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Electricity demand as a function of temperature

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Electricity demand as a function of temperature

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ELECTRICITY DEMAND AS A FUNCTION OF TEMPERATURE
by
Daniel Alexander Bruce
A Thesis Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfilment of the
Requirements for the Degree
MASTER OF SCIENCE
(APPLIED MATHEMATICS)
AUGUST 2001
Copyright 2001 Daniel Alexander Bruce
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UNIVERSITY OF SOUTHERN CALIFORNIA
TH E GRADUATE SCHOOL
U N IV ER S ITY PARK
LOS ANGELES. C A LIFO R N IA SO0O7
This thesis, written by
Daniel Alexander Bruce____________
under the direction of Thesis Committee,
and approved by a ll its members, has been pre
sented to and accepted by the Dean of The
Graduate School, in partial fulfillm ent of the
requirements fo r the degree of
Master of Science
Dms
Tint* August 7 , 2001
THESIS COMMITTEE
t
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TABLE OF CONTENTS
PAGE
LIST OF TABLES iii
SECTION I : INTRODUCTION 1
SECTION II : THE MATHEMATICAL PROBLEM 3
SECTION III: METHOD AND PROCEDURE 3
SECTION IV : REGRESSION RESULTS 11
SECTION V : CONCLUSION 18
BIBLIOGRAPHY 20
APPENDIX 1 21
APPENDIX 2 22
ii
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LIST OF TABLES
PAGE
TABLE 1 TEST FOR DIFFERENCE IN YEARS 5
TABLE 2 TEST FOR DIFFERENCE IN QUARTERS 8
TABLE 3 QUARTERLY LOAD STATISTICS 9
TABLE 4 TEST FOR DIFFERENCE IN HOURS 10
TABLE 5 SECOND QUARTER REGRESSION 12
TABLE 6 THIRD QUARTER REGRESSION 13
TABLE 7 FOURTH QUARTER REGRESSION 13
TABLE 8 MEAN TEMPERATURE 14
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SECTION I
This paper studies the mathematical relationship
between electricity demand and ambient temperature using
data from the City of Anaheim Public Utility Department.
The demand curve for electricity, unlike the demand curve
for most goods and services, is a derived demand in the
sense that consumers do not receive satisfaction from
electricity itself but rather from the many products that
require electricity to operate. Utility companies
generate and distribute electricity in order to make it
possible to consume those goods and services that provide
satisfaction.
One such product is the common air conditioner, and
it is my intention to isolate that particular portion of
electricity demand which is directly related to
controlling environmental conditions. Relief from the
discomfort due to heat and humidity creates a desire for
interior climate control, and the use of electric motors
in air conditioning units produces load demand. This
simple relationship implies that warmer days should
require more electrical power to maintain a given indoor
temperature. Therefore, electricity demand must be some
function of current weather conditions.
l
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Now, if I partition the demand function into the sum
of two distinct parts: hourly load demand as a function
of temperature only, and hourly demand viewed as a random
process with temperature held constant; then statistical
regression theory can be employed to estimate the actual
relationship between temperature and load demand.
Energy economist John Peirson and Andrew Henley at
the University of Kent have already published econometric
estimates of load demand as a function of temperature for
the British electricity industry, see "Electricity load
and temperature" in Energy Economics 1994 16(4) 235-243.
That article discusses dynamic modelling using lagged
temperature variables, where they introduce the process
of temperature transfer from outside the building passing
through the exterior walls and thereby affecting inside
conditions. They proceed under the assumption that
today's weather conditions will have a measurable effect
on tomorrow's air conditioning requirements.
This is an area of continuous research, particularly
with regard to load forecasting models using temperature
as an independent variable. The California Energy
Commission has staff members working on a forecasting
model for the western region of the United States.
2
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SECTION II
From the data set of hourly load demand and
corresponding temperature readings, I will estimate the
differential hourly load required to manage interior
