# Analysis of Controllability in Cyber-Physical Power Systems under a Novel Load-Capacity Model
## 3 Fault Propagation of CPPS under Cascading Failure
### Initial Load and Capacity Model
#### 3.1.1 Load Capacity Model in Power Networks  
In the power grid, power nodes can be classified into three types based on their role:  
* Generation Nodes: These nodes feed power into the grid.
* Load Nodes: These nodes consume power from the grid.
* Transmission Nodes: These nodes neither consume nor contribute power to the grid.  
In previous research, the principle of shortest path propagation is often used to study power networks. However, this approach is not appropriate because the power injected by the generation nodes propagates through all transmission lines.  
Figure 3: Controlling a simple network.  
For a power system with \((N+1)\) nodes, the matrix form of the loop current equations can be given as  
\[\dot{U}_{b}=Z_{b}\dot{I}_{b} \tag{12}\]  
where \(\dot{U}_{b}\) is a column vector of size \(N\times 1\), representing the vector of node voltages; \(Z_{b}\) is a \(N\times N\) matrix of loop impedance; and \(\dot{I}_{b}\) is a column vector of size \(N\times 1\), representing the vector of injection currents. Equation (12) can be expanded as follows:  
\[\begin{bmatrix}\dot{U}_{1}\\ \dot{U}_{2}\\ \vdots\\ \dot{U}_{n}\end{bmatrix}=\begin{bmatrix}z_{11}&z_{12}&\cdots&z_{1n}\\ z_{21}&z_{22}&\cdots&z_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ z_{n1}&z_{n2}&\cdots&z_{m}\end{bmatrix}\begin{bmatrix}\dot{I}_{1}\\ \dot{I}_{2}\\ \vdots\\ \dot{I}_{4}\end{bmatrix} \tag{13}\]  
When power currents are injected between nodes \(i\) and \(j\) (\(\dot{I}_{i}=1,\dot{I}_{j}=-1\)), the voltage at node \(k\) can be determined using the node voltage equations. Assuming that node \(k\) is not the reference node, the equation can be expressed as follows:  
\[\dot{U}_{k}=z_{k1}\dot{I}_{1}+z_{k2}\dot{I}_{2}+\cdots+z_{kn}\dot{I}_{n}=z_{ki} \dot{I}_{i}+z_{kj}\dot{I}_{j} \tag{14}\]  
Unlike complex network theory, the distribution of power in power systems follows Kirchhoff's laws. Therefore, the distance between two points is equivalent to the impedance between those two points. Based on these principles, the impedance between two points in a power system represents the \"distance\" or resistance to power flow between those points. This analysis helps in accurately modeling and understanding the power distribution in the system.  
\[D_{ij}=\dot{U}_{i}-\dot{U}_{j}=\left(z_{ii}-z_{jj}\right)-\left(z_{ji}-z_{jj} \right)=z_{ii}+z_{jj}-2z_{ij} \tag{15}\]  
Therefore, based on the concept of node betweenness centrality, it is possible to calculate the electrical betweenness [28] of node \(i\) in the power network.  
\[B(i)=\begin{cases}\sum\limits_{l\in L,g\in G}\sqrt{W_{l}W_{g}}\sum\limits_{n \in f(n)}\frac{1}{2}\Big{|}I^{\lg}(i,n)\Big{|},&j\
otin L,G\\ \sum\limits_{l\in L,g\in G}\sqrt{W_{l}W_{g}}\sum\limits_{n\in f(n)}\Big{|}I^{ \lg}(i,n)\Big{|},&j\in L,G\end{cases} \tag{16}\]  
where \(L\) and \(G\) are the load and generation nodes, and \(f(i)\) is the set of neighbors of node \(i\). \(W_{l}\) is the actual load or rated load of the load, \(W_{g}\) is the actual power output or rated power of the generator, and \(\Big{|}I^{\lg}(i,n)\Big{|}\) represents the current flowing through the branch \((i,n)\) when a unit current element is injected between the generation node \(G\) and the load node \(L\).  
Therefore, the initial load \(L(i)\) of node \(i\) in the power system can be defined as follows.  
\[L_{p}(i)=B(i) \tag{17}\]  
In actual power systems, each station will allocate a certain amount of reserve capacity to ensure the electricity market demand during equipment maintenance, accidents, and other situations. So, the network capacity parameter \(\alpha\) is introduced in the power network to define the total capacity of each node as follows:  
\[C_{p}(i)=(1+\alpha)L_{p}(i) \tag{18}\]