Overview

This is the first of three fairly intense lessons covering the fundamentals of solar energy. You will need to distill a lot of information and work hard to internalize the key topics. Make sure that you use the discussion forums to communicate with your peers, and be sure to ask questions. Most of us have learned bits and pieces of the following materials, but never together in one setting. And for some of us, this is the very first time we are learning to juggle all the balls that tie together solar energy. You can do it!

Time and Space are related! This lesson will discuss the tools needed to evaluate spatial and temporal relationships:

• of the Sun relative to the Earth;
• of the Observer on the surface of Earth relative to the Sun at any given time;
• while identifying the angles used to describe the orientation of a Solar Energy Conversion System (SECS) surface relative to the moving Sun;
• also identifying the times that local shadows might obscure our SECS. We will use sun charts to analyze shadows with respect to solar collectors.

Just like with navigation in a ship or an airplane, time and space relations are linked together in Solar Energy and can be represented and communicated as geographic information. We input that geographic information in terms of angles and use key relations from spherical trigonometry to make time and space relations easy to calculate with a computer. For our purposes:

angles = coordinates in space and time.

The tools we develop are going to explain the sun’s position relative to any point on the surface of the Earth. Once developed, our useful equations can also be applied to estimate the time and location of shadows that block SECSs or tracking technologies for SECSs.

Figure 2.1: Celestial Sphere - This blue sphere is a simple representation of the Earth within a larger sphere representing the celestial sphere, an imaginary sphere projected out onto the sky around the Earth. The equator is shown as a ring on the celestial sphere, tilted 23.45 degrees away from the Ecliptic ring, which is parallel with the orbit relative to the Sun.

Credit: J. Brownson © Penn State University is licensed under CC BY-NC-SA 4.0 [1]

By the end of this lesson, you should be able to:

• identify and apply the Earth-Sun Angles and the relations/equations between Solar Time and Watch Time (Standard/Daylight Savings);
• identify and apply the Observer-Sun Angles and be able to create a Sun Chart for shading analysis linked with the SAM software;
• identify the Collector Angles used to describe the orientation of a collector surface relative to the moving Sun (fixed or tracking) for power output analysis linked with the SAM software;
• list the important roles of geospatial relations to the three-part goal of Solar Energy Design.

What is due for Lesson 2?

This lesson is loaded with material on Sun-Earth geometry and will take us two weeks to complete. Please refer to the Course Calendar in Canvas for specific timeframes and due dates - those can vary from semester to semester. Specific directions for the assignments below can be found further within this lesson.

Questions?

If you have any questions, please post them to the Lesson 2 General Questions thread in Yellowdig. I will check this forum regularly to respond. Feel free to go through the comments and post your own responses if you are able to help out a classmate.

Solar Energy Journal [3] was established for the International Solar Energy Society (ISES) [4] and has been around for some time now. Solar Energy Journal stands as an important forum for peer to peer sharing of solar research for energy conversion and human applications of solar energy. What I want to establish here is that there is precedent for the complex system of notation used in the solar energy world that has been in use for decades. The original authors have established the following observations:

"Many disciplines are contributing to the literature on solar energy with the result that variations in definitions, symbols and units are appearing for the same terms. These conflicts cause difficulties in understanding which may be reduced by a systematic approach such as is attempted in this paper.

It is recognized that any list of preferred symbols and units will not be permanent nor can it be made mandatory, as new terms will emerge and old ones become less used with the development of the subject. But in the meantime, a list would be appreciated by the many workers who are entering this multi-disciplined field...

...Energy: The S.I. (Systèm International d'Unités) unit is the joule ($J=kg{m}^2{s}^{-2}$ ). The calorie and derivatives, such as the langley (cal cm^-2), are not acceptable.

No distinction between the different forms of energy is made in the S.I. system so that mechanical, electrical and heat energy are all measured in joules. However, the watt-hour (Wh) will be used in many countries for commercial metering of electrical energy...

Power: The S.I. unit is the watt ($W=kg{m}^2{s}^{-3}=J/s$ ). The watt will be used to measure power or energy-rate for all forms of energy and should be used wherever instantaneous values of energy flow are involved. Thus, energy flux density will be expressed as W/m² or specific thermal conductance as $W{m}^{-2}{K}^{-1}$ . Energy-rate should not be expressed as $J/h$.

When energy-rate is integrated for a time period, the result is energy which should be expressed in joules, e.g. an energy-rate of 1.2kW would if maintained for one hour produce 4.3 MJ."

W. A. Beckman, et al.

In summary: received energy flux density (or power density, called irradiance) can be expressed in units of W/m². We also note that the received radiative energy density (called irradiation) can be expressed in units of J/m², or in units of Wh/m². Notice that we did not use radiation, which is an expression of light glowing outward (emitted light, different direction than what we want).

In today's maps of the solar resource, you will often see the units expressed in kWh/m². You should be aware that these are still only representations of solar light energy density, and not the hourly/daily/annual quantity of potential electricity that could be produced. To find that value, we need a simulation tool like SAM (System Advisor Model, which you should have downloaded at the end of Lesson 1), which takes irradiation data and converts it into power data.

I would like you to now take a short self-quiz to see if you recall the common uses of the notation and descriptions for solar energy (used in particular in this class.).

When we want a shorthand to describe spatial relationships on continuous surfaces that are sphere-like, as with the Earth and the surrounding sky and stars, we choose to use Greek letters. In contrast, when we are trying to communicate things like linear distances, lengths, time, or simple Cartesian coordinates, then we will tend to use Roman letters for our shorthand.

You may notice in your reading of older textbooks that several systems of sign convention for the angles have emerged for practical use. Also, the various systems can have different approaches to azimuth that we should be aware of. For instance, the software SAM (System Advisor Model) will use both the $360^\circ$ clockwise standard (from Meteorology) as well as the $\pm180^\circ$ standard used extensively in the component-based models of TRNSYS and SAM. We will be sure to become familiar with both.

Below are four tables showing the Angular Symbols for Standard Solar Relations.

General Angles

Earth-Sun Angles

Sun-Observer Angles

Collector-Sun Angles

Figure 2.2 The axis of the Earth currently tilts approximately 23.5 degrees from the perpendicular (dashed line) to its orbital plane

Credit: D. Babb © Penn State University is licensed under CC BY-NC-SA 4.0 [1]

Earth's Rotation

As we have seen in our reading, the Earth rotates with a roughly constant speed, so that every hour the direct beam (a ray pointing from the surface of the Sun to a spot on Earth) will traverse across a single standard meridian (standard meridians are spaced 15° apart). The implications are that the unit of one hour is equivalent to the rotation of Earth 15 degrees. When Earth rotates such that the beam of the sun shifts +1° of longitude from East to West: it takes 4 minutes of time.

