Multiscale Analysis of Regional Power Load and Its Associations with Meteorological and Socioeconomic Factors: Evidence from Shandong and Inner Mongolia

Abstract

Regional power load is an important basis for renewable energy accommodation, power grid capacity planning, and demand-side resource management. Its variation is jointly associated with meteorological conditions and industrial and consumption-related conditions, which may operate at different temporal scales. To address this issue, this study takes Shandong and Inner Mongolia as study areas and develops an hourly–monthly–annual multiscale analytical framework. Variational mode decomposition is first used to identify low-frequency trends, intraday cycles, intra-annual fluctuations, and high-frequency disturbances in load series. Pearson correlation and mutual information are then applied to characterize linear associations and more general statistical dependencies between load modes and meteorological, industrial, and consumption factors. Annual maximum load, annual electricity consumption, meteorological variables, and key socioeconomic indicators are further examined to describe the long-term regional backgrounds of load variation. The results show that (1) both regions exhibit evident multiscale load structures, and external-factor associations are concentrated in different load modes; (2) separate within-region hourly analyses show that meteorological associations in Shandong are mainly concentrated in the 24 h daily-cycle mode, particularly for radiation- and sunshine-related variables. In Inner Mongolia, meteorological associations are mainly observed in the low-frequency trend and daily-cycle modes; (3) at the monthly scale, meteorological associations in both regions are mainly distributed in the annual and intra-annual components, and many corresponding Pearson associations were retained by the unadjusted block-bootstrap analysis, whereas the socioeconomic associations retained by the same analysis were relatively sparse and localized; and (4) at the annual scale, meteorological and socioeconomic trajectories are presented only as descriptive regional context because only seven observations are available for each region. These findings indicate that decomposition can help identify the temporal components in which overall load–factor associations are concentrated and provide a descriptive basis for subsequent component-specific variable screening and regional load analysis.

1. Introduction

With rapid economic development, urbanization, population concentration, and continuous changes in electricity consumption structures, regional electricity demand has continued to grow [1]. This growth is reflected not only in increasing total electricity consumption but also in increasingly diversified demand structures, more complex consumption patterns, and more pronounced load fluctuations. Existing studies have shown that electricity demand is closely associated with socioeconomic factors such as economic growth, urbanization, and population size, while peak load is also associated with weather conditions and building electricity-use behavior [1,2]. The uncertainty of load variation increases the difficulty of maintaining power supply–demand balance, grid dispatch, and operational security, thereby imposing higher requirements on the safe and stable operation of power systems [3]. In power-system planning and operational management, accurately characterizing load variation at different temporal scales provides an important basis for generation scheduling, equipment maintenance, resource allocation, and demand-side management [4,5,6]. Therefore, identifying the multiscale characteristics of regional power load and its associations with external factors is important for improving grid operation efficiency, supporting load forecasting, facilitating renewable energy accommodation, and informing differentiated load-management strategies.

Complex power-load series often contain trend, periodic, and random disturbance components. Analyzing the original load series as a whole cannot explicitly distinguish structural characteristics occurring at different temporal scales. To separate components with different temporal characteristics, signal decomposition methods have gradually been introduced into load studies [7]. Zhu et al. [8] adopted an EMD–FbProphet–LSTM method to forecast daily electricity consumption at the enterprise level, while Semero et al. [9] used empirical mode decomposition (EMD) to decompose short-term microgrid load and improve forecasting performance. Although EMD can adaptively decompose nonstationary signals, it is susceptible to mode mixing when processing intermittent fluctuations. Ensemble empirical mode decomposition (EEMD) was subsequently developed by adding white noise to the original signal and has been applied to short-term load forecasting [10,11]. However, the introduced noise may also affect the representation of the original fluctuation characteristics. In comparison, variational mode decomposition (VMD) decomposes a signal into several band-limited mode components by constructing and solving a constrained variational problem. Its non-recursive formulation provides relatively stable decomposition performance and improved resistance to mode mixing and noise interference [12], and it has been increasingly applied in power-load analysis and forecasting [13].

In addition to load decomposition, identifying associations between load variation and external factors is essential for interpreting regional electricity-demand patterns. Existing studies commonly use feature-analysis methods to screen load-related variables and reduce the interference of weakly associated or redundant variables in model inputs and result interpretation. Pearson correlation has been widely used to characterize linear associations between power load and candidate variables and to select highly correlated predictors [10]. Information-theoretic methods have also been employed to capture more general statistical dependencies that may not be fully represented by linear correlation. These studies provide an important basis for load-factor identification, but most are primarily oriented toward improving forecasting accuracy. Meteorological, calendar, and socioeconomic variables are generally treated as unified model inputs, while the differences in their associations with specific load components and temporal scales receive comparatively limited attention [10,14]. For regional load, meteorological variables may be associated with both seasonal low-frequency variation and intraday short-cycle fluctuations. Previous studies have shown that variables such as temperature, precipitation, and sunshine are associated with electricity demand across different intraday periods and seasonal conditions [15,16]. Industrial and consumption indicators more commonly characterize changes in monthly economic activity, whereas industrial structure, economic scale, and energy structure provide longer-term contextual information at the annual scale [17]. Without distinguishing the scale-specific association patterns between load components and external variables, relationships occurring at different temporal scales may be conflated, reducing the interpretability of factor analysis.

Recent decomposition-based load studies mainly use EMD, EEMD, VMD, and related methods as preprocessing tools and evaluate their value through forecasting accuracy, while interpretable load studies generally assess feature importance or select external variables within predictive models [18,19]. These approaches provide useful predictive information but rarely compare the associations obtained from the original load series with those identified from frequency-specific load components using the same analytical procedure. Regional electricity-demand studies also tend to focus on aggregate or spatial differences rather than on how meteorological and socioeconomic associations are distributed across temporal scales [20,21]. Studies incorporating ambient temperature into residential electricity-demand forecasting further indicate that the relevance of external variables depends on temporal resolution and electricity-use context [22].

Accordingly, the contribution of this study does not lie in proposing new VMD, Pearson correlation, or mutual-information algorithms individually, but in integrating them into a unified multiscale association framework. The original load series is retained as a common baseline, and the same Pearson correlation and MI procedures are applied to both the original load and the decomposed modes. This design allows the overall associations observed in the original series to be compared with the temporal components in which they are concentrated. Hourly analyses are conducted separately as two within-region cases, whereas regional comparisons are limited to the temporally aligned monthly and annual datasets. The framework is descriptive and is intended to identify scale-specific contemporaneous associations rather than causal effects, lagged responses, or validated forecasting benefits. The objectives are therefore to (1) characterize the multiscale structures of regional power load; (2) identify the temporal components in which meteorological and socioeconomic associations are concentrated; and (3) describe regional differences using the aligned monthly and annual datasets.

2. Study Area, Data, and Technical Framework

2.1. Overview of the Study Area

This study selects Shandong Province and Inner Mongolia Autonomous Region as the study areas, and their locations are shown in Figure 1. Both regions have a strong foundation of industrial electricity consumption and relatively high regional power load levels, but there are obvious differences in meteorological conditions, industrial structure, and socioeconomic activities. Existing studies have shown that temperature variation, cooling demand, and heating demand significantly affect residential electricity consumption and regional load variation, and that such effects differ across climatic zones and between urban and rural electricity consumers [23,24]. Shandong is located in the eastern coastal region of China, with relatively humid climatic conditions and obvious high-temperature and high-humidity characteristics in summer. Cooling load and residential electricity consumption show strong seasonality. Meanwhile, Shandong has a complete industrial system and a solid manufacturing foundation, and socioeconomic activities such as the service industry, urban commercial activities, residential consumption, and transportation logistics are relatively active. Therefore, load variation is more likely to reflect the compound characteristics under the combined effects of industrial production, urban consumption, and meteorological seasonality.

