# A Genetic Algorithm Approach as a Self-Learning and Optimization Tool for PV Power Simulation and Digital Twinning

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## Abstract

**:**

## 1. Introduction

_{mod}). In contrast, due to costs, such measured data is scarce within small-scale and household PV systems.

- Offer a very accurate GA approach to learn and optimize unknown basic parameters of a PV system based on measured PV power data.
- Show the impact different data set sizes have on digital twinning.
- Create a precise digital twin of a PV system using either all-sky or clear-sky conditions as training data for now-casting purposes.

## 2. Previous Work

#### 2.1. Machine Learning and Optimization in the PV Modelling Domain

#### 2.2. Evaluation Metrics

## 3. Methodology and Data

#### 3.1. Methods

#### 3.1.1. PV System Simulation

#### Irradiance Transposition Mode, Albedo, and AOI Effect

#### DC Power

#### AC Power

#### 3.1.2. PV System Parameters Optimization

- The initialization step creates a vector of different PV plants configurations (members), assigning random values (also known as population) based on the initial value to be optimized (initial parameters), with a randomness percentage defined beforehand. The PV power configuration can differ from several elements or only one.
- In the fitness scoring step, every single PV plant configuration of the population is compared with the monitoring data, evaluated, and a score is assigned based on the loss function defined.
- In the fit selection step, some PV plant configurations of the population are stochastically selected based on their scores; the higher the score, the higher the probability to be selected and passed to the next population.
- In the crossover step, some pairs of PV plant configurations resulting from the fit selection step are selected stochastically and their parameters are randomly combined. Hence, new PV plant configurations are added to the new population.
- In the mutation step, some parameters of the PV plant configurations are mutated due to a mutation probability assigned randomly to each parameter of every PV power plant configuration of the next generation.
- Finally, the next population is repopulated based on the parameters of the best PV plant configuration of the current population. The process stops when the stop criteria have been met; the stop criteria is defined in detail further in this section.

#### Irradiance Transposition Mode, Albedo, and AOI Effect

#### DC Power

_{p}. Second, use cross validation optimization (we evaluate every set of three parameters) to select the best fit for the three parameters from the Heydenreich et al. model, and GA optimization can be used to learn the temperature coefficient.

#### AC Power

_{p}nominal power PV system. Based on the Schmidt and Sauer model, we minimized the deviation between simulated PV power AC and normalized measured PV power using GA. PV power AC is calculated using the parameters optimized beforehand with 1% cabling losses assumed.

#### 3.1.3. Clear-Sky Detection

- We used a Python function called detect_clearsky, available in PVLib library [38], to compare the statistical clear-sky curve based on measured PV power, with actual measured PV power. Hence, we were able to identify the clear-sky-like periods in the time series data set. It is important to mention that the parameters of the Python function were tuned by trial and error based on the input data, as suggested by [18].

#### 3.2. Data Used in This Article

#### 3.2.1. Weather Data

#### 3.2.2. PV Power Measured Data

#### 3.2.3. Initial Parameters

- Nominal power = 1 kW
_{p} - Tilt angle = 25°
- Azimuth angle = 180° (south oriented)
- Albedo = 0.2
- Temperature coefficient = −0.43%/°C
- DC to AC ratio = 1

## 4. Results and Discussion

_{p}.

_{p}and −16.90 W/kW

_{p}, respectively. Only daytime values are considered in both examples.

#### 4.1. PV System Parametrization

_{p}, which is equal to −18% of the reported installed capacity. Figure 7b shows less variability in general, with an MAPD of 10.69% for the worst-case scenario using a training data set of 90 days before. It shows that, it is possible to achieve a mean deviation of less than 10% of the reported installed capacity using clear-sky-like periods as training data.

_{p}, when a loss factor of 0.8%/year for the nominal power is considered.

#### 4.2. Digital Twin Now-Casting

_{p}and 1.53 W/kW

_{p}is feasible. Using the GA proposed here, based on satellite-derived weather data, we can now-cast the PV power of a PV system with higher accuracy than previous publications with a MAPD lower than 6%, without evident seasonal effects. MAPD values above 10% have been reported for previous publications by [4,5,6,7,8].