conditions when the prevailing temperature exceeds 72
degrees. The specific hour chosen is the three o'clock
hour for weekday afternoons. The stochastic model I
propose is
D(t) = S + C(t) + e , t > 72 degrees. (1)
In this equation D is hourly load demand as a
function of temperature, S is hourly load demand at
temperatures fixed between 68 and 72 degrees, C is hourly
load demand used for air conditioning purposes only, and
e is a normally distributed random error component with
mean zero and finite variance. Upon taking expectations
we have,
E (D) = E(S) + E(C) . (2)
SECTION III
To estimate these random variables I took raw load
and temperature data from the City of Anaheim covering
the period September 1, 1993 through December 31, 1995,
for a total of twenty eight months (see Appendix 1 & 2).
3
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Ambient temperature was recorded at fifteen minute
intervals at the Lewis substation in Anaheim, and load
demand was recorded every half hour for the entire
system. Holidays were removed from the dataset to
eliminate unusually low load numbers.
The random variable S measures load demand when
temperature is held constant in the range of 68 to 72
degrees, and I consider this to be a normally distributed
stationary random process (Grimmett and Stirzaker, 1995).
While holding temperature constant, the expected load
demand for each individual workday hour is a normal
stationary random process with a specific mean and finite
variance. My paper will study this random process for
the three o'clock hour on weekday afternoons.
The breakdown of energy use by customer
classification in Anaheim shows that over 48 per cent of
electricity generated is consumed by industry and 24 per
cent for commercial use. So on a daily basis, a fairly
constant routeen use of lighting, tools and office
equipment should be expected. Indeed, what we actually
record is the sum total of many individual random
processes, the majority of which are highly stable.
Over the long run this stationary random process is
clearly a function of economic conditions and the current
4
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state of technology. Thus, with economic growth and
technological change, the expected value of S will also
change. During the time period studied, the economy in
Orange County experienced a slow and steady improvement
out of an economic recession. A linear statistical model
is used to test for any significant difference in demand
between the years studied (Montgomery, 1991), Table 1.
TABLE 1
General Linear Models Procedure
Class Level Information
Class Levels Values
YR 2 94 95
Number of observations in data set =217
Dependent Variable: LOAD
Source DF
Model 1
Error 215
Corrected Total 216
Sum of
Squares
3440.264
33366.277
36806.541
Mean
Square
3440.264
155.191
F Value
22.17
Pr > F
0.0001
The table gives the results of a single factor
statistical experiment with two levels of the factor. In
this case I am testing load demand, the single factor, at
two yearly levels. The model postulates load demand as a
linear function of economic activity, defined by year.
5
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Each observation is considered as a linear combination of
three items: a general population mean, a treatment mean
associated with each year, and a random error component.
An analysis of variance test is used to statistically
determine whether the treatment means are significantly
different than zero.
Combining all 217 observations over the two year
period provided a data set from which a sample mean is
calculate. Now looking at Table 1, the mean square error
for the model is the sum of the squared differences
between each yearly mean and the overall mean, divided by
its degrees of freedom. And the mean square error due to
errors is the sum of the squared differences between each
observation and its particular treatment mean, divided by
the degrees of freedom. Under the assumption that each
treatment mean is zero, then both of the above numbers
are point estimates of the same population variance.
When random samples are drawn independently from two
normal populations with equal variances, the ratio of
sample variances is distributed as an F statistic. If
one or both treatment means are different from zero; then
the mean square error for the model, the numerator in the
ratio, will be significantly larger.
6
