Wide-area metering topology modeling and fusion optimization of distributed power trending algorithms in dynamic power networks

Complex network is an effective tool to study the complex relationship, according to the topology composed of interconnections in the factors can be mapped to the complex network in reality, and through the study of the topology of the complex network to reveal the characteristics of the complex relationship.

The sequence of points and edges in a network consisting of two neighboring nodes and their connecting edges is called a chain, and if none of the nodes in the chain are the same, it is called a primitive chain. A graph is a connected graph when there is at least one primitive chain between any two points of the graph. Edges in a network describe the nature of interactions between nodes, and edges can be weighted or directed, and the corresponding networks are also called weighted or directed networks [21].

Complex network models include WS small-world network model and BA scale-free network model.WS small-world network is characterized by a short average path length with a large aggregation coefficient and an exponential degree distribution of nodes.BA scale-free network model is a complex network that obeys a power law distribution, and is constructed based on the two intrinsic mechanisms of growth and preferential connectivity. The algorithmic rules for the BA scale-free network model are as follows. The algorithmic rules are as follows:

First, in initial condition (t = 0), the network consists of m₀ nodes.

Second, at each time step t a new node is added which has m(m≤m0) edges. The newly added node connection takes a merit rule: the probability that the newly added node makes a connection with an existing node i in the network is: (1)Πi=ki∑j=1N0kj

where N₀ is the total number of nodes in the current network.

Finally, after t time, the algorithm generates a network with N = m₀ + t nodes and mt edges, with an average degree of 〈k〉 = 2m and a power-law distribution of degree.

Dependent networks, i.e., interdependent networks (NON), many complex systems in the real world cannot be described by a single complex network, they are more or less interconnected with networks in other domains, thus constituting a non-one-dimensional topology of interdependent complex systems, and the connecting edge relationship between their layer networks represents the interdependent and coupled relationship between different system nodes. Complex networks can be considered as a special case of interdependent networks when the number of network layers is 1. Observing from the breadth involved in the study, the breadth covered by interdependent networks is much larger. Not only the topological characteristics of sub-networks are analyzed, but also the interdependence between sub-networks is considered, which is closer to the large-scale complex systems in reality. The interdependent network mainly includes four models of different relationships, namely, the “one-to-one” relationship, the “one-to-many” relationship, and the “many-to-many” relationship and the “partial dependency” relationship.

A dependency network is composed of two or more complex systems with interdependencies between them, and for the whole dependency network, each complex system is equivalent to a sub-network. Sub-network α can be described by a simple graph i.e. Ga=(Vα,Eα) , where Vα={1,2,…,Nα} is the set of nodes in sub-network α and Eα={eij} is the set of connected edges inside the sub-network [22].

Then subnetwork α can be represented by the following mathematical relation: (2)Vα={v1,v2,…,vNa} (3)Eα={eij=(vi,vj);i≠j,vi,vj∈α}

where v_i, v_f and e_q denote the nodes and internal connecting edges of sub-network α, respectively, and N_α denotes the number of nodes contained in sub-network α.

The connection relationship between nodes in the network can be represented by the adjacency matrix A=(aig)N×N with: (4)aij={ 1 (vi,vj)∈E(α) 0 (vi,vj)∉E(α)i≠j

where E(α) denotes the set of connected edges in subnetwork α.

If there is a dependency between sub-network β and sub-network α, the inter-sub-network coupling edges can be expressed as: (5)ED={ Eα−β=(vp,vq);vp∈α,vq∈β;p∈Nα,q∈Nβ Eβ−α=(vq,vp);vq∈β,vp∈α;q∈Nβ,p∈Nα

Where E_α−β denotes the dependent edge matrix of network α dependent on network β, Eα−β(p,q)=1⇔vq∈β→vp∈α , denotes that the normal operation of node q in subnetwork α requires the support of node p in subnetwork β, otherwise E_α−β(p,q) = 0. Similarly E_β−α denotes the dependent edge matrix of subnetwork β dependent on subnetwork α.

Let the dependency network consists of S layers of sub-networks coupled to each other with g={Ga;α∈{1,…,S}} , then the whole dependency network can be represented by the set ζ=(g,ED) .

