Forecasting Short-Term Electricity Load with Combinations of Singular Spectrum Analysis.

Accurate electricity demand forecasting can provide a timely and effective reference for economic control and facilitate the secure production and operation of power systems. However, electricity data are well known for their nonlinearity and multi-seasonal features, making it challenging to construct forecasting models. This study investigates the combination of singular spectrum analysis to facilitate the construction of decomposition-based forecasting approaches for electricity load. First, we demonstrate and emphasize the importance of separability for specifically extracting different features hidden in the original data; moreover, only by using the separable feature subseries, the constructed individual model can capture the inner and distinct characteristics of original series more effectively. Second, this study decomposes the electricity load into several significant features using singular spectrum analysis. Each feature series is predicted separately to construct aggregate results. In particular, we propose SSA-based period decomposition to not only perform separable decomposition but also overcome the border effect, which has received little attention in previous work. Finally, to verify the effectiveness of the proposed method, we conduct an empirical study and compare the performance of the discussed models. The empirical results show that the proposed approach can obtain the expected forecasting performance and is a reliable and promising tool for extracting different features. © King Fahd University of Petroleum & Minerals 2022.

Introduction

Electricity load demand is an important indicator for enterprise operations and management situations. Therefore, electricity load forecasting, especially short-term load forecasting, can provide a timely and effective reference for not only macro-economy control but also the secure production and operation of power systems. For example, during the COVID-19 pandemic, the production and operation recovery of micro-, small-, and medium-sized enterprises present a great deal of uncertainty, causing an enormous challenge for the production, supply, and operational security of power systems. In this context, effective and accurate load forecasting is especially necessary and urgent.

The natural non-storability of electricity is a distinct characteristic compared to other commodities [1], making it important to balance demand and supply. An alternative solution to achieve this purpose is to perform accurate forecasting for short-term inelastic demand, providing timely and valuable information for the decision-making of generator firms. However, electricity data are well-known for their nonlinearity and multi-seasonal features (e.g., daily cycles due to differences in consumption during the morning, afternoon, and overnight), resulting in considerable uncertainty in the construction of forecasting models. To improve forecasting effectiveness, in recent decades, forecasting approaches for electricity load have been developed from simple to complex approaches. For example, the seasonal autoregressive moving average (ARMA) model is always applied to obtain the seasonality feature of a load series owing to the limits of ARMA [2]. In particular, owing to the limited capability of traditional statistical models in capturing the nonlinear relationships among features, many intelligent computational methods such as artificial neural networks (ANNs) [3, 4] and support vector machines (SVMs) [5, 6] have been increasingly used by researchers and practitioners and widely applied for electricity load forecasting. However, for both intelligent computational methods and traditional statistical models, one concern is whether the constructed model can deal with the seasonal features of the electricity load series. Kolarik and Rodorfer [7], Nelson et al. [8], and Zhang and Qi [9] showed that ANNs have a limited capability to model the seasonality of time series, indicating that the seasonal effect should be tackled through preprocessing analysis. In contrast, Franses and Draisma [10] and Alon et al. [11] confirmed that ANNs can obtain the seasonal mode hidden in series and generate better performance than traditional statistical models. However, no one model is appropriate for all forecasting problems. Computational intelligent methods have been increasingly used in recent research for short-term electricity load forecasting because of their powerful capability for handling nonlinearity. For example, Ding et al. [12] employed a relevance vector machine (RVM) and wavelet transform to predict hour-ahead and day-ahead loads. Jiang et al. [13] developed an optimized SVM for electricity demand forecasting. In particular, computational intelligent models do not focus on the potential interactions between the target and predictor variables; a modeling mechanism minimizes the forecasting error by assigning input vectors to the target.

It is generally recognized that there is no single forecasting method suitable for all situations, and that each method has strengths and weaknesses in solving forecasting problems. Therefore, a highly appreciated idea is to integrate different methods, such as data processing techniques, forecasting models, ensemble methods, and optimization algorithms, to alleviate the resulting uncertainty by complex data. In this regard, load forecasting models can be roughly divided into two categories: combination and hybrid approaches.

The key idea of combination methodologies is that different forecasting models that lead to single forecasts can be integrated to generate more accurate forecasting using appropriate combination weights. It is noteworthy that such a combination should highlight the diversity in single forecasting to obtain distinct time series information [14, 15]. The weight-based combination method is another important factor that affects the forecasting performance of combination models. Nowotarski et al. [16] compared and discussed 11 methods for combining weights for short-term load forecasting, and their study acknowledged that simple and trimmed averaging (TA) presents the best and most robust forecasting.

Compared to combination methodologies, hybrid approaches focus on the potential benefits of different techniques for forecasting effectiveness. For instance, to overcome the instability of machine learning-based methods, intelligent optimization algorithms (IOA) or optimization mechanisms are widely used to tweak the hyperparameters [17], generating the global optimization learned structure of the forecasting method. With the development of the big data era, many IOA algorithms have been proposed and are increasingly commonly applied to hybrid models for load forecasting, such as genetic [18], firefly [19], and cuckoo search algorithms [20]. However, the capability of such hyperparameter tuning-based strategies is also limited to the complex nonlinear mode of data series, and thus, it is difficult to further improve the generalization of forecasting models. To solve this problem, the data-driven concept can be adaptively integrated into the hybrid forecasting framework. In [21], wavelet transforms were employed to decompose the electricity load series into several subseries. Each subseries was used separately to construct individual forecasters. Bento et al. [22] developed a hybrid short-term load forecasting model that integrates a neural network, wavelet transform, and bat algorithm, which includes both an optimization mechanism and data processing. Wu et al. [20] employed fast ensemble empirical mode decomposition (FEEMD) to deal with multiple seasonal patterns of load series, decreasing the uncertainty resulting from mixtures of different features in the original data. Kong et al. [23] proposed an error correction strategy using dynamic mode decomposition (DMD) for short-term load forecasting, and the key role of DMD is to capture the trend feature.

