# The Impact of Data Filtration on the Accuracy of Multiple Time-Domain Forecasting for Photovoltaic Power Plants Generation

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## Abstract

**:**

## 1. Introduction

## 2. The Main Sources of the PVPP Generation Forecasting Errors

- the numerical estimation of cloudiness (the proportion of the sky covered by clouds);
- type of clouds (cirrus, cumulus, stratus, etc.);
- cloud heights from the base to the top;
- the microstructure of clouds (e.g., water content per unit volume);
- distribution of clouds relative to the solar disk position, etc.;

## 3. Data Filtering for Short-Term Forecasting of Photovoltaic Power Plants Generation

#### 3.1. Case Study of the PVPP Generation Short-Term Forecast

- spring weather period: 26 February 2018–11 March 2018, 14 days, 164 observations;
- summer weather period: 21 May 2018–1 June 2018, 12 days, 199 observations;
- autumn weather period: 11 September 2018–20 September 2018, 10 days, 130 observations;
- winter weather period: 28 January 2019–03 February 2019, 7 days, 81 observations.

- measured directly on site:
- ○
- actual PVPP generation, kWh;
- ○
- actual global horizontal irradiance, W/m
^{2}; - ○
- actual ambient temperature, °C;
- ○
- actual wind speed, m/s;

- formed by the meteorological service:
- ○
- actual and forecasted cloudiness, p.u.;
- ○
- forecasted air temperature, °C;
- ○
- forecasted wind speed, m/s;
- ○
- actual and forecasted air humidity, p.u.

#### 3.2. Data Filtration Methods Application

_{Σ}is the total PVPP energy production for the period under consideration, (kWh); E

_{Σ}is the total absolute error for the period under consideration, (kWh); E

_{avg}is the mean absolute error for the period under consideration, (kWh); σ

_{E}is the absolute error standard deviation, (kWh); R

^{2}score is the determination coefficient, (p.u.); SSEn is normalized sum of errors.

^{2}), the SSEn indicator is applied. It is calculated as the sum of squared errors normalized with respect to the square of the installed capacity of the photovoltaic power plant, SSEn = SSE/(P

_{inst})

^{2}, measured in p.u.

#### 3.3. Empirical Data Filtration

#### 3.4. The K-Means Filtration Method

#### 3.5. Filtration Models Comparative Analysis

## 4. Photovoltaic Power Plants Generation Short-Term Forecast Operational Correction

#### 4.1. General Approach to the PVPP Generation Operational Forecasting Models

#### 4.2. PVPP Generation Operational Forecasting Models Description

#### 4.2.1. Persistence Model (Represents the So-Called “Naive” Approach)

#### 4.2.2. Moving Average Model (Is an Advanced Inertial Model)

#### 4.2.3. Autoregressive Model

#### 4.2.4. Autoregressive Moving Average Model

#### 4.2.5. Autoregressive Model with Exogenous Inputs

## 5. Comparison of Operational and Short-Term Forecasting Models

^{2}metrics is introduced as a model quality criterion. As in the previous case (short-term forecast), adjusted R

^{2}also demonstrates how well the model fits the data, but the total score is adjusted according to the number of terms in the model:

## 6. Forecasting System Hard-Ware Implementation for Autonomous Photovoltaic Power Plants

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Flowchart of the photovoltaic power plants (PVPP) generation short-term forecasting algorithm.

