# Generation Data of Synthetic High Frequency Solar Irradiance for Data-Driven Decision-Making in Electrical Distribution Grids

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## Abstract

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## 1. Introduction

- In contrast to recorded measurement data and to approaches used in commercial software [7], a model is more general and non-biased against the occurrence of a specific scenario.
- For stochastic numerical simulation, an infinite amount of data can be generated.
- To investigate the sensitivity of a numerical simulation to specific model parameters, we can change the parameters of the model.

#### 1.1. Literature Review

#### 1.2. Contribution

- We provide a comprehensive model of PV systems power production that is suitable for the numerical simulation of electrical distribution grids.
- The dependency of temporal weather variations, including one-minute, daily, and monthly variations on the PV systems power production, is considered. A model is developed to generate a sequence of PV systems power production with a random average based on the N-state MC model of that day.
- A distinct N-state MC model is proposed for each group of days characterized by cloudy, intermittent cloudy, and clear sky. The proposed model is tested on two locations with ‘‘warm and temperate” and ‘‘tropical wet and dry/savanna” climates.

#### 1.3. Organization

## 2. Proposed Model

#### 2.1. Step 1 (Preprocess Data)

#### 2.2. Step 2 (Train Model)

#### 2.2.1. Pool of Selected Probability Density Functions

#### 2.2.2. Parameter Estimation of Probability Density Function and Goodness of Fitting Test

#### 2.2.3. Markov Chain Training

#### 2.3. Step 3 (Generate Synthetic Data)

#### 2.4. Evaluate Synthetic Data

- similarity of PDFs in different months to evaluate the impacts of monthly weather variation on GHI;
- TAF similarity of average daily GHI for evaluating the impacts of daily weather variation on GHI; and
- TAF similarity of GHI time-series for evaluating the impacts of one-minute-resolution weather variations.

