# Short-Term Forecasting of Energy Production for a Photovoltaic System Using a NARX-CVM Hybrid Model

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Models

#### 2.1. Collinearity Test

#### 2.2. Augmented Dickey–Fuller Test

_{t}is the corresponding value in the time t and Y

_{t−1}is the corresponding value in the time t−1.

#### 2.3. Engle–Granger Causality Test

- ${\mathrm{H}}_{0}:X$ does not Granger cause $Y$;
- ${\mathrm{H}}_{1}:X$ Granger causes $Y$.

- <0.01 $X$ Granger causes $Y$ at the 1%;
- >0.01 $X$ does not Granger cause $Y$ at the 1%.

#### 2.4. Simplified Single Diode Model

#### 2.5. Solar Radiation under Clear Sky Conditions

#### 2.6. Calculation of the Turbidity Factor

- At least one day of each month is completely clear and corresponds to the day that records the maximum SR measured for that month.
- For each month of the year, the maximum extraterrestrial SR value was calculated to each maximum extraterrestrial SR value corresponding to a value of $\mathit{cos}{\theta}_{{z}_{month}}$.

## 3. Methodology for Building the NARX-CVM Hybrid Model

#### 3.1. Step 1: Databases (VARIABLES)

#### 3.2. Step 2: Selecting the Input Variables (INPUTS)

#### 3.2.1. Collinearity Test

#### 3.2.2. Augmented Dickey–Fuller Test (ADF)

#### 3.2.3. Engle–Granger Causality Test Results

**T**, WS), the independent variable WS does not contain useful information to forecast the dependent variable, SR. Then, this variable is discarded from the input variables, and the unit is used.

#### 3.3. Step 3: Lags for the NARX Model (LAGS)

#### 3.4. Step 4: Modeling Photovoltaic Systems

^{2}) time series. A monocrystalline photovoltaic module ISF-250 black [40] was used in order to verify the model’s validity. Mechanical and electrical characteristics are shown in Table 4 and Table 5, respectively.

#### 3.5. Step 5: Multivariable Forecasting Model (NARX)

^{®}program was used.

#### 3.6. Step 6: Output Data Depuration of the Forecasting Model (CVM)

## 4. Performance Tests

^{®}. The input vectors were reported in Table 8. According to the proposed methodology, the best-input vector was formed by SR and T, H-NARX without CVM and H-NARX-CVM once the CVM was applied (see Table 8 and Table 9). The input number of neurons is defined by the input vectors, the hidden layer neurons are set up in 10, the output neuron is one and it is defined by the output vector. The number of lags was obtained using the ACF and PACF. The time series were pre/post-processing using the functions removeconstantrows and mapminmax. The first function removes the rows of the input vector that correspond to input elements that always have the same value because these input elements are not providing any useful information to the network, and the second function transforms input data so that all values fall into the interval [−1, 1]; this can speed up the learning networks. The division of data for training, validation and testing was carried out using dividerand; this function divided data randomly; the sample data were split up into 70%, 15% and 15% for training, validation and testing, respectively. The performance function is the mean squared error (mse). The transfer function is the tan-sigmoid defined as tansig(n) = $\frac{2}{1-\mathrm{exp}\left(-2\xb7n\right)}-1$ and set up in Matlab

^{®}as tansig.

## 5. Results and Discussion

#### 5.1. Comparison between Models with and without CVM

#### 5.2. Comparison of the H-NARX-CVM Model against Other Models

- (1)
- The blind prediction of the power obtained using the proposed methodology;
- (2)
- The blind prediction using the NAR model;
- (3)
- The prediction using the persistence model.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ANN | Artificial neural network |

ADF | Augmented Dickey–Fuller test |

ACF | Autocorrelation function |

CFP | Corrected forecasting power |

CVM | Corrective vector multiplier |

EP | Electric power |

H-NARX-CVM | Hibrid NARX model |

KPI | Key performance index |

NOCT | Nominal operating cell temperature |

NAR | Nonlinear autoregressive |

NARX | Nonlinear autoregressive with exogenous inputs |

PACF | Partial autocorrelation function |

PGM | Photovoltaic generating model |

PVG | Photovoltaic generator |

PVM | Photovoltaic module |

P | Pressure |

RH | Relative humidity |

SR | Solar radiation |

T | Temperature |

WS | Wind speed |

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**Figure 8.**Characteristic of the power curves of the monocrystalline module ISF-250. (

**a**) Manufacturer curves [40], (

**b**) Curves obtained with the algorithm.

**Figure 9.**Development of the NARX models using different input vectors for electrical power prediction (FPOWER).

**Figure 11.**Linear regression analysis for the five study cases: models without CVM at the top and models with CVM at the bottom.

