This study involves forecasting electricity demand for long-term planning purposes. Long-term forecasts for hourly electricity demands from 2006 to 2023 are done with in-sample forecasts from 2006 to 2012 and out-of-sample forecasts from 2013 to 2023. Quantile regression (QR) is used to forecast hourly electricity demand at various percentiles. Three contributions of this study are (1) that QR is used to generate long-term forecasts of the full distribution per hour of electricity demand in South Africa; (2) variabilities in the forecasts are evaluated and uncertainties around the forecasts can be assessed as the full demand distribution is forecasted and (3) probabilities of exceedance can be calculated, such as the probability of future peak demand exceeding certain levels of demand. A case study, in which forecasted electricity demands over the long-term horizon were developed using South African electricity demand data, is discussed.

The aim of the study was: (1) to apply a quantile regression (QR) model to forecast hourly distribution of electricity demand in South Africa; (2) to investigate variabilities in the forecasts and evaluate uncertainties around point forecasts and (3) to determine whether the future peak electricity demands are likely to increase or decrease.

The study explored the probabilistic forecasting of electricity demand in South Africa.

The future hourly electricity demands were forecasted at 0.01, 0.02, 0.03, … , 0.99 quantiles of the distribution using QR, hence each hour of the day would have 99 forecasted future hourly demands, instead of forecasting just a single overall hourly demand as in the case of OLS.

The findings are that the future distributions of hourly demands and peak daily demands would be more likely to shift towards lower demands over the years until 2023 and that QR gives accurate long-term point forecasts with the peak demands well forecasted.

QR gives forecasts at all percentiles of the distribution, allowing the potential variabilities in the forecasts to be evaluated by comparing the 50th percentile forecasts with the forecasts at other percentiles. Additional planning information, such as expected pattern shifts and probable peak values, could also be obtained from the forecasts produced by the QR model, while such information would not easily be obtained from other forecasting approaches. The forecasted electricity demand distribution closely matched the actual demand distribution between 2012 and 2015. Therefore, the forecasted demand distribution is expected to continue representing the actual demand distribution until 2023. Using a QR approach to obtain long-term forecasts of hourly load profile patterns is, therefore, recommended.

Electricity load is the amount of electricity that balances the amount generated with that drawn from the grid. In the absence of black-outs, load-shedding and the availability of electricity generated from renewable electricity sources, the electricity load is equivalent to the electricity demand. Therefore, in this study, the hourly electricity demand is defined as the amount of electricity (load) in kW sent out every hour by Eskom to meet consumers’ demand.

The 1996 census showed that only 57.6% of the South African households had access to electricity for lighting (Statistics South Africa

South Africa experienced an average growth rate of approximately 5% in real terms between 2004 and 2007. However, the period 2008 to 2012 only recorded average growth of just above 2%. (Statistics South Africa

The penetration of other sources of electricity such as renewables for example solar and wind, could also have contributed to a decline in electricity demand from Eskom. In addition, because of the lack of capacity in the generation of electricity experienced by Eskom in 2007 (Inglesi & Pouris

Uncertainties occur in estimation, prediction or in forecasting. When statisticians develop predictions (forecasts) for an uncertain future, they need to quantify the uncertainties around these for those that have to make decisions in the face of those uncertainties. Sigauke (

Hong, Wilson and Xie (

In the late 1880s, when lighting was the sole end use of electricity, the forecasting of electricity demand was straightforward (Hong & Shahidehpour

Electricity demand forecasts can be developed for short, medium- or long-term horizons, and they could be provided as point forecasts, which give one value at each time interval, or as probabilistic forecasts which give a full distribution of future values and therefore allow the assessment of uncertainties around the forecasts. Quantification of uncertainties around forecasts is even more important for long-term forecasts, because, as Sigauke and Chikobvu (

In the literature to date, short-term electricity demand forecasting has attracted substantial attention because of its importance for power system control, unit commitment and electricity markets. Medium- and long-term forecasting have not received much attention, despite their value for system planning and budget allocation (Hyndman & Fan

There are some literature available on long-term forecasting of annual electricity demand as well as peak electricity demand in South Africa (Inglesi-Lotz

