Econometric estimation of peak electricity demands

Journal

of Econometrics

ECONOMETRIC

9 (1979) 119-136.

ESTIMATION

Robert Virginia Polytechnic

M. SPANN

0 North-Holland

Publishing

Company

OF PEAK ELECTRICITY and

Edward

Institute and State University,

DEMANDS*

C. BEAUVAIS Washington, DC 20041, U.S.A.

1. Introduction The majority of existing econometric studies of the demand for eiectricity estimate models of kilowatt-hour usage as a function of prices, income and other economic variables.’ Estimates of kilowatt-hour demand equations, or elasticities of demand for total usage, are only partially useful for utility planning purposes and for analyzing policy alternatives in the electric utility industry. Capacity is built to meet system and monthly peak demands. Utilities are interested not only in kilowatt-hour output, but annual peaks, monthly peaks and the load duration curve.’ In this paper we estimate as econometric model of peak electric demands using time series data for one utility, Virginia Electric Power Company. The estimated econometric model relates system megawatt peaks to marginal electricity prices, temperature, income, and industrial activity. The model is estimated using monthly VEPCO data for the period 1960-1973 for the four summer months, June, July, August and September. *The research on which thts paper is based was supported by the Virginia State Corporation Commission. The opinions and conclusions in this paper are those of the authors and do not necessarily represent the opinion of the Virginia State Corporation Commission. ‘For three exceptions to this rule, see Cargill and Meyer (1971), Uri (1977) or Irish (1976). Recent peak load pricing experiments funded by the FEA should allow estimation of elasticities of demand by time of day. While such experimental data will provide a useful input into the evaluation of time of day rates they will be only partially useful for aggregate peak demand forecasting. The majority of the FEA studies are limited to residential customers. In many utilities non-residential demands are more than fifty percent of system peak demands. ‘For actual planning purposes one should have a forecast of the entire load duration curve (LDC). If one normalized the load duration curve by dividing by peak demand one has a function that is one minus a cumulative density function over the set 1 to 8760 (the number of hours in a normal year). By definition -d(l -F(x))/dx is a probability density function (pdf). Ktlowatt-hour sales is the integral of the pdf or the mean of the pdf times the number of hours in the year. Peak demand is the nth order statistic of the pdf. Thus estimation of kilowatt-hour equations and peak demand equations is equtvalent to using two sampling statistics, the mean and the nth order statistic, to approximate the LDC. While these two statistics may not be the most efticient statisttcs to approximate the pdf that underlies a LDC, they are the two statistics which bear the closest relation to the data usually gathered and analyzed for planning purposes. We are indebted to Allen Meidcma for this sugge\tion.

R.M. Spann and E.C. Beauvais,

120

The estimated results indicate demands are statistically significant

Peak elrcrricir~

dmwtr/s

that price elasticities of from zero. Other important

peak electric results are:

-Price elasticities of peak demand are less than existing estimates of price elasticities of kilowatt-hour sales. This implies that in times of rising real

electricity prices, kilowatt-hour sales may fall relative to kilowatt peaks, or that utility load factors deteriorate, and the optimal capacity mix is altered toward more peaking capacity. -Income elasticities of peak electric demands generally exceed income elasticities of kilowatt-hour sales. This implies that increasing real incomes lead

to greater purchases, and utilization of appliances (such as air conditioning) and capital goods which tend to be operating at the time of electric utility system peaks. -System peaks are sensitive to alternative fuel prices. This result is consistent with engineering data indicating strong substitution possibilities between electricity and other fuels. It also implies that alternative fuels, as well as electricity prices, income, industrial activity and other economic variables should be included in forecasting electric peak demands. The model is utilized to forecast growth rates in peak electric demand under alternative price and income scenarios. The implications of the model for peak load pricing optimal, system planning, and forecasting methodology are discussed. With regard to peak load pricing, the preliminary results presented here would tend to imply that -The

primary

reduced -Peak

effect of peak load pricing

may be reduced fuel costs rather than

capital requirements.3

load pricing

may increase

the total revenues

of electric

utilities.

Simulation results indicate that the variance in the forecast of future demands is not small. This implies that one should forecast not only expected future demands but the variance in expected future demands as well. This is especially important for the use of forecasts in system planning. Often system planning models minimize the cost of meeting a non-stochastic LDC. As the uncertainty m future demands increases, it may be more important to treat system planning as an optimization problem under uncertainty rather than a non-stochastic cost minimization problem. In the final section of the paper we discuss the limitations of econometrics in load forecasting and the reasons for using both econometric and noneconometric techniques in future load forecasting. 3This possibility has been discussed previously in Wenders (1976). As shown below, peak load pricing would reduce the total capacity additions required. However, peak load pricing could shift the LDC in such a manner that the effects of substituting more capital intensive base load capacity additions for additional peaking units makes the utility more capital intensive.

R.M. Spann and E.C. Beauuuis, Peak electricity

2. An econometric

demands

121

model of peak demands

A utility’s peak demand is the sum of all loads placed on the system at the time the peak occurs. The peak demand equation is the sum of the residential, commercial and industrial demand equations for electricity at the time of system peak. As such, the peak demand equation should include all the variables which might enter demand equations for each service class. These variables are (i) the prices residential, industrial and commercial customers4 pay for additional kilowatt-hour of electricity during periods in which the system is likely to peak, (ii) the prices of alternative fuels, (iii) income and industrial activity, and (iv) temperature. In addition, one must recognize that electricity demands do not adjust instantaneously to changes in economic variables. Residential, commercial and industrial customers make current appliance and capital stock additions based on expected future electricity prices. Capital and appliance stocks cannot be adjusted immediately. Thus one would expect the long-run effect of a change in the price of electricity (or other variables influencing electricity demand) to be significantly greater than the short-run effect of a change in price (or other economic variable). This adjustment process is included in the econometric estimations by use of a distributed lag model of consumption adjustment. Each of the variables included and their definitions are discussed in more detail below. 2.1. Marginal

demand

and energy

prices

and thP demand for

electricity

Economic theory predicts that the relevant price to utilize in estimating the demand function for any commodity is the marginal price, or the price the consumer pays for an additional unit of consumption. For most commodities, the consumer faces a constant marginal price independent of his level of consumption. This is not generally true in the case of electricity. Electric utility tariff schedules are usually declining block tariffs in which the amount paid for an additional unit of consumption declines as consumption increases.5 Declining-block tariff structures also imply that the marginal price faced by one consumer within a rate class may not be equal to the marginal price faced by another consumer within the same rate class. For example, the tail block price in a rate structure may affect a large customer but does not affect a very small user. In addition, commercial and industrial customers are generally billed on the basis of both ‘energy consumption’ or the total “Separate prices are discussed for each of these groups of customers since each customer class is billed on a different rate schedule. 51ncreases in fuel adjustment clause levels and a tendency toward ‘rate flattening’ have moved electric rate schedules significantly towards ‘flat’ or constant marginal charges at all levels of consumption within a rate class.

122

R.M.