Based on the current work, a typical hybrid approach generally consists of data decomposition, single forecasting, and a result ensemble. Accordingly, data-driven data decomposition procedures are used to extract different features hidden in the original series, thereby constructing an individual forecasting model [24, 25]. By doing so, a mixture of different modes can be avoided. However, in decomposition-based forecasting, a common and neglected problem is the boundary effect resulting from the missing data at the right end of the observed series, which has a detrimental influence on individual forecasting owing to the intervention of the missing data in the input of the forecast. To address the boundary effect, an alternative and effective solution is to extend the series to substitute for missing observations [26]. For instance, to provide missing data to overcome the edge problem, Rana and Koprinska [27] constructed a neural network-based forecasting method to perform signal extension. Bessec and Fouquau [21] proposed several rules to extend series when investigating combinations with stationary wavelet transforms.

In this study, we focused on the development of a hybrid framework for short-term load forecasting. The proposed approach is based on decomposition-aggregate forecasting. In particular, for decomposition-based forecasting, this study investigates the combinations of singular spectrum analysis to provide effective decomposition. The contributions of this study are as follows:

Discussing and highlighting the importance of separability for decomposition-aggregate forecasting. This study develops a decomposition-based forecasting framework for short-term load. In particular, we emphasize the importance of the separability of the decomposed modes for decomposition-aggregate forecasting, which indicates that separable decomposition is the key factor in effectively implementing decomposition-based forecasting, alleviating the resulting complexity owing to the mixture of different modes hidden in the original data.

Proposing SSA for the decomposition-aggregate load forecasting concerning separability. According to [28], the separability between the decomposed features can be measured by -correlation, and then, the zero -correlation means that the corresponding series are separable and almost w-orthogonal. We propose separability for decomposition-based forecasting considering the multi-seasonal features of the electricity demand load series.

Developing the SSA-based period decomposition method for decomposition-aggregate forecasting of electricity load. It is worth noting that the proposed method can perform separable decomposition using the combination of basic SSA. More importantly, a remarkable effect of SSA-based period decomposition is that it can overcome the influence of the resulting border effect on the forecasting results.

The rest of this paper is organized as follows. Section 2 details the relative and developed methodology. Section 3 describes several benchmark models and presents SSA-based approaches for comparative purposes. Section 4 describes the forecasting implementation and provides an analysis of the results. Finally, the conclusions are presented in Sect. 5.

Methodology

Formulation of Decomposition-Aggregate Load Forecasting

The key idea of decomposition-aggregate (DA) forecasting is to alleviate the complexity resulting from the mixture of different modes hidden in the original data. As a result, the performance of the constructed decomposition-based models depends on the appropriate and effective extraction of independent modes. For the electricity load series, despite their complex nonlinearity, multi-seasonal (e.g., daily, weekly, and yearly) features provide an alternative solution for DA forecasting. To better understand DA forecasting, we consider the following decomposition of the electricity load series:where , ,, and represent the weekly trend, daily trend, season, and residual modes, respectively. The decomposition presented in Eq. (1) is built on prior knowledge; however, there is no forceful evidence to facilitate the decomposition form to improve forecasting effectiveness. Considering Eq. (1) for the real electricity load series, we can build different one-step-ahead forecasting models aiming at different modes, as follows:where and are the structure to be learned and the unknown parameters to be settled, respectively, for the ith mode; is the lag order selected to determine the dimensions of the input vector; and can be determined using historical information. It is worth noting that different models for different modes should be considered to improve the effectiveness of aggregate forecasting. Let be the estimation of ; the aggregate result can be obtained as follows:where represents the weighting approach applied to the aggregate. Generally, the structure of is linear or nonlinear and depends on the effective decomposition and accurate forecasting of individual modes. First, the effective decomposition indicates that the decomposed modes should be independent of each other to avoid mixing modes as much as possible. Another advantage is that using a model constructed for individual modes is more appropriate than using the original series. Second, accurate forecasting of individual modes can significantly weaken the benefits resulting from the weighting approach.

Decomposition-Aggregate Forecasting Based on Separability

The concept of decomposition-based load forecasting is discussed in this section. It is also evident that appropriate and effective decomposition is crucial to performing this idea successfully. Furthermore, a priori knowledge (e.g., daily, weekly, and yearly features) provides the decomposition solution for the electricity load series, which indicates that an appropriate and effective decomposition should result in separability among modes. According to [28], the separability between decomposed modes can be measured by w-correlation, and zero w-correlation means that the corresponding series are separable and almost w-orthogonal. To demonstrate the implications of this separability or w-orthogonal on DA forecasting, the forecasted error at time t for the ith mode can be defined by Eq. (2) for electricity load forecasting as follows:

Using a simple weighting approach for the aggregate result, the weight-based forecasted error at time t for the ith mode can be written aswhere is the estimated weight applied to the forecasted result of the ith mode. Let be the number of testing samples. The sum of the aggregate square error can be obtained as follows:

Let be the weight-based error vector for ith mode. The defined covariance between and can be represented asand another representation of Eq. (6) is presented as

As previously mentioned, zero w-correlation largely indicates w-orthogonality, indicating that for approaches zero when future changes can be forecasted without bias. This discussion emphasizes the importance of the separability of the decomposed modes for DA forecasting. First, the constructed individual model, based on separable series, can effectively capture the inner and distinct characteristics of the original series. Second, separable decomposition largely avoids the mix of different modes, which consequently makes aggregate forecasting more effective by simple weighting approaches. However, the above-discussed condition cannot be met because of the unavoidable limitations of models and inappropriate decomposition. Therefore, a priori knowledge and specific conditions should be considered when selecting the decomposition forms, forecasting models, and weighting approaches for different situations.

SSA-based Period Decomposition (SPD) Method

In Sects. 2.1 and 2.2, we present a general discussion of DA forecasting. In particular, separability is emphasized to avoid the mix of different modes hidden in the original load series. Instead of modeling the original series, the key effect of separable decomposition is to extract the trend, season, and high-frequency modes from the original data, the benefit of which is that we might construct more effective models aiming at more predictable modes. Furthermore, the multi-seasonal features of the electricity load provide an alternative solution for this framework.

