# Time Series ARIMA Model for Prediction of Daily and Monthly Average Global Solar Radiation: The Case Study of Seoul, South Korea

^{1}

^{2}

^{*}

## Abstract

**:**

^{2}) for evaluation. In addition, forecasting was conducted by using the best models with historical data on average monthly and daily solar radiation. The contributions of this study can be summarized as follows: (i) a time series SARIMA model is implemented to forecast the daily and monthly solar radiation of Seoul, South Korea in consideration of the accuracy, suitability, adequacy, and timeliness of the collected data; (ii) the reliability, accuracy, suitability, and performance of the model are investigated relative to those of established tests, standardized residual, autocorrelation function (ACF), and partial autocorrelation function (PACF), and the results are compared with those forecasted by the Monte Carlo method; and (iii) the trend of monthly solar radiation in Seoul for the coming years is analyzed and compared on the basis of the solar radiation data obtained from KMS over 37 years. The results indicate that (1,1,2) the ARIMA model can be used to represent daily solar radiation, while the seasonal ARIMA (4,1,1) of 12 lags for both auto-regressive and moving average parts can be used to represent monthly solar radiation. According to the findings, the expected average monthly solar radiation ranges from 176 to 377 Wh/m

^{2}.

## 1. Introduction

#### 1.1. Motivations of the Study

#### 1.2. Problem Statement

#### 1.3. Literature Review

_{i}) over a designated period of time, where every observation represents a specific time (t) and then predicts future outputs according to previous events. Compared with causal forecasting, time series forecasting is flexible and requires fewer data inputs; thus the technique is easier to implement and does not require much cost. However, the main limitation of time series forecasting is the lack of a deterministic cause [11]. To overcome this limitation, model developers usually depend on large numbers of inputs or stochastic events. (iii) Artificial neural networks (ANNs), which have been used in several studies [12,13,14]. On the other hand, the authors in References [15,16,17,18] showed that a radial basis function neural networks (RBF-NN) can be applied to a wide range of nonlinear equation sets. The authors Baghaee et al. [15] proposed the RBF-NN for nonlinear mapping, which is exploited to solve a nonlinear equation set of load flow analysis. While, References [16,17,18] applied the RBF-NN technique into microgrids. Despite the benefits of ANN, however, a previous study [19] investigated the performance of the time series auto-regressive integrated moving average (ARIMA) model in comparison with ANN models and found that the former generally performs better than the latter due to the effect of weather conditions, such as clouds. Solar radiation concentration is partially dependent on various weather, location, and time factors; thus, it displays a type of serial correlation, which suggests that time series forecasting is appropriate for solar radiation forecasting. The unknown working principle or “black-box” of neural networks limits their applicability in predicting solar radiation. The Numerical Weather Prediction (NWP) is widely available in meteorological organizations. However, NWP is highly dependent on air quality and hydrological characteristics, which strongly vary with time and are sensitive to location [20]. The direct implementation of NWP in solar radiation forecasting has been criticized [21]. Therefore, we considered developing a time series forecasting technique in this study because of its convenience and accurate prediction, low data input requirement, and simple computational process. The forecast procedure can provide a rapid and standard way to generate forecasts for many time series in a single step. In the past, hundreds of series were forecasted at a time, with the series organized into separate variables or across groups. ARIMA is regarded as a smooth technique, and it is applicable when the data is reasonably long and the correlation between past observations is stable [22]. Several studies in the literature have used ARMA and ARIMA models for solar radiation prediction [23,24,25,26]. The ARMA and ARIMA models have also been compared in terms of the goodness-of-fit values produced by the log-likelihood function. As a result, the best statistical models and corresponding parameters for solar radiation prediction can be determined comprehensively. Many feasible comparisons have been conducted for solar radiation prediction. In previous work, the prediction task of many models lacked adequacy and timing in terms of data collection. Here, a time series ARIMA model is built to forecast the daily and monthly solar radiation of Seoul, South Korea in consideration of the accuracy, suitability, adequacy, and timeliness of the collected data, which have been obtained from KMS over 37 years. The reliability, accuracy, suitability, and performance of the model are investigated in comparison with those of established tests, such as standardized residual, ACF, and PACF. Finally, the obtained results are compared with those forecasted by the Monte Carlo method.

