Prediction of electricity energy consumption including COVID‐19 precautions using the hybrid MLR‐FFANN optimized with the stochastic fractal search with fitness distance balance algorithm

Abstract

The increase in energy consumption is affected by the developments in technology as well as the global population growth. Increasing energy consumption makes it difficult to ensure electrical energy supply security. Meeting the energy demand can be achieved with the right planning. Proper planning is critical for both economical use of resources and low cost for the end consumer. On the other hand, erroneous estimation of demand may cause waste of resources and energy crisis. Accurate estimation is possible by accurately modeling the factors affecting electricity consumption. Apart from known factors such as seasonal conditions, days of the week and hours, modeling in extreme events such as pandemics that affect all our behaviors increases the success in modeling the future projection. This ensures that the security of electrical energy supply is carried out effectively with limited resources. For this purpose, in this study, a hybrid multiple linear regression‐feedforward artificial neural network (MLR‐FFANN) based algorithm model was proposed, taking into account the estimated impact of the COVID‐19 pandemic on the energy consumption values of Bursa, an industrial city in Turkey. The aim of the hybrid MLR‐FFANN approach was to simultaneously optimize the β polynomial for multiple linear regression and the weight and bias coefficients for the forward propagation neural network using the adaptive guided differential evolution, equilibrium optimizer, slime mold algorithm, and stochastic fractal search with fitness distance balance (SFSFDB) optimization algorithms. The success of the model whose parameters were optimized using the optimization algorithms was determined according to mean absolute error, mean absolute percentage error, and root mean square error evaluation criteria and statistical analysis of these results. According to the results of the analysis, the MLR‐FFANN approach whose parameters were optimized with the SFSFDB algorithm was more successful in the training of the dataset containing the COVID‐19 precautions.

1. INTRODUCTION

1.1. Background, definitions, and motivation

Today, meeting electrical energy demands is a very difficult problem considering the increasing population, industrialization, depletion of fossil resources, and increasing environmental concerns. In addition to fossil fuels, there is a need to meet increasing energy demands with low‐cost and environmentally friendly renewable resources. However, as renewable energy sources are dependent on nature and exhibit seasonal characteristics, such energy production requires planning. Therefore, today, diversifying energy and ensuring supply security play a critical role for countries. Successful planning of resources means efficient use of energy and preventing waste of resources. By using estimation methods, short‐, medium‐, and long‐term electricity consumption amounts can be determined and necessary measures can be taken to ensure a secure energy supply. The consumption of electric energy is affected by human activities and seasonal factors such as air temperature and humidity. However, the presence of unexpected circumstances also affects the success of the plans. The COVID‐19 pandemic has affected the activities of the global community and as a result, has been reflected in the impact on the electricity consumption sector. The demand for electric energy is affected by human activities and mobility. Therefore, the restrictions brought about by countries during pandemic periods will affect their electricity consumption. ¹ , ²

1.2. Related work

Prediction of electricity consumption over the short, medium, and long term can be carried out in any city, country, building, or structure. Consumption estimation can be performed with mathematical modeling such as least squares, regression analysis, artificial intelligence, or machine learning approaches including the genetic algorithm and fuzzy logic. ³ , ⁴ In the literature, many estimation methods have been presented comparatively for different time periods. Regression methods, Kalman filter‐based forecasting, artificial intelligence techniques, deep recurrent neural networks, and hybrids of these techniques have been used successfully in the forecasting of electricity demand. ⁵ , ⁶ , ⁷ Chen and Lee proposed an adaptive network‐based fuzzy system (ANFIS) method for estimating electricity consumption in public buildings. In their study, a multiple ANFIS approach was presented to predict electricity consumption based on human activities and weather conditions. Temperature, precipitation, sunshine duration, insolation radiation, humidity, working days, school days, and holidays were determined as inputs to the ANFIS model. The performance of the proposed method was compared with the Levenberg–Marquardt back propagation (LM‐BP) neural network, single ANFIS model, and linear and nonlinear regression. Although the method outperformed simple regression models, it underperformed compared to the single ANFIS and LM‐BP. ⁸ Similar to consumption in buildings, electricity consumption estimates have been carried out within cities, countries, and areas directly affected by human use. The support vector regression (SVR) model was used to estimate the daily electricity consumption of a metro station. In this model, similar to the previous model, meteorological data such as air temperature and relative humidity were used as input data that affected consumption. Five‐fold‐cross‐validation with the genetic algorithm was used to optimize the hyper‐parameters of the SVR to improve its performance. ⁹ In addition to the aforementioned input parameters, wind speed and direction, day, and time information have also been used in the estimation of electricity consumption. ¹⁰ Parametric and nonparametric AutoRegressive (AR, NPAR), Smooth Transition AutoRegressive (STAR), and AutoRegressive Moving Average (ARMA) models were used in a study in which the medium‐term electricity consumption estimation of Pakistan was carried out. It was stated that among the proposed methods, ARMA performed better than the other models. ¹¹ A multivariate gray estimation model based on first‐order linear difference equations was proposed for long‐term electricity consumption estimation for Shanxi City (China). In that study, it was stated that electricity consumption was affected by factors such as seasons, holidays, political conditions, technological development level, and the economy. ¹² In addition to daily and annual forecasts, hourly forecasts can be made over a much shorter term. The hourly electricity consumption of a building was estimated using artificial neural network (ANN) and case‐based reasoning (CBR) methods. It was determined that the performance of the ANN was better than that of the CBR. Although a relatively large dataset is needed for ANNs to be successful, it has been stated that CBR can perform estimation with relatively less data. ¹³ It has also been determined in similar studies that in order to provide acceptable accuracy, an ANN should have a sufficient number of samples. ¹⁴ Estimation of electricity consumption was also carried out for different areas such as polymer material production ¹⁵ and electric vehicle charging stations. ¹⁶ Wang et al. proposed a model called ESN‐DE, which uses an improved echo state network (ESN) to perform the estimation of electrical energy consumption. In the proposed model, the differential evolution (DE) algorithm is used to find the optimal values of three important parameters of the ESN. According to the results obtained, it has been determined that the ESN‐DE algorithm gives better results than the basic ESN and ESN‐GA. Therefore, the ESN‐DE is a potential candidate for the estimation of electrical energy consumption due to its easy implementation and stability. ¹⁷ In another study by the authors, the ESN and fruit fly optimization algorithm (FOA) approach were designed to predict medium‐term industrial electricity consumption in China. Numerical results indicate that adaboostsp‐ESN with FOA is successful in predicting future industrial electricity consumption. ¹⁸ Zhou et al. proposed a new model for electricity consumption named panel semi‐parametric quantile regression NN (PSQRNN). The PSQRNN estimation results have been reported to perform better when compared to back propagation NN (BPNN), support vector machine (SVM) and quantile regression neural network (QRNN). ¹⁹ Artificial intelligence algorithms, such as neural network models based on various neural networks and optimization algorithms, have been applied in a variety of science fields. For example, these methods have been successfully used to calculate the viscosity of natural gas, the MMP of pure and impure CO₂ flux, the density of CO₂ adsorption on activated carbon, and the estimation of the viscosity of self‐diverting acids. ²⁰ , ²¹ , ²² , ²³

The COVID‐19 pandemic, which swept throughout the globe before the end of 2019, prompted countries around the world to take immediate anti‐pandemic measures in all sectors of life. The epidemic period, which continues until today, is affected every sector both commercially and scientifically. In the literature, the impact of the pandemic on many sectors in different countries has been examined. In scientific studies, Chen et al., used mixed data sampling (MIDAS) and extension models to estimate the variability of China's crude oil futures with high frequency data. It was determined that the MIDAS‐RV‐CJ model was more successful in the short term, while the MIDAS‐RV‐L model gave better results in long‐term predictions. Furthermore, it has been reported that the leverage effect is the strongest predictor of the COVID‐19 pandemic based on the samples collected during the pandemic. ²⁴ In another study, Wu et al. analyzed the US oil markets based on social media information using convolutional NN (CNN) during the COVID‐19 process. The results obtained from the experimental studies showed that although social media information contributes to the estimation of oil price, production and consumption, the oil inventory does not affect the estimation accuracy. ²⁵ Considering the electricity consumption in terms of the energy sector worldwide, it is seen that electricity consumption is affected like other sectors due to the restrictions applied during the COVID‐19 pandemic period. ¹ In order to examine the impact of COVID‐19 in Turkey, the effect of the number of daily cases on electricity consumption was examined. In the analysis, it was determined that there was an inverse relationship between the number of cases and electricity consumption. ²⁶ Similar to the example of Turkey, the electricity consumption of Spain, one of the countries most affected by the pandemic, has been examined in detail. In the March–April period, a decrease of approximately 13.49% was determined by taking the average of the previous 5 years as a reference. ¹ A similar analysis was carried out in nine European countries and the USA, and the consumption differences between countries were examined. It was determined that the differences were the result of the different restrictions carried out in those countries. ²⁷ From this point of view, the degree of restrictions imposed by countries is an important factor in electricity consumption. Almost no reduction in electricity demand was detected in Norway, which took a much less strict stance in terms of restrictions. Whereas electricity demand decreased by 8% in the USA, decreases by 20%, 25%, and 28%, respectively, were observed in France, Spain, and Italy, which had tight lockdown policies. ²⁸

1.3. Contribution of the article

Accurate realization of demand forecasting is important for economical generation of electric energy. An overestimation of demand may waste resources, whereas underestimation may lead to energy crises. In this context, energy supply security could be ensured through government policies and maintenance‐investment plans. ² In the literature, parameters that affect electricity consumption include the economy, industry, climate, ambient temperature, solar radiation, humidity, wind speed, working days, and the gross domestic product. ²⁹ , ³⁰ However, apart from these classical studies, no examination of extraordinary situations such as pandemics has been carried out. Moreover, for countries that have followed a partial closure system, regional or city‐based studies are important in obtaining critical results.

The main contributions of the study can be listed as follows.

• This study discusses in a comparative way the effect of the COVID‐19 pandemic, which has impacted the whole world with an unexpected speed, on electricity consumption and the estimation of this effect using flexible calculation methods.

• One of the most important contributions of the study is to reveal the electricity consumption behavior under the influence of COVID‐19 in Bursa, an industrial city of Turkey, and to create a strategy for future pandemic periods.

• For the first time, the stochastic fractal search with fitness distance balance (SFSFDB) algorithm was adapted and simulated for the proposed hybrid multiple linear regression‐feedforward artificial neural network (MLR‐FFANN).

• The design of the optimal structure of the feedforward artificial neural network (FFANN) and the determination of the $\beta$ polynomial coefficients of the (multi‐linear regression) MLR were carried out during the training process, and these can be defined as the most important factors for training the proposed hybrid MLR‐FFANN.

• Combinations of hyperbolic tangent and sigmoid activation functions were used in the hidden and output layers of the FFANN and the success of these combinations in the training and learning process of the proposed hybrid MLR‐FFANN approach was investigated.

• The SFSFDB algorithm was broadly compared to and outperformed the adaptive guided differential evolution algorithm (AGDE), ³¹ equilibrium optimizer (EO), ³² and slime mold algorithm (SMA). ³³

• The SFSFDB algorithm successfully trained the hybrid MLR‐FFANN in a minor number of iterations and found the optimal value of weight, bias, and $\beta$ polynomial coefficients of both simultaneously.

• The simulation results were analyzed statistically and according to the results, the SFSFDB algorithm was proven to be more successful in training the MLR‐FFANN algorithm proposed in the prediction dataset compared to the other optimization algorithms.

1.4. Organization of the study

The rest of the study article is organized as follows:

• The progress of the pandemic in Turkey and its effect on energy consumption is discussed in Section 2.

• In Section 3, MLR‐FFANN, SFS, and SFSFDB algorithms are explained.

• Section 4 provides detailed information on the application of the SFSFDB optimization algorithm to the hybrid MLR‐FFANN model.

• Section 5 focuses on the analysis of the experimental results under different operational scenarios.