Based on the above discussion and analysis, this section proposes a robust strategy called the SSA-based period decomposition (SPD) method for DA forecasting of the electricity load. The proposed SPD can perform separable decomposition using an important metric of the separability of basic SSA. More importantly, a remarkable effect of SPD is that it can overcome the influence of the resulting border effect on the forecasting result by decomposition.

Basic SSA

To better understand the proposed method, the four key procedures of basic SSA and related comments are given as follows.

Step 1 Map the original time series to the trajectory matrix using an embedding procedure. Given an observed series , can be transformed into a sequence of multidimensional lagged vectors (trajectory or Hankel matrix) as follows:where denotes the window length of basic SSA, , and .

Step 2 Perform the singular value decomposition (SVD) of the trajectory matrix. Let (d nonzero eigenvalues) be the eigenvalues of the matrix , be the corresponding eigenvectors, and . The SVD of the trajectory matrix is given aswhere and denote the ith eigentriple of SVD.

Step 3 Split the matrices obtained by SVD into several groups using the grouping step. Let be m disjoint groups of indices . The resulting matrix of is defined as . Similarly, the grouping result for can be represented by the following decomposition:

Step 4 Transform the resultant matrices into several additive components of the original series using the diagonal averaging procedure. In general, the original series is decomposed into m additive components:

More details about SSA can be found in [28]. We provide the following discussion to better understand the extraction of separable modes. As mentioned in Sect. 3.1, the electricity load series exhibits obvious multi-seasonal features, and it is advisable to identify and separate these modes for DA forecasting because the decomposition evidently results in the separation of different modes. In SSA, the separability implies the quality of the decomposition, which can be measured by the following quantity (-correlation):where , , and . If , the corresponding reconstructed components and are separable and almost w-orthogonal; otherwise, the quality of this decomposition is poor. Based on -correlation, we can largely confirm the appropriate extraction of different modes with respect to separability. However, for DA forecasting of electricity load, the identification and separation of periodic and quasi-periodic or harmonic components might be unnecessarily specific because there is insufficient prior knowledge to group these components to obtain more predictable modes to construct individual models. In this respect, this study groups the decomposed and reconstructed components of the electricity load series into trend, seasonal, and residual features using additive representation (m = 3 in Eq. (12)). To effectively achieve this objective, the following comments based on the four steps of SSA are provided. First, according to the properties of SSA, the window length is the only parameter at the embedding step. In contrast to other decomposition methods, determining the proper at the decomposition stage of SSA is based primarily on the analysis of specific conditions and prior information about the original series. In particular, if the time series shows an obvious integer period, taking the window length proportional to that period is helpful for extracting different features. For example, considering the daily seasonality of the electricity load series, it is advisable to select = 48. Given the selected window length, the practical conclusion is that the first eigentriple results in a trend feature with a low frequency. Second, as previously mentioned, the decomposed periodic and quasi-periodic or harmonic components are not considered when constructing the individual model. An alternative solution is to identify seasonal features by detecting the main harmonic components with respect to separability.

Proposed SSA-based Period Decomposition (SPD) for Forecasting

SSA decomposes the electricity load into additive, seasonal, and residual features, which is a remarkable and important result with respect to separability. We also highlight the implicit importance of separability for DA forecasting. However, as mentioned in Sect. 2.2, the limitations of forecasting methods and inappropriate decomposition are unavoidable; therefore, the aggregate result might be unexpected. Similar to other popular decomposition techniques, it is challenging to solve the border effect using SSA, which has received little attention in previous studies. To overcome this problem, this paper proposes an SSA-based period decomposition (SPD) method for DA forecasting of the electricity load. It is worth noting that SPD is a decomposition strategy to avoid the resulting border effect instead of reducing border distortion (see [21, 27]). To facilitate the understanding of the proposed SPD method, the following steps were designed to construct the DA forecasting of the electricity load based on the SPD.

Step 1 Select the appropriate. The window length determines the number of reconstructed components applied to the extraction of different features; therefore, the selection of has a great influence the decomposition performance. As mentioned earlier, the selected should be proportional to the period, considering the multi-seasonal features of the electricity load. To facilitate the analysis of the decomposition process, L = 48 is considered in this step for the extraction of different features, and the additive decomposition with trend, season, and residual components can be obtained as follows:

The extraction of the different features above is performed not only in the prior knowledge about the original but also in detecting the decomposed harmonic components. However, as previously discussed, the last point feature at the end of the decomposed features is unavoidably affected by the border effect, which has a significant influence on forecasting performance. To solve this problem, using the additive representation of the original series, another consideration of Eq. (14) can be written aswhere . It is clear that if the equation is previously defined, the remaining task is to estimate the weight based on the decomposition and reconstruction of SSA.

Step 2 Given the observation number T, estimating at time t for each feature is easy to implement based on the decomposed result of SSA but unnecessary for DA forecasting because of the border effect. To overcome the impact of the border effect on the forecasting effect, the following decomposing form is constructed:

The above equation indicates that we can perform a robust decomposition of the daily load series or daily cycles into trend, season, and residual features using the same (the 48 half-hour load), which might result in regular decomposition without distortion at the end of each feature and avoid the border effect because of the same weight applied to the daily load. Based on Eq. (16), the remaining task is to estimate .

Step 3 To perform the decomposition of the electricity load using the weight , the daily regularity is considered as a remarkable pattern applied to the decomposition form. Based on the comments on SSA and the extraction of daily features, can be estimated as follows:where the selected is an integer, indicating that the training series used for decomposition is proportional to the window length. It is observed that the decomposed performance based on the above equation depends on the daily pattern of the electricity load. In addition, despite the remarkable daily pattern, the difference between workday and non-workday loads should be discussed to estimate .

Step 4 Considering the daily load period, steps 1–3 constitute the proposed SPD method. For the purpose of forecasting load, the trend shows a slow tendency of the series, indicating a primarily linear change; therefore, the trend in this step was modeled using the classical autoregressive integrated moving average method. For season and residual features, nonlinear methods, such as neural networks, are employed considering complex nonlinearity. Finally, we obtain the aggregate result by combining the individual forecasts.

