*2.3. Dual Interactions*

Both the interactions among the risk factors and those between the markets constitute the structure of the coupled natural gas–electricity system and serve as the driving force for evolution. These interactions are highly dynamic and vary with stochastic events. For instance, a sudden gas rupture may impair the normal operation of the natural gas market, resulting in a gas supply shortage. In such a scenario, the electricity market may seek alternative energy sources to mitigate the demand–supply gap. As the ratio of gas to electricity declines, the interaction between the two systems decreases accordingly. To elucidate the initial data on the type and strength of the dual interactions, expert judgments were utilized. The cascading effects caused by the intricate interactions exert significant influences on the aggregated risks of the coupled markets.


#### **3. Model Construction**

To comprehensively investigate how multiple risks evolve and affect the coupled natural gas–electricity market considering complex interactions, system dynamics (SD) modeling was used for our analysis. SD is a simulation-based approach with the ability to quantitatively model dynamic and complex problems, offering decision-makers an intuitive interface for experimenting with numerous scenarios and revealing transparent results [34]. The basic principles of SD are that all outcomes of a system are determined by its unique inputs, and the behavior of a system originates from its structure [47,48]. Descriptions in the theoretical framework revealed the core structure of the systems, and physical activities of the coupled markets will be thoroughly investigated in this section, based on which the risk evolution processes, behaviors of multiple entities and state of the system can be characterized by continuously changing variables. These variables are interrelated, constituting feedback loops in response to system changes. Using differential equations, the peculiarity of variables and hypothesized relations can be quantified, which is in turn incorporated by SD software for the simulation.

Figure 2 depicts the SD model for a risk assessment of the coupled markets created using the software tool Vensim DSS. This model was adapted from well-established and verified models, including those developed by [1,8,22]. It portrays the general operation of the coupled two markets, and this model is flexible and can be structured with additional feedback loops representing specific ripple effects in other contexts. State variables such as the natural gas reserve capacity and the total natural gas supply are modeled as stocks. They are symbolized as containers or boxes, representing the accumulation of volume or

capacity at a certain time. There might be inflows to or outflows from the stocks, which are symbolized by valves, inducing variations in the box per unit of time. Auxiliary variables, such as the import shortages and the electricity transmission efficiency, represent constant values or intermediate steps in calculations. The interactions between the variables are depicted as arrows, with "+" signifying a positive causal link and "−" signifying a negative link.

**Figure 2.** System dynamics model for assessing the risk of the coupled natural gas–electricity market (R: Reinforcing loop, B: Balancing loop).

We took China as an example, with China's Energy Statistical Yearbook, China's Statistical Yearbook and other reliable data sources for setting the variables and conducting experiments. As daily data are not accurately accessible for certain variables, we derived daily statistics based on the yearly data and seasonal peaks to preserve reasonableness [8]. The model's timescale was set from September to March (approximately 180 days), which was sufficient to accommodate for understanding a typical risk event, and the time step was set as 1 day. The mathematical settings of variables and links in Vensim DSS are illustrated in the following subsections.