Weron and Misiorek (

Statistical methods differ from ANN in that the former forecast the current value of a variable by using mathematical combination of the previous values of that variable and sometimes the previous values of exogenous factors (Weron & Misiorek

Suganthi and Samuel (

Within the univariate time series framework, the stochastic nature of electricity demand as a function of time has frequently been modelled with seasonal autoregressive integrated moving average (SARIMA) and state space models (Taylor, De Menezes & McSharry

SARIMA models could be extended to a SARIMA-GARCH model to account for the possibility of heteroscedasticity. A GARCH modelling approach could be used to capture potential conditional heteroscedasticity in electricity data (Byström

Seasonal autoregressive integrated moving average with exogenous variables (SARIMAX) models, also known as regression-SARIMA, have been used in load forecasting in order to incorporate important drivers of demand such as calendar variables and temperature (Bunn

Structural time series (STS) models have also been successfully used in demand forecasting. STS modelling was developed by Harvey (

Hyndman and Fan (

OLS regression models model the relationship between covariates _{0} and _{i} which minimise

To apply an OLS regression model, the data must meet stringent assumptions, such as that the residuals should be normally distributed, the observations should be independent and the variance of the residuals should be homoscedastic. As we are dealing with time series data the observations are not independent, and for electricity demand data, the variance is heteroscedastic. Therefore, some assumptions of OLS are violated in the time series data used in this study.

In this article, QR is proposed for developing long-term probabilistic forecasts. QR was developed as an extension of OLS regression for estimating rates of change in all parts of the distribution of the response variable (Cade & Noon

Cornec (_{0} and _{i} which minimises

QR does not require any distribution assumptions regarding the population and can estimate the parameters non-parametrically (Koenker & Bassett

where _{i} is assumed to be zero. The standard QR model is given by:

where

is the conditional τth quantile of the response (_{i}) given the covariate (_{i}) and _{τ}(_{i}

In estimating the QR model for a given quantile, we follow the ideas of Koenker (_{τ}(.), which they called the check-function (Maistre, Lavergne & Patilea

_{τ}(_{τ} are positive and the scale is based on the probability

This concept is extendable to any quantile, such as the 75th, 90th and 99th percentile. The QR estimator for

This is a non-differentiable function and there is no closed-form solution for _{τ}, where the estimate of the

Features that characterise QR and differentiate it from other regression methods are:

QR computes several different regression curves corresponding to the various percentile points of the distribution and thus provides a more complete picture of the relationship between the response variable and the covariates.

Heteroscedasticity can be detected and, if the data are heteroscedastic, median regression estimators can be used instead of mean regression estimators.

Median regression is more robust to outliers than other regression methods that use mean estimators, and it is semi-parametric, therefore, avoiding assumptions about the parametric distribution of the error process.

QR is, therefore, considered to be more suitable than other methods, given the type of data used, as well as its ability to provide the full distribution of forecasted electricity demand.

The hourly electricity demand data for South Africa for the period 1997–2015 was provided by Eskom. For developing the long-term forecasting model, a transformed series was developed using a logarithmic transformation. The logarithmic transformation is convenient for turning a highly skewed variable into one that is more approximately normal (Benoit

Hourly demands were forecasted from 2006 to 2023. The data from 2013 to 2015 were withheld in order to validate the model. The forecasts from 2006 to 2012 were then used as in-sample forecasts, whereas the forecasts from 2013 to 2023 were out of sample forecasts.

Various time-related variables were used as covariates, namely day, public holidays, months, weekends, December break and seasons (see

Variables used in the quantile regression.

