### Reading Assignment

- J.R. Brownson,
*Solar Energy Conversion Systems*(SECS),**Chapter 9**(section on*Utility, Risk, and Return*) - W. Short et al. (1995) Manual for the Economic Evaluation of Energy Efficiency and Renewable Energy Technologies. NREL Technical Report TP-462-5173. (Read pp. 27-34:
*Uncertainty and Risk.*)

### What is Risk?

In solar project design and project management, we would like to 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 discrete 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** (${\sigma}^{2}$ ). 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.