**Figure 2.**Actual versus Forecasted values of PVPP generation for day-ahead time horizon (stages 1–6).

**Figure 5.**Transparency index versus cloudiness for morning/evening altitude angles with empirical filtration.

**Figure 9.**Geometric interpretation of data initialization and clustering results using the k-means method.

Total Absolute Error, kW∙h | Mean Absolute Percentage Error, % | |
---|---|---|

Error of stages 3–6 | 11,590 | 8.7 |

Error of stages 1–6 | 43,112 | 32.3 |

Parameter | 26 February 2018–11 March 2018 | 21 May 2018–1 June 2018 | 11 September 2018–20 September 2018 | 28 January 2019–3 February 2019 | For All Periods |
---|---|---|---|---|---|

${W}_{\mathsf{\Sigma}}$, kW∙h | 391,805.4 | 1,156,028.2 | 601,402.2 | 86,694.7 | 2,235,930.5 |

${E}_{\mathsf{\Sigma}}$, kW∙h | 185,497.2 | 253,367.0 | 258,279.6 | 64,829.7 | 761,973.5 |

${E}_{avg}$, kW∙h | 1131.1 | 1306.0 | 1986.8 | 800.4 | 1339.1 |

${\sigma}_{E}$, kW∙h | 1909.6 | 2201.9 | 2040.6 | 1303.4 | 2210.2 |

${R}^{2}$ | 0.65 | ||||

SSEn | 14.77 |

Solar Altitude Angle Sine | Cloudiness | Solar Inclination Angle | |||
---|---|---|---|---|---|

Description | Value | Description | Value | Description | Value |

morning/evening | 0.1 | almost no clouds | 0.1 | winter | 0.1 |

late night/day | 0.3 | low clouds | 0.5 | off-season closer to winter | 0.33 |

late morning | 0.5 | medium clouds | 0.8 | off-season closer to summer | 0.66 |

midday | 0.7 | heavy clouds | 1 | summer | 1 |

**Table 4.**Results of calculating the parameters for forecast accuracy estimating for various combinations.

$\mathit{X}-\mathit{n}\cdot {\mathit{\sigma}}_{{\mathit{k}}_{\mathit{T}}}\le {\mathit{k}}_{\mathit{T}}\le \mathit{X}+\mathit{n}\cdot {\mathit{\sigma}}_{{\mathit{k}}_{\mathit{T}}}$ | ${\mathit{E}}_{\mathsf{\Sigma}},\text{}\mathbf{kW}\xb7\mathbf{h}$ | ${\mathit{\sigma}}_{\mathit{E}},\text{}\mathbf{kW}\xb7\mathbf{h}$ | ${\mathit{E}}_{\mathit{a}\mathit{v}\mathit{g}},\text{}\mathbf{kW}\xb7\mathbf{h}$ | ${\mathit{E}}_{\mathsf{\Sigma}}^{\mathit{\%}},\text{}\%$ |
---|---|---|---|---|

$X$—arithmetic mean, $n=0.5$ | 464,104.24 | 1560.80 | 815.65 | 20.76 |

$X$—arithmetic mean, $n=1.0$ | 442,004.04 | 1486.48 | 776.81 | 19.77 |

$X$—arithmetic mean, $n=1.5$ | 437,334.47 | 1274.05 | 768.60 | 19.56 |

$X$—arithmetic mean, $n=2.0$ | 496,591.54 | 1470.06 | 872.74 | 22.21 |

$X$—median, $n=0.5$ | 449,945.86 | 1361.23 | 790.77 | 20.12 |

$X$—median, $n=1.0$ | 424,117.13 | 1271.61 | 745.37 | 18.97 |

$X$—median, $n=1.5$ | 417,287.68 | 1214.58 | 733.37 | 18.66 |

$X$—median, $n=2.0$ | 438,152.06 | 1289.76 | 770.04 | 19.60 |

$X$—mode, $n=0.5$ | 495,570.51 | 1466.62 | 870.95 | 22.16 |

$X$—mode, $n=1.0$ | 466,985.74 | 1373.99 | 820.71 | 20.89 |

$X$—mode, $n=1.5$ | 471,971.92 | 1387.26 | 829.48 | 21.11 |

$X$—mode, $n=2.0$ | 530,260.45 | 1583.29 | 931.92 | 23.72 |

**Table 5.**Results of calculating the parameters for estimating the PVPP generation forecasting accuracy using the k-means method for data filtering.