Variable | Type of variable | Scale | Created dummy variables |
---|---|---|---|

Demand | Dependent | Continuous | - |

Newyear | Independent | Dichotomous | 1 if day is 01 January; 0 otherwise |

Humanrights | Independent | Dichotomous | 1 if day is 21 March; 0 otherwise |

FreedomDay | Independent | Dichotomous | 1 if day is 27 April; 0 otherwise |

WorkersDay | Independent | Dichotomous | 1 if day is 01 May; 0 otherwise |

YouthDay | Independent | Dichotomous | 1 if day is 16 June; 0 otherwise |

HeritageDay | Independent | Dichotomous | 1 if day is 24 September; 0 otherwise |

ReconciliationDay | Independent | Dichotomous | 1 if day is 16 December; 0 otherwise |

ChristmasDay | Independent | Dichotomous | 1 if day is 25 December; 0 otherwise |

GoodwillDay | Independent | Dichotomous | 1 if day is 26 December; 0 otherwise |

Month1 | Independent | Dichotomous | 1 if Month is January; 0 otherwise |

Month2 | Independent | Dichotomous | 1 if Month is February; 0 otherwise |

Month3 | Independent | Dichotomous | 1 if Month is March; 0 otherwise |

Month4 | Independent | Dichotomous | 1 if Month is April; 0 otherwise |

Month5 | Independent | Dichotomous | 1 if Month is May; 0 otherwise |

Month6 | Independent | Dichotomous | 1 if Month is June; 0 otherwise |

Month7 | Independent | Dichotomous | 1 if Month is July; 0 otherwise |

Month8 | Independent | Dichotomous | 1 if Month is August; 0 otherwise |

Month9 | Independent | Dichotomous | 1 if Month is September; 0 otherwise |

Month10 | Independent | Dichotomous | 1 if Month is October; 0 otherwise |

Month11 | Independent | Dichotomous | 1 if Month is November; 0 otherwise |

Sin6 | Independent | Continuous | Sine term of Fourier series with Period 6 |

Cos6 | Independent | Continuous | Cosine term of Fourier series with Period 6 |

Sin12 | Independent | Continuous | Sine term of Fourier series with Period 12 |

Cos12 | Independent | Continuous | Cosine term of Fourier series with Period 12 |

Sin18 | Independent | Continuous | Sine term of Fourier series with Period 18 |

Cos18 | Independent | Continuous | Cosine term of Fourier series with Period 18 |

Sin24 | Independent | Continuous | Sine term of Fourier series with Period 24 |

Cos24 | Independent | Continuous | Cosine term of Fourier series with Period 24 |

Lag70128 | Independent | Continuous | The 1st time lag |

Lag70152 | Independent | Continuous | The 2nd time lag |

Lag70176 | Independent | Continuous | The 3rd time lag |

Lag70200 | Independent | Continuous | The 4th time lag |

Lag70224 | Independent | Continuous | The 5th time lag |

Lag70248 | Independent | Continuous | The 6th time lag |

Sun | Independent | Dichotomous | 1 if day is Sunday; 0 otherwise |

Mon | Independent | Dichotomous | 1 if day is Monday; 0 otherwise |

Tues | Independent | Dichotomous | 1 if day is Tuesday; 0 otherwise |

Wed | Independent | Dichotomous | 1 if day is Wednesday; 0 otherwise |

Thurs | Independent | Dichotomous | 1 if day is Thursday; 0 otherwise |

Fri | Independent | Dichotomous | 1 if day is Friday; 0 otherwise |

Fribtwn | Independent | Dichotomous | 1 if Friday preceded by a holiday; 0 otherwise |

Monbtwn | Independent | Dichotomous | 1 if Monday preceded a holiday; 0 otherwise |

Lngwknd | Independent | Dichotomous | Long weekend |

Dec_closure | Independent | Dichotomous | 1 if period between 16 December and 01 January; 0 otherwise |

Winter_schoolholiDay | Independent | Dichotomous | 1 if period is during school closure in June/July; 0 otherwise |

Easter | Independent | Dichotomous | 1 if day is Easter; 0 otherwise |

Winter | Independent | Dichotomous | 1 if period is between June and August |

Sin, Sine term of Fourier series; Cos, Cosine term of Fourier series; Lag, time lag; Fribtwn, Friday preceded by holiday; Monbtwn, Monday preceded by a holiday; Newyear, New year; Humanrights, Human rights day; FreedomDay, Freedom day; WorkersDay, Workers day; YouthDay, Youth day; HeritageDay, Heritage day; ReconciliationDay, Reconciliation day; ChristmasDay, Christmas day; GoodwillDay, Day of Goodwill; Sun, Sunday; Mon, Monday; Tues, Tuesday; Wed, Wednesday; Thurs, Thursday; Fri, Friday; Lngwknd, Long weekend; Dec_closure, December closure; Winter_SchoolholiDay, Winter school holiday.