Inner Mongolia is located in the northern inland region of China and has typical continental climatic characteristics, with cold winters of long duration. Heating demand and the low-temperature background have obvious effects on load variation. In its industrial structure, resource-based industries and energy-intensive industries, such as coal, metallurgy, and energy chemical industries, account for a relatively high proportion, making the industrial base-load characteristics more prominent. Relevant studies have pointed out that Inner Mongolia is an important energy-supply region in China, with obvious characteristics of coal and electricity transmission outward, as well as strong demand from energy-intensive industries. In addition, the types of socioeconomic activities differ between the two regions: Shandong has stronger population agglomeration and urban consumption activities, whereas Inner Mongolia has characteristics involving resource industries, pastoral-area economy, and heating demand. Conducting a multiscale comparison of the two regions is helpful for identifying differences in the association structures of regional power load under different meteorological and socioeconomic backgrounds.

2.2. Data Sources and Preprocessing

This study uses load, meteorological, and socioeconomic data from Shandong and Inner Mongolia at hourly, monthly, and annual resolutions. The load datasets were provided by China Renewable Energy Engineering Institute Corporation Limited (CREEI). The meteorological data were derived from the national meteorological-station database of the China Meteorological Administration (CMA), while the socioeconomic data were obtained from the National Bureau of Statistics of China (NBS).

The meteorological data used in this study were supplied as province-level regional-average series derived from CMA station records. Individual station observations and station metadata were not included in the delivered dataset; therefore, no additional station selection, spatial interpolation, or station-weighted aggregation was performed by the authors. The publicly sourced meteorological and socioeconomic data can be accessed through the China Meteorological Data Service Centre and the National Data platform of the NBS, respectively. The CMA system provides access to surface meteorological datasets, while the NBS platform provides monthly, annual, and regional socioeconomic indicators. The hourly load series represents the province-level load recorded at one-hour intervals and supplied directly by CREEI. The monthly load series consists of one provider-defined province-level load value for each calendar month. It was supplied independently of the hourly series and was not calculated by the authors as a monthly mean, maximum, sum, or other aggregation of the hourly load data. Annual load variation is described using annual maximum load and annual electricity consumption.

Owing to the different availability periods of the provincial hourly load data, the Shandong hourly dataset covers 2024, whereas the Inner Mongolia hourly dataset covers 2015–2021. The hourly analyses are therefore conducted separately within each region. The monthly and annual datasets for both regions cover 2015–2021 and are used for regional comparison. The dataset composition and variable definitions are summarized in

Table 1. Dataset Composition and Time Range.

Region	Type of Data	Time Resolution	Time Range	Data Source
Shandong	Hourly load data	h	2024	CREEI
	Monthly load data	m	2015–2021	CREEI
	Annual load data	y		CREEI
	Meteorological data	h/m/y		CMA
	Socio-economic data	m/y		NBS
Inner Mongolia	Hourly load data	h	2015–2021	CREEI
	Monthly load data	m		CREEI
	Annual load data	y		CREEI
	Meteorological data	h/m/y		CMA
	Socio-economic data	m/y		NBS

,

Table 2. Description of Meteorological Variables in the Initial Dataset.

Variable Name	Units	Comments
SOLAR_LEVEL	/	Characterizes the quality of solar energy resource conditions, with larger values indicating better solar resource conditions.
AVIR	W/m2	The average solar radiation power received per unit area during the statistical period.
IRRA	kWh/m2	The cumulative solar radiation energy received per unit area during the statistical period.
FDIR	kWh/m2	The cumulative radiation energy of solar radiation directly reaching the ground surface.
AVFDIR	W/m2	The average direct solar radiation power received per unit area during the statistical period.
SUNSHINE	h	The cumulative duration of actual sunshine during the statistical period.
STABILITY	/	Reflects the stability of solar energy resources over time.
RATIO	%	The proportion of direct radiation in total solar radiation.
AIRT	°C	The average level of air temperature during the statistical period.
AH	%	The average level of relative humidity during the statistical period.
ATM	hPa	The average level of atmospheric pressure during the statistical period.
DRP	mm	The cumulative depth of precipitation during the statistical period.
WIND_HEIGHT	m	The observation or calculation height corresponding to wind resource parameters such as wind speed and wind direction.
WIND_LEVEL	/	Characterizes the quality of wind energy resource conditions, with larger values indicating better wind resource conditions.
AVWV	m/s	The average level of wind speed during the statistical period.
AVWNDD	°	The angle of wind direction relative to due north.
TURBULENCE	%	The magnitude of wind speed fluctuation relative to the average wind speed.
WIND_EXPONENT	/	The vertical gradient characteristics of wind speed variation with height.
WIND_DENSITY	W/m2	The available wind energy power per unit windward area.
AVWNDD_MAIN	°	The wind direction with the highest frequency during the statistical period.
LD_WIND	m/s	The component of wind speed in the east–west direction, namely the longitudinal direction.
POC	/	Represents the relative photovoltaic output potential associated with solar radiation and sunshine conditions.
WPOF	/	Represents the relative wind power output potential associated with wind speed and wind energy density conditions.
DD_WIND	m/s	The component of wind speed in the north–south direction, namely the latitudinal direction.

,

Table 3. Description of Key Social Factor Variables in the Initial Dataset.

Original Indicator	Units	Comments
GDP	100 million CNY	Regional economic scale indicator; used to describe the macroeconomic background associated with annual load variation.
GDP_PC	CNY/person	Represents regional economic output per resident and the income-related development background.
SEC_IND	100 million CNY	Represents industrial and construction-sector activity; relevant to industrial load background.
TER_IND	100 million CNY	Represents service-sector activity and urban consumption-related economic background.
IND_VA	100 million CNY	Represents industrial production scale and industrial electricity demand background.
CONSTR_OUT	100 million CNY	Reflects construction activity and investment-related economic background.
RETAIL_SALES	100 million CNY	Represents household consumption and commercial activity background.
DISP_INC	CNY/person	Represents overall resident income level and consumption capacity.
URBAN_INC	CNY/person	Represents urban resident income level and urban consumption background.
RURAL_INC	CNY/person	Represents rural resident income level and rural consumption background.
URBAN_POP	10,000 persons	Represents urbanization scale and population-related load background.
POWER_GEN	100 million kWh	Represents regional electricity supply scale and energy production background.
RURAL_ELEC	100 million kWh	Represents electricity consumption in rural areas.
RE_INV	100 million CNY	Represents real estate investment and construction-related economic activity.
FAI_GROWTH	%	Year-on-year growth rate of fixed asset investment excluding rural households.
COMM_HOUSE_SALES	100 million CNY	Represents the transaction scale of commercial housing and real-estate market activity.
COMM_HOUSE_AREA	10,000 m2	Represents the sold floor area of commercial housing and real-estate market activity.

and

Table 4. Description of Monthly Socioeconomic Variables.

Variable	Units	Comments
PPI	Index	Industrial producer price index; represents changes in producer-side industrial prices.
PPIP	Index	Industrial producer purchasing price index; represents changes in industrial input purchase prices.
AQUA_CPI_YOY	Index	Year-on-year consumer price index for aquatic products.
AQUA_CPI_SPLY	Index	Cumulative year-on-year consumer price index for aquatic products.
AQUA_CPI_MOM	Index	Month-on-month consumer price index for aquatic products.
GRAIN_CPI_YOY	Index	Year-on-year consumer price index for grain.
GRAIN_CPI_SPLY	Index	Cumulative year-on-year consumer price index for grain.
GRAIN_CPI_MOM	Index	Month-on-month consumer price index for grain.
EGG_CPI_YOY	Index	Year-on-year consumer price index for eggs.
EGG_CPI_SPLY	Index	Cumulative year-on-year consumer price index for eggs.
EGG_CPI_MOM	Index	Month-on-month consumer price index for eggs.
FRUIT_CPI_YOY	Index	Year-on-year consumer price index for fresh fruit.
FRUIT_CPI_SPLY	Index	Cumulative year-on-year consumer price index for fresh fruit.
FRUIT_CPI_MOM	Index	Month-on-month consumer price index for fresh fruit.
VEG_CPI_YOY	Index	Year-on-year consumer price index for fresh vegetables.
VEG_CPI_SPLY	Index	Cumulative year-on-year consumer price index for fresh vegetables.
VEG_CPI_MOM	Index	Month-on-month consumer price index for fresh vegetables.

.

