I was curious to check the math from my reference against your empirical results, at 11am and 4pm PST on 5/24 at your location. The graphs below imagine that you swing the panels from horizontal down to 45°, and plot the efficiency factor cos(θ). The upper blue graph is the 11 am situation, and it predicts the maximum power will occur when the panel is near 19 degrees below the horizontal, and at that point it will receive better than 95% of what it would if the panel were aimed directly at the sun. At 4pm, the maximum occurs when the panel is near 7 degrees below the horizontal, with a theoretical efficiency of about 63% of what it would receive if it were pointed directly at the sun. The angles 19 degrees and 7 degrees agree with what you found empirically, so I think there is still hope there for the straight out math or look up table, with a real time clock.
I went back and edited my previous post to make the notation agree with the Greek letters in the rreference.
Tracy Allen,
I think that the formulas I use are the same as yours.
I'm not sure if the translation of "ω" is the correct "siderial". In Greek it is called "ωριαία".
φ=the latitude and δ=suns's aberration (In Greek δ= Ηλιακή απόκλιση)
I send you an image to make clear what exactly represent everyone angle.
Tracy Allen, Thank you very much for all the trouble you took to provide this information. I took a slightly different approach in that I worked out a formula that related the solar altitude and azimuth to an array angle that puts the sun in a plane that is perpendicular to the array and coincides with the array axis. I will spend some time comparing your results with mine. A question: what is your source for solar altitude and azimuth? The Greek letters show up fine.
By the way, thank you, that simulation from the University of Nebraska is fascinating, and very informative.
I don't think the math you just posted and the one I posted above from Renewable Energy describe the same situation. The math is similar, as both I think come from the dot product of two vectors characterized by an altitude and an azimuth. In the renewable resources paper, one vector is the one pointing at the sun with magnitude of the solar intensity, and the other a unit vector normal to the plane of the solar panel. The result, θ, is not the complement of the solar altitude, αs. In your diagram, I am not sure what is the angle δ? Where is the normal vector to the surface of the solar panel?
I highly recommend the Excel spreadsheet Twilight by Greg Pelletier at the University of Washington.
It provides VBA macros that implement NOAA calculations for local times of sunrise, solar noon, sunset, dawn, solar azimuth and elevation. For example, in a cell you enter
with reference to the time and position parameters that are held in other cells. The example worksheets are simple calculators, but you can use the formulas, for example, to build up a worksheet of values at your location for every week of the year. I use it in that way to generate data statements that go into a PBASIC program for sunrise and sunset throughout the year at a known location. All I have to do is enter new lat and long, and viol
All this modern techical info begs the question, "Did the ancient archeoastronomers of Stonehenge, Newgrange, Chaco Canyon, etc, prefer using Excel spreadsheets, or doing the trig equations by hand (on a rock)...?"
Tracy,
I still think that we use the same maths with meaningless changes.
The angle "δ" according to your bibliography is the Declination of the Sun (in page 1195 from Renewable Energy it says: δ = 23,44 sin λ. In my algorithm I have replaced λ:
This "n" in the first formula is the number of the day during the year according the given DATE.
For instance: supposing the DATE is the 4th June, the variable n=31+28+31+30+31+4=155 shows the number of days from the begining of the year.
The argorithm above, calculates the Sun's azimuth and Sun's altitude in a specific DATE in a specific PLACE (φ=the place's latitude) in a specific TIME.
I don't use any ventor from the solar panel to calculate Sun's position.
If we know these two angles we can change properly the position of our solar panel in order to be vertically to the Sun.
In my first experiments with this algorithm I used a very simple device with two stepper motor X and Y (X-motor for the azimouth and Y-motor for the altitude) in which I had put a cane (reed) to target the Sun
From the cane's shadow I was able to understand if my device had targeted the sun properly.
If I could see the small bright hole in cane's shadow I was sure that my device was corect.
The results of my tests was satisfactory....
However I'm wondering why "mmoreland" have these different measurments...?
Erco, It is amazing isn't it? I think those ancients sat there under a high desert sky and their curiosity awoke. They were great observers, not just at one point in time, but most important to connect the dots at many points in time, season after season, and that led to the wisdom of prediction. But where did they all go and why? [IMG]file:///Users/thomasallen/Library/Caches/TemporaryItems/moz-screenshot.png[/IMG] What could the people of Chaco done with the energy from solar panels? Those Kivas could be pretty dark and smoky, I think.
I appreciate Michael's approach, good observation to create an annual record of the best panel angle. He is following the path toward the ancient wisdom!
