Applying Sequences, Summation and Probability to Wind Turbine Arrangements
What is the best way to arrange a set number of wind turbines in Shanghai to achieve the maximum generation of energy?
Green energy is the future. As the son of an engineer who works on building wind turbines across the world, I have heard quite a bit about this rapidly growing industry. In a recent discussion with my dad, he told me about the significance of positioning wind turbines on wind farms, which are typically done in grid patterns, in regards to how much energy is generated. This came at a perfect time as I was searching for topics to base my final project on for my Honors Precalculus course at Concordia International School Shanghai. This got me thinking: how can I find the optimal arrangement for a wind farm in Shanghai with a set number of wind turbines using topics from my high school math course, including series, summation and probability? Here are my findings.
Setting Up the Problem
My dad explained that wind turbines experience decay in how much energy they generate based on how they are positioned. For example, if a wind turbine is stationed directly south of another, when the wind comes in from the north, the northern wind turbine will generate more energy as it experiences the force of the wind more directly. This is because wind turbines create turbulence in the incoming wind force, disrupting the wind flow and leading it to dissipate as it passes through the row of wind turbines. Notably, two factors control how much the power generation capability decays when going down a row of wind turbines: rotor diameter and distance between turbines.
I remembered that exponential decay could be represented by a geometric sequence, so taking what my dad had explained about the relationships of these factors, we came up with the equation
“P₁” refers to the amount of power in megawatts generated by the first wind turbine in a row or column, and “Pn” refers to the amount of power in megawatts generated per hour by the nth turbine in that row or column. “R” represents rotor diameter in kilometers, which is the length across the circle swept by the blades of a turbine. Finally, assuming the wind turbines are evenly spaced, “d” represents the distance in kilometers between each wind turbine in a row or column. The entire equation begins to make sense when the proportion of R over d is subtracted from 1. Basically, when there is a very large distance between each wind turbine (d), the effect of turbulence on diminishing the generated power decreases as 1 subtracted by a small decimal yields a common ratio of close to 1. When there is a very large rotor diameter (R), so much so that it approaches the distance between wind turbines (d), the effect of turbulence becomes highly significant as 1 subtracted by a large decimal yields a common ratio of close 0.
As a disclaimer, this formula is overly simplified, but it nonetheless serves as a basis for this mathematical exploration. Usually, the proportionality of R and d should be multiplied by a constant k, which varies depending on the design of the turbine, but this has been omitted for simplicity. Furthermore, when Rd is greater than 0.4, this equation is no longer applicable as the turbulence becomes too significant.
With this formula, the total power generated by a row or column can therefore be calculated by taking the summation of the geometric sequence for n turbines in that row or column. This value is then multiplied by h, for hours at which the wind farm is running at full power. Applying the formula for the sum of a finite geometric sequence, the following summation formula can be derived:
Calculating Power Generation
To understand how to use the above equations, I began with a simple square grid problem. Let’s say that there was a 4km by 4km plot of land that organized 25 wind turbines into a 5-by-5 arrangement, with each one equidistant from each other. In a diagram, this would appear as follows:
As indicated by the yellow line, each wind turbine is 1 km away from another, whether that be a distance in the direction of north, south, east or west, so the d value is equal to 1 km. As a generalization, let’s assume that the wind blowing is constantly 9 miles per hour, the average wind speed in Shanghai, and that each wind turbine can generate 1 megawatt per hour when experiencing a direct blast of 9 mph wind. In this instance, it does not matter which direction the wind is blowing from because it is a square arrangement, meaning the functionality of the wind farm as a whole operates the same regardless of wind direction. If the wind blew for thirty minutes from the north and then blew from thirty minutes from the west, the power generated is still equal to the power generated from an hour of wind from the north. Thus, considerations of wind direction can be omitted from these square based calculations.