Benchmark Models

To validate the effectiveness of the proposed method, we employ three methods to perform electricity load forecasting. First, three baselines (naïve methods) were used to model the original load series. Second, we forecast the electricity load using three widely used forecasting models: seasonal autoregressive moving average, a support vector machine (SVM), and a neural network. Finally, according to [27], this study develops several strategies to deal with border distortion based on SSA, the effectiveness of which will be used for comparative purposes with the proposed SPD.

Naïve Methods

The multi-seasonal pattern is a remark for forecasting purposes. In this regard, we can use previous information to estimate future load without exploring the complex mapping relationship between future changes and historical information. Therefore, the following three naïve methods were considered as baselines:

The nearest information is the most valuable reference for future changes. Therefore, we consider the lag order as the forecast for , which generates the Naïve_lag method.

The electricity load shows an obvious daily cycle, the regularity of which indicates that the load from the previous day can be used to estimate that of the next day. Considering 48 time points per day, the forecast of load at time t can be given by , which is called the Naïve_day method.

Considering the weekly pattern, the workday load might show different fluctuations with non-workdays. Therefore, for a half-hour load, the load at time t might be forecasted by the same time from the previous week, generating the Naïve_week method.

SARIMA and Machine Learning Methods

This study considers two categories of models for forecasting electricity load. Regarding linear and classical statistical methods, the very popular and successful seasonal autoregressive integrated moving average (SARIMA) is first employed for electricity load forecasting. Generally, a SARIMA model can be denoted as SARIMA (p, d, q) (P, D, Q), where (p, d, q) (autoregressive, difference, and moving terms) is a set of arguments to deal with the non-seasonal part, while (P, D, Q) (autoregressive, difference, and moving terms) represents the seasonal part considering the seasonality S. Additionally, we implement the SARIMA model with daily seasonality using the autocorrelation and partial autocorrelation functions (ACF and PACF). The second category is machine learning models. To demonstrate the effectiveness of the proposed method, two popular machine learning models were constructed for electricity load forecasting. The first model is the widely used back-propagation neural network (BPNN), which is a representative ANN because of its performance in solving non-stationary and nonlinear problems. In terms of tackling seasonality, the performance of ANNs has been confirmed by comparison with classical statistical models [29-31]. Additionally, the regularization of the BPNN absolutely depends on the error minimization principle like other neural networks, but employs the error back-propagation mechanism to adjust the weights and bias in the network structure to capture the relationship between the input information and the output. Another model considered is the popular support vector machine (SVM). A remarkable property of SVMs is that the structural risk minimization (SRM) principle is introduced to minimize the training error.

SSA-Based Approaches

This paper presents several processes to deal with border distortion, the key idea of which is to provide the expansion of the original load series with the estimation of future changes using historical information, before applying the SSA to the decomposing process. A difference from the current methods is that the considered strategies extend the load series one day ahead of electricity load forecasting, the aim of which is to make the decomposed load series proportional to the window length, thereby facilitating the extraction of different features. Considering half-hour load forecasting, the detailed processes are as follows.

Previous day or Naïve_day extension: for

Extension with the same day from previous week or Naïve_week method: for

Constant extension with the nearest observation: for

Linear extension: the autoregressive integrated moving average is used to perform one-day-ahead forecasting using the historical information from the same time for each day, where

Combining approaches: simple averaging (arithmetic mean) is used as the combining method, considering its popularity and robust evaluation

Processes - perform the extension using previous information without exploring the mapping relationship between future change and historical information. In particular, compared with process , process might be more effective in terms of weekly seasonality, where non-workday load demand differs from that of workdays in terms of time-varying volatility. To improve the effectiveness of this extension, several strategies have been developed by combining different forecasting models (see Table 1).

Table 1

Treatment of the boundary distortion

Process description	Code
Symmetric approaches: previous day or Naïve_day extension	S_PD
Symmetric approaches: extension based on the same day from the previous week	S_PW
Constant extension	S_C
Using AR to obtain the day-ahead forecast	S_AR
Using ARIMA to obtain the day-ahead forecast	S_AM
AR forecast + S_pd	S_ARPD
ARIMA forecast + S_pd	S_AMPD
AR forecast + S_pw	S_ARPW
ARIMA forecast + S_pw	S_AMPW

Symmetric approaches: day-ahead forecast using the previous day: for ; day-ahead forecast using the same day from last week: for ; Constant approaches: day-ahead forecast using the first-order lag: for ; Linear approaches: based on from the same time for each day, where , the linear model is constructed to perform day-ahead forecasting; Combining approaches: the simple averaging (arithmetic mean) is used as the combining method considering its popularity and robustness.

Forecasting Implementation and Analysis

Experiment Design

Data

This study develops combinations of singular spectrum analysis for short-term electricity load forecasting. Specifically, considering the detrimental influence of the boundary problem on forecasting performance, the SPD method involving the seasonality of the load series is proposed in this study. To verify the effectiveness of the proposed method, the load data in the experimental design consist of three electricity markets: New South Wales (NSW), Tasmania (TAS), and Victoria (VIC). For each market, the 48 half-hour load series covering six weeks are randomly selected for modeling and forecasting, generating three sample datasets. Moreover, to further strengthen the applicability of the proposed method, the sample dataset is considered; consequently, the samples span from 08-Jul-2013 to 08-Aug-2013, 18-Feb-2013 to 31-Mar-2013, and 14-Oct-2015 to 12-Nov-2013 for the NSW, TAS and VIC markets, respectively. Additionally, the weekly and daily seasonality of electricity load is the main features used for short-term load forecasting. To evaluate the forecasting performance, a training set and testing set are set up for each sample. However, there is no universal rule to determine the proportion between the training and testing sets. From experience, selecting 2/3 for training and the remaining 1/3 for testing is the common rule of thumb [32]. Therefore, this study considers the practical split, using four weeks as the training set and two weeks as the testing set for each market.

