# Benchmark Comparison of Analytical, Data-Based and Hybrid Models for Multi-Step Short-Term Photovoltaic Power Generation Forecasting

^{*}

## Abstract

**:**

## 1. Introduction

- Analytical: These methods do not require any prior knowledge regarding the power generation measurements. They deliver the required results using well-known analytical equations that incorporate the technical characteristics of the PV installation along with weather forecasts derived by typical numerical weather prediction (NWP) models.
- Data-based: These models are data-driven, meaning that they are solely dependent on the historical PV power generation data, without any knowledge regarding the PV system itself. The basis of these models is the discovery of patterns and relations within the provided data. This category includes statistical time series models (e.g., autoregressive integrated moving average model—ARIMA), traditional machine learning (ML) models (e.g., artificial neural networks—ANNs) and deep learning (DL) models.
- Hybrid: These models attempt to combine the best characteristics of the other two categories in order to achieve higher forecasting accuracy. Different data-based models merged as one, or data-based models on top of analytical models, or even data-based models using NWP techniques, are some of the combinations identified in the literature. Interestingly enough, hybrid models seem to hold quite the potential in delivering the most accurate forecasting results.

- A comprehensive benchmark comparison between analytical, data-driven and hybrid direct PV power generation forecasting models.
- Extensive experimentation on real-world PV power generation data.
- A novel hybrid short-term PV power generation forecasting model, which outperforms in most cases several well-established analytical and data-based methods.
- Introduction of a new metric designed to accurately quantify the divergence error for the PV power generation forecasting problem.

## 2. Related Work

#### 2.1. Analytical Models

#### 2.2. Data-Based Models

#### 2.2.1. Statistical Time Series Models

#### 2.2.2. Traditional Machine Learning Models

#### 2.2.3. Deep Learning Models

#### 2.3. Hybrid Models

## 3. Materials & Methods

#### 3.1. Field Data

#### 3.1.1. Data Description

#### 3.1.2. Data Segmentation

- Spring: from 24 March 2019 to 31 May 2019
- Summer: from 1 June 2019 to 31 August 2019
- Autumn: from 1 September 2019 to 30 November 2019
- Winter: from 1 December 2019 to 29 February 2020

#### 3.1.3. Data Transformation for Supervised Learning

#### 3.2. PV Power Generation Forecasting Models

#### 3.2.1. Analytical Model

- Location and time: The solar irradiance values, which are used for the calculation of the actual PV power generation values, are calculated by a component of the physical model that estimates the exact position of the sun in terms of installation location and time. Hence, it is important to know the exact location (i.e., latitude and longitude) of the PV installation and to have accurate time measurements in the lowest granularity possible (i.e., hh:mm).
- PV configuration: PV construction details such as type/number of modules, type of inverter, the installation’s tilt and azimuth angle should be defined.
- Weather data: Cloud coverage and temperature forecasts should also be provided as input to the physical model.

#### 3.2.2. Statistical Models

**Persistence model**

**Autoregressive integrated moving average**

#### 3.2.3. Traditional Machine Learning Models

**Support vector regression**

**Gradient boosted trees**

#### 3.2.4. Deep Learning Models

**Deep neural networks**

**Long short-term memory networks**

#### 3.2.5. Hybrid Models

**Hybrid GBT mode—NWP-enriched GBT model**

**AI-Corrected NWP for Enriched Analytical PV Forecast**

#### 3.3. General Experimental Settings

#### 3.3.1. Model Configuration and Hyperparameter Tuning

#### 3.3.2. Forecasting Evaluation Metrics

## 4. Results

#### 4.1. Results According to Season

#### 4.2. Generalized Results

- MAE: $0.010$ kWh - GBT - 8 steps ahead - Sunny Days - Summer
- MAPE: $1.875\%$ - Hybrid - 12 steps ahead - Sunny Days - Spring
- RMSE: $0.021$ kWh - GBT - 4 steps ahead - Sunny Days - Summer
- WRSE: $0.035\%$ - Hybrid - 12 steps ahead - Sunny Days - Spring

- MAE: $0.045$ kWh - Hybrid - 12 steps ahead - Cloudy Days - Summer
- MAPE: $8.752\%$ - Hybrid - 8 steps ahead - Cloudy Days - Summer
- RMSE: $0.067$ kWh - Hybrid - 12 steps ahead - Cloudy Days - Summer
- WRSE: $0.376\%$ - Hybrid - 8 steps ahead - Cloudy Days - Summer