• 1 h = +15° Earth rotation
• 4 min = +1° Earth rotation

Wild fact: a time zone change of one hour is really just 15 degrees of separation between standard meridians.

The axis of rotation of the Earth is tilted at an angle of 23.5 degrees away from vertical, perpendicular to the plane of our planet's orbit around the sun.

The tilt of the earth's axis is important, in that it governs the warming strength of the Sun's energy. The tilt of the surface of the Earth causes light to be spread across a greater area of land, called the cosine projection effect.

Cosine Projection Effect

When you tilt a surface away from a beam of light, you spread the same density of light across a larger area. Recall that irradiance is in units of W/m², so a larger denominator means a smaller value of irradiance, right?

Explore the concept of the cosine projection effect in the following experiment.

Seasons and the Cosine Projection Effect

The sun is about 93 million miles away from the Earth (equivalent to ~150 million km). That is so far away that the photons from solar irradiation effectively travels in parallel rays. So, unlike the flashlight experiment, the tilt of the sun has no bearing on the intensity of the radiation reaching the Earth's surface. Instead, we find that the Earth's tilt controls the intensity of irradiation and the seasons.

Keep in mind that the Earth's axis points to the same position in space (toward the North Star, Polaris). As the Earth travels in a near spherical (a very small eccentricity into an ellipse) orbit around the sun, the Northern Hemisphere can be tilted toward or away from the sun, depending on its orbital position.

Figure 2.4: From late spring to late summer, the sun never sets at Barrow, Alaska (as this multiple exposures of the sun reveals)

Credit: Colin Monteath

Forecasters and meteorologists use different criteria to determine the "meteorological seasons." For example, meteorological winter in PA runs from December 1 to Feb 28/29, a period that statistically includes the three coldest months of the year. This is also centered on a time about 25 days after the Winter Solstice.

Meteorological summer runs from June 1 to August 31, a period that includes the warmest three months of the year. Again, this is a period centered about 25 days from the Summer Solstice.

Figure 2.5: Please review the NASA movie from 2000-2001 above, showing the rhythms of the most intense ultraviolet radiation coinciding with the most direct rays of the sun (around the summer solstices). Again, what may be a surprising observation is that the average air temperatures lag the sun's most direct days.

Credit: NASA

As one more example, review Pittsburgh's plot of annual average high temperatures. The maximum daily temperature occurs in late July, long after the summer solstice.

Figure 2.6: The annual variations of average, record-high, and record-low maximum temperatures at Pittsburgh, PA. Access the annual temperature stats at other cities [6]

Credit: Data/image provided by the NOAA/OAR/ESRL PSL, Boulder, Colorado, USA, from their Website at http://psl.noaa.gov/ [7]

As we see in the reading, time is a critical parameter in solar energy resource assessment, and we use a "different" time from your mobile phone (or if you have one, a watch). The time that we use in solar energy is the apparent time and path of the sun relative to the aperture or collection device, called Solar Time.

• A solar day is 24 hours long, and Solar noon is always used locally as the center of time.
• Solar noon is defined as the moment when the Sun is at its highest point in the sky.
• The hour angle for solar noon is $\omega =0°$ (given the symbol of lowercase omega).
• Time before solar noon counts backward from $\omega =0°$ , and so the angles are negative.
• Time after solar noon counts forward from $\omega =0°$ , and so the angles are positive.

Correcting Time for Big Longitude Changes: Standard Meridians

The time that you are used to using on your laptops, phones, and (if you still have them) watches is called Standard Time, and is referenced back to the Coordinated Universal Time (UTC), the primary time standard by which the world regulates clocks and time.

Our notion of time is also tightly coupled with our system of longitude (the longitude symbol for us is $\lambda$). We have used lines of longitude, or meridians, as a reference for time and position E-W. Because time and angles are all linked together, we cannot escape the sexagesimal (base 60 math) system for geographical locations. As we demonstrated earlier in the lesson, each standard meridian (a major line of longitude) is spaced 15° apart, beginning with the Prime Meridian ($\lambda =0°$) in Greenwich, England, and continuing for 360 degrees, or 24 hours.

I live someplace other than Greenwich. How do I account for Longitude ($\lambda \ne 0°$)?

Your standard time zone will tell you the standard meridian (${\lambda }_{std}$). For example, the EST is -5h from UTC, while Central European Time (CET, like Paris) is +1h from UTC.

• For every standard meridian shift to the West, you will need to subtract 15 degrees per hour.
• For every standard meridian shift to the East of the Prime Meridian, you will need to add 15 degrees per hour.

Correcting for Little Longitude Changes: Inside Time Zones

Where you live, or where your future solar site assessment will occur, will likely be well within the edge of a time zone (meaning ${\lambda }_{std}-{\lambda }_{loc}\ne 0°$ ). We already learned that every 1° of angular rotation on Earth is equal to 4 minutes of time. Standing in one spot on the surface, this means 4 minutes of relative time correction locally per degree of deviation from a Standard Meridian (${\lambda }_{std}$ ). So, locales will have a local longitudinal refinement to account for, in order to account for not living directly on a 15-degree incremental Standard Meridian on Earth.

Standard Meridians define the beginning of a time zone, and not the end of a time zone. So, you are always going to look to the start of a time zone to find the Standard Meridian.

There are a few other cities that actually are well seated for solar time zone correction (close enough for our calculations):

• Philadelphia is fairly close to the EST standard meridian of -75°.
• Denver is fairly close to the MST standard meridian of -105°.

My client lives someplace other than a Standard Meridian, how do I account for that?

First, go to Google and type "<insert city name> longitude". You should get a quick response of both the Longitude (lambda) and the latitude (our symbol for latitude is lowercase Greek "phi": phi), represented in decimal form (more useful to us for trigonometry and angles).

Have you noticed that real time zones are more often political boundaries that zigzag around, rather than following an actual Standard Meridian? So, actually, there can be locales for clients that are East of their own time zone Standard Meridian, instead of the normal relative locations West of the time zone Standard Meridian. This is why, in the reading, you will see $\pm 4$ minutes per degree of local longitudinal shift away from the time zone's Standard Meridian.

Correcting with the Equation of Time: Accounting for Wobbles

Even in Greenwich, where no longitudinal correction is necessary, "noon" UTC will generally not be the time when the sun is directly overhead. We can see in the plot below that watch time and solar time are the same in Greenwich for only 4 days in the year.

• There are deviations of up to $\pm 16$ minutes (regardless of your location on the planet).

As you will have read, our interpretation of watch time assumes an even progression for Earth's planetary rotation, with no weebles or wobbles or precession of the polar axis. However, you will now know that wobbling occurs, and there is great variability in the rotation of the earth throughout the months of the year. This is why we add leap years and leap seconds to our calendars. So, we create a "mean time" based on the length of an average day to keep things simple. Solar time has to correct for this mean time approximation. Equation of time correction versus day of the year is shown in Figure 2.7. As an exercise, you can try to calculate these curves based on the empirical equations (6.10) and (6.11) in Brownson textbook.   

