Incidence Angle Modifier (IAM)

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Summary

The Incidence Angle Modifier (IAM) quantifies optical transmission losses when sunlight strikes a module at non-normal angles. PlantPredict implements five IAM models: None, ASHRAE, Sandia, Physical, and Custom Interpolation. The IAM is applied separately to beam, sky diffuse, and ground-reflected irradiance components after soiling losses and before spectral corrections. Beam IAM depends on the angle of incidence, while diffuse IAM factors are computed by integrating beam IAM over the sky and ground hemispheres using either JS-Bodo or Marion diffuse methods.

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Inputs

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Outputs

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Detailed Description

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Model 1: None

All IAM factors set to unity (no angle-dependent losses): $$UIAMB = UIAMD = UIAMG = 1$$

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Model 2: ASHRAE

The ASHRAE model uses a single parameter $b_0$ for beam IAM: $$UIAMB = 1 - b_0 \left( \frac{1}{\cos(\theta)} - 1 \right)$$ where $\theta$ is the angle of incidence in radians. Diffuse IAM factors are user-specified: $$UIAMD = \text{user-defined sky diffuse IAM}$$ $$UIAMG = \text{user-defined ground diffuse IAM}$$

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Model 3: Sandia

The Sandia model uses a 5th-order polynomial for beam IAM: $$UIAMB = \begin{cases} 1, & \theta < 34° \\ \sum_{i=0}^{5} b_i \theta^i, & \theta \geq 34° \end{cases}$$ where $\theta$ is in degrees and $[b_0, b_1, b_2, b_3, b_4, b_5]$ are Sandia B-factors. Diffuse IAM using JS-Bodo Method: Diffuse IAM factors are calculated using the JS-Bodo sky-view factor method with infinite row spacing: $$UIAMD = \frac{A_{sky}(b_0)}{A_{sky}(0)}$$ $$UIAMG = 1$$ where:

• $A_{sky}(b_0)$ is the sky view area with IAM effects (using ASHRAE $b_0$)
• $A_{sky}(0)$ is the sky view area without IAM effects
• Ground diffuse IAM is set to unity

Parameters for JS-Bodo calculation:

• Module tilt angle $\beta$
• Post-to-post spacing = $\infty$ (to eliminate shading effects)
• Collector bandwidth (row width)
• Table view height $g = 0.5$ (observation point at module center)