Model Design and Assessment

This study focuses on combinations of singular spectrum analysis for short-term load forecasting, the challenging task of which is to extract appropriate features to facilitate DA forecasting. According to the proposed method, although weekly and daily seasonality exists in the used sample, the selected window length is based on the daily cycle rather than the weekly cycle because a larger can result in more decomposition components, making the main features more difficult to identify and extract. In terms of parameter settings, autoregressive integrated moving average models are used based on autocorrelation and partial autocorrelation functions (ACF and PACF). Specifically, SARIMA (p, d, q) (P, D, Q) in this study tackles daily seasonality (S = 48). The default hyperparameters are used for the training process to construct the support vector machine and ANN. It is worth noting that the instability of ANNs is overcome by the average results based on 50 iterations. In addition, the input vector for each machine learning model is determined by the first cross of the ACF of the load series.

Regarding the assessment of forecasting effectiveness, the mean absolute error (MAE) and mean absolute percentage error (MAPE) are defined aswhere N is the number of test samples, is the observed value at time t, and is the corresponding estimated value. For these three indices, smaller values indicate better forecasting performance.

4.0.2 SSA-Based Electricity Load Decomposition.

This section investigates the main features of the electricity load series using the proposed SPD- and SSA-based approaches, as discussed in Sect. 2.3. The key idea of this analysis is not only to generate a multi-solution for DA forecasting by extracting the significant features, but also to emphasize the importance of proper and effective decomposition in DA forecasting. For this target, this section considers the 48 half-hour electricity load to perform SSA-based decomposition. It is worth noting that daily seasonality rather than weekly seasonality is used because of the influence of the window length on the extraction process discussed in Sect. 2.3.1.

Based on the three markets NSW, TAS, and VIC used for the validation, this section discusses the decomposition process to better understand the feature extraction of electricity load series according to the properties of the basic SSA discussed and summarized in Sect. 2.3. First, the window length = 48 is chosen in terms of the daily seasonality and used for the decomposition of the load series, resulting in 48 eigenvalues and eigentriples; the main components can be constructed based on the eigentriples. Second, the trends and seasonal features are identified and extracted using the decomposed eigenvalues and components. To facilitate the extraction process, Fig. 1 shows the movement of the eigenvalues. According to the discussion in Sect. 2.3, the leading eigenvalue is different from the others and is often accompanied by a low-frequency feature dominating the long-term variation of the series, indicating that we can extract the trend feature based on the first eigentriples. However, the decomposition of seasonal features is followed by many intangible harmonic components; therefore, the resultant identification and extraction by eigenvalues must be confirmed using separability measured by -correlation or -orthogonality. Figure 2 shows a plot of the matrix of the absolute values of -correlation considering the positive eigenvalues, which indicates that the 2–11, 2–11, and 2–9 eigentriples that correspond to NSW, TAS, and VIC, respectively, can be clearly separated from the others, which may be associated with high-frequency or noise features. However, as previously mentioned, although the matrix of the -correlation indicates reliable grouping (e.g., paired 2–3, 4–5, 6–7, and 8–9 for VIC), we consider only seasonal features rather than a specific extraction for each paired harmonic component because the larger m in Eq. (12) implies a more diverse error when more single models are constructed for DA forecasting: Therefore, the 2–11, 2–11, and 2–9 eigentriples are used to reconstruct the seasonal features of NSW, TAS, and VIC, respectively, whereas the remaining triples are used to extract residual features.

Fig. 1

Movement of eigenvalues

Fig. 2

Matrix of the absolute values of w-correlation for 48 reconstructed components (RC)

The aforementioned decomposition process constitutes Step 1 of the proposed SPD, which can reveal and extract daily trend, season, and residual features. However, the resultant decomposition may be unhelpful for DA forecasting owing to the border effect. To overcome this problem, we consider the decomposition form presented in Eq. (16). The key aim of this implementation is to obtain a regular decomposition without distortion at the end of each feature. Based on Eq. (17) and the result of Step 1, an estimate of can be obtained. In addition, despite the remarkable daily pattern, we consider the difference between workday and non-workday loads for the estimation of . This argument is supported by the views shown in Fig. 3.

Fig. 3

Decomposed weights of workday and non-workday for trend, season, and residual features

Forecasting and Assessment

SPD-Based DA Forecasting Design

In this study, the SPD approach is developed for short-term electricity load forecasting. To verify the proposed method, based on the DA forecasting framework, we used the classical autoregressive integrated moving average (ARIMA) method for trend feature forecasting and a back-propagation neural network (NN) for season and residual features (Step 4), which generates the expected forecasting model, namely ANS. This implementation suggests that in consideration of the potential overfitting problem of the NN method in the learning process for the linear trend, the linear ARIMA, rather than the NN model alone, should be integrated to guarantee robust aggregate forecasting. In addition, the only NN model for the three extracted features is also constructed to confirm the effect of the ANS, denoting the NNS forecasting model. Furthermore, to strengthen the effectiveness of the proposed method, we also consider SVM and SARIMA as the forecaster with the aforementioned process for DA forecasting based on the proposed SPD method, generating ASS (aggregating ARIMA and SVM forecasts), SVS (aggregating SVM forecasts for the three decomposed features) and SIS (aggregating seasonal ARIMA forecasts for the three decomposed features). All models are executed using MATLAB R2014b on a PC with 8 GB RAM, 64-bit Windows 10, and an Intel® Core™ i5-10,400 CPU @ 2.9 GHZ.

