# Interpolating and Estimating Horizontal Diffuse Solar Irradiation to Provide UK-Wide Coverage: Selection of the Best Performing Models

^{*}

## Abstract

**:**

^{2}mean bias error (MBE), 12 Wh/m

^{2}root mean square error (RMSE)), the Ridley, Boland, Lauret equation (a universal split algorithm) consistently performed well. The combined interpolation/separation RMSE is 86 Wh/m

^{2}).

## 1. Introduction

- (1)
- Interpolate weather station readings of global horizontal irradiation to produce a national map of values.
- (2)
- Compute the solar declination angle, an essential input to the clearness index (k
_{t}). k_{t}is a component of the next stage. - (3)
- Split global horizontal irradiation into its components: beam and diffuse irradiation.
- (4)
- Convert each component from horizontal radiation to the PoA inclination and azimuth.
- (5)
- Sum the results: beam on tilted surface plus diffuse on tilted surface.
- (6)
- Allow for albedo.

## 2. Interpolation Outline

#### 2.1. Why Interpolate?

#### 2.2. Review of Interpolation Techniques

## 3. Decomposition/Separation Model Appraisal

## 4. Methodology

- Use the kriging interpolation method to construct a national map of solar irradiation;
- Separate the global horizontal irradiation into two solar components.

## 5. The Kriging Stage

#### 5.1. Current Progress in the UK and Europe

#### 5.2. Data and Software

#### 5.3. Kriging Operations

_{i}) of known input point values (Z

_{i}) (Equation (1)):

- (1)
- Construct the empirical semi-variogram (see Section 5.5).
- (2)
- Fit a model (see Section 5.6).
- (3)
- For all possible input point pairings, determine the straight-line distance between the points and swap into the chosen theoretical semi-variogram model (see later). Put differently, each point-pair distance is multiplied by the slope of the user-selected semivariance graph. The semi-variogram values obtained fill a data covariance matrix (dcm), to be inverted in preparation for subsequent use (idcm). It is necessary to replace the empirical semi-variogram with a theoretical one to comply with mathematical laws for the kriging equations to be solved.
- (4)
- For each output pixel (Ẑ) whose irradiation value is to be predicted, create a vector of distances between itself and all input points. Again, substitute distance for semi-variance obtained from the semi-variogram graph to create an output pixel semi-variance vector (opsv).
- (5)
- Generate a vector of weight factors (w) by multiplying the inverted input points semi-variogram matrix (idcm from Step 3) by the output pixel semi-variogram vector (opsv from Step 4). This is possible on the grounds that the kriging equation, Ẑ = Sum(W
_{i}× Z_{i}), can be expressed as opsv = w × dcm. Re-arranging the equation gives w = idcm × opsv. - (6)
- Finally, for every output pixel, calculate the predicted irradiation value by multiplying each entry in the weight factors vector w by the original input point measurements and summing the set of products. In this case, the irradiance recorded at each weather station is multiplied by a weight and the results totalled for 85 locations. The weights have to be recalculated for each output pixel because the distances to the input points (weather stations) constantly change as the algorithm moves on to make the next prediction. The dcm matrix stays the same but opsv and therefore w constantly change.

#### 5.4. Forms of Kriging

#### 5.5. Semi-Variogram Type

- (1)
- Measure the distance between two locations.
- (2)
- Reckon half the difference squared between the values at the locations. On the x-axis is the distance between the locations (or simplified distance, grouped into lag bins, h), and on the y-axis is the difference of their values squared, i.e., the semivariance, y(h). Thus, for the purposes of this research, x = distance in km, whilst y = [(irradiation at location i − irradiation at location j)
^{2}] ÷ 2.

#### 5.6. Choice of Theoretical Semi-Variogram and Optimisation of Parameters

- (1)
- Spherical: this plot is linear close to the origin, making it suitable for the depiction of phenomena with close range variability. It demonstrates a progressive decrease of spatial autocorrelation until it reaches the sill (top of the semi-variogram curve), where autocorrelation is zero.
- (2)
- Exponential: this is also linear near the origin but the exponential model differs from the spherical in that it approaches the sill gradually. Autocorrelation only ceases at infinity.
- (3)
- Gaussian: the gaussian model traces a parabolic curve at the origin, representing smoothly varying properties. Like the exponential, it rises gradually to a straight sill at infinite distance.
- (4)
- Linear: this model resembles the side of a trapezoid. It factors in a cease in autocorrelation between point-pairs at a determinable distance.

