Daily average measurements of temperature, relative humidity and dew point were recorded using data loggers calibrated

in Bisley. Attention to their placement was essential as the sensors had to be installed at a height of 1,5m above the ground and could not be sheltered, as this would alter the recordings drastically. Graphs showing the measured maximum and minimum air temperatures for this suburb are provided in Figures 2 and 3. These measurementswere conducted for each day of the time period studied (July 2014 - June 2015). The latitude of Bisley was used to calculate the relevant solar angles using eqs. (3) and (4) which then allowed us to evaluate the extraterrestrial solar radiation (Ho) in Eq. (2). The Hargreaves-Samani (H-S) equation (Eq.(1)), together with the measured air temperature values gave results which are averaged in . The GSR values listed below are the calculated monthly averages based on daily maximum and minimum air temperatures. We then estimated the GSR for Bisley using measured sunshine duration hours and the Ho within the Angstrom-Prescott (A-P) model. The maximum possible sunshine duration Eq. (7) was calculated using the hour angle Eq. (4). These results are represented in .

Figure 2 Graph of measured maximum temperatures for Bisley

Figure 3 Graph of measured minimum temperatures for Bisley

Table 2 Bisley, Ukulinga results using the Hargreaves-Samani Model (July 2014-June 2015)

Table 3 Bisley, Ukulinga results using the Angstrom-Prescott Model (July 2014-June 2015)

The measured values (H_{measured}) listed in and 3, are the actual observed values of GSR for Bisley. The data was supplied by the ARC and enabled us to compare the values calculated by Eqs. (1) and (6) with the measured data. Temperature variations show a similar distribution to the H (both observed and calculated) values which illustrates the relationship between air temperature and GSR.

The accuracy of both models was determined based on the error analysis between the predicted and measured values of GSR. provides the average annual errors for the Hargreaves-Samani (H-S) and Angstrom-Prescott (A-P) models. The mean bias error (MBE) indicates the average deviance of the calculated values from that of the measured and is used to decide the long-term performance of a model [24]. Positive values of MBE correspond to an over estimation, while a negative MBE indicates an under estimation. The RMSE gives insight into the short-term performance of a correlation. Low values for all statistic error measures are desired [25]. Earlier studies suggest that percentage errors between −10% and 10% are acceptable [26]. Statistical analysis reported in was calculated using the below;

**Mean bias error (MBE) and mean absolute bias error (MABE)**

$$\begin{array}{rl}MBE& =\frac{1}{n}\sum _{i=1}^{n}\left({H}_{c}-{H}_{m}\right)\\ MABE& =\frac{1}{n}\sum _{i=1}^{n}\left(\left|{H}_{c}-{H}_{m}\right|\right)\end{array}$$**Mean percentage error (MPE) and mean absolute percentage error (MAPE)**

$$\begin{array}{rl}& MPE=\frac{1}{n}\sum _{i=1}^{n}\left(\frac{{H}_{c}-{H}_{m}}{{H}_{m}}\right)\times 100\%\\ & MAPE=\frac{1}{n}\sum _{i=1}^{n}\left(\frac{{H}_{c}-{H}_{m}}{{H}_{m}}\right)\times 100\%\end{array}$$**Root mean square errors (RMSE)**

$$\begin{array}{r}RMSE=\sqrt{\frac{\sum _{i=1}^{n}{\left({H}_{c}-{H}_{m}\right)}^{2}}{n}}\end{array}$$where *H *_{c} and *H *_{m} are the calculated and measured values of GSR, respectively.

Positive errors indicate that the models under study have overestimated values of GSR for the given period. The MPE falls within the prescribed interval (−10%; 10%), however the MAPE is slightly over the 10% interval (). This is a minor deviation in comparison to the daily sample size. Averaging and rounding of hourly, daily measurements when calculating monthly average values, would have contributed to the error being over the acceptable range. In this regard, the results are still acceptable. The RMSE, MBE and MABE values are moderate and can be lower to show a stronger correlation. The calculated values of H for Bisley conformed well to the shape of the data observed by the ARC, with the exception of a few outliers. This is represented in Figures 4 and 5. Maximum calculated values for H were observed during October - February (Figures 4 and 5) which are the spring and summer months in South Africa. Both the (H-S) and (A-P) models demonstrated the most deviation from the measured GSR values in these spring/summer months, which may indicate over estimation by the selected methods.

