# Sun II - Theoretical Yields
# Introduction
This is the second chapter about sun.
- [I: Sun trajectories]({% post_url 2022-08-11-Sun_1_trajectories %})
- [II: >> **Solar panel orientation and yield** <<]({% post_url 2022-08-11-Sun_2_panel_orientation %})
- [III: True solar panel efficiency]({% post_url 2022-08-11-Sun_3_true_SP_efficiency %})
- [IV: Shade from the trees]({% post_url 2022-08-11-Sun_4_shade_from_trees %})
## Table of Content
- [Energy received](#energy-received)
- [Computing the yield](#computing-the-yield)
- [Optimizing the yield](#optimizing-the-yield)
- [Optimizing the yield when you can rorate your panel](#optimizing-your-yield-when-you-can-rotate-your-panel)
- [If you have a fixed panel](#if-you-have-a-fixed-panel)
- [Surface and shading](#optimizing-installation-surface-area)
- [Illustration of the problem](#illustration-of-shading)
- [Equation for computing spacing](#spacing-panels-to-avoid-shading)
- [Simulator](#solar-panel's-shade-simulator)
- [Conclusion](#conclusion)
- [Sources](#sources-&-other-readings)
---
# Energy received
We are talking more and more about renewable energies, where solar panel is one possible option.
However, the productivity depends on the current latitude, the day and the current time (which governs $$\alpha$$).
If not oriented correctly, the energetical yield is not maximal.
## Computing the Yield
### Flat Panel
Computing the yield at a given date-time is easy.
Take a look at the following figure:
The figure above represents a solar panel put flat on the ground.
You can see that the "sun surface", in orange, is much smaller than the solar panel surface.
The yield is simply this surface ratio.
This yield can be computed for a given $$\alpha$$, assuming the panel is facing the sun (i.e., oriented on the ground according to $$HRA(t)$$).
This yield is:
$$R(0°, t, Lat, N) = \sin(\alpha(t; Lat, N))$$
### Location and Date Matter
Suppose we are located near Paris (latitude `48.52°`), and we look at the yield at noon.
Then $$\alpha$$ is maximal in summer (around `63°`) while minimal in winter (around `19°`).
Because this angle changes day after day, a fixed solar panel cannot give the maximal yield of `100 %` every day.
If we compute the ratio, we have a ratio of `90 %` in summer while only of `35 %` in winter.
The difference is huge !
## Optimizing the Yield
In the previous paragraph, we compared two yields at noon.
However, noon is just a small time-frame during the day.
To optimize the global yield, we have to find the orientation that maximize it all the day long.
To optimize the yield, we first need to understand that $$\alpha$$ is a 3D angle: it is the angle between the sun, our position, and the closest location in the horizon.
Therefore, to obtain the maximal yield of `100 %`, we need to play on two components:
- the **rotation** of the solar panel on the ground,
- the **inclination** of the solar panel.
To optimize the rotation angle of the solar panel, we just need to follow the $$HRA(t)$$ angle, by turning the solar panel at a speed of `15° / hour` so the panel is always facing the sun.
For the inclination angle, looking back at the previous figure, we need to get the solar panel perpendicular to the sun rays.
In other words:
$$\text{Inclination}(h, N, lat) = 90° - \alpha(h, N, lat) = y$$
where $$\text{Inclination}$$ is the angle of the solar panel relatively to the ground, also called the **tilt** angle.
Because the sun moves constantly, these two angles need to be adjusted every minutes.
The best possible option is to add two motors to follow the sun.
However, this option is not always possible:
- You have solar panels on your roof. You cannot turn them. You may have the possibility to tilt them, but that's unlikely, and you might be limited by your roof slope;
- You have a farm of solar panels on the ground. Changing the angle of the panels creates shade on the others. Moving them might be useless if you have a dense setup;
- You don't have electricity (you may have a thermal solar panel).
- ...