The load series were independently normalized to [0, 1] for each region and temporal resolution using Min–Max normalization:

$$x_t^{\mathit{norm}}={\displaystyle \frac{x_t-x_{min}}{{xmin}_{max}}}$$

(1)

where $x_t$ is the original load value at time $t$, $x_{min}$ and $x_{max}$ are the minimum and maximum values of the corresponding load series, respectively. The monthly load data are provider-defined monthly indicators rather than aggregates derived from the hourly series. Annual load variation is described using annual maximum load and annual electricity consumption.

Data-quality control included chronological ordering, alignment of the load and external-variable series using common timestamps, inspection for missing observations, and examination of zero-valued load records. No missing observations were identified. Six dates in the Inner Mongolia hourly dataset contained zero-valued records; these records were retained in the main analysis, and their influence was examined through sensitivity analysis. For the monthly and annual meteorological datasets, cumulative variables, including IRRA, FDIR, SUNSHINE, and DRP, were aggregated by summation, whereas continuous-state variables were aggregated by averaging. Other derived indicators retained their original definitions.

To compare annual variables with different units and magnitudes, Z-score standardization was applied:

$$z_{t,j}={\displaystyle \frac{x_{t,j}-{\mu }_j}{{\sigma }_j}}$$

(2)

where $x_{t,j}$ denotes the value of variable $j$ in year $t$, ${\sigma }_j$ and ${\mu }_j$ denote its mean and standard deviation over 2015–2021, respectively. Z-score standardization was used only for the annual indicators presented in the annual-scale analysis and was not applied to the hourly or monthly VMD.

2.3. Technical Framework and Research Methods

To investigate the multiscale associations between power load and meteorological and socioeconomic factors in Shandong and Inner Mongolia, this study develops an “hourly–monthly–annual” analytical framework. The research procedure is illustrated in Figure 2 and consists of six steps.

(1)

Data preparation: Load, meteorological, and socioeconomic data are collected and organized to construct hourly, monthly, and annual datasets.

(2)

Hourly-scale analysis: VMD is applied to the hourly load series of each region separately to identify low-frequency trends, 24 h and 12 h periodic components, and high-frequency modes. Pearson correlation and mutual information are then used to examine the contemporaneous associations between meteorological variables and the decomposed load modes, followed by a comparison of intraday load profiles across different date types within each region.

(3)

Monthly-scale analysis: VMD is applied to the monthly load series to identify low-frequency trends and intra-annual fluctuation modes. The associations of meteorological, industrial, and consumption-related variables with the monthly load modes are then examined using Pearson correlation and mutual information.

(4)

Annual-scale analysis: Descriptive comparisons are conducted using annual maximum load, annual electricity consumption, meteorological conditions, and socioeconomic indicators to characterize the regional backgrounds associated with long-term load changes. Owing to the limited number of annual observations, this step is used only for descriptive interpretation rather than statistical inference.

(5)

Multiscale association synthesis: The hourly, monthly, and annual results are integrated to summarize how meteorological, industrial, and consumption-related variables are associated with load characteristics at different temporal scales.

(6)

Regional heterogeneity assessment: Regional differences between Shandong and Inner Mongolia are evaluated mainly on the basis of the temporally aligned monthly and annual datasets. The hourly results are interpreted separately as within-region cases and are not used as direct evidence of cross-regional heterogeneity.

2.3.1. VMD Method

Variational mode decomposition (VMD) is a non-recursive signal decomposition method proposed by Dragomiretskiy and Zosso. By constructing a constrained variational model, this method decomposes the original nonstationary signal into several mode components with finite bandwidth, and each mode is distributed around a different center frequency [15]. Compared with recursive decomposition methods such as EMD, VMD shows better performance in suppressing mode mixing and reducing noise interference. In power load studies, VMD is commonly used to decompose complex load sequences, reduce the nonstationarity of the original sequence, and extract load variation components at different frequency scales [13].

VMD first calculates the analytic signal of each mode component $u_k\left(t\right)$ through the Hilbert transform to obtain the unilateral spectrum.

$$\left(\delta (t)+{\displaystyle \frac{j}{\pi t}}\right)\ast u_k(t)$$

(3)

The analytic signal of each mode is then mixed with its corresponding center frequency term $e^{-j{\omega }_{k}t}$, so that the spectrum of each mode is shifted to the corresponding baseband.

$$\left[\left(\delta (t)+{\displaystyle \frac{j}{\pi t}}\right)\ast u_k(t)\right]e^{-j{\omega }_{k}t}$$

(4)

The bandwidth of each mode signal is then estimated by calculating the squared $L$ norm of the gradient of the demodulated signal based on the Gaussian smoothness criterion. Accordingly, the constrained variational model is formulated as follows:

$${\mathrm{m}\mathrm{i}\mathrm{n}}_{\left\{u_k\right\},\left\{{\omega }_k\right\}}\left\{\sum _k{‖{\partial }_t\left[\left(\delta \left(t\right)+{\displaystyle \frac{j}{\pi t}}\right)\ast u_k\left(t\right)\right]e^{-j{\omega }_{k}t}‖}_2^2\right\}$$

(5)

$$s.t.\sum _{k=1}^K{u}_k=f$$

(6)

where ${\partial }_t$ denotes the partial derivative with respect to time (t), $\left\{u_k\right\}$ denotes the set of decomposed mode components, $\left\{{\omega }_k\right\}$ denotes the set of center frequencies corresponding to each mode component, and ∗ denotes the convolution operation.

$$\begin{gathered} L\left(\left\{u_k\right\},\left\{{\omega }_k\right\},\lambda \right)=\alpha \sum _k{‖{\partial }_t\left[\left(\delta (t)+{\displaystyle \frac{j}{\pi t}}\right)\ast u_k(t)\right]e^{-j{\omega }_{k}t}‖}_2^2 \\ +{‖f(t)-\sum _k{u}_k(t)‖}_2^2+⟨\lambda (t),f(t)-\sum _k{u}_k(t)⟩ \end{gathered}$$

(7)

The Lagrange multiplier $l\left(t\right)$ and the quadratic penalty factor $\alpha$ are introduced to transform the constrained variational problem into an unconstrained variational model.

To obtain the optimal solution of Equation (7), VMD adopts the alternating direction method of multipliers. With the cyclic updates in Equations (8) and (9), each decomposed signal $fukg$ and its corresponding center frequency $fwkg$ are iteratively updated.

$${\hat{u}}_k^{n+1}(\omega )={\displaystyle \frac{\widehat{f}(\omega )-{\sum }_{i\ne k}{\widehat{u}}_i(\omega )+\frac{\widehat{\lambda }(\omega )}{2}}{1+2\alpha (\omega -{\omega }_k{)}^2}}$$

(8)

$${\omega }_k^{n+1}={\displaystyle \frac{{\int }_0^{\infty }\omega {\left|{\widehat{u}}_k\left(\omega \right)\right|}^{2}d\omega }{{\int }_0^{\infty }{\left|{\widehat{u}}_k\left(\omega \right)\right|}^{2}d\omega }}$$

(9)

When the convergence criterion in Equation (10) is satisfied, the iterative procedure is terminated.

$$\sum _k{\displaystyle \frac{{‖{\widehat{u}}_k^{n+1}-{\widehat{u}}_k^n‖}_2^2}{{‖{\widehat{u}}_k^n‖}_2^2}}<\epsilon , n<N$$

(10)

The quality of the VMD results was evaluated using the orthogonality index (OI), relative reconstruction error (RRE), and mode mixing index (MMI):

$$\mathrm{OI}={\displaystyle \frac{2}{K(K-1)}}\sum _{i=1}^{K-1}\sum _{j=i+1}^K{\displaystyle \frac{\left|⟨u_i,u_j⟩\right|}{{‖u_i‖}_2{‖u_j‖}_2}}$$

(11)

$$\mathrm{RRE}={\displaystyle \frac{{‖x-{\sum }_{k=1}^K{u}_k‖}_2}{{‖x‖}_2}}$$

(12)

$$\mathrm{MMI}={\displaystyle \frac{2}{K(K-1)}}\sum _{i=1}^{K-1\sum \left[P_i\left(f\right),P_j\left(f\right)\right]}\sum _{j=i+1}^{K\sum }\sum _f^{\sum }min$$

(13)

where $x$ is the original load series, $u_k$ is the mode, $K$ is the number of modes, and $P_k\left(f\right)$ is the normalized one-sided power spectrum of $u_k$. Lower OI, RRE, and MMI values indicate better mode orthogonality, reconstruction accuracy, and spectral separation, respectively.

