3.1 Introduction

Modules and arrays present photovoltaic (PV)’s face to the world, as well as the sun, and the technology’s reputation depends crucially on their technical performance, reliability, and appearance. They must be designed and manufactured for a long and trouble‐free life. The solar cells they contain need careful encapsulation to provide mechanical strength and weatherproofing, and the electrical connections must remain robust and corrosion‐free.

Most PV modules are provided with aluminum frames to give extra protection and simplify mounting on a roof or support structure. Modules without frames, known as laminates, are sometimes preferred for aesthetic reasons, for example, on the façade of a building where reflections from metal frames would be unwelcome. A group of interconnected modules working together in a PV installation is referred to as an array. We mentioned PV modules briefly in Section 2.1, noting the module areas required for a given power output using different cell technologies, and discussed cell and module efficiencies. In this chapter we will focus mainly on electrical characteristics and effective mounting to capture the available sunlight. But first, a few words about module sizes and designs.

For a given level of solar cell efficiency, the rated power output of a module is proportional to its surface area. As we noted in Section 2.1, about 7–8 m² of surface area is required to generate 1 kW_p using crystalline silicon cells, about 16 m² using amorphous silicon, and intermediate areas for thin‐film technologies such as CIGS and CdTe. As more and more PV installations including power plants move into the megawatt range, huge arrays and module numbers are involved. For example, the 550 MW power plant, part of which is shown in Figure 2.3.4, uses more than 5 million CdTe modules each of 0.72 m² area. Recently, First Solar, the manufacturer of these modules, announced a new product, with three times larger area, to speed up installation. The larger the size of a module, the lower the number of electrical interconnections and the associated mounting costs. In response PV manufacturers have steadily increased module sizes and power ratings that now range up to several hundred peak watts. However, such advances must be weighed against the difficulty of handling the larger, and therefore heavier, modules. In addition, for thin‐film PV, the module size is also constrained by the challenge of depositing high‐quality semiconductor films over large areas.

Figure 3.2 shows a cutaway view of the edge of a typical module containing crystalline silicon solar cells. The cells, which are brittle, are cushioned by encapsulation between two layers of ethylene vinyl acetate (EVA) and adhesive that also holds the module parts together. On top is a cover of tempered glass that is sometimes treated with an antireflection coating (ARC) to maximize light transmission. Underneath is a sheet of Tedlar, a light synthetic polymer, acting as a barrier to moisture and chemical attack. The whole “sandwich” is located in a slot in the aluminum frame and fixed with sealant. Other PV module types (e.g., CdTe) are sandwiched between two sheets of glass and do not have frames. This construction must withstand up to 25–30 years of outside exposure in a variety of climates that include desert sands, alpine snows, wind, rain, pollutants, and extremes of temperature and humidity—a highly demanding specification. When things go wrong, it is often due to ingress of moisture or corrosion of electrical contacts rather than faults in the solar cells.

PV module design is by no means static, especially with the new thin‐film technologies. In most cases the films, deposited on glass or other substrates, are scribed to produce the complete pattern of solar cells and interconnections, avoiding the need to handle and mount individual semiconductor wafers. Not only does this reduce manufacturing costs, but it also promises improved electrical reliability within the module. Another area where thin films are having a big impact is in flexible products. The historic market dominance of rigid, relatively heavy glass‐covered modules is increasingly challenged by flexible lightweight designs more easily tailored to the shapes of awkward roofs and unusual structures or the aesthetic demands of architects.

3.2 Electrical Performance

3.2.1 Connecting Cells and Modules

Individual solar cells are hardly ever used on their own. A cell is essentially a low‐voltage, high‐current device with a typical open‐circuit voltage of around 0.5 V, far lower than the operating voltage of most electrical loads and systems. So it is normal for a PV module to contain many series‐connected cells, raising the voltage to a more useful level. For example, many manufacturers offer modules with 36 crystalline silicon cells connected in series, suitable for charging 12 V batteries. These modules have an open‐circuit voltage V_oc of around 20 V and a voltage at the maximum power point (MPP) V_mp of about 17 V, giving a good margin for battery charging, even in weak sunlight. As PV moves increasingly toward high‐power grid‐connected systems, the trend is for more cells per module giving higher output voltages—for example, the modules previously shown in Figure 2.1 each contain 72 cells producing about 35 V at the MPP.

Of course the peak power of a module is also one of its key characteristics. The surface area of a monocrystalline silicon solar cell is limited by the diameter of the original ingot, which in turn restricts its power output. Many cells must therefore be interconnected to produce substantial module power: for example, 72 cells, fitted into a module of about 1.5 m² area, can yield about 200 W_p. Multicrystalline silicon cells, being cut from large cast blocks of silicon, are less restricted in area; thin‐film cells are even less so. But the modules must still incorporate many cells to achieve useful voltage levels and have a substantial surface area to give a reasonable power output.

What happens when many cells are connected in series? The answer would be very straightforward assuming that all the cells are identical and exposed to the same strength of sunlight: with n cells in series, the module voltage would be n times the cell voltage, and the module current would be the same as the cell current. But in practice cells are not functioning exactly the same. There could be small manufacturing tolerances and small temperature differences, depending on where cells are located in the module. If a module becomes partially shaded by buildings or trees, some cells receive more sunlight than others. In all cases the module’s output is limited by the cell with the lowest output—the “weakest link in the chain.” The resulting loss of power is referred to as mismatch loss.

Small mismatch losses are to be expected in commercial modules and are covered by manufacturing tolerances. They need not normally concern us. But significant losses can easily be caused by shading, which should obviously be avoided where possible. The situation can worsen dramatically if one cell in a string becomes truly “bad” and fails to generate current. It then acts as a load for the other cells and starts to dissipate substantial power, which can lead to breakdown in localized areas of its p–n junction. Severe local overheating occurs, possibly causing cracking, melting of solder, or damage to the encapsulating material. This is known as hot‐spot formation.

The hot‐spot condition is illustrated in Figure 3.4. At the top in part (a) is a string of n cells of which n − 1 are “good” and one is “bad.” The string is shown short‐circuited, which is the worst‐case scenario. Since the cells are in series, the current must be the same for all. But whereas the good cells happily generate a solar current I_L, the bad cell cannot do so and is forced into reverse bias. With the string short‐circuited, the bad cell is subjected to the full voltage and power output of the good cells, leading to breakdown and hot‐spot formation.

Part (b) of the figure clarifies the situation from a circuit point of view. The complete current path is shown in red. Each good cell is represented by the simple equivalent circuit previously shown in Figure 2.9(a), consisting of a semiconductor diode in parallel with a current generator. Since the bad cell’s current generator is inactive, the circuit current I_L must pass through its diode in a reverse direction (of course, a diode is not supposed to pass reverse current; but in this case the voltage produced by all the good cells in series is sufficient to cause breakdown and make it conduct). The power produced by the good cells must now be absorbed by the bad cell since none is dissipated in an external load. In conditions other than short circuit, the situation is less severe, but hot spots may still occur.

Hot‐spot failure is avoided by incorporating additional diodes known as bypass diodes, shown in part (c). Here red indicates circuit elements carrying the main current, green indicates inactive (but serviceable) elements, and blue indicates the bad cell. The bypass diodes offer an easy current path around any bad cells. Ideally there would be one bypass diode for each solar cell, but this is rather expensive. Many PV modules therefore incorporate one diode per small group of cells. In our example there are three cells per group. The disadvantage is that each bad cell takes two good cells out of action, preventing them from contributing to the power output of the string. In practice the maximum number of cells in a group that prevents damage is normally reckoned about 10. Bypass diodes are often built into commercial modules, but if they are not, care should be taken to avoid short circuits, especially when there is partial shading by trees, buildings, or other structures.

Many of the ideas we have developed for solar cells also apply to modules. For example, a module, like an individual cell, may be characterized by its open‐circuit voltage, short‐circuit current, and MPP. Indeed we may think of a complete module as a type of “supercell” with higher voltage and power ratings.

