Solar Panels and Optimization  

Image of the Sun. The image was taken from the astrophysics department at Stanford university where you can find more information about solar research.

1. Instructions

This lab is due Wednesday July 7 in lecture. Late labs are not accepted. You are encouraged to work in groups of no more than 3 people, though you may work alone. You are required to submit both a paper copy, and an electronic copy in MS Word or PDF format to the AMESS.

On your title page, clearly mark

• Names
• Student ID numbers
• Section number and professor
• If you do not include this information, as much as 10 points will be deducted from your final score

Format is worth 20% of your grade. Please refer to the following writing guidelines for the expository sections of this report:

Writing Guidelines for Calculus Labs

You are required to know and follow the Writing Guidelines for all labs.

 

2. Background

In this lab you will analyze how the efficiency of a solar panel depends on the season and its orientation.

The solar radiation falling on a tilted plane (such as a solar panel) depends on numerous factors; the orientation of the plane, the time of the day, the season, the weather, etc. In the next few sections you will learn more about how these affect the power we can generate from solar panels.

 

2a. The Earth and the Sun

As we all know the Earth orbits the Sun. The orbit is almost circular so for this lab we will assume that the distance to the Sun is constant. However, the axis of the Earth is tilted relative to the plane in which the Earth moves around the Sun. The angle between the axis and a normal to this plane is roughly $23^\circ$. As indicated in Fig. 1, the north pole points away from the Sun during the northern hemisphere winter, and points towards the Sun during the northern hemisphere summer.

Now let us imagine that we stand on the equator. We introduce the angle s as illustrated in Fig. 2. In words, the angle s determines the angle between you (if you are standing straight up on the equator at noon) and a line from the center of the Earth to the center of the Sun. By ``noon'' we mean the time of the day when the Sun is at its highest position in the sky. As we can see from Fig. 2, $s\simeq -23^\circ$ on December 22 (winter solstice) and $s\simeq 23^\circ$ on June 21 (summer solstice). On March 21 (vernal equinox) and September 23 (autumnal equinox) $s=0^\circ$. Hence, we can think of s as describing the time of the year. This angle s will be of great importance in the lab so let us summarize its values in a table.

Date(s)	s
December 22, 2001	$s=-23^\circ$
December 23, 2001 - March 21, 2002	$-23^\circ<s\leq 0^\circ$ (increasing with time in this period)
March 22, 2002 - June 21, 2002	$0^\circ <s\leq 23^\circ$ (increasing with time in this period)
June 22, 2002 - September 22, 2002	$23^\circ > s \geq 0^\circ$ (decreasing with time in this period)
September 23, 2002 - December 22, 2002	$0^\circ> s \geq -23^\circ$ (decreasing with time in this period)

Note that $23^\circ\simeq 0.4\,\,radians$.

 

2b. Radiation falling on a tilted plane

In this lab we will consider the solar radiation falling on a tilted plane. Let u be the angle between the plane and the ground. If $u=0^\circ$, then the plane is lying on the ground and if $u=90^\circ$ the plane is standing up vertically. (If $u\neq 0$, i.e., it is not just lying on the ground, then we assume that the tilted plane is facing south. However, you do not have to worry about this in the lab.)

Introduce t to be an angle proportional to the time of the day such that $t=-90^\circ(=-\frac{\pi}{2}\,\,radians)$ at 6 am in the morning, $t=0^\circ$ at noon and $t=90^\circ(=\frac{\pi}{2}\,\,radians)$ at 6 pm. If I₀ is the intensity of solar radiation (measured in W/m², where W=J/s) falling on the ground, then the radiation on the tilted plane, I_plane is given by the function

$$I_{plane}(s,u,t)=I_0(\cos s \cos u \cos t - \sin s \sin u).$$ (1)

There are more factors that affect the radiation on the plane. There is also an absorption factor which is strongly dependent on the distance the sun rays have traveled through the atmosphere before it reaches the ground. The law describing absorption is called Beer's law (sic!). Let us describe the absorption with a function A(s,t) such that $0<A(s,t)\leq 1$. Here A(s,t)=1 means that there is no absorption and that the absorption increases with decreasing values of A(s,t) so that a low value ($\simeq 0$) of A(s,t) means that the absorption is very strong.

