Day-Ahead Electricity Demand Forecasting Using a Novel Decomposition Combination Method

Abstract

In the present liberalized energy markets, electricity demand forecasting is critical for planning of generation capacity and required resources. An accurate and efficient electricity demand forecast can reduce the risk of power outages and excessive power generation. Avoiding blackouts is crucial for economic growth, and electricity is an essential energy source for industry. Considering these facts, this study presents a detailed analysis of the forecast of hourly electricity demand by comparing novel decomposition methods with several univariate and multivariate time series models. To that end, we use the three proposed decomposition methods to divide the electricity demand time series into the following subseries: a long-run linear trend, a seasonal trend, and a stochastic trend. Next, each subseries is forecast using all conceivable combinations of univariate and multivariate time series models. Finally, the multiple forecasting models are immediately integrated to provide a final one-day-ahead electricity demand forecast. The presented modeling and forecasting technique is implemented for the Nord Pool electricity market’s hourly electricity demand. Three accuracy indicators, a statistical test, and a graphical analysis are used to assess the performance of the proposed decomposition combination forecasting technique. Hence, the forecasting results demonstrate the efficiency and precision of the proposed decomposition combination forecasting technique. In addition, the final best combination model within the proposed forecasting framework is comparatively better than the best models proposed in the literature and standard benchmark models. Finally, we suggest that the decomposition combination forecasting approach developed in this study be employed to handle additional complicated power market forecasting challenges.

1. Introduction

Before deregulation, the power sector was dominated by state-owned enterprises. In this monopolistic structure, changes in electricity prices are minimal, and chief attention is paid to load forecasting, long-term planning, and investment in the sector. The late 1980s saw major changes in the electricity sector as the national management system was restructured into a deregulated and competitive electricity market. The main objective of the electricity market restructuring was to encourage private investment in the production, supply, and retail sectors, thereby increasing competition among producers, retailers, and buyers [1]. The first electrical reforms were implemented in Chile in early 1981, followed by the global deregulation framework, primarily in Europe. The United Kingdom’s electricity sector was deregulated in 1990; then, in 1992, Norway established its own deregulated electricity sector, leading to a trend of gradual reform in Denmark, Finland, Sweden, etc. Australia, America, China, Canada, the Netherlands, New Zealand, Korea, Japan, and several other developed countries have introduced deregulated electricity markets. The World Bank introduced a range of hybrid markets in other Latin American nations, including Brazil, Colombia, and Peru, during the 1990s, with partial success. The trend towards deregulation of electricity markets is gradually growing worldwide. However, the trend is most visible in Europe [2].

The deregulation of the electricity market also provided a new form of research for efficient and accurate modeling and forecasting of various variables related to electricity markets, e.g., demand, price, production, etc. In many countries, the electricity market consists of different needs, which include a daily market where demand and prices are determined on the day-before delivery by simultaneous (half-hourly or hourly) auctions for the following day. In this market, (on average) every hour of the day, a producer or buyer submits bids or offers for a certain amount of electricity at a specific price. An independent system operator collects these bids and demands to create the aggregated bid and demand curve that determines the price and level of market settlement [3].

Electricity is a secondary energy source obtained by converting primary energy sources such as natural gas, solar. Due to its natural and physical properties, electricity is a commodity that is significantly different from other commodities. For example, electricity cannot be stored in large quantities for long periods and therefore has a short life [4]. To avoid shutdown or complete system failure, the electricity supply and demand must constantly meet. Therefore, a consistent power supply mechanism is needed to shift demand levels between high demand in the short term and moderate demand in the long run. Thus, accurate forecasting is essential for the effective management of the electricity market.

In the past, several methods and models have been considered for electricity demand forecasting based on different time horizons, i.e., very short-term demand forecasting (VSTDF), short-term demand forecasting (STDF), medium-term demand forecasting (MTDF), and long-term demand forecasting (LTDF). VSTDF, with lead times measured in minutes, is often considered another class of forecasting [5]. STDF typically covers the range from minutes to days ahead and is most important for day-to-day market operations. MTDF ranges from days to months and is typically preferred for balance sheet calculations, risk management, and derivatives pricing [6]. This form of modeling has a long history in finance, and an influx of “finance solutions” has been noted. LTDF, which has deadlines defined in months, quarters, or even years, is focused on analyzing and planning returns on investments, such as future power plant sites or fuel supplies [7].

In competitive energy markets, electricity demand time series are characterized by long-term linear (or non-linear) trends, various periodicities (yearly, seasonal, weekly, or daily), non-constant mean and variance, jumps or peaks (extreme values), high volatility, calendar effects, and a tendency to return to average levels. For example, a linear trend and the annual periodicity can be observed in Figure 1 (A), as well as seasonal variation of the electricity demand throughout a year (B), the monthly consumption of electricity demand (C), the weekly and daily electricity demand patterns (D), the box plot for 24 h (E), and the daily hourly pattern (F). The weekly period includes relatively less volatility in the electricity demand time series. Load profiles vary considerably depending on the day of the week, and demand fluctuates throughout the day. Weekdays have more demand than weekends. In addition, electricity demand is affected by time (extended weekends and holidays) and seasonal effects. However, electricity demand for holidays and long weekends (one day between two non-working days) is significantly lower. Likewise, a variety of environmental, topographical, and climatic factors have a direct influence on the time series of electricity demand [8].

In order to capture the specific features of the electricity demand time series mentioned above, researchers have proposed various methods and models for the forecast of electricity demand over the past three decades [9,10,11,12,13,14,15,16,17,18,19]. Typically, these forecasting methods and models can be roughly divided into three categories: statistical methods, machine learning methods, and hybrid models.

In the first category, generally, the statistical forecasting methods are classified into the following types: parametric and nonparametric regression, linear and non-linear time series, Kalman filters, and exponential smoothing models [20]. Parametric and nonparametric regression models include polynomial, sinusoidal, smoothing spline, kernel, and quantile regression. Linear and non-linear time series models consider autoregressive moving average, autoregressive integrated moving average, vector autoregressive moving average, and non-linear time series models containing nonparametric autoregressive conditional heteroscedasticity, generalized autoregressive conditional heteroscedasticity, and threshold autoregressive conditional heteroscedasticity [21,22,23,24]. For example, to forecast one-day-ahead electricity demand, the authors of [5] used various regression and time series techniques with parametric and nonparametric estimation techniques. They used Nord Pool data on hourly electricity demand to evaluate the proposed forecasting methodology. Based on the empirical results, the proposed component-based estimation method was very effective in power demand forecasting. The proposed forecasting methodology, vector autoregressive modeling combined with spline-function-based regression, performs better than other models. Similarly, exponential smoothing (single, double, and triple Holt–Winter models) is also commonly used to forecast electricity demand [25,26,27,28]. For example, the authors of [29] analyzed the parameter space of several seasonal Holt–Winter models applied to power demand in Spain. The authors considered different time frames corresponding to different seasons and distinct training set sizes.

With the rapid development of computational intelligence, machine learning techniques have been extensively applied to different research areas, including short-term electricity load forecasting in the second category. The most commonly used machine learning procedures include artificial neural networks, random forests, deep learning algorithms, decision trees, support vector regression, fuzzy systems, etc. [30,31,32]. For example, the authors of [33] proposed a deep-learning-based framework for electricity demand forecasting. Cluster analysis was used to transform electricity consumption data into segmented data based on seasons. Based on seasonal, date, and duration data, a multi-input, multioutput short-term memory network model was trained to predict power demand and power consumption data for the Union Chandigarh region of India. Other competitive models were also used in the analysis, such as support vector regression models, artificial neural networks, and neural regression networks. For example, the work reported in [34] introduced a profile neural network based on convolutional neural networks, which is fast and promising. The developed architecture of deep neural networks was tested on actual electrical data from buildings on a university campus, notably those with high and low levels of seasonal changes. The novel architecture beat current sophisticated deep learning baseline models in prediction accuracy for 1-day and 1-week prediction horizons, improved the average absolute scale error by up to 25%, and improved the training time–accuracy tradeoff.