Modules are connected together—sometimes in large numbers—to form arrays. Whereas the cells in a single module are usually series‐connected to raise the voltage as much as possible, modules in an array may be connected in series, parallel, or a mixture of the two. This is illustrated by the array of six modules shown in Figure 3.5, consisting of two strings in parallel to amplify the current, each string containing three modules in series to amplify the voltage. According to Ohm’s law (I = V/R), if the modules are perfectly matched, this arrangement produces an array voltage three times the module voltage, an array current twice the module current, and an array power six times the module power.

In practice the array performance will be slightly reduced by mismatch losses between the various modules. It is also worth noting that modules from different manufacturers should not be mixed together in an array, even if they are nominally similar, because differences in I–V characteristics and spectral response are likely to cause extra mismatch losses.

The figure also shows a number of diodes. Those colored green are bypass diodes, one in parallel with each module to provide a current path around the module if it fails or becomes “bad.” The two diodes colored red are referred to as blocking diodes, one in series with each string to ensure that current only flows out of the modules. They are generally used in battery charging systems to prevent the batteries from discharging back through the modules at night. Some manufacturers include blocking diodes within their modules.

An array installed on a domestic roof might typically contain 10–20 modules. A single module is normally sufficient for the type of solar home system (SHS) that provides modest amounts of electricity for a family in a developing country. But the huge PV systems and power plants now being installed in industrialized countries incorporate hundreds of thousands of modules and involve major decisions about how they should be interconnected. A complete PV generator is, of course, a DC generator, and since most large systems feed their solar electricity into an AC grid, it is important to design arrays that can be interfaced safely and efficiently. We will return to this topic in the next chapter.

3.2.2 Module Parameters

Not surprisingly, most of the electrical parameters of a PV module closely reflect those of its solar cells. However the efficiency of a module, measured in standard conditions of bright sunlight (1000 W/m² at 25°C, AM1.5 spectrum), is slightly less than that of the constituent cells because the cells do not completely fill the module’s area and there are small power losses as sunlight passes through the top cover and encapsulant. If blocking diodes are built in, these too will produce small power losses. Average (measured under standard conditions) efficiencies for the most widely used terrestrial modules are 18% for monocrystalline silicon, although modules with efficiencies as high as 21.5% are also in the market, 17% for multicrystalline silicon, approximately 15% for CIS, and 16.5% for CdTe; efficiencies of a‐Si PV are below 10%. Note that these module efficiencies reflect second quarter of 2017 and module efficiencies may be higher by the time you read this text as the PV industry continues to invest in R&D. But as we have emphasized previously, efficiencies in bright sunlight do not tell the whole story. Crystalline silicon modules tend to lose their advantage in weak or diffuse light, or high temperatures, and there is accumulating evidence that the newer thin‐film modules may produce higher annual yields in regions with substantial cloud cover.

We have already described monocrystalline silicon solar cells in some detail and illustrated typical I–V characteristics for a 2 W_p cell at various levels of insolation in Figure 2.10(b). As we pointed out in the previous section, the cells in a module are normally series‐connected, raising the power and voltage (but not the current). As an example we now consider a module containing 72 monocrystalline cells with a peak power rating of 180 W_p—a popular size and power rating. The following module parameters are at the low end of the range of commercial modules in the market today:

Nominal power	180 Wp
Open‐circuit voltage	43.8 V
Short‐circuit current	5.50 A
Voltage at maximum power	35.8 V
Current at maximum power	5.03 A
Power reduction per °C	0.45%
Voltage reduction per °C	0.33%
Length	1600 mm
Width	804 mm
Weight	18 kg
Efficiency	14.0%

As expected, the nominal power of 180 W_p equals the product of voltage and current at the MPP. The efficiency is given by the module power in kW_p divided by the module area in square meters. And the reduction of power at elevated cell operating temperatures, 0.45% per °C, is typical of crystalline silicon—and more serious than for thin‐film technologies. It means, for example, that if the cell temperature is allowed to rise to 65°C, the power output will fall by about 18%, emphasizing the need to keep this type of module as cool as possible with adequate ventilation. For comparison, the loss of power due to heat buildup in CdTe PV is 0.34% per °C and the voltage reduction is 0.29% per °C.

A family of I–V curves for the module is shown in Figure 3.6. Their form is very similar to those for the 2 W_p cell of Figure 2.10(b). The top curve, labeled 1000 W/m², refers to standard insolation and corresponds to the parameters in the previous table. The other three curves confirm that as the level of insolation reduces, the current falls in proportion. Each has its own MPP, labeled P₁–P₄, the operating point at which maximum power output may be obtained.

Other aspects of module performance closely mirror those of the constituent solar cells. For example, since the module is effectively a current source, the actual power output is closely proportional to the voltage at which it is operated, up to the MPP, and the main effect of temperature rise on the I–V curves is a reduction in open‐circuit voltage. You may like to refer back to Figures 2.11 and 2.12 for a discussion of these points.

We have so far concentrated on monocrystalline silicon and you may be wondering how module parameters and I–V characteristics differ for other cell technologies. In fact the differences are very slight for multicrystalline silicon, the main effect being a small decrease in module efficiency (and increase in module area) for a given power rating. The sizes and power ratings of both types of crystalline silicon modules have tended to rise steadily in recent years; most current production is in the range 150–300 W_p. In part this is due to advances in manufacturing processes to satisfy a market increasingly slanted toward large grid‐connected systems. The first commercial module to achieve 500 W_p appeared a few years ago.

When we come to the newer thin‐film modules, the situation changes in two main respects. First, manufacturers generally start testing the market for new cell technologies with relatively small modules. It takes a lot of skill and experience to produce large modules with consistent performance and reliability. CIS and CdTe modules, for example, have worked up from small to medium sizes over a number of years, but maximum power ratings are still behind those of monocrystalline silicon. Large a‐Si modules, a more mature technology, offer power levels into the hundreds of watts, but at amorphous silicon’s relatively low efficiency. Second, the manufacture of thin‐film cells offers considerable flexibility over module voltages. Cells can be scribed in a wide range of sizes within a module; a few large cells give high currents at low voltages, while many small cells give high voltages at low currents. Higher voltages are often preferred for interfacing to an electricity grid and for reducing cable losses when modules are interconnected, a significant factor in large PV systems.

Figure 3.7 illustrates this flexibility with I–V characteristics for four commercial modules based on different technologies; multicrystalline silicon (mc‐Si), CIS, CdTe, and a‐Si. The modules all have the same nominal peak power of 75 W_p but very different voltage and current levels. The curves are all drawn for standard insolation (1000 W/m²), and the MPP for each module is shown by a dot. The figure is meant to be indicative, not definitive, and the situation is of course developing rapidly. For example, the latest Series 4 CdTe PV rated at 117.5 W (16.3% efficiency) gives at PMax both higher voltage and current, 71.2 V and 1.65 A correspondingly. The main point is the increasing range of power levels offered by thin‐film deposition and scribing techniques. Chapter 6 on PV manufacturing discusses processes used by the industry to increase current and voltage of solar cells.

3.3 Capturing Sunlight

The sun is a tremendous fusion reactor, with a volume 1.3 million times larger than of our Earth, at a safe distance from the Earth (Figure 3.8). Forty minutes of sunshine striking the Earth on a clear day is equal to 1 year of global consumption of energy.

Solar radiation at normal incidence received at the surface of the Earth is subject to variations due to change in the extraterrestrial radiation and to two more significant phenomena: (i) atmospheric scattering and (ii) atmospheric absorption by O₃, H₂O, and CO₂. The effects of sunlight scattering and absorption on the spectral distribution of solar beam irradiance were shown in Figures 1.8 and 1.10. Absorption and scattering levels change as the constituents of the atmosphere change. Clouds are the most familiar example of change; they can block most of the direct radiation. Also seasonal variations and trends in ozone layer thickness have an important effect on terrestrial ultraviolet level. Fortunately not only the direct light from the sunlight but also the light that diffuses into the atmosphere and the light that is reflected from surfaces can be useful for creating electrons in photovoltaics.

