# Forecasting of Electrical Generation Using Prophet and Multiple Seasonality of Holt–Winters Models: A Case Study of Kuwait

^{1}

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## Abstract

**:**

^{2}= 0.9899 and MAPE = 1.76%, followed by the double seasonality Holt–Winters model with R

^{2}= 0.9893 and MAPE = 1.83%. Moreover, the Prophet model with multiple regressors was the third-best performing model with R

^{2}= 0.9743 and MAPE = 2.77%. The forecasted annual generation in the year 2030 resulted in 92,535,555 kWh according to the best performing model. The study provides an outlook on the medium- and long-term electrical generation. Furthermore, the impact of fuel cost is investigated based on the five forecasting models to provide an insight for Kuwait’s policymakers.

## 1. Introduction

## 2. Methodologies

#### 2.1. Prophet Forecasting Method

_{t}, denotes any distinctive features of the data that are not fitted by the model.

#### 2.2. Holt–Winters Forecasting Model

_{i}. The term ${\phi}_{AR}^{k}$ is an adjustment for the first autocorrelation error.

- Step 1:
- Compute the yearly average as ${A}_{m}^{i}$ for each seasonality of length ${s}_{m}$ and has a pattern of ${n}_{q}$ times in the dataset$${A}_{m}^{i}=\frac{{{\displaystyle \sum}}_{j=1}^{{s}_{i}}{X}_{\left(m-1\right){s}_{i}+j}}{{s}_{i}}\text{}\mathrm{for}m=1,\text{}2,\text{}\dots ,{n}_{q}$$
- Step 2:
- Divide the observations by the yearly averages as:
**index****1****2****…****q**1 $\frac{{X}_{1}}{{A}_{1}^{i}}$ $\frac{{X}_{{s}_{i}+1}}{{A}_{2}^{i}}$ … $\frac{{X}_{{s}_{i}+1}}{{A}_{q}^{i}}$ … … … … … ${s}_{i}$ $\frac{{X}_{{s}_{i}1}}{{A}_{1}^{i}}$ $\frac{{X}_{2{s}_{i}}}{{A}_{2}^{i}}$ … $\frac{{X}_{{m}_{1}{s}_{i}+1}}{{A}_{{m}_{i}}^{i}}$ - Step 3:
- Write each seasonality as:$${I}_{1-{s}_{i}}^{*\left(i\right)}=\frac{\frac{{X}_{1}}{{A}_{1}^{i}}+\frac{{X}_{{s}_{i}+1}}{{A}_{2}^{i}}+\cdots +\frac{{X}_{({m}_{i}-1){s}_{i}+1}}{{A}_{{m}_{i}}^{i}}}{{m}_{i}}$$$${I}_{1-{s}_{i}}^{*\left(i\right)}=\frac{\frac{{X}_{1}}{{A}_{1}^{i}}+\frac{{X}_{{s}_{i}+1}}{{A}_{2}^{i}}+\cdots +\frac{{X}_{{m}_{i}{s}_{i}+1}}{{A}_{{m}_{i}}^{i}}}{{m}_{i}}$$
- Step 4:
- Write the seasonal indices as:$${I}_{1-{s}_{i}}^{*\left(i\right)}=\{\begin{array}{c}{I}_{t-{s}_{1}}^{*\left(i\right)}i=1,\text{}t=1,\dots ,{s}_{i}\\ \frac{{I}_{t-{s}_{1}}^{*\left(i\right)}}{{{\displaystyle \prod}}_{j=1}^{i-1}{I}_{t-{s}_{j}}^{*\left(i\right)}}i1,\text{}t=1,\text{}\dots ,{s}_{i}\end{array}$$

#### 2.3. Performance Indicators

^{2}), mean absolute error (MAE), and coefficient of variation of root mean square error (CVRMSE) that can be expressed as in the following equations:

## 3. Results and Discussion

#### 3.1. Model Assessments

^{2}. Each of the metrics shed light on one angle of the accuracy of the data. MAPE can be considered one of the utmost used tools for evaluating the accuracy of models. MAE reveals the difference between the estimated value and the real value using the absolute error. RMSE assesses the variability of model response regarding variance and sensitivity to large errors. CVRMSE standardizes the forecasted error and provides a unit-less metric that evaluates the variability of the errors between predicted and real values.