The power physics system mainly consists of generating stations, substations, and transmission lines of various voltage levels, which can be abstracted as a powerless and directionless complex network if the same-pole parallel-frame line is equated to a single line. Then the power physics model can be expressed as: (6){ GP〈EP,NP,g,NP,t,LP〉 NP,g={NP,g,1,⋯,NP,g,i,⋯NP,g,npt} NP,t={NP,t,1,⋯,NP,t,i,⋯NP,t,npt} LP={LP,(a,b),⋯,LP,(c,d),⋯LP,(i,j)}

Where N_P,g is the collection of power station nodes, N_P,t is the collection of substation nodes, L_P is the collection of power lines, E_P is the site (node) adjacency matrix, and n_pg and n_pt represent the number of power stations and substations, respectively [23].

Generation station nodes are equipped with high-frequency cutter devices, and participate in primary and secondary frequency regulation, some substation nodes are equipped with low-frequency load shedding devices, the two types of nodes have a frequency support function, once the node failure or control information failure will lead to a decline in the system’s frequency regulation capability. Therefore, the attribute model of power generation and substation nodes can be expressed by the multivariate group as: (7){ NP,g,i={PG,i,Pof,i,Pf1,i,Pf2,i} NP,t,i={PD,i,Puf,i,Pdr,i}

In the formula, P_G,i, P_of,i, P_f1,i, and P_f2,i represent the conventional active output, high-frequency cut-off capacity, primary FM capacity, and secondary FM capacity of the generating station, respectively; and P_D,i, P_dr,i, and P_uf,i characterize the conventional active load, demand-responsive capacity, and low-frequency load-shedding capacity of the substation, respectively.

For power lines, line reactance, line current and line capacity are used to describe their electrical properties. Then: (8)LP,(i,j)={xi,j,F(i,j)N,F(i,j)M,F0,(i,j)}

Where, x_i,j, F0,(i,j) , F(i,j)N , F(i,j)M characterize the line reactance, line conventional current, normal upper limit value and limit value, respectively.

In order to reflect the influence of frequency control capability on chain faults and the correlation relationship between control information failure and control capability, the system frequency control capability index is established. Namely: (9){ Ef,u=∑i=1npu(ciPf2,i+Pf1,i)+∑i=1npuPuf,i+∑i=1npuciPdr,i Ef,d=∑i=1npu(Pof,i+ciPf2,i+Pf1,i)

In the formula, E_f,u and E_f,d represent the system frequency up-regulation and down-regulation capability respectively; secondary FM P_f2,i and demand response FM P_dr,i are centralized frequency control, and the site control capability is not only related to its own control resources, but also constrained by the controllable state of the corresponding site c_i. If the disturbance exceeds the system frequency control capability, the dispatching center will be forced to remove the station to maintain system stability.

Dynamic power network is a power grid and information network coupled with each other, mutual influence, interdependence, mutual interaction, closely linked network. In order not to be affected by the interference of external information networks, power systems often have their own proprietary power communication network. The power communication network is often laid on the power grid, and the power grid has a similar topology. In this paper, based on the theory of complex networks and dependent networks, the power grid and information network are abstracted as an undirected graph G = (V, E), where V = {v₁, v₂, ..., v_n} represents the set of nodes in the network and E = {e₁, e₂, ..., e_m} represents the set of edges in the network.

The generator nodes, transmission nodes, transformer nodes, and load nodes in the power grid are abstracted as nodes of a single-layer network and the transmission lines of the power grid are abstracted as edges connecting the nodes. Similarly, control stations, switching stations, and equipment control stations in an information network are equated to nodes and fiber optic communication lines in an information network are equated to edges between nodes.