Also, as discussed in Chapter 2, solar cells respond differently to the different wavelengths, or colors, of light, depending on the energy bandgap of the semiconductor materials they employ. For example, crystalline silicon can use most of the visible spectrum, plus some part of the infrared spectrum. But energy in part of the infrared spectrum, as well as longer‐wavelength radiation, is too low to produce electrons. Higher‐energy radiation can produce electrons, but much of this energy is likewise not usable. In summary, light that is too high or low in energy is not usable by a cell to produce electricity and is converted into heat.

Now to measure all this sunlight, we need instruments that are sensitive across the whole solar spectrum, and they can measure all the components of solar irradiation; as we discussed in Chapter 1, these are the direct (beam), diffuse, and albedo (reflected on the ground) solar components. Their total is called global horizontal irradiation (GHI) as is typically measured on horizontal surfaces by instruments called pyranometers. The name is derived from the ancient Greek words pyr “fire” + ouranos “sky,” implying an instrument that measures the strength of the sun’s “fire” from the sky. A pyranometer uses a large number of thermocouples that are connected in series in a pile facing upwards, with each absorbing a part of the solar spectrum. It thus measures voltage induced by thermal energy absorbed at the tip of each thermocouple. A double‐glass dome prevents condensation and reduces heat losses, while its hemispheric shape allows the global irradiation to enter with minimum reflections. Many pyranometers have a cartridge of silica gel to absorb moisture, and this need to be replaced periodically. Also the sensitivity of a pyranometer can decline with time, and it may need to be recalibrated every 2 years or so.

Since pyranometers are rather expensive (costing $1500–2000), irradiation sensors made of solar cells, called reference cells, are an economic alternative. These are precisely calibrated silicon solar cells that are short‐circuited, so the short‐circuit current is converted into a voltage that is being measured in conjunction with temperature measurements with an enclosed thermocouple. Their cost is low, ranging from $150 to 500, but their accuracy is not better than ±5% or ±10%.

When using solar cells with concentrators on sun‐tracking devices, we need to measure the direct (normal on the surface of the PV module) irradiation (DNI). DNI is measured by instruments called pyrheliometers, which track the sun. The name is also derived from the ancient Greek words pyr “fire” + helios “sun” + meter “measure.” Similarly to the pyranometers, sunlight enters the instrument through a window and is directed onto a stack of thermocouples that convert heat to voltage. The signal voltage is amplified and is reported as power per area (W/m²). In contrast to a pyranometer, a pyrheliometer has a very small window allowing only direct sunlight to enter, and it is used with a solar tracking system to keep the instrument aimed at the sun.

Albedo, also called solar reflectance, is defined as the ratio of the reflected to the global irradiation. The solar albedo depends on the directional distribution of incoming radiation and on surface properties at ground level. Albedos of typical surfaces range from about 4% for asphalt and 15% for grass to 90% for snow. This can be measured by an albedometer composed of two identical pyranometers, one facing the sky and one facing the ground (Figure 3.9). Albedo is especially important if the modules are composed of bifacial solar cells; thus solar cells can absorb sunlight from both sides. When conventional PV modules are placed horizontally or with small angles to the horizontal, then the albedo contribution is negligible.

The performance, thus the cost‐effectiveness, of a PV system depends crucially on positioning its solar array to capture as much sunlight as possible. We must therefore appreciate how the sun’s apparent path across the sky varies according to the time of year and the latitude of the site. Of course the variation is the result of the rotation of the Earth around its axis and its trajectory around the sun. Figure 3.10 shows this trajectory, whereas Figure 3.11 shows how a stationary observer on the northern hemisphere sees this trajectory in daily cycles. The sun appears to be at its lowest height from the ground at the winter solstice, around December 22 in the northern hemisphere, and at its highest at the summer solstice, around June 21. In between are the two mid‐season equinoxes around March 22 and September 23, when the sun rises due east and sets due west, giving equal hours of day and night.

The declination shown in Figure 3.10 is the angle between the equator plane and the line connecting the centers of the Earth and the sun. It varies seasonally due to the tilt of the Earth on its axis of rotation and the movement of the Earth around the sun. If the Earth was not tilted on its axis of rotation, the declination would always be 0°. However, the Earth is tilted by 23.45° and the declination angle varies between −23.45° and +23.45°. Only at the spring and fall equinoxes is the declination angle equal to 0°.

The high point is always reached at solar noon when the sun is in the south. In winter, sunrise occurs south of east and sunset occurs south of west; in summer both veer toward the north. In other words the angular span as well as the height of the trajectory varies with the season. This all applies equally well to the southern hemisphere if we swap north for south and interchange the dates of the winter and summer solstices.

Time measured by the apparent motion of the sun is called solar time and fluctuates slightly around the time given by a conventional clock. This is because the Earth’s journey around the sun is slightly elliptical and our distance to the sun varies with the time of year. In this chapter we always refer to solar time (e.g., solar noon) because we are, of course, interested in the apparent movement of the sun in relation to PV installations on Earth. However, the time system used in our everyday lives and shown by our clocks and watches averages out the fluctuations to make every day of equal length and is referred to as mean time. The best‐known example is Greenwich Mean Time, being the time measured at the Greenwich Observatory in London. This is on the prime meridian, reckoned as 0° longitude, thus the longitude where we start measuring east and west. Meridian originates from the Latin word medius “middle” + dies “day.” The use in astronomy is due to the fact that the sun path crosses a local meridian at noon. Fortunately the difference between local solar time and mean time, described by the so‐called equation of time, never exceeds about 17 minutes at any time of year. This is only really significant when designing PV systems that use highly concentrated sunlight and must track the sun very accurately across the sky. We shall meet them later in the chapter.

Latitude also has a big effect—the further we are from the equator, the lower the sun’s path through the sky. On the equinox dates, its elevation angle, also called solar altitude, above the horizon at solar noon, labeled a in Figure 3.11, is equal to 90° minus the latitude. For example, in Madrid, latitude 40°N, the solar altitude at noon on March 22 and September 23 is 50°; in more northerly Berlin, latitude 52°N, it is 38° (and at the North Pole, latitude 90°N, the sun is on the horizon). At the summer solstice, June 21, the noon elevation increases by an angle δ equal to 23.45° and is at its annual peak. At the winter solstice it is reduced by the same amount. In general, a = 90° + δ – ϕ

Thus, at a location of latitude ϕ,

(3.1)

So in Madrid the summer and winter solstice elevations are 73.45° and 26.55°, respectively (intrepid explorers at the North Pole for the winter solstice, in total darkness, are perhaps unaware that the sun is 23.45° below the horizon). These seasonal variations are caused by the offset angle between planet Earth’s axis of rotation and its plane of revolution around the sun.

3.3.1 Aligning the Array

In the previous section we saw how the sun’s trajectory varies according to the time of day, the season, and the latitude. This information suggests how a fixed PV array should be aligned to capture as much sunshine as possible. First, for installations in the Northern hemisphere it should point due south toward the midday sun. In some cases, for example, on existing buildings, this may not be possible, but any deviation from south should preferably not exceed about 30°.

Second, the array should be tilted down from the horizontal so that the sun’s rays at solar noon are normal to its surface. Since the sun’s noon elevation varies continuously through the year, a choice has to be made about when to meet this condition. Very often the two equinoxes (on or around March 22 and September 22) are selected, giving the PV tilt shown in Figure 3.11.

On these two dates the array points “perfectly” at the midday sun but is somewhat too low in the summer and too high in the winter—normally a good compromise. This is shown in Figure 3.13.

In the previous section we noted that the sun’s noon elevation at the equinoxes (denoted by angle a in the figure) equals 90° minus the latitude of the site. It follows that its declination (δ) is equal to the latitude and that the array must be tilted down by this amount (Figures 3.14 and 3.15). For example, in Madrid, latitude 40°N, an array must be tilted down 40° to meet the aforementioned condition. It will then point too high by angle δ (23.45°) at the winter solstice and the same amount too low at the summer solstice.