In the third category, to predict daily electricity demand, many researchers have developed new models by combining the features of two or more models. For instance, the authors of [35] proposed an adaptive probabilistic and interval forecasting system for multilevel electricity price forecasting. The authors presented two case studies and comparative studies to evaluate the performance of the proposal. The results showed that their proposal would be constructive for electricity market planners. Furthermore, for short-term load prediction, the authors of [36] introduced a new association model that combines the seasonal autoregressive integrated moving averages and backpropagation neural network models. The outcomes of the empirical analysis showed that the proposed hybrid model obtained highly accurate and efficient forecasts for short-term load forecasting. Similarly, the authors of [37] showed how to forecast electricity consumption in Turkey using a hybrid model based on the autoregressive integrated moving average model and the least-squares support vector machine model. The results of the proposed approach were evaluated using official forecast data, a single autoregressive integrated moving average model, literature reports, and multiple linear regression approaches. The results show that the model responded better to unexpected responses in the time series.

Another option to enhance performance is to preprocess the original time series data to produce a modified form of the time series that is more reliable [38,39,40,41]. When predicting energy-related time series, it is common to divide the original dataset into various subseries that may be independently anticipated and averaged to create a real-time time series prediction. The objective is to create a new time series with more or less periodic behavior that is easier to anticipate. This assumption depends on the fact that energy-associated variables are intimately associated with and impacted by climatic and societal factors that exhibit periodic behavior. Thus, to be more correctly anticipated, these subseries must be assigned to specific frequencies embedded in the general behavior of the time series. Empirical mode decomposition and variational mode decomposition are two such approaches, which separate time series into numerous subseries, each with correct oscillation behaviors. Both techniques have been used in a variety of forecasting tools, including autoregressive integrated moving averages, convolutional neural networks, multilayer perceptron, support vector regression, and long short-term memory models [42,43,44,45]. For instance, the authors of ref. [46] proposed a “decompose-predict-reconstruct” forecasting model based on empirical modal decomposition and a long short-term memory model. The proposed model effectively improved the load forecast’s efficiency and accuracy. Thus, as reported in these studies, preprocessed models outperform unprocessed models, demonstrating the beneficial effects of including pretreatment in predictive models, representing a fundamental step in the forecasting process because, as proven by previous studies, the performance of forecasting tools improves when correctly chosen and applied to the time series to be forecast.

Motivated by the abovementioned works, in this paper, we propose a novel decomposition combination modeling and forecasting framework that is relatively simple and easy to implement and that boosts the forecasting accuracy of short-term electricity demand. The proposed forecasting technique is built on the basis of a combination of unique proposed decomposition methods and traditional time series models. The proposed forecasting technique consists of the following steps. First, using the three novel decomposition methods, i.e., regression spline decomposition, smoothing spline decomposition, and hybrid decomposition methods, the hourly time series of electricity demand is divided into new subseries, namely the long-run trend series, a seasonal series, and a stochastic series. The decomposed subseries are then predicted using four well-known time series models (three univariate and one multivariate), namely linear autoregressive, non-linear autoregressive, and autoregressive integrated moving averages, as well as vector linear autoregressive and all conceivable combination models. As a result, separate predictive models are immediately merged to provide final forecasts a day ahead for electrical demand. The contributions of this work are summarized in Figure 2.

The rest of this document is formatted as follows. Section 2 describes the general steps of the proposed decomposition combination modeling and forecasting approach. Section 3 presents an experimental application of the proposed decomposition combination framework using Nord Pool hourly electricity demand data. Section 4 provides a detailed discussion of the proposed combination model versus the considered baseline models, including standard univariate and multivariate time series models with and without deterministic components. Finally, Section 5 discusses conclusions, limitations, and directions for future research.

2. The Proposed Decomposition Combination Modeling and Forecasting Procedure

This section describes a proposed decomposition combination modeling and forecasting approach for day-ahead electricity demand forecasting. As explained in the previous section, the electricity demand time series contains the following unique characteristics: linear or non-linear long-term trends, multiple seasonality (daily, weekly, monthly, quarterly, and annual), spikes or jumps (extreme values), high volatility, non-constant mean and variance, and holiday effects. Including these unique features in the model considerably improves the accuracy of predictions. To do this, the electricity demand time series is split into three new subseries—a linear long-term trend series, a seasonal series, and a stochastic series—using the proposed decomposition techniques. Details of these methods are described in the following subsections.

2.1. Modeling the Decomposition for Electricity Demand Time Series

This section describes the general procedure for the decomposition of the time series of electricity demand. To do this, the demand time series (${\mathfrak{d}}_{\mathfrak{n}}$) is decomposed into three new subseries: the long-run $\left({\mathfrak{l}}_{\mathfrak{n}}\right)$ series, a seasonal $\left({\mathfrak{s}}_{\mathfrak{n}}\right)$ series, and a stochastic $\left({\mathfrak{r}}_{\mathfrak{n}}\right)$ series. The mathematical representation of the decomposed subseries expressed as

$${\mathfrak{d}}_{\mathfrak{n}}={\mathfrak{l}}_{\mathfrak{n}}+{\mathfrak{s}}_{\mathfrak{n}}+{\mathfrak{r}}_{\mathfrak{n}}$$

(1)

Here, the long-run series is composed of a linear trend, and the yearly cycles, and the seasonal subseries (${\mathfrak{s}}_{\mathfrak{n}}$) are composed of the weekly (${\mathfrak{w}}_{\mathfrak{n}}$) and quarterly (${\mathfrak{q}}_{\mathfrak{n}}$) series, modeled as:

$${\mathfrak{l}}_{\mathfrak{n}}={\mathfrak{t}}_{\mathfrak{n}}+{\mathfrak{y}}_{\mathfrak{n}}$$

(2)

$${\mathfrak{s}}_{\mathfrak{n}}={\mathfrak{w}}_{\mathfrak{n}}+{\mathfrak{q}}_{\mathfrak{n}}$$

(3)

Therefore, for modeling purposes, the linear trend (${\mathfrak{t}}_{\mathfrak{n}}$) is a function of time (n), and the seasonal subseries (${\mathfrak{s}}_{\mathfrak{n}}$) is encompassed by the weekly (${\mathfrak{w}}_{\mathfrak{j}}$), quarterly (${\mathfrak{q}}_{\mathfrak{j}}$), and yearly ${\mathfrak{y}}_{\mathfrak{j}}$ cycles, which are functions of the (1, 2, 3,…, 7, 1, 2, 3,…, 7,…), (1, 2, 3,…, 90, 1, 2, 3,…, 90,…), and (1, 2, 3,…, 365, 1, 2, 3,…, 365,…) series, whereas the stochastic subseries, which describes the short-run dependence of demand series, is obtained as ${\mathfrak{r}}_{\mathfrak{n}}={\mathfrak{d}}_{\mathfrak{n}}-{\mathfrak{l}}_{\mathfrak{n}}-{\mathfrak{s}}_{\mathfrak{n}}$. However, to account for intraday periodicity, each load period is modeled separately in univariate models. This strategy is common in the literature [47]. The three novel decomposition methods proposed herein, i.e., the regression spline decomposition (RSD), smoothing spline decomposition (SSD), and hybrid decomposition (HD) methods, are explained in [40].

