Extreme daily increases in peak electricity demand: Tail-quantile estimation

Energy Policy 53 (2013) 90–96

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Extreme daily increases in peak electricity demand: Tail-quantile estimation Caston Sigauke a,n, Andre´hette Verster b, Delson Chikobvu b a b

Department of Statistics and Operations Research, University of Limpopo, South Africa Department of Mathematical Statistics and Actuarial Science, University of the Free State, South Africa

H I G H L I G H T S c c c c

Policy makers should design demand response strategies to save electricity. Peak electricity demand is influenced by tails of probability distributions. Both the GSP and the GPD are a good fit to the data. Accurate assessment of level and frequency of extreme load forecasts is important.

a r t i c l e i n f o

a b s t r a c t

Article history: Received 20 April 2012 Accepted 3 October 2012 Available online 22 November 2012

A Generalized Pareto Distribution (GPD) is used to model extreme daily increases in peak electricity demand. The model is fitted to years 2000–2011 recorded data for South Africa to make a comparative analysis with the Generalized Pareto-type (GP-type) distribution. Peak electricity demand is influenced by the tails of probability distributions as well as by means or averages. At times there is a need to depart from the average thinking and exploit information provided by the extremes (tails). Empirical results show that both the GP-type and the GPD are a good fit to the data. One of the main advantages of the GP-type is the estimation of only one parameter. Modelling of extreme daily increases in peak electricity demand helps in quantifying the amount of electricity which can be shifted from the grid to off peak periods. One of the policy implications derived from this study is the need for day-time use of electricity billing system similar to the one used in the cellular telephone/and fixed line-billing technology. This will result in the shifting of electricity demand on the grid to off peak time slots as users try to avoid high peak hour charges. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Load management Generalized Single Pareto distribution Dynamic pricing programme

1. Introduction Modelling of extreme daily increases in peak electricity demand is very important for load forecasters in the electricity sector. Peak electricity demand is an energy policy concern for all economies throughout the world, causing blackouts and increasing the cost of electricity for consumers (Strengers, 2012). This has resulted in many economies in the designing of energy efficient and demand side management strategies to either redistribute or reduce energy demand during peak periods. We define daily increase in peak electricity demand as the positive day-to-day change in daily peak demand (DPD), where DPD is the maximum hourly demand in a 24-hour period. Extreme daily increase in peak electricity demand is therefore positive day-to-day change above a sufficiently high threshold. The demand for electricity forms the basis for power

n

Corresponding author. Tel.: þ27 15 268 2188; fax: þ 27 15 268 3075. E-mail addresses: [email protected] (C. Sigauke), [email protected] (A. Verster), [email protected] (D. Chikobvu). 0301-4215/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.enpol.2012.10.073

system planning, power security and supply reliability (Ismail et al., 2009). This involves finding the optimal day-to-day operation of a power plant. It is therefore important to have an accurate assessment of the level and frequency of future extreme day-today increases in peak electricity demand. Peak electricity demand is subject to a range of uncertainties, including population growth, changing technology, economic conditions, prevailing weather conditions as well as the general randomness in individual usage (Hyndman and Fan, 2010). Extreme value theory (EVT) is used in this paper to investigate whether extreme daily increases such as the one experienced in May 2007 in South Africa is truly an extreme low-probability event or is one which will appear on a regular basis. The distribution of extreme daily increases in peak electricity is modelled using the Generalized Pareto Distribution (GPD). A comparative analysis is then done using the Generalized Single Pareto (GSP) distribution which has one parameter to estimate (Verster and De Waal, 2011). Modelling of extreme peak electricity demand is important for load forecasters and system planners for planning and scheduling of likely maximum daily increases of peak

