dt_nopr_appendix07a_09042011 DRAFT
7

- A

- i APPENDIX 7

- A. TECHNICAL ASPECTS OF THE ENERGY USE AND END

- USE LOAD CHARACTERIZATIO N ANALYSIS

TABLE OF CO NTENTS

7

- A.1 LOADING ANALYSIS FOR LIQUID

- IMMERSED TRANSFORMER S

.......... 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

- 9 7

- A.3 LOADING A NALYSIS FOR DRY

- TYPE TRANSFORMERS

.......................... 7

- A

- 9 7

- A.3.4 No

- load losses

................................ ................................ ................................ ...... 7

- A

- 10 7

- A.3.5 Load Losses

................................ ................................ ................................ ......... 7

- A

- 10 7

- A.3.5.1 Energy Savings

................................ ................................ .................. 7

- A

- 11 7

- A.3.5.2 Demand Savings

................................ ................................ ................ 7

- A

- 13

LIST OF TABLES

Table 7

- A.2.1 Definition of EMM regions and NERC regions in terms of States

........... 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, 1 999, and 2000

................................ ................................ ................. 7

- A

- 7 Figure 7

- A.2.3 Joint Probability Distribution Function for Industrial Load Data; the Function is an Average of T hree Functions Calculated Separately for 1998, 1999, and 2000

................................ ................................ ................. 7

- A

- 8 Figure 7

- A.2.4 Joint Probability Distribution Function for Resid ential Load Data; the Function is an Average of Three Functions Calculated Separately for 1998, 1999, and 2000

................................ ................................ ................. 7

- A

- 8dt_nopr_appendix07a _09042011 DRAFT 7

- A

- 1 APPENDIX 7

- A. TECHNICAL ASPECTS OF THE ENERGY USE AND END

- USE
LOAD CHARACTERIZATIO N ANALYSIS

7

- A.1 LOADING ANALYSIS FOR LIQUID

- IMMERSED TRANSFORMER S

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 pri marily 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 Mark et Module (EMM) regions used in the Energy Information Administration’s National Energy Modeling System. 1 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.

7

- A.2 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 tran sformer experiencing instantaneous load E is:

E T = ∈ NLL + E + ∈ LL � E E max � � 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

- lo ss rate. Because ∈ LL is defined assuming that the transformer is fully loaded, actual losses depend on the size of the scaled load E / E

max , where E

max is the expected peak load on the transformer, here assumed to be equal to it s capacity. The transformer losses are E T - E , and the transformer efficiency rating is defined as E / E T .
For this analysis we assume that each transformer is part of a local system for which either a market

- clearing price or sys tem 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 = EC LL + EC NLL Eq. 7-A.2.2dt_nopr_appendix07a _09042011 DRAFT 7

- A

- 2 where EC NLL is due to the no

- load losses and EC LL to the load losses. Because the no

- load losses are flat,

EC NLL = ∈ 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,

EC LL = ∈ LL ∑ p ( h ) h e 2 ( h )

. Eq. 7-A.2.4

Here we use the variable e ( h ) = E ( h ) E max ⁄ 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 resul t 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 follo ws: 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 e k , with j = 1
, ... N S and
k = 1
, ... N T . 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. Th e shape of the system load can be characterized by a distribution function n j , where

n

j 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

p j = f ( l j ) . The last required term is a function that represents the correlation between the transformer load levels e k and the system load levels l j . Let w jk be the probability that the
transformer load is at level e k when the system load is at level l j . Combining the terms defined above, the hourly sum becomes:

∑ p ( h )

h e 2 ( h ) = ∑ ∑ n j p j w jk e k 2 k j . 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 systemdt_nopr_appendix07a _09042011 DRAFT 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 n j 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:

p j = p �

j + δ j , Eq. 7-A.2.7

where p ̅ j is a constant term and δ j 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 R eliability 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 ) L max ⁄ . The values of l satisfy L min L max ⁄ ≤ l ≤ 1

. 5. The number of bins N S , bin sizes ∆ j , and bin boundaries l j 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 mag nitude, 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 n j , with ∑ n j = 8
, 760 j . 7. The average value of l ( h ) in each bin is calculated; this value is written as l ̅ j . 8. The average value of the price during the hours in which the load is in bin j
, ( p ̅ j ) , is calculated.dt_nopr_appendix07a _09042011 DRAFT 7

