# A Comparative Study of Time Series Forecasting of Solar Energy Based on Irradiance Classification

^{1}

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## Abstract

**:**

## 1. Introduction

^{2}) falling on the PV panel. Therefore, global horizontal irradiance (GHI) is the most important input parameter in most PV power prediction systems, and increasing effort is currently spent on research on forecasting of GHI as a basis for PV power forecasts. Different input data and forecasting models are used depending on the forecast horizon.

## 2. Materials and Methods

#### 2.1. PV System Description

#### 2.2. Forecasting Methodology

^{2}) and temperature (°C) forecasts with hourly resolution and a forecast horizon of 72 h ahead. These meteorological forecasts (“Blue Forecast”) are provided by Blue Sky Wetteranalysen [15]. The Blue Forecast is an automated local weather forecasting tool developed based on statistical methods and meteorological knowhow to correct global GFS-NWP models with historical ground measurements from local weather stations. Local characteristics and conditions such as orography, terrain and climatological factors are used in the model, along with other meteorological features, to enhance the accuracy of global NWP.

#### 2.3. Performance Metrics

#### 2.3.1. Probability Integral Transform

#### 2.3.2. Continuous Ranked Probability Scores

#### 2.3.3. Brier Score

#### 2.4. Database Clustering

## 3. Models

#### 3.1. Linear Regression

#### 3.2. Ridge Regression

#### 3.3. Elastic Net

#### 3.4. SARIMAX

- p: is the order of the AR term.
- d: order of differencing to make the data stationary.
- q: order of the MA term.
- P: order of the seasonal AR term.
- D: order of the seasonal differencing to make the data stationary.
- Q: order of the seasonal MA term.
- S: number of periods in a season.

- ${y}_{t}$ denotes the value of the series at time t.
- ${X}_{1,t}$… ${X}_{k,t}$ denote observations of the exogenous variables.
- ${\beta}_{0}$, ${\beta}_{1}$… ${\beta}_{k}$ is the coefficient value for the k-th exogenous (explanatory) input variable.
- ${\varnothing}_{1}$, ${\varnothing}_{2}$… ${\varnothing}_{p}$ is the weight of the nonseasonal autoregressive terms.
- ${\mathsf{\Phi}}_{1}$, ${\mathsf{\Phi}}_{2}$… ${\mathsf{\Phi}}_{P}$ is the weight of the seasonal autoregressive terms.
- ${\theta}_{1}$, ${\theta}_{2}$… ${\theta}_{q}$ is the weight of the nonseasonal moving average terms.
- ${\mathsf{\Theta}}_{1}$, ${\mathsf{\Theta}}_{2}$… ${\mathsf{\Theta}}_{Q}$is the weight of the seasonal moving average terms.
- ${B}^{S}$denotes the backshift operator such that ${B}^{s}{y}_{t}={y}_{t-s}$
- ${Z}_{t}$denotes the white noise terms.

#### 3.5. Facebook Prophet

- $g\left(t\right)$: piecewise linear or logistic growth curve for modeling non-periodic changes in time series;
- $s\left(t\right)$: periodic changes (e.g., weekly/yearly seasonality);
- $h\left(t\right)$: effects of holidays (user-provided) with irregular schedules;
- ${\epsilon}_{t}$: error term, accounting for any unusual changes not accommodated by the model.

#### 3.6. Random Forest

_{1}, …, x

_{n}, with responses Y = y

_{1}, …, y

_{n}, bagging repeatedly (N times) selects a random sample with replacement of the training set and fits trees to these samples:

- Sample, with replacement, n training examples from X, Y; (X
_{n}, Y_{n}) - Create a classification or regression tree f
_{n}and train it on X_{n}, Y_{n}

- For a classification problem, the default value of m is $\sqrt{p}$, and the smallest node size is 1;
- The default value for m is p/3, and the minimum node size is 5 for a regression problem.