The future hourly electricity demands were forecasted at 0.01, 0.02, 0.03, … , 0.99 quantiles of the distribution using QR, hence each hour of the day would have 99 forecasted future hourly demands, instead of forecasting just a single overall hourly demand as in the case of OLS. To avoid graphs that are too busy and difficult to read, only the 1st, 50th and the 99th percentile graphs are shown and discussed.

The uncertainties in the forecasts are captured by the interval between the 1st and 99th percentiles of the demand distribution, as this is the interval into which 98% of the possible future hourly demands are expected to fall. The wider the interval, the more uncertain we are about the forecasted hourly demand as the variability between the forecasts would be very high. The forecasts at the 50th percentile (median) are important because they could be used as point forecasts, namely, our best guess of demand at that certain hour.

The density functions give the full distribution of the hourly electricity demand. The probability of the hourly demand between the two demand points say, ‘

The South African electricity demand has two daily peaks, especially noticeable in winter, namely a morning demand peak at around 08:00 and an afternoon demand peak at around 19:00. As the winter peak represents the highest annual demand, this is important for planning electricity generation. Therefore, it is important to examine the demand densities of both morning and afternoon peak forecasts over the years. The morning peak density demand is generated from all possible demand forecasts at 07:00, 08:00 and 09:00, whereas the afternoon peak density demand is generated from all possible demand forecasts at 18:00, 19:00 and 20:00. The probability of the future peak electricity demand exceeding a certain value was then calculated by integrating the density functions.

The performance of the QR model is evaluated by comparing the predicted demand density functions with the actual demand density functions. The predicted demand density is generated from all demand forecasts from 1st to 99th percentile of the demand distribution. If the forecasted demand density function closely tracks the actual demand density, then it shows that the model is forecasting well and it is, therefore, reliable. The model is also evaluated by observing the closeness of the actual demand distribution to the predicted lower and upper 99% interval. If the interval is narrow, then the predictions exhibit sharpness. The mean absolute percentage error (MAPE) is used to compare the forecasts at the 50th percentile with the actual demands; this is mainly to determine how far the point forecasts are from the actual demands.

Electricity demand between 1997 and 2015 in South Africa.

Adjusted electricity demand between 1997 and 2015 in South Africa.

The historical demand data from 1997 to 2015 indicate that the demand for electricity increased steadily between 1997 and 2007 (

For each hour, the demand at the 1st to 99th percentiles was forecasted from 2006 to 2023, which translated into the full demand distribution being forecasted.

The actual and predicted demand densities between 2012 and 2015 illustrated in

Comparison of actual and forecasted demand distributions: (a) 2012; (b) 2013; (c) 2014 and (d) 2015.

Distribution of forecasted daily electricity demand: (a) 22 June 2013, (b) 23 June 2013, (c) 24 June 2013, (d) 25 June 2013.

Distribution of forecasted daily electricity demand: (a) 22 June 2014, (b) 23 June 2014, (c) 24 June 2014, (d) 25 June 2014.

Distribution of forecasted daily electricity demand: (a) 22 June 2015, (b) 23 June 2015, (c) 24 June 2015, (d) 25 June 2015.

The MAPE between the hourly demand forecasts at the 50th percentile and the actual hourly demand over the period of 4 years were below 5% and the overall MAPE was 2.77%, as shown in

Further confirmation of the suitability of fit of the QR model is obtained by comparing the actual values to the full range of quantile predictions.

For illustration purposes, estimates for only three of the QR models (at 0.01, 0.5 and 0.99 quantile levels) are given in

For illustration purposes, 4 days in June (22–25 June) were selected and their results from 2013 to 2015 were discussed. (Note that June falls in the high-demand winter period of the year.) The different panels in

In addition to just using the 50th percentile as a ‘best guess’ or point forecast, information contained in the other quantile forecasts produce probabilistic information which may also be useful in the planning process. It can be seen from

The hourly electricity demand distribution in South Africa is bimodal as shown in

Distribution of forecasted hourly demand between 2013 and 2023.

