# Linear-Gompertz Model-Based Regression of Photovoltaic Power Generation by Satellite Imagery-Based Solar Irradiance

^{1}

^{2}

^{*}

## Abstract

**:**

^{2}= 0.85 and nRMSE = 0.09. The Gompertz model coefficients showed normal distributions and equivariance of standard deviations of less than 15% by year and by season. Therefore, it can be conjectured that the Linear-Gompertz model represents the whole country’s PV system performance curve. In addition, the Gompertz coefficient C, which controls the growth rate of the curve, showed a strong correlation with the capacity factor, such that the regression equation for the capacity factor could be derived as a function of the three Gompertz model coefficients with a fitness of R

^{2}= 0.88.

## 1. Introduction

^{2}spatial resolution by employing the geostationary weather satellite COMS (Communication, Ocean, and Meteorological Satellite), which was launched in 2010 and scans the Oceanian hemisphere.

## 2. Data

#### 2.1. Solar Irradiance Data

#### 2.2. PV Power Generation Data

## 3. Methods

#### 3.1. Regression Model

#### 3.1.1. Gompertz Function

^{2}) was expressed as the following equation:

_{N}(kW). Second, a regression analysis was conducted to find the best fitness between the hourly GHI and the normalized PV power output by employing the Gompertz function.

_{N}, as the PV power output generally bended to the nominal capacity.

#### 3.1.2. Linear-Gompertz Conjoint Function

_{j}, but the smallest x

_{j}was chosen for the present study.

_{j}by Equation (8), D by Equation (6), and finally, y

_{j}by Equation (4).

#### 3.2. Evaluation of Regression

_{N}, which was defined as:

_{L-G}and P

_{KPX}stand for the PV power predicted by the Linear-Gompertz (L-G) regression model and the real PV power output data provided by KPX, respectively.

^{2}) was also evaluated for the linear regression (as the reference) and the Linear-Gompertz regression for comparison. The evaluation of regression was carried out by year, by season, and by province to confirm whether the Linear-Gompertz model was suitable as the PV performance curve. The final validation was performed for the 10 sites, which were not used to derive the Gompertz model coefficients.

## 4. Results and Discussions

#### 4.1. Comparison of the Regression Models

^{2}and nRMSE, was evaluated for the 232 PV power plants, excluding the 10 sites which were reserved for validation. Table 2 compares the mean (μ) and the standard deviation (σ) of the fitness measures between the Linear (L) model and the Linear-Gompertz model by year. Of the overall values of the measures, the fitness of L-G was better than L, as R

^{2}was higher and nRMSE was lower for the L-G regression. For the statistical confirmation, t-tests assuming equal means (Equation (10)) were performed, resulting in the rejection of the null hypothesis (p-values < 0.05), which confirms that the means of L and L-G were statistically different. In other words, R

^{2}and nRMSE of L-G were statistically higher than those of L by about 2.5% and 11% respectively.

^{2}) and most of the GHI and PV power output data were concentrated in the linear section (<1000 W/m

^{2}) of the L-G regression model. However, the comparison results obtained after selecting 10 PV power plants of high solar irradiance in excess of 1000 W/m

^{2}verified that the R

^{2}improvement in the L-G model was 6%, which is significantly higher than 2.6%.

^{2}and nRMSE of the Gompertz model over South Korea, in which no obvious spatial pattern can be observed, i.e., a random pattern was present. For reference, the correlation coefficient between R

^{2}and nRMSE was −0.86.

#### 4.2. Gompertz Model Coefficients

^{2}= 0.53, while the Gompertz coefficients A and B showed no correlation with the capacity factor (R

^{2}< 0.02). This can be interpreted to mean that the Gompertz coefficient C controlled the variance of the curve at a given solar irradiance depending on such environmental factors as air temperature. Therefore, it was necessary to analyze the Gompertz coefficient by season in order to identify the effects of air temperature.

^{2}= 0.88, which was higher than that of C and CF, by accommodating the interconnected relationship between C and A, as well as B.

^{2}= 0.88.

#### 4.3. Validation of the Model

^{2}and nRMSE, as shown in Table 2.

_{N}. This error includes not only the error of the regression model, but also that of the input data GHI. According to a study conducted by Kim et al. (2017), the GHI prediction of the UASIBS-KIER model ranged between 7.4%~16.7% in terms of the rRMSE [16]. This level of error is sufficient for the purpose of evaluating the PV potential of South Korea. However, considering that the average rate of error of general one-hour-ahead forecasting models is nRMSE = 7.2% [25], the error in the results of the present study is somewhat large for a PV power forecasting model.

