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The main source of electricity worldwide stems from fossil fuels, contributing to air pollution, global warming, and associated adverse effects. This study explores wind energy as a potential alternative. Nevertheless, the variable nature of wind introduces uncertainty in its reliability. Thus, it is necessary to identify an appropriate machine learning model capable of reliably forecasting wind speed under various environmental conditions. This research compares the effectiveness of Dynamic Architecture for Artificial Neural Networks (DAN2), convolutional neural networks (CNN), random forest and XGBOOST in predicting wind speed across three locations in South Africa, characterised by different weather patterns. The forecasts from the four models were then combined using quantile regression averaging models, generalised additive quantile regression (GAQR) and quantile regression neural networks (QRNN). Empirical results show that CNN outperforms DAN2 in accurately forecasting wind speed under different weather conditions. This superiority is likely due to the inherent architectural attributes of CNNs, including feature extraction capabilities, spatial hierarchy learning, and resilience to spatial variability. The results from the combined forecasts were comparable with those from the QRNN, which was slightly better than those from the GAQR model. However, the combined forecasts were more accurate than the individual models. These results could be useful to decision-makers in the energy sector.

Abstract Modelling of solar irradiation is paramount to renewable energy management. This warrants the inclusion of additive effects to predict solar irradiation. Modelling of additive effects to solar irradiation can improve the forecasting accuracy of prediction frameworks. To help develop the frameworks, this current study modelled the additive effects using non-parametric quantile regression (QR). The approach applies quantile splines to approximate non-parametric components when finding the best relationships between covariates and the response variable. However, some additive effects are perceived as linear. Thus, the study included the partial linearly additive quantile regression model (PLAQR) in the quest to find how best the additive effects can be modelled. As a result, a comparative investigation on the forecasting performances of the PLAQR, an additive quantile regression (AQR) model and the new quantile generalised additive model (QGAM) using out-of-sample and probabilistic forecasting metric evaluations was done. Forecasted density plots, Murphy diagrams and results from the Diebold–Mariano (DM) hypothesis test were also analysed. The density plot, the curves on the Murphy diagram and most metric scores computed for the QGAM were slightly better than for the PLAQR and AQR models. That is, even though the DM test indicates that the PLAQR and AQR models are less accurate than the QGAM, we could not conclude an outright greater forecasting performance of the QGAM than the PLAQR or AQR models. However, in situations of probabilistic forecasting metric preferences, each model can be prioritised to be applied to the metric where it performed slightly the best. The three models performed differently in different locations, but the location was not a significant factor in their performances. In contrast, forecasting horizon and sample size influenced model performance differently in the three additive models. The performance variations also depended on the metric being evaluated. Therefore, the study has established the best forecasting horizons and sample sizes for the different metrics. It was finally concluded that a 20% forecasting horizon and a minimum sample size of 10000 data points are ideal when modelling additive effects of solar irradiation using non-parametric QR.

Dimensionality poses a challenge in developing quality predictive models. Often when modelling solar irradiance (SI), many covariates are considered. Training such data has several disadvantages. This study sought to identify the best variable embedded selection method for different location and time horizon combinations from Southern Africa solar irradiance data. It introduced new variable selection methods into solar irradiation studies, namely penalised quantile regression (PQR), regularised random forests (RRF), and quantile regression forest (QRF). Stability analysis, performance and accuracy metric evaluations were used to compare them with the common lasso, elastic and ridge regression methods. The QRF model performed best in all locations followed by the shrinkage methods on hourly data. However, it was found that QRF is not sensitive to associations through correlations, thereby ignoring the relevance of variables while focusing on importance. Among the shrinkage methods, the lasso performed best in only one location. On the 24-hour horizon, elastic net dominated the performances among the shrinkage methods, but QRF was best in three locations of the six considered. Results confirmed that variable selection methods performed differently on different situational data sets. Depending on the strengths of the methods, results were combined to identify the most paramount variables. Day, total rainfall, and wind direction were superfluous features in all situations. The study concluded that shrinkage methods are best in cases of extreme multicollinearity, while QRF is best on data sets with outliers or/and heavy tails.