#### 5.6.1. Summary of Variogram Selection

#### 5.6.2. Setting Parameter Values

- Sill—the mean of the maximum and median semi-variance values;
- Range—0.1 multiplied by the value read from the diagonal of the bounding box of the map;
- Nugget—the minimum of the semi-variance.

#### 5.7. Synopsis of Kriging Decisions

^{2}, i.e., about 10% difference between weather stations. On the other hand, kriging does have the advantage that it provides the ability to calculate error variance. This provides an indication of where on the map the interpolated values are least trustworthy.

#### 5.8. Success of the Kriging Choices

^{2}(11%) and an average maximum cross-validation RMSE of 211 Wh/m

^{2}(42%). This compares very favourably to PVGIS (yearly average cross-validation RMSE 146 Wh/ m

^{2}/day (4.5%)).

^{2}every 25–30 km distance band from the nearest weather station.

^{2}each, representing a maximum hourly annual range in the UK of 0–1200 Wh/m

^{2}. In practice, only six bands at most appear in one hourly map. That is, in any one hour, global horizontal irradiation is no more than 600 Wh/m

^{2}greater in Cornwall than in Scotland. Figure 3 illustrates a small sample of the 51,164 surfaces created by this technique. Figure 4 is the average of the 5045 hourly irradiation maps for 2013. This masks hourly variations and results in the southwest to northeast irradiation trend expected of Great Britain.

## 6. The Separation Stage

#### 6.1. Data

#### 6.2. UK Weather

- (1)
- North West—cool summer, mild winter, heavy rain;
- (2)
- North East—cool summer, cold winter, moderate rain;
- (3)
- South East—warm summer, mild winter, light rain;
- (4)
- South West—warm summer, mild winter, heavy rain.

#### 6.3. Software Employed for Decomposition Models

#### 6.4. Irradiation Component Separation Models

#### 6.5. Results of Irradiation Component Separation Models

^{2}were filtered out to avoid the inherent inaccuracy in low radiation values. It may be seen that the BRL model delivers the lowest errors for UK locations. This is probably because it has been found to be less dependent on the zenith angle than the other models [28].

^{2}MBE, 12 Wh/m

^{2}RMSE. These errors may possibly be within the range of pyranometer uncertainty. Despite the minor variations in calculated results, it is still necessary to select the separation technique logically, in order to deliver the required data for PV performance modelling. BRL has been identified as the procedure which most accurately reproduces measured values. This finding is to some extent predictable because it is a universal algorithm, whilst the EKD equation was fitted to US data and the CLIMED to Mediterranean information. Given the regional climate of Lerwick, it is reasonable that the CLIMED equation would not be suitable. It is useful to note from the researcher’s point of view, that the same model is effective for all UK sites tested.

## 7. Combination of Stages

^{2}, generated from a dataset of April–December 2015 inclusive. This being a period when both Solys 2 and Met. Office data are available. The overall combined RMSE for Loughborough (from data produced by kriging and the BRL model in sequence and compared to Solys 2 measured diffuse values) was 42 Wh/m

^{2}.

^{2}) and average of Lerwick, Camborne and Loughborough sites’ RMSE for the BRL separation model (64 Wh/m

^{2}calculated from values in Table 4); the expected composite UK national RMSE on an hourly time step is 86 Wh/m

^{2}. The separation stage is responsible for more than half (54%) of the overall error. These results compare well to related work. Burgess [28] obtained a composite RMSE of 46% following comparison of sequential application of IDW interpolation, EKD separation and Perez transposition [48] models to measured data for one site in Cornwall.

## 8. Conclusions

^{2}depending on kriging model used. The same cannot be said of the separation models since there is little to choose between them in terms of error values. Nonetheless, this research logically determines the model which is employed. This results in a newer alternative being chosen, rather than relying on a usual long-standing choice.

## 9. Future Work

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Example of Complex UK Solar Irradiation Trend

**Figure A1.**(

**a**) Graphs and (

**b**) trend plots of global horizontal irradiation (Wh/m

^{2}) plotted against Easting of Weather Station (graph) and Easting/Northing (trend).