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The F-test did reject the hypothesis that both
treatment means are zero. This would indicate that
stationary load had changed between years. The pickup in
economic activity had indeed increased electicity demand.
When modelling stationary load demand over the long run,
economic growth should be considered an important factor.
In that type of model, expected load demand is viewed as
the Siam of a constant base load and a growth factor
representing economic conditions.
However, I did combine both yearly samples to
estimate the mean value of S. The benefit of
consolidating both data sets is the greater number of
observations available for the temperature regression. I
do not feel that this presents a major problem with
regards to estimating the functional relationship between
temperature and load demand. Variation in aggregate air
conditioning demand will not be greatly effected by mild
economic growth, at least in the short run.
With Disneyland situated in Anaheim, and many other
tourist attractions in close proximity, seasonal
variations in hotel occupancy and commercial employment
may cause hourly load demand to differ throughout the
year. To better capture these effects, I partitioned the
calendar year into four quarters and considered each
7
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quarterly sample as coming from a distinct population.
Employing the same methodology as that used in Table 1, I
tested whether the mean hourly load demand for each
quarter was indeed significantly different. The model
tested is once again a linear combination of an overall
population mean plus four separate treatment means and a
random error component. The results of the statistical
test are presented in Table 2.
TABLE 2
General Linear Models Procedure
Class Level Information
Class Levels Values
Q 4 12 3 4
Number of observations in data set = 258
Dependent Variable: LOAD
Source DF
Model 3
Error 254
Corrected Total 257
Sum of
Squares
12611.668
33416.304
46027.973
Mean
Square F Value
4203.889
131.560
31.95
Pr > F
0.0001
The small probability of the F-statistic indicates
that at least one of the treatment means is significantly
different from zero, so I decided to studied each quarter
separately. The data used to estimate the mean and
variance of S was chosen such that all weekday load
8
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observations occurring during the afternoon hours between
one and three o'clock inclusive, and corresponding to
temperatures fixed between 68 and 72 degrees inclusive,
were included, Table 3.
Q1
N Obs Minimum
TABLE 3
Maximum Mean Std Dev
78 300.479 342.959 315.1535 10.0224
Q2 N Obs Minimum Maximum Mean Std Dev
61 312.480 372.806 331.2655 11.6240
Q3 N Obs Minimum Maximum Mean Std Dev
15 329.039 357.839 338.4794 7.4059
Q* N Obs Minimum Maximum Mean Std Dev
104 302.160 352.127 322.2622 12.7738
Using three hours of data for each day, instead of
just the three o'clock hour, gave me more observations
with which to estimate the expected value of S. However,
to justify using all three hours I needed to test for any
significant difference in stationary load demand between
hours. I used an analysis of variance to once again
determine whether each hour's load sample came from the
same overall population. The individual treatment means
used in the linear model are the three distinct hours.
9
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This time the F-test does not reject the hypothesis
that each sample came from the same population. The
probability of getting an F value greater than .21 in a
random sample is greater than eighty per cent. This
tells me that the daily routeen of energy use with
temperature held constant is unchanged over the three
hours chosen. Table 4.
TABLE 4
General Linear Models Procedure
Class Level Information
Class Levels Values
HR 3 13 14 15
Number of observations in data set = 258
General Linear Models Procedure
Dependent Variable: LOAD
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 2 77.3638 38.6819 0.21 0.8070
Error 255 45950.609 180.1984
Corrected Total 257 46027.9734274
There may exist some serial correlation within the
data set over the three hour period for each day. Now if
that is the case, then the data set can not be considered
as a completely ramdom sample of load numbers. Each
observation will not be independent of the others, and
1 0
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this may bias the point estimate for the expected value
