Industrial response to electricity real-time prices: short run and long run.

Schwarz, Peter M. ; Taylor, Thomas N. ; Birmingham, Matthew 等

I. INTRODUCTION

With the emergence of competition in the electric industry, wholesale prices are proving highly volatile. During the summers of 1998 and 1999, prices in the Midwest soared to $7000 or more per megawatt-hour compared to a typical summer price of $30- $50. One factor contributing to this volatility is that relatively few retail customers pay real-time prices (RTPs) that vary with changing supply and demand conditions. As a result, retail use is not deterred by the spikes in wholesale price, exacerbating wholesale volatility. Without providing an opportunity for retail response, there will be pressure on policy makers to curb wholesale volatility with such measures as the price caps approved by the Federal Energy Regulatory Commission in New York and California as a result of high electricity prices during 2000.

Increasingly, electric utilities now offer their largest industrial and commercial customers hourly prices that vary with changes in real-time supply and demand. (1) The purpose of this article is to estimate and evaluate the short-run and long-run factors that influence customer price response to RTPs.

To date, there have been relatively few studies of response to RTP rates, and those have been largely limited to short-run estimates. This study advances the research by evaluating how customer response to RTP changes with experience on the rates. In addition, there is consideration of the sensitivity of price response to changes in the levels of both prices and temperatures. Finally, there is an examination of customer characteristics, particularly on-site generation, that provide flexibility in responding to price changes. The data include 110 industrial customers served by Duke Power, a division of Duke Energy Corporation. Some of the participating customers have been on RTP rates for as long as six years.

Utilities need information on short-run response to know the magnitude of demand relief that price responsiveness may provide when capacity is tight, as well as to determine unit commitment and spinning reserve. They need long-run response to plan additions to capacity. Knowledge of individual customer response can help utilities target those customers who are most likely to opt for RTP. Responsive customers may include those with interruptible (batch) production processes and with on-site generation. In addition to utilities, policy makers need to know price response to determine the role of RTP in achieving efficient electric resource use. (2)

This study adopts the modeling and estimation procedures set forth by Herriges et al. (1993) and King and Shatrawka (1994). Both employed a nested constant elasticity of substitution (CES) functional form. Herriges et al. (1993) estimated response to an RTP rate offered by the Niagara Mohawk Company. At the time, there were only 15 customers with usable data. One strength of the study was the availability of a control group on non-RTP rates. King and Shatrawka (1994) employed a much larger sample of 150 customers served by the Midlands Electric Company (Great Britain) but did not have a control group. Both studies provided shortrun estimates only.

Patrick and Wolak (1997) also used Midlands data to estimate demand elasticities from a Generalized McFadden functional form. Though their findings appear to be consistent with the other studies, it is difficult to compare their results directly given the differing functional form. (3) Their study used several years of data and did include temperature but provided only short-run estimates. Additionally, there were no consistent findings regarding temperature. Their study proposed the analysis of customer learning for future work. (4)

O'Sheasy (1997) and Gupta and Danielson (1998) have raised the possibility that price response depends on the level of price. O'Sheasy (1997) suggests a demand curve with vertical segments over certain price ranges. For example, a company with a batch production process might shut down production above a certain price threshold. Gupta and Danielson (1998) provide modest evidence that customers with self-generation respond significantly to prices only above a threshold point, above which self-generation becomes economical. (5)

Our overall results show elasticities to be somewhat smaller than those of Herriges et al. (1993) and King and Shatrawka (1994). We find statistically significant estimates on the order of 0.04, whereas those studies obtained estimates of 0.1 for both intra and interday elasticities. As in those studies, we find that only some individual customers respond significantly to price. The significant aggregate elasticity is due to only a subset of customers. (6)

On closer inspection, however, we find that some customers reduce their loads dramatically, by as much as 98%, and contribute to a substantial decrease in system peak load. A finding of relatively low estimates for elasticities can mask this response. RTPs can easily increase tenfold over the course of a summer day. With price increases of this magnitude, a small elasticity can be indicative of a relatively large reduction in load. Our other results show that elasticities are lower at high temperatures. However, with overall demand at a high level, absolute reductions associated with price are at their greatest. RTPs therefore promise the largest demand reductions when they are most needed. We also find that response increases with experience on the rates. Finally, we detect a price threshold effect for customers who have generators and also for those who utilize arc furnaces in their production processes. These customers show above-average price responses when price is high enough to justify on-site generati on or delaying production.

II. OVERVIEW OF DATA

Duke Power's RTP rate program has grown from 12 participants in 1994 to 100 in 1999. During this time, ten customers dropped out. The data collected for the study extends over the years 1994 through 1999 and encompasses the four summer months of June, July, August, and September. Duke calls its RTP tariff the HP (hourly pricing) rate and offers it to industrial customers with a demand of at least one megawatt (mW). Table 1 provides an overview of the Duke HP participants. It displays the number of customers associated with each industry classification represented in the HP population.

Table 2 displays the pattern of prices and temperatures faced by Duke's customers for six summers from 1994 to 1999. (7) Most large industrial customers in the sample purchase their electricity from the transmission grid. Approximately 15 customers who receive the electricity at a lower voltage are on the distribution system and pay the higher distribution price. The price includes an energy charge, as well as rationing charges that apply when demand is expected to approach system capacity for generation, transmission, and distribution. When a day-ahead forecast predicts that the system load will be within 10% of the annual forecasted system peak during any hour, the utility applies the transmission and distribution rationing charges for that hour. Such conditions usually apply during extremely hot weather. A generation rationing charge applies during any hour of the next day when it is anticipated that a combustion turbine (peaking unit) will operate. These rationing charges cause prices for transmission ser ved customers to increase by as much as $0.23 and by as much as $0.33 for distribution-served customers.

Transmission prices during the six-year period reached levels between $0.25 and $0.30, and distribution prices were as high as $0.35 to $0.40 per kWh. Though maximum price has fluctuated, average price has risen in each year.

Figure 1 displays load curves for the most recent summer in the study. Specifically, it depicts the aggregate response of all customers at various price levels. Each curve shows the average hourly load for days when the maximum price was within the indicated range. (8)

The figure indicates that customers do reduce load in response to higher prices. Use is higher during early hours, and more so on days of high prices. This higher use may be weather-related, or it may reflect load shifting.