Hence, to graphically show the performance of the proposed decomposition techniques described above, the new decomposed subseries are presented in Figure 3 (left panel, SSD) and Figure 3 (right-panel, RSD). Both panels show the natural log demand time series (${\mathfrak{d}}_{\mathfrak{n}}$), long-run demand series (${\mathfrak{l}}_{\mathfrak{n}}$), and the seasonal demand series (${\mathfrak{s}}_{\mathfrak{n}}$). The figure shows that both methods used to decompose log(${\mathfrak{d}}_{\mathfrak{n}}$) adequately capture both dynamics, i.e., the long-run dynamics (linear trend and yearly cycles) and the seasonality (quarterly and weekly), which adequately capture the electricity demand series.

2.2. Modeling the Decomposed Subseries

After extracting the subseries from the electricity demand series using the proposed decomposition methods mentioned above, each subseries is estimated using various linear and non-linear, univariate, and multivariate time series models, such as linear autoregressive, non-linear autoregressive, autoregressive moving average, and vector linear autoregressive models. Such models are explained in detail in [5].

2.3. Evaluation Measures

Three standard accuracy measures, i.e., mean squared percent error (MSPE), mean absolute percent error (MAPE), Pearson’s correlation coefficient (PCC), and root mean squared error (RMSE), were calculated to validate the performance of all combined models obtained from the proposed decomposition combination modeling and forecasting approach. The formulas for MSPE, MAPE, PCC, and RMSE are presented below.

$$\begin{array}{ccc}\hfill \mathrm{MSPE}& =& \frac{1}{\mathrm{N}}\sum _{\mathrm{n}=1}^{\mathrm{N}}{\left(\frac{|{\mathfrak{d}}_{\mathrm{n}}-{\widehat{\mathfrak{d}}}_{\mathrm{n}}|}{|{\mathfrak{d}}_{\mathrm{n}}|}\right)}^2\times 100,\hfill \end{array}$$

(4)

$$\begin{array}{ccc}\hfill \mathrm{MAPE}& =& \frac{1}{\mathrm{N}}\sum _{\mathrm{n}=1}^{\mathrm{N}}\left(\frac{|{\mathfrak{d}}_{\mathrm{n}}-{\widehat{\mathfrak{d}}}_{\mathrm{n}}|}{|{\mathfrak{d}}_{\mathrm{n}}|}\right)\times 100,\hfill \end{array}$$

(5)

$$\begin{array}{ccc}\hfill \mathrm{PCC}& =& \mathrm{Correlation}\left({\mathfrak{d}}_{\mathfrak{n}}-{\widehat{\mathfrak{d}}}_n\right),\hfill \end{array}$$

(6)

$$\begin{array}{ccc}\hfill \mathrm{RMSE}& =& \sqrt{\sum _{\mathrm{n}=1}^{\mathrm{N}}\frac{{({\mathfrak{d}}_{\mathrm{n}}-{\widehat{\mathfrak{d}}}_{\mathrm{n}})}^2}{\mathrm{N}}},\hfill \end{array}$$

(7)

In the all above equations, ${\mathfrak{d}}_{\mathfrak{n}}$ is observed demand, and ${\widehat{\mathfrak{d}}}_n$ is forecasted demand for the $n^{th}$ observation (n = 1, 2, …, 8760).

In this work, to convince the reader, we represent every combination model with each decomposition method (${}^{\mathfrak{l}}$SSD${}_{\mathfrak{r}}^{\mathfrak{s}}$, top-left (long-run), top-right (seasonal), and bottom-right (residual) subseries). For the estimation models, we assign numbers to each model, e.g., linear autoregressive (1), non-linear autoregressive (2), autoregressive moving average (3), and vector linear autoregressive (4). For example, ${}^1$RSD${}_2^3$ indicates that under the long-run series ($\mathfrak{l}$), a linear autoregressive seasonal series ($\mathfrak{s}$) is modeled by the non-linear autoregressive model, and the residual ($\mathfrak{r}$) is modeled by the autoregressive moving average model. Combining these individual models for the subseries estimates leads us to compare sixty-four $(4^{{\mathfrak{l}}_{\mathfrak{n}}}\times 4^{{\mathfrak{s}}_{\mathfrak{n}}}\times 4^{{\mathfrak{r}}_{\mathfrak{n}}}$ = 64) different combinations for a decomposition method. Therefore, the three proposed decomposition methods (RSD, SSD, and HD) correspond to a total of 192 = ($3\times 64$) models. All forecasting models are directly linked to each other as follows to achieve the final demand forecast one day ahead.

$$\begin{array}{ccc}\hfill {\widehat{\mathfrak{d}}}_{n+1}& =& \left({\widehat{\mathfrak{l}}}_{n+1}+{\widehat{\mathfrak{s}}}_{n+1}+{\widehat{\mathfrak{r}}}_{n+1}\right)\hfill \\ & =& \left(\widehat{\widehat{}}{\mathfrak{t}}_{n+1}+{\widehat{\mathfrak{y}}}_{n+1}+{\widehat{\mathfrak{q}}}_{n+1}+{\widehat{\mathfrak{w}}}_{n+1}+{\widehat{\mathfrak{r}}}_{n+1}\right)\hfill \end{array}$$

(8)

Figure 4 depicts the design of the developed decomposition combination modeling and forecasting technique.

3. Case Study Outcomes

This research work primarily focuses on fluctuations in electricity demand before COVID-19. For this purpose, this work uses the hourly electricity demand (MWh) according to six years of data from the Nordic electricity market collected from 01.01.2014 to 12.31.2019. Each day consists of twenty-four data points, with each data point corresponding to a load period. Therefore, the dataset contains 52,584 data points (2191 days). For modeling and forecasting purposes, we split the data into two parts: a training part (model estimation) and a testing part (out-of-sample forecast). The training part comprised the first five years of data from 01.01.2014 to 12.31.2018 (1826 days or 43,824 observations), spanning five years and accounting for approximately 80% of the total data, with a period of a complete year from 01.01.2019 to 12.31.2019 (365 days or 8760 observations) used for the out-of-sample forecast. Graphical representations of model training and model testing are shown in Figure 5 (top). Moreover, the density plots for the hourly demand time series and log hourly demand series are shown in Figure 5 (bottom right) and Figure 5 (bottom left), respectively. The descriptive statistics of the hourly time series of electricity demand are listed in

Table 1. Descriptive statistics of hourly time series electricity demand in the Nordic electricity market.

Series No.	Statistic	Demand	log(Demand)
1	Minimun	24,739.000	10.120
2	First quartile	35,990.000	10.490
3	Median	40,598.000	10.610
4	Mean	41,882.000	10.620
5	Variance	65,420,529.000	0.037
6	Standard deviation	8088.296	0.192
7	Skewness	0.416	0.036
8	Kurtosis	−0.526	−0.645
9	Third quartile	47,455.000	10.770
10	Maximum	65,311.000	11.090

, where the maximum value is 65,311, and the minimum value is 24,739. However, the log demand series and the series without log demand present significant differences, as confirmed by Table 1.

The following procedures must be taken into account in order to produce a day-ahead forecast of electricity demand using the proposed decomposition combination modeling and forecasting technique. To stabilize the variance of the electricity demand time series, the first natural logarithmic transformation was applied. Second, we used the three proposed decomposition techniques to develop temporal subseries for the linear long-term (${\mathfrak{l}}_{\mathfrak{n}}$), seasonal (${\mathfrak{s}}_{\mathfrak{n}}$), and stochastic (${\mathfrak{r}}_{\mathfrak{n}}$) series. Third, each subseries was subjected to the previously discussed univariate and multivariate models. As a result, the models were estimated using a rolling window technique to achieve a forecast for 365 days (8760 data points). For this purpose, Equation (8) was used to calculate the final one-day-ahead demand forecast. The models’ performances were then evaluated and compared using the MAPE, MSPE, PCC, and RMSE performance metrics, as well as a statistical test (DM test).

