Assessment Method for Dynamic Adjustable Capacity of Distribution Network Feeder Load Based on CNN-LSTM Source–Load Forecasting

Abstract

In response to the demand for flexible regulation resources in distribution networks with high proportion of new energy integration, this study explores the regulation potential of feeder loads. It controls the power of feeder loads through various types of voltage regulation equipment, treating these loads as a key component of virtual power plants (VPPs) to participate in grid security and stability control, demand response, and other fields, thereby enhancing the operational flexibility of the system. This paper focuses on the research of dynamic adjustable capacity evaluation for feeder loads, aiming to provide capacity constraints for their participation in grid interaction. Firstly, a CVR coefficient model is established based on the voltage–power coupling characteristics of feeder loads to characterize their regulation properties. Secondly, an analytical expression for voltage sensitivity is derived using an improved Zbus linearized power flow model, and a system-wide node voltage prediction model is constructed by combining the source–load prediction results from the CNN-LSTM model. On this basis, the dynamic regulation boundaries of each node’s voltage are solved with the constraint of system-wide voltage security. The adjustable capacity for the next 3 h is calculated iteratively by integrating the CVR coefficients of each feeder load, realizing the dynamic evaluation of the regulation capability of feeder loads.

1. Introduction

Faced with the situation of insufficient supply of traditional energy and the challenge of rising global carbon emissions, countries around the world are actively promoting energy revolution and accelerating the construction of a new power system centered on source–grid–load–storage interaction [1,2]. Against this background, new energy industries such as photovoltaic and wind power are developing rapidly and have become the main representatives of energy transformation. China’s new energy installed capacity has grown sharply, and it is expected that the total installed capacity of distributed energy in China will exceed 505GW by 2030 [3]. However, the power generation of new energy shows significant randomness and volatility, leading to a significant increase in the uncertainty on both the source and load sides of the power system [4]. On the short-time scale, the rapid fluctuation of new energy output will cause grid frequency deviation, and in severe cases, it may induce system frequency instability [5]; on the long-term scale, the imbalance between the time-series characteristics of new energy generation and load demand significantly increases the difficulty of system peak regulation [6,7]. The traditional “source following load” dispatching mode can no longer meet the power balance needs under high proportion of new energy access, so it is urgent to build a new regulation system to ensure the safe and stable operation as well as economic and efficient operation of the power system.

The feeder load power control technology realizes active control of the active power of loads by regulating the feeder voltage, and has the advantages of fast response speed, high control accuracy, and large adjustment capacity. This technology relies on series voltage-regulating devices [8,9] or parallel reactive power compensation devices [10,11], and utilizes the active-voltage coupling characteristics of feeder loads [12] to achieve direct control of feeder load power, thereby fully tapping the regulation potential on the load side, enabling it to effectively participate in grid interaction as an important component of virtual power plants, and improving the new energy accommodation capacity and operational safety and stability of power systems. Existing studies mostly use the ZIP model to characterize the regulation characteristics of loads. Among them, constant impedance and constant current loads are the main regulatory objects of this technology due to their strong voltage sensitivity. To reduce the difficulty of identification, this paper uses the CVR coefficient to describe the regulation characteristics of feeder loads, which is defined as the ratio of the load active power change rate to the voltage change rate. Reference [8] proposes a method to simplify and obtain the CVR coefficient based on the ZIP model. Reference [13] develops a composite model that combines different household appliance load curves based on the ZIP model. However, the complexity and time-variability of load composition bring challenges to model parameter identification.

Taking demand response as an example, while participating in grid regulation interaction, virtual power plants need to report their regulation capacity in advance to provide a reliable basis for dispatching decisions. However, due to the fluctuations on both the source and load sides, the system voltage state changes dynamically, which in turn affects the node voltage regulation range and changes the adjustable capacity of feeder loads. Therefore, to accurately evaluate the hourly adjustable capacity of feeder loads, it is necessary to first establish a prediction model for source–load fluctuations, and based on this, predict the spatio-temporal distribution change law of system voltage. Current source–load prediction studies mainly adopt simulation methods [14] and artificial intelligence methods [15,16]. Among them, the simulation method simulates real system behavior through mathematical modeling and simulation, but the credibility of the prediction results is limited [17]. The artificial intelligence method is driven by actual operation data and has high prediction accuracy and reliability. Among artificial intelligence algorithms, traditional algorithms such as support vector machines [18] and BP neural networks, although they have nonlinear fitting and parameter learning capabilities, have obvious shortcomings: on the one hand, they require massive training samples with harsh data requirements; on the other hand, the model complexity is high, which is prone to overfitting and takes a long time for training. In contrast, the Long Short-Term Memory (LSTM) neural network [19] and its improved algorithms, relying on their unique ability to process time-series features, can effectively capture the time-series characteristics of load fluctuations and photovoltaic output, thus showing significant advantages in the field of source–load power prediction and providing a new technical path for improving the prediction accuracy of new energy power systems.

Regarding the issue of evaluating the regulation capability of feeder loads, in recent years, some studies have proposed maximum supply capacity assessment methods based on the N-1 security criterion, which consider the impact of main transformers and feeder interconnection relationships on supply capacity [20,21]. Reference [22] established a TSC (Total Supply Capacity) assessment model considering incentive-based demand response. Reference [23] proposed a method for evaluating the supply capacity of active distribution networks that takes into account reliability and demand response. However, these studies all focus on analyzing the overall supply capacity of distribution networks and fail to conduct in-depth evaluations on the regulation capability of feeder loads specifically. Reference [24] analyzed the voltage regulation range of substation main transformers and calculated the adjustable capacity of feeder clusters by combining CVR coefficients. Nevertheless, this method evaluates the regulation capability from the overall perspective of feeder clusters and fails to fully consider the differentiated regulation effects of user-side voltage regulating equipment on loads at each feeder node. In summary, current research rarely incorporates feeder loads into the regulatory resource pool and lacks assessment methods for evaluating the regulation capacity of feeder loads, making it difficult to support the flexible regulation demands of feeder loads in new power systems.

