Current line parameter identification technologies face several challenges: 1) The low investment and high construction costs of smart devices impede the deployment of PMUs at each bus node, making real-time monitoring of voltage phase angle information difficult. 2) The integration of distributed photovoltaic systems on the user side causes reverse power flow and significant voltage fluctuations, making it difficult to maintain accuracy and robustness in the task of distribution line parameter identification.

The objective of DLPI is to find the mapping between node characteristics and line parameters and to identify the line resistance and reactance of each branch. Given the features of power grid branch, it is achieved using the active power, reactive power, and voltage amplitude provided by measuring equipment in medium-voltage distribution networks. The powerful learning ability of neural networks can be utilized to build a power flow model, mine constraints, and learn historical data to train the model. The power flow calculation using polar form of the nodal power Equation 1 is: P i = P G i − P L i = ∑ j = 1 n V i V j G i j ⁡ cos ⁡ θ i j + B i j ⁡ sin ⁡ θ i j Q i = Q G i − Q L i = ∑ j = 1 n V i V j G i j ⁡ sin ⁡ θ i j − B i j ⁡ cos ⁡ θ i j Z i j = R i j + j X i j = G i j G i j 2 + B i j 2 − j B i j G i j 2 + B i j 2 (1) where P i and Q i are the active and reactive power injected into node i, respectively. P G i and Q G i are the active and reactive power from the power source at node i; P L i and Q L i are the active and reactive power consumed by the load at node i; G i j and B i j are the conductance parameters consisting of g and b between nodes i and j, respectively. V i and V j are the voltage amplitudes of nodes i and j, respectively; θ i j is the difference in the phase angle of the voltage between nodes i and j. In the modeling stage, we use power flow equations expressed in polar coordinates to accurately represent the system’s behavior. However, during the experimental phase, the results are provided in terms of impedance parameters (resistance R and reactance X) as the original IEEE test case data is given in these terms. To facilitate direct comparison with the IEEE standard data, the node admittance matrix is converted into corresponding impedance values.

Meanwhile, parameter identification can be considered as a multi-task regression problem. Considering that the phase angle difference between the two ends of each branch of the distribution network is tiny, we assume that the phase angle difference of adjacent nodes i and node j is 0 for easy analysis. Thus, the linear voltage drop equation for line k is: V i − V j = R k P i j V i + X k Q i j V i (2) where R k and X k are the line resistance and reactance. Equation 2 represents the node connection relationship and line parameters, describing the relationship between node voltage and power. When constructing the system, lines are numbered according to the order of the end nodes of the line. For line k, the input characteristics of the distribution network can be expressed as X k = P i k , P j k , Q i k , P j k , V i k , V i k ∈ R k 6 , allowing the mapping of R and X to be determined from the input X k .

In traditional line parameter identification, the problem can be generalized as a linear regression problem or a quadratic programming problem. However, due to the fitting properties of linear regression, outliers can have significant effects on the regression, resulting in poor robustness. As the distribution network often encounters noise interference, data missing, or other situations due to the complicated operational conditions and numerous measurement devices, the performance of the linear regression model will deteriorate. Therefore, we select the deep learning method to extract the features of nodes on the premise of obtaining reconstituted measurement data samples. Deep learning techniques offer robust feature extraction capabilities, making them well-suited to handle the complexities and noise inherent in medium-voltage distribution networks.

Using network topology as a graph to analyze features allows for a more comprehensive utilization of structural information compared to solely relying on measurement data. Graph data G V , L , consisting of a vertex set V and an edge set L , being non-Euclidean structured, results in better classification accuracy.

In constructing the general distribution network diagram model, the bus is typically regarded as the node and the connecting line as the edge. However, graph learning focuses on node features. Therefore, for parameter identification tasks, we consider parameters as nodes of the graph and common buses between lines as edges to represent the connection relationships between lines. The feature extraction process is illustrated in the following Figure 1.