Nikos, Your panel can rotate around both axes and end up pointing directly toward the sun, whereas, if I understand it correctly Micheal's (mmoreland) panels have only one degree of freedom. They rotate only around a horizontal axis. The panel can point straight up or it can rotate around that axis to any angle down to the horizon. But it can't change its azimuth on the NE plane. Thus his math would need to consider the vector normal to the panel, separate from the direct solar vector. The math minimizes angle between those two vectors. Your mechanical system can minimize the angle to zero, due to your extra degree of freedom in panel motion. I think that is the difference.
I need to make a correction in my graph that I posted above for the situation at 4pm. I dropped a minus sign that comes about because there is a 94.5° angle between the solar azimuth at that time (266.5°) and the panel azimuth (172°, fixed installation 8° east of due south). That results in the optimum angle for the solar panel at that time being -5.7°. (not +5.7° as I had it) It actually should angle back a little bit north. Although not much is lost by having the panel flat horizontal. The sun at that time 4pm PST at Michael's location would be at altitude 38.3° and azimuth 266.5°.
All that is predicated on a clear sky and an open field. The best angle is
β = atan(cos(ϒ - ϒ[SIZE=1]s[/SIZE]) / tan(α[SIZE=1]s[/SIZE]))
β best angle of panel wrt 0=horizontal, positive south (in northern hemisphere)
ϒ azimuth of solar panel, in this situation fixed at 172° , that is 8° east of true south.
ϒ[SIZE=1]s[/SIZE] azimuth of the sun at location and time specified
α[SIZE=1]s[/SIZE] altitude of sun at location and time specified
There is also the interesting and practical problem of how to point a panel that is fixed in one position, no tracking whatsoever. In that case the solar intensity has to be integrated for the entire day or a longer period, and it can to be optimized for one particular season, or for the whole year, or on the basis of some other criterion. I found one quite nice reference that discusses this issue, here.
Thanks for the correction. You got it on the thread just as I was about to ask you about the sign on the tilt. My array won't tilt back past horizontal, but I also saw from the azimuth that the sun would by 4 pm have passed over the array headed to set somewhere behind it.
Hello Tracy Allen, Thank you for the link to Landau's page of tilt and energy collection data. Before I undertook to build an array, I spent a good deal of time doing what Laudau has done comparing the energy collection capability of a couple fixed tilt panels, one that was seasonally adjusted, one that was monthly adjusted and one that tracked. I also compared the capability of a single axis tracker the axis of which was oriented north-south such as the type deployed by RayTracker now a subsidiary of FirstSolar. RayTracker has a nifty set up using GPS whereby, the array is set up, plugged in, it finds itself by means of GPS, builds a set of lookup tables, checks the time and date and begins tracking. It rotates to face due east, rolls to the west as the earth moves. It has other tricky features, but my property wouldn't handle a N-S axis. Property constraints are what led to the 8 degree angle to the east. Now, after half a year of observations, I see I should have done more to orient due south or even somewhat to the west since it is the mid to western sun that yields the greatest generation. I don't know if that's due to atmospheric conditions that change throughout the day, but even the 6-7pm sun has far more generating capability than the 10am sun even with the array angled more toward the east and getting a better angle of incidence.
I performed the calculations alluded to above to check whether or not it were true that it's a better choice to simply use more panels than to track, and I found, if I can trust my math, that over a year, single axis tracking yielded a better energy gain by about 25%, and to add sufficient panels to gain 25% from a fixed array would cost twice as much as to build the array. I also found that at 35-40 north latitude, an east-west axis for a single axis tracker yielded better results than a north-south axis.
It's of great interest to me that you seem to have modified your original best angle equation to give a single answer, best angle result since that was also my approach. I used what I called (in my cowboy mathematician way) vector diagrams to develop the trigonometric relationships that would provide a best angle formula. I could visualize that sun needed to be in a plane that included the array axis and was perpendicular to the array. What I derived, however, differs from what you derived, so I'm thinking that the problem I've been having relating the results of applying my formula to reality may be tied simply to the fact that I made an error in my derivation. I wish I were as competent a mathematician as you appear to be. I've been using the following equation: Tilt = atan (sin(el)/(cos(el) * sin(az)) where el = solar elevation and az = the difference between solar azimuth and the axis azimuth. I'm going to compare results of your equation with more of the data points I've collected over the last six months, and I'm going to go back to try to figure out where I went wrong.
Once again, I thank you for the information, advice and links you've provided. mm
On the Mendocino coast, would those "atmospheric conditions" be frequent coastal fog or haze that burns off sometime around mid day? It is like that here in Berkeley--Fog comes straight across from the Golden Gate bridge. "Sun hours" calculations that I've seen don't weight by hour of day, but intuitively it seems like it would favor a westward bias on the average.