In addition, we should also take different weather patterns into account since the solar radiation on the ground varies with the cloudiness. Let us describe the cloudiness with a function C(s,t) such that $0<C(s,t)\leq 1$. Here C(s,t)=1 means that there are no clouds and that the cloudiness increases with decreasing values of C(s,t) so that a low value ($\simeq 0$) of C(s,t) means that the sky is covered with thick black thunder clouds.

The total amount of energy received per square meter and day is now given by the following equation.

$$W(s,u) = \int_{t_{min}}^{t_{max}} A(s,t)C(s,t)I_{plane}(s,u,t)\ dt$$ (2)

 

Here t_min and t_max is the time of sunrise and sunset respectively. These times vary over the year. However, on the equator these times are relatively constant and to simplify the mathematics we will assume that for our purpose we can use $t_{min}=-\frac{\pi}{2}$ and $t_{max}=\frac{\pi}{2}$ throughout the year. (This is an approximation. For a more careful analysis of a problem, we should let these times depend on the season, i.e., s.)

 

3. Problem Statement

The local company Solar Power Inc. has asked you to do a consulting job for them. They will export solar panels to the pacific island Suluclac which is located on the equator. A meteorologist has described their weather by the ``cloudiness function''

$$C(s,t)= \frac{3-(1+s)\cos^2t}{3}$$ (3)

 

where s (angle related to the time of the year) is defined in section 2a and t (angle proportional to the time of the day) is defined in section 2b.

A physicist has derived the following formula describing the energy (measured in kWh/(m²day) where $1\ kWh=3.6\times 10^6J$) collected by the solar panel each day.

(4)

Note: this equation is only valid when s and u are given in radians!

The variable s is defined in section 2a and u is defined in section 2b, but recall that s determines the time of year and u is the angle between the solar panel and the ground. The energy equation given above, has already incorporated effects due to absorption and cloudiness, so the function W(s,u) is already in the form you will use for this lab.

The company now wants you to answer the following questions:

1. Describe how the weather, or more specifically the cloudiness, varies on Suluclac. In other words, use the information in section 2b and analyze what the function C(s,t) (equation 3) can tell us about how the cloudiness varies over the course of the day and over the course of a year on the island Suluclac.
2. In the domain $-0.4\leq s \leq 0.4$ and $0\leq u \leq 1$ where the angles are given in radians, find all critical points in the domain of the energy function in equation 4. Specify if they are local maxima, local minima or saddle points. Compute the energy at these points.
3. Find all local extreme points of the energy function in its domain (notice that in part 2 above you've already searched for these points in the interior of the domain. All you have left to do is to look for these points on the boundary of the domain of the function. Don't forget to check corner points). Specify if they are local maxima or local minima in the domain of the function. Compute the energy at these points.
4. Use the list of points you found in parts 2 and 3 to find the absolute maximum and minimum of the energy function. Give the values ( s_max,u_max) and ( s_min,u_min) respectively, where the absolute maximum/absolute minimum occur, and the energy obtained for these values of s and u.
5. If we choose the angle (orientation) u_max (as defined in the previous question) for our solar panel, during what time of the year (s) does the maximum occur?
6. Assume there will be a person employed to look after the solar plant on Suluclac on a daily basis, so that he or she can adjust the solar panel once a week. This person will be able to change the orientation (u) of the solar panel so that it gives the maximum energy output for the current season (determined by s). On a contour plot of the energy function over the given domain, draw a curve that describes how the angle u should be chosen for each value of s in order to always give the maximal possible energy output (although you'll draw a curve in the contour plot, you should also look at a 3D plot of the function, in order to have a good idea of how the function behaves inside its domain).
7. One month a year, the solar panels need service and during this period they will be taken down. Based on your path from exercise 6, during what period of the year should we perform the service in order to lose as little energy as possible?

 

Hints: 

• Make sure you are very comfortable with the definitions of section 12.8 in your Calculus book. 
• If you need to check if a point is a local max/min or not, you may do so by looking at a plot of the neighborhood of such point. There's no need for second derivative tests.
• When asked to give ``time of year'', you only need to provide a rough estimate based on the table in section 3a.
• The following links contain Mathematica commands applicable to solving this project: Surface Plots , Application to Optimization