The Management of the Weather Risk on the Electric Energy Market
The aim of this study is to present a new approach for analyzing the effect of temperatures, wind speed and heating degree days index on the variability of the daily demand for electric energy in Silesian region. Specific nature of the time series used in this paper concerning the electric energy co...
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| citation_txt | The Management of the Weather Risk on the Electric Energy Market / A. Wlodarczyk, M. Zawada // Электронное моделирование. — 2009. — Т. 31, № 6. — С. 65-86. — Бібліогр.: 19 назв. — англ. |
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| description | The aim of this study is to present a new approach for analyzing the effect of temperatures, wind speed and heating degree days index on the variability of the daily demand for electric energy in Silesian region. Specific nature of the time series used in this paper concerning the electric energy consumption and the weather variables, such as temperature or heating degree day index, requires application of a special class of models —ARFIMAX—GARCH (Generalized Autoregressive Conditional Heteroscedastic model).
Рассмотрен новый подход к анализу влияния температуры, скорости ветра и индекса степени дневного нагрева на изменчивость суточного потребления электроэнергии в районе Силезии. Особенности используемых временных рядов связаны с потреблением электроэнергии и погодными переменными, такими как температура или индекс степени дневного нагрева, обусловливают применение специального класса моделей — ARFIMAX—GARCH (обобщенная авторегрессионная условная гетероскедастическая модель).
Розглянуто новий підхід до аналізу впливу температури, швидкості вітру та індексу ступеню денного нагріву на змінність добової потреби в електроенергії в районі Сілезії. Властивості використаних часових рядів, пов’язані з споживанням електроенергії та погодними змінними, такими як температура чи індекс ступеню денного нагріву, обумовлюють застосування спеціального класу моделей — ARFIMAX—GARCH (узагальнена авторегресійна умовна гетероскедастична модель).
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A. Wlodarczyk, PhD, M. Zawada, PhD
Czestochowa University of Technology
(Al. Armii Krajowej 19B,42-200 Czestochowa, Poland)
The Management of the Weather Risk
on the Electric Energy Market
The aim of this study is to present a new approach for analyzing the effect of temperatures, wind
speed and heating degree days index on the variability of the daily demand for electric energy in
Silesian region. Specific nature of the time series used in this paper concerning the electric energy
consumption and the weather variables, such as temperature or heating degree day index,
requires application of a special class of models —ARFIMAX—GARCH (Generalized Auto-
regressive Conditional Heteroscedastic model).
Ðàññìîòðåí íîâûé ïîäõîä ê àíàëèçó âëèÿíèÿ òåìïåðàòóðû, ñêîðîñòè âåòðà è èíäåêñà ñòåïåíè
äíåâíîãî íàãðåâà íà èçìåí÷èâîñòü ñóòî÷íîãî ïîòðåáëåíèÿ ýëåêòðîýíåðãèè â ðàéîíå Ñèëåçèè.
Îñîáåííîñòè èñïîëüçóåìûõ âðåìåííûõ ðÿäîâ ñâÿçàíû ñ ïîòðåáëåíèåì ýëåêòðîýíåðãèè è
ïîãîäíûìè ïåðåìåííûìè, òàêèìè êàê òåìïåðàòóðà èëè èíäåêñ ñòåïåíè äíåâíîãî íàãðåâà,
îáóñëîâëèâàþò ïðèìåíåíèå ñïåöèàëüíîãî êëàññà ìîäåëåé — ARFIMAX—GARCH (îáîá-
ùåííàÿ àâòîðåãðåññèîííàÿ óñëîâíàÿ ãåòåðîñêåäàñòè÷åñêàÿ ìîäåëü).
K e y w o r d s: energy market, weather risk management, time series models, ARFIMAX—
GARCH.
The weather has a significant impact on the activity of numerous economic enti-
ties. Estimates of the International Weather Risk Management Association [1]
indicate that approximately 60% of all companies are directly or indirectly de-
pendent on weather conditions. Among the sectors that are exposed to the weather
risk to the greatest extent are: power engineering, agriculture, construction industry,
transport, retail trade, entertainment and tourism industry, and municipal security
personnel. From among the eight sectors listed above, only in case of construction
industry and agriculture a physical reduction of the weather risk is possible to a cer-
tain degree.
Weather risk analysis and management, seen as a possibility to achieve
better or worse financial results due to weather variability, allows one to make
the financial results of companies independent of the weather conditions in ques-
tion. It also enables better financial planning, primarily thanks to the possibility
of improving the sales volume forecast.
It should be emphasized that since it is impossible to store electric energy,
companies dealing with its sales must precisely determine the energy purchase
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2009. Ò. 31. ¹ 6 65
volume. Thus all kinds of methods of the analysis and modeling of the electric
energy demand, taking into account the impact of weather conditions, have be-
come so important. Therefore, the research objective of the authors is to verify
the usability of the ARFIMAX—GARCH models for the description of devel-
opment of electric energy consumption in a selected South Poland region with
reference to weather variables.
Weather risk management. Every enterprise bears the risk characteristic
of the economic activity it pursues. Companies selling gas, fuel oil or electric en-
ergy are exposed to weather deviations from conditions considered standard.
These companies, therefore, require all-year-long protection against such cases.
In winter, protection against temperatures higher than the standard ones is
needed, as higher temperatures cause reduction in the sales of energy-related
products. On the contrary, in summer the same companies need a protection
against temperatures lower than the standard ones, since in this case it also re-
sults in the reduction of the sales volume. Companies providing electric energy
are increasingly seeking for tools of protection against extremely sweltering
heats when the demand for electric energy (used for air conditioning) may ex-
ceed the planned level and force the suppliers to purchase energy on a short-term
market in the period of the highest prices [2].
