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A descriptive model of electricity consumption
O),tEGA, The Int. Jl of Mgmt Sci., Vol. 2, No. 4, 1974
A Descriptive Model of Electricity Consumption JEAN MONETTE Hydro-Quebec, Montreal, Canada (Received February 1974; in revised form March 1974)
The purpose of the model is to give corporate planning personnel, particularly ratemakers, or marketing people, a tool to evaluate quickly how many customers are included in a definite block of energy consumption within some limits of power demand. This will allow them to evaluate the number of customers that will be affected by any proposed rate schedules and, then, to evaluate the resulting effects on utility's revenue. Moreover, this model can, in any time, become a part of a larger model that could be used for financial planning, market analysis, or any other econometric model.
INTRODUCTION T H I S P A P E R discusses the c o n s t r u c t i o n o f a m a t h e m a t i c a l m o d e l used to evaluate the distribution o f electric utility customers included in a block o f energy cons u m p t i o n a n d within some limits o f m a x i m u m p o w e r d e m a n d . 1 F o r this purpose, samples o f c u s t o m e r s ' energy c o n s u m p t i o n s and p o w e r d e m a n d s were taken f r o m the billing c o m p u t e r files. Intervals o f p o w e r d e m a n d s were selected, a n d the d i s t r i b u t i o n o f the n u m b e r o f customers a c c o r d i n g to their energy c o n s u m p t i o n was studied within each interval. It was f o u n d that, for each p o w e r d e m a n d interval, the g a m m a d i s t r i b u t i o n best fitted the data, the ct p a r a m e t e r increasing regularly with the p o w e r d e m a n d . It was also f o u n d t h a t there was a very significant linear c o r r e l a t i o n between the m e a n o f energy c o n s u m p t i o n in each d e m a n d interval, a n d the d e m a n d itself, a n d similarly between the s t a n d a r d d e v i a t i o n o f energy c o n s u m p t i o n a n d the d e m a n d . The a a n d fl p a r a m e t e r s o f the g a m m a d i s t r i b u t i o n for the energy a n d d e m a n d p a r a m e t e r s were then calculated a n d i s o - p r o b a b i l i t y curves o f 5 % , 50%, 95% were drawn, T h e envelope o f these curves, i.e. the 0 % a n d 100% curves, has a quite singular shape. The 0 % one is ' S ' s h a p e d ; the 100% one is a positively sloped straight line, b o t h intersecting at some high p o w e r d e m a n d (see Fig. 1). The logical reasons for this are discussed below. Power demand is the rate of flow of electric energy. 549
Monette--A Descriptive Model of Electricity Consumption
F
-o
Energy c o n s u m p t i o n - - k W h
(log scale)
FIG. ]. [so-probability curves.
DEVELOPMENT Frequency distribution analysis." In order to study the shapes of the frequency distribution of customers, samples of the customers' energy consumption (kilowatt hours) and maximum power demand (kilowatts) figures were taken from the billing computer files. For each rate schedule, the range of power demands was divided into about 15 intervals. Then, inside each of these, the frequency distribution of customers according to their energy consumption was calculated. For low power intervals, the frequency distribution was skewed to the right, and the skewness was found to be diminishing as the power demand increased, up to a point that skewness just vanished. It was found that the probability density function that fitted the data best (according to the results of X z tests) in each power demand interval, was of the g a m m a 2 type, with different ct and fl parameters each time. The ~ was increasing with the power demand, thus explaining the vanishing skewness. In other words, a family of g a m m a probability density functions described the energy consumption of the customers. z Probability =
]3 ~+~ F (~+l)
cae-c;~ dc, where c = energy consumption 550
Omega, Vol. 2, No. 4 Relationship between pdf parameters and power demand." In order to be able to write a general equation of the gamma pdf valid for any power demand interval (this equation being the end result of this work), some relationships between a and/3 parameters, and the power demand had to be found and introduced into the pdf. To achieve this, the expressions of the mean and variance of the gamma pdf were used, i.e. Mean = (a + 1) /3 Variance = (a + 1) /3-" If this relationship between the mean of energy consumption in each power demand interval and the power demand itself could be found, and similarly for the variance, then it would be easy to derive from the equations above a and/3 as functions of the power demands. Consequently regressions were attempted between: M and P, where P = power demand; M = mean of energy consumption in each power demand interval. S and P, where S = standard deviation of energy consumption in each power demand interval. Six regression functions were tried (linear, exponential, etc.), and it was found that the linear function gave a significantly high correlation at all times. Hence a and/3 may be expressed as: O. ~
/3=
+
"--I
5-/+
