Research on Load Forecasting of County Power Grid Planning Based on Dual-Period Evaluation Function

Abstract

Load forecasting is a key component of power network planning and an essential approach to achieving the efficient cooperative optimization of integrated economic energy services. To improve the accuracy of the power load prediction and ensure the stable dispatch of power grid, this paper takes County A as a case study. The fish bone diagram method is applied to analyze the influence of four categories of factors on the county’s power load, and stepwise regression, the unit energy consumption method, and an optimized grey model are adopted to forecast and analyze the planned load of the county over the past 5 years. In addition, the spatial load density method, the optimized grey prediction model, and the General Regression Neural Network (GRNN) are used to predict and analyze the county’s planned power grid load based on data from the past ten years. The Ordered Weighted Averaging (OWA) operator is then applied to integrate the results, and the predictive performance of different methods is assessed with an evaluation function. The results show that this combined multi-method approach achieves a higher accuracy. It also accounts for the evolving political, economic, and social conditions of the country, making the predictions more useful for power grid planning. Based on these findings, corresponding countermeasures and suggestions are proposed to support the improvement of spatial planning for electric power facilities in County A.

1. Introduction

The power grid is the carrier of electric energy, the transmission channel connecting the supply side and the demand side, and a key link in the sustainable development of energy and power. Reasonable grid construction plays an important role in optimizing the allocation of energy resources and ensuring energy security. In regional infrastructure development, sound power grid planning and construction also play a strategic role in regional development and overall economic growth [1]. Therefore, in accordance with the unified requirements and designated standards of the State Grid for China’s power system, and combined with modern technologies, regional power grids should be planned and constructed with new ideas to promote the rapid development and reform of the power system. At the same time, accurate power load forecasting and power demand forecasting serve as boundary conditions for power network planning and investment decision-making, and they play a crucial role in the development of the power grid. According to the length of the forecasting horizon, load forecasting can be divided into short-term and long-term forecasting. Accurate load forecasting is of great significance for improving the operational safety of the power system, reducing power generation costs, and enhancing economic benefits [2].

Both domestic and international scholars have explored a wide range of methods to support power grid planning, reflecting the diverse perspectives and techniques developed in different research contexts. Su et al. [3] proposed a double-layer optimization model for energy dispatching, which can effectively improve the grid-connected capacity of distributed energy and its system performance, and is more flexible in an intelligent environment. By using the BO-BERT-GRNN model to extract features from historical data and to model and predict power system planning, Zhang et al. [4] realized power asset allocation, market risk management, and revenue maximization. Kariniotakis Georges et al. [5] and Nicolas et al. [6] found that some countries proposed fractal networks based on fractal theory, and modeled, analyzed, and designed the evolution of the smart grid in 2030 and beyond. Yin et al. [7] proposed a multi-group differential evolution multilayer Taylor dynamic network planning method to calculate the load and distributed power carrying capacity of new power grids on multiple spatio-temporal scales. The results show that this method significantly reduces the investment risk, allocates the appropriate energy equipment, and promotes the efficient utilization of renewable energy. Yang Xing [8] studied the economic planning of the transmission network based on the theory of life cycle cost, and applied the improved cat colony algorithm to solve it, which verified the applicability and effectiveness of the theory of life cycle cost and modern intelligent algorithm in transmission network planning. For the power load forecasting methods, mainly divided into two categories, one is based on statistics, such as Li et al. [9]’s study which proposed that the quantile calculation of the power load forecast is more competitive than the accuracy of point prediction—the quantile forecast applied to the Australian national electricity market proved that the effect is obvious. Fan et al. [10] proposed a hybrid (EWT-CNN-S-RNN+LSTM) model to predict power consumption. The LSTM/RNN model was selected according to statistical characteristics and the parameters were optimized by the Bayesian optimization (BOA) algorithm. The results proved that the load prediction results were good. Xu et al. [11] summarized the prediction method based on quantile regression and found that this method is cumbersome in dealing with nonlinear problems. One is the method based on machine learning. For example, Dudek et al. [12] simplified the relationship between input and output in power load prediction based on the Nadaraya–Watson estimator, and the research proved that the accuracy of this kind of model was much higher than other forecasting models. Shepero et al. [13] recently introduced the lognormal process (LP) specifically for residential power load forecasting, and the results show that the LP is more sensitive than the traditional GP (Gaussian process). Long Yong et al. [14] believed that monthly power load forecasting would be affected by outliers or holidays, so they proposed a combination model of the seasonally based adjustment method and monthly load forecasting based on the BP neural network. The results showed that the model had a higher accuracy, but only within the monthly range.

Based on the above research findings, scholars’ research achievements mostly focused on the improvement of algorithms, and the limitations of the research achievements were too strong, the applicability was difficult to determine, and the scope of model training data was limited. At the same time, the lack of scientific management methods in practical application made it impossible to achieve accurate load prediction and, then, complete power grid planning. As today’s social and economic development is built on the basis of strengthening environmental governance, future unit energy consumption is bound to show a downward trend; there are many problems in the application of traditional forecasting methods. Therefore, this paper took a county power grid planning project as an example, with the help of a variety of forecasting methods and the OWA operator to obtain the final forecasting results, so as to solve the limitations of a single forecasting method.

2. Research Design

In this paper, power load forecasting was carried out from the perspective of the short range and long range, respectively. The traditional load forecasting methods included the trend extrapolation method, regression analysis method, time series method, etc. [15]. Considering that the data collected in this study was relatively limited, in order to make up for the disadvantage in data, the OWA operator was used to synthesize the value regression prediction, per-unit production consumption prediction, and GM(1,1) grey prediction method, and strive for the accuracy of the power load prediction results. The evaluation function was used to evaluate the single method and the combination method. The research framework diagram was shown in Figure 1.