The following basic types of risk may be distinguished in the energy sales [3]:
risk related to the price of electric energy is considerably dependent on the
energy demand, it is also connected with the fuel price (in Poland mostly coal)
and generally with the costs of production. In those cases, commodity exchange
offers futures and options, allowing one to protect the production and sales;
general systems and political risk related to the economic situation of the
country, social policy (e.g. mining industry) etc.;
transaction and credit risk; each transaction carried out through the com-
modity exchange, significantly reduces this risk due to the use of guaranteed de-
posit mechanisms;
natural environment-related risk, which in the face of the evident impact of
the power station, but also that of numerous energy consumers on the environ-
ment is currently a subject of public pressure and rigorous legal regulations
(green and red certificates);
risk (and opportunity) related to various price areas which may result in ar-
bitration transactions of the use of price differences between the bidders in vari-
ous areas;
volume-related risk, i.e. risk related to the lack of production and consump-
tion balance, which is considerably dependent on weather conditions, commodi-
ty exchange offers options, whose purchasers may exercise the right to purchase
or resign from the right (contracts allowing the salespersons of electric energy to
protect themselves on the balancing market);
A. Wlodarczyk, M. Zawada
66 ISSN 0204–3572. Electronic Modeling. 2009. V. 31. ¹ 6
refinancing-related risk; options offered by commodity exchange allow one
to control the cash flow and to plan revenue and expenditure more securely;
currency risk, covered with currency derivatives;
interest rate risk, similarly to currency risk, covered with interest rate de-
rivatives;
other types of risk in various operational activities, in particular, those re-
lated to managing the risk of changes in the supplier’s prices, which can be con-
trolled in a manner similar to one’s sales control.
For this reason, one of the most essential issues in the risk management pol-
icy is to determine the requirements relating to the expected security of the func-
tioning of the power engineering companies and to search for methods and in-
struments allowing one to move the weather risk out of the company.
The purpose of risk management is not to generate profits, but to reduce
losses. No risk management system ensures a complete elimination of the losses.
However every effort should be made to minimize the risk of their occurrence.
Risk management process should be designed in a manner which excludes the
possibility of bearing losses which may threaten the company existence. Recog-
nition of the risks and implementation of effective tools of their reduction allows
one to take efficient preventive actions in the face of emergency [4].
Weather risk management is not a novelty, as it has been known for many
years, primarily in the context of protection against the after-effects of hurri-
canes, floods or droughts (catastrophic risks). All companies from the energy
sector are characterized by a regularity of the amount of energy consumption in
relation to the air temperature observed outside. The research indicates that, in
case of electric energy, such weather parameters as wind speed, change of cloud-
iness or precipitation have also a considerable impact on the volume [5].
Due to demonopolization in power engineering, the long-term solutions of-
fered by insurance companies have ceased to meet the needs of the energy sec-
tor. The companies of this branch are more exposed to short-term temperature
variations. Every deviation from the mean in the period of summer peak or win-
ter peak may have a direct impact on the reduction in profits. A search for finan-
cial solutions and abandonment of the traditional insurance of many years have
become the basis for the development of a new market of weather risk
management [6].
Weather instruments belong to a dynamically developing part of the capital
market. It appears that financial derivative instruments, i.e. contracts, according
to which the profits (or losses) depend on the value of basic securities, are to a
large extent suitable for the protection against inappropriate weather. Basic se-
curities of weather contracts are generally indices. Basic types of weather indi-
ces have been presented in Table 1.
The Management of the Weather Risk on the Electric Energy Market
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Two indices are most frequently used with reference to power engineering:
heating degree days (HDD) and cooling degree days (CDD). These indices
count up a sum of deviations (negative and positive, respectively) of the average
daily temperature T (calculated as e.g. a mean of the minimum and maximum
daily temperature of the ith day) from a reference temperature, assumed as a pa-
rameter of the contract, in a fixed number of days n.
In the United States, the commonly accepted reference temperature is 65°F,
in Europe it is respectively 18°C — this is the temperature recognized as a con-
ventional border between the period in which air conditioners are used and the
heating period:
HDD � � �
�
�max ( , )018
1
C Ti
i
n
, CDD � � �
�
�max ( , )0 18
1
T Ci
i
n
. (1)
Weather derivative instruments are offered, amongst others, by the Chicago
Mercantile Exchange. However, these are futures contracts and options for fu-
A. Wlodarczyk, M. Zawada
68 ISSN 0204–3572. Electronic Modeling. 2009. V. 31. ¹ 6
Weather index
Weather parameter
constituting the basis
of the weather index
Application of weather index
in a specific economic activity
HDD Air temperature Wide application in various economic
sectors, particularly in power engineer-
ing, agriculture, construction industry,
recreational activity, hotel industry,
transport
CDD " "
Energy degree day (EDD) " " Power engineering
Cumulative average tem-
perature (CAT)
" "
Wide application in various economic
sectors, particularly in power engineer-
ing, agriculture, construction industry,
recreational activity, hotel industry,
transport, etc
Average temperature (AT) " "
Growing degree day (GDD) " " Agriculture
Chilling degree gay (CDH) " "
Frost day (FD) " "
Wind power index (WPI) Wind speed Wind power stations
Critical temperature day
(CTD)
Air temperature Wide application in various economic
sectors, particularly in power engineer-
ing, agriculture, construction industry,
recreational activity, hotel industry,
transport; they are also used by e.g. lo-
cal governments
Critical precipitation day
(CPD)
Precipitation level
(snow, rain)
Table 1. Basic types of weather indices and their application [7]
ture contracts for almost 30 various locations worldwide, including selected re-
gions in Europe. Derivatives issued for weather indices are also placed in the
over-the-counter sales which is characterised by a greater variety of concluded
transactions due to the underlying instrument and a structure of the deriva-
tive instrument.
Majority of forward transactions concluded on the over the counter (OTC) mar-
ket show a great innovativeness, as they are tailored to the individual needs of the
customers who make use of them in the process of weather risk management.
Review of research in the scope of the impact of the climatic factors on
the electric energy consumption. Identification and measurement of the
weather risk is connected with the necessity to isolate from the observable elec-
tric energy consumption a part which is sensitive to the effects of climatic fac-
tors. While analysing historical time series relating to the electric energy de-
mand, containing daily, weekly or monthly data from a dozen years or so, one
may notice a strong long-term tendency, whose occurrence has been affected by
social, demographic and economic factors. The factors might be as follows: po-
pulation growth in a given region, technological progress, growth of industrial-
ization, changes in the market share (measured by the number of customers) of
the companies dealing with energy production and sales, electric energy market
price and prices of alternative energy sources, monthly seasonality connected
with a lower demand of the industry sector for electric energy in the summer
holiday season.
