1.1 Sun Position, Irradiance, and Insolation

Sun Position, Irradiance, and Insolation

Designing and installing a high-performance photovoltaic (PV) system begins with a quantitative understanding of the solar resource. Solar energy reaching the Earth varies constantly based on orbital geometry, daily rotation, geographic coordinates, and atmospheric conditions. System designers and installation professionals must model these dynamic solar angles and radiation components accurately to determine array tilt, orientation, row spacing, and expected energy yield.

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1. Solar Geometry: Altitude, Azimuth, and Solar Noon

The position of the sun in the sky at any given moment and location is defined by two fundamental celestial angles: solar altitude and solar azimuth.

Solar Altitude Angle

The solar altitude angle (often denoted as $\beta$ or $\alpha$) is the vertical angle between the center of the sun's disc and the ideal horizontal plane of the horizon.

• At sunrise and sunset on an ideal unobstructed horizon, solar altitude is $0^\circ$.
• Directly overhead at the zenith point, solar altitude is $90^\circ$.
• The complementary angle to altitude is the solar zenith angle ($\theta_z$), defined as the angle between the sun and the vertical zenith: $\theta_z = 90^\circ - \beta$.

Solar Azimuth Angle

The solar azimuth angle (denoted as $\gamma$ or $\psi$) represents the horizontal compass direction of the sun along the horizon. In solar engineering and PV installation practice throughout the Northern Hemisphere, azimuth is conventionally referenced to True South:

• True South = $0^\circ$ (or $180^\circ$ under standard 360-degree navigational clockwise convention, where North = $0^\circ$, East = $90^\circ$, South = $180^\circ$, and West = $270^\circ$).
• When True South is assigned $0^\circ$, morning positions to the east carry negative values (e.g., due East is $-90^\circ$), and afternoon positions to the west carry positive values (e.g., due West is $+90^\circ$).

Solar Noon and the Solar Declination

Solar noon is the precise instant when the sun crosses the local celestial meridian. At solar noon:

1. The sun reaches its highest altitude angle of the day.
2. The solar azimuth is oriented directly True South in the Northern Hemisphere (or True North in the Southern Hemisphere).
3. Shadows cast by vertical objects point exactly along the true north-south axis.

The sun's maximum altitude at solar noon fluctuates throughout the year because the Earth's rotational axis is tilted at approximately $23.45^\circ$ relative to the plane of its orbit around the sun. This orbital tilt creates the solar declination angle ($\delta$), which cycles between $+23.45^\circ$ at the summer solstice (around June 21) and $-23.45^\circ$ at the winter solstice (around December 21). On the vernal and autumnal equinoxes (around March 21 and September 21), the declination is exactly $0^\circ$.

The maximum solar altitude at solar noon for any latitude in the Northern Hemisphere is calculated using the standard formula:

$\text{Altitude}_{\text{noon}} = 90^\circ - \text{Latitude} + \delta$

Worked Example: Solar Altitude Across Seasons

For an installation located in Denver, Colorado (latitude $40^\circ\text{N}$):

• Summer Solstice ($\delta = +23.45^\circ$): $\text{Altitude} = 90^\circ - 40^\circ + 23.45^\circ = 73.45^\circ$
• Equinoxes ($\delta = 0^\circ$): $\text{Altitude} = 90^\circ - 40^\circ + 0^\circ = 50.00^\circ$
• Winter Solstice ($\delta = -23.45^\circ$): $\text{Altitude} = 90^\circ - 40^\circ + (-23.45^\circ) = 26.55^\circ$

Notice that the solar altitude at noon swings through an arc of $46.9^\circ$ ($2 \times 23.45^\circ$) over the course of the year. This large seasonal swing determines array row-to-row spacing requirements to avoid inter-row shading on the shortest day of the year.

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2. The Solar Window and Solar Time

Photovoltaic arrays do not generate meaningful power during early dawn or late twilight because solar rays must travel through an extensive atmospheric path, and the shallow angle of incidence causes high reflection losses from the module front glass.

