Research on a novel distribution network line loss prediction model based on spatiotemporal feature fusion neural network

1 Introduction

Line loss prediction plays a vital role in power system analysis, since line loss levels serve as an important indicator of both technical performance and operational management in power supply systems (Mahmoud et al., 2016). Therefore, how to effectively reduce line losses and improve the efficiency of electric energy utilization has become increasingly important in practical engineering applications.

Traditional methods for distribution network line loss calculation mainly refer to engineering approximation algorithms summarized from engineering practices and power flow algorithms with explicit analytical expressions (Moghaddam et al., 2020). The engineering approximation algorithms, aiming to simplify computations and facilitate modeling, introduce a considerable number of simplified assumptions into the calculation model, enabling these methods to be easily applied in engineering practice. Such methods typically replace time-varying variables with constant values, use averaged quantities instead of actual operating data, and simplify complex functional relationships. Common engineering approximation methods include the maximum current method, the equivalent resistance method, and the energy method. For example, the composite effects of three-phase imbalance and harmonics in various power quality issues of low-voltage distribution networks on line loss calculation were considered (Tofoli et al., 2006). An approximate harmonic resistance model was adopted to derive a corrected line loss calculation model. The use of maximum demand—defined as the average power consumption of a customer over a certain period within a settlement cycle, with the maximum recorded value retained as the maximum demand of that cycle—for line loss calculation in distribution networks was found to be unreliable due to the unavailability of terminal data (Queiroz et al., 2012). Consequently, a load curve correction factor was introduced, combined with node voltage data, to conduct line loss prediction (Yang et al., 2025).

Compared with engineering approximation algorithms, power flow algorithms require higher parameter accuracy and are mainly applicable when both structural and operational parameters of the power system are known (Sayed and Takeshita, 2014). Common power flow algorithms include the Newton–Raphson method, PQ decomposition method, and forward–backward sweep method. For instance, probabilistic methodologies for comprehensive transmission loss evaluation are explored, providing an alternative perspective to traditional deterministic calculations (Meliopoulos et al., 1991). This approach not only calculates theoretical line losses in the absence of theft or leakage but also, by leveraging electricity consumption data collected through remote automatic meter reading systems, provides a more sensitive means of detecting unidentified losses, thereby improving loss management. However, the approximations inherent in engineering approximation algorithms can lead to insufficient accuracy, while power flow algorithms often involve heavy computational loads and require large amounts of readily available distribution network data. In practice, the available sensor data in distribution networks are often limited, and many measurements at network terminals are unavailable due to economic constraints, thus affecting calculation accuracy (Wu et al., 2020). It is evident that both traditional methods are increasingly challenged in the context of the growing complexity of modern distribution networks (Pengwah et al., 2023).

In contrast, intelligent algorithms for line loss calculation in distribution networks leverage the learning capabilities of artificial intelligence (Ghoddusi et al., 2019), discarding the traditional approach of relying on a specific, explicit functional relationship to compute line losses. Instead, they adopt methods that fit a function representing the relationship between inputs and outputs directly from data (Huang et al., 2020), including regression analysis, support vector machines, and neural networks (Bian et al., 2025). For example, a clustering method based on high-density connected regions and a multi-system fusion-based line loss prediction index system was proposed to address the challenges posed by diverse data sources, data types, and data volumes in line loss analysis (Sun et al., 2019). By analyzing the distribution of line loss rates, a reliable assessment of the regional loss reduction potential was achieved. A grey relational analysis was used to identify electrical indicators correlated with line losses (Zhang et al., 2019). An improved algorithm was developed to address the limitations of the BP neural network, such as the empirical selection of hidden layer nodes and slow convergence, ultimately realizing line loss rate calculation based on the neural network model. A fuzzy C-means clustering was used to classify samples, and a neural network was constructed to model the relationship between deviations in loss indicators and deviations in actual losses, thereby calculating the line loss rate (Ouyang and Wu, 2017). Nevertheless, most of these studies rely heavily on complex, manually selected indicators, and traditional intelligent algorithms such as regression analysis often suffer from poor generalization ability and difficulties in handling large-scale datasets (Wan et al., 2017).

From the above analysis, it is clear that both traditional algorithms and intelligent algorithms have been proposed for distribution network line loss calculation. While existing research has made significant progress (Ming et al., 2017), the aforementioned issues remain to be addressed. Data-driven methods based on machine learning, owing to their ability to fully exploit available data and their computational efficiency, have promising applications in line loss calculation (Yang et al., 2020). For instance, the issue of missing values in load data was addressed through clustering algorithms for imputation, with calculation accuracy further improved by prelayering the network topology (Tan, 2022). A method for identifying the relationship between users and transformers was proposed, based on principal component analysis (PCA) and spectral clustering (Zhao et al., 2022). Voltage data from users in an automated acquisition system were normalized, and PCA was applied to reduce dimensionality and eliminate redundancy. This enabled topology identification, which was then used to calculate line losses with improved accuracy. With sufficient power flow data samples, a neural network was trained to model line loss under varying load conditions (Li et al., 2019). The Levenberg–Marquardt algorithm was introduced to enhance the model’s generalization ability, enabling accurate line loss calculation and electricity theft detection. The concept of user profiling was introduced, integrating multiple parameters such as grid topology and operational modes as standard inputs for line loss prediction. A KNN algorithm was used to correct abnormal data, thereby reducing computational load and improving accuracy (Sun et al., 2023).