Spann and E.C. Beauvais,

Peak electricity

demands

amount of kilowatt-hours used in a month and ‘demand’ or the maximum rate of consumption during a 15, 30 or 60 minute interval6 In order to include the effects of differing marginal prices in the model and to include the two-part nature of electricity tariffs in the industrial and commercial sectors, we define four marginal prices in the industrial and commercial sector: (a) the marginal cost (on a per kilowatt-hour basis) a customer pays if he increases his consumption from one-half of average consumption to average consumption holding his kilowatt billing demand constant at average billing demand for the class as a whole (denoted as E, below); (b) the marginal cost (on a per kilowatt-hour basis) a customer pays if he increases his consumption from average kilowatt-hour consumption to 1.5 times average consumption holding his kilowatt billing demand constant at the average billing demand per commercial customer (denoted as E, below); (c) the marginal cost (on a per kilowatt of billing demand) a customer faces if he increases his kilowatt billing demand from 50% of average billing demand to average billing demand holding constant his kilowatt-hour consumption at the average kilowatt-hour consumption of the commercial class (denoted as D, below), and (d) the marginal cost (on a per kilowatt of billing demand basis) a customer faces if he increases his kilowatt billing demand from average kilowatt billing demand to 1.5 times average kilowatt billing demand holding his kilowatt-hour consumption (denoted as D, below).’ The marginal prices are computed separately for large power and light and small power and light customers.’ Two marginal energy prices are calculated for residential customers: The marginal cost a customer pays, per kilowatt-hour, if he increases his usage from one-half average usage to average usage and the marginal cost he pays if he increases his usage from average usage to 50 percent greater than average usage. The first marginal energy price is denoted as E 1, the second marginal energy price is denoted as E,. All prices are deflated by the consumer price index. The fuel adjustment clause is included in all energy charges. Unfortunately, independent inclusion of all marginal prices leads to multicolinearity problems. Thus a truncated version of the what one might 6For theoretical discussions of multi-part tariffs in electric utilities, see Taylor (1976) or Spann (1976). ‘Commercial and Industrial customers are the F-PC accounting categories of Small Light and Power (predominantly commercial) and Large Light and Power (primarily industrial). The applicable rate schedules are utilized to compute separate marginal energy and demand charges for Small Light and Power and Large Light and Power Customers. In the actual estimations only D, is utilized in the Small Light and Power class. This is due to the fact the small, low load factor customers served from VEPCO’s small general service rate schedules face Lero marginal demand charges. “For Small Light and Power an average usage of IOkW and 2190kWh is assumed. For Large Power and Light an average usage of ‘500 li W and 2.737.500 is assumed.

R.M.

Spann and E.C. Beauvais,

Peak electricity

demands

123

term the full or complete model is estimated. Two price variables, an aggregate energy price and an aggregate demand price are utilized. The marginal energy price is the sum of the logs of the marginal energy prices in the industrial and commercial sectors plus the sum of the logs of two marginal energy prices in the residential sector. More formally, we define

LE=LEy+LE’:+LE;+LE”+LE:+LE;=

t

LE,,

i=l

where the superscripts R represents the residential sector, S small light and power, and L large light and power. An L in front of a variable indicates that the natural log of the variable is utilized. The marginal demand price is defined as9

LD=LDs,+LDk+LDi=

i

LDi.

i=l

2.2. Income and the demand for electricity Obviously income influences the demand for electricity at the time of system peak. Ideally, one should include some measure of value added in various industries or an industrial activity index, income or output of commercial establishments and personal income to appropriately model income effects in the industrial, commercial and residential sectors. Unfortunately, such data does not exist in suitable form for the commonwealth of Virginia, or for VEPCO’s service territory. Thus two proxies variables for income are included in the econometric model. Total taxable income in the observation year is used as an income proxy for commercial and residential customers.” The income or value added in commercial establishments would be greater, the more individuals in a given area have to spend at those establishments. Income is deflated by the consumer price index. ‘Utilization of these price variables is equivalent to the restriction that the elasticities of peak demand with respect to energy prices and with respect to demand prices are the same for all classes of customers. While there are no a priori reasons to believe that this is the case, there are no other restrictions which might be more or less reasonable. In addition, one is interested in estimating the elasticity of system peaks, not individual peaks, so there may not be much distortion induced by the use of aggregate price data. An alternative approach would be to weight the marginal price charged each rate class by percentage of total system sales to that rate class. This weighting scheme could appropriately measure the effects of different rates of change in the prices charged different customer classes and the effects of changes in the mix of customers on peak demands and the price elasticity at time of system peak. “Taxable income is reported by counties in the Commonwealth of Virginia. This study utilized taxable income summed over all counties \er\ed by VEPCO.

124

R.M. Spann and E.C. Beuucuis, Peak electricity

demands

The appropriate proxy for income in the large power and light (predominantly, although not completely, industrial) is more complicated. In addition to considering changes in the output of industries that utilize electricity in VEPCO’s service territory, one must consider the industrial mix served by the utility. A ten percent increase in the level of economic activity in an industry that is extremely electricity intensive has a greater impact on peak demands than a ten percent increase in output or industrial activity in an industry that uses little or no electricity per unit of output. The problem is confounded by the fact that different industries exhibit different cyclical and seasonal patterns of output. The economy as a whole may be entering a recession at the same time an electricity intensive industry is in a cyclical upswing. In order to include these factors in the model, the measure of income calculated for the industrial sector is an electric intensity industrial activity index. The index is derived in the following manner. Employment data for a number of industries was gathered. That data was then weighted by the ratio of electricity consumption to employment for that industry for the nation as a whole in 1971. Summing employment weighted by electricity usage in 1971 per employee leads to the industrial activity index relevant for determining industrial electricity demands.’ ’ The employment data is based on March and September employment reports for Virginia and West Virginia Counties. This measure of industrial activity includes the effects of different cyclical patterns in different industries, and weights employment in those industries which are more electricity-intensive more heavily than those industries which are less electricity-intensive.’ ’ 2.3. Temperature Temperature is included in the model. In genera1 one would summer temperatures to lead to increased summer peaks.

expect higher

2.4. Alternutive fuels und the demund for elec.tricitJ The demand for any commodity depends not only on the price of that commodity, but on the price of substitutes for that commodity. In the case of “This approach to modelling industrial activity suggests an alternative method of including industrial prices in the model. Electrxity is a factor Input in industrial processes. The elasticity of demand for a factor input depends on the share of total costs which is payments to that factor, the elasticity of substitution in production, and the elasticity of demand for the output of the industry. Thus the elasticity of peak demand with respect to the prices charged industrial customers should be dependent on the mix of industrtal customers. Data and time limitations prevented inclusion of this factor tn the model. 12This index is not problem free. The ratio of employment to electricity usage within individual industries is obviously a function of the prices of electricity, labor and capital. Thu> the weights used in deriving the index should be endogenous variables in the model. Again time and data limitations prelented inclusion of thl\ factor in the model.

125

R.M. Spann trtd E.C. Bertr~roi.s. Pruh rlrcrricitp demands

electricity, there are at least three fuels which might act as substitutes: oil, natural gas, and coal. For smaller users, the primary use of alternative fuels, and the point at which decisions are made concerning electricity-using capital investments versus investments which use other fuels, is in the area of space and water heating. Thus, one would expect that alternative fuel prices would have less influence on the peak electric demands of smaller users in the summer peak period. This is not necessarily the case for industrial users of electricity. The majority of larger industrial customers use electricity for a variety of purposes other than space and water heating. Alternative fuels are used for a variety of industrial processes. In the large light and power category, three fuel prices are relevant: the price of natural gas to industrial users, the price of number 6 fuel oil, and the price of coal. Since industrial natural gas usage is curtailed, the price of number 6 oil is taken as a proxy for alternative fuel prices. As in the case of electricity prices, the prices of alternative fuels are deflated by the consumer price index.

2.5. Short-run

versus long-run electricity

demand

The demand for electricity does not adjust instantaneously as the price of electricity, income, or other economic variables change. Capital investment decisions are made on the basis of expected electricity prices, alternative fuel prices, and incomes. It takes time to adjust capital stocks and usage. In order to include this factor in the model, a distributed lag structure is utilized which allows for separate estimation of long-run and short-run elasticities.

2.6. The estimating Formally

equation

the model estimated

for monthly

peak demands

is

where LK W = natural log of peak kilowatt demand, LE = sum of the natural logs of the marginal energy prices, LD=sum natural logs of the marginal demand prices, T = temperature, LI = natural log of the income,

126

R.M.

LlNDEX

Sptrrm und E.C. Bruurcris.

