EME 810
Solar Resource Assessment and Economics

2.15 Applying Shading to a Solar Chart


Reading Assignment

  • SECS, Chapter 7: Applying the Angles to Shadows and Tracking

Here, we are going to work through a problem of plotting critical points of shading onto an existing sun path diagram. Keep in mind that we will use the plots that you developed from the University of Oregon Solar Radiation Monitoring Laboratory on the previous page of this lesson.

Here, I am using a program called Skitch to draw over the top of my PDF files. There is a 30-day free trial of the editing software if you would like to use it to digitally mark up your documents. Otherwise, I recommend that you print out your work, draw on the print directly with a pen, and then take a snapshot of the edited image with your phone or scanner to upload.

Quick review of the Charts:

First, we will go over the key features of the orthographic plots with the arcs of six days plotted out for the first half of the year.

Video: Sunchart Ortho Intro (4:42)

Sunchart Ortho Intro
Click Here for Transcript of Sunchart Ortho Intro video

All right, so, here we have a sun path diagram of State College, Pennsylvania. We've developed this in the University of Oregon's website for our local latitude and local longitude, time zone UTC minus 5 hours. That's Daylight Standard Time, not Daylight Savings Time. And I've done a little bit of modification to this to give it the green background. So, this is not something that normally you would see inside of your normal program.

But, I want to point out a few things. One is that we see that, as discussed before, the sun rises in the east. And the sun sets in the west, right?

So, we're going to have a progression of the sun across the day from left to right. In this particular case, we start out with east on the left, west on the right. South is denoted by 180 degrees, right?

And so, the thing that I wanted to point out here is that here we have December at the bottom and June 21 at the top. So, this December 21 is the winter solstice. June 21 is the summer solstice. And, in between, we have right about March 21 through the 23 is going to be one of the equinoxes. The other equinox, of course, is going to happen in September.

Only half of the year is shown. The arcs that you're seeing here are all the hours of the day interpreted as solar time. So, this is going to be 10 o'clock solar time. We're using a 24-hour system here. So, this is going to be 2 o'clock solar time, right?

And when I look at this, I see that I have two key points-- 90 degrees and 270 degrees. 90 degrees, of course, is going to be due east. And 270 degrees is due west, in this particular meteorological standard for the azimuth directions.

We see that on the left side, we have the altitude angle. That would be alpha. And we're counting down for the zenith angle, the complement to the altitude angle.

Let's go back to this east and west. So, when does east happen? East happens at due east, is 90 degrees. Due west is 270 degrees.

What's really important here is when does that happen? You see that there's an arc right along here for March 21. And that's really close to the equinox. So, during the equinox, we have the sun-- that's the only time of the year-- so, two times during the year when the sun rises due east and sets due west.

And if we count the hours, we have-- one, two, three, four, five, six-- exactly six hours in the morning-- one, two, three, four, five, six-- exactly six hours in the afternoon, so, a 12-hour day, our only 12-hour day that's going to happen. Otherwise, all of the rest of this time if we're looking at-- let's grab a color here. If we're looking at all of this time of the year from December to February and after March, so, before March is down here. After March is up here. So we're going to have short days, shorter than 12. And we're going to have longer than 12 hours up in the summertime.

Gives you a brief breakdown of what's happening on orthographic projection. This is for State College, of course. As we go further north, we're going to see that our sun charts are going to get lower to the ground.

If we were to choose a latitude location that is closer to the equator, we're going to have plots that start to look like this, actually. They're going to look really kind of funny because they're going to spread out up into the 90 degree space because once you're within the tropics, you're going to find that the sun can be to the north or to the south. It's not always in the south as it is when you're beyond the tropics. Anyway, I hope that's a helpful explanation and--

Credit: J. Brownson © Penn State University is licensed under CC BY-NC-SA 4.0

Second, we can compare the key features of the orthographic plots to the polar plots, again with the arcs of six days plotted out for the first half of the year. You should notice the similarities (East is on the left, solar time is represented, the same location is plotted), as well as the distinct differences (the June and December arcs are "flipped").

Video: SunChart Polar Intro (5:49)

SunChart Polar Intro
Click Here for Transcript of SunChart Polar Intro video

All right, and now we got the same location. Again, the location is going to be State College Pennsylvania. Solar time is going to be minus 5 hours, relative to UTC. And latitude longitude is the same.

Again, this is a polar plot of the same data that you just saw in the previous video. And the north of this plot is right down at the bottom. The south is up at the top. And I do this in this particular case, just to keep the arcs in the same general direction. But you're going to notice some differences here.