To address the aforementioned issues, this paper proposes a dynamic assessment method for the adjustable capacity of feeder loads based on source–load power forecasting. The main contributions of this work are reflected in the following aspects: First, a Zbus linearized power flow model is established based on a single fixed-point iteration, and a linear expression for voltage sensitivity is derived, significantly improving the computational efficiency of power flow solutions. Second, a hybrid CNN-LSTM neural network architecture is constructed to achieve accurate source–load power forecasting, which is integrated with the proposed linearized power flow model to establish a system-wide voltage prediction framework. Furthermore, the coupling effects of multiple types of voltage regulation devices on multi-node voltages are fully considered. With the security of system-wide node voltages as the core constraint, an optimization model is developed to maximize the adjustable capacity of feeder loads, comprehensively accounting for the coordinated regulation of multiple types of voltage regulation devices. Finally, the proposed method is systematically validated on the Matlab simulation platform, with results demonstrating its effectiveness and feasibility, thereby filling the current research gap in this area.

2. Modeling of Feeder Regulation Characteristics and Parameter Identification Methods

This section focuses on constructing a mathematical model of feeder load regulation characteristics and proposes a comprehensive parameter identification framework, including effective voltage and power data acquisition methods as well as an efficient identification algorithm. The established model and methodology provide crucial theoretical foundations and technical support for subsequent accurate assessment of adjustable capacity in feeder loads.

2.1. Modeling of Feeder Load Regulation Characteristics

Feeder loads typically include residential, commercial, agricultural, and public service loads. Although the proportions and characteristics of these loads vary, their regulation behaviors exhibit similarities that can be described using a unified load model. Load characteristics are categorized into static and dynamic properties. Static load characteristics emphasize the steady-state response of load power to system voltage and frequency variations, making them suitable for medium-to-long timescale applications. These static characteristics can be represented through polynomial, exponential, or constant impedance models. Taking the widely-used polynomial load model (ZIP model) as an example, its formulation is shown in Equation (1). Since this study focuses on the voltage–power coupling characteristics of feeder loads, the frequency-dependent terms in the load model are not considered.

$$P_u=P_0[A_p{\left({\displaystyle \frac{V}{V_0}}\right)}^2+B_p\left({\displaystyle \frac{V}{V_0}}\right)+C_p]$$

(1)

In the equation, $P_u$ represents the active power corresponding to voltage V; $V_0$ is the initial voltage at the load node, and $P_0$ is the corresponding active power at $V_0$. The three terms in the equation correspond to constant impedance load, constant current load and constant power load, respectively, where $A_p$, $B_p$ and $C_p$ represent the proportion of these three load types, with their sum equal to 1.

The model can be simplified to a ZP load model by neglecting the constant current component, as shown in Equation (2):

$$P_u=P_0[{A_p}^{\prime }{\left({\displaystyle \frac{V}{V_0}}\right)}^2+{C_p}^{\prime }]$$

(2)

In addition to static loads, induction motor loads actually account for a significant proportion in real power systems. Using only static load models cannot accurately reflect the actual variations in system loads. However, dynamic modeling of motor loads is relatively complex and faces difficulties in parameter identification. Moreover, the transient response time of most induction motors in power grids is quite short. Therefore, this study focuses solely on analyzing the steady-state characteristics of induction motor loads, examining the relationship between the active power of motor loads and stator terminal voltage, as shown in Figure 1.

Neglecting the stator resistance and treating the excitation branch as an open circuit yields the following expression for the active power absorbed by the induction motor:

$$P={\displaystyle \frac{U^2{R}_r}{s((\frac{R_r}{s}{)}^2+(X_s+X_r{)}^2)}}$$

(3)

In the equation, X_s and X_r represent the stator and rotor reactances, respectively, R_r denotes the rotor resistance, and s is the slip ratio. Since the speed of induction motors is relatively insensitive to voltage variations, the denominator term can be considered constant when the terminal voltage U fluctuates within a certain range. Consequently, the active power of the motor load exhibits a linear relationship with the square of the terminal voltage U, allowing it to be approximated as a constant impedance load.

The above analysis demonstrates that the three components constituting the active power of feeder clusters—static loads, steady-state induction motor models, and network losses—all exhibit significant coupling relationships with system voltage. To accurately characterize the voltage regulation characteristics of feeder clusters, this study employs the CVR (Conservation Voltage Reduction) coefficient model for representation. The model is defined in Equation (4), with its physical meaning being the ratio of the percentage change in load active power to the percentage change in voltage. As derived from the aforementioned static load model, when the system voltage fluctuates within a reasonable range and the composition of feeder loads remains relatively stable, the CVR coefficient can be approximated as a constant. The CVR coefficient model effectively captures the steady-state voltage regulation characteristics of loads. Its magnitude directly reflects the regulation capability of feeder loads: a larger CVR coefficient indicates stronger voltage regulation capacity. With clear physical significance and practical utility, this model has been widely applied in various medium-to-long-term timescale feeder load power control applications.

$$CV{R}_f=\Delta P\%/\Delta V\%={\displaystyle \frac{\Delta P/P_0}{\Delta V/V_0}}$$

(4)

In the formula, $CV{R}_f$ represents the Conservation Voltage Reduction coefficient of the feeder load; $P_0$ and $V_0$ denote the initial active power and initial voltage of the load, respectively; $\Delta P$ and $\Delta V$ in voltage refer to the variations in power and voltage, respectively.