=natural log large power LO = natural log LK WI =natural log LK W,, = natural log

Peak e/ectricitJ

demands

of the industrial activity-electrical intensity and light customers, of the price of oil, of peak demand in the previous month, of peak demand lagged 12 months.

index

for

The model is in terms of natural logs. Thus the coefficients on the economic variables or /Ii, are elasticities of demand. These coefficients are short-run elasticities of demand since a distributed lag structure is utilized. The coefficients A1 and i, represent adjustment rates. The sum of these two coefficients is the percent of any adjustment to a change in the price of electricity or income, etc. which does not take place after the month a price change occurs. The short-run elasticities are the percentage change in electricity sales due to a one-percent change in a dependent variable the month the change occurs. Long-run elasticities are computed by dividing the short-run elasticity by 1 -3,, -i.,.

3. The econometric

estimates

of peak demand

The peak demand equation is estimated using monthly VEPCO data for summer months for the period 1960-1973. Table 1 lists the estimated summer peak demand equation. In general, the estimations are satisfactory. All the coefficients have the expected sign. The effects of marginal prices, alternative fuel prices and industrial activity are all statistically significant. Table 2 lists the estimated long- and short-run elasticities of demand for VEPCO. These results are discussed individually below.

3.1. Energy price elasticity The estimated short-run elasticity of peak demand with respect to the This coefficient is the short-run summed energy price variable is -0.052. (current month) effect of a one-percent change in one energy price used in the regression. If all six energy prices used to construct the energy price index change by one percent, peak demand changes by 0.31 percent or -0.052 times six. The long-run elasticity with respect to all marginal energy prices is the short-run elasticity (-0.31) divided by the one minus the sum of the lag coefficients (one minus the sum of 0.162 and 0.133).r3 13This estimated elasticity is slightly less than Uri (1977). Using average price data for PG&E, demand of -0.57 and long-run elasticities of reported by Uri may reflect biases in the use of VEPCO.

the price elasticity of peak demand estimated by Uri estimated short-run price elasticities of peak demand of -0.75. The higher price elasticities average prices or differences between PC&E and

R.M.

Spann and E.C. Beuuvais, Peak electricity

demands

127

The estinlated long-run elasticity of peak demand with respect to an equal less percentage increase in all energy prices is -0.44. This is considerably than the existing estimated price elasticities for kilowatt-hour usage. Existing estimates of long-run price elasticities for kilowatt-hour usage are:14

Residential Commercial Industrial

- 1.02 to - 1.89 - 1.36 - 1.25 to -1.82

This result is consistent with economic intuition and the recent experience of utilities throughout the United States. A lower price elasticity of peak demand than kilowatt-hour sales implies that, as real prices increase for all kilowatt-hours, kilowatt-hour sales fall more percentage-wise than kilowattpeaks, As a result, utility load factors deteriorate in times of rising real of this result is that, as the price electricity prices. l5 An intuitive explanation of all kilowatt-hours increases, individuals and business reduce off-peak consumption more than they reduce consumption at the time of the system peak. This result has important implications for utility construction budgets. The amount of capacity required, and the optimal capacity additions, depend on kilowatt-peak demand, kilowatt-hour sales and the load duration curve. In times of increasing real electricity prices, the load duration curve tends to deteriorate and the optimal mix of capacity shifts towards peaking units and away from base load units. Alternatively, if real electricity prices are expected to fall overtime, load factors would tend to improve and the optimal capacity mix shifts toward base load units.

3.2. Demand price elasticities The short-run price elasticity of peak demand with respect to a onepercent change in all demand charges is -0.06. The long-run price elasticity of peak demand with respect to a one-percent change in demand charges is - 0.08. Initially, this result may seem somewhat surprising. The estimated elasticity of peak demand with respect to demand charges is less than estimated price elasticity of peak demand with respect to energy charges. Since demand charges are based on a commercial or industrial customer’s maximum rate of consumption, which would generally occur at or near peak periods, one would expect thdt the price elasticity of peak demand with respect to “This range of estimates is based on Taylor’s (1975) survey. “This proposition assumes all of the factors, including the price and availability of other fuels constant. If one simultaneously observed alternative fuel availability declining and real electricity prices increasing, load factors of summer peaking systems might increase due to substitution of electricity for other fuels in space heating.

128

R.M. Spann and E.C. Beauvais, Peak electricity demands Table Estimated

1

peak demand equation for VEPCO. (t-statistics in parenthesis).

+0.069 LO +O.O127T +0.644 LI (0.94) (6.41) (3.42)

+0.257LINDEX+O.l62LKW, (3.02) (1.28)

Table Estimated

elasticities

Elasticity with respect to All marginal energy prices All marginal demand prices Oil price Income Industrial activity

+133LKW,, (1.23)

2

of peak demand

for VEPCO.

Short-run

Long-run

-0.31 - 0.06 0.069 0.644 0.251

- 0.44 - 0.08 0.10 0.91 0.36

demand charges would exceed the price elasticity of peak demand with respect to energy charges. This argument is only true for customers with demand meters who pay a positive marginal demand charge. Residential energy prices are included in the measure of energy prices. These customers do not have demand meters. Smaller commercial and industrial customers do not have demand meters. Thus a substantial number of customers, who may account for a significant percentage of load at the time of system peak do not pay explicit demand charges and hence are unaffected by increases in those charges. In this context, it is not surprising to observe that the price elasticity of peak demand with respect to demand charges is less than the price elasticity of

R.M. Spann and E.C. Beauvais, Peak electricity demands

129

peak demand with respect to energy charges. This does not imply that elasticities of peak demand with respect to demand charges are less than elasticities of demand with respect to energy charges for customers with demand meters.16

3.3. Elasticities

with respect to alternative fuel prices

Residual fuel oil prices are included in the model as a measure of alternative fuel prices. The short-run elasticity of demand with respect to oil prices is 0.069. The long-run elasticity with respect to alternative fuel prices is 0.10. The result is consistent with previous studies of kilowatt-hour usage which conclude that alternative fuel prices were an important determinant of electricity usage. l7 The implication of this result is that alternative fuel prices are an important determinant of utility demands. It is also consistent with the hypothesis that there is a substantial amount of price induced substitution between electricity and other fuels. The elasticity of summer peak demand with regard to oil prices is less than previous estimates of kilowatthour elasticities with regard to alternative fuel prices. This is consistent with the proposition that the largest component of inter-fuel substitution is in heating markets. 3.4. Income and industrial activity elasticities The estimated short-run elasticity of peak demand with respect to the industrial activity-electric intensity index of 0.257. The long-run elasticity of peak demand with respect to the industrial activity-electric intensity index is 0.36. The estimated short-run elasticity of demand with respect to income is 0.644. The long-run elasticity of demand with respect to income is 0.91. If on sums the elasticities of demand with respect to income and industrial activity to obtain a ‘total income elasticity’, this elasticity is 0.90 in the short run and 1.27 in the long run. It is interesting to note that the relationship between income elasticities and price elasticities differs significantly between the peak demand and the kilowatt-hour demand equations. In existing kilowatt-hour demand equations, long-run price elasticities generally exceeded existing long-run income elasticities indicating that electricity usage is more sensitive to price than to income. This relationship is reversed in the peak demand equation. “Some estimations were conducted which separated out residential, commercial and inprices. Generally these estimations were unsuccessful due to the high degree of

dustrial colinearity “For

between example,

the prices charged see Anderson

(1973).

various

customers.

R.M. Spunn and E.C. Beauuais, Peak electricity

130

demand\

Such a result is not surprising. Consider the effects of a long-run increase in income in the Commonwealth of Virginia (or any other state). At increased incomes, households are more likely to purchase air conditioners, all electric homes, and other appliances which tend to operate during peak periods. At higher incomes, individuals do more shopping and purchase more goods and services at commercial establishments. Such establishments are generally using electricity at the time of the system peak. Finally, one would expect night-shift premiums to increase as incomes increase, leading to a lower likelihood of multi-shift industrial operations and, thus, greater increases in on-peak usage relative to total kilowatt-hour usage.

