Electricity consumption and production forecasting considering seasonal patterns: An investigation based on a novel seasonal discrete grey model

Abstract The electricity forecasting problem is among the prominent issues for policymakers to ensure a reliable and stable electricity supply. Although many studies have been executing effectively to predict China's electricity consumption and production, the results of diverse models are confusing and contradicting because they can hardly identify seasonal fluctuations and provide robust forecasts based on different sample volumes. This work focuses on using a new electricity forecasting tool to present precise results to overcome such shortcomings. Initially, a seasonal discrete grey model is designed based on the characteristics of China's electricity consumption and production. Besides, the rolling mechanism is introduced to improve forecasting accuracy further when performing the verification experiments. Secondly, the DM, SPA, and MCS tests and level accuracy is implemented to measure each competing model's forecasting performance efficiently. Lastly, this new model's robustness over different sample sizes is validated by conducting numerous experiments with diverse sample volumes. Empirical results demonstrate that the technique is convincingly an accurate, robust, and applicable method for China's electricity consumption and production forecasting, outperforming any other prevalent forecasting model.


Introduction
Energy supply and demand management have attracted worldwide attention due to the inadequacy of energy sources, a striking increase in energy consumption, and scarcity of progress in using clean and renewable energies (Ding et al., 2021a;Hu & Jiang, 2017;Steinbuks, 2019). Electricity consumption and production forecasting are essential for operation, generation scheduling, transmission augmentation, power dispatching, and maintenance of the grid utility (Hu, 2017;Zeng et al., 2019). Considering such colossal importance, proposing an accurate and reliable forecasting technique is paramount for electricity prediction. However, seasonal fluctuations and a range of uncertainties, referring to the underlying population changes, economic conditions, and climate factors, make electricity forecasting a challenging task. Furthermore, these complex problems can be particularly challenging in developing countries like China, where historical data are usually sparse and elusive. In light of such severe issues, a simple yet accurate forecasting model is required (Adeoye & Spataru, 2019;Ding et al., 2020b;Mohan et al., 2018).
Recently, considerable studies have been dedicated around the topic of electricity forecasting, and numerous methodologies have been reported, typically classified into three groups: machine learning (ML), statistical approaches, and grey models.
First, machine learning techniques are essential methods to predict power consumption because of their dominant capability to capture nonlinear features hidden in electricity time series, such as artificial neural network (ANN) and support vector machines (SVM), widely used in electricity market studies in recent years (Guo et al., 2017;Jatin & Durga, 2019;Yuan et al., 2018). Though these models have advantages in learning the complex nature of electricity data accurately, their drawbacks are that predicting accuracy is proportional to the data numbers available for model training. Moreover, selecting the optimal hyper-parameters and suitable kernel functions also increases the difficulty and complexity of these models, which potentially costs more calculating time for model calibration and projection. In addition to the single machine-learning-based model, the ensemble or hybrid predicting models combining the advantages of one or more methods are introduced for electricity forecasting. These hybrid techniques can overcome certain shortcomings of the corresponding individual model and yield more accurate forecasts. Besides, most ensemble models are more robust and efficient than the non-ensemble ones (Cao & Wu, 2016;Jiang et al., 2020;. Even though the hybrid methods possess high precision and strong adaptability, they suffer from certain limitations that refer to convergence to optimal local locations, determination of suitable network structures, and high computational complexity (Bai et al., 2021;Zeng et al., 2020;. Second, statistical approaches also predict the future values of electricity time sequences by utilizing a linear function, mainly including linear regression models (Dimitrios et al., 2019;Ergun et al., 2017;Jing et al., 2019), autoregressive moving average (ARMA) (Anwar et al., 2016;Erick & Fernando, 2018) and the extended versions (Guo et al., 2018;Niematallah & Mototsugu, 2018). In general, these statistical models typically possess simple structures and are easy to implement. Although these statistical models have simple structures and broad application fields, they are inadequate to describe the nonlinear and complicated electricity time-series behaviors, failing to estimate future electricity demands accurately. Moreover, they have requirements for stationary time series and large sample sizes. Therefore, they might provide low forecasting accuracy in actual applications (Ding, 2019;Şahin, 2019;Xu et al., 2019;Zhou et al., 2020).