C. Sigauke et al. / Energy Policy 53 (2013) 90–96

electricity demand. Fitting a GPD to exceedances over a sufficiently large threshold is discussed in literature. Castillo and Hadi (1997) discussed the fitting of the GPD to a given set of data. The estimation of the two parameters of the GPD which are the shape and scale parameters is usually not easy. In their paper, Castillo and Hadi (1997) proposed a method for estimating the parameters and quantiles of the GPD. Their proposed method worked well over a wide range of parameter values. Hosking and Wallis (1987) used GPD in modelling annual maximum floods. Recent work includes that of Verster and De Waal (2011) who showed that the tail of a Generalized Burr-Gamma (GBG) distribution can be approximated by a GP-type distribution with one parameter which is the extreme value index. For reviews and references on fitting a GPD to exceedances over a sufficiently large threshold see (Castillo and Hadi, 1997; Beirlant et al., 1999, 2004; Thompson et al., 2009; Bermudez et al., 2010; MacDonald et al., 2011; Chikobvu et al., 2012; among others). The rest of the paper is organized as follows. The GPD and GSP distribution are discussed in Section 2. Bayes estimation of the GPD and GSP distributions is discussed in Section 3. The data set is described in Section 4. The empirical results are discussed in Section 5. A detailed discussion of the significance of the empirical results is given in Section 6. The conclusion and policy implications are discussed in Section 7.

2.2. Generalized single Pareto distribution In Verster and De Waal (2011), it is given that above a reasonably high threshold, t, the tail of a Generalized Burr Gamma (GBG) distribution can be approximated by a GP-type distribution. The GP-type distribution, also a POT distribution, is an approximation of the GPD with the advantage of having only one parameter. The distribution and survival functions of the GP-type distribution which we also refer to as GSP distribution with shape parameter Z (also known as the extreme value index (EVI)) are given as follows:  1=Z Z W Z ðyÞ ¼ 1 1 þ ðytÞ , Z a 0, y 4 t ð3Þ 1 þ tZ  PðY 4 y9tÞ ¼ 1 þ

x

If x 4 0 then W x, s ðyÞ belongs to the heavy-tailed distributions such as Pareto, Student t, Cauchy, loggamma and Frechet whose tails decay like power functions. If x ¼ 0 then W x, s ðyÞ belongs to the Gumbel, normal, exponential, gamma and lognormal whose tails decay exponentially. If x o0, W x, s ðyÞ belongs to the uniform and beta distributions. The survival function of the GPD is given in equation: 8  > xðytÞ 1=x > > 1þ > > > s > >  yt < PðY 4y9tÞ ¼ exp  s > > >  > > x ðy tÞ 1=x > > > 1 þ :

s

if x 4 0,

yt 4 0

if x ¼ 0,

yt 40

if x o 0,

0 o yt o 

ð2Þ

s x

Z a 0,y 4 t

ð4Þ

The two parameters are estimated jointly by considering a Bayesian approach. The joint posterior distribution of x and s is given as follows:  Nt Y 1 xðyi tÞ 1=x1 1þ pðs, xÞ i¼1

s

,

3.1. Bayes estimation of the GPD

pðs, x9yÞp

A peaks over threshold (POT) distribution is considered to model the observations above a sufficiently high threshold. The POT distribution considered here is the GPD with two parameters x, the shape parameter, also known as the Extreme Value Index (EVI) and s, the scale parameter. Balkema and De Haan (1974) and Pickands (1975) showed that the distribution function of the excesses above a high threshold converges to a GPD as the threshold tends to the right endpoint. Let Y 1 ,Y 2 , . . . ,Y n be a sequence of daily increases in peak demand. The increase in peak demand is relative to the previous day. In order to extract upper extremes from this sequence we take the exceedances over a predetermined high threshold t. The distribution function of the GPD is given in the following equation: 8   > xðytÞ 1=x > > 1 1 þ if x 4 0, yt 4 0 > > > s > >  yt < if x ¼ 0, yt 4 0 ð1Þ W x, s ðyÞ ¼ 1exp  s > > >  1=x > > xðytÞ s > > > 1 1 þ if x o0, 0 o yt o :