Parameter | 26 February 2018–11 March 2018 | 21 May 2018–1 June 2018 | 11 September 2018–20 September 2018 | 28 January 2019–3 February 2019 | For All Periods |
---|---|---|---|---|---|

${W}_{\mathsf{\Sigma}}$, kW∙h | 391,805.4 | 1,156,028.2 | 601,402.2 | 86,694.7 | 2,235,930.5 |

${E}_{\mathsf{\Sigma}}$, kW∙h | 113,424.0 | 75,093.5 | 184,327.4 | 45,661.1 | 417,287.7 |

${E}_{avg}$, kW∙h | 691.6 | 377.4 | 1417.9 | 563.7 | 733.4 |

${\sigma}_{E}$, kW∙h | 997.7 | 893.1 | 1724.2 | 733.3 | 1 214.6 |

${E}_{\mathsf{\Sigma}}^{\%}$, % | 29.0 | 6.50 | 30.7 | 52.7 | 18.7 |

${R}^{2}$ | 0.88 |

Parameter | Without Filtration | Simple Filter | K-Means Method |
---|---|---|---|

${W}_{\mathsf{\Sigma}}$, kW∙h | 2,235,930.48 | 2,235,930.48 | 2,235,930.48 |

${E}_{\mathsf{\Sigma}}$, kW∙h | 761,973.46 | 640,411.96 | 417,287.68 |

${E}_{avg}$, kW∙h | 1339.14 | 1151.29 | 733.37 |

${\sigma}_{E}$, kW∙h | 1909.59 | 2242.59 | 1214.58 |

${E}_{\mathsf{\Sigma}}^{\%}$, % | 34.08 | 28.64 | 18.66 |

${R}^{2}$ | 0.65 | 0.70 | 0.88 |

Model | Parameters | ||||
---|---|---|---|---|---|

${\mathit{E}}_{\mathsf{\Sigma}},\text{}\mathbf{kW}\xb7\mathbf{h}$ | ${\mathit{E}}_{\mathit{a}\mathit{v}\mathit{g}},\text{}\mathbf{kW}\xb7\mathbf{h}$ | ${\mathit{\sigma}}_{\mathit{E}},\text{}\mathbf{kW}\xb7\mathbf{h}$ | ${\mathit{E}}_{\mathsf{\Sigma}}^{\mathit{\%}},\text{}\%$ | ${\mathit{R}}_{\mathit{a}\mathit{d}\mathit{j}\mathit{u}\mathit{s}\mathit{t}\mathit{e}\mathit{d}}^{2}$ | |