For undirected graph G V , L , A , V is the set of n vertices, V i ∈ V , L is the set of edges in the graph, and E i j ∈ E . A ∈ R n × n is the adjacency matrix, representing the topology among the nodes. Take X as the input, U is the eigenvector matrix of the normalized Laplace Matrix of the graph, g θ is the response function of the eigenvalue. Original standard of GCN is defined as: X * g θ = U g θ U T X (3)

Use Chebyshev polynomials g θ Λ = ∑ k = 0 K θ k T k Λ to approximate and substitute it into Equation 3 to obtain Equation 4, in which L ∼ = 2 λ max L − I N : X * g θ = ∑ k = 0 K θ k T k L ∼ X (4)

Then the convolution process is approximately defined as Equation 5 by using first-order Chebyshev polynomial to generate the local convolution kernel: X * g θ = θ I N + D − 1 / 2 A D − 1 / 2 X = θ D ∼ − 1 / 2 A ∼ D ∼ − 1 / 2 X (5)

In this formula, A ∼ = A + I N , D i i = ∑ j A ∼ i j . The spectral theory is applied and the output of each layer can be written as Equation 6: X k l + 1 = f X k l , A = σ D ^ − 1 / 2 A D ^ − 1 / 2 X k l W k l (6) where X k l + 1 is the mapping of each layer, l ∈ 0 , … , L − 1 , X k 0 is the input feature X k . X k l represents the feature matrix of the lth layer of the model. σ ∙ is the activation function, W represents the weight matrix in neural network. Since the model’s input is the graph structure data X, including the adjacency matrix and corresponding attributes, the graph construction process must be completed before model training to represent the data itself and uncover the association relationships between the data.

Graph Attention Network (GAT) combines a graph neural network (GNN) with an attention mechanism, specifically tailored for processing graph-structured data by assigning different attention to neighboring nodes on a graph (Velickovic et al., 2017). It can reduce the computational cost and make it more scalable than methods that consider all neighbors equally, such as traditional Graph Convolutional Networks (GCNs). This flexibility is particularly important for large, sparse graphs. In this paper, GAT module is applied to transform the feature of each node into an inter-node attention coefficient through the graph attention layer, producing a new feature that allows for monitoring changes in neighboring nodes. Thus, information about each branch line and the distribution of impedance values between adjacent branches can be learned, improving the accuracy of parameter identification. We input X k 0 and the adjacency matrix into the graph convolution layer to learn the node features and structure.

First, the voltage amplitude of nodes under a single time section is input into the graph attention network to calculate the similarity between each node and its neighbors in the distribution network. For each node i, calculate the corresponding coefficient between node j and itself: e ij k = a W k h i W k h j , j ∈ N i (7) where ⋅ ⋅ is the concatenation operation. W k is the learnable weight matrix for k t h layer of the attention mechanism, finally a ∙ is used to map the concatenated high-dimensional features to a real number. N i indicates the set of nodes adjacent to node i, h i and h j represent the feature value for node i and j respectively.

After obtaining the correlation coefficient for all the neighboring nodes of node i, the attention coefficient is normalized using softmax: α i j k = s o f t m a x e i j exp LeakyReLU e i j k ∑ j ∈ N i exp LeakyReLU e i j k (8) LeakyReLU x = x , x > 0 β x , x ≤ 0 (9)

Where α i j k is the attention coefficient between the k head attention mechanism node i and the adjacent node j. According to Equations 7, 8, new node features are formed by aggregating information using the attention coefficient matrix a.

After the attention weights of all nodes are normalized, the information of nodes is extracted through the graph attention layer. For different features, different attention weights need to be assigned. If only single-layer attention is used, the same attention weights are applied to all attributes of the neighbourhood node, which will weaken the learning ability of the model. The specific calculations in each layer are shown in Equations 9–11.

In each attention layer, we use this to weigh the messages of a node’s neighbours, which are the neighbour’s features multiplied by the same learnable weight matrix W. We do this for each attention head and concatenate the result of the heads together: h i ′ = ‖ k = 1 K σ ∑ j ∈ N i α i j k W k h j (10) σ x = 1 1 + e − x (11)

K is the attention head number and we choose Sigmoid as the activation function. The feature h_i ^' h i ′ , calculated by the multi-head attention mechanism, incorporates the contribution of the features of neighboring node j to node i, therefore having a stronger ability to express features. The feature information is then input into the multi-task module after being learned by the GAT module to identify branch line parameters. Identification tasks for different branch parameters are input into separate expert networks, with each expert responsible for a specific subspace. Moreover, GAT possesses topological extrapolation capabilities. If the topology of the station area changes due to maintenance or other reasons, adaptive anomaly identification can be performed by inputting the new adjacency matrix after training the original adjacency matrix into GAT.

