#
Symmetry in Regression Analysis: Perpendicular Offsets—The Case of a Photovoltaic Cell^{ †}

^{1}

^{2}

^{†}

## Abstract

**:**

## 1. Introduction

#### 1.1. History

#### 1.2. Today

#### 1.3. Aim

## 2. Materials and Methods

#### 2.1. Single-Diode, Double-Diode, and PV Generator Models

#### 2.2. Lambert Function

#### 2.3. Explicit Equations

#### 2.4. Limit Cases

_{4}symmetry operation).

#### 2.5. Minimizing Unexplained Variance (Errors)

#### 2.6. Occam’s Razor: Two Simple Models

#### 2.7. The Experiment and Data Treatment

## 3. Results and Discussion

#### 3.1. Implementation of the Proposed Solution

Algorithm 1: Providing perpendicular offsets. |

//Uses Minimize function solving an optimization problem (Equation (21)) //Implement pf function below Input: m //sample size $(X,Y)$ //sample data ($X=\{{x}_{1},\dots ,{x}_{m}\},Y=\{{y}_{1},\dots ,{y}_{m}\}$) n //number of coefficients C //initial guess for coefficients ($C=\{{c}_{1},\dots ,{c}_{n}\}$) f //function evaluating the model with given coefficients ($f=f(x;C)$) k //data pair index from which to construct the perpendicular offsets Function $\mathrm{PF}(m,X,Y,n,C,f,k)$ $z\leftarrow X\left[k\right]$; $t\leftarrow \mathrm{MINIMIZE}({(z-X\left[k\right])}^{2}+{(f(z;C)-Y\left[k\right])}^{2},z)$ Return ${(t-X\left[k\right])}^{2}+{(f(t;C)-Y\left[k\right])}^{2}$EndFunction$r\leftarrow pf(m,X,Y,n,C,f,k)$ Output: r //the squared perpendicular offset from k to f |

Algorithm 2: Sum of perpendicular offsets. |

//Uses function pf defined in Algorithm 1 & implement function sp below Input: m //sample size $(X,Y)$ //sample data ($X=\{{x}_{1},\dots ,{x}_{m}\},Y=\{{y}_{1},\dots ,{y}_{m}\}$) n //number of coefficients C //initial guess for coefficients ($C=\{{c}_{1},\dots ,{c}_{n}\}$) f //function evaluating the model with given coefficients ($f=f(x;C)$) Function $\mathrm{SP}(m,X,Y,n,C,f)$ $r\leftarrow 0$; For ($k\leftarrow 1,\dots ,m$) $r\leftarrow r+pf(m,X,Y,n,C,f,k)$ EndForReturn rEndFunction$s\leftarrow sp(m,X,Y,n,C,f)$ Output: s //sum of the squared perpendicular offsets |

Algorithm 3: Nonlinear regression with perpendicular offsets. |

//Uses Minimize function solving an optimization problem (Equation (21)) //Uses InitialEstimate function providing an initial guess //Uses function sp defined in Algorithm 2 Input: m //sample size $(X,Y)$ //sample data ($X=\{{x}_{1},\dots ,{x}_{m}\},Y=\{{y}_{1},\dots ,{y}_{m}\}$) n //number of coefficients C //initial guess for coefficients ($C=\{{c}_{1},\dots ,{c}_{n}\}$) f //function evaluating the model with given coefficients ($f=f(x;C)$) $C\leftarrow \mathrm{INITALESTIMATE}(m,X,Y,n,f)$ //or any other good guess initialization $D\leftarrow \mathrm{MINIMIZE}\left(\mathrm{SP}\right(m,X,Y,n,C,f),C)$ Output: D //coefficients minimizing the sum of the perpendicular offsets |

#### 3.2. The Numerical Results

## 4. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

PV (cell) | Photovoltaic |

PN (junction) | Positive (P)/negative (N) semiconductor interface |

$\eta $, ${\eta}_{1}$, ${\eta}_{2}$ | Diffusion and recombination diode ideality factor(s) |