The aforementioned “equinox criterion” for tilting a PV array is widely adopted, but, as we shall see, it is not essential. Minor variations of tilt angle have very little effect on an array’s annual yield, and in any case there are some situations where a different choice of tilt may prove beneficial. A good example is a stand‐alone PV system in high latitude, required to provide a steady supply of electricity throughout the year. The winter months are the most difficult and will determine the size of array required, for if the system is able to cope in the winter, it will certainly do so in the summer. So the downward tilt of the array is often increased to make the most of winter sunshine. Unusual climatic conditions may also favor different amounts of tilt, for example, in parts of South Asia typified by hot humid summers with overcast skies followed by clear cool winters with plenty of sunshine. If it is required to maximize the annual electricity yield of a grid‐connected system, a larger downward tilt may well be helpful. Conversely, in a system required to optimize electricity yield in the summer months, the PV array may be aligned closer to the horizontal. A good example is the summer holiday home described in Section 5.4.1.

In the case of rooftop arrays, tilt is often predetermined by roof geometry, giving little or no flexibility. However it is worth noting that buildings in high‐latitude countries such as Norway and Sweden often have high‐tilt roofs to encourage snow to slide off easily, whereas roofs in Morocco or Egypt are much more likely to be flat, or nearly so. In this way vernacular architecture tends to suit the sun’s trajectory and the preferences of PV system designers.

So far we have concentrated on capturing as much of the sun’s direct radiation as possible. This is certainly important, but, as our discussion of the solar resource in Section 1.2 made clear, there is also diffuse and albedo radiation to consider (see Figure 1.10). What happens when we start considering the actual radiation falling on a PV array, taking into account scattered light?

You may find it helpful to refer back to Figure 1.7 showing the large‐scale effects of climate on insolation at the Earth’s surface. In temperate regions with plenty of “cloudy‐bright” weather, the diffuse component can make a surprisingly large contribution to the annual total; for example, in Western Europe it is often over 50%. One effect is to make array alignment less critical because diffuse light tends to come from all over the sky. Another is to reduce the overall importance of shadows in determining the annual energy yield.

To narrow these general ideas down to a particular PV system, we need more detailed local information. Fortunately, the great surge of interest in solar energy in recent years has spawned data on average sunlight conditions for many cities and locations around the world (one valuable source of information is provided by NASA¹). The data is often presented in the form of 12 monthly mean values of global (direct and diffuse) daily radiation on a horizontal surface, expressed in kWh/m². Albedo radiation is not included since it does not affect horizontal surfaces and anyway is highly site dependent. Sometimes the proportions of direct and diffuse light are found by practical measurements with specialized instruments; sometimes they are inferred from the global figure and a clearness index summarizing the amount of light scattering caused by clouds and particles in the local climate. Figure 3.16(a) shows a typical distribution for a West European city such as London or Amsterdam with a temperate climate giving plenty of “sunshine and showers” in summer and cloudy skies in winter. The height of each bar represents global radiation, composed of direct (yellow) and diffuse (orange) components. The daily average over the whole year is about 2.8 kWh/m², giving an annual total of about 1050 kWh/m². Part (b) of the figure is for the Sahara Desert. Here the extremely sunny, hot, and reliable climate produces a daily average of about 6 kWh/m² and an annual total of about 2200 kWh/m² on a horizontal surface. Most of the radiation is direct.

Given such figures it is quite easy to make a rough estimate of the annual output from a PV module or array using the concept of peak sun hours. If the energy received throughout the year is compressed into an equivalent duration of standard “bright sunshine” (1 kW/m²), then the number of peak sun hours is the same as the global annual figure. For example, London has about 1050 peak sun hours in a year, so a PV module rated at 200 W_p can be expected to produce around 1050 × 200 = 210 000 Wh = 210 kWh/year. However this is for a horizontally mounted module—unlikely in London. It also assumes ideal “bright sunshine” conditions, whereas much of London’s sunlight is diffuse. Direct light and diffuse light have different spectral distributions, and solar cells do not generally respond equally to them, nor are most cells equally efficient in bright and low‐level light. So estimates of array output based on peak sun hours are only very approximate, especially for locations with a large proportion of diffuse light. In most cases actual annual yield is considerably lower. For example, our 200 W_p module in London is more likely to produce 150–160 kWh/year.

It must be emphasized that distributions such as those in Figure 3.16 are normally based on data collected over many years. In a given year, especially in unpredictable climates, they may look very different; it is not unusual to see a 10% variation in annual figures or a 30% variation in monthly ones, and this must be taken into account when making predictions or designing a PV system.

We have so far considered sunlight falling on a horizontal surface. What happens when a PV array is tilted downward to take account of the latitude? How are the figures for global, direct, and diffuse radiation affected? Below we discuss these effects; the methodology and equations for determining the irradiation components on tilted surfaces will be presented later in the Appendix 3.A.

Figure 3.17 shows some estimates for south‐facing tilted PV panels in London (latitude 52°N) and the Sahara Desert (latitude 24°N). In each case three different tilt angles are illustrated: 0° (horizontal), shown by blue bars; an angle equal to the latitude, shown by red bars; and 90° (vertical), shown green. You may be surprised at the choice of 0° and 90°, but actually results for angles closer to the latitude are often almost indistinguishable. And remember that horizontal PV arrays may be installed on flat roofs and vertical ones on building façades (Figure 3.18). The results illustrate several interesting points:

• London: The results for 0° and 52° tilt are quite similar, mainly due to the large proportion of diffuse sunlight, but 0° receives slightly more radiation in the summer and less in the winter; 90° tilt (as on a vertical building façade) receives much less irradiation in the summer months.
• Sahara Desert: The results for 0° and 24° tilt are very similar since we are much closer to the equator. The really big effect is the reduction in radiation for a 90° tilt in summer; when the sun is high in the sky, the radiation is almost all direct and the surface receives very little of it.

Of course such graphs are estimates that cannot take account of fine variations in local climate—for example, different amounts of cloud and shade. As PV enters the multi‐gigawatt era, with systems of all shapes and sizes installed around the world, system designers will no doubt have access to ever more performance data collected from working systems.

3.3.2 Sunshine and Shadow

When positioning a PV array, it is very important to avoid shadows as far as possible for two main reasons. Shading can greatly reduce the output of the modules, and in severe cases it runs the risk of hot‐spot formation. What may be termed “occasional shadows” caused by bird droppings, dust layers, or snow on PV modules can obviously be reduced by proper maintenance and cleaning. The situation becomes more complicated if shadows are cast by nearby obstructions. For example, PV roofs may be partially shaded by dormer windows, satellite dishes, chimneys, or ventilation pipes; a small, ill‐positioned pipe at 2 m can cause more trouble than a skyscraper at 2 km! Of course small local obstructions should be easier to control and perhaps eliminate. A newly designed roof should always take special care to avoid them.

The degree of shading at different times of year depends upon the sun’s trajectory and may be assessed by recasting Figure 3.11 in two‐dimensional form, called sun path chart, and adding the outlines of buildings, trees, and high terrain that cast shadows over the PV array. Thus a sun path diagram is a graphical representation of the sun’s altitude and azimuth angles during the solstice and the equinox dates; often it also shows average monthly sun paths. Figure 3.19 shows such a diagram for latitude 40°N, relevant to world cities including Madrid, New York, and Beijing, and shading obstructions on a particular site. It shows that there are detrimental shading effects in the winter, but shading is not significant during the rest of the year. Sun paths, thus diagrams of the solar altitude α, can be created from the latitude ϕ, the sun declination δ, and the hour angle ω according to the equation

(3.2)

In practice, predicting shading effects requires placing a panoramic image of obstructions on a solar path diagram correctly aligned for the location, day, and time of taking the image. There are a number of devices on the market for predicting potential shading problems, ranging from handheld viewers to photographic instruments supported by computer software.

The simplest device is the Solar Pathfinder, which has a translucent and highly polished convex plastic dome set on top of a solid base and a compass. The user looks down onto the dome to see panoramic reflection of the obstructions restraining the sunlight at the location. A sun path paper diagram representing the sun path for the location is placed underneath the dome. The sun path diagram then is rotated to its local declination and is pointed to the magnetic south. Slots in the side of the dome allow the user to trace the outline of the reflected obstructions onto the diagram; alternatively a photograph can be taken and superimposed on the sun path diagram. This way we can determine the obstructions that will shade the selected installation site and the times when shading may occur. Figure 3.20 shows the top of the instrument and the outline of the traced obstructions; shading occurs above the white line of the shade tracing shown in the figure on the right.