Each mode was further characterized by its center frequency, dominant period, amplitude, and standard deviation. The center frequency was obtained from the converged frequency estimate of the VMD algorithm. The remaining indicators were calculated as

$$T_k={\displaystyle \frac{1}{f_{k,\mathrm{peak}}}}$$

(14)

$$A_k={\max }_t\left[u_k(t)\right]-{\min }_t\left[u_k(t)\right]$$

(15)

$${\sigma }_k=\sqrt{{\displaystyle \frac{1}{N}}\sum _{t=1}^N{\left[u_k(t)-{\overline{u}}_k\right]}^2}$$

(16)

where $f_k,peak$ is the frequency corresponding to the largest non-zero spectral peak, $N$ is the sequence length, and ${\overline{U}}_k$ is the mean of mode $u_k$. The dominant periods are expressed in hours for hourly data and months for monthly data. Detailed parameter settings and sensitivity analyses are presented in Section 2.4.

2.3.2. Pearson Correlation and Mutual Information Analysis

To characterize the associations between different load modes and external factors, the Pearson correlation coefficient and mutual information (MI) are jointly employed. The Pearson correlation coefficient measures the direction and strength of the linear association between two variables, whereas MI measures their general statistical dependence, including both linear and nonlinear dependence. Let the decomposed load-mode sequence be $X=\{x_1, x_2, \dots , x_N\}$ and the influencing factor sequence be $Y=\{y_1, y_2, \dots , y_N\}$. The Pearson correlation coefficient is calculated as follows:

$$r_{XY}={\displaystyle \frac{{\displaystyle \sum _{i=1}^N(x_i-\overline{x})}(y_i-\overline{y})}{\sqrt{{\displaystyle \sum _{i=1}^N{(x_i-\overline{x})}^2}}\sqrt{{\displaystyle \sum _{i=1}^N{(y_i-\overline{y})}^2}}}}$$

(17)

where $r_{XY}\in [-1, 1]$ When $r_{XY}>0$, the two variables are positively correlated; when $r_{XY}<0$, the two variables are negatively correlated. The closer $\left|r_{XY}\right|$ is to 1, the stronger the linear correlation.

MI measures the general statistical dependence between two variables and is defined as follows:

$$I(X;Y)={\displaystyle \iint p}(x,y)\log {\displaystyle \frac{p(x,y)}{p(x)p(y)}}dxdy$$

(18)

where $p\left(x,y\right)$ is the joint probability density function, and $p\left(x\right)$ and $p\left(y\right)$ are the marginal probability density functions, respectively. When $I\left(X;Y\right)=0$, the two variables are independent of each other. The larger the mutual information value, the stronger the degree of information dependence between them. Since the research variables are mostly continuous sequences, mutual information can be estimated using a nonparametric method based on $k$ nearest neighbors to reduce the influence of binning methods on the results [25].

Considering the serial dependence of the monthly series, the uncertainty of the monthly Pearson correlation coefficients was further evaluated using a circular moving-block bootstrap [26]. For each load–factor pair, the two series were resampled jointly to preserve their contemporaneous relationship and local temporal dependence. Let $Z_t=(X_t, Y_t)$$t=1, \dots , N$ denote the paired monthly observations. A circular block of length $L$ beginning at position $i$ is defined as

$$B_i^{\left(l\right)}=\left(Z_i, Z_{i+1}, \dots , Z_{i+l-1}\right),\hspace{1em}{Z}_{N+j}=Z_j$$

(19)

For the $b$-th bootstrap sample, $M=[N/L]$ blocks were randomly selected with replacement and concatenated, after which the resulting sequence was truncated to the original sample length $N$. The Pearson correlation coefficient was recalculated for each bootstrap sample:

$$r^{\ast \left(b\right)}=\mathit{corr}\left(X^{\ast \left(b\right)}, Y^{\ast \left(b\right)}\right),\hspace{1em}b=1, 2, \dots ,B$$

(20)

The percentile confidence interval was obtained from the empirical distribution of the bootstrap correlation coefficients:

$$\mathrm{C}{\mathrm{I}}_{1-\alpha }=\left[Q_{\alpha /2}\left(r^{\ast }\right), Q_{1-\alpha /2}\left(r^{\ast }\right)\right]$$

(21)

where $Q_p(r\ast )$ denotes the $p$-th quantile of the bootstrap distribution. A Pearson association was recorded as retained by the unadjusted block-bootstrap analysis when its unadjusted 95% confidence interval did not include zero. The bootstrap procedure was applied only to the monthly Pearson coefficients, whereas the MI results were retained as descriptive measures of statistical dependence.

2.3.3. Calendar-Related Statistical Analysis

Calendar-related differences were evaluated among weekdays, ordinary weekends, and statutory holidays. The mean 24 h load profiles and their 95% confidence intervals were estimated using 4000 nonparametric bootstrap resamples at the daily-profile level. The daily peak-to-valley range was calculated as

$$R_d={\max }_{1\le h\le 24}{L}_{d,h}-{\min }_{1\le h\le 24}{L}_{d,h}$$

(22)

where $L_{d,h}$ denotes the load at hour $h$ on day $d$. Differences among the three date categories were assessed using the Kruskal–Wallis test [27]. When the overall test was significant, Dunn’s pairwise test with the Holm correction was applied, and epsilon squared was used to quantify the effect size:

$${\epsilon }^2={\displaystyle \frac{H-g+1}{N-g}}$$

(23)

where $H$ is the Kruskal–Wallis statistic, $g$ is the number of groups, and $N$ is the total number of daily profiles. A sensitivity analysis was conducted after excluding the six zero-valued dates in the Inner Mongolia dataset.

2.4. Parameter Settings and Sensitivity Analysis

The number of modes K was selected separately for each load series by jointly considering the orthogonality index (OI), relative reconstruction error (RRE), mode mixing index (MMI), and the interpretability of the center-frequency and dominant-period structures. The other VMD parameters were fixed at α = 2000, τ = 0, DC = 0, init = 1, a convergence tolerance of 1 × 10⁻⁷, and a maximum of 500 iterations. Based on the overall evaluation, K = 6 was selected for the Shandong hourly load, K = 10 for the Inner Mongolia hourly load, and K = 6 for both monthly load series. The detailed results are reported in

Table 5. Sensitivity Analysis of the Number of VMD Modes for Hourly and Monthly Load Series.

Data Series	K	OI	RRE	MMI
Shandong hourly load	5	0.0046	0.0126	0.0022
	6	0.0049	0.0089	0.0017
	7	0.0063	0.0083	0.0027
Inner Mongolia hourly	9	0.0149	0.0059	0.0037
	10	0.0142	0.0054	0.0032
	11	0.0154	0.0050	0.0036
Shandong monthly	5	0.0101	0.0282	0.0089
	6	0.0035	0.0252	0.0142
	7	0.0108	0.0222	0.0144
Inner Mongolia monthly	5	0.0072	0.0186	0.0248
	6	0.0074	0.0139	0.0180
	7	0.0114	0.0135	0.0415

Note: OI denotes the orthogonality index, RRE denotes the relative reconstruction error, and MMI denotes the mode mixing index. Lower values indicate better performance.

.

After determining K, the penalty parameter α was further tested at 1500, 2000, and 2500. The dominant periods of the corresponding modes remained unchanged under the three settings. Considering the overall performance of OI, RRE, and MMI, as well as the need for consistent parameter settings across regions and temporal scales, α = 2000 was retained for the final decomposition. The detailed results are presented in

Table 6. Sensitivity Analysis of the VMD Penalty Parameter α with Fixed Mode Numbers.

Data Series	Alpha	OI	RRE	MMI
Shandong hourly load	1500	0.0063	0.0075	0.0022
	2000	0.0049	0.0089	0.0017
	2500	0.0048	0.0098	0.0027
Inner Mongolia hourly	1500	0.0163	0.0044	0.0037
	2000	0.0142	0.0054	0.0032
	2500	0.0127	0.0062	0.0028
Shandong monthly	1500	0.0114	0.0220	0.0139
	2000	0.0035	0.0252	0.0142
	2500	0.0075	0.0284	0.0116
Inner Mongolia monthly	1500	0.0089	0.0121	0.0200
	2000	0.0074	0.0139	0.0180
	2500	0.0063	0.0154	0.0184

.