Figure 2.7: Equation of time function represented over all the days of a year. Note how the minimum and maximum do not exceed 16 minutes.

Credit: Jeffrey R. S. Brownson © Penn State University is Licensed under CC BY-NC-SA 4.0 [1]

The following picture was a composite of images, taken at the same watch time every few days for an entire year, to record the position of the sun. We call the shape an analemma. Notice how there is a big loop and a little loop, and compare the same big waves and little waves in the first image of the Equation of Time correction above. If you were to draw a line down the center, you would have removed the error from watch time, and you would be one step closer to solar time.

Figure 2.8: Analemma photograph taken from Bell Laboratories, circa 1998.

Credit: Analemma Fishburn by J. Fishburn from Wikipedia [9] is licensed under CC BY-SA 3.0 [10]

For views of amazing solar analemma photography by Anthony Ayiomamitis, also, please visit the Stanford SOLAR Center [11].

These images were taken at the same time and location every day for one year. You will see the curve described by the Sun over that year. An analemma is a beautiful way to capture both the range of declination $\delta$ (along the length of the analemma) and the Equation of Time Et (the expansion or width of the analemma) in a graphical format.

Putting Time Correction Together:

As we have seen in the reading, we have a method to correct local watch time to solar time, to correct for the true time of day according to the sun relative to the meantime.

The time correction factor is a function of the equation of time, the meridian of the local standard time zone (L_st), and the longitude of the collector/observer (L_loc), and the Equation of Time (E_t). The equation of time correction accounts for the wobble and is the first step. Then we need to correct for the change in longitude leading to time zones (standard meridians occur every 15 degrees) and the change of time for locales that are not located along standard meridians.

In these conversions, each year is assumed to begin just after midnight, Dec. 31, and time counts up from there. The time corrections here are in terms of minutes, not hours. The Equation of Time corrects the time to mean solar time (by adding or subtracting up to 16 minutes of correction). The Time Correction Factor corrects time for your shift in longitude from each Standard Meridian (which occurs every 15 degrees away from the Prime Meridian). Daylight Savings Time is an optional correction, as there is a +60 minute difference from Standard Time between March and November in the USA (and some other regions of the world).

Now, let's convert "time" into an angle, for our future trigonometric relations:

When we convert time to an angular value, we can no longer use a 24 hr format. We need to convert hourly time into a useful angle based on the properties of a sphere, again using spherical trigonometry.

Video: Hour Angles (5:03)

Finishing Step for Time Conversions! The Hour Angle ($\omega$) in decimal degrees.

We represent the apparent displacement of the sun away from solar noon, either as a negative or positive angle. An $\omega$ of zero indicates that the sun is at its highest point for that given day.

• The sign of the hour angle before noon is negative, because we have to count backward from the "zero hour" of solar noon.
• The sign of the hour angle after noon is positive, because we are counting forward from the "zero hour" of solar noon.

Another way of looking at it is the angular difference between the local meridian of the observer/collector and the meridian that the beam of the sun is intersecting at a given moment.

$$\omega =\frac{{360}^{\circ }}{24\text{hr}}(t_{sol}-12)=\frac{{15}^{\circ }}{\text{hr}}(t_{sol}-12)$$ $$\omega =\{\begin{array}{c}-0-180\text{degrees},\text{if before noon (morning)}\\ +0-180\text{degrees, if after noon (evening)}\end{array}$$

Now, the day won't really begin at -180 degrees, anywhere between the arctic circles. However, I wanted to emphasize that the day only begins at -90° on two days a year. Those days occur during the Equinox moments in the orbit of the Earth about the Sun, and the length of those days is actually 12 hours long. All other days are either shorter than 12 hours, or longer than 12 hours. As such, they end either less than 90° or greater than 90°.

Figure 2.9: The Sun Path projected onto graphed sphere for a latitude of 50 degrees, and the day of an Equinox (12 hours). The hours for 8h00, 12h00, and 14h00 are listed in solar time, along with the associated hour angles of -60°, 0°, and +30°.

Credit: Original [12] by Tau'olunga from Wikimedia Commons licensed under CC BY-SA 2.5 [13] - Modified by J. Brownson. 

The images below demonstrate singular arcs of the Sun for each hour in solar time, at five different latitudes on Earth (no analemma correction necessary). The peak hour (or the hour with the highest solar altitude angle) is defined as solar noon. You will notice that the solar equinox has twelve sun spots for latitudes below the arctic circle, and that the sun rises due East, while setting due West. During the solar solstices, you see multiple arcs: one for winter and one for summer. Notice that there are more hours of the day in the summer, and the sun rises farther from the equator in the Summer (sun rise in the northeast for the Northern Hemisphere).

What else do you notice in comparing the four critical times of year at different latitudes?

Solar Equinox

[14] [15] [16] [17] [18]

Figure 2.10: Solar Equinox

Click on the images above to see full-size versions with descriptions.

Credit: All images [12] are by Tau'olunga from Wikimedia Commons licensed under CC BY-SA 2.5 [13]

Solar Solstice

[19] [20] [21] [22] [23]

Figure 2.11: Solar Solstice.

Click on the images above to see full-size versions with descriptions.

Credit: All images [24] are by Tau'olunga from Wikimedia Commons licensed under CC BY-SA 2.5 [13]

Now, we are ready to use hour angle to find out positions in our solar design projects!

Remember that the Earth rotates at 15 degrees per hour, and 0.25 degrees per minute. It can get confusing when you're comparing spatial seconds with temporal seconds, right? That's why we will stick with decimal notation in all of our references to latitude ($\phi$), longitude ($\lambda$$$ ), and angles (degrees).

Step 1: Correcting for Standard Time (Standard Meridian) in Hours: Find the time zone of your client. The hourly longitude correction is plus or minus X hours from the Prime Meridian, where UTC = 0h. Multiply the positive or negative hour value by 15°, and you will know your standard meridian for the Time Zone.

Key: Your standard meridian for your time zone begins on the East side of the time zone. (Sun rises in the East, right?)

Step 2: Correcting for Time from the Local Longitude Relative to Standard Meridian in Minutes:

You need to know the longitude for the standard meridian of the client ( ${\lambda }_{std}$) from their local time zone, and the local longitude of the client's location in question (${\lambda }_{loc}$). We calculate the longitudinal correction, $t_{\lambda }$ from these two.