of S (Mendenhall and Wackerly, 1990). However, there is
no reason to expect any of the individual load demands to
be correlated, and I am working under the assumption that
stationary demand is very stable for weekday afternoons.
For example, the standard deviation for the third quarter
is less than 2.3 per cent of the mean load value of
338.48 megawatt/hour, Table 3.
SECTION IV
Once the expected value of S was determined, I ran
separate regressions for each set of quarterly data to
estimated the functional relationship between load demand
and temperature. First, I selected all load observations
where the temperature exceeded 72 degrees during the
three o'clock hour. Load values are reported every half
hour for the Anaheim system, so I took the average of the
3:00 and 3:30 numbers as the megawatt/hour load demand.
After subtracting the value of S, load demand was fixed
at zero for temperature readings of 72 degrees and below.
L = (hourly load) - (mean stationary load) . (3)
Temperature values were reported at fifteen minute
intervals. So beginning at three o'clock, I took the
11
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average of four readings to obtain my hourly number.
Temperature was also normalized to zero for use in the
regression data set by the equation
t = (mean hourly temperature) - 72 (4)
Regression results are presented for the second,
third, and fourth quarter as Table 5 through 7.
TABLE 5
QUARTER 2 L = load - 331.27 t = temp - 72
Dependent Variable: L
Analysis of Variance
Source DF
Sum of
Squares
Mean
Square F Value Prob>F
Model 2 78357.776 39178.888 41.845 0.0001
Error 48 44942.231 936.296
Total 50 123300.0084
R-square 0.6355
Variable DF
t 1
t2 1
Parameter
Estimate
10.141754
-0.475701
Standard
Error
1.63009866
0.14179410
T for HO:
Parameter=0
6.222
-3.355
Prob > |T|
0.0001
0.0016
12
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TABLE 6
QUARTER 3 L = load - 338.48 t = temp - 72
Dependent Variable: L
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Prob>F
Model 2 979971.100 489985.550 706.541 0.0001
Error 142 98476.840 693.498
Total 144 1078447.9417
R-square 0.9087
Parameter Standard T for HO:
Variable DF Estimate Error Parameter=0 Prob > |T|
t 1 8.363933 0.53593714 15.606 0.0001
t2 1 -0.125273 0.02998927 -4.177 0.0001
TABLE 7
QUARTER 4 L = load - 322.26 t = temp - 72
Dependent Variable: L
Analysis of Variance
Sum of Mean
Source DF Squares Square
Model 2 72984.131 36492.065
Error 81 23939.814 295.553
Total 83 96923.94609
R-square 0.7530
F Value
123.470
Prob>F
0.0001
Variable DF
Parameter
Estimate
Standard
Error
T for HO:
Parameter=0 Prob > | T |
t 1 1.960868 0.58626796 3.345 0.0012
t2 1 0.111881 0.03948172 2.834 0.0058
13
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The best regression results were found using data
from the third quarter, where the mean temperature was
just under 83 degrees, Table 8. Also, the summer quarter
provided the largest data set for use in the regression
with 145 temperature observations that exceeded 72
degrees.
TABLE 8
MEAN TEMPERATURE FOR EACH QUARTER
Q1
N Obs Minimum Maximum Mean Std Dev
387 49.5325 89.6325 66.9689 8.0124
Q2 N Obs Minimum Maximum Mean Std Dev
390 52.6200 89.6975 70.2425 7.4886
Q3 N Obs Minimum Maximum Mean Std Dev
459 68.4200 100.5400 82.8635 6.3530
Q4 N Obs Minimum Maximum Mean Std Dev
588 54.1775 97.8600 71.5770 7.5348
For the summer quarter the expected air conditioning
load demand is given as a second order polynomial in t,
C(t) = 8.363933*t - 0.125273*tA2 . (5)
The intercept is forced to be zero because temperature
and load are both normalized to zero for the regression
data set. After subtracting a constant stationary demand
of 338.48 megawatt/hour from each load observation, then
14
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The best regression results were found using data
from the third quarter, where the mean temperature was
just under 83 degrees, Table 8. Also, the summer quarter
provided the largest data set for use in the regression
with 145 temperature observations that exceeded 72
degrees.
TABLE 8
MEAN TEMPERATURE FOR EACH QUARTER
Q1 N Obs Minimum Maximum Mean Std Dev
387 49.5325 89.6325 66.9689 8.0124
Q2 N Obs Minimum Maximum Mean Std Dev
390 52.6200 89.6975 70.2425 7.4886
Q3 N Obs Minimum Maximum Mean Std Dev
459 68.4200 100.5400 82.8635 6.3530
Q4 N Obs Minimum Maximum Mean Std Dev
588 54.1775 97.8600 71.5770 7.5348
