7-A-i

APPENDIX 7-A. TECHNICAL ASPECTS OF THE ENERGY USE AND END-USE 

LOAD CHARACTERIZATION ANALYSIS

TABLE OF CONTENTS

7-A.1 LOADING ANALYSIS FOR LIQUID-IMMERSED TRANSFORMERS .......... 7-A-1

7-A.2 ENERGY LOSSES ................................................................................................ 7-A-1

7-A.2.1 Price-Load Function............................................................................................... 7-A-2

7-A.2.2 Joint Distribution of System and Transformer Loads ............................................ 7-A-6

7-A.2.3 Transformer Peak Responsibility Factor ............................................................... 7-A-8

7-A.3 LOADING ANALYSIS FOR DRY-TYPE TRANSFORMERS .......................... 7-A-8

7-A.3.1 No-load losses ........................................................................................................ 7-A-9

7-A.3.2 Load Losses ......................................................................................................... 7-A-10

7-A.3.2.1 Energy Savings .................................................................................. 7-A-10

7-A.3.2.2 Demand Savings ................................................................................ 7-A-12

LIST OF TABLES

Table 7-A.2.1 Definition of EMM regions and NERC regions ........................................ 7-A-4

LIST OF FIGURE

Figure 7-A.2.1 System Load and Price for the SPP Region. Fifteen Load Bins are 

Defined in the Figure, Indicated by the Different Colors .......................... 7-A-4

Figure 7-A.2.2 Joint Probability Distribution Function for Commercial Load Data; the 

Function is an Average of Three Functions Calculated Separately for 

1998, 1999, and 2000 ................................................................................. 7-A-7

Figure 7-A.2.3 Joint Probability Distribution Function for Residential Load Data; the 

Function is an Average of Three Functions Calculated Separately for 

1998, 1999, and 2000 ................................................................................. 7-A-7
7-A-1

APPENDIX 7-A. TECHNICAL ASPECTS OF THE ENERGY USE AND END-USE 

LOAD CHARACTERIZATION ANALYSIS

7-A.1 LOADING ANALYSIS FOR LIQUID-IMMERSED TRANSFORMERS 

This section provides technical details regarding the methodologies the U.S. Department 

of Energy (DOE) used to estimate the energy savings and coincident peak demand reductions 

associated with higher efficiency for liquid-immersed transformers. These types of transformers 

are owned primarily by utility companies. From the utility perspective, the economic value of 

transformer energy losses is determined by (1) the marginal price for electricity and (2) the

utility’s avoided capacity costs. The marginal price for electricity is both time-dependent and a 

property of the system or control area to which the utility belongs. For this analysis, we assign 

each utility to a geographic region, for which we calculate a price that varies hourly. The regions 

used here are the set of Electricity Market Module (EMM) regions used in the Energy 

Information Administration’s National Energy Modeling System.1

7-A.2 ENERGY LOSSES

Regarding a utility’s capacity 

costs, the type of generation capacity avoided depends on the shape of the load duration curve 

for the losses, while the amount of capacity avoided depends on the value of the transformer load 

when the system load is at its peak. Hence, correct estimation of the value of transformer 

efficiency requires an understanding of the load shape of the energy losses. 

Transformer energy losses are the sum of two terms: the no-load losses (NLL), which are 

approximately constant in time and occur whenever the transformer is energized, and the load 

losses (LL), which are proportional to the square of the instantaneous load on the transformer. 

Including losses, the total energy used by a transformer experiencing instantaneous load E is: 

ET =∈NLL+ E +∈LL �E

Emax � �

2

. Eq. 7-A.2.1 

Here ∈NLL is a parameter that represents the constant (or no-load) loss rate, and ∈LL is a 

parameter that expresses the load-loss rate. Because ∈LL is defined assuming that the transformer 

is fully loaded, actual losses depend on the size of the scaled load E/Emax, where Emax is the 

expected peak load on the transformer, here assumed to be equal to its capacity. The transformer 

losses are ET

-E, and the transformer efficiency rating is defined as E/ET. 