Probe | Sensor | Range | Accuracy |
---|---|---|---|

CS500 Temperature probe | $1000\mathsf{\Omega}$ platinum resistance, DIM43760B | −40.0 °C to +60.0 °C | ±0.5 °C |

CS500 Relative humidity probe | Vaisala INTERCAP | 0 to 100% | ±3% |

R.M. Young wind sentry anemometer | Cups Wheel Assembly | 0.0 to 50.0 m/s | ±0.5 m/s |

PTB110 Barometer | Vaisala BAROCAP | 500.0–1100.0 hPa | ±0.3 hPa |

WXT510 Weather transmitter | Ultrasonic Signal BAROCAP THERMOCAP Sensor HUMICAP Sensor | 0 to 60 m/s 600 to 1100 hPa −52.0 °C to 60.0 °C 0 to 100% RH | 3% ±0.5 hPa ±0.3 °C ±3% RH |

Variable | Durbin–Watson Statistic | Critical Value | T–Statistic | p–Valor |
---|---|---|---|---|

SR | 2.00 | −1.94 | −1.31 | 0.18 |

T | 2.00 | −1.94 | −0.40 | 0.54 |

RH | 2.00 | −1.94 | −1.46 | 0.13 |

WS | 1.99 | −1.94 | −1.71 | 0.08 |

P | 1.99 | −1.94 | −0.02 | 0.67 |

- (a)
- Group of variables (SR, T, WS)→ Dependent variable SR
| |

Variable | Probability |

T | 0.00 |

WS | 0.12 |

- (b)
- Group of variables (SR, RH, WS)→ Dependent variable SR
| |

Variable | Probability |

RH | 0.00 |

WS | 0.00 |

- (c)
- Group of variables (SR, P, WS)→ Dependent variable SR
| |

Variable | Probability |

P | 0.00 |

WS | 0.00 |

**Table 4.**Mechanical characteristics of the monocrystalline module ISF-250 [40].

Parameter | Characteristics |
---|---|

Solar cell | Monocrystalline silicon–156 mm × 156 mm (6 inches) |

Number of cells | 60 cells (6 × 10) |

Dimensions | 1667 × 994 × 45 mm (65.63 × 39.13 × 1.77 in) |

Weight | 19 kg (41.89 pounds) |

Glass | High transmittance, patterned, tempered, 3.2 mm (EN-12150) |

Frame | Anodized aluminium, grounding drills |

Maximum mechanical load | 5400 Pa (112.78 psf) (Snow load) |

Junction box | IP 65 with three bypass diodes |

Cables, plug | Solar cable 1 m (39.37 in), four mm^{2} (12 AWG). MC4 or LC4 |

**Table 5.**Electrical characteristics of the monocrystalline Module ISF-250 [40]. Performance at STC: Irradiance, 1000 W/m

^{2}; cell temperature, 25 °C (77 °F); AM, 1.5.

Parameter | Characteristics |
---|---|

Rated power (P_{max}) | 250 W |

Open-circuit voltage (V_{oc}) | 37.8 V |

Short-circuit current (I_{sc}) | 8.75 A |

Maximum power point voltage (V_{max}) | 30.6 V |

Maximum power point current (I_{max}) | 8.17 A |

Efficiency | 15.1% |

Power tolerance (% P_{max}) | 0/+3% |

Variable | Estimated | Actual | Error |
---|---|---|---|

${P}_{max}\left(W\right)$ | 248.1 | 250.0 | 0.75% |

${V}_{oc}\left(V\right)$ | 37.5 | 37.8 | 0.83% |

${V}_{max}\left(V\right)$ | 30.8 | 30.6 | −0.78% |

${I}_{sc}\left(A\right)$ | 8.8 | 8.8 | −0.12% |

${I}_{max}\left(A\right)$ | 8.1 | 8.2 | 1.52% |

Models | Lags $\left(\mathit{L}\right)$ | Input $\mathit{x}\left(\mathit{t}\right)$ | Output $\mathit{y}\left(\mathit{t}\right)$ | $\mathbf{Hidden}\mathbf{Neurons}\left(\mathit{h}\mathit{n}\right)$ | $\mathbf{Output}\mathbf{Neurons}\left(\mathit{m}\right)$ | Tests |
---|---|---|---|---|---|---|

