum of 130 percent. Standard engineering

practice for sizing distribution transformers selects a transformer based on the expected annual

peak of the load being served, with some provision made for load growth. Because of the

provision for load growth, initial peak loading will usually be less than 100 percent. However, in

practice, there is also some error in estimating the annual peak load that will be served by a

transformer, and engineers generally use a discrete set of transformer ratings that are imperfectly

matched with the expected peak load. Distribution transformers generally are manufactured in

discrete kilovolt-ampere (kVA) ratings and, on average, the next larger kVA rating is 50 percent

larger than the previous kVA rating (measured relative to the smaller size). An initial peak

loading as high as 130 percent can occur because transformers can be loaded for short periods to

levels of more than 130 percent of nameplate capacity.5 Figure 6.2.1 illustrates the distribution

of initial peak loading the Department used.

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Figure 6.2.1 Histogram of the Peak Loading Distribution Generated in the

Hourly Load Analysis

6.2.3.4 Load Factor and Inverse Load Factor

The load factor is the ratio of the average load to the peak load. Inverse load factor is the

ratio of the peak load to the average load. The transformer load simulation provides hourly

simulated loads using a distribution of load factors that depends on the initial peak load. 

Consistent with data on electrical customer loads, transformers serving larger annual peak loads

tend to serve loads with larger load factors. 

The load factor of customer and transformer loads has a wide range of variation, but

correlates with the average size of the load and the annual peak load. The initial annual peak

load on a transformer is the product of the transformer initial peak loading, the transformer

capacity, and the power factor, assumed to be 1 W/kVA. 

The Department analyzed hourly load data from ELCAP and confirmed its results with

confidential utility data to develop an aggregate distribution that models the inverse load factor as

a function of annual peak load. The Department fit a power-law regression to the relationship

between inverse load factor and peak load. The power-law regression produced the following

equation for inverse load factor as a function of peak load: 

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0

1

2

3

4

5

6

7

8

1 10 100 1000

Peak Load (kW)

Inverse Load Factor

Figure 6.2.2 Scatterplot of Inverse Load Factor versus Peak Load

from Hourly ELCAP Data

InvLoadFact = 1 + 9.59 × (PeakLoad)-0.418 Eq. 6.2

Figure 6.2.2 shows a scatterplot of the inverse load factor-versus-peak load relationship.

The figure plots both the ELCAP data and the regression function used to relate inverse load

factor and initial peak loading for the hourly load model that the Department used for liquidimmersed transformers. Also shown is the plot of the regression equation used to model inverse

load factor as a function of peak load for the hourly load model. 

6.2.3.5 Load Distribution Function

The load distribution function determines the probability that a customer load

(transformer load) will be a particular value during some hour of the year. The Department

ensured that the simulated transformer load has a specified load factor by selecting a consistent

load distribution function. A consistent load function is one that can be mathematically shown to

result in loads with the given load factor. In the simulation, the Department calculated hourly

loads using the selected load distribution function.

The Department approximated the load distribution function with a one-parameter

exponential fit that depended on the load factor. Specifically, the Department used a load

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selected from a function of the normalized load: Load/Lmax, where Lmax is the annual

maximum load. The following three equations define the load distribution function:

LoadProb = BLoad × ( exp(-ALoad × Load/Lmax)) when Load/Lmax < x0 Eq. 6.3

LoadProb = CLoad × ( exp(-DLoad × (Load/Lmax - x0 ))) when

Load/Lmax > x0, Eq. 6.4

LoadProb = 1/8760 when Load/Lmax = 1 Eq. 6.5

The function is continuous, and ALoad, BLoad, CLoad, and DLoad are constants. The

variable x0 is a parameter that defines when the load probability function (LoadProb) begins

decreasing rapidly to the value of 1/8760. The Department selected the approximate value of x0

by examining the load distribution functions from available hourly load data. Lmax is the

maximum or peak annual load. The constants are determined by the following conditions: 

1. The integral from 0 to 1 of LoadProb with respect to Load/Lmax is 1; 

2. The inverse load factor is set to a particular, given value; 

3. The load probability function is continuous; and 

4. LoadProb(1) = 1/8760.

The load distribution function is, therefore, a probability function of Load/Lmax that

depends on the parameters of inverse load factor and the constant parameter x0. For each

transformer in the hourly simulation model, once the Department selected the inverse load factor 

for the transformer, the values of ALoad, BLoad, CLoad, and DLoad were calculated given the

four conditions that the probability function satisfies. The Department selected the value of the

turning-point parameter x0 as 0.95, based on examination of load distribution functions from the

ELCAP hourly load data. On a logarithmic scale, the empirical load distribution functions

showed a point at which they would descend rapidly toward the value of 1/8760 at

Load/Lmax = 1. 