Three proposed decomposition techniques were used in this work when decomposing the hourly electricity demand time series (${\mathfrak{d}}_{\mathfrak{n}}$) into three new subseries: a linear long-term (${\mathfrak{l}}_{\mathfrak{n}}$), seasonal (${\mathfrak{s}}_{\mathfrak{n}}$), and stochastic (${\mathfrak{r}}_{\mathfrak{n}}$) subseries. Four distinct univariate and multivariate time series models were used to forecast these subseries. For univariate and multivariate models, three main combinations are possible: univariate–univariate, univariate–multivariate, and multivariate–multivariate. Combining the models for the subseries estimates led us to compare sixty-four $(4^{{\mathfrak{t}}_{\mathfrak{n}}}\ast 4^{{\mathfrak{s}}_{\mathfrak{n}}}\ast 4^{{\mathfrak{r}}_{\mathfrak{n}}}$ = 64) combinations for each presented decomposition technique, corresponding to a total of 192 (192 = $3\times 64$) models for the three presented decomposition techniques (SSD, RSD, and HD). The out-of-sample predictions for one-day-ahead accuracy metrics (MAPE, MSPE, PCC, and RMSE) for these 192 models are presented in

Table 2. Nordic electricity market day-ahead out-of-sample accuracy measures for all models combined with the SSD decomposition technique.

Series No.	Model	MSPE	MAPE	PCC	RMSE	Series No.	Model	MSPE	MAPE	PCC	RMSE
1	${}^1$SSD${}_1^1$	0.4330	4.8380	0.9200	2519.0680	33	${}^3$SSD${}_1^1$	0.4350	4.8440	0.9190	2522.3280
2	${}^1$SSD${}_2^1$	0.4350	4.8400	0.9190	2522.7700	34	${}^3$SSD${}_2^1$	0.4360	4.8460	0.9190	2526.0140
3	${}^1$SSD${}_3^1$	0.3990	4.5790	0.9270	2368.8810	35	${}^3$SSD${}_3^1$	0.4000	4.5850	0.9270	2373.0780
4	${}^1$SSD${}_4^1$	0.3980	4.5370	0.9280	2384.3290	36	${}^3$SSD${}_4^1$	0.3990	4.5430	0.9280	2387.9700
5	${}^1$SSD${}_1^2$	0.1690	3.0230	0.9660	1625.6310	37	${}^3$SSD${}_1^1$	0.1690	3.0190	0.9660	1623.0640
6	${}^1$SSD${}_2^2$	0.1680	3.0160	0.9660	1621.5420	38	${}^3$SSD${}_2^2$	0.1680	3.0130	0.9670	1618.9200
7	${}^1$SSD${}_3^2$	0.1380	2.6980	0.9730	1402.7460	39	${}^3$SSD${}_3^2$	0.1380	2.6960	0.9730	1401.1680
8	${}^1$SSD${}_4^2$	0.1460	2.7690	0.9710	1502.6650	40	${}^3$SSD${}_4^2$	0.1460	2.7690	0.9720	1500.8010
9	${}^1$SSD${}_1^3$	0.2870	4.0410	0.9460	2076.6530	41	${}^3$SSD${}_1^2$	0.2880	4.0420	0.9460	2077.2260
10	${}^1$SSD${}_2^3$	0.2870	4.0390	0.9460	2075.5240	42	${}^3$SSD${}_2^3$	0.2880	4.0410	0.9460	2076.0680
11	${}^1$SSD${}_3^3$	0.2550	3.7590	0.9530	1899.5340	43	${}^3$SSD${}_3^3$	0.2550	3.7620	0.9530	1901.1900
12	${}^1$SSD${}_4^3$	0.2570	3.7510	0.9530	1940.4910	44	${}^3$SSD${}_4^3$	0.2580	3.7540	0.9530	1941.5300
13	${}^1$SSD${}_1^4$	0.1070	2.4360	0.9770	1347.9260	45	${}^3$SSD${}_1^3$	0.1070	2.4320	0.9770	1343.9180
14	${}^1$SSD${}_2^4$	0.1060	2.4290	0.9770	1343.4840	46	${}^3$SSD${}_2^4$	0.1060	2.4250	0.9770	1339.4120
15	${}^1$SSD${}_3^4$	0.0770	2.0510	0.9830	1087.7100	47	${}^3$SSD${}_3^4$	0.0770	2.0490	0.9830	1084.6090
16	${}^1$SSD${}_4^4$	0.0850	2.1330	0.9820	1201.7080	48	${}^3$SSD${}_4^4$	0.0850	2.1300	0.9820	1198.6600
17	${}^2$SSD${}_1^1$	0.4320	4.8340	0.9200	2515.2640	49	${}^4$SSD${}_1^4$	0.4280	4.8150	0.9210	2489.3260
18	${}^2$SSD${}_2^1$	0.4340	4.8350	0.9200	2519.3450	50	${}^4$SSD${}_2^1$	0.4310	4.8200	0.9210	2494.8200
19	${}^2$SSD${}_3^1$	0.4000	4.5870	0.9270	2369.1690	51	${}^4$SSD${}_3^1$	0.3940	4.5520	0.9290	2345.3670
20	${}^2$SSD${}_4^1$	0.3980	4.5330	0.9280	2380.9650	52	${}^4$SSD${}_4^1$	0.3930	4.5070	0.9290	2354.5170
21	${}^2$SSD${}_1^2$	0.1680	3.0140	0.9670	1618.4510	53	${}^4$SSD${}_1^1$	0.1640	3.0010	0.9680	1579.8900
22	${}^2$SSD${}_2^2$	0.1670	3.0070	0.9670	1615.0480	54	${}^4$SSD${}_2^2$	0.1630	2.9970	0.9680	1579.1090
23	${}^2$SSD${}_3^2$	0.1380	2.7010	0.9730	1400.9060	55	${}^4$SSD${}_3^2$	0.1330	2.6730	0.9740	1363.4370
24	${}^2$SSD${}_4^2$	0.1450	2.7670	0.9720	1496.0220	56	${}^4$SSD${}_4^2$	0.1410	2.7440	0.9730	1455.4610
25	${}^2$SSD${}_1^3$	0.2870	4.0390	0.9470	2072.5530	57	${}^4$SSD${}_1^3$	0.2820	4.0160	0.9480	2037.9720
26	${}^2$SSD${}_2^3$	0.2860	4.0380	0.9470	2071.8590	58	${}^4$SSD${}_2^3$	0.2820	4.0180	0.9480	2039.0850
27	${}^2$SSD${}_3^3$	0.2550	3.7690	0.9530	1899.9340	59	${}^4$SSD${}_3^3$	0.2490	3.7290	0.9550	1867.1160
28	${}^2$SSD${}_4^3$	0.2570	3.7510	0.9530	1936.7190	60	${}^4$SSD${}_4^3$	0.2510	3.7140	0.9550	1900.6590
29	${}^2$SSD${}_1^4$	0.1050	2.4240	0.9770	1336.5310	61	${}^4$SSD${}_1^4$	0.1020	2.4200	0.9780	1292.7420
30	${}^2$SSD${}_2^4$	0.1040	2.4180	0.9770	1332.9160	62	${}^4$SSD${}_2^4$	0.1010	2.4150	0.9780	1292.6320
31	${}^2$SSD${}_3^4$	0.0770	2.0510	0.9830	1081.8090	63	${}^4$SSD${}_3^4$	0.0720	2.0280	0.9850	1037.4010
32	${}^2$SSD${}_4^4$	0.0830	2.1260	0.9820	1190.0170	64	${}^4$SSD${}_4^4$	0.0790	2.1070	0.9830	1141.1880

,

Table 3. Nordic electricity market day-ahead out-of-sample accuracy measures for all models combined with the RSD decomposition technique.

