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Explicit Expressions for Solar Panel Equivalent Circuit Parameters Based on Analytical Formulation and the Lambert W-Function^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Methods

**Figure 2.**I-V curve of a solar panel. The three characteristic points (short circuit, maximum power, and open circuit points) are indicated on the curve.

_{pv}is the photocurrent delivered by the constant current source; I

_{0}is the reverse saturation current corresponding to the diode; R

_{s}is the series resistor that takes into account losses in cell solder bonds, interconnection, junction box, etc.; R

_{sh}is the shunt resistor that takes into account the current leakage through the high conductivity shunts across the p-n junction; and a is the ideality factor that takes into account the deviation of the diodes from the Shockley diffusion theory. V

_{T}is not an unknown parameter; it is the thermal voltage of the diode and depends on the charge of the electron, q, the Boltzmann constant, k; the number of cells in series, n; and the temperature, T:

#### 2.1. Parameter Calculation

- Short circuit equation:
- Open circuit equation:
- Maximum power point circuit equation:
- Zero derivative for the power at maximum power point circuit equation:

- Equation for R
_{s}: - Equation for R
_{sh}: - Equation for I
_{0}: - Equation for I
_{pv}:

_{s}. This equation is not difficult to solve with numerical calculation programs (e.g., Matlab or Maple). Nevertheless, it is possible to transform it into an explicit equation using the Lambert W-function [47]. The Lambert W-function, W(z), is defined as:

^{W(z)}

_{0}(x) for W(x) ≥ −1 and W

_{−1}(x) for W(x) ≤ −1 in the aforementioned bracket. The general strategy to apply the Lambert W-function in solving exponential equations is to use the following equivalence:

^{Y}⇔ Y = W(X)

_{−1}is the negative branch of the Lambert W-function (as the left part of Equation (14) is lower than −1 for typical cells and solar panels). And then, an explicit expression for R

_{s}is obtained:

_{s}= A(W

_{−1}(Bexp(C)) − (D + C))

_{s}from the remaining implicit equation after obtaining explicit equations for the other parameters. Then, a fully decoupled set of explicit equations for the equivalent circuit parameters can be obtained. Data from two solar cells, included in Table 1, have been used to validate the results obtained following the procedure described in the text. Also, the accuracy of the method can be checked. In Table 2, the values of the equivalent circuit parameters obtained are compared to the ones from [17], obtained through a curve fitting technique and used as reference for the parameter a, and to the results from another method based on the Lambert W-function, with the difference that this other one is iterative (and not explicit) [50].

**Table 1.**Characteristic of blue solar cells and grey solar cells [17].

Blue Solar Cell | Grey Solar Cell | ||
---|---|---|---|

a | 1.51 ± 0.07 | a | 1.72 ± 0.08 |

I_{mp} (A) | 0.0934 | I_{mp} (A) | 0.485 |

V_{mp} (V) | 0.433 | V_{mp} (V) | 0.387 |

I_{sc} (A) | 0.1023 | I_{sc} (A) | 0.561 |

V_{oc} (V) | 0.536 | V_{oc} (V) | 0.524 |

T_{r} (K) | 300 | T_{r} (K) | 307 |

**Table 2.**Characteristic of blue solar cells and grey solar cells [17].

Blue Solar Cell | Grey Solar Cell | ||||||
---|---|---|---|---|---|---|---|

Equivalent Circuit Parameters | Benchmark [17] | Iterative Method [50] | Present Method | Equivalent Circuit Parameters | Benchmark [17] | Iterative Method [50] | Present Method |

R_{s} (Ω) | 0.07 ± 0.009 | 0.0671 | 0.0652 | R_{s} (Ω) | 0.08 ± 0.01 | 0.0784 | 0.0781 |

R_{sh} (Ω) | 1000 ± 50 | 977 | 1093 | R_{sh} (Ω) | 26 ± 1 | 26.09 | 26.25 |

I_{pv} (A) | 0.1023 ± 0.0005 | 0.1023 | 0.1023 | I_{pv} (A) | 0.5625 ± 0.0005 | 0.561 | 0.5627 |

I_{0} (A) | 110 ± 50 × 10^{−9} | 111 × 10^{−9} | 111 × 10^{−9} | I_{0} (A) | 6 ± 3 × 10^{−6} | 5.6 × 10^{−6} | 5.4 × 10^{−6} |

#### 2.2. Parameter Dependence on Environmental Conditions

_{r}is the reference temperature; βV

_{oc}and βV

_{mp}are respectively the percentage variation of the open circuit and maximum power point voltages when the temperature increases one degree; finally, αI