III. THE ECONOMETRIC MODEL

The Basic Model

The econometric model employs a procedure set out by King and Shatrawka (1994), a variant on the approach of Herriges et al. (1993). This model incorporates a nested CBS functional form to characterize customer demand for electricity. Consumption within days is weakly separable from consumption across days. This functional form allows for flexibility in electricity use within the day in response to hourly price changes and between days in response to a daily price index. The estimating equation is

(1) ln([E.sub.dh]/[E.sup.g.sub.th] = [summation over (t)] [A.sub.t] - [[sigma].sub.H] ln([P.sub.dh]/[P.sup.g.sub.th])

-([[sigma].sub.D] - [[sigma].sub.H] ln([D.sub.d]/[D.sup.g]),

where [E.sub.dh] is electricity usage during hour h on day d, [P.sub.dh] is the electricity price during hour h on day d, and

ln([E.sup.g.sub.th]) = 1/[N.sub.t][summation over (4/m=1)][summation over (30/d[member of]t)]ln([E.sub.mdh])

and

ln([P.sup.g.sub.th]) = 1/[N.sub.t][summation over (4/m=1)][summation over (30/d[member of]t)]ln([P.sub.mdh])

are the logs of geometric means over the four months m of a season, and

ln([D.sub.d]/[D.sup.g]) = 1/2[summation over (24/h=1)]([w.sub.dh] + [w.sub.th])ln([P.sub.dh]/[P.sup.g.sub.th])

is a daily price index where

[w.sub.dh] = [P.sub.dh][E.sub.dh]/[[summation over (24/k=1)][P.sub.dk][E.sub.dk]]

is the share of electricity expenditure during hour h of day d and

[w.sub.th] = 1/[N.sub.t][summation over (4/m=1)][summation over (30/d[member of]t)][w.sub.dh]

is the arithmetic mean of the expenditure share for hour h over all days of type t. (9) Let

E [equivalent to] ln([E.sub.dh]/[E.sup.g.sub.th])

A [equivalent to] [summation over (t)][A.sub.t]

P [equivalent to] ln([P.sub.dh]/[P.sup.g.sub.th])

D [equivalent to] ln([D.sub.d]/[D.sup.g]).

Substituting these expressions into (1) and rearranging terms yields our estimating model for [[sigma].sub.H] and [[sigma].sub.D]:

(2) E = A + [[sigma].sub.H](D - P) + [[sigma].sub.D](- D)

The a priori expectation for the signs of both estimated coefficients is positive. As in King and Shatrawka (1994), the additional variable A represents a vector of binary variables used to control for any influence by days of the week. There is an additional binary variable for the first two-week period of July, a traditional vacation period for industry in the area.

The data set consists of a time series of observations for each customer over the summer months, beginning with the first summer that the customer was billed on the HP rate. There are estimates for individual customers and also for the aggregate of all customers. The aggregate is the sum of individual customer loads. At the aggregate level, there are separate estimates for customers served from the transmission and distribution systems. These groups faced substantially different prices as was shown in Table 1. We follow Herriges et al. (1993) in assuming an error term with a first-order autoregressive process. (10)

Controlling for Temperature

To avoid producing biased results, Herriges et al. (1993) point out that it is necessary to control for weather conditions. (11) Accordingly, the model is augmented with temperature as follows:

(3) [[sigma].sub.H] = [b.sub.H]+[m.sub.H][T.sub.H] and [[sigma].sub.D] = [b.sub.D]+[m.sub.D][T.sub.D],

where [T.sub.H] is hourly temperature in the region and [T.sub.D] is average daily temperature in the region. Substituting (3) into (2) and arranging terms:

(4) E = A+([b.sub.H])(D-P)+([m.sub.H])[T.sub.H](D-P)+[b.sub.D](-D)+[m.sub.D][ T.sub.D](-D).

This equation is also estimated with a first-order autoregressive process for the error term. (12)

IV. FINDINGS

We begin by presenting aggregate elasticities using data from the summer of 1999, the most recent summer for which data are available. Next, we disaggregate to consider individual customer response. We then examine the effects of temperature, the level of prices, and the length of time customers have been on the rate.

Aggregate Results

Table 3 portrays system-wide estimated elasticities segmented by customers served from the distribution and transmission systems.

Estimates are from equation (2). Results are similar for both groups. All elasticities are significant at the 1% level. (13) These estimates are smaller than those found in Herriges et al. (1993) and in King and Shatrawka (1994) who found elasticities on the order of 0.1. A second difference is that we find hourly elasticities to be slightly larger than daily elasticities; those studies found daily elasticities to be slightly larger. (14)

We use these estimates to obtain predicted loads for the aggregate of RTP customers at the highest and lowest price ranges. (15) The aggregate is the sum of transmission and distribution system loads. By comparing aggregate loads at these prices, we can obtain an estimate of capacity savings associated with customer response to the RTP rate.

Figure 2 shows predicted consumption. With maximum demands typically placed on the utility system during the late afternoon hours, it appears that load reductions are about 70 mW. At Duke's most recently filed avoided cost of $39 per kW per year for generating capacity, this reduction would translate to a long-term savings of some $2.7 million per year. In recent years, there have been wholesale prices as high as $7000 per mWh in Midwest wholesale markets. Even with price caps as low as the $150 recently imposed in California, short-term savings associated with the response to RTP can be substantial.

Individual Customer Response

Table 4 contains a summary of individual customer response using data from each customer's complete history on Duke's hourly pricing program. The table includes all customers with positive and significant elasticities. Elasticities are estimated using equations (3) and (4) and the assumption that temperature is 75[degrees]F, the average temperature in the region.

Several characteristics are notable among the customers showing the largest responses. Of the top 12 responders, 7 have generators that may be dispatched against the hourly price, and these customers tend to show both daily and hourly response. (16)

A second factor that appears to favor a large price response is a discrete production process that allows adjustments when prices are high. In addition to owning a generator, the paper manufacturer also has large grinders for making pulp. Grinders can be operated in discrete increments to avoid hours with high prices, producing pulp to be stored in inventory for use throughout the week.

Two of the top responders that do not self-generate electricity are electrode manufacturing and steel. They utilize processes involving arc furnaces that can be scheduled to be price-responsive with day-ahead notification of prices. The feed mill is able to schedule its grinding operation around high-priced hours, and the natural gas storage facility performs pumping operations to avoid high-priced hours.

Even though the estimates in Table 4 are for individual customers, it is possible to draw some information for industry groups. Textiles, chemical fiber, and pipelines have multiple representatives in the Duke data, and each group shows relatively low response.