^{2}) for the used models. Five different models were used: The single Prophet, multiple Prophet models, single, double, and triple seasonality Holt–Winters models. Abbreviations are used to denote the different models used: PSR denotes a Prophet single regressor model. PMR denotes Prophet multiple regressors model, HWSS denotes Holt–Winters with single seasonality model, HWDS denotes Holt–Winters with double seasonality model, and HWTS denotes Holt–Winters with triple seasonality model. The triple seasonality Holt–Winters model achieved a superior value of accuracy in comparison with other model’s performance metrics. The double seasonality Holt–Winters model was the second with all performance measures comparable to the triple seasonality model. MAPE was proposed as a reference indicator for assessing energy forecasting performance at different horizons [2]. All models achieved a low MAPE value of less than 5%, which indicates the models’ high accuracy. It is benchmarked [40] that When MAPE is less than 10%, it is considered a highly accurate model. Highly accurate models were then categorized into four levels, with the best model denoted by I level when MAPE is ≤1.2%. Other levels are as follows: II (1.2–2.8%), III (2.8–4.6%), and IV (4.6–10%) [2]. Three of the used models, namely the multiple Prophet model and the double and triple seasonality of Holt–Winters models, fall in the II level of this rating, whereas the other two, namely single Prophet and single seasonality of Holt–Winters model, fall in the III level. The triple seasonality Holt–Winters model outperforms all the other models in MAPE value. Superiority is also observed for both double and triple seasonality Holt–Winters models in MAPE values compared to other models. The MAPE values were 1.76% and 1.83% for triple and double seasonality Holt–Winters models, respectively.

^{2}, of the five models are plotted with excellent values in Figure 3, showing small differences among the forecasted and real data values. The best-fitting was for the Holt–Winters model’s triple seasonality with R

^{2}= 0.9899, then to the Holt–Winters model’s double seasonality R

^{2}= 0.9893 followed by multiple Prophet model with R

^{2}= 0.9743. The least values of R

^{2}were 0.9709 and 0.9641 for the single Prophet model and single seasonality of the Holt–Winters model, respectively. The MAE of the best model, which is the triple seasonality of Holt–Winters, was found to be 47.82, whereas the value of MAE for the least model, which was the single seasonality Holt–Winters model, was 127.82, which is approximately three times of the triple model.

^{2}by studying the random distributed Gaussian white noise on the training dataset. Table 4 shows the variance of R

^{2}affected by different noise intensities.

^{2}of the triple seasonality of the Holt–Winters model with the least value of 0.9816 at 80% noise intensity. The Holt–Winters model’s single seasonality showed the least robustness among the other models with a minimum value of 0.9552. The other models showed excellent robustness across the different noise intensities. These results drive to the same previous conclusion that the triple seasonality then double seasonality of the Holt–Winters model outperformed the other models in terms of accuracy, generalization, and robustness.

#### 3.2. Future Generation Forecasting

#### 3.3. Implications of Elecrical Generation on Fuels Cost

## 4. Conclusions

^{2}. According to the different performance indicators, the double seasonality of the Holt–Winters model was the second-best performing model with superior accuracy. The Prophet model, with multiple regressors, has shown comparable performance to the Holt–Winters double seasonality model. In contrast, the least performing models were the single regressor Prophet model followed by the single seasonality Holt–Winters model. The estimated annual electrical generation for the year 2030 of the Holt–Winters models’ triple and single seasonality models were 92,535,555 and 102,262,507 kWh, respectively. The generation of electricity in Kuwait during the upcoming ten years was then presented and discussed. The results reveal that the higher electrical generation period occurs between May and September of the upcoming two years. The five models’ long-term electrical generation forecast was then used to estimate the fuel’s total cost to provide an overview for policymakers to support the execution of planning decisions.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

GDP | Gross domestic product |

AI | Artificial intelligence |

ML | Machine learning |

DL | Deep learning |

GA | Genetic algorithm |

ANN | Artificial neural networks |

SVR | Support vector regression |

IPSO | Improved particle swarm optimization |

PCA-FFNN | Principal component analysis and fuzzy feed-forward neural network |

PCFI-RBF | Partial-consensus fuzzy intersection and radial basis function network |

