This page explains how to evaluate the probability of achieving different levels of wind production that I refer to as P90, P99 etc. I also address the painful question of defining the difference between a one-year P90 and a ten-year P90 which is often a very confusing issue. Measuring the probability of achieving electricity from a wind project is a central part financing a wind project and are prepared before the financial close of a project because the P90 or P99 often drive the level of debt. In describing P90, P99 etc. I begin with evaluating some wind studies and describing two classes of uncertainty that drive the probability analysis. The first source of uncertainty is the variation in wind and the resulting energy production. The second is modelling error which is a lot more difficult to understand and define. Then I explain how to combine standard deviations (often called uncertainty) with one another using mean squared error. Once the mean squared error is defined the one-year and the ten-year (or twenty-year) P90 and P99 is established. At the bottom of this page I discuss incorporating the P90 and P50 in a financial model where the tariff may be derived from a target IRR on the P50 case and the financing could be derived from either a P50 or a P99 or a P90 case and the financing could further be derived from either debt to capital or from DSCR.
The tricky part of the P90, P99 etc. is to understand the sources of variation in wind production estimates that are mean reverting and that are related to modelling errors that do not self-correct. This webpage also describes how to size debt with alternative parameters. This is used to demonstrate the P90, P50 etc. without incorporating permanent and and modelling errors. The third section demonstrates some studies of 1-year and 10-year P50 and P90 and how to use the NORMINV function and the mean square error statistics.
Examples of How P99 or P90 Affects Debt in Term Sheets
To illustrate how P99 or P95 or P90 can affect the size of debt I have included selected examples of terms from actual term sheets. The first screenshot is from a term sheet that was used for negotiation. At the bottom of the excerpt, you can see that the debt size is constrained by the P99 case where the DSCR for a one-year P99 case is 1.0. In the P50 case, the DSCR constraint is 1.35. This means that as long as the percent difference between P99-one year and P50 is less than 25.9% (DSCR-1)/DSCR, the P99 case will drive the debt sizing from the the DSCR. The second screenshot below the term sheet excerpt is from various different lender bids. This has very similar terms with respect to DSCR’s on the P50 and one-year P99 in case you do not believe me.
The use of one-year P99 just means bankers look around for the worst possible number in the wind study. The term 1-year P99 versus 10-year P99 does not have anything to do with how many historic years were used in estimating the probabilities. What it means is that the 1-year includes two things. The first is potential wind variation that can occur in one year. The second thing it includes is modelling errors. When you make a modelling error, it is permanent. When there is variation in wind, this is mean reverting and not as much to worry about. A 10-year or 20-year P99 means that instead of worrying about fluctuations every year, you average out the wind variation that will move up and down and instead you average out the wind for 10 or 20 years. All of this means that the one-year P99 has two effects: (1) cyclical effects and, (2) permanent modelling errors. The 20-year P99 has one effect: the permanent modelling error.
Selected Examples of P90, P95 etc. Estimates from Wind Studies
I have included selected examples of wind studies below in terms of the percent difference between the one-year P99 and the P50 case below so you can get an idea of whether the P99 or the P50 drives debt sizing from the DSCR. The first case presents a case where there was no good reference site and the difference between the P50 case and the one-year p99 case is large. When computing the P99 or the P90 in the above table, you can use the NORMINV function. The inverse part of the function implies that you begin with the probability and derive the X value. You need the mean and the standard deviation along with the probability. With the standard deviation defined in line 15 and line 16 you can compute the lower values for the P99 and P90. For example, to compute the 1-year P99 of 78.31 in the first column, you can enter the function as follows:
Analysis of Multiple Wind Farms and Evaluation of One-year and Ten-year P90
The screenshot below shows data from the credit report for different wind farms as well as the P95 and other cases. The screenshot shows the one-year P95 etc. as well as the 10-year P95 etc. for the different wind farms. Note first that the P50 case is the same for the 1-year case and the 10-year case. For the other cases, there is a bigger variation in the 1-year case than the 10-year case. Note that P50 case is the same in all of the tables as the mean does not change. For example, in the case of the first farm, LB II, the one-year P95 is 270 while the 10-year case is 300, which is closer to the P50 case of 333. The 10-year case does not include cyclical changes and only includes uncertainty related to modelling error. When there a reduction in energy production because of modelling error, this error continues and does not have any reversion to the P50 case. The one-year case includes this permanent modelling error as well as cyclical differences that can make one year have a lower value because of cyclical wind patterns.
The percent difference between the P50 and one-year P95 is shown in the table. Note that percent difference varies between 82% for SW Mesa and 56% for Montfort. This implies that the required DSCR for SW Mesa to cover 1-year P95 wind variation would be 1/.82 or 1.22 for SW Mesa. The DSCR to cover the 1-year P95 risk would be 1/.56 or 1.79. The ratio of the 10-year to 1-year variation shows how much of the variation comes from temporary or cyclical wind speed changes versus how much comes from permanent modelling errors. If the percent 1-year to 10-year is 100%, then all of the error comes from modelling permanent error and none comes from cyclical effects. If the percent is lower, then the cyclical effects are relatively greater.