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

Wp | watt-peak |

DNN | Deep Neural Network |

LSTM | Long Short-Term Memory |

PV | Photovoltaic |

NWP | Numerical Weather Prediction |

RMSE | Root Mean Square Error |

MAPE | Mean Absolute Percentage Error |

WRSE | Weighted Relative Square Error |

RES | Renewable Energy Source |

DER | Distributed Energy Resource |

ML | Machine Learning |

NOCT | Nominal Operating Cell Temperature |

ANN | Artificial Neural Networks |

SVR | Support Vector Regression |

ARIMA | Autoregressive Integrated Moving Average |

DL | Deep Learning |

ANFIS | Adaptive Neuro-Fuzzy Inference Systems |

RBFN | Radial Basis Function Network |

CNN | Convolutional Neural Networks |

RNN | Recurrent Neural Networks |

SVM | Support Vector Machines |

kNN | k-Nearest Neighbors |

DIH | Digital Innovation Hubs |

CERTH | Centre for Research and Technology Hellas |

API | Application Programming Interface |

UTC | Coordinated Universal Time |

OLS | Ordinary Least Squares |

GBT | Gradient Boosted Trees |

ReLU | Rectified Linear Unit |

EXTRA | Extremely Randomised Tree Regression |

kW | Kilowatts |

GW | Gigawatt |

TWh | Terawatt-hours |

MW | Megawatt |

Wp | Watt-peak |

AI-PVF | AI-Corrected NWP for Enriched Analytical PV Forecast Hellas |

PDE | Partial Differential Equation |

ELM | Extreme Learning Machines |

SOM | Self-Organizing Map |

GA | Genetic Algorithms |

MAE | Mean Absolute Error |

## References

- Taylor, A.J.W. Low Carbon Energy Observatory Photovoltaics Technology Market Report 2018—Public Version, EUR 29935 EN, European Commission. 2019. Available online: https://publications.jrc.ec.europa.eu/repository/bitstream/JRC118307/jrc118307_1.pdf (accessed on 10 September 2020).
- Jäger-Waldau, A. PV Status Report 2019, EUR 29938EN, Publications Office of the European Union, Luxembourg. 2019. Available online: https://ec.europa.eu/jrc/sites/jrcsh/files/kjna29938enn_1.pdf (accessed on 10 September 2020).
- European Commission. Energy Roadmap 2050. 2012. Available online: https://ec.europa.eu/energy/sites/ener/files/documents/2012_energy_roadmap_2050_en_0.pdf (accessed on 24 August 2020).
- Qi, B.; Hasan, K.N.; Milanović, J.V. Identification of Critical Parameters Affecting Voltage and Angular Stability Considering Load-Renewable Generation Correlations. IEEE Trans. Power Syst.
**2019**, 34, 2859–2869. [Google Scholar] [CrossRef] [Green Version] - IRENA. Future of Solar Photovoltaic: Deployment, Investment, Technology, Grid Integration and Socio-Economic Aspects (A Global Energy Transformation: Paper). 2019. Available online: https://www.irena.org/-/media/Files/IRENA/Agency/Publication/2019/Nov/IRENA_Future_of_Solar_PV_2019.pdf (accessed on 24 August 2020).
- Mellit, A.; Massi Pavan, A.; Ogliari, E.; Leva, S.; Lughi, V. Advanced Methods for Photovoltaic Output Power Forecasting: A Review. Appl. Sci.
**2020**, 10, 487. [Google Scholar] [CrossRef] [Green Version] - Han, Y.; Wang, N.; Ma, M.; Zhou, H.; Dai, S.; Zhu, H. A PV power interval forecasting based on seasonal model and nonparametric estimation algorithm. Sol. Energy
**2019**, 184, 515–526. [Google Scholar] [CrossRef] - Wang, K.; Qi, X.; Liu, H. A comparison of day-ahead photovoltaic power forecasting models based on deep learning neural network. Appl. Energy
**2019**, 251, 113315. [Google Scholar] [CrossRef] - Antonanzas, J.; Osorio, N.; Escobar, R.; Urraca, R.; Martinez-de Pison, F.J.; Antonanzas-Torres, F. Review of photovoltaic power forecasting. Sol. Energy
**2016**, 136, 78–111. [Google Scholar] [CrossRef] - Graditi, G.; Ferlito, S.; Adinolfi, G. Comparison of Photovoltaic plant power production prediction methods using a large measured dataset. Renew. Energy
**2016**, 90, 513–519. [Google Scholar] [CrossRef] - Das, U.K.; Tey, K.S.; Seyedmahmoudian, M.; Mekhilef, S.; Idris, M.Y.I.; Van Deventer, W.; Horan, B.; Stojcevski, A. Forecasting of photovoltaic power generation and model optimization: A review. Renew. Sustain. Energy Rev.
**2018**, 81, 912–928. [Google Scholar] [CrossRef] - Akhter, M.N.; Mekhilef, S.; Mokhlis, H.; Shah, N.M. Review on forecasting of photovoltaic power generation based on machine learning and metaheuristic techniques. IET Renew. Power Gener.
**2019**, 13, 1009–1023. [Google Scholar] [CrossRef] [Green Version] - Wang, H.; Lei, Z.; Zhang, X.; Zhou, B.; Peng, J. A review of deep learning for renewable energy forecasting. Energy Convers. Manag.
**2019**, 198, 111799. [Google Scholar] [CrossRef] - Shi, J.; Lee, W.J.; Liu, Y.; Yang, Y.; Wang, P. Forecasting power output of photovoltaic systems based on weather classification and support vector machines. IEEE Trans. Ind. Appl.
**2012**, 48, 1064–1069. [Google Scholar] [CrossRef] - Alessandrini, S.; Delle Monache, L.; Sperati, S.; Cervone, G. An analog ensemble for short-term probabilistic solar power forecast. Appl. Energy
**2015**, 157, 95–110. [Google Scholar] [CrossRef] [Green Version] - Lingwei, Z.; Zhaokun, L.; Junnan, S.; Chenxi, W. Very short-term maximum Lyapunov exponent forecasting tool for distributed photovoltaic output. Appl. Energy
**2018**, 229, 1128–1139. [Google Scholar] - Pelland, S.; Galanis, G.; Kallos, G. Solar and photovoltaic forecasting through post-processing of the Global Environmental Multiscale numerical weather prediction model. Prog. Photovoltaics Res. Appl.
**2013**, 21, 284–296. [Google Scholar] [CrossRef] - Masa-Bote, D.; Castillo-Cagigal, M.; Matallanas, E.; Caamaño-Martín, E.; Gutiérrez, A.; Monasterio-Huelín, F.; Jiménez-Leube, J. Improving photovoltaics grid integration through short time forecasting and self-consumption. Appl. Energy
**2014**, 125, 103–113. [Google Scholar] [CrossRef] [Green Version] - Dolara, A.; Grimaccia, F.; Leva, S.; Mussetta, M.; Ogliari, E. A physical hybrid artificial neural network for short term forecasting of PV plant power output. Energies
**2015**, 8, 1138–1153. [Google Scholar] [CrossRef] [Green Version] - Celik, A.N.; Acikgoz, N. Modelling and experimental verification of the operating current of mono-crystalline photovoltaic modules using four- and five-parameter models. Appl. Energy
**2007**, 84, 1–15. [Google Scholar] [CrossRef] - Tossa, A.K.; Soro, Y.; Azoumah, Y.; Yamegueu, D. A new approach to estimate the performance and energy productivity of photovoltaic modules in real operating conditions. Sol. Energy
**2014**, 110, 543–560. [Google Scholar] [CrossRef] - Ma, T.; Yang, H.; Lu, L. Development of a model to simulate the performance characteristics of crystalline silicon photovoltaic modules/strings/arrays. Sol. Energy
**2014**, 100, 31–41. [Google Scholar] [CrossRef] - Ciulla, G.; Brano, V.L.; Di Dio, V.; Cipriani, G. A comparison of different one-diode models for the representation of I–V characteristic of a PV cell. Renew. Sustain. Energy Rev.
**2014**, 32, 684–696. [Google Scholar] [CrossRef] - Brano, V.L.; Orioli, A.; Ciulla, G.; Di Gangi, A. An improved five-parameter model for photovoltaic modules. Sol. Energy Mater. Sol. Cells
**2010**, 94, 1358–1370. [Google Scholar] [CrossRef] - Fuentes, M. A simplified thermal model of photovoltaic modules. In Sandia National Laboratories Report, SAND85-0330; Sandia National Labs.: Albuquerque, NM, USA, 1985. [Google Scholar]
- International Electrotechnical Commission. Terrestrial Photovoltaic (PV) Modules—Design Qualification and Type Approva—Part 1: Test Requirements; Standard IEC 61215-1:2016; International Organization for Standardization: Geneva, Switzerland, 2016. [Google Scholar]
- Toledo, C.; López-Vicente, R.; Abad, J.; Urbina, A. Thermal performance of PV modules as building elements: Analysis under real operating conditions of different technologies. Energy Build.
**2020**, 223, 110087. [Google Scholar] [CrossRef] - De Leone, R.; Pietrini, M.; Giovannelli, A. Photovoltaic energy production forecast using support vector regression. Neural Comput. Appl.
**2015**, 26, 1955–1962. [Google Scholar] [CrossRef] [Green Version] - Stengel, M.; Lindskog, M.; Undén, P.; Gustafsson, N. The impact of cloud-affected IR radiances on forecast accuracy of a limited-area NWP model. Q. J. R. Meteorol. Soc.
**2013**, 139, 2081–2096. [Google Scholar] [CrossRef] - Voyant, C.; Motte, F.; Notton, G.; Fouilloy, A.; Nivet, M.L.; Duchaud, J.L. Prediction intervals for global solar irradiation forecasting using regression trees methods. Renew. Energy
**2018**, 126, 332–340. [Google Scholar] [CrossRef] [Green Version] - Perez, R.; Lorenz, E.; Pelland, S.; Beauharnois, M.; Van Knowe, G.; Hemker, K., Jr.; Heinemann, D.; Remund, J.; Müller, S.C.; Traunmüller, W.; et al. Comparison of numerical weather prediction solar irradiance forecasts in the US, Canada and Europe. Sol. Energy
**2013**, 94, 305–326. [Google Scholar] [CrossRef] - Pierro, M.; Bucci, F.; De Felice, M.; Maggioni, E.; Moser, D.; Perotto, A.; Spada, F.; Cornaro, C. Multi-Model Ensemble for day ahead prediction of photovoltaic power generation. Sol. Energy
**2016**, 134, 132–146. [Google Scholar] [CrossRef] - Zhang, Y.; Beaudin, M.; Taheri, R.; Zareipour, H.; Wood, D. Day-ahead power output forecasting for small-scale solar photovoltaic electricity generators. IEEE Trans. Smart Grid
**2015**, 6, 2253–2262. [Google Scholar] [CrossRef] - Al-Dahidi, S.; Ayadi, O.; Adeeb, J.; Alrbai, M.; Qawasmeh, B.R. Extreme learning machines for solar photovoltaic power predictions. Energies
**2018**, 11, 2725. [Google Scholar] [CrossRef] [Green Version] - Chen, C.; Duan, S.; Cai, T.; Liu, B. Online 24-h solar power forecasting based on weather type classification using artificial neural network. Sol. Energy
**2011**, 85, 2856–2870. [Google Scholar] [CrossRef] - Das, U.K.; Tey, K.S.; Seyedmahmoudian, M.; Idris, I.; Yamani, M.; Mekhilef, S.; Horan, B.; Stojcevski, A. SVR-based model to forecast PV power generation under different weather conditions. Energies
**2017**, 10, 876. [Google Scholar] [CrossRef] - Tahasin, S.; Chenhui, S.; Hui, W.; Jingjing, L.; Xi, Z.; Mingyang, L. Iterative multi-task learning for time-series modeling of solar panel PV outputs. Appl. Energy
**2018**, 212, 654–662. [Google Scholar] - De Giorgi, M.G.; Congedo, P.M.; Malvoni, M. Photovoltaic power forecasting using statistical methods: Impact of weather data. IET Sci. Meas. Technol.