In addition, the forecasted hourly density demands (in

The morning peak demand distributions.

The afternoon peak demand distributions.

Finally, the forecasts obtained from QR can be used to investigate the expected future peak demands and their probability of exceedance over the years. The annual peak demands are very important for planning purposes, as these represent the maximum that would need to be supplied in an hour and if the power generating company could meet the daily peak hourly demand, it could meet any hourly demand.

The daily electricity demand in South Africa generally has two peaks, more noticeable during winter than summer seasons. The morning demand peak occurs at around 08:00 and the afternoon demand peak at around 19:00. OLS would most likely underestimate the peaks as it models the mean of the demand distribution while QR models demand at all percentiles of the demand distribution and therefore can provide better peak forecasts. In addition, as QR gives the full hourly demand distribution, the uncertainties around the forecasts are quantifiable. While the best guess of the future hourly electricity demand can be obtained from forecasted demands at the 50th percentile, QR gives forecasts at all percentiles of the distribution, allowing the potential variabilities in the forecasts to be evaluated by comparing the 50th percentile forecasts with the forecasts at other percentiles. Additional planning information, such as expected pattern shifts and probable peak values, could also be obtained from the forecasts produced by the QR model, while such information would not easily be obtained from other forecasting approaches.

The first important finding presented in this article is that the demand forecasts at the 50th percentile from the QR model closely estimate the actual hourly demands (see the red and blue circles in

The forecasted electricity demand distribution closely matched the actual demand distribution between 2012 and 2015 as shown in

The research was conducted during a contract research project funded by Eskom. The researchers would therefore like to thank Eskom for funding the research, and we would also like to thank the following Eskom staff members for the supply of data: Moonlight Mbata and Ntokozo Sigwebela. I would also like to thank the following CSIR staff members for their technical support: Nontembeko Dudeni-Tlhone and Nosizo Sebake.

The authors declare that they have no financial or personal relationship(s) that may have inappropriately influenced them in writing this article.

P.M. was the project leader. Did the analysis and wrote the paper. P.D, J.G. and V.S.S. were supervisors for P.M.’s PhD and gave guidance and critically read the paper. All authors edited the article. R.K. gave guidance, assisted in the writing of conclusions, proofread and did the final editing of the paper. S.S. helped with cleaning the data and searching for some literature.

Selected quantile regression models.

Variables | 1st percentile level |
50th percentile level |
99th percentile level |
|||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Estimates | 95% confidence limits | Pr > |t| | Estimates | 95% confidence limits | Pr > |t| | Estimates | 95% confidence limits | Pr > |t| | ||||

Intercept | 8.1435 | 8.0515 | 8.2355 | <0.0001 | 9.3791 | 9.351 | 9.4071 | <0.0001 | 9.7052 | 9.6011 | 9.8093 | <0.0001 |

Newyear | −0.1577 | −0.1909 | −0.1246 | <0.0001 | −0.1189 | −0.1328 | −0.105 | <0.0001 | −0.0308 | −0.0661 | 0.0045 | 0.0872 |

Humanrights | −0.0626 | −0.0999 | −0.0252 | 0.001 | −0.051 | −0.0614 | −0.0407 | <0.0001 | −0.0236 | −0.0376 | −0.0095 | 0.001 |

FreedomDay | −0.0612 | −0.1149 | −0.0076 | 0.0251 | −0.0315 | −0.0412 | −0.0219 | <0.0001 | −0.0443 | −0.0602 | −0.0285 | <0.0001 |

WorkersDay | −0.0386 | −0.055 | −0.0221 | <0.0001 | −0.0699 | −0.0773 | −0.0625 | <0.0001 | −0.064 | −0.0817 | −0.0463 | <0.0001 |

YouthDay | −0.0903 | −0.1129 | −0.0676 | <0.0001 | −0.0566 | −0.0695 | −0.0436 | <0.0001 | −0.0485 | −0.0641 | −0.033 | <0.0001 |

HeritageDay | −0.0711 | −0.0924 | −0.0498 | <0.0001 | −0.0532 | −0.0618 | −0.0447 | <0.0001 | −0.0355 | −0.0604 | −0.0107 | 0.0051 |