(Where P = power demand in kilowatts and D,E,F,G are constants of the linear regression curves.) and a single equation representing a family of probability density curves was introduced. By calculating all the points of 95% probability, for example, and joining them we get the 95}/0 iso-probability curve, in other words, the limit below which lie 95% of all the customers for a given electricity consumption pattern: energy and maximum power demand during the billing period. DISCUSSION AND INTERPRETATION ISO-PROBABILITY CURVES
OF
THE
A few iso-probability curves are illustrated in Fig. 1, overlaid by a few probability density functions, each representing the distribution of customers 551
Monette--A Description Model of Electricity Consumption in a power demand interval that is coincident with the horizontal axis of the pdf. For lower power demands, the p d f is skewed to the right and widely stretches over the energy scale. It is a consensus a m o n g electric utility people that the lower the m a x i m u m demand is, the lower the load factor 3 is, and the more dispersed are the energy consumption figures (and therefore, the larger is the standard deviation). This is reflected in the model and can be seen in Fig. 1. The reason for skewness is that most customers have to consume a minimum essential quantity of energy to operate; whereas less and less customers are likely to consume larger and larger amounts of energy, always in low demand intervals. Analysis of customers' bills also reveals that, for increasing power demand, the load factor is likely to increase, thus reducing the dispersion of energy consumption figures (and therefore, the smaller the standard deviation). Indeed, high power demand customers pay high demand charges, whether they use energy or not, so they try to make a more regular use of energy, thus increasing the average demand more than the m a x i m u m demand. This explains why the load factor increases with the power demand, thus reducing the standard deviation of consumption figures as the demand increases. This, in turn, explains the convergence of the iso-probability curves. Now consider the 100~o iso-probability curve in Fig. 1. This line represents the energy consumption when the m a x i m u m power demand is used all the time, each kilowatt of demand consuming a constant 730 kilowatthours per month (the average number of hours in a m o n t h taken over one year). Clearly the relationship between power demand and energy consumed is now linear. The accuracy of the model has been successfully tested against actual billing figures. Moreover, it is believed that a similar model would always apply to a consumption phenomenon, be it electricity, or gas, or else. 3 The load factor is the percentage of the time that the maximum power demand is used during the billing period. It may be expressed as the ratio of the average demand over the maximum demand.
A Descriptive Model of Electricity Consumption JEAN MONETTE Hydro-Quebec, Montreal, Canada (Received February 1974; in revised form March 1974)
The purpose of the model is to give corporate planning personnel, particularly ratemakers, or marketing people, a tool to evaluate quickly how many customers are included in a definite block of energy consumption within some limits of power demand. This will allow them to evaluate the number of customers that will be affected by any proposed rate schedules and, then, to evaluate the resulting effects on utility's revenue. Moreover, this model can, in any time, become a part of a larger model that could be used for financial planning, market analysis, or any other econometric model.
INTRODUCTION T H I S P A P E R discusses the c o n s t r u c t i o n o f a m a t h e m a t i c a l m o d e l used to evaluate the distribution o f electric utility customers included in a block o f energy cons u m p t i o n a n d within some limits o f m a x i m u m p o w e r d e m a n d . 1 F o r this purpose, samples o f c u s t o m e r s ' energy c o n s u m p t i o n s and p o w e r d e m a n d s were taken f r o m the billing c o m p u t e r files. Intervals o f p o w e r d e m a n d s were selected, a n d the d i s t r i b u t i o n o f the n u m b e r o f customers a c c o r d i n g to their energy c o n s u m p t i o n was studied within each interval. It was f o u n d that, for each p o w e r d e m a n d interval, the g a m m a d i s t r i b u t i o n best fitted the data, the ct p a r a m e t e r increasing regularly with the p o w e r d e m a n d . It was also f o u n d t h a t there was a very significant linear c o r r e l a t i o n between the m e a n o f energy c o n s u m p t i o n in each d e m a n d interval, a n d the d e m a n d itself, a n d similarly between the s t a n d a r d d e v i a t i o n o f energy c o n s u m p t i o n a n d the d e m a n d . The a a n d fl p a r a m e t e r s o f the g a m m a d i s t r i b u t i o n for the energy a n d d e m a n d p a r a m e t e r s were then calculated a n d i s o - p r o b a b i l i t y curves o f 5 % , 50%, 95% were drawn, T h e envelope o f these curves, i.e. the 0 % a n d 100% curves, has a quite singular shape. The 0 % one is ' S ' s h a p e d ; the 100% one is a positively sloped straight line, b o t h intersecting at some high p o w e r d e m a n d (see Fig. 1). The logical reasons for this are discussed below. Power demand is the rate of flow of electric energy. 549
Monette--A Descriptive Model of Electricity Consumption
F
-o
Energy c o n s u m p t i o n - - k W h
(log scale)
FIG. ]. [so-probability curves.