**2014**, 8, 90–97. [Google Scholar] [CrossRef] - Ogliari, E.; Grimaccia, F.; Leva, S.; Mussetta, M. Hybrid predictive models for accurate forecasting in PV systems. Energies
**2013**, 6, 1918–1929. [Google Scholar] [CrossRef] [Green Version] - Yang, H.T.; Huang, C.M.; Huang, Y.C.; Pai, Y.S. A weather-based hybrid method for 1-day ahead hourly forecasting of PV power output. IEEE Trans. Sustain. Energy
**2014**, 5, 917–926. [Google Scholar] [CrossRef] - Landelius, T.; Andersson, S.; Abrahamsson, R. Modelling and forecasting PV production in the absence of behind-the-meter measurements. Prog. Photovoltaics Res. Appl.
**2019**, 27, 990–998. [Google Scholar] [CrossRef] [Green Version] - Lee, W.; Kim, K.; Park, J.; Kim, J.; Kim, Y. Forecasting Solar Power Using Long-Short Term Memory and Convolutional Neural Networks. IEEE Access
**2018**, 6, 73068–73080. [Google Scholar] [CrossRef] - Qing, X.; Niu, Y. Hourly day-ahead solar irradiance prediction using weather forecasts by LSTM. Energy
**2018**, 148, 461–468. [Google Scholar] [CrossRef] - Zhou, H.; Zhang, Y.; Yang, L.; Liu, Q.; Yan, K.; Du, Y. Short-Term Photovoltaic Power Forecasting Based on Long Short Term Memory Neural Network and Attention Mechanism. IEEE Access
**2019**, 7, 78063–78074. [Google Scholar] [CrossRef] - Colak, I.; Yesilbudak, M.; Genc, N.; Bayindir, R. Multi-period Prediction of Solar Radiation Using ARMA and ARIMA Models. In Proceedings of the 2015 IEEE 14th International Conference on Machine Learning and Applications (ICMLA), Miami, FL, USA, 9–11 December 2015; pp. 1045–1049. [Google Scholar]
- Raza, M.Q.; Nadarajah, M.; Ekanayake, C. On recent advances in PV output power forecast. Sol. Energy
**2016**, 136, 125–144. [Google Scholar] [CrossRef] - Fernandez-Jimenez, L.A.; Muñoz-Jimenez, A.; Falces, A.; Mendoza-Villena, M.; Garcia-Garrido, E.; Lara-Santillan, P.M.; Zorzano-Alba, E.; Zorzano-Santamaria, P.J. Short-term power forecasting system for photovoltaic plants. Renew. Energy
**2012**, 44, 311–317. [Google Scholar] [CrossRef] - Han, S.; Qiao, Y.H.; Yan, J.; Liu, Y.Q.; Li, L.; Wang, Z. Mid-to-long term wind and photovoltaic power generation prediction based on copula function and long short term memory network. Appl. Energy
**2019**, 239, 181–191. [Google Scholar] [CrossRef] - Li, G.; Wang, H.; Zhang, S.; Xin, J.; Liu, H. Recurrent Neural Networks Based Photovoltaic Power Forecasting Approach. 2019, pp. 1–17. Available online: https://www.researchgate.net/publication/334157021_Recurrent_Neural_Networks_Based_Photovoltaic_Power_Forecasting_Approach (accessed on 16 November 2020).
- Qian, L.; Wu, X.X. Estimate and characterize PV power at demand-side hybrid system. Appl. Energy
**2018**, 218, 66–77. [Google Scholar] - Yanting, L.; Yong, H.; Yan, S.; Shu, L. Forecasting the daily power output of a grid-connected photovoltaic system based on multivariate adaptive regression splines. Appl. Energy
**2016**, 180, 392–401. [Google Scholar] - Liu, H.; Cocea, M. Traditional Machine Learning. In Granular Computing Based Machine Learning: A Big Data Processing Approach; Springer International Publishing: Cham, Switzerland, 2018; pp. 11–22. [Google Scholar]
- Barbieri, F.; Rajakaruna, S.; Ghosh, A. Very short-term photovoltaic power forecasting with cloud modeling: A review. Renew. Sustain. Energy Rev.
**2017**, 75, 242–263. [Google Scholar] [CrossRef] [Green Version] - Mosaico, G.; Saviozzi, M. A hybrid methodology for the day-ahead PV forecasting exploiting a Clear Sky Model or Artificial Neural Networks. In Proceedings of the IEEE EUROCON 2019-18th International Conference on Smart Technologies, Novi Sad, Serbia, 1–4 July 2019; pp. 1–6. [Google Scholar]
- Sideratos, G.; Hatziargyriou, N.D. A distributed memory RBF-based model for variable generation forecasting. Int. J. Electr. Power Energy Syst.
**2020**, 120, 106041. [Google Scholar] [CrossRef] - Nespoli, A.; Ogliari, E.; Leva, S.; Pavan, A.M.; Mellit, A.; Lughi, V.; Dolara, A. Day-ahead photovoltaic forecasting: A comparison of the most effective techniques. Energies
**2019**, 12, 1621. [Google Scholar] [CrossRef] [Green Version] - Gao, M.; Li, J.; Hong, F.; Long, D. Day-ahead power forecasting in a large-scale photovoltaic plant based on weather classification using LSTM. Energy
**2019**, 187, 115838. [Google Scholar] [CrossRef] - Abdel-Nasser, M.; Mahmoud, K. Accurate photovoltaic power forecasting models using deep LSTM-RNN. Neural Comput. Appl.
**2019**, 31, 2727–2740. [Google Scholar] [CrossRef] - Ouyang, W.; Yu, K.M.; Sodsong, N.; Chuang, K.H. Short-term solar PV forecasting based on recurrent neural network and clustering. In Proceedings of the 2019 International Conference on Image and Video Processing, and Artificial Intelligence, International Society for Optics and Photonics, Shanghai, China, 23–25 August 2019; Volume 11321, p. 113212U. [Google Scholar]
- Ghimire, S.; Deo, R.C.; Raj, N.; Mi, J. Deep solar radiation forecasting with convolutional neural network and long short-term memory network algorithms. Appl. Energy
**2019**, 253, 113541. [Google Scholar] [CrossRef] - Kim, K.; Lee, D. Photovoltaic (PV) Power Output Prediction Using LSTM Based Deep Learning. Adv. Nat. Appl. Sci.
**2019**, 13, 25–31. [Google Scholar] - De, V. Photovoltaic Power Forecasting using LSTM on Limited Dataset. In Proceedings of the 2018 IEEE Innovative Smart Grid Technologies—Asia (ISGT Asia), Singapore, 22–25 May 2018; pp. 710–715. [Google Scholar]
- Bracale, A.; Caramia, P.; Carpinelli, G.; Di Fazio, A.R.; Ferruzzi, G. A Bayesian method for short-term probabilistic forecasting of photovoltaic generation in smart grid operation and control. Energies
**2013**, 6, 733–747. [Google Scholar] [CrossRef] [Green Version] - Luyao, L.; Yi, Z.; Dongliang, C.; Jiyang, X.; Zhanyu, M.; Qie, S.; Hongyi, Y.; Ronald, W. Prediction of short-term PV power output and uncertainty analysis. Appl. Energy
**2018**, 228, 700–711. [Google Scholar] - Holmgren, W.; Hansen, C.; Mikofski, M. pvlib python: A python package for modeling solar energy systems. J. Open Source Softw.
**2018**, 3, 884. [Google Scholar] [CrossRef] [Green Version] - King, D.L.; Kratochvil, J.A.; Boyson, W.E. Photovoltaic Array Performance Model; United States Department of Energy, Sandia National Labs.: Albuquerque, NM, USA, 2004.
- Box, G.E.P.; Jenkins, G. Time Series Analysis, Forecasting and Control; Holden-Day, Inc.: San Francisco, CA, USA, 1990. [Google Scholar]
- Chen, T.; Guestrin, C. XGBoost. In Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, San Francisco, CA, USA, 13–17 August 2016. [Google Scholar] [CrossRef] [Green Version]
- Cybenko, G. Approximation by superpositions of a sigmoidal function. Math. Control Signals Syst. (MCSS)
**1989**, 2, 303–314. [Google Scholar] [CrossRef] - Hochreiter, S.; Schmidhuber, J. Long Short-Term Memory. Neural. Comput.
**1997**, 9, 1735–1780. [Google Scholar] [CrossRef] [PubMed] - Williams, R.J.; Zipser, D. Gradient-Based Learning Algorithms for Recurrent Networks and Their Computational Complexity. In Backpropagation: Theory, Architectures, and Applications; Psychology Press, Taylor & Francis Group: London, UK, 1995; pp. 433–486. [Google Scholar]
- Geurts, P.; Ernst, D.; Wehenkel, L. Extremely randomized trees. Mach. Learn.
**2006**, 63, 3–42. [Google Scholar] [CrossRef] [Green Version] - Timplalexis, C.; Bezas, N.; Bintoudi, A.; Zyglakis, L.; Pavlopoulos, V.; Tsolakis, A.; Krinidis, S.; Tzovaras, D. In Proceedings of the 12th Mediterranean Conference on Power Generation, Transmission, Distribution and Energy Conversion, Dubrovnik, Croatia, 12–15 December 2018; Available online: https://www.iti.gr/iti/publications/MedPower_2020_01.html (accessed on 25 August 2020).
- Sorjamaa, A.; Hao, J.; Reyhani, N.; Ji, Y.; Lendasse, A. Methodology for long-term prediction of time series. Neurocomputing
**2007**, 70, 286–2869. [Google Scholar] [CrossRef] [Green Version] - Hamzacebi, C.; Akay, D.; Kutay, F. Comparison of direct and iterative artificial neural network forecast approaches in multi-periodic time series forecasting. Expert Syst. Appl.
**2009**, 36, 3839–3844. [Google Scholar] [CrossRef] - Seabold, S.; Perktold, J. statsmodels: Econometric and statistical modeling with python. In Proceedings of the 9th Python in Science Conference, Austin, TX, USA, 28 June–3 July 2010. [Google Scholar]
- Pedregosa, F.; Varoquaux, G.; Gramfort, A.; Michel, V.; Thirion, B.; Grisel, O.; Blondel, M.; Prettenhofer, P.; Weiss, R.; Dubourg, V.; et al. Scikit-learn: Machine Learning in Python. J. Mach. Learn. Res.
**2011**, 12, 2825–2830. [Google Scholar] - Abadi, M.; Agarwal, A.; Barham, P.; Brevdo, E.; Chen, Z.; Citro, C.; Corrado, G.S.; Davis, A.; Dean, J.; Devin, M.; et al. TensorFlow: Large-Scale Machine Learning on Heterogeneous Distributed Systems. arXiv
**2016**, arXiv:1603.04467. [Google Scholar] - Chollet, F. Keras. 2015. Available online: https://keras.io (accessed on 10 October 2020).