Series No.	Model	MSPE	MAPE	PCC	RMSE	Series No.	Model	MSPE	MAPE	PCC	RMSE
1	${}^1$RSD${}_1^1$	0.4270	4.7800	0.9210	2511.9550	33	${}^3$RSD${}_1^1$	0.4270	4.7790	0.9210	2511.8960
2	${}^1$RSD${}_2^1$	0.4340	4.8230	0.9200	2529.8830	34	${}^3$RSD${}_2^1$	0.4340	4.8210	0.9200	2529.2340
3	${}^1$RSD${}_3^1$	0.3950	4.5550	0.9280	2390.6800	35	${}^3$RSD${}_3^1$	0.3950	4.5530	0.9280	2390.5720
4	${}^1$RSD${}_4^1$	0.4000	4.5400	0.9280	2390.5580	36	${}^3$RSD${}_4^1$	0.3990	4.5370	0.9280	2389.5070
5	${}^1$RSD${}_1^2$	0.1580	2.9520	0.9690	1598.9190	37	${}^3$RSD${}_1^1$	0.1580	2.9510	0.9690	1598.7270
6	${}^1$RSD${}_2^2$	0.1590	2.9630	0.9680	1606.5720	38	${}^3$RSD${}_2^2$	0.1590	2.9610	0.9680	1605.5820
7	${}^1$RSD${}_3^2$	0.1230	2.5970	0.9760	1402.7120	39	${}^3$RSD${}_3^2$	0.1230	2.5960	0.9760	1402.4370
8	${}^1$RSD${}_4^2$	0.1370	2.7100	0.9730	1477.3220	40	${}^3$RSD${}_4^2$	0.1370	2.7080	0.9730	1475.8490
9	${}^1$RSD${}_1^3$	0.2790	3.9810	0.9480	2055.9620	41	${}^3$RSD${}_1^2$	0.2790	3.9820	0.9480	2056.4100
10	${}^1$RSD${}_2^3$	0.2830	4.0090	0.9480	2069.6610	42	${}^3$RSD${}_2^3$	0.2830	4.0090	0.9480	2069.4310
11	${}^1$RSD${}_3^3$	0.2460	3.6980	0.9550	1911.4470	43	${}^3$RSD${}_3^3$	0.2460	3.6980	0.9550	1911.8340
12	${}^1$RSD${}_4^3$	0.2540	3.7250	0.9540	1932.3820	44	${}^3$RSD${}_4^3$	0.2540	3.7230	0.9540	1931.6680
13	${}^1$RSD${}_1^4$	0.1030	2.3950	0.9780	1336.5860	45	${}^3$RSD${}_1^3$	0.1030	2.3930	0.9780	1335.3890
14	${}^1$RSD${}_2^4$	0.1050	2.4170	0.9770	1349.3580	46	${}^3$RSD${}_2^4$	0.1040	2.4150	0.9770	1347.2450
15	${}^1$RSD${}_3^4$	0.0700	1.9880	0.9850	1109.6090	47	${}^3$RSD${}_3^4$	0.0700	1.9850	0.9850	1108.1630
16	${}^1$RSD${}_4^4$	0.0840	2.1270	0.9820	1199.0310	48	${}^3$RSD${}_4^4$	0.0830	2.1240	0.9820	1196.1610
17	${}^2$RSD${}_1^1$	0.4270	4.7770	0.9210	2508.8050	49	${}^4$RSD${}_1^4$	0.4240	4.7670	0.9220	2495.4730
18	${}^2$RSD${}_2^1$	0.4330	4.8200	0.9200	2526.7790	50	${}^4$RSD${}_2^1$	0.4310	4.8090	0.9210	2513.5320
19	${}^2$RSD${}_3^1$	0.3950	4.5570	0.9280	2389.0680	51	${}^4$RSD${}_3^1$	0.3920	4.5360	0.9290	2374.1880
20	${}^2$RSD${}_4^1$	0.3990	4.5350	0.9280	2387.2300	52	${}^4$RSD${}_4^1$	0.3960	4.5180	0.9290	2370.3870
21	${}^2$RSD${}_1^2$	0.1570	2.9490	0.9690	1594.7440	53	${}^4$RSD${}_1^1$	0.1550	2.9430	0.9690	1577.0550
22	${}^2$RSD${}_2^2$	0.1590	2.9600	0.9680	1602.6560	54	${}^4$RSD${}_2^2$	0.1570	2.9520	0.9690	1585.7200
23	${}^2$RSD${}_3^2$	0.1230	2.5980	0.9760	1400.6370	55	${}^4$RSD${}_3^2$	0.1210	2.5820	0.9770	1380.0170
24	${}^2$RSD${}_4^2$	0.1370	2.7070	0.9730	1472.9740	56	${}^4$RSD${}_4^2$	0.1340	2.6910	0.9740	1449.0940
25	${}^2$RSD${}_1^3$	0.2790	3.9780	0.9480	2052.6990	57	${}^4$RSD${}_1^3$	0.2760	3.9680	0.9490	2034.8400
26	${}^2$RSD${}_2^3$	0.2820	4.0070	0.9480	2066.5130	58	${}^4$RSD${}_2^3$	0.2800	3.9960	0.9490	2048.9300
27	${}^2$RSD${}_3^3$	0.2460	3.6990	0.9550	1909.8130	59	${}^4$RSD${}_3^3$	0.2420	3.6810	0.9560	1889.8340
28	${}^2$RSD${}_4^3$	0.2540	3.7210	0.9540	1928.7870	60	${}^4$RSD${}_4^3$	0.2500	3.6970	0.9550	1906.1540
29	${}^2$RSD${}_1^4$	0.1020	2.3900	0.9780	1330.2080	61	${}^4$RSD${}_1^4$	0.1000	2.3890	0.9790	1311.2700
30	${}^2$RSD${}_2^4$	0.1040	2.4130	0.9780	1343.4130	62	${}^4$RSD${}_2^4$	0.1020	2.4080	0.9780	1325.7590
31	${}^2$RSD${}_3^4$	0.0700	1.9890	0.9850	1105.3920	63	${}^4$RSD${}_3^4$	0.0680	1.9740	0.9860	1081.6590
32	${}^2$RSD${}_4^4$	0.0830	2.1230	0.9820	1192.1570	64	${}^4$RSD${}_4^4$	0.0800	2.1100	0.9830	1163.9230

and

Table 4. Nordic electricity market day-ahead out-of-sample accuracy measures for all models combined with the HD decomposition technique.