For the mutual-information analysis, MI was estimated using the k-nearest-neighbor estimator implemented in mutual_info_regression in scikit-learn, with k = 3 and random_state = 42.

All variables were independently rescaled to [0, 1] before estimation, and the same settings were applied to all external-variable–load-mode pairs. Pearson correlation describes linear association, whereas MI measures general statistical dependence. In the joint plots, the numerical labels represent Pearson coefficients and bubble size represents MI; MI was rescaled by the maximum value only for visualization. For the monthly Pearson association analysis, the circular moving-block bootstrap was performed using the 84 monthly observations from January 2015 to December 2021. The block length was set to 12 months to retain the annual dependence structure of the monthly series, and 4000 bootstrap samples were generated. The 2.5th and 97.5th percentiles were used to construct the 95% confidence intervals. Associations whose 95% confidence intervals excluded zero were retained in the block-bootstrap summary. The same bootstrap settings were applied to the meteorological and socioeconomic variable groups in both regions.

3. Results and Discussion

3.1. Hourly-Scale Within-Region Associations Between Load Modes and Meteorological Factors

This section examines the associations between hourly load and meteorological factors separately for Shandong and Inner Mongolia. VMD is applied to the hourly load series of each region, and the Pearson correlation coefficient and mutual information (MI) are calculated for both the original load series and the resulting modes. The original load is used as a baseline to represent the overall association, whereas the decomposed modes are used to identify the temporal scales at which these associations are concentrated. The Pearson correlation coefficient characterizes the direction and strength of linear associations, whereas MI measures general statistical dependence. Because the hourly datasets for Shandong and Inner Mongolia cover different periods and have different record lengths, the results are interpreted only within each region and are not used for direct cross-regional comparison or regional heterogeneity inference.

3.1.1. Hourly-Scale Results for Shandong

The VMD results of the hourly load in Shandong are shown in

Table 7. VMD Mode Characteristics of Shandong Hourly Load.

Mode	Center Frequency	Dominant Period (h)	Amplitude	Standard Deviation
IMF1	0.00004	4380	0.4109	0.0700
IMF2	0.04165	24.00	0.2889	0.0721
IMF3	0.08337	12.00	0.1115	0.0258
IMF4	0.16533	6.00	0.0323	0.0064
IMF5	0.32960	3.00	0.0108	0.0019
IMF6	0.41492	2.40	0.0101	0.0018

. These results reveal an evident multiscale structure in the hourly load sequence. IMF1 mainly characterizes low-frequency changes in the overall load level, while IMF2 and IMF3 correspond to the dominant periods of 24 h and 12 h, respectively. IMF4–IMF6 have dominant periods of 6 h and below and mainly represent short-period fluctuations. Overall, the hourly load in Shandong presents a hierarchical structure consisting of low-frequency variation, intraday periodic components, and short-period fluctuations.

Figure 3 compares the associations of the original load and the decomposed modes with meteorological variables. The original load shows moderate positive Pearson correlations with radiation- and sunshine-related variables. Specifically, the correlation coefficients of AVIR, IRRA, FDIR, AVFDIR, SUNSHINE, and RATIO range from 0.501 to 0.563. Their MI values range from 0.237 to 0.301, indicating that these variables retain observable overall statistical associations with the original hourly load. After decomposition, these associations become more clearly localized in IMF2. The Pearson correlation coefficients of AVIR, IRRA, FDIR, AVFDIR, SUNSHINE, and RATIO with IMF2 range from 0.704 to 0.811, while the corresponding MI values range from 0.493 to 0.641. Since IMF2 has a dominant period of 24 h, the results indicate that the overall associations observed in the original load are mainly concentrated in the daily-cycle component. The decomposition therefore provides additional scale-specific information rather than merely increasing the association strength. Because the common diurnal periodicity was not removed from the load and meteorological series, these results are interpreted as contemporaneous statistical associations and do not establish independent meteorological effects.

The low-frequency IMF1 presents a different association pattern. Its Pearson correlations with most meteorological variables are relatively limited, ranging from −0.179 for wind-related variables to 0.230 for precipitation. However, IMF1 shows relatively high MI values with AIRT, ATM, AVWNDD, DD_WIND, AVWV, WIND_DENSITY, and LD_WIND. In particular, the MI values for AIRT and ATM reach 1.009 and 0.656, respectively, compared with 0.311 and 0.057 for the original load. This indicates that the statistical dependence of temperature, atmospheric pressure, and several wind-field variables is mainly concentrated in the low-frequency load component and is not fully represented by linear correlation alone. However, these MI results do not by themselves establish a specific nonlinear functional relationship. The associations of IMF3–IMF6 with meteorological variables are substantially weaker and less concentrated than those of IMF1 and IMF2. Although IMF3 represents the 12 h semi-daily component, its Pearson correlation coefficients remain below 0.10 for most variables. The Pearson correlations of IMF4–IMF6 are generally close to zero, while their MI values are also lower than those observed for the low-frequency and daily-cycle components. These results indicate that the contemporaneous associations between individual meteorological variables and short-period load fluctuations are relatively limited.

3.1.2. Hourly-Scale Results for Inner Mongolia

The VMD results of the hourly load in Inner Mongolia are shown in

Table 8. VMD Mode Characteristics of Inner Mongolia Hourly Load.

Mode	Center Frequency	Dominant Period (h)	Amplitude	Standard Deviation
IMF1	0.00001	8760	0.4108	0.0489
IMF2	0.04132	24.00	0.5819	0.0225
IMF3	0.08342	12.00	0.5164	0.0177
IMF4	0.12650	8.00	0.3352	0.0072
IMF5	0.20330	4.80	0.2142	0.0040
IMF6	0.24866	4.00	0.2164	0.0039
IMF7	0.29005	3.43	0.1989	0.0038
IMF8	0.33347	3.00	0.1580	0.0029
IMF9	0.37989	2.67	0.2091	0.0026
IMF10	0.43025	2.40	0.3133	0.0026

. These results reveal a clear multiscale structure in the hourly load series of Inner Mongolia. IMF1 mainly characterizes low-frequency trend changes, reflecting variations in the overall load level and stage-specific fluctuations. IMF2 and IMF3 correspond to dominant periods of 24 h and 12 h, respectively, characterizing the principal intraday periodic components of the load. IMF4–IMF10 are concentrated in shorter-period ranges and mainly reflect local high-frequency fluctuations. Overall, the hourly load in Inner Mongolia exhibits a hierarchical structure consisting of low-frequency trend variations, intraday periodic fluctuations, and short-period high-frequency fluctuations.

The associations between Inner Mongolia hourly load modes and meteorological factors show scale differentiation characteristics, mainly concentrated in the low-frequency trend term and intraday cycle terms, and rapidly weakened in high-frequency modes. For the original load series, the Pearson correlation coefficients range from −0.085 to 0.153, while the MI values are generally low, with the highest value of 0.091 observed for the temperature-related variable. Overall, the Pearson correlation coefficients between meteorological variables and load modes are generally not high, indicating that there is no strong linear synchronous relationship between meteorological factors and Inner Mongolia hourly load. However, the MI results indicate that several low-frequency and intraday modes retain statistical dependence with meteorological variables that is not fully represented by linear correlation alone, as shown in Figure 4.