#### 4.3. Limitations

#### 4.4. Future Directions

## 5. Conclusions

_{p}have been estimated for the digital twin now-casting accuracy of the GA approach of this work.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**General methodology for the Photovoltaic (PV) system parameters optimization. Small dotted lines represent the interactions between the software blocks. Large dotted lines represent measured PV power data as input. Continuous lines represent additional data required as input, namely meteorological data.

**Figure 4.**Heydenreich et al. model fitted curves of 107 different Fraunhofer CalLab measurement results for module efficiency at different irradiance levels.

**Figure 5.**Monthly distribution of clear-sky moments from a PV system located in south-west Germany in the year 2017.

**Figure 6.**Exemplary results of a clear-sky day (

**a**) and an overcast day (

**b**) of 2017 digital twin PV power simulation trained with the 30 days before. Deviations between simulated PV power and measured PV power of a clear-sky day and an overcast day are shown in subfigures (

**c**,

**d**), respectively.

**Figure 7.**Parametrization results of the GA optimization. Left hand side subfigures (

**a**,

**c**,

**e**,

**g**,

**i**,

**k**) show the parametrization results considering all-sky conditions as training data and right hand side subfigures (

**b**,

**d**,

**f**,

**h**,

**j**,

**l**) show the parametrization results considering only clear-sky conditions as training data.

**Figure 8.**Digital twin now-casting results based on GA optimization. 52 chosen days of year 2017. Six different training data sets for each day were considered: all-sky and clear-sky conditions with 30, 60, and 90 days of training data sets. The lines represent the mean value and the shadows show a confidence interval of 95%, only daytime values are considered.

Parameter | Value |
---|---|

Tilt angle (°) | 20 |

Azimuth angle (°) | 181 |

Albedo | 0.2 |

Temperature coefficient (%/°C) | −0.43 |

Heydenreich a | 0.001084 |

Heydenreich b | −7.247061 |

Heydenreich c | −156.5457 |

DC to AC ratio | 1.04 |

Nominal power (kW_{p}) | 553 |

**Table 2.**Parametrization results of the genetic algorithm (GA) optimization including all-sky and clear-sky conditions.

