In this document
We will explain how probabilistic scenarios, such as P50, P90, and in general PXX, help quantify risks by incorporating uncertainties from irradiance estimates, simulation models, and interannual variability, thereby ensuring reliable energy assessments.
Overview
Probabilistic scenarios, such as P50, P90, and in general PXX values, are indispensable for quantifying and communicating the uncertainties in energy production forecasts. These metrics provide a probabilistic understanding of potential energy outputs, enabling stakeholders to assess risks and make informed decisions about system performance and financial viability.
To calculate these probabilistic scenarios, it is essential to identify and account for all sources of uncertainty. In PV energy production, key contributors include satellite-based irradiance estimates, inaccuracies in PV simulation models, and interannual variability in solar resources. Each of these factors introduces variability that can significantly influence the accuracy of energy predictions.
To ensure reliable results, these uncertainties must be systematically integrated into a unified framework using statistical methods such as the root-sum-square approach. This method assumes that the uncertainty sources are independent and follow a normal distribution, allowing for a coherent quantification of their combined effect. By applying this structured approach, energy forecasts can better represent the inherent variability, providing confidence levels that support robust decision-making.
Meaning of P50, P90 or any PXX value
Probabilistic scenarios are critical for quantifying and communicating the uncertainty in energy production forecasts, allowing stakeholders to balance risk and reliability in decision-making.
Most common values are:
P50 value: The annual P50 value represents the most likely energy production estimate for a single year in the future. There is a 50% probability that the actual production will be higher than P50 value, and a 50% probability that it will be lower.
P90 value: The annual P90 value indicates the energy production level that is expected to be exceeded in 90% of cases. It reflects a more conservative estimate, typically used to assess risk for financing or investment purposes.
In general we can talk about PXX Value as a generalized form, representing the energy production level that would be exceeded in X% of cases. For example:
P75 value: The value exceeded in 75% of cases.
P99 value: The value exceeded in 99% of cases, representing an extremely conservative estimate.
In solar resource assessment, both P50 and P90 values can refer to solar irradiation (usually GHI) or directly to expected PV output (PVOUT).
Identification of sources of uncertainty
Accurately predicting photovoltaic (PV) energy output requires understanding and addressing various sources of uncertainty that can affect performance estimates. These uncertainties stem from multiple aspects of the energy modeling process:
Satellite-based model irradiance uncertainty: The error in solar irradiance estimates derived from satellite data, caused by limitations in spatial resolution, temporal updates, and atmospheric modeling assumptions.
PV Simulation uncertainty: The inaccuracies in predicting PV system performance due to assumptions about components, system losses in simulation models and environmental effects.
Interannual variability: The year-to-year fluctuations in solar energy availability caused by natural climatic changes, weather patterns, and large-scale phenomena like ENSO (El Niño–Southern Oscillation).
Uncertainty propagation and related assumptions
To provide reliable energy estimates, it is crucial to integrate all sources of uncertainty into a cohesive and unified framework that accurately captures their collective impact (uncertainty propagation).
Uncertainty propagation involves quantifying how individual uncertainties—such as those from satellite-based irradiance models, PV simulations, and interannual variability—interact and propagate throughout the energy estimation process.
It is assumed that:
Values follow a normal (Gaussian) distribution.
Uncertainty sources are independent of one another
This allows for the use of statistical methods, such as the root-sum-square approach, to combine uncertainties into an overall estimate of variability.
If the uncertainties are independent, they can be combined using standard statistical methods e.g., root-sum-square method, where each uncertainty is squared, summed, and then square-rooted to obtain the total uncertainty.
However, if uncertainties are dependent, more advanced techniques like Monte Carlo simulations would be needed to properly account for the correlation and generation of multiple scenarios (e.g., thousands of simulations) would be required instead. By varying the input parameters (e.g., solar irradiance, component efficiencies) according to their probability distributions.
Besides, the combined uncertainty can be easily expressed with confidence intervals (e.g., P50, P90 values), providing a comprehensive understanding of the risks and reliability of energy forecasts.
Description of a Normal Distribution
When describing a normal distribution, three main parameters define its shape and characteristics.
Represented by the mean (μ\mu), this parameter determines the location of the peak or the "center" of the distribution.
The mean is the average value, around which the data points are symmetrically distributed.
In the context of probability and statistics, the mean indicates the most likely value for the dataset.
