Electrical Effect of Shading

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Summary

When modules in a string are partially shaded, electrical mismatch losses occur beyond the geometric beam shading. Shaded modules produce less current than unshaded modules, and because strings operate in series, the lowest-current module limits the entire string’s output. PlantPredict implements four electrical shading models: None (no mismatch losses), Linear (electrical loss equals beam shading), Fractional (fixed percentage loss when any shading occurs), and Step-Fractional (loss quantized by bay partitions). The electrical shading fraction is applied as an additional loss factor after geometric shading.

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Inputs

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Outputs

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Detailed Description

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Model 0: None

No electrical mismatch losses are calculated. Only geometric beam shading is applied. $$f_{elec} = 0$$ Use cases:

• Conservative baseline analysis
• Systems with module-level power electronics (MLPE)
• Preliminary design phase

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Model 1: Linear

Electrical shading fraction equals the beam shading fraction. This assumes electrical losses scale linearly with shaded area. $$f_{elec} = f_{beam}$$ Physical interpretation: Assumes each shaded module in a string reduces string current proportionally to its shaded fraction. This is a simplified approximation that underestimates mismatch losses in non-uniform shading conditions. Use cases:

• General engineering analysis
• Uniform shading patterns
• Conservative estimate of mismatch losses

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Model 2: Fractional

When any shading occurs (threshold: $f_{beam} > 0.01$), a fixed fractional loss is applied: $$f_{elec} = \begin{cases} 1 - (1 - f_{beam}) \times (1 - P_{frac}/100), & \text{if } f_{beam} > 0.01 \\ 0, & \text{otherwise} \end{cases}$$ where $P_{frac}$ is the user-specified fractional shading percentage (default: 50%). Expanded form: $$f_{elec} = f_{beam} + (1 - f_{beam}) \times \frac{P_{frac}}{100} \quad \text{if } f_{beam} > 0.01$$ Physical interpretation: This model assumes that when partial shading occurs, the electrical mismatch loss is disproportionately higher than the geometric shading. The fractional parameter represents the additional loss beyond beam shading. Example: If $f_{beam} = 0.10$ (10% beam shading) and $P_{frac} = 50\%$: $$f_{elec} = 0.10 + (1 - 0.10) \times 0.50 = 0.55$$ Total shading loss is 55% (10% from beam shading + 45% from electrical mismatch). Use cases:

• Systems with severe mismatch losses
• Non-uniform shading patterns
• Worst-case scenario analysis

Note: This model is described as “not physical” in the source code, meaning it does not derive from first-principles electrical modeling but rather represents an empirical upper bound on mismatch losses.

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Model 3: Step-Fractional

The bay is divided into $N_{frac}$ partitions (fractions). When shading affects any portion of a partition, the entire partition is considered lost. This creates quantized (stepped) electrical losses. Partition Calculation: $$f_{partition} = \frac{1}{N_{frac}} \times \lceil N_{frac} \times f_{beam} \rceil$$ where $\lceil \cdot \rceil$ is the ceiling function (round up to nearest integer). Electrical Shading Fraction: $$f_{elec} = \begin{cases} f_{partition}, & \text{if } f_{beam} > 0.005 \\ 0, & \text{otherwise} \end{cases}$$ Physical interpretation: Represents string/bay architecture where modules are grouped into partitions with bypass diodes or optimization. When any module in a partition is shaded, the entire partition operates at reduced output. Example: If $N_{frac} = 4$ (bay divided into 4 partitions) and $f_{beam} = 0.15$ (15% beam shading): $$f_{partition} = \frac{1}{4} \times \lceil 4 \times 0.15 \rceil = \frac{1}{4} \times \lceil 0.6 \rceil = \frac{1}{4} \times 1 = 0.25$$ Total shading loss is 25% (one full partition lost). Use cases:

• Systems with defined string partitioning
• Module-level or optimizer-based systems
• Bay-level bypass diode configurations

Sensitivity to $N_{frac}$:

• $N_{frac} = 1$: Entire bay lost if any shading (worst case)
• $N_{frac} = 2$: Half-bay partitions
• $N_{frac} = 4$: Quarter-bay partitions (typical)
• $N_{frac} \to \infty$: Approaches linear model

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Application to Energy Calculations

The electrical shading fraction is an additional loss beyond beam shading: $$f_{total} = \max(f_{beam}, f_{elec})$$ This ensures that total shading loss is at least as large as the beam shading loss (electrical losses cannot reduce total losses). The total shading factor applied to irradiance is: $$U_{total} = 1 - f_{total}$$ Calculation Sequence:

1. Calculate geometric beam shading: $f_{beam}$
2. Apply beam shading to POA irradiance: $I_{beam,shaded} = I_{beam,POA} \times (1 - f_{beam})$
3. Calculate electrical shading: $f_{elec} = g(f_{beam}, \text{model parameters})$
4. Compute total shading: $f_{total} = \max(f_{beam}, f_{elec})$
5. Apply electrical effect to DC power calculation (after IAM, spectral, temperature corrections)

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Aggregation Across Bays

For site-level shading results, bay-level electrical shading fractions are aggregated: Average per Inverter: $$\bar{f}_{elec,inv} = \frac{\sum_{i \in \text{inverter}} A_i \cdot f_{elec,i}}{\sum_{i \in \text{inverter}} A_i}$$ where $A_i$ is the area of bay $i$. Average per Timestamp: $$\bar{f}_{elec,t} = \frac{\sum_{i=1}^{N_{bays}} A_i \cdot f_{elec,i}}{\sum_{i=1}^{N_{bays}} A_i}$$ Aggregation is area-weighted to account for different bay sizes.

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References

• Deline, C., Sekulic, W., Stein, J., Barkaszi, S., Yang, J., & Kahn, S. (2014). Evaluation of maxim module-integrated electronics at the DOE Regional Test Centers. Proceedings of the IEEE 40th Photovoltaic Specialist Conference, 3259–3264.
• Stein, J. S., Hansen, C. W., & Reno, M. J. (2012). The Sandia Array Performance Model (SAPM). SAND2012-2389, Sandia National Laboratories.
• King, D. L., Boyson, W. E., & Kratochvil, J. A. (2004). Photovoltaic array performance model. SAND2004-3535, Sandia National Laboratories.