Since the type and quantity of loads carried by each feeder line vary across different time periods, it is necessary to select an appropriate identification cycle to update the Conservation Voltage Reduction (CVR) coefficient. Only in this way can the load regulation capacity of the feeder line be accurately evaluated.

2.2. Parameter Identification Methods for Regulation Characteristic Models

Not all voltage and power measurement data can be effectively used for CVR coefficient identification. Only when load power variations are explicitly caused by proactive voltage adjustments—specifically during Load-to-Voltage (LTV) processes—does the corresponding data possess identification value. Therefore, the typical practice involves selecting voltage and power measurements collected before and after OLTC tap changes, supplemented by correlation analysis to filter data demonstrating significant relevance to LTV processes as valid observation datasets.

However, in addition to the influence of voltage, the feeder load power is also affected by slow time-varying environmental factors such as ambient temperature and relative humidity. In this paper, the Savitzky–Golay (SG) filtering method is used to filter the measured voltage and power data, eliminating the low-frequency trend terms and extracting the voltage-sensitive components in the load power, so as to accurately analyze the relationship between load power and voltage. Based on the principle of local polynomial fitting, SG filtering obtains polynomial weighting coefficients through sliding window and least square method fitting, retaining trend components and suppressing noise [25]. For the Savitzky–Golay filter, the window length can be initially set to the number of data points that cover the Full Width at Half Maximum (FWHM) of the signal’s primary peaks. The polynomial order is typically chosen as 3 or 4 to strike a balance between smoothing effectiveness and waveform preservation.

The width of the filtering window is set to 2m + 1, and a (k − 1)-th degree polynomial is used to fit the data points in the window, resulting in the following polynomial fitting model:

$$y=a_0+a_{1}x+a_2{x}^2+\dots +a_{k-1}{x}^{k-1}$$

(5)

In the formula, $x$ is the non-voltage trend term data; $y$ is the fitted output data, and $a$ is the parameter to be solved.

The width of the filtering window is 2m + 1, which contains 2m + 1 data points, thus forming a k-dimensional linear system of equations consisting of 2m + 1 equations. These equations are written in matrix form as follows:

$$Y=XA+B$$

(6)

The least squares solution for parameter $A={\left(a_0,\dots ,a_{k-1}\right)}^T$ is determined by the least squares method:

$$\widehat{A}={\left(X^{T}X\right)}^{-1}{X}^{T}Y$$

(7)

thus obtaining the model filtering value, i.e., the fitting result of the non-voltage trend term:

$$\widehat{Y}=XA=X{\left(X^{T}X\right)}^{-1}{X}^{T}Y$$

(8)

By subtracting each fitting data of the non-voltage trend term from the original power data, the voltage-sensitive power data sequence can be obtained.

After obtaining valid identification parameters, this study employs the least squares method for parameter estimation. As a classical identification algorithm, the least squares method offers significant advantages including computational efficiency, theoretical rigor, and strong noise immunity. The method achieves optimal parameter estimation by minimizing the sum of squared residuals between model-predicted values and actual observations. The mathematical formulation of this approach is presented in Equation (9), with a one-hour data window length selected for the identification process.

$$\min J={\displaystyle \sum _{\mathrm{i}=1}^N{\left(P_i\%-\theta U_i\%\right)}^2}$$

(9)

In the equation, N represents the length of the data sequence, J denotes the objective function, P_i% and U_i% are the percentages of the measured active load power and voltage relative to their initial values, respectively, and θ represents the parameter to be identified (i.e., the CVR coefficient). When the objective function J is minimized, the optimal fitting of parameter θ is achieved. This problem is therefore transformed into finding the minimum value of the objective function. By taking the derivative of J with respect to θ and setting it equal to zero, we obtain the solution for θ—the feeder cluster’s CVR coefficient—as shown in Equation (10):

$$CV{R}_i=\theta ={\displaystyle \frac{{\displaystyle \sum _{\mathrm{i}=1}^N{U_i\%}^2{\displaystyle \sum _{\mathrm{i}=1}^N{P}_i\%-{\displaystyle \sum _{\mathrm{i}=1}^N{U}_i\%{\displaystyle \sum _{\mathrm{i}=1}^N{U}_i\%P_i\%}}}}}{n{\displaystyle \sum _{\mathrm{i}=1}^N{U_i\%}^2-{\left({\displaystyle \sum _{\mathrm{i}=1}^N{U}_i\%}\right)}^2}}}$$

(10)

The CVR factor exhibits significant time-varying characteristics, requiring dynamic selection of appropriate time windows based on the control command cycle and online rolling updates of this parameter using real-time voltage and power measurement data.

3. CNN-LSTM-Based Source–Load Forecasting Model

The stochastic fluctuations on both generation and load sides can trigger dynamic variations in the grid-wide voltage profile. These time-varying voltage distribution characteristics directly impact the adjustable capacity of feeder-load virtual power plants (VPPs). Given that static assessments at single time points cannot adequately reflect the dynamic nature of system regulation capabilities, it becomes essential to conduct accurate and continuous evaluations of feeder load adjustable capacity over future time periods. To achieve this objective, the establishment of a high-precision forecasting model for source–load fluctuations is prerequisite. This section accordingly develops a hybrid CNN-LSTM prediction model that integrates the feature extraction capability of Convolutional Neural Networks (CNNs) with the temporal modeling strengths of Long Short-Term Memory (LSTM) networks. The model delivers precise predictions of source–load fluctuations over forthcoming time horizons, thereby laying the foundation for subsequent dynamic assessments of feeder load adjustable capacity.