In the parameter identification work of distribution networks, multiple branches typically require parameter identification, with each branch having multiple target parameters to identify. The magnitudes of branch resistance and reactance are generally quite different, but their characteristics depend on the same factors. Moreover, branch data are strongly interrelated, and variations in the electrical variables of one transmission branch often affect the measurement data of the entire distribution network. Multi-task learning leverages the correlations between multiple tasks to optimize the performance of multi-parameter identification. In this paper, a multi-task strategy is employed to identify multiple targets simultaneously, thereby reducing computational effort. According to previous works, different branches are identified as independent tasks.

The overall framework of the proposed GAT-MMoE for distribution line parameter identification is depicted in Figure 3 As shown in Figure 3, a multi-task learning model based on an attention graph is constructed. The input of GAT-MMoE we proposed is feature matrix X and the adjacency matrix A, which means that our input features contain n nodes, each node containing six features. The final output of the whole model can be expressed as Equation 20: Y ∼ k = M T k X j (20) where M T k represents the mapping function of the k-th branch. X j is the j-th expert output.

To overcome the DLPI problems, the system graph is established according to the description in Section 2, with each node corresponding to a physical bus in the distribution network. The input feature of the distribution system is expressed as P i , P j , Q i , Q j , V i , V j ∈ X k . The number of nodes is decided by the scale of distribution network, which means that our input features contain n nodes, each node contains the above six features.

The GAT module extracts the characteristics of the system and pays attention to different branch information in different subspaces by using multi-head attention mechanism. Then the different subspaces are concatenated to infuse the learnt information. The features information is then input into the multi-task module for training. The multi-task module is an MMoE-backboned MTL module, which contains multiple expert subnetworks. Gate control units are used to calculate the loss for different tasks during the training process and update the parameters related to each task based on the loss. The model’s outputs are the estimated branch line impedance.

The power flow formula of power system is as follows: P i = V i ∑ j = 1 n V j G ij ⁡ cos ⁡ θ ij + B ij ⁡ sin ⁡ θ ij Q i = V i ∑ j = 1 n V j G ij ⁡ sin ⁡ θ ij ‐ B ij ⁡ cos ⁡ θ ij (21)

According to Equation 21, Node voltage, active power and reactive power data are simulated by pandapower toolkit (Thurner et al., 2018). The active power injected by nodes in the load data is sampled by the Latin hypercube sampling method at 0.8 P s , 1.2 P s , where Ps represents the standard active power of the distribution network. Reactive power data set is generated by active power data set and power factor. The power factor cos ⁡ φ satisfies the uniform distribution of parameters 0.85 , 0.95 . Reactive power is calculated by power factor through Equation 22. The formula for calculating reactive power injected by nodes is as follows: Q i t = P i t tan arccos ⁡ φ t (22) where φ t ∈ 0.85 , 0.95 , P i t and Q i t are the active power and the reactive power injected into node i at time t. φ t represents the power factor of the system at time t.

In this paper, 20 types of radiative network topologies are selected, then we add Gaussian noise with σ of 0.01, 0.03, 0.5, 1, 2 to each load level, and sample each noise 6 times. A total of 69,120 sets of samples are obtained and divided, including 70% data as training set, 20% data as test set, 10% data as verification set. The hyperparameters of the model, such as the attention coefficient α, and learning rates are determined by 10% of the data set.

A suite of metrics is employed to manifest the performance of the model proposed in this work. Specific calculation formula as shown in Equations 23, 24:

Root mean square error (RMSE): E R M S E = 1 K ∑ i = 1 K y i − y ^ i 2 (23)

Mean absolute percentage error (MAPE): E M A P E = 1 K ∑ i = 1 K y i − y ^ i y i × 100 % (24)

Where y i and y ^ i respectively represent the true value and the predicted value, K represents the sampling number.

In order to prove the validity of our proposed model, we adopt the following methods as baselines:

1) LR: By minimizing the sum of squares of errors, the linear regression model is used to provide coefficients that quantify the contribution of each feature to the target variable, but may face the problem of overfitting.