**Figure 1.**CERTH/ITI Smart House and the roof-based PV Installation (only the rooftop PVs are used for the current study).

**Figure 4.**Boxplot of residuals for all PV power generation forecasting models. The data partition examined in this case is the sunny days subperiod of the summer season and the forecasting horizon is 1 step ahead.

**Figure 5.**Boxplot of residuals for all PV power generation forecasting models. The data partition examined in this case is the cloudy days subperiod of the winter season and the forecasting horizon is 1 step ahead.

**Figure 6.**Boxplot of residuals for all PV power generation forecasting models. The data partition examined in this case is the sunny days subperiod of the summer season and the forecasting horizon is 12 steps ahead.

**Figure 7.**Boxplot of residuals for all PV power generation forecasting models. The data partition examined in this case is the cloudy days subperiod of the winter season and the forecasting horizon is 12 steps ahead.

Data | Forecasting | ${\mathit{N}}_{\mathit{u}}$ | ${\mathit{N}}_{\mathit{u}}$ | ${\mathit{N}}_{\mathit{u}}$ | ${\mathit{N}}_{\mathit{u}}$ | # |
---|---|---|---|---|---|---|

Partitions | Horizons | Input | Hidden Layer 1 | Hidden Layer 2 | Output | Epochs |

Spring, sunny days | 1 | 3 | 4 | 8 | 1 | 30 |

2 | 3 | 8 | 8 | 1 | 50 | |

4 | 3 | 8 | 8 | 1 | 40 | |

8 | 3 | 8 | 8 | 1 | 100 | |

12 | 3 | 4 | 16 | 1 | 80 | |

Spring, cloudy days | 1 | 3 | 4 | 8 | 1 | 10 |

2 | 3 | 8 | 8 | 1 | 30 | |

4 | 3 | 8 | 8 | 1 | 40 | |

8 | 3 | 8 | 8 | 1 | 40 | |

12 | 3 | 4 | 8 | 1 | 25 | |

Summer, sunny days | 1 | 3 | 4 | 8 | 1 | 25 |

2 | 3 | 8 | 8 | 1 | 40 | |

4 | 3 | 8 | 8 | 1 | 40 | |

8 | 3 | 8 | 8 | 1 | 100 | |

12 | 3 | 8 | 8 | 1 | 50 | |

Summer, cloudy days | 1 | 3 | 4 | 8 | 1 | 30 |

2 | 3 | 8 | 8 | 1 | 40 | |

4 | 3 | 8 | 8 | 1 | 40 | |

8 | 3 | 16 | 8 | 1 | 100 | |

12 | 3 | 8 | 8 | 1 | 50 | |

Autumn, sunny days | 1 | 3 | 8 | 4 | 1 | 50 |

2 | 3 | 8 | 16 | 1 | 50 | |

4 | 3 | 4 | 8 | 1 | 40 | |

8 | 3 | 16 | 8 | 1 | 60 | |

12 | 3 | 8 | 16 | 1 | 100 | |

Autumn, cloudy days | 1 | 3 | 4 | 4 | 1 | 10 |

2 | 3 | 4 | 4 | 1 | 20 | |

4 | 3 | 4 | 4 | 1 | 20 | |

8 | 3 | 4 | 4 | 1 | 40 | |

12 | 3 | 4 | 8 | 1 | 50 | |

Winter, sunny days | 1 | 3 | 4 | 4 | 1 | 30 |

2 | 3 | 4 | 4 | 1 | 40 | |

4 | 3 | 4 | 8 | 1 | 50 | |

8 | 3 | 8 | 4 | 1 | 50 | |

12 | 3 | 4 | 8 | 1 | 100 | |

Winter, cloudy days | 1 | 3 | 8 | 16 | 1 | 100 |

2 | 3 | 8 | 16 | 1 | 50 | |

4 | 3 | 16 | 8 | 1 | 100 | |

8 | 3 | 8 | 16 | 1 | 50 | |

12 | 3 | 16 | 16 | 1 | 100 |

Spring, Sunny Days | Forecasting Horizons | Autumn, Sunny Days | Forecasting Horizons | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 4 | 8 | 12 | 1 | 2 | 4 | 8 | 12 | ||

${N}_{u}$ Input | 3 | 3 | 3 | 3 | 3 | ${N}_{u}$ Input | 3 | 3 | 3 | 3 | 3 |

${N}_{u}$ Hidden Layer 1 | 2 | 4 | 8 | 16 | 4 | ${N}_{u}$ Hidden Layer 1 | 8 | 16 | 16 | 16 | 32 |

${N}_{u}$ Hidden Layer 2 | 4 | 8 | 8 | 8 | 8 | ${N}_{u}$ Hidden Layer 2 | 8 | 8 | 8 | 8 | 8 |