In recent years, there has been increasing interest in the joint modelling of compound extreme events such as high temperatures and low rainfall. The increase in the frequency of occurrence of these events in many regions has necessitated the development of models for estimating the concurrent probabilities of such compound extreme events. The current study discusses an application of copula models in predicting the concurrent probabilities of compound low rainfall and high-temperature events using data from the Lowveld region of the Limpopo province in South Africa. The second stage discussed two indicators for monitoring compound high temperature and low rainfall events. Empirical results from the study show that elevations ranging from 100–350 m, 350–700 m and 700–1200 m exhibit varying probabilities of experiencing drought, with mild droughts having approximately 64%, 66%, and 65% chances, moderate droughts around 36%, 39%, and 38%, and severe droughts at approximately 16%, 19%, and 18%, respectively. Furthermore, the logistic regression models incorporating the southern oscillation index as a covariate yielded comparable results of copula-based models. The methodology discussed in this paper is robust and can be applied to similar datasets in any regional setting globally. These findings could be useful to disaster management decision makers, helping them formulate effective mitigation strategies and emergency response plans.

Precise tourism estimates for tourism destination sites are crucial for decision-making. The objective of the study is to model and project Great Zimbabwe National Monuments (GZNM) tourist arrivals by combining hierarchical tourism forecasts. The approach improves tourism forecasting accuracy. GZNM monthly tourist arrivals are grouped according to tourism sources. A logarithm transformation is applied to tame the volatile data. Forecasting accuracy of the Simple Average Combination Method (SACM) and three hierarchical forecasting approaches (top-down, bottom-up, and optimal combination) were compared. The SACM under Autoregressive Integrated Moving Average (ARIMA) outperformed the other models, according to Root Mean Square Error (RMSE) measure. SACM is used to combine future tourist arrivals for the following 60 months and show a slow increase in tourist arrivals at GZNM. The data used in modeling are outside the COVID-19 pandemic period. Tourism stakeholders are encouraged to adopt the SACM in future tourism projections as it improves forecasting accuracy. Tourism stakeholders could carefully strategise and plan a recovery and ensure improvement in the tourism sector beyond the COVID-19 pandemic period. The COVID-19 pandemic is significantly affecting the tourism industry, reducing tourist arrivals to zero in some cases. The study revealed a fresh line of inquiry into how combining projections can increase forecasting accuracy.

The global horizontal irradiance data recorded at the earth's horizontal surface is a mixture of deterministic (extra-terrestrially) and stochastic components due to the ever-changing atmospheric conditions. The Box-Jenkins short memory stochastic models and their hybrid versions have been used successfully to forecast solar irradiance data. However, these models lack robustness and faulter for distant horizon forecasting such as more than 2 hours in the hourly case. Using a quantile regression model as both a robust benchmark and robustification model, we remedy this drawback by using long memory models and their hybrid versions in 24 hour ahead irradiance forecasting at three sites in South Africa (RVD, UPR and SUN). Results show that the inherent long memory is anti-persistent in all models but one that shows persistence. The forecasts from these long memory-based models are generally within the prediction interval of the benchmark model forecasts for two sites. Based on the RMSE, rRMSE, MAE, rMAE and the pinball loss function at the central quantile level and two extreme quantile levels, forecasts obtained through robustification via forecasts combinations with the quantile regression yield the best performance. The predictive ability of the best model before forecasts combination against the quantile regression model combined (robustified) model based Diebold-Mariano test rejected the null hypothesis of the same accuracy. As a follow up to distinguish which has a superior predictive accuracy, “murphy diagrams” were used. For the RVD and UPR sites, the model based on forecast combination has the best predictive accuracy while the quantile regression based benchmark model has the best accuracy for the SUN data.