Based on the experimental design, the forecasting performance of the proposed models is presented in Table 2. First, in terms of individual forecasting models, it is remarkable that the seasonal ARIMA in this study significantly outperforms machine learning models such as ANN and SVM, which indicates that seasonal ARIMA obtains a better and more robust learned structure with the treatment of seasonal features, while the unexpectedly unsuccessful performance of ANN and SVM may be due to the uncertainty of hyperparameters and input vectors. When considering DA forecasting with the proposed SPD method, DA forecasting models such as ANS, NNS, ASS, SVS, and SIS outperform the corresponding single forecasting models overall. However, the forecasting improvement of the machine learning model seems to be more significant than that of the seasonal ARIMA, indicating that linear statistical models might be limited in tackling nonlinear season and residual features. Additionally, in comparison, it is obvious that ANS (ASS) can provide a more satisfactory forecasting accuracy than NNS (SVS), which means that the aggregate forecasting performance using ARIMA for the trend feature is superior to that of the NN method and that there may be a potential overfitting problem using the NN method for linear trend forecasting. Third, from the forecasting performance presented in Table 2, DA forecasting yielded a more effective improvement for non-workdays in comparison with load forecasts for workdays. This finding indicates that the DA strategy based on separable decomposition can provide an alternative solution to handle the inner features hidden in the original series, consequently avoiding the uncertainty caused by the mix of different modes and generating a more accurate final forecast.

Table 2

Forecasting performance of the discussed models

Dataset	Metric	ANS	NNS	ANN	ASS	SVS	SVM	SIS	SARIMA
NSW
Global	MAE	49.2433	54.8302	62.9436	63.6588	79.6984	87.9621	50.2935	53.8947
	MAPE	0.5948	0.6693	0.7637	0.7754	0.9842	1.0573	0.6256	0.6723
Workday	MAE	48.8706	48.6887	60.8519	60.9504	62.7371	86.0200	45.2802	48.1192
	MAPE	0.5708	0.5682	0.7185	0.7199	0.7418	1.0013	0.5365	0.5737
Non-workday	MAE	50.1750	70.1839	68.1726	70.4297	122.1018	92.8176	62.8269	68.3334
	MAPE	0.6546	0.9220	0.8767	0.9143	1.5902	1.1972	0.8485	0.9189
TAS
Global	MAE	14.4873	17.2457	19.6281	14.4981	18.7473	23.6515	14.7431	15.9992
	MAPE	1.4987	1.8307	2.0848	1.4957	2.0094	2.5703	1.5257	1.6602
Workday	MAE	15.6735	17.3385	19.8115	16.0391	18.2242	23.0812	15.7695	16.4396
	MAPE	1.6112	1.8182	2.0800	1.6424	1.9208	2.4773	1.6191	1.6857
Non-workday	MAE	11.5221	17.0137	19.1695	10.6458	20.0551	25.0773	12.1771	14.8981
	MAPE	1.2175	1.8620	2.0970	1.1288	2.2307	2.8029	1.2922	1.5964
VIC
Global	MAE	36.7025	37.0597	49.0538	39.5231	42.8709	66.5394	38.8780	40.3551
	MAPE	0.6953	0.7001	0.9349	0.7500	0.8088	1.2691	0.7483	0.7757
Workday	MAE	37.3746	37.7471	49.0035	39.7576	43.6667	69.4458	37.2291	38.9007
	MAPE	0.6765	0.6806	0.8914	0.7201	0.7858	1.2744	0.6761	0.7056
Non-workday	MAE	35.0225	35.3411	49.1796	38.9369	40.8815	59.2734	43.0002	43.9912
	MAPE	0.7423	0.7488	1.0437	0.8248	0.8661	1.2558	0.9288	0.9511

In summary, through proper and separable decomposition and extraction for different features, the constructed ANS, NNS, ASS, SVS, and SIS achieve more effective forecasting and present potentially competitive power in terms of MAE and MAPE on average, indicating that the proposed SSA-based period decomposition method is a reliable and promising tool for the decomposition and extraction of different features. Moreover, it is worth highlighting that the DA strategy based on separable decomposition is a suitable solution for overcoming the uncertainty and instability of single forecasts.

SSA-Based Forecasting and Comparison

This section presents the SSA-based approaches, including all the processes listed in Table 1 and no treatment (S_NO) for border distortion, the purpose of which is to highlight the superiority and necessity of the proposed SPD and separable decomposition. Specifically, we implement the aggregate ARIMA + machine learning methods with a simple average considering the real nonlinearity of the season and residual features. All the discussed processes and models are implemented in the environment described in Sect. 4.3.1. Table 3 reports the forecasting performance in terms of MAE and MAPE.