1=Z

Z ðytÞ 1 þ tZ

3. Bayes estimation

2. EVT and modelling of peak electricity daily changes 2.1. Generalized Pareto distribution

91

s

ð5Þ

s

where pðs, xÞpð1=sÞex is the maximal data information (MDI) prior (Zellner, 1977) and N t is the number of observations above the threshold. The two parameters are estimated by simulating a large number of s’s and x’s values from the posterior distribution and taking the mean of the simulated values to obtain estimates. To simulate a set of ðs, xÞ’s from the posterior we make use of the Gibbs sampling method by simulating alternatively s from its conditional density function given a fixed x. The parameter x is then simulated from its conditional density given the selected s. This process is repeated a large number of times. Future posterior predictive tail probabilities of a future observation, Y0, can be predicted through the following posterior predictive density:

PðY 0 4y0 9y, tÞp

Z Z



x

pðx, s9yÞ 1 þ ðy0 tÞ s

1=x

1 o x o1

ds dx, ð6Þ

Eq. (6) cannot be computed analytically, but can be approximated easily by simulation. We simulate a large number of x’s and s’s from the posterior distribution (Eq. (5)) which are then substituted into Eq. (6). The average over all the tail probabilities is then used to estimate the posterior predictive tail probability. 3.2. Bayes estimation of the GSP distribution The tail index parameter Z is estimated by considering a Bayesian approach and obtaining the posterior distribution of Z. The posterior distribution of Z is  Nt Y 1 Zðyi tÞ 1=Z1 pðZ9yÞp 1þ pðZÞ ð7Þ 1þ Zt 1 þ Zt i¼1 where pðZÞpeZ =ð1 þ ZtÞ is the MDI prior (Zellner, 1977) and Nt is the number of observations above the threshold. The mode of

C. Sigauke et al. / Energy Policy 53 (2013) 90–96

the posterior distribution for various values of Z is taken as the estimate of Z. Future posterior predictive tail probabilities of a future observation, Y0, can be predicted through the following posterior predictive density:  1=Z Z Z PðY 0 4 y0 9y, tÞp pðZ9yÞ 1 þ ðy0 tÞ dZ, 1 o Z o 1 1 þ Zt ð8Þ As is the case of Eq. (6), Eq. (8) cannot be computed analytically, but can be approximated easily by simulation. Again we simulate a large number of Z’s from the posterior distribution which is then substituted into Eq. (8). The average over all the tail probabilities is then used to estimate the posterior predictive tail probability.

36000

Electricity demand (megawatts)

92

34000

32000

30000

28000

26000

5

4. Data set

10

15

20

Time (Hours)

Table 1 Descriptive statistics of daily changes. Mean

Standard deviation

Fig. 2. Hourly load profile for 24 May 2007.

4000

Daily increases in peak demand (MW)

Aggregated DPD data is used for the industrial, commercial and residential sectors of South Africa. Modelling of daily increases in peak electricity demand is done using data for the period, years 2000–2011. In total we have 4271 observations from which we calculate 4270 daily changes. We define daily change as follows: let zt be DPD on day t and zt1 be DPD on day t1, then daily change is defined as day-to-day change in peak electricity demand, i.e. ðzt zt1 Þ, t ¼ 1, . . . ,n. The data is provided by Eskom, South Africa’s power utility company. The distribution of the daily changes is nonnormal as evidenced by the skewness, kurtosis and Jarque–Bera test given in Table 1. A visual inspection of Fig. 1 shows that the largest daily change in peak electricity demand is experienced in 2007 over the sampling period. A typical hourly load profile is given in Fig. 2. The hourly load profile given in Fig. 2 is for 24 May 2007 when South Africa experienced the highest daily peak electricity demand of 37 158 MW over the sampling period 2000–2011.

3000

2000

1000

0

Skewness

Kurtosis

Jarque-Bera

0

500

1000

1500

Number of observations 2.8475

1251.752

0.9299

3.5152

662.66 (0.0000)

5. Empirical results

4000

Daily changes in DPD (megawatts)

Fig. 3. Daily increases in peak demand.

5.1. Fitting the GPD 2000

0

−2000

2000

2002

2004

2006

2008

Year Fig. 1. Daily changes in DPD (2000–2011).