The adjacency matrix of graph theory is a two-dimensional matrix which is used to describe the connection relationship between network nodes in a network. Matrices A and B are used to define the connectivity relationships between the information network and the grid nodes, respectively, and the adjacency matrices A and B are: (10)A={aij}=( a11 ⋯ a1n ⋮ ⋱ ⋮ an1 ⋯ amn),1≤i,j≤n (11)B={bij}=( b11 … bm ⋮ ⋱ ⋮ bm1 ⋯ bmm),1≤i,j≤m

where matrices A and B are square matrices of order n and m, respectively, and n and m are the number of nodes in the information grid and the power grid. The adjacency matrix is a symmetric matrix with 0 elements on the diagonal and non-diagonal elements defined according to the following equation: (12)Aij={ 1,(Vi,Vj)∈E 0,(Vi,Vj)∉E (13)Bij={ 1,(Vi,Vj)∈E 0,(Vi,Vj)∉E

where (Vi,Vj) is the connecting edge between nodes V_i and V_j in the graph.

Combined with the four different relationships of the interdependent network proposed above, this paper carries out the Wide Area Metering Topology Model (DPG) in the dynamic power network based on the interdependent network model of the “partial dependence” relationship, and adopts the “degree-between” coupling mode, and its specific structure is shown in Figure 1. “Degree - median” coupling refers to the nodes in the information network and power grid are arranged according to the degree and median from large to small, and then according to the control requirements of the information network in the degree of the largest number of nodes designated as the central control nodes and distributed control nodes and these information nodes are eliminated, and the rest of the nodes are coupled one-to-one correspondence. The “degree-moderate” coupling method can have strong robustness and stability, which can make the dynamic power network maintain normal operation to a maximum extent after malicious attacks.

Figure 1.

The dynamic power network of the “partial dependency” relationship

There are two specific types of dependencies in the wide-area metering topology in dynamic power networks, one is the communication dependency of the grid nodes on the information nodes and the other is the power dependency of the information network nodes on the grid nodes. The set of system dependency edges is obtained on the basis of the “partial dependency” dependency network model. E_C−P is the dependency edge matrix of the information network dependent on the power grid, E_P−C is the dependency edge matrix of the power grid dependent on the information network, and the dependency edge matrix is defined according to the following equation: (14)EC−p(i,j)={ 1,Vp(j)→Vc(i) 0, Other (15)EP−c(i,j)={ 1,VC(j)→Vp(i) 0, Other

Where V_P(j) → V_C(i) denotes that node j in the grid provides power support to node i in the information network, and others are similar to this. The dependency matrix of the grid and the information network can be expressed as: (16)EC−P=[ 010 001 100] (17)EP−C=[ 001 100 010]

The power physical network is the core of the smart grid system, which undertakes the functions of power generation, transmission, and distribution, etc. Based on the critical impact of the stable supply of electricity on the overall operation of the society, the study of the operational risk for the power physical network has always been an important part of the grid-related research [24]. In the power physical network, the flow of power follows Kirchhoff’s law, and by obtaining the topology and physical characteristics of the power physical network (such as bus distribution, voltage level of the generating station, voltage phase difference between the generating stations, impedance of the power line, load of the terminal station, etc.), the current power distribution in the power physical network can be effectively estimated based on the tidal current equation.

The general form of the power system tidal equation for a N-node network is: (18)Si=Pi+jQi=UiIi*=Ui∑j=1NYij*Uj*

where the conductance Y_ij = G_ij + jB_ij of node i and node j, and substitution into the above equation expansion yields the active power injected into node i as: (19)Pi=|Ui|∑j=1N|Uj|(Gijcosθij+Bijsinθij)

Based on the active power P_i injected into node i, the line current P_ij can be derived as: (20)Pij=Ui2(Gi0+Gij)−UiUj(Bijsinθij+Gijcosθij)

Limited by the complexity of the AC current calculation model and the convergence uncertainty problem, an approximation process is introduced to simplify the nonlinear equations of the AC current model to the linear equations of the DC current model. 1)

Since the branch reactance B is much greater than the branch resistance G, depending on the branch resistance G ≈ 0, then: (21)G+jB=jB=1R+jX=1jX=−j1X

Therefore, the line conductance is Bij≈−1Xij , where X_ij is the line reactance. 2)

Since the phase difference of voltage between nodes is small, there is sin θ_ij ≈ θ_ij = θ_i − θ_j.