The Design Solar Window

The solar window is the critical daylight period during which the overwhelming majority of usable solar energy reaches the Earth's surface. Industry standards define the prime solar window as:

• 9:00 AM to 3:00 PM Solar Time (a 6-hour window centered on solar noon), or alternatively whenever the solar altitude exceeds $15^\circ$ to $20^\circ$.
• Approximately 80% to 90% of total daily radiant energy is concentrated within this 6-hour window.
• A site shading survey that identifies zero obstructions between 9:00 AM and 3:00 PM across all seasons guarantees that the array will capture the primary solar harvest.

Solar Time vs. Clock (Civil) Time

Solar time does not match standard local clock time. Solar time is based strictly on the sun's position relative to the local observer's meridian. Two adjustments convert civil time to solar time:

1. Longitude Correction: Standard time zones are established around reference meridians spaced every $15^\circ$ of longitude (e.g., Eastern Standard Time at $75^\circ\text{W}$, Central at $90^\circ\text{W}$, Mountain at $105^\circ\text{W}$, Pacific at $120^\circ\text{W}$). Because the Earth rotates at $15^\circ$ per hour, every $1^\circ$ of longitude discrepancy between the site and its time zone reference meridian equals a 4-minute time adjustment ($60 \text{ minutes} / 15^\circ = 4 \text{ min/degree}$).
2. Equation of Time (EoT): Because Earth's orbit is elliptical rather than circular and its axis is tilted, solar days vary slightly in length throughout the year. The Equation of Time provides a correction factor ranging from approximately $-14$ minutes in February to $+16$ minutes in November.
3. Daylight Saving Time (DST): Local civil clocks are advanced by 1 hour during DST, shifting solar noon on civil clocks closer to 1:00 PM.

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3. Atmospheric Attenuation and Air Mass (AM 1.5)

Before sunlight strikes a collector on the roof or ground, it passes through the Earth's atmosphere, where gases, water vapor, ozone, and suspended particulates absorb and scatter the incoming photons.

The Air Mass Ratio

Air Mass (AM) is the relative path length of sunlight through the Earth's atmosphere compared to the shortest possible path when the sun is directly at the zenith ($90^\circ$ altitude at sea level):

$\text{Air Mass (AM)} = \frac{1}{\cos(\theta_z)} = \frac{1}{\sin(\beta)}$

• AM 0 (Extraterrestrial): Sunlight in outer space outside the atmosphere, with an irradiance equal to the solar constant ($G_{sc} \approx 1,367 \text{ W/m}^2$). Used for satellite and aerospace PV rating.
• AM 1.0 (Zenith): Sunlight passing perpendicularly through the atmosphere at sea level when the sun is directly overhead ($\beta = 90^\circ$).
• AM 1.5 (Standard Test Conditions): The universal terrestrial rating standard for PV modules. AM 1.5 corresponds to a solar zenith angle of approximately $48.2^\circ$ (solar altitude of $41.8^\circ$), representing an atmospheric path length 1.5 times thicker than zenith overhead sunlight. This standard corresponds to an irradiance of $1,000 \text{ W/m}^2$ and reflects average temperate continental conditions.
• AM 2.0+ (Low Sun): Early morning, late evening, or high-latitude winter conditions where the slant path exceeds twice the vertical thickness, producing significant spectral reddening and lower energy density.

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4. Radiation Components: Irradiance vs. Insolation

Field professionals must maintain a strict technical distinction between irradiance (instantaneous power) and insolation (accumulated energy).