#### 3.7. Gradient Boosting

_{1}, x

_{2}, …, x

_{n}), related to each other by some probabilistic distribution function, the objective is to reconstruct functional dependence x $\stackrel{f}{\to}$ y with an estimate, $\widehat{f}\left(x\right)$, such that the specified loss function, $\mathsf{\Psi}\left(y,f\left(x\right)\right)$, is minimized:

- Computation of so-called pseudo variables or the negative gradient:

- Fitting a base learner (or a week learner, e.g., tree) regression model, $g\left(x\right)$, to pseudo-residuals, i.e., training the model using the training dataset ${\left({x}_{i},{z}_{i}\right)}_{i=1}^{N}$ and predicting z
_{i}from the covariates of x_{i}; - Computation of multiplier γ by solving the following one-dimensional problem to choose a gradient descent step size as:

- Updating of the model estimate of $f\left(x\right)$ as:

#### 3.8. Extreme Gradient Boosting

- ▪
- Regularization: XGB offers additional regularization hyperparameters that provide added protection against overfitting.
- ▪
- Early stopping: XGB implements early stopping, allowing for stopping of the process of modelling when additional trees offer no improvement.
- ▪
- Parallel Processing: It is particularly difficult to parallelize GBM because of its sequential nature. XGB has introduced methods to support GPU and Spark compatibility, which enables the use of distributed parallel processing to fit gradient boosting.
- ▪
- Loss functions: Using specific objective and evaluation criteria, users can define and optimize gradient boosting models in XGB.
- ▪
- Continue with existing model: A user can train an XGB model, save the results and later return to that model and continue building onto the results, thereby allowing continued training of the model without starting from scratch.
- ▪
- Different base learners: Most GBM implementations are built with decision trees, but XGB also provides boosted generalized linear models.

- Computation of gradients and hessians as:

- Fitting a base learner (or a week learner, e.g., tree) regression model, $g\left(x\right)$, to pseudo-residuals, i.e., training the model using the training dataset ${({x}_{i}-\frac{{g}_{i}}{{h}_{i}})}_{i=1}^{N}$ by solving the following optimization problem:

- Update the model estimate of $f\left(x\right)$ as:

- Model output:

#### 3.9. GEKKO

- Variable type: fixed variable;
- Equation: user-defined;
- Optimization simulation type: non-dynamic steady state;
- Estimation: model parameter update $\left(IMODE=2\right)$;
- Measure power value to provide an initial guess: $\left(y.STATUS=1\right)$;
- APOPT solver: $\left(m.SOLVER=1\right)$, as the number of degrees of freedom is less than 2000.

#### 3.10. Ensemble Model

## 4. Results and Discussion

#### 4.1. Overview and Analysis

#### 4.2. Probabilistic Forecast

- Rare event R1, where ${K}_{t}$ is lower than its 10% percentile;
- Rare event R2, where ${K}_{t}$ is higher than its 90% percentile;
- Normal event N1, where ${K}_{t}$ is higher than its 50% percentile.

## 5. Conclusions and Outlook

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

PV | Photovoltaics |

GHI | Global Horizontal Irradiance |

K_{t} | Clear Sky Index |

ML | Machine Learning |

ANN | Artificial Neural Network |

GFS | Global Forecast System |

ECMWF | European Centre for Medium Range Forecasts |

NAM | North American Mesoscale Model |

WRF | Weather Research Forecast |

NWP | Numerical Weather Prediction |

ARIMA | Auto Regressive Integrated Moving Average |

SARIMAX | Seasonal Auto Regressive Integrated Moving Average Exogenous |

RR | Ridge Regression |

LASSO | Least Absolute Shrinkage Selector Operator |

EN | Elastic Net |

FBP | Facebook Prophet |

RF | Random Forest |

GB | Gradient Boosting |

XGB | Extreme Gradient Boosting |

FHOOE | Fachhochschule Oberösterreich |

CIS | Copper Indium Selenium |

CdTe | Cadmium Telluride |

a-Si | Amorphous Silicon |

p-Si | Polycrystalline Silicon |

c-Si | Monocrystalline Silicon |

MBE | Mean Bias Error |

MAE | Mean Absolute Error |

rMAE | Relative Mean Absolute Error |

nMAE | Normalized Mean Absolute Error |

IMAE | Inverse of Mean Absolute Error |

RMSE | Root Mean Square Error |

rRMSE | Relative Root Mean Square Error |

nRMSE | Normalized Root Mean Square Error |

CRPS | Continuous Ranking Probability Score |

rCRPS | Relative Continuous Ranking Probability Score |

PIT | Probability Integral Transform |

BS | Brier Score |

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**Figure 1.**PV system and solar radiation measurement station on the rooftop of the University of Applied Science Upper Austria (FHOOE) at Wels Campus, Austria. Two pyranometers and a pyrheliometer are mounted on a tracker for the measurement of global, diffuse and direct irradiance.