- A

- 4 Table 7

- A.2.1 Definition of EMM regions and NERC regions in terms of States 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 37000Lambda $/MWh

Load MWhdt_nopr_appendix07a _09042011 DRAFT 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 fi gure. 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 ma y differ in the details of this relationship. To capture this effect, the price model includes a random increment **δ** j , 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 )

− p ̅ j ; 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. Mat hematically, the distribution is defined by three parameters: a j , b j , and c j , where:

- a

j = is the point at which the triangular distribution intersects the negative z

- axis,
b

j = is th e point at which the triangular distribution intersects the positive z

- axis, and

c j = 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 follo wing expression for c j :

c j = 2 � b j + a j � �

. 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 a j and b j are defined so that the average positive (or negative) value of the difference z = P ( h )

− p ̅ j is the same for the triangular model distribution as it is for the real distribution. These constraints can be written:

S

j − = c j a j 6 � ; S j + = c j a j 6 �

, Eq. 7-A.2.9

where S j − is the sum of all negative differences z in bin j , and S j + is the sum of all positive differences. These equations allow us to determine a j , b j , and c j 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.dt_nopr_appendix07a _09042011 DRAFT 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 , e x ( 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 w jk 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 i ndividual building loads:

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

The function w jk is estimated by distributing the hourly load pairs � e x ( h )

, pl ( h ) � into a set of N T by N S 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 )

, e x ( 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 describ ed in section 7

- A.2.1 above. There are N S bins having index j

. 3. Define the bins for the individual transformer loads; here the number of bins is N T , the bin index is k , and the bin widt h is constant and equal to 1/N T

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

. 5. Calculate the average of the m jk x for all the transformer load time series and divide by 8,760 to convert this count to a probability:

w

jk = ( 1 / 8760 ) ( 1 / M ) ∑ m jk x x . Eq. 7-A.2.11

The number w jk 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 residential

, commercial and industrial buildings for three data years. Because the correlation patter ns may depend on building type, separate JPDFs were calculated for residential, commercial and industrial buildings. The proxy system load is always defined as the sum of residential, commercial , and industrial building loads. To make use of multiple years of building data, we estimated w jk 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 i nstallations are correlated loosely with system load when system loads aredt_nopr_appendix07a _09042011 DRAFT 7

- A

- 7 low, and are correlated more tightly when system loads are high. The industrial and residential JPDF, shown in f igure 7

- A.2.3 and figure 7

- A.2.3 , shows that transformers in residential and industrial installations generally are not st rongly 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 1 2 3 4 5 6 7 8 9 10

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

Proxy System Load Bins 0.020

- 0.025

0.015

- 0.020

0.010

- 0.015

0.005

- 0.010

0.000

- 0.005dt_nopr_appendix07a _09042011 DRAFT 7

- A

- 8 Figure 7

- A.2.3 Joint Probability Distribution Function for Industrial Load Data; the Function is an Average of Three Functions Calculated Separately for 1998, 1999, and 2000

Figure 7

- A.2.4 Joint Probability Distribution Function for Residential Load Data; the Function is an Average of Three Functions Calculated Separately fo r 1998, 1999, and 2000

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 1 2 3 4 5 6 7 8 9 10

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

Proxy System Load Bins 0.030

- 0.040

0.020

- 0.030

0.010

- 0.020

0.000

- 0.010

1 3 5 7 9

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

- 0.005dt_nopr_appendix07a _09042011 DRAFT 7

- A

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

7

- A.2.3 Transformer P eak 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 w jk . We define the probability r k that the transformer load level is e k when the system load is at a peak as:

r k = w N S , k ∑ � w N S , k � 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 = N S ) . 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 r egarding 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 e conomic 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

comme rcial 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 = (ΔE LL + ΔE NLL ) * MPE + (ΔD LL + ΔD NLL ) *MPD Eq. 7-A.3.1
Where:

ΔB = the total change in the electricity bill for the transformer owner;
MPE = the marginal price for building electricity consumption (dollars per kilowatt

- hour [$/k Wh]);

a The L1 norm is equal to the absolute value of the difference between the two functions.dt_nopr_appendix07a _09042011 DRAFT 7

- A

- 10 MPD = the marginal price for building electricity demand ($/kW)
ΔE LL = the change in electricity consumption due to load losses (kWh);
ΔE NLL = the change in electricity consumption due to no

- load losses (kWh);
ΔD LL = the change in electricity billing demand due to load losses (kW); and
ΔD 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 ch ange 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 sta ndard, 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). Fo r 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 i n the LCC spreadsheet.