Solar irradiance forecasting is essential in renewable energy grids amongst others for back-up programming, operational planning, and short-term power purchases. This study focuses on forecasting hourly solar irradiance using data obtained from the Southern African Universities Radiometric Network at the University of Pretoria radiometric station. The study compares the predictive performance of long short-term memory (LSTM) networks, support vector regression and feed forward neural networks (FFNN) models for forecasting short-term solar irradiance. While all the models outperform principal component regression model, a benchmark model in this study, the FFNN yields the lowest mean absolute error and root mean square error on the testing set. Empirical results show that the FFNN model produces the most accurate forecasts based on mean absolute error and root mean square error. Forecast combination of machine learning models' forecasts is done using convex combination and quantile regression averaging (QRA). The predictive performance we found is statistically significant on the Diebold Mariano and Giacomini-White tests. Based on all the forecast accuracy measures used in this study including the statistical tests, QRA is found to be the best forecast combination method. QRA was also the best forecasting model compared with the stand-alone machine learning models. The median method for combining interval limits gives the best results on prediction interval widths analysis. This is the first application of LSTM on South African and African solar irradiance data to the best of our knowledge. This study has shown that providing adequate and detailed evaluation metrics, including statistical tests in forecasting gives more insight into the developed forecasting models.

Extreme value modelling of peak load process is critical to the reliable specification of power generation, distribution and maintenance purposes during both peak and off-peak periods. In this study, a frequency assessment of extreme peak electricity demand for the four seasons of the year using South African data for the period, January 1997 to December 2013 is carried out. A point process approach from extreme value theory is proposed as an ingenious extreme value theory approach. The data are made stationary by using a time-varying threshold which has a positive shift factor. The non-linear detrended datasets are then grouped into summer, spring, winter and autumn according to the calendar dates in the Southern Hemisphere. The datasets were declustered to keep the series relatively independent using Ferro and Segers automatic declustering method. A stationary point process model is then fitted to each of the cluster maxima. The modelling framework, which is easily extensible to other peak load parameters, assumes that peak power follows a Poisson point process. The parameters of the developed model are estimated using the maximum likelihood method. Empirical results show that daily peak electricity demand could be experienced approximately 27, 16, 7 and 15 days per year in winter, spring, summer and autumn, respectively. The modelling approach could assist system operators of utility companies in scheduling maintenance of generating units including long term planning.

Challenges in utilising fossil fuels for generating energy call for the adoption of renewable energy sources. This study focuses on modelling and nowcasting optimal tilt angle(s) of solar energy harnessing using historical time series data collected from one of South Africa’s radiometric stations, USAid Venda station in Limpopo Province. In the study, we compared random forest (RF), K-nearest neighbours (KNN), and long short-term memory (LSTM) in nowcasting of optimum tilt angle. Gradient boosting (GB) is used as the benchmark model to compare the model’s predictive accuracy. The performance measures of mean absolute error (MAE), mean square error (MSE), root mean square error (RMSE) and R2 were used, and the results showed LSTM to have the best performance in nowcasting optimum tilt angle compared to other models, followed by the RF and GB, whereas KNN was the worst-performing model.

Accurate global horizontal irradiance (GHI) forecasting promotes power grid stability. Most of the research on solar irradiance forecasting has been based on a single-site analysis. It is crucial to explore multisite modeling to capture variations in weather conditions between various sites, thereby producing a more robust model. In this research, we propose the use of spatial regression coupled with Gaussian Process Regression (GP Spatial) and the GP Autoregressive Spatial model (GP-AR Spatial) for the prediction of GHI using data from seven radiometric stations from South Africa and one from Namibia. The results of the proposed methods were compared with a benchmark model, the Linear Spatial Temporal Regression (LSTR) model. Five validation sets each comprised of three stations were chosen. For each validation set, the remaining five stations were used for training. Based on root mean square error, the GP model gave the most accurate forecasts across the validation sets. These results were confirmed by the statistical significance tests using the Giacommini–White test. In terms of coverage probability, there was a 100% coverage on three validation sets and the other two had 97% and 99%. The GP model dominated the other two models. One of the study’s contributions is using standardized forecasts and including a nonlinear trend covariate, which improved the accuracy of the forecasts. The forecasts were combined using a monotone composite quantile regression neural network and a quantile generalized additive model. This modeling framework could be useful to power utility companies in making informed decisions when planning power grid management, including large-scale solar power integration onto the power grid.