${N}_{u}$ Output | 1 | 1 | 1 | 1 | 1 | ${N}_{u}$ Output | 1 | 1 | 1 | 1 | 1 |

# Epochs | 50 | 100 | 50 | 100 | 150 | # Epochs | 20 | 15 | 15 | 25 | 50 |

Spring, cloudy days | Forecasting Horizons | Autumn, cloudy days | Forecasting Horizons | ||||||||

1 | 2 | 4 | 8 | 12 | 1 | 2 | 4 | 8 | 12 | ||

${N}_{u}$ Input | 3 | 3 | 3 | 3 | 3 | ${N}_{u}$ Input | 3 | 3 | 3 | 3 | 3 |

${N}_{u}$ Hidden Layer 1 | 2 | 4 | 4 | 8 | 4 | ${N}_{u}$ Hidden Layer 1 | 8 | 4 | 4 | 4 | 4 |

${N}_{u}$ Hidden Layer 2 | 4 | 4 | 4 | 4 | 8 | ${N}_{u}$ Hidden Layer 2 | 8 | 8 | 8 | 8 | 8 |

${N}_{u}$ Output | 1 | 1 | 1 | 1 | 1 | ${N}_{u}$ Output | 1 | 1 | 1 | 1 | 1 |

# Epochs | 50 | 20 | 100 | 100 | 30 | # Epochs | 15 | 15 | 10 | 25 | 25 |

Summer, sunny days | Forecasting Horizons | Winter, sunny days | Forecasting Horizons | ||||||||

1 | 2 | 4 | 8 | 12 | 1 | 2 | 4 | 8 | 12 | ||

${N}_{u}$ Input | 3 | 3 | 3 | 3 | 3 | ${N}_{u}$ Input | 3 | 3 | 3 | 3 | 3 |

${N}_{u}$ Hidden Layer 1 | 4 | 4 | 16 | 8 | 4 | ${N}_{u}$ Hidden Layer 1 | 8 | 8 | 8 | 16 | 16 |

${N}_{u}$ Hidden Layer 2 | 4 | 4 | 8 | 8 | 8 | ${N}_{u}$ Hidden Layer 2 | 8 | 8 | 8 | 8 | 8 |

${N}_{u}$ Output | 1 | 1 | 1 | 1 | 1 | ${N}_{u}$ Output | 1 | 1 | 1 | 1 | 1 |

# Epochs | 50 | 20 | 20 | 50 | 50 | # Epochs | 30 | 30 | 30 | 40 | 40 |

Summer, cloudy days | Forecasting Horizons | Winter, cloudy days | Forecasting Horizons | ||||||||

1 | 2 | 4 | 8 | 12 | 1 | 2 | 4 | 8 | 12 | ||

${N}_{u}$ Input | 3 | 3 | 3 | 3 | 3 | ${N}_{u}$ Input | 3 | 3 | 3 | 3 | 3 |

${N}_{u}$ Hidden Layer 1 | 2 | 4 | 8 | 8 | 4 | ${N}_{u}$ Hidden Layer 1 | 16 | 8 | 8 | 8 | 16 |

${N}_{u}$ Hidden Layer 2 | 4 | 4 | 4 | 4 | 8 | ${N}_{u}$ Hidden Layer 2 | 8 | 4 | 8 | 16 | 16 |

${N}_{u}$ Output | 1 | 1 | 1 | 1 | 1 | ${N}_{u}$ Output | 1 | 1 | 1 | 1 | 1 |

# Epochs | 50 | 50 | 50 | 30 | 30 | # Epochs | 20 | 50 | 70 | 50 | 100 |

Steps | Models | Sunny Days | Cloudy Days | ||||||
---|---|---|---|---|---|---|---|---|---|

MAE | MAPE | RMSE | WRSE | MAE | MAPE | RMSE | WRSE | ||

(kWh) | (%) | (kWh) | (%) | (kWh) | (%) | (kWh) | (%) | ||

1 | Analytical | 0.073 | 4.043 | 0.092 | 0.153 | 0.398 | 79.437 | 0.495 | 18.816 |

Persistence | 0.103 | 6.229 | 0.155 | 0.361 | 0.154 | 25.021 | 0.233 | 2.565 | |

ARIMA | 0.063 | 3.452 | 0.157 | 0.123 | 0.152 | 28.758 | 0.227 | 2.668 | |

SVR | 0.151 | 9.743 | 0.165 | 0.823 | 0.17 | 40.005 | 0.233 | 3.572 | |

GBT | 0.076 | 4.142 | 0.156 | 0.172 | 0.145 | 23.677 | 0.228 | 2.213 | |

DNN | 0.082 | 4.502 | 0.17 | 0.209 | 0.159 | 28.27 | 0.238 | 2.792 | |

LSTM | 0.061 | 3.382 | 0.149 | 0.118 | 0.151 | 22.582 | 0.226 | 2.339 | |

HGBT | 0.071 | 4.153 | 0.151 | 0.174 | 0.141 | 24.332 | 0.231 | 2.28 | |

AI-PVF | 0.053 | 2.898 | 0.092 | 0.076 | 0.294 | 88.793 | 0.442 | 15.022 | |

2 | Analytical | 0.077 | 3.683 | 0.101 | 0.135 | 0.433 | 80.351 | 0.512 | 18.875 |

Persistence | 0.167 | 11.213 | 0.206 | 1.04 | 0.233 | 46.222 | 0.319 | 6.672 | |

ARIMA | 0.076 | 4.278 | 0.175 | 0.181 | 0.225 | 52.178 | 0.307 | 6.671 | |

SVR | 0.2 | 12.018 | 0.206 | 1.335 | 0.226 | 52.823 | 0.305 | 6.515 | |

GBT | 0.094 | 5.158 | 0.142 | 0.246 | 0.211 | 44.42 | 0.323 | 5.89 | |

DNN | 0.105 | 6.234 | 0.147 | 0.363 | 0.225 | 41.985 | 0.309 | 5.535 | |

LSTM | 0.063 | 3.576 | 0.16 | 0.126 | 0.223 | 45.488 | 0.305 | 6.04 | |

HGBT | 0.084 | 4.578 | 0.142 | 0.202 | 0.212 | 42.091 | 0.321 | 5.371 | |

AI-PVF | 0.052 | 2.714 | 0.113 | 0.069 | 0.296 | 90.961 | 0.453 | 15.4 | |

4 | Analytical | 0.072 | 3.667 | 0.998 | 0.14 | 0.411 | 83.484 | 0.512 | 19.968 |

Persistence | 0.299 | 18.964 | 0.351 | 3.124 | 0.382 | 88.03 | 0.475 | 19.471 | |

ARIMA | 0.075 | 4.337 | 0.105 | 0.179 | 0.355 | 92.93 | 0.438 | 17.597 | |

SVR | 0.195 | 11.314 | 0.204 | 1.217 | 0.338 | 83.996 | 0.437 | 15.654 | |

GBT | 0.101 | 5.467 | 0.143 | 0.291 | 0.291 | 86.234 | 0.421 | 12.595 | |

DNN | 0.229 | 13.107 | 0.237 | 1.65 | 0.333 | 78.374 | 0.421 | 13.724 | |

LSTM | 0.073 | 4.1 | 0.112 | 0.166 | 0.277 | 80.339 | 0.382 | 11.446 | |

HGBT | 0.086 | 4.611 | 0.13 | 0.2 | 0.287 | 74.771 | 0.412 | 10.611 | |

AI-PVF | 0.056 | 2.876 | 0.112 | 0.081 | 0.311 | 94.644 | 0.457 | 16.313 | |

8 | Analytical | 0.101 | 5.121 | 0.133 | 0.266 | 0.43 | 91.348 | 0.531 | 23.041 |

Persistence | 0.511 | 28.45 | 0.599 | 7.688 | 0.618 | 199.45 | 0.744 | 58.724 | |