When shading is unavoidable, it may be possible to reduce its effects by careful planning of module interconnections in the PV array. As we noted in the previous section, a single “bad” or shaded cell in a series‐connected string affects all the other cells and can seriously reduce the string’s output. The same applies to strings of modules. So it is important to try and prevent one module in a string from becoming shaded at the expense of the others. And if a shadow is large enough to fall on several modules at the same time, it is best if all are members of the same string.

These ideas are illustrated in Figure 3.21 for a PV roof containing 30 modules arranged as 10 parallel strings of three modules each. The various strings, indicated by different green tints, have been arranged to take account of two shadows, S₁ and S₂. The first of these is a narrow stripe formed by a nearby electricity pole, and its effects are reduced by a vertical arrangement of modules. As the shadow moves laterally in line with the sun, it is mainly confined to a single string and affects three modules equally. The second shadow, cast by a neighboring tree, is roughly triangular in shape and falls on the lower right‐hand corner of the array. Assuming trees and neighbors are to remain undisturbed, modules may be connected in triangular strings. Once again this minimizes the number of strings affected by the shadow as it moves onto the corner of the roof. And finally there is an unfortunate chimney pipe, labeled c, near the middle of the roof that cannot be moved. Its nuisance value is reduced by a string of “dummy” modules, shown white, which preserves the array’s appearance but avoids using expensive real modules that would produce little electricity. In practice it would probably be economical to connect all the unshaded modules in one or two longer strings, a matter we shall return to later.

Although such “array design” is only partly effective, it is virtually cost‐free—an important benefit since recurring shadows can degrade an array’s output over its entire working life. Finally, it is worth noting that a number of manufacturers now offer module‐level power equipment (MLPE) technologies as an alternative to conventional “string” inverters/power optimizers. These technologies aim to reduce shading and mismatch losses by allowing each module in a string to operate at its MPP, regardless of what the others are doing. In principle such devices should be able to overcome many of the shading problems discussed previously. There are essentially two MLPE options that you may come across if you are shopping for a solar system: microinverters (separate inverters for each module) and power optimizers. Both solutions offer similar benefits: better performance when your solar panels are in shade for part of the day. At the same time, solar systems that use these technologies tend to be more expensive than those that use string inverters.

3.4 One‐Axis Tracking

Recently, with the evolution of high‐efficiency modules, utilities started preferring one‐axis tracking systems for ground‐mount installations in sunny regions. One‐axis tracking systems follow the sun throughout the day to increase energy output and are expected to generate up to 25% more energy than fixed latitude tilt systems. Furthermore, they provide a much flatter power output and can satisfy more of the afternoon peak load than fixed‐tilt modules. It appears that they are cost‐effective in many high irradiation locations. The only disadvantage is a larger land profile as they need wider spacing between rows and more maintenance than fixed tilt as they have moving parts.

Two common types of one‐axis tracking are illustrated in Figure 3.22. In part (a) a PV array rotates from east to west on a horizontal axis, oriented either north or south. In Figure 3.22(b) tracking takes place about a polar axis aligned with the Earth’s axis of rotation; this configuration is called polar or tilted single‐axis tracking. This limits the sun’s offset angle to a maximum of 23.45° from the plane of illumination, giving more efficient overall energy collection than with a horizontal axis (for an explanation of the angle, you may like to refer back to Figure 3.10 and Section 3.3.1). However the high profile means that more ground area is required to avoid shading of adjacent trackers, wind loading tends to be more serious, and the mounting is more awkward.

The horizontal axis also allows for mounting more PV modules on a given tracking system. The most obvious is to make the tracker larger. A single axis could be made longer or the PV could be stacked in double rows to make each row contain more PV. For a tilted axis tracker, making the rows longer or increasing the width of the PV on a row becomes impractical rather quickly. This is because one end of the tilted axis is sloped up into the air; the impacts of longer rows not only complicates the support structure required to elevate the row but also increases the wind loads, which increase exponentially with height above the ground. As a result, increasing row length as a way to increase array size is not an option for polar axis trackers; thus SunPower also uses long one‐axis tracking systems (like Figure 3.20(a)) in their large utility‐scale solar farms.

By virtue of having moving machinery and requiring a less dense configuration than fixed‐tilt systems, trackers virtually always come at an added cost relative to fixed systems. In order for a tracker to make economic sense, the increased energy harvest must exceed the added cost of installing and maintaining trackers over the lifetime of the system. An additional factor to be considered in the decision to use trackers or fixed systems is land use; tracking systems tend to use additional land because they must be spaced out in order to avoid shading one another as they track the sun. Also fixed racking systems offer more field adjustability than single‐axis tracking systems. Fixed systems can generally accommodate up to 20% slopes in the E/W direction, while tracking systems typically offer less of a slope accommodation unless the ground is tilted in the N/S direction facing the equator.

It is also worth noting that a very basic form of tracking can be achieved manually. By adjusting the orientation of a flat‐plate module just three or four times a day, in line with the sun’s trajectory, over 90% of the electricity yielded by a fully automatic tracker may be obtained. This is an interesting possibility for small systems with just one or two modules, for example, SHS in developing countries where family members are generally on‐site and can easily make the adjustments.

Of course the most accurate tracking of the sun requires rotation on two axes to handle both the hourly during a day and the daily during seasonal variations. Such systems are considerably more expensive and are deserved only for concentrating systems where accurate sun tracking is essential.

3.5 Concentration and Two‐Axis Tracking

Ever since the dawn of the modern PV age, the PV community has pondered the attractions of concentrated sunlight. After all, if the price of solar cells is very high (and it certainly was in the early days) and is closely related to their surface area, it should make sense to focus the sun’s light onto cells of very small area using materials that are less expensive than the solar cells. Furthermore, specialized cells designed to work under concentrated sunlight can achieve considerably higher conversion efficiencies than conventional cells. For example, in our discussion of gallium arsenide cells in Section 2.5.1, we noted that efficiencies around 40% make them suitable candidates for high‐concentration PV systems. But the approach is only viable if efficiency improvements and cost savings on the cells more than offset the additional costs of lenses or mirrors plus, in most cases, equipment to track the sun on its daily journey across the sky. Unsurprisingly, there are plenty of skeptics, not least because the cost of conventional solar cells and modules continues to fall. But the jury is still out, and it will be fascinating to see how the market develops in the coming decade.

To summarize, such systems offer two main attractions:

• The area of the solar cells can be greatly reduced.
• Cells designed for high‐intensity concentrated sunlight can achieve better conversion efficiencies than standard cells.

However the disadvantages and challenges appear rather onerous:

• Lenses or mirrors must be used to concentrate the light.
• Above a certain level of concentration, it becomes essential to accurately track the sun across the sky, keeping the focused light exactly aligned on the solar cells.
• High concentration is effective for the direct component of sunlight, but not the diffuse and albedo components.
• Focusing and tracking equipment must be robust and properly maintained to match the expected lifetime—say, 25 years—of solar cells and modules.
• Tracking systems are unsuitable for building‐integrated PV, including rooftop arrays.

We may therefore expect to see high‐concentration tracking systems largely restricted to power plants in areas with a high percentage of direct sunlight. So how is sunlight concentrated and focused onto small‐area solar cells? There are two basic approaches: using transparent lenses or reflective mirrors. The first of these is illustrated in Figure 3.23. Part (a) shows a circular lens, normally made of plastic, which concentrates the direct sunlight onto a small solar cell. Simple refractive lenses become very thick if their diameters exceed about 10 cm, so a special form known as the Fresnel lens is widely used. Rather than allowing the lens to get thicker and thicker toward its center, the convex surface is collapsed back to a thinner profile in a series of steps. A family of these lenses, each focusing sunlight onto a single solar cell, can be built up as a parquet to make a large flat PV module.

Whereas the circular Fresnel lens is point focus, the linear domed form of Fresnel lens shown in part (b) of the figure produces a line focus onto a long array of cells. Once again the lens profile is collapsed in a series of steps, keeping its thickness reasonably constant around the curve. The curvature increases mechanical strength and avoids optical problems that can arise with more flexible, flat lenses.