## 5. Conclusions

- (1)
- The Linear-Gompertz model successfully expressed the sigmoidal characteristics of the PV system performance countrywide as a single function of GHI, which is the simplest regression form adequate for machine learning needed to develop a forecasting model. The nonphysical trend of the Gompertz model in the low GHI range was fixed by combining a linear equation having the same slope at the conjoint point. The fitness of the Linear-Gompertz regression was R
^{2}= 0.85, and the nRMSE of normalized power output ratio was 0.09. - (2)
- The three Gompertz coefficients A, B, and C were calculated by year, by season, and by province, and it was found that they had normal distributions and equivariances, meaning that the Gompertz coefficients were the general parameters for the entire country. Moreover, the Gompertz coefficient of the growth rate C showed a strong correlation (R
^{2}= 0.53) with the capacity factor of the PV power plant. Therefore, it was possible to derive the capacity factor equation as a function of A, B, and C, that showed a fitness of R^{2}= 0.88. - (3)
- In order to use the Linear-Gompertz model to obtain South Korea’s general PV performance curve for PV power forecasting, it will be necessary to increase the fitness of the model to over R
^{2}> 0.9 by including significant environmental variables such as ambient temperature. Future research will consist of securing long-term PV power output data and analyzing the aging effect of the PV panel to correct the degradation effect. In addition, the accuracy of the Linear-Gompertz model will be improved by calculating and applying the POA, the primary input variable, instead of GHI. To that end, the solar irradiance decomposition algorithm should be improved in the UASIBS-KIER model, and verification and compensation steps using actual measurement data should be implemented in advance. - (4)
- Because the solar irradiance in most regions of South Korea is less than 1300 W/m
^{2}, an additional verification of the conditions of high solar irradiance is needed to apply this result to regions with high solar irradiance. In addition, since PV power generation is significantly influenced by climate conditions, there will be some differences compared to regions in which the climate zone is completely different from that of South Korea. However, South Korea has four distinct seasons and a wide temperature distribution ranging from −15 °C to +35 °C throughout the year. Thus, the effect of climate conditions on PV power generation is relatively significant, which means the prediction error that occurred when the present regression model was applied to other climate zone is expected to be smaller in a relative sense.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Provincial distribution of the sample PV power plants (the blue symbols correspond to the 10 validation plants).

**Figure 2.**The scatterplot between the PV power output and the global horizontal irradiance (GHI; Jeollanam-do, Youngkwang-gun, 300 kW).

**Figure 3.**Effects of the Gompertz model coefficient: (

**a**) effect of A (B = 1.10, C = 4·10

^{−3}), (

**b**) effect of B (A = 0.77, C = 4·10

^{−3}), and (

**c**) effect of C (A = 0.77, B = 1.10).

**Figure 4.**Conjoint process of a linear line (dashed line) and a Gompertz curve (solid line). The line and Gompertz curve appear separated in (

**a**) and conjoined in (

**b**).

**Figure 5.**Distribution of the Gompertz model fitness measures: (

**a**) coefficient of determination (R

^{2}) distribution, and (

**b**) normalized root mean square error (nRMSE) distribution.

**Figure 7.**The relationship between the Gompertz coefficient C and the capacity factor (mean capacity factor = 14.8 ± 1.4%).

**Figure 8.**Comparison of the PV system performance curves between summer and winter (Jeollabuk-do Gochang-gun, 500 kW).

Province | Sampling | PV Plants (>50 kW_{p}) | Total No. of PV Plants |
---|---|---|---|

Jeollanam-do | 86 | 218 | 6728 |

Jeollabuk-do | 28 | 71 | 9040 |

Gyeongsangnam-do | 29 | 72 | 2398 |

Gyeongsangbuk-do | 37 | 91 | 3847 |

Chungcheongnam-do | 16 | 39 | 4323 |

Chungcheongbuk-do | 6 | 13 | 2144 |

Gyeonggi-do | 17 | 39 | 3435 |

Gangwon-do | 9 | 23 | 2320 |

Jeju-do | 14 | 34 | 590 |

Sum | 242 | 600 | 34,825 |

Year | 2014 | 2015 | 2016 | 2014~2016 | ||||
---|---|---|---|---|---|---|---|---|

