# Power System Reliability Assessment Considering Coal-Fired Unit Peaking Characteristics
## 3 Multi-Stage Probabilistic Model
### Coal-Fired Unit Multi-State Probabilistic Model
#### 3.2.1 State-Space Model of Coal-Fired Units  
In the process of PPS, three types of multi-state models for coal-fired units have been established based on the unit's operating conditions and load positions. These include the base-load unit model, the flexible load-following unit model, and the start-stop load-following unit model. The specific characteristics of these models are shown in Table 1. The models take into account constraints such as the depth of load-following, load change rate, load change time, and start-up failures. The impact of load-following depth and load change rate on high-temperature components within the unit is approximated as the overall impact on the coal-fired unit. The specific correlation and relationships are based on the research results provided in reference [28], which were used for fitting.  
By considering the load-following capabilities of different units, along with system constraints and the time-series load demand characteristics, the state transition of coal-fired units is updated in real time. The specific state-space diagram is shown below:  
Model 1: Base-load state-space model, as shown in Figure 2.  
In Figure 2, \(\lambda\) and \(\mu\) represent the unit's repair rate and failure rate, respectively, which are calculated as shown in Equations (8) and (9). Here, mean time to failure (MTTF) denotes the average time between failures, and mean time to repair (MTTR) represents the average repair time. These parameters are typically obtained from historical operational data of the unit. During the base-load period, the unit needs to maintain rated operation to meet the basic load demand. Therefore, a two-state model is used, consisting of the forced outage state and the rated operating state.  
\[\lambda=\frac{1}{MTTR} \tag{8}\]  
\[\mu=\frac{1}{MTTF} \tag{9}\]  
Model 2: Flexible load-following state-space model, as shown in Figure 3.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Model** & **Features** \\ \hline Base-load unit model & Maintaining stable operation of the power system by continuously providing a minimum supply of power to operate at base-load at rated capacity. \\ Flexible peaking unit model & Responding quickly to power system load demand and performing peaking operations at waist load. \\ Start-stop peaking unit model & Meeting short-term load demand during peak load periods, promoting wind power consumption, and operating at peak loads. \\ \hline \hline \end{tabular}
\end{table}
Table 1: Classification and characteristics of unit model.  
Figure 2: Base-load state-space model.  
repair rate \(\lambda\). The failure rate is adjusted based on the equipment's lifecycle conditions during operation. Further details can be found in Section 4.  
\[\
u_{k-1\sim k}^{+}=\frac{\rho^{+}(k-1,k)}{TR_{k-1\sim k}} \tag{13}\]  
\[TR_{k-1\sim k}=\frac{P_{k}-P_{k-1}}{L_{rate}} \tag{14}\]  
where \(P_{k}\) represents the output corresponding to state \(k\), \(L_{rate}\) is the load change rate, and \(\
u_{k-1\sim k}^{+}\) is the transition rate between states.  
The flexible load-following unit typically operates in the mid-load range and adjusts the unit's output within a certain load range to meet the load demand. A multi-state model is used, where the unit's output is discretized between the minimum technical output and the rated output. The transition characteristics between different states are considered to determine the load-following positions of the flexible load-following unit.  
Model 3: Start-stop load-following state-space model, as shown in Figure 4.  
In Figure 4, compared to the flexible load-following unit model, the start-stop load-following model includes the consideration of the unit's active shut-down state. This model is typically applied during peak-load periods to primarily satisfy short-term load demand responses of the power system. Similarly, a multi-state model is used to determine the unit's start-stop operations and load-following positions.