3.1. Strongly Correlated Feature Selection Method

This paper utilizes a CNN-LSTM hybrid model for ultra-short-term forecasting of distributed generation output and nodal load demand to capture the dynamic characteristics of source–load temporal data. In practical power systems, distributed energy resources such as photovoltaic systems and load demand are subject to multiple influencing factors, particularly meteorological parameters including irradiance, temperature and humidity, which significantly impact PV module performance and electricity consumption patterns. To improve the input feature selection for PV power forecasting models, this paper develops a joint analytical method combining Pearson Correlation Coefficient and Mutual Information to comprehensively assess both linear and nonlinear dependencies between meteorological factors and power generation. For each meteorological variable, the method systematically computes its correlation coefficient and mutual information value with historical power generation sequences, constructing a two-dimensional correlation plane for integrated feature evaluation. This approach enables effective identification of key meteorological variables strongly associated with PV output fluctuations while filtering out weakly relevant features, thereby enhancing the prediction model’s input quality and ultimately improving forecasting accuracy.

$${\rho }_{X,Y}={\displaystyle \frac{Cov(X,Y)}{{\sigma }_X{\sigma }_Y}}={\displaystyle \frac{{\displaystyle \sum _{i=1}^n(x_i-\overline{x})(y_i-\overline{y})}}{\sqrt{{\displaystyle \sum _{i=1}^n{(x_i-\overline{x})}^2}}\sqrt{{\displaystyle \sum _{i=1}^n{(y_i-\overline{y})}^2}}}}$$

(11)

$$I(X;Y)={\displaystyle \sum _{x\in X}{\displaystyle \sum _{y\in Y}p(x,y)log(\frac{p(x,y)}{p(x)\cdot p(y)})}}$$

(12)

In Equation (11), x_i and y_i represent the variable values at the i-th time step, $\overline{x}$ and $\overline{y}$ denote the sample means, ${\sigma }_X$ and ${\sigma }_Y$ are the standard deviations of the variables, and $Cov(X,Y)$ is the covariance. In Equation (12), $p(x,y)$ represents the joint probability density distribution of variables x and y, while $p(x)$ and $p(y)$ are their marginal probability distributions, respectively.

In the two-dimensional plane, the horizontal axis represents the Pearson correlation coefficient, reflecting the degree of linear correlation between variables and PV power output, while the vertical axis denotes the mutual information value, revealing the strength of nonlinear informational dependency between variables and generation power. Each point in the plot corresponds to a meteorological variable, and its distance from the origin indicates the overall correlation strength with power generation—variables positioned farther from the origin demonstrate stronger comprehensive relationships and thus higher modeling value. Based on this spatial distribution, key meteorological factors can be systematically selected as inputs for the CNN-LSTM prediction model, achieving effective feature dimensionality reduction.

3.2. LSTM Prediction Method

The LSTM network replaces the hidden layer of traditional Recurrent Neural Networks (RNNs) with specialized memory cell structures. The structure of the LSTM is shown in Figure 2. Through three gating mechanisms, it dynamically regulates the cell state: the forget gate determines how much historical information to retain, the input gate controls the extent of new information written to memory, and the output gate selects which memories to use for the current output. This architecture effectively addresses the issues of gradient vanishing and memory loss. By employing selective memory retention and forgetting mechanisms, it significantly enhances temporal modeling capabilities. The fundamental computational formulas of this network are as follows:

$$f_t=\sigma (W_f[h_{t-1},x_t]+b_f)$$

(13)

$$i_t=\sigma (W_i[h_{t-1},x_t]+b_i)$$

(14)

$${\tilde{C}}_t=tanh(W_C[h_{t-1},x_t]+b_C)$$

(15)

$$C_t=f_t\ast C_{t-1}+i_t\ast {\tilde{C}}_t$$

(16)

$$o_t=\sigma (W_o[h_{t-1},x_t]+b_o)$$

(17)

$$h_t=o_t\ast \tanh (C_t)$$

(18)

In the formulation, x_t represents the current input (including historical power data and meteorological forecast values), h_t−1 denotes the output from the previous timestep (predictions of photovoltaic and load power), while f_t, i_t, and o_t correspond to the forget gate, input gate, and output gate, respectively, controlling the retention, writing, and output of information. C_t is the current memory cell state, and ${\tilde{C}}_t$ is the candidate memory state. W and b are model parameters, σ is the sigmoid activation function, and tanh denotes the hyperbolic tangent function.

3.3. CNN-LSTM Hybrid Forecasting Model

A Convolutional Neural Network (CNN) is a deep learning model, widely used in image recognition, computer vision and other fields. Its core design idea is to automatically learn multi-level representations of data through local connectivity, weight sharing, and hierarchical feature extraction. A CNN is mainly composed of three key components: a convolutional layer locally senses input data by sliding multiple learnable filters (convolutional kernels), which simulates the local sensing mechanism of biological vision, thus extracting the underlying features to the higher level; a pooling layer reduces the computational effort by aggregating the features of the local region and reducing the feature map size, which reduces the size of the feature map and reduces the amount of computation, and can achieve translation invariance, so that the features are insensitive to small displacements of the input data, and enhance the model robustness; the final fully connected layer maps the extracted distributed features to the sample labeling space to complete the classification or regression task.

Although the traditional LSTM network performs well in processing time series data and can effectively capture the long-term dependencies between data, its main limitation lies in the difficulty of effectively extracting spatial features from the input data, which restricts its performance when dealing with data with spatiotemporal correlations. Therefore, this paper considers adopting the CNN-LSTM combined model. By integrating the spatial feature extraction capability of a CNN and the time series modeling advantage of LSTM, it can capture both the local spatial patterns and temporal dynamic changes in the data. This enables more accurate modeling and prediction in tasks such as time series forecasting, thereby improving the performance of the prediction model and the accuracy of the results. Its structure is shown in Figure 3.