ARIMA | 0.151 | 7.964 | 0.17 | 0.634 | 0.53 | 195.031 | 0.608 | 44.886 | |

SVR | 0.21 | 10.776 | 0.233 | 1.197 | 0.507 | 201.05 | 0.597 | 44.164 | |

GBT | 0.153 | 7.625 | 0.181 | 0.598 | 0.412 | 133.64 | 0.551 | 25.152 | |

DNN | 0.367 | 18.927 | 0.393 | 3.673 | 0.492 | 162.751 | 0.585 | 38.321 | |

LSTM | 0.096 | 4.829 | 0.132 | 0.244 | 0.404 | 146.537 | 0.531 | 27.372 | |

HGBT | 0.131 | 6.67 | 0.165 | 0.443 | 0.37 | 129.471 | 0.512 | 22.256 | |

AI-PVF | 0.061 | 3.432 | 0.127 | 0.112 | 0.32 | 104.34 | 0.482 | 19.154 | |

12 | Analytical | 0.109 | 5.227 | 0.121 | 0.277 | 0.441 | 97.856 | 0.542 | 24.983 |

Persistence | 0.681 | 34.433 | 0.817 | 11.64 | 0.819 | 291.016 | 0.95 | 109.336 | |

ARIMA | 0.237 | 11.668 | 0.27 | 1.371 | 0.598 | 210.935 | 0.683 | 57.454 | |

SVR | 0.238 | 11.614 | 0.338 | 1.374 | 0.576 | 213.133 | 0.672 | 56.045 | |

GBT | 0.293 | 14.098 | 0.603 | 2.01 | 0.478 | 148.314 | 0.622 | 31.059 | |

DNN | 0.351 | 17.022 | 0.456 | 2.964 | 0.578 | 194.592 | 0.662 | 50.619 | |

LSTM | 0.107 | 5.178 | 0.16 | 0.274 | 0.577 | 178.01 | 0.664 | 45.815 | |

HGBT | 0.281 | 13.542 | 0.651 | 1.853 | 0.51 | 197.48 | 0.672 | 46.675 | |

AI-PVF | 0.039 | 1.875 | 0.055 | 0.035 | 0.318 | 111.339 | 0.484 | 20.213 |

Steps | Models | Sunny Days | Cloudy Days | ||||||
---|---|---|---|---|---|---|---|---|---|

MAE | MAPE | RMSE | WRSE | MAE | MAPE | RMSE | WRSE | ||

(kWh) | (%) | (kWh) | (%) | (kWh) | (%) | (kWh) | (%) | ||

1 | Analytical | 0.081 | 9.328 | 0.111 | 0.397 | 0.181 | 31.021 | 0.207 | 3.706 |

Persistence | 0.083 | 9.039 | 0.121 | 0.452 | 0.106 | 33.198 | 0.122 | 2.012 | |

ARIMA | 0.039 | 3.821 | 0.089 | 0.085 | 0.095 | 44.976 | 0.119 | 2.215 | |

SVR | 0.135 | 12.223 | 0.148 | 0.968 | 0.114 | 70.364 | 0.137 | 3.851 | |

GBT | 0.043 | 3.425 | 0.093 | 0.082 | 0.048 | 24.812 | 0.079 | 0.503 | |

DNN | 0.034 | 3.558 | 0.092 | 0.07 | 0.107 | 58.002 | 0.133 | 2.904 | |

LSTM | 0.035 | 3.329 | 0.09 | 0.066 | 0.069 | 27.623 | 0.086 | 0.976 | |

HGBT | 0.045 | 3.402 | 0.097 | 0.082 | 0.049 | 19.778 | 0.071 | 0.39 | |

AI-PVF | 0.079 | 7.874 | 0.113 | 0.368 | 0.077 | 9.602 | 0.121 | 0.497 | |

2 | Analytical | 0.068 | 6.513 | 0.102 | 0.277 | 0.171 | 29.728 | 0.199 | 3.626 |

Persistence | 0.154 | 15.327 | 0.183 | 1.44 | 0.213 | 83.045 | 0.232 | 9.395 | |

ARIMA | 0.058 | 4.746 | 0.133 | 0.17 | 0.184 | 106.125 | 0.22 | 9.859 | |

SVR | 0.158 | 11.869 | 0.179 | 1.179 | 0.166 | 84.252 | 0.19 | 6.867 | |

GBT | 0.059 | 4.327 | 0.132 | 0.161 | 0.071 | 52.834 | 0.114 | 1.681 | |

DNN | 0.05 | 3.845 | 0.119 | 0.118 | 0.168 | 100.353 | 0.201 | 8.333 | |

LSTM | 0.044 | 3.531 | 0.113 | 0.095 | 0.074 | 79.6 | 0.117 | 2.968 | |

HGBT | 0.061 | 4.681 | 0.142 | 0.183 | 0.069 | 51.021 | 0.131 | 1.61 | |

AI-PVF | 0.079 | 6.012 | 0.108 | 0.281 | 0.071 | 8.891 | 0.109 | 0.467 | |

4 | Analytical | 0.064 | 4.651 | 0.080 | 0.168 | 0.174 | 30.622 | 0.193 | 3.771 |

Persistence | 0.303 | 24.683 | 0.344 | 4.505 | 0.428 | 210.696 | 0.458 | 48.987 | |

ARIMA | 0.092 | 6.428 | 0.204 | 0.355 | 0.321 | 235.41 | 0.389 | 40.863 | |

SVR | 0.17 | 10.894 | 0.223 | 1.137 | 0.229 | 159.897 | 0.274 | 19.293 | |

GBT | 0.094 | 5.132 | 0.021 | 0.302 | 0.131 | 229.187 | 0.221 | 14.834 | |

DNN | 0.073 | 4.738 | 0.178 | 0.21 | 0.197 | 165.124 | 0.254 | 17.557 | |

LSTM | 0.07 | 4.523 | 0.195 | 0.192 | 0.11 | 142.994 | 0.17 | 8.403 | |

HGBT | 0.088 | 6.392 | 0.182 | 0.363 | 0.121 | 147.149 | 0.198 | 8.256 | |

AI-PVF | 0.066 | 4.728 | 0.077 | 0.195 | 0.071 | 8.832 | 0.113 | 0.449 | |

8 | Analytical | 0.054 | 3.018 | 0.062 | 0.086 | 0.163 | 35.072 | 0.183 | 4.937 |

Persistence | 0.66 | 42.765 | 0.712 | 16.466 | 0.836 | 515.241 | 0.88 | 255.806 | |

ARIMA | 0.1 | 6.37 | 0.176 | 0.372 | 0.492 | 463.461 | 0.602 | 140.732 | |

SVR | 0.141 | 8.39 | 0.157 | 0.692 | 0.385 | 392.454 | 0.484 | 92.777 | |

GBT | 0.010 | 6.193 | 0.202 | 0.375 | 0.181 | 315.132 | 0.333 | 34.51 | |

DNN | 0.069 | 4.101 | 0.1 | 0.164 | 0.278 | 340.524 | 0.379 | 59.836 | |

LSTM | 0.065 | 3.923 | 0.196 | 0.15 | 0.179 | 277.139 | 0.281 | 28.147 | |

HGBT | 0.091 | 6.188 | 0.200 | 0.28 | 0.217 | 395.192 | 0.393 | 55.491 | |

AI-PVF | 0.066 | 3.968 | 0.062 | 0.151 | 0.051 | 8.752 | 0.089 | 0.376 | |

12 | Analytical | 0.047 | 2.619 | 0.057 | 0.067 | 0.171 | 46.22 | 0.186 | 8.59 |

Persistence | 1.003 | 57.301 | 1.056 | 31.744 | 1.151 | 849.754 | 1.208 | 622.292 | |

ARIMA | 0.157 | 9.201 | 0.243 | 0.795 | 0.547 | 556.519 | 0.647 | 198.784 | |

SVR | 0.104 | 6.367 | 0.155 | 0.365 | 0.541 | 588.298 | 0.649 | 211.344 | |

GBT | 0.084 | 4.578 | 0.152 | 0.211 | 0.254 | 401.342 | 0.393 | 94.995 | |

DNN | 0.074 | 4.494 | 0.132 | 0.183 | 0.525 | 553.897 | 0.624 | 191.69 | |

LSTM | 0.062 | 3.892 | 0.125 | 0.132 | 0.436 | 459.917 | 0.512 | 125.204 | |

HGBT | 0.094 | 4.708 | 0.141 | 0.221 | 0.303 | 592.422 | 0.488 | 145.237 | |

AI-PVF | 0.066 | 3.743 | 0.077 | 0.136 | 0.045 | 10.58 | 0.067 | 0.448 |

Steps | Models | Sunny Days | Cloudy Days | |||||||
---|---|---|---|---|---|---|---|---|---|---|