One is that the June arc is at the bottom. Whereas the December arc is at the top. So, this polar projection is what you would get if you were effectively lying on the ground and looking at the sky with a fisheye lens. And then trying to project that flat.

So, we see here that we've got the arcs of the day the morning begins over here. The evening ends over here, right in June time. And the progression of the day is going to be across the arc, again, from left to right, the same thing as we had before. Left to right in the winter months as well.

But here, you're really seeing the differences in the length of the day. It's probably a lot more apparent here that the length of the December 21st day is much shorter of an arc then the summer solstice on June 21st. Again, our arrows just pointing out, these green arcs are days. And the red lines are the hours of the day in solar time.

So that this top location here at 90 degrees, is going to be the top of the sky, the zenith. So, the zenith angle is basically any angle down from here to one of these circles. Whereas the altitude angle is going to be the angle up from the ground, which is going to be in our case the edge of the ring up.

So, we're going to see in a zenith angle going down or going outward, two rings. An altitude angle coming up or inward, basically coming along the edge of that skydome. And any one of these points of these green arcs are going to be a combination of an altitude angle and a azimuth angle.

And here, the azimuth angles are going to rotate from North which is zero degrees. North right here, is zero degrees. Rotating along to plus 30, plus 60, to finally when we're due east we are at 90 degrees. When we are due west, we're at 270 degrees. South in this case, is going to be 180 degrees. So, the azimuth rotates around clockwise and 180 degrees is in the meteorological standard going to the south.

Again, I want you to pay attention to the one day of the year when the sun rises due east and sets due west. And that's going to be around this, March 21st through the 23rd. It's kind of a flexible date depending on the year.

But it basically is defined as the day when within which the equinox occurs. And so, it's going to be one of the few days, or the only official day that you're going to have twelve hours of sunlight. So, we can count again one, two, three, four, five, six hours in the morning; that's going to mirror to the six hours in the evening making it a 12 hour day.

And again, that means that we're going to have everything in the summer is going to be longer, whereas everything in the winter months is going to be shorter. And that's the flip that I'm talking about in the notes, that the arcs flip back and forth. So, long days are on the bottom, short days are on the top, or short days are towards the south.

This should make sense when we think about the fact that the sun is low in the sky, low in the sky is going to be closer to the outermost rings. The sun is low in the sky in the winter. The sun is high in the sky, especially around the noon hour, during the summer.

And you're seeing that right here, is that the closer I am to this center ring, the closer I am to right here. Which is 90 degrees, the higher in the sky that I am. And so, in the winter time, I'm close to the perimeter, which is close to zero degrees altitude angle. This up at the top is close to 90 degrees altitude angle.

Credit: J. Brownson © Penn State University is licensed under CC BY-NC-SA 4.0

How to integrate shading as an overlay

Now, we need to add an additional layer of information to the plot. I'm going to talk through the addition of critical points on the sun chart, followed by connecting those points and shading the areas correctly. In each of the following examples, we will use orthographic projections, but there is no reason why you couldn't use polar plots instead.


We are not plotting what the shadow "looks like." Instead, we are plotting the times over the course of a year (or half-year) for which a shadow exists on our receiving points of interest. We know that the actual shadows on a building are rectangular in shape (or have straight edges). However, the plot of the times in which shading occurs will not necessarily be rectangular.

The first example will come from the textbook. We will be plotting shading relative to a single receiving point: X, with three critical points of shading (A, B, and C).

Video: Shading Wing Demo (5:52)

Shading Wing Demo
Click Here for Transcript of Shading Wind Demo video

So now, we want to plot some critical points. And I'm going to go back to what you already encountered in the textbook in Chapter 7, applying the angles to shadows and tracking. So, let's focus on shadows. The first example that we've got here is the example of a wing wall, a wing wall being an extension out from the side of a building that might be blocking the sun. In the case that we've got in the textbook, we've got a wall that is to the west of the site.

And so, in our case, that means that on this plot, it's going to be a shadow that's going to be occurring on the west side in the afternoon sun. And we'd like to plot when is that shading going to happen on our central point, our point of interest, x. In this case, x was in the center of the building. So, we can imagine that this is a window that we wanted to understand-- or a central point in the window-- and we wanted to understand when is it being shaded versus not shaded.

And so, we did some basic trigonometry in the textbook to give us critical points between the central point, x, and three other shadowing critical points, A, B, and C. And those points are listed in the textbook. And so, what I'm going to do right now is just start plotting those out. And so, we had our points in that case listed as 0 being the south.

And so, when I had plus 54 degrees, what I was having is in addition of 54 degrees onto the original 180 degrees. And so, we're going to end up with some points at the bottom. And they're going to go up to our critical point of A, which was 0 degrees-- altitude angle-- 0 degrees altitude angle and 54 degrees to the west, to the afternoon azimuth angle.

