The portfolio optimization of power grid infrastructure projects does not only focus on one single project, but comprehensively considers the regional grid as a whole. Some of the projects may play a crucial role in the safety and reliability of the regional grid, and should be mandatorily selected, regardless of the comprehensive evaluation results. Such projects must be constructed and put into operation, and would certainly be of the highest priority. The mandatory projects cover three voltage levels of 500kV, 220kV for regional and provincial main grids and 110kV for distribution network. Furthermore, the project properties cover power supply delivery projects, electric railway supporting projects, UHV supporting projects and new energy collection stations and other power grid infrastructure projects.

The coexistence relation means that the two projects need to cooperate with each other to make sense, that is, both projects either going into production or not being selected at all. While building a new power transmission and transformation project, substations and transmission lines in the corresponding area will be constructed. In order to ensure the delivery of electric energy, it is necessary to construct supporting transmission projects corresponding to each voltage level. For example, a 220kV power transmission and transformation project and the 110kV transmission project of the 220kV substation are coexistent projects.

The interdependence relation refers to the fact that there is a sequential construction sequence between two projects in the aspects of time sequence or space for construction. One project must be arranged after another project is put into operation. On one hand, due to the large scale, technical difficulty and long construction period of power grid infrastructure projects, in order to avoid and reduce risks, the power supply delivery and transmission line projects of the regional and provincial main grids are implemented in two or three phases, so that there is an interdependence relation between the phased projects. On the other hand, multi-circuit lines are established for the newly-started and renovation transmission line projects of the regional and provincial main grids and part of the 110kV distribution network, which are spatially consistent. These projects are interdependent, working together to improve the security, stability and reliability of power grids.

The mutual exclusion relationship means that two projects are conflicting and cannot be selected simultaneously. Due to the huge number of power grid infrastructure projects, there may be risks that projects will be recorded repeatedly, the coverage regions will overlap, and projects with the same function may exist. In order to avoid unnecessary waste of resources caused by repeated construction, such projects should be selected on merit.

The input of the R-GCN-based identification method is defined as the original triples of the entity node feature vector of one project-the relation-the entity node feature vector of another project, which is essentially composed of limited power grid infrastructure projects and limited linkages among these power grid projects. Therefore, the input f t r i ( y ) can be summarized as the following expression: f t r i ( y ) = { V p r o , E l i n , X V , A V } (1) Where V p r o = { v i } ∈ R n denotes the set of massive power grid project entity nodes and n is the number of all power grid project entity nodes of V p r o . Correspondingly, E l i n is the set of defined linkages among massive power grid infrastructure projects. And X V = { x i , j } ∈ R n × d is the feature matrix of power grid project entity nodes of V p r o , with d being the sum of the number of existing project properties and the number of existing engineering attributes of massive power grid infrastructure projects (Wang et al., 2021). That is to say, x i , j denotes the value of the j-th attribute of the power grid project entity node i of V p r o . And A V is the adjacency matrix of V p r o , which represents the linkage between every two power grid project entity nodes. The definition of the adjacency matrix A V = { a i , j } ∈ R n × n is shown as follows: a i , j = { 1 ,     if project entity node   v i  is connected with project entity node  v j   0 ,     if project entity node   v i  is not connected with project entity node  v j    (2)

Based on the above definition, the input f t r i ( y ) can be converted into a spectral signal f ^ t r i ( y ) by Graph Fourier Transform, as shown in the formula 3. f ^ t r i ( y i ) = U V T f t r i ( y ) (3) Where f t r i ( y ) is the input of defined original power grid project triples and f ^ t r i ( y ) is corresponding spectral input. U V T is the transposed eigenvector matrix which originates from the eigen-decomposition of the normalized Laplacian matrix L V which corresponds to the adjacency matrix A V (Ngo et al., 2020).

Based on the graph convolution methodology and the Graph Fourier Transform, the graph convolution of the input of defined original power grid project triples can be realized in the standard orthogonal space of the spectral domain, as shown below: f t r i ^ ( y ) ∗ G g = F − 1 ( F ( f t r i ^ ( y ) ) ⊙ F ( g ) ) = U V ( U V T f t r i ^ ( y ) ⊙ U V T g ) (4) Where g is the graph convolution kernel, ∗ G is the graph convolution operator, and ⊙ denotes the Hadamard product.

After converting the Hadamard product into the matrix multiplication, the graph convolution of the input of original power grid project triples is changed into the following formulas: f t r i ^ ( y ) ∗ G g = U V g θ U V T f t r i ( y ) (5) U V T f t r i ( y ) = [ θ 1 , θ 2 , ... , θ n ] T (6) g = d i a g ( [ θ 1 , θ 2 , ... , θ n ] ) (7) Where θ 1 , θ 2 , ... , θ n are the parameters of the graph convolution kernel g .