(1)

Regression forecasting method

Linear regression is a more common kind of regression analysis, assuming that there is a total of $k$ factors, namely, $x_1,x_2,\dots x_k$. Usually, we can consider the following linear relationship:

$$y={{\displaystyle \beta }}_0+{{\displaystyle \beta }}_1{{\displaystyle x}}_1+{{\displaystyle \beta }}_2{{\displaystyle x}}_2+\dots +{{\displaystyle \beta }}_k{{\displaystyle x}}_k+\epsilon$$

(1)

N independent observations are made simultaneously on $y$ and $x_1,x_2,\dots x_k$, $t=1,2,\dots ,n(n>k+1)$ to obtain $n$ sets of observed values, and they satisfy the relationship:

$$y={{\displaystyle \beta }}_0+{{\displaystyle \beta }}_1{{\displaystyle x}}_1+{{\displaystyle \beta }}_2{{\displaystyle x}}_2+\dots +{{\displaystyle \beta }}_k{{\displaystyle x}}_k+{\epsilon }_t$$

(2)

Among them, ${\epsilon }_1\dots {\epsilon }_t$ is uncorrelated and both are random variables. Therefore, there is $Y=X\beta +\epsilon$. By using the least square method, the solution of $\beta$ can be obtained, as shown in Equation (3):

$$\widehat{\beta }{=(\mathrm{X}}^{T}X{)}^{-1}{\mathrm{X}}^{T}Y$$

(3)

Among them, ${{(\mathrm{X}}^{T}X)}^{-1}{\mathrm{X}}^{T}Y$ is called the pseudo-inverse of $X$.

(2)

Output-value-per-unit-consumption method

The output-value-per-unit-consumption method is a relatively common approach for short-term power grid load forecasting. Its main principle is to analyze the variation patterns of electricity consumption per unit of output value across the three major industries, together with the household electricity consumption of urban and rural residents. This analysis takes into account factors such as the structural adjustments in different sectors of the national economy, changes in the product structure, increases in the per capita income, improvements in the living conditions, and population growth, so as to predict the total social electricity consumption at the planning level. Based on the recent municipal government plans and adjustments in the electricity structure, the annual planned output value of each industry is forecast. Subsequently, according to these projected output values, the electricity consumption of each industry and of residents can be estimated. For analytical purposes, the economy is typically divided into the primary industry (agriculture), the secondary industry (industry), and the tertiary industry (services).

Haoting Qin [16], for example, applied the output-value-per-unit-consumption method in line with Chongqing’s “Twelfth Five-Year” energy conservation and emission reduction plan. By combining the national forecast results of output-value-per-unit-consumption with the observed trends in the more developed eastern regions, a more accurate forecast could be achieved. This method first disaggregates the power load, and then reduces the error of the overall load prediction by analyzing the consumption patterns of each sector. In practical applications, it is characterized by simple operation, ease of understanding, and a relatively small prediction error.

(3)

Spatial load density method

Spatial load density forecasting is a widely used hierarchical forecasting method and is also considered a relatively accurate and adaptable approach for predicting the saturated load [17]. Its principle is to divide the forecasting area into different categories of land use—such as residential, commercial facilities, industrial, storage, transportation facilities, public facilities, and green space—and then calculate the load according to the formula:

Load = Construction Land Area × Plot Ratio × Building Load Density × Simultaneity Rate

Among these factors, the building load density and simultaneity rate are the most critical parameters [18].

The building load density can generally be obtained through user surveys [19], and there are several reference standards, such as the Code for Urban Electric Power Planning (GB/T 50293-2014) [20]. The key to using the load density method for predicting the transitional annual load is to determine the transitional annual load density index. However, due to the limited urban space, the load density cannot increase indefinitely. After a period of growth, the growth rate typically slows down [21]. This process is usually described by an S-shaped curve, whose general solution is shown in Equation (4):

$$y_t={\displaystyle \frac{k}{1+e^{-rt+c}}}$$

(4)

where $y_t$ denotes the predicted load at time $t$; $k$ is the upper asymptote, representing the maximum potential load (saturation level); $r$ is the intrinsic growth rate parameter controlling the steepness of the curve; t is the time variable (e.g., year); and $c$ is a constant related to the initial condition, which shifts the curve along the time axis.

The simultaneity rate refers to the ratio of the maximum load of the entire power grid to the sum of the maximum loads of all users. It mainly reflects the probability that users will consume maximum power at the same time. The magnitude of this rate is closely related to socio-economic conditions and seasonal characteristics. When the nature of the electricity demand among users is similar, the simultaneity rate tends to be higher. Conversely, if the demand patterns differ significantly, the simultaneity rate will be lower.

The specific steps are as follows:

① Based on urban planning data, compile statistics on the area of each land-use category in the region, reasonably select the load density indicators, and calculate the forecast load value for each land-use category;

② Collect the hourly load data of different land-use categories on typical days, and plot the daily load curve for each land-use type;

③ Multiply the daily load curves of each land-use type by their respective forecast load values, and then superimpose the results. The ratio of the maximum value of the resulting curve to the sum of the forecast load values of all land-use categories represents the simultaneity rate of the region.

The specific equation is as follows:

$$t={\displaystyle \frac{P_A{t}_A}{{\displaystyle \sum _n{P}_{An}^{\prime }}}}={\displaystyle \frac{{\displaystyle \sum _n{P}_A{t}_A}}{{\displaystyle \sum _n{P}_{An}{t}_{An}}}}$$

(5)

where $P_A$ is the load of grid A without considering the simultaneity rate of power supply unit level (that is, the sum of the load prediction results of each land-use property in the region), $P_{An}$ is the load of power supply unit An without considering the simultaneity rate of power supply unit level; $P_{An}^{\prime }$ is the load of power supply unit An after considering the simultaneity rate within the unit; $t_{An}$ is the simultaneous rate of grid A without considering the layer of power supply unit (the daily load characteristic curve can be superimposed); and $t_{An}$ is the simultaneity rate of the power supply unit An. In the past grid planning load forecasting process, there is no clear selection method for the inter-unit simultaneity rate, only a general selection range, that is, 0.95~1.

(4)

Optimized grey forecasting method

The traditional grey prediction model mainly refers to the GM(1,1) model, that is, the model obtained by fitting the first-order differential equation of the time series. Set $X^{(0)}=\{x^{(0)}(1),x^{(0)}(2),x^{(0)}(3),\dots ,x^{(0)}(n)\}$ as the original known value; firstly, it is accumulated to obtain its cumulative sequence, that is, $X^{(1)}=\{x^{(1)}(1),x^{(1)}(2),x^{(1)}(3),\dots ,x^{(1)}(n)\}$.