2010

2012

We define daily increase in peak demand on day t, as Y t ¼ maxðzt zt1 ,0Þ, t ¼ 1, . . . ,n, where zt is defined in Section 4. In this definition decreases in demand are ignored. An initial threshold is set at zero after taking the first differences of DPD. Only observations above zero are considered giving us a total of 1718 data points. A plot of daily increases in peak demand is shown in Fig. 3. The data is transformed by taking natural logarithms to reduce the impact of heteroskedasticity that may be present because of the large data set and its high frequency. If Yt denotes the daily increase in peak demand, then the log demand Yt guarantees that the forecasted demand will never be negative. A Pareto quantile plot is used to obtain the threshold. The observation on the y-axis where the plot starts to follow a horizontal line is taken as the threshold. In this case t ¼ expð8Þ ¼ 2980:958. The Pareto quantile plot is shown in Fig. 4.

C. Sigauke et al. / Energy Policy 53 (2013) 90–96

We now consider the observations greater than t to be Generalized Pareto distributed. x and s are then estimated by simulating from the joint posterior distribution given in Eq. (5). Fig. 5 shows a scatter plot of the simulated x’s against s’s for the data. The means of the simulated x’s and s’s are calculated as 0.0079 and 269.164 respectively and are considered as the estimates of x and s. The negative sign of x, the extreme value index (EVI), indicates that the data has an upper bound. This is probably due to the fact that the day-to-day increase in daily peak demand between any two consecutive days cannot exceed the maximum in the daily peak demand between the two days concerned and also that demand cannot exceed supply. For observations above the threshold the empirical cumulative distribution function (cdf) and theoretical cdf of the GPD with the estimated parameter values are constructed and are shown in Fig. 6. The theoretical cdf in Fig. 6 is close to the empirical cdf, indicating that the GPD can be used to model the observations above the threshold. A QQ plot can be constructed to indicate the goodness of fit. The theoretical quantiles from the GPD with the estimated parameter are plotted against the empirical quantiles. If the plot lies on the 451 line it indicates a good fit as is the case in Fig. 7.

93

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 2800

3000

3200

3400

3600

3800

4000

4200

4400

Fig. 6. Graphical plot of the empirical cdf (dotted curve) and the theoretical cdf of the GPD (solid curve) on the exceedances. The x-axis represents the exceedances above the threshold of 2981 and the y-axis represents the cumulative probabilities.

4400

5.2. Fitting the GSP 4200

The GSP has one parameter instead of two as in the GPD case. Consider again the threshold obtained previously t ¼ expð8Þ ¼

3800

9

Log of the observations

4000

8

3600

7

3400

6

3200

5 3000 4 2800 2800

3

3000

3200

3400

3600

3800

4000

4200

4400

Fig. 7. QQ plot of GPD above the threshold. The horizontal axis represents the standard theoretical quantiles while the empirical quantiles are plotted on the vertical axis.

2 1 0 0

1

2

3

4

5

6

7

8

−log[1−i/(n+1)] Fig. 4. Generalized Pareto quantile plot on the positive observations. The x-axis represents the standard theoretical quantiles. 450

400

σ

350

300

2980:958. For different values of Z the posterior of Z is constructed in Fig. 8. The mode of the posterior is an estimate of Z. In this study Z^ ¼ 0:08. The empirical cdf and theoretical cdf of the GSP with the estimated parameter values Z^ ¼ 0:08 are constructed for observations above the threshold and shown in Fig. 9. It is evident from Fig. 9 that the theoretical cdf is close to the empirical cdf, indicating the appropriateness of using the GSP distribution to model the observations above the threshold. A QQ plot can be constructed to indicate the goodness of fit. The theoretical quantiles from the GSP distribution with the estimated parameter are plotted against the empirical quantiles. If the plot lies on the 451 line then it indicates a good fit as is the case in Fig. 10.

250

6. Comparative analysis and discussion

200

150 −0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

ξ Fig. 5. Plot of 3000 simulated ðx, sÞ values simulated through Gibbs sampler.