Then the feature matrix of power grid project entity nodes output by the graph convolutional neural network at layer l + 1 can be obtained and represented by the following formula (Peng, 2020): H j l + 1 = τ ( U V ∑ i = 1 s g i , j l U V T H i l ) ,       j = 1,2 , ... , t (8) Where H i l ∈ R n is the value of the i-th input attribute of all the power grid project entity nodes output at layer l , s is the number of dimensions of input attributes of all the power grid project entity nodes at layer l + 1 , and similarly t is the number of dimensions of output attributes of all the power grid project entity nodes at layer l + 1 . g i , j l is the spectral graph convolution kernel and τ denotes the typical non-linear activation function.

Although the above graph convolutional neural network could be applied to form a multi-layer convolutional neural network, it is not reliable enough and eigen-decomposition is required in the above-mentioned calculation process and might cause the high complexity of calculation. In order to make up for the above shortcomings, the Chebyshev neural network is introduced to parameterize all the parameters to be learned of the graph convolution kernel g (Li et al., 2020). g = g ( Λ ) ≈ ∑ k = 0 K − 1 θ k T k ( Λ ^ ) (9) Λ ^ = 2 Λ λ max − I n (10) Where θ k denotes the coefficients of Chebyshev polynomial.

If the Chebyshev polynomial of the eigenvalue block-diagonal matrix is defined as the graph convolution kernel, the graph convolution of the input of original power grid project triples can be computed by the following formula: f t r i ( y ) ∗ G g = U V ( ∑ k = 0 K − 1 θ k T k ( L ^ ) ) U V T f t r i ( y ) = ∑ k = 0 K − 1 θ k T k ( U V L ^ U V T ) f t r i ( y ) = ∑ k = 0 K − 1 θ k T k ( L ˜ ) f t r i ( y ) (11)

Moreover, after the introduction of the Chebyshev neural network, the feature matrix of power grid project entity nodes output by the graph convolutional neural network at layer l + 1 is shown below: H j l + 1 = τ ( U V ∑ i = 1 s ∑ k = 0 K − 1 ( θ k T k ( L ^ ) ) U V T H i l ) = τ ( ∑ i = 1 s ∑ k = 0 K − 1 ( θ k T k ( L ˜ ) ) H i l ) ,     j = 1,2 , ... , t (12)

In order to further simplify the calculation process, the first-order approximation is also introduced to the above graph convolutional neural network. Fixing the maximum eigenvalue of the normalized Laplacian matrix L V as constant two (Li et al., 2020; Jalali et al., 2021), the formula 11 and formula 12 can be simplified as follows: f t r i ( y ) ∗ G g = θ 0 f t r i ( y ) − θ 1 D V − 1 / 2 A V D V − 1 / 2 f t r i ( y ) (13) H j l + 1 = τ ( ∑ i = 1 s ( θ 0 − θ 1 ( L V − I n ) ) H i l ) ,     j = 1,2 , ... , t (14)

To avoid the problem of overfitting, let θ = θ 0 = − θ 1 . And then the formula 13 and formula (14) can be further simplified, as shown below: f t r i ( y ) ∗ G g = θ ( I n + D V − 1 / 2 A V D V − 1 / 2 ) f t r i ( y ) (15) H j l + 1 = τ ( ∑ i = 1 s θ D ˜ V − 1 / 2 A ˜ V D ˜ V − 1 / 2 H i l ) ,     j = 1,2 , ... , t (16)

On the basis of the above simplified graph convolutional neural network, the relational graph convolutional neural network comprehensively considers the connection mode with neighbor power grid project entity nodes under different types of defined linkages and adds a special self-connection to each power grid project entity node so that the information about all the power grid project entity nodes at each layer can be effectively transmitted (Gusmao et al., 2021). Consequently, the feature matrix of power grid project entity nodes output by the relational graph convolutional neural network at layer l + 1 is defined as follows: H j l + 1 = σ ( ∑ r ∈ R ∑ m ∈ N i r 1 c i , r W r l H m l + W o l H i l ) ,     j = 1,2 , ... , t (17) Where σ is the activation function, W r l is the regularization weight matrix of the corresponding power grid project entity nodes, and W o l is their own weight matrix. r ∈ R represents the r-th linkage of the set of all defined linkages between related power grid project entity nodes and m ∈ N i r denotes the set of neighbor nodes of the specific power grid project entity node i at layer l + 1 under the specific linkage r. Specially, c i , r is a normalized constant that can either be learned or chosen in advance, here let c i , r = | N i r | .

However, when applying the formula 17 to the input of defined original power grid project triples which is essentially a multi-relational dataset, the number of parameters of the relational graph convolutional neural network will increase rapidly with the increase of the number of defined linkages among massive power grid infrastructure projects, which can easily lead to the problem of overfitting. To address this issue, we introduce one method—basis-decomposition—for regularizing the weights of R-GCN-layers (Schlichtkrull et al., 2017). With the basis-decomposition, each W r l in formula 17 is defined as follows: W r l = ∑ b = 1 B a r b l C b l (18) Where W r l is a linear combination of the basic transformations C b l with the coefficients a r b l which are only related to the corresponding linkage r between specific power grid infrastructure projects.