Model P | 346,892.60 | 1063.18 | 609.65 | 15.51 | 0.874 |

Model MA(2) | 337,843.22 | 1035.45 | 593.75 | 15.11 | 0.887 |

Model MA(3) | 347,495.89 | 1065.03 | 610.71 | 15.54 | 0.874 |

Model MA(4) | 355,941.97 | 1090.92 | 625.56 | 15.92 | 0.862 |

Model MA(5) | 364,991.34 | 1118.65 | 641.46 | 16.32 | 0.851 |

Model AR(1) | 325,777.39 | 998.47 | 572.54 | 14.57 | 0.904 |

Model AR(2) | 301,645.74 | 924.51 | 530.13 | 13.49 | 0.940 |

Model AR(3) | 331,810.31 | 1016.96 | 583.15 | 14.84 | 0.895 |

Model ARMA(1,2) | 328,793.85 | 1007.71 | 577.85 | 14.71 | 0.900 |

Model ARMA(2,2) | 337,843.22 | 1035.45 | 593.75 | 15.11 | 0.887 |

Model ARMA(3,2) | 358,958.43 | 1100.16 | 630.86 | 16.05 | 0.858 |

Model ARMA(2,3) | 352,925.51 | 1081.67 | 620.26 | 15.78 | 0.866 |

Model ARX(2,1) | 327,180.05 | 1002.77 | 575.01 | 14.63 | 0.902 |

Model ARX(2,2) | 330,196.50 | 1012.01 | 580.31 | 14.77 | 0.898 |

Model ARX(2,3) | 333,212.96 | 1021.26 | 585.61 | 14.90 | 0.893 |

**Table 8.**The parameters calculating results for assessing the PVPP generation operational forecast accuracy.

Parameter | 26 February 2018–11 March 2018 | 21 May 2018–1 June 2018 | 11 September 2018–20 September 2018 | 28 January 2019–3 February 2019 | For All Periods |
---|---|---|---|---|---|

${W}_{\mathsf{\Sigma}}$, kW∙h | 391,805.4 | 1,156,028.2 | 601,402.2 | 86,694.7 | 2,235,930.5 |

${E}_{\mathsf{\Sigma}}$, kW∙h | 82,951.0 | 76,503.8 | 119,241.6 | 23,406.3 | 301,645.8 |

${E}_{avg}$, kW∙h | 505.8 | 384.4 | 917.2 | 289.0 | 530.1 |

${\sigma}_{E}$, kW∙h | 722.5 | 867.6 | 1011.8 | 440.0 | 924.5 |

${E}_{\mathsf{\Sigma}}^{\%}$, % | 21.17 | 6.62 | 19.83 | 27.00 | 13.5 |

${R}^{2}$ | 0.94 |

**Table 9.**The parameters comparison for assessing the short-term and operational PVPP generation forecast accuracy.

Parameter | STF, K-Means Methodology | Operational Forecast, AR(2) |
---|---|---|

${W}_{\mathsf{\Sigma}}$, kW∙h | 2,235,930.48 | 2,235,930.48 |

${E}_{\mathsf{\Sigma}}$, kW∙h | 417,287.68 | 301,645.74 |

${E}_{avg}$, kW∙h | 733.37 | 530.13 |

${\sigma}_{E}$, kW∙h | 1214.58 | 924.51 |

${E}_{\mathsf{\Sigma}}^{\%}$, % | 18.66 | 13.49 |

${R}^{2}$ | 0.88 | 0.94 |

**Table 10.**The reliability parameters comparison of PVPP generation operational and short-term forecast.

Confidence Interval | The Share of the Forecasts within the Interval | |
---|---|---|

STF, K-Means Method | Operational Forecast, AR(2) | |

±1 MW (6.7% from PVPP P_{inst}) | 78.9% | 88.5% |

±2 MW (13.3% from PVPP P_{inst}) | 92.2% | 97.5% |

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**MDPI and ACS Style**

Eroshenko, S.A.; Khalyasmaa, A.I.; Snegirev, D.A.; Dubailova, V.V.; Romanov, A.M.; Butusov, D.N.
The Impact of Data Filtration on the Accuracy of Multiple Time-Domain Forecasting for Photovoltaic Power Plants Generation. *Appl. Sci.* **2020**, *10*, 8265.
https://doi.org/10.3390/app10228265

**AMA Style**

Eroshenko SA, Khalyasmaa AI, Snegirev DA, Dubailova VV, Romanov AM, Butusov DN.
The Impact of Data Filtration on the Accuracy of Multiple Time-Domain Forecasting for Photovoltaic Power Plants Generation. *Applied Sciences*. 2020; 10(22):8265.
https://doi.org/10.3390/app10228265

**Chicago/Turabian Style**

Eroshenko, Stanislav A., Alexandra I. Khalyasmaa, Denis A. Snegirev, Valeria V. Dubailova, Alexey M. Romanov, and Denis N. Butusov.
2020. "The Impact of Data Filtration on the Accuracy of Multiple Time-Domain Forecasting for Photovoltaic Power Plants Generation" *Applied Sciences* 10, no. 22: 8265.
https://doi.org/10.3390/app10228265