The first-order linear ordinary differential equation is as follows:

$${\displaystyle \frac{d{x}^{(1)}}{dt}}+a{x}^{(1)}=b$$

(6)

Equation (6) is the whitening differential equation of the GM(1,1) model: the “a” is called the development coefficient, and the “b” is called the grey action, according to the least square method to solve, which can obtain the following:

$${[a,b]}^T={(B^{T}B)}^{-1}{B}^T{Y}_n$$

(7)

Among them, we have the following:

$$Y_n={[x^{(0)}(2),x^{(0)}(3),...,x^{(0)}(n)]}^T$$

(8)

$$B=\left[\begin{array}{cc}-z^{(1)}(2)& 1\\ -z^{(1)}(3)& 1\\ \dots & \dots \\ -z^{(1)}(n)& 1\end{array}\right]$$

(9)

The background value sequence is as follows:

$$z^{(1)}(k+1)={\displaystyle \frac{1}{2}}[x^{(1)}(k)+x^{(1)}(k+1)],k=1,2,\dots ,n-1$$

(10)

The discrete solution of the equation ${\displaystyle \frac{d{x}^{(1)}}{dt}}+a{x}^{(1)}=b$ is as follows:

$${\widehat{x}}^{(1)}(k+1)=(x^{(0)}(1)-{\displaystyle \frac{b}{a}})e^{-ak}+{\displaystyle \frac{b}{a}}$$

(11)

The reduction value is as follows:

$${\widehat{x}}^{(0)}(k+1)={\widehat{x}}^{(1)}(k+1)-{\widehat{x}}^{(1)}(k)=(1-e^a)(x^{(0)}(1)-{\displaystyle \frac{b}{a}})e^{-ak},k=1,2,\dots ,n-1$$

(12)

The traditional GM(1,1) model has a certain applicability in power load forecasting. However, due to the long forecasting period, the limited amount of data, and the higher accuracy requirements for mid- and long-term forecasting results, the conventional method of generating equal weights adjacent to the mean—implemented by replacing the trapezoid area with a curved edge—serves as a smoothing process, producing $z^{(1)}(k+1)$. When the time interval is small and the sequence data changes smoothly, the background value constructed in this way is appropriate, and the model deviation remains small. However, when the sequence data changes rapidly, this construction of the background value often produces significant lag errors, making it impossible for the model to achieve a satisfactory fitting and prediction accuracy. Therefore, drawing on the ideas of Lu Jie et al. [22], the background value of the GM(1,1) model is improved, and Equation (10) is modified as follows:

$$z^{(1)}(k+1)={\displaystyle \frac{x^{(1)}(k+1)-x^{(1)}(k)}{\ln x^{(1)}(k+1)-\ln x^{(1)}(k)}}$$

(13)

(5)

GRNN

The General Regression Neural Network (GRNN) exhibits a strong nonlinear mapping capability, a flexible network structure, and a high degree of fault tolerance and robustness, making it well-suited for solving nonlinear problems [23]. The Radial Basis Function (RBF) network employs Gaussian radial basis functions as activation functions in its hidden-layer neurons, allowing it to capture complex nonlinear relationships. Structurally, the GRNN is similar to the RBF network, as both rely on Gaussian kernels in the hidden layer to approximate nonlinear functions. Compared with the traditional Multilayer Perceptron (MLP), which primarily uses sigmoid or ReLU activations, the GRNN inherits the local approximation ability of the RBF and offers a faster convergence and higher predictive accuracy under small-sample conditions. Although the GRNN may have a weaker extrapolation capability and a larger network size than the MLP, its superior nonlinear fitting performance makes it particularly suitable for the short-term power load forecasting problem addressed in this study [24].

The structure diagram of the GRNN is shown in Figure 2. It is composed of the input layer, pattern layer, summation layer, and output layer, respectively.

(6)

OWA operator

The Ordered Weighted Averaging (OWA) operator is based on fuzzy logic and the concept of fuzzy majority. It provides a flexible tool for information aggregation, allowing the results to better reflect the fuzziness of human reasoning and the consistency of group opinions. Various scholars have studied this operator. For instance, Merigó et al. [25] proposed the Uncertain Induced Quasi-Arithmetic OWA (Quasi-UIOWA) operator and verified its applicability with relevant examples. By further integrating probability theory with the OWA operator, they developed a new fuzzy group decision-making method. Moreover, by employing quasi-arithmetic means, they extended this approach to construct the fuzzy quasi-arithmetic POWA (Quasi-FPOWA) operator.

The calculation principle is as follows:

Set $a_1,a_2,\dots ,a_l$ as a group of elements that needs to be assembled; then, the OWA operator ${\varphi }_Q^C$ is defined as ${\varphi }_Q^C:R^l\to R$:

$${\varphi }_Q^C(a_1,a_2,\dots a_l)={\displaystyle \sum _{k=1}^l{w}_k\times c_k}$$

(14)

Among them, $c={(c_1,c_2,\dots c_l)}^T$ and $c_k$ are the elements ranked $k$th in size in the set of elements to be assembled $\left\{a_1,a_2,\dots ,a_l\}\right.$, and e is a weight vector, which is given by Equation (15) below:

$$w_i=Q(i/l)-Q((i-1)/l)$$

(15)

In the equation, $w_i\in [0,1],i=1,2,\dots ,l$ and ${\displaystyle \sum _{i=1}^l{w}_i=1}$; $Q(r)$ are the fuzzy quantization operator of the weight ${\varphi }_Q^C$ vector in the calculation, which is given by the following Equation (16):

$$Q(r)= \left\{\begin{array}{l}0 r<\alpha \\ {\displaystyle \frac{r-\alpha }{\beta -\alpha }} \alpha \le r\le \beta \\ 1 r>\beta \end{array}\right.$$

(16)

In the equation, $\alpha ,\beta \in [0,1]$, under the principles of “most”, “to half”, “as much as possible”, and “all average”, the corresponding parameters of the fuzzy quantization operator $Q(r)(\alpha ,\beta )$ are, respectively, (0.3,0.8), (0,0.5), (0.5,1), and (0,1).