I'm mostly concerned with small 3 W to 20 W panels to run instrumentation. Snow shedding is important and favors the winter angle, as does energy needs, but placement is constrained by the presence of hills or vegetation. It is seldom optimum. We also have to discourage perching birds and curious raccoons too.
As to the math, I like plane geometry/trig, but struggle to make sense of the spherical. I think you have it right that there is a plane defined by the axis of the solar panel and the line normal to its face, and the sun should be in that plane. You don't have the extra degree of freedom to bring the panel to face the sun exactly, but having the sun in that plane is the next best thing. It does look like your formula is different from mine. I have tan(αs) in the denominator, whereas in your formula it (as sin/cos) is in the numerator. The other term might be the same, if by "axis azimuth" you mean 82 degrees. I was using 172 degrees for the perpendicular to the axis.
cos(ϒ - ϒs) = sin((ϒ-90) - ϒs)
Sometimes although formulas are absolutely correct they give unexpected results because of the wrong use of the angles. Trigonometric functions sin(x), cos(x) and tan(x) in the most programming languages accept the angle x in “rad” form. It is very important to convert degrees into rad using the formula: rad/π=degrees/180 . (π=3.14)
If you write in BASIC the command: Print SIN(90), COS(90) you will have the result: 0,8939967 , -0,4480736 instead of the correct (expected) 1 , 0. The mistake is in the way we write the 90 degrees.
The correct command is SIN(3.14*90/180)=1 and COS(3.14*90/180)=0
Tracy Allen, I see you don't mean that. You mean cos(172-solar azimuth). I've got a list of collected times and angles, and I'm checking them against your equation. Again, thanks!
Michael,
It is very easy to make these calculations using a board of education. I believe that is more difficult to use a calculator than the basic stamp board and Pbasic.
Don't hesitate to try Pbasic and basic stamp. This forum has very helpful people who they can help you!
How about a Real Time Clock chip. Many examples of interfacing a DS1302 or DS1307 can be found on this site along with sample code. The chip, a crystal, 2 resistors, and a watch battery are all you need to get it going, total cost 10 bucks.
For 3.3V operation substitute DS1338Z-33 for the DS1307. The DS1307 can't handle a supply voltage that's less/equal to the backup Li battery, the DS1338 can...
I'm attaching the Excel spreadsheet that I used to draw my graphs. It has a worksheet titled, "panel tilt" that is uses Greg Pelletier's solar position macros. You can input time, position, and the horizontal azimuth of the panel, and it will calculate the optimum tilt and also draw a graph of solar power vs tilt.
Nikos, I for one will be very interested in what you come up with for the Stamp. I've always resorted to tables and interpolation for this sort of thing, thinking that the Stamp would not do so well on direct calculations.
Thank you, Tracy Allen. The results from the worksheet don't match my hourly measurements at the array, but they are close enough, and the difference could be attributed to inaccuracy on my part as much as anything else. This will be a far easier way to build look-up tables. Fortunately I have an older iBook with a version of Excel that runs macros. Our newer iMac wouldn't do it.
I'm still running Office 2004 on my recent MacBook, and it does run macros. Microsoft in their great wisdom eliminated VBA macro support from the 2007 Mac release, but my understanding is that it is back with Office Mac 2010. I haven't sprung for that yet.
I was looking back at what you said about your panels' installation. Is the horizontal axis completely level wrt the horizon, or is there a tilt to that too?
Yes, the axis is level; no tilt. We acquired a new iMac this year, but they included only "Pages," "Numbers," and "Keynote" which appear to date from '09. No sign of Office Mac 2010. I could feel a little gypped by that. Your business/website is very interesting, and I'll be spending more time there exploring.
So, here’s what I’ve got: 28501 GPS from Parallax, Memsic 2125 Dual-axis Accelerometer, Basic Stamp 2, a formula for buiding lookup tables for axis tilt, and the wild hope that I’ll be able to build a controller for the array described in the first entry in this thread. I’ve got a copy of _Programming and Customizing the Basic Stamp Computer_ by Scott Edwards and the _Basic Stamp Manual Vol.2_, but those are both pretty old. Any recommendations for books to help me learn how to wire and write code for a controller using the above parts? Maybe this needs to be a new thread...
Any recommendations for books to help me learn how to wire and write code for a controller using the above parts?
http://www.parallax.com/tabid/440/Default.aspx is where to start then look at the other downloadable PDFs on sensors and industrial integration available for free from the Parallax site.