Series No.	Model	MSPE	MAPE	PCC	RMSE	Series No.	Model	MSPE	MAPE	PCC	RMSE
1	${}^1$HD${}_1^1$	0.4290	4.7940	0.9210	2002.9227	33	${}^3$HD${}_1^1$	0.4300	4.8000	0.9210	2004.9415
2	${}^1$HD${}_2^1$	0.4310	4.7940	0.9200	2004.1631	34	${}^3$HD${}_2^1$	0.4330	4.8000	0.9200	2006.3179
3	${}^1$HD${}_3^1$	0.3880	4.4910	0.9290	1900.3824	35	${}^3$HD${}_3^1$	0.3890	4.4970	0.9290	1902.7926
4	${}^1$HD${}_4^1$	0.3960	4.5220	0.9280	1880.3001	36	${}^3$HD${}_4^1$	0.3980	4.5280	0.9280	1882.6063
5	${}^1$HD${}_1^2$	0.1630	2.9660	0.9680	1245.8358	37	${}^3$HD${}_1^1$	0.1630	2.9630	0.9680	1243.5373
6	${}^1$HD${}_2^2$	0.1630	2.9630	0.9680	1242.9326	38	${}^3$HD${}_2^2$	0.1630	2.9610	0.9680	1240.9328
7	${}^1$HD${}_3^2$	0.1250	2.5670	0.9760	1114.1985	39	${}^3$HD${}_3^2$	0.1250	2.5650	0.9760	1112.8599
8	${}^1$HD${}_4^2$	0.1420	2.7310	0.9720	1147.4354	40	${}^3$HD${}_4^2$	0.1420	2.7310	0.9720	1146.8686
9	${}^1$HD${}_1^3$	0.2840	4.0070	0.9470	1665.0723	41	${}^3$HD${}_1^2$	0.2840	4.0090	0.9470	1665.2460
10	${}^1$HD${}_2^3$	0.2840	4.0050	0.9470	1664.5225	42	${}^3$HD${}_2^3$	0.2850	4.0070	0.9470	1664.9015
11	${}^1$HD${}_3^3$	0.2430	3.6650	0.9560	1553.5288	43	${}^3$HD${}_3^3$	0.2440	3.6680	0.9560	1554.9985
12	${}^1$HD${}_4^3$	0.2560	3.7360	0.9540	1549.1223	44	${}^3$HD${}_4^3$	0.2560	3.7380	0.9540	1549.7197
13	${}^1$HD${}_1^4$	0.1040	2.3910	0.9780	1005.1318	45	${}^3$HD${}_1^3$	0.1030	2.3880	0.9780	1002.8055
14	${}^1$HD${}_2^4$	0.1030	2.3870	0.9780	1001.9463	46	${}^3$HD${}_2^4$	0.1030	2.3850	0.9780	999.8675
15	${}^1$HD${}_3^4$	0.0670	1.9210	0.9850	849.9637	47	${}^3$HD${}_3^4$	0.0670	1.9200	0.9860	848.8607
16	${}^1$HD${}_4^4$	0.0840	2.1100	0.9820	883.7530	48	${}^3$HD${}_4^4$	0.0830	2.1070	0.9820	881.8987
17	${}^2$HD${}_1^1$	0.4280	4.7920	0.9210	2000.7327	49	${}^4$HD${}_1^4$	0.4230	4.7690	0.9220	1991.9499
18	${}^2$HD${}_2^1$	0.4310	4.7920	0.9200	2001.7646	50	${}^4$HD${}_2^1$	0.4260	4.7730	0.9220	1994.6764
19	${}^2$HD${}_3^1$	0.3890	4.5010	0.9290	1903.5566	51	${}^4$HD${}_3^1$	0.3840	4.4720	0.9310	1887.4206
20	${}^2$HD${}_4^1$	0.3960	4.5180	0.9280	1878.0788	52	${}^4$HD${}_4^1$	0.3910	4.4950	0.9300	1865.4914
21	${}^2$HD${}_1^2$	0.1630	2.9610	0.9680	1242.3638	53	${}^4$HD${}_1^1$	0.1570	2.9400	0.9690	1234.9295
22	${}^2$HD${}_2^2$	0.1620	2.9580	0.9680	1239.5933	54	${}^4$HD${}_2^2$	0.1570	2.9380	0.9690	1233.0628
23	${}^2$HD${}_3^2$	0.1250	2.5730	0.9760	1115.2920	55	${}^4$HD${}_3^2$	0.1210	2.5500	0.9770	1101.3930
24	${}^2$HD${}_4^2$	0.1420	2.7310	0.9730	1146.7243	56	${}^4$HD${}_4^2$	0.1360	2.7030	0.9740	1135.0276
25	${}^2$HD${}_1^3$	0.2830	4.0080	0.9470	1665.1848	57	${}^4$HD${}_1^3$	0.2770	3.9790	0.9490	1653.4191
26	${}^2$HD${}_2^3$	0.2840	4.0060	0.9470	1664.5514	58	${}^4$HD${}_2^3$	0.2780	3.9810	0.9490	1654.2733
27	${}^2$HD${}_3^3$	0.2440	3.6750	0.9560	1558.2125	59	${}^4$HD${}_3^3$	0.2380	3.6420	0.9570	1539.5891
28	${}^2$HD${}_4^3$	0.2560	3.7370	0.9540	1549.3139	60	${}^4$HD${}_4^3$	0.2490	3.7000	0.9550	1531.2569
29	${}^2$HD${}_1^4$	0.1020	2.3810	0.9780	999.1348	61	${}^4$HD${}_1^4$	0.0980	2.3660	0.9790	996.2893
30	${}^2$HD${}_2^4$	0.1020	2.3780	0.9780	996.8985	62	${}^4$HD${}_2^4$	0.0980	2.3640	0.9790	994.1642
31	${}^2$HD${}_3^4$	0.0670	1.9270	0.9860	848.7548	63	${}^4$HD${}_3^4$	0.0630	1.9080	0.9870	837.6524
32	${}^2$HD${}_4^4$	0.0820	2.1050	0.9830	880.1453	64	${}^4$HD${}_4^4$	0.0780	2.0790	0.9840	870.6488

. The combined univariate–multivariate models outperform the univariate–univariate and multivariate–multivariate models according to the performance metrics. Although multivariate–multivariate combination models produced smaller forecast errors than univariate–univariate models, the mean errors are still larger than those of univariate–multivariate models. In addition, the ${}^4$SSD${}_3^4$ model is confirmed to provide better predictions than all other combination models using the SSD decomposition technique. The best forecasting model (${}^4$SSD${}_3^4$) achieved values of 2.028, 0.072, and 0.985 1037.4010 for MAPE, MSPE, PCC, and RMSE, respectively. The ${}^3$SSD${}_3^4$, ${}^2$SSD${}_3^4$, and ${}^1$SSD${}_3^4$ models gave the second, third, and fourth best results within all combination using the SSD method. Using the RSD decomposition method, the combined prediction model (${}^4$RSD${}_3^4$) achieved the lowest prediction errors, with values of 1.974, 0.068 0.986, and 1081.6590 for MAPE, MSPE, PCC, and RMSE, respectively, with the second, third, and fourth best results achieved by the ${}^3$RSD${}_3^4$, ${}^2$RSD${}_3^4$, and ${}^1$RSD${}_3^4$ models, respectively. On the other hand, the combined forecasting model (${}^4$HD${}_3^4$) achieved the lowest prediction errors using the HD decomposition method, with values of 1.908, 0.063, 0.989, and 994.1642 for MAPE, MSPE, PCC, and RMSE, respectively. The second, third, and fourth best results were achieved by the ${}^3$HD${}_3^4$, ${}^2$HD${}_3^4$, and ${}^1$HD${}_3^4$ models, respectively, within all combination models using the HD method. Among all 192 models, the ${}^4$HD${}_3^4$ model achieved the best forecast using the three decomposition approaches (SSD, RSD, and HD). Based on these conclusions, it is confirmed that univariate–multivariate combinations achieve more accurate predictions competitive model combinations. In addition, the HD decomposition method achieved the lowest mean error among the proposed decomposition methods.

Among the three proposed decomposition techniques, the best four combination models from each decomposition technique were chosen and compared. The mean performance metrics are listed in

Table 5. Nordic electricity market day-ahead out-of-sample accuracy measures for the best twelve models among all one hundred and ninety-two combination models.