The influence of the demographic factors on the energy demand may be
eliminated by dividing the total energy consumption in a given region by the
population, i.e. by considering the average energy consumption per capita [8, 9].
In order to isolate the electric energy demand which is sensitive to weather fac-
tors, various ways of data filtration may be used. In empirical research on model-
ing the above relation the following methods are used.
I. Method of analytical trend function adjusted data [10, 11]. Robinson sug-
gested calculating the demand sensitive to the effects of weather factors as a
fraction of a demand determined by economic and demographic factors, with
specified weather conditions:
E NE NE w NE w NE Wt t t p t p t p� � � � �( )1 ,
where Et — total electricity load; wp — weather factors (i.e. air temperature,
wind speed); Wp — normalized weather sensitive load; NEt — nonweather sen-
sitive load, computed from analytic tendency function,
NE tt j
j
j
m
t� � �
�
�� � �
0
1
.
The Management of the Weather Risk on the Electric Energy Market
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2009. Ò. 31. ¹ 6 69
On the other hand, Moral-Carcedo and Vicens-Otero modified Robinson’s
method by taking into account the seasonality occurring in the energy demand in
the business sector, which is related to the reduction in or cessation of the pro-
duction over the weekends and holiday seasons
1
:
E t I WD FEt j
j
j
m
t t t� � � � �
�
�� �
0
1
Aug.,
,
where I tAug.,
is a dummy variable taking the value 1 for observations made in
August, and the value 0 for others; introduced for the purpose of eliminating the
effect of the reduced demand for energy in the business sector during the holiday
season;WDt — a variable that takes the value 1 for Wednesday, whereas for all
other days of the week it equals a ratio of electric energy consumption on a given
day in relation to the energy consumption on Wednesday (in a week in which the
given observation was made); a variable describing the calendar effect of the
«working day» on the energy demand; FEt — electricity load, which is free of
the influence of demographic and economic factors.
II. Index-related equalization of the long-term tendencies which do not re-
sult from weather conditions in terms of the electric energy demand [8, 9, 12]:
FE
E
E E
t
r t
r
�
,
/
,
where Er t,
— monthly electricity consumption for month t in year r; Er —
monthly average electricity consumption in year r; E — monthly average elec-
tricity consumption in all analyzed period.
The suggested method of data filtration allows one to determine a monthly
seasonal structure in the formation of the demand for the electric energy. It does
not however eliminate all seasonal effects that do not result from climatic fac-
tors. Indices for daily data in relation to a weekly seasonal cycle may be con-
structed in a similar manner [12].
After the estimation of the electric energy demand which is sensitive to cli-
matic factors, the strength and nature of the causa link between the weather vari-
ables and the electric energy consumption should be assessed. Previous research
[13] has proved that among various weather variables, the air temperature has
the most significant impact on the electric energy demand. Linear regression
method may be used for the measurement of the weather risk:
FE xt l l t
l
r
tp
� � �
�
�� � �
0
1
,
(2)
A. Wlodarczyk, M. Zawada
70 ISSN 0204–3572. Electronic Modeling. 2009. V. 31. ¹ 6
1
Sailor D. J. and Munoz J. R. [8] estimated linear regression models for the whole period, as well
as for the winter and summer period separately, due to the seasonality in the mean, typical of
the climatology-related processes.
where x l tp
— the value of l’s-weather index in the t period;� l — model parame-
ter informing about the sensitivity of the electric energy demand to the effect of
l’s weather factor.
Those types of models were used in the research by [8] who made the elec-
tric energy consumption dependent, in a linear manner, on air temperature, wind
speed and a relative air humidity as well as HDD and CDD indices
1
. Robinson,
on the other hand, in addition to considering the linear regression function of the
electric energy demand in relation to the air temperature, also considered the de-
pendence of a non-linear nature of the form:
FE Tt t� � �� � �
��
��
�
0 1 2 3
3
.
A summary of the research conducted by the above-mentioned authors is an in-
dex of elasticity of the electric energy demand in relation to a specified weather
factor, suggested by [12]:
�
HDD
HDD
HDD
HDD� �
E
f ( ),
where �
HDD
is elasticity of energy consumption in relation to the HDD
2
index;
E f
HDD
HDD� ( ) — theoretical value of the function of the energy demand for a
determined value of the HDD index.
A description of the seasonal structure (for seasonal cycles of various
length) occurring in the mean of the climatic processes may be obtained by intro-
ducing (2) harmonic elements or dummy variables to the equation
3
:
FE x D M Ht l l t
l
r
i it
i
k kt
k
t tp
� � � � � �
� � �
� � �� � � � �
0
1 1
6
1
11
, (3)
where Dit — dummy variable for daily data (D t1
1� for Monday, D t1
0� for
other days of the week); M it — dummy variable for monthly data (M t1
1� for
January, M t1
0� for other months of the year); Ht — dummy variable for holi-
days (Ht = 1 for holidays, Ht = 0 for other days of the year).
The Management of the Weather Risk on the Electric Energy Market
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2009. Ò. 31. ¹ 6 71
1
Sailor D. J. and Munoz J. R. [8] estimated linear regression models for the whole period, as well
as for the winter and summer period separately, due to the seasonality in the mean, typical of
the climatology-related processes.
2
Indicator of the flexibility of the electric energy demand can be estimated for any weather factor
which has a significant impact on the analysed phenomenon.
3
Due to the occurrence of collinearity between the dummy variables and the constant term, the
variables for Sunday and December were omitted in the equation (3). Lacking estimations of
the parameters are calculated from applicable identities (compare to [14]).