• Finally, Section 6 provides the conclusions drawn from this study and ideas for future works.

2. PROGRESS OF THE PANDEMIC IN TURKEY AND ITS EFFECT ON ENERGY CONSUMPTION

In this section, Turkey's energy outlook and Bursa's electric energy production and consumption are detailed. The course of the pandemic in Turkey and the measures taken by the government against the pandemic are examined.

2.1. Situation of the city of Bursa

According to the 2020 data, the population of Bursa is 3,101,832, making it the fourth largest city in Turkey in terms of population. For a clearer understanding of Bursa's position in Turkey, data on industry and electricity consumption are given in Table 1. ³⁴

TABLE 1.

Indicators of Bursa and Turkey

		2013	2014	2015	2016	2017	2018	2019
Exports (thousand $)	Bursa	9,970,943	9,140,463	10,364,727	11,066,414	11,716,921	10,371,117	9,076,069
	Turkey	166,504,862	150,982,114	149,246,999	164,494,619	177,168,756	171,464,945	160,514,811
The number of enterprises	Bursa	126,554	128,169	131,407	136,322	140,633	147,618	153,435
	Turkey	3,397,724	3,434,912	3,498,586	3,608,470	3,696,004	3,845,951	3,954,698
Gross domestic product per person ($)	Bursa	14,282	13,829	12,537	12,270	12,132	11,451	10,382
	Turkey	12,582	12,178	11,085	10,964	10,696	9792	9213
Total electricity consumption per person (kWh)	Bursa	3427	3455	3733	3726	4106	4246	3094
	Turkey	2583	2669	2760	2897	3082	3149	4155
Industrial electricity consumption per person (kWh)	Bursa	2135	2087	2310	2251	2574	2649	–
	Turkey	1216	1258	1315	1357	1441	1435	–

The total electricity consumption value per person in Bursa is above the national average. Similarly, industrial electricity consumption per person for Bursa is high in all of the years mentioned. The number of enterprises, as an important parameter of the industry indicator, realized in Bursa corresponds to approximately 3.88% of the number of enterprises realized in Turkey. In addition, an average of 6.29% of the exports produced in Turkey for the specified years originated from Bursa. Considering all these indicators, Bursa is seen as an important industrial city.

2.2. Electricity energy outlook in Turkey

Turkey has a growing population and strong economic growth. This is also reflected in the energy requirements. In the last 10 years, the Turkish economy has grown around 5% on average. This growth also increases energy consumption and makes it difficult to ensure energy supply security. Consequently, in order to ensure energy supply security, installed power has been continuously increasing, especially after 2000. The installed power, which was 27,264.1 MW in 2000 and 49,524.1 MW in 2010, reached 96,709.6 MW as of February 2021. ² , ³⁵ As of February 2021, the installed power values according to their sources are given in Figure 1. ³⁵

FIGURE 1.

Distribution of installed power by resources

Between 2000 and 2009, oil prices increased by 76% and natural gas prices by 114%. According to the projections made for the years 2015–2040, an increase of 186.3% in oil prices and 85.7% in natural gas prices is expected. ³⁶ Considering the price instability in fossil fuels and the depletion of resources, renewable energy sources come to the fore as an alternative. Figure 1 shows that 50.05% of Turkey's installed power consists of renewable resources. Among renewable resources, hydraulic resources take the first place, with 31,177.9 MW. With the licensed power plant investments that were temporarily accepted in 2020, the amount of additional installed power is 5430.31 MW. Only 60.80 MW of this additional power amount is from fossil fuel. ³⁷ The investments made can be evaluated as an indicator of the importance given to renewable energy in Turkey. In addition, with these investments, the energy supply will not be secured with imported resources, but with clean and renewable resources without fuel costs.

According to the market development report published by the Energy Market Regulatory Authority, the total licensed electricity generation amount in Turkey in 2019 was 294,251,318.65 MWh, with Bursa contributing at a rate of 2.49%, or 7,322,483.15 MWh. Whereas the total licensed installed power of Turkey was 84,957.72 MW as of 2019, that of Bursa constituted 3.07% of this power, with 2607.03 MW. The unlicensed installed power was 6309.27 MW in Turkey for 2019, whereas in Bursa it was 70.54 MW, making up 1.12%. Unlicensed electricity generation in Turkey amounted to 9,829,447.73 MWh in total for 2019, with Bursa contributing 97,251.79 MWh (0.99%). When the electricity consumption values are examined, the total consumption billed for 2019 was 229,597,913.65 MWh in Turkey and 11,813,549.28 MWh in Bursa. With a share of 5.15%, Bursa was the city with the fourth highest electricity consumption in Turkey. Consumption amounts billed on a consumer basis are given in Table 2. ³⁸

TABLE 2.

Amount of billed consumption by type of consumer

	Lighting (MWh)	Residential (MWh)	Industry (MWh)	Agricultural irrigation (MWh)	Commercial (MWh)	Total (MWh)
Bursa	126,061.66	1,969,089.31	7,260,805.16	125,251.54	2,332,341.62	11,813,549.28
Turkey	5,041,682.96	56,389,775.22	94,462,698.78	8,553,367.43	65,150,389.26	229,597,913.65

Considering the consumption amounts for Bursa, the highest was realized in the industrial section, followed by the commercial establishments. Considering its share in total consumption and the amounts by type of consumer, Bursa is seen as a robust industrial city.

2.3. Course of the COVID‐19 pandemic

The World Health Organization (WHO) gave the first notification for the coronavirus pandemic on December 31, 2019, by reporting that the disease was seen in China. On March 11, WHO declared the outbreak a pandemic. As of March 13, 2021, according to the data announced by WHO, the number of cases in the world was 118,754,336 and the number of deaths was 2,634,370. ³⁹ These figures forced governments to take strict actions. The measures taken caused difficulties for people and countries both socially and economically.

There were significant changes in the work, education, shopping, and entertainment habits during the pandemic period. Societies were forced to carry out many activities remotely or in accordance with social distancing rules. It was important to provide uninterrupted health services during these critical measures. In order for all these services to be carried out safely, electric energy had to be provided without any problems. Gradual restrictions were introduced in Turkey with the occurrence of cases in March 2020. Due to these restrictions, periodic closures or capacity reductions were experienced in the manufacturing sector. These restrictions and slowdowns in activities were also reflected in the electricity consumption. ⁴⁰ Figure 2 presents the change in consumption for the years 2019 and 2020. ³⁷

FIGURE 2.

Electricity consumption amounts

Especially in April and May, a significant decrease was observed in consumption amounts, followed by a recovery. In August, as a continuation of this recovery, 2020 consumption increased compared to the same time period of the previous year, and this trend continued until the end of the year.

COVID‐19 has caused a decrease in electricity demand in many countries and an increase in the share of renewable energy sources in electricity production. In a study carried out in Germany, energy production from renewable sources increased by 8% in the first quarter of 2020, taking the same period of 2019 as a reference. ⁴¹ Similarly, in a study conducted in Italy, the effect of COVID‐19 on electricity consumption was examined by taking into account its environmental dimensions. It was determined that the significantly reduced electricity consumption was supported by wind‐based generation, resulting in a 30% decrease in wholesale energy prices. This has also had a positive impact on reducing carbon dioxide emissions. In addition, the researchers observed that the decrease in consumption affected the amount of energy supplied from traditional (thermal) sources. ⁴²

In order to reduce and control the impact of the pandemic, governments have applied different restrictions. These policies can cause disadvantageous situations in many sectors. The detection of these situations is important, especially in the later stages of the pandemic or similar pandemics that may occur later, in terms of precautions to be taken. The first case in Turkey was seen on March 11, 2020, after which, various measures were taken to control the pandemic. The steps taken during the pandemic process are summarized in Table 3.

TABLE 3.

Milestones of COVID‐19 and the steps taken in Turkey

Date
February 3, 2020	Turkey suspends all flights from China.
March 11, 2020	The first coronavirus case was announced in Turkey.
March 14, 2020	Passenger transportation banned with 16 countries, 9 of which were European countries.
March 18, 2020	The first death due to coronavirus occurred.
March 20, 2020	With the Presidential circular, all scientific, cultural, artistic, and similar meetings and activities were stopped until the end of April.
March 21, 2020	Curfew restrictions were imposed on citizens aged 65 and over.
March 22, 2020	With the Presidential circular, flexible and remote working was allowed in public institutions and organizations.
March 23, 2020	Formal education was interrupted and distance education programs were started.
March 27, 2020	International flights were completely suspended. Intercity transportation was subject to the permission of the Governorship. Entrance to places such as picnic areas, forests, and historical sites was prohibited at the weekend.
April 3, 2020	30 metropolitan areas and 1 province were closed to entry and exit of vehicles with some exceptions. Curfews were imposed on those under 20 years old throughout the whole country.
April 11–12, 2020	The Government declared curfew in 30 metropolitan areas and 1 province (2 days)
April 18–19, 2020	The Government declared curfew in 30 metropolitan areas and 1 province (2 days)
April 23–26, 2020	The Government declared curfew in 30 metropolitan areas and 1 province (4 days)
May 1–3, 2020	The Government declared curfew in 30 metropolitan areas and 1 province (3 days)
May 10, 2020	The curfew for 65 years and older has been lifted between 12:00 and 18:00.
May 13, 2020	The curfew for children aged 0–14 has been lifted between 11:00 and 15:00.
May 15, 2020	The curfew for young people between the ages of 15 and 20 has been lifted between 11:00 and 15:00.
May 23–26, 2020	A curfew was imposed during the Ramadan Holiday (4 days).
May 24, 2020	The curfew for 65 years and older has been lifted between 12:00 and 18:00.
June 1, 2020	The intercity travel restriction has been lifted. Flexible and remote working in the public sector has come to an end.
June 3, 2020	The curfew imposed on people over the age of 65 has been lifted.
June 20, 2020	A curfew was imposed during the university entrance exam (first step).
June 27–28, 2020	A curfew was imposed during the university entrance exam (second step).
August 26, 2020	Flexible and remote working in public institutions and organizations was allowed.
November 17, 2020	A curfew will be imposed on weekends outside of 10.00–20.00. Restaurants will only provide takeaway service, shopping malls and markets will be closed at 20:00. (First starting on 04.12.2020.)
November 30, 2020	A curfew will be imposed, starting at 21:00 on Fridays and ending at 05:00 on Mondays.
November 30, 2020	Until a new decision, a curfew will be applied between 21.00 and 05.00 on weekdays throughout the country. (First started on 01.12.2020.)
December 31, 2020–January 4, 2021	A curfew will be imposed, starting from 21:00 on Thursday, December 31, 2021, and ending at 05:00 on Monday, January 4, 2021.
February 2, 2021	Face‐to‐face education will begin gradually in village schools as of February 15, 2021 and for other classes as of March 1, 2021.
February 3, 2021	The South African and Brazilian variants were also seen in Turkey.
March 1, 2021	As of March 2, 2021, pre‐school education institutions, primary school, secondary school 8th and 12th grades will start education in all cities and will start at other levels, including secondary and high schools, in low and medium risk provinces.
	The scope of the curfews has been changed according to the provinces with the new Controlled Normalization Process announced on March 1, 2021. The provinces have been divided into four different risk groups (low, medium, high, and very high) and the level of precaution has been determined according to the risk group.
March 2, 2021	BURSA: In the medium risk group. According to this: • A curfew will be applied between 21.00 and 05.00 on weekdays. • A curfew will be applied between 21.00 and 05.00 on weekends. • For our citizens aged 65 and over and under the age of 20, the curfew will be lifted outside the hours stated above. • The food and beverage sector, which will operate between 07:00 and 19:00, will serve with a 50% capacity limitation.
	Restriction		Inform		Rule bending

3. METHODS

3.1. Multi‐linear regression

Regression analysis is a statistical technique used to investigate and model the relationship between variables in order to obtain an approximate function between the relevant response (output) and independent variables (inputs). ⁴³ If there is more than one independent variable in the regression analysis, it is called MLR. In most MLR problems, it is not known how the independent variables relate to the dependent variables. Therefore, the first step and the main problem are to find an approximation method to relate them. ⁴⁴ Generally, first‐order and second‐order polynomial models are used, as given in Equations (1) and (2).

$$y={\beta }_0+\sum _{i=1}^k{\beta }_i{x}_i+\epsilon ,$$ (1)

$$y={\beta }_0+\sum _{i=1}^k{\beta }_i{x}_i+\sum _{i=1}^k{\beta }_{\mathit{ii}}{x}_i^2+\sum _{i=1}^{k-1}\sum _{j>1}^k{\beta }_{\mathit{ij}}{x}_i{x}_j+\epsilon ,$$ (2)

where β denotes the polynomial coefficients and x the independent variables, y is defined as the dependent variable, and ε identifies other forms of variation, such as errors or lack of numerical convergence. The k index represents the total number of variables. The i and j indices represent a certain variable between 1 and k.

