A Two-Level Electricity Demand Model

Hausman, Kinnucan, and Mcfadden

Introduction

• Analyzed the Connecticut Peak Load Pricing Test– Oct 1975 to Oct 1976– 199 Households– Meters installed to measure electricity

consumption– Faced a peak load pricing system with 3 prices• 16 cents / kwh peak, 3 cents / kwh intermediate, 1 cent

/ kwh off-peak.

Procedure

• Determine household preference for amount to spend on electricity

• Estimate the cost of that electricity based on time of consumption (TOD pricing) or previous rate structure (controls)

• Determine from relative demand and pricing, the total demand

Technique

• Estimate consumer demand w/ two stage budget– Treat electricity demand within each period as a

separate commodity• Estimate demand across periods conditioned

on relative prices, appliance stock, household characteristics, and environmental effects

Technique

• Estimate a price index for electricity– Household consumption determined by price of

electricity and prices for other goods– Absolute demand is derived from relative demand

and price index

• Two Level Budgeting Model imposes seperability between consumption of electricity and other commodities– Decides how much to spend on electricity and all

other goods

Derivation of Time of Day Model

• Distribution of relative load across households more stable than consumption levels– Two level budgeting model

• Reduces complication of controls (due to declining block tariff)• Requires linear homogeneity

• Considers household electricity consumption in a representative day.– Appliance mix is assumed exogenous, therefore no effect on

consumption patterns– Seasonality ignored– Interday effects ignored– Rates do not change with consumption only time of day.

Derivation of time of day model

• Assume:– Utility determined by vector of electricity

consumption at different times of day, and consumption of all other goods (x0)

– HH decides how much to spend on electricity and how much to spend on other commodities

– Total consumption then depends on price of electricity

Derivation of Time of Day Model

• Divide day into series of 15 minute periods– Vector of electricity rates with a price

corresponding to each period– Indirect utility function:

– Utility function from with assumptions:

– Utility function for electricity• Where , total electricity expendature

Derivation

• Indirect utility function

• Use Roy’s identity, linear homogeneity and Eulers theorem to get demand function

Derivation

• Proportion of total consumption in base period

• Consumption in any period relative to base period is

Derivation

• Taylor expansion gives

• Full accounting gives single period cons. func

Derivation

• Price index to use in consumption function determined by total electricity consumption and total consumption.

Derivation

• Used to derive demand function for total daily demand:

• Pre-experimental period, price constant on all periods, but is a function of monthly demand due to declining block rate structure.

Estimation

• Sample of 150 households used– 20 not used to estimate, but used to test

• Two levels of demand estimated– Estimated with daily data using demand equation– Estimated price index (post exp) or insturmental

variables (pre exp) used to estimate consumption equation

Problems

• Not a random sample– Chosen based on an endogenous variable

• Residuals surely correlated with RHS vars• Inverse sample weights (WLS) used to correct for this issue

– Estimates not efficient– Downward biased standard errors

– Voluntary participation • Unobserved attributes determine participation, cannot be

checked• Aigner and Hausman (1978) find that elasticity lower when

not subject to voluntary choice, thus here, elasticity may be overstated.

Estimation

• First level– Hourly demand equations– Estimated from 4 periods, weekdays and

weekends during 2 winter months and 2 summer months, corresponding to system peak demand periods.

– 11pm – 7am is base period

Results

• Reported results for 3 representative periods, peak, off peak and intermediate

• Coefficients from demand equation are all relative to base periods– Not accounting for long run substitution

Findings

• Appliance coefficients have correct sign– Range / clothes dryer should be positive, since they will be

operated during the day• Range use lower when facing peak pricing, moved to intermediate

period

– Freezer should have negative coefficient since it operates all the time, this increases the base period demand for the household.

• Heater has spillover effects– Households used less heat energy during peak periods, lower

demand spills over to non peak periods because people don’t readjust the thermostat

Findings

• Elasticities are negative – 9-11 am more elastic than 5-7 pm– Intermediate period elasticities are lowest near

peak periods– Day of the week variables suggest demand

changes occur more during the day and less between days

• First state equation is reasonable

Price index estimate

• Compared pre experimental rates to post experimental price index– Treat declining block rate structure as linear two

part tariff• Lump sum $6.46, marginal price is 3.54 cents / kwh

– Experimental period prices come from price index equation, using estimated coefficients from demand equation estimate• Winter weekdays: 3.89 cents / kwh• High: 4.66 cents/kwh; Low: 3.25 cents/kwh

Price Index Estimate

• Comparing experimental period to pre experimental period:– Households faced higher marginal cost by ~16%

but no lump sum payment during experiment– Decrease in consumption did occur• 1% from previous year and 5% decline from controls

– Increase in marginal prices outweighs the removal of lump sum payment

Estimation

• Second Level– Daily demand equations– 4 demand estimates

• Weekday / weekend, winter / summer

– Less precise than first level estimates– Coefficients generally have correct total effect

• Presence of electric heat adds consumption during winter• 1% rise in MP leads to reduction of weekday consumption by

~4.3%• Freezer adds to consumption, but elasticity is not significant,

indicating non-discretionary nature of this appliance

Validation• Used 20 households from original sample to

test predictive capacity of model

• Forecasts do well in peak pricing periods forecasting relative load, though not significant

• Large forecast errors for other periods, insignificant results, less important

Validation

• When aggregated by period, forecasts appear accurate– Peak usage in validation group occurs in non-peak

pricing period. Model correctly predicts this

Validation

• Next, use forecasted price indices to forecast demand– Over prediction for low demand customers– Under prediction for high demand customers– Even when authors correct for sample issues,

results are not significant– Second level (daily demand) not as reliable as first

level (hourly demand) equations

Validation

• Load forecasts better in peak than non peak periods– Smaller errors in peak periods forecast load

reasonably well