For this analysis we assume that each transformer is part of a local system for which 

either a market-clearing price or system lambda is defined. The hourly price is denoted p(h). The 

annual energy cost associated with transformer energy losses is the sum of two terms: 

EC = ECLL + ECNLL Eq. 7-A.2.2 
7-A-2

where ECNLL is due to the no-load losses and ECLL to the load losses. Because the no-load 

losses are flat, 

ECNLL =∈NLL< p > 8,760, Eq. 7-A.2.3 

where <p> is the average over all hours of the hourly marginal production cost. For the 

load losses, 

ECLL =∈LL ∑h p(h) e2(h). Eq. 7-A.2.4 

Here we use the variable e(h) = E(h)⁄Emax to represent the hourly scaled transformer 

load. This term depends on the correlation between the transformer’s hourly load and the system 

hourly price. Because we expect individual transformer loads to be correlated with the system 

load, it follows that they also will be correlated with the system price. Failure to correctly 

represent this correlation will result in underestimating the value of the load losses.

 The sum over hours in equation Eq. 7-A.2.4 can be converted to a sum over load levels as 

follows: Let L(h) be the hourly system load, and l(h) the hourly scaled system load (the hourly 

system load divided by the annual system load maximum). Both the transformer loads and the 

system loads can be represented as a set of discrete load levels l

j

 and ek, with j = 1, ... NS and 

k = 1, ... NT. This means that in each hour we replace the actual load value with the closest 

discrete value. This procedure does not introduce a bias and will not lead to a significant loss of 

precision. The shape of the system load can be characterized by a distribution function nj, where 

nj is the number of times the system load is at level l

j

. It is also reasonable to assume that the 

system price P can be represented as a function f of the system load: 

p(h) = f(l(h)). Eq. 7-A.2.5 

 This function is equivalent to assuming that variation in the system price is driven by 

variation in the system load. Given the function f and the load level l

j

, a price is defined as 

pj = f(lj). The last required term is a function that represents the correlation between the 

transformer load levels ek and the system load levels l

j

. Let wjk be the probability that the 

transformer load is at level ek when the system load is at level l

j

. Combining the terms defined 

above, the hourly sum becomes: 

∑h p(h) e2(h) = ∑ ∑ njpjwjkek

2

kj . Eq. 7-A.2.6 

7-A.2.1 Price-Load Function

 This section describes how DOE developed a function that expresses the system price as 

a function of system load. For each EMM region, we calculated hourly time series for system 
7-A-3

loads and system prices based on 2008 hourly load and price data for individual utilities and 

control areas, obtained from Federal Energy Regulatory Commission (FERC) Form 714 filings. 

Then we calculated the system load distribution function nj by defining a set of bins to contain 

the load levels, and counting the number of times the system load falls into each bin. The system 

price function is estimated assuming: 

pj = p�j + δj, Eq. 7-A.2.7 

where p̅j is a constant term and δj is a random increment that may be positive or 

negative.

The calculation steps are described in more detail below. 

1. Each load and price time series obtained from FERC is assigned to an EMM region, 

based either on the appropriate North American Electric Reliability Corporation 

(NERC) region or the set of states in which the utility operates. Table 7-A.2.1 lists the 

EMM regions, the NERC regions they belong to. 

2. The load time series for an EMM region is defined as the sum of the load data for 

each of the utilities or control areas in that region. 

3. The price time series is defined as the average load-weighted sum of the price data for 

each utility or control area in the region. 

4. The minimum and maximum system loads are calculated, followed by the scaled 

system load l(h) = L(h)⁄Lmax. The values of l satisfy Lmin⁄Lmax ≤ l ≤ 1. 

5. The number of bins NS, bin sizes ∆j, and bin boundaries lj are defined. The bin widths 

may vary with j. Widths are chosen to satisfy two criteria: (1) that the number of 

points in each bin is of the same order of magnitude, and (2) that the range of 

variation in price within each bin is not too large. 

6. The number of hourly values l(h) that fall into each bin is counted; this number is 

defined as nj

, with ∑j nj = 8,760. 

7. The average value of l(h) in each bin is calculated; this value is written as l̅

j. 

8. The average value of the price during the hours in which the load is in bin j, (p̅j), is 

calculated. 
7-A-4

Table 7-A.2.1 Definition of EMM regions and NERC regions

EMM Region NERC Region

NE NPCC

NY NPCC

SPP SPP

ERCOT TRE

FL FRCC

RA WECC

RA WECC

NPP WECC

CA WECC

ECAR RFC

MAAC RFC

MAIN RFC

MAIN MRO

MAPP MRO

MAIN SERC

ECAR SERC

SERC SERC

The price load-relationship, and the way the data are sorted into bins, is illustrated in 

Figure 7-A.2.1. 