For further simplicity, let’s assume that the rotor diameter is 0.1 km wide and the wind farm is active for one hour. Now, the calculations for the power generated by each turbine becomes a matter of plugging in the values. Let’s assume the wind is blowing from the north, so all wind turbines in the same row theoretically generate the same amount of power as they experience the same amount of decay. Their row number, as labeled in the diagram, becomes their n value in the sequence. With the values of the aforementioned assumptions plugged in, the sequence is now written as follows:
Therefore, by plugging in the n for each row number, the power generated per wind turbine in each row is:
Row 1 = P1 = 1 megawatt
Row 2 = P2 = 0.9 megawatts
Row 3 = P3 = 0.81 megawatts
Row 4 = P4 = 0.729 megawatts
Row 5 = P4 = 0.6561 megawatts
To get the power generated by a column of wind turbines in this example, simply plug in the correct values in the summation formula, which now looks like this:
This yields 4.095 megawatts of power. However, this square wind farm has a total of 5 columns all producing the same amount of power, so the wind farm as a whole yields 5 times that amount: 20.476 megawatts.
Although the calculations for this example were simple, the same process can be applied for any other values of R, d, h and n. In fact, square wind farm arrangements can go a step further in their equation. Let’s assume that each side of the square plot of land is represented by the variable l, for length, which is equal to 4km in the sample problem. Reexamining the diagram, it becomes apparent that d, the distance between wind turbines, is equal to l/(n-1) because the distance d is equal to a fraction of the entire distance l. The reason that the denominator is (n-1) is because for every n wind turbines in a row or column, there are n-1 spaces between them, as shown in the diagram.
Therefore, new equations for the power generated at the nth row and the total power generated can be derived.
These new equations directly relate land size to power generated, but they are only representative of square plots of land and square arrangements of wind turbines. The summation formula is now also multiplied by n for each row or column that generates that amount of energy, allowing Pgenerated to be the sum of the power generated by every row and column.
To test understanding, how much power would a 15-by-15 km wind farm with a 12-by-12 arrangement of wind turbines generate in 6 hours with constant wind speeds of 9 mph? What if the same wind farm experienced a slower constant speed of wind, and the highest generating wind turbine only operated at a rate of generating 0.75 megawatts? The solutions are outlined below:
For a max. performance of one megawatt generated per hour:
For a max performance of 0.75 megawatts generated per hour:
Optimizing the Design of a Square Wind Farm
The above section discussed how to calculate a wind farm that has already been built. In this section, the focus will be on how to optimize the power generated by building a square arrangement of turbines. To begin, the dimensions of the plot of land must be fixed. For consistency, let’s explore the same 4km-by-4km plot of land that was given in a diagram above.
In a square design, the wind turbines must be built in an n-by-n design, meaning there will be n columns and n rows of wind turbines on the plot of land. How can we calculate the best number of wind turbines in each row and column (n) and the best distance in between each wind turbine (d) to optimize the power generation of a square plot of land?
Recall from the previous section that the total energy generated within a span of time by a square wind farm is written in the following formula:
Let’s assume that, just like the previous problem, the time span is 1 hour at full power, the P1 is 1 megawatt per hour, the R is 0.1 kilometers, and the length per side of the land is 4 km. Adding in these values, we are left with this equation:
One of the best ways to find the maximum of an equation is to graph it. After slightly simplifying the expression and substituting Pgenerated for y and n for x, we are now left with the graphable equation:
It is important to note that this equation is not graphing the equation for the summation of power generated by a single arrangement over a span of time. Rather, since the ratio of the geometric sequence is constantly changing due to the changes in x within the ratio, each y-value is the summation of a different geometric sequence, representing a different arrangement of wind turbines. This allows us to compare the total power generated across many different arrangements of wind farms.
The graph’s maximum falls at an x-value of 13.388. Rounding this to the nearest integer, the optimal square wind farm for a 4km-by-4km plot of land is 13-by-13, which generates 42.913 megawatts of power. Recalling the equation relating distance between turbines (d) to land dimensions (l) and number of turbines per row (n), the distance between each turbine on this plot of land can be calculated as follows:
Now, we can map out the wind farm like this:
Applying Wind Direction
The previous problem allowed for essentially unlimited wind turbines within the 4km-by-4km land as long as it had a square arrangement. However, in the real world, companies who build wind turbines have a budget, meaning they will have a rough number of wind turbines that they can afford to build within a certain plot of land.