The grey models have also been widely used in predicting electricity consumption because of their advantages of high accuracy and usage of few data. Li et al. (2012) used the adaptive grey-based approach to forecast short-term electricity consumption based on the Asia-Pacific economic cooperation energy database. Results demonstrated that this proposed model outperforms the backpropagation neural networks (BPPN) and support vector regression (SVR). Hamzacebi and Es (2014) developed the optimized grey Model to forecast Turkey's total electricity energy demand from 2011 to 2025 directly and iteratively. They found that the direct forecasting approach can achieve better predictive performance. To better achieve the principle of new information priority, Ding et al. (2018) advanced a rolling NOGM(1,1). They used the PSO algorithm to optimize the generating parameters of the new initial condition, which has higher accuracy than other competing models for predicting electricity consumption. Liu et al. (2020) put forward a novel fractional grey polynomial model with time power items to project China's and India's electricity consumption. The quantum genetic algorithm is utilized to determine the parameter values efficiently. Xu et al. (2017) introduced a new model optimized by the nonlinear optimization method with the PSO algorithm to predict China's short-term power consumption and obtain good prediction accuracy. To enhance grey models' forecasting capability further, J. Z.  proposed a new hybrid model by combining the multi-objective ant lion optimizer algorithm and dynamic choice rolling-GM (1, 1) model, then verified its availability by comparing its prediction results with other benchmarks. Subsequently, considering population, a novel multivariate grey model was proposed by Wu et al. (2018), which has been applied in predicting electricity consumption in Shandong Province. To grasp the seasonality in electricity data, Z. X.  put forward a seasonal GM(1,1) model to predict the primary economic sectors' electricity consumption, illustrating superior performance. Further, Zhu et al. (2020) proposed a self-adaptive grey fractional weighted model to overcome the drawbacks of the previous seasonal grey model, whose generating parameters are estimated by using PSO. Experimental comparisons highlighted the superiority of the proposed model over other competitors in predicting Jiangsu's electricity consumption.
Even though many previous techniques possess high forecasting precision, they usually fail to provide adaptability and generality for electricity prediction. Although they have an outstanding ability to dig for load data dynamics, machine learning methods often have complicated structures that are not easy for beginners to understand. Moreover, their learning architectures are parameterized functions. Hence their excellent forecasting performance is proportional to appropriate hyper-parameter pick referring to network topologies and parameters (Ding et al., 2021b;Ye et al., 2021;. Additionally, the statistical models typically encounter issues in addressing difficult and nonlinear time series, which might strikingly limit their application domains. Moreover, their stationary assumptions and requirements for large sample sizes are not easy to satisfy (Comert et al., 2021;Ofosu-Adarkwa et al., 2020). Besides, the existing grey prediction models do have particular merits in describing the dynamic features of electricity data.
However, on the one hand, they often have fixed structures to recognize the emotional seasonal fluctuations with limited adaptability. On the other hand, their generality over different sample sizes has rarely been discussed, especially for small and large sample sizes. Moreover, the currently-used accuracy testing methods that concentrate on single measuring criteria, such as APE, can hardly evaluate a forecasting method from a comprehensive perspective, which would not provide a reliable evaluation on a specific forecasting model (Hansen et al., 2011).
Thus, a systematic and extensive evaluation method is necessary for validating the efficacy and practicability of the theoretical models. In general, there is a requirement for developing an adaptative and generalized model for electricity forecasting.
Motivated by the existing studies, we put forward a new seasonal discrete grey model combing with a rolling mechanism for electricity consumption and production predictions, whose details will be elaborated on in the next section. Moreover, Model Confidence Set (MCS), Diebold-Mariano (DM), and Superior Predictive Ability (SPA) tests are utilized for testing the validity of this model. To the best of our knowledge, this is the first work that uses these techniques to test grey models' effectiveness. Subsequently, we verify our method on two realworld datasets, namely China's electricity consumption and production. The empirical results illustrate that the SDGM(1,1) model can deliver more accurate forecasts than the other benchmark models in the presence of both electricity consumption and production predictions.