#### Appendix A.2. Selection of Theoretical Semi-Variogram Model

#### Appendix A.2.1. Model Selection Based on Spatial Autocorrelation

**Figure A2.**Semi-variogram cloud of global irradiation for 11:00 a.m. on 1 November 2012. Sites with very different irradiation values (end-to-end of UK mainland, Scottish mountains and south coast) linked by lines on the map and highlighted on the semi-variogram.

#### Appendix A.2.2. Data Visualisation and Variance

**Figure A3.**Output surfaces of hourly global irradiation for 11:00 a.m. on 1 January 2012 produced by three kriging semi-variograms: (

**a**) spherical; (

**b**) exponential; and (

**c**) gaussian.

^{2}between the highest and lowest value. The exponential algorithm created a wider range (343 Wh/m

^{2}), with the gaussian falling between the two (300 Wh/m

^{2}). The difference between the exponential and spherical surfaces is obvious in Figure A4. The neutral colour represents the lower differences (0–60 Wh/m

^{2}). Note these roughly correspond to where the weather stations dots are more clustered. Clustering is displayed by constructing 25 km distance bands around the weather stations. Where these join together, weather stations are closer. Differences are greatest (red in Figure A4, 125–190 Wh/m

^{2}) where weather stations are further apart (single, circular distance bands) and in coastal regions.

**Figure A4.**Difference between the exponential and spherical models of hourly global irradiation for 11:00 a.m. on 1 January 2012.

#### Appendix A.2.3. Manual Fitting of Semivariogram Graphs

#### Appendix A.2.4. Cross-Validation

- Remove a known point from data set;
- Use remaining points to estimate the value at the point removed;
- Compare the estimated to known value;
- Repeat for all points and calculate root mean squares.

^{2}) and the spherical 33% of times (minimum 0, maximum 237, average 58 Wh/m

^{2}). Whilst the average difference between the exponential and spherical RMSE was 0.96 Wh/m

^{2}, the maximum was 21 Wh/m

^{2}. Larger differences are recorded randomly throughout the year; there is no period when one algorithm performs better than the other. For the most part the exponential model delivers the lowest RMSE. Hence, this analysis recommends the exponential semi-variogram.

#### Appendix A.2.5. Ability of Model to Represent Reality/Not Fail When Automated

^{2}was interpolated in the London area. The probable cause was the fact that one London weather station recorded an irradiation value of 46 Wh/m

^{2}whilst the adjacent one is over 400 Wh/m

^{2}. In scientific terms, there are outliers in the data that cause high semi-variance at a short distance. These can most likely be explained by rapidly passing clouds.

#### Appendix A.3. Separation Stage Details

#### Features of Irradiation Component Separation Models

Separation Model | Clearness Index | k_{d} | k_{t} |
---|---|---|---|

EKD/Erbs (1982) Model | Low clearness index, completely overcast, almost all irradiation diffuse | k_{d} = 1 − 0.09k_{t} | k_{d} ≤ 0.22 |

Intermediate | k_{d} = 0.9511 − 0.1604k_{t} + 4.388k_{t}^{2} − 16.638k_{t}^{3} + 12.336k_{t}^{4} | k_{t} > 0.22 or k_{t} < 0.8 | |

High clearness index, bright, sunny, nearly all irradiation direct | k_{d} = 0.165 | k_{t} ≥ 0.8 | |

CLIMED/De Miguel (2001) Model | Low clearness index, completely overcast, almost all irradiation diffuse | k_{d} = 0.995 − 0.08k_{t} | k_{t} ≤ 0.21 |

Intermediate | k_{d} = 0.724 − 2.738k_{t} + 8.32k_{t}^{2} − 4.967k_{t}^{3} | k_{t} > 0.21 or k_{t} < 0.76 | |

High clearness index, bright, sunny, nearly all irradiation direct | k_{d} = 0.180 | k_{t} ≥ 0.76 |

_{t}is daily clearness index; and $\mathsf{\psi}$ is persistence factor.

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**Figure 2.**A typical example of semivariogram showing different components (Creative Commons [41]).

**Figure 3.**Kriged hourly global horizontal irradiation (bands of 100 Wh/m

^{2}) of two sample days (21 June 2013 and 21 December 2013).