There are 51 textile customers among the 110 total customers who participated at some time during the period. Excluding the customer with the generator, six have significant elasticities, four hourly and two daily. These significant elasticities tend to be relatively small. This finding of a lack of general responsiveness for textiles is consistent with the continuous nature of their production process, which is costly to interrupt or curtail. However, it is conceivable that those customers who are responding are finding discretionary loads in their plants, such as lighting, air conditioning, and compressors, that can be moderated when prices are high.

A similar conclusion may be assumed for the chemical fiber group, which has 13 customers in this population. Again setting aside the customer with a generator, only 3 of the remaining 12 have significant elasticities, and these are relatively small.

The pipeline group, with nine representatives, is the only other industry with substantial representation. Only one pipeline delivery has a significant elasticity. The operation of individual deliveries must be coordinated with others along the pipeline to maintain proper flow volumes and pressures.

Most pipelines cross through the service areas of many utilities, each with different rate structures, making it difficult to respond to price signals.

Temperature

We are able to utilize data for the summer of 1999 to investigate the extent to which the level of temperature influences response to hourly prices. Table 5 contains the estimated coefficients for the temperature model presented in equation (4). (17)

The estimates of primary interest are [m.sub.H] and [m.sub.D], the coefficients of the temperature terms in equation (4). A negative coefficient indicates that elasticity decreases with an increase in temperature. Although the coefficients are negative, they approach significance only for distribution customers.

Further analysis indicates that this significant response is attributable to a university included in that population. With a hypothesis that residential air conditioning usage diminishes price response as temperatures increase, equation (4) was estimated for the university alone. The temperature coefficient for the hourly elasticity was found to be negative and significant. A similar analysis was performed for the single university customer in the transmission-served population. In this case, the temperature coefficients for both hourly and daily elasticities were negative and significant. There is some evidence, then, for the hypothesis that higher temperatures do reduce customer elasticities through the demand for electricity associated with air conditioning usage.

We also examine the effect of temperature from a different perspective. Those days with an average temperature of 80[degrees]F or higher are defined as hot days, and equation (2) is modified as follows:

E = A + ([[sigma].sub.H] + [[delta].sub.H,HOT]HOT)(D - P) + ([[sigma].sub.D] + [[delta].sub.D,HOT]HOT)(- D),

which can be rewritten

(5) E = A + [[sigma].sub.H](D - P) + [[delta].sub.H,HOT]HOT(D - P) + [[sigma].sub.D](- D) + [[delta].sub.D,HOT]HOT(- D),

where the binary variable HOT is equal to one when average daily temperature is 80[degrees]F or higher and zero otherwise. The signs of the interaction parameters ([[delta].sub.H,HOT], [[delta].sub.D,HOT]) indicate the effect of hot days on customer response.

Table 6 results show that coefficients of the interaction terms are negative for both transmission and distribution customers and significant for the transmission customers. (18) These results provide evidence that response to prices may diminish somewhat in hot weather.

Another aspect to the effect of temperature is customer response during a hot spell, a prolonged period of hot weather. During the initial days of a hot spell, the customer may suppress use, expecting to increase production when temperatures cool and, accordingly, when prices drop. As the hot spell continues, however, the customer is more likely to run into production deadlines and less likely to continue to delay production. Therefore, elasticity may decrease as the hot spell continues.

To test this hypothesis, the summer 1999 data offered 71 relatively hot days with average temperature of at least 77.5[degrees]F. A binary variable identifies 25 late-in-a-hot-spell days defined as the period July 26-August 19 when the Carolinas were more than a week into a hot spell. Equation (5) was estimated with this binary variable and results are in Table 7.

Generally, the signs of the interaction terms are mixed and coefficients are not statistically significant. Results do not change when the analysis is repeated with the hot spell defined as August 1-19.

Though there is some indication that estimates are reduced by hot weather, there is little statistically significant evidence of such an effect in general for all customers in our population. Nor is there consistent evidence that customers alter their response during a hot spell.

Price Level

In addition to estimating elasticities over the entire price range, we consider the possibility that price response depends on the level of price. The nested CES function imposes constant elasticity. Gupta and Danielson (1998) provide evidence for a small sample of customers that response increases above a threshold price where on-site generation becomes cost-effective.

To examine this hypothesis, we develop a series of data sets by successively deleting those days when prices exceed a given level; that is, the first data set includes all data, the second eliminates all days when prices exceed $0.25, the third eliminates days where price exceeds $0.20, and so on. For a given customer, the data include all years that the customer was on the RTP rate. For each data set, we estimate equation (4). Results are in Table 8.

Customers with generation do indeed respond to price, and the evidence suggests a threshold effect for hourly response at a price above $0.05. This result is consistent with the findings of Gupta and Danielson (1998). A threshold appears to exist for the daily response also, but at a slightly higher price, based on the number of significant responses. Note that the substitution elasticities for customers with generation are considerably greater than the aggregate estimates of Tables 3 and 5. Also, customers with generation demonstrate a daily response that is larger than hourly response, a result that is in contrast with our aggregate estimates but in agreement with the studies of other researchers.

We look separately at two customers who utilize arc furnaces in their production processes. They produce electrodes and steel, and, as indicated in Table 4, they demonstrate a significant response to hourly prices. The results of Table 8 suggest an hourly threshold effect above $0.11. In addition to a threshold effect, these customers show a larger response at higher price ranges. With a discrete production process, these customers can transfer greater amounts of load the higher the price. Although overall results showed a tendency toward smaller elasticities at high temperatures, exactly when price is likely to be high, customers with discrete production processes are likely to be able to increase their response at such times.

Finally, we analyzed evidence for the 67 customers who do not use generation or arc furnaces. These results show relatively small price response and provide no evidence of a threshold effect.