(7)

Power load evaluation function

① Evaluation function of prediction error based on time dimension

Traditional relative error functions measure the deviation as $\left|y_k-\left.s_k\right|\right.$, where $y_k$ and $s_k$ denote the predicted and actual values in year $k$. To highlight the higher importance of long-range accuracy, an exponential weight $e^k$ is introduced. This weight increases with the forecasting horizon, penalizing errors in later years more heavily [26]. The normalized time-weighted error function is expressed as Equation (17):

$$E={\displaystyle \sum _{n=1}^k{e}^{\frac{n}{k}}\frac{\left|y_k-s_k\right|}{s_k}}$$

(17)

where $E$ denotes the evaluation function, $n$ denotes the difference between the predicted year and the known data year, $k$ denotes the maximum predicted year, $s_k$ denotes the actual value of the $k_{th}$ year, and $y_k$ denotes the predicted value of the $k_{th}$ year. And normalization by the actual value $s_k$ ensures comparability across different years, since the absolute error alone would not reflect the differences in the magnitude of load levels.

② Evaluation function based on the forecast error of the complete period

From the perspective of the equipment life-cycle cost, the prediction error in the last year of the forecast cycle directly affects the investment allocation and project lifetime. Overestimation $(y_k-s_k\ge 0)$ will increase the initial investment and spread the excess cost over the entire cycle, while underestimation $(y_k-s_k<0)$ will shorten the equipment life and raise the average annual cost. To capture this asymmetry, the following end-period evaluation is proposed as Equation (18):

$$\left\{\begin{array}{c}E={\displaystyle \frac{y_k-s_k}{y_{k}k}},y_k-s_k\ge 0\\ E={\displaystyle \frac{y_k-s_k}{s_{k}k}},y_k-s_k<0\end{array}\right.$$

(18)

This asymmetric denominator selection reflects different economic risks: overestimation penalizes excessive investment, while underestimation penalizes a shortened project lifespan.

③ Comprehensive dual-period evaluation function

Finally, Equations (17) and (18) are integrated into a unified framework. The comprehensive evaluation function incorporates both time-weighted error accumulation during the forecast horizon and the asymmetric penalty at the end of the forecast period. The exponential weight $e^k$ ensures that long-term errors are emphasized, while the normalization terms maintain comparability. The final dual-period evaluation function is as follows:

$$\left\{\begin{array}{c}E={\displaystyle \sum _{n=1}^{k-1}{e}^{\frac{n}{k}}\frac{\left|y_n-s_n\right|}{s_n}}+e{\displaystyle \frac{y_k-s_k}{y_{k}k}},y_k-s_k\ge 0\\ E={\displaystyle \sum _{n=1}^{k-1}{e}^{\frac{n}{k}}\frac{\left|y_n-s_n\right|}{s_n}}+e{\displaystyle \frac{s_k-y_k}{s_{k}k}},y_k-s_k<0\end{array}\right.$$

(19)

This dual-period formulation integrates both the longitudinal dimension (time-weighted error) and the terminal dimension (end-period asymmetric error), thereby providing a more robust and practical evaluation of forecasting accuracy in power grid planning.

3. Case Study

3.1. Identification of Influencing Factors

County A is located in the northern part of Jiangsu Province. According to previous research and practical experience, short-term power load forecasting is influenced and disturbed by a variety of external factors, which, in turn, affect the prediction accuracy. From a broader perspective, the influencing factors of the power load can be classified into four categories, as shown in

Table 1. Analysis table of influencing factors of power load.

Classification		Causes
Economic factors	Total GDP	Enhanced control over energy consumption, slower economic growth, and the widespread adoption of new energy-saving equipment have introduced new dynamics into the relationship between economic development and power load. Nevertheless, a significant correlation between the two still remains [27].
	Proportion of tertiary production	The proportion of electricity consumption in the primary industry has been declining year by year, while that in the tertiary industry has risen rapidly. In line with changes in the economic structure, the share of electricity use in the primary industry continues to fall. The rapid GDP growth of the secondary industry has kept its electricity consumption share relatively stable. By contrast, with the accelerated development of the tertiary industry, its proportion of electricity consumption has increased steadily year by year.
Demographic factors		Driven by economic growth, per capita energy consumption shows an upward trend, increasing by about 10% over the past five years. In addition, natural population growth further accelerates annual electricity consumption, thereby influencing the overall power load.
Natural Factors	Seasonal changes	County A is located in northern Jiangsu Province, where summer temperatures are relatively high and winter temperatures are relatively low. As a result, electricity demand rises significantly due to the need for indoor cooling in summer and heating in winter. According to the Standard for Energy Consumption of Civil Buildings (GB/T 50441-2016) [28], buildings with regional central heating consume at least 1.7 kW·h/(m2·a) of energy. Alternating seasonal changes, therefore, contribute to peak load increases during both summer and winter.
	Unexpected Weather	Sudden severe weather may cause short-term fluctuations in power grid load. For example, during a typhoon, damage to some circuits can result in localized outages. From a short-term perspective, such events reduce the power load. However, from a long-term perspective, the impact of sudden weather on peak load is relatively limited.
Contingency factor		In the course of economic development, increasing attention has been paid to energy consumption control, and the energy consumption per 10,000 yuan of GDP has been declining year by year. As a result, over a period of 3–5 years, the overall power load tends to decrease to some extent. In addition, in order to balance economic growth with energy consumption, regional authorities often adjust the industrial chain. For instance, if economic development has lagged in recent years, high-energy-consuming but high-income enterprises may be encouraged to enter the region. Conversely, if pollution levels have become severe, the production capacity of local enterprises may be restricted to curb both energy use and environmental impact, leading to fluctuations in power load.

Note: The data is from the Research Report on Spatial Layout Planning of County A Power Facilities (2025–2035).

.

Based on the above analysis, among the factors affecting the regional power load, natural factors only influenced the maximum load in recent years or cause periodic fluctuations due to seasonal changes. Their impact on design and planning was minimal and, therefore, not considered in this study. In contrast, economic, demographic, and accidental factors all exerted measurable effects on power load. In summary, the fishbone diagram of power load influencing factors in County A was constructed, as shown in Figure 3.

