Development of a Short-Term Electrical Load Forecasting in Disaggregated Levels Using a Hybrid Modified Fuzzy-ARTMAP Strategy

Abstract

In recent years, electrical systems have evolved, creating uncertainties in short-term economic dispatch programming due to demand fluctuations from self-generating companies. This paper proposes a flexible Machine Learning (ML) approach to address electrical load forecasting at various levels of disaggregation in the Peruvian Interconnected Electrical System (SEIN). The novelty of this approach includes utilizing meteorological data for training, employing an adaptable methodology with easily modifiable internal parameters, achieving low computational cost, and demonstrating high performance in terms of MAPE. The methodology combines modified Fuzzy ARTMAP Neural Network (FAMM) and hybrid Support Vector Machine FAMM (SVMFAMM) methods in a parallel process, using data decomposition through the Wavelet filter db20. Experimental results show that the proposed approach outperforms state-of-the-art models in predicting accuracy across different time intervals.

1. Introduction

Electrical load forecasting in deregulated electricity markets is a valuable tool for managing Power Systems [1,2]. Managing such systems, whether in operation, maintenance, or planning, is a complex and challenging task [3] that results from the technological evolution of electrical systems [4]. This evolution [5] is due to the incorporation of non-conventional renewable resources [6] and the participation of industrial users in self-supplying.

To ensure the efficient management of current electrical systems, it is important to optimize their operation. This can be achieved through economic dispatch, which aims to minimize costs while ensuring safety and maximizing the utilization of energy resources [2]. As a result, it becomes crucial to accurately monitor the balance between supply and demand at various busbars of electrical systems in the short, medium, and long term.

Predicting this demand in the short term [7], following the dispatch schedule for the next day, at different busbars is a complex and challenging task, given the large variety of disaggregated loads [8], self-supply from users, and energy injection from non-conventional sources [5]. Therefore, the next-day market programming needs to have load profiles with high efficiency for various levels of disaggregation, so that optimization of the programming can ensure security, avoiding the operation of inflexible power plants which generate high variable costs and, consequently, high operating costs [2].

The complexity of the forecasts is quantified by the level of load aggregation. Elevated levels of aggregation, which use traditional forecasting methods, achieve high efficiency, thus surpassing uncertainties. However, for various levels of disaggregation, given the current conditions involving electrical systems, the methods become complex in modeling the characteristic load patterns [8]. For this reason, fluctuating consumption patterns motivate the exploration of new proposals for forecasting tools with more efficient and robust methods to minimize errors.

ML is a field of Artificial Intelligence (AI) that encompasses a set of methods that can automatically detect and generalize patterns in datasets, thus supporting the prediction of the future or decision-making under various conditions of uncertainty [9]. The AI field has been used worldwide to solve problems in various areas of knowledge [10]. In the last two decades, several models have been applied to energy systems to solve energy prediction problems, solar irradiation, wind speed, urban heating, classification of power quality disturbances, electricity market price prediction, among others [11].

This paper is motivated by the application of ML techniques to model, design, and forecast electric loads at various levels of disaggregation. We propose a flexible methodology that can adapt to the current dynamics of the electrical system at various levels of disaggregation.

In addition to this introduction, this paper includes a section on related works for various levels of disaggregation in the day-ahead load forecasting field, along with our contributions. Section 2 presents the materials and methods employed. After that, Section 3 presents the methodology process. The results are shown in Section 4, providing a detailed comparison of the load forecasting between our proposal and its determinants. The conclusion is presented in Section 5, along with limitations and future recommendations.

1.1. Related Works

The load forecasting models in the literature, as outlined in [12], can be categorized into three groups: (i) Average Method: involves extrapolating time series data using simple moving average and exponential smoothing techniques. (ii) Mathematical Model-based method: uses statistical techniques such as regression and autoregressive methods to forecast demand by establishing a relationship between load consumption and variables that can affect demand, such as weather, consumer behavior, and occupancy. (iii) Artificial Intelligence method: learns consumption patterns from historical load data and is commonly used to discover nonlinear relationships between inputs and outputs without prior assumptions about the correlation between data. Therefore, based on this categorization, the current state-of-the-art in this field is focused on enhancing forecast accuracy at various levels of disaggregation using robust methods that propose an efficient utilization of all available natural resources to achieve an optimal supply schedule.

In the Average Method approach, the moving average and exponential smoothing techniques are simple forms to extrapolate the time series data for forecasting problems, as observed in the literature [13,14,15,16]. In recent research, structural combinations that use the average method as base models have been proposed to outperform competitive benchmarks [17,18]. Therefore, model uncertainty, data uncertainty, and parameter uncertainty are associated with the model base to guarantee excellent performance.

Mathematical Model-based methods are also applied in load forecasting studies, such as simple and multiple linear regression models [19,20,21,22], multivariate linear regressions [23,24] such as multivariate adaptive regression splines (MARS), and autoregressive models [25,26,27]. All these mathematical methods are extensively used for accuracy in load forecasting problems. One of their highlights compared to average methods is their capacity to establish relationships between load consumption and external variables that can affect demand.

In recent years, there has been a growing demand for load forecasting models that utilize Machine Learning methods applied in diverse load-disaggregation levels [8,28]. This is primarily due to their efficiency and flexibility in discovering non-linear associations between data, which are caused by the electrical system’s evolution, unlike the Average Method and Mathematical Model-based methods.

Among Machine Learning (ML) methods applied to day-ahead load-disaggregation forecasting are Artificial Neural Network [29,30,31], Support Vector Machine (SVM) [32,33], Fuzzy Logic (FL) [34], Ensemble Learning [35,36,37], and Decision Tree [38]. These ML methods offer various advantages in identifying and modeling nonlinear relationships in the context of the evolving electrical system and have demonstrated superior performance compared to traditional approaches as shown in

Table 1. Related Works.