Table 3

Forecasting performance of the SSA-based approaches

Method	Metric	S_PD	S_PW	S_C	S_AR	S_AM	S_ARPD	S_AMPD	S_ARPW	S_AMPW	S_NO
NSW ARIMA + NN
Global	MAE	342.21	183.66	171.38	352.99	327.02	338.02	315.08	224.18	200.19	365.09
	MAPE	4.1660	2.1996	2.0878	4.2826	3.9816	4.1080	3.8447	2.7310	2.4502	4.5655
Workday	MAE	273.40	170.08	179.06	288.11	289.25	274.28	258.19	182.68	161.73	225.45
	MAPE	3.1183	1.9537	2.1233	3.2854	3.3338	3.1301	2.9599	2.0905	1.8614	2.6128
Non-work	MAE	514.24	217.60	152.17	515.21	421.47	497.38	457.30	327.92	296.34	714.19
	MAPE	6.7852	2.8145	1.9992	6.7758	5.6010	6.5530	6.0568	4.3320	3.9221	9.4472
TAS ARIMA + NN
Global	MAE	28.09	56.06	18.42	31.33	31.10	27.79	27.94	39.65	39.22	33.33
	MAPE	2.8725	5.9117	1.8889	3.2275	3.1968	2.8543	2.8688	4.1561	4.1072	3.4648
Workday	MAE	31.32	53.87	20.25	35.44	33.53	31.02	30.33	41.20	39.96	34.17
	MAPE	3.1805	5.6745	2.0558	3.6228	3.4322	3.1616	3.0969	4.3036	4.1807	3.4988
Non-work	MAE	20.03	61.53	13.83	21.03	25.04	19.72	21.97	35.77	37.36	31.22
	MAPE	2.1026	6.5049	1.4718	2.2392	2.6081	2.0860	2.2986	3.7875	3.9232	3.3798
VIC ARIMA + NN
Global	MAE	245.08	183.59	103.31	252.52	241.60	251.06	240.05	196.49	198.94	267.27
	MAPE	4.6364	3.4148	2.0299	4.7542	4.5805	4.7291	4.5559	3.6743	3.7403	5.2938
Workday	MAE	218.65	208.50	111.85	232.57	218.14	232.50	217.36	199.96	201.22	143.78
	MAPE	3.8694	3.7250	2.1368	4.1071	3.8680	4.1130	3.8696	3.5441	3.5825	2.5897
Non-work	MAE	311.17	121.33	81.97	302.40	300.26	297.45	296.76	187.80	193.24	575.98
	MAPE	6.5540	2.6394	1.7624	6.3718	6.3615	6.2695	6.2715	3.9999	4.1349	12.0539
NSW ARIMA + SVM
Global	MAE	284.38	158.85	153.36	293.83	280.54	287.36	272.80	198.27	177.72	303.52
	MAPE	3.4711	1.9168	1.8626	3.5723	3.4286	3.5005	3.3358	2.4219	2.1843	3.7810
Workday	MAE	238.88	147.51	160.46	252.10	257.78	243.83	236.71	167.54	149.01	208.23
	MAPE	2.7503	1.7109	1.8990	2.9008	3.0042	2.8062	2.7413	1.9336	1.7334	2.4151
Non-work	MAE	398.13	187.21	135.61	398.14	337.45	396.16	363.05	275.10	249.51	541.74
	MAPE	5.2729	2.4316	1.7714	5.2513	4.4899	5.2363	4.8218	3.6428	3.3115	7.1959
TAS ARIMA + SVM
Global	MAE	25.19	48.29	17.81	27.85	27.70	25.38	25.22	34.75	34.89	29.56
	MAPE	2.5849	5.1030	1.8268	2.8701	2.8467	2.6132	2.5922	3.6429	3.6478	3.0628
Workday	MAE	27.61	47.13	19.51	31.08	29.69	27.93	27.11	36.14	35.73	30.33
	MAPE	2.8151	4.9717	1.9831	3.1790	3.0373	2.8544	2.7720	3.7768	3.7292	3.0995
Non-work	MAE	19.11	51.18	13.55	19.76	22.74	19.02	20.47	31.25	32.79	27.63
	MAPE	2.0096	5.4313	1.4362	2.0978	2.3703	2.0102	2.1428	3.3082	3.4445	2.9710
VIC ARIMA + SVM
Global	MAE	198.96	144.21	95.10	205.70	197.44	201.64	186.83	164.58	156.54	196.17
	MAPE	3.7577	2.6754	1.8384	3.8728	3.7197	3.8023	3.5271	3.0702	2.9220	3.8516
Workday	MAE	176.93	165.70	103.89	185.45	180.10	180.28	164.72	166.89	157.92	127.74
	MAPE	3.1190	2.9579	1.9548	3.2578	3.1670	3.1714	2.8922	2.9507	2.7864	2.3110
Non-work	MAE	254.02	90.47	73.12	256.32	240.79	255.05	242.10	158.80	153.09	367.26
	MAPE	5.3544	1.9690	1.5474	5.4102	5.1015	5.3797	5.1143	3.3690	3.2609	7.7031

Further discussion and analysis of the forecasting effectiveness are presented as follows. First, we assess the impact of the border extension on forecasting effectiveness. According to the results listed in Table 4, it is particularly obvious that the forecasting performance by the extension with SSA-based approaches is better than that with no treatment for overall border distortion. Moreover, from the comparison of all the given processes, we find that the S_C with SSA and constant extrapolation outperforms the others, which demonstrates that constant extrapolation in this study might be more robust than one-day-ahead forecasting. This result is not surprising because one-day forecasting is a difficult task; therefore, the last observation associated with the input of the forecasting cannot be tackled effectively. Next, we compare the results for workdays and non-workdays. It can be found that the DA strategy with the treatment of the border distortion generates a better improvement for non-workday load forecasting, which is supported by considering the difference between workday and non-workday loads and is also consistent with previous conclusions. Third, despite the good improvement of the SSA-based approaches compared with cases with no treatment, it is surprising and unacceptable that the performance with the help of these processes cannot exceed that of single forecasting (see Table 2). This implies that these methods fail to provide a better capability for DA forecasting because they cannot obtain the expected treatment of the last observation of the original series. Finally, as presented in Tables 2 and 3, the forecasting models based on the proposed SPD were significantly superior to the discussed SSA-based approaches. This result is expected, as the proposed method emphasizes regular and separable decomposition using historical information rather than one-day forecasts.

Table 4

Results of the DM test

Model	S_PD	S_PW	S_C	S_AR	S_AM	S_ARPD	S_AMPD	S_ARPW	S_AMPW	S_NO
NSW
ANS	12.988a	11.073 a	12.429 a	13.395 a	14.916 a	13.619 a	13.520 a	13.132 a	12.785 a	12.634 a
NNS	12.944 a	10.921 a	11.920 a	13.349 a	14.840 a	13.569 a	13.459 a	13.050 a	12.676 a	12.637 a
ASS	13.499 a	11.709 a	11.934 a	13.699 a	17.389 a	13.544 a	15.227 a	14.261 a	14.005 a	13.681 a
SVS	13.111 a	10.479 a	9.647 a	13.292 a	16.583 a	13.143 a	14.640 a	13.433 a	12.771 a	13.514 a
TAS
ANS	6.834 a	18.110 a	5.060 a	13.083 a	11.022 a	9.625 a	8.496 a	15.537 a	14.775 a	9.474 a
NNS	5.839 a	18.409 a	1.946c	11.963 a	10.107 a	8.111 a	7.284 a	15.678 a	14.941 a	8.497 a
ASS	7.812 a	17.141 a	5.022 a	12.427 a	11.164 a	9.967 a	9.009 a	15.082 a	14.743 a	8.986 a
SVS	4.945 a	17.434 a	− 1.273	9.144 a	8.517 a	6.464 a	5.945 a	14.471 a	14.220 a	6.964 a
VIC
ANS	13.116 a	12.650 a	12.786 a	12.900 a	17.452 a	12.566 a	13.692 a	16.606 a	17.153 a	10.380 a
NNS	13.112 a	12.623 a	12.703 a	12.897 a	17.455 a	12.563 a	13.688 a	16.600 a	17.141 a	10.376 a
ASS	13.113 a	12.381 a	11.399 a	13.275 a	17.554 a	13.154 a	14.405 a	15.916 a	16.215 a	11.764 a
SVS	13.040 a	12.240 a	11.040 a	13.207 a	17.506 a	13.084 a	14.324 a	15.799 a	16.130 a	11.695 a

Discussion

The previous assessment confirmed the effectiveness and superiority of the proposed method in terms of the evaluation metric. To intensify the comparison of forecasting performance, further analysis based on the forecasting error is presented in terms of three aspects.