3.5. Short-run

versus long-run

elasticities

The estimated lag structure is relatively short, indicating a more rapid adjustment in peak demand to prices than observed in existing econometric estimates of kilowatt-hour demand equations. It is not clear whether this result is incorrect, or indicates biases introduced by the use of pooled crosssectional and time-series data in existing studies.

3.6. Temperature

sensitivity

The coefftcient of temperature in the peak demand equation is 0.013. Since temperature is expressed in logarithmic form, this coefficient indicates that for each one degree increase in average temperature, system peaks increase by about 1.3 %.

4. Forecasts of peak demands: 1976-1985 Forecasting is both an art and a science. One method of forecasting is to simply apply estimates of future variables to an estimated econometric model. This seemingly straightforward method of forecasting is neither simple nor problem-free. Even if one assumes the parameters of the econometric model are unbiased, one must forecast, or use forecasts of, the independent variables (price, income, population, etc.) to forecast the dependent variable, (kilowatt peaks or kilowatt-hour sales by customer classification). As will be discussed below, variations in the assumptions one makes concerning forecasts of independent variables, in particular electricity prices and income, lead to considerable variations in the forecast of electric utility peak demands. In addition, econometric estimates and econometric forecasts are not ‘cast in stone’ nor should forecasting be a pure mechanical exercise. Considerable judgement in use of econometric models must be applied if accurate and reliable forecasts are to be developed.

R.M. Spann and E.C. Beauvais, Peak electricity demands

131

The econometric model of peak demands estimated in the previous section of this paper can be used to forecast peak demands under various assumptions. Forecasting peak demand growth rates under ‘normal’ weather is relatively straightforward. The model is in logarithmic form and thus can be expressed in terms of growth rates. More formally, the forecasting equation is

GPK= --0.052 ;

GE,-0.097 i

i=l

CD,

i=l

+0.069GO+0.644GI+0.257GINDX +0.162

GPK, +0.133 GPK,,,

where G/X =the annual growth rate in peak in the forecast year over the previous year’s peak, GE, = the annual growth rate in the ith marginal energy price including the fuel adjustment clause in the forecast year (i.e. the percentage by which prices have increased from the previous year to the forecast year), GDi= the annual growth rate in the ith marginal demand charge from the previous year to the forecast year, GO = the annual growth rate in oil prices from the previous year to the forecast year, GI= the annual growth rate in income from the previous year to the forecast year, GIN DX = the annual growth rate in the electric-intensiti.Aty industrial activity index, GPK,, = the annual growth rate in the peak one year prior to the forecast GPK,, year.

Since the VEPCO is equally likely to peak in any summer month, we assume GPK, = GPK for forecasting. The growth rate of all prices and price (or income are expressed in real terms; that is the rate of nominal income) increase less the rate of inflation. Long-term forecasting is further simplified by noting that one would expect a close relationship between industrial activity and income. For the period 1962-1973, the relationship between the growth rate in real disposable income and the growth rate of the industrial activity index is

GINDX = - 1.55f0.96

GI.

132

R.M. Spann and E.C. Beauvais,

Peak electricity

demands

Substituting this relationship into the previous equation express the model in terms of growth rate in real income, prices and the real price of alternative fuels or

GPK,=

-0.52

; i=l

allows one to real electricity

GE,-0.097 i GDi+0.069GO+0.644GI i=l

+0.162(-1.55+0.96GI)+(0.162+0.133)GPK,_,

An alternative approach would be to forecast employment in each of the industries used to construct the industrial activity index. This method could be utilized if one wanted to determine the impact of changes in Virginia employment patterns. For simplicity, we utilize the relationship between income and industrial activity. Before discussing the actual forecasts it is important to discuss the input data, in particular the income growth assumptions. Official state forecasts assume a substantial increase in the rate of real income growth in Virginia after 1979. In the 1980’s official state forecasts assume a growth rate in real income of about 9% per year. In order to evaluate the effects of alternative income growth scenarios in Virginia, we include forecasts assuming that the growth rate in real income in Virginia is 5.5% per year from 1979 to 1985. Three growth rates of electricity prices are used for forecasting, 2%, 6.8% and 10%. A long-run rate of inflation of 4.8 oA (consistent with official state forecasts) is assumed. While this rate of inflation may seem low relative to previous year’s rates of inflation it does not materially affect the model. The model is in terms of real prices. Thus a 2 % growth rate in electricity prices is prices. If one wanted to equivalent to a -3 % growth in real electricity assume a higher rate of growth in inflation, say 8 %, a 5 % growth rate in nominal electricity prices is a - 3 a/o growth rate in real electricity prices. Two assumptions are made concerning alternative fuel prices. The first is that oil prices will not change in real terms, i.e., nominal oil prices will rise at the rate of inflation, 4.8 %. The second is that real oil prices will increase at a rate of 5 %, or a nominal increase in the price of oil of approximately 10% per year. Table 3 lists predicted long-run equilibrium growth rates of peak demand under alternative assumptions. That table illustrates the importance of the inputs into a forecasting model and the necessity to constantly update forecasts. Alternative assumptions concerning price and income changes can and will lead to substantially different forecasts. Thus it may be as important for forecasters to concentrate on accurate input data as it is for forecasters to be concerned with improving estimated coefficients or econometric techniques.

R.M.

Spann and E.C. Beauvais,

Table Long-run

Peak elecrricity

133

demands

3

equilibrium growth rates for VEPCO under sumptions, 19761980 (in percent).

various

Real income

Nominal electricity prices

Nominal oil prices

Peak demand

1.6 7.6 7.6 5.5 5.5 5.5 7.6 7.6 7.6 5.5 5.5 5.5

2.0 6.8 10.0 2.0 6.8 10.0 2.0 6.8 10.0 2.0 6.8 10.0

10.0 10.0 10.0 10.0 10.0 10.0 4.8 4.8 4.8 4.8 4.8 4.8

10.40 7.51 5.82 7.38 4.85 3.17 9.53 7.00 5.31 6.87 4.34 2.66

‘All growth rates are compound annual growth An inflation rate of 4.8%, consistent with official assumed.

as-

rates thru 1985. state forecasts is

The actual forecast of peak demand growth for the 197&85 time period for VEPCO is 5.8-7.0 percent per year. This forecast is based on estimates made by the Virginia State Corporation Commission staff concerning future trends in electricity and energy prices. This forecast is similar to that prepared by VEPCO using alternative techniques. 5. Implications of the model for peak load pricing The model presented in this paper, when combined with other econometric results has important implications concerning the effects of peak load pricing. The fact that kilowatt peak demand price elasticities are significantly less than previously estimated price elasticities for kilowatt-hours implies that the primary effect of increasing peak prices and decreasing off-peak prices will be to increase kilowatt-hour sales both absolutely and relative to system peaks. This implies a shift in the mix of capacity additions towards higher capital cost base load capacity and a reduction in required turbine and pump storage capacity additions. Thus the primary effect of peak load pricing probably may be significantly reduced fuel costs with little change in capital costs. ’ 8 “This does not imply that peak load pricing is not cost effective. Both utilities and their customers are concerned with total costs (i.e., the sum of capital and fuel costs). Thus so long as the load shifts induced by peak load pricing lead to increases in the efficiency of resource allocation, it does not matter whether the cost savings due to implementing peak load pricing result from reduced capital of reduced fuel costs. These differences are only relevant when regulatory rules lead to differential treatment of the timing of the revenue realization of changes in capital or fuel cost changes of when financing constraints are dominating capacity expansion plans.

In addition the model implies that peak load pricing may increase the total revenues of electric utilities. In the areas in which demands are fairly inelastic, i.e. peak periods, prices would be increased. Existing estimates of the price elasticity of demand for kilowatt-hours indicate price elasticities slightly in excess of unity. Since peak load pricing would probably reduce the overall average price of kilowatt-hours, prices are being increased in an area where demand is inelastic and lowered in an area where demand is elastic. An increase in prices in the inelastic portion of a demand curve and a decrease in prices in an elastic portion of the demand curve increases total revenues.