Ref.	Proposals	Dataset	Horizon	Performance Metrics	Contributions
[29]	Artificial Neural Network (ANN)	District building.	Very-short-term Short-term	Mean Absolute Percentage Error (MAPE)	(1) Assessment of various neural networks and learning methods for forecasting load consumption accuracy in a district of buildings. (2) Evaluation of forecast performance depending on the time horizon used for generating the forecast. (3) Examination of forecast performance for single buildings and aggregated buildings.
[30]	Hybrid Convolution Neural Network (CNN) and Gated Recurrent Units (GRU)	Appliances Energy Prediction. Individual Household Electric Power Consumption.	Short-Term	Mean Squared Error (MSE) Root Mean Squared Error (RMSE) Mean Absolute Error (MAE)	(1) Application and comparison of diverse machine learning and deep learning models for prediction accuracy. (2) Development of a hybrid CNN–GRU model for hourly electricity consumption prediction. (3) Evaluation of the proposed model’s high performance on the dataset.
[31]	Combines both bagging and boosting to train Bagged–Boosted ANNs (Bag-BoostNN)	The New England Pool region.	Short-Term	MAPE	(1) Improved short-term load forecasting technique using Bag-BoostNN. (2) Demonstrated reduction in bias and variance using real data, compared to single ANN, Bagged ANN, and Boosted ANN.
[32]	Improved Seagull Optimization Algorithm (ISOA) that optimizes Support Vector Machines (SVM) (ISOA-SVM).	Power plant in eastern Slovakia.	Short-Term	MSE	(1) Constructed a power load forecasting model based on the Improved Seagull Optimization Algorithm (ISOA) that optimizes Support Vector Machines (SVM) (ISOA-SVM). (2) Utilized ISOA to optimize SVM’s internal parameters, addressing the issue of random parameter selection affecting the model’s performance. (3) Proposed three strategies to improve the optimization performance and convergence accuracy of the Seagull Optimization Algorithm (SOA). (4) Developed the Improved Seagull Optimization Algorithm (ISOA) with better optimization performance and higher convergence accuracy. (5) Established a load forecasting model based on ISOA-SVM using Mean Square Error (MSE) as the objective function. (6) Demonstrated better prediction performance and accuracy of the ISOA-SVM model compared to other models, providing guidance for power generation and power consumption planning.
[33]	Hybrid model that integrates the multivariate empirical modal decomposition (MEMD) and adaptive differential evolution (ADE) algorithm with a Support Vector Machine (SVM).	Independent System Operator (ISO) New England (ISO-NE) energy sector	Short-Term	Directional Accuracy (DA) MAPE RMSE Coefficient of determination (R2)	(1) Transition towards hybridization in the approach. (2) Utilized ADE technique for hyperparameters modification/adaption. (3) Implemented innovative performance evaluation criteria.
[34]	Improved complete ensemble empirical mode decomposition with adaptive noise (ICEEMDAN)	Victoria in Australia	Short-Term	The average error (AE) MAE RMSE MAPE Direction Accuracy (DA) Fractional Bias (FB) Theil’s Inequality Coefficient (TIC) Internal Forecasting Coverage Probability (IFCP) Internal Forecasting Average Width (IFAW)	(1) Conducted data cleaning to eliminate noise and mine hidden characteristics using data de-noising methods and FTS. (2) Applied a multi-objective optimization algorithm to optimize ANNs for enhanced forecasting stability and accuracy. (3) Considered seasonal and workday–weekend temporal patterns in the analysis.
[35]	A scalable and flexible deep ensemble learning framework	Irish Commission for Energy Regulation	Short-Term	The pinball loss score The Winkler score	(1) Proposed an end-to-end deep ensemble learning model for probabilistic load forecasting without the need for additional feature extraction and selection; well-suited for distributed computing and large-scale industry applications. (2) Formulated a LASSO-based quantile forecast combination strategy for the deep ensemble learning model to elevate performance by refining individual forecasts.
[36]	Fuzzy-based ensemble model that uses hybrid deep learning neural networks	Hellenic interconnected power system The isolated power system of Crete	Short-Term	MAPE MAE RMSE	(1) Developed a novel, hybrid structure for week-ahead load forecasting combining ensemble forecasting, artificial neural networks, and deep learning architectures. (2) Implemented a new Fuzzy clustering method to create an ensemble prediction by clustering input data. (3) Applied a new regression approach to model the load forecasting problem locally for each cluster. (4) Utilized a two-stage approach, involving training a radial basis function neural network (RBFNN) using three-fold cross-validation, followed by using a convolutional neural network (CNN) with the transformed input data. (5) Constructed a neural network with an RBF, a convolutional, a pooling, and two fully-connected layers, trained using the Adam optimization algorithm within the Tensorflow deep learning framework. (6) Designed the model to predict hourly load for the next seven days and evaluated its effectiveness in two different case studies.
[37]	Stacked generalization ensemble-based hybrid Light Gradient Boosting Machine (LGBM), eXtreme Gradient Boosting machine (XGB), and Multi-Layer Perceptron (MLP) (LGBM-XGBMLP)	Power supply industry in the city of Johor.	Short-Term	MAE RMSE R2 Mean Squared Logarithmic Error (MSLE) Median Absolute Error (MdAE) MAPE	(1) Proposed a novel Stacked XGB-LGBM-MLP model to improve overall regression performance. (2) Explored and developed a novel short-term load forecasting (STLF) technique. (3) Conducted a comprehensive comparative analysis of five hybrid optimization (HO) algorithms for STLF. (4) Assessed the proposed technique using two real datasets. (5) Performed a comparative study with recent benchmark techniques.

. Besides, various hybrid methods have been applied to address similar problems. These approaches combine multiple techniques to enhance the performance of load forecasting models. The references mentioned below showcase the application of such hybrid methods in the context of load disaggregation forecasting. Barman and Dev Choudhury [39] proposed an innovative approach to improve short-term load forecasting by incorporating seasonality effects using a season-specific similarity concept (SSSC) based on a firefly algorithm (FA) and SVM. Eseye et al. [40] used a Feedforward Artificial Neural Network (FFANN) based on a Binary Genetic Algorithm and Gaussian Process Regression (BGA-GPR) to forecast aggregate customer classes, which is a decentralized Energy System from Finland. Yan et al. [41] proposed a hybrid ensemble deep learning framework that combines Long Short-Term Memory (LSTM) and Stationary Wavelet Transform (SWT) to accurately forecast energy consumption for individual households. Wang et al. [42] proposed a novel ensemble Hidden Markov Model (e-HMM) framework based on bagging ensemble learning to improve the accuracy of Short-Term Load Forecasting (STLF) for individual industrial customers. Amorim et al. [43] proposed a reverse training concept to Fuzzy-ARTMAP (FAM) ANN, applying it to the global and multinodal load forecasting up to 24 h ahead. Müller et al. [44] proposed a method to improve forecasting electrical load in disaggregated levels up to 24 h ahead by using a combination of singular spectrum analysis (SSA) and FAM ANN. Jin et al. [45] proposed a hybrid model combining system singular spectrum analysis (SSA) and Parallel Long Short-Term Memory (PLSTM) ANN for STLF of residential electricity consumption.

From the related works, it is clear that hybrid methods have been investigated and employed for disaggregated load forecasting. However, only a limited number of studies have specifically focused on adapting ML models for multi-level disaggregated load forecasting. This paper aims to present a versatile approach that can be tailored to various levels of disaggregation.

1.2. Contributions

The paper presents several contributions to multi-level disaggregated load forecasting. Firstly, it introduces SVMFAMM, a hybrid ML framework that provides reliable and accurate load forecasting. The internal parameters of SVMFAMM are set using a cross-validation process applied in FAMM, which generates a geometric region for the best parameters related to multi-level feature disaggregated load data.

In addition, the paper proposes a methodology based on Wavelet Parallel Training between FAMM and SVMFAMM, called WPT-SVMFAMM. This methodology manages and organizes inputs through multi-resolution analysis via Wavelet decomposition, resulting in accurate electricity load-disaggregation forecasting.

To validate the effectiveness of WPT-SVMFAMM for reliable load forecasting, it is applied to various levels of disaggregation using real data acquired from the Peruvian electrical system operator (COES) website, and meteorological data are taken into consideration as input patterns. The paper also assesses the performance of WPT-SVMFAMM compared to FAMM and the Hybrid Method between Least Squares and FAMM (MMQFAMM) for load forecasting models, in terms of RMSE, MSE, MAE, and MAPE for the day-ahead electricity market. Overall, the contributions of this paper provide a valuable framework for improving the accuracy and reliability of multi-level disaggregated load forecasting.

2. Material and Methods

In this section, we will provide a detailed explanation of the data and methods utilized in the present paper.

2.1. Data Description

The publicly available data sources employed in this study include electrical load and meteorological data, gathered from September 2017 to December 2019.

Electrical load data were obtained from the COES’s Daily Operation Evolution Report (IEOD) [46], which includes global loads, northern area, southwest sub-area, and large free users in central and southern areas. The global aggregation level shows predictable weekly patterns, while disaggregated levels introduce fluctuations and complexities.

The National Service of Meteorology and Hydrology of Peru (SENAMHI), a Peruvian public institution, provides real-time meteorological variables on its website [47], such as temperature, humidity, precipitation, and wind speed. In this paper, we utilized temperature and humidity data as exogenous variables due to their degree of correlation with disaggregated electrical loads, unlike precipitation and wind speed data, which show a weaker relationship.