7

- A.3.4 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 r ate and the standards

- case loss rates, times the number of hours per year the transformer is energized:

ΔE NLL = (NLL BaseCase

– NLL StandardsCase )*HPY Eq. 7-A.3.2

Where:

NLL BaseCase = the no

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

- load loss rate in the standards case (kW); and
HPY = the hours per year that the transformer is energized, equal to 8,760.

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:

ΔD NLL = (NLL BaseCase

– NLL StandardsCase ).

7

- A.3.5 Load Losses

This section describes the load losses for distributi on transformers, used to calculate both the energy and demand savings in the LCC spreadsheet.dt_nopr_appendix07a _09042011 DRAFT 7

- A

- 11 7

- A.3.5.1 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 t he load loss rate times the square of the hourly load L(h) , summed over all hours in the year:

ΔE LL = (LL BaseCase

– LL StandardsCase )*[ ∑ h (L(h)/PL) 2 ]*(PL/CAP) 2 Eq. 7-A.3.3

Where:

LL BaseCase = the load loss rate in the base ca se (kW),
LL StandardsCase = the load loss rate in the standards case (kW), L(h) = the hourly transformer load h,
PL = the annual peak load on the transformer, and
CAP = the transformer capacity.

The above equation follows the convention whereby hourly lo ads 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:

LSF M = [ ∑ h (L(h)/PL) 2 ]/NH
Where:

LSF M = the monthly transformer loss factor, and
NH = the number of hours during the billing period that the transformer is energized, defined here as 8,760/12

A statistical model is used to estimate LSF M as a function of the building’s monthly load factor. This approach is based on the well

- known “rule

- o f

- thumb”: 3

LSF = α*LF + (1

- α)*LF 2

Where:

α = alpha, a numerical parameter defined so that 0 < α < 0.5;
LSF = the transformer loss factor defi ned 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

.dt_nopr_appendix07a _09042011 DRAFT 7

- A

- 12 The equation can be rearranged to give:

α = (LSF

– LF 2 )/(LF

- LF 2 ) 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 LF M and loss factor LSF M . For each building and each month, we use the values of LF M and LSF M 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 < LF M ≤ 1. 3), medium (1/3 < LF M ≤ 2/3), and high (2/3 < LF M ≤ 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 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. 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 alphafrequency Low LF Medium LF High LFdt_nopr_appendix07a _09042011 DRAFT 7

- A

- 13 7

- A.3.5.2 Demand Savings
The billing demand savings associated with each possible standard are equal to the change in the transformer loa d loss rate times the square of the transformer load during the hour of the building peak load:

ΔD LL = (LL BaseCase – LL StandardsCase )*(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:

ΔD LL = (LL BaseCase – LL StandardsCase )*CPL 2 *(P L/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 mo nth, 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 distrib ution function for CPL as well as several powers of CPL.dt_nopr_appendix07a _09042011 DRAFT 7

- A

- 14 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 se nsitive 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 m odel 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 ( LF M , CPL 4 ). Both variables have a range from zero to one.
2. We distributed the data into a set of 10 x 10 bins, according to the values of CPL 4 and LF M . The bin sizes are constant for each variable. An example of the distribution for summer data is shown in Figure 7

- A.3.3

. 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 CPL 4 is in a particular bin, given that the load factor is in a given bin.
4. We converted the probabilities into a distribution for CPL 2 by taking the square root of the bin limits defined for CPL 4

.

Within the LCC, the building’s monthly load factor is used as a proxy for the transformer load factor. Fo r 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^4dt_nopr_appendix07a _09042011 DRAFT 7

- A

- 15 defined above to select a random value for CPL 2 , which then was used to calculate the demand savings from the load losses.

Figure 7

- A.3.4 Distribution of Values of the Pairs CPL 4 (Ho rizontal Axis) and LF M (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 S10dt_nopr_appendix07a _09042011 DRAFT 7

- A

- 16 REFERENCES

1. Energy Information Administration

- Office of Integrated Analysis and Forecasting The National Energy Modeling System . (U.S. Department of Energy: Washington DC, 2009).at <http://www.eia.doe.gov/oiaf/aeo/overview/>

2. Coughlin, K., Bolduc, C., Van Buskirk, R., Rosenquist, G. & McMahon, J. Tariff

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