Length of Time on the RTP Rate

In the long run, it is expected that price response increases as customers gain experience with real-time pricing. We evaluate the influence of experience using a two-stage procedure set out in Taylor and Schwarz (1990). (19) The first stage utilizes equation (2) to estimate elasticities for each customer, for each month on the rate. The second stage regresses these estimates on the number of years that the customer has been on the hourly pricing program. The stage 2 estimating equation is:

(6) [[sigma].sup.H.sub.jmy] = [a.sup.H] + [b.sup.H][Y.sub.jmy] + [summation over (i)] [c.sup.H.sub.i] [X.sub.jmy],

where [[sigma].sup.H.sub.jmy] is the hourly elasticity calculated for customer j in month m of year y, and [Y.sub.jmy] is the number of years that customer j had been on the HP program as of month m of year y. Similarly,

(7) [[sigma].sup.D.sub.jmy] = [a.sup.D] + [b.sup.D][Y.sub.jmy] + [summation over (i)] [C.sup.D.sub.i] [X.sub.jmy]

The variables X control for other effects including the presence of an on-site generator, an arc furnace, the levels of price and temperature as measured by average price and temperature for the summer, and customer size measured in m W. Binary variables are included for the summer months. As in short-run results, it is possible that changes in price or temperature conditions from one year to the next contribute to a change in price responsiveness. It is also possible that the representation of responsive customers, particular those with on-site generation or arc furnaces, has changed over time.

Table 9 shows the estimated coefficients. The initial ordinary least squares estimates were subject to both heteroskedasticity and autocorrelation. The standard errors are corrected for heteroskedasticity using White's (1980) approach. This method corrects for heteroskedasticity but not autocorrelation. The second set of estimated coefficients adjusts for autocorrelation but not heteroskedasticity. The key finding is unaffected: Elasticities increase with experience. (20)

The stage 1 estimates of both [[sigma].sub.H] and [[sigma].sub.D] are positively and significantly related to the variable "years on HP" suggesting that experience with the hourly pricing rate leads to increases in customer response. As experience with the rate increases, it is reasonable to believe that customers discover options to avoid high prices.

With each additional year of experience, estimated elasticities increase on the order of 0.03. The positive and significant coefficients for variables for generator and arc furnace indicate a considerably higher elasticity associated with those technologies. Monthly prices, temperatures, and binary variables do not have significant coefficients. (21)

Although the inclusion of variables for generation, arc furnaces, etc., controls for the responsiveness associated with these technologies and effects over time, it is possible that the results could be biased by customer participation patterns. For example, those firms that are best able to respond to RTP tariffs may have been the first to opt for the rate. Similarly, those who dropped off may have had less ability to respond to the rate. The combined effect could lead to an increasing elasticity for the entire group that might be interpreted as an increasing response by individual customers over time.

We note that during the initial years of the HP rate program, Duke accepted a diverse customer mix, including a number of customers with technologies that may limit response. For example, 10 of the 21 customers who participated at any time during the first two years were either textile or chemical fiber and did not have self-generation. As shown in Table 4, these customers demonstrated relatively low price responsiveness.

The criteria for participation in the program are (1) an ability to respond to hourly prices, or (2) the potential to add new production that might not be possible at the higher average levels of other rates. Those with an ability to respond have always been encouraged to participate. In addition, the relatively low level of prices during many hours has been attractive to some seeking locations for additional production facilities, even though their price responsiveness may be limited.

Looking at the first two years of the program, there were 17 customers who began participation and remained on the rate until the end of 1999. Eight of those customers demonstrated positive and significant elasticities over the period, including five of the seven firms with an hourly elasticity of substitution greater than 0.2. Of those eight, four have some self-generation capabilities and one has an arc furnace, effects that are controlled for in equations (6) and (7).

Of the ten that have discontinued the rate, the decision does not appear to be due to a lack of price responsiveness. Three customers dropped off the rate as a result of plant closings, three others dropped off due to changes in ownership, three did not realize additional production as planned, and one changed its operation, leading to a reduction in electrical load. Of these ten, two were actually price responsive, ranking 8th and 13th in Table 4.

To further investigate the potential for self-selection bias, we analyzed the 17 customers who were participating in summer 1994 or 1995 and remained on the rate through 1999. By studying only these customers, we remove the possibility that participation patterns over time might influence results. Of the 17 early participants, 8 were identified in Table 4 as responders, including 5 of the 7 firms with an hourly elasticity greater than 0.2.

We analyzed the data in panel form, with each customer weighted equally, controlling for experience by substituting [[sigma].sub.H] = [b.sub.H] + [m.sub.H] Y and [[sigma].sub.D] = [b.sub.D] + [m.sub.D] Y into equation (2). The results of the estimate are as follows:

[[sigma].sub.H] = 0.1911 + 0.0108 Y

[[sigma].sub.D] = 0.1378 + 0.0111 Y.

Data are from 1995-99 only, and Y = 0 for 1995, Y = 1 for 1996, and so on. All coefficients are significant at greater than 0.001 levels. A customer with an elasticity just under 0.2 in 1995 would show an increase to 0.25 by 1999. From our knowledge of the participation criteria, the reasons for customers dropping off the rate, and our analysis of the responsiveness of early participants, there is persuasive evidence that price response increases with experience on the rate.

V. SUMMARY AND CONCLUSIONS

With the emergence of competition in the electric industry, utilities are expanding their use of real-time rates. There have been several studies of customer real-time elasticities, typically using one year of data. This study has customer data for as many as six years, from 1994 to 1999.

The data cover the four summer months of June, July, August, and September. In all, 110 customers participated. There are approximately 20 industries represented, with the largest representation in textiles, followed by pipelines, chemical fibers, and paper.

Our results are directly comparable to real-time estimates of Herriges et al. (1993) and King and Shatrawka (1994). We adopt their choice of a nested CES function to obtain intraday and interday price responses.

We go beyond existing studies in several ways. First, we consider the possibility that elasticity may be different for different price ranges, evaluating evidence that a threshold effect may exist. Second, we consider temperature, and the possibility that elasticity may vary in different temperature ranges. Third, we consider the length of time that the customer has been billed under RTPs to determine if response increases with experience. Finally, we investigate the characteristics of customers that affect their ability to respond to prices.

RTP customers as a group substantially reduce their load during high-priced hours. Perhaps surprisingly, this large response is not necessarily associated with a large elasticity. Prices vary by five- or tenfold on a hot summer day. Absolute quantity changes are largest at peak hours, but these changes are dwarfed by percentage changes in price that may be as large as 500% on a very hot day.

We obtain within-day elasticities on the order of 0.04, slightly smaller than the 0.1 figure obtained in earlier studies. As in those earlier studies, we also find that only a subset of customers respond significantly to hourly rates. Those who do respond are likely to have their own generators or have a production process that can be interrupted.

Gupta and Danielson (1998) suggest that customers with generation display a threshold effect, responding significantly to realtime rates above the price where it becomes worthwhile to use their generators. Consistent with their study, we find a threshold effect for these customers. We also find a threshold effect plus a price effect whereby customers who use arc furnaces in their operation delay more of their production as price increases.