The construction of the input structures in the proposal is based on a mobile window approach [48]. In this sense, the input vector a is defined as follows:

The input for the global, north area, and southwest subarea electrical load levels will be as follows; these inputs consider meteorological values due to their correlation with each other:

$$\mathit{a}={\left[t d_S d_{At} {\mathit{n}}_{\mathbf{1}}{}^T {\mathit{n}}_{\mathbf{2}}{}^T L_{\left(t-3\right)} L_{\left(t-2\right)} L_{\left(t-1\right)} L_{\left(t\right)}\right]}^T, \mathit{a}\in {\mathbb{R}}^m$$

(1)

The input for the electrical load levels of the large free users, large free users of the northern area, and the southern area will be as follows. In this situation, there is no correlation with meteorological data:

$$\mathit{a}={\left[t d_S d_{At} {\mathit{n}}_{\mathbf{2}}{}^T L_{\left(t-3\right)} L_{\left(t-2\right)} L_{\left(t-1\right)} L_{\left(t\right)}\right]}^T, \mathit{a}\in {ℝ}^{m_1}$$

(2)

where $t$—time encoding in 30-min intervals; $d_S$—day of the week; $d_{At}$—coding of normal and atypical days; $L_{\left(t-n\right)}$—load belonging to instantly $t$ − $n$, $n_1$—vector that stores temperature and humidity; $n_2$—vector that stores $L_{\left(t-1334\right)}$; $L_{\left(t-1008\right)}$; $L_{\left(t-672\right)}$;$L_{\left(t-336\right)}$ The output vector b corresponds to the time (t + 1), this represents the future value from 30 min onwards.

$$\mathit{b}=\left[ L_{\left(t+1\right)}\right], \mathit{b}\in {\mathbb{R}}^1$$

(3)

2.2. Wavelet Transform

The Wavelet transform (WT) [49,50,51] enables a multiresolution analysis [52,53], which adapts the signal resolution at various frequencies. It decomposes the signal into different components, changing time and frequency resolution to address signal-processing problems with varying time-frequency resolutions. Thus, the WT is designed to produce high time resolution and low-frequency resolution for high-frequency signals and good frequency resolution for low-frequency signals. The two most used classes of WT are Continuous Wavelet Transform (CWT) and Discrete Wavelet Transform (DWT) [54].

The development of the DWT algorithm led to the theory of multiresolution analysis [52,53], which forms the basis for implementing filter banks [55]. These filters consist of a low-pass and high-pass filter, and when the original signal passes through these filters, the resulting output coefficients are $c_{j,k}$ and $d_{j,k},$ respectively. Decomposing a signal into various levels can be achieved by passing the scaling coefficients obtained in the previous filtering stage through a pair of identical filters, resulting in the coefficients of the next level.

The multiresolution analysis of signals, which uses the orthogonal dyadic Wavelet Transform, is a multi-step WT process. This corresponds to sequences of nested subspaces in the band-pass filters. In this process, the coefficients are divided into approximation (A) and detail (D) coefficients, which are the results of sampling operators such as downsampling in the decomposition process and upsampling in the signal reconstruction [55]. The approximation coefficients represent the high-scale values, corresponding to the low-frequency components of the signal and are associated with the scaling function determined with a low-pass filter. Conversely, the detail coefficients are the low-scale values corresponding to the high-frequency components and are associated with the Wavelet function determined as a high-pass filter.

2.3. Modified Fuzzy ARTMAP

The modified Fuzzy ARTMAP (FAMM) model [56] is a special proposal of the Adaptive Resonance Theory (ART) neural networks [57,58] that provides a rational and economical form to improve the training process. The principles of this proposal are based on a process analogous to how the brain reacts to new knowledge. The brain contains a wide number of inactive neurons, which become active because of learning. This process occurs when new knowledge is required. Additionally, FAMM retains a special property of ART that solves the plasticity/stability dilemma faced by all intelligent systems where the plasticity assumes the new patterns’ capacity without losing the previously acquired knowledge and the stability brings the ability to respond appropriately to irrelevant events [59].

Management of the plasticity/stability dilemma contains three internal modules based on supervised training that provides an associative memory with an auto-regulatory mechanism, as shown in Figure 1. The Fuzzy $AR{T}_a$ and $AR{T}_b$ modules receive arbitrary input data to create stable recognition categories, where $AR{T}_a$ processes the input vector and $AR{T}_b$ processes the expected output vector. The final module, the associative module Inter-ART verifies an active category correspondence between the $AR{T}_a$ and $AR{T}_b$ modules. This module works with an auto-regulator match-tracking mechanism that permits $AR{T}_a$ and $AR{T}_b$ modules to match in a special tuning parameter condition [57].

Moreover, to become a more rational and economical training process, FAMM has an algorithm that fires internal neurons according to the new inputs [56]. The process starts with just a cluster, whose data constitute the first entry data, and during the training process, will be expanding to accommodate new patterns. As a result, the calculation of the activation function is performed for a reduced number of clusters, and for an established vigilance criterion.

The FAMM structure consists of various parameters that require prior tuning. These parameters are inherently linked to the problem and thus have a significant impact on the system’s learning speed, efficiency, and effectiveness [58].

The model’s internal parameters consist of the chosen parameter ($\alpha$), responsible for controlling the search sequence between nodes of layer $F2$. The value of α must be positive $\alpha >0$, according to Carpenter et al. [58]. The training rate ($\beta$), within the interval (0,1], determines the speed at which the weights can adapt in the network. Lower values result in slower learning. The vigilance parameter ($\rho$), within the interval (0,1], controls resonance in the neural network and is related to the number of categories created. Higher values result in more categories, but less generalization of the network, according to Carpenter et al. [58]. The vigilance criterion or analysis of resonance occurrence applies to the ARTa(${\rho }_a$), ARTb(${\rho }_b$), and Inter-ART(${\rho }_{ab}$) modules.

In the following paragraph, we outline the training algorithm for the modified Fuzzy ART [56]. To assist with the explanation, Figure 2 depicts the flowchart utilized during the training of the FAMM, which comprises two Fuzzy ART modules and an Inter-ART.

Step 1: Reading input data:

The following vector represents the input data:

$$\mathit{a}={\left[a_1 a_2 \dots a_M\right]}^T;$$

(4)

$M$-dimensional. The input vector is normalized to avoid the proliferation of categories, as follows:

$$\overline{\mathit{a}}=\frac{\mathit{a}}{\left|\mathit{a}\right|}$$

(5)

where $\overline{\mathit{a}}$—normalized vector.

$$\left|a\right|={\displaystyle \sum }_i^M{a}_i;$$

(6)

Step 2: Input vector encoding:

Complement coding is performed to preserve the breadth of information.

$${\overline{a}}_i^c=1-{\overline{a}}_i;$$

(7)

where vector ${\overline{a}}_i^c$ is complementary to the normalized input vector. In this way, the encoding vector will have 2$M$-dimensional which will be represented by:

$$\mathit{I}={\left[\overline{\mathit{a}}{ }^T {\overline{\mathit{a}}}^c{}^T\right]}^T;$$

(8)

Note that:

$$\left|\mathit{I}\right|={\displaystyle \sum }_i^M{\overline{a}}_i+{\displaystyle \sum }_i^M{\overline{a}}_i^c=M$$

(9)

All input vectors after normalization and coding will have the same magnitude $M$.

Step 3: Activity vector:

$F_2$, activity vector will be represented by the following relationship:

$$\mathit{y} ={\left[ y_1 y_2 \dots y_N\right]}^T;$$

(10)

where $N$—number of categories created in $F2$, so the following relationship is presented.

$$y_j=\left\{\begin{array}{c}\hspace{1em}1,\hspace{1em}If node j of F_2 is active,\hfill \\ \hspace{1em}0,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}Otherwise\hfill \end{array}\right.;$$

(11)

Step 4: Network parameters:

The parameters used in the processing of the modified Fuzzy ART network are:

• Chosen parameters: $\alpha >0$
• Training rate: $\beta \in \left[0,1\right]$
• Vigilance parameter: $\rho \in \left[0,1\right]$

Step 5: Initialization of weights:

Compared to conventional Fuzzy ART, this network starts with only one cluster that contains the data of the first input vector, thus:

$$w_{11}\left(0\right)=I_{11}; w_{12}\left(0\right)=I_{12}; \dots w_{jM}\left(0\right)=I_{1,2M};$$

(12)

This means that there is only one cluster active as the first pattern data.

Step 6: Cluster counter initialization:

$$N=1;$$

(13)

where $N$—indicates the number of active clusters.

Start of Training.

Step 7: Counter initialization:

In order to verify that all clusters have been tested to receive the said entry and, in the event that no existing cluster is able to allow the current entry, a temporary cluster is created to solve this problem, so it was necessary to include a counter:

$$Cont=1;$$

(14)

Each time an entry is presented to the network, this counter is initialized.

Step 8: Calculation of the choice function:

Having vector $\mathit{I}$ in $F_1$, for each node in $F_2$, the choice function is determined by:

$$T_j=\frac{\left|{\mathit{I}}^T\wedge {\mathit{w}}_j\right|}{\alpha +\left|{\mathit{w}}_j\right|};$$

(15)

where $\wedge$—the Fuzzy intersection or conjunction (AND).

Cluster search.