6.2.3.6 Hourly System Load

The hourly system load is the hourly load in the control area or electricity market where

the transformer is located. 

Electricity hourly load and cost data come from FERC and electricity market web sites. 

The source of data for regulated utilities was FERC Form 714,4 which provides data on hourly

system loads, hourly system lambdas, and forecasts of peak demand for transmission control

areas. In a control area, there is usually “one electric utility charged with operating the control

area” and other utilities “which are bound together through pooling contracts, holding company

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operations or other contractual arrangements” (see:

http://www.ferc.gov/docs-filing/eforms/form-714/instructions.asp). Each utility with a planning

area that has an annual peak demand greater than 200 megawatts must file Form 714. The

system lambda file in Form 714 is derived from the economic dispatch function associated with

automatic generation control. It is the incremental fuel cost for a set of on-line and loaded

thermal generating units “when each unit is loaded and operating at the same incremental fuel

cost ($/MWh) with the sum of the unit loadings (MW) equal to the system demand plus the net

of interchange with other control areas.” The Department used the system lambda as the

marginal fuel cost of electricity for regulated utilities. 

For utilities that sell into regional electricity markets, the hourly system load and

electricity cost data are obtained from the Independent System Operator of that market. These

data include both the actual system load and the hourly market price. 

For electricity markets, the system load is the “Actual System Load” as published by the

operator of that market. The Department used the actual market system load as the system load

for utilities in areas where there are electricity markets. The Department assumes that the

marginal wholesale electricity economics of a utility are characterized by the economics shown

in the local control area or electricity market. This assumption means that some geographically

small-scale variations in electricity economics may not be captured in the analysis.

6.3 MONTHLY LOAD MODEL

This section describes the model the Department developed to estimate monthly

transformer loads for use in analyzing potential efficiency standards for dry-type distribution

transformers, which are owned mainly by C&I entities.

6.3.1 Overview of Monthly Load Model

For dry-type transformers, the Department developed a monthly load model that estimates

the impacts of transformer loads and resultant transformer losses on the monthly electricity

usage, demand, and electricity bills of customers.

Transformer losses have a constant loss component (the no-load or core losses) and a loss

component that depends on the square of the load on the transformer (the load or coil losses). 

The economic value of transformer losses is a function of the load on the transformer and the

timing of the load with respect to variable energy costs and peak demand. To the extent there is a

correlation between transformer losses and variable energy costs, then the cost of the electricity

supplying the transformer losses will be different from the average cost of electricity. The LCC

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analysis for dry-type transformers estimated the economic impact of losses by estimating changes

in monthly peak demand and the corresponding demand charges for C&I customers. 

6.3.2 Monthly Load Estimation Steps

The Department used monthly load estimates to calculate the LCC savings due to

efficiency standards for dry-type distribution transformers. Commercial and industrial customers

are usually billed for electricity according to monthly demand, usage, and fixed charges. There

are a great variety of electricity tariff structures, including time-of-use rates and real-time prices,

that are used for billing electricity to C&I customers. For simplicity in estimating the monthly

bills for C&I customers, the Department used only the commercial tariffs that have monthly

demand, usage, and/or fixed charges in the tariff formula (this process is described in more detail

in section 8.3.5.2). Load estimates of monthly customer usage and demand are inputs to the bill

calculator in the LCC analysis.

To estimate the impact of transformer losses on C&I electricity bills, the Department

modeled the relationship between monthly transformer load characteristics and customer demand

and usage. The model used by the Department estimates the load factor, coincident peak load,

and RMS load on the transformer, relative to the total customer load. The Department derived

distributions of load parameters from hourly load data from ELCAP and confirmed the results

with proprietary utility data. It used these same load-parameter distributions as inputs to estimate

the monthly transformer energy and coincident peak demand losses. When used as inputs into a

monthly electricity bill calculator, monthly transformer energy demand and energy increments

provide the estimate of monthly bill impacts for transformer owners.

