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IEC 61725: Analytical Expression for Daily Solar Profiles
Tip: IEC 61725 defines a standardised mathematical expression for the daily profile of solar irradiance. It provides a simple yet powerful tool for engineers to model solar radiation distribution throughout the day without requiring complex radiative transfer models or site-specific historical data.
1. Scope and Mathematical Formulation
IEC 61725 specifies an analytical expression that describes the variation of solar irradiance on a horizontal surface over the course of a clear day. The standard was developed to address a practical need in photovoltaic system design: when only the daily total solar irradiation (or monthly average daily irradiation) is known, how can the engineer reconstruct a realistic instantaneous irradiance profile for energy yield calculations? The analytical expression provides a repeatable, standardised method for this purpose, eliminating the variability that would arise from using different empirical models or site-specific data fitting approaches.
The core of IEC 61725 is the following analytical expression for the global solar irradiance G(t) at time t (solar time, in hours) on a horizontal surface:
IEC 61725 Analytical Solar Profile:
G(t) = Gmax · sinπ(π · (t − tr) / (ts − tr))
Where:
G(t) = global solar irradiance at time t (W/m2)
Gmax = maximum irradiance at solar noon (W/m2)
tr = sunrise time (solar hour)
ts = sunset time (solar hour)
π = exponent (dimensionless), typically in the range 1.0–1.5
The daily irradiation H (kWh/m2) is obtained by integrating G(t) from tr to ts.
The exponent π in the sine function determines the shape of the daily profile. When π = 1.0, the profile is a pure sine half-wave, symmetric about solar noon. This corresponds approximately to clear-sky conditions with uniform atmospheric turbidity throughout the day. As π increases above 1.0, the profile becomes more “peaked” — the irradiance rises more steeply in the morning, peaks more sharply at noon, and falls more rapidly in the afternoon. Typical values of π range from 1.0 for very clear, dry atmospheres to 1.5 for hazy or humid conditions where increased atmospheric scattering broadens the angular distribution of diffuse radiation. The standard provides guidance on selecting π based on climatological data, but also notes that π = 1.25 is a reasonable default for many temperate climates when no specific information is available.
Limitation: IEC 61725 models the clear-sky solar profile. It does not account for cloud cover, which introduces stochastic variability that cannot be captured by a deterministic analytical function. For energy yield calculations, the standard’s profile should be used with daily irradiation values that already incorporate the statistical effects of cloud cover (e.g., monthly average daily irradiation from meteorological databases), rather than attempting to model clouds analytically.
2. Application in PV System Design and Simulation
IEC 61725 serves as a fundamental building block in photovoltaic system simulation. When combined with the daily total irradiation H (kWh/m2) for a given location and month, the analytical profile enables the calculation of instantaneous power output at any time of day. This is particularly valuable for: (1) sizing inverters relative to array capacity, since the peak irradiance determines the maximum instantaneous power; (2) evaluating the match between PV generation and load profiles for self-consumption optimisation; and (3) calculating array self-shading losses at different sun angles throughout the day.
The standard provides a direct relationship between the daily irradiation H and the peak irradiance Gmax. For a given day length (ts − tr) and exponent π, the peak irradiance can be derived analytically:
| Exponent π | Gmax / H Ratio (per hour) | Atmospheric Condition | Typical Application |
|---|---|---|---|
| 1.00 | 1.571 / (ts − tr) | Pure sine wave, very clear atmosphere | High-altitude desert locations |
| 1.25 | 1.700 / (ts − tr) | Moderate turbidity (default) | Temperate climates, default value |
| 1.50 | 1.812 / (ts − tr) | Hazy or humid atmosphere | Tropical lowland, urban areas |
This relationship has direct engineering significance. Consider a PV system at a location with a 12-hour day length and a daily irradiation of 5.0 kWh/m2. With π = 1.25, the peak irradiance at solar noon is approximately 708 W/m2. With π = 1.0, it would be approximately 655 W/m2, and with π = 1.5, approximately 755 W/m2. The choice of π thus directly affects the predicted peak power and, consequently, inverter sizing decisions. An inverter sized based on an underestimated peak irradiance may clip more frequently than expected, reducing energy yield.
Design Insight: For inverter sizing, using IEC 61725 with π = 1.0 (conservative) will predict a lower peak irradiance and may lead to undersized inverters. Using π = 1.5 will predict higher peaks and may lead to oversized inverters with lower utilisation. The standard recommends that engineers consider the site-specific atmospheric conditions when selecting π, and for critical applications, validate against measured data from a local weather station.