Step 9: Category choice:

The category is chosen to be the node $J$ active, that is:

$$J=\arg \max \left\{T_j:\hspace{1em}j=1, 2, \dots N\right\};$$

(16)

When using this equation, it is possible to find more than one active category; the selected category will be the one with the lowest index.

Step 10: Vigilance test:

The vigilance criterion is represented by:

$$\frac{\left|{\mathit{I}}^T\wedge {\mathit{w}}_j\right|}{\left|{\mathit{I}}^T\right|}\ge \rho ;$$

(17)

If the criterion defined in this equation is satisfied, go to Step 13.

Step 11: Verification that all clusters have been tested.

Checking whether all clusters have already been tested is performed using the equation:

$$N<Cont;$$

(18)

If the defined criterion presented in the equation is not satisfied, go to Step 12. Otherwise, the counter is updated as follows:

$$Con{t}^{new}=Con{t}^{old}+1 ;$$

(19)

Then, the reset takes place. On reset, node $J$ of $F_2$ is excluded from the search process given by $J$ through the following relation:

$$T_j=0;$$

(20)

Then, return to Step 9.

Step 12: Creating a Cluster:

For the case in which relation (18) is not satisfied, it means that none of the already existing clusters supports the current entry. In this way, it is necessary to create a new cluster, so the cluster counter is updated:

$$N^{new}=N^{old}+1;$$

(21)

and

$$J=N;$$

(22)

Therefrom,

$${\mathit{w}}_J={\mathit{I}}^T;$$

(23)

After creating the cluster, go to Step 14.

Step 13: Update of weights (Training):

In this step, the training takes place, the weight vector is updated in this way:

$${\mathit{w}}_J^{new}=\beta \left({\mathit{I}}^T\wedge {\mathit{w}}_J^{old}\right)+\left(1-\beta \right){\mathit{w}}_J^{old} ;$$

(24)

where $J$—active category; ${\mathit{w}}_J^{old}$—updated weight vector; ${\mathit{w}}_J^{new}$—weight vector referring to the previous update.

Step 14: Activity vector:

This activity vector $F_2$ is symbolized by:

$$\mathit{y}={\left[ y_1 y_2 \dots y_N\right]}^T;$$

(25)

In this case, $N$ will be the number of clusters created in $F_2$. In this way, the following relation is presented:

$$y_j=\left\{\begin{array}{c}\hspace{1em}1,\hspace{1em}If node j of F_2 is active,\hfill \\ \hspace{1em}0,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}Otherwise\hfill \end{array}\right.;$$

(26)

2.4. Support Vector Machines

The Support Vector Machine (SVM) is a universal method for solving pattern recognition problems, including nonlinear problems, with good generalization capability, as evidenced in studies such as that of Cortes and Vapnik [60]. This method demonstrated that regression vectors are a valuable tool in regression estimation [61]. The generic estimating function of SVM in regression is expressed with the following equation:

$$\widehat{y}=\widehat{w_0}+{\widehat{\mathit{w}}}^{\mathit{T}}\mathit{x}$$

(27)

where $\widehat{y}$—estimating function; $\widehat{w_0}$—bias scalar; $\widehat{\mathit{w}}$—weight vector; $\mathit{x}$—support vectors.

Therefore, the SVM in the regression process is a useful technique for solving multidimensional function estimation problems [61]. The technique consists of in minimizing the quadratic function of $\mathit{w}$ subject to the linear constraints. The corresponding objective function is usually written with the following equation:

$$J=C{\displaystyle \sum }_{i=1}^N{L}_{\in }\left(y_i,\widehat{y_{ i}}\right)+\frac{1}{2}{\Vert \mathit{w}\Vert }^2$$

(28)

where $J$—objective function; $y_i$—the set of label training patterns;$\widehat{y}$—estimating function; $C$. is a regularization constant. Vapk [62], who rendered the estimation not only robust but also sparse, proposed the epsilon insensitive loss function, defined by:

$$L_{\in }\left(y,\widehat{y}\right)\triangleq \left\{\begin{array}{c}0\hspace{1em}\hspace{1em}se \left|y-\widehat{y}\right|<ϵ\\ \left|y-\widehat{y}\right|-ϵ\hspace{1em} otherwise\end{array}\right.$$

(29)

where $ϵ\ge 0$—epsilon represents an acceptable error margin.

This function is the most used cost function which means that any point lying outside the $ϵ$-insensitive tube is penalized. Therefore, the objective problem represented is convex and unconstrained, but not differentiable as a result of $ϵ$-insensitive. A popular approach is to introduce slack variables to represent the degree to which each point lies outside the tube:

$$J=C{\displaystyle \sum }_{i=1}^N\left({\xi }_i^{+}+{\xi }_i^{-}\right)+\frac{1}{2}{\Vert \mathit{w}\Vert }^2$$

(30)

Subject to

$$y_i\le f\left({\mathit{x}}_{\mathit{i}}\right)+ϵ+{\xi }_i^{+}$$

(31)

$$y_i\ge f\left({\mathit{x}}_{\mathit{i}}\right)-ϵ-{\xi }_i^{-}$$

(32)

$${\xi }_i^{+}>0;{ \xi }_i^{-}>0$$

(33)

where ${\xi }_i^{+}$—points above the $ϵ$-insensitive tube; ${\xi }_i^{-}$—points below the $ϵ$-insensitive tube.

The performance of the generalized algorithm depends on correctly adjusting the regularization parameter C and the epsilon parameter, as well as the parameters related to the kernel [63]. The optimal solution for minimizing the objective function takes the following form:

$$\widehat{\mathit{w}}={\displaystyle \sum }_i{\alpha }_i{\mathit{x}}_{\mathit{i}}$$

(34)

where ${\alpha }_i\ge 0$—is the weight of support vector in the feature space; replacing the solution in the generic estimating function, we obtain the following equation:

$$\widehat{y}=\widehat{w_0}+{\displaystyle \sum }_i{\alpha }_i{\mathit{x}}_{\mathit{i}}{}^{\mathit{T}}\mathit{x}$$

(35)

Finally, by replacing ${\mathit{x}}_{\mathit{i}}{}^{\mathit{T}}\mathit{x}$ with the kernel function, $K\left({\mathit{x}}_{\mathit{i}},\mathit{x}\right)$ we obtain a kernelized solution form:

$$\widehat{y}=\widehat{w_0}+{\displaystyle \sum }_i{\alpha }_i K\left({\mathit{x}}_{\mathit{i}},\mathit{x}\right)$$

(36)

There are basic kernel functions used in SVM [64] such as the Linear, the Gaussian or Radial-Basis Function, and the Sigmoid kernel. The Linear kernel that computes the dot product is expressed by the following equation:

$$K\left({\mathit{x}}_{\mathit{i}},\mathit{x}\right)=\delta \left({\mathit{x}}_{\mathit{i}}.\mathit{x}\right)+k$$

(37)

where $\delta >0$; $k$—these values are constant.

The Gaussian kernel that computes the dot product is expressed by the following equation:

$$K\left({\mathit{x}}_{\mathit{i}},\mathit{x}\right)=\exp \left(-\sigma {\Vert \mathit{x}}_{\mathit{i}}-{\mathit{x}\Vert }^2\right)$$

(38)

where $\sigma >0$—this value is a constant.

The Sigmoid kernel that computes the dot product is expressed by the following equation:

$$K\left({\mathit{x}}_{\mathit{i}},\mathit{x}\right)=tanh\left(\delta \left({\mathit{x}}_{\mathit{i}}.\mathit{x}\right)+k\right)$$

(39)

where $\delta >0;k<0$—these values are constant.

2.5. SVMFAMM Forecasting Framework

To address the challenge of disaggregated multi-level load forecasting, our proposal utilized a hybrid model called SVMFAMM. This forecasting framework combines two methods, which are FAMM and SVM. We used FAMM in the training process to adapt the neuronal network, and the SVM regression in the diagnostic process to adjust the internal categories data and improve the load forecasting.

The training process for SVMFAMM was the same as that used for FAMM, as shown in Figure 3. In this case, exogenous variables and historical loads were used as inputs in the training process using the windows mobile process. After the training was complete, we moved on to the diagnostic process, which focuses on forecasting new load values. This process is important because it allows us to save the best candidates between categories for use in the SVM regression. We iteratively searched and stored the categories created during the diagnostic process for values greater than a cutoff of the vigilance parameter ${\rho }_a$.