The first thing is to get the Memsic 2125 and the GPS running with the demo software and tutorials from the Parallax web site. Find those on the product pages for the 28501 and the 28017. I'd suggest that you start with the Memsic, because there is a wealth of good material. In your final version, you will be extracting only the pieces of code that you need, but the tutorials are fun and educational and will show fast results. The GPS code on the Parallax product page is written for the BS2p, so it will take a few little changes to make it work on your BS2. It is a large program, but don't let that deter you. All you really need from the GPS will be the time, right? The panel isn't going anywhere.
The tilt angles will probably be held in a table of bytes in the memory. I forget how many changes of tilt you want per day, but suppose something like a table with one row for each of 52 weeks in the year, and columns with 4 times of day, 52 x 4 table, 208 bytes. The program might interpolate between those values. There are other parametric approaches too.
Alright; thank you all for your helpful suggestions. I'm currently working with the "twilight-tta" spreadsheet, and it has afforded me the insight that my measurement of the axis azimuth is incorrect. It is 6 degrees north of east instead of 8. At least those are my preliminary findings. I'm trying to get a better measurement, but when I adjust the azimuth on the spreadsheet, the generated results exactly match the actual tilt measurements made every half hour throughout the day. It's a bit of backward logic, and I'd like to prove my findings by getting an accurate measurement of the axis orientation, but this has all been incredibly helpful. I'll begin work on the controller following the suggested procedures soon. Again, thanks. mm
Comments
I was curious to check the math from my reference against your empirical results, at 11am and 4pm PST on 5/24 at your location. The graphs below imagine that you swing the panels from horizontal down to 45°, and plot the efficiency factor cos(θ). The upper blue graph is the 11 am situation, and it predicts the maximum power will occur when the panel is near 19 degrees below the horizontal, and at that point it will receive better than 95% of what it would if the panel were aimed directly at the sun. At 4pm, the maximum occurs when the panel is near 7 degrees below the horizontal, with a theoretical efficiency of about 63% of what it would receive if it were pointed directly at the sun. The angles 19 degrees and 7 degrees agree with what you found empirically, so I think there is still hope there for the straight out math or look up table, with a real time clock.
I went back and edited my previous post to make the notation agree with the Greek letters in the rreference.
I think that the formulas I use are the same as yours.
I'm not sure if the translation of "ω" is the correct "siderial". In Greek it is called "ωριαία".
φ=the latitude and δ=suns's aberration (In Greek δ= Ηλιακή απόκλιση)
I send you an image to make clear what exactly represent everyone angle.
Nikos Giannakopoulos
By the way, thank you, that simulation from the University of Nebraska is fascinating, and very informative.
I don't think the math you just posted and the one I posted above from Renewable Energy describe the same situation. The math is similar, as both I think come from the dot product of two vectors characterized by an altitude and an azimuth. In the renewable resources paper, one vector is the one pointing at the sun with magnitude of the solar intensity, and the other a unit vector normal to the plane of the solar panel. The result, θ, is not the complement of the solar altitude, αs. In your diagram, I am not sure what is the angle δ? Where is the normal vector to the surface of the solar panel?
I highly recommend the Excel spreadsheet Twilight by Greg Pelletier at the University of Washington.
It provides VBA macros that implement NOAA calculations for local times of sunrise, solar noon, sunset, dawn, solar azimuth and elevation. For example, in a cell you enter with reference to the time and position parameters that are held in other cells. The example worksheets are simple calculators, but you can use the formulas, for example, to build up a worksheet of values at your location for every week of the year. I use it in that way to generate data statements that go into a PBASIC program for sunrise and sunset throughout the year at a known location. All I have to do is enter new lat and long, and viol
http://www.world-mysteries.com/alignments/mpl_al1.htm#Mesoamerican
http://archaeoastronomy.com/
http://en.wikipedia.org/wiki/Archaeoastronomy
http://terpconnect.umd.edu/~tlaloc/archastro
The more I learn, the more amazed I am!
I still think that we use the same maths with meaningless changes.
The angle "δ" according to your bibliography is the Declination of the Sun (in page 1195 from Renewable Energy it says: δ = 23,44 sin λ. In my algorithm I have replaced λ:
This "n" in the first formula is the number of the day during the year according the given DATE.
For instance: supposing the DATE is the 4th June, the variable n=31+28+31+30+31+4=155 shows the number of days from the begining of the year.
The argorithm above, calculates the Sun's azimuth and Sun's altitude in a specific DATE in a specific PLACE (φ=the place's latitude) in a specific TIME.
I don't use any ventor from the solar panel to calculate Sun's position.
If we know these two angles we can change properly the position of our solar panel in order to be vertically to the Sun.