Reconciliation Day | 0.0251 | 0.0002 | 0.05 | 0.0478 | −0.0008 | −0.0093 | 0.0077 | 0.8546 | 0.0287 | 0.0059 | 0.0515 | 0.0137 |

ChristmasDay | −0.1253 | −0.1635 | −0.0872 | <0.0001 | −0.0844 | −0.0981 | −0.0706 | <0.0001 | −0.0586 | −0.0782 | −0.0389 | <0.0001 |

GoodwillDay | −0.1014 | −0.1377 | −0.0652 | <0.0001 | −0.0941 | −0.105 | −0.0831 | <0.0001 | −0.0882 | −0.1077 | −0.0686 | <0.0001 |

Month1 | −0.0703 | −0.0784 | −0.0622 | <0.0001 | −0.0095 | −0.0117 | −0.0073 | <0.0001 | 0.0026 | −0.0006 | 0.0058 | 0.1065 |

Month2 | 0.003 | −0.0022 | 0.0082 | 0.2596 | 0.0078 | 0.0059 | 0.0096 | <0.0001 | 0.0037 | 0.0008 | 0.0066 | 0.0116 |

Month3 | −0.0015 | −0.0074 | 0.0043 | 0.6131 | −0.0006 | −0.0026 | 0.0014 | 0.5641 | 0.0083 | 0.0048 | 0.0117 | <0.0001 |

Month4 | −0.0063 | −0.0132 | 0.0006 | 0.0741 | −0.0102 | −0.0121 | −0.0084 | <0.0001 | 0.0103 | 0.0063 | 0.0143 | <0.0001 |

Month5 | −0.0087 | −0.016 | −0.0014 | 0.0203 | 0.0019 | −0.0009 | 0.0047 | 0.1837 | 0.0376 | 0.032 | 0.0431 | <0.0001 |

Month6 | −0.0196 | −0.0314 | −0.0079 | 0.001 | 0.0212 | 0.0172 | 0.0251 | <0.0001 | 0.0453 | 0.0378 | 0.0528 | <0.0001 |

Month7 | −0.0158 | −0.0283 | −0.0033 | 0.0133 | 0.0115 | 0.0072 | 0.0158 | <0.0001 | 0.0336 | 0.0244 | 0.0428 | <0.0001 |

Month8 | −0.0499 | −0.06 | −0.0398 | <0.0001 | −0.0016 | −0.0053 | 0.0021 | 0.3961 | 0.0199 | 0.0102 | 0.0295 | <0.0001 |

Month9 | 0.0097 | 0.0054 | 0.014 | <0.0001 | 0.0127 | 0.0111 | 0.0144 | <0.0001 | 0.0155 | 0.0118 | 0.0192 | <0.0001 |

Month10 | 0.0168 | 0.0125 | 0.021 | <0.0001 | 0.0105 | 0.0088 | 0.0121 | <0.0001 | 0.0057 | 0.0027 | 0.0088 | 0.0002 |

Month11 | 0.0106 | 0.0057 | 0.0156 | <0.0001 | 0.0106 | 0.0088 | 0.0125 | <0.0001 | 0.0113 | 0.0083 | 0.0144 | <0.0001 |

Sin6 | 0.0048 | 0.0035 | 0.0061 | <0.0001 | 0.0113 | 0.0109 | 0.0118 | <0.0001 | 0.0178 | 0.0169 | 0.0188 | <0.0001 |

Cos6 | −0.0079 | −0.0092 | −0.0065 | <0.0001 | −0.0059 | −0.0063 | −0.0054 | <0.0001 | −0.0026 | −0.0036 | −0.0017 | <0.0001 |

Sin12 | −0.0594 | −0.0611 | −0.0577 | <0.0001 | −0.0689 | −0.0694 | −0.0684 | <0.0001 | −0.0681 | −0.0694 | −0.0669 | <0.0001 |

Cos12 | 0.0004 | −0.0011 | 0.0019 | 0.5896 | −0.0124 | −0.0128 | −0.0119 | <0.0001 | −0.0218 | −0.0228 | −0.0208 | <0.0001 |