6. Implications

of the model for capacity planning

The previous section of this paper indicates that forecasts of future peak demand growth rates are sensitive to alternative assumptions concerning growth rates in electricity prices, income and other variables which influence peak demands. In addition, any forecast involves both a mean and a variance. Such uncertainties in forecasting future loads are sufficiently large that they should be included in capacity planning analysis. Rather than approaching capacity planning as a problem of minimizing the costs of meeting future load duration curve, the relevant problem may be minimizing the costs of meeting a stochastic load duration curve and evaluating risk rate of return trade-offs in investment planning. Flexibility and the degree to which a construction program can be ‘slipped’ (if demand fails to grow as expected) or accelerated (if demand grows more rapidly than anticipated) may become more important in choosing among alternative construction or capacity expansion plans. Alternatives which allow for greater flexibility and lower cost responses to unanticipated changes in demand growth may have higher costs in a world of certainty, but lower risk adjusted costs at reasonable levels of risk aversion. It is, of course, true that system planners already include a number of uncertainties in their analysis. Uncertainties such as prices and availability of fuels, environmental regulations, etc. are currently included in the capacity planning analysis of many utilities. Given that these uncertainties will not necessarily decrease, and the uncertainties associated with forecasting future peak demands, approaching system planning from a stochastic optimization framework will become increasingly important in the future.

7. The limitations

of econometrics

in load forecasting

The body of this p;~pcr has ~-cpc~rted and discussed an econometric model pcah dctnands. Rather than conclrtding this paper by highfor forecasting

R.M.

Spann and E.C. Beauvais,

Peak electricity

demands

135

lighting the virtues of such a model it is more appropriate to discuss the limitations of econometrics in load forecasting. The actual determination of peak demand depends on a large number of variables and complex, potentially nonlinear relationships. Econometric models are constrained in that the number of variables which can be included in any one equation must not be ‘overly large’ if one is to avoid multicolinearity problems. In addition econometric modeling requires stringent assumptions concerning the functional forms of estimating equations. Use of functional forms which allow for variable elasticities, interactions between price and income elasticities etc. are feasible. However, in many cases, the data are simply not rich enough to support estimation of such models. An alternative to econometrics that should be investigated in more detail in load forecasting is the ‘non-econometric’ or ‘building block’ approach. Within this approach one specifies engineering relationships between electricity usage and other variables. Such an approach does not necessarily require the price elasticities be assumed to be zero but rather, one can build a model that assumes consumers make economic choices with regard to space heating, space conditioning, building design etc. Within specific industrial sectors, one can use process-engineering models to determine electricity demands under alternative price assumptions. Such models have a disadvantage in that they require substantial use of judgment. However that judgment can be quantified and iticluded in a mode1 which utilized on a large number of technological relationships to forecast peak demands. Both approaches have strength and weaknesses. Because of this factor it is important that both approaches be employed in forecasting peak demands. Economists should not concentrate solely on econometric models simply because economists are more familiar with econometric than noneconometric techniques.” “‘An additional reason for utilizing both econometric and non-econometric techniques in load forecasting is the uncertainties involved in both approaches. A model which includes both econometric and non-econometric approaches allows the forecaster to utilize one methodology as to check the other methodology. Alternatively, if one treats forecasts as investments, increased uncertainty generally implies that a more diversified portfoli/is optimal. In this case, use of more than one approach to forecasting is equivalent to investing in a diversified portfolio.

References Anderson, K.P., 1971, Toward econometric estimation of industrial energy demand: An experimental application to the primary metals industry, Rand Corporation Report R-719NSF, Dec. Cargill, T.E. and R.A. Meyer, 1971, Estimating the demand for electricity by time of day, Applied Economics 3, 233-246. Irish. W.. 1976, Direct testimony, Docket E-100, Sub 22 North Carolina Utilities Commission, Dec. Spann, R.M.. 1976, Industrial and commercial rate structures and econometric estimation of the demand for electricity, in: James Boyd. ed., Proceedmgs on forecasting methodology for time of day and seasonal electric utility loads (Electric Power Research Institute, Palo Alto, CA).

136

R.M. Spann and E.C. Beauvais, Peak electricity demands

Taylor, L.D., 1975, The demand for electricity: A survey, The Bell Journal of Economics 6, no. 1, Spring, 74-l 10. Uri, N., 1978, A mixed time-series/econometric approach to forecasting peak system load, this issue. Wenders, J.T., 1976, Peak load pricing in the electric utility industry, Bell Journal of Economics and Management Science, Spring.

Journal

of Econometrics

9 (1979) 137-153.

0 North-Holland

Publishing

Company

AN APPROACH TO MODELING SEASONALLY STATIONARY TIME SERIES* Emanuel

PARZEN

and

Marcello

PAGAN0

Texas A & M University, College Station, TX 77843, U.S.A. Harvard School of Public Health, Boston, MA, U.S.A.

1. Introduction An individual or household generates many time series Y(t), t =O, 5 1,. . . Examples of variables Y(t) are the demand for electricity of the household in an hour ending at time t (measured in hours) or some medical characteristic, such as blood pressure, of an individual at time t. One usually has two aims: to forecast future values of the series Y(t), and to explain how the behavior of the series Y( ) depends on various possible explanatory variables. The regression analysis and econometric modeling approach attempt to treat these two problems simultaneously by building a linear model for Y(t) in terms of the explanatory variables. We would like to propose a different approach. One builds a time-series model for Y(t) which can be used to forecast it. The parameters and other characteristics of the model fitted to each household’s time series can be used to describe a finite number of model types to which a random household’s time series will belong. The explanatory variables are used to predict the type of the time series Y( . ) rather than its values. This proposal is fully nonparametric, since it does not postulate a linear model for Y(t) in terms of the explanatory variables. The aim of this paper is only to describe our approach to the first half of the above program. More precisely, we will discuss how to build time-series models for time series (observed hourly, daily, or monthly) which are periodically stationary in the sense that the mean varies with the hour, day, or month, and the correlation between two values at times t and t su depends on v and the index j when one writes t = kd + j, where d is the period (24 for hourly values, 7 for daily values, 12 for monthly values). The basic approach to modeling a time series Y(t), t =O, + 1,. . . is to decompose the value Y(t) into the sum of two components as follows: Y(t)= *Research

supported

Y”(t)+

YY(t),

in part by the Offke

of Naval

Research.

E. Ptrrxn

138

und M. Pagano,

Seasonull~

stationary

time series

where Y”(t) is the explained or predictable part of Y(t), and Y”(t), is the error, or unexplained, or unpredictable part of Y(t). We use letter v as a superscript because Y”(t) is the ‘new’ part of Y(t), and we use p as a superscript to connote ‘mean’, since Yp(f) represents an average or smoothed value about which Y(t) fluctuates. One takes for Y”(t) the conditional mean (which is usually more computationally and theoretically tractable than the conditional median), so we define Yp(f)=E[Y(t)JY(t-l),Y(t-2),...]. Further, we assume Y( ) is a normal combination of Y (t - 1), Y (t - 2), . We call

process;

then

Y”(t)

the innovation at time t: the innovation series Y”(t) consists zero-mean random variables. Their variances are denoted

is a lineur

of independent

0,2(t) =E[lY”(t)12]. Y( ) is a stationary time series (defined below) it holds that a:(t) a constant independent of t called the infinite-memory innouation tiariance, or infinite-memory one-step-ahead prediction error. Then Y ( . ) is an independent, identically distributed sequence, and we call it white noise. Finding formulas for Y@(t) or Y”(t) are equivalent problems. We define the time-series modeling problem as finding the filter (called the whitening filter) which transforms Y(r) to Y”(r),

When 2

=(Jlc,

Y(t)

-i_k

YV(C).