In my first experiments with this algorithm I used a very simple device with two stepper motor X and Y (X-motor for the azimouth and Y-motor for the altitude) in which I had put a cane (reed) to target the Sun
From the cane's shadow I was able to understand if my device had targeted the sun properly.
If I could see the small bright hole in cane's shadow I was sure that my device was corect.
The results of my tests was satisfactory....
However I'm wondering why "mmoreland" have these different measurments...?
Nikos Giannakopoulos
I appreciate Michael's approach, good observation to create an annual record of the best panel angle. He is following the path toward the ancient wisdom!
All that is predicated on a clear sky and an open field. The best angle is There is also the interesting and practical problem of how to point a panel that is fixed in one position, no tracking whatsoever. In that case the solar intensity has to be integrated for the entire day or a longer period, and it can to be optimized for one particular season, or for the whole year, or on the basis of some other criterion. I found one quite nice reference that discusses this issue, here.
I performed the calculations alluded to above to check whether or not it were true that it's a better choice to simply use more panels than to track, and I found, if I can trust my math, that over a year, single axis tracking yielded a better energy gain by about 25%, and to add sufficient panels to gain 25% from a fixed array would cost twice as much as to build the array. I also found that at 35-40 north latitude, an east-west axis for a single axis tracker yielded better results than a north-south axis.
It's of great interest to me that you seem to have modified your original best angle equation to give a single answer, best angle result since that was also my approach. I used what I called (in my cowboy mathematician way) vector diagrams to develop the trigonometric relationships that would provide a best angle formula. I could visualize that sun needed to be in a plane that included the array axis and was perpendicular to the array. What I derived, however, differs from what you derived, so I'm thinking that the problem I've been having relating the results of applying my formula to reality may be tied simply to the fact that I made an error in my derivation. I wish I were as competent a mathematician as you appear to be. I've been using the following equation: Tilt = atan (sin(el)/(cos(el) * sin(az)) where el = solar elevation and az = the difference between solar azimuth and the axis azimuth. I'm going to compare results of your equation with more of the data points I've collected over the last six months, and I'm going to go back to try to figure out where I went wrong.
Once again, I thank you for the information, advice and links you've provided. mm
I'm mostly concerned with small 3 W to 20 W panels to run instrumentation. Snow shedding is important and favors the winter angle, as does energy needs, but placement is constrained by the presence of hills or vegetation. It is seldom optimum. We also have to discourage perching birds and curious raccoons too.
As to the math, I like plane geometry/trig, but struggle to make sense of the spherical. I think you have it right that there is a plane defined by the axis of the solar panel and the line normal to its face, and the sun should be in that plane. You don't have the extra degree of freedom to bring the panel to face the sun exactly, but having the sun in that plane is the next best thing. It does look like your formula is different from mine. I have tan(αs) in the denominator, whereas in your formula it (as sin/cos) is in the numerator. The other term might be the same, if by "axis azimuth" you mean 82 degrees. I was using 172 degrees for the perpendicular to the axis.
cos(ϒ - ϒs) = sin((ϒ-90) - ϒs)
If you write in BASIC the command: Print SIN(90), COS(90) you will have the result: 0,8939967 , -0,4480736 instead of the correct (expected) 1 , 0. The mistake is in the way we write the 90 degrees.
The correct command is SIN(3.14*90/180)=1 and COS(3.14*90/180)=0
It is very easy to make these calculations using a board of education. I believe that is more difficult to use a calculator than the basic stamp board and Pbasic.
Don't hesitate to try Pbasic and basic stamp. This forum has very helpful people who they can help you!
The correct is:
P.s. I'm waiting a DS1302 Timekeeping Chip from my latest order from Parallax and I will be able to provide a solar tracking project in forum!
For 3.3V operation substitute DS1338Z-33 for the DS1307. The DS1307 can't handle a supply voltage that's less/equal to the backup Li battery, the DS1338 can...
Nikos, I for one will be very interested in what you come up with for the Stamp. I've always resorted to tables and interpolation for this sort of thing, thinking that the Stamp would not do so well on direct calculations.
I was looking back at what you said about your panels' installation. Is the horizontal axis completely level wrt the horizon, or is there a tilt to that too?
Solar tracker uses green LEDs as photovoltaic sensors: http://www.redrok.com/led3xassm.htm
http://www.redrok.com/main.htm
http://www.redrok.com/electron.htm more general electronics
The tilt angles will probably be held in a table of bytes in the memory. I forget how many changes of tilt you want per day, but suppose something like a table with one row for each of 52 weeks in the year, and columns with 4 times of day, 52 x 4 table, 208 bytes. The program might interpolate between those values. There are other parametric approaches too.