Some of the posterior predicted tail probabilities for various extreme daily increases in peak electricity demand are given in Table 2. For the GPD, 1000 x’s and s’s were simulated and substituted into Eq. (6). Similarly for the GSP distribution, 1000 Z’s were simulated and substituted into Eq. (8). Empirical results

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C. Sigauke et al. / Energy Policy 53 (2013) 90–96

1

Table 2 Posterior predictive tail probabilities.

0.9 0.8 0.7 0.6

y0 (in MW)

PðY 0 4 y0 9y, tÞ GPD

PðY 0 4 y0 9y, tÞ GSP

3400 3700 4000 4300

0.2004 0.0628 0.0195 0.0060

0.1966 0.0688 0.0261 0.0106

0.5 0.4 0.3 4200

0.1 0 −0.05

0

0.05

η

0.1

0.15

0.2

Fig. 8. The posterior distribution of Z.

1 0.9

Exceedances above 2981 MW

0.2

4000

3800

3600

3400

3200

0.8 0.7

3000

0.6 0

0.5

20

40

60

80

100

Number of observations

0.4

Fig. 11. Exceedances above the threshold t ¼ 2981.

0.3 0.2 0.1 0 2800

3000

3200

3400

3600

3800

4000

4200

4400

Fig. 9. Graphical plot of the empirical cdf (dotted curve) and the cdf of the GSP distribution (solid curve) on the exceedances. The x-axis represents the exceedances above the threshold of 2981 and the y-axis represents the cumulative probabilities. 4400 4200 4000 3800 3600 3400 3200 3000 2800 2800

3000

3200

3400

3600

3800

4000

4200

4400

Fig. 10. QQ plot of GSP distribution above the threshold. The horizontal axis represents the standard theoretical quantiles while the empirical quantiles are plotted on the vertical axis.

show that both the GSP distribution and the GPD are a good fit to the data. However, the QQ plot of the GSP distribution given in Fig. 10 incorporates most extreme observations in the tail slightly

better than the GPD. It is more attractive to use the GSP distribution since it has one parameter to estimate. The plot of 100 exceedances above the threshold of 2981 MW over the sampling period, 2000–2011 is given in Fig. 11. Over this sampling period the maximum extreme daily increase is 4327 MW and the minimum is 2983 MW. All the exceedances of the extreme daily increases occur on Mondays as expected since Sunday has the lowest demand of electricity. The maximum extreme daily increase is on Monday 21 May 2007. It is interesting to note that on Thursday 24 May 2007 South Africa had the highest daily peak demand over the sampling period 2000–2011, which is the same week the country experienced the maximum extreme daily increase. Modelling of extreme daily increases in peak electricity demand improves the reliability of a power network if an accurate assessment of the level and frequency of future extreme load forecasts is carried out (Hor et al., 2008). The bar chart of the monthly frequency of occurrence of exceedances is given in Fig. 12. The months May, June and July have the highest frequencies above the threshold. This is an indication that South Africa experiences a higher peak electricity demand in winter than in summer. This is in line with similar studies for the influence of temperature on electricity demand in South Africa. Sigauke and Chikobvu (2010) showed that electricity demand in South Africa is highly sensitive to temperature fluctuations during the winter periods than the summer periods. In South Africa the winter period experiences a higher systemwide demand in energy than the summer period as shown in Fig. 13. The bar chart of the yearly frequency of occurrence of exceedances is given in Fig. 14. Over the sampling period the year 2004 has the largest number of exceedances. This analysis is important for prediction of return period of this largest number of exceedances. This will help decision makers in long term strategic

C. Sigauke et al. / Energy Policy 53 (2013) 90–96

planning which will include capacity expansion to ensure that there is enough energy during periods of abnormal peak demand. Fig. 14 shows that there is an upward increase in peak electricity demand. This poses a challenge to decision makers in Eskom who have to meet this demand in a reliable and cost reflective way. Eskom has put in place an integrated demand management programme for all sectors of the economy (Eskom website). This is meant to shift the load curve to lower demand levels and also shift between loads from one energy system to another. This is in line with the International Energy Agency (IEA) Demand-side Management (DSM) programme which seeks to reduce the demand peaks, shift the loads between times of day or even seasons, fill the demand valleys to better utilize existing power resources and reduce overall demand in the context of delivering the required energy services by use of less energy (IEA DSM website).