For the summer quarter the expected air conditioning
load demand is given as a second order polynomial in t,
C(t) = 8.363933*t - 0.125273*tA2 . (5)
The intercept is forced to be zero because temperature
and load are both normalized to zero for the regression
data set. After subtracting a constant stationary demand
of 338.48 megawatt/hour from each load observation, then
14
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the expected air conditioning load demand must be zero at
72 degrees (t = 0).
The negative coefficent for the second polynomial
term is consistant with the notion of a decreasing
marginal use of electricity as the thermometer rises.
Although expected load demand is increasing in
temperature, the rate of change is decreasing. The
derivative of this function is,
Cr (t) = 8.363933 - 0.250546*t . (6)
The regression uses the least squares method whereby
the total sum of squared errors, which is the squared
difference between each observation and its predicted
value, is minimized over the estimated coefficients
(Mendenhall and Wackerly, 1990; Watson et al., 1993). A
function of the two coefficients is found by substituting
the second order polynomial for the predicted value in
the sum of squared errors. The minimum value of this
function is determined by setting the partial derivatives
with respect to the two coefficients in the polynomial
equal to zero and solving for their values.
The analysis of variance table presents the results
of a statistical test done on the regression
coefficients. Assuming that each coefficicent is zero,
and hence temperature must not be a factor in load
15
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error for the model is very large relative to the mean
squared error for error.
The bottom portion of the regression table tests for
the significance of each coefficient separately using a
T-statistic. The test hypothesis assumes that the
coefficient being analyzed is equal to zero. Dividing
the parameter estimate by its sample standard diviation
gives a T-statistic with which to perform a significance
test. For example, Table 6 for the third quarter shows
both coefficients to be significantly different than
zero.
The sample parameter estimate, or coefficient for
the regression model, is a normally distributed random
variable about the true population coefficient. So, if I
could hold all load demand effects constant and then
obtain a new data set from the City of Anaheim, I would
most likely get a different parameter estimate for each
coefficient. Given the Central Limit Theorem, as the
data set used for the regression becomes larger, the more
accurate will be the point estimate for each coefficient.
The r-square number for the third quarter is .9087,
and this high value indicates that the equation has a
very good predictive value. In other words, the equation
provides a better prediction of air conditioning load
1 7
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demand than simply using the average load number from the
regression data set.
The coefficient of determination, or r-square
number, is equal to one minus the following quantity: the
sum of the squared difference of each observation from
its predicted value over the sum of the squared
difference of each observation from the data set mean.
If the estimated equation were completely accurate, then
the predicted value would equal the observed value for
each temperature reading, and the r-square number would
be equal to one minus zero. Therefore, an r-square value
of .90 indicates that most of the variation between
observed values within the data set can be explained by
the regression equation.
SECTION V
By examining load demand as a linear combination of
some fixed seasonal demand, and a separate function of
temperature only, I found a good estimate of the effects
of ambient temperature on hourly demand. Holding all
other demand effects constant through the stationary
random variable, S, allowed the pure temperature effect
to be isolated for study.
18
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Utility companies can use this mathematical
relationship to more accurately determine future
generating needs. Combining sohisticated weather
forecasts with open market operations will provide a more
efficient and cost effective use of our energy resources.
Also, using the methodology presented in this paper a
larger model can be built to study and coordinate