RMSE | Root mean square error |

MAPE | Mean absolute percentage error |

R^{2} | Coefficient of determination |

MAE | Mean absolute error |

CVRMSE | Coefficient of variation of root mean square error |

PSR | Single regressor for Prophet Model |

PMR | Multiple regressors for Prophet Model |

HWSS | Single seasonality Holt-Winters model |

HWDS | Double seasonality Holt-Winters model |

HWTS | Triple seasonality Holt-Winters model |

HFO | Heavy fuel oil |

LNG | Liquefied natural gas |

KPC | Kuwait Petroleum Corporation |

## Appendix A

**Table A1.**The real data and the forecasted annual generation of electricity for Kuwait from 2015–2030.

Annual Total Electrical Generation (kWh) | ||||||
---|---|---|---|---|---|---|

Actual | PSR | PMR | HWSS | HWDS | HWTS | |

2015 | 68,286,350 | 70,208,828 | 68,449,290 | 68,935,381 | 68,299,370 | 68,047,664 |

2016 | 70,084,727 | 71,309,681 | 69,977,535 | 71,082,353 | 70,008,887 | 69,798,293 |

2017 | 72,787,590 | 73,359,898 | 72,903,860 | 72,910,975 | 72,810,502 | 72,665,778 |

2018 | 74,430,304 | 74,872,001 | 74,156,996 | 76,173,631 | 74,432,331 | 74,326,541 |

2019 | 75,069,410 | 76,179,317 | 75,040,336 | 77,113,949 | 75,008,034 | 74,916,365 |