Series No.	Model	MSPE	MAPE	PCC	RMSE
1	${}^4$SSD${}_3^4$	0.0719	2.0282	0.9845	1081.6590
2	${}^2$SSD${}_3^4$	0.0766	2.0506	0.9833	1105.3920
3	${}^3$SSD${}_3^4$	0.0768	2.0492	0.9832	1108.1630
4	${}^1$SSD${}_3^4$	0.0771	2.0513	0.9831	1109.6090
5	${}^4$RSD${}_3^4$	0.0678	1.974	0.9856	1081.659
6	${}^2$RSD${}_3^4$	0.07	1.9891	0.985	1108.163
7	${}^3$RSD${}_3^4$	0.0701	1.9853	0.9849	1109.609
8	${}^1$RSD${}_3^4$	0.0702	1.9876	0.9849	1105.392
9	${}^4$HD${}_3^4$	0.063	1.9082	0.9867	1037.4010
10	${}^2$HD${}_3^4$	0.0671	1.9272	0.9856	1081.8090
11	${}^3$HD${}_3^4$	0.067	1.9205	0.9856	1084.6090
12	${}^1$HD${}_3^4$	0.0673	1.921	0.9855	1087.7100

. The ${}^4$HD${}_3^4$ model produced the optimal accuracy metric values (MSPE = 0.063, MAPE = 1.908, PCC = 0.989, and RMSE = 1037.4010) when comparing the results of this model with those of the best remaining model using the proposed decomposition method. Based on the results presented above, we found that the RSD method has the highest prediction accuracy among the proposed decomposition methods.

Once the performance of the proposed best model was determined according to mean accuracy errors, in the next stage, the Diebold and Mariano (DM) test [48] was used to examine the importance of the differences in prediction abilities between these models. The DM test is a statistical approach widely used in the literature to evaluate predictions from two models [49,50,51]. As a consequence, Table 5 shows a DM test on each pair of models to confirm the superiority of the model outcomes (mean performance measure). The DM test results (p values) for day-ahead out-of-sample metrics for the best twelve models are shown in

Table 6. Nordic electricity market outcomes (p value) of the DM test for the twelve best models.

Model	${}^4$SSD${}_3^4$	${}^2$SSD${}_3^4$	${}^3$SSD${}_3^4$	${}^1$SSD${}_3^4$	${}^4$RSD${}_3^4$	${}^2$RSD${}_3^4$	${}^3$RSD${}_3^4$	${}^1$RSD${}_3^4$	${}^4$HD${}_3^4$	${}^2$HD${}_3^4$	${}^3$HD${}_3^4$	${}^1$HD${}_3^4$
${}^4$SSD${}_3^4$	-	0.7413	0.6996	0.7056	0.1441	0.1515	0.1555	0.1598	0.2180	0.2131	0.2510	0.2585
${}^2$SSD${}_3^4$	0.2587	-	0.4813	0.5048	0.1429	0.0840	0.0855	0.0872	0.1868	0.0961	0.1322	0.1362
${}^3$SSD${}_3^4$	0.3004	0.5187	-	0.5493	0.1664	0.1096	0.0930	0.0920	0.2011	0.1125	0.1123	0.1195
${}^1$SSD${}_3^4$	0.2944	0.4952	0.4507	-	0.1642	0.1067	0.0889	0.0869	0.1988	0.1099	0.1131	0.1119
${}^4$RSD${}_3^4$	0.8559	0.8571	0.8336	0.8358	-	0.5507	0.5137	0.5313	0.5765	0.5201	0.5421	0.5498
${}^2$RSD${}_3^4$	0.8485	0.9160	0.8904	0.8934	0.4493	-	0.4167	0.4695	0.5307	0.4891	0.5260	0.5374
${}^3$RSD${}_3^4$	0.8445	0.9145	0.9070	0.9111	0.4863	0.5833	-	0.6056	0.5522	0.5179	0.5551	0.5687
${}^1$RSD${}_3^4$	0.8402	0.9128	0.9080	0.9131	0.4688	0.5306	0.3944	-	0.5376	0.5007	0.5394	0.5530
${}^4$HD${}_3^4$	0.7821	0.8132	0.7989	0.8012	0.4235	0.4693	0.4478	0.4624	-	0.4599	0.4963	0.5066
${}^2$HD${}_3^4$	0.7869	0.9039	0.8875	0.8902	0.4800	0.5109	0.4821	0.4993	0.5402	-	0.5669	0.5883
${}^3$HD${}_3^4$	0.7490	0.8679	0.8877	0.8869	0.4579	0.4740	0.4449	0.4606	0.5037	0.4331	-	0.5511
${}^1$HD${}_3^4$	0.7415	0.8639	0.8805	0.8882	0.4502	0.4626	0.4313	0.4470	0.4934	0.4117	0.4489	-

. All twelve models listed in Table 5 are statistically significant at the 5% significance level, while using the HD method, the best four models have high p values compared to the two other proposed decomposition methods. Therefore, according to the accuracy mean errors and a statistical test, we confirm the superiority of the HD model.

Once the models were tested using the accuracy metrics (MSPE, MAPE, PCC, and RMSE) and a statistical test (the DM test), a graphic analysis was conducted. For instance, a graphical representation of the MSPE, MAPE, PCC, and RMSE for all one hundred and ninety-two models is presented in Figure 6. Figure 6 shows that the lowest mean error and the highest PCC values were produced by the ${}^4$SSD${}_3^4$ models, and the ${}^3$SSD${}_3^4$, ${}^2$SSD${}_3^4$, and ${}^1$SSD${}_3^4$ models produced the second, third, and fourth best results using SSD method, respectively, while the lowest mean errors and highest Pearson correlation coefficient (MAPE, MSPE, PCC, and RMSE) values were obtained by the ${}^4$RSD${}_3^4$, ${}^3$RSD${}_3^4$, ${}^2$RSD${}_3^4$, and ${}^1$RSD${}_3^4$ models (Figure 6). In the same way, the lowest mean errors and highest correlation coefficient (MAPE, MSPE, PCC, and RMSE) values were obtained by the ${}^4$HD${}_3^4$, ${}^3$HD${}_3^4$, ${}^2$HD${}_3^4$, and ${}^1$HD${}_3^4$ models (Figure 6).

On the other hand, Figure 7, Figure 8 and Figure 9 show the scatter plots of the twelve best models (four for each decomposition approach), as well as their PCC values. Figure 7, Figure 8 and Figure 9 illustrate that the best models have the largest PCC values, with a substantial correlation between actual and predicted values. Furthermore, the hourly MAPE and MSPE for the twelve best models determined using the proposed decomposition techniques (SSD, RSD, and HD) are plotted in Figure 10 for MPAE (A, C, and E) and MSPE (B, D, and F). The mean hourly errors are lower during the starting hours of the day and slowly increase, with the first peak around 8:00 a.m. After the first peak, the mean error values monotonically decrease, then gradually increase, with the second peak around 4:00 p.m. Finally, Figure 11 depicts the original and forecasted values for the twelve best models. According to this visualization, the best models’ forecasts closely match observed demand. In addition, this figure confirms that the weekly period includes relatively less volatility in the electricity demand time series, load profiles vary considerably depending on the day of the week, and demand fluctuates throughout the day. Weekdays have more demand than weekends. Therefore, we can determine that the twelve best models outperformed all others of the one hundred and ninety-two models. Finally, based on these descriptive and graphical results, we conclude the univariate–multivariate combined forecasting models achieve the least forecasting errors relative to their counterparts.

4. Discussion

In this section, we compare the results of the best combination model identified in this study with those reported in the literature, as well as those of considered benchmark models. We found our model to be highly comparable with the considered methods.

Table 7. Nordic electricity market day-ahead accuracy measures of the proposed models versus those presented in the literature.