Due to specific properties of meteorological time series, such as
1
trend and
seasonality in the mean, long process memory, seasonality in variation, as well
as the autoregressive conditional heteroscedastic (ARCH) effect, models of the
ARFIMAX(P, D, Q, r) —FIGARCH (p, d, q) class dependencies between the
weather variables and the energy demand
2
:
� �� � �( ) ( ) ( )B E BD
t t t� � , (4)
� � � � �t l l t
l
r
i it
i
k kt
k
tx D M H
p
� � � � �
� � �
� � �0
1 1
6
1
11
, (5)
� t t tz h� , z IIDt ~ ( , )01 , (6)
� � � � � � �( ) [ ( )](B D M H Bd
t i it
i
k kt
k
t t�
2
1
6
1
11
1� � � � � �
� �
� �
2
�ht ), (7)
where �
D
— difference filter of D order (–1 < D < 0.5),
�
D D j j
j
B
D
j
B� � �
�
�
�
�
�
� �
�
�
�( ) ( )1 1
0
;
�
d
— difference filter of d order (0 < d < 1),
�
d d s s
s
B
d
s
B� � �
�
�
�
�
�
� �
�
�
�( ) ( )1 1
0
;
B — lag operator; B y ys
t t s�
�
; � � �( ) ...B B BP
P
� � � �1
1
, � �( ) ...B B� � �1
1
...��Q
QB , � � �( ) ...B B Bq
q
� � � �1
1
, � � �( ) ...B B Bp
p
� � �
1
.
The assumed specification of equations (4), (5) allows one to describe the
effect of the long and short memory in conditional mean process, seasonality of
various length of the cycle in the time series of electric energy consumption
which result from, among other things, changeability of weather conditions or
different structure of energy consumers on working days and holidays. More-
over, relations (6), (7) allow one to consider in the process of modelling the elec-
tric energy demand the heteroscedasticity effect as well as stochastic seasonal-
A. Wlodarczyk, M. Zawada
72 ISSN 0204–3572. Electronic Modeling. 2009. V. 31. ¹ 6
1
More information about modeling the meteorological data can be found in the paper of [15].
2
In order to ensure the stationarity of the analysed time series models one assumes that the
polynomial roots � ( )B � 0, � ( )B � 0 are beyond the unit circle (compare to [5, 16]).
ity, long memory of the conditional variance process and the «thick tails» effect
of the distribution of random component
1
. In particular, when D = 0 and d = 0,
the process of the electric energy consumption is described by the ARMAX(P,
Q, r) — GARCH (p, q) models [17].
Statistical analysis of properties of the analysed time series. The follow-
ing information was used in this study: information related to electric energy
consumption (in kWh), air temperature (in °C) and wind speed (in m/s) in one of
the South Poland regions. The initial time series connected with particular vari-
ables contained observations from a period of time from September 1, 2005 to
June 30, 2008 with the reading frequency of 1 hour. Analyses and further calcu-
lations were based on the daily data (1034 observations) created from the initial
database (the hour one) in the following way:
the energy consumption on each day constitutes a sum of 24 information
from each hour of the day;
The Management of the Weather Risk on the Electric Energy Market
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1
While modeling high frequency data one assumes that scaled residuals {zt} are subject to «thick
tail» distribution which includes generalized error distribution (GED), t-Student distribution
and �-stable distribution.
a
b
c
Fig. 1. Daily consumption of electric energy, kWh, (a), air temperature, �C, (b) and wind speed,
m/s, (c) in the South Poland region in the period of 1000 days (time from September 1 2005 to
June 30 2008)
air temperature is expressed as the average temperature calculated from 24
information from each hour of the day;
wind speed is expressed as the average speed calculated from 24 informa-
tion from each hour of the day;
HDD (heating degree days) were calculated from the relation (1) whereTi is
the average daily air temperature on the ith day;
CDD (cooling degrees days) were calculated from the relation (1) where Ti
is the average daily air temperature on the ith day.
Fig. 1 proves the occurrence of similar periodic structure (for periodic cy-
cles of various length) in the mean of analysed processes and the effect of vola-
tility clustering. On figures 1—6 we have development of daily consumption of
electric energy (in kWh) shown on the axes abscissas.
In order to analyse the properties of electric energy consumption distribu-
tion and particular weather variables distribution, the basic descriptive statistics
were identified (Table 2).
While analysing the results presented in Table 1 one can notice that the rela-
tion between the standard deviation and the mean for each weather variable is
A. Wlodarczyk, M. Zawada
74 ISSN 0204–3572. Electronic Modeling. 2009. V. 31. ¹ 6
a
ACF
b
c
Days
Fig. 2. ACF functional (delay k=365) for consumption of electric energy (a), ACF-HDD (b), air
temperature (c)
very high, especially in case of air temperature. This proves that the daily
volatitlity of climate factors and thus the weather risk to which a power company
is exposed, is very big. Values of skewness and kurtosis as well as the conducted
Jarque-Bera normality test indicate significant differences between empirical
distribution of the analysed variables and the normal distribution
1
. Moreover,
statistics in the ADF test examining the occurrence of unit root for each variable
are significant at the significance level � = 0.01, which allows one to reject the
zero hypothesis in favour of the alternative hypothesis, so the analysed time se-
ries are stationary in variance.
A characteristic phenomenon in the electric energy market is the periodicity
(at various cycle lengths: daily, weekly, yearly) in the development of the elec-
tric energy demand. Meteorological data have a similar property (compare to
[15]). For the electric energy demand and the weather variables considered in
this paper, the diagrams of ACF functions were prepared in order to identify sig-
nificant autocorrelation dependencies and to assess the occurrence of the effect
of the long process memory (compare to Fig. 2). The significance of autocor-
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Statistics Load Temperature Wind HDD CDD
Average 4532400 7.5579 3.4495 10.670 0.22747
Standard deviation 504420 8.4172 1.5052 8.0799 0.84236
Skewness –0.23758 –0.37406 0.90855 0.53073 4.5151
Kurtosis –0.46885 –0.39923 1.3241 –0.39649 22.787
Minimum 3204400 –20.208 0.45833 0.00000 0.00000
Maximum 5657800 26.042 11.917 38.208 8.0417
L– B(36) 10188.3 19372.2 539.61 19103.6 3401.58
L– B
2
(36) 10402.1 16759.9 498.929 12120.5 1334.8
J– B 19.198
**
30.980
**
217.79
**
55.316
**
25883
**
ADF –12.09
**
–5.278
**
–18.19
**
–5.257
**
–13.22
**
Source: own calculations in PcGive package; symbol ** indicates the significance of the result at
the 0.01 level.
Table 2. Descriptive statistics for the analysed variables
1
Descriptive statistics for the CDD variable take atypical values due to relatively small number of
non-zero values of this index. It is advisable to conduct subsequent research related to another
way of calculating the HDD and CDD index for Poland — for temperatures lower than the
assumed 18 �C threshold value of the air temperature within the definition of this index.
