EME 810
Solar Resource Assessment and Economics

8.3 Uncertainty and Risk

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Reading Assignment

This section is the setup for why we use prediction and forecasting. As you will note in the reading, the actual world of meteorology and markets is full of uncertainty with respect to the time horizons of interest---with uncertainty comes risk. Events that occur in the future have importance to our clients, often as much as current events. We have established that our time horizon for financial analysis is on the order of decades, and we are effectively predicting the financial metrics to help our clients to encounter less risky scenarios than if they were to design and build a SECS themselves. So, what is risk?

What is Risk?

In solar project design and project management, we would like work for our clients to minimize risk. We describe risk as the dispersion of outcomes around an expected value. Something is riskier if the spread of possible outcomes is bigger around an expected value. The more specific terms of variance and standard deviation describe the spread of data (the dispersion) about an expected value. Events that occur in the future will have a spread of possible outcomes, because we cannot know the final value for the future with 100% confidence until it actually occurs.

When we really know what to expect, what we imply is that the dispersion of possible outcomes is clustered tightly about that expectation. From that information, we can adapt or make changes for the future appropriately. However, knowing the spread of possible outcomes about an expected value is deeply important, even if you know that the spread of outcomes is very broad. The greater the dispersion of outcomes, the higher the risk. In our reading, we comment that “riskier” scenarios in solar project development, or systems operation and management, will have a larger dispersion of outcomes around the expected value.

Risk is often framed as the probability of an uncertain event occurring in the future multiplied by the expected loss should the event occur. We call the model of the probability distribution the pdf, or the probability distribution function. Note that there are specific applications for continuous or discreet distributions (cf. probability density function, and probability mass function). If we know the probability distribution of all possible outcomes, then we also know the expected value of the outcome, surrounded by the dispersion of outcomes around that expected value. If the pdf is normalized, then the probability of any event can be evaluated by integrating a section under the curve. If we integrate under the entirety of a normalized pdf, then we are integrating across all possible outcomes. The total probability is then equal to 1.

Variance in Data

A common measure for the spread of data can be the variance ( σ 2 ). Given a sample of multiple events, the variance is a measure of the spread of the data about an expected value or outcome. We have also discovered in our reading that a portfolio of renewable energy can potentially be used to reduce the variance of the power generation coupled to the same grid. This is something to bear in mind in the future of large, distributed PV.

In Figure 8.1, below, we present a normal probability distribution function (also called a Gaussian). Now, many distributions in solar energy are not normally distributed, but this is a starting point. The data tends to be strongly skewed toward clear days (more clear days than overcast), or bimodal in nature. Quantitative analytics often use available statistical software such as R (The R Project for Statistical Computing) to estimate density functions based on discrete real data (e.g. a histogram). This is called density estimation. The peak(s) in a pdf represent the highest likelihood of expected values.

Ideal Gaussian distribution of data with boxplot and standard deviations shown.
Figure 8.1 Ideal Gaussian distribution of data with boxplot and standard deviations shown.
Click Here for a Text Alternative of Figure 8.1
This figure shows a normal distribution cast as a probability distribution function, with bands of probability labelled. The first part of the figure demonstrates that the “interquartile range,” or IQR, represents the band in which 50% of occurrences are expected to fall. The IQR is centered around the median of the normal distribution and extends to +/- 0.6745 standard deviations. The left and right edges of the IQR are labelled Q1 and Q3 respectively. Two additional bands are shown outside the IQR. To the left of the IQR, a whisker extends to Q1 – 1.5 * IQR and to the right the whisker extends to Q3 + 1.5 * IQR. Each of these bands is shown to have a probability of 24.65%. A second normal distribution below contrasts the IRQ with probabilities associated with moving one standard deviation relative to the median. We expect 15.73% of normally distributed occurrences to take values less than one standard deviation below the median, 15.73% of occurrences to occur with values greater than one standard deviation above the median, and 68.27% to occur within the +/- 1 standard deviation band surrounding the median.
Source: Box plot vs PDF by Jhguch Wikimedia is licensed under CC BY-SA 2.5

Self-check questions:

1. What does "pdf" stand for in the assessment of data regarding the solar resource?

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ANSWER: For a continuous distribution, pdf represents the probability density function, and reveals a model of the spread of data. The peak(s) in a pdf represent the highest likelihood of expected values.

2. How can we represent the spread of data about an expected value?

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ANSWER: The spread of data is represented by the variance, or the square of the standard deviation ($\sigma^2$) for the pdf.

3. Are solar resource data (e.g. global irradiance, direct normal irradiance) distributed normally?

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ANSWER: No. The data tends to be strongly skewed toward clear days (more clear days than overcast), or bimodal in nature.