Figure 7-A.2.1 System Load and Price for the SPP Region. Fifteen 

Load Bins are Defined in the Figure, Indicated by 

the Different Colors

0

50

100

150

200

250

12000 17000 22000 27000 32000 37000

Lambda $/MWh

Load MWh
7-A-5

Figure 7-A.2.1 shows a scatter plot of hourly prices (on the vertical axis) and loads (on 

the horizontal axis) for the SPP region. The load data, and corresponding price values, have been 

distributed into a set of fifteen bins, which are given different colors in the figure. The plot 

shows that there is a large range of price variability within each bin, and that the range also 

varies with the bin index. In this region the price variability is lowest in the low and high bins, 

and highest near the center. Other regions may differ in the details of this relationship. To 

capture this effect, the price model includes a random increment δj, which may be positive or 

negative, and which is chosen from an empirically determined probability distribution function 

(PDF). Each region and bin has its own parameters for the PDF. The data used to define this PDF 

are the differences z = P(h) − p̅j; there is one z for each hourly price P(h) in bin j. For 

simplicity, the PDF is assumed to be triangular and centered at zero. Mathematically, the 

distribution is defined by three parameters: aj

, bj, and cj, where: 

-aj = is the point at which the triangular distribution intersects the negative z-axis, 

bj = is the point at which the triangular distribution intersects the positive z-axis, and 

cj = is the value of the probability distribution function at z = 0. 

 By definition, the area under the distribution is equal to one, which leads to the following 

expression for cj

: 

cj = 2

��bj + aj�

. Eq. 7-A.2.8 

 There generally are various possible ways to map data onto a triangular distribution. 

Here, the primary concern is to include the effect of variability without introducing any price 

bias into the model. To this end, the parameters aj

 and bj are defined so that the average positive 

(or negative) value of the difference z = P(h) − p̅j is the same for the triangular model 

distribution as it is for the real distribution. These constraints can be written: 

Sj

− = cjaj

6� ; Sj

+ = cjaj

6� , Eq. 7-A.2.9 

where Sj

− is the sum of all negative differences z in bin j, and Sj

+ is the sum of all positive 

differences. These equations allow us to determine aj

, bj, and cj from the data. The model is 

validated by using the PDF to generate a series of δ-values for each bin, then comparing the 

standard deviation of the δ’s for the original data and for the simulated data. For each region and 

each bin, the standard deviations for the model data typically are within about 10 percent (higher 

or lower) of the standard deviation values calculated for the original data.
7-A-6

7-A.2.2 Joint Distribution of System and Transformer Loads

 In this section we describe how DOE calculated the joint probability distribution function 

(JPDF) of transformer loads and system loads. The data set available at the time of this analysis 

was a set of hourly building loads indexed by x, ex(h) , x=1,...M. Each hourly load is scaled by 

its annual maximum, so that they all range in magnitude between zero and one. We assume that 

the loads on individual transformers are similar in shape to the loads on buildings. The 

relationship expressed by the function wjk is the correlation between an individual load and the 

total load of the system of which the individual load is a part. To estimate this relationship from 

the available data, we defined a (scaled) proxy system load pl(h) as the sum of the individual 

building loads: 

pl(h) = �1� � M ∑ exx (h) Eq. 7-A.2.10 

The function wjk is estimated by distributing the hourly load pairs �ex(h), pl(h)� into a 

set of NT by NS bins and counting the number of points in each bin. The modeling steps are 

described in more detail below.

 

1. Construct the set of scaled data pairs �pl(h), ex(h)�, h = 1,2 ... ,8760. 

2. Define the bins for the proxy load pl; these are identical to the bins used in the system 

load analysis described in section 7-A.2.1 above. There are NS bins having index j. 

3. Define the bins for the individual transformer loads; here the number of bins is NT, 

the bin index is k, and the bin width is constant and equal to 1/NT. 

4. Count the number of points (j, k) in each bin; this count is defined as mjk

x . 

5. Calculate the average of the mjk

x for all the transformer load time series and divide by 

8,760 to convert this count to a probability: 

wjk = (1/8760)(1/M) ∑ mjk

x x . Eq. 7-A.2.11 

 The number wjk is an estimate of the probability of finding a transformer load in bin k, 

given a system load in bin j. The value of the transformer load in bin k is estimated as the 

average value for all points in the bin, irrespective of the value of j. 