In summary, further research into feature extraction and model construction is essential for developing data-driven distribution network line loss prediction models (Liu et al., 2019). This paper proposes a comprehensive prediction framework based on the novel spatiotemporal feature fusion neural network architecture.

The primary contributions of this work are threefold:

• A minimum Redundancy Maximum Relevance (mRMR) feature selection mechanism is introduced to systematically identify the most informative inputs while eliminating redundancy, thereby enhancing model efficiency and reducing overfitting risk.
• A hybrid CNN-LSTM architecture is designed, where convolutional layers extract spatial patterns from grid-structured data to capture topological dependencies, while LSTM layers model temporal dynamics to handle load fluctuations and distributed generation intermittency.
• Bayesian optimization is incorporated to automate hyperparameter tuning, ensuring optimal model configuration across varying operating conditions.

2 Data-driven framework for distribution network line loss prediction

The data-driven framework for distribution network line loss prediction is shown in Figure 1. Focusing on the full process of data preprocessing–model construction, the framework systematically addresses the temporal characteristics, multi-factor correlations, and dynamic topology of distribution network line losses from the following aspects. Feature selection based on the mRMR criterion ensures the quality of input data, the CNN-LSTM model captures temporal patterns, adapts to dynamic scenarios involving changes in the distribution network topology. This enables accurate prediction of line losses in distribution networks. Ultimately, the framework provides data-driven decision support for loss management, including loss reduction strategy formulation and abnormal loss localization, thereby enhancing the economic operation and refined management of distribution networks.

The core objective of this work is to develop a high-fidelity, data-driven model that maps the high-dimensional, time-varying operational state of a distribution network to its corresponding total line loss. Given the inherent spatiotemporal dependencies and nonlinearities, we formulate this as a sequential supervised learning problem.

Let the distribution network be represented by a graph G = V , ε, where V = 1 , 2 , . . . , N denotes the set of N nodes and ε denotes the set of branches. The operational state at time t is characterized by a multivariate time series X t ∈ R N × D, where D is the number of measured features per node. Over a historical window of T time steps, we obtain an input sequence χ = X t − T + 1 , X t − T + 2 , . . . , X t.

The target variable is the total active power line loss P t + 1 l o s s for the subsequent time period t + 1. The goal is to learn a nonlinear function F Θ parameterized by Θ, such that as formulated in Equation 1:where P ⌢ t + 1 l o s s is the predicted line loss, the function F Θ explicitly acknowledges the underlying grid topology G, R i j and X i j are the resistance and reactance of branch i , j, respectively.

P ⌢ t + 1 l o s s = F Θ X t − T + 1 , X t − T + 2 , . . . , X t ⏟ Historical State Sequence ; V , ε ⏟ Network Topology ; R i j , X i j ∀ i , j ∈ ε ⏟ Branch Impedance Set (1)

The model must capture two fundamental, intertwined types of dependencies inherent in the distribution network. Spatial dependencies arise from the physical interconnection via ε, which creates complex, non-local correlations between nodal states, such as voltage gradients and power flow patterns across the grid. Concurrently, temporal dependencies are governed by the system’s time-continuous dynamics, including evolving load profiles and the intermittent ramps of distributed generation outputs, which impose long-range sequential relationships on the state evolution.

Therefore, the learning problem can be formalized as Equation 2 finding the optimal parameters Θ * that minimize the expected prediction error over the joint distribution of historical states and future losses:where L · is a loss function, P is the true data distribution, Ω Θ is a regularization term penalizing model complexity, and λ is a regularization coefficient. The proposed spatiotemporal feature fusion neural network framework is designed as a principled solution to this formulation, integrating mechanisms for feature selection, spatial modeling, temporal modeling, and automated hyperparameter optimization for Θ.

Θ * = arg ⁢ min Θ ⁢ E χ , P l o s s ∼ P ⁢ L ⁡ P l o s s , F Θ ⁢ X t − T + 1 , X t − T + 2 , . . . , X t ; V , ε ; R i j , X i j ∀ i , j ∈ ε + λ · Ω ⁡ Θ (2)

3 A hybrid spatiotemporal feature fusion neural network model for line loss prediction

This section focuses on the data-driven modeling problem for line loss prediction and explores three key aspects: feature selection based on mRMR, the BO-CNN-LSTM neural network structure, and prediction metrics for the model.