Our analysis of temperature effects provides modest evidence that elasticity decreases under hot conditions. However, the very high prices at these temperatures bring about substantial reductions in load.

Finally, we evaluate long-run response by testing for the effect of experience on customer response to the RTP rate. Results show an increase in response as the length of time on the rate increases. Customers who signed up early, such as in 1994 and 1995, show a larger than average response that increases over time.

Overall, our results show that RTPs can dramatically reduce load during the hours where load relief is most desirable. Customers with on-site generators or with the ability to delay production are particularly likely to reduce their use during the highestpriced hours.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

TABLE 1

Hourly Pricing Customers by Industry Classification, 1999

 Number of
Industry Classification Customers

Air separation 1
Aluminum metal 1
Electrodes 1
Auto equipment, mfg., parts 4
Battery manufacturing 2
Brewery 1
Chemical, chemical fiber 12
Computer manufacturing 1
Fiber cable 1
Food product 1
Glass fabrics, tempering 3
Metal fabrication, products 2
Natural gas storage 1
Paper, paper products 5
Pipeline 9
Plastics 3
Printing 1
Retail 1
Steel 1
Textiles 46
Tobacco 1
University 2

Note: Industry classifications as reported by Duke Power Company of
North Carolina.
TABLE 2

Prices and Temperatures for Customers on Hourly Pricing Rate

 Transmission Distribution
 North Carolina North Carolina
 ($/kWh) ($/kWh)
 Average Maximum Average Maximum

1994 0.028 0.277 N/A N/A
1995 0.030 0.231 N/A N/A
1996 0.031 0.250 0.037 0.354
1997 0.033 0.289 0.037 0.394
1998 0.037 0.305 0.042 0.409
1999 0.045 0.278 0.053 0.382

 System
 Temperature
 ([degrees]F)
 Average Maximum

1994 74.0 92.3
1995 75.1 97.7
1996 74.1 93.3
1997 74.1 95.0
1998 76.8 97.0
1999 75.3 99.0

Note: N/A means Not applicable.
TABLE 3

Aggregate Hourly and Daily Elasticities

Elasticity Standard Approx.
(Equation [1]) Estimate Error t-Ratio Prob>t

Distribution system
 Hourly ([[sigma].sub.H]) 0.0328 0.00119 27.6 0.0001
 Daily ([[sigma].sub.D]) 0.0282 0.00338 8.36 0.0001
Transmission system
 Hourly ([[sigma].sub.H]) 0.0466 0.00227 20.5 0.0001
 Daily ([[sigma].sub.D]) 0.0293 0.00547 5.35 0.0001
TABLE 4

Industrial Customer Hourly and Daily Elasticities


 Start
Industry System Date Generation

University Distribution 1995 Yes
Textiles Transmission 1994 Yes
Electrodes Transmission 1999 No

Paper Transmission 1994 Yes
Glass tempering Distribution 1996 No
Steel Transmission 1994 No
Chemical fiber Transmission 1994 Yes
Feed mill Distribution 1996-7 No
Brewery Transmission 1997 Yes
University Transmission 1997 Yes
Natural gas storage Transmission 1999 No
Food products Transmission 1997 Yes
Textiles Transmission 1997-8 No
Textiles Transmission 1999 No
Computer mfg. Transmission 1995 No
Pipeline Transmission 1998 No
Chemical fiber Distribution 1999 No
Chemical Transmission 1997 No
Textiles Transmission 1994 No
Textiles Transmission 1996 No
Metal fabric Transmission 1995 No
Chemical fiber Transmission 1994 No
Chemical fiber Transmission 1996 No
Textiles Transmission 1995 No
Textiles Transmission 1998 No

 Elasticities
 Hourly F Daily
Industry ([[sigma].sub.H]) statistic ([[sigma].sub.D])

University 1.484 *** 6853 1.366 ***
Textiles 0.851 *** 1944 0.852 ***
Electrodes 0.459 *** 155 0.327 ***
 0.161 *** 72.4 0.128 ***
Paper 0.390 *** 1925 0.286 ***
Glass tempering 0.364 *** 45.4 0.326 ***
Steel 0.294 *** 129 0.122 ***
Chemical fiber 0.202 *** 2201 0.216 ***
Feed mill 0.107 *** 7.43
Brewery 0.070 ** 116 0.008
University 0.059 *** 159 0.115 ***
Natural gas storage 0.053 ** 4.41 0.066
Food products 0.051 *** 193 0.061 ***
Textiles 0.024 1.68 0.076
Textiles 0.024 * 3.15 0.008
Computer mfg. 0.017 *** 113 0.017 ***
Pipeline 0.012 0.192 0.169 ***
Chemical fiber 0.012 *** 44.2
Chemical 0.009 *** 7.05 0.008
Textiles 0.009 *** 7.54
Textiles 0.009 ** 4.01
Metal fabric 0.009 * 3.05
Chemical fiber 0.008 *** 81.4 0.005 *
Chemical fiber 0.006 *** 12.3 0.010 *
Textiles 0.002 * 4.87
Textiles 0.010 ***

 Elasticitie
 s
 F
Industry statistic

University 2456
Textiles 273
Electrodes 25.1
 11.7
Paper 245
Glass tempering 12
Steel 19.6
Chemical fiber 453
Feed mill
Brewery 0.33
University 124
Natural gas storage 1.02
Food products 50.7
Textiles 3.69
Textiles 0.0416
Computer mfg. 22.1
Pipeline 6.618
Chemical fiber
Chemical 1.04
Textiles
Textiles
Metal fabric
Chemical fiber 3.24
Chemical fiber 6.5
Textiles
Textiles 12.7

Notes: Table entries are limited to those customers who show a positive
and significant response. A blank entry indicates that the elasticity is
less than 0.001. There is an F statistic, rather than a t statistic
because elasticity is a linear combination (equations [3] and [4]),
whereas elasticity was estimated directly in Table 4 (equation [2]). Two
sets of results are shown for the electrodes customer, who had a split
in service periods.