Moreover, the research conducted by other authors [9] prove that in the climatic zone in which
Poland is located, the heating season effect in the development of electric energy demand is
more visible than the effect connected with the use of air-conditioning devices.
relation factors to the level 36 was examined with the use of Ljung-Box test (Ta-
ble 2). All determined test statistics were significant at the significance level
0.001 [18].
Another property characteristic of the considered time series is the notion of
volatility clustering, which is indicated by strong autocorrelation of squares of
values of a given series (compare to Fig. 3) and the significance of the statistics
in the McLeod and Li test (compare to the values L-B
2
(36) in Table 2). On the
basis of plots presenting the daily volatility of the energy demand and the HDD
index one can also formulate an assumption about the occurrence of a similar pe-
riodicity in the volatility of these time series.
In the next stage of the analysis, the Pearson correlation coefficients were
assessed in order to point out which of the initially proposed weather variables
has a significant impact on the development of the electric energy demand and
what is the direction of such correlation (compare to Table 3).
The temperature and HDD have the biggest impact on the energy consump-
tion, whereas the wind speed and CDD were meaningless. Thus only the temper-
A. Wlodarczyk, M. Zawada
76 ISSN 0204–3572. Electronic Modeling. 2009. V. 31. ¹ 6
b
c
d
a
Fig. 3. Daily variations of electrical load, kWh, (a), HDD index, °C, (b), wind speed, m/s, (c) air
temperature, °C, (d) in a South Poland region in the period of 1000 days time from September 1,
2005 to June 30, 2008
ature and HDD were included in further considerations and in the constructed
econometric models. Similar conclusions can be drawn while analysing the scat-
ter plots presented in Fig. 4.
It is worth mentioning that the direction of the correlation between the HDD
index and electric energy consumption is positive, whereas the variable energy
consumption and air temperature are correlated negatively.
An analysis of the properties of time series conducted in this part of the paper
constitutes an important stage of econometric modeling, because identification of a
regularity in the development of the examined variables brings effects in the form of
a proper specification of equations of conditional mean and conditional variance of
The Management of the Weather Risk on the Electric Energy Market
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2009. Ò. 31. ¹ 6 77
a
c
Air temperature C�
b
Fig. 4. Scatter plots of electricity load vs HDD, �C, (a), electricity load vs air temperature, °C,
(b), electricity load vs wind speed, m/s, (c)
Pearson correlation coefficient Load
HDD 0.6737
Temperature –0.6740
Wind 0.1235
CDD �0.2727
Table 3. Value of the Pearson correlation coefficients between analyzed variables
the process of energy demand. In other words it enabled to construct a consistent ec-
onometric model according to the concept of Z. Zielinski.
Estimation and verification of electrical energy consumption models. Is-
sues connected with the modeling of electric energy demand include three areas of
searching for a proper specification of initial model of ARFIMAX(P, D, Q, r) —
FIGARCH(p, d, q) class defined by the relations (4) — (7), namely:
specification of the deterministic part of the model connected with the con-
ditional mean of the process � t t tE E�
�
( | )
1
;
specification of the stochastic part of the model including
equation of conditional variance of the process h Et t t�
�
var ( | )
1
;
the choice of the form of the density function with zero mean and unit
variance for innovation process z z iid Dt t: ~ ( . )01
1
At the first stage of the research the authors tried to indicate a deterministic
trend connected with the impact of the demographic, economic and social fac-
tors on the electric energy demand. The polynomial trend of the third degree was
selected among the estimated various models of the development trend for daily
electricity load, taking into account the value of the coefficient of determination
and the significance of the assessments of the structural parameters of the mod-
els. Due to the occurrence of a strong linear correlation between the load and the
HDD index, this weather variable was included as the explaining variable in the
equation of conditional mean of the process of the energy demand.
Additionally, the classical equation of linear regression was extended with
dummy variables which aim at describing the weekly periodicity, annual sea-
sonality and the impact of holidays on the development of the electric energy de-
mand. Finally, the following specification of the first equation of the I model of
energy consumption was proposed:
E t t t D Mt t i it
i
j jt
j
� � � � � � � �
� �
� �� � � � ! �
0 1 2
2
3
3
1
1
6
2
12
HDD
� � � �
� �
"
S S S ut t t t2 1 3 1
, (8)
where t — time variable; i — parameter measuring the periodic effect in a
given phase of weekly cycle (day);� j — parameter measuring the seasonal ef-
fect in a given phase of annual cycle (month); Dit — dummy variable which
equals 1 in day i and 0 otherwise; M jt — dummy variable which equals 1 in
month j and 0 otherwise; St — dummy variable which equals 1 on a holiday and
0 otherwise; St–1 — dummy variable which equals 1 on the day preceding the
A. Wlodarczyk, M. Zawada
78 ISSN 0204–3572. Electronic Modeling. 2009. V. 31. ¹ 6
�
1
Set t �1
includes all information available till the t�"moment; D (0.1) symbol usually means in
practice normal distribution, t-Student distribution or GED distribution.
holiday and 0 otherwise; St +1 — dummy variable which equals 1 on the day fol-
lowing the holiday and 0 otherwise.
Due to the fact that the residuals ut in the equation (8) show strong
autocorrelation and the occurrence of the ARCH effect, the ARMA(P, Q) —
GARCH(p, q) models was used in their modeling. The order of the ARMA(P,
Q) model was chosen on the basis of the information criteria: Akaike’s (AIC),
Bayes extension Akaike’s (BIC), Schwartz’s (SC), Hannan — Quinn’s (HQ),
which minimize the volatility of the remaining element of the model and the sig-
nificance of the model parameters [19]. For the time series analysed in this paper
it was assumed that P = Q = 1:
1
u ut t t t� � �
� �
� � # �
" 1 1 1
,
where � t — random parameter in the ARMA (1.1) model.