#### 2.5. Benchmark Model

## 3. Exemplary Applications of the Proposed Model

## 4. Experimental Results

#### 4.1. Input Data

#### 4.2. Model Training on Data of Location (a)

#### 4.3. Synthetic Data Generation at Location (a)

#### 4.4. Evaluation of Synthetic Data of Location (a)

#### 4.5. Model Training on Data of Location (b)

## 5. Conclusions and Future Works

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

BSRN | Baseline Surface Radiation Network |

COP21 | 21st Conference of Parties |

CSI | Clear Sky Index |

DC | Determination Coefficient |

EV | Electric Vehicle |

GEV | Generalized Extreme Value |

GHI | Global Horizontal Irradiance |

GoF | Goodness of Fitting |

HMM | Hidden Markov Model |

Kuma | Kumaraswamy |

LogitN | Logit-Normal |

Logist | Logistic |

Log-Log | Log-Logistic |

LV | Low-Voltage |

MC | Markov Chain |

MLE | Maximum Likelihood Estimation |

MTAF | Modified Temporal Autocorrelation Function |

MV | Medium-Voltage |

Probability Density Function | |

PLF | Probabilistic Load Flow |

PV | Photovoltaic |

SCSI | State of Clear Sky Index |

SFOE | Swiss Federal Office of Energy |

TAF | Temporal Autocorrelation Function |

WRMC | World Radiation Monitoring Center |

## Nomenclature

Indices | |

m | Month |

d | Day |

t | Minute time-step |

$\tau $ | Lag of TAF in minutes |

Functions | |

${f}_{Beta}\left(x\right|a,b)$ | Beta PDF |

${f}_{Kuma}\left(x\right|c,d)$ | Kumaraswamy PDF |

${f}_{LogitN}\left(x\right|\mu ,\sigma )$ | Logit-Normal PDF |

${f}_{Logist}\left(x\right|\lambda ,s)$ | Logist PDF |

${f}_{Log-Log}\left(x\right|\tau ,v)$ | Log-Log PDF |

${f}_{GEV}\left(x\right|k,\rho ,c)$ | GEV PDF |

$\mathrm{TAF}\left(\tau \right)$ | Standard TAF |

$\mathrm{MTAF}\left(\tau \right)$ | Modified version of TAF |

Variables | |

$G(m,d,t)$ | Measured GHI at month m, day d, and time step t |

$\widehat{G}(m,d,t)$ | Synthetic GHI at month m, day d, and time step t |

${G}_{c}(m,d,t)$ | Clear sky GHI at month m, day d, and time step t |

$\mathrm{CSI}(m,d,t)$ | Clear Sky Index at month m, day d, and time step t |

$\mathrm{SCSI}(m,d,t)$ | State of CSI at month m, day d, and time step t |

$\underline{G}$, $\overline{G}$ | Minimum and maximum of tolerance band |

${G}^{<0.25>}$, ${G}^{<0.75>}$ | 0.25 and 0.75 quantile of the measured GHI |

$\mathrm{Type}(m,d)$ | Type of day d of month m |

$\eta (m,d)$ | Average of CSI over day d at month m |

$\sigma (m,d)$ | Standard deviation of CSI over day d at month m |

${r}_{\eta}(m,d)$ | Residual of CSI average from the centers of clusters |

${r}_{\sigma}(m,d)$ | Residual of CSI standard deviation from the centers of clusters |

$P\left(\mathrm{Type}\right)$ | Transition matrix of a day type |

Parameters | |

${D}_{m}$ | Number of days in month m |

${\eta}_{th1}$, ${\eta}_{th2}$ | Average thresholds to cluster the days |

${\sigma}_{th1}$, ${\sigma}_{th2}$ | Standard deviation thresholds to cluster the days |

a, b | Shape parameters of Beta PDF |

c, d | Shape parameters of Kumaraswamy PDF |

$\mu $, $\sigma $ | Average and standard deviation of of Logit-N PDF |

$\lambda $, s | Average and scale parameter of of Logist PDF |

$\tau $, v | Log average and log scale parameter of Log-Log PDF |

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**Figure 1.**Flow diagram of the proposed model for generation and evaluation of synthetic data of a PV system’s power production.