3.1 Feature selection based on mRMR

Feature selection is a critical step in predictive data processing and model construction. An excessively high-dimensional and redundant feature space makes it difficult to uncover the mapping relationships between meteorological information and output power, while also increasing the training difficulty and computational complexity of the model. From an information-theoretic perspective, the objective of feature selection is to choose a subset of features that contains as much information as possible about the output variable while having the smallest possible dimensionality, that is, to satisfy both max ⁡ I X 1 , X 2 , · · · X n ; Y and min n, where the multivariate mutual information can be computed as Equation 3:where S n is the selected feature subset consisting of n features. However, due to the limited number of samples and the computational complexity of inverting high-dimensional matrices, the joint probability density function p x 1 , x 2 , · · · , x n cannot be accurately obtained, making direct computation of multivariate mutual information impractical. To address this, the mRMR algorithm approximates the feature selection process as Equation 4 the following optimization problem:

I S n , Y = ∫ · · · ∫ p x 1 , x 2 , · · · , x n , y log p x 1 , x 2 , · · · x n , y p x 1 , x 2 , · · · x n p y d x 1 d x 2 · · · d x n d y (3)

max 1 n ∑ X i ∈ S n I X i ; Y − 1 n 2 ∑ X i , X j ∈ S n I X i ; X j (4)

The first term in the above equation represents the relevance between the selected meteorological feature set S and the photovoltaic output Y, while the second term indicates the redundancy among the meteorological features. Typically, the mRMR algorithm adopts an incremental search strategy, in which the optimal feature at each step is selected and added to the candidate feature subset. Specifically, the newly added feature x k is chosen to satisfy (Equation 5):

max X j ∈ S − S k − 1 I X j ; Y − 1 k − 1 ∑ X i ∈ S k − 1 I X j ; X i (5)

3.2 Long short term memory neural network model

Long Short-Term Memory networks (Hochreiter and Schmidhuber, 1997) are a special type of Recurrent Neural Network (RNN) (Bengio et al., 2002) capable of learning long-term dependencies in sequential data. LSTMs combine temporal and spatial information, making them highly effective for modeling complex patterns in time series with long delays.

As illustrated in Figure 2, the typical structure of an LSTM cell includes forget, input, and output gates, as well as a memory cell. These gates control the flow of information, enabling the network to mitigate the vanishing and exploding gradient problems common in standard RNNs.

LSTM enhances traditional recurrent neural networks by introducing a forget gate, input gate, output gate, and memory cell, enabling finegrained control over information flow within the model. These control units allow LSTM networks to effectively mitigate issues such as gradient explosion commonly encountered in standard RNNs.

3.3 Bayesian optimized CNN-LSTM hybrid model

To address the limitations of individual neural network architectures and leverage the complementary strengths of different approaches, this paper proposes a Bayesian Optimized Convolutional Neural Network - Long Short-Term Memory (BO-CNN-LSTM) hybrid model. This integrated architecture combines spatial feature extraction, temporal pattern learning, and automated hyperparameter optimization into a unified framework, providing a effective solution for distribution network line loss prediction.

Convolutional Neural Networks (CNNs) are particularly effective for extracting spatial patterns from grid-structured data. In distribution networks, the spatial relationships among nodes and branches convey critical information regarding power flow distribution and loss mechanisms. The spatial organization of voltage measurements, load profiles, and distributed generation outputs naturally forms a grid-like structure that can be effectively exploited by CNN architectures. The CNN module employs convolutional layers to extract hierarchical spatial features from the input data. This spatial feature extraction enables the model to identify voltage gradients across the network, clustering patterns of high-load regions, and spatial correlations between distributed generation outputs and local voltage profiles. Accordingly, the CNN component of the proposed hybrid model is formulated as follows (Equation 6):where H l represents the feature maps at layer l, W l are the convolutional filters, * denotes the convolution operation, and σ is the activation function. The CNN architecture employs 1D convolutional layers to process the spatial arrangement of measurement data across different network nodes.

H l = σ W l * H l − 1 + b l (6)

While the CNN component captures spatial correlations within the distribution network, the temporal evolution of system states presents equally important information for accurate line loss prediction. Load variations, distributed generation fluctuations, and network switching operations all exhibit temporal dependencies that must be modeled effectively. The LSTM component addresses this challenge by maintaining long-term memory of system dynamics through its gated architecture. The forget gate selectively retains relevant historical information, the input gate incorporates new observations, and the output gate controls the flow of information to subsequent layers. This gating mechanism allows the model to learn both short-term fluctuations and long-term trends in the system behavior, which is crucial for accurate line loss prediction under varying operating conditions. The combined CNN-LSTM architecture can be represented as Equations 7, 8: where X t is the input at time t, X t s p a t i a l are the spatial features extracted by CNN, and h t , c t are the hidden state and cell state of the LSTM.