The main contributions of this paper contain: A novel seasonal discrete grey model is used for electricity forecasting by embedding the seasonal dummy variables into the conventional discrete grey model structure. This new method can keep the advantages of the discrete grey model for describing time series with insufficient information and uncertainty and mine the data to identify the latent seasonal characteristics. For further improvements in forecasting accuracy, the rolling mechanism is integrated with the model. This technique metabolism can update input data by removing old data for each loop, ensuring that the next prediction's data is the most recent. Accordingly, it can be used to enhance prediction performance. Moreover, in addition to the conventional level accuracy indicators, the MCS, DM, and SPA tests are employed to comprehensively illustrate the model's superiority over other competitors in forecasting electricity consumption and production. The performance advantage of the newlydesigned model is evaluated through two comparative experiments on forecasting China's electricity consumption and production. Moreover, this novel technique's expertise to forecast demand and production data with various sample sizes is investigated. Experimental results demonstrate that the model can be considered a powerful tool to accurately predict electricity consumption and production with high efficiency and great generality.
The subsequent sections of this paper are organized below. The mathematical concepts of the conventional grey models are introduced in Sec. 2. Besides, Sec. 3 presents the procedures of the discrete grey model and provides the evaluation rules for measuring the modeling accuracy. Section 4 investigates the real applications of forecasting the seasonal electricity consumption and production in China, verifying the efficacy and applicability of this model. Lastly, several conclusions will be drawn in Sec. 5.

Necessary concepts
This section mainly focuses on the introduction of the conventional grey prediction models related to the proposed model. As previous literature shows, grey system theory is widely used for prediction and modeling, especially for sparse data and insufficient information (Ding et al., 2021a;Xu et al., 2017). With the grey accumulation formation technique's help, a new generalized accumulation sequence having less randomness and enhanced exponential trends can be obtained. Subsequently, based on the accumulated series, the grey differential equation is built, and the predicted values of the original time series are generated.
In this theory, GM(1,1) is the most popular and widely-used method, as it has a simple structure, less calculating time, and relatively fewer requirements for data amount. Moreover, based on this model's conventional forms, many derived versions have been put forward by other scholars to further expand its application fields, referring to seasonal GM(1,1) (Z. X.  and discrete GM(1,1) (Xie and Liu 2009). Owing to spatial confinement, the mechanism and drawbacks are presented in the Supplementary material.

Proposed approach
Considering the inherent shortcomings in the conventional GM(1,1) and DGM(1,1) models, discussed in previous Sec. 2, a novel seasonal discrete grey model is elaborated on in Sec. 3.1 to address such hindrance. Moreover, for further testing the new model's effectiveness, the level accuracy and model tests are employed in Sec. 3.2. Detailed introductions about these techniques will be presented below.
3.1. The seasonal discrete grey model Similar to the conventional grey models, the modeling procedures of the proposed model can be outlined as the following steps.
Step 2: Building the seasonal discrete grey model (abbreviated as SDGM 1, 1 ð Þ) to describe the time series having seasonal effects, whose formula is provided by where s is the number of seasons in one year. For instance, s ¼ 4 means the quarterly time series, while s ¼ 12 represents the monthly seasonal cycles. Additionally, ux 1 ð Þ k ð Þ stands for the influence of historical data on the trends. Moreover, P sÀ1 i¼1 c i I i represents the seasonal dummy variables, which can accurately describe the seasonal fluctuations of the original data. Besides, for any i 2 1, 2, :::, s À 1 f g in Eq. (1), Therefore, from the structural components of the proposed SDGM 1, 1 ð Þmodel, it can be found that this new model can effectively describe the influences of historical observations and seasonal fluctuations on the development trends of a given system. What is more, for easy explanation purposes, we take the quarterly time series (s ¼ 4, n ¼ 8) as an example to illustrate the model process, which can be modeled by the following equations.
Transforming Eq. (5) into the matrix form, it equals Q ¼ Pg: By utilizing the least square method, the estimated values of g can be obtained byĝ ¼ P T P ð Þ P T Q: Subsequently, the forecasting function of Eq. (1) can be given in Step 4.
Step 4: Obtaining the fitted and predicted values in the transformed domain, which can be provided bŷ Equation (6) can be proved by using the mathematical induction method.
Subsequently, supposing Eq. (6) (1), we can obtain Thus, Eq. (6) is proved to be correct for generating forecasts in the transformed domain.
Step 5: Obtaining the fitted and predicted values in the original domain by using the inverse 1-AGO, In addition, for further enhancing the forecasting ability of specific grey models, the rolling mechanism is usually combined with the grey model. Moreover, the superiority of this technique is verified by many authors, such as Sahin (2019), Xu et al. (2019), and Ding et al. (2018). Therefore, this rolling mechanism is integrated with the new seasonal discrete grey model as well, whose modeling procedure is displayed in Figure 1.