Figure 4 Graph comparing the measured and calculated values of H using the H-S Model

Figure 5 Illustration comparing the measured and calculated values of H using the A-P Model

The over prediction may be a consequence of; the accuracy and competence of the equipment used, the effects of wind, or other temperature invasion factors such as pollution. Observed values could be better validated by adjusting the temperature based model (H-S) to account for short wave radiation. The sunshine duration model may be modified by introducing a non-linear relationship between the GSR and sunshine duration ratio. For optimal prediction, this work suggests that a new model be devised to include both sunshine and temperature variables. This is a consequence of both models being able to sufficiently estimate the GSR in Bisley (based on the annual average errors), while each model performed differently when we consider each individual month. Other meteorological factors such as relative humidity, wind speed and air pressure can also be included for improved prediction. During the autumn, winter and parts of spring months, both models performed considerably well in estimating the amount of GSR. Overall, the distribution and monthly variation of the calculated values of GSR show great similarities when compared to the observed values.

The annual average GSR values obtained for Pietermaritzburg, show close similarities to the results presented by Maluta *et al*. [20], for the Limpopo Province in South Africa. In the study conducted by [20], stations which have an altitude close to that of Pietermaritzburg, had an annual average H value in the range: [14.71-17.82] MJ/m^{2}, whilst H_{calculated} for Bisley is in the range: [16.54-16.81] MJ/m^{2}. The main contributing difference in these locations is the site’s latitude.

The clearness index, being the ratio of GSR to extraterrestrial radiation provides information on the degree of transparency of the atmosphere. Using eq. (5), *K*_{T} values were calculated and interpreted by the following work conducted in [6];

The calculated values of clearness index for each model as shown in our results suggest that Pietermaritzburg experiences a high number of partially cloudy days, with not many days being classified as cloudy according to . The discrepancies experienced in the calculated H values may be a result of the influence of cloudiness on the air temperature and sunshine duration data. On average, the monthly data gives a clearness index which falls into the partially cloudy category for both prediction models as well as the measured data in question ().

Table 5 Clearness index (K_{T}) results using each of the models

Table 6 Classification of day by clearness index, K_{T}

In this study we have used the prescribed A-P coefficients of *a* = 0.25 and *b* = 0.5. The Angstrom coefficients vary with each geographical location depending on the amount of relative sunshine received. Other factors such as the geographical locations of the site and atmospheric effects may also introduce deviations in the clearness index. For a more consistent set of Angstrom coefficients, this study should be extended to analyze data over a longer period of time for the chosen location.

The results obtained in this study indicate that the city of Pietermaritzburg receives sufficient GSR for the use of solar powered technologies such as solar panels, solar heating and cooling technologies for industrial, commercial and residential areas. Prediction models may be used to identify which areas are optimal for the harnessing of GSR. Accurate GSR predictions for this city will also enable a better understanding of the climate experienced and its effects. Clear to partially cloudy days are experienced throughout the year, including during the winter months, making GSR easy to acquire. The H-S and A-P models are suitable for the calculation of GSR, however the accuracy of results during the summer season can be improved. Comprehensive models including both meteorological variables can be introduced to account for this.

Evaluation of the A-P coefficients (a and b) as well as the H-S (K_{r}) coefficient can be conducted via the study of historic meteorological data for enhanced prediction models. These coefficients give insight into the transmissivity and transparency of the atmosphere. Prediction of the type of day and clearness index can also be made, provided the Angstrom coefficients are well-established. Though solar radiation data in the city of Pietermaritzburg is not readily available, the amount of GSR incident in this city can be sufficiently estimated using the Hargreaves-Samani and Angstrom-Prescott models. This study has shown the suitability of this interior region to contribute to the decrease in demand of grid energy by making use of the incident GSR in Pietermaritzburg. Furthermore, the forecasting method described above can be easily implemented for GSR prediction within any location of the world where air temperature and sunshine duration are measurable quantities.

Figure 6 Graph comparing calculated and measured values of K_{T}

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