The Management of the Weather Risk on the Electric Energy Market
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2009. Ò. 31. ¹ 6 79
a
b
c
Fig. 5. The diagram of empirical and theoretical properties for the ARMA (1.1) model (a), scaled
residuals of the ARMA (1.1) model (b) and the function of the density of scaled residuals of the
ARMA (1.1) model (c)
1
The authors also assessed the ARFIMA (P, D, Q) model for the residuals of the model (8),
however the D parameter responsible for the occurrence of the effect of long memory process
to be statistically insignificant.
A. Wlodarczyk, M. Zawada
80 ISSN 0204–3572. Electronic Modeling. 2009. V. 31. ¹ 6
Parameter
Equation (1) Equation (2) Equation (3)
Coefficient p-value Coefficient p-value Coefficient p-value
�
0
4933810 0.0000
***
4933810 0.0000
***
4841330 0.0000
***
�
1 �1147.36 0.0000
***
�1147.36 0.0000
***
�1179.82 0.0000
***
�
2
1.99800 0.0000
***
1.99800 0.0000
***
2.04483 0.0000
***
�
3 �0.000803 0.0080
***
�0.000803 0.0080
***
�0.000822 0.0052
***
!
1
19004.3 0.0000
***
19004.3 0.0000
***
6935.10 0.0016
***
!
2
9098.65 0.0022
***
!
3
7989.53 0.0003
***
1 �214921 0.0000
***
�214921 0.0000
***
�217467 0.0000
***
2 �760854 0.0000
***
�760854 0.0000
***
�765886 0.0000
***
3 �205569 0.0000
***
�205569 0.0000
***
�215784 0.0000
***
4 47048.1 0.0227
**
47048.1 0.0227
**
�57524.7 0.0040
***
5 �9088.51 0.6604 �9088.51 0.6604 �13289.0 0.5060
6
1844.44 0.9290 1844.44 0.9290 438.427 0.9825
�
2 �165255 0.0000
***
�165255 0.0000
***
�153460 0.0000
***
�
3 �168648 0.0000
***
�168648 0.0000
***
�143445 0.0000
***
�
4 �270178 0.0000
***
�270178 0.0000
***
�221569 0.0000
***
�
5 �555488 0.0000
***
�555488 0.0000
***
�477985 0.0000
***
�
6 �674895 0.0000
***
�674895 0.0000
***
�584053 0.0000
***
�
7 �561600 0.0000
***
�561600 0.0000
***
�465870 0.0000
***
�
8 �551100 0.0000
***
�551100 0.0000
***
�449232 0.0000
***
�
9 �410895 0,0000
***
�410895 0,0000
***
�327723 0,0000
***
�
10 �225479 0.0000
***
�225479 0,0000
***
�168370 0,0000
***
�
11 �66476.4 0.0146
**
�66476.4 0,0146
**
�33547,1 0,2071
�
12
36979.9 0.1572 36979.9 0,1572 36569,6 0,1473
1 �789853 0.0000
***
�789853 0,0000
***
�778551 0,0000
***
2 �182665 0.0000
***
�182665 0,0000
***
�177341 0,0000
***
3 �359959 0.0000
***
�359959 0,0000
***
�372871 0,0000
***
!r2
15825.0 0,0000
***
!r3 �4021.01 0,0000
***
!r4
9686.06 0,0000
***
!r5
20713.3 0,0000
***
!r6 2798.30 0,0000
***
!r7 �14294.1 0,0000
***
�
1 �81339.7 0,0000
***
� 249.234 0,0000
***
Table 4. Results of estimation of parameters of electric energy consumption models (8)
Since the conducted diagnostic tests for the random component in the
ARMA (1.1) model indicate a lack of its autocorrelation and occurrence of the
ARCH effect, in the next phase the GARCH(1.1) model was proposed with the
t-Student distribution to the description of the phenomenon of volatility cluster-
ing and thick tails of empirical distribution (Fig. 5):
� t t tz h� , z IIDt ~ ( . )01 ,
h ht t t� �
�
� � � ��
0 1
2
1 1
, (9)
where�
0
0$ ,�
1
0% , �
1
0% .
Assessments of model I parameters and results of diagnostic tests related to
properties of the random component is placed in Table 4. While assessing the ad-
justment of the model to the empirical data, the following criteria were taken into
account: the significance of the model parameters verified by the t-Student test,
determination coefficient (R2
), heteroscedasticity — adjusted mean square error
(HMSE), Akaike’s criterion (AIC).
Model II constitutes a modification of model I by taking into account the pe-
riodic structure of the random component of the ARMA(1.1) model and intro-
ducing in the equation a conditional variance (9) of additional regressor in the
The Management of the Weather Risk on the Electric Energy Market
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2009. Ò. 31. ¹ 6 81
Parameter
Equation (1) Equation (2) Equation (3)
Coefficient p-value Coefficient p-value Coefficient p-value
�
1
0.730415 0.0000
***
0.739969 0.0000
***
#
1 �0.158129 0.0000
***
�1069.63 0.0000
***
�
0
2.62155 0.0000
***
2626830000 0.0000
***
6030390000 0.0000
***
�
1
0.135355 0.0000
***
0.134292 0.0000
***
0.238152 0.0000
***
�
1
0.721565 0.0000
***
0.726271 0.0000
***
0.397007 0.0000
***
(t –Student) df 4.94570 0.0000
***
4.43292 0.0000
***
5.65441 0.0000
***
Adjusted R2
0.877352 0.877352 0.885469
HMSE 8.13597 10.1826 9.18767
AIC 26.279748 26.2787753 26.3823341
� + � 0.85692 0.860563 0.635159
Effect
ARCH
0.52449
[0.5920]
0.14178
[0.8678]
0.26156
[0.7699]
L�B (32) 33.742
[0.3833]
33.272
[0.4051]
29.064
[0.6159]
Source: own calculation in PcGive; p-value in brackets.
Table 4 (continued)
form of weather variable HDD:
1
� ! �t ri ri
i
r r t tD S z h� � �
�
�
2
7
, z IIDt ~ ( . )01 ,
(10)
h ht t t t� � �
�
� � � �� �
0 1
2
1 1
HDD , (11)
where Dri — the dummy variable which describes the periodicity of the random
component with weekly cycle (e. g. Dr2
1� if r is Tuesday and Dr2
0� other-
wise); Sr — dummy variable which describes the effect of holidays in the devel-
opment of the random component (Sr = 1 if r is a holiday, Sr = 0 otherwise),
�
0
0$ ,�
1
0% , �
1
0% ,� %0.