Terrestrial Radiation Components

Total solar radiation reaching a plane surface consists of three distinct components:

1. Direct Normal Irradiance (DNI): Sunlight traveling in a straight line from the sun's disc directly to the receiver without being scattered. It is measured perpendicular to the incoming rays.
2. Diffuse Horizontal Irradiance (DHI): Sunlight that has been scattered by atmospheric molecules (Rayleigh scattering), aerosols, clouds, and dust, reaching the receiver from all quadrants of the sky vault.
3. Albedo (Ground-Reflected Radiation): Sunlight reflected from the surrounding terrain onto the collector surface. The albedo coefficient ($\rho$) represents the fraction of incident light reflected:
   • Fresh snow: $\rho = 0.70 - 0.85$
   • White membrane roof (cool roof): $\rho = 0.55 - 0.70$
   • Green vegetation / grass: $\rho = 0.20 - 0.25$
   • Weathered asphalt shingle: $\rho = 0.15 - 0.20$
   • Wet dark soil: $\rho = 0.08 - 0.12$

On a horizontal surface, Global Horizontal Irradiance (GHI) combines direct and diffuse radiation:

$GHI = DNI \times \cos(\theta_z) + DHI$

For an array tilted toward the sun, the total radiation on the collector plane is called Plane of Array (POA) irradiance, which incorporates direct beam, diffuse sky light, and ground-reflected albedo.

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5. Peak Sun Hours (PSH) and Energy Production Sizing

Solar irradiance fluctuates continuously from zero at dawn to peak values near $1,000 \text{ W/m}^2$ at solar noon, and back to zero at dusk. Integrating these changing bell-shaped curves directly into system calculations is mathematically cumbersome. The solar industry simplifies this through Peak Sun Hours (PSH).

Defining the Peak Sun Hour

One Peak Sun Hour is defined as the equivalent time during which solar irradiance remains constant at the standard test intensity of $1,000 \text{ W/m}^2$ ($1 \text{ kW/m}^2$):

$1 \text{ PSH} = 1,000 \text{ W/m}^2 \times 1 \text{ hour} = 1 \text{ kWh/m}^2$

Therefore, daily solar insolation expressed in $\text{kWh/m}^2/\text{day}$ is numerically identical to Peak Sun Hours per day:

• An insolation of $4.8 \text{ kWh/m}^2/\text{day}$ equals $4.8 \text{ PSH}$.
• An insolation of $6.2 \text{ kWh/m}^2/\text{day}$ equals $6.2 \text{ PSH}$.

Estimating Array Energy Yield from PSH

With known PSH, a system's gross daily energy yield can be calculated using the direct sizing formula:

$E_{\text{daily}} = P_{\text{array, STC}} \times \text{PSH} \times \text{PR}$

Where:

• $E_{\text{daily}}$ is daily AC electrical energy output ($\text{kWh}$).
• $P_{\text{array, STC}}$ is array DC nameplate capacity at STC ($\text{kW}$).
• $\text{PSH}$ is local peak sun hours for the design month or annual average ($\text{hours}$).
• $\text{PR}$ is the Performance Ratio (system derate factor, typically $0.75 - 0.85$ accounting for inverter efficiency, temperature derating, wire resistance, soiling, and mismatch).

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6. Worked Solar Geometry and Seasonal Insolation Table

The following table illustrates solar altitude, noon zenith angle, daylight hours, and average horizontal versus tilted ($35^\circ$ tilt facing True South) insolation for a representative mid-latitude location ($40^\circ\text{N}$):

Analytical Insights from the Data

1. Winter Tilt Advantage: On the winter solstice, tilting the collector at $35^\circ$ doubles the collected daily energy ($4.2 \text{ kWh/m}^2$ vs. $2.1 \text{ kWh/m}^2$) because the lower sun angle strikes the tilted plane far closer to perpendicular ($90^\circ$).
2. Summer Tilt Trade-Off: On the summer solstice, a $35^\circ$ tilt yields slightly less energy than a horizontal surface ($6.5 \text{ kWh/m}^2$ vs. $7.2 \text{ kWh/m}^2$) because the sun rises and sets north of east/west, reaching high midday altitudes where a flatter plane captures more diffuse and direct high-zenith rays.
3. Annual Production Optimization: Fixed residential and commercial rooftop arrays are typically installed flush to the roof structure. When ground or ballast mounting allows tilt selection, tilting the array equal to the local latitude optimizes annual energy yield. Tilting at latitude $+15^\circ$ favors winter off-grid performance, while latitude $-15^\circ$ maximizes summer net metering revenue.