Model | L | L-G | L | L-G | L | L-G | L | L-G |

R^{2} | 0.83 | 0.85 | 0.84 | 0.86 | 0.82 | 0.85 | 0.83 | 0.85 |

nRMSE | 0.10 | 0.09 | 0.10 | 0.09 | 0.10 | 0.09 | 0.10 | 0.09 |

Year | 2014 | 2015 | 2016 | 2014~2016 | |
---|---|---|---|---|---|

A | μ | 0.77 | 0.78 | 0.76 | 0.77 |

σ | 0.05 | 0.05 | 0.05 | 0.05 | |

B | μ | 1.09 | 1.10 | 1.10 | 1.10 |

σ | 0.06 | 0.06 | 0.06 | 0.06 | |

C | μ (×10^{−3}) | 4.20 | 4.09 | 4.15 | 4.14 |

σ (×10^{−3}) | 0.64 | 0.62 | 0.60 | 0.61 |

Coefficient | Season | 2014 | 2015 | 2016 | 2014~2016 |
---|---|---|---|---|---|

A | Winter | 0.82 | 0.87 | 0.83 | 0.84 |

Spring | 0.87 | 0.86 | 0.86 | 0.86 | |

Summer | 0.74 | 0.78 | 0.79 | 0.77 | |

Autumn | 0.70 | 0.75 | 0.73 | 0.73 | |

B | Winter | 1.16 | 1.17 | 1.15 | 1.16 |

Spring | 1.06 | 1.07 | 1.09 | 1.07 | |

Summer | 0.96 | 0.93 | 0.96 | 0.95 | |

Autumn | 0.84 | 1.00 | 1.11 | 0.98 | |

C (×10^{−3}) | Winter | 4.85 | 4.62 | 4.71 | 4.73 |

Spring | 3.57 | 3.63 | 3.55 | 3.59 | |

Summer | 3.72 | 3.33 | 3.27 | 3.44 | |

Autumn | 4.57 | 4.21 | 4.50 | 4.43 |

Year | 2014 | 2015 | 2016 | 2014~2016 | ||||
---|---|---|---|---|---|---|---|---|

Code | R^{2} | nRMSE | R^{2} | nRMSE | R^{2} | nRMSE | R^{2} | nRMSE |

1730 | 0.83 | 0.10 | 0.85 | 0.09 | 0.84 | 0.09 | 0.84 | 0.09 |

1837 | 0.87 | 0.09 | 0.89 | 0.09 | 0.87 | 0.09 | 0.88 | 0.09 |

1882 | 0.88 | 0.08 | 0.89 | 0.08 | 0.87 | 0.08 | 0.88 | 0.08 |

1947 | 0.87 | 0.09 | 0.88 | 0.09 | 0.87 | 0.09 | 0.87 | 0.09 |

1996 | 0.81 | 0.10 | 0.82 | 0.10 | 0.82 | 0.10 | 0.82 | 0.10 |

8877 | 0.86 | 0.10 | 0.86 | 0.10 | 0.84 | 0.10 | 0.85 | 0.10 |

8907 | 0.85 | 0.09 | 0.89 | 0.09 | 0.86 | 0.09 | 0.87 | 0.09 |

9577 | 0.88 | 0.10 | 0.88 | 0.09 | 0.87 | 0.09 | 0.88 | 0.09 |

9617 | 0.89 | 0.08 | 0.89 | 0.08 | 0.90 | 0.08 | 0.89 | 0.08 |

9927 | 0.89 | 0.09 | 0.89 | 0.09 | 0.89 | 0.09 | 0.89 | 0.09 |

Mean | 0.86 | 0.09 | 0.87 | 0.09 | 0.86 | 0.09 | 0.87 | 0.09 |

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**MDPI and ACS Style**

Vilanova, A.; Kim, B.-Y.; Kim, C.K.; Kim, H.-G.
Linear-Gompertz Model-Based Regression of Photovoltaic Power Generation by Satellite Imagery-Based Solar Irradiance. *Energies* **2020**, *13*, 781.
https://doi.org/10.3390/en13040781

**AMA Style**

Vilanova A, Kim B-Y, Kim CK, Kim H-G.
Linear-Gompertz Model-Based Regression of Photovoltaic Power Generation by Satellite Imagery-Based Solar Irradiance. *Energies*. 2020; 13(4):781.
https://doi.org/10.3390/en13040781

**Chicago/Turabian Style**

Vilanova, Alba, Bo-Young Kim, Chang Ki Kim, and Hyun-Goo Kim.
2020. "Linear-Gompertz Model-Based Regression of Photovoltaic Power Generation by Satellite Imagery-Based Solar Irradiance" *Energies* 13, no. 4: 781.
https://doi.org/10.3390/en13040781