Finally, the categories vector was used in the SVM regression process to obtain a generic estimation function. This estimation function uses input data to generate load forecasting. The flowchart for the diagnostic process is shown in Figure 4.

3. Methodology

In summary, the methodology utilized in this paper involved dividing the raw data into smoothed and noise data via Wavelet filters. In a parallel training approach, we used the smoothed data as the training set for FAMM, and the noisy data as the training set for SVMFAMM. The load forecasting results are obtained by superimposing the outputs from both models.

3.1. Wavelet Decomposition

In this step, we utilized the multiresolution analysis methodology, a powerful method for extracting specific features from data. To decompose the electricity load data, we divided the raw data into smoothed and noise components using the Wavelet filter. The smoothed component contains the approximation part, while the noise component contains the detail coefficients up to the third level.

To accomplish this, we opted to use the Wavelet Daubechies in db20, applied to three decomposition levels, due to its ability to form an orthonormal basis, have compact support, and maintain low computational cost [11]. The resulting coefficients were then reconstructed. This procedure is illustrated in Figure 5.

The reconstructed coefficients were stored in vectors and used as inputs for the proposed models of this methodology. It is worth noting that we repeated this process for all six levels of disaggregation proposed in this paper, ensuring that the electricity load data were analyzed in detail at all levels.

To summarize, the Wavelet transform allows for the analysis of time series databases in multiresolution, making it a critical component of this methodology.

3.2. Hybrid FAMM Methodology Implementation Strategy

The hybrid methodology consists of a parallel process of training and testing between the FAMM and SVMFAMM methods, shown in Figure 6. Wavelet decomposition is used to generate inputs based on reconstructed data at various levels of disaggregation, to use them in the parallel process. The approximation data from the Wavelet reconstruction are applied in the training process linked to FAMM, while all the detailed data obtained are added and applied in the training process linked to SVMFAMM. Once the parallel training process is complete, the testing process begins for both methods, in accordance with their respective established procedures.

3.3. Perform Evaluation

To evaluate the performance, in the literature there are several statistical measures [65]. The forecasting results are compared using RMSE, MSE, MAE, and MAPE. Additionally, the Pearson correlation coefficient (PCC) is used to measure the linear correlation between the real load data and the forecasted values [66]:

$$\mathrm{RMSE}=\sqrt{\frac{1}{n}{\displaystyle \sum }_{i=1}^n{\left({{\rm Y}}_i-{\widehat{{\rm Y}}}_i\right)}^2}$$

(40)

$$\mathrm{MSE}=\frac{1}{n}{\displaystyle \sum }_{i=1}^n{\left({{\rm Y}}_i-{\widehat{{\rm Y}}}_i\right)}^2$$

(41)

$$\mathrm{MAE}=\frac{1}{n}{\displaystyle \sum }_{i=1}^n\left|{{\rm Y}}_i-{\widehat{{\rm Y}}}_i\right|$$

(42)

$$\mathrm{MAPE}=\frac{1}{n}{\displaystyle \sum }_{i=1}^n\left|\frac{{{\rm Y}}_i-{\widehat{{\rm Y}}}_i}{{{\rm Y}}_i}\right|\times 100$$

(43)

$$\text{PCC}=\frac{cov\left({\rm Y},\widehat{{\rm Y}}\right)}{{\sigma }_{{\rm Y}}{\sigma }_{\widehat{{\rm Y}}}}$$

(44)

In the above equations, ${\widehat{{\rm Y}}}_i$ represent the forecasted electricity load; ${{\rm Y}}_i$ the real load value. Additionally, $n$ represents the quantity of inputs presented during the diagnosis phase.

4. Results

In this section, we present the results based on the information provided in Section 2 and Section 3, as shown in

Table 2. Information on the methods used for the evaluation process.

Methodologies	ML Methods	Day-Ahead Forecast			Trained Data
		Day	Month	Year
Benchmark 1	FAMM	12 to 18	August	2019	4032
Benchmark 2	MMQFAMM [67]
Proposal	WPT-SVMFAMM (Linear/Gaussian/Optimized)

. The evaluation of the forecasts was carried out for the day-ahead horizon in 30-min intervals, covering forty-eight load values for the day. The methodologies were trained with 4032 items of historical data from 20 May to 11 August 2019. To assess the performance of the established methods in Section 3.3, appropriate metrics were applied. Furthermore, all simulations were conducted on a computer equipped with an Intel(R) Core (TM) i7-8550U CPU @ 1.80 GHz 2.00 GHz and 8.00 GB of RAM using MATLAB software with Wavelet and SVM Toolboxes.

The methodologies in Table 2 used the FAMM training process shown in Figure 2. Internal parameter selection in the training process was important to achieve accurate results; the parameter ${\rho }_a$ was tuned using the cross-validation method $k$-fold [68] equal $5$ in the interval of 0.85 to 0.99 with steps of 0.01. Other parameters were considered fixed as the following values 0.99 for ${\rho }_a$, 0.99 for ${\rho }_{ab}$, 1 for $\beta$, 0.05 for $\alpha$, 0.001 for $\epsilon$. The cutoff in the testing process of SVMFAMM was optimized in the interval of 0.90 to 0.99 with steps of 0.01 for ${\rho }_a$. Furthermore,

Table 3. Computational time for the analysis during the training and diagnostic process.

ML Methods	Day-Ahead Average Computational Time (s)
	Global	North Area	Southwest Sub-Area	Large Free Users (LFU)	Central Area LFU	Southern Area LFU
FAMM	70	77	391	74	63	280
MMQFAMM	124	120	340	120	115	265
WPT-SVMFAMM Linear	150	205	282	53	63	234
Gaussian	152	209	285	54	65	230
Optimized	192	266	458	110	88	277

presents the computational time required for this process that yielded the best result. Detailed insights into the evaluation process and its relation to the computational time for each level of disaggregation will be provided in the following.

Table 4. Evaluation metrics by methods—Global loads (SEIN).

DAY	METRICS	FAMM	MMQFAMM	WPTSVMFAMM Opt	WPTSVMFAMM Linear	WPTSVMFAMM Gauss
		0.93	0.97	0.98A/0.99D0.91	0.98A/0.99D0.90	0.98A/0.99D0.90
12 August 2019	RMSE	110.60	97.98	104.67	105.86	103.65
	PCC	0.99	0.99	0.99	0.99	0.99
	MSE	12,233.32	9600.33	10,955.43	11,205.51	10,742.97
	MAE	83.51	77.92	82.76	83.54	79.79
	MAPE	1.51	1.36	1.45	1.46	1.41
13 August 2019	RMSE	107.88	154.57	108.25	105.30	97.86
	PCC	0.98	0.96	0.98	0.97	0.98
	MSE	11,638.32	23,893.32	11,717.63	11,087.34	9576.81
	MAE	85.07	118.00	87.80	87.10	79.31
	MAPE	1.42	1.94	1.47	1.45	1.32
14 August 2019	RMSE	136.09	143.85	117.46	117.57	122.27
	PCC	0.98	0.97	0.98	0.97	0.98
	MSE	18,521.24	20,693.31	13,797.94	13,822.64	14,951.15
	MAE	112.73	117.12	92.08	93.14	99.78
	MAPE	1.88	1.98	1.54	1.56	1.68
15 August 2019	RMSE	94.03	109.42	85.49	85.23	84.25
	PCC	0.98	0.97	0.98	0.98	0.98
	MSE	8842.22	11,972.21	7309.18	7264.90	7097.56
	MAE	72.96	85.52	70.56	69.84	67.16
	MAPE	1.20	1.40	1.18	1.17	1.12
16 August 2019	RMSE	104.56	92.56	59.31	59.78	63.47
	PCC	0.97	0.98	0.99	0.99	0.99
	MSE	10,932.11	8566.66	3517.12	3573.13	4028.41
	MAE	85.36	76.59	49.42	49.69	50.64
	MAPE	1.41	1.26	0.81	0.81	0.82
17 August 2019	RMSE	71.80	156.16	123.88	126.01	129.44
	PCC	0.98	0.95	0.98	0.98	0.98
	MSE	5155.33	24,385.94	15,345.90	15,878.37	16,754.35
	MAE	53.22	131.39	107.59	109.89	111.61
	MAPE	0.89	2.19	1.78	1.82	1.84
18 August 2019	RMSE	94.34	106.51	98.71	95.76	94.52
	PCC	0.97	0.97	0.97	0.97	0.97
	MSE	8900.91	11,345.20	9742.98	9170.90	8934.71
	MAE	76.60	87.30	79.52	76.35	76.91
	MAPE	1.39	1.60	1.46	1.40	1.41

,

Table 5. Evaluation metrics by methods—North area.