Forecasting electricity demand in South Africa remains an increasingly national challenge as the government does not sufficiently take into account the impact of the electricity prices in their electricity demand forecast. Effective measures to rapidly reduce the demand of electricity are urgently needed to deal with future electricity prices and government policies uncertainties within the energy industry. Moreover, long-term peak electricity demand forecasting methods are needed to quantify the uncertainty of future electricity demand for better electricity security management. The prediction of long-term electricity demand assists decision makers in the electricity sector in planning for capacity generation. This paper presents an application of quantile regression averaging (QRA) approach using South African monthly and quarterly data ranging from January 2007 to December 2014. Variable selection is done in a comparative manner using ridge, least absolute shrinkage and selection operator (Lasso), cross validation (CV) and elastic net. We compare the forecasting accuracy of monthly peak electricity demand (MPED) and quarterly peak electricity demand (QPED) forecasting models using generalised additive models (GAMs) and QRA. The coefficient estimates for ridge, Lasso and elastic net are estimated and compared using MPED and QPED data.

The interaction between global horizontal irradiance (GHI) and temperature helps determine the maximum amount of solar power generated. As temperature increases, GHI increases up to the point that it increases at a decreasing rate and then decreases. Therefore, system operators need to know the maximum possible solar power which can be generated. Using the multivariate adaptive regression splines, extreme value theory and copula models, the present paper seeks to determine the maximum temperature that will result in the generation of the maximum GHI ceteris paribus. The paper also discusses extremal dependence modelling of GHI with temperature and relative humidity (RH) at one radiometric station using South African data from 16 November 2015 to 16 November 2021. Empirical results show that the marginal increases of GHI converge to 0.12 W/m2 when temperature converges to 44.26 °C and the marginal increases of GHI converge to −0.1 W/m2 when RH converges to 103.26%. Conditioning on GHI, the study found that temperature and RH variables have a negative extremal dependence on large values of GHI. Due to the nonlinearity and different structure of the dependence on GHI against temperature and RH, unlike previous literature, we use three Archimedean copula functions: Clayton, Frank and Gumbel, to model the dependence structure. The modelling approach discussed in this paper could be useful to system operators in power utilities who must optimally integrate highly intermittent renewable energies on the grid.

With the use of empirical data, this paper focuses on solving financial and investment issues involving extremal dependence of 10 pairwise combinations of the 5 BRICS (Brazil, Russia, India, China, and South Africa) stock markets. Daily closing equity indices from 5 January 2010 to 6 August 2018 are used in the study. Unlike previous literature, we use bivariate point process and conditional multivariate extreme value models to investigate the extremal dependence of the stock market returns. However, it is observed that the point process was able to model many more extreme observations or exceedances that contribute to the likelihood estimation. It gives more information than the threshold excess method of the CMEV model. This study shows varying levels of low extremal dependence structure whose outcomes are highly beneficial to investors, portfolio managers and other market participants interested in maximising investment returns and financial gains.

Background: In South Africa, entrepreneurship literature demonstrates that three out of four businesses collapse within 3 years of their inception. A plethora of research effort identifies factors such as the lack of finance and access to markets as the leading causes for the high attrition rates amongst emerging businesses. This study finds the narrative to be limiting and inadequate as it does not address the possible gap between entrepreneurs’ expectations and their realities of managing their businesses. Aim: To present the entrepreneurship gaps framework (EGF), an early-stage business diagnostic tool that seeks to assess entrepreneurs’ preparedness. Setting: This study focused on emerging entrepreneurs operating within the limits of developing economies. The framework can be used by emerging entrepreneurs, capacity development institutions and lenders. Methods: A descriptive research design supported by a mixed-method research approach was employed. This was coupled by a two-phase data collection procedure which took place within Limpopo province with 215 participants. Explorative data analysis based on discrete choice models was further implemented. Results: Findings on the EGF illustrated the ability of the framework to act as a more comprehensive diagnostic mechanism that improves early-stage entrepreneurship survival. Conclusion: Entrepreneurship gaps framework is a decision-making tool that can be used by lenders and capacity development institutions to evaluate the emerging entrepreneur with respect to specific areas of business. This results in the necessary support for improving entrepreneur preparedness being provided to entrepreneurs. Secondly, entrepreneurs are likely to benefit from the EGF, if used as a self-diagnostic tool to measure their business preparedness and experience.