In high‐concentration systems the necessary focus is sometimes achieved in two stages (Figure 3.24). The main lens performs an initial concentration of the light, directing it onto a secondary optical element for further concentration. This also offers an opportunity to ensure that the intensity of light striking the active area of the solar cell is as uniform as possible.

Reflective mirrors provide an alternative to lenses. You are probably aware that a parabolic dish mirror receiving light parallel to its axis brings the light to a point at its focus. This is shown in Figure 3.25(a) with a solar cell mounted at the focus. Another effective configuration is the linear parabolic trough shown in part (b) that focuses the incoming light onto a linear array of cells.

The degree of concentration achieved by a lens or mirror is commonly expressed in suns. This is the ratio between the intensity of the incoming sunlight, normally taken as the standard insolation of 1000 W/m², and the average intensity of the light focused onto the active area of the solar cell, or cells. For example, a concentration of 100 suns produces a nominal 100 kW/m² or 10 W/cm² at the cell surface. Note, however, that in practice the insolation of 1000 W/m² is not all direct sunlight, even under clear skies. Typically 85% is direct, and the other 15% diffuse. So a nominal 100‐sun concentrator would more likely produce about 8.5 W/cm² at the cell surface in strong sunlight—and systems are sometimes rated on this basis. The amount of concentration in practical systems varies from as little as 2 or 3 suns in static systems that do not need to track the sun up to 1000 suns in high‐concentration tracking systems, some of which now employ multi‐junction gallium arsenide solar cells.

Unlike solar cells used in conventional PV modules that must be illuminated over their entire area for efficient performance, small concentrator cells are often designed with an “active area” surrounded by a non‐illuminated edge carrying bus bars and connections. This means that the reduction in cell area, one of the main advantages of a concentrator, is less than its number of suns. Furthermore, when many small cells are cut from a semiconductor wafer, there is quite a lot of wastage at the edges. The net reduction in wafer usage is substantially less than might be expected from a simple consideration of the concentration ratio.

Two more aspects of concentration optics should be mentioned briefly. The first of these is acceptance angle, the angular range over which a concentrator can accept light from the sun. Clearly, in the case of a tracking system, the greater the acceptance angle, the better, because it minimizes the tracking accuracy required and hence the complexity and cost of the tracking equipment. But perhaps not surprisingly there is a fundamental trade‐off between acceptance angle and concentration: increasing the acceptance angle reduces the amount of concentration attainable, and vice versa. An engineering compromise is required.

The second aspect is nonimaging optics. As any student who has played with a magnifying glass knows, light from distant objects can be brought to a focus on a sheet of white paper, producing an inverted image of the scene. If the magnifying glass is used to focus the sun’s rays onto the paper in an attempt to set it alight, the brilliant circular dot of light is an image of the sun. These are examples of classical imaging optics. But in the case of PV concentrators, there is no particular virtue in obtaining an image of the sun. It is more important to illuminate the active area of the solar cell, or cells, as uniformly as possible using a lens or mirror system with as large an acceptance angle as possible. Such design considerations have led to big advances in nonimaging optics applied to PV systems.²

High‐concentration optics generally demand tracking about two axes (to follow both the azimuth and the altitude of the sun’s trajectory during the day) so that the focused light always falls accurately on the solar cells. Two schemes are illustrated in Figure 3.26. Part (a) shows the widely used pedestal form of tracker with rotation about a horizontal (elevation) axis and also a vertical (azimuth) axis. This scheme is simple to install but tends to suffer from high wind loads, producing large torques on the drive system. Large trackers with surface areas up to 250 m² or more (as the system shown in Figures 3.27 and 3.28) normally adopt a horizontal position in very high winds. Part (b) of Figure 3.26 shows a less common form, known as the roll‐and‐tilt tracker. Wind loading is generally less serious, but more bearings and supports are needed.

The development of efficient two‐axis concentrator PV systems is a very interesting challenge as it requires optimization of four dimensions: the cell efficiency, the module cost, the optics cost and accuracy, and the tracking accuracy and reliability (Figure 3.29).

Appendix 3.A

3.A.1 Converting Global Horizontal Irradiation Data to Tilted and Sun‐Tracking Surfaces

Predicting solar radiation on an inclined (tilted) south‐facing PV array involves the following steps:

1. Solar data collection.
2. Calculation of extraterrestrial radiation.
3. Calculation of clearness index.
4. Determining the beam and diffuse components to the global irradiation on the horizontal surface.
5. Where appropriate, estimate the albedo contribution using reflectivity values for typical ground surfaces. In many cases the albedo is small or insignificant, but snow cover can be particularly relevant.
6. Using a model to calculate the energy incident on the inclined surface.

Typically hourly values of solar data are available and steps 2–6 are conducted on hourly time increments.

3.A.1.1 Solar Data Collection

The main source for 45‐year historical horizontal global irradiation data in the United Sates is the National Solar Radiation Database (NSRDB). The NSRDB is a serially complete collection of hourly values of the three most common measurements of solar radiation (global horizontal, direct normal, and diffuse horizontal) over a period of time adequate to establish means and extremes and at a sufficient number of locations to represent regional solar radiation climates. It also includes meteorological data. The database provides data for 1454 stations covering the entire United States for the years 1991–2010 and data for about 300 stations for the previous 25 years (http://rredc.nrel.gov/solar/old_data/nsrdb/1991‐2010).

The following table shows the type of data listed in the NSRDB.

Global horizontal radiation in Wh/m2	Atmospheric pressure in millibars
Direct normal radiation in Wh/m2	Wind direction in increments of 10°
Diffuse horizontal radiation in Wh/m2	Wind speed in m/s
Extraterrestrial radiation (ETR) in Wh/m2	Horizontal visibility in km
Direct normal ETR in Wh/m2	Ceiling height in decameters
Total sky cover in tenths	Present weather
Opaque sky cover in tenths	Total precipitable water in mm
Dry‐bulb temperature in °C	Aerosol optical depth
Dew‐point temperature in °C	Snow depth in cm
Relative humidity in percent	Number of days since last snowfall

Solar data for Europe, Africa, and Asia can be found in the PVGIS database provided by the European Commission, Joint Research Centre, Ispra, Italy (http://re.jrc.ec.europa.eu/pvgis).

3.A.1.2 Calculation of Extraterrestrial Radiation

The extraterrestrial radiation is reported in the NSRDB. In any case it is instructive to derive it from the trigonometric relationships describing the movement of the Earth. For this we need to refer to the solar declination angle discussed earlier and the angles shown in Figure 3.A.1.

Solar declination δ, approximately given in radians, is

(3.A.1)

where n = number of the day of the year (1 ≤ n ≤ 365).

As shown in Figure 3.12, the declination angle is zero at the equinoxes vernal (spring, about 22 March) and autumnal (22 September). On these days the sun rises exactly in the east and sets exactly in the west. At the summer solstice (~22 June), δ = 23.45° and at the winter solstice (~22 December), δ = −23.45° (Figure 3.A.2):

We define as I_o and H_o the hourly and daily, respectively, extraterrestrial radiation on a horizontal surface at the top of the atmosphere; these are given as follows:

(3.A.2)

(3.A.3)

where

• θ: angle of incidence
• ϕ: latitude
• δ: declination
• ω: the hour angle

It is the angular displacement of the sun east or west of the local meridian due to rotation of the Earth on its axis at 15°/h: morning negative, afternoon positive,

(3.A.4)

where t is the decimal, 24 hours time, for example, 8 : 45 p.m. is 20.75 and the angles are expressed in radians.

