# Mathematical Model for Regular and Irregular PV Arrays with Improved Calculation Speed

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## Abstract

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## 1. Introduction

## 2. Mismatching Conditions and Internal Connections in PV Arrays

- Design the fixed connection of a PV array subjected to shading conditions.
- Reconfigure the PV array in real-time to adjust the best connections, depending on the shading pattern, which changes, depending on the sun position.
- Design a simulation platform to estimate the power production of any PV array with regular or irregular connections.

## 3. Inflection Point Concept

- If the string current is lower than the short-circuit current of both modules, i.e., ${i}_{st}<{i}_{sc2}<{i}_{sc1}$, which means that both modules could produce the string current, i.e., ${i}_{pv1}={i}_{pv2}={i}_{st}$, which imposes null current to both bypass diodes, i.e., ${i}_{d1}={i}_{d2}=0\phantom{\rule{3.33333pt}{0ex}}\left[A\right]$. Therefore, both bypass diodes are inactive, and both modules are active to produce usable power. For the example of Figure 4, when ${i}_{st}<{i}_{sc2}$, both of the modules are active, despite having different short-circuit currents.
- Instead, for the same irradiance and shading pattern, if the string current is higher than the short-circuit current of some modules, i.e., ${i}_{sc2}<{i}_{st}<{i}_{sc1}$, this means that some modules are not able to produce the string current, i.e., ${i}_{pv2}={i}_{sc2}<{i}_{st}$ and ${i}_{pv1}={i}_{st}$, which forces the activation of the corresponding bypass diodes, i.e., ${i}_{d1}={i}_{st}-{i}_{sc2}>0$ and ${i}_{d2}=0$. Therefore, some of the modules are inactive to produce usable power, i.e., in the example of Figure 4 the module 2 is short-circuit (inactive) when ${i}_{st}>{i}_{sc2}$.
- Finally, if the string current is higher than the short-circuit current of all the modules, i.e., ${i}_{st}>{i}_{sc1}>{i}_{sc2}$, this means that all of the modules are not able to produce the string current, which forces the activation of all the bypass diodes. Therefore, the PV array is inactive in producing usable power.

## 4. Inflection Points Calculation

#### 4.1. PV Module Representation

#### 4.2. Matrix Representation a PV Array and Its Sub-Arrays

#### 4.3. Inflection Voltage Calculation for a Sub-Array

Algorithm 1 Calculation of ${F}_{o}(\overrightarrow{{V}_{n}})$ |

INPUT:$\overrightarrow{{V}_{n}}$, ${N}_{n}$, ${M}_{nvo}^{sa}$, ${M}_{conn}^{sa}$, ${M}_{Isc}^{sa}$, ${i}_{o}$, ${j}_{o}$, N, ${M}_{sa}$, sub-array parameters’ matricesOUTPUT: ${F}_{o}(\overrightarrow{{V}_{n}})$1: Set ${i}_{f}=1$ 2: for columns $j=1$ to ${M}_{sa}$ do3: for rows $i=1$ to $N-1$ do4: if $j=1$ OR ${M}_{conn}^{sa}(i,j-1)=0$ then:5: Identify ${l}_{m}$ and ${r}_{m}$ from ${M}_{conn}^{sa}$ 6: for modules in the node ${j}_{c}={l}_{m}$ to ${r}_{m}$ do7: Set ${n}_{n}={M}_{nvo}^{sa}(i,{j}_{c})$ and ${e}_{a}(i,{j}_{c})=\overrightarrow{{V}_{n}}\left({n}_{n}\right)$ 8: Set ${n}_{n}={M}_{nvo}^{sa}(i+1,{j}_{c})$ and ${e}_{b}(i,{j}_{c})=\overrightarrow{{V}_{n}}\left({n}_{n}\right)$ 9: if $i={i}_{o}$ AND $j={j}_{o}$ then:10: Set ${I}_{i,{j}_{c}}={M}_{Isc}^{sa}(i,{j}_{c})$ 11: else12: Calculate ${I}_{i,{j}_{c}}$ with (1) 13: end if14: Calculate ${F}_{o}\left({i}_{f}\right)={F}_{o}\left({i}_{f}\right)+{I}_{i,{j}_{c}}$ 15: Set ${n}_{n}={M}_{nvo}^{sa}(i+1,{j}_{c})$, ${e}_{a}(i+1,{j}_{c})=\overrightarrow{{V}_{n}}\left({n}_{n}\right)$ 16: if $i=N-1$ then:17: ${e}_{b}(i+1,{j}_{c})=0\phantom{\rule{0.277778em}{0ex}}\left[\mathrm{V}\right]$ 18: else:19: Set ${n}_{n}={M}_{nvo}^{sa}(i+2,{j}_{c})$ and ${e}_{b}(i+1,{j}_{c})=\overrightarrow{{V}_{n}}\left({n}_{n}\right)$ 20: end if21: if $i+1={i}_{o}$ AND $j={j}_{o}$ then:22: Set ${I}_{i+1,{j}_{c}}={M}_{Isc}^{sa}(i,{j}_{c})$ 23: else24: Calculate ${I}_{i+1,{j}_{c}}$ with (1) 25: end if26: Calculate ${F}_{o}\left({i}_{f}\right)={F}_{o}\left({i}_{f}\right)-{I}_{i+1,{j}_{c}}$ 27: end for modules in the node28: Set ${i}_{f}={i}_{f}+1$ 29: end if30: end for rows 31: end for columns32: if ${i}_{o}<N$ then:33: Set ${n}_{n1}={M}_{nvo}^{sa}({i}_{o},{j}_{o})$ and ${n}_{n2}={M}_{nvo}^{sa}({i}_{o}+1,{j}_{o})$ 34: Calculate ${F}_{o}({N}_{n}+1)=\overrightarrow{{V}_{n}}\left({n}_{n1}\right)-\overrightarrow{{V}_{n}}\left({n}_{n2}\right)$ 35: else:36: Set ${n}_{n}={M}_{nvo}^{sa}({i}_{o},{j}_{o})$ 37: Calculate ${F}_{o}({N}_{n}+1)=\overrightarrow{{V}_{n}}\left({n}_{n}\right)$ 38: end if39: Return ${F}_{o}\left(\overrightarrow{{V}_{n}}\right)$ |