It is important to predict extreme electricity demand in power utilities as the uncertainties in the future of electricity demand distribution have to be taken into consideration to achieve the desired goals. The study focused on the prediction of extremely high conditional quantiles (between 0.95 and 0.9999) and extremely low quantiles (between 0.001 and 0.05) of electricity demand using South African data. The paper discusses a comparative analysis of the additive quantile regression model with an extremal mixture model and a nonlinear quantile regression model. The estimated quantiles at each level were then combined using the median approach. The comparisons were carried out using daily peak electricity demand data ranging from January 1997 to May 2014. Proper scoring rules were used to compare the three models, and the model with the smallest score was preferred. The results could be useful to system operators including decision-makers in power utility companies by giving insights and guidance for future electricity demand patterns. The prediction of extremely high quantiles of daily peak electricity demand could help system operators know the possible largest demand that will enable them to supply adequate electricity to consumers and shift demand to off-peak periods. The prediction of extreme conditional quantiles of daily peak electricity demand in the context of South Africa using additive quantile regression, nonlinear quantile regression, and extremal mixture models has not been performed previously to the best of our knowledge.

In this paper, seasonal autoregressive integrated moving average (SARIMA) and regression with SARIMA errors (regression-SARIMA) models are developed to predict daily peak electricity demand in South Africa using data for the period 1996 to 2009. The performance of the developed models is evaluated by comparing them with Winter’s triple exponential smoothing model. Empirical results from the study show that the SARIMA model produces more accurate short-term forecasts. The regression-SARIMA modelling framework captures important drivers of electricity demand. These results are important to decision makers, load forecasters and systems operators in load flow analysis and scheduling of electricity.

The paper discusses the modelling of the influence of temperature on average daily electricity demand in South Africa using a piecewise linear regression model and the generalized extreme value theory approach for the period - 2000 to 2010. Empirical results show that electricity demand in South Africa is highly sensitive to cold temperatures. Extreme low average daily temperatures of the order of 8.20C are very rare in South Africa. They only occur about 8 times in a year and result in huge increases in electricity demand.

Probabilistic solar power forecasting has been critical in Southern Africa because of major shortages of power due to climatic changes and other factors over the past decade. This paper discusses Gaussian process regression (GPR) coupled with core vector regression for short-term hourly global horizontal irradiance (GHI) forecasting. GPR is a powerful Bayesian non-parametric regression method that works well for small data sets and quantifies the uncertainty in the predictions. The choice of a kernel that characterises the covariance function is a crucial issue in Gaussian process regression. In this study, we adopt the minimum enclosing ball (MEB) technique. The MEB improves the forecasting power of GPR because the smaller the ball is, the shorter the training time, hence performance is robust. Forecasting of real-time data was done on two South African radiometric stations, Stellenbosch University (SUN) in a coastal area of the Western Cape Province, and the University of Venda (UNV) station in the Limpopo Province. Variables were selected using the least absolute shrinkage and selection operator via hierarchical interactions. The Bayesian approach using informative priors was used for parameter estimation. Based on the root mean square error, mean absolute error and percentage bias the results showed that the GPR model gives the most accurate predictions compared to those from gradient boosting and support vector regression models, making this study a useful tool for decision-makers and system operators in power utility companies. The main contribution of this paper is in the use of a GPR model coupled with the core vector methodology which is used in forecasting GHI using South African data. This is the first application of GPR coupled with core vector regression in which the minimum enclosing ball is applied on GHI data, to the best of our knowledge.