Combining the above DC current equation with the information of power parameters of the power physics network, the network line current distribution can be calculated. When the structure of the power physics network is changed, trend redistribution will occur.

The DPG model is used as a basis for analyzing the cascade failure of the dynamic power network in conjunction with the power trend calculation. When a cascade failure occurs in the DPG model, the control center at the information layer performs optimal scheduling of the power layer based on the collected grid operation data. The whole process of the information layer uses the DC Optimal Current Flow (DCOPF) scheduling algorithm to make the grid operate in the most economical way by adjusting the output of generators while the grid maintains the balance between supply and demand. This process can be transformed into the following linear programming problem with the objective function of minimizing the generation cost. Namely: (23)min∑i∈gfg,iPg,i (24)F=IP (25)∑ingPg,i=∑jndPd,j,i∈Gj∈M (26)Pg,imin≤Pg,i≤Pg,imax (27)|Fl|≤Flmax,l∈Π

Where equation (23) is the objective function, where f_g,i, P_g,i denote the cost coefficient and output power of generator i, respectively. Equation (24) is the DC power flow (DCPF) equation, where F is the branch power matrix, I is the branch conductance matrix, and is P the node power injection matrix. Equation (25) is the power balance equation of the grid, where P_g,i is the active power output of generator node i, and P_d,j represents the load demand of node j. Equation (26) is the upper and lower bound constraints on the active power output of the generator node. Equation (27) is the branch power limitation constraint, Flmax represents the power limitation constraint of each power branch l, and Π is the set of all branches in the power network. The power limiting constraints of power branches can be computed as: (28)P(i)=(1+α)L0

Where, α is the redundancy parameter and L₀ indicates the initial load of the branch. A higher value of redundancy α indicates a higher upper limit of power passing through the transmission line.

Fig. 2 shows the propagation process of a metering fault occurring in the DPG model with the following steps:

Figure 2.

The cascade failure process of DPG

Step1 Input the parameters of power and information networks, including system topology, load demand of each node, type and output power of generators, transmission lines, etc. Step2 Establish the DPG model, in which the “partial dependency” degree coupling mode is adopted between power nodes and information nodes.

Step2 Establish the DPG model, in which the power nodes and information nodes are coupled with each other in a “partial dependency” mode.

Step3 Set the power limit of each power branch, where two times of the power of each branch calculated by DCOPF at the initial time of the grid is taken as the upper limit of its power.

Step4 Select the initial fault node.

Step5 Update the topology of the DPG model and check whether the power grid is unlisted into multiple islands.

Step6 Check whether there is a generator node in each power island, if there is not a generator node on a power island, remove all components on the island and return to Step5, otherwise execute Step7.

Step7 Perform DCOPF computation on the power island and check whether the computation converges or not. If it converges, perform Step8. Otherwise, perform DCPF computation for the power island.

Step8 After all the power islands are computed, compute the final results of the dynamic power network wide-area metering topology model, including the system topology, the power output of generators, the total system load, and the load loss of each node.

During a cascading fault, when a transformer, transmission line or generator in the power system exceeds its designed capacity, the dispatch control center receives the operational data and issues an optimization command based on the optimal current flow (OPF), which restores the system to a stable operating state by adjusting the generator output and removing part of the overloaded load. The system’s ability to withstand disturbances and attacks can be enhanced by applying protection measures to highly critical nodes.

Due to the uneven distribution of tidal currents in the power network, nodes with the same number of meshes or degrees in the power network have different degrees of criticality in the overall system. By combining the topological centrality of the nodes and the active power injected by the nodes the power network node criticality index is proposed, viz: (29)KPower(i)=αPTPower*(i)+βPLPower*(i) (30)TPower*(i)=φPCPower,B*(i)+λPCPower,D*(i)

Where K_Power(i) denotes the criticality of the power network node, TPower∗(i) denotes the normalized topological centrality of the node, LPower∗(i) denotes the normalized active power injected by the node, α_P denotes the weight of topological centrality in the criticality K_Power(i). β_P denotes the weight of the active power injected by the node, φ_P denotes the weight of the centrality of the degree in TPower∗(i) , and λ_P denotes the weight of the centrality of the degree in TPower∗(i) .