 The building data set available for this analysis included both residential and commercial 

buildings for three data years. Because the correlation patterns may depend on building type, 

separate JPDFs were calculated for residential and commercial buildings. The proxy system load 

is always defined as the sum of both residential and commercial building loads. To make use of 

multiple years of building data, we estimated wjk separately for each data year, then averaged the 

results. An example of the output is shown in Figure 7-A.2.2. The plot shows the JPDF 

calculated for the commercial building data. In this plot, lower bin indices correspond to lower 

load levels. The plot shows the expected feature that transformer loads in commercial 

installations are correlated loosely with system load when system loads are low, and are 
7-A-7

correlated more tightly when system loads are high. The residential JPDF, shown in Figure 7-

A.2.3, shows that transformers in residential installations generally are not strongly correlated 

with system loads.

Figure 7-A.2.2 Joint Probability Distribution Function for 

Commercial Load Data; the Function is an Average 

of Three Functions Calculated Separately for 1998, 

1999, and 2000 

 Figure 7-A.2.3 Joint Probability Distribution Function for 

Residential Load Data; the Function is an Average 

of Three Functions Calculated Separately for 1998, 

1999, and 2000 

1

2

3

4

5

6

7

8

9

10

1 2 3 4 5 6 7 8 9 101112131415

Transformer load bin

Proxy sytem load bin

Joint probability distribution average 1998-2000 

commercial

0.015-0.020

0.010-0.015

0.005-0.010

0.000-0.005

1

3

5

7

9

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Transformer load bin

Proxy sytem load bin

Joint probability distribution average 1987-1988 residential

0.020-0.025

0.015-0.020

0.010-0.015

0.005-0.010

0.000-0.005
7-A-8

There are insufficient data to validate the JPDF directly. We examined the robustness of 

the JPDFs by calculating test JPDFs using subsets of the full set of available hourly loads. The 

total number of buildings in a single sample varies with year and with building type, from about 

50 to about 300. The difference between the JPDF calculated using a fraction of the full data set 

and a JPDF calculated using all buildings was quantified using the L1 norm.a

7-A.2.3 Transformer Peak Responsibility Factor 

 The tests showed 

that the JPDF is insensitive to the subset of buildings chosen as long as there are about 100 

buildings or more in the data set. The JPDFs calculated with different subsets of 100 or more 

buildings do not vary significantly. 

 Reductions in transformer losses can reduce the system peak load, and hence avoid 

capacity costs. The size of the reduction in system peak load depends on the size of the 

transformer load loss during the hour of the system peak. This value is known as the transformer 

peak responsibility factor. A probability distribution for the responsibility factor can be estimated 

easily from the JPDF wjk. We define the probability r

k that the transformer load level is ek

 when 

the system load is at a peak as: 

rk = wNS,k

∑k�wNS,k� � . Eq. 7-A.2.12 

Strictly speaking, r

k

 gives the probability that the transformer load is in bin k when the 

system load is in its highest bin (j=NS). When averaged over the lifetime of the transformer, this 

value should give a reasonably estimate of the distribution of the responsibility factor. 

7-A.3 LOADING ANALYSIS FOR DRY-TYPE TRANSFORMERS

 This appendix provides technical details regarding the methodologies the U.S. 

Department of Energy (DOE) used to estimate the energy use and peak demand for dry-type 

transformers. This type of equipment is used primarily in commercial buildings and is owned by 

the building owner or operator. The economic value of energy losses therefore is determined by 

the marginal price of electricity for the building, which is set by the prevailing electricity tariff. 

In this analysis, the Department draws on a previous, detailed study of energy prices for 

commercial buildings.2

 That study showed that each building’s electricity costs can be 

represented as a marginal price for energy (MPE) and a marginal price for demand (MPD), 

which vary by region and by season. In an economic analysis, these prices are used as follows: 

ΔB = (ΔELL + ΔENLL ) * MPE + (ΔDLL + ΔDNLL

Where:

) *MPD Eq. 7-A.3.1 

 

a

The L1 norm is equal to the absolute value of the difference between the two functions.
7-A-9

ΔB = the total change in the electricity bill for the transformer owner;

MPE = the marginal price for building electricity consumption (dollars per 

kilowatt-hour [$/kWh]);

MPD = the marginal price for building electricity demand ($/kW)

ΔELL

ΔE

 = the change in electricity consumption due to load losses (kWh); 

NLL

ΔD

 = the change in electricity consumption due to no-load losses (kWh); 

LL

ΔD

 = the change in electricity billing demand due to load losses (kW); and 

NLL = the change in electricity billing demand due to no-load losses (kW). 