In these cases, you need to choose the best orientation angles which maximize your yield.
### Defining your Use-case
If you cannot move your panel (or very infrequently), you have first to define clearly the usage of this panel:
- All year long to produce electricity / water
- During the cold season to produce water (you don't want hot water in summer)
- To get hot water on the evening (assuming you cannot store hot water without energy loss from the morning)
- To dry fruits in August-September [^1]
- ...
If you are selling your electricity to the network, you prefer to maximize the global yield if you have a flat price.
There are already online calculators [^2] that gives you this optimal angle.
But if it is for personal use, you are likely to use more electricity in winter (for heating, for preparing food) than in summer (where you prefer to eat salads).
[^1]: [Fabriquer un séchoir solaire](https://www.solarbrother.com/creation/fabriquer-un-sechoir-solaire/)
[^2]: [Solar panel tilt angle calculator](https://footprinthero.com/solar-panel-tilt-angle-calculator)
Because the yield is not a linear function, the tilt angle is not $$90° - \bar{\alpha}$$ (where $$\bar{\alpha}$$ is the average value of $$\alpha$$).
## Optimizing your Yield when you can rotate your Panel
If you can at least add a motor to rotate your panel on the ground at a constant speed, you can already have a good yield.
We will move progressively from:
1. The hourly yield
2. The daily yield
3. The yearly yield
### Yield over a Day Hour by Hour
The yield hour after hour is just the instantaneous yield:
$$\text{Yield}(t, N, lat, \beta) = \sin(\beta + \alpha(t, N, lat))$$
{% include solar_panel_orient_yield.html %}
When changing the tilt angle, you can see "jumps", as moving your panel doesn't help to rise the sun earlier.
### Average Yield over the Day VS Tilt Angle
To get the average yield over a day, we need to integrate over the daytime.
$$\text{Yield}(N, lat, \beta) = \frac{1}{t_{set} - t_{rise}}\int_{t_{rise}}^{t_{set}} \text{Yield}(t, N, lat, \beta) \,dt$$
Daytime is longer in summer, and shorter in summer.
Here, we normalize by the daytime to compare the average yield.
Nevertheless, we have to keep in mind that all days do not have the same importance.
{% include sun_yield_day_fixed_panel.html %}
You may seem that there is a glitch for high latitude in winter (`>70°`), when the yield move from `100%` to `0%`.
This is because there is no "day", the sun does not rise, even for one minute.
As this plot is the average yield over daytime (not over 24 hours), this might be misleading.
### Total Yield over a Day vs Tilt angle
As discussed previously, days are shorter in winter and longer in summer.
If we compute the yield relatively to one day, we get the "true" yield.
$$\text{Yield}^*(N, lat, \beta) = \frac{1}{24}\int_{t_{rise}}^{t_{set}} \text{Yield}(t, N, lat, \beta) \,dt$$
{% include sun_yield_day_fixed_panel_time.html %}
### Yield over the Year
When we have a photovoltaic solar panel, the goal is to maximize the yield over the year if we sell electricity.
Therefore, we are interested by the angle that maximize this value.
$$\text{Yield}^*(lat, \beta) = \frac{2}{24 \times 365}\int_{365}^{0}\int_{t_{rise}}^{t_{set}} \text{Yield}(t, N, lat, \beta) \,dt\,dN$$
Here we multiply by $$2$$ as there are $$12$$ hours **in average** in a day.
This value is more realistic, as it corresponds to the best possible yield given the total daytime.
{% include sun_yield_year.html %}
What you can see is that even you have a fixed tilt angle, this is not that bad.
If you selected the optimal orientation, if you can turn your panel, yields are above `80%`, without the need to be precise on the tilt angle.
## If you have a Fixed Panel
When you cannot turn your panel over the day, the best rotation angle is facing south (unless you are in a mountain).
For the tilt angle, there are many online calculator [^0], [^3].
