CHAPTER 6. ENERGY USE AND END-USE LOAD CHARACTERIZATION 

TABLE OF CONTENTS 

6.1 INTRODUCTION ................................................... 6-1


6.1.1 Transformer Losses ............................................. 6-1


6.1.2 Transformer Loading ............................................ 6-2


6.2 HOURLY LOAD MODEL ............................................ 6-2


6.2.1 Overview of Hourly Load Model ................................... 6-2


6.2.2 Hourly Load Simulation Method ................................... 6-3


6.2.3 Hourly Load Simulation Inputs .................................... 6-5


6.2.3.1
 Utility Sample ......................................... 6-5


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 ................................... 6-14


6.3.3
 Monthly Load Model Inputs ..................................... 6-15


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


Figure 6.3.3
 Scatterplot of Peak Responsibility Factor versus Load Factor ............ 6-22


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


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


Figure 6.3.1
 Histogram of Annual Load Factors Derived from Monthly CBECS 1995 Bill 


Data ....................................................... 6-18


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


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


6.3.3.1 Customer Demand and Usage Data ........................ 6-15


6.3.3.2 Customer Sampling Weights ............................. 6-16


6.3.3.3 Initial Peak Load ...................................... 6-17


6.3.3.4 Transformer Load Factor ................................ 6-18


6.3.3.5 Root-Mean-Square-to-Average Load Ratio .................. 6-20


6.3.3.6 Transformer Peak Responsibility Factor ..................... 6-21


LIST OF TABLES 

LIST OF FIGURES 

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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 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, DOE developed a statistical transformer load 

simulation model to estimate the hourly characteristics of liquid-immersed transformer loads. For 

dry-type transformers, DOE 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 DOE 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, which 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 I2

R 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 percent for loads less than rated load) temperature correction. Considering both load and 

no-load losses, DOE 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 

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, 

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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 percent of 

liquid-immersed transformers, commercial/industrial entities use mainly dry-type), DOE 

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

designed one analysis (based on hourly load and price data) for liquid-immersed transformers that 

are used mainly by electric utilities, and the second analysis (based on monthly load data) for drytype 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 DOE developed to estimate transformer 

loads for use in analyzing potential efficiency standards for liquid-immersed transformers. 

6.2.1 Overview of Hourly Load Model 

DOE 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. 

DOE performed the hourly load simulation with a data-processing program that generated 

load inputs for the life-cycle cost (LCC) spreadsheet. DOE estimated 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 

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the transformer.a

 DOE derived a scaling relationship between peak load and load factor from 

available hourly load monitoring data. It determined a load distribution function with the 

estimated load factor and simulated the transformer load consistent with the load distribution 

function. DOE 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. It 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 

DOE 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 

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. 

a

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

years), then the average peak loading should be somewhat below the rating stated on the nameplate. DOE 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 of 130 

percent of nameplate. The median value that DOE selected 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. DOE 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 th e movement of electricity in all or parts of Delaware, Maryland, New Jersey, Oh io, 

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

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DOE 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-hours 

(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. It 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. It 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 they are evenly distributed between 1 and 8760 (the number of hours 

in a year), 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 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. 

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. 

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6.2.3 Hourly Load Simulation Inputs 

The following sections describe the inputs used in the hourly load simulation. 

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, DOE began with hourly system load and economic data from the Nation's 

electric utilities. DOE compiled the full list of the Nation’s electric utilities by combining all 

utilities that report retail data using DOE's Energy Information Administration (EIA)’s 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. DOE 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), MidAtlantic 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), MidAtlantic 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), MidAtlantic 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), MidAtlantic 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. DOE statistically simulated a set of transformer 

loads for each utility that were correlated with the utility’s hourly system load. DOE also 

calculated average hourly price profiles using hourly system lambdas (or market prices). It 

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). 

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6.2.3.2 Customer Type 

DOE 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 DOE 

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, DOE selected the customer 

type for the transformer after it had selected the sample utility. It simulated an equal number of 

sample transformers for each customer type and each utility. DOE 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 

DOE 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. DOE 

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 DOE 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 watt (W)/kVA. 

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

peak load. 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 

individual loads is smaller than the relative fluctuations in the sum of a few individual loads. DOE 

used data from the 1995 Commercial Building Energy Consumption Survey (CBECS) to 

determine the relationship between peak load and load factor. The variation of the load factor 

with the annual peak demand is illustrated in Figure 6.3.2 (in the next section) 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. 

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DOE used the following equation to model the relationship between annual peak load and mean 

load factor: 

LF = 0.0504Ln(x) + 0.1212 Eq. 6.2 

where: 

Ln = the natural logarithm,


x = the annual peak load, and


LF = the load factor. 


DOE used same relationship between annual peak load and load factor in the monthly 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. DOE 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, DOE calculated hourly loads using the selected load distribution 

function. 

DOE approximated the load distribution function with a one-parameter exponential fit that 

depended on the load factor. Specifically, DOE used a load 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. DOE 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 

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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 DOE selected the inverse load factor, the values 

of ALoad, BLoad, CLoad, and DLoad were calculated given the four conditions that the 

probability function satisfies. DOE 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, data on 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 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 (MWs) 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.” DOE used the system lambda as the marginal fuel cost of 

electricity for regulated utilities. 