Keep in mind: Time zone borders are political boundaries, and can be constructed on both sides of a Standard Meridian. A locale to the east of the Standard Meridian would still be input as a positive value (the resulting minutes would be positive).

$$t_{\lambda }=-4(\frac{min}{deg})({\lambda }_{std}-{\lambda }_{loc})[min]$$

Note: we use the sign convention that longitudes are positive valued for to the East of the Prime Meridian, and negative valued to the West of the Prime Meridian. This equation is valid for both sides of the Prime Meridian. The exterior negative sign is there to make the time correction algorithm work correctly.

Step 3: Calculating Equation of Time (Analemma) Correction in Minutes (E_t): We begin with the two simple coefficients of n (the day of the year, from 1-365), and B (see the first equation). Our assumption is that the year begins at midnight of the new year, and the trigonometric portions of the equation of time will take an argument of "B" in degrees.

$$B=(n-1)\frac{{360}^{\circ }}{365}$$
$$\begin{gathered} E_t=229.2(0.000075)+229.2(0.001868\cos B-0.032077\sin B)- \\ 229.2(0.014615\cos 2B+0.04089\sin 2B)[min] \end{gathered}$$

You have seen that the Equation of Time has a graphical representation, the analemma. Once we determine a correction in the scale of minutes, we can use it in the Time Correction Factor, TC.

Step 4: Completing the Time Correction Factor ($TC$): The time correction factor is a time shift in minutes.

$TC=t_{\lambda }+E_t$

Recall that 0.25° of longitudinal rotation (of the planet) will consume 1 minute of time. That would make each degree of change equivalent to four minutes. Hence, we need to multiply our longitudinal correction by a factor of four $$\frac{\min }{\mathrm{deg}}$$ to yield a consistent unit of minutes in time. This is shown as the value of the longitudinal correction (L_c) in units of minutes (temporal, not geospatial).

Step 5: Accounting for Daylight Savings Time (DST) in 60 minutes.

The equations do not tell you when DST occurs from country to country. There is a +60 minute difference between March and November in the USA. So, if we had to correct for DST, then we would need to subtract that 60 minute addition back out.

$$LocalSolarTime(tsol)=\{\begin{array}{c}StandardTime+TC\text{ if Standard time.}\\ StandardTime+TC-60\text{ if Daylight Savings time.}\end{array}$$

Again, make sure that the data is presented in minutes, rather than hours.

Continuing Conversations in Yellowdig!

I hope you are getting the hang of Yellowdig conversation platform and are ready to pick up the pace! You can access the discussion platform through the Canvas menu.

Lesson 2 Discussion: Reflection on Time Conversions

Tagging

Yellowdig Tip: When you create a post in the Yellowdig discussion space, you are required to choose a topic tag. For Lesson 2 discussions, use any of the following:

Know that you can tag your post with one or several of these topics depending on its content. It gives you flexibility to discuss these things cumulatively or, if you prefer, to break your writing into several smaller posts.

Importance of interaction

Once again: the more you participate, the more opportunities for your discussion score to grow through the week. Comment, ask questions, react, throw in additional thoughts – it all is in your benefit!

Grading

Yellowdig points you earn over the weekly point earning period (from Saturday to next Friday) will count towards 1000 pts. weekly target. But you can go above it (to 1350 pts. max) if you are feeling active. That may help you to make up for some low-activity weeks. Yellowdig discussions will account for 15% of the total grade in the course. Check back the Orientation Yellowdig page in Canvas for more details on the points earning rules.

Deadline

There is no hard deadline for participating in these discussions, but I encourage you to create your posts in the middle of the study week (Sunday) to allow others to engage and respond while we are learning specific topics in the lesson.

The Greek notation used below is now fairly standard for the field, but you may see different approaches in informal settings or unusual textbooks. I follow the established nomenclature defined by Beckman, et al., in the research journal Solar Energy in 1978, reprinted in Solar Energy in 1997. As noted earlier, the document is available to you from the Library E-Reserves.

Figure 2.12: We are showing a vector S' (S-prime) originating at the center of Earth (point C) and directed toward the Sun. The position of the collector (point Q) is set on the surface of the Earth. The location of Point Q defines the meridional axis, based on the collector meridian, and serves as a reference for the hour angle $\omega$.

Credit: J. Brownson © Penn State University is licensed under CC BY-NC-SA 4.0 [1]

Hour Angle ($\omega$):

Again, this is the way we represent the apparent displacement of the sun away from solar noon. An hour angle of zero degrees indicates that the sun is directly above, and the sign of the hour angle is determined by occurring either before noon (negative) or after noon (positive). Another way of looking at it is the difference in angle between the local meridian of the observer/collector and the meridian that the beam of the sun is intersecting at a given moment.

Angle of Declination ($\delta$)

This is not an observed angle relative to the surface of the Earth and the Collector. Declination is the observed angle (due to the polar tilt) between the plane of the Earth’s equator and the plane of the ecliptic (the plane in which the Earth's orbit about the Sun lies). Declination has a maximum angle of 23.45° at either solstice. In this case of coordinates, the sun observed in the North is positive, and in the South is negative. One can imagine this as a series of sun paths over the course of a year. In contrast, the declination reaches a midpoint at either of the equinoxes. The range of declination is limited to Earth’s tilt: −23.45° (winter solstice) $\le \delta \le +23.45°$ (summer solstice).

Declination is independent of location on the planet!

The declination can be calculated by a simple approximation (first equation) or a Fourier series (second equation).

$$\delta =23.45\sin \left(\frac{360}{365}(284+n)\right)$$ $$\begin{array}{cc}\delta & =(180/\pi )\times (0.006918-0.399912\cos B+0.070257\sin B\\ & -0.006758\cos 2B+0.000907\sin 2B\\ & -0.002679\cos 3B+0.00148\sin 3B)\end{array}$$ where:
$$B=(n-1)\times \frac{360}{365}$$

Angle of Latitude ($\varphi$) and Longitude ($\lambda$)

The complement of longitude ($\lambda$) in geospatial coordinates is the latitude. When combined together, we have identified a singular point on the surface of Earth. However, separately, they refer to arcs or circles spanning huge distances. We can look up the latitude and longitude of a site on a search engine like Google.

Key latitude angles of interest are:

• Arctic Circle $\varphi$ = +66.56° North of this, the sun is above/below the horizon for 24 continuous hours at least once per year (during the solstices).
• Tropic of Cancer $\varphi$ = +23.45° The maximum tilt of the North Pole to the Sun.
• Equator $\varphi$ = 0°
• Tropic of Capricorn $\varphi$ = −23.45° The maximum tilt of the South Pole to the Sun.
• Antarctic Circle $\varphi$ = −66.56° South of this, the sun is above/below the horizon for 24 continuous hours at least once per year (during the solstices).