In this figure, for simplicity, each original input datum is represented by the time of its occurrence. For example, 1, 2, :::, n describe the raw data x 0 ð Þ 1 ð Þ, x 0 ð Þ 2 ð Þ, :::, x 0 ð Þ n ð Þ, respectively. Moreover, supposing that h data points are used to establish the SDGM 1, 1 ð Þmodel and m data points mean the forecast horizons at each rolling step. Subsequently, the architectural diagram of the proposed SDGM 1, 1 ð Þframework with the rolling mechanism can be constructed as explained below in conjunction with the corresponding steps portrayed in Figure 1.
Step 1: Firstly, use the raw observations Step 2: Remove the oldest data point Þwill be forecasted, as portrayed in the second graph from the top of Figure 1.
Step 3: Repeat Step 2 by removing the oldest data and employing the most recent h data points for predicting the next group of m data points.
Step 4: Stop the procedure until all the forecasts are available for future discussion and analysis.

Evaluation of the modeling accuracy
In this section, four popular measuring indicators are employed for calculating the level accuracy of the competing models, whose definitions are available as follows.
Mean absolute error (MAE): MAE can tell the average discrepancy between the raw and predicted values, whose expression is defined as: Mean absolute relative error (MARE): MARE measures the average ratio of the MAE to the collected values, which represents the predicting error in percentage: Mean squared relative error (MSRE): MSRE is defined to measure the relative differences between the predicted and original data with the quadratic scoring rule over the whole sample. As the relative forecasting errors are squared before they are averaged, the MSRE puts relatively high weight on larger errors. Therefore, this indicator is compelling when large errors are particularly undesirable.
Although these above four measuring indicators obtain high popularity among the applications of time series forecasting, they might have certain shortcomings. For instance, if several outliers are hidden in the forecasted values, the forecasting errors may be enormously enlarged. Moreover, single measuring criteria, such as APE, for determining the optimal model may be inferior to a multi-criteria test. Accordingly, the conclusions that justify which model performs better in the real application, which is obtained from these enlarged errors and single criteria, will be plainly misguided. Thus, in addition to the above four level-accuracy indicators, the MCS (Hansen et al., 2011;Seri et al., 2021.) is introduced to compare each model's performance to determine the statistically-superior group of best models. In this approach, the number of models trimmed is not fixed exogenously by the econometrician but determined by a statistical test comparing model accuracy (Samuels & Sekkel, 2017). In general, this method can allow users to rank competing models based on their closeness to the benchmark and establish a plausible set of alternative specifications of parameters that cannot be differentiated from the selected one, as least not in a statistical sense (Amendola et al., 2020;  Cummins et al., 2017). In this respect, this technique is employed in China's electricity forecasting applications, from which the gains are larger and more robust compared to often-used approaches.
In addition, before using MCS, the loss function should be determined for measuring forecasting deviations. However, scholars have no agreement about which loss function can be the most appropriate one to take such above missions. Therefore, Hansen and Lunde (2005) recommended that diverse forms of loss functions can be employed as the standards for testing forecasting performance. Accordingly, four measuring indicators, namely MAE, MARE, MSE, and MSRE, are selected as the loss functions (L 1 , L 2 , L 3 , L 4 ) that can measure forecasting errors from different perspectives in this work. Subsequently, based on these loss functions, the MCS procedures are outlined as follows (Wei et al., 2015).