#### 1.4. Contributions of the Study

- A time series ARIMA model is built to forecast the daily and monthly solar radiation of Seoul, South Korea, considering the accuracy, suitability, adequacy, and timeliness of the collected data.
- The reliability, accuracy, suitability, and performance of the model are investigated in comparison with those of established tests, such as standardized residual, ACF, and PACF, and the results are compared with the results forecasted by the Monte Carlo method.
- The trend of monthly solar radiation in Seoul for the coming years is analyzed and compared based on solar radiation data obtained from the KMS over 37 years.

#### 1.5. Paper Organization

## 2. Case Study: Seoul, South Korea

## 3. Data Collection

^{2}is observed in May; the lower adjacent is 251.1 Wh/m

^{2}and the upper adjacent is 377 Wh/m

^{2}. The red cross represents outliers, which mostly belonged to the lower adjacent. Thus, these outliers should not be considered to reduce their effect on the collected data. The two lower outliers were 209.2 and 224.8 Wh/m

^{2}, which occurred in May 1990 and 1981, respectively. The lowest average solar radiation, 166.1 Wh/m

^{2}, was observed in December. The lower adjacent was 129 Wh/m

^{2}in December 1992, and the upper adjacent was 207.7 Wh/m

^{2}in December 1985. The two lower outliers, 98.3 Wh/m

^{2}and 113.4 Wh/m

^{2}, occurred in December 1990 and 1991, respectively. The largest variation in solar radiation was observed in September, which may be attributed to seasonal weather fluctuations in this month of the year.

## 4. ARIMA Forecasting Model

_{t}is as follows:

_{p}(B) is an auto-regressive operator of order p, θ

_{q}(B) is a moving average operator of order q, and W

_{t}= ΔdX

_{t}.

^{2}) [37].

_{t}is the forecasted observation and X

_{o}is the actual observation.

## 5. Result and Discussion

^{2}= 79%.

^{2}= 68%.

^{12})y

_{t}= (1 − 0.676L)(1 − 0.656L

^{12})ε

_{t}

_{t}is the daily average of solar radiation Wh/m

^{2}at day (t).

^{2}. The general fluctuation trend was maintained, which can be explained by variations in solar radiation due to changes in weather conditions throughout the year. The general trend of monthly solar radiation increased with time, as shown in Figure 7, which may be expected due to increasing UV radiation levels as a result of climate change and ozone layer depletion [45]. The expected monthly solar radiation is expected to reach 377 Wh/m

^{2}. Figure 7b shows the expected average daily radiation for a month in advance. The fluctuation is existing for real average daily radiation, and this fluctuation is kept with the forecasted values. The fluctuations for forecasted average daily radiation seemed larger in comparison with real inputs or forecasted monthly solar radiation.

^{2}. The lowest monthly solar radiation was expected in December (184.3Wh/m

^{2}), which represents the lowest month of solar radiation in actual readings. However, the general trend of monthly solar radiation increased with time. For instance, an increase in average solar radiation of about 2.5% was recorded in May, while the increment in average solar radiation for December is about 10%.

## 6. Conclusions

^{2}/coefficient of determination (share of explained variance), Phillips–Perron test, and Jarque–Bera test, from which the standardized residuals indicate that the residuals of the model are non-correlated and normally distributed. The model has passed the tests, and the results demonstrate the capability of ARIMA to provide accurate monthly and daily solar prediction, especially with the availability of solar radiation data from the previous 37 years. The RMSE was equal to 33.18 and the coefficient of determination (R

^{2}) was equal to 79% for the monthly solar radiation model. Meanwhile, the RMSE was equal to 104.26 and the R