Renewable energy forecasts are critical to renewable energy grids and backup plans, operational plans, and short-term power purchases. This paper focused on short-term forecasting of high-frequency global horizontal irradiance data from one of South Africa’s radiometric stations. The aim of the study was to compare the predictive performance of the genetic algorithm and recurrent neural network models with the K-nearest neighbour model, which was used as the benchmark model. Empirical results from the study showed that the genetic algorithm model has the best conditional predictive ability compared to the other two models, making this study a useful tool for decision-makers and system operators in power utility companies. To the best of our knowledge this is the first study which compares the genetic algorithm, the K-nearest neighbour method, and recurrent neural networks in short-term forecasting of global horizontal irradiance data from South Africa.

Wind offers an environmentally sustainable energy resource that has seen increasing global adoption in recent years. However, its intermittent, unstable and stochastic nature hampers its representation among other renewable energy sources. This work addresses the forecasting of wind speed, a primary input needed for wind energy generation, using data obtained from the South African Wind Atlas Project. Forecasting is carried out on a two days ahead time horizon. We investigate the predictive performance of artificial neural networks (ANN) trained with Bayesian regularisation, decision trees based stochastic gradient boosting (SGB) and generalised additive models (GAMs). The results of the comparative analysis suggest that ANN displays superior predictive performance based on root mean square error (RMSE). In contrast, SGB shows outperformance in terms of mean average error (MAE) and the related mean average percentage error (MAPE). A further comparison of two forecast combination methods involving the linear and additive quantile regression averaging show the latter forecast combination method as yielding lower prediction accuracy. The additive quantile regression averaging based prediction intervals also show outperformance in terms of validity, reliability, quality and accuracy. Interval combination methods show the median method as better than its pure average counterpart. Point forecasts combination and interval forecasting methods are found to improve forecast performance.

Natural hazards (events that may cause actual disasters) are established in the literature as major causes of various massive and destructive problems worldwide. The occurrences of earthquakes, floods and heat waves affect millions of people through several impacts. These include cases of hospitalisation, loss of lives and economic challenges. The focus of this study was on the risk reduction of the disasters that occur because of extremely high temperatures and heat waves. Modelling average maximum daily temperature (AMDT) guards against the disaster risk and may also help countries towards preparing for extreme heat. This study discusses the use of the r largest order statistics approach of extreme value theory towards modelling AMDT over the period of 11 years, that is, 2000–2010. A generalised extreme value distribution for r largest order statistics is fitted to the annual maxima. This is performed in an effort to study the behaviour of the r largest order statistics. The method of maximum likelihood is used in estimating the target parameters and the frequency of occurrences of the hottest days is assessed. The study presents a case study of South Africa in which the data for the non-winter season (September–April of each year) are used. The meteorological data used are the AMDT that are collected by the South African Weather Service and provided by Eskom. The estimation of the shape parameter reveals evidence of a Weibull class as an appropriate distribution for modelling AMDT in South Africa. The extreme quantiles for specified return periods are estimated using the quantile function and the best model is chosen through the use of the deviance statistic with the support of the graphical diagnostic tools. The Entropy Difference Test (EDT) is used as a specification test for diagnosing the fit of the models to the data.

The paper discusses an application of generalised additive models (GAMs) in predicting medium-term hourly electricity demand using South African data for 2009 to 2013. Variable selection was done using least absolute shrinkage and selection operator (Lasso) via hierarchical interactions, resulting in a model called GAM-Lasso. The GAM-Lasso model was then extended by including tensor product interactions to yield a second model, called GAM- -Lasso. Comparative analyses of these two models were done with a gradient-boosting model to act as a benchmark model and the third model. The forecasts from the three models were combined using a forecast combination algorithm where the average loss suffered by the models was based on the pinball loss function. The results showed significantly improved accuracy of forecasts, making this study a useful tool for decision-makers and system operators in power utility companies, particularly in maintenance planning including medium-term risk assessment. A major contribution of this paper is the inclusion of a nonlinear trend. Another contribution is the inclusion of temperature based on two thermal regions of South Africa.