20

15

Frequency

95

10

5

0 Jan Feb Mar Apr May Jun

Jul Aug Sep Oct Nov Dec

Month Fig. 12. Bar chart of the monthly frequency of occurrence of exceedances.

Daily peak demand (MW)

36000

34000

32000

30000

28000

26000

0

100

200

300

Day Fig. 13. Daily peak demand data for South Africa, 2007.

14

12

Frequency

10

8

6

4

2

0 00

01

02

03

04

05

06

07

08

09

10

11

Year Fig. 14. Bar chart of the yearly frequency of occurrence of exceedances.

7. Conclusion and policy implications In this paper extreme daily increases in peak electricity demand in South Africa are modelled using a Generalized Pareto Distribution (GPD). A comparative analysis of the GPD is done with the generalized Pareto-type (GP-type) also known as the Generalized Single Pareto (GSP) distribution. A Pareto quantile plot is used to obtain the optimum threshold. Empirical results show that both the GSP distribution and the GPD are a good fit to the data. One of the main advantages of the GSP distribution is the estimation of only one parameter instead of two as is the case with GPD. The simultaneous simulations of two parameters can become quite difficult and time consuming. Modelling of extreme daily increases in peak electricity demand improves the reliability of a power network if an accurate assessment of the level and frequency of future extreme load forecasts is carried out. Policy implications derived from this study are that policy makers and demand side managers should play a pivotal role of achieving behavioural change in electricity usage particularly during peak periods (Newsham and Bowker, 2010). Policy makers should design demand response strategies where users are exposed to time-of-day electricity pricing incentives (Strapp et al., 2007). One of these incentives could be in the form of a tax rebate to users who save electricity during peak hours (South African National Energy ACT, 2008). The South African National Energy Development Institute (SANEDI) has already put in place legislation for tax rebate incentives to those users who save electricity (SANEDI, 2012). This could also simply mean lower electricity charges for off peak users of electricity. Use of energy efficient technologies should be encouraged to all energy users. There is need for a realistic day-time electricity billing system similar to the one used in the cellular telephone/fixed line billing technology. This technology is being used in other countries (Newsham and Bowker, 2010; Newsham et al., 2011; APS, 2012). This will result in the shifting of electricity demand on the grid into off peak time slots as users try to avoid high peak charges. The time of use of electricity based billing system will inevitably have an impact on the life pattern of individuals and economic activity of the country as there are cost efficiencies to be gained by shifting demand from peak to off peak periods. In the context of the South African power utility company, Eskom, independent power producers, municipalities and provincial government entities can benefit from investigating and understanding the potential socioeconomic impact of the time of use of electricity billing system could have. It is important that public entities are cognisant of the potential implications of the time of use electricity billing system so that they are able to monitor risks and associated costs, and are able to engage effectively with the relevant stakeholders.

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Areas for further study would include the use of other methods to determine an optimum threshold. Modelling the tail of residual distributions to understand how extremes may affect forecasts (Hor et al., 2008) and development of peak load reduction strategies in all sectors of the economy (Newsham and Bowker, 2010; Dlamini and Cromieres, 2012). Another interesting area for further study will be the use of robust Bayesian analysis in which a class of prior distributions is used instead of a single prior distribution. These areas will be studied elsewhere.