^{2}was equal to 68% for the daily solar radiation model. An R

^{2}value higher than 50% indicates excellent performance of the prediction model. Moreover, the Jarque–Bera test was implemented to investigate the null hypothesis of the normal distribution of the standardized residual. The results support the null hypothesis at P-value = 0.313, which indicates the normal distribution of the standardized residual and its goodness of fit. The standardized residual also shows that the model can effectively predict solar radiation on a monthly basis. In addition, a comparison of the ARIMA model with the Monte Carlo simulations of monthly and daily solar radiation was conducted. The results show that the average monthly solar radiation fluctuates by approximately 250 Wh/m

^{2}, which can be considered a reference figure for estimating potential solar power and in building a method for feasibility calculation. Furthermore, the expected average monthly solar radiation ranges from 176 Wh/m

^{2}(December) to 377 Wh/m

^{2}(May), which is compatible with the general trends of the highest and lowest monthly values and daily fluctuations. Considering these findings is essential in sustainable and proper planning, especially in the field of solar power generation.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. List of Abbreviations

Abbreviation | Meaning |
---|---|

ACF | Autocorrelation Function |

ANNs | Artificial Neural Networks |

AR | Auto-regression |

ARIMA | auto-regressive Integrated Moving Average |

KMA | Korean Meteorological Administration |

MA | Moving Average |

NWP | Numerical Weather Prediction |

PACF | Partial Autocorrelation Function |

PV | Photovoltaic |

RMSE | Root Mean Square Error |

SARIMA | Seasonal Autoregressive Integrated Moving Average |

UV | Ultraviolet |

## Appendix B. List of Symbols

Symbols | Meaning |
---|---|

d | Number of non-seasonal differences needed for stationarity |

p | Number of autoregressive terms |

q | Number of lagged forecast errors in the prediction equation |

B | Backshift operator. |

Φ_{p}(B) | An autoregressive operator of order p |

θ_{q}(B) | A moving average operator of order q |

Xt | Forecasted observation |

Xo | Actual observation |

yt | Daily average of solar radiation |

## References

- Alsharif, M.H.; Kim, J.; Kim, J.H. Green and sustainable cellular base stations: An overview and future research directions. Energies
**2017**, 10, 587. [Google Scholar] [CrossRef] - Kim, K.-G. Evolution of Climate Resilience and Low-Carbon Smart City Planning: A Process. In Urban Governance and Informal Settlements; Springer Nature: Berlin, Germany, 2017; pp. 1–76. [Google Scholar]
- Lee, J.-S.; Kim, J.-W. South Korea’s urban green energy strategies: Policy framework and local responses under the green growth. Cities
**2016**, 54, 20–27. [Google Scholar] [CrossRef] - Alsharif, M.H. A solar energy solution for sustainable third generation mobile networks. Energies
**2017**, 10, 429. [Google Scholar] [CrossRef] - Kwon, T.-H. Is the renewable portfolio standard an effective energy policy?: Early evidence from South Korea. Util. Policy
**2015**, 36, 46–51. [Google Scholar] [CrossRef] - Korean Energy Agency (KEA). Annual Report 2015. Available online: http://www.energy.or.kr/renew_eng/resources/resources_view.aspx?no=12&page=1 (accessed on 22 January 2019).
- Sindhu, S.; Nehra, V.; Luthra, S. Solar energy deployment for sustainable future of India: Hybrid SWOC-AHP analysis. Renew. Sustain. Energy Rev.
