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

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## 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 |

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**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 |

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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