***, **, *, significant at 1%, 5%, and 10%, respectively.
TABLE 5

Effect of Temperature on Aggregate Hourly and Daily Elasticities

 Elasticity [sigma] = b + mT

Coefficient Estimate

Distribution customers
[b.sub.H] 0.0526
[m.sub.H] -0.000247
[[sigma].sub.H] (hourly elasticity) 0.0341
[b.sub.D] 0.0651
[m.sub.D] -0.000454
[[sigma].sub.D] (daily elasticity) 0.0311

Transmission customers
[b.sub.H] 0.0542
[m.sub.H] -0.000094
[[sigma].sub.H] (hourly elasticity) 0.0471
[b.sub.D] 0.0521
[m.sub.D] -0.000289
[[sigma].sub.D] (daily elasticity) 0.0304

 Elasticity
 [sigma] =
 b + mT
 Standard t Ratio or
Coefficient Error F statistic

Distribution customers
[b.sub.u] 0.01 5.226
[m.sub.H] 0.0001 -1.987
[[sigma].sub.H] (hourly elasticity) 630
[b.sub.D] 0.03 2.136
[m.sub.D] 0.0004 -1.17
[[sigma].sub.D] (daily elasticity) 72.1

Transmission customers
[b.sub.H] 0.02 2.773
[m.sub.H] 0.0002 -0.386
[[sigma].sub.H] (hourly elasticity) 345
[b.sub.D] 0.06 0.921
[m.sub.D] 0.0007 -0.397
[[sigma].sub.D] (daily elasticity) 26.8



Coefficient Prob > t or F

Distribution customers
[b.sub.u] 0.0001
[m.sub.H] 0.047
[[sigma].sub.H] (hourly elasticity) 0.0001
[b.sub.D] 0.0328
[m.sub.D] 0.242
[[sigma].sub.D] (daily elasticity) 0.0001

Transmission customers
[b.sub.H] 0.0056
[m.sub.H] 0.699
[[sigma].sub.H] (hourly elasticity) 0.0001
[b.sub.D] 0.3573
[m.sub.D] 0.6912
[[sigma].sub.D] (daily elasticity) 0.0001

Notes: Data for 1999 analyzed, [[sigma].sub.H] and [[sigma].sub.D]
estimated using Equation (4) at 75[degrees]F. F statistic reported for
[[sigma].sub.H] and [[sigma].sub.D].
TABLE 6

Influence of Hot Days on Customer Response

 Elasticity = [sigma] +
 [[delta].sub.HOT]HOT
Coefficient Estimate

Distribution customers
 [[sigma].sub.H] (hourly 0.0335
 elasticity)
 [[delta].sub.H,HOT] -0.0013
 [[sigma].sub.D] (daily elasticity) 0.0295
 [[delta].sub.D,HOT] -0.00311

Transmission customers
 [[sigma].sub.H] (hourly 0.0616
 elasticity)
 [[delta].sub.H,HOT] -0.0248
 [[sigma].sub.D] (Daily elasticity) 0.0484
 [[delta].sub.D,HOT] -0.0354

 Elasticity = [sigma]+
 [[delta].sub.HOT]HOT
Coefficient Standard Error t Ratio

Distribution customers
 [[sigma].sub.H] (hourly 0.00175 19.1
 elasticity)
 [[delta].sub.H,HOT] 0.00239 -0.545
 [[sigma].sub.D] (daily elasticity) 0.00388 7.61
 [[delta].sub.D,HOT] 0.00467 -0.665

Transmission customers
 [[sigma].sub.H] (hourly 0.00317 19.4
 elasticity)
 [[delta].sub.H,HOT] 0.00443 -5.598
 [[sigma].sub.D] (Daily elasticity) 0.0064 7.566
 [[delta].sub.D,HOT] 0.00871 -4.068


Coefficient Prob > t

Distribution customers
 [[sigma].sub.H] (hourly 0.0001
 elasticity)
 [[delta].sub.H,HOT] 0.586
 [[sigma].sub.D] (daily elasticity) 0.0001
 [[delta].sub.D,HOT] 0.5061

Transmission customers
 [[sigma].sub.H] (hourly 0.0001
 elasticity)
 [[delta].sub.H,HOT] 0.0001
 [[sigma].sub.D] (Daily elasticity) 0.0001
 [[delta].sub.D,HOT] 0.0001

Notes: [[delta].sub.H,HOT] and [[delta].sub.D,HOT] are coefficients of
interaction terms with the hot days binary variable HOT. Hot days are
defined as days when average daily temperature is 80[degrees]F or
higher.
TABLE 7

Influence of a Hot Spell on Customer Response

 Elasticity = [sigma]
 +[[delta].sub.SPELL]SPELL
Coefficient Estimate

Distribution customers
 [[sigma].sub.H] (hourly 0.033
 elasticity)
 [[delta].sub.H,SPELL] 0.00123
 [[sigma].sub.D](daily elasticity) 0.00454
 [[delta].sub.D,SPELL] 0.0225

Transmission customers
 [[sigma].sub.H] (hourly 0.0344
 elasticity)
 [[delta].sub.H,SPELL] -0.012
 [[sigma].sub.D](daily elasticity) 0.0116
 [[delta].sub.D,SPELL] -0.00381

 Elasticity = [sigma]
 +[[delta].sub.SPELL]SPELL
Coefficient Standard Error t Ratio

Distribution customers
 [[sigma].sub.H] (hourly 0.0028 11.8
 elasticity)
 [[delta].sub.H,SPELL] 0.0037 0.332
 [[sigma].sub.D](daily elasticity) 0.0069 0.66
 [[delta].sub.D,SPELL] 0.0118 1.91

Transmission customers
 [[sigma].sub.H] (hourly 0.0047 7.38
 elasticity)
 [[delta].sub.H,SPELL] 0.0063 -1.915
 [[sigma].sub.D](daily elasticity) 0.009 1.29
 [[delta].sub.D,SPELL] 0.0145 -0.263


Coefficient Prob > t

Distribution customers
 [[sigma].sub.H] (hourly 0.0001
 elasticity)
 [[delta].sub.H,SPELL] 0.74
 [[sigma].sub.D](daily elasticity) 0.5091
 [[delta].sub.D,SPELL] 0.0564

Transmission customers
 [[sigma].sub.H] (hourly 0.0001
 elasticity)
 [[delta].sub.H,SPELL] 0.0558
 [[sigma].sub.D](daily elasticity) 0.1964
 [[delta].sub.D,SPELL] 0.7923

Notes: The hot spell is defined as the period from July 26,1999, to
August 19, 1999, and a binary variable SPELL is set equal to one for
data from those days. [[delta].sub.H,SPELL] and [[delta].sub.D,SPELL]
are coefficients of interaction terms with the hot spell binary
variable.
TABLE 8