**2017**, 72, 1138–1151. [Google Scholar] [CrossRef] - Park, E.; Yoo, K.; Ohm, J.Y.; Kwon, S.J. Case study: Renewable electricity generation systems on Geoje Island in South Korea. J. Renew. Sustain. Energy
**2016**, 8, 015904. [Google Scholar] [CrossRef] - Yadav, A.K.; Chandel, S. Solar radiation prediction using Artificial Neural Network techniques: A review. Renew. Sustain. Energy Rev.
**2014**, 33, 772–781. [Google Scholar] [CrossRef] - Benmouiza, K.; Cheknane, A. Forecasting hourly global solar radiation using hybrid k-means and nonlinear autoregressive neural network models. Energy Convers. Manag.
**2013**, 75, 561–569. [Google Scholar] [CrossRef] - Beaumont, C.; Makridakis, S.; Wheelwright, S.C.; McGee, V.E. Forecasting: Methods and Applications. J. Oper. Res. Soc.
**1984**, 35, 79. [Google Scholar] [CrossRef] - Koca, A.; Öztop, H.F.; Varol, Y.; Koca, G.O. Estimation of solar radiation using artificial neural networks with different input parameters for Mediterranean region of Anatolia in Turkey. Expert Syst. Appl.
**2011**, 38, 8756–8762. [Google Scholar] [CrossRef] - Voyant, C.; Muselli, M.; Paoli, C.; Nivet, M.-L. Optimization of an artificial neural network dedicated to the multivariate forecasting of daily global radiation. Energy
**2011**, 36, 348–359. [Google Scholar] [CrossRef] - Ramedani, Z.; Omid, M.; Keyhani, A.; Shamshirband, S.; Khoshnevisan, B. Potential of radial basis function based support vector regression for global solar radiation prediction. Renew. Sustain. Energy Rev.
**2014**, 39, 1005–1011. [Google Scholar] [CrossRef] - Baghaee, H.R.; Mirsalim, M.; Gharehpetian, G.B.; Talebi, H.A. Generalized three phase robust load-flow for radial and meshed power systems with and without uncertainty in energy resources using dynamic radial basis functions neural networks. J. Clean. Prod.
**2018**, 174, 96–113. [Google Scholar] [CrossRef] - Baghaee, H.R.; Mirsalim, M.; Gharehpetian, G.B.; Talebi, H.A. Three-phase AC/DC power-flow for balanced/unbalanced microgrids including wind/solar, droop-controlled and electronically-coupled distributed energy resources using radial basis function neural networks. IET Power Electron.
**2017**, 10, 313–328. [Google Scholar] [CrossRef] - Baghaee, H.R.; Mirsalim, M.; Gharehpetian, G.B.; Bagahee, H.R.; Gharehpetian, G.B.B. Power Calculation using RBF Neural Networks to Improve Power Sharing of Hierarchical Control Scheme in Multi-DER Microgrids. IEEE J. Emerg. Sel. Top. Power Electron.
**2016**, 4, 1. [Google Scholar] [CrossRef] - Baghaee, H.R.; Mirsalim, M.; Gharehpetan, G.B.; Talebi, H.A. Nonlinear Load Sharing and Voltage Compensation of Microgrids Based on Harmonic Power-Flow Calculations Using Radial Basis Function Neural Networks. IEEE Syst. J.
**2018**, 12, 1–11. [Google Scholar] [CrossRef] - Reikard, G. Predicting solar radiation at high resolutions: A comparison of time series forecasts. Sol. Energy
**2009**, 83, 342–349. [Google Scholar] [CrossRef] - Bauer, P.; Thorpe, A.; Brunet, G. The quiet revolution of numerical weather prediction. Nature
**2015**, 525, 47–55. [Google Scholar] [CrossRef] - Shamim, M.A.; Remesan, R.; Bray, M.; Han, D. An improved technique for global solar radiation estimation using numerical weather prediction. J. Atmos. Sol. -Terr. Phys.
**2015**, 129, 13–22. [Google Scholar] [CrossRef] - Farhath, Z.A.; Arputhamary, B.; Arockiam, D.L. A Survey on ARIMA Forecasting Using Time Series Model. Int. J. Comput. Sci. Mobile Comput.
**2016**, 5, 104–109. [Google Scholar] - 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), Institute of Electrical and Electronics Engineers (IEEE), Miami, FL, USA, 9–11 December 2015; pp. 1045–1049. [Google Scholar]
- Ferrari, S.; Lazzaroni, M.; Piuri, V.; Cristaldi, L.; Faifer, M. Statistical models approach for solar radiation prediction. In Proceedings of the 2013 IEEE International Instrumentation and Measurement Technology Conference (I2MTC), Institute of Electrical and Electronics Engineers (IEEE), Minneapolis, MN, USA, 6–9 May 2013; pp. 1734–1739. [Google Scholar]
- Boualit, S.B.; Mellit, A. SARIMA-SVM hybrid model for the prediction of daily global solar radiation time series. In Proceedings of the 2016 International Renewable and Sustainable Energy Conference (IRSEC), Institute of Electrical and Electronics Engineers (IEEE), Marrakesh, Morocco, 14–17 November 2016; pp. 712–717. [Google Scholar]
