Perez Model

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Summary

The Perez transposition model estimates sky diffuse irradiance on a tilted surface using empirical coefficients derived from extensive sky radiance measurements. The model decomposes sky diffuse into three components: isotropic background, circumsolar brightening near the solar disk, and horizon brightening near the horizon band. Sky clearness and brightness indices determine which coefficient set is applied. PlantPredict supports multiple Perez coefficient sets derived from different locations and time periods to optimize accuracy for specific climatic conditions.

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Inputs

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Outputs

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Detailed Description

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Sky Clearness and Brightness Indices

The Perez model uses two atmospheric parameters to characterize sky conditions: Sky Clearness Index ($\varepsilon$): Represents the clarity of the atmosphere, accounting for the ratio of total to diffuse horizontal irradiance and solar zenith angle: $$\varepsilon = \frac{\frac{DHI + DNI}{DHI} + \kappa \theta_z^3}{1 + \kappa \theta_z^3}$$ where $\theta_z$ is in radians and $\kappa = 1.041$ is an empirical constant. Sky Brightness Index ($\Delta$): Represents the amount of diffuse irradiance relative to extraterrestrial irradiance, normalized by air mass: $$\Delta = \frac{DHI \cdot m}{DNI_{extra}}$$ where $m$ is the relative (non pressure-corrected) air mass calculated using the selected air mass model.

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Perez Coefficient Lookup

The clearness index $\varepsilon$ is binned into 8 categories corresponding to sky conditions from overcast to clear. Each bin has associated empirical coefficients used to compute brightness coefficients $F_1$ and $F_2$: For each bin, coefficients $f_{11}, f_{12}, f_{13}, f_{21}, f_{22}, f_{23}$ are defined. PlantPredict supports multiple coefficient sets, each tailored to specific climatic conditions and derived from empirical measurements: Composite Coefficient Sets:

• PlantPredict: Default coefficient set based on Perez 1987/1988 formulation.
• All Sites Composite 1990: Comprehensive set derived from data across multiple locations using 1990 methodology.
• All Sites Composite 1988: Earlier composite set based on 1988 methodology from various sites.
• Sandia Composite 1988: Developed using data from Sandia National Laboratories.
• USA Composite 1988: Based on data collected from various U.S. locations.

Location-Specific Coefficient Sets (1988):

• France 1988: Derived from data collected in France.
• Phoenix 1988: Based on data from Phoenix, Arizona.
• El Monte 1988: Derived from data collected in El Monte, California.
• Osage 1988: Based on data from Osage, Iowa.
• Albuquerque 1988: Derived from data collected in Albuquerque, New Mexico.
• Cape Canaveral 1988: Based on data from Cape Canaveral, Florida.
• Albany 1988: Derived from data collected in Albany, New York.

The selection of coefficient set can influence model accuracy depending on site-specific climatic conditions. Composite sets provide general applicability, while location-specific sets may offer improved accuracy for sites with similar climate characteristics.

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Brightness Coefficients

The brightness coefficients $F_1$ and $F_2$ are computed using the binned coefficients: $$F_1 = f_{11} + f_{12} \Delta + f_{13} \theta_z$$ $$F_2 = f_{21} + f_{22} \Delta + f_{23} \theta_z$$ where $\theta_z$ is in radians. A quality control constraint is applied: $F_1 \geq 0$.

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Irradiance Components

Isotropic Component: Uniform diffuse irradiance from the sky dome, reduced by the fraction attributed to circumsolar and horizon brightening: $$E_{isotropic} = DHI \cdot \frac{1 - F_1}{2} \cdot (1 + \cos(\beta))$$ Circumsolar Component: Directional diffuse irradiance concentrated near the solar disk: $$E_{circumsolar} = DHI \cdot F_1 \cdot \frac{\max(\cos(\theta), 0)}{\max(\cos(\theta_z), \cos(85°))}$$ The denominator is constrained to prevent numerical instabilities near the horizon. Horizon Component: Diffuse irradiance from the horizon band: $$E_{horizon} = DHI \cdot F_2 \cdot \sin(\beta)$$ The horizon component can be negative, representing reduced diffuse irradiance when the tilted surface views less of the bright horizon band.

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Circumsolar Allocation

The circumsolar component can be allocated to beam or diffuse POA irradiance: Direct Allocation: $$E_{sky} = E_{isotropic} + E_{horizon}$$ $$E_{beam} = DNI \cdot \cos(\theta) + E_{circumsolar}$$ Diffuse Allocation: $$E_{sky} = E_{isotropic} + E_{circumsolar} + E_{horizon}$$ $$E_{beam} = DNI \cdot \cos(\theta)$$

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Quality Control

Version-dependent quality controls are applied:

• Version 11 and later: $E_{isotropic} \geq 0$, $E_{circumsolar} \geq 0$, $E_{horizon}$ is allowed to be negative.
• Version 10 and earlier: No minimum constraints on components.

Sky diffuse irradiance is capped at 800 W/m² to prevent unrealistically high values in edge cases.

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References

• Perez, R., Seals, R., Ineichen, P., Stewart, R., & Menicucci, D. (1987). A new simplified version of the Perez diffuse irradiance model for tilted surfaces. Solar Energy, 39(3), 221–231.
• Perez, R., Ineichen, P., Seals, R., Michalsky, J., & Stewart, R. (1990). Modeling daylight availability and irradiance components from direct and global irradiance. Solar Energy, 44(5), 271–289.
• Holmgren, W. F., Hansen, C. W., & Mikofski, M. A. (2018). pvlib python: A python package for modeling solar energy systems. Journal of Open Source Software, 3(29), 884.