Average of Customer Elasticities by Level of Price

 Customers with Generation
 Elasticities
Price Hourly Number Daily
Range ([[sigma].sub.H]) Significant ([[sigma].sub.D])

<$0.25 0.265 7 0.319
<$0.20 0.26 7 0.321
<$0.15 0.272 6 0.401
<$0.13 0.283 6 0.422
<$0.11 0.281 5 0.388
<$0.09 0.323 5 0.404
<$0.07 0.258 4 0.353
<$0.05 -0.02 1 0.171
<$0.03 -0.063 1 0.107

 Customers Customers with Arc Furnaces
 with
 Generation
 Elasticities Elasticities
Price Number Hourly Number
Range Significant ([[sigma].sub.H]) Significant

<$0.25 7 0.423 2
<$0.20 6 0.328 2
<$0.15 6 0.213 2
<$0.13 6 0.184 1
<$0.11 6 0.06 0
<$0.09 5 0.054 0
<$0.07 2 -0.018 0
<$0.05 1 -0.011 0
<$0.03 1 0.085 0

 Customers with Arc Furnaces
 Elasticities Other
Price Daily Number Hourly
Range ([[sigma].sub.D]) Significant ([[sigma].sub.H])

<$0.25 0.157 0 0.009
<$0.20 0.154 0 0.009
<$0.15 0.196 0 0.008
<$0.13 0.565 0 0.001
<$0.11 -0.146 0 0.002
<$0.09 0.292 0 0.005
<$0.07 0.143 0 0.022
<$0.05 0.137 0 0
<$0.03 -0.277 0 -0.024

Notes: Estimates developed from equation (4) assume a temperature of
90[degrees]F. To assure that there are a reasonable number of
observations in each price range, customers with only one summer on the
rate are not included. There are 8 customers with generation, 2 with are
furnaces, and 67 that do not have generation.
TABLE 9

Change in Customer Response with Time on the Rate

 Ordinary Least Squares
Coefficient Estimate White SE t Ratio

[[sigma].sub.H]
(hourly elasticity)
Constant 0.5927 0.4431 1.3376
Years on HP 0.0274 0.0053 5.1446
Generator 0.3659 0.0456 8.0151
Arc furnace 0.2813 0.0534 5.2730
mW -0.0002 0.0005 -0.3630
Price level 0.3509 0.7115 0.4932
Temp level -0.0085 0.0060 -1.4146
JULY 0.0016 0.0177 0.0913
AUGUST 0.0138 0.0172 0.8014
SEPT -0.0047 0.0157 -0.2967
 Chi-sq (White test) = 161.75
 P (White test) = 0.0001

[[sigma].sub.D]
(daily elasticity)
Constant -0.5395 0.5234 -1.0308
Years on HP 0.0305 0.0059 5.2134
Generator 0.3377 0.0456 7.4020
Arc furnace 0.1712 0.0437 3.9228
mW -0.0001 0.0005 -0.1719
Price level -0.8774 0.8543 -1.0270
Temp level 0.0071 0.0070 1.0151
JULY -0.0260 0.0186 -1.3989
AUGUST 0.0118 0.0190 0.6195
SEPT -0.0349 0.0190 -1.8409
 Chi-sq (White test) = 164.37
 P (White test) = 0.0001

 Ordinary Least Squares Yule-Walker
Coefficient Prob > t Estimate

[[sigma].sub.H]
(hourly elasticity)
Constant 0.1813 -0.0238
Years on HP 0.0001 0.0255
Generator 0.0001 0.2692
Arc furnace 0.0001 0.2942
mW 0.7167 0.0007
Price level 0.6220 -0.4434
Temp level 0.1575 0.0002
JULY 0.9272 0.0020
AUGUST 0.4231 0.0150
SEPT 0.7667 -0.0013
 Durbin-Watson statistic = 0.7899
 P (Durbin-Watson) = 0.0001

[[sigma].sub.D]
(daily elasticity)
Constant 0.3029 -0.7660
Years on HP 0.0001 0.0310
Generator 0.0001 0.3207
Arc furnace 0.0001 0.1874
mW 0.8635 0.0001
Price level 0.3047 -0.9699
Temp level 0.3103 0.0102
JULY 0.1621 -0.0269
AUGUST 0.5358 0.0123
SEPT 0.0659 -0.0311
 Durbin-Watson statistic = 1.444
 P (Durbin-Watson) = 0.0001

 Yule-Walke
 r
Coefficient Prob > t

[[sigma].sub.H]
(hourly elasticity)
Constant 0.9592
Years on HP 0.0001
Generator 0.0001
Arc furnace 0.0001
mW 0.1341
Price level 0.7240
Temp level 0.9755
JULY 0.8397
AUGUST 0.1758
SEPT 0.8923



[[sigma].sub.D]
(daily elasticity)
Constant 0.1843
Years on HP 0.0001
Generator 0.0001
Arc furnace 0.0003
mW 0.8634
Price level 0.4536
Temp level 0.1906
JULY 0.0975
AUGUST 0.4791
SEPT 0.0542



Notes: Linear regression results are from stage 2 estimating equation.
The White test indicates heteroskedasticity. The Durbin- Watson test
indicates autocorrelation. Yule-Walker estimates are from an AR(1)
model.

(1.) Generally, there is advance notice of prices by the end of the previous business day. See Taylor and Schwarz (2000).

(2.) These motivations are given in O'Sheasy et al. (1996, 6-1, 6-2).

(3.) Other studies that have used different functional forms are Zarnikau (1990), Aubin et al. (1995), Woo et al. (1996), and Kim (1998).

(4.) They indicated that allowing for a longer time horizon for substitution was a computationally burdensome but not insurmountable problem. They would also have to collect more detailed information on customer characteristics. See pp. 7-1 and 7-2 of their report.

(5.) Gupta and Danielson (1998) examine data from only four customers and find three of the four respond only above a threshold price.

(6.) To date, utilities have offered RTP rates on an optional basis. To infer the response if rates were mandatory, it would be necessary to adjust for the self-selection aspect of optional rates. Kim (1998) provides methodology and estimates that correct for self-selection.

(7.) The table contains North Carolina prices. South Carolina prices are similar.

(8.) Textile plants in the area traditionally take their vacation during the first two weeks in July. These dates are omitted in Figure 1.

(9.) The expression ln ([D.sub.d]/[D.sup.g]) is the daily price index formed using a Tornqvist price index. Usage in each hour, relative to the average level, is a function of relative price in that hour and the daily aggregate price index. King and Shatrawka (1994) provide the Tornqvist index in note 3.