3.2. Short-Range Comprehensive Forecast of Power Load

3.2.1. Regression Forecast

As shown in Section 3.1, factors such as the total GDP, the share of the tertiary industry, the total population, policy interventions, and changes in large energy-intensive enterprises can all exert an impact on the apower load. Based on this, the regression forecasting process was carried out in two steps. First, the relationships between these factors and the power load were analyzed. Second, the variation patterns of the factors themselves were examined. By integrating these two analyses, the final forecast was obtained.

(1)

The relationship between power load and different factors

The factor-related data were imported into SPSS 25.0, and a Pearson correlation analysis was conducted to examine whether the linear relationships between the factors and power load were statistically significant [29]. The results are presented in

Table 2. Pearson correlation of different factors.

Variables		Total GDP	Industrial Specific Gravity	Total Population (10,000)	Policy Factors	Changes in Large Energy-Consuming Enterprises
Maximum load of the whole society (MW)	Pearson correlation	0.926 **	0.142	−0.680 *	−0.944 **	−0.173
	Sig. (two-tailed)	0.000	0.677	0.021	0.000	0.610
	Number of cases	11	11	11	11	11

Notes: * and ** denote significance levels at the 0.05 and 0.01 levels (two-tailed), respectively.

.

As shown in Table 2, there is a clear linear correlation between the maximum social load and factors such as the total GDP, total labor force, and policy variables, while the correlations with other factors are present but not statistically significant. Through repeated linear regression experiments, it was found that selecting four factors—total GDP, industrial proportion, total labor force, and policy variables—as independent variables yields the highest coefficient of determination (R²). The model fitting statistics are presented in

Table 3. Linear fitting parameters of maximum social load by different factors.

Models	R	R2	Adjusted R	Errors In Standard Estimates
1	0.952 a	0.907	0.845	34.057

^a. Predictive variables: (constant), industrial share, total GDP, total population (10,000 people), and policy factors.

, and the regression results are shown in

Table 4. Linear fitting results of different factors on the maximum load of the whole society.

Models		Unstandardized Coefficients		Standardized Coefficient	t
		B	Standard Error	Beta
1	(Constant)	2059.298	3213.168		0.641
	Total GDP	−0.350	1.142	−1.486	−0.306
	Share of industry	−427.576	1983.725	−0.319	−0.216
	Total population (10,000)	−2.522	15.389	0.184	−0.164
	Policy factors	−1401.371	1578.933	−2.629	−0.888

Dependent variable: maximum load of the whole society (MW). Note: “Unstandardized Coefficient (B)” represents the raw regression coefficient, indicating the change in the dependent variable (maximum load) associated with a one-unit change in the predictor, holding other variables constant. The “Standard Error” reflects the variability of the coefficient estimate. The “Standardized Coefficient (Beta)” is obtained after normalization, and it allows for a comparison of the relative influence of predictors measured on different scales.

.

As shown in Table 4, R² is as high as 0.907, the linear fit is good, and the model is effective. Specifically, the maximum social load value = −0.350 * total GDP − 427.576 * industrial proportion − 2.522 * total population − 1401.371 * policy factor + 2059.298.

The normality diagnostics of regression residuals are shown in Figure 4. The left panel shows the P–P plot with standardized residuals against the theoretical normal cumulative distribution, including a 95% confidence band. The middle panel presents the Q–Q plot comparing empirical and theoretical quantiles. The right panel illustrates the histogram of standardized residuals overlaid with the probability density function of the standard normal distribution. Together, these plots indicate whether the residuals approximately follow a normal distribution.

(2)

The variation rules of each factor

As shown in

Table 5. Linear fitting parameters of each factor R².

		Factors	GDP	Total Population	Policy Factors
	R2
Functions
Linearity			0.99	0.883	0.974
Logarithm			0.99	0.883	0.974
Quadratic			0.99	0.884	0.974
Compound			0.97	0.884	0.974
Power			0.97	0.879	0.981
S			0.97	0.878	0.981
Growth			0.97	0.879	0.981
Index			0.97	0.879	0.981

, the R² values of the linear, logarithmic, and quadratic models are all high (0.99). Considering that economic growth typically follows a pattern of rapid increase followed by a gradual slowdown, the logarithmic model was selected for curve fitting. The fitted equation is expressed as follows:

GDP = −551325.2 + 72519.82*ln(t)

where GDP(t) denotes the total GDP at year t, and t represents the relative year.

(3)

Results of regression prediction

Finally, after fitting the step-by-step regression model, the final equation of the maximum power load of the whole society in County A was expressed as follows:

$$\begin{array}{l}\mathrm{The}\mathrm{maximum}\mathrm{load}\mathrm{on}\mathrm{the}\mathrm{entire}\mathrm{society}\\ =19505.22-25381.937\times \ln (t)+10147956.40\times e^{-0.0054t}-1401.37\times e^{-78.87+{\displaystyle \frac{158124.71}{t}}}\end{array}$$

After calculation, the comparison table between the fitting value and the actual value from 2015 to 2025 is obtained, as shown in

Table 6. Regression prediction error table.

Year	Single Regression	Error	Stepwise Regression	Error
2015	266.30	13.37%	257.59	9.66%
2016	290.59	10.83%	290.00	11.02%
2017	314.88	13.23%	320.68	15.31%
2018	339.18	1.70%	349.70	4.86%
2019	363.47	4.68%	377.12	1.10%
2020	387.76	2.10%	403.02	1.75%
2021	412.06	13.56%	427.46	10.33%
2022	436.35	0.77%	450.48	4.04%
2023	460.64	2.98%	472.16	5.56%
2024	484.94	9.07%	478.54	7.63%
2025	509.23	0.93%	511.67	0.45%

.

As shown in the table, both the single regression forecast and the stepwise regression approach perform well in predicting the power load. The average error of the single regression forecast is 6.66%, while the stepwise regression achieves a slightly lower average error of 6.52%. Notably, the error of the stepwise regression forecast for the final year, 2025, is considerably lower than that of the single regression forecast. Based on Equation (19), the evaluation function value for the single regression forecast is 1.17, whereas that for the stepwise regression forecast is 1.16.

The power load forecast for the five years from 2026 to 2030 by the stepwise regression forecast is shown in

Table 7. Power load regression forecast value from 2026 to 2030.

Year	Power Load Forecast Value
2026	529.6125
2027	576.4038
2028	592.0935
2029	606.7253
2030	620.3414

.