First, to highlight the significant difference between the two models and confirm the best, we conduct the popular Diebold–Mariano (DM) test [33] for comparison. It is worth stating that this test focuses on the impact of the border effect on the forecasting performance. Therefore, to simplify the demonstration of the results, ANS, NNS, ASS, and SVS are employed for comparison. Table 4 presents the results of the DM tests. As indicated, at the 1% significance level, all the computed values of the statistics of the DM test, except the comparative test between NNS, SVS, and S_C for the TAS market, are greater than the upper bound, which implies that the null hypothesis that there is no difference in forecasting effectiveness is rejected, and there is a statistically significant difference. More importantly, this comparison suggests that the proposed SPD is superior to the processes used in the literature.

Next, based on the error analysis, we demonstrate the importance of the weakened border distortion obtained by using the proposed method. As previously mentioned in Sect. 2.2, zero -correlation largely indicates -orthogonality, indicating for and indicating that the sum of the aggregate square error can be represented as the addition of the single. However, the aforementioned assumption is impossible because of inappropriate decomposition and unavoidable limitations of the forecasting method. Therefore, from this perspective, we can highlight the effect of dealing with border distortion by comparing the DA forecasting error and the error of single forecasting. For this purpose, the RSSE (root mean square error ratio) is defined as follows:where , , , and are the sum of the square errors of the DA, trend, season, and residual forecasts, respectively. The denominator in Eq. (8) is given by the above assumption , which indicates that an close to 1 implies better DA forecasting based on separable decomposition. Table 5 reports the computed results for RSSE. It can be observed that both the constructed ANS and NNS obtain better results, indicating that border distortion can be overcome based on SPD and that separable decomposition should be introduced to reduce the impact of the mixture of different features on DA forecasting.

Table 5

Estimated results of the RSSE

Data	S_PD	S_PW	S_C	S_AR	S_AM	S_ARPD	S_AMPD	S_ARPW	S_AMPW	NNS	ANS
NSW	76.82%	71.04%	20.26%	75.81%	75.67%	76.25%	76.37%	75.09%	72.61%	85.30%	89.19%
TAS	54.95%	65.25%	24.50%	55.91%	59.00%	54.69%	54.88%	57.49%	58.83%	86.44%	99.40%
VIC	73.16%	65.94%	20.45%	70.42%	68.88%	70.26%	70.33%	67.79%	65.09%	82.14%	83.78%

Finally, as previously discussed, the decomposition of seasonal features is followed by many intangible harmonic components; therefore, we take the separability measured by the w-correlation and eigenvalues to determine the season and residual components for DA forecasting, which provides an alternative solution for the significant decomposition of electricity load with multiple seasonal patterns. However, the sensibility of the proposed method across different groups for the extraction of seasonal features is a concern. For this purpose, we traverse the number of harmonic components to reconstruct the potential seasonal features for DA forecasting, resulting in 2–1, 2–2 and 2–3 eigentriples for NSW, TAS, and VIC, respectively. Additionally, 47 ≥ 1 > 11, 47 ≥ 2 > 11, and 47 ≥ 3 > 9 are specified to guarantee that the extracted feature contains the harmonic components associated with the seasonal cycles. Using the constructed model ANS and the metric MAPE, Fig. 4 depicts the forecasting performance considering different numbers of harmonic components. It can be found that the MAPE presents an increasing trend overall, indicating that the uncertainty of DA forecasting increases as more high-frequency components are included in the seasonal feature. Meanwhile, despite not obtaining the best accuracy, the decomposition implemented in Sect. 3.2 can provide a relatively reliable reference for DA forecasting. It is also worth noting that the number of harmonic components applied to the extraction of seasonal features should be considered to improve forecasting accuracy.

Fig. 4

The evolution of MAPE with the harmonic component increase for seasonal features

Conclusion

In this study, we propose SSA-based period decomposition to facilitate the construction of a DA forecasting framework. With the introduction of multiple seasonality and volatility of the electricity load, this study first demonstrates a feasible solution for DA forecasting for short-term load forecasting. Second, we emphasize the importance of the separability of the decomposed modes for DA forecasting, the key idea of which is that separable decomposition can largely avoid the mix of different modes, consequently building proper single forecasts and making aggregate forecasts more effective. To do so, we propose SSA-based period decomposition to not only perform separable decomposition but also overcome the border effect, which has received little attention in previous work. To verify the effectiveness of the proposed SSA-based period decomposition, the classical autoregressive integrated moving average method and neural network model are employed to achieve DA forecasting. Finally, the empirical results demonstrate that the proposed approach can achieve the expected forecasting performance. In particular, we perform SSA-based approaches and no treatment for border distortion, which emphasizes that the proposed SSA-based period decomposition method is a reliable and promising tool for the decomposition and extraction of different features. Moreover, it is worth highlighting that the DA strategy based on separable decomposition is a suitable solution for overcoming the uncertainty and instability of single forecasts.

Developing a separable and robust decomposition method is the key to performing DA forecasting, which can not only further minimize and analyze the inner features of time series but also improve forecasting performance. Although the proposed approach provides an alternative solution for DA forecasting of electricity load, it can be applied to the forecasting of other economic time series with the introduction of the seasonal pattern of real data series.