# The Ångström–Prescott Regression Coefficients for Six Climatic Zones in South Africa

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

^{2}) was 0.910. The statistical validation metrics results show that there is a strong correlation and linear relation between observed and estimated GHI values. The AP model coefficients calculated in this study can be used with quantitative confidence in estimating daily GHI data at locations in South Africa where daily observation sunshine duration data are available.

## 1. Introduction

**a**and

**b**in different climatic zones could be due to variations in latitude, altitude, aerosols, and water vapor concentration, surface albedo, and mean solar altitude [14]. The study by Tsung et al. [15] focused on one location using n and GHI data collected from two different stations because of the unavailability of both GHI and n from one location. In this study, eight stations located in all six climatic zones of South Africa, and where both n and GHI data were collected from the same location, are considered so that the respective AP coefficients are representative of a climatic zone in the country and not the whole country.

## 2. Materials and Methods

_{TOA}) and theoretical sunshine duration (N) were calculated using Equations (1)–(9), from Iqbal [13], and the solar angles were calculated using the Solar Position Algorithm (SPA) on Python PVLIB [27,28] and Microsoft Excel. The coefficients

**a**and

**b**of the AP model were calculated by using the linear regression analysis between the irradiance fraction or clearness index, $\frac{GHI}{GH{I}_{TOA}}$ and daily sunshine fraction, $\frac{n}{N}$ for each day, based on a linear relationship shown by Equation (1) proposed by Ångström [11] and then, modified by Prescott [12].

^{2}.

- Mean Bias Error (MBE), which estimates the average error in the prediction. A positive MBE indicates that the prediction is overestimated and vice versa; the lower values of MBE indicate a strong correlation between the prediction and observation. A relative Mean Bias Error (rMBE), which measures the size of the error in percentage terms, was also calculated. The metrices are expressed as:$$MBE=\frac{1}{n}{\displaystyle \sum}_{i=1}^{n}\left(Pi-Oi\right)$$$$rMBE=100\times \frac{1}{n}{\displaystyle \sum}_{i=1}^{n}\frac{\left(Pi-Oi\right)}{\overline{O}i}$$
- Mean Absolute Error (MAE), which measures the absolute value of the differences between the observed and the predicted values, gives a better idea of the prediction accuracy; relative Mean Absolute Error (rMAE), which measures the size of the error in percentage terms, was also calculated. The caution with MBE and rMBE is with the cancelling of positive and negative bias, which can lead to a false interpretation. The metrics are expressed as:$$MAE=\frac{1}{n}{\displaystyle \sum}_{i=1}^{n}\left|Pi-Oi\right|$$$$rMAE=100\times \frac{1}{n}{\displaystyle \sum}_{i=1}^{n}\frac{\left|Pi-Oi\right|}{\overline{O}i}$$
- Root Mean Square Error (RMSE), which compares the predicted and observed datasets, measures the statistical variability of the prediction accuracy and is expressed as shown in Equation (14), while Equation (15) shows the relative Root Mean Square Error (rRMSE), which measures the size error in percentage terms. The RMSE and rRMSE are also indifferent to the direction of the error. They are considered in this study since these put extra weight on large errors. The metrices are expressed as:$$RMSE=\sqrt{\frac{1}{n}{\displaystyle \sum}_{i=1}^{n}{\left(Pi-Po\right)}^{2}}$$$$rRMSE=\frac{100}{\overline{Oi}}\times \sqrt{\frac{1}{n}{\displaystyle \sum}_{i=1}^{n}{\left(Pi-Po\right)}^{2}}$$
- Coefficient of Determination (R
^{2}), which is a statistical measure of the strength of the relationship between the movement of predicted and observed. R^{2}also measures how well the regression line represents the data. The value of R^{2}is such that $0\le {\mathrm{R}}^{2}\le 1$. The closer R^{2}is to 1, the better the prediction. The metric is expressed as:$${\mathrm{R}}^{2}=1-\frac{{{\displaystyle \sum}}_{i=1}^{n}\text{}{\left(Pi-Oi\right)}^{2}}{{{\displaystyle \sum}}_{i=1}^{n}{\left(Pi-\overline{Oi}\right)}^{2}}$$

^{2}to MJ m

^{−2}d

^{−1}by dividing by 11.57415, a methodology used by Almorox et al. [5] to allow for easy comparison with other literature studies. Monthly averages of each metric were calculated and then aggregated to annual averages, and where observation data were not available, data were replaced by NaN. The annual AP coefficients

**a**and

**b**were calculated.

## 3. Results and Discussion

#### 3.1. Annual AP Results

**N**were used as inputs. The calculated a and b were then used together with daily

**n**and

**N**to estimate daily GHI, which was then compared to corresponding observed daily GHI. Statistical metrics in Equations (10)–(16) were used to quantify the errors between the two datasets; the results are shown in Table 2 and Figure 4, Figure 5, Figure 6 and Figure 7.

