The Graph Neural Network (GNN) [25] is a deep neural network model that processes structured data. Unlike traditional neural networks, GNN considers the relationships between nodes, where the core is the mutual aggregation of features between each node, then its surrounding nodes can form new node features to complete information transmission. After iteration, the GNN can obtain all the structural information of the graph. Graph Attention Network (GAT) is an important branch of GNN that considers the topological relationships between nodes in the graph and applies attention to updating feature vectors.

1)

Topological structure of a power grid

Our model adapts topological structure same as a power grid, A graph is a representation of entities through nodes, edges, and their connections. It is comprised of three elements, namely, nodes, edges, and global information. The global information is based on nodes and edges according to a certain topological structure. Power grids are an entity consisting of transmission lines between user nodes, so their composition may correspond entirely to the graph structure. Therefore, we set the same number of nodes in the model as the power grid, then set the edges between nodes according to the distribution of power lines in the power grid. In this way, nodes can aggregate the information of neighboring nodes to fit power flows in the power grid. The carbon emission factors of the power grid are essentially the power flow of nodes, which represents the information transmission between nodes. This makes GAT suitable for structures of power grids and enables it to properly reflect the transmission characteristics of power grids. 2)

Method of updating parameters

We use multilayer GAT to build the model, which can increase the receptive area of the network [26] so that the information of the marginal nodes can be aggregated to the central node. The attention mechanism specifically integrates the information of the neighboring nodes through the attention correlation coefficient between the feature vectors. The structure of the model is shown in Fig. 1 and the specific propagation mode between the layers is as Eqs. (8)–(12) [19].

(8)H(l+1)=(h1(l+1),h2(l+1),h3(l+1)⋯hn(l+1)) (9)hi(l+1)=σ(∑j∈NiαijW(l)hj(l)) (10)αij=exp⁡(LeakyReLu(aT[Whi||Whj])∑c∈Niexp⁡(LeakyReLu(aT[Whi||Whc]) (11)hi(l+1)=σ(1E∑e=1E∑j∈NiαijWe(l)hj(l)) (12)H(m+1)=σ(W(m)H(m)+B(m))

where H(l+1) is the output feature matrix of layer l and H(l) is the input feature matrix of layer l, n is the total number of nodes in the power grid; GAT update parameters are calculated point-by-point, and Eq. (9) shows the process of this calculation, σ is the non-linear activation function with optional ReLU, softmax, etc. Ni is the set of nodes connected to the node i, αij is the attention correlation coefficient, which reflects the importance of each other by measuring the correlation of the feature vectors of two nodes, that will determine the weighted value of different connections when the node features are aggregated. W is a learnable weight matrix whose function is to convert the dimension of the input feature vector into the output dimension; αij can be calculated by Eq. (10), aT is a weight vector and it can transform the two feature vectors into a one-dimensional correlation coefficient by multiplying with the splice vector Whi||Whj, then we can introduce LeakyReLu for nonlinear activation. To compare the correlation coefficients, different correlation coefficients connected to node i are regularized by the softmax; Eq. (11) introduces a multi-head mechanism, which adds E-1 channel relative to Eq. (9), and the average value of all channels is taken as the final output result.

After the last graph attention layer, the full connection layer is connected to organize the output characteristics and adjust the output dimensions for the prediction task. Eq. (12) is the mode of propagation between the fully connected layers, H₁ is the output of the input layer of the fully connected layer and the last layer of GAT. W(m) is the weight vector of layer m, B(m) is the bias vector of layer m, the former is a non-shared parameter while the latter is a shared parameter. 3)

Evaluating indicator

The backpropagation algorithm is a common and effective parameter updating method based on supervised loss. Because the predicted carbon emission factors of the power grid are small, the error value is generally less than 1, which makes the mean square error function (MSE) slow in this range. Therefore, the mean absolute error (MAE) can be trained as a loss function, and MAPE is an indicator to evaluate the effect of the model as follows: (13)MAE=1n∑i=1n|y~i−yi|

(14)MAPE=100%n∑i=1n|y^i−yiyi|

where y~i and yi indicate the predicted and true values of the carbon factors at node i, respectively, n is the number of samples of the data set.

In this paper, the gradient descent method of the small batch is used to train the prediction model. For the convenience of training, setting the batch size of samples, and then the corresponding network parameters updated according to the gradient descent direction of a batch can avoid the fluctuation of the descent direction.

The generator typesetting of the IEEE-39 bus system is shown in Table 1 and the network training parameters are shown in Table 2.

The effectiveness of GAT in predicting the node carbon emission factors is tested using the IEEE-39 bus system. Due to the excessive number of nodes in the system, typical nodes are selected based on their representativeness. The degree (degree represents the number of neighboring nodes) of each node in the IEEE-39 bus system is shown in Fig. A1 of Appendix A. Node 16 has the highest degree, indicating that it is the central node, this kind of node is a hub node in the actual power grid, and its power flow is easily influenced by the surrounding nodes, so its carbon emission factors fluctuate greatly. The node 39 is connected to the generator and has a large load, which makes it a complex node. This kind of node is close to the generator in the actual power grid, when its load is greater than the input power of the connected generator, the carbon emission factors will also be affected by the carbon emission factors of other nodes, otherwise, the carbon emission factors of this node will be consistent with the power generation carbon emissions intensity of the generator. Additionally, there is a typical class of nodes (such as node 10), whose carbon emission factors do not change with the load. Therefore, they are selected as typical nodes for analysis.

Fig. 3 presents a comparison between the predicted values and the true values (the top 50 samples of the test set) in the dataset. The prediction error rates of nodes 39 and 10 are overall at a low level, but that of node 16 is slightly larger. Specifically, the MAPE of the three typical nodes is 4.63%, 1.20%, and 0.06%, respectively.

To illustrate the prediction results more intuitively, a 3D scatter plot is drawn in Fig. 4, where blue and red scatter points are test samples and predicted carbon emission factors values, respectively. Due to a large number of nodes, only some nodes are displayed. Fig. 5 is a violin figure that presents the distribution of the prediction error rates for all the nodes, The wider width of the x-axis direction represents the greater corresponding error rate, and four lines in the graph represent the maximum, mean, median, and minimum. It shows that most node’s prediction error values are below 10%, the maximum error rate is 17.5% and the relative average error between them is only 2.46%. In conclusion, This model achieves an average node prediction error rate of less than 5% and a maximum node prediction error rate of less than 20% under the dataset of this paper.

The number of generator nodes and complexity of nodes in the IEEE-118 bus system have been greatly improved compared with the 39 system. Types and settings of generators in the system are given in Table A1 of Appendix A. The network training parameters are shown in Table 3, and the training parameters of the contrast algorithm are shown in parentheses.

To objectively evaluate the effectiveness of the proposed method, Table 4 statistics the MAPE of MLP, GCN [30,31], and GAT with the same dataset. We can see that using the GAT model yielded a minimum MAPE of 2%, The largest is the MLP model whose value reaches 4.63%.

Similar to the IEEE-39 bus system, the degrees of each node in the IEEE-118 bus system are shown in Fig. A2 of Appendix A. The selected typical nodes are 59 (complex node), 49 (central node), and 62 (carbon emission factors constant node). MAPE of the three typical nodes is 0.55%, 0.20%, and 1.82%, respectively, The prediction results of each typical node are similar to those in the IEEE-39 bus system, as shown in Fig. 6, the scatter plots of some nodes are shown in Fig. 7.

Fig. 8 shows that the prediction error values of most nodes are below 10%, the maximum error rate is 30% and the relative average error between them is only 2.51%, which indicates that the GAT is still effective in large-scale node systems.