From the linear correlation results, IMF2 is the mode with the clearest meteorological association. This mode corresponds to the 24 h daily-cycle term and is generally positively correlated with radiation-, sunshine-, and photovoltaic-output-related variables, reflecting a certain synchronous variation between the intraday load rhythm and the solar radiation process. Compared with the original load series, these associations are more clearly concentrated in IMF2. However, its correlation strength is generally limited, indicating that although the daily-cycle fluctuation of Inner Mongolia load shows contemporaneous associations with intraday meteorological variables, the observed associations remain limited and do not establish independent meteorological effects. The Pearson correlation of IMF1 is not prominent, but the mutual information results show that it has relatively stronger information dependence with temperature-, atmospheric-pressure-, and wind-speed-related variables. The MI values of the temperature- and atmospheric-pressure-related variables increase from 0.091 and 0.039 in the original load to 0.439 and 0.240 in IMF1, respectively. This difference between weak linear correlation and high mutual information indicates that the low-frequency trend term and meteorological background conditions are not fully represented by linear correlation alone during slow-changing processes. IMF3 corresponds to the 12 h semi-daily cycle term and still shows certain associations with radiation-, sunshine-, and photovoltaic-output-related variables, but the overall strength is weaker than that of IMF2. The Pearson correlations between IMF4 and subsequent modes and most meteorological variables are close to zero, and their mutual information levels are also significantly reduced, indicating that short-cycle high-frequency disturbances show limited contemporaneous associations with individual meteorological variables and are more manifested as local fluctuation components in the load sequence.

3.1.3. Calendar-Related Differences in Intraday Load Profiles

Calendar-related differences were examined by classifying complete daily load profiles as weekdays, ordinary weekends, or statutory holidays according to the national public-holiday calendar. Adjusted weekend workdays were classified as weekdays, while weekend days within continuous holiday periods were classified as holidays. Mean 24 h profiles and their 95% confidence intervals were estimated using 4000 bootstrap resamples of complete daily profiles. The daily peak–valley range was calculated as the difference between the maximum and minimum hourly load within each day. Differences among date categories were assessed separately for each region using the Kruskal–Wallis test, followed by Dunn’s pairwise test with the Holm correction. The effect size was measured using ε². As shown in Figure 5a,c, the Shandong dataset comprised 250 weekdays, 85 ordinary weekends, and 30 statutory holidays. The holiday profile remained below the weekday and ordinary-weekend profiles throughout the day. The median daily peak–valley ranges were 0.2304, 0.2124, and 0.1717, respectively. Significant differences were found among the three date categories (H = 78.59, p < 0.001, ε² = 0.212), and all pairwise comparisons remained significant after the Holm correction.

For Inner Mongolia, 1746 weekdays, 602 ordinary weekends, and 207 statutory holidays were included. The holiday profile was generally lower than the other two profiles, whereas weekday and ordinary-weekend profiles were similar over most hours. Their median daily peak–valley ranges were 0.0868, 0.0900, and 0.0848, respectively. Although the overall difference was significant (H = 12.52, p = 0.0019), the effect size was small (ε² = 0.004). Significant differences were identified between weekdays and ordinary weekends and between ordinary weekends and statutory holidays, while the weekday–holiday comparison was not significant after the Holm correction.

Six Inner Mongolia dates contained zero-valued hourly records and accounted for the extreme peak–valley observations in Figure 5d. These records were retained because they were present in the source dataset. A sensitivity analysis excluding these dates produced similar results (H = 13.02, p = 0.0015, ε² = 0.004). Because the two hourly datasets cover different periods, the results represent separate within-region calendar-related differences rather than a direct regional comparison.

3.2. Monthly-Scale Associations Between Load Modes and Meteorological and Socioeconomic Factors

Monthly-scale load differs from hourly load in temporal resolution and variation characteristics. Its variation is associated with both seasonal meteorological conditions and medium- and low-frequency socioeconomic indicators related to industrial activity and consumption. To characterize these associations, VMD is applied to the monthly load series, and the Pearson correlation coefficients and mutual information (MI) between the external variables and both the original load and decomposed modes are calculated. The original load is included as a baseline for comparison. The Pearson correlation coefficient characterizes the direction and strength of linear associations, whereas MI measures general statistical dependence.

3.2.1. Monthly-Scale Results for Shandong

The decomposition results are shown in

Table 9. VMD Mode Characteristics of Shandong Monthly Load.

Mode	Center Frequency	Dominant Period (m)	Amplitude	Standard Deviation
IMF1	0.00002	84.00	0.0512	0.0129
IMF2	0.08016	12.00	0.1379	0.0377
IMF3	0.16731	6.00	0.1883	0.0557
IMF4	0.24994	4.00	0.1080	0.0329
IMF5	0.38569	2.63	0.0519	0.0104
IMF6	0.43033	2.27	0.0584	0.0119

. These results indicate that the monthly load series contains both low-frequency and intra-annual fluctuation components. The dominant period of IMF1 covers the entire study period, characterizing slow variations in the overall level of monthly load. IMF2 exhibits a clear 12-month period, corresponding to the annual periodic component of the load series. IMF3 and IMF4 have dominant periods of 6 months and 4 months, respectively, representing shorter intra-annual fluctuations. The dominant periods of IMF5 and IMF6 are further shortened to 2.63 months and 2.27 months, respectively. Overall, the Shandong monthly load series presents a multiscale structure consisting of low-frequency variation, an annual periodic component, intra-annual fluctuations, and short-period intermonthly fluctuations.

The associations between the original Shandong monthly load, its decomposed modes, and meteorological factors are mainly observed in the annual and intra-annual fluctuation components. As shown in Figure 6, the original load exhibits overall associations with precipitation-, temperature-, wind-speed-, and wind-power-related variables, while the decomposition results further identify the temporal scales at which these associations are concentrated. IMF1 represents the long-term low-frequency variation of monthly load, but its Pearson correlation coefficients and MI values with most meteorological variables are relatively low, indicating limited contemporaneous associations at this scale. In contrast, IMF2, with a dominant period of 12 months, shows clearer associations with temperature-, wind-speed-, wind-level-, and wind-power-output-related variables, indicating that the annual periodic component contains the main seasonal meteorological associations. IMF3, with a dominant period of 6 months, shows relatively high MI values with temperature-, atmospheric-pressure-, radiation-, and sunshine-related variables, whereas its Pearson correlation coefficients are relatively dispersed. This pattern indicates that the statistical dependence between the intra-annual load component and these meteorological variables is not fully represented by linear correlation alone. The associations of IMF4–IMF6 with most meteorological variables are generally weaker. Overall, compared with the original load series, the decomposed results more clearly distinguish the meteorological associations distributed across the annual and intra-annual components.

The associations between Shandong monthly load modes and socioeconomic factors are shown in Figure 7, and the results exhibit scale-specific patterns that differ from those observed for meteorological factors. For the original load series, the strongest Pearson correlations are observed with the month-on-month indices of eggs and fresh vegetables, reaching 0.521 and 0.520, respectively. Its MI values are generally low, with the highest value of 0.211 observed for the month-on-month egg price index. Combined with the VMD results, the descriptive Pearson and MI results show associations in the low-frequency and several intermonthly fluctuation components rather than in the annual and semiannual periodic components where meteorological associations are more evident. Compared with the original load series, the associations with the year-on-year and cumulative year-on-year price indicators are more clearly concentrated in IMF1. IMF1, which represents the low-frequency variation of monthly load, shows relatively high Pearson correlation coefficients and MI values with the year-on-year and cumulative year-on-year indicators of the industrial producer price index and several consumer price indices. These variables are treated as indicators of industrial and consumer price conditions and are considered collectively in the association analysis. These descriptive Pearson and MI results indicate associations between IMF1 and several slowly varying price indicators; however, most corresponding Pearson associations were not retained in the block-bootstrap analysis.

The associations between IMF2 and socioeconomic factors are generally weak. IMF2 has a relatively clear 12-month period and shows more evident contemporaneous associations with seasonal meteorological variables such as temperature and wind speed, whereas its associations with price-related socioeconomic indicators are relatively limited. Accordingly, the annual periodic component of Shandong monthly load shows clearer associations with seasonal meteorological variation than with the selected price indicators. The intra-annual fluctuation components, including IMF3 and IMF4, show certain associations with several month-on-month consumer price indicators, particularly fresh vegetables, aquatic products, and eggs. The original load also shows positive correlations with the month-on-month indices of eggs and fresh vegetables, while the decomposed results distinguish these associations across IMF2–IMF4. These indicators exhibit pronounced intermonthly fluctuations and are interpreted collectively as indicators of short-term changes in the consumer price environment.

3.2.2. Monthly-Scale Results for Inner Mongolia

The decomposition results for Inner Mongolia are shown in

Table 10. VMD Mode Characteristics of Inner Mongolia Monthly Load.