- David, M.; Ramahatana, F.; Trombe, P.; Lauret, P. Probabilistic forecasting of the solar irradiance with recursive ARMA and GARCH models. Sol. Energy
**2016**, 133, 55–72. [Google Scholar] [CrossRef] [Green Version] - Korean Statistical Information Service (KOSIS). Available online: http://kosis.kr/statHtml/statHtml.do?orgId=111&tblId=DT_1B040A4&vw_cd=&list_id=&scrId=&seqNo=&lang_mode=ko&obj_var_id=&itm_id=&conn_path=E1 (accessed on 22 January 2019).
- Byrne, J.; Taminiau, J.; Kurdgelashvili, L.; Kim, K.N. A review of the solar city concept and methods to assess rooftop solar electric potential, with an illustrative application to the city of Seoul. Renew. Sustain. Energy Rev.
**2015**, 41, 830–844. [Google Scholar] [CrossRef] - Nematollahi, O.; Kim, K.C. A feasibility study of solar energy in South Korea. Renew. Sustain. Energy Rev.
**2017**, 77, 566–579. [Google Scholar] [CrossRef] - Korean Ministry of Trade, Industry and Energy (MOTIE) & Korea Energy Economics Institute (KEEI). Available online: https://www.keei.re.kr/main.nsf/index_en.html (accessed on 22 January 2019).
- Alsharif, M.H.; Kim, J.; Kim, J.H. Opportunities and Challenges of Solar and Wind Energy in South Korea: A Review. Sustainability
**2018**, 10, 1822. [Google Scholar] [CrossRef] - Korea Meteorological Administration (KMA), Synoptic weather observation. Available online: https://data.kma.go.kr/data/grnd/selectAsosList.do?pgmNo=34 (accessed on 22 January 2019).
- Khashei, M.; Bijari, M. A novel hybridization of artificial neural networks and ARIMA models for time series forecasting. Appl. Soft Comput.
**2011**, 11, 2664–2675. [Google Scholar] [CrossRef] - Chung, S.S. Projecting municipal solid waste: The case of Hong Kong SAR. Resour. Conserv. Recycl.
**2010**, 54, 759–768. [Google Scholar] [CrossRef] - Xu, L.; Gao, P.; Cui, S.; Liu, C. A hybrid procedure for MSW generation forecasting at multiple time scales in Xiamen City, China. Waste Manag.
**2013**, 33, 1324–1331. [Google Scholar] [CrossRef] - Younes, M.K.; Nopiah, Z.M.; Basri, N.E.A.; Basri, H. Medium term municipal solid waste generation prediction by autoregressive integrated moving average. Stat. Oper. Res. Int. Conf. (SORIC 2013)
**2014**, 427–435. [Google Scholar] [CrossRef] - Antanasijević, D.Z.; Pocajt, V.V.; Povrenović, D.S.; Ristić, M.Đ.; Perić-Grujić, A.A. PM10 emission forecasting using artificial neural networks and genetic algorithm input variable optimization. Sci. Total Environ.
**2013**, 443, 511–519. [Google Scholar] [CrossRef] - Miswan, N.H.; Said, R.M.; Anuar, S.H.H. ARIMA with regression model in modelling electricity load demand. J. Telecommun. Electr. Comput. Eng.
**2016**, 8, 113–116. [Google Scholar] - MathWorks, Phillips-Perron Test. Available online: https://www.mathworks.com/help/econ/pptest.html (accessed on 31 January 2019).
- Kleiner, B. Time Series Analysis: Forecasting and Control. Technometrics
**1977**, 19, 343–344. [Google Scholar] [CrossRef] - Chatfield, C.; Weigend, A.S. Time series prediction: Forecasting the future and understanding the past. Int. J. Forecast.
**1994**, 10, 161–163. [Google Scholar] [CrossRef] - Mishra, A.K.; Desai, V.R. Drought forecasting using stochastic models. Stoch. Environ. Res. Risk Assess.
**2005**, 19, 326–339. [Google Scholar] [CrossRef] - Badran, A.; Dwaykat, B. Prediction of Solar Radiation for the Major Climates of Jordan: A Regression Model. J. Ecol. Eng.
**2018**, 19, 24–38. [Google Scholar] [CrossRef] [Green Version] - MathWorks, Jarque-Bera Test. Available online: https://www.mathworks.com/help/econ/pptest.html (accessed on 31 January 2019).
- Häder, D.-P.; Wängbergke, S.Å.; Rose, K.C.; Helbling, E.W.; Sinha, R.P.; Worrest, R.; Williamson, C.E.; Rautio, M.; Gao, K. Effects of UV radiation on aquatic ecosystems and interactions with other environmental factors. Photochem. Photobiol. Sci.
**2015**, 14, 108–126. [Google Scholar] [Green Version]