If Equations (1) and (2) are written in a matrix form, Equation (3) is obtained.

$$\mathit{Y}=\mathit{X}{\mathit{\beta }}^{\prime }+\mathit{\epsilon }.$$ (3)

The related matrices and vectors can be elaborated as in Equation (4).

$$\begin{array}{cc}& \mathit{Y}\mathbf{=}\left[\begin{array}{c}{y}_1\\ \begin{array}{c}{y}_2\\ ⋮\\ y_n\end{array}\end{array}\right],\mathit{\epsilon }\mathbf{=}\left[\begin{array}{c}{\mathit{\epsilon }}_0\\ \begin{array}{c}{\mathit{\epsilon }}_1\\ ⋮\\ {\mathit{\epsilon }}_n\end{array}\end{array}\right],\\ & \mathit{\beta }\mathbf{=}{\beta }_0{\beta }_1\begin{array}{cc}\dots & {\beta }_k\end{array}\begin{array}{ccc}{\beta }_{11}& \dots & {\beta }_{\mathit{kk}}\end{array}\begin{array}{ccc}{\beta }_{12}& \dots & {\beta }_{1k}\end{array}\begin{array}{ccc}{\beta }_{23}& \dots & {\beta }_{2k}\end{array}\begin{array}{cc}\cdots & {\beta }_{(k-1)k}\end{array},\\ & \mathit{X}\mathbf{=}\left[\begin{array}{c}\begin{array}{ccc}1& x_{11}& \cdots \\ 1& x_{21}& \cdots \end{array}\\ \begin{array}{ccc}⋮& ⋮& \ddots \\ 1& x_{n1}& \cdots \end{array}\end{array}\begin{array}{c}\begin{array}{ccc}{x}_{1k}& x_{11}^2& \cdots \\ x_{2k}& x_{21}^2& \cdots \end{array}\\ \begin{array}{ccc}⋮& ⋮& \ddots \\ x_{\mathit{nk}}& x_{n1}^2& \cdots \end{array}\end{array}\begin{array}{c}\begin{array}{ccc}{x}_{1k}^2& x_{11}{x}_{12}& \cdots \\ x_{2k}^2& x_{21}{x}_{22}& \cdots \end{array}\\ \begin{array}{ccc}⋮& ⋮& \ddots \\ x_{\mathit{nk}}^2& x_{n1}{x}_{n2}& \cdots \end{array}\end{array}\begin{array}{c}\begin{array}{ccc}{x}_{11}{x}_{1k}& x_{12}{x}_{13}& \cdots \\ x_{21}{x}_{2k}& x_{22}{x}_{23}& \cdots \end{array}\\ \begin{array}{ccc}⋮& ⋮& \ddots \\ x_{n1}{x}_{\mathit{nk}}& x_{n2}{x}_{n3}& \cdots \end{array}\end{array}\begin{array}{c}\begin{array}{ccc}{x}_{12}{x}_{1k}& \cdots & x_{1(k-1)}{x}_{1k}\\ x_{22}{x}_{2k}& \cdots & x_{2(k-1)}{x}_{2k}\end{array}\\ \begin{array}{ccc}⋮& \ddots & ⋮\\ x_{n2}{x}_{n3}& \cdots & x_{n(k-1)}{x}_{\mathit{nk}}\end{array}\end{array}\right]\mathit{.}\end{array}$$ (4)

To estimate the coefficients of MLR models, the least squares method given for the vector sample β in Equation (5) is generally used. Thus, MLR models can be developed according to the existence of errors. ⁴⁵

$$\mathit{\beta }={\left({\mathit{X}}^{\prime }\mathit{X}\right)}^{-\mathbf{1}}{\mathit{X}}^{\prime }\mathit{Y}.$$ (5)

3.2. Feedforward artificial neural networks

ANNs are a distributed computing technology, inspired by the information processing technique of the human brain that can record the experimental information of systems. The ANN topology is divided into two types: feedforward (FF) and back propagation neural networks (BPNN). ⁴⁶ The FFANN is used more widely because it requires less memory in the implementation phase. The FFANN allows a one‐way signal flow. In addition, feedforward neural networks are organized in multiple layers: the input layer, the hidden layer, and an output layer. The neurons of each layer are interconnected by the weights and bias terms of the other neurons in the previous layers. ⁴⁷ This system is represented by the mathematical expression given in Equation (6).

$$y_j=\sum _{i=1}^n{\omega }_{\mathit{ij}}{x}_i+{\theta }_j,j=1,2,3,\dots ,h,$$ (6)

where n is the number of input nodes and ${\omega }_{\mathit{ij}}$ is the connection weight of the jth node in the hidden layer from the ith node in the input layer; θ denotes the jth hidden node bias and x _i denotes the ith input. Accordingly, Equation (7) is used to calculate the output of each hidden node.

$$\begin{array}{cc}& Y_j=\text{Hyperbolic tangent}\left(y_j\right)=\frac{e^{y_j}-e^{-y_j}}{e^{y_j}+e^{-y_j}},j=1,2,3,\dots ,h,\\ & Y_j=\text{Sigmoid}\left(y_j\right)=\frac{1}{1+e^{-y_{\mathrm{j}}}},j=1,2,3,\dots ,h.\end{array}$$ (7)

The final outputs are defined as given in Equations (8) and (9).

$$o_k=\sum _{i=1}^n{\omega }_{\mathit{jk}}{Y}_i+{\theta }_k,k=1,2,3,\dots ,m,$$ (8)

$$\begin{array}{cc}& O_k=\text{Hyperbolic tangent}\left(o_k\right)=\frac{e^{o_k}-e^{-o_k}}{e^{o_k}+e^{-o_k}},k=1,2,3,\dots ,m,\\ & O_k=\text{Sigmoid}\left(o_k\right)=\frac{1}{1+e^{-o_k}},k=1,2,3,\dots ,m,\end{array}$$ (9)

where ${\omega }_{\mathit{jk}}$ is the connection weight from the jth hidden node to the kth output node and ${\theta }_k$ represents the bias value of the kth output node.

As can be understood from the equations, the most important parts of these structures are the weight and bias values because they determine the final value of the output. In order to find these values at the optimum value, the training of the neural network is very important. ⁴⁸ The aim is to obtain the minimum error value at the end of both the training process and the testing process of the neural network. Based on calculating the difference between the actual and predicted values, the mean square error (MSE) is used to evaluate the forward propagation neural network. This mathematical expression is given in Equation (10).

$$\mathrm{MSE}=\frac{1}{n}\sum _{i=1}^n{\left(e_i\right)}^2=\frac{1}{n}\sum _{i=1}^n{\left(x_i-o_i\right)}^2,$$ (10)

where $e_i$ represents the error value of the ith data, $x_i$ and $o_i$ are defined as the actual and estimated value of the ith data, respectively, and n is the total number of data. The FFANN model with the structure of 4‐5‐1 is illustrated in Figure 3.

FIGURE 3.

FFANN scheme with the structure of 4‐5‐1

3.3. Proposed optimization algorithm

3.3.1. Stochastic fractal search algorithm

The stochastic fractal search (SFS) algorithm was developed because there was no information exchange between fractals and the many parameters that needed to be determined according to the problem to be optimized by the fractal search algorithm. ⁴⁹ The SFS algorithm is constituted of two basic processes: the propagation and update processes. The update process also consists of two parts: the first and second update processes. The propagation process performs the local search feature in the algorithm, and the update processes perform the global search functions. ⁵⁰ The two functions given in Equations (11) and (12) are proposed for the propagation process of fractals.

$${\mathrm{GW}}_1=\text{Gaussian}\left({\mu }_{\mathrm{BP}},\sigma \right)+\left(\mathit{\epsilon x}\mathrm{BP}-\mathit{\epsilon x}{P}_i\right),$$ (11)

$${\mathrm{GW}}_2=\text{Gaussian}\left({\mu }_P,\sigma \right).$$ (12)

In Equation (12), μ _BP represents the position of the fractal that gives the best fitness value found up to that generation, BP is the fractal that gives the best fitness value, and P _i denotes the propagating fractal; the ε expression defines randomly generated numbers in the range of [0, 1]. In addition, μ _P in Equation (12) represents the position of the spreading fractal. The σ value is the standard deviation value that expresses the step length of the Gaussian walking function. The standard deviation expression is calculated as given in Equation (13).

$$\sigma =\left|\frac{\log (g)}{g}x\left(P_i-\mathrm{BP}\right)\right|.$$ (13)

Equation (13) shows that the difference between the fractal with the best fitness value and the spreading fractal is obtained by multiplying the expression log(g)/g. The g in the log(g)/g expression added to Equation (13) represents the generation value and creates a damping effect in the standard deviation function, that is, the value of the standard deviation decreases as the number of generations progresses. Although this causes a more local search, it increases the probability of finding a solution with an even better fitness value in the immediate vicinity of the found solution point. After the propagation process is performed, the first and second update processes are run. For these operations, first of all, a probabilistic value is given to each fractal by taking into account the order of the fitness values, using the expression given in Equation (14).

$$P_{a_i}=\frac{\mathrm{rank}\left(P_i\right)}{N},$$ (14)

where N is the number of fractals in the generation and the expression rank(P _i ) expresses the rank of the fractal in the generation according to its fitness value.

The first update process starts after the probabilistic value is calculated. Depending on whether the randomly generated ε expression is smaller or larger than the probabilistic value, the relevant dimension of the relevant fractal is updated. This process is as given in Equation (15).

$$P_i^{\prime }(j)=\left\{\begin{array}{cc}{P}_r(j)-\mathit{\epsilon x}\left(P_t(j)-P_i(j)\right),& \epsilon >P_{a_i},\\ P_i(j),& \epsilon \le P_{a_i}.\end{array}\right.$$ (15)

The r and t subindices given in Equation (15) represent randomly selected fractals from within the generation.

After the first update process, Equation (14) is reordered and the fractals are assigned probabilistic values. For fractals entering the second update process, the position of the fractal is changed according to whether a randomly generated number between 0 and 1 is smaller or larger than 0.5. The second update process is performed with the expression given in Equation (16).

$$P_i^{\prime \prime }=\left\{\begin{array}{cc}{P}_i^{\prime }-\widehat{\epsilon }x\left(P_t^{\prime }-\mathrm{BP}\right),& {\epsilon }^{\prime }\le 0.5,\\ P_i^{\prime }-\widehat{\epsilon }x\left(P_t^{\prime }-P_r^{\prime }\right),& {\epsilon }^{\prime }>0.5.\end{array}\right.$$ (16)

The algorithm continues to generate a new generation and generate new solution points until it meets the termination criterion.

