Combining model uncertainty and interannual variability

For understanding the solar radiation model performance key indicators, it is important to consider the several factors that influence the accuracy of the values, provided both by satellite-based modelling and on-site ground sensors.

On the modelling side, accuracy will be determined by cloud estimate (e.g. intermediate clouds), under/over estimation of atmospheric input data, terrain, microclimate and other effects, which are not captured by the model. Part of this discrepancy is natural - as satellites monitor large area (of approx. 3x4 km) while ground sensors see only micro area of approx. 1 squared centimetre. Due to higher complexity of the model, bias of satellite-based DNI is higher than GHI.

On the measurement side, the discrepancy will be determined by accuracy/quality and errors of the instrument, pollution of the detector, misalignment, data loggers, insufficient quality control, etc. Only quality-controlled measurements from high-standard sensors should be used for reliable validation of satellite-based solar models. Any issues in the ground measured data result in a skewed evaluation.

The most required value by project developers, technical consultants and finance industry is the uncertainty of the long-term yearly GHI or DNI estimate for the project site. The model uncertainty can be calculated from the validation statistics (Bias).

However, the uncertainty values should be also taking into account the fact that the measurements also include an uncertainty component itself. In addition, when assessing the uncertainty of one single year, inter-annual variability due to the climate factors should be evaluated as well.

In conclusion, assuming that the solar radiation values can be described using a normal probability distribution (similarly as we have done when characterizing the model bias distribution), the total combined uncertainty is calculated from:

• Uncertainty of the Solargis model estimate
• Uncertainty of the ground measurements
• Inter-annual weather variability.

The influence of these three factors in the final uncertainty is calculated through the square root of the quadratic sum of each uncertainty.

Estimate of the long-term uncertainty of ground measurements can be a bit subjective – it can be based on combination of the theoretical uncertainty of the instrument, results of quality control procedures and comparison of the redundant measurements.

Considering that the absolute majority of the validation data have been collected using high-accuracy instruments and applying the best measurement practices and strict quality control procedures, it is considered that a ±2.0% should be added from GHI measurements from pyranometers and ±1.0% from DNI measurements from pyrheliometers. This could serve as a good starting point for assessing annual uncertainty of solar instruments. It is known from other comparisons that these values could be exceeded in standard operating conditions.

On the other hand, utilization of the state-of-the-art instruments does not alone guarantee good results. Any measurements are subject to uncertainty and the information is only complete, if the measured values are accompanied by information on the associated uncertainty. Sensors and measurement process has inherent features that must be managed by quality control and correction techniques applied to the raw measured data.

The lowest possible uncertainties of solar measurements are essential for accurate determination of solar resource. Uncertainty of measurements in outdoor conditions is always higher than the one declared in the technical specifications of the instrument. The uncertainty may dramatically increase in extreme operating conditions and in case of limited or insufficient maintenance. Quality of measured data has significant impact on validation and regional adaptation of satellite models.

Theoretically-achievable daily uncertainty of GHI at 95% confidence level

Theoretically-achievable daily uncertainty of DNI at 95% confidence level

Weather changes in cycles and it has also a stochastic nature. Therefore annual solar radiation in each year can deviate from the long-term average in the range of few percent. This is expressed by interannual variability, i.e. the magnitude of the year-by-year change.

The interannual variability for selected sites is calculated from the unbiased standard deviation of the yearly values over the available period of years, considering a simplified assumption of normal distribution of the annual sums. All sites show similar patterns of variation over the recorded period. This analysis can be made for longer periods, i.e. the uncertainty at different confidence levels expected for average values within more than one-year period. 

The historical period used for calculating the inter-annual variability may have some influence, although it is observed to be quite small (e.g. if we compare results from 10 years of data with results from 20 years of data). The expected difference for GHI would be less than 1% (depends on the climate zone). A higher influence may be found in data sets representing occurrence of large stratospheric volcano eruptions.

Table of GHI interannual variability of a period of 1, 5, 10 and 25 years for several sample sites

Uncertainty of yearly GHI estimate due to interannual variability: is being considered for a period of N years, then the STDEV is to be divided by the square root of N (typically one year, 10 years, or the total expected lifetime of the solar energy asset). For a single year, this uncertainty is highest, and it decreases with the number of years. It is given in the sample report in P50 confidence interval or should be calculated as follows:

To calculate the uncertainty related to the interannual variability at different Pxx scenarios and the Pxx value for the interannual variability, the following steps can be performed: 

1. Yearly STDEV is to be divided by the square root of N: In this case, 41 / sqrt(27) = 7.89 (full years in our time series is 27 years in this case).
2. Then, divide 7.89 by the long-term average GHI (P50 value) to get the variability at STDEV level (probability of occurrence 68.7%) = 7.89 / 1883 = 0.0042
3. Lastly, convert this to any Pxx level from STDEV level (following a Gaussian distribution) 0.0042 * 1.282 (for P90)=  0.0054 = ±0.54%

Other Pxx can be calculated as shown below if the P50 uncertainty is given:

Good models have known and lowest possible uncertainty. On the other hand, the expected uncertainty for a specific site can be derived from the analysis of validation statistics for a sufficient number of validation points.

Assuming that the solar resource estimates from two different models follow a normal distribution, the combined uncertainty can be compared and represented in charts. The most expected value (P50) and its uncertainty will determine the position of the center and the width of the probability distribution respectively. The value exceeded with 90% probability (P90) can be calculated and identified in the probability distribution as well.

Using a solar model with proven higher accuracy like Solargis provides more probable estimates and less weather-related risk for the project. In other words, the distance between P50 and P90 values is smaller in such case.

In the sample below, the model A has a higher uncertainty than model B. Therefore the distribution of expected values in model A will be spread within a wider range. In other words, values expected by model B will occur with a higher probability.

The sample also shows that even in the case that P50 value provided by model B is lower than model A, the difference in terms of uncertainty results in model B providing more favorable financial conditions for the project, with a P90 value that is higher.

Uncertainty of GHI values from two models at a sample site located in Slovakia.

Pxx combined uncertainty for solar parameters represents the total uncertainty, it is calculated as shown below, where two sources of uncertainty are considered: uncertainty of the solar model and interannual variability for N years period.