In order to facilitate the analysis and comparison the node mediator centrality C_Power,B, node degree centrality C_Powar,D and the active power injected by the node LPower∗ are normalized, then: (31){ CPower,B*(i)=CPower,B(i)−CPower,BminCPower,Bmax−CPower,Bmin(1−X)+X CPower,D*(i)=CPower,D(i)−CPower,DminCPower,Dmax−CPower,Dmin(1−X)+X LPower*(i)=L(i)−LminLmax−Lmin(1−X)+X

Where C_PowerBmax, C_PowerDmax and L_max denote the maximum values of mediator centrality, degree centrality and active power injected into the node for all nodes, and C_PowerBmin, C_PowerDmin and L_min denote the minimum values of mediator centrality, degree centrality and active power injected into the node for all nodes. Since the difference in node importance should not be too large, X = 1 is taken so that the importance is within the interval of [0.1,1].

In an information network, data is generally transmitted along the shortest path, and the data stream is sent through the shortest link, which can substantially improve the utilization efficiency of the information branch. When the starting point as well as the end point of data transmission is determined, it can be transformed into a graph theoretic problem to find the shortest path through an algorithm, which is mathematically described as follows:

In a entitled undirected graph G = (V, E), noting that W_ij is the weight of an edge aij=(Vi,Vj) in graph G and P is a path from the start point S to the end point T in G, define the value W_P of the path P to be the sum of the weights of all of its edges, which is computed by the formula: (32)W(P)=∑Vi,Vj∈eWij

If W(P)^min is the smallest value in the path from S to T, then W(P)^min is the shortest path from S to T in Figure G.

Floyd’s algorithm is an algorithm to solve the shortest path between any two points, which can correctly deal with the shortest path problem in the graph, and its algorithm has a time complexity of O(n3) , which is a typical dynamic programming algorithm. Then: (33)D(i,j)=min{D0(i,k)+D0(k,j),D0(i,j)}

The effective distance solves the limitation that the intuitive spatial location distance cannot effectively characterize the closeness of the connection between the nodes, if the nodes are connected by edges between the node levy node j. The formula is: (34)dij=(1−lnWij)

where d_ij denotes the effective distance between node i and node j, and W_ij denotes the weight of the connecting edges of node i and node j, characterizing the amount of information transmitted in the information branch.

The calculation of the DPG model vulnerability can be done using the ICS formula in the IEC 61508 standard, where vulnerability R is equal to the product of the loss of value V caused by the attack event and the probability of the risk occurring P, ie: (35)R=V×P

This paper quantitatively evaluates the vulnerability of DPG models from the attacker’s perspective. Considering the cascading fault vulnerability of the DPG model, this paper constructs the Load Proportional Reduction Probability (LPRP) as a vulnerability evaluation metric for the DPG model by optimizing the traditional Electricity Shortage Expectation (ENDS) metric. The index can be expressed as: (36)LPRP(j)=P(i)×C(j)C

Where LPRP(j) represents the vulnerability value of the grid node, the larger the value, the higher the vulnerability degree of the grid node, P(i) is the risk probability of the distribution electronic station node i in the information layer, C(j) is the total amount of load cut by the relay protection mechanism in the case of generator failure of the jth node in the physical layer, and C is the total active load.

Combined with the indicator decomposition method, this paper decomposes the LPRP to obtain three sub-indicators, i.e., the optimal trend load shedding percentage probability, the voltage collapse load shedding percentage probability and the islanding removal load shedding percentage probability. Then: (37){ Pvels=P(i)×CVCLSC Pti=P(i)×CPIC Popt=P(i)×COPFC

where C_VCLS, C_PI, C_OPF is the voltage collapse load shedding, islanding removal load shedding, and optimal current load shedding, respectively.

By using the above three decomposition sub-indicators of LPRP, it is possible to analyze the vulnerability factors affecting the DPG model in multiple dimensions, and then analyze and verify the main factors affecting the vulnerability of each node in the DPG model.