(10.) Herriges et al. (1993, p. 453). Other lags were tried, but there was little difference between the estimates assuming an AR(1) process and other more complex lags.

(11.) Herriges et al. (1993, p. 453). They control for this effect by including both test and control customers and so do not actually include weather conditions. In the model and in estimation, temperature is our proxy for weather conditions. Results were virtually unchanged when cooling degree days were used instead of temperature.

(12.) As suggested to us by Steve Braithwait and an anonymous referee, there could be additional temperature effects. For example, higher temperatures could simply increase electricity use. We examined this hypothesis by interacting temperature with the intercept term. The inclusion of a temperature intercept did not change the results appreciably. To not overly complicate the current model, we leave more complete consideration of possible temperature effects for future work.

(13.) For customers served by the transmission system, the difference between hourly and daily elasticity estimates is significant at the 1% level, whereas for distribution-served customers, hourly and daily elasticity estimates are significantly different only at the 15% level.

(14.) The mix of industrial customers differs in each study, which would affect aggregate elasticities. Our disaggregated results produce daily elasticities for some customers that are larger than hourly elasticities, consistent with the earlier studies.

(15.) Predicted loads, developed from equation (2), are for weekdays, not including the traditional vacation period during the first two weeks of July.

(16.) There is one additional customer with a generator. That customer began participation on the HP rate in summer 1999, and, during this relatively short period of time, has not shown a significant price responsiveness.

(17.) As mentioned in note 12, the addition of a temperature intercept did not change the results materially. Coefficient magnitudes were similar. The significant coefficient of [m.sub.H] in the distribution equation drops from 5% to 10% significance.

(18.) Note that in Table 5 there were only four distribution customers with significant response, and one of these, the feed mill, was not on the hourly pricing rate during summer 1999. This small population of responders may contribute to the lack of significance for the interaction term for distribution customers.

(19.) Taylor and Schwarz (1990), 435, amended to correct for heteroskedasticity and autocorrelation. An anonymous referee pointed out that the second-stage standard error would reflect both the original error of the postulated elasticity relationship and the measurement error and would likely be both heteroskedastic and correlated across observations.

(20.) See White (1980), for the heteroskedasticity correction. We tried a GARCH procedure that corrects for both heteroskedasticity and autocorrelation. It gave similar results for [[sigma].sub.H] but did not converge for [[sigma].sub.D]

(21.) The lack of significance was not due to multicollinearity.

REFERENCES

Aubin, C., D. Fougere, E. Husson, and M. Ivaldi. "Real-Time Pricing of Electricity for Residential Customers: Econometric Analysis of an Experiment." Journal of Applied Econometrics, 10, 1995, S171-91.

Gupta, N., and A. Danielson. "Real-Time Pricing: Ready for the Meter? An Empirical Study of Customer Response." Public Utilities Fortnightly, 136(20), 1998, 50-61.

Herriges, J, S. M. Baladi, D. W Caves, and B. F Neenan. "The Response of Industrial Customers to Electric Rates Based upon Dynamic Marginal Costs." Review of Economics and Statistics, 75(3), 1993, 446-54.

Kim, J. "Customer Response to Real-Time Pricing of Electricity." Ph.D. diss., Department of Economics, University of Colorado, 1998.

King, K., and P. Shatrawka. "Customer Response to Real-Time Pricing in Great Britain." ACEEE 1994 Summer Study on Energy Efficiency in Buildings, Panel 2 Demand and Load Shapes, 1994, 2194-203.

O'Sheasy, M. "Real-Time Pricing--Supplanted by PriceRisk Derivatives?" Public Utilities Fortnightly, March 1, 1997, 31-35.

O'Sheasy, M., S. Braithwait, D. Glyer, K. King, and S. Baladi. "Accounting for RTP Load Response in Utility Operations and Planning." Proceedings: 1996 EPRI Conference on Innovative Approaches to Electricity Pricing: Managing the Transition to Market-Based Pricing, California, March 1996, 6-1-9.

Patrick, R., and F Wolak. "Customer Load Response to Spot Prices in England: Implications for Retail Service Design." EPRI TR-109143 Final Report, November 1997.

Taylor, T, and P. Schwarz. "Advance Notice of Real-Time Electricity Prices." Atlantic Economic Journal, 28(4), 2000, 478-88.

-----. "The Long-Run Effects of a Time-of-Use Demand Charge." RAND Journal of Economics, 21(3), 1990, 431-45.

White, H. "A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity." Econometrica, 48(4), 1980, 817-38.

Woo, C. K., P. Chow, and I. Horowitz. "Optional Real-Time Pricing of Electricity for Industrial Firms." Pacific Economic Review, 1(1), 1996, 79-92.

Zarnikau, J. "Customer Responsiveness to Real-Time Pricing of Electricity." Energy Journal, 11(4), 1990, 99-116.

RELATED ARTICLE: ABBREVIATIONS

CES: Constant Elasticity of Substitution

HP: Hourly Pricing

RTP: Real-Time Prices

SHANA L. DARDAN *

* We wish to acknowledge useful comments from Steve Braithwait, Jen Troyer, and two anonymous referees, as well as valuable assistance from Kevin Lewis, former graduate assistant, Department of Economics, UNC-Charlotte. We also appreciate comments from Andy Kleit, Herb Thompson, and other participants at the Nineteenth Annual Advanced Workshop in Regulation and Competition sponsored by Rutgers University in May 2000.

Schwarz: Professor, Economics Department, University of North Carolina-Charlotte, Charlotte, NC 28223-0001, and Senior Economist, Energy Resource Planning, Research Triangle Institute, Research Triangle Park, NC 27709. Phone 1-704-687-2666, Fax 1-704-687-6442, E-mail [email protected]

Taylor: Senior Economist, Rate Department, Duke Power, A Division of Duke Energy Corporation, PB01A, 422 South Church Street, P.O. Box 1244, Charlotte, NC 28201-1244. Phone 1-704-382-7042, Fax 1-704-382-4671, E-mail [email protected]

Birmingham: 3712 Bingham Drive, Concord, NC 28027-8539. Phone 1-704-786-8999, Fax 1-704-786-7397, E-mail [email protected]

Dardan: Department of Information and Operations Management, University of North Carolina Charlotte, Charlotte, NC 28223-0001. Phone 1-704-687-2320, Fax 1-704-687-6330, E-mail [email protected]