#### 4.4. Calculation Example

## 5. Sub-Array and Array Current Calculation Using Inflection Voltages

#### 5.1. Sub-Array Current Calculation

Algorithm 2 Calculation of $F(\overrightarrow{V})$ |

INPUT:${V}_{arr}$, $\overrightarrow{V}$, ${N}_{n}$, ${M}_{nv}^{sa}$, ${M}_{conn}^{sa}$, N, ${M}_{sa}$, sub-array parameters’ matricesOUTPUT: $F(\overrightarrow{{V}_{n}})$1: Set ${i}_{f}=1$ 2: for columns $j=1$ to ${M}_{sa}$ do3: for rows $i=1$ to $N-1$ do4: if $j=1$ OR ${M}_{conn}^{sa}(i,j-1)=0$ then: Identify $lm$ and $rm$ from ${M}_{conn}^{sa}$5: for column ${j}_{c}=lm$ to $rm$ do6: Set ${n}_{n}={M}_{nv}^{sa}(i,{j}_{c})$, ${e}_{b}(i,{j}_{c})=\overrightarrow{V}\left({n}_{n}\right)$ 7: if $i=1$ then: ${e}_{a}(i,{j}_{c})={V}_{arr}$8: else: Set ${n}_{n}={M}_{nv}^{sa}(i-1,{j}_{c})$, and ${e}_{a}(i,{j}_{c})=\overrightarrow{V}\left({n}_{n}\right)$9: end if10: Calculate ${I}_{i,{j}_{c}}$ with (1) and ${F}_{i}\left({f}_{i}\right)={F}_{i}\left({f}_{i}\right)+{I}_{i,{j}_{c}}$ 11: Set ${n}_{n}={M}_{nv}^{sa}(i,{j}_{c})$, ${e}_{a}(i+1,{j}_{c})=\overrightarrow{V}\left({n}_{n}\right)$ 12: if $i=N-1$ then: ${e}_{b}(i+1,{j}_{c})=0\phantom{\rule{0.277778em}{0ex}}\left[\mathrm{V}\right]$13: else: Set ${n}_{n}={M}_{nv}^{sa}(i+1,{j}_{c})$, ${e}_{b}(i+1,{j}_{c})=\overrightarrow{V}\left({n}_{n}\right)$.14: end if15: Calculate ${I}_{i+1,{j}_{c}}$ with (1) and ${F}_{i}\left({f}_{i}\right)={F}_{i}\left({f}_{i}\right)-{I}_{i+1,{j}_{c}}$ 16: end for17: Set ${i}_{f}={i}_{f}+1$ 18: end if19: end for rows20: end for columns21: Return $F(\overrightarrow{{V}_{n}})$ |

#### 5.2. Calculation of the Array Current

#### 5.3. Calculation Example

## 6. Performance Evaluation

#### 6.1. Errors in the Current Prediction

#### 6.2. Calculation Time for Different Number of Rows

## 7. Application Example: Reconfiguration of Pv Arrays

## 8. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 8.**Graphical example of the calculation of ${M}_{Vo}$, ${M}_{Vno}^{1}$, ${M}_{Vno}^{2}$, and ${M}_{Vno}^{3}$ for $SA1$.