X t s p a t i a l = C N N X t (7)

h t , c t = L S T M X t s p a t i a l , h t − 1 , c t − 1 (8)

The performance of deep learning models is highly sensitive to hyperparameter configurations, and manual tuning often requires extensive trial-and-error experimentation. To overcome this challenge, Bayesian optimization provides a principled approach for automated hyperparameter search. The method constructs a probabilistic model of the objective function and uses this model to select the most promising hyperparameters to evaluate next. This approach balances exploration of unknown regions of the parameter space with exploitation of known good configurations. For the CNN-LSTM architecture, Bayesian optimization searches for optimal values of learning rate, number of layers, filter sizes, dropout rates, and LSTM hidden dimensions, among other parameters. The optimization process minimizes the validation loss while considering computational constraints, resulting in a model configuration that achieves high accuracy with reasonable computational requirements. The optimization problem is formulated as Equation 9:where θ represents the hyperparameters, Θ is the search space, and L θ is the validation loss.

θ * = arg min ⁡ L θ θ ∈ Θ (9)

Bayesian optimization uses a surrogate model to approximate the objective function and an acquisition function to guide the search (Equation 10):

α θ ; D = Ε max 0 , L min − L θ | D (10)

The complete BO-CNN-LSTM architecture integrates the three components into a cohesive framework. The architecture begins with an input layer that receives multi-dimensional time-series data from the distribution network, including voltage magnitudes, power injections, network topology indicators, and environmental factors. The input data first passes through the feature processing module, where multiple CNN layers extract spatial patterns from the grid-structured measurements. These convolutional layers are interspersed with batch normalization layers to stabilize training and dropout layers for regularization. Pooling operations reduce the spatial dimensionality while preserving important features, preparing the data for temporal modeling.

The temporal modeling module receives the spatially-processed features and captures their evolution over time through stacked LSTM layers. Residual connections between layers facilitate gradient flow during backpropagation, mitigating the vanishing gradient problem and enabling deeper architectures. The final stages of the network consist of fully connected dense layers that combine the spatial and temporal features to produce the line loss prediction.

The optimization module operates in parallel with the training process, continuously monitoring model performance and adjusting hyperparameters through Bayesian optimization. This module implements an early stopping mechanism based on validation performance, preventing overfitting while ensuring efficient use of computational resources. The complete mathematical formulation of the integrated model can be expressed through a composite function that combines the spatial processing of CNN with the temporal modeling of LSTM, followed by final regression layers for prediction.

Building upon the BO-CNN-LSTM foundation, this paper further enhances the model by integrating mRMR feature selection. The mRMR algorithm operates as a preprocessing stage, identifying the most informative subset of features while minimizing redundancy among them. This feature selection process is guided by mutual information criteria, ensuring that the selected features contain maximum information about the target variable while maintaining minimal interdependence. The resulting spatiotemporal feature fusion neural network model represents a comprehensive approach that addresses three critical aspects of distribution network line loss prediction: feature relevance through mRMR selection, spatial-temporal pattern recognition through CNN-LSTM architecture, and parameter optimization through Bayesian methods. This integrated approach provides significant advantages in terms of prediction accuracy, computational efficiency, and model interpretability, making it particularly suitable for practical applications in distribution system operations and management.

3.4 Evaluation metrics for line loss prediction models

In data-driven distribution network line loss prediction (Liu et al., 2024), a highly discussed issue is determining the most suitable model to establish and, for a given model, identifying the optimal parameter configuration. When assessing the performance of a model, one cannot rely solely on the loss value output at each step during training, as focusing only on training loss may lead to overfitting (Jiang, 2018). This, in turn, results in poor performance on unseen data and a loss of generalization ability. The loss merely reflects the difference between the predicted and actual values in the training set, without indicating the prediction error on the test set.

Therefore, in the context of line loss prediction for distribution networks, the training set error should be incorporated into the evaluation process. For example, when comparing different parameter settings for the same model, the Mean Absolute Error (MAE) can be adopted as the evaluation metric. This approach helps avoid situations where the Mean Absolute Percentage Error (MAPE) becomes excessively large due to small actual sample values, thereby diminishing its ability to assess model performance. The MAE and MAPE are defined as Equations 11, 12.

M A E = 1 n ∑ i = 1 n y i − y ^ i (11)

M A P E = 1 n ∑ i = 1 n y i − y ^ i y i × 100 % (12)

When comparing the performance of different types of models for line loss prediction, the Root Mean Square Error (RMSE) between the predicted and actual line losses is directly used as the evaluation metric. This avoids cases where a very small absolute value of line loss causes the percentage error to increase sharply, thereby undermining the purpose of the metric. The RMSE is defined as Equation 13:where y i and y ^ i denote the actual and predicted line loss values respectively, and n is the number of samples.

R M S E = 1 n ∑ i = 1 n y i − y i ^ 2 (13)

Beyond conventional accuracy metrics, robustness to system variations represents a critical consideration for practical deployment in distribution networks. Operational conditions frequently change due to load fluctuations, distributed generation intermittency, and network reconfigurations. To quantify model stability across varying conditions, a Robustness Score is defined as Equation 14 based on performance consistency across multiple topological configurations and operating scenarios:where μ M A E represents the mean absolute error across all tested scenarios, and σ M A E denotes the standard deviation of MAE values. This formulation produces scores in the range [0,1], with higher values indicating more consistent performance despite system variations. A robustness score of one represents perfect consistency, while lower scores indicate greater sensitivity to changing conditions.