Mode	Center Frequency	Dominant Period (m)	Amplitude	Standard Deviation
IMF1	0.00001	84.00	0.0460	0.0158
IMF2	0.07923	12.00	0.1264	0.0331
IMF3	0.16751	6.00	0.0642	0.0180
IMF4	0.24772	4.00	0.0416	0.0121
IMF5	0.32438	3.11	0.0208	0.0051
IMF6	0.41192	2.40	0.0432	0.0076

. The results indicate that the monthly load series contains a low-frequency trend component, an annual periodic component, and several intra-annual fluctuation components. The dominant period of IMF1 covers the entire study period, characterizing slow variations in the overall level of the monthly load series. IMF2 exhibits a clear 12-month period and corresponds to the principal annual seasonal component. The dominant periods of IMF3–IMF6 are further shortened, mainly corresponding to intra-annual fluctuations of 6 months and below. Overall, the Inner Mongolia monthly load series presents a multiscale structure consisting of low-frequency variation, an annual seasonal component, and short-period intermonthly fluctuations.

The associations between Inner Mongolia monthly load modes and meteorological factors are shown in Figure 8. For the original load series, negative Pearson correlations are observed with radiation-, sunshine-, and photovoltaic-output-related variables, with the strongest correlation observed for the photovoltaic-output-related variable at −0.580. Its MI values are relatively higher for radiation- and sunshine-related variables, with the maximum value of 0.265. The results indicate that these associations are not evenly distributed across the VMD modes but are mainly concentrated in the annual periodic component and several intra-annual fluctuation components. IMF1, which characterizes the low-frequency variation of monthly load, shows relatively weak Pearson correlations and low MI values with most meteorological variables, indicating limited contemporaneous associations between the long-term load variation and individual meteorological variables. IMF2, with a dominant period of 12 months, exhibits a clearer meteorological association pattern. Radiation-, sunshine-, and photovoltaic-output-related variables are mostly negatively correlated with IMF2, whereas atmospheric-pressure- and several wind-field-related variables show positive correlations, revealing directional differences in the associations between the annual load component and different groups of seasonal meteorological variables. Compared with the original load series, these linear associations are more clearly concentrated in IMF2. Relative to the Shandong results, where temperature- and wind-speed-related variables show broader associations with intra-annual load fluctuations, the Inner Mongolia annual periodic component displays a clearer contrast between solar-radiation-related variables and atmospheric-pressure- and wind-field-related variables. The Pearson correlations of IMF3 are generally weak, whereas its MI values with radiation- and sunshine-related variables are relatively high, indicating that these statistical dependencies are not fully represented by linear correlation alone. The higher MI values in IMF3 further distinguish these associations from the overall pattern observed in the original load series. The meteorological associations of the shorter-period intermonthly components are generally weaker.

The associations between Inner Mongolia monthly load modes and socioeconomic factors are shown in Figure 9. For the original load series, the Pearson correlations with socioeconomic indicators are generally weak, with the highest coefficient of 0.309 observed for the month-on-month fresh-vegetable price index. Its MI values are relatively higher for the cumulative year-on-year indices of grain, eggs, and aquatic products, ranging from 0.214 to 0.256. These associations are mainly concentrated in the low-frequency component and several local intermonthly fluctuation components. Combined with the VMD results, IMF1, which characterizes the low-frequency variation of monthly load, shows relatively high MI values with the industrial producer price index, the purchasing price index, and several food-related consumer price indices. Compared with the original load series, these associations are more clearly concentrated in IMF1. This indicates that the statistical dependence between the low-frequency load component and industrial and consumer price conditions is not fully represented by contemporaneous linear correlation alone. Compared with the meteorological results, IMF1 shows relatively stronger statistical dependence with the selected socioeconomic indicators than with most meteorological variables.

For the annual periodic and intra-annual fluctuation components, the associations with meteorological and socioeconomic factors partly occur at similar temporal scales. The preceding meteorological results show that IMF2 and IMF3 retain associations with radiation-, sunshine-, atmospheric-pressure-, and wind-field-related variables. Among the socioeconomic indicators, the month-on-month or year-on-year indices of fresh vegetables, fresh fruits, and eggs also show certain associations with several intra-annual fluctuation components. The original load also shows positive correlations with the month-on-month indices of fresh vegetables and eggs, while the decomposed results distinguish these associations across different modes. These indicators are considered collectively as indicators of short-term changes in the consumer price environment, and their associations with the corresponding load modes describe contemporaneous statistical dependence at the intermonthly scale.

3.2.3. Block-Bootstrap Analysis of Monthly Pearson Associations

Shandong

To account for serial dependence in the monthly series, circular moving-block bootstrap was applied to the Pearson correlation coefficients. For the meteorological variables, 15 of the 17 associations with the original load had 95% confidence intervals excluding zero. Among the decomposed modes, 10 and 13 associations were retained for IMF2 and IMF3, respectively, whereas none were retained for IMF1 or IMF5. Five associations were retained across IMF4–IMF6. These results are consistent with the preceding analysis, indicating that the monthly meteorological associations were mainly distributed in the annual and intra-annual components.

Fewer socioeconomic associations were retained. Only 11 of the 119 tested variable–load-component pairs had confidence intervals excluding zero. IMF1 retained one association, while several additional associations were distributed across IMF2–IMF4 and mainly involved individual month-on-month food-price indices. Therefore, the socioeconomic results are interpreted as sparse and localized associations rather than as a systematic pattern.

Inner Mongolia

The block-bootstrap results for Inner Mongolia show that the retained meteorological Pearson associations were mainly concentrated in IMF2 and IMF3. Among the 17 meteorological variables, 9 associations with the original load had 95% confidence intervals excluding zero. The corresponding numbers were 11 for IMF2 and 12 for IMF3, whereas no association was retained for IMF1 or IMF5. Six associations were retained across IMF4–IMF6.

Fewer socioeconomic associations were retained. Only 6 of the 119 tested variable–load-component pairs had confidence intervals excluding zero. These associations were distributed across the original load, IMF2, IMF4, and IMF6 and mainly involved individual month-on-month price indices. No association was retained for IMF1, IMF3, or IMF5. The socioeconomic results are therefore interpreted as sparse and localized associations rather than as a systematic pattern across the monthly load components.

3.2.4. Monthly-Scale Regional Comparison

Both regions exhibit similar multiscale monthly load structures consisting of low-frequency, annual, and intra-annual components. The original Pearson and MI results indicate that meteorological associations are mainly distributed in the annual and intra-annual components. The block-bootstrap analysis further showed that the meteorological Pearson associations with 95% confidence intervals excluding zero were concentrated primarily in IMF2 and IMF3 in both regions. Fewer socioeconomic associations met this criterion: 11 variable–component pairs in Shandong and 6 pairs in Inner Mongolia, most of which involved individual month-on-month price indices. The socioeconomic results are therefore interpreted as sparse and localized contemporaneous associations rather than broad regional patterns. These findings describe differences in the distribution of monthly load–factor associations but do not establish independent or causal effects. A summary of the retained monthly Pearson associations is presented in

Table 11. Summary of Monthly Pearson Associations Retained by the Circular Moving-Block Bootstrap Analysis.

Load Component	Shandong Meteorological	Shandong Socioeconomic	Inner Mongolia Meteorological	Inner Mongolia Socioeconomic
Original load	15/17	2/17	9/17	1/17
IMF1	0/17	1/17	0/17	0/17
IMF2	10/17	2/17	11/17	2/17
IMF3	13/17	3/17	12/17	0/17
IMF4	2/17	2/17	3/17	2/17
IMF5	0/17	0/17	0/17	0/17
IMF6	3/17	1/17	3/17	1/17
Total	43/119	11/119	38/119	6/119

Note: Values indicate the number of associations whose 95% circular moving-block bootstrap confidence intervals excluded zero relative to the total number tested. The analysis used 12-month blocks and 4000 resamples. No multiple-comparison adjustment was applied. MI results are descriptive.

.