 The electricity billing demand is the building peak load during the billing period, which 

is assumed to be one calendar month. Hence, the change in demand is equal to the change in 

transformer losses at the time of the building peak load. The life-cycle (LCC) analysis (chapter 8 

of this preliminary TSD) calculates the change in the bill for each month in a calendar year for 

each efficiency standard, and totals those changes to estimate the annual operating cost savings 

for a given transformer owner. The load profiles for both the building and the transformer vary 

by month, but the marginal MPE and MPD vary by season only (summer and winter). For both 

no-load and load losses, the change in electricity consumption and demand depend on the 

difference between the base-case transformer loss rates, and the standards-case loss rates. The 

rest of this appendix explains how those changes are calculated in the LCC spreadsheet.

7-A.3.1 No-load losses 

No-load losses are independent of the load on the transformer and thus have a perfectly 

flat load shape. The change in the transformer no-load losses is equal to the difference between 

the base-case transformer loss rate and the standards-case loss rates, times the number of hours 

per year the transformer is energized:

ΔENLL = (NLLBaseCase – NLLStandardsCase)*HPY Eq. 7-A.3.2 

Where:

NLLBaseCase

NLL

 = the no-load loss rate in the base case (kW);

StandardsCase

HPY = the hours per year that the transformer is energized, equal to 8,760. 

 = the no-load loss rate in the standards case (kW); and

Because the no-load losses are flat (constant in every hour), the change in billing demand 

is equal only to the change in the no-load loss rate: 

ΔDNLL = (NLLBaseCase – NLLStandardsCase).
7-A-10

7-A.3.2 Load Losses 

 This section describes the load losses for distribution transformers, used to calculate both 

the energy and demand savings in the LCC spreadsheet. 

7-A.3.2.1 Energy Savings 

Load-dependent losses are proportional to the square of the load on the transformer. The 

change in transformer losses is equal to the change in the load loss rate times the square of the 

hourly load L(h), summed over all hours in the year: 

ΔELL = (LLBaseCase – LLStandardsCase)*[ ∑h (L(h)/PL)2 ]*(PL/CAP)2 Eq. 7-A.3.3 

Where:

LLBaseCase

LL

 = the load loss rate in the base case (kW),

StandardsCase

L(h) = the hourly transformer load h,

 = the load loss rate in the standards case (kW), 

PL = the annual peak load on the transformer, and 

CAP = the transformer capacity.

 The above equation follows the convention whereby hourly loads are expressed as a 

fraction of the annual transformer peak load PL, and the peak load is expressed relative to the 

transformer capacity. The annual PL is equal to the initial peak load times an annual growth 

factor, both of which parameters are inputs to the spreadsheet. Load shape information is 

contained in the sum of squared hourly loads. For the LCC, the sum should be calculated for 

each monthly billing period: 

LSFM = [ ∑h (L(h)/PL)2

Where:

]/NH

LSFM

NH = the number of hours during the billing period that the transformer is 

energized, defined here as 8,760/12 

 = the monthly transformer loss factor, and 

A statistical model is used to estimate LSFM as a function of the building’s monthly load 

factor. This approach is based on the well-known “rule-of-thumb”:3

LSF = α*LF + (1-α)*LF

Where:

2

α = alpha, a numerical parameter defined so that 0 < α < 0.5;
7-A-11

LSF = the transformer loss factor defined for a given, fixed period; and 

LF = the load factor, equal to the average load divided by the peak load, using 

the same period as for the LSF. 

 The equation can be rearranged to give: 

α = (LSF – LF2)/(LF - LF2

) Eq. 7-A.3.4 

A distribution of values for the parameter α is estimated using hourly building load data. 

First we process the data to produce monthly values of the load factor LFM and loss factor LSFM. 

For each building and each month, we use the values of LFM and LSFM to calculate a value of α. 