A general model for a time series Y(r) is that there is a deterministic curve m(t) representing mean values about which Y(t) is fluctuating with variances 02(t) which may be time-varying. One thus assumes the representation

where m(t) is the mean function, m(r)=EY(t);

139

E. Parzen and M. Pagano, Seasonally stationary time series

o’(t) is the variance

function,

a2(t)=var

Y(r);

and Z(t) is the fluctuating

Z(t) =

function,

Y(t)--m(t) a(t)

The model of a stationary time series assumes that m(t) and g2(t) are constant. This is not a realistic assumption for data that clearly have a natural period d in the sense that, for all t=O, _t l,..., m(t+d)=m(t), a2(t+d)=a2(t). Some periods are: for monthly data, d= 12; for daily data, d=7; and for hourly data, d =24. The mean value function m(t) then has parameters m(l),m(2),. . .,m(d), and the variance function has d parameters a’(l), . . ., a’(d). To model the fluctuation function Z(r), we assume it is normally distributed with zero means and variance 1, which is either (a) stationary in the sense that there exists a correlation function p(n) satisfying

E[Z(t)Z(t+c)]=p(v), or (b) periodically stutionarp (with period d) in the sense correlation functions P,(U), . ., ~~(0) satisfying

that

there

exist

where j(t) is the index j satisfying j= 1,. . .,d and j(r) =t modulo d, or equivalently t =j(t) + kd for some integer k. When Z(t) is stationary, the whitening filter, which transforms Z( ) to its innovation series Z’( ), is time-invariant,

ZY(t)=Z(f)+tlz(l)Z(t-l)+...+tlW(m)Z(t-m)+..., 0: =E[IZ’(t)l2].

When

Z(t)

is periodically

stationary

with

period

d, there

are d whitening

140

E. Parzen and M. Pagano,

Seasonally

stationary

filters, depending on the time t (more precisely, is equal to t modulo d),

ZY(t)=Z(t)+

f k=l

There are also d innovation

time series

on the value j= 1,. . .,d, which

a,,,(k)Z(t-k). variances

0:. 50,. ., CT:,~ such that

The notation is now at hand to state the problems seasonally stationary time series:

that arise in modeling

(a) Estimate and model the time-series means m(l), . . ., m(d). (b) Estimate and model the time-series variances o’(l), . . ., a*(d). (c) If m(j) and O(j) denote the fitted means and variances, estimate

Z(r) by

Y(t)-fiU(t)l

z+)=

dxt)l (d) Estimate

and model the correlations

filter(s) of Z(

and whitening

The primary method we use to estimate whitening filters order autoregressive schemes whose order is determined from an order-determining criterion. The fitted ‘schemes’ can also as moving-average and mixed autoregressive moving-average desired; such representations are useful for decompositions of sums of signal plus noise, or trend plus seasonal plus irregular.

2. Stationary

autoregressive

).

is to fit linitethe data using be represented schemes if the series into

schemes and their estimation

A normal zero-mean stationary time series Z(t), t = 0, k 1, rfr2,. . ., is said to obey an autoregressive scheme of order p if there exist coefficients a(j), j=l , . . ., p, such that

Z(t)+

i

atj)Z(t-j)=&(t),

j=l

where E( . ) represents independent variance o2 ; the polynomial,

zero-mean

normal

random

variables

with

E. Parzen and M. Pagano, Seasonally stationary time series

141

has all its roots in the complex z-plane outside the unit circle; and p is an integer called the orderof the scheme. When the order p is known, the parameters a’, a(l), . . ., cc(p), have asymptotically efficient estimators ai, a(l), . . ., B(p), obtained by solving the normal equations

j$oet_dPT(.-j)=o, v=l,..., P. i wbT(-_j)=~;, j=O where B(O)= 1, and T-u

1 z(r)z(t+u)

PTk)=

1=1

>/(iZ2(4

is the sample correlation function. The asymptotic joint distribution,

T+[oicj)- afj)l, is normal

j=l

as T tends to co, of , . . .1P>

with zero mean and covariance

matrix.

o’{p(j-k)J--1. The determination of the order m approximates a true infinite-order extensive current research. We use which CAT(m) is minimized, where Parzen (1977).

CAT(O)=

-

CAT(m)=&

> i c$;'-$;',

P-l

where T “2 fJ---(p p T-eP

,

1,; (

of an autoregressive scheme which best autoregressive scheme is a subject of as an estimator for m the value & at CAT(m) is the criterion introduced by

p'

m=1,2,...,

142

E. Parzen

und M. Paguno,

3. Periodic autoregressive

Seasonal/J: stutionur!,

time series

schemes and their estimation

A normal zero-mean periodically stationary time series Z(r), t =O, t_ 1,. ., is said to obey a periodic autoregressive scheme of order p ifthere exist coefficients a,(k), k= 1,2,. . .,p,such that Z(t)+

a,(k)Z(t-

i

k)=E(t),

k=l

where I:( ) are independent zero-mean normal random variables with variance 0,‘. For an autoregressive scheme, 0,’ is a constant a’; here we assume o_! depends on the value of t modulo d, so that there are d variances 2 CT: such that Cl,..., 2 Or

=

2 uj(t)

In words, crj’ is the mean square prediction error when one predicts the value of Z(t) in month j (that is, t = j modulo d) ‘using past values in the most recent p months. The coefficients of this predictor are denoted aj( l), . . ., aj(p); in general, the autoregressive coefficients a,(k) satisfy

When one fits an autoregressive scheme of order p as a means of determining an optimum formula for the predictor, denoted Y“(t), of Y(t) given the infinite past Y (t - 1), Y (t - 2), . ., one permits the order p to vary in order to determine the optimum memory m (which is precisely the problem of greatest interest). For a stationary time series, we denote by C: the mean square prediction error using p past values. For a periodic stationary time series, we denote by $, p the mean square prediction error of the value in month j using p past values. The normal equations determining the coefficients r,,,,(k) are found by multiplying both sides of the equation by Z(I -I.): then for ,j = 1,. ., rl and l’= I . .. I’.

Pj(,,(--V)+ f: a,‘CC’ (k)Pj(t,-tc(k-c)=() kzl

Pjct,(O)+i k=

1

aj(r)(k)pjc,)-k(k)=aj:1,.

When one solves these equations for a given value of j, one can permit the order p to depend on j (in which case one writes pj as the upper limit of summation). If one does so, the model is called non-constant-order periodic autoregression.

E. Parzen

The periodic variance matrix

and M. Paguno.

Seasonally

stationary

correlations pj(,,(u) can be expressed T(u) of the multiple time series,

143

time series

in terms

of the cross-

Z(1 +nd)

1 1. 2(2+nd)

V(n)=

...

Z(d+nd)

We write function

V*(n)

to denote

the transpose

of V(n).

The

matrix

covariance

has (j, k)th entry rj,(u)=E[z(j+nd)z(k+nd+ud)] =pj(k-j+ud). Computationally, one first forms estimators of the matrices f(u)= order to form estimators of pj(u) for j = 1,. . ., d and u = 0, _+1,. . . . The asymptotic joint distributions, as T tends to co, of

are independent and covariance

for j, #j,; matrix

rTf{pj-&

time-series

A normal

with zero mean

very pleasant

facts are very deep

and

are given

modeling

zero-mean

stationary

I

d-dimensional

VI(n) V(n)=

in

-k,))-?