And then point B was going to be a separate altitude angle upwards of 35 degrees. So, let's grow to approximately 35 degrees. We'll plot a second point right there. And our third point-- so, this was point XA. This point was point XB.

And now we're going to need a third point that's going to be point X relative to the critical point, C. And that was going to be 45 degrees up, so, we count up 45 degrees. And we're going to go up 45 degrees. And we need to go into the west in the azimuth 114 degrees. So, first, we had 100 degrees. And then, we're going to get about 114 degrees across to finally get our critical point, C.

So, if I want to connect these guys together, the first thing I know in an orthographic projection-- and I'm just telling you this-- is that it's a vertical drop down. So, lines connected together vertically will basically just have a vertical line down. The connection for the points here is going to be more like an arc. And so, we make sure that we have a nice arc tying these points together.

And I'm just going to extend continuously out towards the north. And then, I'm going to shade that data in under here. So, all of this region is going to be under shadows or our wing wall. So, we understand that in December, we get shading happening at about 4 o'clock in the afternoon. So, 4 o'clock solar time shading happens for the rest of the day.

On March 21, shading is going to happen at about 2:50. And by the time we get to the other extreme, the summer solstice of June 21, that time is going to be about 3:45. So, from 3:45 on to the end of the day, we're going to have shadowing occurring for this particular point, X in the problem.

And now, if I were to take that same plot and show you what we did inside the textbook, it's going to look just like this. So, it's something very much like what the textbook is just by plotting the points out and connecting those points together.

Credit: J. Brownson © Penn State University is licensed under CC BY-NC-SA 4.0

In the next example, we will look at setting up the problem to assess a PV array shading problem. Our intent is to plot shading relative to multiple receiving points: A, B, and C, with three critical points of shading (1, 2, and 3). The result is nine values for altitude coordinates ( α , no subscript) and nine values for azimuth coordinates ( γ , no subscript).

Video: Shading Array Demo 1 (6:08)

Shading Array Demo 1
Click Here for Transcript of Shading Array Demo 1 video

So, here we have the scenario of an array of photovoltaic that are set up one row behind the next. They're each going to have a certain tilt. That tilt is represented by beta and each one is going to have a common collector azimuth of gamma, represented down here. And that gamma again, is that plane or rotation. In this case, the array that you're seeing is rotated 9 degrees towards the east. So, minus 9 degrees of rotation or 9 minus 180 degrees to give us our azimuth.

The distance between the panels, right now it's just specified as D. And the panels themselves are going to have a shadow. And that shadow is going to change over the course of the year, as the sun is high in the sky, and low in the sky.

And what we'd really like is for these panels to be spaced appropriately. Such that, they do not block each other. Because this is one of our goals, one of our mechanisms for the goal of solar design. You want to maximize the solar utility for the client or stakeholders in a given locale. And in this locale, want to know how far apart we can space these to collect the energy to basically avoid, or remove shading from the spacing of these panels.

So, what you're seeing is a system that we're going to define in terms of critical points. We're going to take those critical points and we're going to plot them on a diagram. So, the first thing is, how do we list these critical points. Well, now, we don't have a central point X.

Now, we actually have three points for each one of the panels across the top and across the bottom. This guy is going to be behind here, you won't see it. But you're going to have three points along the bottom, three points along the top. The points along the top are ultimately going to shade these critical points along the bottom.

So, I'm going to name these critical points A, B, and C. And the points that we will be referring to in terms of what kind of shading are we expecting, we're going to label 1, 2, and 3.

So, now, going into this, you're looking at this from the side and you're seeing a system like this; there's going to be a certain tilt beta. The beta is going to be the same from both collectors and they're going to be separated by a certain distance D. That's either going to be the spacing from top to top or from bottom to bottom, that's the same spacing with D.

So, looking at this, we want to basically compare any point 1. And what we really like to see is, how does 1 compare to point C, point C is down here. One to point C, 1 to point B, and 1 to point A. That's one of our first questions.

And then, after we've done that, we're going to look at how this point 2 compares to critical point C, critical point B, and A. And then, we'll finally finish that with 3 C, 3 B, and 3 A. And what we should be able to see is that because of similar geometries, we're going to find some kind of similar responses, in terms of all of these geometric relationships.

And I can show you that, again this is in the textbook, but if I bring this up right here, you're going to see that I've got a table of points 1.A, 2.A, 3.A, just like what we were talking about. And 1.B, 2.B, 3.B, 1.C, 2.C, 3.C. They each have their own set of altitude angles, and you're going to notice that there are certain 21 degree common altitude angles. Just due to common geometries. Similarly, you're going to see common 41 degree angles and two 12 degree angles.