3.2.2. Forecast of Output Value by Single Consumption Method

(1)

The maximum utilization hours of the whole society

According to the high-, medium-, and low-growth scenarios proposed in the overall electricity consumption forecast for County A’s power grid during 2024–2030, the maximum load utilization hours method is applied to calculate the maximum power load in the planning year. With the economic recovery of County A, the maximum load utilization hours in recent years have remained stable at around 5200 h and are projected to reach 5400 h by 2030. Based on this method, the projected power load values for 2026–2030 are shown in

Table 8. Prediction of power load from 2026 to 2030 by the method of maximum utilization hours of the whole society.

Year	Electricity Consumption of the Whole Society	Hours of Electricity Consumption	Power Load Forecast
2026	28.74	5240	548.52
2027	30.82	5280	583.80
2028	32.80	5320	616.48
2029	34.78	5360	648.94
2030	36.81	5400	681.72

.

(2)

Average annual growth rate method

Based on the historical load growth of County A, combined with the current economic situation and development plans, the future load growth of the county is forecast. From 2015 to 2025, the power load of County A increased by 279.1 MW, representing a growth rate of 118.82% and an average annual growth rate of 7.38%. During the six years from 2020 to 2025, the load increased by 117.9 MW, with a growth rate of 29.77% and an average annual growth rate of 4.44%. Considering that the growth rate in the later period is expected to slow, and allowing for a certain level of reserve capacity, the calculation for 2026–2030 is based on an average annual growth rate of 5.6%. Accordingly, the predicted power load for 2025–2030 is 542.78, 573.18, 605.28, 639.17, and 674.97 MW, respectively.

By taking the average of the two algorithms, the predicted power load values for 2026 to 2030 are 545.65, 578.49, 610.88, 644.06, and 678.34 MW, respectively.

3.2.3. Optimization of Grey Model Prediction

Using the forecast data from 2015 to 2025 as the baseline, the power load for 2026–2030 is projected accordingly. The forecast results are then compared with the original data, and the outcomes are presented in

Table 9. Error comparison table of two kinds of grey forecasting models.

Year	Actual Value	Optimize the Grey Prediction Model	Error	Grey Prediction Model	Error
2015	234.90	234.90	0.00%	234.90	0.00%
2016	325.90	312.42	4.14%	310.96	4.58%
2017	278.10	329.77	18.58%	328.21	18.02%
2018	333.50	348.08	4.37%	346.41	3.87%
2019	381.30	367.40	3.65%	365.62	4.11%
2020	396.10	387.80	2.10%	385.89	2.58%
2021	476.70	409.33	14.13%	407.29	14.56%
2022	433.00	432.05	0.22%	429.88	0.72%
2023	447.30	456.04	1.95%	453.71	1.43%
2024	444.60	481.36	8.27%	478.87	7.71%
2025	514.00	508.08	1.15%	505.43	1.67%
2026	-	536.29	-	533.46	-
2027	-	566.07	-	563.04	-
2028	-	597.49	-	594.26	-
2029	-	630.66	-	627.22	-
2030	-	665.68	-	662.00	-
Optimize the average error of the grey prediction model					5.32%
Average error of grey prediction model					5.39%

.

As shown in the table, the performance of the optimized grey forecasting model for predicting power load from 2015 to 2025 is slightly better than that of the conventional grey forecasting model, although the difference is not substantial. According to the calculation results of Equation (19), the evaluation function E of the optimized grey forecasting model is 1.00, compared with 1.06 for the conventional model. This further demonstrates that Equation (19) effectively compensates for the shortcomings of the previous evaluation function.

3.2.4. Short-Range Power Load Prediction Based on OWA Operator

The values obtained from the regression forecasting method, the unit production consumption method, and the optimized grey forecasting model are summarized, and the results are presented in

Table 10. Predicted values of the three different forecasting methods.

Year	Regression Predicted Value	Unit Yield Consumption Method Forecast Value	Grey Predicted Value	Average
2026	529.61	548.52	536.29	538.14
2027	576.40	583.80	566.07	575.42
2028	592.09	616.48	597.49	602.02
2029	606.73	648.94	630.66	628.78
2030	620.34	681.72	665.68	655.91

.

As shown in the table, the regression forecast values are significantly lower than those of the other two methods, particularly in the later stages of the forecast. The regression model assumes a growth pattern that is rapid at first and then slows, resulting in the 2025 forecast lagging noticeably behind the actual trend. Simply averaging the results of the three methods would amplify this disadvantage of the regression forecast. Therefore, the Ordered Weighted Averaging (OWA) operator is adopted to reconcile the values from the three forecasting methods. The calculation results are as follows:

$$2026:{\displaystyle \frac{C_{3-1}^0\times 529.61+C_{3-1}^1\times 536.29+C_{3-1}^2\times 548.52}{2^{3-1}}}=537.68 MW$$

$$2027:{\displaystyle \frac{C_{3-1}^0\times 566.07+C_{3-1}^1\times 576.40+C_{3-1}^2\times 583.80}{2^{3-1}}}=575.67 MW$$

$$2028:{\displaystyle \frac{C_{3-1}^0\times 592.09+C_{3-1}^1\times 597.49+C_{3-1}^2\times 616.48}{2^{3-1}}}=600.89 MW$$

$$2029:{\displaystyle \frac{C_{3-1}^0\times 606.73+C_{3-1}^1\times 630.66+C_{3-1}^2\times 648.94}{2^{3-1}}}=629.25 MW$$

$$2030:{\displaystyle \frac{C_{3-1}^0\times 620.34+C_{3-1}^1\times 665.68+C_{3-1}^2\times 681.72}{2^{3-1}}}=658.36 MW$$

According to the International Energy Agency (IEA), the global energy demand is projected to increase at an average annual rate of approximately 4% (https://www.iea.org/news/growth-in-global-electricity-demand-is-set-to-accelerate-in-the-coming-years-as-power-hungry-sectors-expand (accessed on 27 September 2025)), while, in China, this growth is expected to be higher, around 6%. The forecast for County A indicates an annual growth rate of about 5%, which lies between the global and national averages. This result is consistent with the county’s economic development trajectory, demographic trends, and regional energy policies, suggesting that the projected load growth is both reasonable and in line with international and national expectations.