${V}_{T}$ | Thermal voltage (${V}_{T}=\frac{{\mathrm{k}}_{\mathrm{B}}}{{\mathrm{e}}^{-}}T=\frac{\mathrm{R}}{\mathrm{F}}T$, with T the temperature, in Kelvin (K)) |

k_{B} | Boltzmann’s constant (k_{B} = $1.380649\xb7{10}^{-23}\phantom{\rule{3.33333pt}{0ex}}\mathrm{J}\xb7{\mathrm{K}}^{-1}$) |

e^{−} | Electron (elementary) electric charge constant (e^{−} = $1.602176634\xb7{10}^{-19}\phantom{\rule{3.33333pt}{0ex}}\mathrm{C}$) |

e | Euler’s number, e = 2.71828182845904523... (e = ${\sum}_{k=0}^{\infty}k{!}^{-1}$) |

R | Regnault’s constant (R = $8.31446261815324\phantom{\rule{3.33333pt}{0ex}}\mathrm{J}\xb7{\mathrm{mol}}^{-1}\xb7{\mathrm{K}}^{-1}$) |

${R}_{h}$, ${R}_{s}$ | Shunt (${R}_{h}$) and series (${R}_{s}$) resistances (see §2.1) |

F | Faraday’s constant (F = $96485.3321233100184\phantom{\rule{3.33333pt}{0ex}}\mathrm{C}\xb7{\mathrm{mol}}^{-1}$) |

${I}_{\mathrm{sc}}$ | Short-circuit intensity |

${U}_{\mathrm{oc}}$ | Open-circuit voltage |

${I}_{\mathrm{xp}}$ | Current intensity at maximum power point |

${U}_{\mathrm{xp}}$ | Voltage at maximum power point |

${P}_{\mathrm{xp}}$ | Power at maximum power point (${P}_{\mathrm{xp}}$ = ${I}_{\mathrm{xp}}\xb7{U}_{\mathrm{xp}}$) |

$RSS$ | Residual sum of squares (statistics) |

${r}_{{\mathrm{adj}}^{2}}$ | Adjusted determination coefficient (statistics) |

F | F (Fisher’s) value (statistics) |

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**Figure 1.**Voltaic pile design: Silvered copper (Ag), water (or washing soda solution), and zinc (Zn) or tin (Sn), and a metallic wire); (+) Zn(s) | Zn

^{2+}(aq) || H

^{+}(aq) | H

_{2}(g) (−), $E\approx 0.8\phantom{\rule{3.33333pt}{0ex}}\mathrm{V}$.

**Figure 7.**Vertical (“|”), horizontal (“−”), and perpendicular (“⊥”) offsets in calculating the experimental errors ($\u03f5$).

**Figure 8.**Rational function ${f}_{1}\left(x\right)$ from Equation (2) with some convenient coefficients.

**Figure 9.**Power function ${f}_{2}\left(x\right)$ from Equation (3) with some convenient coefficients.

I (mA) | 1.147 | 1.187 | 1.257 | 1.312 | 1.362 | 1.406 | 1.48 | 1.493 | 1.556 | 1.609 | 1.672 | 1.742 | 1.776 | 1.785 | 1.812 | 1.821 | 1.834 |

U (mV) | 1132 | 1110 | 1080 | 1038 | 1010 | 973 | 930 | 900 | 845 | 772 | 703 | 593 | 493 | 405 | 332 | 254 | 163 |

n | $\mathit{f}(\mathit{x};\mathit{C})$ | Initial C | Statistics | PV Cell Parameters |
---|---|---|---|---|

4 | ${f}_{1}$ from Equation (11) | ${a}_{1}$ = −3350 ${a}_{2}$ = 2689 ${a}_{3}$ = −476 ${a}_{4}$ = −1.93 | $RSS$ = 0.0015674 ${r}_{\mathrm{adj}}^{2}$ = 0.9976 F = 2396 | ${I}_{\mathrm{sc}}=1.85482\phantom{\rule{3.33333pt}{0ex}}\mathrm{mA}$ ${U}_{\mathrm{oc}}=1735.75\phantom{\rule{3.33333pt}{0ex}}\mathrm{mV}$ ${I}_{\mathrm{xp}}=1.3921\phantom{\rule{3.33333pt}{0ex}}\mathrm{mA}$ ${U}_{\mathrm{xp}}=982.02\phantom{\rule{3.33333pt}{0ex}}\mathrm{mV}$ ${P}_{\mathrm{xp}}=1.3671\phantom{\rule{3.33333pt}{0ex}}\mathrm{mW}$ |