3.3.2. Fitness distance balance selection method

The purpose of the fitness distance balance (FDB) method is to effectively guide meta‐heuristic search (MHS) algorithms in the search process. ⁵⁰ The feature that distinguishes the FDB method from other selection methods is the calculation of the score values of the solution candidates and the selection process according to the score values. In the calculation of the score, the fitness values of the solution candidates and their distance from the best solution candidate in the population are taken into account. This ensures that the solution candidate with a high fitness value is selected. On the other hand, this prevents the selection of a solution candidate that is very close to the best solution in the population. This selection strategy of the FDB method contributes to the solution of the early convergence problem frequently encountered in the MHS process. ⁵¹ , ⁵² , ⁵³ The steps to implement the FDB method are as follows:

1. The score values of the solution candidates in the 𝑃 population should be calculated. If 𝑋𝑏𝑒𝑠𝑡 is assumed to be the best solution in population 𝑃, the scores of the solution candidates in population 𝑃 are expressed by the vector 𝑆_𝑝 given in Equation (17).

   $$S_P\equiv {\left[\begin{array}{c}{s}_1\\ ⋮\\ s_n\end{array}\right]}_{n\times 1}.$$ (17)
2. While calculating the score of each solution candidate, the values of fitness and distance are taken into account. In population 𝑃, the distance between the ith solution candidate and 𝑋𝑏𝑒𝑠𝑡 (the best solution in 𝑃) is calculated using the Euclidean metric as given in Equation (18).

   $${\forall }_{i=1}^n,P_i\ne P_{\mathit{\text{best}}},D_{P_i}=\sqrt{{\left(x_{1P_i}-x_{1P_{\mathit{\text{best}}}}\right)}^2+{\left(x_{2P_i}-x_{2P_{\mathit{\text{best}}}}\right)}^2+\dots +{\left(x_{m{P}_i}-x_{m{P}_{\mathit{\text{best}}}}\right)}^2}.$$ (18)
3. The distance vector 𝐷𝑝 of the solution candidates in the population 𝑃 can be represented as given in Equation (19).

   $$D_P\equiv {\left[\begin{array}{c}{d}_1\\ ⋮\\ d_n\end{array}\right]}_{n\times 1}.$$ (19)
4. Another parameter used in calculating the scores of the solution candidates is the fitness value. The fitness values of the solution candidates in the population 𝑃 are represented by the vector 𝐹. The 𝐹 and 𝐷𝑝 vectors are normalized so that the two parameters do not dominate each other in the score calculation. The score calculation of the ith solution candidate is given in Equation (20).

   $${\forall }_{i=1}^n{P}_i,S_{P_i}=\omega *\mathrm{norm}{F}_{P_i}+(1-\omega )*\mathrm{norm}{D}_{P_i}.$$ (20)

The ω parameter given in Equation (20) is used to weight the effects of 𝐹 and 𝐷𝑝 parameters. The ω parameter can have an infinite number of values in the range [0, 1]. A larger value of the ω parameter increases the effect of the fitness value on the score. In this case, the effect of the distance value on the score decreases. When ω = 1, the score value of the solution candidate depends only on the fitness value. When ω = 0, the solution candidate's score depends only on the distance value. This ensures that the solution candidate farthest from P _best is selected in the population. Therefore, the variation in the population increases as the ω parameter is close to zero. This provides discovery capability for MHS algorithms. This parameter gives the FDB the ability to adapt. The ω parameter has the important function of switching and balancing between operating and reconnaissance tasks.

3.3.3. FDB based stochastic fractal search algorithm

A step‐by‐step explanation can be given for improving the search performance of MHS algorithms using the FDB method. Before applying the FDB method to an MHS algorithm, the search process lifecycle of the MHS algorithm should be analyzed and the process in which the algorithm provides diversity should be determined. Therefore, the SFS algorithm needs to be rearranged based on the general steps of the MHS process. ⁵⁴ The purpose of rearranging the algorithm is to reveal the exploration and exploration processes, and the step in which the FDB method is applied. After the exploration process, the diffusion process is carried out, and the tasks of searching for a neighborhood and providing diversity in the update process are fulfilled. In the next step, reference positions that guide the search process lifecycle in the SFS algorithm are examined. Thus, reference positions and the methods used to select them are determined. In this process, three reference positions are selected, one using the turn‐based method and the other two randomly. According to the information obtained here, the FDB selection method was applied. Thus, the process of ensuring diversity in the population was realized. Equations ((17), (18), (19), (20)) are taken into account in the FDB selection method. The SFSFDB flowchart is given in Figure 4.

FIGURE 4.

Flowchart of SFSFDB algorithm

4. SFSFDB ALGORITHM APPLIED TO HYBRID MLR‐FFANN

In the training of ANNs, a large number of uncertainties occur with the relationship between different datasets and the combination of many solutions. This uncertainty and different relationship status make their training process difficult. Moreover, ANNs have a nonlinear structure, which affects the behavior of the network in the learning process. Therefore, the importance is increasing of determining the most appropriate values of the weight and bias coefficients used in the training of the network and the activation functions used in the neurons. Considering this situation, the weight and bias coefficients of FFANN and the β polynomial coefficients of MLR in the proposed MLR‐FFANN model were determined simultaneously by optimization algorithms. In this study, the application of the SFSFDB algorithm described in Section 3.3 to the training of the proposed MLR‐FFANN model is explained step by step as follows.

Step 1: In this step, the prepared dataset is read from the relevant file and the values of the dataset are normalized between 0 and 1 using Equation (21).

$$x_{\mathrm{norm}}=\frac{x-x_{\min }}{x_{\max }-x_{\min }}.$$ (21)

The maximum number of iterations of the SFSFDB algorithm, the size of the population and the number of dimensions are entered into the system by the user. In addition to this step, the initial population is created within the minimum and maximum values determined for the parameters (weight, bias, and β polynomial coefficients) of the proposed MLR‐FFANN model to be optimized.

Step 2: The fitness function values of the initial population are calculated and the position value with the best fitness value is determined.

Step 3: The algorithm checks whether the current iteration number has reached the determined maximum number of iterations. If not, the natural process of the algorithm continues. If reached, the best value and best position values of the fitness function are displayed.

Step 4: Depending on the structure of the algorithm, the maximum diffusion number is determined. Depending on the maximum diffusion number, the steps of the diffusion process in Figure 4 are applied.

Step 5: According to Step 4, after the condition in the diffusion process is met and the necessary operations are performed, the algorithm performs the first updating process step in the flow diagram. The algorithm checks whether each ordered P _i value is less than the randomly generated number. Depending on this condition, the necessary processing is carried out.

Step 6: When the condition step in Step 5 is completed, the second updating process is started. In this step, all solution candidates in the population are ranked. Depending on the specified condition, either the selection method of the SFS algorithm is applied or the FDB selection method is applied, depending on Equations ((17), (18), (19), (20)). After the selection steps are performed depending on the specified conditions, the position of the P _i solution candidate in the population is updated.

Step 7: The current iteration number of the algorithm is increased and the algorithm goes to Step 3 and checks whether the condition there is met.

Figure 5 shows the application of the SFSFDB algorithm to the training of the proposed MLR‐FFANN model. Here, the input dataset is expressed as environmental conditions, days of the week, COVID‐19 pandemic precautions, the average electrical energy consumption values for the MLR method, and the average electric energy consumption estimated error values for FFANN, whereas the output dataset shows the daily average energy consumption of the city of Bursa. By using the normalization expression shown in Equation (21) for the input and output datasets, the values of the data are transformed into a dataset varying between 0 and 1.

FIGURE 5.

Flowchart of SFSFDB algorithm applied to MLR‐FFANN

5. EXPERIMENTAL RESULTS

In this study, a multiple linear regression and feedforward neural network‐based hybrid algorithm was proposed to predict the electric energy consumption of the city of Bursa during the COVID‐19 pandemic, and the design of this algorithm was carried out using AGDE, EO, SMA, and SFSFDB optimization algorithms. Although the optimization algorithms used minimized the mean squared error, they aimed to find the most appropriate weight and bias coefficients of the FFANN model and the β polynomial coefficients of the multiple linear regression algorithm.

Furthermore, the success of the following design structures in training of the FFANN model was examined in the optimization process.

The case studies of the FFANN structural designs were carried out as follows.

• Case 1: Using the hyperbolic tangent function in the hidden and output layers for the FFANN model.

• Case 2: Using the hyperbolic tangent function in the hidden layer and the sigmoid function in the output layer for the FFANN model.

• Case 3: Using the sigmoid function in the hidden layer and the hyperbolic tangent function in the output layer for the FFANN model.

• Case 4: Using the sigmoid function in the hidden and the output layers for the FFANN model.

Each of the structures for the FFANN model case studies is shown in Table 4. The simulation studies were conducted according to these network structures.

TABLE 4.

FFANN structures used in study cases

		Case 1		Case 2		Case 3		Case 4
		Hidden layer	Output layer	Hidden layer	Output layer	Hidden layer	Output layer	Hidden layer	Output layer
		Hyperbolic tangent	Hyperbolic tangent	Hyperbolic tangent	Sigmoid	Sigmoid	Hyperbolic tangent	Sigmoid	Sigmoid
FFANN structures	21 × 5 × 1	✓		✓		✓		✓
	21 × 6 × 1	✓		✓		✓		✓
	21 × 7 × 1	✓		✓		✓		✓
	21 × 8 × 1	✓		✓		✓		✓
	21 × 9 × 1	✓		✓		✓		✓
	21 × 10 × 1	✓		✓		✓		✓

The input and output parameters of the proposed hybrid MLR‐FFANN model are as follows.

Input parameters of the hybrid MLR‐FFANN model

• Days of the week

• COVID‐19 pandemic process precautions

• Environmental conditions

• Average electric energy consumption values for MLR $\{y(t-1),y(t-2)\}$

• Predicted error values of average electric energy consumption for FFANN $\{\mathit{er}(t-1),\mathit{er}(t-2)\}$

Output parameters of the hybrid MLR‐FFANN model

• Electricity energy consumption values of the Bursa Metropolitan Province

During the COVID‐19 pandemic, 80% of the dataset created for the estimation of the electric energy consumption of the Bursa Metropolitan Province was used in the training of the hybrid MLR‐FFANN model and 20% in the testing of the model. The mean absolute error (MAE), mean absolute percentage error (MAPE), and root mean square error (RMSE) criteria were used to evaluate the success of the proposed model in training and testing data. These evaluation criteria are shown in Equations ((22), (23), (24)). The MAE, MAPE, and RMSE evaluation criteria results from the training and testing data of different FFANN structures for all working cases of the AGDE algorithm are given in Table 5. Depending on these results, all working cases and the ranking of the network structures are shown in Table 6 in order to identify which working condition among the network structures used in the different working cases was more successful in the FFANN training process of the AGDE algorithm.

$$\mathrm{MAE}=\frac{\sum _{i=1}^n\left|e_i\right|}{n}=\frac{\sum _{i=1}^n\left|x_i-o_i\right|}{n},$$ (22)

$$\text{MPAE}=\frac{100\%}{n}\sum _{i=1}^n\left|\frac{x_i-o_i}{x_i}\right|,$$ (23)

$$\text{RMSE}=\sqrt{\frac{1}{n}\sum _{i=1}^n{e}_i^2}=\sqrt{\frac{1}{n}\sum _{i=1}^n{\left(x_i-o_i\right)}^2}.$$ (24)

TABLE 5.

