Probabilistic renewable energy modelling in South Africa

Abstract

The variability of solar power creates problems in planning and managing power system operations. It is critical to forecast accurately in order to maintain the safety and stability of large-scale integration of solar power into the grid. Accurate forecasting is vital because it prevents transmission obstruction and maintains a power equilibrium. This thesis uses robust models to solve this problem by addressing four main issues. The first issue involves the construction of quantile regression models for forecasting extreme peak electricity demand and determining the optimal number of units to commit at minimal costs for each period using the forecasts obtained from the developed models. The bounded variable mixed-integer linear programming (MILP) model solves the unit commitment (UC) problem. This is based on priority constraints where demand is first met from renewable energy sources followed by energy from fossil fuels. Secondly, the thesis discusses the modelling and prediction of extremely high quantiles of solar power. The methods used are a semi-parametric extremal mixture (SPEM), generalised additive extreme value (GAEV) or quantile regression via asymmetric Laplace distribution (QR-ALD), additive quantile regression with covariate t (AQR-1), additive quantile regression with temperature variable (AQR-2) and penalised cubic regression smoothing spline (benchmark) models. The predictions from this study are valuable to power utility decision-makers and system operators in knowing the maximum possible solar power which can be generated. This helps them make high-risk decisions and regulatory frameworks requiring high-security levels. As far as we know, this is the first application to conduct a comparative analysis of the proposed robust models using South African solar irradiance 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 temperature range in which the maximum possible solar power can be generated. The study used the multivariate adaptive regression splines and extreme value theory to determine the maximum temperature to generate the maximum GHI ceteris paribus. Lastly, the study discusses extremal dependence modelling of GHI with temperature and relative humidity (RH) using the conditional multivariate extreme value (CMEV) and copula modes. 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. This work was then extended by constructing a mixture copula model which combined the Frank and Gumbel models. One of the contributions of this thesis is the construction of additive quantile regression models for forecasting extreme quantiles of electrical load, which are then used in solving the UC problem with bounded MILP with priority constraints. The other contribution is developing a modelling framework that shows that GHI converges to its upper limit if temperature converges to the upper bound. Another contribution is constructing a mixture of some copulas for modelling the extremal dependence of GHI with temperature and RH. This thesis reveals the following key findings: (i) the additive quantile regression model is the best-fitting model for hours 18:00 and 19:00. In contrast, the linear quantile regression model is the best-fitting model for hours 20:00 and 21:00. The UC problem results show that using all the generating units, such as hydroelectric, wind power, concentrated solar power and solar photovoltaic is less costly. (ii) the AQR-2 was the best-fitting model and gave the most accurate prediction of quantiles at τ = 0.95, 0.97, 0.99 and 0.999, while at 0.9999- quantile, the GAEV model had the most accurate predictions. (iii) 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. (iv) the dependence structure between GHI and variable temperature and RH is asymmetric. Furthermore, the Frank copula is the best-fitting model for variable temperature and RH, implying the presence of extreme co-movements. The modelling framework discussed in this thesis could be useful to decisioniii makers in power utilities, who must optimally integrate highly intermittent renewable energies on the grid. It could be helpful to system operators that face uncertainty in GHI power production due to extreme temperatures and RH, including maintaining the minimum cost by scheduling and dispatching electricity during peak hours when the grid is constrained due to peak load demand.