The above figure shows the prediction model based on the CNN-LSTM hybrid architecture proposed in this paper, which can be used for accurate prediction of photovoltaic power generation and load power fluctuations. The model extracts local spatiotemporal features of the input time series through a one-dimensional convolutional neural network (1D-CNN), including key patterns such as the fluctuation characteristics of photovoltaic output and load change trends. Subsequently, the long short-term memory network (LSTM) is used to model the long-term temporal dependencies between features, achieving high-precision prediction of volatile renewable energy output and electrical loads. The prediction results can provide reliable data support for the full-node voltage prediction of power systems and the evaluation of adjustable capacity of feeder loads, thereby supporting the efficient and safe participation of distribution network feeder loads in grid interaction.

4. Feeder Load Adjustable Capacity Assessment Method

This section develops a system-wide nodal voltage prediction model using a linearized Zbus power flow model [26], based on source–load power forecasting results. While ensuring all nodal voltages operate within safe limits, it calculates the voltage regulation boundaries at the distribution network’s low-voltage bus. Building upon this foundation and incorporating CVR coefficient identification results, the method quantitatively assesses the adjustable capacity range of feeder loads. This provides operational constraints for subsequent participation of feeder loads, as virtual power plant components, in grid interactions.

4.1. Full-System Node Voltage Prediction Model

To address the issue of low computational efficiency in the traditional Newton–Raphson method, this paper proposes a Zbus linearized power flow model based on single fixed-point iteration. The traditional Zbus power flow model assumes that the voltage at each system node is composed of the superposition of the voltage V₁ generated by the slack bus and the voltage V₂ produced by the remaining PQ nodes. Building upon this foundation, the literature [26] establishes a linearized relationship between nodal voltages and injected power through a single fixed-point iteration approach, resulting in the linearized Zbus power flow model as expressed in Equations (19)–(21):

$$V=A\overline{S}+W$$

(19)

$$A={Y_{LL}}^{-1}diag{({\overline{V}}^{\mathrm{o}})}^{-1}$$

(20)

$$W=-{Y_{LL}}^{-1}{Y}_{\mathit{L}\mathbf{0}}{V}_0$$

(21)

The equations define the following variables: $S$ represents the injected power at each PQ node, $\overline{S}$ denotes the conjugate matrix of $S$, $V_0$ indicates the slack bus voltage, $V$ corresponds to the voltage phasors of PQ nodes, $V^o$ signifies the reference power flow voltage state at the current time which requires iterative updates, $Y_{LL}$ stands for the admittance matrix of PQ nodes, and $Y_{L\mathbf{0}}$ represents the mutual admittance matrix between the slack bus and other nodes.

With the node voltages of the system at the initial moment set as the reference voltage values, and combined with the source–load forecasting curve, the voltage distribution at each time interval can be solved step-by-step through the linearized power flow model to construct the full-system node voltage prediction model. This is specifically expressed as follows:

$$S_t=S_{t\mathbf{-}\mathbf{1}}+\Delta P_{pv,t\mathbf{-}\mathbf{1}}+\Delta P_{load,t\mathbf{-}\mathbf{1}}+\Delta Q_{load,t\mathbf{-}\mathbf{1}}$$

(22)

$$V_t=A{\overline{S}}_t+W$$

(23)

$S_t$ and $S_{t\mathbf{-}\mathbf{1}}$ represent the power injections at PQ nodes during the $t$ and $t-1$ minutes, respectively, $V_t$ denotes the voltage vector of all PQ nodes at the current time, $\mathit{\Delta }{P}_{pv,t\mathbf{-}\mathbf{1}}$ indicates the active power injection increment at the i-th PV node during the $t-1$ minute, $\Delta P_{load,t\mathbf{-}\mathbf{1}}$ and $\Delta Q_{load,\mathit{t}\mathbf{-}\mathbf{1}}$ correspond to the active and reactive power injection increments of the load node.

4.2. Calculation of Feeder Load Adjustable Capacity

The volatility of source and load leads to a significant time-varying characteristic in the adjustable capacity of feeder load, and the evaluation result at a single moment cannot fully reflect the system’s regulatory capability. Based on the CNN-LSTM source–load prediction results, this paper uses a linearized power flow model to iteratively calculate the full-system node voltage values within the next 3 h at a 15-min resolution. Relying on these results, it evaluates the adjustable capacity of feeder load, captures its dynamic changes, and provides a reliable basis for dispatching decisions. Additionally, the prediction data and evaluation results are updated on a rolling basis every 15 min to reflect the latest system status in a timely manner and avoid decision-making errors caused by information lag.

This paper considers the coordinated control of grid-side series voltage regulation equipment (including OLTC) and user-side shunt voltage regulation equipment (including SVC and new energy inverters). The calculation of feeder load adjustable capacity mainly consists of two key steps: identifying the CVR coefficient of each feeder load and determining the voltage regulation boundary of each feeder node. The adjustable capacity of feeder load is the sum of the products of these two factors. To ensure that the full-system node voltages meet safety constraints during the regulation process, based on the aforementioned linearized power flow model, the partial derivatives of node voltages with respect to the injected power of each node and the slack bus voltage are calculated to obtain the voltage sensitivity expression, which is used to analytically determine the voltage changes in each node during the regulation process.

$${\displaystyle \frac{\partial V}{\partial P}}={Y_{LL}}^{-1}diag{({\overline{V}}^o)}^{-1}$$

(24)

$${\displaystyle \frac{\partial V}{\partial Q}}=-j{Y_{LL}}^{-1}diag{({\overline{V}}^o)}^{-1}$$

(25)

$${\displaystyle \frac{\partial V}{\partial V_0}}=-{Y_{LL}}^{-1}{Y}_{L0}$$

(26)

The aforementioned expression establishes a linear relationship between node voltages, injected power, and changes in the slack bus voltage. Based on this linearized model, the evaluation of the coordinated control effect of two types of key voltage regulation equipment can be realized: on one hand, relying on the slack bus voltage expression, it is possible to quantitatively analyze the impact of tap position adjustment of series-type voltage regulation equipment (such as OLTC) on the voltage distribution across the entire network; on the other hand, based on the reactive power–voltage sensitivity expression, the voltage regulation effect of shunt-type reactive power compensation equipment (such as photovoltaic inverters and SVC) can be evaluated.