**Figure 10.**I-V curves of irregular arrays with different sizes with Circuital model (black line), Proposed model (blue line), and Reference model (dashed red line).

**Figure 11.**Current errors for irregular arrays with different sizes with Proposed model (dashed blue line) and Reference model (dotted red line) regarding Circuital model.

**Figure 12.**Calculation time of proposed and reference model for an array with three columns and rows from three to fifteen.

**Table 1.**The calculation time of the proposed and reference models for an array with three columns and rows from three to fifteen.

Rows | Proposed Model | Reference Model | ||
---|---|---|---|---|

Average (s) | Std (s) | Average (s) | Std (s) | |

3 | 6.396 | 0.509 | 5.362 | 0.169 |

4 | 8.462 | 0.079 | 8.934 | 0.166 |

5 | 10.893 | 0.463 | 12.190 | 0.157 |

6 | 13.616 | 0.336 | 15.160 | 0.296 |

7 | 16.413 | 0.180 | 19.735 | 0.199 |

8 | 18.025 | 0.792 | 25.353 | 0.767 |

9 | 20.232 | 0.229 | 31.533 | 0.357 |

10 | 24.212 | 0.307 | 46.348 | 0.456 |

11 | 27.771 | 0.121 | 59.698 | 0.352 |

12 | 34.096 | 0.294 | 77.808 | 2.059 |

13 | 38.008 | 0.310 | 89.932 | 1.159 |

14 | 42.471 | 0.224 | 100.773 | 0.531 |

15 | 46.995 | 0.214 | 122.431 | 0.555 |

**Table 2.**${I}_{ph}$ currents (in $\left[\mathrm{A}\right]$) of the reconfigurable modules of the array.

Shading Profile 1 | Shading Profile 2 | Shading Profile 2 | |||
---|---|---|---|---|---|

5.1337 | 5.1337 | 5.1337 | 3.5936 | 5.1337 | 5.1337 |

3.5936 | 5.1337 | 3.5936 | 5.1337 | 3.5936 | 5.1337 |

3.5936 | 3.5936 | 1.5401 | 1.5401 | 3.5936 | 3.5936 |

1.5401 | 3.5936 | 5.1337 | 3.5936 | 1.5401 | 1.5401 |

1.5401 | 1.5401 | 1.5401 | 5.1337 | 1.5401 | 5.1337 |

5.1337 | 0.5134 | 5.1337 | 1.5401 | 5.1337 | 5.1337 |

**Table 3.**Calculation time (in $\left[\mathrm{min}\right]$) of the proposed and reference models to evaluate the 32 configurations for each shading profile.

Shading Profile | Proposed Model | Reference Model |
---|---|---|

1 | 18.2180 | 38.0085 |

2 | 19.1983 | 34.7788 |

3 | 17.9341 | 38.2907 |

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**MDPI and ACS Style**

Trejos-Grisales, L.A.; Bastidas-Rodríguez, J.D.; Ramos-Paja, C.A.
Mathematical Model for Regular and Irregular PV Arrays with Improved Calculation Speed. *Sustainability* **2020**, *12*, 10684.
https://doi.org/10.3390/su122410684

**AMA Style**

Trejos-Grisales LA, Bastidas-Rodríguez JD, Ramos-Paja CA.
Mathematical Model for Regular and Irregular PV Arrays with Improved Calculation Speed. *Sustainability*. 2020; 12(24):10684.
https://doi.org/10.3390/su122410684

**Chicago/Turabian Style**

Trejos-Grisales, Luz Adriana, Juan David Bastidas-Rodríguez, and Carlos Andrés Ramos-Paja.
2020. "Mathematical Model for Regular and Irregular PV Arrays with Improved Calculation Speed" *Sustainability* 12, no. 24: 10684.
https://doi.org/10.3390/su122410684