Figure 2.13: We are showing azimuth orientation Convention I, where the vector S is pointing from the surface of Earth toward the Sun, given the zero basis for the azimuth is directed toward the Equator (±180°). Convention I follows the vectorial form of S = S_z i + S_e j + S_sk. Convention II (the meteorological standard) is where the zero basis is directed to the North (0-360° clockwise).

Credit: J. Brownson © Penn State University is licensed under CC BY-NC-SA 4.0 [1]

The Solar Altitude (${\alpha }_s$) measures the angle between the central ray from the Sun (beam radiation), and a horizontal plane containing the observer. Note that the subscript "$_s$'' is there to indicate that the altitude relative to the observer of the Sun. This will become important in evaluating the altitude angles of other objects projected onto the sky dome, like buildings, overhangs, wing walls, and arrays of solar receivers.

$$\sin {\alpha }_s=\sin \delta \sin \varphi +\cos \delta \cos \varphi \cos \omega$$

$$\cos {\alpha }_s\sin {\gamma }_s=-\cos \delta \sin \omega$$ $${\alpha }_s={\sin }^{-1}[\sin \delta \sin \varphi +\cos \delta \cos \varphi \cos \omega ]$$

The Zenith Angle (${\theta }_z$) is the geometric complement of the solar altitude angle. We direct your attention to the use of $\theta$ here, as the general concept of the angular deviation of the Sun's ray from the normal projection of a surface is called the angle of incidence, $\theta$. In effect, the zenith angle represents the angle of incidence for a horizontal surface.

Recall: "normal" means perpendicular to a surface.

$$\cos {\theta }_z=\sin \delta \sin \varphi +\cos \delta \cos \varphi \cos \omega$$$$$$ $${\theta }_z={\cos }^{-1}[\sin \delta \sin \varphi +\cos \delta \cos \varphi \cos \omega ]$$

The Solar Azimuth ($\gamma s$) measures the angle on the horizontal plane between the meridian of the 0 degrees axis (South, for the Northern Hemisphere, opposite Down Under) and the meridional projection of the Sun's central beam (the Sun's meridian). The convention we will use is also used in the advanced design tools for solar energy, TRNSYS (UW-Madison: Transient eNergy Simulation Software), and PVSyst. The angle varies from 0 degrees at the South-pointing coordinate axis to $$\pm 180$$ degrees. East (earlier than noon) is negative and west (later than noon) is positive in this basis. This is a well-used standard taken from the original text of M. Iqbal.\cite{iqbal83}

$${\gamma }_s=\text{sign(}\omega \text{)}\left|{\cos }^{-1}\frac{\cos {\theta }_z\sin \varphi -\sin \delta }{\sin {\theta }_z\cos \varphi }\right|$$

Note: the function "sign($\omega$)" is specifically defined as a cases form of positive and negative notation (meaning there are two alternate cases to choose from).

$$\gamma s=\{\begin{array}{c}+{\cos }^{-1}\frac{\cos {\theta }_z\sin \varphi -\sin \delta }{\sin {\theta }_z\cos \varphi }\text{ if }\omega \text{>0,}\\ -{\cos }^{-1}\frac{\cos {\theta }_z\sin \varphi -\sin \delta }{\sin {\theta }_z\cos \varphi }\text{ if }\omega \text{<0.}\end{array}$$

There are several texts and software available that use a solar azimuth, measuring clockwise on the horizontal plane from a North-pointing (0 degrees) coordinate axis. This is also the convention for the field of meteorology and for the wind industry. This azimuth convention uses 360 degrees for the meridional projection of the Sun's central beam. SAM Software, PVWATTs, and University of Oregon's Sun path tool - all use this convention as well, so you must be aware of the difference, as the math changes with the two methods.

The following angles describe the orientation of the collector surface and the relation of the collector surface to the Sun. As these three angles describe our primary surface of interest, the SECS, they do not have a subscript for the two coordinate angles of collector tilt and collector azimuth.

• Tilt: the angle between the plane of the collector (or aperture) and the horizontal. Denoted by the symbol beta, β.
• Azimuth: the planar rotation East or West that an aperture will have. Denoted by the symbol gamma with no subscript, $\gamma$.
• Angle of Incidence: the angle between the vector perpendicular to the collector plane, called the normal of the plane, and the projection of the Sun’s central beam to the collector surface. Denoted as theta with no subscript ($\theta$).

Figure 2.14: We depict the SECS Collector-Sun Orientation here: the collector tilt (beta, $\beta$) and azimuth (gamma, $\gamma$) are positioned to reflect a site somewhere in the Northern Hemisphere. The Sun's position is described fully using the solar altitude angle (${\alpha }_s$) and the solar azimuth angle (${\gamma }_s$).

Credit: Jeffrey R. S. Brownson © Penn State University is licensed under CC BY-NC-SA 4.0 [1]

For general cases, where the collector has a non-horizontal orientation ($\beta \ne 0$), the angle of incidence is not the same as the zenith angle ($$θ_z). In fact, the zenith angle is a special case of an angle of incidence for horizontal surfaces, where the zenith is referenced to the aperture as a normal projection.

The first and second angles of tilt and surface azimuth ($\beta$and γ$$) are typically known for fixed surfaces. The third key angle is the angle of incidence (θ$$), which uses the following rather lengthy equation:

$$\begin{gathered} \cos \theta =\sin \varphi \sin \delta \cos \beta -\cos \varphi \sin \delta \sin \beta \cos \gamma +\cos \varphi \cos \delta \cos \beta \cos \omega \\ +\sin \varphi \cos \delta \sin \beta \cos \gamma \cos \omega +\cos \delta \sin \beta \sin \gamma \sin \omega \end{gathered}$$

In order to generate an actual value for theta, we will need to also take the arccosine of the long equation. For your calculators and math programs, all of the arguments are in terms of degrees, not radians. You will need to convert degrees to radians in most programs.

However, this is a really long equation that can actually be broken down into parts. We will break up the equation for the angle of incidence (theta, $\varphi$) into three lines. Take a look at them, and look for common arguments to the sine and cosine functions.

• ($\varphi$ , $\delta$ , $\beta$: latitude, declination, and tilt)
• ($\varphi$ , $\delta$ , $\beta$ , $\gamma$: latitude, declination, tilt, and then collector azimuth)
• ($\varphi$ , $\delta$ , $\beta$ , $\gamma$ , $\omega$: latitude, declination, tilt, collector azimuth, and then the hour angle)

$$\cos \theta =\sin \varphi \sin \delta \cos \beta -\cos \varphi \sin \delta \sin \beta \cos \gamma +$$ $$\cos \varphi \cos \delta \cos \beta \cos \omega +\sin \varphi \cos \delta \sin \beta \cos \gamma \cos \omega +$$ $$\cos \delta \sin \beta \sin \gamma \sin \omega$$

Mechanisms to Maximize Solar Utility:

In the introduction of the textbook, there is a reference to the goal of solar energy design: to maximize the solar utility for a client in a given locale.