An objective function is constructed with the goal of maximizing feeder load, and optimization calculations are performed under different scenarios. The objective function is as follows:

$$F_1=\max \left({\displaystyle \sum _{i=1}^{N_{node}}{f}_i^{CVR}\Delta V_{i,up}^{\mathrm{Feeder}}}\right)$$

(27)

$$F_2=\max \left({\displaystyle \sum _{i=1}^{N_{node}}{f}_i^{CVR}\Delta V_{i,down}^{\mathrm{Feeder}}}\right)$$

(28)

In the objective function, $f_i^{CVR}$ represents the CVR coefficient of the load at feeder node i; $\Delta V_{i,up}^{\mathrm{Feeder}}$ and $\Delta V_{i,down}^{\mathrm{Feeder}}$ respectively denote the allowable voltage upward and downward adjustment ranges of each node under the coordination of various types of voltage regulation equipment, on the premise of meeting the voltage safety constraints of all system nodes; and N_node is the total number of feeder load nodes.

The optimization process must satisfy the following constraints:

(1)

Node voltage constraints

$$V_{\min }\le |V_{i,t,0}+\Delta V_{i,t}^{Feeder}|\le V_{\max }$$

(29)

In the equation, $V_{i,t,0}$ denotes the initial measured voltage at node i for the current time step, $\Delta V_{i,t}^{feeder}$ represents the voltage adjustment increment at node i after regulation, $V_{\max }$ and $V_{\min }$ are the maximum and minimum values of the node voltage, respectively.

(2)

Operational Constraints of Regulation Equipment

On-Load Tap Changer (OLTC):

$$V_t^{out}=V_t^{in}+X_t^{OLTC}\Delta V_{tap}$$

(30)

$$X_{\min }^{OLTC}\le X_t^{OLTC}\le X_{\max }^{OLTC}$$

(31)

$${\displaystyle \sum _{t\in T}\left|X_t^{OLTC}-X_{t-1}^{OLTC}\right|\le }{N}_{\max }^{OLTC}$$

(32)

In the formula, $V_t^{out}$ and $V_t^{in}$ correspond to the pre-regulation and post-regulation voltages, respectively; $X_t^{OLTC}$ represents the tap position at time step t; $X_{\max }^{OLTC}$ and $X_{\min }^{OLTC}$ denote the operational limits of tap positions; while $N_{\max }^{OLTC}$ defines the maximum permissible number of daily tap change operations.

Photovoltaic (PV) Systems: Regulate node voltage by controlling the reactive power output of PV equipment.

$$P_t^{PV}=P_t^{{PV}_{pre}}$$

(33)

$$\left|{Q_t}^{PV}\right|\le \sqrt{{\left(S^{PV}\right)}^2-{\left(P_t^{PV}\right)}^2}$$

(34)

$$Q^{PV,\min }\le Q_t^{PV}\le Q^{PV,\max }$$

(35)

In the equation, $P_t^{PV}$ and $Q_t^{PV}$ represent the active and reactive power outputs of the photovoltaic system at time t, respectively; $Q^{PV\max }$ and $Q^{PV\min }$ denote the maximum and minimum limits of PV reactive power output; $S^{PV}$ indicates the rated capacity, while $P_t^{PVpre}$ stands for the forecasted value of PV active power generation.

Static Var Compensator (SVC) Equipment:

$$Q^{SVC \min }\le Q_t^{SVC}\le Q^{SVC \max }$$

(36)

In the equation, $Q_t^{SVC}$ represents the reactive power injected by the Static Var Compensator (SVC) at time t, while $Q^{SVC\max }$ and $Q^{SVC\min }$ denote the maximum and minimum limits of SVC reactive power injection capability, respectively.

Capacitor Bank (CB) equipment: Capacitor banks are commonly used reactive power compensation devices in electrical power systems, typically employed to improve power factor and optimize voltage regulation. They consist of multiple capacitor units connected in parallel or series configurations to provide the required capacity and performance. Compared to other reactive power compensation devices, capacitor banks have limitations on the number of switching operations they can undergo.

$$Q_{i,t}^{CB}=X_{i,t}^{CB}{Q}_i^{CBR}$$

(37)

$${\displaystyle \sum _{t\in T}\left|X_{i,t}^{CB}-X_{i,t-1}^{CB}\right|}\le N_{\max }^{CB}$$

(38)

In the equation, $Q_i^{CBR}$ represents the rated capacity of the shunt capacitor bank connected to node i; $X_{i,t}^{CB}$ denotes the switching state of the capacitor bank group, where $X_{i,t}^{CB}=1$ when the capacitor bank group is in operation, and during this period, the reactive power $Q_{i,t}^{CB}$ injected by the capacitor bank group at node i is $Q_i^{CBR}$. Conversely, $X_{i,t}^{CB}=0$ indicates the capacitor bank is not in operation, thus $Q_{i,t}^{CB}=0$. $N_{\max }^{CB}$ is the daily threshold for the maximum number of switching operations for the capacitor bank.