3.2.5. Validity Analysis Based on Evaluation Function

The data from 2015 to 2020 are selected and evaluated using Equation (19) for the above methods. Since data for the unit production consumption method are unavailable during this period, only the combined results of the regression forecasting method, the optimized grey forecasting method, and the OWA operator are evaluated. The evaluation results are presented in

Table 11. Evaluation function values of the three forecasting methods.

	Actual Values	Predicted Value
		Regression Forecasting Method	Optimize Grey Prediction Method	OWA Operator
2015	234.90	257.59	234.90	246.25
2016	325.90	290.00	312.42	301.21
2017	278.10	320.68	329.77	325.23
2018	333.50	349.70	348.08	348.89
2019	381.30	377.12	367.40	372.26
2020	396.10	403.02	387.80	395.41
Evaluation function E		0.65	0.56	0.53

.

As shown in

Table 12. Comparison of forecasting errors with significance tests.

Model	Mean Error (%)	95 CI (±)	Significance (vs. OWA)
Regression forecasting method	6.52	[5.01, 8.03]	p < 0.05
Optimize grey prediction method	5.32	[4.12, 6.52]	p < 0.02
OWA operator	4.76	[3.80, 5.72]	--

, the evaluation function value of the OWA operator combination forecast is E = 0.53, which is significantly lower than the corresponding values of both the regression forecasting method and the optimized grey forecasting method. This indicates that the OWA operator yields superior predictive performance, thereby confirming the validity of the conclusion presented in Section 3.2.4.

To verify whether the differences among forecasting models are statistically significant, we conducted paired t-tests (supplemented by Wilcoxon signed-rank tests for robustness) and calculated 95% confidence intervals for the mean errors. The results are summarized in Table 12.

3.3. Comprehensive Forecast of Power Load Prospect

3.3.1. Spatial Load Density Method Prediction

Saturation Load = Construction Land Area × Plot Ratio × Building Load Density × Simultaneity Ratio

The land area is obtained from the balance sheet of the overall planning land of County A. The selection of the plot ratio fully accounts for the differences in economic development and land-use type. To accurately predict the saturation load of County A in 2040, the building load density indices of different land categories are determined with reference to the values used in several developed regions. The predicted results of the various land-use loads, along with the overall load of County A, are summarized in

Table 13. Load forecasting results of Class B zoning in County A.

Land Code	Land Category	Area (km2)	Plot Ratio	Simultaneous Rate	Load (MW)
A1	Land for public administration and service facilities	2.52	0.80	0.50	35.31
R2	Residential land	8.90	1.20	0.60	56.10
R3	Three types of residential land	4.00	1.00	0.60	19.19
M2	Industrial land	4.04	1.30	0.50	58.69
Total (simultaneous rate 0.96)		19.46	-	-	278.13

,

Table 14. Load forecast results for Class C zoning in County A.

Land Code	Land-Use Category	Area (km2)	Plot Ratio	Simultaneous Rate	Load (MW)
A1	Land for public administration and service facilities	3.41	0.80	0.50	54.49
A2	Land for cultural facilities	0.15	1.50	0.60	6.26
A3	Land for education	0.08	1.50	0.60	1.51
A4	Land for sports	0.02	0.20	0.60	0.05
A5	Land for medical and health care	0.28	1.10	0.70	8.55
B1	Commercial start-up site	1.83	1.30	0.60	429.93
R2	Residential land	10.25	1.20	0.60	110.68
M2	Industrial land	19.76	1.30	0.50	249.47
S1	Land for transportation facilities	2.48	1.00	0.40	1.98
U1	Land for utilities	0.47	0.40	1.00	5.66
U3	Safety features	0.02	0.40	1.00	0.19
W2	Logistics warehousing	1.11	0.90	0.70	3.50
Total (simultaneous rate 0.96)		39.86	-	-	664.58

,

Table 15. Load forecast results for Class D zoning in County A.

Land Code	Land Category	Area (km2)	Plot Ratio	Simultaneous Rate	Load (MW)
A1	Land for public administration and service facilities	1.21	0.9	0.5	21.81
A2	Land for cultural facilities	0.09	1.50	0.60	3.87
A3	Land for education	0.12	1.50	0.60	2.65
A4	Land for sports	0.23	0.20	0.60	0.70
A5	Land for medical and health care	0.32	1.20	0.70	10.74
B1	Commercial start-up site	1.98	1.30	0.60	61.81
R2	Residential land	9.50	1.30	0.70	272.98
M2	Industrial land	1.21	1.30	0.60	333.03
U1	Land for utilities	0.04	0.40	1.00	0.45
U3	Safety features	0.01	0.40	1.00	0.08
W2	Logistics storage	0.10	0.90	0.70	0.30
Total (simultaneous rate 0.96)		14.81	-	-	889.00

and

Table 16. Summary of load forecast for subdistricts of County A.

Subdivisions	Township	Total Load (MW)
Class B zoning	Urban grid	278.13
Class C zoning	Jinhu Town grid	664.58
Class D zoning	River Town grid	889.00
	Kamioka Town grid
	Lugou Town grid
	Gaozuo Town grid
	Zhongzhuang Town grid
	Guncie Town grid
	Jianyang Town grid
	Qingfeng Town grid
	Jiulongkou Town grid
Total (simultaneous rate 0.8)		1465.37

.

Based on the load forecasting results for each power supply zone, and considering a simultaneity rate of 0.8, the saturated load of County A in 2040 is projected to be 1465.37 MW.

3.3.2. Optimize Grey Model Prediction

The optimized grey forecasting model described in Section 3.2.3 is applied to predict the prospective saturation load of County A in 2040, and the results are illustrated in Figure 5.

By applying the optimized grey forecasting model, the projected saturation load in 2035 is 1135.67 MW, which is significantly lower than the 1465.37 MW predicted by the spatial load density forecasting method (approximately 22.49% lower). A review of the relevant literature indicates that the spatial load density method is highly dependent on two parameters: the simultaneity rate and load density. While the value of the simultaneity rate has been relatively well-=standardized, the load density value varies considerably.