R = 1 − σ M A E μ M A E (14)

The robustness metric complements traditional accuracy measures by addressing practical concerns about model reliability in dynamic distribution network environments. Models achieving both high accuracy and high robustness are preferable for operational deployment, as they maintain reliable performance despite the inherent variability of power distribution systems. This comprehensive evaluation framework, combining accuracy metrics with robustness assessment, provides a more complete characterization of model suitability for distribution network applications.

4 Case study

To substantiate the theoretical framework and quantitatively evaluate the performance of the proposed spatiotemporal feature fusion neural network model, comprehensive case studies are conducted on two benchmark distribution systems.

4.1 IEEE-33 system parameter settings

For the IEEE 33-bus system shown in Figure 3 (Baran and Wu, 1989), considering the practical situation of large-scale distributed generation (DG) integration into the power grid, six distributed generators are added at buses 12, 17, 21, 24, 28, and 32. These include three wind turbines and three photovoltaic (PV) units, each adopting a typical output profile for their respective technology. Considering that the availability of measurement information in practical distribution networks is constrained by economic factors, smart meters are installed only at buses 1, 4, 6, 9, 12, 16, 17, 18, 19, 21, 24, 25, 28, 31, and 32.

To account for possible topological changes in the operation of distribution networks, two additional dashed-line connections are introduced: one between buses 17 and 32, and another between buses 11 and 21. These represent possible network reconfigurations, resulting in a total of four typical topologies.

In data-driven model training, a large number of samples is essential; insufficient sample size can cause the model to inadequately learn certain features, making it incapable of accurately identifying them. Incomplete samples also weaken the model’s generalization ability (Zhang, 2013). The selection of training samples is therefore crucial to model performance. To ensure the rationality of the training dataset, this study considers three factors and enriches the dataset by referencing actual operating conditions.

Considering that the operating conditions vary from day to day in practice, a normal distribution disturbance is superimposed on the generated data to enhance its randomness (Liu, 2010). The magnitude of this disturbance reflects the scale of load fluctuations during operation. The formula for the disturbance model is given as Equation 15:where Y is the variable with disturbance applied; X is the original variable; Z is a normally distributed random disturbance with mean μ and standard deviation σ.

Y = X + Z ∼ μ , σ (15)

To obtain experimental samples for various operating conditions, the load at each bus is first scaled from the base load level by a factor ranging between 80% and 120%. The generator outputs are then adjusted accordingly to match the changed load conditions. To construct the input feature set for the IEEE 33-node system, we follow practical economic constraints commonly encountered in real distribution networks. Specifically, power measurements are available at 18 selected nodes, while voltage measurements are retained only at several critical nodes with high connectivity degrees. This configuration simulates the realistic scenario where voltage sensors are less frequently installed than power meters due to economic limitations. The remaining 18 nodes' measurement data are deliberately omitted to evaluate model performance under incomplete information. Applying the mRMR feature selection algorithm to this initial feature set yields an approximately 60% dimensionality reduction, with the retained features predominantly consisting of power injections at DG-integrated nodes and voltage magnitudes at critical nodes.

The key implementation details of the proposed framework are as follows. The CNN module employs two one-dimensional convolutional layers with 64 and 128 filters, respectively, each with a kernel size of three and ReLU activation, followed by max-pooling and batch normalization. The LSTM module consists of two stacked layers with 128 hidden units each, using a dropout rate of 0.3. The model is trained using the Adam optimizer with an initial learning rate of 0.001 and a batch size of 32, with a maximum of 500 epochs and early stopping patience of 20 epochs based on validation loss. The training, validation, and test sets are partitioned in a ratio of 7:1.5:1.5. All experiments are conducted on a workstation equipped with an AMD Ryzen 7 CPU, 16 GB RAM, and an NVIDIA GeForce RTX 4060 GPU. The total training time is approximately 19.2 min, and the inference time per sample is approximately 2.3 m. The model contains approximately 1.47 million trainable parameters.

4.2 Spatiotemporal feature fusion neural network line loss prediction with different input lengths

According to Table 1, the influence of input sequence length on prediction accuracy is systematically evaluated using the complete BO-CNN-LSTM framework, with comparative analysis against baseline LSTM models. This investigation establishes the optimal temporal context for distribution network line loss prediction while quantifying the performance advantages achievable through architectural optimization. The prediction spans sequence lengths from 3 to 18 time-steps, corresponding to historical windows ranging from 45 min to 4.5 h at 15-min resolution.

TABLE 1. Performance evolution across input sequence lengths for optimized and baseline models.

TABLE 2. Comparative performance of architectural paradigms.

The BO-CNN-LSTM framework demonstrates consistently better performance across all evaluated sequence lengths, with Mean Absolute Error decreasing from 0.0782kW at t = 3 to 0.0685kW at t = 18. This represents an 12.4% overall improvement as sequence length increases, though with diminishing returns observed beyond 12 time-steps. The marginal accuracy gain diminishes substantially after t = 12, with only 1.9% additional improvement from t = 12 to t = 18, suggesting that very distant historical information contributes minimally to current loss estimation when processed through the optimized architecture.