DEVELOPMENT Frequency distribution analysis." In order to study the shapes of the frequency distribution of customers, samples of the customers' energy consumption (kilowatt hours) and maximum power demand (kilowatts) figures were taken from the billing computer files. For each rate schedule, the range of power demands was divided into about 15 intervals. Then, inside each of these, the frequency distribution of customers according to their energy consumption was calculated. For low power intervals, the frequency distribution was skewed to the right, and the skewness was found to be diminishing as the power demand increased, up to a point that skewness just vanished. It was found that the probability density function that fitted the data best (according to the results of X z tests) in each power demand interval, was of the g a m m a 2 type, with different ct and fl parameters each time. The ~ was increasing with the power demand, thus explaining the vanishing skewness. In other words, a family of g a m m a probability density functions described the energy consumption of the customers. z Probability =
]3 ~+~ F (~+l)
cae-c;~ dc, where c = energy consumption 550
Omega, Vol. 2, No. 4 Relationship between pdf parameters and power demand." In order to be able to write a general equation of the gamma pdf valid for any power demand interval (this equation being the end result of this work), some relationships between a and/3 parameters, and the power demand had to be found and introduced into the pdf. To achieve this, the expressions of the mean and variance of the gamma pdf were used, i.e. Mean = (a + 1) /3 Variance = (a + 1) /3-" If this relationship between the mean of energy consumption in each power demand interval and the power demand itself could be found, and similarly for the variance, then it would be easy to derive from the equations above a and/3 as functions of the power demands. Consequently regressions were attempted between: M and P, where P = power demand; M = mean of energy consumption in each power demand interval. S and P, where S = standard deviation of energy consumption in each power demand interval. Six regression functions were tried (linear, exponential, etc.), and it was found that the linear function gave a significantly high correlation at all times. Hence a and/3 may be expressed as: O. ~
/3=
+
"--I
5-/+
(Where P = power demand in kilowatts and D,E,F,G are constants of the linear regression curves.) and a single equation representing a family of probability density curves was introduced. By calculating all the points of 95% probability, for example, and joining them we get the 95}/0 iso-probability curve, in other words, the limit below which lie 95% of all the customers for a given electricity consumption pattern: energy and maximum power demand during the billing period. DISCUSSION AND INTERPRETATION ISO-PROBABILITY CURVES
OF
THE
A few iso-probability curves are illustrated in Fig. 1, overlaid by a few probability density functions, each representing the distribution of customers 551
Monette--A Description Model of Electricity Consumption in a power demand interval that is coincident with the horizontal axis of the pdf. For lower power demands, the p d f is skewed to the right and widely stretches over the energy scale. It is a consensus a m o n g electric utility people that the lower the m a x i m u m demand is, the lower the load factor 3 is, and the more dispersed are the energy consumption figures (and therefore, the larger is the standard deviation). This is reflected in the model and can be seen in Fig. 1. The reason for skewness is that most customers have to consume a minimum essential quantity of energy to operate; whereas less and less customers are likely to consume larger and larger amounts of energy, always in low demand intervals. Analysis of customers' bills also reveals that, for increasing power demand, the load factor is likely to increase, thus reducing the dispersion of energy consumption figures (and therefore, the smaller the standard deviation). Indeed, high power demand customers pay high demand charges, whether they use energy or not, so they try to make a more regular use of energy, thus increasing the average demand more than the m a x i m u m demand. This explains why the load factor increases with the power demand, thus reducing the standard deviation of consumption figures as the demand increases. This, in turn, explains the convergence of the iso-probability curves. Now consider the 100~o iso-probability curve in Fig. 1. This line represents the energy consumption when the m a x i m u m power demand is used all the time, each kilowatt of demand consuming a constant 730 kilowatthours per month (the average number of hours in a m o n t h taken over one year). Clearly the relationship between power demand and energy consumed is now linear. The accuracy of the model has been successfully tested against actual billing figures. Moreover, it is believed that a similar model would always apply to a consumption phenomenon, be it electricity, or gas, or else. 3 The load factor is the percentage of the time that the maximum power demand is used during the billing period. It may be expressed as the ratio of the average demand over the maximum demand.