Looking at the azimuth angles, the 0 degree azimuth corresponds to 180 degrees in the meteorological standard, and so on down. So, we're seeing that 76 degrees is equivalent to 250 degrees. And minus 64 degrees is equivalent to 116 degrees.

So, we're going to take these points this 180, 244, 256 for the azimuth angles of the collector. And we're going to plot those in the next block and we'll plot the alpha angles. And what we're going to come up with is basically something that looks like the cross section of a loaf of bread. It's going to have two vertical sides and it's going to have an arc in the middle.

Credit: J. Brownson © Penn State University is licensed under CC BY-NC-SA 4.0

You will notice how the horizontal surfaces of the building that create shadows are transposed to the projection as an arc, not as a horizontal line, while the vertical surfaces remain vertical. This has to do with the manner in which spherical data is distorted in an orthographic projection. Hence, the plot of a building shading a point on a window will look a lot like a slice of bread, flat on the sides with a soft curve across the top. The same will be found for this example, where the receiving points A, B, and C are shaded at different times by critical points 1, 2, and 3.

Video: Shading Array Demo 2 (5:57)

Shading Array Demo 2
Click Here for Transcript of Shading Array Demo 2 video

Now, we transition to plotting those points for the two rays onto our sun path chart. And so, here we have our sun path. It's again, I'm just using it for State College. You could use it for your location, in which case, the times at which you're actually going to be shadowing each other are going to be different.

So, if I go to my table from the textbook and I look at 1a, 2a, 3a-- I'm going to do those in blue first-- I'm going to get alpha values at, first, 180 degrees azimuth. And I'm going to go up 41 degrees for 1a. Then I'm going to go to 244 degrees azimuth, or 64 degrees from the convention in the book, up 21 degrees.

All right, 244 and 21. And I'm going to go down to 256. So, 256 is 76 degrees, and down to 12. So, somewhere right around here is where I'm looking at.

That plots the three points for 1a, 2a, and 3a. If I go over to 1b, 2b, 3b, I'm going to need to go to 116 degrees and then up 21 degrees. That's going to be for 1b. I'm going to need to go 180 degrees and up to 41 degrees, the same point as 2a. And then, my final point is going to be 21 degrees up and 244 degrees over.

So, again, we've got common overlap there. We're going to see this a couple of times. So, now, let's go to our last set of points. And that was going to be the 1c, 2c, and 3c. We're going to do those in green. And that's going to be 12 degrees up at 104.

So, that's probably going to be the mirror image of this point right over here, point 3a. And we're going to also look at 116 degrees 21 degrees up. That's an equivalent point, also.

And finally, our third point is going to be 180 degrees and 41 degrees up, which again, as we saw, is a common point for all of these. And so, when I connect these together, I'm going to go to my farthest point on each edge and drop a vertical line down. This is vertical, should be vertical. And then, I'm going to draw this loaf of bread arc between all of them.

It might not be pretty. In fact, I might do that one again. Let me just connect these points. Oops. Like this, back down. And then, finally, connect it together.

So, everything under this curve is going to be in shadow. And so, what we are seeing is that the months for this particular array, the way that it was designed, you're going to see that-- if I just look at the analysis of this-- you're going to see that even up to March in the winter months, definitely throughout the entire month of December, you're going to have the photovoltaic array shaded, which is not a good sign. And, again, into the afternoon of March, we're going to have shading.

So, we're going to see some distinct shading possibilities for this array, suggesting that when we actually want to develop this array, we'd want to space the array further apart. And to what degree would we want to space it further apart, we'd effectively want to look into how do we get this array spacing to be far enough apart so that the loaf of bread top fits underneath this area where it's not blocking any of the months in the hours of the day. And we can do that with effective design.

So, right now, as it's designed, the front array is going to shade the rear array. And that's going to create a problem. And we can count the number of hours that shading is occurring in that period. And we could enter that data into our system adviser model into SAM and then run the simulation to find out what the losses would be relative to no shading in that system. And we're going to do that in the next page.

Credit: J. Brownson © Penn State University is licensed under CC BY-NC-SA 4.0

Next, we will be interpreting our results, and inputting the shadow data into SAM.


In your homework assignment, you will be asked to do a similar procedure for the solar array, but with the use of the polar projection of the SunChart, which provides somewhat better accuracy for determining the shading factors. While the solar coordinates are positioned differently in the polar projection, the principle of plotting critical points is the same - you find the positions for the points and connect them to define the shade/no-shade boundary. See an additional video demo on this in Canvas.