The Department took the following steps in calculating the monthly load:

1. The Department obtained a statistical sample of monthly demand and usage data for C&I

buildings from the Energy Information Administration (EIA)’s 1995 Commercial

Building Energy Consumption Survey (CBECS)6 and other data sources. 

2. The Department estimated an annual peak transformer load from a transformer peak

loading distribution, derived from an estimate of the impacts of conservative transformer

sizing practice that tends to over-size distribution transformers.

3. The Department estimated the load factor (i.e., the ratio of the average load to the annual

peak load) of the load on the transformer from load factor distributions found in CBECS.

4. The Department derived distributions for the ratio of the RMS load to the average load as

a function of the transformer load factor, through an analysis of hourly building load data.

5. The Department estimated monthly peak transformer load and monthly RMS transformer

load by scaling annual peak transformer values. The scaling estimates monthly values of

transformer peak and RMS load as being proportional to customer monthly demand and

energy usage data, respectively.

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6. Using an analysis of available hourly building load data, the Department estimated a

transformer peak responsibility factor (PRF) distribution that is a function of load factor.

7. The Department applied the average PRF to monthly transformer peaks to estimate

monthly transformer demand.

8. The Department’s model calculates incremental monthly customer usage and demand

from no-load and load loss coefficients, transformer RMS load, and transformer

coincident peak load in the LCC spreadsheet.

6.3.3 Monthly Load Model Inputs

The monthly load model inputs are:

C customer demand and usage data

C customer sampling weights

C initial peak load

C transformer load factor

C root mean square load to average load ratio

C transformer peak responsibility factor

6.3.3.1 Customer Demand and Usage Data

Customer demand and usage are the monthly peak load and electrical energy

consumption, in units of kilowatts and kilowatt-hours, respectively. Usage is sometimes also

called energy, or consumption. The Department used these data to calculate the base monthly

customer electricity bills that may be affected by changes in transformer efficiency. Commercial

and industrial customers may have many distribution transformers on their premises. An

individual transformer will make an incremental impact on the customer electricity bill by

making incremental changes in the customer demand and usage.

For commercial customers, the Department obtained a statistically representative sample

of monthly customer demand and usage from data collected in the 1995 CBECS. The

Department did not have at its disposal a sample of monthly demand and usage data for industrial

customers. The Department applied the CBECS commercial customer monthly data to estimate

the monthly bills for industrial customers by adjusting the customer weights to compensate for

the larger size and the corresponding larger load factors of industrial customers.

The Department assumed that monthly demand and usage for larger industrial and large

commercial customers are similar. It verified this assumption by comparing load factor

distributions for C&I customers for a utility in the southeastern U.S. The Department found that

the differences between customer classes were much smaller than the variability within each

class, although the average load factor for groups of industrial customers with peak loads less

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than 500 kW can be 0–30 percent less than the load factor for commercial customers with the

same annual peak load. The Department explored the potential impact of this assumption on the

analysis by examining the sensitivity of the LCC results to changes in the average transformer

loading.

6.3.3.2 Customer Sampling Weights

The customer sampling weight is a number that is proportional to the probability that a

particular customer sample will be selected during a national LCC calculation. It should be

proportional to the probability that a transformer of a particular design line may be owned by a

customer with load characteristics similar to the customer sample data. The Department adjusted

the sample weights for commercial customers provided by CBECS to produce sample weights

for equivalent industrial customers and to ensure that customers with loads too small for a

particular transformer design were not included. The Department adjusted the CBECS weights

by a power of the customer load to produce a distribution of customers that has the same average

load as industrial customers. The Department assigned a zero weight to those customers who

had an average demand less than the transformer capacity.

To produce a distribution of monthly demand and usage data to represent industrial

customers, the Department examined the EIA’s Form 861 data2 from 1995 to calculate the

average size ratio between industrial and commercial customers for 1995. The Department

found that, on average, industrial customers had loads that were 26.2 times larger than those of

commercial customers in 1995. Then for the CBECS distribution of customers, for those

buildings with monthly usage and demand data, the Department constructed a new set of weights

that had the same total sum, but which had an average load 26.2 times larger than the CBECS

distribution. It constructed the new distribution by multiplying the CBECS weight by a power of

the average load of the CBECS customer. The Department chose the power to provide the

desired change in average customer size.

