## 5 Key Technologies Supporting Deliberate Attack Defense for IECPS
### Theoretical Foundations of IECPS
#### 5.1.2 Model Solving of IECPS  
The IECPS deliberate attack defense optimization model is a large-scale, nonlinear, multi-stage optimization model, which is mathematically challenging to solve for a global optimum. Typically, the optimization model needs to be processed before solving. Specifically, there are three methods for solving the model: analytic methods [192], heuristic algorithms [188], and machine learning methods [193; 194; 195].  
* **Analytic methods** The analytic method involves using techniques such as linearization, convex relaxation, and duality to convert the IECPS defense optimization problem into a solvable linear programming or mixed-integer programming problem. The linearization method can transform the nonlinear equations of the gas or thermal systems into linear equations. Common linearization methods include piecewise linearization, logarithmic transformation linearization, Big-M method, and convex relaxation [196]. To convert the deliberate attack model into a single-layer model for solving, duality or Karush-Kuhn-Tucker conditions are commonly used for the transformation.  
The analytic method can be specifically categorized into two approaches: unified solving and decomposition-coordination solving. Unified solving involves integrating all constraints and variables of the IECPS defense optimization problem into a single model for solution. This method faces difficulties in solving complex systems. Decomposition-coordination solving involves optimizing and solving each subsystem individually, then exchanging the results of each subsystem's solution. The results are used to iteratively update the subsystems until the optimal solution with the required precision is found. This method ensures the confidentiality of each subsystem's information during the decision-making process while achieving a global or near-global optimum for the entire system.  
* **Heuristic algorithms** Considering that some defense problems are highly complex, nonlinear, and non-convex, traditional analytic methods are difficult to model and solve, heuristic algorithms can be employed for handling them. Heuristic algorithms are optimization algorithms constructed based on intuition or experience, capable of providing a feasible solution for defense optimization within an acceptable time. Currently, commonly used heuristic algorithms are swarm intelligence algorithms, such as particle swarm optimization and ant colony optimization [193].
* **Machine learning methods** AI methods do not focus on the type and characteristics of the problem when solving optimization problems, making them effective in solving IECPS defense optimization problems with non-linear and non-convex properties. Compared to analytic methods, AI methods can flexibly adapt to different attack scenarios without the need for model reconstruction, and they offer more efficient solution speeds. Compared to heuristic algorithms, they can handle most deliberate attack problems. However, AI methods require high-quality training data to ensure the accuracy of the solution. Although unsupervised learning methods that do not require attack cases have been developed, these methods still lack interpretability. Common machine learning methods include support vector machines, k-nearest neighbors, principal component analysis, Q-learning, and deep reinforcement learning [194; 196; 197; 198].  
In practical applications, the above methods can be combined based on the specific problem at hand. For instance, in robust optimization problems involving attack uncertainty, particle swarm optimization can be used to generate defense strategies, or machine learning methods can be employed to optimize attack strategies, while the remaining parts of the problem can be solved using analytical methods. This combined approach improves both the solution efficiency and the accuracy of the results.