3.3. Annual-Scale Load Changes and Regional Structural Context

Because only seven annual observations are available for each region, VMD and Pearson/MI-based association analyses are not conducted at the annual scale. Instead, the annual analysis is based on the original annual series and is used only to descriptively compare load changes and the corresponding meteorological and socioeconomic contexts of Shandong and Inner Mongolia. All monetary indicators are reported in nominal terms, and Z-score standardization is applied solely for visual comparison of temporal trajectories.

3.3.1. Characteristics of Annual Load Changes

At the annual scale, both the annual maximum load and annual electricity consumption of Shandong and Inner Mongolia show an overall upward trend from 2015 to 2021, as shown in Figure 10. The annual maximum load of Shandong increased from 64,369 MW to 109,970 MW, representing an increase of 70.84%, whereas that of Inner Mongolia increased from 27,206 MW to 38,842 MW, representing an increase of 42.77%. Annual electricity consumption in Shandong increased from 511.7 billion kWh to 738.3 billion kWh, while that in Inner Mongolia increased from 201.962 billion kWh to 284.297 billion kWh. Overall, both regions experienced an expansion in annual load scale during the study period, with Shandong showing a larger proportional increase in annual maximum load.

3.3.2. Differences in Annual Meteorological and Socioeconomic Contexts

Figure 11 presents the standardized interannual variations in annual meteorological variables for Shandong and Inner Mongolia. After Z-score standardization, positive values indicate that the value in a given year is above the multi-year mean, whereas negative values indicate that it is below the mean. In Shandong, radiation-, sunshine-, and photovoltaic-output-related indicators show relatively high standardized values in 2017 and 2019, while several indicators are relatively low in 2020 and 2021. Precipitation-, atmospheric-pressure-, wind-speed-, and wind-density-related variables alternate between positive and negative values across the study period. In Inner Mongolia, radiation- and sunshine-related indicators are relatively high in 2017 and 2020, whereas wind-speed-, wind-density-, and wind-power-output-related indicators show relatively high values in selected years. Overall, the annual meteorological variables in both regions exhibit interannual fluctuations without a consistent unidirectional trend.

Figure 12 presents the standardized annual trajectories of selected socioeconomic indicators in the two regions. In Shandong, gross regional product, per capita gross regional product, the added value of the secondary and tertiary industries, total retail sales of consumer goods, and household income generally show upward trajectories from 2015 to 2021. In Inner Mongolia, gross regional product, secondary-industry value added, industrial value added, and household income also exhibit broadly increasing trajectories. However, these nominal monetary indicators are affected by inflation, nonstationarity, and common temporal trends. The 2020–2021 period is treated as a distinct shock-and-recovery interval because the COVID-19 pandemic disrupted economic activity in 2020, followed by a broad recovery in 2021. Therefore, Figure 12 is used only to describe the socioeconomic background of annual load changes rather than to establish statistical dependence or observed association.

Overall, the annual results indicate that load growth in Shandong and Inner Mongolia occurred against different regional socioeconomic backgrounds. In Shandong, indicators related to economic scale, industrial activity, household income, and consumption generally increased during the study period. In Inner Mongolia, indicators related to economic scale, secondary industry, industrial activity, and household income also showed broadly upward trajectories. The annual-scale analysis therefore complements the hourly- and monthly-scale results by providing a descriptive background for the observed load changes.

3.4. Discussion

3.4.1. Multiscale Identification of Regional Load–Factor Associations

The results indicate that the associations between load and external variables are not uniformly distributed across temporal scales but are concentrated in specific load components. Direct correlation analysis based on the original load series may therefore obscure differences among low-frequency, periodic, and short-term fluctuation components. At the hourly scale, meteorological associations are mainly observed in the low-frequency and intraday periodic components. At the monthly scale, meteorological and socioeconomic variables exhibit different association patterns across seasonal, low-frequency, and intermonthly fluctuation components. These results indicate that multiscale decomposition separates the load series into components with distinct temporal characteristics and helps identify the components in which contemporaneous associations with external variables are concentrated. Accordingly, regional load analysis may consider not only the number of candidate variables but also whether their associations are concentrated in particular load components. Meteorological variables, socioeconomic indicators, and calendar information may exhibit different contemporaneous association patterns across load components. In subsequent forecasting or association analyses, the identified component-specific association patterns may be used as preliminary information for candidate-variable screening. The practical value of such a strategy would require independent forecasting validation.

3.4.2. Potential Implications for Load Forecasting and Demand-Side Analysis

For the Shandong hourly case, the 24 h load component shows clear contemporaneous associations with radiation- and sunshine-related variables, while the average intraday profiles also vary across calendar categories. These results suggest that solar-related meteorological variables and calendar information can be considered as candidate inputs in future short-term load forecasting studies. For Inner Mongolia, the monthly and annual results show that load variations occur alongside changes in seasonal meteorological conditions and industrial and socioeconomic indicators. These variables may therefore provide candidate information for medium- and long-term load analysis. However, the present study does not conduct forecasting experiments or demand-response simulations. The above implications should therefore be understood as guidance for subsequent variable selection and model design rather than as evidence that these variables necessarily improve forecasting accuracy or demand-side management performance.

3.4.3. Applicability Boundaries of Regional Difference Analysis

The regional comparison is subject to several methodological limitations. The hourly datasets for Shandong and Inner Mongolia cover different periods and record lengths, so the hourly results are interpreted as separate within-region cases rather than direct cross-regional evidence. The monthly analysis contains 84 observations and includes potentially collinear socioeconomic indicators. Serial dependence in the monthly Pearson coefficients was partially addressed using a circular moving-block bootstrap, whereas multiple-comparison correction and uncertainty assessment for MI were not conducted. The annual analysis contains only seven observations for each region and is therefore limited to descriptive comparison. Moreover, only the load series was decomposed, whereas the external variables were retained in their original forms; associations involving the 24 h or 12-month modes may partly reflect shared periodicity. Province-level averages may also obscure spatial and sectoral differences within each region. The Pearson and MI results should therefore be understood as contemporaneous statistical associations rather than independent or causal effects. Future studies could use longer and temporally aligned datasets, spatially or sectorally disaggregated load data, decomposed external variables, and lagged or causal analytical methods.

4. Conclusions

This study takes Shandong and Inner Mongolia as the study areas and constructs an hourly–monthly–annual multiscale analysis framework to compare the scale-dependent characteristics of load changes under different meteorological, industrial, and consumption backgrounds. The main conclusions are as follows.

(1)

The hourly loads of both regions exhibit evident multiscale structures. The load series contain low-frequency variations, a 24 h daily-cycle component, a 12 h semi-daily-cycle component, and shorter-period high-frequency fluctuations. Regional hourly load is jointly characterized by overall level changes, intraday rhythms, and local short-period fluctuations.

(2)

The associations between meteorological factors and hourly load modes show clear scale differences. In Shandong, meteorological associations are mainly concentrated in the 24 h daily-cycle mode, with radiation- and sunshine-related variables being more prominent. In Inner Mongolia, several low-frequency and daily-cycle components show general statistical dependence with meteorological variables, although these associations are not fully represented by linear correlation alone. Because the hourly datasets of the two regions cover different periods and have different record lengths, these results are interpreted separately within each region rather than used for direct cross-regional comparison. Calendar-related analysis further shows that the 24 h load profiles and daily peak–valley differences vary across weekdays, weekends, and holidays, reflecting calendar-related regularities in hourly load.

(3)

At the monthly scale, meteorological associations in both regions were mainly distributed in the annual and intra-annual components. The block-bootstrap results showed that the meteorological Pearson associations with confidence intervals excluding zero were concentrated primarily in IMF2 and IMF3. In contrast, fewer socioeconomic associations met this criterion, and these were mainly localized to individual price indices and load components.

(4)

At the annual scale, both regions showed overall growth in annual maximum load and electricity consumption from 2015 to 2021, with Shandong exhibiting a larger proportional increase in annual maximum load. Because only seven annual observations were available for each region, the annual results were interpreted descriptively. The meteorological and socioeconomic indicators were used only to characterize the interannual and regional contexts accompanying load changes.

Overall, these findings suggest that examining load–factor associations across decomposed temporal components can provide more specific descriptive information than analysis based only on the original load series. The multiscale framework helps identify the load components in which meteorological, industrial, and consumption-related associations are concentrated and provides a descriptive reference for regional load analysis and subsequent variable screening.