Based on the DPG model established in this paper, the calculation of grid topology statistical characteristic parameters is carried out. IEEE24, IEEE62, IEEE120, Central China 500kV grid, CCD grid, and Western M grid are selected as examples for topology modeling, grid statistical characterization and comparison of each grid. The corresponding random networks are generated based on the actual grids, and the number of nodes (N), the number of edges (M), and the average degree (k) of all grids are calculated to obtain the average path lengths and aggregation coefficients of the grids and random networks, respectively. Table 1 shows the topological characteristics of different power grids, where L1 and L2 are the average path lengths of the actual power grids and the DPG model, and C1 and C2 are the aggregation coefficients of the actual power grids and the DPG model, respectively.

Analyzing the data in the table, it can be seen that as the grid size increases, the average path lengths of both the actual grid and the DPG model show an increasing trend, and the average degree fluctuates within the range of [2.72,4.24], which does not increase with the increase of the grid size. Among them, the western grid of country M, which has the largest network size (4859), instead has the smallest average degree (2.72), indicating that the average degree is a regional characteristic of the area where the grid is located rather than the size of the grid. This indicates that the number of substations with lower voltage levels increases and most of the degree values are 1 or 2, so the grid size increases while the average degree decreases. The average path length of the western M grid is much higher than the other grids. The western M grid has the largest average path length of 19.52 due to the geographic factors and the development of the city, which has a very long distance between towns and cities, and the sparse land in rural areas.

In addition, the network size obtained from the DPG model is consistent with the actual grid, i.e., the number of edges and nodes in the DPG model is consistent with the grid, but the network nodes and edges are randomly connected. Combining the data in the table, it can be seen that the average path length (L1) of the actual grid is larger than the average path length (L2) of the DPG model, and the overall gap fluctuates between 0.95% and 13.78%. And it can be seen that the aggregation coefficient of the actual grid is less than 0.24, which shows the relative coefficient of connection between grid plants and stations and is much higher than the aggregation coefficient of the DPG model, which indicates that the grid is relatively more tightly connected than the network of the DPG model, and the grid is relatively regular. The average path length of the grid is slightly larger than that of the DPG model, and the aggregation coefficient is much larger than that of the DPG model, indicating that the grid is a small-world network.

Analyzing the main characteristic parameters of the power grid leads to the conclusions: 1)

The average degree, aggregation coefficient and average path length of the actual power grid vary within a certain range.

Dynamic power network in the wide-area metering topology which contains the power network and communication network, the essence of its distributed power grid system, and distributed power grid in the operation process of the output will be affected by the external environment, which has a random change characteristics, which in turn makes the trend of the DPG model has a stochastic nature. For this reason, this paper takes the reactive power configuration of the substation of 500kV power grid in central China as an example, and combines the power physical network model given in the previous section to simulate and analyze the distributed power trend. For the sake of simplicity, the results of the distributed power trend calculation of the DPG model are given as an example for the BLS primary substation, and the line of the BLS primary substation is shown in Fig. 3.

Figure 3.

Circuit diagram of BLS substation

Based on the equivalent line diagram of Baoyi substation, each 60kV/110kV substation is equivalent to a node to carry out distributed power trend calculation, Table 2 and Table 3 show the node input data and line input data of Baoyi substation, and Table 4 and Table 5 show the node and line output data of Baoyi substation. P and Q represent active power and reactive power respectively, VR, VI, VGW, GNW are voltage real part, voltage imaginary part, voltage, given active power, given reactive power respectively, LW, LF, AP, AL, RL and LL are line active power, line reactive power, apparent power, active loss, reactive power loss and line loss rate respectively.

The results in the table show the simulation results of the BLS primary variable for the grid current calculation with distributed generation. In the simulation process, the first need to calculate the trend in the program needs to be uniquely identified number of each node, generally the root node, that is, with the previous level of voltage network grading point as the balance point. Among them, Baolongshan wind power as a power source to access the grid operation, its power is a negative value indicates that the power issued. By analyzing and comparing the simulation results, in the output data of Baoyi transformer line, the line loss percentage of seven lines is controlled below 4.5%, the reactive power loss is below 0.05 and the reactive power loss of some line output data is 0. This shows that the distributed power trend calculation results are accurate, the trend trend and direction are correct, and the error is within the permissible range. This also reflects that considering the distributed power trend changes in the dynamic power network can further identify the line loss and node loss in the distributed dynamic power network and provide data support for optimizing the power network transmission.