DAY	METRICS	FAMM	MMQFAMM	WPTSVMFAMM Opt	WPTSVMFAMM Linear	WPTSVMFAMM Gauss
		0.96	0.96	0.99A/0.99D0.94	0.99A/0.99D0.90	0.99A/0.99D0.90
12 August 2019	RMSE	38.34	40.32	23.91	24.71	20.22
	PCC	0.96	0.93	0.97	0.97	0.98
	MSE	1470.19	1625.58	571.92	610.70	408.88
	MAE	31.35	33.23	18.76	19.43	15.98
	MAPE	3.93	4.12	2.39	2.47	2.05
13 August 2019	RMSE	45.68	45.13	45.89	46.77	46.79
	PCC	0.91	0.87	0.92	0.92	0.93
	MSE	2086.79	2036.37	2106.16	2187.40	2189.29
	MAE	36.38	33.28	37.42	38.21	38.25
	MAPE	4.26	3.86	4.30	4.39	4.40
14 August 2019	RMSE	31.51	40.10	38.85	39.21	36.54
	PCC	0.95	0.86	0.93	0.93	0.94
	MSE	992.78	1607.84	1509.60	1537.39	1335.07
	MAE	26.06	28.17	31.13	31.50	29.20
	MAPE	3.04	3.24	3.53	3.58	3.32
15 August 2019	RMSE	34.70	39.07	24.73	24.18	24.31
	PCC	0.95	0.90	0.95	0.95	0.95
	MSE	1203.99	1526.44	611.64	584.73	590.76
	MAE	25.44	31.17	17.62	17.71	17.82
	MAPE	3.08	3.66	2.10	2.11	2.13
16 August 2019	RMSE	23.93	27.96	25.46	24.21	23.05
	PCC	0.96	0.93	0.95	0.95	0.96
	MSE	572.85	781.96	648.41	585.98	531.42
	MAE	18.39	20.97	20.11	19.43	18.92
	MAPE	2.14	2.39	2.32	2.24	2.20
17 August 2019	RMSE	36.50	37.78	26.18	25.01	23.66
	PCC	0.96	0.92	0.95	0.95	0.96
	MSE	1331.99	1427.54	685.17	625.63	559.93
	MAE	30.29	28.24	21.13	19.90	19.25
	MAPE	3.55	3.25	2.45	2.31	2.25
18 August 2019	RMSE	47.88	51.71	33.61	33.93	31.17
	PCC	0.94	0.90	0.94	0.94	0.95
	MSE	2292.74	2673.43	1129.43	1151.01	971.86
	MAE	38.77	39.18	25.43	25.21	23.87
	MAPE	4.65	4.63	3.03	3.02	2.86

,

Table 6. Evaluation metrics by methods—Southwest sub-area.

DAY	METRICS	FAMM	MMQFAMM	WPTSVMFAMM Opt	WPTSVMFAMM Linear	WPTSVMFAMM Gauss
		0.99	0.99	0.99A/0.99D0.90	0.99A/0.99D0.97	0.99A/0.99D0.90
12 August 2019	RMSE	52.70	50.91	45.94	45.89	48.12
	PCC	0.73	0.76	0.82	0.82	0.80
	MSE	2777.79	2592.17	2110.79	2105.83	2315.67
	MAE	43.41	39.97	35.34	35.29	36.63
	MAPE	3.67	3.40	3.00	3.00	3.10
13 August 2019	RMSE	58.58	56.39	55.88	56.79	55.96
	PCC	0.19	0.01	0.48	0.49	0.46
	MSE	3431.62	3180.35	3122.73	3225.57	3131.07
	MAE	47.77	44.58	47.36	48.53	47.73
	MAPE	4.04	3.77	4.00	4.09	4.03
14 August 2019	RMSE	58.24	56.03	57.11	58.18	55.90
	PCC	0.24	0.22	0.18	0.20	0.20
	MSE	3391.74	3139.00	3261.45	3384.44	3124.71
	MAE	50.19	48.52	50.44	50.61	48.39
	MAPE	4.38	4.23	4.40	4.42	4.23
15 August 2019	RMSE	108.97	103.43	119.86	120.57	121.14
	PCC	0.24	0.26	0.22	0.18	0.17
	MSE	11,874.81	10,697.21	14,366.61	14,535.96	14,673.70
	MAE	77.39	73.51	97.02	96.63	97.40
	MAPE	7.25	6.90	9.02	8.99	9.06
16 August 2019	RMSE	63.04	55.94	62.33	62.15	62.17
	PCC	0.77	0.80	0.82	0.82	0.80
	MSE	3973.52	3129.39	3885.61	3862.08	3864.70
	MAE	52.88	46.86	54.44	54.47	53.54
	MAPE	4.48	3.97	4.61	4.61	4.54
17 August 2019	RMSE	44.16	39.46	31.30	31.24	31.36
	PCC	0.57	0.59	0.83	0.83	0.83
	MSE	1950.10	1556.93	979.68	975.83	983.43
	MAE	37.03	33.55	26.81	26.68	26.78
	MAPE	3.11	2.81	2.23	2.22	2.22
18 August 2019	RMSE	68.21	66.81	35.59	36.00	35.90
	PCC	0.28	0.10	0.47	0.48	0.47
	MSE	4652.06	4463.64	1266.36	1295.65	1288.71
	MAE	56.40	53.95	28.96	29.78	29.42
	MAPE	4.93	4.73	2.50	2.57	2.54

,

Table 7. Evaluation metrics by methods—Large free users (Global).

DAY	METRICS	FAMM	MMQFAMM	WPTSVMFAMM Opt	WPTSVMFAMM Linear	WPTSVMFAMM Gauss
		0.92	0.97	0.92A/0.99D0.90	0.92A/0.99D0.90	0.92A/0.99D0.97
12 August 2019	RMSE	79.65	110.09	79.76	81.00	82.67
	PCC	0.87	0.65	0.78	0.78	0.80
	MSE	6344.24	12,118.91	6361.46	6560.81	6835.11
	MAE	61.14	93.24	59.73	60.58	58.83
	MAPE	3.10	4.73	3.01	3.05	2.96
13 August 2019	RMSE	86.68	88.16	83.55	84.38	73.06
	PCC	0.85	0.82	0.81	0.81	0.86
	MSE	7513.26	7772.86	6981.36	7120.61	5337.67
	MAE	61.86	66.75	63.81	63.54	59.01
	MAPE	3.11	3.39	3.24	3.22	2.98
14 August 2019	RMSE	104.35	104.33	59.68	60.22	58.41
	PCC	0.65	0.61	0.86	0.86	0.87
	MSE	10,889.54	10,885.64	3561.63	3626.22	3412.13
	MAE	86.89	82.11	47.20	47.64	47.13
	MAPE	4.63	4.45	2.50	2.52	2.48
15 August 2019	RMSE	88.44	91.25	77.99	79.22	81.40
	PCC	0.79	0.78	0.79	0.79	0.77
	MSE	7822.02	8326.97	6081.93	6275.57	6625.84
	MAE	64.01	72.91	60.64	62.20	62.37
	MAPE	3.34	3.86	3.18	3.26	3.27
16 August 2019	RMSE	87.84	74.55	79.93	76.75	81.46
	PCC	0.87	0.86	0.89	0.89	0.90
	MSE	7715.70	5557.55	6388.44	5889.87	6635.45
	MAE	75.21	60.12	66.99	63.83	69.96
	MAPE	3.88	3.02	3.39	3.22	3.54
17 August 2019	RMSE	81.23	86.19	91.00	90.67	84.17
	PCC	0.70	0.66	0.57	0.58	0.66
	MSE	6599.12	7429.01	8281.00	8220.69	7084.54
	MAE	61.88	62.77	70.03	70.07	64.30
	MAPE	3.11	3.28	3.63	3.63	3.33
18 August 2019	RMSE	61.10	49.09	52.79	56.99	56.88
	PCC	0.50	0.57	0.56	0.54	0.57
	MSE	3733.37	2410.21	2786.54	3247.58	3234.86
	MAE	48.01	40.79	44.16	47.50	48.20
	MAPE	2.31	1.96	2.12	2.28	2.31

,

Table 8. Evaluation metrics by methods—Large free users of the central area.