MAE, MAPE and RMSE values obtained from the different FFANN structures optimized by AGDE algorithm

Algorithm	FFANN structures	Data type	Evaluation criteria	Case 1		Case 2		Case 3		Case 4
				Hidden layer	Output layer	Hidden layer	Output layer	Hidden layer	Output layer	Hidden layer	Output layer
				Hyperbolic tangent	Hyperbolic tangent	Hyperbolic tangent	Sigmoid	Sigmoid	Hyperbolic tangent	Sigmoid	Sigmoid
AGDE	21 × 5 × 1	Training data	MAE	27.3941		26.1891		30.2360		30.1717
			MAPE	5.2678		5.1011		5.9940		5.9849
			RMSE	38.7474		38.8531		44.0428		44.1025
		Test data	MAE	56.7584		53.1980		44.8230		42.6288
			MAPE	10.8761		10.5233		8.9435		8.5632
			RMSE	98.5928		78.2590		64.4790		61.7684
	21 × 6 × 1	Training data	MAE	27.5425		27.7760		28.6354		30.0492
			MAPE	5.3205		5.3031		5.5539		5.9714
			RMSE	39.7643		38.4424		41.2921		44.0109
		Test data	MAE	52.5854		54.7330		49.2259		42.6781
			MAPE	10.3154		10.7155		9.8232		8.5095
			RMSE	78.6412		76.5316		73.4692		61.5986
	21 × 7 × 1	Training data	MAE	25.3562		26.9770		27.3865		25.8093
			MAPE	4.8113		5.3183		5.3252		4.9777
			RMSE	35.1984		42.2227		39.6723		36.9277
		Test data	MAE	58.7017		42.7280		45.3613		50.1409
			MAPE	11.4637		8.6879		9.2956		9.9568
			RMSE	93.8435		60.8729		68.7305		71.9617
	21 × 8 × 1	Training data	MAE	26.4724		27.7247		25.8503		25.2653
			MAPE	5.0011		5.3367		4.9074		4.9397
			RMSE	36.9890		39.8150		36.2223		36.9891
		Test data	MAE	51.1938		52.9556		52.8339		60.1406
			MAPE	10.1769		10.3547		10.4806		11.5400
			RMSE	79.5222		75.2103		78.2415		97.8941
	21 × 9 × 1	Training data	MAE	26.6057		26.2527		27.5657		31.0802
			MAPE	5.0962		5.0866		5.3265		6.1767
			RMSE	37.7601		38.1095		39.3924		45.5558
		Test data	MAE	47.7024		53.5154		54.5175		41.0055
			MAPE	9.4646		10.4809		10.7926		8.3016
			RMSE	73.3299		80.3011		80.9939		59.6515
	21 × 10 × 1	Training data	MAE	24.3042		26.2617		26.8161		28.0184
			MAPE	4.5708		5.0730		5.1697		5.4671
			RMSE	34.7527		36.8862		38.2322		39.9554
		Test data	MAE	63.1113		50.4191		51.8507		50.8396
			MAPE	12.3587		9.8081		10.1927		10.0942
			RMSE	99.3556		76.0028		76.3146		74.0462

TABLE 6.

Ranking of results of simulation studies of the AGDE algorithm for different operational cases

Algorithm	Cases	Layers	Activation functions	Data type	Evaluation criteria	FFANN structures
						21 × 5 × 1	21 × 6 × 1	21 × 7 × 1	21 × 8 × 1	21 × 9 × 1	21 × 10 × 1
AGDE	Case 1	Hidden layer/output layer	Hyperbolic tangent/hyperbolic tangent	Training data	MAE	5	6	2	3	4	1
					MAPE	5	6	2	3	4	1
					RMSE	5	6	2	3	4	1
				Test data	MAE	4	3	5	2	1	6
					MAPE	4	3	5	2	1	6
					RMSE	5	2	4	3	1	6
	Case 2	Hidden layer/output layer	Hyperbolic tangent/sigmoid	Training data	MAE	1	6	4	5	2	3
					MAPE	3	4	5	6	2	1
					RMSE	4	3	6	5	2	1
				Test data	MAE	4	6	1	3	5	2
					MAPE	5	6	1	3	4	2
					RMSE	5	4	1	2	6	3
	Case 3	Hidden layer/output layer	Sigmoid/hyperbolic tangent	Training data	MAE	6	5	3	1	4	2
					MAPE	6	5	3	1	4	2
					RMSE	6	5	4	1	3	2
				Test data	MAE	1	3	2	5	6	4
					MAPE	1	3	2	5	6	4
					RMSE	1	3	2	5	6	4
	Case 4	Hidden layer/output layer	Sigmoid/sigmoid	Training data	MAE	6	4	2	1	5	3
					MAPE	5	4	2	1	6	3
					RMSE	5	4	1	2	6	3
				Test data	MAE	2	3	4	6	1	5
					MAPE	3	2	4	6	1	5
					RMSE	3	2	4	6	1	5

Figure 6 shows that the training of the proposed MLR‐FFANN model was carried out in 250 iterations by all optimization algorithms. During the training of the MLR‐FFANN model in different operational situations, it is clearly seen that the SFSFDB algorithm in Cases 1 and 3, and the AGDE algorithm in Cases 2 and 4 compared to the other algorithms, were more successful in finding the optimal value of the objective function. Moreover, it can be seen from Figure 6B that the SFSFDB algorithm was as successful as the AGDE algorithm in the training method proposed for Case 2.

FIGURE 6.

(A–D) Convergence curves of optimization algorithms for different operational cases

The results of MAE, MAPE, and RMSE evaluation criteria obtained from the training and test data of different FFANN structures for all working cases of EO, SMA, and SFSFDB algorithms are given in Tables 7, 8, 9, respectively. The ranking of the evaluation criteria results obtained for the training and testing data at the end of the optimization process of the FFANN design structures by the EO, SMA, and SFSFDB optimization algorithms in the different study cases is shown in Tables 10, 11, 12.

TABLE 7.

MAE, MAPE, and RMSE values obtained from the different FFANN structures optimized by the EO algorithm

Algorithm	FFANN structures	Data type	Evaluation criteria	Case 1		Case 2		Case 3		Case 4
				Hidden layer	Output layer	Hidden layer	Output layer	Hidden layer	Output layer	Hidden layer	Output layer
				Hyperbolic tangent	Hyperbolic tangent	Hyperbolic tangent	Sigmoid	Sigmoid	Hyperbolic tangent	Sigmoid	Sigmoid
EO	21 × 5 × 1	Training data	MAE	30.4668		29.8874		31.3811		31.8208
			MAPE	5.9711		5.8785		6.2732		6.3167
			RMSE	43.8322		43.3388		46.7406		45.9720
		Test data	MAE	44.7642		43.8510		39.0741		41.4298
			MAPE	9.0075		8.8721		8.0824		8.4246
			RMSE	66.3296		62.0308		59.1625		59.7926
	21 × 6 × 1	Training data	MAE	29.3265		30.3324		30.6505		30.9127
			MAPE	5.7169		5.9751		5.9525		6.1542
			RMSE	41.6963		42.6786		44.3530		45.5809
		Test data	MAE	50.8371		49.8149		42.1645		40.7968
			MAPE	10.1827		9.9533		8.4256		8.2865
			RMSE	74.7142		71.3203		63.2783		59.3978
	21 × 7 × 1	Training data	MAE	28.2143		27.7485		30.0383		29.5706
			MAPE	5.4786		5.4395		5.9537		5.8674
			RMSE	41.0958		40.2583		43.4762		43.6306
		Test data	MAE	42.5659		53.8792		48.8363		42.6071
			MAPE	8.5576		10.4588		9.7582		8.6709
			RMSE	63.3066		79.3093		69.8616		64.3107
	21 × 8 × 1	Training data	MAE	28.6195		29.0233		30.5674		30.9093
			MAPE	5.4598		5.6943		5.9648		6.1330
			RMSE	38.5721		40.6563		42.8045		44.4658
		Test data	MAE	61.6132		51.9173		45.9266		45.8902
			MAPE	12.1829		10.3261		9.1488		9.2047
			RMSE	86.7060		73.3204		67.5272		66.7979
	21 × 9 × 1	Training data	MAE	28.6827		27.5769		29.9710		30.0967
			MAPE	5.5448		5.3586		5.8727		5.8982
			RMSE	40.0931		39.1470		42.8298		42.6977
		Test data	MAE	54.3271		44.2227		45.5620		49.3985
			MAPE	10.5901		8.9883		9.1701		9.6140
			RMSE	77.5188		66.2420		68.7613		69.9643
	21 × 10 × 1	Training data	MAE	26.3307		24.9453		26.8807		29.1426
			MAPE	5.0676		4.7770		5.1726		5.7892
			RMSE	37.2934		36.1464		37.8466		42.4553
		Test data	MAE	49.0802		47.4105		55.0736		44.3604
			MAPE	9.7212		9.3795		10.7414		8.9299
			RMSE	74.5325		70.9335		86.7372		65.7630

TABLE 8.

MAE, MAPE, and RMSE values obtained from the different FFANN structures optimized by the SMA

Algorithm	FFANN structures	Data type	Evaluation criteria	Case 1		Case 2		Case 3		Case 4
				Hidden layer	Output layer	Hidden layer	Output layer	Hidden layer	Output layer	Hidden layer	Output layer
				Hyperbolic tangent	Hyperbolic tangent	Hyperbolic tangent	Sigmoid	Sigmoid	Hyperbolic tangent	Sigmoid	Sigmoid
SMA	21 × 5 × 1	Training data	MAE	33.7515		32.1509		34.9322		35.0882
			MAPE	6.7271		6.4573		6.9716		7.0292
			RMSE	49.4351		46.5082		50.2342		50.9672
		Test data	MAE	38.6627		46.7917		41.1690		37.9265
			MAPE	7.9884		9.4161		8.4393		7.8526
			RMSE	57.0402		65.4499		58.6201		55.8728
	21 × 6 × 1	Training data	MAE	36.0985		34.1528		35.2396		36.2255
			MAPE	7.2939		6.8609		7.0943		7.2331
			RMSE	53.2378		50.1897		51.1227		51.4780
		Test data	MAE	41.1084		40.7327		42.4998		43.9866
			MAPE	8.4412		8.3030		8.6026		8.8966
			RMSE	56.4548		58.7742		59.1442		61.4382
	21 × 7 × 1	Training data	MAE	33.0708		32.6873		32.7321		32.6362
			MAPE	6.6120		6.5353		6.5624		6.5491
			RMSE	48.9362		47.4181		48.3851		48.8187
		Test data	MAE	39.6327		41.7810		39.6390		37.0926
			MAPE	8.1020		8.4235		8.1305		7.6234
			RMSE	58.5108		59.8181		58.1576		54.8335
	21 × 8 × 1	Training data	MAE	33.0936		34.9420		33.9971		34.3282
			MAPE	6.6271		6.9643		6.8000		6.8296
			RMSE	48.1543		49.2945		49.3698		49.5032
		Test data	MAE	42.0874		44.1913		40.3400		38.3903
			MAPE	8.6223		9.1382		8.2148		7.8880
			RMSE	61.7824		63.0710		58.8103		56.4119
	21 × 9 × 1	Training data	MAE	32.5330		32.6000		34.0248		33.0650
			MAPE	6.4707		6.4883		6.8125		6.6219
			RMSE	46.9966		47.2959		49.1337		48.5220
		Test data	MAE	44.2997		42.8522		39.2549		38.5194
			MAPE	8.9143		8.7290		8.0704		7.8535
			RMSE	62.6968		60.9679		57.0847		56.4597
	21 × 10 × 1	Training data	MAE	31.6898		32.4855		35.2936		33.8145
			MAPE	6.3566		6.4431		7.0546		6.7522
			RMSE	46.9098		45.7092		50.2625		48.8474
		Test data	MAE	36.7966		47.8138		40.2164		42.9139
			MAPE	7.6497		9.4573		8.3951		8.7180
			RMSE	55.3233		69.9524		59.6641		60.5452

TABLE 9.