MAE | MAPE | RMSE | WRSE | MAE | MAPE | RMSE | WRSE | |||

(kWh) | (%) | (kWh) | (%) | (kWh) | (%) | (kWh) | (%) | |||

1 | Analytical | 0.086 | 26.798 | 0.142 | 5.43 | 0.189 | 163.776 | 0.275 | 67.238 | |

Persistence | 0.054 | 94.033 | 0.097 | 6.092 | 0.108 | 72.636 | 0.218 | 17.661 | ||

ARIMA | 0.051 | 69.031 | 0.069 | 2.881 | 0.114 | 92.745 | 0.21 | 20.602 | ||

SVR | 0.118 | 189.461 | 0.125 | 11.001 | 0.169 | 208.859 | 0.229 | 55.079 | ||

GBT | 0.027 | 40.836 | 0.055 | 1.225 | 0.122 | 83.312 | 0.227 | 21.532 | ||

DNN | 0.029 | 62.957 | 0.052 | 1.74 | 0.12 | 105.383 | 0.21 | 23.229 | ||

LSTM | 0.051 | 82.019 | 0.071 | 3.053 | 0.113 | 80.737 | 0.208 | 18.913 | ||

HGBT | 0.04 | 64.383 | 0.072 | 3.006 | 0.123 | 82.62 | 0.232 | 22.025 | ||

AI-PVF | 0.025 | 33.549 | 0.059 | 0.894 | 0.224 | 257.481 | 0.312 | 134.189 | ||

2 | Analytical | 0.099 | 22.467 | 0.161 | 4.237 | 0.19 | 161.755 | 0.276 | 66.511 | |

Persistence | 0.081 | 305.741 | 0.142 | 19.126 | 0.148 | 133.543 | 0.27 | 40.318 | ||

ARIMA | 0.078 | 248.732 | 0.097 | 8.254 | 0.162 | 160.097 | 0.264 | 47.179 | ||

SVR | 0.171 | 316.716 | 0.185 | 15.014 | 0.199 | 214.459 | 0.284 | 71.138 | ||

GBT | 0.043 | 175.97 | 0.086 | 3.882 | 0.167 | 135.6 | 0.265 | 45.264 | ||

DNN | 0.066 | 202.241 | 0.079 | 4.797 | 0.157 | 145.991 | 0.266 | 43.466 | ||

LSTM | 0.068 | 264.427 | 0.091 | 9.705 | 0.152 | 143.015 | 0.268 | 43.695 | ||

HGBT | 0.059 | 183.313 | 0.15 | 6.402 | 0.164 | 144.939 | 0.271 | 48.531 | ||

AI-PVF | 0.021 | 26.354 | 0.041 | 0.337 | 0.225 | 260.527 | 0.312 | 134.089 | ||

4 | Analytical | 0.121 | 21.509 | 0.184 | 4.086 | 0.195 | 167.746 | 0.282 | 69.431 | |

Persistence | 0.137 | 398.498 | 0.236 | 32.803 | 0.22 | 183.546 | 0.351 | 73.561 | ||

ARIMA | 0.131 | 154.094 | 0.163 | 4.955 | 0.248 | 230.569 | 0.348 | 91.645 | ||

SVR | 0.15 | 160.519 | 0.173 | 6.711 | 0.279 | 260.503 | 0.388 | 118.426 | ||

GBT | 0.072 | 16.73 | 0.125 | 1.126 | 0.235 | 227.827 | 0.336 | 98.012 | ||

DNN | 0.135 | 27.090 | 0.176 | 0.498 | 0.242 | 226.255 | 0.342 | 89.965 | ||

LSTM | 0.131 | 169.734 | 0.164 | 5.564 | 0.256 | 250.655 | 0.361 | 107.936 | ||

HGBT | 0.089 | 34.12 | 0.17 | 3 | 0.228 | 215.417 | 0.333 | 89.113 | ||

AI-PVF | 0.027 | 19.879 | 0.051 | 0.315 | 0.231 | 294.799 | 0.317 | 142.091 | ||

8 | Analytical | 0.168 | 17.434 | 0.214 | 3.057 | 0.202 | 192.095 | 0.294 | 72.831 | |

Persistence | 0.381 | 28.950 | 0.473 | 7.955 | 0.343 | 298.781 | 0.501 | 143.47 | ||

ARIMA | 0.153 | 5.592 | 0.215 | 0.306 | 0.387 | 432.485 | 0.463 | 215.61 | ||

SVR | 0.186 | 5.592 | 0.267 | 0.314 | 0.361 | 385.300 | 0.461 | 184.385 | ||

GBT | 0.306 | 18.786 | 0.449 | 3.539 | 0.322 | 318.336 | 0.42 | 148.522 | ||

DNN | 0.119 | 3.915 | 0.173 | 0.158 | 0.373 | 380.117 | 0.44 | 185.254 | ||

LSTM | 0.202 | 6.960 | 0.264 | 0.472 | 0.434 | 486.760 | 0.514 | 291.991 | ||

HGBT | 0.309 | 18.94 | 0.458 | 3.584 | 0.311 | 270.281 | 0.412 | 120.602 | ||

AI-PVF | 0.038 | 3.831 | 0.06 | 0.152 | 0.237 | 280.743 | 0.326 | 145.337 | ||

12 | Analytical | 0.227 | 17.227 | 0.249 | 2.988 | 0.205 | 194.313 | 0.303 | 83.789 | |

Persistence | 0.738 | 218.253 | 0.904 | 46.433 | 0.473 | 338.102 | 0.637 | 181.524 | ||

ARIMA | 0.224 | 69.777 | 0.274 | 5.071 | 0.503 | 555.070 | 0.564 | 338.556 | ||

SVR | 0.263 | 71.057 | 0.33 | 4.86 | 0.454 | 465.749 | 0.54 | 246.662 | ||

GBT | 0.429 | 26.776 | 0.556 | 7.189 | 0.451 | 549.927 | 0.561 | 337.949 | ||

DNN | 0.269 | 81.495 | 0.356 | 5.327 | 0.46 | 517.506 | 0.542 | 294.673 | ||

LSTM | 0.472 | 89.491 | 0.605 | 11.239 | 0.505 | 576.146 | 0.576 | 357.323 | ||

HGBT | 0.422 | 25.38 | 0.557 | 6.473 | 0.431 | 471.48 | 0.534 | 278.867 | ||

AI-PVF | 0.056 | 4.122 | 0.074 | 0.176 | 0.241 | 316.105 | 0.335 | 169.131 |

Steps | Models | Sunny Days | Cloudy Days | ||||||
---|---|---|---|---|---|---|---|---|---|

MAE | MAPE | RMSE | WRSE | MAE | MAPE | RMSE | WRSE | ||

(kWh) | (%) | (kWh) | (%) | (kWh) | (%) | (kWh) | (%) | ||

1 | Analytical | 0.249 | 74.62 | 0.304 | 8.831 | 0.316 | 93.759 | 0.469 | 32.029 |

Persistence | 0.167 | 47.431 | 0.229 | 4.152 | 0.213 | 68.038 | 0.392 | 15.565 | |

