ETD Collection

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    Modelling temperature in South Africa using extreme value theory
    (2018) Nemukula, Murendeni M.
    This dissertation focuses on demonstrating the use of extreme value theory in modelling temperature in South Africa. The purpose of modelling temperature is to investigate the frequency of occurrences of extremely low and extremely high temperatures and how they influence the demand of electricity over time. The data comprise a time series of average hourly temperatures that are collected by the South African Weather Service over the period 2000−2010 and supplied by Eskom. The generalized extreme value distribution (GEVD) for r largest order statistics is fitted to the average maximum daily temperature (non-winter season) using the maximum likelihood estimation method and used to estimate extreme high temperatures which result in high demand of electricity due to use of cooling systems. The estimation of the shape parameter reveals evidence that the Weibull family of distributions is an appropriate fit to the data. A frequency analysis of extreme temperatures is carried out and the results show that most of the extreme temperatures are experienced during the months January, February, November and December of each year. The generalized Pareto distribution (GPD) is firstly used for modelling the average minimum daily temperatures for the period January 2000 to August 2010. A penalized regression cubic smoothing spline is used as a time varying threshold. We then extract excessesabovethecubicregressionsmoothingsplineandfitanon-parametricmixturemodel to get a sufficiently high threshold. The data exhibit evidence of short-range dependence and high seasonality which lead to the declustering of the excesses above the threshold and fit the GPD to cluster maxima. The estimate of the shape parameter shows that the Weibullfamilyofdistributionsisappropriateinmodellingtheuppertailofthedistribution. The stationary GPD and the piecewise linear regression models are used in modelling the influence of temperature above the reference point of 22◦C on the demand of electricity. The stationary and non-stationary point process models are fitted and used in determining the frequency of occurrence of extremely high temperatures. The orthogonal and the reparameterizationapproachesofdeterminingthefrequencyandintensityofextremeshave i been used to establish that, extremely hot days occur in frequencies of 21 and 16 days per annum, respectively. For the fact that temperature is established as a major driver of electricity demand, this dissertation is relevant to the system operators, planners and decision makers in Eskom and most of the utility and engineering companies. Our results are furtherusefultoEskomsinceitisduringthenon-winterperiodthattheyplanformaintenance of their power plants. Modelling temperature is important for the South African economy since electricity sector is considered as one of the most weather sensitive sectors of the economy. Over and above, the modelling approaches that are presented in this dissertation are relevant for modelling heat waves which impose several impacts on energy, economy and health of our citizens.
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    Modelling heavy rainfall over time and space
    (2011-06-06) Khuluse, Sibusisiwe Audrey
    Extreme Value Theory nds application in problems concerning low probability but high consequence events. In hydrology the study of heavy rainfall is important in regional ood risk assessment. In particular, the N-year return level is a key output of an extreme value analysis, hence care needs to be taken to ensure that the model is accurate and that the level of imprecision in the parameter estimates is made explicit. Rainfall is a process that evolves over time and space. Therefore, it is anticipated that at extreme levels the process would continue to show temporal and spatial correlation. In this study interest is in whether any trends in heavy rainfall can be detected for the Western Cape. The focus is on obtaining the 50-year daily winter rainfall return level and investigating whether this quantity is homogenous over the study area. The study is carried out in two stages. In the rst stage, the point process approach to extreme value theory is applied to arrive at the return level estimates at each of the fteen sites. Stationarity is assumed for the series at each station, thus an issue to deal with is that of short-range temporal correlation of threshold exceedances. The proportion of exceedances is found to be smaller (approximately 0.01) for stations towards the east such as Jonkersberg, Plettenbergbay and Tygerhoek. This can be attributed to rainfall values being mostly low, with few instances where large amounts of rainfall were observed. Looking at the parameters of the point process extreme value model, the location parameter estimate appears stable over the region in contrast to the scale parameter estimate which shows an increase towards in a south easterly direction. While the model is shown to t exceedances at each station adequately, the degree of uncertainty is large for stations such as Tygerhoek, where the maximum observed rainfall value is approximately twice as large as the high rainfall values. This situation was also observed at other stations and in such cases removal of these high rainfall values was avoided to minimize the risk of obtaining inaccurate return level estimates. The key result is an N-year rainfall return level estimate at each site. Interest is in mapping an estimate of the 50-year daily winter rainfall return level, however to evaluate the adequacy of the model at each site the 25-year return level is considered since a 25 year return period is well within the range of the observed data. The 25-year daily winter rainfall return level estimate for Ladismith is the smallest at 22:42 mm. This can be attributed to the station's generally low observed winter rainfall values. In contrast, the return level estimate for Tygerhoek is high, almost six times larger than that of Ladismith at 119:16 mm. Visually design values show di erences between sites, therefore it is of interest to investigate whether these di erences can be modelled. The second stage is the geostatistical analysis of the 50-year 24-hour rainfall return level The aim here is to quantify the degree of spatial variation in the 50-year 24-hour rainfall return level estimates and to use that association to predict values at unobserved sites within the study region. A tool for quantifying spatial variation is the variogram model. Estimation of the parameters of this model require a su ciently large sample, which is a challenge in this study since there is only fteen stations and therefore only fteen observations for the geostatistical analysis. To address this challenge, observations are expanded in space and time and then standardized and to create a larger pool of data from which the variogram is estimated. The obtained estimates are used in ordinary and universal kriging to derive the 50-year 24-hour winter rainfall return level maps. It is shown that 50-year daily winter design rainfall over most of the Western Cape lies between 40 mm and 80 mm, but rises sharply as one moves towards the east coast of the region. This is largely due to the in uence of large design values obtained for Tygerhoek. In ordinary kriging prediction uncertainty is lowest around observed values and is large if the distance from these points increases. Overall, prediction uncertainty maps show that ordinary kriging performs better than universal kriging where a linear regional trend in design values is included.