Training Data Set Length | |||||||
---|---|---|---|---|---|---|---|

Parameter | Error Metric | All-Sky Conditions | Clear-Sky Conditions | ||||

30 Days | 60 Days | 90 Days | 30 Days | 60 Days | 90 Days | ||

Nominal Power | Reported (kW_{p}) | 553 | 553 | ||||

Mean (kW_{p}) | 468.76 | 890.13 | 954.05 | 509.19 | 517.5 | 503.17 | |

MBD (kW_{p}) | −84.24 | 337.13 | 401.05 | −43.81 | −35.5 | −49.83 | |

MAPD (%) | 16.94 | 88.14 | 90.73 | 9.92 | 9.95 | 10.69 | |

RMSD (kW_{p}) | 125.46 | 2073.5 | 1995.17 | 64.33 | 66.2 | 65.31 | |

Azimuth | Reported (°) | 181 | 181 | ||||

Mean (°) | 185.79 | 183.09 | 204.09 | 187.37 | 185.06 | 183.47 | |

MBD (°) | 4.79 | 2.09 | 23.09 | 6.37 | 4.06 | 2.47 | |

MAPD (%) | 6.63 | 8.55 | 13.51 | 4.58 | 4.24 | 2.53 | |

RMSD (°) | 22.18 | 33.43 | 50.81 | 13.91 | 15.79 | 9.3 | |

Tilt | Reported (°) | 20 | 20 | ||||

Mean (°) | 29.06 | 29.95 | 22.65 | 17.71 | 17.13 | 17.71 | |

MBD (°) | 9.06 | 9.95 | 2.65 | −2.29 | −2.87 | −2.29 | |

MAPD (%) | 66.16 | 77.56 | 52.12 | 28.23 | 25.83 | 17.71 | |

RMSD (°) | 25.72 | 38.88 | 26.02 | 7.29 | 6.72 | 5.1 | |

Albedo | Reported | 0.2 | 0.2 | ||||

Mean | 1.46 | 0.26 | 1.43 | 0.19 | 0.18 | 0.18 | |

MBD | 1.26 | 0.06 | 1.23 | −0.01 | −0.02 | −0.02 | |

MAPD (%) | 659.04 | 59.42 | 633.46 | 30.58 | 36.15 | 31.15 | |

RMSD | 8.02 | 0.34 | 6.85 | 0.08 | 0.09 | 0.07 | |

Temperature coefficient | Reported (%/°C) | −0.43 | −0.43 | ||||

Mean (%/°C) | −1.87 | −97.69 | −0.28 | −0.42 | −0.41 | −0.34 | |

MBD (%/°C) | −1.44 | −97.26 | 0.15 | 0.01 | 0.02 | 0.09 | |

MAPD (%) | 428.71 | 22702.5 | 46.15 | 39.04 | 35.87 | 28.89 | |

RMSD (%/°C) | 11.52 | 496.7 | 0.25 | 0.28 | 0.23 | 0.18 | |

DC to AC | Reported | 1.04 | 1.04 | ||||

Mean | 0.74 | 0.73 | 0.9 | 0.67 | 0.73 | 0.68 | |

MBD | −0.3 | −0.31 | −0.14 | −0.37 | −0.31 | −0.35 | |

MAPD (%) | 38.05 | 37.94 | 69.62 | 47.12 | 44.84 | 44.01 | |

RMSD | 0.5 | 0.53 | 1.72 | 0.57 | 0.58 | 0.57 |

**Table 3.**Optimized parameters for Heydenreich et al. model considering all-sky and clear-sky conditions.

Parameter | All-sky | Clear-Sky |
---|---|---|

Heydenreich a | 0.011515 | 0.003708 |

Heydenreich b | −11.160905 | −14.834605 |

Heydenreich c | −173.888934 | −208.739158 |

**Table 4.**Basic parameters reported from the PV plant located in south-west Germany and mean bias deviation (MBD) of 10 days optimized with a training data set comprised of clear-sky moments from 90 days before.

Parameter | Reported Value | Optimized Value (MBD) |
---|---|---|

Tilt angle (°) | 20 | 19.6 |

Azimuth angle (°) | 181 | 180.24 |

Albedo factor | 0.2 | 0.13 |

Temperature coefficient (%/°C) | −0.43 | −0.41 |

Heydenreich a | 0.001084 | 0.003708 |

Heydenreich b | −7.247061 | −14.834605 |

Heydenreich c | −156.5457 | −208.739158 |

DC to AC ratio | 1.04 | 0.87 |

Nominal power (kW_{p}) | 553 | 482.9 |

**Table 5.**Accuracy results of the digital twins created for 52 randomly chosen days in the year 2017. Six different training data sets including all-sky and clear sky-conditions with lengths of 30, 60, and 90 days.

Error Metric | Training Data Set Length | ||
---|---|---|---|

30 Days | 60 Days | 90 Days | |

All-sky conditions | |||

MBD (W/kW_{p}) | −9.01 | 28.12 | −7.79 |

MAPD (%) | 6.18 | 10.59 | 5.74 |

RMSD (W/kW_{p}) | 44.04 | 45.92 | 48.67 |

Clear-sky conditions | |||

MBD (W/kW_{p}) | 1.53 | −1.3 | −0.39 |

MAPD (%) | 5.2 | 5.3 | 5.4 |

RMSD (W/kW_{p}) | 41.32 | 41.96 | 41.95 |

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**MDPI and ACS Style**

Guzman Razo, D.E.; Müller, B.; Madsen, H.; Wittwer, C.
A Genetic Algorithm Approach as a Self-Learning and Optimization Tool for PV Power Simulation and Digital Twinning. *Energies* **2020**, *13*, 6712.
https://doi.org/10.3390/en13246712

**AMA Style**

Guzman Razo DE, Müller B, Madsen H, Wittwer C.
A Genetic Algorithm Approach as a Self-Learning and Optimization Tool for PV Power Simulation and Digital Twinning. *Energies*. 2020; 13(24):6712.
https://doi.org/10.3390/en13246712

**Chicago/Turabian Style**

Guzman Razo, Dorian Esteban, Björn Müller, Henrik Madsen, and Christof Wittwer.
2020. "A Genetic Algorithm Approach as a Self-Learning and Optimization Tool for PV Power Simulation and Digital Twinning" *Energies* 13, no. 24: 6712.
https://doi.org/10.3390/en13246712