The proofs of these Pagan0 (1976). Multiple

for j, = j, = j, they are normal

{fjk(u)}

. .. ; v,(f~)

,

n=o, &I,...

multiple

time series

in

144

E. Parzen and M. Pagano, Seasonally stationary rime series

with covariance

is often modeled

matrix

as a multiple

autoregressive

scheme

V(n)+A(l)I’(n-l)+...+A(p)V(n-p)=&(t), where c(t) is d-dimensional

inultiple

white noise with covariance

matrix

c=E[&(t)&*(t)]. The number of parameters to be estimated in this model is often too large to be estimated from the available sample size N. Thus, if d= 12 and p= 1 and one has N =20, one would have Nd=240 data points to estimate d2 = 144 parameters in A( 1) and d(d - 1)/2 = 66 parameters in C. One can reduce the number of parameters to be estimated by reformulating the model as a periodic autoregression. Suppose one can order the components of V(n) so that V,(n) is ‘caused’ by V,(n),. . ., V,_ I(n) as well as by values of V(n-1), V(n-2),...; V,_,(n) is ‘caused’by VI(n),...,V,_,(n) as well as by V(n-j), j=l,2,...; and so on. Thus V,(n) is ‘caused’ by V, (n) as well as by V(n - l), V(n - 2), . . . . The scalar time series

Z(t)= r/;(n),

if

t=j+nd,

is periodically stationary and often can be modeled using fewer parameters than would be needed for a multiple time-series model of the multiple time series V(n).

4. Fitting seasonal means and variances The mean value function m(t) of a periodic-stationary time series with period d has parameters m(l), m(2), . ., m(d), and the variance function has d parameters g2(l), . . ., a’(d). Given a sample {Y(t), t = 1,. . ., T} where T = nd, estimators m(j) and a2(j), for a fixed j= 1,. . .,d, are given by

E. Parzen

and M. Pagano,

Seasonally

stationary

time series

145

We call m,(j), j= 1,. . ., d, the seasonal means and a:(j), j= 1,. . ., n, the seasonal variances. The possibility that m(j) equals a common constant m for j= 1,. . .,d, as well as other smooth relations among m(j), is treated by fitting a harmonic representation (assuming d = 12),

+a(6)cosnj. Estimators m,,a,(k), b,,(k) of these mators m,(j) by the formulas

coefficients

can

be obtained

from

esti-

,

In order to ‘test’ the estimators a,,(k) and b,(k) for ‘significant difference’ from zero, one needs to determine their probability distribution. However, this depends on the properties of the fluctuation function Z(c) whose estimation is the aim of the investigation. Initially, one tests the a,(k) and b,,(k) under the assumption that Z(r) is white noise (a sequence of independent, identically distributed random variables). A statistic for testing the significance from zero of a,(k) and b,(k) is R,(k)=n

ai(k)+bi(k) 0;

’

where

which has a probability distribution that is chi-square freedom. We call the index k significant if R,(k) is above (often taken to be 4).

with 2 degrees of a suitable threshold

We define the seasonal

The seasonal

fitted means

fitted variances

to be

are defined

by

The seasonal fitted means m,(j) and the seasonal means m,(j) are not expected to be significantly different and, therefore, the seasonal variances a:(j) and the seasonal fitted variances 0,2(j) are not expected to be significantly different. We use the fitted means and variances in our modeling because it seems preferable to fit as smooth a periodic mean value function as possible to periodically varying time series. To help guide us in choosing between the stationary and the periodicstationary model for fluctuations, we test the hypothesis that the d seasonal variances o*(j) are equal. No satisfactory test is known for testing the equality of variances; two statistics that one might use as a rough guide are the Bartlett statistic for homogeneity of variances, and the ratio (Hartley’s test) max

5,2(j)+

j=l....,d

min

O.Z(j).

j=l,...,d

5. Empirical seasonal stationary

time-series

analysis

In Parzen (1976), a philosophy of empirical time-series modeling was illustrated on hourly electricity data. We here consider the daily totals of this time series. We thus consider the time series Y(t), t= 1,2,. . ., 365 graphed in fig. 1, where t is measured in days, and Y(t) is the total electricity generated over the 24-hour period by a certain utility in the western U.S.A. The graph of the time series clearly exhibits a seven-day periodicity. One expects the whitening filter to consist of three filters in series, respectively representing detrending, deseasonalizing, and innovations (other time-series plots are shown in figs. 2 4). time series

From S-8).

y( t )

the data

detrend

deseasonal

filter

filter

analysis

we obtain

the conclusions

discussed

I

white noise

below

(see figs.

E. Parzen and M. Pagano, Seusonally stationary time series

? P

I .bo Lx.

147

E. Parzen and M. Pagano, Seasonally stationurg rime series

148

B 3 f

4 2-

Q642I

1

0.00

0.06

0.16

0.24

I

I

I

0.40

0.48

0.56

0.64

0.40

0.46

0.56

0.64

0.32 CAT ARSP 1

Fig. 5 ‘r -

Ekctrkity Demand, 1973 STAT CAT Spectra From 22

0.16

0.24

0.32

CAT ARSP 2

Fig. 6

Approach 1. Fit autoregressive schemes to tuations of the time series about its overall mean.

Zl(t)=

Y(t)-

x

the

fluc-

Using the CAT criterion, one finds the optimal order is 22 (with only coefficients of lag 1, 7, 8, 14, 15, 21, and 22 being above 0.1 in magnitude). Using SELECT (as a subset autoregression algorithm), one finds the representation

E. Parzen

and M. Pagano,

Seasonally

stationary

149

time series

f

P

I 0.00

0.06

0.16

0.32

0.24

0.40

0.48

0.58

0.64

0.48

0.56

0.64

SEL ARSP 1

Fig. 7

i6 i

e

c

Ekcbicity Demand. 1973 STAT SEL Spectra From 22

4 2 1 t

f

4

3

2 1 e8 I\

?z f 0.00

0.06

0.16

0.24

0.32

0.40

SEL ARSP 2

Fig. 8

where el(t) is not quite white noise. Indeed, one should permit the filter to include lags 21 and 22. However, our aim here is not to explore this approach in detail, but only to indicate that it readily suggests that the autoregressive filter be factorized to represent two filters in series, El(t)= Approach

tuations

(Z-0.62L)(I-0.45L?-0.43L’4)Z(t).

2. Fit autoregressive schemes to Zl (t)= Y(t)of the time series about its seasonal (daily) means.

P(t),

the fluc-

150

E. Parzen and M. Pagano, Seasonally stationary time series

Using the CAT criterion on the time series of length an autoregressive scheme of order 6,

365, one fits Z2(. ),

omitting coefficients less than 0.02 in magnitude. If we reduce the time-series length to T =364, a multiple of 7, we obtain for Z2(. ), an autoregressive scheme of order 2,

The time-series model can again be viewed respectively deseasonalize and detrend.

as a series

of filters

which

Approach

tuations tuations

3. Fit stationary schemes to Zl (t)= Y(t)- P(t), the flucabout seasonal means, and Z2(t) = {Y(t) - F(t)} + a[ Y (t)], the flucabout seasonal means normalized by seasonal standard deviations.

We consider the time series Y( .) over the first 280 days of the year (t=l,..., 280); over this period, it has mean 65, variance (about total mean) 15.3, and variance (about daily means) 14.9. The daily variances are Monday (day 1) Tuesday (day 2) Wednesday (day 3) Thursday (day 4) Friday (day 5) Saturday (day 6) Sunday (day 7)

21.8 14.5 16.0 16.2 15.3 11.3 9.2

These variances are not significantly different by the inapplicable tests of equality of variances developed under the assumption of normal white noise. The autoregressive scheme fitted by CAT has order 2 for Zl and order 6 for 22. SELECT fits to both Zl and 22 an autoregressive scheme with lags 1, 2, and 8:

The innovation

variances

are as follows:

JqEl@)I21

-q~2(~)~21

CAT

0.29

0.24

SELECT

0.28

0.25

Note that SELECT SELECT. We next generate

chose a longer

order than CAT, since lag 8 is chosen

the normalized

mean square

by

innovation,

over the fitted period; it approximately equals 0.28 for each of the above four methods, although it is slightly less for the model generated by SELECT autoregression of Zl ( . ). Approuch

4.

Fit periodic

autoregressive

schemes to Zl( . ) and Z2(. ).