Sin18 | −0.0013 | −0.0025 | −0.0001 | 0.033 | 0.0003 | −0.0002 | 0.0007 | 0.2228 | −0.0004 | −0.0012 | 0.0004 | 0.305 |

Cos18 | 0.0008 | −0.0005 | 0.002 | 0.2318 | 0.0002 | −0.0003 | 0.0006 | 0.5197 | 0.0008 | 0.0001 | 0.0016 | 0.0358 |

Sin24 | −0.0648 | −0.0663 | −0.0632 | <0.0001 | −0.0746 | −0.0751 | −0.074 | <0.0001 | −0.0719 | −0.0731 | −0.0708 | <0.0001 |

Cos24 | −0.0779 | −0.0792 | −0.0766 | <0.0001 | −0.0822 | −0.0827 | −0.0817 | <0.0001 | −0.0846 | −0.0862 | −0.083 | <0.0001 |

Lag70128 | 0.0086 | −0.0196 | 0.0367 | 0.5496 | −0.0835 | −0.0945 | −0.0724 | <0.0001 | −0.1164 | −0.1331 | −0.0997 | <0.0001 |

Lag70152 | −0.0002 | −0.0282 | 0.0279 | 0.9912 | −0.0421 | −0.0537 | −0.0306 | <0.0001 | 0.0071 | −0.0147 | 0.0289 | 0.524 |

Lag70176 | −0.091 | −0.1134 | −0.0686 | <0.0001 | −0.1039 | −0.1165 | −0.0913 | <0.0001 | −0.1735 | −0.194 | −0.1529 | <0.0001 |

Lag70200 | −0.0629 | −0.0871 | −0.0386 | <0.0001 | −0.0194 | −0.0309 | −0.0078 | 0.001 | 0.076 | 0.0543 | 0.0977 | <0.0001 |

Lag70224 | 0.4198 | 0.3919 | 0.4476 | <0.0001 | 0.4573 | 0.444 | 0.4705 | <0.0001 | 0.2638 | 0.2464 | 0.2811 | <0.0001 |

Lag70248 | −0.08 | −0.1024 | −0.0575 | <0.0001 | −0.127 | −0.1378 | −0.1161 | <0.0001 | −0.0007 | −0.0166 | 0.0152 | 0.9309 |

Sun | −0.012 | −0.0166 | −0.0073 | <0.0001 | −0.0164 | −0.0181 | −0.0146 | <0.0001 | −0.0201 | −0.0227 | −0.0174 | <0.0001 |

Mon | 0.0215 | 0.0172 | 0.0258 | <0.0001 | 0.0096 | 0.0074 | 0.0118 | <0.0001 | 0.0211 | 0.0173 | 0.0249 | <0.0001 |

Tues | 0.0314 | 0.0264 | 0.0363 | <0.0001 | 0.0198 | 0.018 | 0.0217 | <0.0001 | 0.0146 | 0.0116 | 0.0176 | <0.0001 |

Wed | 0.033 | 0.0281 | 0.038 | <0.0001 | 0.0153 | 0.0133 | 0.0174 | <0.0001 | 0.0089 | 0.0055 | 0.0123 | <0.0001 |

Thurs | 0.0341 | 0.0298 | 0.0385 | <0.0001 | 0.0159 | 0.0141 | 0.0177 | <0.0001 | 0.0105 | 0.0069 | 0.0141 | <0.0001 |

Fri | 0.0314 | 0.0274 | 0.0354 | <0.0001 | 0.0155 | 0.0137 | 0.0173 | <0.0001 | 0.0062 | 0.0035 | 0.0088 | <0.0001 |

Fribtwn | −0.1295 | −0.1758 | −0.0833 | <0.0001 | −0.0673 | −0.0738 | −0.0608 | <0.0001 | −0.0348 | −0.0686 | −0.001 | 0.0438 |

Monbtwn | −0.0115 | −0.0416 | 0.0187 | 0.4561 | −0.0289 | −0.0357 | −0.0222 | <0.0001 | 0.0006 | −0.0136 | 0.0148 | 0.933 |