Model III is the modification of model I consisting in the introduction of the
second order delays of the HDD index, while assuming the impact of air temper-
ature from previous days on the development of electric energy consumption:
E t t t D Mt k t k
k
i it
i
j� � � � � � �
� �
� �
� �� � � � ! �
0 1 2
2
3
3
1
0
2
1
6
HDD jt
j�
� �
2
12
� � � �
� �
"
S S S ut t t t2 1 3 1 2
,
u ut t t t2 2 1 2 1 2 1
� � �
� �
� � # �
"
,
�
2 2t t tz h� , z IIDt2
01~ ( . ) ,
h ht t t� �
�
� � � ��
0 1 2
2
1 1
,
where�
0
0$ ,�
1
0% , �
1
0% .
While analyzing the results placed in Table 4, especially while taking into
account the considered comparative criteria and tests verifying the properties of
scaled residuals, one can indicate that model I described in the best way the de-
velopment of electric energy consumption in the South Poland region.
Introduction of the GARCH structure with the conditional t-Student distri-
bution (characterized by tails thicker than in the normal distribution) resulted
each time in the elimination of volatility clustering effect which was present in
the residuals of the ARMA model. By including dummy variables in the equa-
tion (10) and the weather variable in the equation of conditional volatility (11) it
was possible to model the periodicity of the volatility of the random component,
A. Wlodarczyk, M. Zawada
82 ISSN 0204–3572. Electronic Modeling. 2009. V. 31. ¹ 6
1
The daily volatility of the energy consumption and of the HDD index (compare to Fig. 3) is
characterized by periodicity, which can be explained by the weekday effect, holiday effect
orannual seasonality. Seasonality in volatility for time series of energy consumption can also
develop under the influence of the seasonality in the volatility of air temperature hence the idea
of including in the stochastic part of the model II the dummy variables and the weather factor –
HDD index.
however it did not cause any significant improvement of the properties of model
II in relation to the specification I or III from the perspective of AIC, HMSE cri-
terion. Also the introduction of the autoregressive structure for the HDD weather
variable in model III did not bring any effects in the form of significant reduction
of the values of Akaike criterion or HMSE criterion.
The extension of ARMA-GARCH models with the cause and effect relation
which occurs between air temperature and energy consumption made it possible
to assess the sensitivity of the electric energy demand in respect of the weather
factor, such as the HDD index. Assessments of parameters at dummy variables
which model the periodicity of a weekly cycle in the electric energy demand, ex-
cept for Friday and Saturday, are statistically significant and indicate that the
days with the greatest energy consumption in a given region of Poland are:
Thursday, Saturday and Sunday. All parameters at dummy variables which re-
late to holidays and days preceding and following holidays are statistically sig-
nificant and negative, which proves that the electric energy consumption is
lower in the holiday period than on weekdays. In case of dummy variables
related to monthly seasonal effects, all parameter assessments are significant and
negative, except for coefficients related to December. Such results allow one to con-
clude that the electric energy demand in the analyzed region of Poland is the greatest
The Management of the Weather Risk on the Electric Energy Market
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2009. Ò. 31. ¹ 6 83
a
b
Fig. 6. Conditional volatility (a) and the function of the density of empirical distribution of stan-
dardized residuals (b) of the GARCH (1.1) model with t-Student distribution
r: Residual ARMA (1.1)
---- N (1.01)
in January and December. It should be also emphasized that the trend, dummy vari-
ables modeling calendar effects and seasonal fluctuations as well as the weather in-
dex explain the variability of the electric energy consumption by 87.7 %.
Moreover, the assessment of the ARMA-GARCH models on the basis of the
residuals of the model (8) made it possible to assess the conditional volatility of
the process of electric energy demand. With the use of conditional volatility one can
measure the volatility of the electric energy demand, i.e. risk related to unpredictable
change in the energy consumption under the influence of e.g. changing weather con-
ditions. Periods which can be seen in Fig. 6 and correspond to a high value of condi-
tional volatility should be understood as the periods in which the energy consump-
tion was significantly changing under the influence of unpredictable factors (i.e. all
the variables omitted in the equation (8)).
Conclusions. Demonopolization in power industry has forced the power
engineering companies to prepare and implement internal procedures of manag-
ing the risk involved in the energy sales. Such activities do not serve the purpose
of generating income but rather the purpose of limiting potential losses resulting
from other demand for the electric energy than it was planned by the company.
Companies of this industry branch are more exposed to short-term fluctuations
of the weather conditions, since each deviation of the given weather factor from
conditions considered normal may have a direct impact on the energy consump-
tion by final consumers and may thus result in the deterioration of their financial
results. That is why the companies are increasingly using weather derivative in-
struments to protect themselves against weather risk consequences, since such
actions allow them to make their financial results independent of the change in
weather conditions. The decisions concerning the security level are taken on the
basis of, among others, breakdown of the costs involved in the hedge transac-
tion, using weather derivative instruments as well as current and forecast situa-
tion in the energy market, especially information on the amount of energy de-
mand in the future.
While reviewing the research on modeling the electric energy demand, one
can notice a lot of interest, both on the part of scientists and practitioners in ana-
lysing the sensitivity of the energy demand to weather factors. Analyses of the
impact of selected weather factors on the electric energy consumption conducted
by the authors, were related only to a selected region of the South Poland. Unfor-
tunately, in Polish conditions, gaining access to this type of data involves high
purchase costs, whereas in many countries databases concerning weather vari-
ables are made available free of charge on the weather stations’ websites or state
units responsible for the collection of this type of data.