Maximum Absolute Error shows greater sensitivity to sequence length than mean error metrics. This 21.1% reduction in worstcase prediction error indicates that the optimized framework particularly improves handling of extreme operating conditions when provided with sufficient historical context. The substantial improvement in maximum error suggests that the architectural enhancements not only improve average performance but also enhance robustness during system transients, contingency conditions, and rapid load changes.

Comparative analysis with baseline LSTM performance reveals that the BO-CNN-LSTM framework achieves an average improvement of 16.1% across all sequence lengths, with differential improvement patterns across the evaluated range.

Greater relative improvements are observed at shorter sequences compared to longer sequences, suggesting that architectural optimization provides greater benefit when temporal context is limited. This pattern indicates that the enhanced feature processing capabilities of the BO-CNN-LSTM framework partially compensate for insufficient historical information through superior spatial feature extraction and optimized parameter configuration.

4.3 CNN-LSTM hybrid architecture enhancement

The integration of convolutional neural network layers prior to LSTM processing represents a significant architectural advancement that enables explicit spatial feature extraction from grid-structured measurement data. This hybrid approach addresses a fundamental limitation of pure temporal models by incorporating spatial awareness into the learning process, recognizing that distribution network measurements inherently possess topological relationships that conventional sequence models cannot fully exploit. The CNN component employs one-dimensional convolutional filters operating across the spatial dimension of input data, capturing local correlations between adjacent measurement points and extracting hierarchical features including voltage gradients, load concentration patterns, and power flow distributions.

The basic LSTM architecture establishes baseline performance with MAE of 0.0852 kW and RMSE of 0.1158 kW. The hybrid CNN-LSTM architecture yields a substantial performance gain over the pure LSTM baseline, confirming the critical value of explicit spatial feature extraction for processing grid-structured measurement data. The convolutional layers effectively distill raw measurements into spatially-aware representations that encapsulate local patterns in voltage distribution and power flow, thereby providing a more informative input for subsequent temporal modeling by the LSTM. Further integrating bidirectional LSTM processing provides incremental improvement, suggesting that bidirectional temporal modeling offers complementary benefits when applied to these spatially-enhanced features. The integration of Bayesian optimization delivers additional accuracy improvements by systematically identifying optimal hyperparameter configurations. This automated process ensures robust model performance across varying conditions while eliminating the extensive trial-and-error typically required for manual tuning.

The complete spatiotemporal feature fusion neural network framework achieves the highest accuracy. This final enhancement incorporates systematic feature selection that identifies the most informative measurements while eliminating redundant information, resulting in both improved accuracy and reduced model complexity. The 60% dimensionality reduction through mRMR selection provides additional benefits in computational efficiency and generalization capability.

Maximum Absolute Error analysis reveals even more pronounced architectural improvements. This substantial improvement in worst-case prediction error demon-strates that architectural enhancements are particularly valuable for handling extreme operating conditions, system transients, and rapid topology changes. The progressive reduction in Max-AE across architectural variants confirms that each component contributes to enhanced robustness.

The prediction accuracy of the proposed framework is first quantified through the temporal evolution of prediction errors, as shown in Figure 4. This plot directly compares the error trajectories of the proposed spatiotemporal feature fusion neural network model and the baseline LSTM model over the multi-day evaluation period. It clearly demonstrates that the proposed model maintains errors consistently closer to zero with lower volatility, whereas the baseline model exhibits significantly larger and more variable deviations, especially during periods of high load or generation fluctuation.

To statistically attribute these performance differences to specific operating conditions, Figure 5 presents the distribution of prediction errors across different load intervals. The boxplot analysis confirms that the proposed model achieves not only a lower median error but also superior robustness—indicated by a narrower interquartile range and fewer outliers—across all load levels. This statistical evidence directly links the architectural advancements, including spatial feature extraction via convolutional layers and robust hyperparameter tuning via Bayesian optimization, to enhanced model stability under the dynamic conditions of modern distribution networks.

The multi-day analysis further validates the model’s ability to capture both short-term variations and longer-term patterns in distribution network losses. The consistent tracking across approximately three full days provides evidence of the framework’s stability and reliability for extended operational deployment. The close alignment between predicted and actual values throughout the observation period confirms the model’s capability to generalize across different days, load conditions, and operational scenarios.

4.4 Performance comparison between spatiotemporal feature fusion neural network and other ML models

The proposed spatiotemporal feature fusion neural network framework is evaluated against three representative models: Multilayer Perceptron (MLP) as a baseline, Gradient Boosting Machine with feature engineering (GBM-FE) for advanced tree-based methods, and Temporal Convolutional Network with skip connections (TCN-Skip) for modern convolutional approaches.