Finally, we calculate a frequency distribution for α from the set of monthly values. Because 

electricity prices are seasonal, we examined the data to evaluate whether the α-distributions 

varied with season, but found no significant dependence. The α-distributions do vary as a 

function of load factor, however. To capture this effect, we calculated three separate distributions 

for three ranges of load factor: low (0 < LFM ≤ 1.3), medium (1/3 < LFM ≤ 2/3), and high (2/3 < 

LFM ≤ 1). The distributions are shown in Figure 7-A.3.2. 

Figure 7-A.3.2 Frequency Distributions for the LSF

Within the LCC, the consumer data include monthly values for the building load factor, 

which are used as proxies for the transformer load factor. For each month, the distributions 

0%

5%

10%

15%

20%

25%

30%

35%

40%

45%

50%

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

alpha

frequency

Low LF Medium LF High LF
7-A-12

shown in Figure 7-A.3.1 are used to select a value of α, and Eq. 5-P.1.4 is used to estimate the 

loss factor. 

7-A.3.2.2 Demand Savings 

 The billing demand savings associated with each possible standard are equal to the 

change in the transformer load loss rate times the square of the transformer load during the hour 

of the building peak load: 

ΔDLL = (LLBaseCase – LLStandardsCase)*(L(hmax)/PL)2 *(PL/CAP)2

 Eq. 7-A.3.5 

Where:

hmax = the hour of the building peak load, and 

L(hmax) = the transformer load during hour hmax.

 The ratio L(hmax)/PL is defined as the coincident peak load (CPL) for the building. 

Using this parameter, the equation for the billing demand savings becomes: 

ΔDLL = (LLBaseCase – LLStandardsCase)*CPL2 *(PL/CAP)2 Eq. 7-A.3.6

 The square of the CPL is known as the peak responsibility factor (RF). The LCC 

calculation uses a statistical model to estimate monthly values of CPL/RF. The data available for 

this study included only whole building loads, not individual transformer loads. To approximate 

the behavior of a building containing several transformers, we manipulated the building data as 

follows. 

1. We summed the individual hourly loads to create a single aggregate load. 

2. For each month, we calculated the hour hmax of the peak aggregate load. 

3. For each individual hourly load and each month, we calculated the value of the individual 

load during hour hmax. 

4. From this procedure, we derived a set of monthly values of CPL. 

 

 The distribution of values of the coincident peak loads calculated in this way is illustrated 

in Figure 7-A.3.2, which shows the cumulative distribution function for CPL as well as several 

powers of CPL. 
7-A-13

Figure 7-A.3.3 Cumulative Distribution Function for the Coincident Peak 

Load 

 Roughly 80 percent of the CPL values in the sample are greater than 0.8. Examination of 

the data showed that the value of CPL is sensitive to season; this makes sense as space 

conditioning should lead to higher coincidence and therefore higher values for CPL in summer 

than in winter. The values of CPL also are sensitive to the building’s monthly load factor. We 

defined a statistical model capturing these effects, as follows. 

1. To even out the distribution, we use the fourth power of the CPL (square of the 

responsibility factor); the data for each building and each month provide a set of sample 

pairs (LFM, CPL4

2. We distributed the data into a set of 10 x 10 bins, according to the values of CPL

). Both variables have a range from zero to one. 

4 and 

LFM

Figure 7-A.3.3

. The bin sizes are constant for each variable. An example of the distribution for 

summer data is shown in . 

3. We used the number of values in each bin, divided by the total number of values in the 

sample, as an estimate of the probability that CPL4

4. We converted the probabilities into a distribution for CPL

 is in a particular bin, given that the 

load factor is in a given bin. 

2

 by taking the square root of 

the bin limits defined for CPL4. 

 Within the LCC, the building’s monthly load factor is used as a proxy for the transformer 

load factor. For each month, we used the value of this load factor and the probability distribution 

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0% 20% 40% 60% 80% 100%

CPL CPL^2 CPL^3 CPL^4
7-A-14

defined above to select a random value for CPL2

, which then was used to calculate the demand 

savings from the load losses. 

Figure 7-A.3.4 Distribution of Values of the Pairs CPL4 (Horizontal Axis) and 

LFM (Vertical Axis), Summer

4-11

1 2 3 4 5 6 7 8 9 1 11

S1

S2

S3

S4

S5

S6

S7

S8

S9

S10
7-A-15

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3. M. E. de Oliveira, A. Padilha-Feltrin & F. J. Candian Investigation of the Relationship 

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

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7-A-16

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