## 5 Discussion  
In this paper, a preventive control model based on the finite automaton theory is designed, which is a six-tuple to describe preventive actions for blocking the propagation of cross-domain cascading failures in an active distribution network of cyber-physical system, and cross-domain cascading failures sequences should be as short as possible. This model is helpful for guiding the trajectory data collection and learning policy selection.  
Then, this paper proposes a methodology (named as PCMD) for constructing a preventive control policy to block the propagation of cross-domain cascading failures in an active distribution network of cyber-physical system. This methodology is based on the deep deterministic policy gradient idea to train the deep neural networks with the trajectory data samples originated from simulations and does not need to consider the specific power flow equations. In addition, the parameter adjustments are proposed to guarantee the convergence of the construction process for generating a preventive control policy against cross-domain cascading failures. The gradient theorem of deterministic control policy and the compatibility condition of the fitted state-action function and the fitted preventive control policy have been given to ensure the suboptimality of the generated preventive control policy.  
Finally, the proposed PCMD has been verified in the CEPR-CPS provided by the China Electric Power Research Institute. It is shown that the PCMD is better than the multi-agent method in terms of reducing the number of failure nodes and avoiding the state space explosion problem. So, the scalability of the multi-agent method is poor and not suitable for large-scale networks due to the curse of dimensionality. The space complexity of PCMD is very high, and the PCMD has a defect in storing a lot of data. In addition, the proposed PCMD is compared with the theoretical analysis methods and the physical experimental methods in terms of time, cost, reliability, and accuracy. It is shown that the PCMD is better than theoretical analysis methods and physical experimental methods except in terms of reliability.  
In future works, there is also a point we should consider. The propagation process of cross-domain cascading failures is time-dependent, and it also takes time for a preventive control policy to work. Therefore, how to reduce the deviation between the time when the preventive control policy works and the time of failure occurrence, that is, the control actions applied by the preventive control policy, really work before a next failure occurs in a cross-domain cascading failures propagation sequence so as to achieve an accurate preventive control effect.  
Conceptualization, P.S. and Y.D.; methodology, P.S.; software, P.S. and S.Y.; validation, P.S., Y.D.; formal analysis, P.S.; investigation, P.S. and Y.D.; resources, Y.D.; data curation, P.S., S.Y. and C.W.; writing--original draft preparation, P.S.; writing--review and editing, Y.D.; visualization, P.S.; supervision, Y.D.; project administration, Y.D.; funding acquisition, Y.D. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the National Key Research and Development Program of China (Basic Research Class)--Basic Theories and Methods of Analysis and Control of the Cyber-Physical Systems for Power Grid under Grant No. 2017 YFB0903000.  
The authors declare that they have no conflicts of interest.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Property** & **PCMD** & **Theoretical Analysis Methods** & **Physical Experimental Methods** \\ \hline time & short & long & long \\ cost & low & high & high \\ reliability & medium & high & high \\ accuracy & high & high & medium \\ \hline \hline \end{tabular}
\end{table}
Table 3: Comparison between different methods.