2020 | 80,036,707 | 78,329,573 | 81,683,193 | 77,607,284 | 76,441,652 | |

2021 | 85,131,998 | 81,893,161 | 83,971,685 | 79,070,098 | 77,576,721 | |

2022 | 87,184,615 | 83,945,450 | 87,453,223 | 80,757,167 | 78,803,803 | |

2023 | 88,888,944 | 84,818,623 | 89,208,193 | 83,518,864 | 80,801,914 | |

2024 | 90,819,983 | 86,916,798 | 92,262,607 | 85,265,977 | 83,624,082 | |

2025 | 91,833,231 | 88,556,264 | 93,165,837 | 85,770,309 | 85,417,389 | |

2026 | 92,878,920 | 89,406,883 | 93,465,256 | 86,293,807 | 85,933,949 | |

2027 | 94,350,541 | 90,451,648 | 95,353,865 | 88,667,276 | 86,469,671 | |

2028 | 95,636,495 | 91,515,320 | 96,037,295 | 90,582,478 | 88,899,766 | |

2029 | 97,623,149 | 94,049,623 | 98,668,062 | 92,751,443 | 90,860,715 | |

2030 | 99,677,375 | 95,421,752 | 102,262,507 | 94,313,533 | 92,535,555 |

Monthly Total Generation (kWh) | |||||
---|---|---|---|---|---|

PSR | PMR | HWSS | HWDS | HWTS | |

21 January | 4,719,195 | 4,506,418 | 4,723,899 | 4,154,973 | 4,160,775 |

21 February | 4,265,083 | 4,085,284 | 4,253,269 | 3,812,010 | 3,819,632 |

21 March | 5,407,054 | 5,148,962 | 5,413,977 | 4,797,070 | 4,828,850 |

21 April | 6,207,547 | 5,940,924 | 6,196,026 | 5,609,673 | 5,679,516 |

21 May | 8,150,344 | 7,765,217 | 8,169,369 | 7,655,697 | 7,802,910 |

21 June | 9,445,881 | 9,051,420 | 9,446,366 | 8,798,820 | 8,997,350 |

21 July | 10,276,964 | 9,856,725 | 10,275,340 | 9,567,496 | 9,791,745 |

21 August | 10,152,294 | 9,738,065 | 10,157,051 | 9,287,612 | 9,500,731 |

21 September | 9,048,569 | 8,676,132 | 9,050,778 | 8,229,272 | 8,404,845 |

21 October | 7,912,560 | 7,534,398 | 7,904,534 | 6,922,840 | 7,040,503 |

21 November | 5,160,468 | 4,926,339 | 5,163,376 | 4,584,972 | 4,613,376 |

21 December | 4,884,369 | 4,663,277 | 4,878,559 | 4,186,849 | 4,194,054 |

22 January | 4,841,008 | 4,639,763 | 4,849,182 | 4,240,582 | 4,251,986 |

22 February | 4,423,749 | 4,236,537 | 4,423,660 | 3,890,423 | 3,903,769 |

22 March | 5,532,091 | 5,310,500 | 5,534,806 | 4,884,470 | 4,934,148 |

22 April | 6,248,921 | 5,960,376 | 6,239,217 | 5,639,118 | 5,802,449 |

22 May | 8,455,250 | 8,067,020 | 8,473,173 | 7,812,368 | 7,970,160 |

22 June | 9,582,494 | 9,164,224 | 9,567,936 | 8,978,583 | 9,189,225 |

22 July | 10,355,563 | 9,995,653 | 10,348,046 | 9,762,630 | 10,000,339 |

22 August | 10,534,098 | 10,092,835 | 10,528,398 | 9,414,655 | 9,637,299 |

22 September | 9,431,675 | 9,043,116 | 9,461,242 | 8,355,913 | 8,540,640 |

22 October | 7,942,639 | 7,593,484 | 7,936,245 | 7,039,516 | 7,165,731 |

22 November | 5,376,527 | 5,160,420 | 5,371,959 | 4,683,238 | 4,718,244 |

22 December | 4,912,699 | 4,681,523 | 4,909,340 | 4,271,941 | 4,285,130 |

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**Figure 2.**The generation of daily loads data in Kuwait: the trained, predicted, and forecasted data using the five models of (

**a**) two types of the Prophet model and (

**b**) three types of the Holt-Winters model.

**Figure 3.**The coefficient of determination R

^{2}and the fitting characteristics of real and simulated data for all the models.

**Figure 5.**Variations of the coefficient of determination at different noise intensities for the five models.

**Figure 8.**Forecasted fuel budget in millions of Kuwaiti Dinars for the upcoming ten years based on the power generation prediction of all proposed models.

**Figure 9.**Fuel contribution cost as a percentage of total fossil fuel cost for the years 2020 and 2030.

**Table 1.**The growing population and per capita consumption in three decades [21].

Year | Population | Per Capita Consumption kWh/Person | Mean Annual Rate of Growth during 10 Years % |
---|---|---|---|

1989 | 2,048,000 | 10,295 | - |

1999 | 2,148,032 | 12,552 | 3.87% |

2009 | 3,484,881 | 13,372 | 0.65% |

2019 | 4,776,407 | 14,002 | 0.51% |

Method and Seasonality | $\mathit{\alpha}$ | $\mathit{\gamma}$ | ${\mathit{\delta}}^{\mathit{i}}$ |
---|---|---|---|

HWSS—yearly | 0.048 | 0.023 | ${\delta}^{y}$ = 0.3 |

HWDS—weekly, yearly | 0.039 | 0.029 | ${\delta}^{y}$ = 0.32 ${\delta}^{w}$ = 0.28 |

HWTS—daily, weekly, yearly | 0.042 | 0.025 | ${\delta}^{y}$ = 0.3 ${\delta}^{q}$ = 0.29 ${\delta}^{w}$ = 0.35 |

Indicator | PSR | PMR | HWSS | HWDS | HWTS |
---|---|---|---|---|---|

MAPE | 3.18% | 2.77% | 3.29% | 1.83% | 1.76% |

MAE | 120.10 | 78.23 | 127.82 | 54.01 | 46.82 |

RMSE | 153.81 | 104.38 | 165.08 | 75.09 | 67.05 |

CVRMSE | 22.92 | 16.26 | 24.50 | 11.22 | 10.45 |

R^{2} | 0.9709 | 0.9743 | 0.9641 | 0.9893 | 0.9899 |

**Table 4.**The accuracy reduction in coefficient of determination for different models under various noise intensities.

Noise Intensity | PSR | PMR | HWSS | HWDS | HWTS |
---|---|---|---|---|---|

0% | 0.9709 | 0.9743 | 0.9641 | 0.9893 | 0.9899 |

20% | 0.9697 | 0.9710 | 0.9619 | 0.9853 | 0.9872 |

40% | 0.9671 | 0.9705 | 0.9592 | 0.9847 | 0.9867 |

60% | 0.9670 | 0.9684 | 0.9587 | 0.9820 | 0.9858 |

80% | 0.9667 | 0.9682 | 0.9552 | 0.9798 | 0.9816 |

Year | Date | Peak Load | Generation of the Same Date | Peak Generation | Date |
---|---|---|---|---|---|