Represented by the standard deviation (σ\sigma), this parameter determines the spread or "width" of the distribution.
A smaller standard deviation results in a narrower, steeper curve, indicating that data points are closely clustered around the mean.
A larger standard deviation results in a wider, flatter curve, indicating greater variability in the data.
A normal distribution is symmetric around its mean, with equal probabilities on either side. The curve follows a bell shape, where approximately:
68% of the data lies within ±1σ.
95% lies within ±2σ
99.7% lies within ±3σ
Calculation of confidence intervals
Assuming a normal distribution, the P50 value corresponds to the mean of the data series, representing the most likely outcome.
The standard deviation, which reflects a 68% probability of occurrence, serves as the foundation for constructing other confidence intervals:
Probability of occurrence | Formula | |
|---|---|---|
One standard deviation | 68.3% | ± STDEV |
Two standard deviations | 95.5% | ± 2STDEV |
Three standard deviations | 99.7% | ± 3STDEV |
P75 uncertainty | 50% | ± 0.675STDEV |
P90 uncertainty | 80% | ± 1.282STDEV |
P95 uncertainty | 90% | ± 1.645STDEV |
P97.5 uncertainty | 95% | ± 1.960STDEV |
P99 uncertainty | 98% | ± 2.326STDEV |
These intervals allow for the calculation of various probability scenarios, providing insights into exceedance and non-exceedance levels across different confidence thresholds.
Probability of exceedance | Probability of non-excedance | Formula | |
|---|---|---|---|
P50 value | 50% | 50% | Mean |
P75 value | 75% | 25% | Mean - 0.675STDEV |
P90 value | 90% | 10% | Mean - 1.282STDEV |
P95 value | 95% | 5% | Mean - 1.645STDEV |
P97.5 value | 97.5% | 2.5% | Mean - 1.960STDEV |
P99 value | 99% | 1% | Mean - 2.326STDEV |
Sample calculations
Calculation of annual GHI P90 value for a sample site in Almeria (Spain)
GHI P50 value:
1879 kWh (calculated as the average of GHI time series)
Uncertainty Sources:
Satellite-based model GHI uncertainty (1): ±3.5% (for P90 confidence interval).
Interannual variability of annual GHI (2): ±2.6% (for P90 confidence interval).
Combination of uncertainties using root-sum-square method:
SQRT[0.0352 + 0.0262] = ±4.36% (for P90 confidence interval).
GHI P90 value:
1879 * (1 - 0.0436) = 1797 kWh
(1) Uncertainty depends on several factors and it can be modeled and estimated after comprehensive research and expertise on the models.
(2) We can follow these steps to calculate interannual variablity:
Calculate the standard deviation of the yearly values over the available period of N years.
Divide it by the square root of N. If we are calculating the variability of one single year, N=1.
Divide the result by the average value from the whole series to get the value in percentage.
Convert the factor to any P90 confidence level from the standard deviation level by multiplying by 1.282.
Calculation of annual PVOUT P90 value for a sample site in Almeria (Spain)
PVOUT P50 value: :
1705 kWh (calculated as the average of PVOUT time series)
Uncertainty Sources:
Satellite-based model GHI uncertainty (1): ±3.5% (for P90 confidence interval).
PV simulation uncertainty: ±5% (for P90 confidence interval).
Interannual variability of annual PVOUT (2): 3.2% (for P90 confidence interval).
Combination of uncertainties root-sum-square method:
SQRT[0.0352 + 0.052 + 0.0322] = ±6.89% (for P90 confidence interval).
PVOUT P90 value:
1705 * (1 - 0.0689) = 1588 kWh
(1) Uncertainty depends on several factors and it can be modeled and estimated after comprehensive research and expertise on the models.
(2) We can follow these steps to calculate interannual variablity:
Calculate the standard deviation of the yearly values over the available period of N years.
Divide it by the square root of N. If we are calculating the variability of one single year, N=1.
Divide the result by the average value from the whole series to get the value in percentage.
Convert the factor to any P90 confidence level from the standard deviation level by multiplying by 1.282.
In some calculations, a simplification is made by assuming that PVOUT interannual variability is equal to GHI interannual variability. However, this approximation has limitations, as it overlooks critical factors such as the influence of cell temperature, system inefficiencies, and the non-linear effects of partial shading on module performance.