Lngwknd | −0.0579 | −0.0666 | −0.0492 | <0.0001 | −0.0264 | −0.0288 | −0.0239 | <0.0001 | −0.0175 | −0.0222 | −0.0128 | <0.0001 |

Dec_closure | −0.1011 | −0.1117 | −0.0906 | <0.0001 | −0.0563 | −0.0593 | −0.0533 | <0.0001 | −0.0405 | −0.0449 | −0.0361 | <0.0001 |

Winter_SchoolholiDay | −0.0069 | −0.0117 | −0.0021 | 0.0051 | 0.0045 | 0.0028 | 0.0062 | <0.0001 | 0.0078 | 0.0044 | 0.0112 | <0.0001 |

Easter | −0.0826 | −0.0996 | −0.0656 | <0.0001 | −0.0502 | −0.0546 | −0.0458 | <0.0001 | −0.0495 | −0.0578 | −0.0412 | <0.0001 |

Winter | 0.0411 | 0.029 | 0.0532 | <0.0001 | 0.0341 | 0.0303 | 0.038 | <0.0001 | 0.042 | 0.0332 | 0.0508 | <0.0001 |

Sin, Sine term of Fourier series; Cos, Cosine term of Fourier series; Lag, time lag; Fribtwn, Friday preceded by holiday; Monbtwn, Monday preceded by a holiday;

Newyear, New year; Humanrights, Human rights day; FreedomDay, Freedom day; WorkersDay, Workers day; YouthDay, Youth day; HeritageDay, Heritage day; ReconciliationDay, Reconciliation day; ChristmasDay, Christmas day; GoodwillDay, Day of Goodwill; Sun, Sunday; Mon, Monday; Tues, Tuesday; Wed, Wednesday; Thurs, Thursday; Fri, Friday; Lngwknd, Long weekend; Dec_closure, December closure; Winter_SchoolholiDay, Winter school holiday.

Mean absolute percentage error – Forecasts at the 50th percentile.

Hours | 2012 | 2013 | 2014 | 2015 | Average |
---|---|---|---|---|---|

0 | 2.92 | 2.23 | 3.29 | 3.93 | 3.09 |

1 | 2.75 | 1.83 | 3.02 | 3.67 | 2.82 |

2 | 2.67 | 1.65 | 2.82 | 3.58 | 2.68 |

3 | 2.58 | 1.54 | 2.64 | 3.37 | 2.53 |

4 | 2.46 | 1.72 | 2.35 | 3.02 | 2.39 |

5 | 3.85 | 3.67 | 3.68 | 3.84 | 3.76 |

6 | 4.30 | 4.15 | 3.99 | 4.32 | 4.19 |

7 | 2.76 | 3.44 | 3.88 | 4.62 | 3.68 |

8 | 2.15 | 2.55 | 2.89 | 3.47 | 2.77 |

9 | 2.15 | 2.26 | 2.54 | 2.65 | 2.40 |

10 | 2.35 | 1.88 | 1.97 | 2.38 | 2.15 |

11 | 2.59 | 1.79 | 1.91 | 2.48 | 2.19 |

12 | 2.47 | 1.72 | 2.21 | 2.83 | 2.31 |

13 | 2.50 | 1.79 | 2.49 | 3.15 | 2.48 |

14 | 2.55 | 1.79 | 2.55 | 3.20 | 2.52 |

15 | 2.58 | 1.75 | 2.30 | 2.96 | 2.40 |

16 | 2.41 | 1.94 | 2.14 | 2.93 | 2.36 |

17 | 2.85 | 3.23 | 2.80 | 3.60 | 3.12 |

18 | 3.34 | 4.29 | 3.67 | 3.67 | 3.74 |

19 | 2.19 | 2.42 | 2.12 | 2.55 | 2.32 |

20 | 2.42 | 2.22 | 1.98 | 2.83 | 2.36 |

21 | 2.19 | 2.21 | 2.22 | 3.31 | 2.48 |

22 | 2.15 | 2.22 | 2.84 | 3.81 | 2.76 |

23 | 2.60 | 2.19 | 3.23 | 3.93 | 2.99 |

Average | 2.66 | 2.35 | 2.73 | 3.34 | 2.77 |

The autocorrelation plot: Correlation of hourly electricity demand data.