Background

P-values

What is a P90?

Designing, financing, building, commissioning and operating a PV power plant requires regular estimates of the plant’s energy yield.

It is not sufficient to predict a single value, like 4.8 GWh. That would only be possible if one knew the future solar irradiance and weather experienced by the plant, and the future behaviour of the plant itself.

Rather than a single value, engineers therefore forecast the energy yield as a probability distribution, like in the image below. This distribution incorporates the uncertainty in the weather, and the plant behaviour, and even in the decisions of the electricity regulator.

The probability distribution quantifies the likelihood of producing a particular energy yield. The wider the distribution, the less confident we are in our forecast.

A common way to quantify this uncertainty is with ‘P-values’, such as P90. As shown in the figure, P90 represents the energy yield that has a 90% probability of being exceeded.

Other P-values

All stakeholders care about P50 because it represents the best estimate of a plant’s energy yield, and many software tools determine P50, with PVsyst being the best known.

Most stakeholders, however, would also like a conservative estimate of the yield --- like P90 or P95. Conservative forecasts are particularly desirable for those that need to quantify and mitigate financial risk. For example, the financiers of a PV plant are concerned that the plant’s revenue will be sufficient to make the loan repayments during every year of operation, particularly in the early years. Thus, they are interested in the low-side of the distribution, with P-values like P90, P95 and even P99. These are the yields that might arise in a year of poor weather, or an unreliable system; or it might arise from an average year but where the modeller was overoptimistic.

On the other hand, an owner that operates a system over a long period, will be interested in the upside and the downside, that is, in forecasts that account for the good and the bad years, as well as the good and the bad plants in their portfolio. Thus, they tend to be interested in all P-values.

---

Calculating yield uncertainty

The uncertainty in an energy-yield forecast can arise from many constituent sources of uncertainty.

Some of the major sources of uncertainty relate to the solar radiation, to modelling accuracy, to soiling, degradation, availability and curtailment.

The challenge for the yield forecaster is (i) to accurately quantify those major sources of uncertainty, and (ii) to combine them correctly.

We’ll now describe the two main ways to combine uncertainties: the simple sum-of-squares method and the Monte Carlo method. SunSolve P90 applies the latter.

Sum-of-squares

A common way to combine uncertainties is the ‘sum-of-squares’ method.

It’s great advantage is simplicity. It can also be applied independently of the P50 calculation. That is, whether the P50 was determined using PVsyst, SunSolve Yield or any other program, the yield uncertainty can be calculated afterwards without repeating the P50 calculation.

The sum-of-squares method assumes that all significant sources of uncertainty follow a Gaussian distribution.

In its simplest application, these standard deviations, $\sigma_1$, $\sigma_2$, …, $\sigma_n$, combine to give a Gaussian yield distribution with a standard deviation of

$$\sigma_Y = \sqrt{\sigma_1^2 + \sigma_2^2 + \cdots + \sigma_n^2}.$$

This approach, however, contains three important assumptions that are not always justified.

The first assumption is that all parameters with significant uncertainty contribute to the yield as simple multipliers; i.e., the yield depends on those parameters as

$$Y = P_1 \times P_2 \times \cdots \times P_n.$$

In reality, however, some aspects of yield behaviour cannot be well represented as multipliers. For example, the module power $P_m$ depends on the irradiance $\Phi$ and thermal $U$ value as roughly $P_{m,} \propto U - C \cdot \Phi / U$ at a given ambient temperature where $C$ is a constant (see §8 of equations). Both $\Phi$ and $U$ tend to have significant uncertainty. Although the sum-of-squares method can be applied to incorporate more complicated equations such as this, it is not trivial.

The second assumption is that all significant sources of uncertainty can be well represented by Gaussian distributions. In fact, many sources of uncertainty in a yield forecast have a notable asymmetry (and are therefore non-Gaussian). Examples of asymmetric uncertainties are discussed in the tutorials.

The third assumption is that the uncertainties are independent. In reality, some are not. For example, if the solar irradiation is higher than expected, the temperature is more likely to be hotter than expected. Or if the albedo is higher than expected, then snow-shading might be higher than expected. We emphasise here that SunSolve P90 currently doesn’t account for such interdependencies between uncertainties.

In any case, we can avoid making these three assumptions by applying the Monte Carlo method.

Monte Carlo

SunSolve P90 uses the Monte Carlo method to handle uncertainty.

More information on Monte Carlo simulations coming soon.

Determining P-values from a Gaussian yield distribution

It is worth noting that when a yield distribution is Gaussian, the P-values can be rapidly determined from the P50 and $\sigma_Y$ as

$$\text{P-value} = P_{50} \times (1 + z \cdot \sigma_Y),$$

where the z-values are given in the table below.

P-valuez-valueP-valuez-valueP99, –2.326, P1, +2.326P95, –1.645, P5, +1.645P90, –1.282, P10, +1.282P75, –0.674, P25, +0.674

Also, the 95% confidence interval is the range $\pm 1.960 \cdot \sigma_Y$.

Significant and negligible uncertainties

Some uncertainties are significant and others are negligible.

Take, for instance, the contributions of efficiency and area when computing the power of a module. Say the module’s STC efficiency is $\eta = (20.0 ± 0.8)$ and its area is $A = 2.00 ± 0.01$ m², amounting to relative errors of 4% and 0.5%, respectively.

At first glance, 0.5% does not seem negligible compared to 4%, but remember that errors combine as the sum-of-squares, when they’re independent and Gaussian.

Here, the STC module power is $P = \eta \cdot A \cdot$ 1000 W/m2, and so the uncertainties combine to give a relative percentage error of

$$\epsilon_P = \sqrt{4^2 + 0.5^2} = 4.03$$

Thus, in this example, the uncertainty in the module area contributes just 0.03% uncertainty to the module power. That is, the module power is (400.0 ± 16.0) W when one includes uncertainty in the area, or (400.0 ± 16.1) W when one excludes it. For most purposes, this additional error would be negligible.