It should be emphasized that from the analysed weather variables only the
air temperature and the HDD index determined on its basis have proved to have a
A. Wlodarczyk, M. Zawada
84 ISSN 0204–3572. Electronic Modeling. 2009. V. 31. ¹ 6
substantial impact on the level of energy consumption. In subsequent research
studies the authors will verify the impact of the remaining weather variables, i.e.
clouds or precipitation on the electric energy consumption and will perform
modeling for the HDD index defined in relation to another threshold tempera-
ture. Specific nature of the time series used in this paper concerning the electric
energy consumption and the weather variables, requires application of a special
class of models — ARMAX—GARCH. Dummy variables were applied in order
to eliminate the deterministic seasonality. However in further research studies,
alternative methods of elimination of periodicity should be applied (analysis of
periodicity indices, harmonic analysis, differentiation method with an adequate
length of delay, rolling volatility). In order to confirm the research results pre-
sented in this paper, the analysis should be extended to cover the area of the
whole country and the data frequency should be increased (e.g. by introducing
hourly data) due to the specific nature of the energy sales. One can also take into
consideration the distinction of hourly profiles in the energy sales and create
models for each hour of the day separately; such an approach renders unneces-
sary the consideration of models with the double periodic component: daily and
weekly.
While extending the analysis of the impact of weather factors on the func-
tioning of a power industry branch company, one should apply the Value at Risk
methodology to measure the weather risk. Such an approach will make compa-
nies dealing with the energy production and sales aware of the potential losses
they may suffer as a result of unexpected change of weather factors.
Ðîçãëÿíóòî íîâèé ï³äõ³ä äî àíàë³çó âïëèâó òåìïåðàòóðè, øâèäêîñò³ â³òðó òà ³íäåêñó ñòóïåíþ
äåííîãî íàãð³âó íà çì³íí³ñòü äîáîâî¿ ïîòðåáè â åëåêòðîåíåð㳿 â ðàéîí³ Ñ³ëå糿. Âëàñòè-
âîñò³ âèêîðèñòàíèõ ÷àñîâèõ ðÿä³â, ïîâ’ÿçàí³ ç ñïîæèâàííÿì åëåêòðîåíåð㳿 òà ïîãîäíèìè
çì³ííèìè, òàêèìè ÿê òåìïåðàòóðà ÷è ³íäåêñ ñòóïåíþ äåííîãî íàãð³âó, îáóìîâëþþòü
çàñòîñóâàííÿ ñïåö³àëüíîãî êëàñó ìîäåëåé — ARFIMAX—GARCH (óçàãàëüíåíà àâòîðåã-
ðåñ³éíà óìîâíà ãåòåðîñêåäàñòè÷íà ìîäåëü).
1. Pre J. Zarz dzanie ryzykiem pogodowym. — Warszawa: CeDeWu, 2007. — 12 p.
2. Kaliszewski I. Ryzyko w obrocie energia.— www. cire.pl (1.06.2009).
3. Jadwiszczok A., Zawi a-Niedzwiecki J. Zarz dzanie ryzykiem zmiany ceny energii elekt-
rycznej z wykorzystaniem derywatow mi dzynarodowej gie dy towarowej. — www.al-
pha.biz.pl (25.04.2009).
4. Weron R. Bezpieczenstwo energetyczne: Ryzyko, Zarz dzanie ryzykiem, Bezpieczenstwo,
II Ogolnopolska Konferencja «Polska Elektroenergetyka — Realia, Problemy, Dylematy». —
www.cire.pl (15.06.2009).
5. Pre J. Zarz dzanie ryzykiem pogodowym. —Warszawa: CeDeWu, 2007. — 35 p.
6. Burnecki K., Kukla G. Instrumenty finansowe na ryzyko pogodowe. — www.im.pwr.
wroc.pl (12.05.2009).
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ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2009. Ò. 31. ¹ 6 85
�
� �
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Submitted on 25.08.09
A. Wlodarczyk, M. Zawada
86 ISSN 0204–3572. Electronic Modeling. 2009. V. 31. ¹ 6
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| id | nasplib_isofts_kiev_ua-123456789-101529 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0204-3572 |
| language | English |
| last_indexed | 2025-11-27T18:27:06Z |
| publishDate | 2009 |
| publisher | Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України |
| record_format | dspace |
| spelling | Wlodarczyk, A. Zawada, M. 2016-06-04T13:16:50Z 2016-06-04T13:16:50Z 2009 The Management of the Weather Risk on the Electric Energy Market / A. Wlodarczyk, M. Zawada // Электронное моделирование. — 2009. — Т. 31, № 6. — С. 65-86. — Бібліогр.: 19 назв. — англ. 0204-3572 https://nasplib.isofts.kiev.ua/handle/123456789/101529 The aim of this study is to present a new approach for analyzing the effect of temperatures, wind speed and heating degree days index on the variability of the daily demand for electric energy in Silesian region. Specific nature of the time series used in this paper concerning the electric energy consumption and the weather variables, such as temperature or heating degree day index, requires application of a special class of models —ARFIMAX—GARCH (Generalized Autoregressive Conditional Heteroscedastic model). Рассмотрен новый подход к анализу влияния температуры, скорости ветра и индекса степени дневного нагрева на изменчивость суточного потребления электроэнергии в районе Силезии. Особенности используемых временных рядов связаны с потреблением электроэнергии и погодными переменными, такими как температура или индекс степени дневного нагрева, обусловливают применение специального класса моделей — ARFIMAX—GARCH (обобщенная авторегрессионная условная гетероскедастическая модель). Розглянуто новий підхід до аналізу впливу температури, швидкості вітру та індексу ступеню денного нагріву на змінність добової потреби в електроенергії в районі Сілезії. Властивості використаних часових рядів, пов’язані з споживанням електроенергії та погодними змінними, такими як температура чи індекс ступеню денного нагріву, обумовлюють застосування спеціального класу моделей — ARFIMAX—GARCH (узагальнена авторегресійна умовна гетероскедастична модель). en Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України Электронное моделирование The Management of the Weather Risk on the Electric Energy Market Влияние риска изменения погодных условий на электро-энергетический рынок Article published earlier |
| spellingShingle | The Management of the Weather Risk on the Electric Energy Market Wlodarczyk, A. Zawada, M. |
| title | The Management of the Weather Risk on the Electric Energy Market |
| title_alt | Влияние риска изменения погодных условий на электро-энергетический рынок |
| title_full | The Management of the Weather Risk on the Electric Energy Market |
| title_fullStr | The Management of the Weather Risk on the Electric Energy Market |
| title_full_unstemmed | The Management of the Weather Risk on the Electric Energy Market |
| title_short | The Management of the Weather Risk on the Electric Energy Market |
| title_sort | management of the weather risk on the electric energy market |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/101529 |
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