(3)

Voltage Variation Constraint

The voltage regulation quantity at node i can be rapidly calculated based on voltage sensitivity:

$$\Delta V_{i,t}^{\mathrm{Feeder}}=\left[\begin{array}{ccc}{\displaystyle \frac{\partial V_i}{\partial Q}}& {\displaystyle \frac{\partial V_i}{\partial P}}& {\displaystyle \frac{\partial V_i}{\partial V_0}}\end{array}\right]\left[\begin{array}{c}\Delta Q_t\\ \Delta P_t\\ \Delta V_{0,t}\end{array}\right]$$

(39)

The first matrix on the right-hand side of the equation represents the voltage sensitivity matrix, which comprises the sensitivity of voltage $V_i$ at feeder node i to variations in active and reactive power injections at each node and to changes in root node voltage. The second matrix consists of reactive power injection increments $\Delta Q_t$ from voltage-regulating devices in the network, active power injection increments $\Delta P_t$ from distributed photovoltaics and loads, and root node voltage increments $\Delta V_{0,t}$.

5. Case Analysis

5.1. Case Configuration

In this chapter, an improved IEEE 33-node distribution network model is built based on the MATLAB platform, with its topological structure shown in Figure 4. An OLTC is installed at the head end of the system, with each tap adjustment of 0.0125 p.u. and a total of 11 taps. The total load of the system is 3715 kW + j2300 kvar. It includes 2 distributed photovoltaic (PV) systems connected to nodes 22 and 32, respectively, with a rated capacity of 500 kW each and initial powers of 300 kW and 350 kW, respectively. In addition, the system is equipped with 3 reactive power compensation devices connected to nodes 8, 17, and 24, respectively, with an adjustable range of [−280, 280] kvar. It should be noted that in the original IEEE 33-node system, all nodes except the slack node are PQ nodes, and the voltage–power coupling characteristics of feeder loads are not considered. Therefore, it is necessary to modify the original node admittance matrix and node injection power matrix in combination with the CVR coefficient of each node.

5.2. Accuracy Analysis of CNN-LSTM Prediction Model

In this section, verification and analysis are conducted based on the measured operation data of a photovoltaic power station in Hebei Province, China. A historical operation dataset of 30 consecutive days is selected, with the first 25 days serving as the training set and the last 5 days as the test set. A sliding time window is adopted, where the 6-h preceding time-series data of photovoltaic output and the corresponding meteorological forecast data are used as inputs. An ultra-short-term photovoltaic output prediction model for the next 3 h (with a resolution of 15 min) is constructed based on the CNN-LSTM prediction model. Before prediction, a feature selection algorithm is first applied to perform correlation analysis on multi-dimensional meteorological indicators, so as to extract key meteorological feature variables that have a significant correlation with photovoltaic output. The results of feature extraction are shown in Figure 5.

Points of different colors in the figure correspond to various meteorological variables. Among them, the three variables of total radiation, direct radiation, and scattered radiation are significantly far from the origin in the two-dimensional plane, which means they have strong linear and nonlinear correlations with photovoltaic power fluctuations and play an important role in the photovoltaic power prediction model. In contrast, the three variables of air pressure, wind speed, and wind direction are close to the origin, indicating that their linear correlation and nonlinear information association with photovoltaic power are both weak, and thus they have low importance in the modeling process. The temperature and humidity variables are in a relatively intermediate position: they show a certain degree of linear correlation with photovoltaic power, but their nonlinear association is not very prominent. Based on the above analysis, the prediction data of five meteorological factors—total radiation, direct radiation, scattered radiation, air temperature, and humidity—are selected as the model inputs. Meanwhile, to evaluate the model performance, prediction models based on Support Vector Machine (SVM) and Long Short-Term Memory (LSTM) are adopted as comparative cases.

The neural network was trained in the MATLAB R2023a environment using an NVIDIA RTX 5070 GPU. The network structure employs a 1D convolutional layer to extract local temporal features (filter width 3, number of channels 32), followed by an LSTM layer (100 hidden units) to capture temporal dependencies, and finally a fully connected layer to output multi-step prediction results. During training, the Adam optimizer was used with an initial learning rate of 0.005, a maximum of 100 iterations, and a batch size of 32. Gradient clipping was set to prevent gradient explosion. All input data were normalized using z-score to improve training stability and prediction accuracy. The prediction curves of a typical day under the three strategies are shown in Figure 6, and the absolute and relative errors between the predicted values and the true values are shown in Figure 7.

As shown in Figure 6 and Figure 7, the CNN-LSTM model shows significant advantages in photovoltaic output prediction. Compared to traditional SVM and LSTM prediction models, the CNN-LSTM prediction curve has a higher degree of fit with the real curve at most times and can accurately capture the power change trend. Meanwhile, to more intuitively compare the performance of the two prediction methods, MAE (Mean Absolute Error) and RMSE (Root Mean Square Error) are adopted as evaluation indicators. MAE calculates the average absolute deviation between the predicted values and the true values, and its formula is shown in Equation (40), where $y_{i,pre}$ is the photovoltaic output prediction value in each period, and $y_{i,true}$ is the measured value. This indicator is insensitive to outliers and its results are easy to interpret. RMSE processes the mean of squared errors through square root, and its specific calculation method is shown in Equation (41), which can highlight significant deviations in prediction. The combination of the two can comprehensively evaluate the accuracy and stability of the model. The calculation results of the two indicators for the three methods are shown in Figure 8.

$$MAE={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n\left|y_{i,pre}-y_{i,true}\right|}$$

(40)

$$RMSE=\sqrt{{\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{\left(y_{i,pre}-y_{i,true}\right)}^2}}$$

(41)

Meanwhile, as can be seen from Figure 8, the Root Mean Square Error (RMSE) and Mean Absolute Error (MAE) of the CNN-LSTM prediction results are both lower than those of traditional SVM and LSTM models, validating the practicality and accuracy of the CNN-LSTM forecasting model.