Because $$\alpha$$ varies between $$[0, \alpha_{max}]$$, a rule of thumb is to select the tilt angle in between.
In some websites, they recommend $$\beta \approx 90 - (\alpha - 10)$$, where $$\alpha$$ is the value at noon in summer, as you have longer daytime.
[^0]: [Solar Panel Angle Calculator](http://www.solarelectricityhandbook.com/solar-angle-calculator.html)
[^3]: [Solar Reviews: best solar panel angle](https://www.solarreviews.com/blog/best-solar-panel-angle)
### Computing the yield
#### Solar panel description
The solar panel with tilt angle $$\beta$$ and sides of length $$a$$ and $$b$$ can be described by $$4$$ points: $$i=(0, 0, 0)$$, $$j=(0, a, 0)$$, $$k=(b\cos\beta, a, b\sin \beta)$$, and $$\ell=(b\cos\beta, 0, b\sin\beta)$$.
One vector perpendicular to the solar panel is $$(-\tan (\beta), 0, 1)$$.
#### Sun plane
The sun rays have an angle $$\alpha$$ over the `z`-axis, and an angle $$HRA$$ within the $$(x, y)$$ plane.
One vector perpendicular to the sun plane is $$\overrightarrow{v_{sun}} = (\cos(HRA), -\sin(HRA), - \tan(\alpha))$$.
The equation of this plane is $$x\cos(HRA) - y\sin(HRA) - z\tan(\alpha) = 0$$.
#### Images
We want to compute the image of all points of the solar panel within the sun plane.
In other words, we want to find $$\lambda$$ such as:
$$j' = j + \lambda \overrightarrow{v_{sun}}$$
and $$j'$$ is in the sun plane.
For a point in the panel plan $$(x, y, z)$$, the corresponding $$\lambda$$ is:
$$\lambda = \frac{-x \cos(HRA) + y \sin(HRA) + z\tan(\alpha)}{1 + \tan^2(\alpha)}$$
For simplicity of the equation, we denote $$X = 1 + \tan^2(\alpha)$$ and $$Y= \sin(\beta) \tan(\alpha) - \cos(\beta)\cos(HRA)$$
The image of $$i$$ is itself.
The image of $$j$$ is $$j'=a(\frac{\cos(HRA)\sin(HRA)}{X}, 1 - \frac{\sin^2(HRA)}{X}, - \frac{\tan(\alpha)\sin(HRA)}{X})$$.
The image of $$\ell$$ is $$\ell'= b(\cos(\beta) + \frac{Y\cos(HRA)}{X}, -\frac{\sin(HRA) Y}{X}, \sin(\beta) - \frac{\tan(\alpha)Y}{X})$$.
#### Area
The area of the parallelogram $$(i', j', k', \ell')$$ can be expressed as:
$$\mathcal{A} = \sqrt{1 - \left(\frac{\overrightarrow{ij}.\overrightarrow{i\ell}}{\|\overrightarrow{ij}\|\|\overrightarrow{i\ell}\|}\right)^2}\times \|\overrightarrow{ij}\|\|\overrightarrow{i\ell}\|$$
$$\|\overrightarrow{ij}\| = a\sqrt{1 - \frac{\sin^2(HRA)}{X}}$$
$$\|\overrightarrow{i\ell}\| = b\sqrt{1 - \frac{Y^2}{X}}$$
$$\overrightarrow{ij}.\overrightarrow{i\ell} = -ab\frac{\sin(HRA)Y}{X}$$
To get the yield, we need to divide the area by the solar panel surface $$a\times b$$.
### Simulator
#### Day
On the next figure, you can explore the yield as a function of time.
{% include yield_fixed_tilt.html %}
The best yield are concentrated near noon, and sharply drop for the rest of the day.
If the panel is well oriented, the yield in winter is as good as in summer, as all the production is made around noon in both cases.
{% include yield_day_fixed_panel_norot.html %}
With a non-moving panel, you can see how the yield is affected.
