I didn't get a chance to write a program to make the calculations, so I followed Wayne's "paper" calculations. When I arrived on-site, the owner had decided he wanted to mount the panels on the South-of-East-facing roof, even though the rails were too long for that roof, so I redid the calculations for that case. A Google Earth plot gave me an azimuth of 28 degrees South of East versus the 30 I had previously guesstimated (not that it really makes a difference versus 30 degrees).
Here are the calculations and their somewhat-surprising results:
wwhitney wrote:I think the way to approach this is to use vector math on the vector orthogonal to the face of the panels. Vector length will be arbitrary, as we are only interested in direction.
So let's have the x-axis point east, the y-axis point north, and the z-axis point up.
If the 4:12 pitch roof faced due south, the orthogonal vector would be (0,-4,12) or (0,-1,3).
Since the building is turned 120 degree CCW from above, this become (-sin 120, -cos 120, 3) = (-sqrt(3)/2 , 0.5, 3)
Actually, looking at this, I think the building was 150 degrees CCW from East. No matter, here is what I calculated for the South-of-East-facing roof:
Since the building is turned 28 degrees CW from above, this become (cos 28, -sin 28, 3) = (0.88 , -0.46, 3)
wwhitney wrote:Say you want the panels to face south at an elevation of 40 degrees. That vector would be (0,-sin 40, cos 40).
We decided to shoot for 45 degrees elevation angle, so this is what are target was:
That vector would be (0,-sin 45, cos 45) = (0, -0.707, 0.707).
wwhitney wrote:The cosine of the angle between the vectors is their dot product divided by their lengths. The angle is
arccos((0 + cos120 sin40 + 3 cos 40)/sqrt(10)) = arccos(0.625) = 51 degrees.
So you would need to an elevation kit that would rotate your panels 51 degrees, not sure if that is practical. We still need to find what direction to run the rails.
We did some experimentation with the tilt kit on the ground and determined that with a spacing of 53" we could hit the bottom of the panel close enough from the bottom to keep the panels above the ground and 1/3 of the distance from the top. That gave an elevation angle right around 50 degrees. More on this later.
In any case, the I calculated for the tilt angle was as follows:
Tilt angle = arccos((0 + sin 45 sin 28) + 3 sin 45)/sqrt(10)) = 39 degrees.
That's good, since it is LOWER than the maximum we could achieve with the tilt kit which was purchased.
wwhitney wrote:The rails would need to run perpendicular to both the pre and post rotation orthogonal vectors, i.e. perpendicular to both (-sin120, -cos120, 3) and (0, -sin 40, cos 40). That would be in the direction of their cross product. So the direction is given by
(-cos 120 * cos 40 - 3 * (-sin 40), 3 * 0 - (-sin 120) * cos 40, -sin 120* (-sin 40) - (-cos 120) * 0) = (2.31, 0.766, 0.557)
For my calculations, I found the direction of the rails as follows:
Direction of rails = (-sin 28 cos 45 - 3 (-sin 45), 3 * 0 - (cos 28 cos 45, (- cos 28)(-sin45) - (-sin 28)(0))
= (1.65, -0.624, 0.624)
wwhitney wrote:If the rails ran horizontally across the roof, their direction would be (cos 120, -sin 120, 0).
In the chosen case: (-sin 28, -cos 28, 0)
wwhitney wrote:So the angle between this and the desired rail direction is
arccos( (2.31 * cos 120 - 0.766 * sin 120 + 0) / sqrt (2.31^2 + 0.766^2 + .557^2) ) = 137 degrees.
Since the rail directions are only defined up to +-1, this is really 180 - 137 = 43 degrees.
Rail rotation angle = arccos((1.165*(-sin28) + 0.628*cos 28 + 0)/sqrt(1.65^2 + 0.624^2 + 0.624^2)) = 83 degrees.
wwhitney wrote:I haven't calculated whether that's CW or CCW, but from the roof it should be obvious which way to go.