There are three main design mechanisms that will increase the solar utility of a SECS for a client or group of stakeholders.

1. Decrease the cosine projection effect: This is done by tilting the collector toward the Sun's average annual noontime position. The higher the latitude of the locale, the more tilt a collector will need. Seasonal tilt changes can also slightly improve the solar gains. We also direct the collector toward the equator, although there is significant flexibility in both tilt and azimuth.
2. Minimize the angle of incidence: This is a refinement of the first mechanism over the course of a given day, particularly relevant to solar tracking systems. By "pointing" the collector normal at the Sun during the day, the angle of incidence is minimized, and more light can be collected.
3. Minimize or remove shading effects from the collector: We will cover this in the next few pages. It should make sense that when shadows cover a collector, then the majority of the Sun's light for that unexposed area is no longer providing power. Photovoltaics will be much more susceptible to shading issues than solar thermal systems, due to the near instantaneous generation of charge carriers in PV vs. the slower reaction from a thermal response of fluids.

In a spherical coordinate system, the angles are the coordinates. So, if I were standing in a field in North Dakota, looking at something tall like an enormous wind turbine, I could define the position of the top of the nacelle relative to me by stating the general azimuth angle ($\gamma$, the rotation across the horizon from due South) and the general altitude angle ($\alpha$, the rotation up from the horizon). Effectively, this is my x ($\gamma$) and y ($\alpha$) coordinates on an orthographic projection of the sky dome on a flat surface.

Figure 2.15: A common example of spherical coordinates (r, $\theta$, $\varphi$) commonly used in physics: radial distance r, polar angle $\theta$ (theta), and azimuthal angle $\varphi$ (phi). The symbol $\rho$ (rho) is often used instead of r.

Credit: 3D Spherical Coordinates [25] by Andeggs from Wikimedia (Public Domain)

Many of you will be familiar with Cartesian coordinates in space (x, y, z). However, when dealing with spatial relations of spherical objects like the Earth and the Celestial Sphere, we find that working with basic spherical coordinate systems makes trigonometry available to us to solve for space and time equations. For spherical coordinates, we need information of radial distance, zenith angle, and azimuth. However, in solar positioning studies, a radius of one (unit radius) is all we need to establish a unit vector, and we are left with equations for only the zenith angle and azimuth (and the complement of the zenith angle, the altitude angle). Note how the zenith angle in Figure 2.15, above (the generic $\theta$ angle), is congruent with the solar zenith (${\theta }_z$) of the Sun, and the generic azimuth angle ($\phi$) is congruent with the solar azimuth (${\gamma }_s$).

The following equations describe the Cartesian coordinates (x, y, z) for unit vectors, followed by the equivalent functionals using the complementary altitude angle.
$$\begin{array}{c}\begin{array}{ccccc}z& =& \cos \theta & =& \sin \alpha \\ x& =& \sin \theta \cos \phi & =& \cos \alpha \cos \phi \\ y& =& \sin \theta \sin \phi & =& \cos \alpha \sin \phi \end{array}\end{array}$$

As seen in Figure 2.16, we need to pull back to our old trigonometry mnemonics! Recall "Soh Cah Toa" when looking at the figure:

Figure 2.16: Right Triangle with side lengths [a (opposite), b (adjacent), h (hypotenuse)] and angles [A, B, C].

Credit: J. Brownson © Penn State University is licensed under CC BY-NC-SA 4.0 [1]

• Sine(A): Opposite over Hypotenuse (a/h)
• Cosine(A): Adjacent over Hypotenuse (b/h)
• Tangent(A): Opposite over Adjacent (a/b)

Sun Charts: Projections of Solar Events and Shadowing from the Sky Dome

The emphasis of this lesson is the Sun Chart tool (or Sun Path). These flat diagrams are found in many solar design tools, but may look completely foreign to the new student in solar energy. How do we interpret the arcs and points plotted on a sun chart? Why do we have two different types of plots (one looks like a rectangle, and one looks like a circle)? Why do some plots go from 0-360°, while others go from -180° to +180°?

Video: Angles Sunchart 1 (4:45)

What are Sun Charts?

If we want to visually convert our observations of the sky-dome onto a two-dimensional medium, we can either use an orthographic projection or a spherical projection on a polar chart. These projections are useful for calculating established times of solar availability or shadowing for a given point of solar collection.

The Sun Path describes the arc of the sun across the sky in relation to an earth-bound observer at a given latitude and time.

Figure 2.17: We display the path of the Sun across two days for the Northern Hemisphere. One day in summer and one in winter, where the trail of the beam has been projected onto the sky dome using angular coordinates of solar azimuth (${\gamma }_s$) and solar altitude (${\alpha }_s$).

Credit: J. Brownson © Penn State University is licensed under CC BY-NC-SA 4.0 [1]

All light incident upon Earth's surface must pass through the atmosphere and be attenuated (lost from absorption or back scattering). In order to simplify the many points of origin of light, we divide the sky and the Earth's surface into components, or spatial blocks of an imaginary hemispherical projection on the sky. The Sky Dome refers to the sum of the components for the entire sky from horizon to zenith, and in all azimuthal directions. In our following sections, a collecting surface is assumed to be horizontal first, as a pyranometer measuring device is mounted horizontally and facing the sky to measure the Global Irradiance/Irradiation in the shortwave band for the sky dome. Most of our solar collectors will be tilted up from horizontal in some way (PV, solar hot water, windows, walls, even your eyes). Those surfaces oriented otherwise are termed a Plane of Array measurement (POA), requiring specific tilt and azimuth information in the description. For those solar collecting surfaces that are not horizontal, the reflectance of the ground is an additional source of light, through the albedo effect. The beam, sky diffuse, and ground diffuse light sources incident upon the tilted collector are estimated using models of light source components.

Projections

The sky dome can be projected onto flat surfaces for analysis of shading and sky component behavior.

Figure 2.18: The sky dome as projected to the right in orthographic form, and as projected upward in polar form.

Credit: J. Brownson © Penn State University is licensed under CC BY-NC-SA 4.0 [1]

Video: Angles Sunchart 2 (6:01)

1. Orthographic Projection: takes the sky dome and projects altitude and azimuth values outward onto a surrounding vertical cylinder. The cylinder is then opened flat. Figure 2.19, below, shows the sun rising in the East (to the left) and setting in the West (to the right). Proper observation shows that the largest arc in the chart at the top, June 21, is the Northern Solstice, while the smallest, December 21, is the Southern Solstice.
   

   Figure 2.19: Orthographic Projection.