DAY	METRICS	FAMM	MMQFAMM	WPTSVMFAMM Opt	WPTSVMFAMM Linear	WPTSVMFAMM Gauss
		0.93	0.97	0.93A/0.99D0.90	0.93A/0.99D0.93	0.93A/0.99D0.99
12 August 2019	RMSE	67.55	108.92	52.43	53.45	46.69
	PCC	0.83	0.78	0.90	0.89	0.92
	MSE	4563.15	11,864.41	2748.95	2856.94	2179.92
	MAE	47.45	94.49	44.05	43.38	38.07
	MAPE	5.80	11.49	5.50	5.37	4.73
13 August 2019	RMSE	68.34	177.03	56.61	56.80	51.45
	PCC	0.82	0.45	0.82	0.82	0.85
	MSE	4670.64	31,338.28	3204.34	3226.61	2647.07
	MAE	47.39	159.98	48.41	48.47	42.22
	MAPE	5.96	19.43	6.07	6.10	5.38
14 August 2019	RMSE	74.19	83.74	64.51	65.14	67.26
	PCC	0.79	0.75	0.85	0.85	0.85
	MSE	5504.00	7011.69	4161.09	4242.72	4524.39
	MAE	60.47	72.22	52.73	53.18	55.16
	MAPE	8.19	9.86	7.37	7.44	7.65
15 August 2019	RMSE	75.22	95.21	67.05	67.43	68.23
	PCC	0.68	0.68	0.70	0.71	0.71
	MSE	5657.31	9065.31	4495.28	4547.31	4655.71
	MAE	61.53	78.56	54.78	55.32	55.63
	MAPE	8.07	10.51	7.15	7.23	7.23
16 August 2019	RMSE	67.09	233.96	74.03	75.68	73.16
	PCC	0.80	0.71	0.86	0.86	0.88
	MSE	4501.46	54,736.32	5480.10	5728.02	5351.75
	MAE	52.55	223.80	62.13	63.85	61.73
	MAPE	7.04	29.42	7.95	8.17	8.02
17 August 2019	RMSE	66.75	181.29	75.94	77.03	69.66
	PCC	0.70	0.41	0.69	0.71	0.73
	MSE	4455.70	32,867.84	5767.22	5933.83	4852.88
	MAE	52.41	163.36	61.53	64.49	55.18
	MAPE	7.12	20.95	8.15	8.40	7.25
18 August 2019	RMSE	72.11	138.44	60.83	63.36	63.28
	PCC	0.06	−0.05	−0.14	−0.06	−0.07
	MSE	5200.44	19,166.67	3700.12	4014.96	4004.97
	MAE	56.70	123.39	49.42	51.09	49.73
	MAPE	6.72	14.17	5.81	6.02	5.84

and

Table 9. Evaluation metrics by methods –Large free users of the southern area.

DAY	METRICS	FAMM	MMQFAMM	WPTSVMFAMM Opt	WPTSVMFAMM Linear	WPTSVMFAMM Gauss
		0.99	0.99	0.99A/0.99D0.91	0.99A/0.99D0.93	0.99A/0.99D0.93
12 August 2019	RMSE	83.71	82.38	72.54	72.38	72.67
	PCC	−0.10	−0.15	−0.03	−0.03	−0.01
	MSE	7007.12	6787.23	5262.14	5239.26	5280.70
	MAE	70.35	69.31	55.87	55.76	55.92
	MAPE	7.02	6.88	5.61	5.59	5.61
13 August 2019	RMSE	68.30	66.73	59.96	59.99	62.15
	PCC	0.70	0.70	0.76	0.77	0.75
	MSE	4665.28	4452.95	3595.45	3598.50	3862.63
	MAE	55.99	54.38	45.77	45.82	47.92
	MAPE	5.68	5.51	4.66	4.67	4.88
14 August 2019	RMSE	86.13	66.99	51.33	51.63	50.85
	PCC	0.09	0.34	0.49	0.49	0.50
	MSE	7418.55	4487.51	2634.77	2666.15	2585.51
	MAE	63.39	51.11	44.04	44.21	43.37
	MAPE	6.19	5.00	4.48	4.50	4.42
15 August 2019	RMSE	55.96	55.75	54.33	54.09	54.99
	PCC	0.61	0.60	0.62	0.62	0.61
	MSE	3131.54	3107.81	2951.32	2926.22	3024.07
	MAE	37.14	37.83	38.00	37.98	38.81
	MAPE	3.79	3.86	3.87	3.86	3.96
16 August 2019	RMSE	50.87	46.24	49.31	49.35	49.44
	PCC	0.45	0.54	0.50	0.50	0.51
	MSE	2587.48	2138.16	2431.76	2435.45	2443.85
	MAE	42.36	38.54	41.91	41.72	41.82
	MAPE	4.13	3.76	4.07	4.05	4.06
17 August 2019	RMSE	20.41	18.11	22.61	23.02	22.54
	PCC	0.72	0.72	0.53	0.54	0.55
	MSE	416.47	327.84	511.43	530.07	508.21
	MAE	17.19	14.67	17.51	17.69	17.73
	MAPE	1.62	1.38	1.66	1.67	1.68
18 August 2019	RMSE	65.31	62.62	65.86	66.85	67.71
	PCC	−0.25	−0.32	−0.16	−0.15	−0.16
	MSE	4264.99	3921.36	4337.85	4468.56	4584.65
	MAE	51.22	48.56	52.70	53.80	54.61
	MAPE	4.97	4.72	5.11	5.21	5.29

provide a comparison of the results obtained for diverse levels of load forecasting. For each comparison, the best result obtained from tuned vigilance parameters is presented. The vigilance parameters ${\rho }_a$ are indicated in the header of all tables.

Table 4 presents the result of the global load forecast results, which demonstrate that the WPT-SVMFAMM with a Gaussian kernel outperforms the benchmark methods. The weekly average of MAPE improved slightly, with a decrease of 1% compared to FAMM and a drop of 18% compared to MMQFAMM. In daily comparisons, our proposed method exhibited acceptable accuracy for 12–16 August, with a maximal reduction of 42% in terms of MAPE when compared to FAMM. Furthermore, the daily comparison between our method and MMQFAMM showed a high performance for 13–18 August, with an up to 35% drop in terms of MAPE.

On the other hand, when comparing the computational time between methods for this level, Table 3 indicates that the FAMM method requires less effort compared to the others, due to a non-exhaustive search for patterns within the data related to the vigilance parameter.

The results presented in Table 5 demonstrate the superior performance of WPT-SVMFAMM with a Gaussian kernel over the benchmark methods for load forecasting in the north area. The weekly average of MAPE indicated a slight improvement, with a 22% decrease compared to FAMM and a 24% drop compared to MMQFAMM. Daily comparisons showed that our proposed method achieved acceptable accuracy for 12, 15, 17, and 18 August, with a maximal reduction of 48% in terms of MAPE compared to FAMM. Furthermore, the daily comparison between our method and MMQFAMM demonstrated a remarkable performance for 12 August and the period of 15–18 August, with an up to 50% drop in terms of MAPE.

Alternatively, when examining the computational time for this level among different methods, Table 3 reveals that the FAMM method demands less effort compared to the others. This is a result of a less comprehensive search for patterns in the data, which is related to the vigilance parameter.