3. Engineering Design Insights and Practical Considerations
Beyond its direct application in irradiance profile generation, IEC 61725 provides a framework for several important engineering analyses. One key application is the clear-sky index estimation for PV system monitoring. By comparing measured instantaneous irradiance against the IEC 61725 profile for the same time of day, the clear-sky index (ratio of actual to clear-sky irradiance) can be calculated. This index is a fundamental input to PV performance analysis, enabling the separation of weather-related variability from system-induced performance issues. A sudden drop in the clear-sky index that does not correspond to meteorological observations may indicate sensor degradation or soiling.
The standard’s analytical expression also enables tilted surface irradiance calculation when combined with transposition models. The horizontal irradiance profile from IEC 61725 can be decomposed into direct beam and diffuse components using standard separation models (such as the Erbs or Reindl correlations), and then transposed to a tilted plane using geometric relationships. This three-step process (profile generation, separation, transposition) forms the core of many PV simulation tools and is explicitly referenced in IEC 61724-1 for monitoring applications.
| Parameter | Symbol | Unit | Source / Determination |
|---|---|---|---|
| Daily irradiation (horizontal) | H | kWh/m2 | Meteorological database or measurement |
| Day length | ts − tr | hours | Solar geometry (latitude & declination) |
| Profile exponent | π | — | Climatological data or default (π = 1.25) |
| Peak irradiance | Gmax | W/m2 | Derived from H, day length, and π |
| Instantaneous irradiance | G(t) | W/m2 | Analytical expression from IEC 61725 |
| Hourly irradiation | Hh | Wh/m2 | Integral of G(t) over the hour |
Important: IEC 61725 is a standardised model, not a measurement standard. The uncertainty of the analytical expression relative to actual clear-sky irradiance varies with atmospheric conditions. At low sun angles (early morning and late afternoon), the model tends to overestimate irradiance because it does not fully capture enhanced atmospheric path length effects. For energy calculations where morning and evening contributions are significant (e.g., high-latitude installations), engineers should be aware of this systematic bias and consider corrections for low sun-angle conditions.
For PV engineers performing energy yield assessments, IEC 61725 offers a computationally efficient alternative to TMY (Typical Meteorological Year) data for preliminary design studies. While TMY data provides more accurate results by incorporating real weather variability, the IEC 61725 approach requires only the monthly average daily irradiation — a parameter available from numerous global solar atlases and databases (NASA SSE, Solargis, Global Solar Atlas). The standard enables rapid iteration over multiple design options during the conceptual design phase, reserving detailed TMY-based simulation for the final design stage. An example workflow might use IEC 61725 to compare 20 different tilt/orientation configurations in minutes, then validate the top 3 candidates with full TMY simulation.
Finally, the standard has found application beyond its original scope in solar thermal collector testing and building energy simulation. In solar thermal testing per ISO 9806, the IEC 61725 profile is used to define the standard diurnal temperature and irradiance cycle for collector performance characterisation. In building energy simulation, the profile provides the solar heat gain input for dynamic thermal models, enabling hourly calculation of cooling and heating loads without coupling to a full meteorological time series.
Frequently Asked Questions
Q1: Can IEC 61725 be used for cloudy or overcast days?
No. IEC 61725 explicitly models clear-sky (or average) conditions. For cloudy conditions, the instantaneous irradiance is dominated by stochastic cloud attenuation, which cannot be represented by a deterministic analytical function. The standard’s intended use is with monthly average daily irradiation values, where cloud effects are already integrated into the daily total. For sub-hourly cloudy-time simulation, stochastic models or measured time series are required.
Q2: How is day length (ts − tr) determined for the IEC 61725 profile?
Day length is calculated from standard solar geometry using the site latitude and solar declination (which depends on the day of year). The standard references the solar geometry equations from IEC 61724-1 and ISO 9488. For monthly average calculations, the day length on the 15th or 21st day of the month is typically used as representative. At the equator, day length is approximately 12 hours year-round; at 45° latitude, it varies from about 9 hours (winter solstice) to 15 hours (summer solstice).
Q3: Does IEC 61725 account for the difference between solar time and clock time?
The standard defines the profile in solar time, not clock time. The conversion between solar time and local clock time requires the equation of time correction (which accounts for the Earth’s elliptical orbit and axial tilt) and the longitude correction (4 minutes per degree of longitude difference from the local time zone meridian). Engineers should apply these corrections when comparing the modelled profile against clock-time measurements from a PV system.
Q4: How should the exponent π be determined for a specific location?
IEC 61725 recommends three approaches: (1) Use the default value of π = 1.25 when no site-specific information is available; (2) Derive π from the Linke turbidity factor or Angstrom turbidity coefficient, which are available from global maps and satellite-derived databases; or (3) Calibrate π by fitting the analytical expression to measured clear-sky irradiance profiles from a local weather station, using least-squares regression over multiple clear days. Approach 3 provides the highest accuracy for critical applications.