   Credit: J. Brownson © Penn State University is licensed under CC BY-NC-SA 4.0 [1]
2. Polar Projection: takes the sky dome and projects altitude and azimuth values down onto a circular plane. However, in the polar projection, the arc for December 21 is at the top while the arc for June 21 is at the bottom. This happens because we are effectively lying on the ground with our heads facing south, and holding that large piece of paper straight up to the sky.
   

   Figure 2.20: Polar Projection

   Credit: Jeffrey R. S. Brownson © Penn State University is licensed under CC BY-NC-SA 4.0 [1]

How do We Make and Read Sun Charts?

Go to the University of Oregon Solar Radiation Monitoring Laboratory [26] website. The scientists at the Laboratory have provided an excellent tool for plotting sun paths onto orthographic projections or polar/spherical projections. The default page is for creating an orthographic projection of your site of interest. The alternate page for polar projections [26] will use the same data you can input, but will output the alternate form. Note that both use the meteorological standard for azimuth angles, where North is set at 0°, South will correspond to 180, and the azimuth values are increasing clockwise to 360°.

So how do you use this tool? Go through the following steps:

1. Specify the location by $\varphi$ and $\lambda$. (Latitude is important for our calculations of sun-observer angles).
2. Specify the time zone (this is optional - for most time zones the software makes this correction from UTC to local time zone automatically based on the coordinates provided).
3. Choose months to be plotted - you can do the full year or isolate a certain month or even day. Note that the plots are symmetrical between the first half and second half of the year versus summer solstice (when calculated in solar time). December through June data will look the same as June through December. 
4. Choose projection: orthographic (Cartesian) or polar.
5. Further you can specify a few additional parameters. We do need time contours and labels, so keep those checked.
6. Once you click "Submit", the plot will be generated, and you will be given options to download the image and data.

What about Shadows?

When designing a solar energy conversion system for any application, we must pay special attention to the occurrence of shadows throughout the year. Later in this lesson, we will discuss a method to assess the shading using 2-D projections. Specifically for one of your assignments, you will use the polar SunChart projection for shading analysis. 

The next page gives you an opportunity to print and analyze your own Sun Chart.

Now, based on your understanding of the SunCharts, work through the self-check questions below. While you work through these steps, try to think about all the calculations that went into each plotted point on the curves. You should reflect on the fact that the SunChart tool is simply plotting points and lines based on solar calculations of time and longitude. The same calculations that you are mastering now in class. Pretty exciting!

Self-Check Questions - based on the Cartesian SunChart for State College, PA

Answer the same questions for your own locale. Are the answers different or the same compared to State College, PA?

If you check out the polar projection of the Sun Chart, you will see that it looks quite different, but the plot is based on virtually the same data. Here is the polar chart for State College, PA, location. Note that in this case the coordinates used are solar azimuth and solar zenith angle (SZA) as opposed to the solar elevation angle in the cartesian projection (I wish there were an option to switch between those two). If you need to get the solar elevation angles from this plot, just remember that those angles are complementary:

$${\alpha }_s=1-{\theta }_z$$

This is the projection you will be using for shading analysis in your homework. So take some time studying this plot.

• What is each blue curve represents here?
• Pick a point on a curve. Can you determine the solar azimuth and solar zenith angle for this point from the plot? What would be solar elevation angle at that point?
• Which point on the plot would correspond to 3 pm on February 21st (tip: look for month and hour labels)

Going forward, you can always check your solar coordinate calculations versus the Sun Chart or use them for quick estimates of the Sun position.

Quick review of the Charts:

First, we will go over the key features of the orthographic plots with the arcs of six days plotted out for the first half of the year.

Video: Sunchart Ortho Intro (4:42)

Second, we can compare the key features of the orthographic plots to the polar plots, again with the arcs of six days plotted out for the first half of the year. You should notice the similarities (East is on the left, solar time is represented, the same location is plotted), as well as the distinct differences (the June and December arcs are "flipped").

Video: SunChart Polar Intro (5:49)

How to integrate shading as an overlay

Now, we need to add an additional layer of information to the plot. I'm going to talk through the addition of critical points on the sun chart, followed by connecting those points and shading the areas correctly. In each of the following examples, we will use orthographic projections, but there is no reason why you couldn't use polar plots instead.

The first example will come from the textbook. We will be plotting shading relative to a single receiving point: X, with three critical points of shading (A, B, and C).

Video: Shading Wing Demo (5:52)

In the next example, we will look at setting up the problem to assess a PV array shading problem. Our intent is to plot shading relative to multiple receiving points: A, B, and C, with three critical points of shading (1, 2, and 3). The result is nine values for altitude coordinates ($\alpha$ , no subscript) and nine values for azimuth coordinates ($\gamma$ , no subscript).

Video: Shading Array Demo 1 (6:08)

You will notice how the horizontal surfaces of the building that create shadows are transposed to the projection as an arc, not as a horizontal line, while the vertical surfaces remain vertical. This has to do with the manner in which spherical data is distorted in an orthographic projection. Hence, the plot of a building shading a point on a window will look a lot like a slice of bread, flat on the sides with a soft curve across the top. The same will be found for this example, where the receiving points A, B, and C are shaded at different times by critical points 1, 2, and 3.

Video: Shading Array Demo 2 (5:57)

Next, we will be interpreting our results, and inputting the shadow data into SAM.

System Advisor Model

I hope you were successful installing SAM software and getting it ready to use in the last lesson. Now you will be applying your interpretation of the shading diagram to determine shading factors and input them into SAM for system energy performance simulations where shading interferes with the solar resource.

We have come a long way in interpreting sun path diagrams and plotting shaded areas as overlays onto the sun path charts. How can we input those shade/no-shade conditions to the PV performance model in SAM? The video below provides a demo of how shading conditions can be applied in SAM simulations. 

Video: SAM Shading Intro (6:26)

So, the important take-aways are that we can use SunCharts or other geometric shading data to get us to a shade-based performance adjustment of PV performance simulations, and that shading matters in solar energy. How significant can it be? Quite significant, but you will get a more accurate answer after performing your homework assignment in this lesson! We use these SAM simulations in solar project design, and this will be the way to assess the shading losses for any specific scenario (and in your final course project, eventually). 

When working with SAM shading table, we can input partial values for hours that are on the shade/no-shade borderline (e.g. 50% etc.). SAM conveniently grades the color scheme from white (full sun) to red (full shade) in the process. Also, note that the locale matters! Do not forget to set your location in the Location and Resource tab first: the default for SAM is often in the Southwest USA, and you will need a correct solar resource file before running your simulations.

This activity allows you to apply the concepts of time and space coordinates you learned in this lesson to a specific example of a solar energy conversion system. In the process, you will work with both Sun Chart and SAM software to model the performance of a PV array. You will be required to demonstrate your learned skills for Sun Chart interpretation, shadow tracking, and including shading information into SAM simulations.