Initially, we assume that there are m 0 forecasting models (m 0 ¼ 8in this paper), which is noted as X 0 ¼ 1, 2, :::, m 0 f g : Subsequently, each model can generate out-of-sample forecasts (x 0 ð Þ k ð Þ , k ¼ nþ 1, n þ 2, :::, n þ M), among which n stands for the sample size and M represents the forecasting horizons. Based on these forecasted values, we can obtain these four loss functions' values by utilizing Eqs. (30)-(33), which can be marked as L i, w, t , i ¼ 1, 2, 3, 4; ð w ¼ 1, 2, :::, m 0 ; t ¼ n þ 1, n þ 2, :::, n þ MÞ: Thus, we will calculate the relative differences of the loss functions between random two forecasting models (w, v w, v 2 X 0 ð Þ ), which is expressed as Subsequently, we will define the set of superior objects M Ã , whose formula can be expressed as The main principle of MCS is to get rid of the competing model that has inferior forecasting performance by conducting a series of significance tests among the collection of X 0 : Meanwhile, the test basis of MCS is the equivalence test and elimination rule. The former base is used to test the null hypothesis that these two compared models have the same predicting capability, which can be obtained by In contrast, the elimination rule is utilized to remove the model that refuses the null hypothesis, which can be denoted as where d i, wÁ ¼ 1 n P v2X d i, wv : During this process, the test statistic is the Range Statistic and Semi-quadratic Statistic, which are defined as where d i, wv represents the average relative differences of the loss functions between the model w and v: Besides, due to the complex distributions of the statistics T R and T SQ , their values and corresponding p values are obtained by using the Bootstrap technique (Hansen & Lunde, 2005). Lastly, we can perform the MCS test to determine the optimal forecasting model by using the following steps. First, we can set the collection X ¼ X 0 : Second, we can use the equivalence test to check the null hypothesis H 0, X under the determined significance level a: Third, if we accept the null hypothesis H 0, X , we will obtain the optimal forecasting model collection, noted as X Ã 1Àa ¼ X: Otherwise, we will remove the forecasting model that refuses the null hypothesis from the collection X based on the elimination rule. Subsequently, repeat these above three steps until the null hypothesis is no longer present. Then, we can obtain the surviving objects X Ã 1Àa , which contain the optimal forecasting model under the significance level a: For the given model w w 2 X 0 ð Þ, if its p value under the MCS test is over the significance level a, then this model belongs to X Ã 1Àa : In other words, the larger p value means better forecasting performance for the given model. According to the experience of Hansen and Lunde (2005), the significance level is customarily set as a ¼ 0:1: In addition to the MCS test, DM (Diebold & Mariano, 1995) and SPA (Hansen, 2005) tests are also employed to demonstrate the efficacy and robustness of the proposed model. Detailed analysis of these test techniques will be illustrated in the following sections.

Experiments and discussions
For illustration and comparison purposes, the proposed model is performed to forecast electricity consumption and production in China. Data description and experiment design are initially provided in Sec. 4.1. Secondly, Sec. 4.2 further discusses the comparative forecasting performance of the new model over a range of other benchmarks. Finally, Sec. 4.3 gives detailed experiments conducted to highlight the robust performance advantage of the proposed method in terms of electricity consumption and production data.

Data description and experiment design
In this study, the monthly electricity consumption and production onto China's electric market are utilized as data samples to construct two empirical cases gathered from the Wind Database (https:// www.Wind.com.cn). It is worth mentioning that because the Chinese Spring Festival usually occurs in January or February, people typically consume a large amount of electricity in these two months. This phenomenon may bring much uncertainty and changes in electricity consumption and production in January and February. Thus, to reduce this lunar year's influence, the Chinese government typically publishes the total electricity consumption and production of these first two months (January and February). Accordingly, the electricity data for eleven months are collected for experiments each year.
Additionally, owing to the data availability, the electricity consumption covers the period from February 1990 to December 2018 (1990. M1-2018. M11) with a total of 319 observations, while the electricity production is available from February 2004 to December 2018 (2004. M1-2018. M11) with a total of 165 observations. Prior to conducting a detailed analysis, data characteristics must be discussed to provide a basis for model selection. Figure  2 records the shape of data series in different months, revealing strong seasonal fluctuations in China's electricity consumption and production data. As discussed before, these seasonal patterns are positively correlated with the data combination of January and February. It might bring enormous challenges for future estimations. Thereby, a superior forecasting technique is necessary for conducting electricity consumption with various complex and seasonal features. Subsequently, these collected data are divided into two subsets, i.e., the training and  testing subsets. For this proposed model, due to the rolling mechanism's introduction, the rolling window size is set to 33, and the forecasting horizon is 11. As for the competing models, the last 11 observations are treated as the testing set for evaluating forecasting performance, while the other preceding data are used as a training set for model calibration. Moreover, two forecasting horizons (m ¼ 11 and 33) are adopted in this article to illustrate the forecasting performance of the proposed model. The predictions with 11 horizons are discussed thoroughly, while those with 33 horizons are provided in the supplementary material.
For competition and verification purposes, the proposed technique is performed to forecast China's electricity consumption and production embedding with substantial seasonal deviations, in comparison with seven paramount benchmark models, involving the grey prediction model (SGM(1,1)), traditional econometric technology (SARIMA, Holt-Winters (HW), and Snaive), and artificial intelligence (LSSVR, MLP, and LSTM). These competing models represent the most widely-used tools for electricity forecasting in recent studies.

