Single Diode Model

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Summary

The Single Diode Model calculates DC power output using a five-parameter equivalent circuit representing a PV module. The model solves the circuit equation iteratively to find the maximum power point voltage and current. PlantPredict implements three variants: OneDiode (standard), OneDiodeRecombination (with Martin & Ruiz recombination), and OneDiodeRecombinationNonLinear (with polynomial gamma). The solution uses Newton-Raphson iteration with convergence tolerance of 0.0001 V, typically converging in less than 2000 iterations. The output includes maximum power point values ($V_{mp}$, $I_{mp}$, $P_{mp}$) and open-circuit voltage ($V_{oc}$).

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Inputs

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Outputs

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Detailed Description

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Standard Single Diode Model (OneDiode)

Circuit equation: $$I = I_{ph} - I_0 \left( e^{\alpha V} - 1 \right) - \frac{V}{R_{sh}}$$ where $V$ is internal voltage: $V = V_{terminal} + I R_s$ Maximum power condition: Find voltage $V$ where $\frac{dP}{dV} = 0$ using Newton-Raphson iteration: $$V_{n+1} = V_n - \frac{f'(V_n)}{f''(V_n)}$$ where $f(V) = \frac{dP}{dV}$ and derivatives calculated from circuit equation. Iterative solution: Starting from $V_2 = 1$ V, iterate: $$V_2 = V_1 - \frac{\text{numerator}}{\text{denominator}}$$ Numerator: $$\left( \frac{(-1 + e^{V_1 \alpha}) I_0 R_s R_{sh} - I_{ph} R_s R_{sh} + (R_s + R_{sh}) V_1}{R_{sh}^2} \right) \left( 1 + e^{V_1 \alpha} I_0 R_{sh} \alpha \right)$$ $$+ \left( e^{V_1 \alpha} I_0 R_{sh} - (I_0 + I_{ph}) R_{sh} + V_1 \right) \left( R_s + R_{sh} + e^{V_1 \alpha} I_0 R_s R_{sh} \alpha \right)$$ Denominator: $$(\alpha^2 I_0 R_s e^{V_1 \alpha}) (I_{ph} - I_0 (e^{V_1 \alpha} - 1) - V_1 / R_{sh})$$ $$+ (1 + \alpha I_0 R_s e^{V_1 \alpha} + R_s / R_{sh}) (-I_0 \alpha e^{V_1 \alpha} - 1 / R_{sh})$$ $$+ (1 + I_0 R_s \alpha e^{V_1 \alpha} + R_s / R_{sh}) (-I_0 \alpha e^{V_1 \alpha} - 1 / R_{sh})$$ $$+ (V_1 - I_{ph} R_s + I_0 R_s e^{V_1 \alpha} - I_0 R_s + V_1 R_s / R_{sh}) (-I_0 \alpha^2 e^{V_1 \alpha})$$ Convergence: Continue until $|V_1 - V_2| < 0.0001$ V Failure handling: If $V_2 = \infty$ or NaN or 2000 iterations reached:

• Reset: $V_2 = hold + 1$
• If $hold > 500$, exit with error

Current calculation: $$I_{mp} = I_{ph} - I_0 (e^{\alpha V_2} - 1) - \frac{V_2}{R_{sh}}$$ Round to 4 decimal places. Validation: If $I_{mp} > I_{ph}$ or $I_{mp} < 0$, restart iteration with $V_2 = hold + 1$ Terminal voltage: $$V_{mp} = V_2 - I_{mp} R_s$$ Round to 4 decimal places. Power: $$P_{mp} = V_{mp} \times I_{mp}$$ Open-circuit voltage: Solve for $V$ when $I = 0$ using Lambert W function method.

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Advanced Diode Model (OneDiodeRecombination)

Circuit equation with recombination: $$I = I_{ph} - I_0 (e^{\alpha V} - 1) - \frac{V}{R_{sh}} - \frac{di_2/u_t \cdot I_{ph}}{V_{bi,scaled} - V}$$ where:

• $V_{bi,scaled} = V_{bi} \times N_s$
• $di_2/u_t$ is recombination parameter
• $V_{bi}$ is built-in voltage per cell

Newton-Raphson iteration: $$V_2 = V_1 - \frac{f'(V_1)}{f''(V_1)}$$ First derivative: $$f' = \left( (-1 + e^{V_1 \alpha}) I_0 R_s + \frac{(R_s + R_{sh}) V_1}{R_{sh}} + I_{ph} R_s \left(-1 + \frac{di_2/u_t}{V_{bi,scaled} - V_1}\right) \right)$$ $$\times \left( -\frac{1}{R_{sh}} - \frac{di_2/u_t \cdot I_{ph}}{(V_{bi,scaled} - V_1)^2} - e^{V_1 \alpha} I_0 \alpha \right)$$ $$+ \left( I_0 - e^{V_1 \alpha} I_0 + I_{ph} - \frac{V_1}{R_{sh}} - \frac{di_2/u_t \cdot I_{ph}}{V_{bi,scaled} - V_1} \right)$$ $$\times \left( 1 + R_s \left( \frac{1}{R_{sh}} + \frac{di_2/u_t \cdot I_{ph}}{(V_{bi,scaled} - V_1)^2} + e^{V_1 \alpha} I_0 \alpha \right) \right)$$ Second derivative (similar expansion with additional recombination terms). Current calculation: $$I_{mp} = I_{ph} - I_0 (e^{\alpha V_2} - 1) - \frac{V_2}{R_{sh}} - \frac{di_2/u_t \cdot I_{ph}}{V_{bi,scaled} - V_1}$$ Round to 5 decimal places. Convergence and validation: Same as standard model. Open-circuit voltage: Solve using advanced diode Lambert W method.

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Nonlinear Gamma Model (OneDiodeRecombinationNonLinear)

Uses same recombination equations but with gamma calculated using polynomial temperature dependence before solving.

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References

• King, D. L., Boyson, W. E., & Kratochvil, J. A. (2004). Photovoltaic array performance model. SAND2004-3535, Sandia National Laboratories.
• De Soto, W., Klein, S. A., & Beckman, W. A. (2006). Improvement and validation of a model for photovoltaic array performance. Solar Energy, 80(1), 78–88.
• 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.