MAE, MAPE, and RMSE values obtained from the different FFANN structures optimized by the SFSFDB algorithm

Algorithm	FFANN structures	Data type	Evaluation criteria	Case 1		Case 2		Case 3		Case 4
				Hidden layer	Output layer	Hidden layer	Output layer	Hidden layer	Output layer	Hidden layer	Output layer
				Hyperbolic tangent	Hyperbolic tangent	Hyperbolic tangent	Sigmoid	Sigmoid	Hyperbolic tangent	Sigmoid	Sigmoid
SFSFDB	21 × 5 × 1	Training data	MAE	29.7848		26.3936		25.0693		25.8692
			MAPE	5.7796		5.0850		4.8339		5.0302
			RMSE	41.6004		37.3705		35.9982		38.1307
		Test data	MAE	51.5971		49.7136		54.0009		55.3014
			MAPE	10.3010		9.7947		10.7137		10.8689
			RMSE	77.2815		69.1236		88.6259		85.7601
	21 × 6 × 1	Training data	MAE	26.4507		27.2546		27.7269		29.7217
			MAPE	5.0356		5.2810		5.3298		5.8510
			RMSE	37.8495		39.0741		39.7307		42.3777
		Test data	MAE	59.9968		53.4792		53.6484		48.2257
			MAPE	11.6997		10.5107		10.6577		9.7180
			RMSE	88.5541		78.1406		80.7798		69.5811
	21 × 7 × 1	Training data	MAE	25.3758		26.7794		27.3811		28.3706
			MAPE	4.8527		5.2326		5.2293		5.5572
			RMSE	35.4393		38.7671		38.5420		40.9998
		Test data	MAE	60.1612		48.0063		51.6829		47.8107
			MAPE	11.9726		9.6096		10.0116		9.4816
			RMSE	89.8189		70.8651		84.3398		70.8205
	21 × 8 × 1	Training data	MAE	24.7913		26.6539		27.2681		27.8048
			MAPE	4.6964		5.1637		5.1986		5.4324
			RMSE	35.3336		37.5864		37.9751		40.4902
		Test data	MAE	66.8222		52.5460		55.5138		45.6329
			MAPE	13.3227		10.3486		11.0176		9.0612
			RMSE	106.6947		75.4689		81.1272		69.7324
	21 × 9 × 1	Training data	MAE	23.3997		23.2679		22.7836		23.5089
			MAPE	4.3995		4.4133		4.2657		4.4659
			RMSE	33.7388		33.2138		32.3052		33.6666
		Test data	MAE	63.8802		62.8841		56.8832		54.1425
			MAPE	12.6739		12.2250		11.3105		10.6307
			RMSE	105.3484		95.0279		86.8698		84.1150
	21 × 10 × 1	Training data	MAE	22.3310		21.9726		22.0975		21.7047
			MAPE	4.0906		4.1294		4.1256		4.1347
			RMSE	32.1979		31.0354		31.2159		32.0118
		Test data	MAE	69.5505		61.4864		55.8286		56.7811
			MAPE	13.6395		11.9117		11.2382		11.1619
			RMSE	103.0115		92.5567		85.6887		82.4051

TABLE 10.

Ranking of results of simulation studies of the EO algorithm for different operational cases

Algorithm	Cases	Layers	Activation functions	Data type	Evaluation criteria	FFANN structures
						21 × 5 × 1	21 × 6 × 1	21 × 7 × 1	21 × 8 × 1	21 × 9 × 1	21 × 10 × 1
EO	Case 1	Hidden layer/output layer	Hyperbolic tangent/hyperbolic tangent	Training data	MAE	6	5	2	3	4	1
					MAPE	6	5	3	2	4	1
					RMSE	6	5	4	2	3	1
				Test data	MAE	2	4	1	6	5	3
					MAPE	2	4	1	6	5	3
					RMSE	2	4	1	6	5	3
	Case 2	Hidden layer/output layer	Hyperbolic tangent/sigmoid	Training data	MAE	5	6	3	4	2	1
					MAPE	5	6	3	4	2	1
					RMSE	6	5	3	4	2	1
				Test data	MAE	1	4	6	5	2	3
					MAPE	1	4	6	5	2	3
					RMSE	1	4	6	5	2	3
	Case 3	Hidden layer/output layer	Sigmoid/hyperbolic tangent	Training data	MAE	6	5	3	4	2	1
					MAPE	6	3	4	5	2	1
					RMSE	6	4	5	2	3	1
				Test data	MAE	1	2	5	4	3	6
					MAPE	1	2	5	3	4	6
					RMSE	1	2	5	3	4	6
	Case 4	Hidden layer/output layer	Sigmoid/sigmoid	Training data	MAE	6	5	2	4	3	1
					MAPE	6	5	2	4	3	1
					RMSE	6	5	3	4	2	1
				Test data	MAE	2	1	3	5	6	4
					MAPE	2	1	3	5	6	4
					RMSE	2	1	3	5	6	4

TABLE 11.

Ranking of results of simulation studies of SMA for different operational cases

Algorithm	Cases	Layers	Activation functions	Data type	Evaluation criteria	FFANN structures
						21 × 5 × 1	21 × 6 × 1	21 × 7 × 1	21 × 8 × 1	21 × 9 × 1	21 × 10 × 1
SMA	Case 1	Hidden layer/output layer	Hyperbolic tangent/hyperbolic tangent	Training data	MAE	5	6	3	4	2	1
					MAPE	5	6	3	4	2	1
					RMSE	5	6	4	3	2	1
				Test data	MAE	2	4	3	5	6	1
					MAPE	2	4	3	5	6	1
					RMSE	3	2	4	5	6	1
	Case 2	Hidden layer/output layer	Hyperbolic tangent/sigmoid	Training data	MAE	1	5	4	6	3	2
					MAPE	2	5	4	6	3	1
					RMSE	2	6	4	5	3	1
				Test data	MAE	5	1	2	4	3	6
					MAPE	5	1	2	4	3	6
					RMSE	4	1	2	4	3	6
	Case 3	Hidden layer/output layer	Sigmoid/hyperbolic tangent	Training data	MAE	4	5	1	2	3	6
					MAPE	4	6	1	2	3	5
					RMSE	4	6	1	3	2	5
				Test data	MAE	5	6	2	4	1	3
					MAPE	5	6	2	3	1	4
					RMSE	3	5	2	4	1	6
	Case 4	Hidden layer/output layer	Sigmoid/sigmoid	Training data	MAE	5	6	1	4	2	3
					MAPE	5	6	1	4	2	3
					RMSE	5	6	2	4	1	3
				Test data	MAE	2	6	1	3	4	5
					MAPE	2	6	1	4	3	5
					RMSE	2	6	1	3	4	5

TABLE 12.

Ranking of results of simulation studies of the SFSFDB algorithm for different operational cases

Algorithm	Cases	Layers	Activation functions	Data type	Evaluation criteria	FFANN structures
						21 × 5 × 1	21 × 6 × 1	21 × 7 × 1	21 × 8 × 1	21 × 9 × 1	21 × 10 × 1
SFSFDB	Case 1	Hidden layer/output layer	Hyperbolic tangent/hyperbolic tangent	Training data	MAE	6	5	4	3	2	1
					MAPE	6	5	4	3	2	1
					RMSE	6	5	4	3	2	1
				Test data	MAE	1	2	3	5	4	6
					MAPE	1	2	3	5	4	6
					RMSE	1	2	3	6	5	4
	Case 2	Hidden layer/output layer	Hyperbolic tangent/sigmoid	Training data	MAE	3	6	5	4	2	1
					MAPE	3	6	5	4	2	1
					RMSE	3	6	5	4	2	1
				Test data	MAE	2	4	1	3	6	5
					MAPE	2	4	1	3	6	5
					RMSE	1	4	2	3	6	5
	Case 3	Hidden layer/output layer	Sigmoid/hyperbolic tangent	Training data	MAE	3	6	5	4	2	1
					MAPE	3	6	5	4	2	1
					RMSE	3	6	5	4	2	1
				Test data	MAE	3	2	1	4	6	5
					MAPE	3	2	1	4	6	5
					RMSE	6	1	3	2	5	4
	Case 4	Hidden layer/output layer	Sigmoid/sigmoid	Training data	MAE	3	6	5	4	2	1
					MAPE	3	6	5	4	2	1
					RMSE	3	6	5	4	2	1
				Test data	MAE	5	3	2	1	4	6
					MAPE	5	3	2	1	4	6
					RMSE	6	1	3	2	5	4

The comparative analysis and ranking of FFANN structures in simulation cases for all optimization algorithms are given in Table 13. These rankings are shown depending on the sum of the ranking results, mean values of these results, and standard deviation values given in Tables 6 and 10, 11, 12. According to these ranking results, the FFANN structure that completed the training and testing process the most successfully was determined among the simulation cases for each optimization algorithm. The identified FFANN structures are denoted in Table 13 in different colors for each optimization algorithm.

TABLE 13.

Comparison analysis of the ranking results of the FFANN structures for optimization algorithms

Algorithm	Cases	FFANN structures	Final rank	Sum	Mean	Standard deviation	Algorithm	Cases	FFANN structures	Final rank	Sum	Mean	Standard deviation
AGDE	Case 1	21 × 5 × 1	6	28	4.6667	0.5164	EO	Case 1	21 × 5 × 1	3	24	4.0000	2.1909
		21 × 6 × 1	5	26	4.3333	1.8619			21 × 6 × 1	6	27	4.5000	0.5477
		21 × 7 × 1	3	20	3.3333	1.5055			21 × 7 × 1	2	12	2.0000	1.2649
		21 × 8 × 1	2	16	2.6667	0.5164			21 × 8 × 1	4	25	4.1667	2.0412
		21 × 9 × 1	1	15	2.5000	1.6432			21 × 9 × 1	5	26	4.3333	0.8165
		21 × 10 × 1	4	21	3.5000	2.7386			21 × 10 × 1	1	12	2.0000	1.0954
	Case 2	21 × 5 × 1	4	22	3.6667	1.5055		Case 2	21 × 5 × 1	3	19	3.1667	2.4014
		21 × 6 × 1	6	29	4.8333	1.3292			21 × 6 × 1	6	29	4.8333	0.9832
		21 × 7 × 1	2	18	3.0000	2.2804			21 × 7 × 1	5	27	4.5000	1.6432
		21 × 8 × 1	5	24	4.0000	1.5492			21 × 8 × 1	4	27	4.5000	0.5477
		21 × 9 × 1	3	21	3.5000	1.7607			21 × 9 × 1	1	12	2.0000	0.0000
		21 × 10 × 1	1	12	2.0000	0.8944			21 × 10 × 1	2	12	2.0000	1.0954
	Case 3	21 × 5 × 1	4	21	3.5000	2.7386		Case 3	21 × 5 × 1	5	21	3.5000	2.7386
		21 × 6 × 1	5	24	4.0000	1.0954			21 × 6 × 1	2	18	3.0000	1.2649
		21 × 7 × 1	1	16	2.6667	0.8165			21 × 7 × 1	6	27	4.5000	0.8367
		21 × 8 × 1	3	18	3.0000	2.1909			21 × 8 × 1	3	21	3.5000	1.0488
		21 × 9 × 1	6	29	4.8333	1.3292			21 × 9 × 1	1	18	3.0000	0.8944
		21 × 10 × 1	2	18	3.0000	1.0954			21 × 10 × 1	4	21	3.5000	2.7386
	Case 4	21 × 5 × 1	6	24	4.0000	1.5492		Case 4	21 × 5 × 1	4	24	4.0000	2.1909
		21 × 6 × 1	2	19	3.1667	0.9832			21 × 6 × 1	3	18	3.0000	2.1909
		21 × 7 × 1	1	17	2.8333	1.3292			21 × 7 × 1	2	16	2.6667	0.5164
		21 × 8 × 1	4	22	3.6667	2.5820			21 × 8 × 1	6	27	4.5000	0.5477
		21 × 9 × 1	3	20	3.3333	2.5820			21 × 9 × 1	5	26	4.3333	1.8619
		21 × 10 × 1	5	24	4.0000	1.0954			21 × 10 × 1	1	15	2.5000	1.6432