Since the Department estimated that 60 percent of dry-type transformers go into

commercial buildings, while 40 percent go into industrial buildings, it used a 60/40 weighted

average of the C&I building weights for the total building weight, when the average building

demand is larger than the transformer capacity. If the average building demand is less than the

transformer capacity, then the CBECS data for that building have no weight or influence on the

LCC calculation. This is because dry-type transformers are typically used for customer

component loads for particular circuits or equipment. These circuits have demand that is less

than the total building demand, and the transformers for the circuits are sized to meet this smaller

demand. 

The approach that the Department took was based on the fact that, when the transformer

is larger than the entire customer demand, the customer will own the transformer. In addition,

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the Department believes that the load characteristics of industrial customers are similar to those

of large commercial customers. 

6.3.3.3 Initial Peak Load

Initial peak load is the annual, per-unit peak load on the transformer during the first year

of operation. The initial peak load is estimated as a percentage of the rated peak load of the 

transformer. The Institute of Electrical and Electronics Engineers, Inc. (IEEE) Draft Guide for

Distribution Transformer Loss Evaluation7 defines a similar but distinct measure of peak

transformer loading called an “Equivalent Annual Peak Load” that accounts for changes in peak

load over the life of the transformer. IEEE’s Draft Guide refers to the initial peak loading as

“Initial Transformer Loading” and uses values of 0.9 and 0.95 in its example calculations.

Rather than use the equivalent annual peak load method, the Department accounted for

annual changes in peak load by applying an annual rate of change in transformer load in the LCC

calculation to account for year-to-year changes in transformer load. The Department

characterized a range of possible initial peak loads by defining a distribution of initial peak loads. 

Distribution transformers generally are manufactured in discrete kVA ratings representing

their power handling capacity. On average, the each kVA rating is 50 percent larger then the

previous kVA rating (measured relative to the smaller rating). Transformers can be loaded above

their kVA rating (or “nameplate capacity”) for short periods of time. However, transformers are

often sized conservatively to avoid the possibility of an overload. The initial peak load on a

transformer ranges from 60 percent to 90 percent if electrical engineers accurately size dry-type

transformers conservatively with a 10 percent safety margin relative to the nameplate capacity. 

The high end of the range, i.e., 90 percent initial peak loading, is the maximum initial peak load

with a 10 percent safety margin. The low end of the range of 60 percent initial peak loading

reflects the threshold peak load where a smaller kVA rating can be selected with 90 percent peak

loading. The Department therefore selected a distribution for the initial peak load that has a

constant probability between 60 percent and 90 percent of nameplate capacity.

The Department believes that, in selecting an appropriate kVA rating for an application,

engineers choosing dry-type transformers are conservative and do not take advantage of the fact

that transformers can be safely overloaded for short periods of time. The National Electrical

Code8 encourages conservative transformer sizing behavior by requiring a transformer that is

serving a secondary circuit of less than 600 V to be rated at not less than 80 percent of the total

amperage of the secondary circuit protection (Table 450.3(A) of the Code).

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0

50

100

150

200

250

300

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95

Load Factor

CBECS Record Count

Figure 6.3.1 Histogram of Annual Load Factors Derived from Monthly CBECS

1995 Bill Data

6.3.3.4 Transformer Load Factor

The transformer load factor is the ratio of the average load to the annual peak load. To

estimate the load factor on the transformer, the Department examined the load factors of the

loads in the CBECS 1995 monthly building demand and usage data and derived distributions

consistent with these data. Figure 6.3.1 shows an aggregate histogram of the load factors from

these data, for those buildings with both demand and usage data.

From Figure 6.3.1, the Department estimated that the minimum load factor is

approximately 0.01 and the maximum load factor is approximately 0.85, with only a few data

points outside this range. The Department modeled the distribution of transformer load factors

as a triangular probability distribution with the values of 0.01 and 0.85 as the minimum and

maximum of the distribution, respectively. The Department set the maximum probability point

of the triangular distribution at 0.32 so that the mean load factor of the triangular distribution

matched the mean load factor of the CBECS data.

