Optimal Tilt Angle of Solar Array

[drogerssolar]

Hello!

I have a difficult problem I have been working with recently. I have been trying to model the ever changing position of a solar PV array system. The system is a single axis tracker and orientates itself to have the smallest angle of incidence(AOI) to the suns beam as possible (the best performance is when the suns beam is directly perpendicular to the array). Also to clarify the AOI is the angle between the array planes "normal" and the suns direct beam.

I was able to determine how to calculate the AOI using the following formula:

COS[AOI] = {COS[X]*COS[Y]*COS[Z]} + {SIN[ACOS[COS[X]*COS[Y]]*SIN[Z]*COS[W-ACOS[ATAN[SIN[Y]/TAN[X]]*COS[Y]]]}

Where:
X = Axis Tilt = 35 degrees (for my application)
Y = Rotation about the axis (This is my unknown)
Z = Zenith angle of sun for a particular time of day
W = Azimuth angle of sun for the same time of day
AOI = Angle of incidence

Now my goal here is to determine what value of Y will give me the smallest possible value of AOI.
I think that I need to take a derivative of the equation with respect to AOI and then set the resulting derivative = 0.
Does that sound right? I know that this is a Maxima/Minima problem but the size of the equation frightens me.
Any guidance would be very much appreciated.

 

[berkeman]

Hello!

I have a difficult problem I have been working with recently. I have been trying to model the ever changing position of a solar PV array system. The system is a single axis tracker and orientates itself to have the smallest angle of incidence(AOI) to the suns beam as possible (the best performance is when the suns beam is directly perpendicular to the array). Also to clarify the AOI is the angle between the array planes "normal" and the suns direct beam.

I was able to determine how to calculate the AOI using the following formula:

COS[AOI] = {COS[X]*COS[Y]*COS[Z]} + {SIN[ACOS[COS[X]*COS[Y]]*SIN[Z]*COS[W-ACOS[ATAN[SIN[Y]/TAN[X]]*COS[Y]]]}

Where:
X = Axis Tilt = 35 degrees (for my application)
Y = Rotation about the axis (This is my unknown)
Z = Zenith angle of sun for a particular time of day
W = Azimuth angle of sun for the same time of day
AOI = Angle of incidence

Now my goal here is to determine what value of Y will give me the smallest possible value of AOI.
I think that I need to take a derivative of the equation with respect to AOI and then set the resulting derivative = 0.
Does that sound right? I know that this is a Maxima/Minima problem but the size of the equation frightens me.
Any guidance would be very much appreciated.

How is this related to your other thread that you have going on this subject?

https://www.physicsforums.com/showthread.php?t=497570

.