Algorithm	Cases	FFANN structures	Final rank	Sum	Mean	Standard deviation	Algorithm	Cases	FFANN structures	Final rank	Sum	Mean	Standard deviation
SMA	Case 1	21 × 5 × 1	3	22	3.6667	1.5055	SFSFDB	Case 1	21 × 5 × 1	5	21	3.5000	2.7386
		21 × 6 × 1	6	28	4.6667	1.6330			21 × 6 × 1	4	21	3.5000	1.6432
		21 × 7 × 1	2	20	3.3333	0.5164			21 × 7 × 1	3	21	3.5000	0.5477
		21 × 8 × 1	5	26	4.3333	0.8165			21 × 8 × 1	6	25	4.1667	1.3292
		21 × 9 × 1	4	24	4.0000	2.1909			21 × 9 × 1	1	19	3.1667	1.3292
		21 × 10 × 1	1	6	1.0000	0.0000			21 × 10 × 1	2	19	3.1667	2.4833
	Case 2	21 × 5 × 1	3	19	3.1667	1.7224		Case 2	21 × 5 × 1	1	14	2.3333	0.8165
		21 × 6 × 1	4	19	3.1667	2.4014			21 × 6 × 1	6	30	5.0000	1.0954
		21 × 7 × 1	2	18	3.0000	1.0954			21 × 7 × 1	3	19	3.1667	2.0412
		21 × 8 × 1	6	29	4.8333	0.9832			21 × 8 × 1	4	21	3.5000	0.5477
		21 × 9 × 1	1	18	3.0000	0.0000			21 × 9 × 1	5	24	4.0000	2.1909
		21 × 10 × 1	5	22	3.6667	2.5820			21 × 10 × 1	2	18	3.0000	2.1909
	Case 3	21 × 5 × 1	4	25	4.1667	0.7528		Case 3	21 × 5 × 1	3	21	3.5000	1.2247
		21 × 6 × 1	6	34	5.6667	0.5164			21 × 6 × 1	6	23	3.8333	2.4014
		21 × 7 × 1	1	9	1.5000	0.5477			21 × 7 × 1	2	20	3.3333	1.9664
		21 × 8 × 1	3	18	3.0000	0.8944			21 × 8 × 1	4	22	3.6667	0.8165
		21 × 9 × 1	2	11	1.8333	0.9832			21 × 9 × 1	5	23	3.8333	2.0412
		21 × 10 × 1	5	29	4.8333	1.1690			21 × 10 × 1	1	17	2.8333	2.0412
	Case 4	21 × 5 × 1	3	21	3.5000	1.6432		Case 4	21 × 5 × 1	5	25	4.1667	1.3292
		21 × 6 × 1	6	36	6.0000	0.0000			21 × 6 × 1	6	25	4.1667	2.1370
		21 × 7 × 1	1	7	1.1667	0.4082			21 × 7 × 1	4	22	3.6667	1.5055
		21 × 8 × 1	4	22	3.6667	0.5164			21 × 8 × 1	1	16	2.6667	1.5055
		21 × 9 × 1	2	16	2.6667	1.2111			21 × 9 × 1	2	19	3.16667	1.3292
		21 × 10 × 1	5	24	4.0000	1.0954			21 × 10 × 1	3	19	3.16667	2.4833

In Table 14, the average values of the evaluation criteria obtained for the FFANN structures used in all cases are calculated and the sorting studies of the algorithms according to these values are indicated in both the training and test datasets. In this table, the most successful cases of the AGDE algorithm can be expressed as Cases 2 and 3. The most successful cases for EO, SMA, and SFSFDB algorithms are defined as Cases 2, 4, and 3, respectively. The results of the evaluation criteria obtained in both training and test data for each case (Table 14) are shown in detail in the bar graphs in Figure 7.

TABLE 14.

Comparison analysis of the ranking results of study cases for optimization algorithms

AGDE							EO
Cases	Mean values of the results for all of cases						Cases	Mean values of the results for all of cases
	Training data results			Test data results				Training data results			Test data results
	MAE	MAPE	RMSE	MAE	MAPE	RMSE		MAE	MAPE	RMSE	MAE	MAPE	RMSE
Case 1	26.279183	5.011283	37.201983	55.008833	10.7759	87.2142	Case 1	28.60675	5.5398	40.430483	50.531283	10.040333	73.851283
Case 2	26.863533	5.203133	39.054816	51.258183	10.095066	74.529616	Case 2	28.2523	5.5205	40.3709	48.515933	9.663016	70.52605
Case 3	27.748333	5.37945	39.809016	49.768716	9.921366	73.7047	Case 3	29.914833	5.864916	43.00845	46.106183	9.221083	69.22135
Case 4	28.399016	5.58625	41.2569	47.905583	9.494216	71.153416	Case 4	30.408783	6.02645	44.133716	44.080466	8.8551	64.337716

Ranking of the results for all of cases							Ranking of the results for all of cases
Cases	Training data results			Test data results			Cases	Training data results			Test data results
	MAE	MAPE	RMSE	MAE	MAPE	RMSE		MAE	MAPE	RMSE	MAE	MAPE	RMSE
Case 1	1	1	1	4	4	4	Case 1	2	2	2	4	4	4
Case 2	2	2	2	3	3	3	Case 2	1	1	1	3	3	3
Case 3	3	3	3	2	2	2	Case 3	3	3	3	2	2	2
Case 4	4	4	4	1	1	1	Case 4	4	4	4	1	1	1

Comparison analysis of the ranking results for all of cases					Comparison analysis of the ranking results for all of cases
Cases	Final rank	Sum	Mean	Standard deviation	Cases	Final rank	Sum	Mean	Standard deviation
Case 1	2	15	2.5000	1.6432	Case 1	4	18	3.0000	1.0954
Case 2	1	15	2.5000	0.5477	Case 2	1	12	2.0000	1.0954
Case 3	1	15	2.5000	0.5477	Case 3	2	15	2.5000	0.5477
Case 4	2	15	2.5000	1.6432	Case 4	3	15	2.5000	1.6432

SMA							SFSFDB
Cases	Mean values of the results for all of cases						Cases	Mean values of the results for all of cases
	Training data results			Training data results				Training data results			Test data results
	MAE	MAPE	RMSE	MAE	MAPE	RMSE		MAE	MAPE	RMSE	MAE	MAPE	RMSE
Case 1	33.372866	6.681233	48.944966	40.43125	8.286316	58.634716	Case 1	25.522216	4.809066	36.027033	60.334666	12.26073	95.118183
Case 2	33.16975	6.624866	47.735933	44.027116	8.911183	63.005583	Case 2	25.387000	4.884166	36.174550	54.685933	10.733383	80.197133
Case 3	34.3699	6.882566	49.751333	40.51985	8.308783	58.580166	Case 3	25.387750	4.830483	35.961183	54.592966	10.824883	84.571866
Case 4	34.192933	6.83585	49.689266	39.804883	8.138683	57.59355	Case 4	26.163316	5.078566	37.946133	51.315716	10.153716	77.069033

Ranking of the results for all of cases							Ranking of the results for all of cases
Cases	Training data results			Test data results			Cases	Training data results			Test data results
	MAE	MAPE	RMSE	MAE	MAPE	RMSE		MAE	MAPE	RMSE	MAE	MAPE	RMSE
Case 1	2	2	2	2	2	3	Case 1	3	1	2	4	4	4
Case 2	1	1	1	4	4	4	Case 2	1	3	3	3	2	2
Case 3	4	4	4	3	3	2	Case 3	2	2	1	2	3	3
Case 4	3	3	3	1	1	1	Case 4	4	4	4	1	1	1

Comparison analysis of the ranking results for all of cases					Comparison analysis of the ranking results for all of cases
Cases	Final rank	Sum	Mean	Standard deviation	Cases	Final rank	Sum	Mean	Standard deviation
Case 1	2	13	2.1667	0.4082	Case 1	4	18	3.0000	1.2649
Case 2	3	15	2.5000	1.6432	Case 2	2	14	2.3333	0.8165
Case 3	4	20	3.3333	0.8165	Case 3	1	13	2.1667	0.7528
Case 4	1	12	2.0000	1.0954	Case 4	3	15	2.5000	1.6432

FIGURE 7.

(A–D) Evaluation criteria for all optimization algorithms

The results of the evaluation criteria of the best cases in each optimization algorithm (Figure 7 and Table 14) are shown in bar graph form in Figure 8 for both training and test data so that the results can be better interpreted.

FIGURE 8.

Evaluation criteria for the best case of all optimization algorithms

After determining the best cases used in the training process of FFANN for each optimization algorithm (Table 14), the results of the evaluation criteria of the best FFANN structures in the cases, the ranking of these results for both training and test data, and the sum of these ranking results, mean value, and standard deviation values are shown in Table 15. In other words, the final ranking in Table 15 clearly shows that the EO algorithm was the first in the training of the dataset, followed by the SFSFDB and SMA algorithms.

TABLE 15.

Determination of the best network structures for all optimization algorithms

The evaluation criteria of the best FFANN structures for the training and test process
Algorithms	Cases	FFANN structures	Training data results			Test data results
			MAE	MAPE	RMSE	MAE	MAPE	RMSE
AGDE	Case 3	21 × 7 × 1	27.3865	5.3252	39.6723	45.3613	9.2956	68.7305
EO	Case 2	21 × 9 × 1	27.5769	5.3586	39.1470	44.2227	8.9883	66.2420
SMA	Case 4	21 × 7 × 1	32.6362	6.5491	48.8187	37.0926	7.6234	54.8335
SFSFDB	Case 3	21 × 10 × 1	22.0975	4.1256	31.2159	55.8286	11.2382	85.6887

Ranking of FFANN structures according to evaluation criteria for the training and test process
Algorithms	Cases	FFANN structures	Training data results			Test data results
			MAE	MAPE	RMSE	MAE	MAPE	RMSE
AGDE	Case 3	21 × 7 × 1	2	2	3	3	3	3
EO	Case 2	21 × 9 × 1	3	3	2	2	2	2
SMA	Case 4	21 × 7 × 1	4	4	4	1	1	1
SFSFDB	Case 3	21 × 10 × 1	1	1	1	4	4	4

Final ranking of all optimization algorithms for the training and testing process
Algorithms	Cases	FFANN structures	Final rank	Sum	Mean	Standard deviation
AGDE	Case 3	21 × 7 × 1	3	16	2.6667	0.5164
EO	Case 2	21 × 9 × 1	1	14	2.3333	0.5164
SMA	Case 4	21 × 7 × 1	2	15	2.5000	1.6432
SFSFDB	Case 3	21 × 10 × 1	2	15	2.5000	1.6432

The FFANN structures determined for all optimization algorithms were adapted to the proposed hybrid MLR‐FFANN algorithm. By using MLR‐FFANN algorithms, where the most appropriate weight, bias, and β polynomial coefficients were determined by the optimization algorithms, the electric energy consumption of the Bursa Metropolitan Province was predicted during the COVID‐19 pandemic period until September 2020. As a result of the prediction process, the evaluation criteria, the ranking of these criteria, and the final ranking of the algorithms are shown in Table 16. According to the final ranking of the algorithms, the MLR‐FFANN method trained with the SFSFDB algorithm was more successful than other algorithms in estimating the electric energy consumption values of Bursa during the COVID‐19 pandemic. When Tables 15 and 16 were evaluated together for the final rankings of the optimization algorithms, the MLR‐FFANN, whose training process was completed with the SFSFDB algorithm, was more successful than the MLR‐FFANN trained with the other (EO, SMA, and AGDE) algorithms.

TABLE 16.

Evaluation criteria and ranking results for all optimization algorithms for the year 2020

Prediction of the year 2020 for all optimization algorithms
Algorithms	Cases	FFANN structures	MAE	MAPE	RMSE
AGDE	Case 3	21 × 7 × 1	52.5230	11.2477	72.7393
EO	Case 2	21 × 9 × 1	41.1935	8.2490	56.9251
SMA	Case 4	21 × 7 × 1	67.5097	14.6852	92.2592
SFSFDB	Case 3	21 × 10 × 1	33.2284	7.0958	48.9234

Ranking of evaluation criteria of the year 2020 for all optimization algorithms
Algorithms	Cases	FFANN structures	MAE	MAPE	RMSE
AGDE	Case 3	21 × 7 × 1	3	3	3
EO	Case 2	21 × 9 × 1	2	2	2
SMA	Case 4	21 × 7 × 1	4	4	4
SFSFDB	Case 3	21 × 10 × 1	1	1	1

Final ranking of all optimization algorithms for prediction of the year 2020
Algorithms	Cases	FFANN structures	Final rank	Sum	Mean	Standard deviation
AGDE	Case 3	21 × 7 × 1	3	9	3	0.0000
EO	Case 2	21 × 9 × 1	2	6	2	0.0000
SMA	Case 4	21 × 7 × 1	4	12	4	0.0000
SFSFDB	Case 3	21 × 10 × 1	1	3	1	0.0000

In other words, the SFSFDB algorithm had a value of 33.2284 for the MAE evaluation criterion, which is 36.7355%, 19.3358%, and 50.7798% less than the results of the AGDE, EO, and SMA algorithms, respectively. When the results were evaluated in terms of the MAPE criterion, the SFSFDB algorithm had a value of 7.0958, a value that is 36.9133%, 13.9798%, and 51.6825% less than the results of the system trained by the other optimization algorithms. Considering the RMSE values, the result of the model designed with the SFSFDB optimization algorithm was 23.8159%, 8.0017%, and 43.3358% less than the results of the other algorithms, respectively.