In our implementation, the GAT module utilizes three attention heads, with a hidden representation dimensionality set to 128 and a dropout rate of 0.3. For optimization, we use a batch size of 128 and a learning rate of 0.005. The MMoE module is trained with the Adam optimizer and the learning rate is grid searched from [0.0001, 0.001, 0.01]. To prevent overfitting in both the expert and gating networks, the dropout rate is 0.2. Different weights of each task are assigned using homoscedastic uncertainty.

According to Figure 6, We can find that the gradient of our proposed model decreases rapidly and converges fast after around the 20th epoch with little change in accuracy, which demonstrates the superiority of our model in deep learning-based algorithms.

From Figure 7, it can be noticed that the proposed GAT-MMoE model can effectively identify the branch line parameters. For case 1, the max relative errors lie in branch 7 and 1 respectively for R and X, reaching 3.83% and 4.35%. Table 1 shows the parameter identification errors of branch resistance and reactance, the corresponding average errors are 3.84% and 2.67% for R and X. For case 2, the relative max errors are 9.63% and 9.87%. From Table 2, average errors for R and X are 6.69% and 7.24%. The GAT-MMoE model can achieve the lowest error in most cases. The deviation comes from the Gaussian noise we add and as the resistance and reactance have different orders of magnitude, the errors are within the allowable range.

Compared with the baseline model, our proposed GAT-MMoE model demonstrates higher accuracy and better robustness considering measurement error. The results in Tables 1–4 indicate that the proposed GAT-MMoE method outperforms all other models in terms of RMSE and MAPE. Network-based methods generally show superior performance compared to linear regression and traditional machine learning methods, underscoring the value of graph attention learning in extracting high-quality features for DLP prediction. The superior performance of GAT-MMoE can be attributed to its effective utilization of related knowledge between neighboring nodes in the graph. Most baseline methods do not specifically address the issue of sparsity in their models, resulting in suboptimal performance. Our model leverages multiple data sources to construct the information network and employs a multi-task learning framework to address the specific task of predicting branch line parameters. Consequently, GAT-MMoE outperforms the selected baselines. Additionally, the model demonstrates good robustness when photovoltaic power supply is integrated into the system. Once the model training is completed, it can simultaneously predict all parameters of the power grid branch. Despite the lengthy training process, the method’s robustness and accuracy compensate for this drawback. Moreover, the trained neural network model can be easily and rapidly deployed to the required locations, making it a practical solution for real-world applications.

From Tables 3, 4, different we can find that our proposed GAT-MMoE model achieves the best identification results in most cases. The indexes of multi-task learning model are better than that of single task learning model. By comparing the machine learning methods, we can find that the LR method achieves very good identification results without noise, but when the input features contain disturbance and noise, the accuracy of branch parameter identification is greatly reduced. Deep learning method like FCN fail to balance accuracy between resistance and reactance, as the loss function is only simple addition. Also, the method is more inclined to the identification result of line resistance R and ignores the branch reactance X.

The influence of distributed photovoltaic access on the proposed identification method is further explored. We incorporated multiple distributed photovoltaic (PV) systems into the distribution network, with the power data of the PV sources derived from the Desert Knowledge Australia Solar Centre (DKA Solar Center, 2024) We integrated PV1 and PV2 at nodes 7 and 12 in the IEEE 14-bus system, and at nodes 22 and 33 in the IEEE 33-bus system. Based on the sampling frequency of every 15 min, the system node data is obtained by power flow calculation. After collecting system P, Q, and V data over multiple time profiles, we input them into the model for parameter identification. Figure 8 shows that the identification error increases when a distributed power supply is present in the network. This increase is due to the changes in power flow direction caused by the integration of distributed photovoltaics. However, the identification errors remain within the acceptable range, demonstrating the robustness of the method.

For most existing distribution power system, the complexities and dynamic conditions present unique challenges and opportunities for parameter identification. Given that most existing power grid branch parameter identification methods are model-driven, resulting in low accuracy and poor reliability, our proposed model leverages a large volume of multi-source power grid operation data. It is constrained by the grid topology while integrating both local and global information. This approach allows for the comprehensive use of historical data to more accurately identify branch parameters, which can be then fed back to the power grid dispatch center. As a result, dispatch operators gain a clearer understanding of the changing trends in branch parameters, ensuring the safe and stable operation of the power grid.