#### 3.2. Validation Results

^{2}from 0.910 to 0.948. De Aar, Irene, and Thohoyandou had a positive MBE, meaning that the model overestimated GHI, while Upington, Durban, Mthatha, George, and Polokwane had a negative MBE, meaning that the model underestimated GHI values at these locations. The values of MBE and rMBE for all the stations were less than 1, indicating that there was a strong correlation between the predicted and observed GHI values. The worst case R

^{2}value was 0.910, suggesting that there is a very strong linear relation between observed and predicted values.

^{−2}d

^{−1}was less than 1.94 and 1.9 MJ m

^{−2}d

^{−1}that De Medeiros et al. [4] and Tsung et al. [14], respectively, determined. The maximum MAE of 1.425 MJ m

^{−2}d

^{−1}was less than 1.8 MJ m

^{−2}d

^{−1}that Tsung et al. [14] determined. The maximum MBE of 0.733 MJ m

^{−2}d

^{−1}was less than 1.040 and 0.85 MJ m

^{−2}d

^{−1}that De Medeiros et al. [4] and Tsung et al. [14], respectively, determined, and the worst case R

^{2}of 0.910 was greater than 0.875, 0.74, and 0.58 that Zhang et al. [3], Adamala et al. [2], and De Medeiros et al. [4], respectively, determined. As shown in Table 2, the results across the range of climate zones (including the ones from the literature), all differ significantly.

## 4. Conclusions

^{2}). The results were in good agreement with what other studies found.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Abbreviation | Full Description | Units |

SAWS | South African Weather Service | |

AP | Ångström–Prescott | |

a | Ångström–Prescott regression coefficient | |

b | Ångström–Prescott regression coefficient | |

GHI | Global Horizontal Irradiance | W/m^{2} |

$GH{I}_{TOA}$ | Daily extraterrestrial or Top of the atmosphere global horizontal irradiance | W/m^{2} |

TOA | Top of the Atmosphere | W/m^{2} |

N | Daily astronomical day length | Hours |

n | Daily measured sunshine duration | Hours |

BSRN | Baseline Solar Radiation Network | |

QC | Quality control | |

SPA | Solar Position Algorithm | |

${\mathrm{I}}_{SC}$ | Solar constant | W/m^{2} |

${\mathrm{E}}_{o}$ | Eccentricity factor | Degrees |

${\mathsf{\omega}}_{s}$ | Sunset hour angle | Degrees |

$\varnothing $ | Degrees of latitude | Degrees |

δ | Solar declination | Degrees |

MBE | Mean Bias Error | MJ m^{−2}d^{−1} |

rMBE | relative Mean Bias Error | Percentage (%) |

MAE | Mean Absolute Error | MJ m^{−2}d^{−1} |

rMAE | relative Mean Absolute Error | Percentage (%) |

RMSE | Root Mean Square Error | MJ m^{−2}d^{−1} |

rRMSE | relative Root Mean Square Error | Percentage (%) |

R^{2} | Correlation coefficient | |

NaN | Not a Number | |

D | Day of the year | |

CMP11 | Secondary standard Kipp & Zonen pyranometers |

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**Figure 2.**The flowchart of the approach used to calculate Ångström–Prescott regression coefficients a and b.

**Figure 3.**A Python code or script that was written and used to calculate and plot regression coefficients

**a**and

**b**from $X=\frac{n}{N}$ and $Y=\frac{GHI}{GH{I}_{TOA}}$.

**Figure 4.**Regression lines and AP (the Ångström–Prescott) coefficients for Upington station (top left), De Aar (top right), George (bottom left), and Thohoyandou (bottom right).

**Figure 5.**Regression lines and AP coefficients for Mthatha station (top left), Durban station (top right), Irene station (bottom left), and Polokwane (bottom right).

**Figure 6.**Comparison between measured and predicted monthly GHI values in De Aar (top left), Upington (top right), George (bottom left), and Thohoyandou (bottom right). Blue dotted line represents observed, GHI red solid line represents estimated GHI.