2015 | 30 August 2015 | 12,810 | 279,228 | 279,228 | 30 August 2015 |

2016 | 15 August 2016 | 13,390 | 288,058 | 290,304 | 2 August 2016 |

2017 | 26 July 2017 | 13,800 | 290,288 | 299,694 | 14 August 2017 |

2018 | 10 July 2018 | 13,910 | 284,718 | 300,231 | 12 July 2018 |

2019 | 27 June 2019 | 14,420 | 318,531 | 318,531 | 27 June 2019 |

2020 | 30 July 2020 | 14,960 | 317,677 | 326,437 | 31 July 2020 |

Year | Natural Gas (KSCF) Standard Cubic Feet | Gas Oil (Barrels) | Crude Oil (Barrels) | Heavy Oil (Barrels) | Total Cost (KD) |
---|---|---|---|---|---|

2012 | 264,080,165 | 11,913,629 | 16,566,894 | 38,557,558 | 2,423,012,351 |

2013 | 253,461,108 | 9,237,306 | 11,323,855 | 46,967,101 | 2,327,992,356 |

2014 | 313,936,191 | 11,153,661 | 14,409,093 | 37,954,682 | 2,435,107,934 |

2015 | 350,979,921 | 8,570,450 | 4,849,437 | 46,722,496 | 1,288,525,905 |

2016 | 378,535,102 | 5,731,758 | 4,057,944 | 48,460,342 | 1,010,903,300 |

2017 | 374,964,177 | 5,196,552 | 9,194,665 | 41,591,383 | 1,297,642,956 |

2018 | 403,438,524 | 3,623,846 | 6,236,988 | 42,956,365 | 1,694,031,251 |

2019 | 433,605,520 | 5,376,675 | 3,439,839 | 39,107,701 | 1,442,738,912 |

Fuel | Price | Unit | ||
---|---|---|---|---|

Low | Average | High | ||

Natural Gas | 0.294 | 0.375 | 0.456 | KD/MSCF |

Gas Oil | 11.846 | 18.904 | 25.963 | KD/bbl |

Crude Oil | 8.084 | 14.085 | 20.085 | KD/bbl |

H.F.O. | 6.496 | 13.398 | 20.300 | KD/bbl |

LNG | 1.265 | 2.626 | 3.987 | KD/MSCF |

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**MDPI and ACS Style**

Almazrouee, A.I.; Almeshal, A.M.; Almutairi, A.S.; Alenezi, M.R.; Alhajeri, S.N.; Alshammari, F.M.
Forecasting of Electrical Generation Using Prophet and Multiple Seasonality of Holt–Winters Models: A Case Study of Kuwait. *Appl. Sci.* **2020**, *10*, 8412.
https://doi.org/10.3390/app10238412

**AMA Style**

Almazrouee AI, Almeshal AM, Almutairi AS, Alenezi MR, Alhajeri SN, Alshammari FM.
Forecasting of Electrical Generation Using Prophet and Multiple Seasonality of Holt–Winters Models: A Case Study of Kuwait. *Applied Sciences*. 2020; 10(23):8412.
https://doi.org/10.3390/app10238412

**Chicago/Turabian Style**

Almazrouee, Abdulla I., Abdullah M. Almeshal, Abdulrahman S. Almutairi, Mohammad R. Alenezi, Saleh N. Alhajeri, and Faisal M. Alshammari.
2020. "Forecasting of Electrical Generation Using Prophet and Multiple Seasonality of Holt–Winters Models: A Case Study of Kuwait" *Applied Sciences* 10, no. 23: 8412.
https://doi.org/10.3390/app10238412