Figure 9A gives the comparison of the most suitable MLR‐FFANN structures, determined according to the results obtained from all optimization algorithms, in estimating the electric energy consumption values of the city of Bursa until September 2020 during the COVID‐19 pandemic process, according to the evaluation criteria. Figure 9B shows the difference between the actual and estimated electric energy consumption values. Actual and predicted electricity energy consumption data are detailed in Figure 9C.

FIGURE 9.

For the year 2020: (A) Prediction for the year 2020, (B) error values between actual and predicted consumption values, and (C) actual and predicted electric energy consumption values of the city of Bursa for the year 2020

5.1. Accuracy and reliability of the developed SFSFDB based MLR‐FFANN

5.1.1. Outlier detection

Outlier detection is critical in the development of appropriate models for recognizing individual data that may differ from the data chunk contained in a dataset. ⁵⁵ In the literature, the methods consisting of both numerical and graphical algorithms have been proposed for this purpose. ⁵⁶ , ⁵⁷ , ⁵⁸ Leverage approach is an algorithm that is considered quite good for detecting these different values. This method considers the deviations of a model from the experimental data, a matrix of experimental data known as the Hat matrix, and the predicted values. The main criterion in the application of this method is to use a model that can calculate or estimate the relevant data within acceptable limits. Leverage or hat indices (hat matrix (H)) are calculated using Equation (25) ⁵⁵ , ⁵⁶ , ⁵⁷ , ⁵⁸ , ⁵⁹ , ⁶⁰ :

$$H=X{\left(X^{t}X\right)}^{-1}{X}^t,$$ (25)

where X is a (n × k) matrix consisting of n data (rows) and k parameters (columns) of the model and t denotes the transpose matrix. Hat values of the data form the diagonal elements of the H value. The Williams plot is created by calculating the H values in Equation (25). Outliers or suspicious data are graphically identified in this plot. This graph shows the correlation of the Hat indices with the standardized cross‐validated residuals (R). A warning leverage (H*) is usually fixed at a value equal to 3p/n; where n represents the number of data points and p represents one more than the number of model parameters. A leverage set to three is normally considered a cut‐off value to accept points within ±3 standard deviations from the mean. The fact that the majority of data points are in the range of 0 ≤ H ≤ H* and −3 ≤ R ≤ 3 indicates that both the model development and its predictions are within the applicability limits. Thus, a statistically valid model is presented. “Good high leverage” points must be in the H* ≤ H and −3 ≤ R ≤ 3 fields. Points greater than H* or within the specified value range (R < −3 or 3 < R) are defined as model outliers or “bad high leverage” points. These erroneous representations/predictions are called as the doubtful data.

In this study, the statistical hat matrix, Williams plot, and leverage approach, which enables outliers to be recognized in the model results, were used to check the reliability of the proposed model. ⁵⁶ H values were calculated with Equation (25) and the evaluation steps were followed step by step according to the procedure mentioned above. Thus, the Williams plot for the results from the SFSFDB model is plotted in Figure 10. The cyan color line in Figure 10 is the critical H* value or the leverage constraint which is computed as 0.1317 in this study. Moreover, the red line of R = ±3 is defined as the suspected limits. It can be seen that the majority of data points lie in the range 0 ≤ H ≤ H* and −3 ≤ R ≤ 3. As a result, the points corresponding to 96.3068% of the total number of data in the data set fall into the domain, while the remaining 3.6932% can be expressed as points outside the domain. These results show that the applied model is statistically acceptable and valid.

FIGURE 10.

Detection of potentially suspect data and the domain of applicability of the developed SFSFDB model

5.1.2. Analysis of the cumulative frequency versus absolute percent relative error

The cumulative frequency graph can be consulted for further comparison of the accuracy of the proposed models. ⁵⁵ , ⁵⁶ , ⁵⁷ , ⁵⁸ , ⁵⁹ , ⁶¹ , ⁶² , ⁶³ , ⁶⁴ , ⁶⁵ , ⁶⁶ This statistical methodology gives an idea of the average absolute percent relative error or absolute percent relative error models below a certain value. Equation (26) is explained as a relative percent error:

$$E_i=\left[\frac{x_i-o_i}{x_i}\right]\times 100\to i=1,2,3,\dots ,n,$$ (26)

where E _i is identified as the relative deviation of a represented/predicted value from an experimental value. x _i and o _i are defined as the actual and estimated/predicted value of the ith data, respectively, and n is the total number of data. The mean absolute percentage relative errors given in Equation (27) are used to calculate the absolute deviation of the experimental data.

$$E_a=\frac{1}{n}\sum _{i=1}^n\left|E_i\right|.$$ (27)

In this study, the cumulative frequency curve given in Figure 11, in which absolute percent relative error values are plotted against the cumulative frequency values, provides further comparisons regarding the accuracy of the proposed models. As can be seen from Figure 11, 90% of the predictions made by the optimized models by the SFSFDB, SMA, AGDE, and EO algorithms have absolute percent relative errors smaller than 27.1328%, 79.6028%, 53.9706%, and 32.6391%, respectively. These results represent another validation analysis of the superiority of the SFSFDB model over other models. Moreover, when the results of all proposed soft calculation methods are evaluated in themselves, they can be expressed as methods that provide fair enough accuracy for estimating electrical energy consumption data involving COVID‐19 precautions.

FIGURE 11.

For the cumulative frequency versus absolute percent relative error: (A) The results of the different models and (B) zoom version of the results

5.1.3. Sensitivity analysis

The robust predictions can be made as a result of the proposed algorithm being a sufficiently successful model. Therefore, it is necessary to determine whether it is a sufficiently successful model. The important thing here is that accuracy should not be evaluated alone when determining the adequacy of the model. Considering this situation, the degree of effectiveness of the independent variables affecting the model should be investigated by performing a sensitivity analysis. The cosine amplitude method (CAM), which is defined as one of the best sensitivity analyzes in the literature, was used in this study. ⁶⁷ The variable (R _ij ), which affects the electrical energy consumption the most, is used to measure the degree of sensitivity among the independent variables. The higher the R _ij , the more the independent variables affect the electrical energy consumption. If the relationship between the independent variables and electrical energy consumption is positive, the R _ij value is positive, and if the relationship between them is negative, the R _ij value is also negative. X array with n arguments is X = {x ₁, x ₂ , x ₃ , …, x _n }, each element of this array as “x _i ” and if there are “m” of each argument, the elements of the data set are X _i  = {x _i1 , x _i2 , x _i3 ,…,x _im }. If the relationship between the dependent and independent variables in the data set is called x _i and x _j , the R _ij value is calculated as in Equation (28) ⁶⁸ ;

$$R_{\mathit{ij}}=\left(\sum _{k=1}^m{x}_{\mathit{ik}}{x}_{\mathit{jk}}\right)/\left(\sqrt{\sum _{k=1}^m{x}_{\mathit{ik}}^2\sum _{k=1}^m{x}_{\mathit{jk}}^2}\right),0\le R_{\mathit{ij}}\le 1.$$ (28)

Figure 12 shows the results of the sensitivity analysis according to the CAM performed within the scope of this study. er(t − 1) is defined as the error value between the previous prediction and the true value, and er(t − 2) represents the error value between the two previous prediction and the true value. As can be seen from the results of the analysis, the factor that affects the electrical energy consumption estimation the most is the average humidity value with 0.9271. Average wind speed value follows the average humidity value with 0.9206. Average pressure with 0.8555 and average temperature with 0.8326, which are very close to these values, are seen as the most the other influencing factors. It is seen that the previous error value, er(t − 1) and the two previous error values, er(t − 2), also highly affect the estimation process. It is seen that the days of the week and COVID‐19 precautions have a less effect than the others.

FIGURE 12.

The sensitivity analysis of dataset

6. CONCLUSION

In this study, the recently proposed SFSFDB algorithm was used for the first time as a trainer of the hybrid MLR‐FFANN algorithm for the estimation of the energy consumption values of the city of Bursa during the COVID‐19 pandemic. The SFSFDB algorithm has the ability to search both locally and globally and optimizes the weight, bias, and β polynomial coefficients of the proposed hybrid MLR‐FFANN model. The design of the network and the activation functions to be used in the training of the FFANN are among the factors affecting the success percentage of the network in training. Considering this situation, network structures with different hidden layer numbers were considered in the FFANN design of the hybrid model. Moreover, the activation functions in the hidden and output layers used in these network structures were discussed in different study cases as combinations of hyperbolic tangent and sigmoid functions. Whereas the training process of traditional ANNs is carried out by the subjective experience of the researcher or through many iterations, the training process of the hybrid model proposed for different cases in this study was carried out in 250 iterations. In order to show the success of the proposed hybrid MLR‐FFANN approach in the training process, the simulation results obtained from the SFSFDB algorithm were compared with the AGDE, EO, and SMA heuristic optimization methods used in the optimization problems of different science fields in the literature. The MAE, MAPE, and RMSE evaluation criteria obtained from the comparison results were statistically examined to evaluate the success of the algorithms in the training of the proposed method. Among the heuristic optimization approaches used in the training process of the hybrid MLR‐FFANN, the SFSFDB algorithm was more successful in Cases 1 and 3, the SFSFDB and AGDE algorithms for Case 2, and the AGDE algorithm in Case 4, compared to other algorithms. The simulation results in terms of the evaluation criteria in both training and testing processes and the success ranking of the algorithms showed that the EO and SFSFDB algorithms were more successful than the other algorithms. Furthermore, during the 2020 COVID‐19 pandemic, the electric energy consumption values of Bursa were estimated and at the end of the estimation process, the hybrid MLR‐FFANN, which was trained by optimizing the parameters via the SFSFDB algorithm, was more successful in the prediction process than the MLR‐FFANN structures trained with the other optimization algorithms. The following conclusions were reached as a result of this research:

• When the results of the proposed method based on SFSFDB are examined, statistically MAE, MAPE and RMSE values have the best values with 33.2284, 7.0958, and 48.9234, respectively.

• The validity of the suggested method was demonstrated by outlier analysis, which shows that all data points are within an acceptable range.

• Environmental factors had a more pronounced effect on the estimation of electrical energy consumption than days of the week, error values, and COVID‐19 precautions, according to the results of the sensitivity analysis.

• As a result, COVID‐19 pandemic precautions were used as input data in the dataset for the estimation of electric energy consumption, and the hybrid MLR‐FFANN approach, whose parameters were optimized with the SFSFDB algorithm, successfully carried out the training and learning processes.

CONFLICT OF INTEREST

The authors declare that they have no conflict of interest.

Dalcali A, Özbay H, Duman S. Prediction of electricity energy consumption including COVID‐19 precautions using the hybrid MLR‐FFANN optimized with the stochastic fractal search with fitness distance balance algorithm. Concurrency Computat Pract Exper. 2022;34(15):e6947. doi: 10.1002/cpe.6947

DATA AVAILABILITY STATEMENT

The data that support the findings of this study are available on request from the corresponding author. The data are not publicly available due to privacy or ethical restrictions.