NARX I | 24 | SR, T, RH, WS, P | FPOWER | 10 | 1 | All variables |

NARX II | 24 | SR, T, WS | FPOWER | 10 | 1 | Collinearity and causality |

NARX III | 24 | SR, RH, WS | FPOWER | 10 | 1 | Collinearity and causality |

NARX IV | 24 | SR, WS, P | FPOWER | 10 | 1 | Collinearity and causality |

H-NARX | 24 | SR, T | FPOWER | 10 | 1 | Collinearity and causality |

Model | Lag | Input | Output | MBE (W) | MSE (W^{2}) | RMSE (W) | R^{2} |
---|---|---|---|---|---|---|---|

NARX I | 24 | SR, T, RH, WS, P | FPower | 0.45 | 210.30 | 14.50 | 0.95 |

NARX II | 24 | SR, T, WS | FPower | 0.70 | 147.83 | 12.16 | 0.97 |

NARX III | 24 | SR, RH, WS | FPower | −0.27 | 149.81 | 12.24 | 0.97 |

NARX IV | 24 | SR, WS, P | FPower | 0.72 | 145.15 | 12.05 | 0.97 |

H-NARX | 24 | SR, T | FPower | −0.18 | 131.42 | 11.46 | 0.97 |

Model | Lag | Input | Output | cMBE (W) | cMSE (W^{2}) | cRMSE (W) | cR^{2} |
---|---|---|---|---|---|---|---|

NARX-CVM I | 24 | SR, T, RH, WS, P | CPower | −0.45 | 184.80 | 13.59 | 0.96 |

NARX-CVM II | 24 | SR, T, WS | CPower | −0.01 | 142.78 | 11.95 | 0.97 |

NARX-CVM III | 24 | SR, RH, WS | CPower | −0.57 | 145.41 | 12.06 | 0.97 |

NARX-CVM IV | 24 | SR, WS, P | CPower | 0.40 | 143.96 | 12.00 | 0.97 |

H-NARX-CVM | 24 | SR, T | CPower | −0.41 | 130.07 | 11.40 | 0.97 |

Model | RMSE (W) | cRMSE (W) | Improvement |
---|---|---|---|

NARX I vs. NARX-CVM I | 14.50 | 13.59 | 6.7% |

NARX II vs. NARX-CVM II | 12.16 | 11.95 | 1.8% |

NARX III vs. NARX-CVM III | 12.24 | 12.06 | 1.5% |

NARX IV vs. NARX-CVM IV | 12.05 | 12.00 | 0.4% |

H-NARX vs. H-NARX-CVM | 11.46 | 11.40 | 0.5% |

Models | Performance Tests | |||
---|---|---|---|---|

MBE | MSE | RMSE | R^{2} | |

H-NARX-CVM | −0.41 | 130.07 | 11.40 | 0.97 |

NAR | −1.12 | 300.57 | 17.34 | 0.94 |

Persistence | 0.00 | 386.12 | 19.65 | 0.92 |

Models | Performance Tests | |||
---|---|---|---|---|

MBE | MSE | RMSE | R^{2} | |

H-NARX-CVM | 0.08 | 142.59 | 11.94 | 0.97 |

NAR | 0.56 | 220.54 | 14.85 | 0.95 |

Persistence | 1.48 | 330.01 | 18.17 | 0.93 |

Models | Performance Tests | |||
---|---|---|---|---|

MBE | MSE | RMSE | R^{2} | |

H-NARX-CVM | −0.29 | 329.36 | 18.15 | 0.90 |

NAR | 1.12 | 291.60 | 17.08 | 0.91 |

Persistence | −6.89 | 803.77 | 28.35 | 0.78 |

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**MDPI and ACS Style**

Rangel-Heras, E.; Angeles-Camacho, C.; Cadenas-Calderón, E.; Campos-Amezcua, R.
Short-Term Forecasting of Energy Production for a Photovoltaic System Using a NARX-CVM Hybrid Model. *Energies* **2022**, *15*, 2842.
https://doi.org/10.3390/en15082842

**AMA Style**

Rangel-Heras E, Angeles-Camacho C, Cadenas-Calderón E, Campos-Amezcua R.
Short-Term Forecasting of Energy Production for a Photovoltaic System Using a NARX-CVM Hybrid Model. *Energies*. 2022; 15(8):2842.
https://doi.org/10.3390/en15082842

**Chicago/Turabian Style**

Rangel-Heras, Eduardo, César Angeles-Camacho, Erasmo Cadenas-Calderón, and Rafael Campos-Amezcua.
2022. "Short-Term Forecasting of Energy Production for a Photovoltaic System Using a NARX-CVM Hybrid Model" *Energies* 15, no. 8: 2842.
https://doi.org/10.3390/en15082842