Table 6 presents the results of the load forecast for the southwest sub-area, highlighting the superior performance of WPT-SVMFAMM using a Gaussian kernel over the benchmark methods. The weekly average of MAPE showed a slight improvement, with a 7% decrease compared to FAMM. In the daily comparisons, our proposed method exhibited acceptable accuracy for 12, 14, 17, and 18 August, achieving a maximal reduction of 49% in terms of MAPE compared to FAMM. Additionally, the daily comparison between our method and MMQFAMM demonstrated remarkable performance for 12, 17, and 18 August, with a drop of up to 46% to the MAPE metric.

On the other hand, when assessing the computational cost of the methods, Table 3 reveals that the computational effort of our WPT-SVMFAMM Linear and Gaussian proposal surpasses the benchmark methods. In this specific analysis, the methods were assessed using closely related high vigilance parameters, which led to longer processing times.

Table 7 highlights the superior performance of WPT-SVMFAMM with a Gaussian kernel over the benchmark methods in the load forecast for the north area. The weekly average of MAPE showed a slight improvement, with an 11% decrease compared to FAMM and a 15% drop compared to MMQFAMM. Regarding daily comparisons, our proposed method exhibited acceptable accuracy for 12–16 August, achieving a maximal reduction of 46% in terms of MAPE compared to FAMM. Moreover, the daily comparison between our method and MMQFAMM showed a remarkable performance for 12–15 August, with an up to 44% drop in terms of MAPE.

On the other hand, when assessing the computational cost of the methods, Table 3 reveals that the computational effort of our WPT-SVMFAMM Linear and Gaussian proposal outperforms the benchmark methods. In this analysis, the MMQFAMM method exhibited a high computational cost due to its direct relationship with the vigilance parameter, which resulted from an exhaustive search for characteristic patterns. In contrast, the other methods, with lower parameters, were able to significantly improve their speed in the training and diagnostic process.

In Table 8, the load forecast results for large free users in the central area are presented, showing the superior performance of WPT-SVMFAMM with a Gaussian kernel over the benchmark methods. The weekly average of MAPE demonstrated a slight improvement, with a 6% decrease compared to FAMM and a 60% drop compared to MMQFAMM. For daily comparisons, our proposed method achieved acceptable accuracy during 12–15 August and 18 August, with a maximal reduction of 19% in terms of MAPE compared to FAMM. Furthermore, the comparison between our method and MMQFAMM showed a remarkable performance during 12–18 August, with an up to 73% drop in terms of MAPE.

Alternatively, when evaluating the computational costs of the various methods, Table 3 demonstrates that our WPT-SVMFAMM Linear and Gaussian approach surpasses the benchmark methods in terms of computational effort. In this analysis, the MMQFAMM method displayed a high computational cost, stemming from its close connection with the vigilance parameter and the exhaustive search for distinctive patterns. On the other hand, the remaining methods, possessing lower parameters, managed to enhance their speed during the training and diagnostic phases.

The results of the load forecast for large free users in the southern area are presented in Table 9, highlighting the superior performance of WPT-SVMFAMM using an optimization process over the benchmark methods. The weekly average of MAPE showed a slight improvement, with a decrease of 11% compared to FAMM and a drop of 5% compared to MMQFAMM. Regarding daily comparisons, our proposed method showed acceptable accuracy for the period of 12–14 August and 16 August, achieving a maximal reduction of 28% in terms of MAPE compared to FAMM. Furthermore, the daily comparison between our method and MMQFAMM showed a remarkable performance for the period of 12–14 August, with an up to 19% drop in terms of MAPE.

In contrast, when examining the computational expenses associated with the different methods, Table 3 shows that the computational effort required by our WPT-SVMFAMM Linear and Gaussian approach shows better performance than the benchmark methods. In this specific analysis, the methods were assessed using closely related high vigilance parameters, which led to longer processing times.

The load curves exhibit a high degree of correlation with the real data, enabling us to observe that WPT-SVMFAMM provides better visual agreement with the actual values. Figure 7 also indicates that the FAMM method demonstrates greater fluctuations, exceeding the real values, thus raising doubts about the system’s response to unusual peaks during certain periods.

In Figure 8, our proposed method outperformed all benchmark methods in terms of curves related to Pearson correlation. All benchmark methods exhibited large fluctuations in load forecasts in daily profile curves. Additionally, the MMQFAMM methods in the central area lost their ability to generate stable results due to insufficient data in the storage vector linked to the diagnostic process. In summary, our proposed methodology, which employs a Gaussian kernel, allows for more accurate tracking of the curve in the day-ahead horizon in 30-min intervals.

Additionally, the statistical significance of the error measures shown in Table 2, Table 3, Table 4, Table 5, Table 6 and Table 7 was verified through a Diebold–Mariano (DM) Test [69]. The results of the test for the best-performing method are shown in Figure 9, which represents the DM test statistic value. A 5% significance level was used in this test, and the corresponding critical values are 1.96. The interpretation of these DM values is carried out using the following intervals: a DM test $<-1.96$ implies significance outperformance, a DM test $>1.96$ implies significance underperformance, and no significance is observed for out- or underperformance otherwise. These results are visually represented in Figure 9, where green signifies both outperformance and underperformance, while gold indicates a lack of significant difference in performance.

The test process considered the results of the weekly projection carried out on a 24-h day-ahead forecasting horizon, with 30-min intervals. According to Figure 9, the proposed methodology, in general, has an acceptable level of outperformance compared to the FAMM and MMQ-FAMM methods. However, in the southwest sub-area, the statistical test does not guarantee a significant performance difference between the compared methods because it is not within the critical value.

5. Conclusions, Limitations, and Future Direction

This paper proposes a hybrid ML methodology that combines FAMM and SVMFAMM models in a parallel process to address the multi-level disaggregation load forecasting problem. Our proposed methodology achieved the best results by using approximation and detailed data from the Wavelet reconstruction process, outperforming the benchmark methods established for day-ahead forecasting in a weekly window. The complexity of the data presents a challenge for modeling due to the variability in user load requirements at various levels of disaggregation and the occurrence of atypical operations in the electrical power system. To address this challenge, we reconstructed the historical data using Wavelet filters, resulting in a flexible electrical forecasting model that can forecast the electrical loads at various levels of disaggregation for the following day. We evaluated our proposed methodology against two benchmarks using FAMM and MMQFAMM models at all the levels of disaggregation established. We carefully analyzed the internal parameters of the models via cross-validation and used various performance evaluation metrics such as RMSE, PCC, MSE, MAE, and MAPE to assess the results of the forecasts; additionally, the DM Test was used for comparing methods.

Addressing concerns about the computational effort and performance of the proposed implementation, our methodology demonstrates a relatively low computational cost. This is achieved through a combination of efficient ML techniques and a parallel processing approach, allowing for improved performance without adding significant computational overhead. Furthermore, the flexible nature of the methodology allows for the easy modification of internal parameters, enabling optimization of the trade-off between computational effort and forecasting accuracy. The results of our study indicate that the proposed approach not only achieves high accuracy across various time intervals but also maintains a manageable computational burden, making it a suitable solution for electrical load forecasting in fluctuating contexts. After comparing and analyzing the results, as well as the computational effort, the proposed methodology utilizing the Gaussian kernel was determined to be the most efficient and flexible forecasting model that can effectively adapt to the various levels of electrical load disaggregation in this study.

In summary, this paper contributes to improving day-ahead forecasting for a weekly window at various levels of disaggregation, identifying crucial factors, developing pre-processing data methodologies, and evaluating methodologies for diverse kernels and benchmarks. Our proposed methodology can be applied to other electrical systems to improve operational efficiency and reduce associated costs.

Although the methodology proposed in this paper has achieved satisfactory load forecasting results, there are still some limitations that need to be addressed. First, the optimal selection of internal parameters to improve performance requires further study. Second, additional research is required to determine the optimal selection of exogenous variables and the appropriate quantity of past electrical load data for the input. Third, the evaluation of various Wavelet functions in the filter should be explored. Finally, the prediction ability of the methodology needs to be further strengthened and its universality tested.

As future recommendations, we suggest the following: The current study does not consider the optimal permutation of input values between exogenous variables that can potentially improve the performance of our methodology. Investigating this effect could be a future research direction. Additionally, while our methodology employs the FAMM method, there are various other Fuzzy ART methods in the literature that can be hybridized with SVM to further enhance our proposed methodology.