Prediction performance comparisons
According to the experimental scheme in Sec. 4.1, this section focuses on describing the experiments conducted to examine the forecasting performance of the newly-proposed technique. To show the merits of performance, the electricity consumption and production data from the Chinese energy market are considered. This operation highlights the potentiality and generality of the new model for accurate electricity forecasting. Subsequently, the prediction performance comparisons are conducted from two perspectives, including the comparisons over measuring indicators and comparisons over the MCS, DM, and SPA tests.

Comparisons over the measuring indicators
Initially, the predicted values of electricity consumption and production are visualized in Figures 3 and  4. It can be seen from these two graphic comparisons that the proposed SDGM(1,1) model follows the pattern of the actual electricity consumption and production sequences with minor deviation, thereby achieving an excellent prediction. This is because of the inherent ability of the seasonal dummy variables to grasp the underlying seasonal features of electricity data. Following the proposed model, the SGM(1,1) obtains the second-highest forecasting precision, which infers that the grey prediction models have advantages in describing seasonal fluctuations. Additionally, although the SARIMA, LSSVR, and LSTM show a good fitted performance, the complexity of these methods is high, and the parameter tuning must attain good accuracy. The HW and Snavie models have achieved good results, a little inferior to the proposed model. Due to their simple structure, they can be the candidate for predicting electricity consumption.
In contrast, the predicted line of the MLP model deviates the largest from the actual values, inferring that this model performs the worst predictions. The proposed technique generally discovers the electricity characteristics with less complexity and higher accuracy than the other statistical and data-driven models. The accurately predicted electricity consumption and production results can be further used to guide the grid management system to make decisions associated with electricity generation and scheduling, electricity purchase, and consumption, resource allocation, etc. Besides, the predicted values can reveal the gap between electricity consumption and production and thus ensure the balance of electricity supply and demand.
Moreover, similar conclusions can also be drawn from the perspective of the four indicators, whose compared results are tabulated in Table 1. The best performance values are present in bold. It can be seen that the proposed model obtains the best predicting performance in terms of MAE, MARE, MSE, and MSRE in both the processes of electricity consumption and production forecasting. In contrast, the MLP model provides the most significant forecasting errors, no matter forecasting electricity consumption or predicting electricity production. For illustration purposes, we take the MSRE as an example to tell the differences between these two models. For the electricity consumption and production forecasting, the MSRE of the proposed model reaches 0.0039 and 0.0023, respectively, while that of the MLR model obtains 0.0237 and 0.0278, respectively. The MLP model is nearly ten times the errors of the proposed model in both two case studies. The SDGM(1,1) model generally improves performance due to its capability to learn seasonal and nonlinear behavior than other competing models. Moreover, this grey model discovers the electricity characteristics with less complexity while other deep-learning-based techniques are high.
Additionally, to highlight the stability and practicality of the proposed SDGM(1,1) model, the box graph of the four measuring indicators of each competing model when predicting the electricity consumption and production is presented in Figures 5  and 6, respectively. As seen in these two figures, it is evident that the SDGM(1,1) model makes more reliable and accurate predictions than SGM(1,1), SARIMA, LSSVR, LSTM, MLP, HW, and Snaive. We take the MAE of predicting electricity consumption and production as an example for illustrations in these two case studies. It is evident that the proposed model has much fewer outliers than the other competitors. In contrast, the LSSVR and MLP obtain many outliers in both two cases. As for the other three measuring indicators, similar conclusions can also be drawn from these two figures. Thus, we can conclude that the proposed method can exhibit higher accuracy and stability as the  dummy variables effectively capture the latent seasonality. Moreover, the forecasted error of the competing models when m ¼ 33 are available in Table S2 in the supplementary material. A similar conclusion can be found that the proposed model is superior to the other competitors in terms of level accuracy. Therefore, the SDGM(1,1) model is a promising and reliable technique for predicting electricity consumption and production.

Comparisons over the MCS, DM, and SPA tests
This section is dedicated to describing other comparative experiments conducted to highlight the performance advantage of the novel SDGM(1,1) model with a range of existing dominant electricity forecasting models. This experiment refers to conducting MCS, DM, and SPA tests to validate the effectiveness and superiority of the proposed model.