_{sc}and αI

_{mp}are the percentage variation of the short circuit and maximum power point currents when the temperature increases one degree. Some manufacturers include the percentage variation of the maximum power with temperature, γ, instead of specific variations for current and voltage at that point. If αI

_{mp}or βV

_{mp}(one of them) is missing, then the variation of the boundary condition can be calculated using the parameter γ and the remaining condition, for example: I

_{mp}value in terms of γ and βV

_{mp}is:

_{mp}and βV

_{mp}are not included in the manufacturer’s datasheet and only the parameter γ is provided, it can be assumed that βV

_{mp}≈ βV

_{oc}before using Equation (18). With these relations it is possible to directly relate the equivalent circuit parameters and the rate of variations with T [22,44]. Nevertheless, a different approximation is used in the present work: the equivalent circuit parameters are calculated for different temperatures with Equation (17), using the values of the characteristic points at those temperatures. Then, the variation of the parameters as a function of temperature T is empirically determined based on these results. This procedure is very accurate, as the calculated equivalent circuits directly meet the manufacturers’ data.

_{sc}linearly and V

_{oc}exponentially, whereas R

_{s}remains not affected for temperature variations [51]. Those conditions lead to the following equation [44]:

_{pv}

_{,G}is the photocurrent delivered by the current source of the equivalent circuit, and G

_{r}and I

_{pv}

_{,Gr}are the reference values. When data about irradiation dependence is available, it is generally shown as experimental I-V curves, in this case it would be possible to extract information about the characteristic points graphically, and therefore define the dependence on irradiation with a similar procedure to the one followed in case of temperature variations.

## 3. Results and Discussion

- Estimate the value of the parameter a. In this method this parameter is considered as a constant, independent of the temperature and the irradiation. In this case the chosen value is a = 1.1.
- Calculate V
_{t}for the panel at that temperature level with Equation (2). - Determine the boundary conditions at that temperature and irradiation level using manufacturer’s data and Equations (17) and/or (18).
- Calculate R
_{s}with Equation (15). - Calculate R
_{sh}with Equation (8). - Calculate I
_{0}with Equation (9). - Calculate I
_{pv}with Equation (10).

MSP290AS-36.EU (Multicrystalline) | MSMD290AS-36.EU (Monocrystalline) | ||||||
---|---|---|---|---|---|---|---|

n | 72 | T_{r} (°C) | 25 | n | 72 | T_{r} (°C) | 25 |

P_{mp} (W) | 290 | γ (%/°C) | −0.45 | P_{mp} (W) | 290 | γ (%/°C) | −0.44 |

I_{mp} (A) | 7.82 | αI_{mp} (%/°C) | - | I_{mp} (A) | 7.70 | αI_{mp} (%/°C) | - |

V_{mp} (V) | 37.08 | βV_{mp} (%/°C) | −0.35 | V_{mp} (V) | 37.66 | βV_{mp} (%/°C) | −0.35 |

I_{sc} (A) | 8.37 | αI_{sc} (%/°C) | +0.04 | I_{sc} (A) | 8.24 | αI_{sc} (%/°C) | +0.04 |

V_{oc} (V) | 44.32 | βV_{oc} (mV/°C) | −0.33 | V_{oc} (V) | 44.68 | βV_{oc} (mV/°C) | −0.31 |

^{2}) and T

_{r}= 25 °C, are included in Table 4.

**Table 4.**Parameters of MSP290AS-36.EU and MSMD290AS-36.EU (München Solarenergie GmbH) solar panels equivalent circuits at STC (1000 W/m

^{2}irradiance, 25 °C cell temperature, AM1.5g spectrum).

MSP290AS-36.EU (Multicrystalline) | MSMD290AS-36.EU (Monocrystalline) | ||
---|---|---|---|

a | 1.10 | a | 1.10 |

I_{pv} (A) | 8.37 | I_{pv} (A) | 8.24 |

I_{0} (A) | 2.86 × 10^{−9} | I_{0} (A) | 2.36 × 10^{−9} |

R_{s} (Ω) | 0.162 | R_{s} (Ω) | 0.130 |

R_{sh} (Ω) | 331 | R_{sh} (Ω) | 316 |

_{s}turns negative for high temperatures (the value of parameter R

_{sh}could also turn negative under some conditions). As previously suggested, those mathematically correct but not physically valid solutions at high temperatures are derived in cases where the boundary conditions (slopes of the I-V curve at short circuit and open circuit points), are not compatible with the curvature determined by the chosen ideality factor.