**Figure 5.**Comparison between Diffuse Irradiation values from three separation models and measured values: Camborne 2013.

**Figure 6.**Diffuse irradiation at Loughborough: Solys 2 measured value compared to calculated value. Calculation by interpolation of hourly global horizontal irradiation from Met Office weather stations (Meteorological Office Integrated Data Archive System (MIDAS) data) followed by application of the BRL model to separate out the diffuse value. (10 weeks, from the end of March 2015 to mid-June 2015 illustrated).

**Figure 7.**Diffuse irradiation UK-wide: produced by application of the BRL model to separate the diffuse value from global horizontal irradiation provided by Meteorological (Met.) Office weather stations (MIDAS data).

Method | Preferred Model | Reason for Choice |
---|---|---|

Spatial autocorrelation | Spherical | Mirrors local variation |

Data visualisation | Exponential or Gaussian | Provides detailed results |

Manual fitting of semi-variograms | Gaussian | Best fit to data points |

Cross-validation | Exponential | On average, generates the smallest errors |

Representation of reality | Exponential | Delivers plausible values |

**Table 2.**Results of five methods of Selection of Grid Resolution. ESRI: Environmental Systems Research Institute.

Method | Author | Resultant Grid Size | Scientific Basis | |
---|---|---|---|---|

The shorter of the map width or height divided by 250 | ESRI software | 2.5 km | None—computationally feasible | |

Divide the diagonal width of the map by 250 | MapInfo software | 5 km | ||

Half the average spacing between the closest point pairs | Nyquist frequency | 1 km | Well-known mathematical rule | |

Effective Mapping Scale on Ordnance Survey 1:10,000 series | Best: 0.0005 × 10,000 | Hengl [45] | Best:5 km | Cartographic rule |

Finest: 0.0001 × 10,000 | Finest: 1 km | |||

Inspection density: number of points/area | Best: 0.0791 × √ (density) | Best: 4.2 km | ||

Finest: 0.05 × √ (density) | Finest: 2.7 km |

Location | Latitude | 21 June Solar Noon Zenith Angle | 21 December Solar Noon Zenith Angle |
---|---|---|---|

Camborne | 50.22 | 26.72 | 73.72 |

Loughborough | 52.77 | 31.96 | 77.43 |

Lerwick | 60.14 | 36.64 | 83.64 |

**Table 4.**Mean bias error (MBE) and root mean square error (RMSE) between modelled global horizontal diffuse and World Radiation Data Centre (WRDC)/Loughborough measured value. BRL: Boland, Ridley, Lauret; CLIMED: Climatic Synthetic Time Series for the Mediterranean Belt; and EKD: Erbs, Klein, Duffie.

Location | Model | MBE Wh/m^{2} | RMSE Wh/m^{2} | |
---|---|---|---|---|

Lerwick | 1 | BRL | 25.55 | 59.68 |

2 | CLIMED | 35.84 | 69.39 | |

3 | EKD | 34.07 | 70.33 | |

Camborne | 1 | BRL | 26.90 | 81.00 |

2 | CLIMED | 36.60 | 86.33 | |

3 | EKD | 36.47 | 88.55 | |

Loughborough | 1 | BRL | 8.50 | 53.29 |

2 | CLIMED | 24.34 | 56.34 | |

3 | EKD | 22.55 | 58.21 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ( http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Palmer, D.; Cole, I.; Betts, T.; Gottschalg, R.
Interpolating and Estimating Horizontal Diffuse Solar Irradiation to Provide UK-Wide Coverage: Selection of the Best Performing Models. *Energies* **2017**, *10*, 181.
https://doi.org/10.3390/en10020181

**AMA Style**

Palmer D, Cole I, Betts T, Gottschalg R.
Interpolating and Estimating Horizontal Diffuse Solar Irradiation to Provide UK-Wide Coverage: Selection of the Best Performing Models. *Energies*. 2017; 10(2):181.
https://doi.org/10.3390/en10020181

**Chicago/Turabian Style**

Palmer, Diane, Ian Cole, Tom Betts, and Ralph Gottschalg.
2017. "Interpolating and Estimating Horizontal Diffuse Solar Irradiation to Provide UK-Wide Coverage: Selection of the Best Performing Models" *Energies* 10, no. 2: 181.
https://doi.org/10.3390/en10020181