Acknowledgments The authors are grateful to Eskom for providing the data and to the numerous people for helpful comments on this paper. References APS, 2012 /http://www.aps.com/aps_services/residential/rateplans/ResRate Plans_28.htmlS (Accessed 3 August 2012). Balkema, A., De Haan, L., 1974. Residual life time at great age. Annals of Probability 2, 792–804. Beirlant, J., De Waal, D.J., Teugels, J.L., 1999. The generalized Burr-Gamma family of distributions with application in extreme value analysis. Limit Theorems in Probability and Statistics 1, 113–132. Beirlant, J., Goedgebeur, Y., Segers, J., Teugels, J., 2004. Statistics of Extremes Theory and Applications. Wiley, England. Bermudez, P., DE, Z., Kotz, S., 2010. Parameter estimation of the generalized Pareto distribution-Part II. Journal of Statistical Planning and Inference 140, 1374–1388. Castillo, E., Hadi, A.S., 1997. Fitting the generalized Pareto distribution to data. Journal of the American Statistical Association 92 (440), 1609–1620. Chikobvu, D., Sigauke, C., Verster, A., 2012. Winter peak electricity load forecasting in South Africa using extreme value theory. South African Statistical Journal 46, 185–201. Dlamini, N.G., Cromieres, F., 2012. Implementing peak load reduction algorithms for household electrical appliances. Energy Policy 44, 280–290. Eskom Integrated Demand Management /http://www.eskomidm.co.za/home/ aboutS.

Hor, C., Watson, S., Infield, D., Majithia, S., 2008. Assessing load forecast uncertainty using extreme value theory. In: 16th PSCC, Glasgow, Scotland, 14–18 July /http://www.pscc-central.org/uploads/tx_ethpublications/pscc2008_571. pdfS (Accessed on 27 January 2012). Hosking, J.R.M., Wallis, J.R., 1987. Parameter and quantile estimation for the generalized Pareto distribution. Technometrics 29 (3), 339–349. Hyndman, R.J., Fan, S., 2010. Density forecasting for long-term peak electricity demand. IEEE Transactions on Power Systems 25 (2), 1142–1153. Ismail, Z., Yahya, A., Mahpol, K.A., 2009. Forecasting peak load electricity demand using statistics and rule based approach. American Journal of Applied Sciences 6 (8), 1618–1625. MacDonald, A., Scarrott, C.J., Lee, D., Darlow, B., Reale, M., Russell, G., 2011. A flexible extreme value mixture model. Computational Statistics and Data Analysis 55, 2137–2157. National Energy ACT, 2008. 16 September 2011, Regulations on the Allowance for Energy Efficiency Savings /http://www.info.gov.za/view/DownloadFileAction?id= 150701S (Accessed on 29 May 2012). Newsham, G.R., Bowker, B.G., 2010. The effect of utility time-varying pricing and load control strategies on residential summer peak electricity use: a review. Energy Policy 38, 3289–3296. Newsham, G.R., Birt, B.J., Rowlands, I.H., 2011. A comparison of four methods to evaluate the effect of a utility residential air-conditioner load control program on peak electricity use. Energy Policy 39 (10), 6376–6389. Pickands, J., 1975. Statistical Inference using extreme order statistics. The Annals of Statistics 3, 119–131. SANEDI, 2012 /http://www.sanedi.org.za/S (Accessed on 29 May 2012). Sigauke, C., Chikobvu, D., 2010. Daily peak electricity load forecasting in South Africa using a multivariate non-parametric regression approach. ORiON 26 (2), 97–111. Strapp, J., King, C., Talbott, S., 2007. Ontario Energy Board Smart Price Pilot Final Report. Prepared by IBM Global Business Services and eMeter Strategic Consulting for the Ontario Energy Board /http://www.oeb.gov.on.ca/AF9CF86 B-333A-4635-B899-C02E2D5B6508/FinalDownload/DownloadId-B9DDC 81F75CF02F29E22108EA3370D59/AF9CF86B-333A-4635-B899- C02E2D5B6508/ documents/cases/EB-2004-0205/smartpricepilot/OSPP%20Final%20Report%20-% 20Final070726.pdfS. Strengers, Y., 2012. Peak electricity demand and social practice theories: reframing the role of change agents in the energy sector. Energy Policy 44, 226–234. Thompson, P., Cai, Y., Reeve, D., Stander, J., 2009. Automated threshold selection methods for extreme wave analysis. Coastal Engineering 56, 1013–1021. Verster, A., De Waal, D.J., 2011. A method for choosing an optimum threshold if the underlying distribution is generalized Burr-Gamma. South African Statistical Journal 45, 273–292. Zellner, A., 1977. Bayesian Analysis in Econometrics and Statistics. Edward Elgar, Lyme US.