As summarized in Table 3, the proposed framework demonstrates clear superiority, achieving 26.8% improvement over MLP and maintaining advantages over more sophisticated alternatives. The performance differential is particularly notable for robustness, where the integrated architecture achieves a score of 0.969 compared to 0.782 for MLP.

TABLE 3. Methodological comparison summary.

The comparative analysis reveals that while MLP provides basic neural network performance, it lacks capacity for complex spatio-temporal relationships. GBM-FE demonstrates the strengths of gradient boosting with engineered features but faces limitations in capturing intricate temporal dependencies. TCN-Skip offers efficient convolutional processing for sequence modeling but exhibits reduced adaptability to irregular temporal patterns. To provide a more comprehensive comparison, we further include Gated Recurrent Unit (GRU) and Graph Attention Network (GAT) as additional benchmarks (Liu et al., 2026). GRU is a simplified recurrent architecture that captures temporal dependencies with a gating mechanism similar to LSTM but with fewer parameters. GAT is a graph neural network that employs attention mechanisms to learn spatial dependencies directly from the topological structure of the distribution network. Both models yield satisfactory performance, with GAT achieving an MAE of 0.0765 kW. Nevertheless, the proposed framework consistently outperforms all alternative models, demonstrating the synergistic benefits of its integrated design.

The proposed framework addresses these limitations through systematic integration: CNN layers extract spatial patterns from grid-structured data, LSTM components model complex temporal dynamics, mRMR optimizes feature selection to eliminate redundancy, and Bayesian methods ensure optimal parameter configuration. This holistic approach provides the best balance of accuracy and robustness.

The results confirm that the proposed modeling, combined with systematic optimization, provides significant advantages for distribution network line loss prediction compared to specialized approaches from different methodological paradigms.

4.5 Analysis of performance enhancement by the spatiotemporal feature fusion neural network model

The consistent performance superiority of the proposed spatiotemporal feature fusion neural network model, as demonstrated across all evaluation metrics and comparative scenarios, stems from its integrated architecture that systematically addresses the multifaceted nature of distribution network line loss prediction. The model begins with the mRMR feature selection mechanism, which operates as an intelligent filter that distills high-dimensional monitoring data into a compact yet highly informative feature subset.

This preprocessing stage not only reduces computational burden but, more importantly, enhances the signal-to-noise ratio for downstream learning by eliminating redundant and irrelevant variables. The effectiveness of the mRMR module is quantitatively demonstrated by comparing the full model against a variant without mRMR selection. The comparison shows that the mRMR module improves the MAE by approximately 7.5% while reducing the input dimensionality by approximately 60% on the IEEE 33-node system and 65% on the IEEE 123-node system, confirming its critical role in improving both accuracy and generalization.

The refined features then flow into the CNN component, which is specifically designed to exploit the inherent spatial structure of distribution network measurements. Through its convolutional operations, the model captures localized patterns—such as voltage gradients across adjacent nodes and spatial correlations between distributed generation outputs and local loads—that traditional sequence-based models fail to recognize explicitly.

These spatially enriched representations are subsequently processed by the LSTM layers, which excel at modeling temporal dynamics by learning long-term dependencies and remembering critical patterns from historical sequences. This temporal modeling capability is essential for tracking the diurnal load cycles, intermittent renewable generation profiles, and slower network reconfigurations that characterize modern distribution systems.

Finally, the entire architecture is underpinned by Bayesian optimization, which automates the search for optimal hyperparameters—from learning rates and dropout probabilities to convolutional filter sizes and LSTM cell dimensions. This automated tuning ensures that the model operates at its peak configuration across diverse operating conditions and topological variations, thereby achieving both high accuracy and exceptional robustness.

The synergy of these components creates a holistic prediction system where feature selection enhances data quality, spatial and temporal modules provide complementary pattern recognition, and optimization ensures model stability—collectively explaining the framework’s significant improvements over conventional and even advanced standalone methods.

4.6 Scalability and robustness validation on a IEEE-123-node system

To rigorously validate the scalability and robustness of the proposed spatiotemporal feature fusion neural network framework for practical large-scale applications, an extended case study is conducted on a more complex IEEE-123-node radial distribution test feeder. This network, comprising 123 buses and 122 branches with a total load of 3.48 MW and 2.32 Mvar, represents a realistic urban or suburban distribution scale. To emulate modern grid challenges, eight distributed generators—including four photovoltaic units and four wind turbines—are integrated at strategic nodes, resulting in a peak DG penetration exceeding 40%. The network also features a more meshed topology with several normally open tie-lines, allowing consideration of three distinct topological configurations during data generation.

A comprehensive dataset is constructed, following the methodology above but scaled appropriately. It spans 2 weeks at a 15-min resolution, incorporating diversified time-varying load profiles across residential, commercial, and small industrial customer types, as well as stochastic DG output curves based on realistic solar irradiance and wind speed profiles to introduce higher intermittency. The mRMR feature selection process reduces the input dimension by approximately 65%, effectively high-lighting key state variables such as voltage magnitudes at critical junctions and power flows through tie-lines.