Uncertainty in electricity demand is caused by many factors. Large changes are usually attributed to extreme weather conditions and the general random usage of electricity by consumers. More understanding requires a detailed analysis using a stochastic process approach. This paper presents a Markov chain analysis to determine stationary distributions (steady state probabilities) of large daily changes in peak electricity demand. Such large changes pose challenges to system operators in the scheduling and dispatching of electrical energy to consumers. The analysis used on South African daily peak electricity demand data from 2000 to 2011 and on a simple two-state discrete-time Markov chain modelling framework was adopted to estimate steady-state probabilities of two states: positive inter-day changes (increases) and negative inter-day changes (decreases). This was extended to a three-state Markov chain by distinguishing small positive changes and extreme large positive changes. For the negative changes, a decrease state was defined. Empirical results showed that the steady state probability for an increase was 0.4022 for the two-state problem, giving a return period of 2.5 days. For the three state problem, the steady state probability of an extreme increase was 0.0234 with a return period of 43 days, giving approximately nine days in a year that experience extreme inter-day increases in electricity demand. Such an analysis was found to be important for planning, load shifting, load flow analysis and scheduling of electricity, particularly during peak periods.

Due to its variability, solar power generation poses challenges to grid energy management. In order to ensure an economic operation of a national grid, including its stability, it is important to have accurate forecasts of solar power. The current paper discusses probabilistic forecasting of twenty-four hours ahead of global horizontal irradiance (GHI) using data from the Tellerie radiometric station in South Africa for the period August 2009 to April 2010. Variables are selected using a least absolute shrinkage and selection operator (Lasso) via hierarchical interactions and the parameters of the developed models are estimated using the Barrodale and Roberts’s algorithm. Two forecast combination methods are used in this study. The first is a convex forecast combination algorithm where the average loss suffered by the models is based on the pinball loss function. A second forecast combination method, which is quantile regression averaging (QRA), is also used. The best set of forecasts is selected based on the prediction interval coverage probability (PICP), prediction interval normalised average width (PINAW) and prediction interval normalised average deviation (PINAD). The results demonstrate that QRA gives more robust prediction intervals than the other models. A comparative analysis is done with two machine learning methods—stochastic gradient boosting and support vector regression—which are used as benchmark models. Empirical results show that the QRA model yields the most accurate forecasts compared to the machine learning methods based on the probabilistic error measures. Results on combining prediction interval limits show that the PMis the best prediction limits combination method as it gives a hit rate of 0.955 which is very close to the target of 0.95. This modelling approach is expected to help in optimising the integration of solar power in the national grid.

Short-term hourly load forecasting in South Africa using additive quantile regression (AQR) models is discussed in this study. The modelling approach allows for easy interpretability and accounting for residual autocorrelation in the joint modelling of hourly electricity data. A comparative analysis is done using generalised additive models (GAMs). In both modelling frameworks, variable selection is done using least absolute shrinkage and selection operator (Lasso) via hierarchical interactions. Four models considered are GAMs and AQR models with and without interactions, respectively. The AQR model with pairwise interactions was found to be the best fitting model. The forecasts from the four models were then combined using an algorithm based on the pinball loss (convex combination model) and also using quantile regression averaging (QRA). The AQR model with interactions was then compared with the convex combination and QRA models and the QRA model gave the most accurate forecasts. Except for the AQR model with interactions, the other two models (convex combination model and QRA model) gave prediction interval coverage probabilities that were valid for the 90 % , 95 % and the 99 % prediction intervals. The QRA model had the smallest prediction interval normalised average width and prediction interval normalised average deviation. The modelling framework discussed in this paper has established that going beyond summary performance statistics in forecasting has merit as it gives more insight into the developed forecasting models.