**Figure 7.**Comparison between measured and predicted GHI values in Mthatha (top left), Durban (top right), Irene (bottom left), and Polokwane (bottom right). Blue dotted line represents observed, red solid line represents estimated GHI.

**Table 1.**South African Weather Services radiometric station location, altitude, period covered, and climatic zones. (The climate zones correspond with the regions shaded in Figure 1).

Station | Latitude (°) | Longitude (°) | Altitude (m) | Period | Climatic Zone |
---|---|---|---|---|---|

Upington | −28.48 | 21.12 | 848 | 2014-02-01 to 2019-11-30 | Arid Interior |

De Aar | −30.67 | 23.99 | 1284 | 2014-05-01 to 2019-12-31 | Cold Interior |

Irene | −25.91 | 28.21 | 1524 | 2014-03-01 to 2019-12-31 | Temperate Interior |

Mthatha | −31.55 | 28.67 | 744 | 2014-07-01 to 2019-12-31 | Subtropical Coastal |

George | −34.01 | 22.38 | 192 | 2015-01-01 to 2019-12-31 | Temperate Coastal |

Durban | −29.61 | 31.11 | 91 | 2015-03-01 to 2019-12-31 | Subtropical Coastal |

Polokwane | −23.86 | 29.45 | 1233 | 2015-03-01 to 2019-12-31 | Temperate Interior |

Thohoyandou | −23.08 | 30.38 | 619 | 2015-03-01 to 2017-10-31 | Hot Interior |

Station | a | b | RMBE | rMBE (%) | MAE | rMAE (%) | RMSE | rRMSE (%) | R^{2} |
---|---|---|---|---|---|---|---|---|---|

Upington | 0.243 | 0.549 | −0.360 | −0.120 | 0.841 | 0.311 | 1.061 | 0.393 | 0.930 |

De Aar | 0.191 | 0.600 | 0.733 | 0.371 | 1.136 | 0.506 | 1.375 | 0.598 | 0.930 |

Irene | 0.224 | 0.546 | 0.689 | 0.353 | 1.328 | 0.608 | 1.618 | 0.729 | 0.912 |

Mthatha | 0.210 | 0.562 | −0.104 | −0.013 | 1.168 | 0.582 | 1.474 | 0.735 | 0.951 |

George | 0.215 | 0.560 | −0.270 | −0.036 | 1.261 | 0.636 | 1.520 | 0.769 | 0.948 |

Durban | 0.207 | 0.540 | −0.322 | −0.106 | 1.425 | 0.745 | 1.741 | 0.910 | 0.915 |

Polokwane | 0.243 | 0.515 | −0.286 | −0.085 | 1.272 | 0.488 | 1.572 | 0.606 | 0.910 |

Thohoyandou | 0.188 | 0.571 | 0.286 | 0.168 | 1.071 | 0.550 | 1.433 | 0.746 | 0.937 |

Almorox et al. | 0.287 | 0.452 | −0.002 | - | - | - | 1.260 | - | - |

De Medeiros et al. | 0.39 | 0.29 | 1.040 | 6.29 | - | - | 1.94 | - | 0.58 |

Tsung et al. | 0.5 | 0.11 | 0.85 | 3.4 | 1.8 | - | 1.9 | - | - |

Zhang et al. | 0.214 | 0.552 | - | - | 2.249 | - | 0.214 | - | 0.875 |

Adamala et al. | 0.28 | 0.52 | - | - | - | - | 7.04 | - | 0.74 |

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**MDPI and ACS Style**

Mabasa, B.; Lysko, M.D.; Tazvinga, H.; Mulaudzi, S.T.; Zwane, N.; Moloi, S.J.
The Ångström–Prescott Regression Coefficients for Six Climatic Zones in South Africa. *Energies* **2020**, *13*, 5418.
https://doi.org/10.3390/en13205418

**AMA Style**

Mabasa B, Lysko MD, Tazvinga H, Mulaudzi ST, Zwane N, Moloi SJ.
The Ångström–Prescott Regression Coefficients for Six Climatic Zones in South Africa. *Energies*. 2020; 13(20):5418.
https://doi.org/10.3390/en13205418

**Chicago/Turabian Style**

Mabasa, Brighton, Meena D. Lysko, Henerica Tazvinga, Sophie T. Mulaudzi, Nosipho Zwane, and Sabata J. Moloi.
2020. "The Ångström–Prescott Regression Coefficients for Six Climatic Zones in South Africa" *Energies* 13, no. 20: 5418.
https://doi.org/10.3390/en13205418