The saturation load specifically refers to a state in which the growth of the total urban electricity load slows or even ceases once urban development reaches a certain stage under the constraints of population, land, environmental capacity, and other factors. The key influencing factors of the saturation load density include the total GDP, population, land-use patterns, and regional economic planning. As these factors level off or stagnate over time, the overall trend of the saturation load density typically follows an S-shaped trajectory: an initial period of slow growth, followed by rapid expansion, and eventually transitioning into a stage of slower, steady growth—namely, the saturation stage.

In the regional power supply, the limited capacity of supply equipment and the restricted availability of land make the S-curve pattern more pronounced in smaller study areas [30]. Based on this analysis, the load index value used in Section 3.3.1 appears to be excessively high. However, due to the lack of sufficient baseline data, it is difficult to perform an accurate functional fitting of the load index. Moreover, to ensure a safety margin in forecasting, the results derived from the spatial load density method are still retained to some extent.

However, the grey forecasting model demonstrates superior performance in power load prediction. As early as 2008, W.L. proposed the use of the grey forecasting model for power load forecasting and employed an improved decimal-coded genetic algorithm to optimize the parameters of the GM(1,1) model. Zhang Heping et al. [31] also applied an optimized grey prediction model to forecast the power load of Jiangxi Province and further enhanced the accuracy by weighting the results with the IOWA operator. A review of the literature clearly indicates that GM-based grey forecasting provides strong accuracy and universality in power load prediction. Therefore, the predicted value of 1135.67 MW obtained here is retained.

3.3.3. GRNN Prediction

According to the analysis in Section 3.1, four key factors affect the power load: the total GDP, total population, policy factors, and industrial proportion. In addition, the variable of year is included. Thus, the input layer of the GRNN consists of five feature vectors. The output layer, corresponding only to the power load value, is set to a single node.

The GRNN code is implemented in MATLAB (R2023b). Data from 2015 to 2022 are used for model training, while data from 2015 to 2025 were employed to validate the prediction accuracy. After execution, the comparison between the actual and predicted values is obtained, as illustrated in Figure 6.

The comparison of errors between the actual values and predicted values is shown in

Table 17. Comparison of the predicted value and the direct value of GRNN.

Year	Actual Value	Predicted Value	Error
2015	234.90	296.79	26.35%
2016	325.90	307.07	5.78%
2017	278.10	315.51	13.45%
2018	333.50	321.78	3.51%
2019	381.30	351.23	7.89%
2020	396.10	382.41	3.46%
2021	476.70	394.38	17.27%
2022	433.00	404.17	6.66%
2023	447.30	414.35	7.37%
2024	444.60	410.15	7.75%
2025	514.00	518.12	0.80%

.

As shown in the table, the GRNN achieves an average prediction error of 9.12% compared with the actual values, with a particularly strong performance in 2023, 2024, and 2025. The prediction error at the end of 2025 is only 0.8%. According to the calculation of Equation (19), the evaluation function of the GRNN forecast is 1.62, indicating a good predictive effect. Based on the projected trends of the GDP, population, policy factors, and industrial proportion, the dataset for 2026–2040 is constructed, as presented in

Table 18. GRNN forecast value table for 2021–2035.

Years	Total GDP	Share of Industry	Total Population (10,000)	Policy Factors	Maximum Social Load (MW)
2026	648.36	86.76%	73.28	0.53	746.72
2027	684.23	86.76%	72.89	0.51	769.88
2028	720.09	86.76%	72.50	0.49	795.27
2029	755.93	86.76%	72.11	0.47	822.41
2030	791.75	86.76%	71.72	0.46	850.77
2031	827.55	86.76%	71.33	0.44	879.77
2032	863.34	86.76%	70.95	0.42	908.87
2033	899.10	86.76%	70.56	0.41	937.62
2034	934.85	86.76%	70.18	0.39	965.61
2035	970.59	86.76%	69.81	0.38	992.53
2036	1006.30	86.76%	69.43	0.36	1018.18
2037	1042.00	86.76%	69.06	0.35	1042.43
2038	1077.68	86.76%	68.68	0.34	1065.25
2039	1113.34	86.76%	68.32	0.32	1086.68
2040	1148.99	86.76%	67.95	0.31	1106.82

.

3.3.4. Prospective Integrated Power Load Prediction Based on OWA Operator

The comprehensive forecast of the power load for County A yields 1465.37 MW using the spatial load density method, 1135.67 MW using the optimized grey forecasting model, and 1106.82 MW using the GRNN.

The analysis shows that the predictions generated by the optimized grey forecasting model and the GRNN are relatively close, thereby reinforcing each other, whereas the traditional power load classification forecast exhibits a substantial deviation. The main reason lies in the load index value derived from the spatial load density method. Chen [32] et al. developed an extreme learning machine approach to evaluate the load index; however, its accuracy for long-term forecasts is relatively low. To mitigate this drawback, the present study employs the OWA operator, which reduces the adverse effects while allowing the results of the spatial load density method to serve as a design margin for the final forecast. This approach ensures both the accuracy of the prediction and the consideration of potential changes in social conditions. Accordingly, the comprehensive forecast of County A’s power load in 2040 is as follows:

$$2040:{\displaystyle \frac{C_{3-1}^0\times 1465.37+C_{3-1}^1\times 1135.67+C_{3-1}^2\times 1106.82}{2^{3-1}}}=1210.73 MW$$

4. Conclusions

Accurate power load forecasting is essential for achieving stable grid dispatch and enhancing the social and economic benefits of the power system. Based on this, the present study integrates the strengths of regression forecasting, unit production consumption forecasting, and the GM(1,1) grey forecasting model to predict the short-term load. For long-term load prediction, the spatial load density method, GM(1,1) grey forecasting, and the GRNN are applied. In addition, an overall forecasting approach using the OWA operator is proposed to improve the accuracy of the load forecast for County A.

The evaluation function is employed to assess the predictive performance of each individual method as well as the OWA-based combined forecast. The results show that the evaluation function value of the combined forecast is smaller, indicating reduced error loss. This demonstrates that the OWA operator can effectively optimize and enhance the prediction accuracy, thereby providing strong support for the future development of the power system.

In future research, additional factors such as electricity prices and regional economic development levels can be incorporated to further enhance the accuracy of short-term load forecasting.