I wasn't quite sure which way this angle went, but we discussed things and decided that this was close enough to perpendicular that we would take advantage of that fact and simply go with the perpendicular for the installation. Further, we decided to go with a 48-inch spacing between the feet of the tilt kit rather than the 53 inches we had previously determined. Together these two decisions allowed us to mount the feet directly into the rafters and therefore minimize the amount of time and effort spent in the attic (particularly close to the bottom of the roof where space is very limited).
The resulting installation achieved a result very close to what was desired. I wasn't really sure how to accurately measure either azimuth or elevation, but the azimuth appeared to be very close to South and my attempt to measure the elevation with my iPad yielded 40 degrees. That elevation result was a bit of a surprise to me, since we had been targeting 45 degrees and had INCREASED the tilt angle on the kit by spacing the two feet closer than needed to get 50 degrees tilt. (I'm wondering if I misinterpreted the measurement and the real number is 50 degrees. I'm just not sure.)
Some thoughts on the resulting installation:
1) We couldn't be happier with the resulting pointing angle. I was more than a bit surprised that running the rails vertically up the roof was sufficient to achieve the desired pointing angle.
2) Because we moved the feet of the tilt kit to be 48 inches apart, we cantilevered more of the PV module off the top of the panel than is specified by LG. The final result is that nearly one meter of the panels is above the top rail. That will mean additional stresses on both the PV module, the roof and the tilt kit which we used to mount it.
3) I really don't think the tilt kit is intended to be used in any application other than on a flat roof or with rails running horizontally along the roof. Hopefully the PV modules are able to take most of the odd diagonal stresses which were created via this approach. The owner is also considering adding some cross-bracing between the tilt kits to give additional rigidity.
4) We had there people working when mounting the panels on the roof. We concluded that this could not have been reasonably been done with only two people. This coming from the owner and his son, who are both builders who have done major projects with just two people. The reason is that there were simply two many moving parts and strange forces all acting at one time until the parts could be tightly connected together.
5) One interesting result of mounting the modules in this orientation is that they cast REALLY long shadows in the afternoon. This is because the top of the western-most module is very high and casts a shadow which hits to bottom of the eastern-most panel even when they are very far apart. This would require about 20 feet of spacing between rows to achieve a decent amount of shadow-free operation around the winter solstice. This array was mounted on the far North end of the roof, so there IS room for another row, but the owner wants to keep the rest of the roof shadow-free for a possible addition of a grid-tied array. As a result, we decided that the second array for this off-grid application will go on the western roof and will not point directly South, but will be more easterly to better catch the rays of the morning Sun. The owner will purchase some square aluminum bars to allow us to extend the remaining tilt kilts significantly. We may also need to purchase another tilt kit (or two) to accommodate the longer rails.
6) Since the rails were about four feet longer than the Eastern roof, we cantilevered about four feet off the top of the roof. At the end of the day, we decided to cut off these rails and NOT mount a fifth panel up there. There were two main reasons for this: A) Since the second array will point in a different direction, we need to group the PV modules in pairs to go with the two charge controllers that are being used. That necessitates having one array of four panels and one array of six panels. (They were already going to be grouped that way ELECTRICALLY, but the different angles require them to also be grouped that way PHYSICALLY.) We all felt like the four tilt kits ALREADY have their work cut out for them to support the four modules up there. Adding another one cantilevered off the end would likely result in the entire structure being ripped from the roof in high winds. The owner intends to cut off the ends of the rails and we will add them to the other rails to be able to accommodate all six panels in the other array.
7) It will be interesting to see how long this system lasts. The equipment which we installed is OLD. The charge controllers are RV Power Products Solar Boost 3048s which were built in 1999 (operated about four years total) and the inverter is a Trace Engineering SW5548 built in 2002 (operated about nine years total). I stepped up the voltage on the battery side slowly in 12V steps to anneal the electrolytic capacitors before exposing them to the full 48 Volts. I really like both of these products and hope the owner gets a long time out of them. We'll see.
wwhitney wrote:Cheers, Wayne
Thanks again, Wayne! I was stuck without your assistance.
I forgot to take photographs of the final result, but I will try to do that and post them here when I have a chance.