ARIMA | 0.098 | 44.299 | 0.159 | 2.042 | 0.212 | 71.493 | 0.382 | 15.226 | |

SVR | 0.181 | 122.508 | 0.208 | 8.866 | 0.256 | 163.040 | 0.368 | 30.275 | |

GBT | 0.139 | 36.331 | 0.234 | 2.389 | 0.233 | 64.294 | 0.41 | 14.96 | |

DNN | 0.126 | 51.247 | 0.17 | 2.921 | 0.215 | 65.626 | 0.371 | 15.103 | |

LSTM | 0.125 | 40.571 | 0.185 | 2.255 | 0.213 | 68.582 | 0.36 | 14.837 | |

HGBT | 0.149 | 37.244 | 0.25 | 2.646 | 0.226 | 65.674 | 0.396 | 14.669 | |

AI-PVF | 0.169 | 74.214 | 0.217 | 5.379 | 0.289 | 128.983 | 0.422 | 36.397 | |

2 | Analytical | 0.252 | 69.149 | 0.306 | 8.586 | 0.323 | 90.641 | 0.475 | 31.334 |

Persistence | 0.219 | 102.493 | 0.308 | 10.601 | 0.304 | 115.356 | 0.49 | 33.604 | |

ARIMA | 0.149 | 79.513 | 0.21 | 5.269 | 0.292 | 112.678 | 0.463 | 29.84 | |

SVR | 0.209 | 128.066 | 0.242 | 11.106 | 0.305 | 154.637 | 0.448 | 35.904 | |

GBT | 0.145 | 82.728 | 0.216 | 4.595 | 0.299 | 138.026 | 0.479 | 30.728 | |

DNN | 0.167 | 62.902 | 0.231 | 4.747 | 0.276 | 94.667 | 0.424 | 25.184 | |

LSTM | 0.14 | 54.189 | 0.199 | 3.207 | 0.283 | 101.472 | 0.44 | 26.615 | |

HGBT | 0.14 | 51.017 | 0.209 | 3.252 | 0.303 | 136.07 | 0.485 | 31.001 | |

AI-PVF | 0.169 | 74.214 | 0.217 | 5.379 | 0.289 | 128.983 | 0.422 | 36.397 | |

4 | Analytical | 0.258 | 43.074 | 0.313 | 6.776 | 0.332 | 84.045 | 0.486 | 30.213 |

Persistence | 0.388 | 211.354 | 0.493 | 33.507 | 0.445 | 245.596 | 0.632 | 80.702 | |

ARIMA | 0.227 | 108.414 | 0.264 | 9.894 | 0.421 | 230.086 | 0.605 | 66.24 | |

SVR | 0.243 | 100.962 | 0.279 | 10.249 | 0.425 | 249.162 | 0.598 | 69.102 | |

GBT | 0.213 | 61.045 | 0.28 | 6.122 | 0.394 | 157.075 | 0.558 | 49.665 | |

DNN | 0.191 | 89.520 | 0.247 | 7.099 | 0.364 | 191.311 | 0.526 | 51.159 | |

LSTM | 0.221 | 46.635 | 0.279 | 5.576 | 0.366 | 189.151 | 0.521 | 49.572 | |

HGBT | 0.212 | 60.904 | 0.275 | 5.997 | 0.388 | 194.132 | 0.557 | 52.459 | |

AI-PVF | 0.176 | 44.863 | 0.225 | 3.824 | 0.306 | 124.321 | 0.437 | 35.163 | |

8 | Analytical | 0.27 | 24.019 | 0.324 | 5.071 | 0.357 | 82.051 | 0.51 | 29.465 |

Persistence | 0.72 | 410.840 | 0.814 | 87.531 | 0.608 | 299.521 | 0.763 | 119.64 | |

ARIMA | 0.333 | 136.454 | 0.385 | 14.302 | 0.517 | 191.974 | 0.684 | 66.893 | |

SVR | 0.322 | 116.509 | 0.378 | 12.556 | 0.497 | 173.045 | 0.662 | 57.626 | |

GBT | 0.308 | 92.87 | 0.37 | 11.231 | 0.477 | 203.216 | 0.637 | 62.127 | |

DNN | 0.293 | 90.705 | 0.356 | 9.347 | 0.452 | 165.597 | 0.609 | 47.894 | |

LSTM | 0.308 | 100.780 | 0.356 | 11.803 | 0.482 | 205.350 | 0.632 | 60.168 | |

HGBT | 0.319 | 103.987 | 0.381 | 12.66 | 0.456 | 216.769 | 0.637 | 62.127 | |

AI-PVF | 0.187 | 25.552 | 0.236 | 2.64 | 0.328 | 126.362 | 0.457 | 34.471 | |

12 | Analytical | 0.291 | 23.355 | 0.342 | 4.894 | 0.378 | 82.389 | 0.533 | 28.664 |

Persistence | 0.988 | 297.119 | 1.096 | 88.263 | 0.786 | 650.118 | 0.961 | 259.727 | |

ARIMA | 0.4 | 53.756 | 0.499 | 10.63 | 0.56 | 269.605 | 0.735 | 82.885 | |

SVR | 0.436 | 54.642 | 0.552 | 12.372 | 0.552 | 193.198 | 0.741 | 63.351 | |

GBT | 0.385 | 128.341 | 0.47 | 14.261 | 0.511 | 237.448 | 0.678 | 59.922 | |

DNN | 0.444 | 50.802 | 0.557 | 12.81 | 0.499 | 138.776 | 0.674 | 46.83 | |

LSTM | 0.414 | 84.491 | 0.5 | 13.106 | 0.51 | 166.407 | 0.696 | 49.02 | |

HGBT | 0.524 | 173.485 | 0.614 | 25.594 | 0.516 | 247.258 | 0.671 | 61.988 | |

AI-PVF | 0.207 | 26.995 | 0.252 | 2.902 | 0.341 | 130.629 | 0.472 | 32.946 |

**Table 7.**Results of Kruskal-Wallis statistical test performed for the sunny days subperiod of the summer season and the cloudy days subperiod of the winter season for all forecasting horizons. The rejection of the null hypothesis is highlighted by green color, while its acceptance by red. The best performing model in each case in highlighted by bold letters.

Forecasting | Summer | Winter |
---|---|---|

Steps | Sunny Days | Cloudy Days |

Step 1 | Analytical, LSTM, HGBT | Analytical, GBT, HGBT |

Statistical value = 111.073 | Statistical value = 9.662 | |

Step 2 | Analytical, LSTM, HGBT | Analytical, DNN, AI-PVF |

Statistical value = 77.278 | Statistical value = 0.433 | |

Step 4 | Analytical, LSTM, AI-PVF | Analytical, GBT, AI-PVF |

Statistical value = 50.986 | Statistical value = 7.655 | |

Step 8 | Analytical, LSTM, AI-PVF | Analytical, DNN, AI-PVF |

Statistical value = 38.674 | Statistical value = 12.483 | |

Step 12 | Analytical, LSTM, AI-PVF | Analytical, DNN, AI-PVF |

Statistical value = 16.261 | Statistical value = 12.338 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Salamanis, A.I.; Xanthopoulou, G.; Bezas, N.; Timplalexis, C.; Bintoudi, A.D.; Zyglakis, L.; Tsolakis, A.C.; Ioannidis, D.; Kehagias, D.; Tzovaras, D.
Benchmark Comparison of Analytical, Data-Based and Hybrid Models for Multi-Step Short-Term Photovoltaic Power Generation Forecasting. *Energies* **2020**, *13*, 5978.
https://doi.org/10.3390/en13225978

**AMA Style**

Salamanis AI, Xanthopoulou G, Bezas N, Timplalexis C, Bintoudi AD, Zyglakis L, Tsolakis AC, Ioannidis D, Kehagias D, Tzovaras D.
Benchmark Comparison of Analytical, Data-Based and Hybrid Models for Multi-Step Short-Term Photovoltaic Power Generation Forecasting. *Energies*. 2020; 13(22):5978.
https://doi.org/10.3390/en13225978

**Chicago/Turabian Style**

Salamanis, Athanasios I., Georgia Xanthopoulou, Napoleon Bezas, Christos Timplalexis, Angelina D. Bintoudi, Lampros Zyglakis, Apostolos C. Tsolakis, Dimosthenis Ioannidis, Dionysios Kehagias, and Dimitrios Tzovaras.
2020. "Benchmark Comparison of Analytical, Data-Based and Hybrid Models for Multi-Step Short-Term Photovoltaic Power Generation Forecasting" *Energies* 13, no. 22: 5978.
https://doi.org/10.3390/en13225978