The overall performance of different models on this IEEE-123-node system is summarized in Table 4. As anticipated, absolute prediction errors for all models increase compared to the 33-node system due to the heightened system complexity and dimensionality. However, the proposed spatiotemporal feature fusion neural network framework maintains its superior performance, achieving the lowest MAE and RMSE.

TABLE 4. Performance metrics of different models on the ieee-123-node system.

Table 4 illustrates how model performance scales with system complexity by comparing the MAE on the IEEE-33-node and the IEEE-123-node systems. While all models exhibit increased error on the larger system, the spatiotemporal feature fusion neural network model shows the most graceful degradation, with the smallest relative increase in MAE. This validates the framework’s excellent scalability, attributable to its ability to extract es-sential spatiotemporal features while filtering redundant information.

To specifically test the model’s robustness under conditions indicative of high renewable integration, we analyze its performance across three representative and challenging operational scenarios. These include the ‘High-Load Night’ scenario with minimal DG contribution, the ‘Baseline’ scenario with moderate and balanced conditions, as well as the ‘High-PV Midday’ scenario which simulates the critical challenge of high solar generation during low-load periods. The model is trained on a composite dataset encompassing these scenarios. Figure 6 presents the distribution of instantaneous prediction errors for each scenario.

A key observation from Figure 6 is that the median prediction error remains consistently close to zero across all scenarios. More importantly, the interquartile range and the overall spread of the individual errors do not exhibit systematic expansion under the ‘High-PV Midday’ scenario compared to the others. This indicates that the spatiotemporal feature fusion neural network model’s accuracy does not degrade under the high intermittency and potential reverse power flow conditions induced by substantial PV integration. Its stable performance across these distinct regimes underscores the framework’s strong generalization capability and its suitability for future distribution networks with high shares of variable renewable generation.

To gain deeper insights into the performance characteristics of different models, particularly their ability to handle the spatial complexities introduced by high DG penetration, a spatial analysis of prediction errors was conducted on the IEEE-123-node system. The analysis focuses on how prediction accuracy varies with electrical distance from the nearest DG unit, where electrical distance is defined as the shortest path length in the network graph. A comprehensive dataset comprising 7,200 samples was analyzed to ensure statistical reliability.

Figure 7 presents the spatial distribution of prediction errors for four representative models. The shaded region highlights the DG electrical neighborhood where renewable intermittency has the most pronounced impact. All models exhibit decreasing prediction errors with increasing electrical distance from DG units, confirming the localized impact of renewable intermittency on grid state estimation. This pronounced spatial heterogeneity indicates a fundamental limitation of purely temporal models in capturing the spatial correlations induced by distributed generation. Moreover, the architectural enhancements progressively mitigate this spatial dependence. The CNN-LSTM hybrid reduces the average error in the DG vicinity to 2.15 kW and decreases the spatial sensitivity from −0.085kW/hop to −0.062kW/hop. The BO-CNN-LSTM variant further improves performance, achieving 1.68 kW in the DG vicinity with a spatial gradient of −0.048kW/hop.

The spatial analysis further reveals that the impact of distributed generation on prediction uncertainty is highly localized, with approximately 85% of the performance degradation concentrated within a 2-hop electrical neighborhood of DG units. This finding offers important guidance for practical applications. Monitoring and control efforts can be strategically focused on these critical regions where renewable intermittency induces the most significant voltage fluctuations and reverse powerflow patterns. Beyond the 2-hop boundary, the influence of DG on line-loss predictability diminishes rapidly and approaches background levels at distances greater than 10 hops.

5 Conclusion

Traditional theoretical line loss prediction methods struggle to fully leverage the growing volume of monitoring data in distribution networks and often exhibit limited accuracy when applied to systems with high renewable penetration and dynamic topologies. To address these challenges, this paper proposes a data-driven line loss pre-diction model based on an mRMR-BO-CNN-LSTM spatiotemporal feature fusion neural network framework. The model employs mRMR for feature selection to reduce redundancy and retain the most relevant inputs, uses a CNN-LSTM hybrid network to capture both spatial and temporal dependencies in grid data, and applies Bayesian optimization to automatically tune hyperparameters. Validation on the IEEE-33 node distribution system demonstrates that the proposed model significantly outperforms conventional machine learning models and baseline neural networks, achieving a mean absolute error of 0.0655 kW and showing strong robustness across varying operating conditions and network topologies. The model’s exceptional scalability and practical potential are further confirmed by its consistent performance on a large-scale IEEE-123-node system with high renewable integration. The results confirm that the integrated approach offers an accurate, reliable, and adaptable solution for line loss assessment, supporting enhanced efficiency and smarter management in modern distribution networks.

Generative AI statement

The author(s) declared that generative AI was not used in the creation of this manuscript.

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Footnotes

Edited by: Jun Liu, Xi’an Jiaotong University, China

Reviewed by: Xiaoming Liu, Dalian University of Technology, China

Jiacheng Liu, Xi’an Jiaotong University, China