The screenshot below uses the table above and derives the P99 case. To do this, the standard deviation must be derived. You can derive the production for a P95 or P90 or P75 case with the NORMINV function. To compute production from the NORMINV function, you need the probability (which is given), the mean (which is the P50 case) and the standard deviation. The standard deviation is not known. But you can put in some arbitrary number for the standard deviation. Then you can compare the computed production using the NORMINV function with the given production in the table. With the difference between the computed production and the given production for P95, P90 and P75, you can use the goal seek tool to derive the implied standard deviation.
Once the standard deviation is computed with the goal seek the P99 can be evaluated. Note that for all of the wind farms except the last Monfort plant, the standard deviation is the same whether the goal seek is made of P95, P90, or P75. This should be the case as the standard deviation is a given number. For the Monfort case, the change in standard deviation does not make sense and suggests there is something wrong with the data (note the change from 15 to 11 to 7 in column J and lines 13-15). The manner in which the goal seek can work across multiple rows and columns is demonstrated in the insert below.
Sub Mult_goal_seek() ' ' 1. Works for cells rather than range name ' 2. Can make loop with zero ' 3. replace row and col ' For i = 0 To 3 row1 = i + 23 row2 = i + 13 For k = 0 To 6 col = k + 4 Cells(row1, col).GoalSeek Goal:=0, ChangingCell:=Cells(row2, col) Next k Next i End Sub
Modelling uncertainty are not “mean revering”. The videos and files also cover a subject that I find one of the most difficult issues to explain — i.e. the difference between one year P90 and ten-year or twenty-year P90. I have tried to explain this with file named “Wind Study” listed below. This file uses a nice old financial analysis report that listed P50, P75, P90 and P95 for a series of different wind farms. It also reported the production statistics on an 1-year basis and on a 10-year basis. Using the P90 etc. production statistics you can back out the standard deviation that is related to wind variation only as well as the variation that is only related to permanent effects. I have also compiled an analysis of the variability in wind after projects are operating relative to before they are operating. My key theme is that standard deviations underlying the ten year P90 are subjective. I demonstrate how to use the NORMINV function in excel to understand data in wind studies.
Wind Uncertainty Analysis from Analysis of Actual Plants
I have created a database of actual wind projects that contains actual historic production data on operating wind projects in MWH. In the U.S. the Energy Information Agency collects data on every power plant production by month (some plants do not seem to report in some years). I have used the month by month data and converted the data to annual uncertainty in order to evaluate the actual year to year uncertainty. There are some natural problems with this analysis. First, the configuration of the project may change. Second, there are not a lot of projects with long-term data. I am in the process of adding capacity to the database so that you can evaluate the capacity as well as the production data so you can see the capacity factor. I have put a similar file in the solar analysis section and the hydro analysis section. The hydro section has much more data. You can download the file by pressing the button below. I have put a couple of screenshots below the button to download the file. The screenshots illustrate the data collected from the EIA and the analysis made with the data.
Dissecting Wind Variation in Analysis with MSE
Study in which you can dissect the one-year and ten-year P90, P99 etc.
In the case with 1-year and 10-year P90, P95 etc. you car given selected results. But you cannot evaluate how much of the production of electricity from wind comes from the wind variation by itself. In this section I will demonstrate how this can be done. Part of the reason for this is because you can have a better understanding of what causes variability and what causes the variation. After the variation is understood you can compare the wind only variation to the analysis discussed in the prior page.
The general idea is that standard deviations underlying the ten year P90 are very subjective where standard deviation in things like wake effect, availability, turbulence, correlation to historic site, wind shear, losses and other factors. One of the main tools in analysis of wind production with different probabilities is use the NORMINV function in excel to understand data in wind studies.
Debt Sizing with P50 and P99 etc.
It has become standard in the industry to apply different debt service coverage ratios to different wind production cases. A typical scenario is that a 1.35x coverage ratio is applied to the P50 case while either a 1.2x coverage is applied to a P90 ten-year case or a 1.0x coverage is applied to the P99 one year case. The modelling issues can be a little difficult as the debt may be sized on one scenario but the equity IRR is computed from a different scenario. The exercise below applies these concepts.
Debt Sizing in a Financial Model from Either P50 or P90 while Deriving Prices from the P50 Case
The first case has a single debt issue and does not include a process which goes to the P50 case, finds the tariff and then tests the debt size in both the P50 case and the P99 case. The second file does this with multiple debt issues. It is one of the tricky issues in project finance I think.
P90, P99 and DSCR Debt Sizing.xlsm
P90, P99 and DSCR Constraint.xlsm