5.3. Evaluation Results of Feeder Load Adjustable Capacity

Each three hours constitutes an evaluation cycle. At the initial moment of the cycle, the full-system node voltages are predicted for every 15 min based on the source–load prediction results. In this example, one evaluation cycle is selected, with the slack node voltage set to 1.05 p.u. at the initial moment. To simplify the calculation, it is assumed that the power fluctuation trends of all load nodes in the network are consistent, and the active power output fluctuation characteristics of each distributed photovoltaic system are also consistent. A unified power fluctuation coefficient is used to characterize their variation law, and its mathematical expressions are shown in Equations (42) and (43). This simplified processing method effectively reduces the simulation complexity while ensuring calculation accuracy.

$$P_{loadi,t}=K_{load,t}\ast P_{load\_i,0}$$

(42)

$$P_{PVj,t}=K_{PV,t}\ast P_{PVj,0}$$

(43)

In the formula, $P_{loadi,t}$ represents the active load power of node i at time t, and $P_{PVj,t}$ represents the active output of photovoltaic j at time t. $P_{loadi,0}$ and $P_{PVj,0}$ are their respective reference values, while $K_{load,t}$ and $K_{pv,t}$ are the feeder load fluctuation coefficient and photovoltaic fluctuation coefficient, respectively, which characterize the variation trends of their active power. The actual active power at each time period is equal to the reference power value multiplied by the fluctuation coefficient at the corresponding time. The reference value of load power at each node is the corrected original power value of the IEEE 33-node system. The predicted curves of source–load variation trends for the next 3 h obtained based on the CNN-LSTM model are shown in Figure 9.

To simplify the analysis process, this example ignores the dynamic changes in the CVR coefficient at each load node during the evaluation period, and uses the CVR coefficient obtained in the initial identification cycle as the constant value for each node throughout the evaluation process. The calculation results are as

Table 1. Conservation voltage reduction factor of network node.

Node Number	${\mathit{f}}_{\mathit{i}}^{\mathbf{CVR}}$	Node Number	${\mathit{f}}_{\mathit{i}}^{\mathbf{CVR}}$
2	1.699	18	1.732
3	1.730	19	1.507
4	1.838	20	1.548
5	1.795	21	1.845
6	1.734	22	1.694
7	1.599	23	1.838
8	1.767	24	1.584
9	1.533	25	1.721
10	1.750	26	1.752
11	1.764	27	1.513
12	1.792	28	1.746
13	1.856	29	1.645
14	1.893	30	1.520
15	1.808	31	1.696
16	1.733	32	1.577
17	1.871	33	1.549

.

Based on the results of power grid voltage prediction and sensitivity analysis, under the constraint of ensuring the safe operation of all system node voltages, the adjustable capacity of feeder loads in each evaluation cycle is calculated, and three comparative calculation cases are set as follows:

Case 1: Only the grid-side series voltage regulation equipment (OLTC) is used as the regulation means to analyze the overall regulation capacity of the feeder cluster.

Case 2: Only the user-side parallel voltage regulation equipment is used as the main regulation body, and the feeder load regulation capacity is analyzed considering the participation of regulatory equipment such as SVC and new energy inverters.

Case 3: Coordinate series-parallel voltage regulation equipment to achieve in-depth voltage regulation and maximize the adjustable capacity of feeder loads.

The optimization results of the feeder load adjustable capacity for the three cases are shown in Figure 10 and Figure 11.

Figure 10 and Figure 11 respectively show the dynamic change processes of the downward and upward adjustable capacities of feeder loads in the three cases under the condition of ensuring the voltage safety constraints of all system nodes. It is not difficult to find from the figures that regardless of upward or downward capacity, Case 3 always demonstrates the maximum adjustable capacity in each evaluation cycle, followed by Case 1, while the adjustable capacity of Case 2 is generally smaller. This phenomenon is mainly because the OLTC (On-Load Tap Changer) can centrally regulate the voltage level of the entire feeder, thereby achieving effective control over the power of all load nodes within the entire feeder range. In contrast, although user-side parallel reactive power compensation devices have certain voltage regulation capabilities, their scope of action is usually limited to the vicinity of the installation node, and their individual capacity is limited, making it difficult to achieve global voltage optimization at the feeder scale. In the case of coordinated operation of multiple types of voltage regulation equipment on the grid side and user side, on the one hand, user-side reactive power equipment can perform supplementary regulation on nodes that still have regulation margins; on the other hand, their reactive power output can also be reversely regulated, which effectively expands the regulation range of the discrete voltage regulation equipment OLTC under the premise of ensuring system voltage safety, thereby significantly improving the overall adjustable capacity of feeder loads.

Figure 12 and Figure 13 respectively show the scatter plots of the system-wide voltage distribution under the operating conditions where various types of voltage regulation equipment are adjusted to their upper and lower limits. The data in the figures indicate that during the evaluation period, after various types of voltage regulation equipment are adjusted to their limits, all node voltages are strictly controlled within the safe range of 0.93–1.07 p.u. [27]. This shows that the dynamic voltage sensitivity evaluation model proposed in this paper can accurately predict the system voltage distribution, and can effectively identify and prevent the risk of node voltage exceeding limits during the voltage regulation process, ensuring that the system voltage is always maintained within the safe operating range.

6. Conclusions

This paper proposes a method for evaluating the dynamic adjustable capacity of feeder loads based on source–load prediction. First, a model for the regulation characteristics of the feeder load CVR coefficient is established, and the single fixed-point iteration method is used to linearize the traditional Zbus power flow model. The voltage sensitivity expression is derived, and a system-wide node voltage prediction model is constructed by combining the source–load prediction results of CNN-LSTM. Then, with the safety of all system node voltages as the constraint condition, the maximum adjustable capacity of feeder loads under the coordinated control of various types of voltage regulation equipment is calculated based on the CVR coefficient identification results. The simulation results show that the ultra-short-term source–load prediction of CNN-LSTM has higher accuracy than traditional methods, and the proposed method can accurately evaluate the adjustable capacity of feeder loads on the premise of ensuring system voltage safety.