To find the best rectangular arrangement for a set number of wind turbines, we must consider the wind patterns of a geographical area, such as Shanghai. In order to do this, I needed sufficient data to back up my analysis, so I first found a wind “rose” graph detailing wind direction trends in Shanghai:
I took the hours of wind per direction and manually added them into a Google Sheet. As studied in my math class, I can then take the number of hours per direction and divide that by the total number of hours within the dataset (roughly one year) to get the probability that the wind will be blowing from one of the directions at any given moment. This is outlined in the table below:
With that said, these 16 directions dramatically overcomplicate the factors to consider for a rectangular grid. For simplicity, I reduced the directions into just north, south, east and west by combining the hours of intermediate directions to their closest one of the four. For northeast, southeast, southwest and northwest directions, which fall right in the middle of two main directions, I divided their hours in half and added them to their corresponding main directions. This yields a new, simpler table:
Taking it a step further, it actually does not matter which exact direction the wind is coming from for the purposes of calculation. Because the automation of wind turbines allows power generation to be equally efficient from all directions of wind, what really matters is the axis the wind is blowing from. Regardless of wind from the east or west, they are blowing horizontal to the orientation of the wind farm. Similarly, wind from the north and south are always blowing vertical to the orientation of the wind farm. Therefore, numerically combining north with south and east with west, an even simpler table can be used:
With North-South representing the vertical axis and East-West representing the horizontal axis, we now recognize that the wind will approach along the vertical axis slightly more often than from the horizontal axis. Thus, we must tailor the wind farm to be slightly wider than it is long to take advantage of the winds from the north and south.
Let’s say that a wind turbine company wants to build approximately 50 wind turbines on their 4km-by-4km plot of land in Shanghai and maximize the power they generate. Different from previous problems, the turbine arrangement is no longer square, so we now must distinguish between rows and columns. We can use variables for their numbers: n rows and m columns.
Since the company wants 50 turbines in total, we know that nm=50. Therefore, m=50/n.
The key to incorporating wind trends into our calculations is creating a weighted equation for power generation. Since winds come from the north and south 54.44% of the time, the entire wind farm will be generating power based off of n-rows 54.44% of the time. This is because the distance between each turbine and the amount of decay that occurs depends on the number of rows of turbines (or how many turbines are in a single column). With the same reasoning, the wind farm will be generating power from the m-columns for the remaining 45.56% of the time, which corresponds to the winds from the east and west. We can now create a new equation by piecing together this weather data with the previous equation.
Notice how the decay sequence for one row, calculated using n-rows, is multiplied by m-columns to account for the total wind turbines on the wind farm. The same logic applies to multiplying the decay sequence for one column by n-rows.
Let’s assume that, just like the previous problems, the time span is 1 hour, the P1 is 1 megawatt per hour, the R is 0.1 kilometers, and the length per side of the land is 4 km. Substituting in these values, we get:
Recalling that m=50/n, we can substitute 50/n in for m in the equation to keep only one variable:
To find the maximum power generated from these values, we’ll need to convert Pgenerated into y and n into x. This yields a final equation that we can plug into a graphing calculator:
The maximum of the graph falls at 6.333, which can be rounded to the nearest integer, 6. As proven by the points at x=6 and x=7, having an n-value of 6 generates more power (32.158 megawatts) than an n-value of 7 (32.068 megawatts). With 6 as the n-value, meaning 6 rows, then the m-value must be 50/6, or 8.333. This is rounded to the nearest integer, 8, creating a final arrangement of 6-by-8. While a total of 48 wind turbines is not exactly 50, the company will be grateful that they can build less turbines and generate more power, saving money altogether. The exact amount of power can be calculated by replugging these integer values into the original equation:
In a diagram, this layout appears as follows:
From a logical standpoint, this makes sense because if the wind was coming from the vertical axis more often, you would want there to be less rows to minimize decay and more columns to generate energy at this level of decay.