3 | ${f}_{2}$ from Equation (12) | ${b}_{1}$ = 1.83 ${b}_{2}$ = −22 ${b}_{3}$ = 3.07 | $RSS$ = 0.0019093 ${r}_{\mathrm{adj}}^{2}$ = 0.9986 F = 9228 | ${I}_{\mathrm{sc}}=1.83000\phantom{\rule{3.33333pt}{0ex}}\mathrm{mA}$ ${U}_{\mathrm{oc}}=1576.51\phantom{\rule{3.33333pt}{0ex}}\mathrm{mV}$ ${I}_{\mathrm{xp}}=1.3804\phantom{\rule{3.33333pt}{0ex}}\mathrm{mA}$ ${U}_{\mathrm{xp}}=998.00\phantom{\rule{3.33333pt}{0ex}}\mathrm{mV}$ ${P}_{\mathrm{xp}}=1.3776\phantom{\rule{3.33333pt}{0ex}}\mathrm{mW}$ |

n | $\mathit{f}(\mathit{x};\mathit{C})$ | Value C | Statistics | PV Cell Parameters |
---|---|---|---|---|

4 | ${f}_{1}$ from Equation (11) | ${a}_{1}$ = −3349.81 ${a}_{2}$ = 2689.40 ${a}_{3}$ = −475.164 ${a}_{4}$ = −1.92823 | $RSS$ = 0.0013094 ${r}_{\mathrm{adj}}^{2}$ = 0.9977 F = 2526 | ${I}_{\mathrm{sc}}=1.85071\phantom{\rule{3.33333pt}{0ex}}\mathrm{mA}$ ${U}_{\mathrm{oc}}=1737.24\phantom{\rule{3.33333pt}{0ex}}\mathrm{mV}$ ${I}_{\mathrm{xp}}=1.3866\phantom{\rule{3.33333pt}{0ex}}\mathrm{mA}$ ${U}_{\mathrm{xp}}=985.94\phantom{\rule{3.33333pt}{0ex}}\mathrm{mV}$ ${P}_{\mathrm{xp}}=1.3671\phantom{\rule{3.33333pt}{0ex}}\mathrm{mW}$ |

3 | ${f}_{2}$ from Equation (12) | ${b}_{1}$ = 1.82622 ${b}_{2}$ = −22.2914 ${b}_{3}$ = 3.11345 | $RSS$ = 0.0011128 ${r}_{\mathrm{adj}}^{2}$ = 0.9987 F = 9528 | ${I}_{\mathrm{sc}}=1.82622\phantom{\rule{3.33333pt}{0ex}}\mathrm{mA}$ ${U}_{\mathrm{oc}}=1561.10\phantom{\rule{3.33333pt}{0ex}}\mathrm{mV}$ ${I}_{\mathrm{xp}}=1.3823\phantom{\rule{3.33333pt}{0ex}}\mathrm{mA}$ ${U}_{\mathrm{xp}}=991.19\phantom{\rule{3.33333pt}{0ex}}\mathrm{mV}$ ${P}_{\mathrm{xp}}=1.3701\phantom{\rule{3.33333pt}{0ex}}\mathrm{mW}$ |

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**MDPI and ACS Style**

Jäntschi, L.
Symmetry in Regression Analysis: Perpendicular Offsets—The Case of a Photovoltaic Cell. *Symmetry* **2023**, *15*, 948.
https://doi.org/10.3390/sym15040948

**AMA Style**

Jäntschi L.
Symmetry in Regression Analysis: Perpendicular Offsets—The Case of a Photovoltaic Cell. *Symmetry*. 2023; 15(4):948.
https://doi.org/10.3390/sym15040948

**Chicago/Turabian Style**

Jäntschi, Lorentz.
2023. "Symmetry in Regression Analysis: Perpendicular Offsets—The Case of a Photovoltaic Cell" *Symmetry* 15, no. 4: 948.
https://doi.org/10.3390/sym15040948