For dynamic power networks, cascading faults in the physical and communication layers of the grid may cause large fluctuations in the grid currents, bringing impacts to the operation of dynamic power networks. Based on this, this section analyzes the scale and difference analysis of cascading faults between uncontrolled and controlled systems based on the IEEE24 standard power network simulated by the DPG model constructed in the previous section. Exploring the stability performance and difference between the uncontrolled system and the controlled system when cascading faults occur, Fig. 4 shows the comparison results of the stability of different types of cascading faults. Among them, Fig. 4(a)~(b) shows the stability of the information layer and the physical layer under different line fault ratios, respectively.

Figure 4.

Comparison of cascade fault stability

By observing the graph of the proportion of failures of the lines in the information layer, it can be obtained that the proportion of failures of the communication network after cascading failures of the controlled system is basically smaller than that of the uncontrolled system. With a low capacity setting of the power lines (capacity setting around 0.2), the information layer of the uncontrolled system basically fails in its entirety, while the information layer of the controlled system still retains a portion of functional nodes. In addition, it can be observed that the proportion of faults in the communication layer nodes of both controlled and uncontrolled systems is characterized by a stepwise decrease with the increase of power line capacity. For the information layer line failure proportion graph, it can be observed that it exists two threshold capacities, which are around 0.1 and 0.7, respectively, less than 0.1 when the information layer is basically loss of function, between 0.1 and 0.7 when the information layer is semi-loss of function, and greater than 0.7 when it is basically functional. It can be seen that near the capacity threshold, the percentage of failures in the controlled system is significantly lower than in the uncontrolled system. This suggests that the choking effect of the information system on faults is more clearly demonstrated when the capacity is at a critical point near the threshold. Meanwhile, observing the fault proportion graphs of the physical layer lines, the proportion of faults of the controlled system is obviously lower than that of the uncontrolled system and is more obvious at the capacity threshold, which reflects the positive role played by the information layer control system on the stability of the power system from another perspective. This also shows that distributed power flow control based on distributed power flow control can significantly improve the stability of dynamic power networks, so much so that the role brought by the information layer control mechanism cannot be ignored in modeling cross-layer network cascading faults.

According to the IEEE24 node system model established based on the DPG model in the previous section, combined with the cascading fault model incorporating distributed power trend calculation in this paper, the vulnerability identification of the power grid with generator buses in the IEEE24 node system arithmetic example is carried out and the vulnerable nodes in it are identified. The transmission capacity of each line in the IEEE24 arithmetic example is set to be 15 MVA.Fig. 5 shows the vulnerability identification results of the IEEE24 nodal system.

Figure 5.

Vulnerability identification results of the IEEE24 node system

Based on the vulnerability identification results of the IEEE24 node system, the following conclusions are obtained: 1)

When the generator of bus node No. 10 fails, its vulnerability index LPRP is 0.0364, which is the maximum value among the 10 generator nodes, so that node No. 10 can be recognized as the vulnerability node in the IEEE24 node arithmetic. In the LPRP, its main sub-indicators are the optimal tidal current load shedding percentage probability (0.0136) and the islanding load shedding percentage probability (0.0233), which indicates that these two sub-indicators are the main reasons affecting the vulnerability. Meanwhile, among these two sub-indicators of LPRP, the value of the islanded load shedding share probability has the highest percentage, which indicates that there is a large optimal tidal current shedding in the power network when the generator on this bus is removed, which has the greatest impact on the vulnerability of the node. According to the calculation results of risk probability, node 10 is in the control area of the highest risk S5 distribution station, indicating that the risk probability has a large impact on the vulnerability of the node.

In summary, on the basis of considering the trend fluctuation in the dynamic power network, combining the distributed power trend settlement results can clarify its vulnerability nodes, so as to better ensure the stable operation of the dynamic power network.