## 2 The Proposed Cascading Failure Model  
In Section 2, we propose a novel load-redistribution-based cascading failure model, considering both active and reactive loads, which pays particular attention to the load model and its role in modelling the cascade. Its derivation and development are shown in the following equations and deductions.  
At each bus-bar, the loading condition at two consecutive sampling times can be described as follows [10],\[\left\{\begin{array}{l}P_{i}(t+1)=P_{i}(t)+\sum\limits_{j\in\Phi_{j}(t)}\Delta P_{ ji}(t),\\ \\ Q_{i}(t+1)=Q_{i}(t)+\sum\limits_{j\in\Phi_{j}(t)}\Delta Q_{ji}(t),\end{array}\right. \tag{1}\]  
where \(t\) represents the number of time instances, \(i\) indicates the \(i-\)th node and \(P_{i}(t)\) and \(Q_{i}(t)\) are the active and reactive power at the \(i-\)th node at time instance \(t\). Terms \(\Delta P_{ji}(t)\) and \(\Delta Q_{ji}(t)\) are the load increments, which result from the redistribution from node \(j\) to node \(i\) at \(t\), and \(\Phi_{i}(t)\) stands for the index set of node \(i\)'s neighbours at \(t\). If we use \(\Omega(t)\) to represent the configuration of the power system at \(t\), then we can write \(i\in\Omega(t)\), indicating the \(i-\)th node or bus inside the connection \(\Omega\) at time instance \(t\). If a node is overloaded at time instance \(t\), then this node is removed from \(\Omega(t)\) instantaneously. Power variations \(\Delta P_{ji}(t)\) and \(\Delta Q_{ji}(t)\) are given by:  
\[\left\{\begin{array}{l}\Delta P_{ji}(t)=p_{j}(t)P_{j}(t)q_{j}(t),\\ \\ \Delta Q_{ji}(t)=p_{j}(t)Q_{j}(t)q_{j}(t),\end{array}\right. \tag{2}\]  
where \(p_{j}(t)\) describes the state of node \(j\) at \(t\) with \(p_{j}(t)=0\) implying that node \(j\) is in the normal working condition, and \(p_{j}(t)=1\) means that node \(j\) breaks down as:  
\[p_{j}(t)=\left\{\begin{array}{l}0,\text{ if }P_{j}(t)\leq PM_{i}\text{ and }Q_{j}(t)\leq QM_{i},\\ \\ 1,\text{ otherwise},\end{array}\right. \tag{3}\]  
where \(PM_{i}\) and \(QM_{i}\) are the active and reactive power load capacities, which are given by:  
\[\left\{\begin{array}{l}PM_{i}=(1+\beta)P_{j}(0),\\ \\ QM_{i}=(1+\gamma)Q_{j}(0),\end{array}\right. \tag{4}\]  
where \(\beta\) and \(\gamma\) denote the tolerance parameters of the active and reactive power loads, respectively, and \(P_{j}(0)\) and \(Q_{j}(0)\) are the initial active and reactive power values, which are given by:  
\[\left\{\begin{array}{l}P_{j}(0)=k_{j}^{\alpha}(0),\\ \\ Q_{j}(0)=\frac{1}{10}\cdot\rho\cdot P_{j}(0),\end{array}\right. \tag{5}\]  
where \(k_{j}(0)\) is the degree (the number of its connected neighbours) of node \(j\) at \(t=0\), \(\alpha\) (>0) is a tunable parameter, which determines the load strength of the node, and:  
\[q_{j}(t)=\frac{k_{j}^{\alpha}(t)}{\sum\limits_{i\in\Phi_{j}(t)}k_{i}^{\alpha} (t)} \tag{6}\]  
is the proportional function of the failed node's load at \(t\), with \(k_{j}(t)\) being the degree of node \(j\) at \(t\). Equation (6) is substituted into Equation (2) for power change calculation. According to the industrial standard, the reactive power load is normally assumed as 10% of its active power load. In real power systems, however, not all reactive loads have the same relation to their active loads, and therefore, a modification factor \(\rho\) is employed in (5). In a power grid, it is reasonable to assume that a node's loading capacity is proportional to its degree \(k_{j}\), i.e., a node with more connections needs to have a larger loading capacity [21].  
It should be mentioned that power systems will perform islanding when line overloading happens. When small islanded grids are formed, the power flow will be recalculated in the islanded grids. It is noted that the proposed method is still valid, regardless of the size of the grid. A small grid simulation will be the same as the simulation in the study, but with a reduced number of buses.