**Figure 4.**ACF and PACF of the first difference of monthly and daily solar radiation data. (

**a**) Month; (

**b**) Daily.

**Figure 8.**Comparison of ARIMA forecasts versus the Monte Carlo simulation model. (

**a**) Month; (

**b**) Day.

Variables | Daily Average Value | Monthly Average Value |

Number of readings | 13,513 | 444 |

Minimum (Wh/m^{2}) | 9.4 | 85.6 |

Maximum (Wh/m^{2}) | 676.2 | 377 |

Mean (Wh/m^{2}) | 244.6 | 244 |

Median (Wh/m^{2}) | 244.8 | 244.5 |

Standard Deviation | 117.25 | 58.6 |

Range | 666.8 | 291.4 |

Parameter | Value | Standard Error | T Statistic |
---|---|---|---|

Constant | 0.08 | 0.201 | 0.395 |

AR(4) | −0.152 | 0.044 | −3.441 |

SAR(12) | −0.296 | 0.037 | −8.073 |

MA(1) | −0.676 | 0.032 | −21.018 |

SMA (12) | −0.656 | 0.041 | −16.128 |

Variance | 1101.03 | 71.413 | 15.417 |

Parameter | Value | Standard Error | T Statistic |
---|---|---|---|

Constant | −0.002 | 0.081 | −0.027 |

AR(1) | 0.0565 | 0.039 | 1.446 |

MA (1) | −0.767 | 0.040 | −19.662 |

MA (2) | −0.148 | 0.034 | −4.247 |

Variance | 10,870.5 | 146.211 | 74.347 |

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**MDPI and ACS Style**

Alsharif, M.H.; Younes, M.K.; Kim, J.
Time Series ARIMA Model for Prediction of Daily and Monthly Average Global Solar Radiation: The Case Study of Seoul, South Korea. *Symmetry* **2019**, *11*, 240.
https://doi.org/10.3390/sym11020240

**AMA Style**

Alsharif MH, Younes MK, Kim J.
Time Series ARIMA Model for Prediction of Daily and Monthly Average Global Solar Radiation: The Case Study of Seoul, South Korea. *Symmetry*. 2019; 11(2):240.
https://doi.org/10.3390/sym11020240

**Chicago/Turabian Style**

Alsharif, Mohammed H., Mohammad K. Younes, and Jeong Kim.
2019. "Time Series ARIMA Model for Prediction of Daily and Monthly Average Global Solar Radiation: The Case Study of Seoul, South Korea" *Symmetry* 11, no. 2: 240.
https://doi.org/10.3390/sym11020240