Available data indicate that for buildings, load factors increase with increasing peak load

on average. This relationship is due to diversity effects, where the fluctuations of many smaller

loads tend to average out over time, such that the relative fluctuations in the sum of many

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y = 0.0504Ln(x) + 0.1212

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 10 100 1000 10000 100000

Annual Peak Demand (kW)

Load Factor 

Load Factor

Log. (Load Factor)

Figure 6.3.2 Scatterplot of Annual Load Factor versus Annual Peak Demand

from Monthly CBECS 1995 Bill Data

individual loads is smaller that the relative fluctuations in the sum of a few individual loads. The

variation of the load factor with the annual peak demand is illustrated in Figure 6.3.2 for the

CBECS 1995 buildings that have both monthly usage and demand data. The relationship

between load factor and peak load illustrated here is insensitive to whether monthly or annual

data are used. The Department used the following equation to model the relationship between

annual peak load and mean load factor:

LF = 0.0504Ln(x) + 0.1212 Eq. 6.6

where: 

Ln = the natural logarithm,

x = the annual peak load, and

LF = the load factor. 

Figure 6.3.2 also shows the logarithmic fit to the load factor/annual demand relationship for the

CBECS data. 

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Application of this analysis to the estimation of transformer load factors assumes that the

statistical behavior of transformer loads is similar to the statistical behavior of whole building

loads.

6.3.3.5 Root Mean Square-to-Average Load Ratio

The root mean square-to-average load ratio is the ratio of the root mean square (RMS)

load on the transformer to the average load on the transformer. Distribution transformer load

losses are proportional to the square of the RMS load on the transformer. For loads with large

amounts of variability, the RMS and the average load on the transformer may be significantly

different. The Department analyzed hourly load data in order to characterize the ratio of the

RMS load to the average load on the transformer. This characterization included an estimate of

the variability in the RMS-to-average load ratio, and the dependence of this ratio on the load

factor of the load being served by the transformer.

The Department analyzed both older, public data from an ELCAP study of hourly

electrical loads in the Northwest in 1987 and 19889 and more recent (proprietary) hourly load

data from the Southeast from 1998, 1999, and 2000. The Department found consistent behavior

between ELCAP commercial building loads and both industrial and commercial loads from the

more recent southeastern data. The Department found the differences between the average

RMS/average load ratios from the data sets to be less than 10 percent.

The Department found that the RMS-to-average load ratio depended on the load factor. 

The Department fit the relationship between load factor and (RMS/Avg -1) to an exponential

trend line with an R-square of 0.8. This relationship can be expressed by the following equation:

(RMS/Avg - 1) = 1.4 × e -7*LF Eq. 6.7

where:

RMS/Avg = the ratio of the RMS load to the average load, and 

LF = the annual load factor.

In its treatment of RMS-to-average load ratios, the Department assumed that:

1. The statistical behavior of a set of whole-building loads of a given average load is similar

to the statistical behavior of a set of individual transformer loads with the same average

load;

2. The RMS load analyzed from hourly data is the same as the RMS load analyzed from

data of higher time resolution (in reality, RMS values from higher-time-resolution data

may be higher than RMS values from hourly data); and 

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3. The load losses on the transformer vary as the square of the load being served by the

transformer.

In this part of the analysis, the Department made no assumptions about the load factor of

the load being served by the transformer, but only estimated the relationship between load factor,

average load, and RMS load.

6.3.3.6 Transformer Peak Responsibility Factor

The transformer PRF is the square of the ratio of the transformer load at the time of the

customer peak load to the transformer peak load. The PRF of a load depends on how well the

load is correlated with the system load; it also depends on the load factor (LF). A load with an

LF close to one is likely to also have a PRF close to one. The Department estimated the

distribution of PRF as a function of LF from available hourly load data. The Department used a

distribution for PRF values that varies with LF. Figure 6.3.3 shows a scatterplot of PRF versus

LF for hourly commercial data from the ELCAP study. The Department also examined more

recent C&I hourly load data and found the results of the ELCAP data analysis to be consistent

with the more recent hourly load data from the Southeast from 1998, 1999, and 2000.

The Department fit the mean value of the PRF as a function of LF to the following curve,

which is also illustrated as the solid center line in Figure 6.3.3:

PRFmid = (1 + e C

PRF

 )/(1+ e C

PRF

 /LF) Eq. 6.8

where: 

PRFmid = the mean PFR,

CPRF = a calibrated constant equal to 0.6, and

LF = the annual load factor.