The mean square prediction error can be reduced from 0.28 using stationary autoregressive models for Z2(. ) to 0.20 using periodic-stationary autoregressive models (of order 6, or variable orders for each day determined by PCAT, a periodic version of CAT). While the periodic-stationary autoregressive models using only a few lags chosen by SELECT perform no better than the stationary models, they do illustrate the structure of the periodicstationary predictors: Z’(Mon)=0.7 Z(Sat), Z’(Tue) =056Z(Sun), ZV(Wed) = 0.79 Z(Tue), = 0.56 Z(Wed) + 0.41 Z(Tue), Z’(Thu) Z”(Fri) =0.98 Z(Thu), ZP(Sat) =0.95 Z(Fri), Z”(Sun) = 0.94 Z(Sat). These low-order autoregressions are perhaps reasonable, since when one fits a robust stationary autoregression for Z( . ), one obtains the model Zr(t) =0.99Z(t1).

6. Change detection To detect changes in the system generating a time series, one approach is to choose a time t, near which the change would have occurred; tit a model to the values Y(t), ts t,; and then form the one-step-ahead predictions YP(t) at times t > t,. Two useful statistics for detecting change are PRER and NEGER, defined as follows. PRER is the mean square prediction error ratio PRER =

PREDERRFORE PREDERRFIT

’

152

E. Parzen and M. Pagano,

Seasonally

stationary

time series

WhCXe

PREDERRFZT=5 i IY(t)- rq#, t-1 PREDERRFORE

NEGER is the ratio -YY’(t),t>t,.

= &,=$+I

of negative

0

values

[Y(t)-

among

YW12.

the forecast

errors

Y(t)

Using the stationary models fitted to the first 280 days of our 1973 electricity demand time series, the forecast errors for the last 84 days have PRER ~2 and NEGER=0.63; for 84 forecasts, 95% significance levels of PRER and NEGER are greater than 1.4 and 0.61, .respectively. Therefore, one might conclude that there has been a downward shift in electricity demand in the last 84 days of 1973, compared with the first 280 days. For a theoretical justification of PRER as a statistic for detecting changes in the probability distribution of a time series, see Box and Tiao (1976).

Appendix:

Data analysis summary for periodically

Y(t)

stationary time series

Fitting means and variances From the sample, one computes its seasonal means and fitted means Y(t), diagnostics for the equality of variances.

overall mean and variance, the seasonal variances a(t),

the and

Correlations One computes the correlations of the seasonally mean-adjusted = Y(t)- Y(t) and the seasonally mean- and variance-adjusted =Zl (t)/a(t). Autoregressive autoregression

coefficients for

order

determined

by CAT

series Zl (t) series 22(t)

and for

subset

One computes CAT1 and CAT2, the CAT criteria for Zl and 22. The autoregressive coefficients of the order minimizing CAT are denoted ALPHl(Z) and ALPH2(l). Then one computes autoregressive coefficients for Zl and 22 using a subset autoregressive approach. Day-dependent correlations of 22 For each day t of the week, where t = 1,2,. . ., 7 (here day 1 is Monday day 7 is Sunday), compute the correlation between Z2(t) and Z2(t-v).

and

E. Parzen and M. Pagano, Seasonally stationary time series

Periodic

autoregressive

153

coefficients

For day t of the week, compute the coefficients of the best autoregressive predictor of the value on that day given values on preceding days. Summary

of one-step-ahead

prediction

error

mean squares

For the eight possible models we could build, we compute the mean squares of one-step-ahead prediction errors over the stretch of time series used to tit the model, and also over the next stretch of time series (over which the model would actually be used for prediction, or which would be used to determine whether the model had changed).

References Box, G.E.P. and G.C. Tiao, 1976, Comparison of forecast and actuality, Applied Statistics 25, 1955200. Pagano, M., 1976, On periodic and multiple autoregression, Technical Report no. 44 (Statistical Science Division, State University of New York, Buffalo, NY). Parzen, E., 1977, Multiple time series: Determining the order of approximating autoregressive schemes, in: P. Krishnaiah, ed., Multivariate analysis-IV (North-Holland, Amsterdam). Parzen, E., 1976, An approach to time series modeling and forecasting illustrated by hourly electricity demands, Technical Report no. 37 (Statistical Science Division, State University of New York, Buffalo, NY).

Journal

of Econometrics

9 (1979) 155-171.

0 North-Holland

Publishing

A MIXED TIME-SERIES/ECONOMETRIC TO FORECASTING PEAK SYSTEM Noel Department

Company

APPROACH LOAD*

D. URI

of Energy, Washington,

DC 20461, U.S.A

1. Introduction The need for accurate intermediateto long-term forecasts of peak system loads for electric utilities continues unabated. In fact, the past few years have witnessed an increase in the intensity of electric utility planners’ desire to refine their modeling efforts as the cost of poor forecasts has escalated. Accurate forecasts are an essential ingredient of planning. Based on estimates of future energy requirements, decisions are made about whether to contract for power purchases or sales with other utilities, where and what kind of plant to build, what licenses must be acquired, and so on. An underforecast of peak system load, for example, results in insufficient capacity. The consequence is that a utility will have a reserve margin below the desired 20% (historical average), which will lower the minimum level of reliability and imply potentially serious problems in meeting system demand. Given the critical nature of forecasts, a wide variety of methodologies exist to obtain these forecasts. For example, researchers engaged in forecasting have employed various modeling techniques in economic forecasting. These techniques encompass a wide variety of models, ranging from the univariate autoregressive and/or moving-average scheme to complex econometric models. The evidence points out that the simple time-series models can often outperform large econometric models [Steckler (1968)]. Pure time-series approaches have been extensively criticized, however, because if the resulting forecasts are poor, one is at a loss to provide an explanation since they have no basis in economic theory [Naylor et al. (1972)]. Furthermore, if one desires to explain the behavior of an economic system and not merely to provide forecasts, then pure time-series methods are not acceptable. Since they are void of economic theory, they cannot be used to test hypotheses about economic phenomena.

*The views expressed are those of the author and do not necessarily represent the Department of Energy (DOE) or the views of other DOE staff members.

the policies of

N.D. Uri, Forecasting

156

peak system load

This paper takes a pure time-series approach (Box-Jenkins time-series analysis) and integrates it into an econometric system. The results should mitigate the intensity of the criticisms of a pure time-series approach as well as indicate an improved method of forecasting peak system load for an electric utility.

2. A mixed time-series/econometric

model’

Let z, denote the peak system load of a specific utility and assume, following Uri (1976) that (1 -B)(l

-B”)lnz,=

(l-OB)(l

at time t (monthly),

-4B12)a,,

where B is the backward shift operator (i.e., RPz, =z,-,) and a, represents independent random deviates with mean zero and variance g2. Assume, next, that the parameters 8 and 4 are estimated, using a BoxJenkins procedure, from a historical data set consisting of 48 monthly peaksystem-load observations (four years of 12 monthly observations). In particular, assume that forecasts made at time t are based on parameters estimated from the preceding 48 observations. Additionally, assume that the forecasting equation is updated (i.e., its parameters reestimated) at intervals of three months. The procedure just described will yield a sequence of estimates for 0 and 4% ..., ‘B(t-3),~(t-3)),(B(t),~(r):, (

(O(t+3),O(t+3)),...,

(2)

that, in general, will vary through time. A time-series approach could be employed to analyze the evolution of this sequence, but what is proposed here is an econometric approach instead. In particular, one can realistically hypothesize that 0 and C#Iare functions of the price of electrical energy, income, temperature, and so on and specify a regression model accordingly. With this objective, let 8 and 4 represent vectors of ‘observations’ on 0 and 4, respectively, and let X denote a set of economic and weather-related variables. Next, it is hypothesized that e=XLY+E1,

9=xp+c,, where c( and B are stochastic terms.

(4) appropriate

‘This model is based on the sugestion\

vectors

of Taylor

of constants

(1975).

and

E, and

Ed are