More Test Cases
To show the extent to which different weather conditions can lead to different optimal arrangements, I’ll illustrate some additional test cases below. Let’s pick two new, imaginary locations: Neutralia and Eastwestia.
Neutralia happens to have perfectly even winds coming from the north, south, east and west, meaning 50% coming from the horizontal axis and 50% coming from the vertical axis. Assuming all the same conditions of the previous question except for the wind conditions, how will the wind farm in Neutralia look different?
The “weighted” equation would look like this:
The graphable equation looks like this:
The maximum falls right around 7, which yields an optimal design of 7-by-7. In fact, the n-value at the maximum pictured on the graph is actually the square root of 50. This proves that perfectly square wind farms are ideal when there are no differences in the probabilities of wind directions. If you do the power calculations for this wind farm, you will see that it generates approximately 31.706 megawatts. On the other hand, the 8-by-6 wind farm that was ideal for Shanghai would only generate approximately 31.181 megawatts under these wind conditions. The 7-by-7 would look like this:
Meanwhile, Eastwestia has 80% of its winds coming from the east and west and only 20% of its winds coming from the north and south.To find the ideal wind turbine arrangement for this case, we can write the following weighted equation:
Notice how the 0.2 goes in front of the sequence with the n term because that applies to the north and south winds, which decay based on the number of rows and not the number of columns. This yields the graphable equation:
The graph for Eastwestia looks different from the others as there is no “hump” to find the maximum. Rather, the maximum occurs as it plateaus, at x=32.284, indicating that as the arrangement becomes increasingly tall and thin, there is minimal difference in power generation. With that said, since the wind turbine company wants 50 wind turbines in total, the closest factor of 50 is 25. The best course of action would therefore be creating a 25-by-2 arrangement. If the company chooses, they may want to add more wind turbines, but as the graph shows, the trade-off for the energy gained may not be worth the investment.
With a 25-by-2 arrangement, the wind farm could generate 33.733 megawatts. Taking the ideal 8-by-6 arrangement for Shanghai, which has 48 total turbines, and placing it in Eastwestia, the wind farm would only generate 28.222 megawatts. The 25-by-2 layout looks like this:
Conclusions
With all factors considered, the power generated by a wind farm with n columns of wind turbines on a plot of land with dimensions l*w can be calculated with the formula:
where:
R = rotor diameter length in kilometers
P1 = power generated by the first wind turbine in a row or column in megawatts
k = proportionality constant based on the design of wind turbine
s = total desired number of wind turbines on the farm
W1 = percent of time where wind blows along vertical axis
W2 = percent of time where wind blows along horizontal axis
l = the length (vertical distance) of the rectangular plot of land in kilometers
w = the width (horizontal distance) of the rectangular plot of land in kilometers
Notice how this equation now includes the proportionality constant k. As mentioned previously, different wind turbine designs may be affected by rotor diameter length (R) and distance between turbines (d) to different extents. Thus, k ensures that the proportional relationship between R and d in how those two variables influence Pgenerated is accurately accounted for in real world scenarios.
Also notice how the equation that once had purely l to represent land measurements now has both l and w, length and width. This was done to be inclusive of plots of land that may be rectangular but not square. Since the distance between each wind turbine (d) would be affected if the dimensions of the land were changed, it makes sense to now separate length and width in power calculations.
Though this equation seems daunting and convoluted, when it is applied to the real world, most if not all of the values should already be known. Companies should have data for wind patterns (W1 and W2) within their geographic location and already understand their design (P1, k and R). Moreover, companies should know the land they are building on (l) and already have a budget for a set number of wind turbines (s). Finding the maximum for n therefore becomes a simple matter of plugging it into a graphing calculator. With that said, even if you are not finding an arrangement but looking for some other unknown, this formula can still be helpful.
In the onset of climate catastrophe, this information is becoming increasingly important. Ultimately, the discoveries outlined in this paper are critical to ensuring not only that green energy companies and governments can make good investments, but also that the world’s ever depleting resources and ever shrinking land space is used to the best of its capability for the future of the planet.