**Figure 4.**Comparing the pool of different models for average daily CSI: (

**a**) PDFs and (

**b**) CDFs at location (a) in February.

**Figure 5.**Comparing the pool of different models for average daily CSI: (

**a**) PDFs and (

**b**) CDFs at location (a) in June.

**Figure 6.**Surface plot for transition matrices of days with (

**a**) cloudy sky, (

**b**) intermittent cloudy sky, and (

**c**) clear sky.

**Figure 7.**Surface plot for transition matrix of the state-of-the-art model [31].

**Figure 8.**(

**a**) Real data of G and ${G}_{c}$ for month June 2017; (

**b**) Synthetic data of G and ${G}_{c}$ for month June 2016 based on state-of-the-art model [31]; (

**c**) Synthetic data of G and ${G}_{c}$ for month June 2016 based on our proposed model.

**Figure 9.**Comparing the proposed model, state-of-the-art model, and real data: (

**a**) Standard TAF; (

**b**) Modifed TAF.

**Figure 10.**Comparing the pool of different models for average daily CSI: (

**a**) PDFs and (

**b**) CDFs at location (b) in June.

**Figure 11.**Comparing real data and synthetic data generated by the proposed model and state-of-the-art one: (

**a**) Real data of G and ${G}_{c}$ for month June 2011 in location (b); (

**b**) Synthetic data of G and ${G}_{c}$ for month June 2016 based on state-of-the-art model [31]; (

**c**) Synthetic data of G and ${G}_{c}$ for month June 2016 based on our proposed model.

**Figure 12.**Comparing the proposed model, state-of-the-art model, and real data: (

**a**) Standard TAF; (

**b**) Modifed TAF.

Month | DC | |||||
---|---|---|---|---|---|---|

Beta | LogitN | Kuma | Logist | Loglog | GEV | |

Jan. | 0.9481 | 0.9541 | 0.9515 | 0.8664 | 0.9288 | 0.8974 |

Feb. | 0.9490 | 0.9574 | 0.9503 | 0.8954 | 0.9173 | 0.8921 |

Mar. | 0.9854 | 0.9803 | 0.9856 | 0.9485 | 0.9236 | 0.9454 |

Apr. | 0.9518 | 0.9763 | 0.9384 | 0.8877 | 0.9494 | 0.9150 |

May | 0.9917 | 0.9845 | 0.9916 | 0.9493 | 0.9291 | 0.9590 |

Jun. | 0.9796 | 0.9591 | 0.9813 | 0.9609 | 0.8950 | 0.9591 |

Jul. | 0.9911 | 0.9810 | 0.9912 | 0.9667 | 0.9268 | 0.9648 |

Aug. | 0.9725 | 0.9669 | 0.9713 | 0.9317 | 0.8493 | 0.9289 |

Sep. | 0.9814 | 0.9812 | 0.9803 | 0.9295 | 0.9211 | 0.8068 |

Oct. | 0.9706 | 0.9757 | 0.9679 | 0.9026 | 0.8676 | 0.8902 |

Nov. | 0.9234 | 0.9639 | 0.9274 | 0.8854 | 0.9389 | 0.9236 |

Dec. | 0.8260 | 0.8924 | 0.8339 | 0.7704 | 0.8855 | 0.9307 |

Month | DC | |||||
---|---|---|---|---|---|---|

Beta | LogitN | Kuma | Logist | Loglog | GEV | |

Jan. | 0.9534 | 0.9695 | 0.9612 | 0.9604 | 0.9196 | 0.9756 |

Feb. | 0.9697 | 0.9726 | 0.9689 | 0.9562 | 0.9508 | 0.9748 |

Mar. | 0.9275 | 0.9433 | 0.9477 | 0.9577 | 0.9144 | 0.9572 |

Apr. | 0.9889 | 0.9870 | 0.9871 | 0.9828 | 0.9798 | 0.9858 |

May | 0.9726 | 0.9816 | 0.9824 | 0.9770 | 0.9561 | 0.9940 |

Jun. | 0.9851 | 0.9887 | 0.9931 | 0.9896 | 0.9786 | 0.9871 |

Jul. | 0.9948 | 0.9944 | 0.9933 | 0.9915 | 0.9872 | 0.9948 |

Aug. | 0.9945 | 0.9932 | 0.9924 | 0.9930 | 0.9903 | 0.9877 |

Sep. | 0.9927 | 0.9929 | 0.9921 | 0.9888 | 0.9826 | 0.9896 |

Oct. | 0.7810 | 0.6872 | 0.8091 | 0.9905 | 0.7960 | 0.9898 |

Nov. | 0.9652 | 0.9749 | 0.9825 | 0.9769 | 0.9592 | 0.9779 |

Dec. | 0.9239 | 0.9405 | 0.9233 | 0.9245 | 0.8780 | 0.9305 |

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## Share and Cite

**MDPI and ACS Style**

Rayati, M.; Falco, P.D.; Proto, D.; Bozorg, M.; Carpita, M.
Generation Data of Synthetic High Frequency Solar Irradiance for Data-Driven Decision-Making in Electrical Distribution Grids. *Energies* **2021**, *14*, 4734.
https://doi.org/10.3390/en14164734

**AMA Style**

Rayati M, Falco PD, Proto D, Bozorg M, Carpita M.
Generation Data of Synthetic High Frequency Solar Irradiance for Data-Driven Decision-Making in Electrical Distribution Grids. *Energies*. 2021; 14(16):4734.
https://doi.org/10.3390/en14164734

**Chicago/Turabian Style**

Rayati, Mohammad, Pasquale De Falco, Daniela Proto, Mokhtar Bozorg, and Mauro Carpita.
2021. "Generation Data of Synthetic High Frequency Solar Irradiance for Data-Driven Decision-Making in Electrical Distribution Grids" *Energies* 14, no. 16: 4734.
https://doi.org/10.3390/en14164734