• n: number of the day in the year (1 on 1st of January and 365 on 31st December)
• ω_s: the sunset angle (cosω_s=‐tanϕ. tanδ)
• θ_z: zenith angle, the angle between the vertical and the line to the sun, that is, the angle of incidence of beam radiation on a horizontal plane

The angle of incidence θ and the zenith angle θ_Z can be determined from the latitude and the sun declination as follows:

(3.A.5)

(3.A.6)

3.A.1.3 Determining the Diffuse and the Direct Components of the Global Horizontal Irradiation

We use the top of the atmosphere (extraterrestrial) hourly data (I_o) and hourly data of the global horizontal irradiation (GHI) (I) on the surface of the Earth, and from their ratio we determine what is called hourly clearness index (k_T), a measure of the losses in the atmosphere of sunlight intensity:

(3.A.7)

k _T can be correlated with the ratio of diffuse radiation (I_d) to the global horizontal one (I) using one of the three following empirical equations:

(3.A.8)

Typically data are divided into small ranges of k_T and then are averaged to produce single points. A k_T of 0.5 may be produced by skies with thin cloud cover, resulting in a high diffuse fraction, or by skies that are clear for part of the hour and heavily clouded for part of the hour, leading to a low diffuse fraction.

The beam radiation (I_b) can be calculated if we subtract the diffuse radiation (I_d) from the global; thus I_b = I −I_d.

3.A.1.4 Using a Model to Calculate the Energy Incident on the Inclined Surface per Time Increment

The simplest and most conservative model for calculating solar irradiation on a tilted surface is the isotropic model, which assumes that the diffuse radiation from the whole sky is of the same strength. In reality the sky around the sun is much brighter than in the whole horizon, and this is described with anisotropic models, which are more complicated, yet more accurate.

A common anisotropic model is called HDKR from the names of its developers (Hay–Davies–Klucher–Reindl). More recently, an anisotropic model developed by Richard Perez is found to be even more accurate in high irradiation conditions. The anisotropic models assume that diffuse radiation is composed of three parts: (i) an isotropic part received uniformly from the entire sky dome, (ii) a circumsolar diffuse part resulting from forward scattering of solar radiation and concentrated in the part of the sky around the sun, and (iii) the horizon brightening part (it is concentrated near the horizon and is most pronounced in clear skies).

• Isotropic model:where
  • I_T: total hourly radiation incident on a surface tilted at slope β
  • I_b: beam radiation
    
  • θ: angle of incidence
  • θ_Z: zenith angle
  • ρ_g: albedo of the ground
  • I: hourly radiation incident on a horizontal surface
  • I_b,T: hourly beam radiation on the tilted surface
  • I_d: diffuse radiation
  It is calculated from any of the I_d/I versus k_T correlations.
• Anisotropic models:
  HDKR model:
  

  where

  • Ai: anisotrophy index I_b/I_o. It is a function of the transmittance of the atmosphere for beam radiation.
  • f: modulating factor. f = 

  Perez model:

  

  where F₁ and F₂ are coefficient describing circumsolar and horizon anisotropy, respectively; a = max(0, cos θ), and b = max(0.087, cos θ_Z).

3.A.1.5 Comparisons of Different Configurations

Let us now see the impact of tilting the surface of PV modules and of putting them on one‐axis trackers. Figure 3.A.3 shows this for a location in New York with moderate irradiation. The red lines show the daily range over the course of a year of the GHI, and blue shows the global irradiation on a plane tilted at latitude tilt. The later configuration receives a lot more sunlight in the winter and the fall, whereas the irradiation in the summer is somewhat lower. Over the course of the year, the latitude tilt configuration receives significantly more sunlight than the horizontal plane.

Figure 3.A.4 shows the irradiation over different angles for a high solar irradiation site in northern Chile. Note that the location is in the south hemisphere, so January is summer and July is winter. When we installed the modules at a 27° facing north, that is, latitude tilt in this location, we get a noticeable improvement in the irradiation that can be harvested especially during the winter. One‐axis tracking extends the duration of the power output and slightly tilting its axis toward the north and optimizes tracking operation for this location.

Self‐Assessment Questions

1. Q3.1 Explain the phenomenon of hot‐spot formation in a PV module and how it may be prevented.
2. Q3.2 Figure 3.5 shows an array of six PV modules. If each module has a voltage 20 V and current 5 A at the maximum power point (MPP), (a) what is the array voltage and current at the MPP; (b) how are these values altered if the six modules are reconnected in series?
3. Q3.3 A monocrystalline silicon PV module has dimensions 1640 mm × 1000 mm and is rated at 300 W_p. It contains 60 cells connected in series.
   1. Calculate the module efficiency.
   2. Approximately what module voltage and current do you expect at the maximum power point?
   3. If the open‐circuit voltage is 39 V in standard conditions and the temperature coefficient is −0.3% per °C, what is the voltage at a temperature of 0°C?
4. Q3.4 A CdTe PV module has dimensions 1860 mm × 1200 mm and is rated at 360 W_p. Its voltage at the maximum power point (MPP) is 177 V.
   1. Estimate the module efficiency and the current at the MPP.
   2. If the power temperature coefficient is −0.34% per °C, what is the maximum power output at −10°C?
5. Q3.5 The latitude of New York City is 40.71°N. What is the elevation of the sun above the horizon at solar noon on dates corresponding to (a) the equinoxes, (b) the summer solstice, and (c) the winter solstice?
6. Q3.6 Define the Arctic Circle. What is its latitude, and why?
7. Q3.7 The average daily solar irradiation on a horizontal surface in Las Vegas, Nevada, is 5.7 kWh/m². Estimate (a) the peak sun hours in a year and (b) the approximate annual output of a horizontally mounted PV module rated at 250 W_p.
8. Q3.8 Why is a PV array often pointed toward the south (or north in the southern hemisphere) and tilted down from horizontal by an angle equal to the latitude of the site? What tilt angle would you expect for an array located in (a) Rio de Janeiro, Brazil; (b) Lisbon, Portugal; and (c) Berlin, Germany?
9. Q3.9 A PV array in London, England, consists of 20 modules rated at 180 W_p each, facing south and tilted at an angle equal to London’s latitude. (a) Use Figure 3.17(a) to estimate the approximate annual solar radiation received by the array; (b) using the peak sun hours concept, estimate the approximate annual output of the array in megawatt‐hours (MWh). Why is the latter figure likely to be overoptimistic?
10. Q3.10 Place the following six city locations in order of their suitability for high‐concentration PV tracking systems: (a) Palm Springs, California; (b) Palo Alto, California; (c) Cape Town, South Africa; (d) Durban, South Africa; (e) Crucero, Chile; and (f) Santiago, Chile.
12. Q3.11 What is “solar declination” and what is its range?
13. Q3.12 What are the three components of solar irradiance on a terrestrial plane?
14. Q3.13 What is the average solar irradiance in an early afternoon on a clear day?
15. Q3.14 Why does concentration of sunlight improve efficiency of a solar cell? Why is there a practical upper limit?
16. Q3.15 What is the difference between irradiance and irradiation?
17. Q3.16 Give approximate ranges of energies for different regions of the electromagnetic spectrum. Specifically, give ranges for ultraviolet, visible, near infrared, mid infrared, and far infrared in terms of commonly used units of energy (eV), wavelength (nm and µm), and frequency (GHz and THz).

Problems

Solar irradiation

1. 3.1 The sun is at an altitude of 35° from the horizontal. What is the corresponding air mass?
2. 3.2 Calculate the sun’s altitude at solar noon on 21 June in New York (latitude 40.71°N); in Athens, Greece (latitude 37.98°N); and in Antofagasta, Chile (latitude 23.65°S).
3. 3.3 Estimate the zenith angles needed to produce AM1.5, AM2.0, and AM3.0 (hint: AM1.0 occurs at 0°).
4. 3.4 Calculate the zenith angle at solar noon in New York (latitude 40.72°) on July 1 (121th day of the year).
5. 3.5 Use a sun path (Figure 3.19 or 3.20 or a different one) and make a table of shaded times for each month of the year.
6. 3.6 A PV system comprises four rows of south facing modules tilted so that the top of the modules is 3 ft off the ground. If the site is at a latitude of 41°N, determine the spacing between the rows needed to prevent shading from one row to the other. How does this change in a latitude of 30°?
7. 3.7 What are the maximum and the minimum solar elevations at a site with latitude 40.71 and when do they occur?
8. 3.8 The monthly mean daily solar irradiation on a horizontal plane at New York (latitude 40.7°N) and Chania, Greece (latitude 35.5°N), are listed in the table in units of Wh/m²/day:

   	Jan	Feb	Mar	April	May	June	July	Aug	Sept	Oct	Nov	Dec
   New York	1.9	2.7	3.8	4.9	5.7	6.1	6.0	5.4	4.3	3.2	2.0	1.6
   Chania	2.4	3.10	5.0	6.2	7.3	8.4	8.3	7.6	5.8	4.1	2.7	2.0

   1. Determine the mean daily solar radiation outside the Earth atmosphere for each month of the year and the corresponding clearness index k_T.
   2. Determine the daily global radiation for angles of incidence in the range of 0–90°, assuming a ground reflectivity of 0.2 (create a table showing monthly values).
   3. What angle of inclination would you recommend in each location to maximize input to the PV system during the summer months and to obtain the maximum energy during the whole year?
9. 3.9 Assume that we want maximum PV output in the summer in New York between the hours of 2 and 5 P.M. Calculate the orientation and tilt of PV modules that will produce this maximum on July 15.
10. 3.10 Do the same calculation as in Problem 3.9 for Phoenix, Arizona, and Seattle, Washington, and compare the results.