PV Array System

• A fixed-axis array of PV modules installed in the Baltimore MD area.
• The array consists of two rows of 8 modules stacked together (see more description and picture in Chapter 7 of the SECS Book), each module is 1.25 m in height and 1 m wide.

Figure 2.21: Sample array

Credit: Mark Fedkin © Penn State University is licensed under CC BY-NC-SA 4.0 [1]

• Orientation: azimuth -9^o (i.e. it is facing a few degrees East of South - in standard 360^o degree notation, surface azimuth = 171^o)
• Tilt: 35^o (fixed)
• The array is assumed to be part of a larger solar installation, meaning that a similar array positioned in the front can cast shadow to our array of interest.
• Spacing between the arrays in the field is 3 m.
• Nominal array capacity 4 kW
• Assume PV cells and modules are connected in series, so partial shading along the base will affect the entire array (simplified case).

Task

Model the effect of shading on the PV array performance (kWh of energy generated) over a year. So, in the end, you will need to produce the energy output data from the array for the full-sun and shaded scenarios and observe the difference.

Directions

1. SunChart. Create the Sun Chart for sun trajectory of the Baltimore, MD, location using the University of Oregon Solar Radiation Monitoring Laboratory solar tool. [27] You are to use the polar sun chart in this activity. Download your Sun Chart diagram as a PDF file. You have seen examples of this in the videos included in this Lesson. You will use this chart further to put your shading information on.

2. Apply shading. Study the “Array Packing” example in Chapter 7 of the J. Brownson’s SECS Book, understanding how shading coordinates can be obtained for a specific system geometry. Plot on your Sun Chart all the critical points by shading coordinates given in Table 7.3. List of critical points to plot is also provided in Canvas for your convenience. Connect the points with a smooth line to show the shaded zone on your diagram (see lesson videos and textbook for more guidance on how to do it). You can use a graphical software for this purpose or do it by hand if that is easier.

   NOTE: the solar azimuth coordinates used by SunChart are different from those Brownson's book. The former considers 180^o as true south, while the latter considers 0^o as true south. So, you would need to convert the solar azimuths from the book by adding 180 in order to correctly plot the points on the SunChart.  

3. SAM Simulaton A (you have already downloaded SAM as instructed in Lesson 1):

   • Start a new PV project in SAM – choose Detailed PV Model - Residential Owner option when initiating a project
   • Under Location and Resource tab, input Baltimore, MD
   • Under System Design tab, input specifications for the system (4 kW nameplate capacity, array tilt and azimuth)
   • For the first round, do not include shading data
   • Run the simulation by pressing the blue 'Simulate' button at the bottom left of the screen
   • To get the monthly energy data in table, click on  DataTables tab on top. Then choose output variables: you need Monthly Energy (KWh) and total Annual Energy (kWh) for a single year. You can also find the latter by just summing up your monthly values.
   • Export your table to an Excel spreadsheet
4. Sam Simulation B. Repeat the same simulation as described in Step 3 now including the shading data. This is how: Save your project as a new file. Under 'Shading Layout' tab, click 'Edit Shading' button. Then, in the editing window, choose the option ‘Enable month by hour beam irradiance shading losses’. You will be given a table to input shading factors for each hour in each month. For simplicity, we will consider the same (averaged) shading factor for all days in the same month. Looking at your Sun Chart, input 0 for no shading, 100 for full shade, and 50 for borderline or changing conditions. Once done, run the SAM simulation and export data table as described in Step 3. (See demo video in Canvas for this step)

5. Discussion. Provide a brief discussion comparing your shaded and non-shaded results and what you learned from this activity. 

Submitting Your Work

Prepare a written report including the following results: 

• Sun Chart for Baltimore, MD with critical points and shaded zone plotted on it.
• Table of SAM simulation results for full sun and shaded scenarios. This table should contain monthly energy generated as well as total annual energy delivered by the array. In your energy performance table, your numbers should be rounded to the nearest whole kWh (no decimals).
• Screenshot of shading factor table from SAM.
• A paragraph with a discussion of your results.

Upload your report file to the Lesson 2 Learning Activity DropBox in PDF or docx format.

Grading Criteria

• Sun Charts (10 pt). This part will be graded based on your ability to plot the calculated critical points and identify shaded versus non-shaded conditions for the system at the specific time and location.
• Energy Tables (10 pt). You will be given 1 pt per month based on the reasonable correctness of the numerical values. The purpose is not necessarily to get the exact "right" answer, but rather to have done the process and understand what you are doing. So discrepancies within +/- 30 kWh will not cause point deductions.
• Discussion (5pt).
• Screenshot of the Table with shading factors used for SAM simulation (5 pt). 

Deadline

See the Canvas Calendar for specific due dates.

Summary

So far, we have delved deep into the world of angles, time, and shading. Why have we done this? The solar resource is all about timing and placement of the SECS relative to the Sun, and the use of angles as coordinates in time and place are fundamental building blocks to the solar expert. We now have computational tools that make the work of calculating details trivial, but one must always know what the foundations are underneath those tools.

Who knows, maybe you will be the next open software developer to create a simple solar tool for resource assessment. You now have the keys to access many of those computational tools. More to the point, these fundamentals are not shrouded in mystery, and you don't necessarily need to pay for software to get access to them!

Looking ahead, I can tell you that students in residence at Penn State have already found a solar systems design plugin for Trimble SketchUp [29] called Skelion [30], which does shading analysis just like you performed, but on the fly inside of the SketchUp design software. Both are free to use, and I'd recommend you take a look at each in preparation for your end of semester design projects! This is an untapped resource for you to explore, now that you know "how" the shading calculations work.

Knowing these principles of angles and time also exposes us to the strengths and potential weaknesses of purely geometric relations in the solar resource. As we shall see in the next lesson, the role of the atmosphere and meteorology will tend to muck up our ideal angles of beams of light, and produce anisotropic (uneven) intensities of irradiance over the day, which will of course influence our solar resource estimations.

On to the next lesson--let's learn about the role of weather and the sky dome in estimating the solar resource!

The Goal of Solar Design and our Three New Mechanisms

The Goal of Solar Design is to:

1. Maximize the solar utility
2. for the client
3. in a given locale.

Given the goal of solar design, we have learned of three mechanisms to leverage during the design process that will increase the solar utility for our client in their locale of interest:

1. Reduce the cosine projection effect on an aperture/receiver (the extreme angles of incidence, $\theta$ or low glancing angles $\alpha$),
2. Reduce the angle of incidence on an aperture/receiver (refinement on minimizing $\theta$),
3. Reduce losses from shadowing on an aperture/receiver.

In order to leverage these three tools, we have demonstrated that one needs to know where and when the Sun will be relative to our collector system.