supply/demand situations in different regions.
All statistical work was done using SAS software,
version 6 for personal computers. A sample of the data
set is present as Appendix 1 and Appendix 2.
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BIBLIOGRAPHY
G. R. Grimmett, and D. R. Stirzaker
PROBABILITY AND RANDOM PROCESSES
SECOND EDITION, 1995
Oxford University Press Walton Street, Oxford 0X2 6DP
William Mendenhall, Dennis D. Wackerly,
and Richard L. Scheaffer
MATHEMATICAL STATISTICS WITH APPLICATIONS
FOURTH EDITION, 1990
Dusbury Press Belmont, California
Douglas C. Montgomery
DESIGN AND ANALYSIS OF EXPERIMENTS
THIRD EDITION, 1991
John Wiley & Sons Toronto, Canada
Watson, Billingsley, Croft, and Huntsberger
STATISTICS FOR MANAGEMENT AND ECONOMICS
FIFTH EDITION, 1993
Prentice Hall Englewood Cliffs, New Jersey
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APPENDIX 1
SAMPLE OF DATA SET FROM ANAHEIM
LOAD NUMBERS AS REPORTED, KILOWATT HOURS
year 1995
JAN FEB MAR APR
30 194400.25 207360.59 206881.03 209278.81
01:00 188640.64 201600.98 201600.98 201118.97
01:30 183360.59 197280.05 196800.5 194400.25
02:00 178080.55 195838.91 195359.36 189599.75
02:30 173280.05 196800.5 195838.91 185760.84
03:00 169920.69 194879.8 192479.56 182881.03
03:30 166558.86 190079.31 191040.89 181439.91
04:00 163199.5 188640.64 193441.14 180001.23
04:30 162719.95 192000 195838.91 180480.8
05:00 162719.95 196800.5 205439.91 181919.45
05:30 164161.09 208799.27 218400.25 184799.27
06:00 166558.86 223680.3 229439.91 181439.91
06:30 169920.69 252001.23 256799.27 185278.81
07:00 169441.14 264000 271200.75 191520.44
07:30 167520.44 275519.22 282719.94 201118.97
08:00 171838.91 281760.84 294239.16 208319.7
08:30 177600.98 295200.75 306240.41 220320.94
09:00 181919.45 303360.59 310558.88 228001.23
09:30 186240.39 310558.88 314879.81 234719.95
10:00 189120.2 318721.19 319200.75 238558.86
10:30 191040.89 325919.47 322560.09 242879.8
11:00 191520.44 330240.41 322080.53 245759.61
11:30 191040.89 334079.31 323039.66 246721.19
12:00 191040.89 336479.56 321118.97 245759.61
12:30 188161.09 336959.13 319680.28 244320.94
13:00 187679.06 337920.69 315359.34 243359.36
13:30 187199.5 340800.5 316320.94 242400.25
14:00 186240.39 343200.75 317280.06 241920.69
14:30 185760.84 344159.84 314400.25 240000
15:00 185760.84 341280.06 312000 238079.31
15:30 184799.27 336000 308161.09 235199.5
16:00 186719.95 331199.5 307679.06 233760.84
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APPENDIX 2
SAMPLE OF DATA SET FROM ANAHEIM
TEMPERATURE NUMBERS AS REPORTED
15 minute intervals for all months
from September 1995 to 1996 current
Call X4285 if you have any questions
Tim 1995 1995 1995 1995
Sep Oct Nov Dec
1 71.98 63.81 58.21 52.62
2 71.67 63.65 58.16 51.99
3 71.19 62.43 58.27 51.72
4 70.56 63.17 58.16 51.04
5 70.08 62.96 58.21 51.09
6 69.5 62.33 58.37 51.35
7 69.13 61.64 58.58 51.14
8 68.66 61.27 58.69 50.98
9 69.08 61.33 58.74 50.67
10 69.4 61.27 58.64 50.09
1 1 69.35 61.33 58.48 50.09
12 69.19 61.17 58.69 50.3
13 68.5 59.8 58.79 50.04
14 67.76 58.58 58.53 49.77
15 67.18 60.32 58.58 49.72
16 68.13 59.48 58.64 49.09
17 68.55 59.58 58.58 49.24
18 68.4 59.58 58.53 48.93
19 68.29 59.22 58.64 48.82
20 67.71 59.16 58.53 49.3
21 67.29 58.58 58.53 49.67
22 66.29 59.22 58.16 49.61
23 66.29 58.64 57.84 49.51
24 66.13 58.16 57.21 49.45
25 65.81 58.53 56.68 49.77
26 66.92 58.37 56.42 48.03
27 68.34 58.37 57.16 48.45
28 70.51 58.05 58.27 50.25
29 69.5 59.11 59.16 51.2
22
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Asset Metadata

Creator Bruce, Daniel Alexander (author)

Core Title Electricity demand as a function of temperature

School Graduate School

Degree Master of Science

Degree Program Applied Mathematics

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag Energy,engineering, civil,Mathematics,OAI-PMH Harvest

Language English

Contributor Digitized by ProQuest (provenance)

Advisor Baxendale, Peter (committee chair), Alexander, Kenneth S. (committee member), Rosen, Gary (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c16-42864

Unique identifier UC11341151

Identifier 1409579.pdf (filename),usctheses-c16-42864 (legacy record id)

Legacy Identifier 1409579.pdf

Dmrecord 42864

Document Type Thesis

Rights Bruce, Daniel Alexander

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the au...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus, Los Angeles, California 90089, USA