PV performance

1. 3.11 Calculate the daily horizontal extraterrestrial irradiation at New York City and Los Angeles on 21 January by using the formulas listed in Appendix 3.A. (NYC latitude: 40.71°; LA latitude: 34.05°.)
2. 3.12 How many kW (DC rated) would be needed on a rooftop to deliver 6000 kWh/year to a home in at Long Island, NY, when the derated factor (performance ratio) is 0.78? Assume (a) horizontal roof and (b) south‐facing rooftop with an inclination of latitude at 15°. The global horizontal irradiation (GHI) at the location (latitude is 40.789°N) is 1630 kWh/m²/year, and the average insolation on a 25.789° south‐facing plane is 1790 kWh/m²/year.
3. 3.13 Go to the NSRDB website (http://rredc.nrel.gov/solar/old_data/nsrdb/1991‐2010/) and download the .csv file with hourly solar data for station with USAF number 723860 (Las Vegas International Airport) for year 1991. You need to choose the column containing global horizontal irradiation (GHI) data produced by the METSTAT model.
   1. What is the horizontal extraterrestrial energy (kWhr) for each of the 24 hours of the 18th day of June? Use the hourly formula given by Equation 3.A.2 to perform your calculations. Sum up the hourly calculations to figure out the daily radiation incident on the surface. Do the same with the extraterrestrial data from your .csv file. Compare the results. You can use Excel or an equivalent spreadsheet program. Note 1: You will get same negative values for some hours. That means nighttime. Equal all negative values to zero. Note 2: Evaluate ω at the midpoint of each hour.
   2. Repeat (a) performing only one calculation using the daily formula given by Equation 3.A.3. The result has to be given in kWh.
   3. What day of the year a horizontal surface located on the equator and at the top of the atmosphere would receive the maximum solar radiation?
   4. The owner of a Las Vegas casino has decided to install a PV system on the roof of his hotel to reduce his annual electricity bill and what to decide between putting the PV horizontally (flat) on his roof or putting them facing south tilted at latitude angle. Use the HDKR model given in class to break global radiation into its beam and diffuse components for the same date as in (b). Calculate hour by hour the energy incident per square meter on the tilted PV panels using the isotropic sky model. Sum up the hourly values to get the daily energy for the same date as in (b) and compare with the daily energy received by a horizontal surface. Explain why there is a difference and what will be the best choice for the casino owner.
4. 3.14 Repeat the aforementioned calculation by using data from National Solar Radiation Database (NSRDB) (http://rredc.nrel.gov/solar/old_data/nsrdb/1991‐2010/) and (a) the isotropic model, (b) the HDKR anisotropic model, and (c) the Perez anisotropic model (formulas in Appendix 3.A).
5. 3.15 You are asked to size a PV system for an AC load appliance in a location of 23.45°. Assume that the air condition only operates from May to October. You don’t want to oversize the system and so you decide to size for month June rather than August (the hottest month of the year). Explain what do you think that the optimum tilt of your PV system would be.

Ohm’s law

1. 3.16 A lamp is designed to consume 60 W when it is connected to a 12 V power source. What is the resistance of the lamp, and what current will flow through and what energy will consume over the course of a month if it operates for 5 hours a day?
2. 3.17 Estimate the total resistance of the network shown below:

   

3. 3.18 Suppose that a battery is modeled as a 12 V source in series with a 0.12 Ω internal resistance. (a) What would be the output voltage when the battery delivers a current of 10A? (b) What would be the output voltage when the battery is connected to a 2 Ω load?
4. 3.19 A 12 V battery is connected with a 100 W lightbulb at a distance of 50 ft. If no. 14 (d = 0.163 cm) copper wire is used, what would be the power loss on the wire? Copper Wire Resistance and Amperage Ratings

   Wire Gauge Number	Diameter (inches)	Resistance (Ohm/100 ft)	Max Current (A)
   14	0.0641	0.2525	15
   12	0.0808	0.1588	20
   10	0.1019	0.0999	39
   8	0.1285	0.0628	40

1. 3.20 A system is delivering 15 A of current through 12‐gauge wire to a load 100 ft away. a) find the voltage drop in the wires; b) what percent of the power generated by the PV is lost in the connected wire?
2. 3.21 Based on the wire specifications shown in the Table above, what wire gauge would you have chosen to minimize losses?
3. 3.22 Use a 120 V to connect over a 50 ft distance a 100 W lightbulb with the same wire as above. What would be the power loss on the wire?
4. 3.23 A PV module comprises 72 solar cells 1.5 V each wired in series. Draw the I–V curve when (a) all cells are exposed to the sun and (b) when two cells are shaded; assume i) no bypass diodes and ii) bypass diodes for every cell.
5. 3.24 For the same conditions, estimate the current, voltage, and power in the PV module above connected to a 10 Ω load.

Answers to Questions

1. Q3.1 Explain the phenomenon of hot‐spot formation in a PV module and how it may be prevented.
2. Q3.2 (a) 60 V, 10 A (b) 120 V, 5 A
3. Q3.3 (a) 18.3% (b) about 30 V and 10 A (c) 41.9 V
4. Q3.4 (a) 16.1%, 2.03 A (b) 428 W
5. Q3.5 (a) 49.29° (b) 72.74° (c) 25.84°
6. Q3.6 About 66°N
7. Q3.7 (a) 2080 hours (b) 520 kWh/year
8. Q3.8 (a) 23° (b) 39° (c) 53°
9. Q3.9 (a) 1200 kWh/m² (b) 4.3 MWh
10. Q3.10 (e), (a), (c), (b), (f), (d)
11. Q3.11 It is the angle between the equator plane and the line connecting the centers of the Earth and the sun. It is measured in degrees.
12. Q3.12 Direct, diffuse, and albedo
13. Q3.13 1 kW/m²
14. Q3.14 Concentration increases photon density making them more energetic; however it also increases the temperature in the cell and after a certain concentration limit cooling is not practical and the thermal losses overcome the gains from concentration.
15. Q3.15 Irradiance is a power flux (W/m²), whereas irradiation is an energy flux (kWh/m²) over a course of time, typically a day or a year.
16. Q3.16 Answer:

    Spectrum	Wavelength	Frequency (Hz)	Energy (eV)
    Ultraviolet	10 nm–380 nm	8×1014–3.0×1016	124–3.1
    Visible	380 nm–750 nm	4.0×1014–8×1014	3.1–1.77
    Near Infrared	750 nm–10 μm	3.0×1013–4.0×1014	1.77–0.124
    Mid Infrared	10 μm–100 μm	3.0×1012–3.0×1013	0.124–0.0124
    Far Infrared	100 μm–1 mm	3.0×1011–3.0×1012	0.0124–0.00124

References

1. 1 NASA. Surface meteorology and Solar Energy Tables (2010). eosweb.larc.nasa.gov/sse (Accessed on August 28, 2017).
2. 2 A. Luque. Solar Cells and Optics for Photovoltaic Concentration, Adam Hilger: Bristol, Philadelphia (1989).