In the best best cases, you get a yield of `40%` in summer, and `30%` in winter (over 24 hours).
Compare to the `60%` in summer and `40%` in winter when you can rotate your panel, you have a drop of `-33%` of efficiency, which is a lot.
If you plan/have a **thermal** solar panel (for heating water), it is more interesting to use winter optimal tilt angle:
you may prefer to take a hot bath in winter than in summer when it is 30°C outside !
---
# Optimizing Installation Surface Area
If you have a 2-axis rotative solar panel, you can get `100 %` yield all the day long.
However, this is for one panel.
If you have several solar panels, you need to space them appropriately, otherwise their shadows may reduce the yield.
## Illustration of Shading
When the sun is high or/and when the panels have a low tilt angle,
there is no shade. There is even land that receive sunlight.
When panel are optimally oriented and sufficiently spaced, they can all reach the maximal yield.
But when the sun is too low, or if there isn't enough space between the panels,
the first one in the row can reach the `100 %` yield, but not the others.
You have a trade-off between available land, the number of solar panels, and the total electricity production.
- If most of the land is covered by solar panel, it is useless to add motors.
- If land is almost free / unlimited, it is better to space solar panels to avoid shading.
## Spacing Panels to avoid Shading
To space correctly the panel, to avoid shading, the distance between panels (the distance from left edge of panel i to left edge of panel i+1), for a panel of length $$D$$ is:
$$D_1 = S \cos(\beta)$$
$$H = S \sin(\beta)$$
$$D_2 = \frac{H}{\tan(\alpha^{min})} = S \frac{\sin(\beta)}{\tan(\alpha^{min})}$$
Therefore, the total distance $$D=D_1 + D_2$$ between solar panels is:
$$D = S \left[\cos(\beta) + \frac{\sin(\beta)}{\tan(\alpha^{min})}\right]$$
where $$\alpha$$ is the **minimal** sun angle we want to capture.
In theory, $$\alpha^{min}$$ is minimal at sunrise and sunset, with a value of `0°`.
However, this leads to infinite distance between panels, which is not the best strategy to cover an area.
And next, we have to consider the environment: relief, trees, building, which will prevent from seeing the sun until it reaches some specific $$\alpha$$.
Unless you are on the top of a mountain, this angle is greater than $$0$$.
Also, if your panel has a fixed angle, yield when sun has a low elevation angle is low, so this worst case is not that impacting.
In any case, if you want to build a farm of solar panel, you need to have access to each of them, in case of maintenance.
### Solar Panel's Shade Simulator
{% include panel_shade.html %}
This is a simple simulator, where you can play on the panel size, and angles.
Given the minimal angle you want capturing energy, you can design your system by spacing sufficiently the panels.
For very small angles (lower than `20°`), you can see that the shade goes very far away.
For an angle of `30°` ($$sin(30°) = 0.5$$), the size of the shade is equal to the area occupied by the solar panel.
# Conclusion
In this blog post, we presented the theoretical yields of a solar panel.
We made the assumption that the tilt angle was fixed, but we can rotate it.
Even if the tilt angle is not very precise, just following the sun over the day is enough to get a good yield.
Nevertheless, for making a solar panel farm, we have to consider the shade made by solar panels on the others.
This require a non-negligible space between solar panels.
In the next post, we will look at "true" or "real life" solar panels.
You can go by clicking here:
[III: True solar panel efficiency]({% post_url 2022-08-11-Sun_3_true_SP_efficiency %})
# Sources & Other Readings
- [Solar Equations](https://www.pveducation.org/pvcdrom/properties-of-sunlight/elevation-angle)
- [Other Tools](https://www.sunearthtools.com/dp/tools/pos_sun.php?lang=fr)
- [Land-Use Requirements for Solar Power Plants in the United States](https://www.nrel.gov/docs/fy13osti/56290.pdf)