The Department fit the maxima and minima of the scatterplot distribution as a function of

LF using the following equations:

PRFmax = Min(1, 6 × PRFmid) Eq. 6.9

PRFmin = Max(0, PRFmid - (1.2 - PRFmid)) Eq. 6.10

where:

PRFmax and PRFmin = the maximum and minimum PFRs, respectively,

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0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Load Factor

Peak Responsibility Factor

Figure 6.3.3 Scatterplot of Peak Responsibility Factor versus Load Factor

Min = the function that returns the minimum of two arguments,

and 

Max = the function that returns the maximum of two arguments.

The upper and lower solid lines on Figure 6.3.3 also show the maximum and minimum

bounds on the ELCAP data scatterplot, respectively.

In its treatment of PRF, the Department assumed that building loads and transformer

loads with the same load factor and annual peak amplitude have statistically similar behavior for

the peak responsibility factor, and that industrial and commercial loads of the same average size

and load factor have statistically similar behavior for the peak responsibility factor.

The Department found the aforementioned assumptions to be consistent with hourly

ELCAP and proprietary utility data analyzed by the Department.

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REFERENCES

1. Pratt, R. G., M. A. Williamson, E. E. Richman, and N. E. Miller. Commercial Equipment

Loads: End-Use Load and Consumer Assessment Program (ELCAP). July, 1990. Pacific

Northwest Laboratory. Richland, WA. Report No. DOE/BP-13795-24. Prepared for the

Bonneville Power Administration. 

2. U.S. Department of Energy - Energy Information Administration. Form EIA-861: Annual

Electric Utility Data. 2001. (Last accessed July, 2005.) This material is available in

Docket #83. Contact Ms. Brenda Edwards-Jones, U.S. Department of Energy, Building

Technologies Program, Mailstop EE-2J, 1000 Independence Avenue, SW, Washington,

DC, 20585-0121, telephone (202) 586-2945 for more information.

<http://www.eia.doe.gov/cneaf/electricity/page/eia861.html>

3. U.S. Department of Energy - Federal Energy Regulatory Commission. Form No. 1 -

Electric Utility Annual Report. 2002. (Last accessed February 11, 2005.) This material is

available in Docket #83. Contact Ms. Brenda Edwards-Jones, U.S. Department of

Energy, Building Technologies Program, Mailstop EE-2J, 1000 Independence Avenue,

SW, Washington, DC, 20585-0121, telephone (202) 586-2945 for more information.

<http://www.ferc.gov/docs-filing/eforms/form-1/form1_blank.pdf>

4. U.S. Department of Energy - Federal Energy Regulatory Commission. Form No. 714 -

Annual Electric Control and Planning Area Report. 2002. (Last accessed February 18,

2005.) This material is available in Docket #83. Contact Ms. Brenda Edwards-Jones,

U.S. Department of Energy, Building Technologies Program, Mailstop EE-2J, 1000

Independence Avenue, SW, Washington, DC, 20585-0121, telephone (202) 586-2945 for

more information. <http://www.ferc.gov/docs-filing/eforms/form-714/data.asp>

5. U.S. Department of the Interior - Bureau of Reclamation. Permissible Loading of OilImmersed Transformers and Regulators. 1991. Facilities Branch, Denver Office, Denver,

CO. (Last accessed March 25, 2005.) <http://www.usbr.gov/power/data/fist_pub.html>

6. U.S. Department of Energy-Energy Information Administration. Information on the

Commercial Building Sector. 1995. (Last accessed June, 2005.) This material is available

in Docket #83. Contact Ms. Brenda Edwards-Jones, U.S. Department of Energy,

Building Technologies Program, Mailstop EE-2J, 1000 Independence Avenue, SW,

Washington, DC, 20585-0121, telephone (202) 586-2945 for more information.

<http://www.eia.doe.gov/emeu/cbecs/>

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7. Institute of Electrical and Electronics Engineers Inc. Draft Guide for Distribution

Transformer Loss Evaluation. October, 2001. 345 East 47th Street, New York, NY.

Report No. IEEE PC57.12.33/D8. 

8. National Fire Protection Association. National Electrical Code, NFPA 70. 2002 ed.

2002. Quincy, MA.

9. Taylor, Z. T. and R. G. Pratt. Description of Electric Energy Use in Commercial

Buildings in the Pacific Northwest. 1989. Bonneville Power Administration. Portland,

OR. Report No. DOE/BP-13795-22. 

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