Diffuse-Direct Decomposition

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Summary

Diffuse-Direct decomposition models separate global horizontal irradiance (GHI) into its direct normal irradiance (DNI) and diffuse horizontal irradiance (DHI) components. PlantPredict implements three decomposition models: Erbs, Reindl, and DIRINT. These models use empirically derived relationships based on the clearness index and atmospheric parameters to estimate the diffuse fraction of GHI under varying sky conditions.

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Inputs

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Outputs

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Detailed Description

All decomposition models begin by computing the clearness index, representing the fraction of extraterrestrial irradiance reaching the ground: $$K_t = \frac{GHI}{DNI_{extra} \cos(\theta_z)}$$ where $\theta_z$ is the solar zenith angle. $K_t$ is set to 0 if $\theta_z \geq 87.9°$. The clearness index characterizes sky conditions from overcast ($K_t < 0.3$) to clear ($K_t > 0.7$).

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Erbs Model

The Erbs model uses only clearness index to compute diffuse fraction $f_d$ using a piecewise polynomial function: Once $f_d$ is determined, the DHI is calculated, then the DNI is obtained from the closure equation: $$DHI = f_d \times GHI$$ $$DNI = \frac{GHI - DHI}{\cos(\theta_z)}$$

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Reindl Model

The Reindl model extends the clearness index approach by incorporating solar zenith angle, air temperature, and relative humidity, accounting for sun position and atmospheric effects on scattering. When relative humidity is provided in the weather file (fraction from 0 to 1): When relative humidity is not provided in the weather file: Physical limits: The empirical correlations can produce non-physical values at edge cases. The following bounds from Reindl et al. (1990) are applied to clamp results: PlantPredict uses 0.83 as the upper clearness index threshold (modified from 0.78 in the original published model). Once $f_d$ is determined, the DHI is calculated, then the DNI is obtained from the closure equation: $$DHI = f_d \times GHI$$ $$DNI = \frac{GHI - DHI}{\cos(\theta_z)}$$

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DIRINT Model

The DIRINT (Direct Insolation Radiation INTegration) model is an enhancement of the DISC model. It first computes an initial DNI estimate using DISC, then applies a correction factor based on temporal stability and atmospheric conditions. This allows DIRINT to distinguish between steady hazy conditions and variable cloudy conditions that have similar instantaneous clearness indices. DIRINT uses pressure-corrected air mass $AM'$ as an input, calculated internally using the Bird-Hulstrom formula (see Air Mass for details). Step 1: DISC Initial Estimate The DISC (Direct Insolation Simulation Code) model computes an initial DNI estimate using direct normal transmittance factors:

• $K_{nc}$ = Clear-sky transmittance: theoretical maximum under clear conditions, decreasing with air mass
• $\Delta K_n$ = Transmittance reduction: correction for clouds, aerosols, and turbidity (derived from $K_t$)
• $K_n$ = Actual transmittance: net transmittance after atmospheric effects

The clear-sky transmittance is a polynomial function of air mass: $$K_{nc} = 0.866 - 0.122 AM' + 0.0121 AM'^2 - 0.000653 AM'^3 + 0.000014 AM'^4$$ The transmittance reduction uses coefficients $A$, $B$, $C$ that depend on $K_t$: $$\Delta K_n = A + B e^{C \times AM'}$$ where each coefficient $X \in \{A, B, C\}$ is computed as: $$X = c_0 + c_1 K_t + c_2 K_t^2 + c_3 K_t^3$$ The actual transmittance and DNI are then: $$K_n = K_{nc} - \Delta K_n$$ $$DNI_{DISC} = K_n \times DNI_{extra}$$ $DNI_{DISC} = 0$ if any of the following conditions are met:

• $GHI < 1$ W/m²
• $DNI_{DISC} < 0$
• $\theta_z > 87°$ (Version 09 and earlier) or $\theta_z > 90°$ (Version 10 and later)

Step 2: DIRINT Correction Coefficient Lookup DIRINT improves upon DISC by using four parameters to look up a correction coefficient $C_{DIRINT}$: modified clearness index ($K'_t$), temporal stability ($\Delta K'_t$), zenith angle ($\theta_z$), and precipitable water ($W$, not used by PlantPredict). The modified clearness index normalizes for air mass effects, isolating the influence of clouds and excess atmospheric turbidity: $$K'_t = \frac{K_t}{1.031 \exp\left(-\frac{1.4}{0.9 + 9.4/AM'}\right) + 0.1}$$ with constraint $K'_t \leq 0.82$. Temporal stability captures cloud transients by comparing clearness across adjacent timestamps: $$\Delta K'_t = 0.5 \left( |K'_t - K'_{t,next}| + |K'_t - K'_{t,prev}| \right)$$ A four-dimensional lookup table retrieves the DIRINT correction coefficient $C_{DIRINT}$ based on binned values of these four parameters. PlantPredict does not use precipitable water as an input for the model and defaults to the “precipitable water unavailable” bin. Step 3: Final DNI and DHI Calculation The final DNI is the DISC estimate scaled by the DIRINT correction coefficient: $$DNI = DNI_{DISC} \times C_{DIRINT}$$ DHI is then derived using the closure equation: $$DHI = GHI - DNI \cos(\theta_z)$$

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Physical Constraints

All decomposition models apply the following physical constraints:

• If $GHI \leq 0$, then $DHI = 0$ and $DNI = 0$
• $DHI \leq GHI$
• $DNI \geq 0$
• If $\theta_z \geq 90°$, then $DNI = 0$

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References

• Erbs, D. G., Klein, S. A., & Duffie, J. A. (1982). Estimation of the diffuse radiation fraction for hourly, daily and monthly-average global radiation. Solar Energy, 28(4), 293–302.
• Reindl, D. T., Beckman, W. A., & Duffie, J. A. (1990). Diffuse fraction correlations. Solar Energy, 45(1), 1–7.
• Perez, R., Ineichen, P., Maxwell, E., Seals, R., & Zelenka, A. (1992). Dynamic global-to-direct irradiance conversion models. ASHRAE Transactions, 98(1), 354–369.
• Maxwell, E. L. (1987). A quasi-physical model for converting hourly global horizontal to direct normal insolation. Technical Report SERI/TR-215-3087, Solar Energy Research Institute.