# Uncertainty Quantification in Energy

## Introduction

 Fig. 1: Wind turbine in Germany. (Source: Wikimedia Commons)

The energy business is at the center of human societies and powers the advancements in technology and overall human well being. However, with the steady increase in the global population, projected to reach almost 10 billion individuals by 2050, the energy supply has to align with the demand. [1] Consequently, decisions about and management of resources have become critical, as they can have a huge economic impact or can cause energy shortage if poorly handled.

These concerns lead to the consideration of uncertainty quantification in Energy. Here, uncertainty quantification does not refer to probability states in quantum mechanics, nor will it deal with uncertainties in commodity pricing. We will first define what uncertainty quantification means in the current context, then give two examples of applications in the Energy business.

## What Is Uncertainty Quantification?

Energy production often relies on phenomena dictated by nature that are not fully controllable or only partially understood. Yet, even if often a physical model is available, we usually have access to sparse data about their key parameters (think about the weather forecast). Uncertainty quantification consists in taking into consideration these sparse data to assess the variability and potential risks in energy production. Hence, the aim is to give a measure of confidence in energy production scenarios given that we only partially capture the parameters of our model.

This methodology is critical for every major energy sources. For solar, the performance of panels obviously depends on the solar exposition, which has variations at different time scales (day, month or year), but the longevity of the panels also depends on how long they can withstand weather changes (heat and cold cycles, rain, etc.). For hydraulic, even though the power produced by a hydropower resource

 Fig. 2: Example of an oil reservoir. Courtesy of Matthias Cremon.
P = η ρ Q g h

has parameters (η = turbine efficiency, ρ = density of water, Q = flow rate, g = acceleration of gravity and h = height) that can be robustly estimated, its efficiency in a longer term may depend on the weather variability (cf. the severe draught in California). [2] For batteries, we can, for instance, question their lifetime or their probability of failure given the components that we are using. This uncertainty framework typically involves the use of probability, coupled with the study of partial differential equations.

## Two Examples

Let us take wind and hydrocarbon based energy sources as examples:

1. Wind: Wind is a very clean source of energy that requires a minimal human intervention to function, except obviously for building the wind turbines (see Fig. 1). However, because of wind intermittence and non-storability, wind power forecasts can be unreliable. For this reason, uncertainty quantification of wind power forecast can be valuable to mitigate shortage risks and be useful to take informed decisions. According to Petrone et al., the wind power business encounters numerous sources of uncertainty. [3] First, the wind turbines can suffer from wear and tear that can affect the production quality. This issue can be caused by various weather changes (e.g. snow) or by insect contamination. Second, maybe prior to installing wind turbines, we need to have a estimate of the wind probability distribution in the area where the wind turbines are installed. This distribution is often modeled as a Weibull distribution. [4] (see Fig. 3) Finally, the wind power also has a probability distribution, with a peculiar distribution as wind turbines provide no power outside of cut-in and cut-out wind speed. [5] For all of these uncertainties, we can assign probability distributions and then apply statistical and numerical methods to obtain confidence intervals of the wind power output.

2.  Fig. 3: Fitting Weibull distribution to observed wind data in Plymouth Mount Batten, UK. (Source: F. Ibrahima. Adapted from Bradbury. [7])
3. Oil Reservoirs: Reservoir management deals with efficiently operating an oil field to optimize the oil recovery under certain constraints (see Fig. 2). This recovered oil is then ultimately used as a source of energy to power most of nowadays transportation means. According to Zaydullin, oil reserves can be estimated by the use of decline curve analysis and material balance methods. [6] However, and as pointed out in the article, modern stochastic approaches should be used together with these previous analyses because of uncertain reservoir parameters. Indeed, since reservoir simulation gives a prediction of phase flow in the subsurface, how can I be sure about the composition of the subsurface when I only have access to sparse measurements (essentially around the drilled wells)? The answer is simple: we cannot. Therefore, we have to take these uncertainties into account when we simulate oil recovery predictions. This is generally done by assigning a probability distribution to the subsurface properties (such as its porosity and permeability fields) and then estimate the flow response, which ultimately leads to an estimate of oil recovery.

## Conclusion

We have seen through two different examples that uncertainty quantification indeed plays an important role in the energy world. The ability to obtain reliable confidence intervals of key factors to energy production has been a major research topic. Thanks to the improvement of high performance computing and the development of novel and efficient mathematical methods, we can better measure the impact of uncertain factors to energy production and consequently take better informed decisions. Finally, these two examples show that data play a huge role in the uncertainty quantification. And if predicting the physical models may not be sufficient because of uncertainties, predictions with data seem to be a seducing option. Questioning the capacity of predicting and providing energy based on data with the use of machine learning will be the topic of the next article.

© Fayadhoi Ibrahima. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.

## References

[1] "World Population Prospects, The 2015 Revision: Key Findings and Advance Tables", United Nations Working Paper No. EAS/P/WP.241, 2015.

[2] D. A. Chin, Water-Resources Engineering, 2nd Ed. (Prentice Hall, 2006), p.863.

[3] G. Petrone et al., "Wind Turbine Performance Analysis Under Uncertainty," American Institute of Aeronautics and Astronautics, AIAA-2011-0544, 49th AIAA Aerospace Sciences Meeting, 4 Jan 11.

[4] K. Conradsen, L. B. Nielsen, and P. P. Prahm, "Review of Weibull Statistics for Estimation of Wind Speed Distributions," J. Clim. Appl. Meteorol. 23, 1173 (1984).

[5] R. Miller, "Wind Energy: Why We Don't Use It", Physics 240, Stanford University, Fall 2014.

[6] R. Zaydullin, "Oil Recovery Prediction", Physics 240, Stanford University, Fall 2013.

[7] L. J. S. Bradbury, "A Critique of the Sherford Wind Turbine Plan", November 2011.