The detailed procedure of MCS is introduced in Sec. 3.2. According to Hansen and Lunde (2005), the significance level is set as a ¼ 0:1, which means that the prediction model will be kept due to its excellent forecasting capability if its corresponding P-value is over 0.1. Otherwise, the model will be removed due to its lousy forecasting precision.
The MCS test results when performing China's electricity consumption and prediction forecasting when m ¼ 11 and m ¼ 33 are provided in Table 2 in  this article and Table S3 in the supplementary, respectively. We would take Table 2 as an example to discuss the comparisons of the competing models. As Table 2 shows, the four measuring indicators  Figure 6. Visualization of forecasting error on electricity production data using box graph.
represent the loss function (explained in Sec. 3.2) and the numbers stand for the values of the statistics T R and T SQ of each competing model under different loss functions. As seen in this table, several findings can be drawn as follows: 1. The proposed SDGM(1,1) model will always be kept, no matter which loss function is selected when conducting electricity consumption and production forecasting. Because its P values all equal one that is over the significance level and those of other competing models. In other words, the forecasting performance of the proposed model is much superior to other benchmarks in terms of predicting electricity consumption and production. 2. The LSSVR, LSTM, MLP, and Snaive models will be removed because their P values are lower than the significance level. Especially for the LSTM, Snaive, and MLP, the P values of the statistics T R and T SQ under the four loss functions are close to zero, indicating inferior forecasting ability. Thus, these four models cannot handle the forecasting missions of China's electricity consumption and production. 3. As for the SGM(1,1), HW, and SARIMA models, their forecasting performance is unstable. On the one hand, when predicting the electricity consumption, the P values of the SGM(1,1) and SARIMA under the MAE and MSE loss function are over the significance level, while those under the MARE and MSRE are less than the significance level. On the other hand, when predicting electricity production, the P values of these two models are all over the significance level under the four-loss functions, except that the P values of the SARIMA model under the MSRE loss function. Additionally, the HW model shows less stability when predicting electricity consumption and production because the p values are sometimes over or less than the significance level. Thus, the SGM(1,1) delivers a little higher forecasting precision than the SARIMA model.
Additionally, as Table S3 shows, the proposed model still produces the overall best forecasting performance. The new model is superior to the others in projecting electricity production, although it is second to the SGM(1,1) model in predicting electricity consumption. Generally, the novel model is validated to have better forecasting capability according to the MCS test with 11 and 33 horizons.
Subsequently, the results of the DM test (m ¼ 11) are available in Table 3. It shows that the p-values comparing the proposed model with the SARIMA, LSSVR, LSTM, MLP, HW, and Snaive models are all less than 0.1 in terms of MAE, MARE, MSE, and MSRE, revealing that the forecasting precision of the proposed model is quietly different from those of the competing models. Considering the error analysis in Table 1, the SDGM(1,1) model is superior to the others. Additionally, similar conclusions can also be found from the DM test results when m ¼ 33, which are available in Table S4 in the supplementary material, that the proposed model performs better than the other competing models.
Last but not least, the SPA test is performed, and the results are presented in Tables S5 and S6 in the supplementary material. In these two tables, we find that each forecasting model will be selected as the benchmark model for running the SPA test.
As Table S5 shows, when the SDGM(1,1) model is chosen as the benchmark model, its p-values all equal one in terms of MAE, MARE, MSE, and MSRE, implying that the novel model performs better than the other seven competitors. Moreover, the SGM(1,1) model is superior to the remaining nongrey models because its p-values are equivalent to one when selected as the benchmarks. Furthermore, when the SARIMA, LSSVR, LSTM, MLP, HW, and Snaive models are chosen as the benchmarks independently, the p-values of the proposed model are mostly less than 0.1, which means these benchmarks possess weak forecasting capability in predicting electricity consumption and production. Moreover, compared with the LSSVR, LSTM, and MLP, the SARIMA, HW, and Snaive models have a slight superiority.
As Table S6 displays, a similar conclusion can be found that the proposed model still achieves the best forecasting ability in forecasting electricity production with 33 forecast horizons. However, the SGM(1,1) performs best when predicting electricity consumption, followed by SDGM(1,1). Moreover, SARIMA is superior to the LSSVR, LSTM, MLP, Snaive, and HW models when predicting electricity consumption and production.
