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Summary

The angle of incidence is the angle between incoming solar rays and the normal (perpendicular) to the module surface. Accurate calculation of the angle of incidence is essential for determining direct beam irradiance on the plane-of-array, applying incidence angle modifiers, and optimizing tracker positions. PlantPredict calculates incidence angles for both fixed-tilt and tracking systems using vector mathematics and spherical trigonometry.

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Inputs

NameSymbolUnitsDescription
Solar Zenith Angleθz\theta_zdegreesAngle between sun and local vertical
Solar Azimuth Angleγs\gamma_sdegreesHorizontal angle of sun, clockwise from North
Surface Tilt Angleβ\betadegreesAngle of module surface from horizontal
Surface Azimuth Angleγ\gammadegreesHorizontal angle of surface normal, clockwise from North

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Outputs

NameSymbolUnitsDescription
Angle of Incidenceθ\thetadegreesAngle between sun’s rays and surface normal
Projection Factorcos⁡(θ)\cos(\theta)—Cosine of AOI, used for beam irradiance calculation

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Detailed Description

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Vector Formulation

The angle of incidence is calculated using the dot product of the solar position vector and the surface normal vector. Solar position unit vector: s⃗=[sin⁡(θz)sin⁡(γs)cos⁡(θz)sin⁡(θz)cos⁡(γs)]\vec{s} = \begin{bmatrix} \sin(\theta_z) \sin(\gamma_s) \\ \cos(\theta_z) \\ \sin(\theta_z) \cos(\gamma_s) \end{bmatrix} Surface normal unit vector: n⃗=[sin⁡(β)sin⁡(γ)cos⁡(β)sin⁡(β)cos⁡(γ)]\vec{n} = \begin{bmatrix} \sin(\beta) \sin(\gamma) \\ \cos(\beta) \\ \sin(\beta) \cos(\gamma) \end{bmatrix} Angle of incidence: cos⁡(θ)=s⃗⋅n⃗\cos(\theta) = \vec{s} \cdot \vec{n} cos⁡(θ)=sin⁡(β)sin⁡(θz)cos⁡(γs−γ)+cos⁡(β)cos⁡(θz)\cos(\theta) = \sin(\beta) \sin(\theta_z) \cos(\gamma_s - \gamma) + \cos(\beta) \cos(\theta_z) θ=arccos⁡(cos⁡(θ))\theta = \arccos(\cos(\theta))

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Beam Irradiance Application

Direct beam irradiance on the tilted surface: IbPOA=DNI×max⁡(0,cos⁡(θ))I_b^{POA} = DNI \times \max(0, \cos(\theta)) The max⁡(0,cos⁡(θ))\max(0, \cos(\theta)) ensures zero irradiance when sun is behind the module.

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References

  • Duffie, J. A., & Beckman, W. A. (2013). Solar Engineering of Thermal Processes (4th ed.). John Wiley & Sons.
  • Marion, B., & Dobos, A. (2013). Rotation Angle for the Optimum Tracking of One-Axis Trackers. NREL/TP-6A20-58891.