For utilities that sell into regional electricity markets, DOE obtained the hourly system 

load and electricity cost data 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. DOE used the actual market system load as the system load for utilities 

in areas where there are electricity markets. DOE assumed 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. 

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6.3 MONTHLY LOAD MODEL 

This section describes the model DOE 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, DOE 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 

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 

DOE 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 is 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, DOE 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 Chapter 8, 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, DOE modeled the 

relationship between monthly transformer load characteristics and customer demand and usage. 

The model that DOE used estimates the load factor, coincident peak load, and RMS load on the 

transformer, relative to the total customer load. DOE 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 provided the estimate of monthly bill 

impacts for transformer owners. 

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DOE took the following steps in calculating the monthly load: 

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

from the EIA’s 1995 CBECS6

 and other data sources. 

2.	 DOE 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.	 DOE 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.	 DOE 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.	 DOE estimated monthly peak transformer load and monthly RMS transformer load by 

scaling annual peak transformer values. The scaling resulted in estimates of monthly 

values of transformer peak and RMS load that were proportional to customer monthly 

demand and energy usage data, respectively. 

6.	 Using an analysis of available hourly building load data, DOE estimated a transformer peak 

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

7.	 DOE applied the average PRF to monthly transformer peaks to estimate monthly 

transformer demand. 

8.	 DOE’s model calculated 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 (kW) and kWh, respectively. Usage is sometimes also called energy, or 

consumption. DOE 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. 

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For commercial customers, DOE obtained a statistically representative sample of monthly 

customer demand and usage from data collected in the 1995 CBECS. DOE did not have at its 

disposal a sample of monthly demand and usage data for industrial customers. It 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. 

DOE 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. DOE found that the differences among 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 than 500 kW can be 0–30 percent less than 

the load factor for commercial customers with the same annual peak load. DOE 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. DOE 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. DOE 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. DOE 

assigned a zero weight to those customers whose average demand was less than the transformer 

capacity. 

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

customers, DOE examined the EIA’s Form 861 data2 from 1995 to calculate the average size ratio 

between industrial and commercial customers for 1995. DOE found that, on average, industrial 

customers had loads 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, DOE 

constructed a new set of weights that had the same total sum, but 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. DOE chose the power 

to provide the desired change in average customer size. 

Since DOE 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 

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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 DOE 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, DOE 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)’s 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, DOE 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. DOE 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, 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, especially for low-voltage, 

dry-type transformers. DOE received stakeholder comments that medium-voltage, dry-type 

transformers are loaded more heavily than low-voltage, dry-type transformers,8 and thus DOE 

selected higher initial peak loading for medium-voltage, dry-type transformers compared to 

low-voltage dry-types. 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. In response to stakeholder 

comments, DOE adjusted this assumption for medium-voltage, dry-type transformers and selected 

initial peak loadings 10 percent higher than the low-voltage, dry-type transformer case. DOE 

therefore selected a distribution for the initial peak load that has a constant probability between 60 

percent and 90 percent of nameplate capacity for low- voltage, dry-type transformers and a 

constant probability between 70 percent and 100 percent of nameplate for medium-voltage, 

dry-type transformers. 

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DOE 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 Electric Code9 

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). 

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, DOE 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. 

Figure 6.3.1	 Histogram of Annual Load Factors Derived from Monthly CBECS 

1995 Bill Data 

From Figure 6.3.1, DOE 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. DOE 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. DOE set the maximum probability point of the triangular distribution at 0.32 so that 

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the mean load factor of the triangular distribution matched the mean load factor of the CBECS 

data. 

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

peak load. 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 

individual loads are smaller than 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. DOE 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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Figure 6.3.2	 Scatterplot of Annual Load Factor versus Annual Peak Demand from 

Monthly CBECS 1995 Bill Data 

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 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 variability, 

the RMS and the average load on the transformer may be significantly different. DOE analyzed 

hourly load data 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. 

DOE analyzed both older, public data from an ELCAP study of hourly electrical loads in 

the Northwest in 1987 and 1988,10 and more recent (proprietary) hourly load data from the 

Southeast from 1998, 1999, and 2000. It found consistent behavior between ELCAP commercial 

building loads and both industrial and commercial loads from the more recent southeastern data. 

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DOE found the differences between the average RMS-to-average load ratios from the data sets to 

be less than 10 percent. 

DOE found that the RMS-to-average load ratio depended on the load factor. DOE 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, DOE 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 

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, DOE 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. DOE estimated the distribution of PRF as a 

function of LF from available hourly load data. DOE 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. DOE 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. 

DOE 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: 

/LF PRF ) mid = (1 + e C )/(1+ e C	

PRF PRF Eq. 6.8

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where: 

C

PRFmid = the mean PFR,


PRF = a calibrated constant equal to 0.6, and


LF = the annual load factor.


DOE 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, 

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. 

Figure 6.3.3 Scatterplot of Peak Responsibility Factor versus Load Factor 

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In its treatment of PRF, DOE 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. 

DOE found the aforementioned assumptions to be consistent with hourly ELCAP and 

proprietary utility data analyzed by DOE. 

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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 

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Docket #86. Contact Ms. Brenda Edwards-Jones, U.S. Department of Energy, Building 

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

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

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Report No. IEEE PC57.12.33/D8. 

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