Model	MSPE	MAPE	PCC	RMSE
${}^4$HD${}_3^4$	0.0630	1.9082	0.9867	1037.401
Stoc-AR [52]	0.2310	3.0134	0.9491	2845.412
RS-VAR [5]	0.1878	2.5587	0.9622	1245.324
${\mathrm{VAR}}_{\mathrm{E}}^{\mathrm{NP}}$ [3]	0.1778	2.5587	0.9702	1207.531

and Figure 12 show numerical and graphical comparisons of the best (${}^4$HD${}_3^4$) model identified in this study with the models proposed by other researchers in the literature. Our best (${}^4$HD${}_3^4$) model produces the smallest mean errors and the highest correlation coefficient (MSPE = 0.0630, MAPE = 1.9082, RMSE = 1037.401, and PCC = 0.9867) in comparison with the best models reported in the literature. For instance, the best stochastic autoregressive (Stoc-AR) proposed in [52] was applied to the dataset used in this study, and its accuracy measure (MSPE = 0.2310, MAPE = 3.0134, PCC = 0.9491, and RMSE = 2845.412) was determined and found to be significantly greater than that of our best (${}^4$HD${}_3^4$) model. The best model proposed in another study (see [5]), the regression spline vector autoregressive (RS-VAR) model, was applied to dataset used in the present study and obtained mean errors and Pearson correlation coefficients (MSPE, MAPE, PCC, and RMSE of 0.1878, 2.5587, 0.9622, and 1245.32, respectively) that were also relatively higher than those of our best (${}^4$HD${}_3^4$) model. Similarly, in reference [3], the best proposed nonparametric vector autoregressive model with an elastic net component (VAR${}_{\mathrm{E}}^{\mathrm{NP}}$) applied to working dataset used in the present study obtained performance metrics (MSPE = 0.1778, MAPE = 2.5587, PCC = 0.9702, and RMSE = 1207.53) worse than those obtained with our best combination model (${}^4$HD${}_3^4$). In this sense, we conclude that the best final model (${}^4$HD${}_3^4$) in this study showed high efficacy and high accuracy compared to the best models reported in the literature.

Furthermore, to assess the performance of our best combination model (${}^4$HD${}_3^4$), we also compared it with different standard benchmark models, including the autoregressive (AR), nonparametric autoregressive (NPAR), autoregressive integrated moving average (ARIMA), and vector autoregressive (VAR) models, all of which either included or exclude a deterministic component. In the deterministic component, we included a linear or non-linear long-run trend and yearly, quarterly, and weekly periodicities. Thus, we considered models without deterministic components, such as AR, ARIMA, NPAR, and VAR, as well as models with deterministic components, such as AR-X, ARIMA-X, NPAR-X, and VAR-X. The numerical and graphical comparison results are presented in

Table 8. Nordic electricity market day-ahead accuracy measures of the proposed models versus benchmarks.

Model	MSPE	MAPE	PCC	RMSE
${}^4$HD${}_3^4$	0.063	1.908	0.987	1037.401
AR	0.499	5.252	0.932	2845.410
ARIMA	0.414	4.902	0.936	2623.213
NPAR	0.456	5.048	0.940	2709.567
VAR	0.343	4.382	0.949	1567.987
ARX	0.248	3.211	0.945	1345.324
ARIMX	0.245	3.173	0.946	1350.234
NPARX	0.242	3.147	0.947	1335.987
VARX	0.218	2.859	0.952	1196.098

and Figure 13, respectively. Within the non-deterministic component models, the VAR model produces the best results (MSPE = 0.3425, MAPE = 4.3823, PCC = 0.9492, and RMSE = 1567.987). When comparing this best model reported in the literature to our best model ((${}^4$HD${}_3^4$)), the ${}^4$HD${}_3^4$ model shows significant accurate and efficient outcomes (MSPE = 0.0630, MAPE = 1.9082, PCC = 0.9867, and RMSE = 1037.401). In the same way, the best outcomes are obtained within the deterministic models by the VAR-X model, such as 0.2178, 2.8587, 0.9522, and 1196.068 for MSPE, MAPE, PCC, and RMSE, respectively. However, when compared to the best combination model (${}^4$HD${}_3^4$) identified in the current work, it shows significantly worse results. These comparisons are displayed Figure 13, confirming the superiority of the best combination model (${}^4$HD${}_3^4$) identified in the current work. In summary, the best combination model identified in this work obtained high accuracy compared to all benchmark models.

Overall, based on the comparison results of the best proposed models with those presented in the literature using standard univariate and multivariate time series models with and without deterministic components, the best proposed combination model exhibits high efficiency and high accuracy in forecasting day-ahead electricity demand for the Nord Pool electricity market. Precise and efficient day-ahead load forecasting offers various benefits, including effective short-term strategic planning for decreased operating and maintenance costs, optimized supply and demand management, improved system dependability, and future investments. In addition, day-ahead electric power demand forecasting assists in reducing risks and making effective economic decisions that affect profit margins, revenue, distribution of resources, enlargement potential, inventory accounting, operational expenses, personnel, and total expenditure.

To conclude this section, it is worth noting that all implementations were completed using ’R’, a statistical computing language and environment. The GAM library was used to create decomposition techniques and for modeling and forecasting. The forecast, tsDyn, and vars libraries were used to estimate and forecast univariate and multivariate time series models. Furthermore, all computations were run on an Intel (R) Core (TM) i5-6200U CPU running at 2.40 GHz.

5. Conclusions

In today’s liberalized energy markets, electricity demand forecasting plays an important role in planning of generation capacity and required resources. However, electricity demand time series exhibit unique characteristics that corresponding to linear or non-linear long-term trends, with multiple seasonalities (daily, weekly, monthly, quarterly, and annually), jumps or spikes (extreme values), high volatility, non-constant mean and variance, and effects of public holidays. Thus, including these unique characteristics in the model greatly improves predictive accuracy. In order to achieve this, we first split the hourly electricity demand time series into three new subseries—a linear long-term series, a seasonal series, and a stochastic (irregular) series—using the three proposed decomposition methods: regression spline decomposition, smooth spline decomposition, and hybrid decomposition techniques. Next, to forecast each subseries, we used different univariate and multivariate time series models, including autoregressive, non-linear autoregressive, autoregressive integrated moving average, vector autoregressive, and all feasible combinations thereof. In summary, in this work, we compared three decomposition methods and all their possible combination models with univariate and multivariate models. The individual forecasting models were combined directly to obtain the final result of a one-day-ahead electricity demand forecast. From 1 January 2014 through 31 December 2019, the proposed modeling and forecasting technique was applied to hourly electricity demand based on Nord Pool power market data. Three accuracy measures (MSPE, MAPE, and PCC), a statistical test (the DM test), and graphical representations (line plot, bar plot, correlation plot, and dot plot) were used to validate the performance of the proposed modeling and forecasting framework. The empirical findings indicate that the proposed decomposition combination modeling framework is highly effective in short-term electricity demand forecasting. It is also confirmed that among the proposed decomposition methods, the hybrid decomposition method outperforms the other two proposed methods and increases the performance of final model forecasts. Additionally, our forecasting outcomes are comparatively better than those achieved using methods mentioned in the literature. Lastly, we recommend that the modeling and forecasting approach described in this study be applied to tackle other electrical market forecasting difficulties.

The main drawback of this research is that it only includes hourly electrical demand data, without incorporating additional exogenous factors, such as electricity pricing, temperature, wind speed, natural gas prices, etc., which might enhance day-ahead electricity demand forecasting. On the other hand, in the current work, we used only data from the Nord Pool electricity market. Therefore, the results reported herein can be extended to other electricity markets to evaluate the performance of the proposed decomposition-combination modeling and forecasting approach. Furthermore, since only linear and non-linear univariate and multivariate time series models were used in this work, machine learning models such as deep learning and artificial neural networks can also be considered within the current decomposition combination forecasting framework. It can also be extended and applied to other approaches and datasets (for example, energy [53,54,55], air pollution [56,57,58,59], solid waste [60] and academic performance [61]).