Additionally, these two case studies also highlight the computing speed of the proposed SDGM(1,1) model for electricity consumption and production forecasting. To perform predictions at a faster rate, the forecasting technique should be computationally effective and faster with a comparatively simple structure. The running time of these competing models has been recorded in Table 4, which is performed on the same computer with 6 G RAM and i7-8750 processors. As Table 4 reveals, the proposed model has efficiently forecasted electricity consumption and production, as the running time required for model calibration is much less than the other non-grey models, which illustrates the proposed model's convenience and practicability. In contrast, the LSTM and MLP models consume more than one hour to tackle these forecasting issues. Thus, the merits in computing time can facilitate the real-time applications of this new model for electricity consumption and production forecasting. Further, the proposed SDGM(1,1) model possesses a simple structure with no requirement for parameter settings, unlike the other competing models for electricity forecasting.
In general, these two experiments confirm the competitive performance of the SDGM(1,1) model for electricity consumption and production forecasting. The efficient capability of this model to identify the hidden seasonal patterns from electricity consumption and production series contributes to accurate future predictions. Further, the rolling mechanism can significantly improve its precision because it helps to grasp the most recent data for model calibration. Subsequently, the satisfactory performance of the proposed SDGM(1,1) model for electricity consumption and production forecasting task is certified by lower predicting errors, such as MAE, MASE, MSE, MSRE, and running time. Therefore, its excellent competing performance suggests that this new model can be employed as a promising technique for electricity forecasting and even other energy sources (e.g., gas, oil, and solar). This extension can be considered as a direction for future research.

Discussions on the robustness over diverse sample sizes
In addition to verifying forecasting accuracy in the above two cases with large sample sizes, this section describes comparative experiments performed to highlight the robustness and applicability of the proposed model dealing with small sample sizes. The original observations of electricity consumption and production in China are used in this section. For electricity consumption and production forecasting, the sample size varies from 13 to 48, and the forecasting horizon is 11. The MARE for the predicted values is recorded in Figure 7. As shown in Figure 7, the proposed model can still achieve excellent forecasting performance when addressing electricity consumption and production sequences with sparse data. To be specific, the MARE values for electricity consumption vary between 0.031 and 0.043, while those for electricity production change between 0.041 and 0.060. These experiments demonstrate the great potential of the proposed model for predicting electricity consumption and production with limited observations. Therefore, the conclusion can be safely drawn that the proposed model is convincingly an accurate, robust, and applicable technique for China's electricity consumption and production forecasting.

Conclusions
Electricity consumption and production forecasting play a key role in making future strategies and planning available resources better. Especially for developing countries, such as China and India, accurate forecasts can be helpful to run the grid energy management system and maintain the balance between supply and demand of electricity. To this end, a novel seasonal discrete grey model is proposed by incorporating dummy variables into the model structure. With the support of the rolling mechanism, this new model can efficiently mine the latent seasonal dynamics of the electricity consumption and production data. Further, a range of tests and level accuracy is implemented to measure each competing model's forecasting performance efficiently. After comparisons with the benchmarks in terms of predicting China's electricity consumption and production, the key advantages of the proposed technique are concluded as follows: This proposed model is adaptive to identify multiple seasonal fluctuations in electricity consumption and production sequences, thereby offering an enhanced generalization capability for electricity projections of diverse seasons. This new method is demonstrated to provide accurate forecasts under the circumstances of both sufficient and insufficient data. Thus, it has strong robustness over different sample sizes, which is validated in Sec. 4.3. Also, this model can use the latest information to guarantee accurate forecasts with the support of the rolling mechanism. The complexity of this proposed technique in terms of model structure and computing speed is much less than that of the conventional statistical models and artificial intelligent models (explained in Sec. 4.2), which will facilitate the real-time applications of the novel model.
The competence of the proposed SDGM(1,1) model has been thoroughly investigated through various experiments on predicting China's electricity consumption and production with diverse sample sizes. Its predicting accuracy and comprehensibility are satisfactory in comparison with other prevailing competitors. The strikingly lower predicting errors confirm the excellent performance of the proposed method. Thus, it can be safely concluded that the SDGM(1,1) model tends to be a practicable and robust tool for electricity consumption and production forecasting so as to build an efficient energy supply-and-demand system. For future work, this model can be used to predict other interdependent variables in the energy system, such as renewable energy sources, natural gas consumption, etc. Moreover, the multivariable grey model can also be established by considering the influences of the factors.