Quality control: $UIAMD = \max(UIAMD, 0)$

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Model 4: Physical

The Physical IAM model uses Fresnel equations and Beer-Lambert law (available in Version 11+). Fresnel Reflection at First Interface: Incidence angle: $\theta_1 = \theta$ (in radians) $$\cos(\theta_1) = \max(0, \cos(\theta))$$ $$\sin(\theta_1) = \sqrt{1 - \cos^2(\theta_1)}$$ Refraction angle at first interface (air to AR coating or glass): $$\sin(\theta_2) = \frac{n_1}{n_2} \sin(\theta_1)$$ $$\cos(\theta_2) = \sqrt{1 - \sin^2(\theta_2)}$$ where $n_1 = 1$ (air), $n_2 = n_{AR}$ if AR coating present, otherwise $n_2 = n$ (glass). Reflectance for s-polarized and p-polarized light: $$\rho_{12,s} = \left( \frac{n_1 \cos(\theta_1) - n_2 \cos(\theta_2)}{n_1 \cos(\theta_1) + n_2 \cos(\theta_2)} \right)^2$$ $$\rho_{12,p} = \left( \frac{n_1 \cos(\theta_2) - n_2 \cos(\theta_1)}{n_1 \cos(\theta_2) + n_2 \cos(\theta_1)} \right)^2$$ Reflectance at normal incidence: $$\rho_{12,0} = \left( \frac{n_1 - n_2}{n_1 + n_2} \right)^2$$ Transmittance through first interface: $$\tau_s = 1 - \rho_{12,s}$$ $$\tau_p = 1 - \rho_{12,p}$$ $$\tau_0 = 1 - \rho_{12,0}$$ AR Coating Internal Reflections: If AR coating present ($n_{AR} \neq n_1$), calculate second interface (AR to glass) with $n_3 = n$ (glass): $$\sin(\theta_3) = \frac{n_2}{n_3} \sin(\theta_2)$$ $$\cos(\theta_3) = \sqrt{1 - \sin^2(\theta_3)}$$ Second interface reflectance: $$\rho_{23,s} = \left( \frac{n_2 \cos(\theta_2) - n_3 \cos(\theta_3)}{n_2 \cos(\theta_2) + n_3 \cos(\theta_3)} \right)^2$$ $$\rho_{23,p} = \left( \frac{n_2 \cos(\theta_3) - n_3 \cos(\theta_2)}{n_2 \cos(\theta_3) + n_3 \cos(\theta_2)} \right)^2$$ $$\rho_{23,0} = \left( \frac{n_2 - n_3}{n_2 + n_3} \right)^2$$ Account for multiple internal reflections: $$\tau_s \leftarrow \tau_s \cdot \frac{1 - \rho_{23,s}}{1 - \rho_{23,s} \rho_{12,s}}$$ $$\tau_p \leftarrow \tau_p \cdot \frac{1 - \rho_{23,p}}{1 - \rho_{23,p} \rho_{12,p}}$$ $$\tau_0 \leftarrow \tau_0 \cdot \frac{1 - \rho_{23,0}}{1 - \rho_{23,0} \rho_{12,0}}$$ Beer-Lambert Absorption in Glass: $$\tau_s \leftarrow \tau_s \cdot e^{-K L / \cos(\theta_3)}$$ $$\tau_p \leftarrow \tau_p \cdot e^{-K L / \cos(\theta_3)}$$ $$\tau_0 \leftarrow \tau_0 \cdot e^{-K L}$$ where $K$ is extinction coefficient (m⁻¹) and $L$ is glass thickness (m). Beam IAM Factor: $$UIAMB = \frac{\tau_s + \tau_p}{2 \tau_0}$$ If $\theta \geq 90°$ and no AR coating ($n_2 = 1$), then $UIAMB = 0$. Diffuse IAM using Marion Method: Calculate beam IAM at discrete angles: $[0°, 30°, 50°, 60°, 70°, 75°, 80°, 85°, 90°]$ Integrate using Marion diffuse method: $$UIAMD = \text{MarionDiffuse}(\text{Sky}, \beta, \{\theta_i\}, \{UIAMB(\theta_i)\})$$ $$UIAMG = \text{MarionDiffuse}(\text{Ground}, \beta, \{\theta_i\}, \{UIAMB(\theta_i)\})$$ Results cached by tilt angle (rounded to nearest degree). Quality control: $UIAMD = \max(UIAMD, 0)$ if NaN; $UIAMG = 0$ if NaN.

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Model 5: Custom Interpolation

The Custom Interpolation model uses user-supplied IAM factor pairs. Beam IAM: For $0° < \theta < 90°$: $$UIAMB = \text{CubicSpline}(\{(\theta_i, f_i)\}, \theta)$$ $$UIAMB \leftarrow \min(UIAMB, 1)$$ For $\theta \leq 0°$ or $\theta \geq 90°$: $UIAMB = 0$ Diffuse IAM: Version 08 and Earlier: Uses JS-Bodo method with ASHRAE $b_0$ (same as Sandia diffuse calculation). Version 09 and Later: Uses Marion integration method with user-defined IAM table: $$UIAMD = \text{MarionDiffuse}(\text{Sky}, \beta, \{\theta_i\}, \{f_i\})$$ $$UIAMG = \text{MarionDiffuse}(\text{Ground}, \beta, \{\theta_i\}, \{f_i\})$$ Results cached by tilt angle. Quality control: $UIAMD = 0$ if NaN, $UIAMG = 0$ if NaN.

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Application to Irradiance

IAM factors are applied to irradiance components after soiling: $$E_{beam,IAM} = E_{beam,soiled} \times UIAMB$$ $$E_{sky,IAM} = E_{sky,soiled} \times UIAMD$$ $$E_{ground,IAM} = E_{ground,soiled} \times UIAMG$$

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References

• ASHRAE Standard 93-2010. Methods of Testing to Determine the Thermal Performance of Solar Collectors. American Society of Heating, Refrigerating and Air-Conditioning Engineers.
• King, D. L., Boyson, W. E., & Kratochvil, J. A. (2004). Photovoltaic array performance model. SAND2004-3535, Sandia National Laboratories.
• Martin, N., & Ruiz, J. M. (2001). Calculation of the PV modules angular losses under field conditions by means of an analytical model. Solar Energy Materials and Solar Cells, 70(1), 25–38.
• Marion, B. (2017). Numerical method for angle-of-incidence correction factors for diffuse radiation incident photovoltaic modules. Solar Energy, 147, 344–348.