**Figure 5.**Box plot of aggregate power forecast comparison of different models for a 24 h horizon. (y: measured power; yhat: forecast power; slr: sklearn linear; rr: ridge regression; en: elastic net; rf: random forest, sar: sarimax; fbp: Facebook Prophet; rfff: random forest with Fourier; entf: elastic net with Fourier; xgb: extreme gradient boost; gb: gradient boost).

**Figure 7.**Metrics: aggregate and under different clear sky index clusters and monthly for 24 h horizons.

**Figure 9.**Variation of aggregate nMAE and nRMSE of different models for different forecast horizons.

**Figure 10.**Intraday forecast for 24 h horizon-time-series plot forecast power (green) with 95% confidence interval measured power (black dots) for GEKKO, QGB and weighted ensemble models.

PV Technology: Silicon mono crystalline | Manufacturer: Sharp |

Rated Power: 185 Wp | No. of Modules: 12 |

Maximum Power of the String STC: 2200 Wp | Inverter: SMA Sunnyboy 1.5/2.0/2.5 |

Azimuth Angle: 0° south direction | PV Panel Tilt Angle (β): 32° |

Cluster Class | CSI Range | Variability |
---|---|---|

Overcast | $csi<0.3$ | low |

Highly Cloudy | $0.3\le csi0.5$ | high |

Cloudy | $0.5\le csi0.7$ | high |

Almost Clear | $0.7\le csi0.9$ | low |

Clear | $csi\ge 0.9$ | low |

Date (from) | Date (to) |
---|---|

1 January 2021 | 10 July 2021 |

22 July 2021 | 30 July 2021 |

20 August 2021 | 25 August 2021 |

1 September 2021 | 3 September 2021 |

21 September 2021 | 30 September 2021 |

Coefficients | Clear Sky Conditions | ||||
---|---|---|---|---|---|

Clear | Almost Clear | Cloudy | Highly Cloudy | Overcast | |

a | 1.067684687 | −0.461843016 | −2.828685511 | 11.89417827 | 12.06287803 |

b | 0.00025502 | 0.007193564 | 0.009622766 | −0.022698139 | −0.09306182 |

c | −1.04 × 10^{−6} | −5.42 × 10^{−6} | −5.35 × 10^{−6} | 2.63 × 10^{−5} | 0.000287555 |

d | −1.19666093 | 101.6616766 | 20.19126047 | 55.41016311 | 122.701323 |

e | 0.139029555 | −6.15386442 | −1.77577578 | −6.238438444 | −10.7498668 |

f | −0.00652831 | 0.101729804 | 0.040920868 | 0.184678849 | 0.287472093 |

g | −890.843258 | 830.2304169 | 800.9744453 | −2954.010321 | −892.164004 |

h | 4878.199184 | −1303.08823 | 1879.956472 | 1135.067326 | 435.5724609 |

i | −2772.24594 | 985.3253282 | −1807.3081 | −8.943386893 | 231.1234833 |

j | 29.6447275 | 4332.245064 | −1373.59256 | 1948.511962 | −6901.24578 |

k | 3.28850648 | −13063.3550 | 3732.719303 | −7003.472994 | 33939.9029 |

l | −6.25419753 | 8462.759824 | −2893.75276 | 4896.689361 | −58542.1699 |

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**MDPI and ACS Style**

Thaker, J.; Höller, R.
A Comparative Study of Time Series Forecasting of Solar Energy Based on Irradiance Classification. *Energies* **2022**, *15*, 2837.
https://doi.org/10.3390/en15082837

**AMA Style**

Thaker J, Höller R.
A Comparative Study of Time Series Forecasting of Solar Energy Based on Irradiance Classification. *Energies*. 2022; 15(8):2837.
https://doi.org/10.3390/en15082837

**Chicago/Turabian Style**

Thaker, Jayesh, and Robert Höller.
2022. "A Comparative Study of Time Series Forecasting of Solar Energy Based on Irradiance Classification" *Energies* 15, no. 8: 2837.
https://doi.org/10.3390/en15082837