- For MSP290AS-36.EU:I
_{pv}(T) = 8.37 + 3.62·10^{−3}∆T − 3.38·10^{−6}∆T^{2}− 7.58·10^{−8}∆T^{3}

R_{s}(T) = 1.62·10^{−1}− 3.21·10^{−3}∆T + 7.05·10^{−7}∆T^{2}− 3.01·10^{−8}∆T^{3}

R_{sh}(T) = 1/(3.03·10^{−3}+ 2.65·10^{−4}∆T + 1.50·10^{−6}∆T^{2}+ 1.56·10^{−8}∆T^{3})

I_{0}(T) = exp(−1.97·10^{1}+ 1.44·10^{−1}∆T − 4.80·10^{−4}∆T^{2}+ 1.15·10^{−6}∆T^{3}) - For MSMD290AS-36.EU:I
_{pv}(T) = 8.24 + 3.49·10^{−3}∆T − 1.68·10^{−6}∆T^{2}− 2.41·10^{−8}∆T^{3}

R_{s}(T) = 1.30·10^{−1}− 1.97·10^{−3}∆T + 2.53·10^{−6}∆T^{2}− 1.07·10^{−8}∆T^{3}

R_{sh}(T) = 1/(3.18·10^{−3}+ 2.33·10^{−4}∆T + 1.27·10^{−6}∆T^{2}+ 1.33·10^{−8}∆T^{3})

I_{0}(T) = exp(−1.98·10^{1}+ 1.41·10^{−1}∆T − 4.69·10^{−4}∆T^{2}+ 1.13·10^{−6}∆T^{3})

_{pv}and can be described with Equation (19), including this variation the following equations can be then derived for the five parameters of the equivalent circuit:

- For MSP290AS-36.EU:
- For MSMD290AS-36.EU:

**Figure 3.**Calculated values (using the explained methodology) of the equivalent circuit parameters R

_{s}, R

_{sh}, I

_{pv}, and I

_{0}, for multicrystalline (MSP290AS-36.EU) and monocrystalline (MSMD290AS-36.EU) solar cell panels as a function of the temperature, T. Calculated points are indicated with symbols whereas the polynomial approximations fitted to those data (Equations (20) and (21)) have been included in each case as solid lines.

## 4. Experimental Section

**Figure 6.**I-V curves of panel MSP290AS-36.EU simulated with LTSpice calculation of the equivalent circuits for different irradiation levels, from 200 W/m

^{2}to 1000 W/m

^{2}, at 25 °C.

**Figure 7.**Power curves of panel MSP290AS-36.EU simulated with LTSpice calculation of the equivalent circuits for different temperature levels, from 15 °C to 70 °C, with 1000 W/m

^{2}irradiation level.

**Figure 8.**I-V curves of panel MSMD290AS-36.EU simulated with LTSpice calculation of the equivalent circuits for different irradiation levels, from 200 W/m

^{2}to 1000 W/m

^{2}, at 25 °C.

**Figure 9.**Power curves of panel MSMD290AS-36.EU simulated with LTSpice calculation of the equivalent circuits for different temperature levels, from 15 °C to 85 °C, with 1000 W/m

^{2}irradiation level.

**Figure 10.**Variation of multicrystalline (MSP290AS-36.EU) and monocrystalline (MSMD290AS-36.EU) solar panel characteristic data, I

_{sc}, V

_{oc}, P

_{mp}, and V

_{mp}as a function of the temperature, T. Symbols represent the characteristic points of calculated equivalent circuits (parameters defined in Equations (22) and (23)), whereas solid lines represent the experimental temperature dependence of these points included in the manufacturer datasheet (see Table 3 and Equations (17) and (18)).

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Cubas, J.; Pindado, S.; De Manuel, C.
Explicit Expressions for Solar Panel Equivalent Circuit Parameters Based on Analytical Formulation and the Lambert W-Function. *Energies* **2014**, *7*, 4098-4115.
https://doi.org/10.3390/en7074098

**AMA Style**

Cubas J, Pindado S, De Manuel C.
Explicit Expressions for Solar Panel Equivalent Circuit Parameters Based on Analytical Formulation and the Lambert W-Function. *Energies*. 2014; 7(7):4098-4115.
https://doi.org/10.3390/en7074098

**Chicago/Turabian Style**

Cubas, Javier, Santiago Pindado, and Carlos De Manuel.
2014. "Explicit Expressions for Solar Panel Equivalent Circuit Parameters Based on Analytical Formulation and the Lambert W-Function" *Energies* 7, no. 7: 4098-4115.
https://doi.org/10.3390/en7074098