## 2 Circuit Breaker Overload Protection Working Principle and Its Service Status
### Overload Protection Service State Considering Safety Margins for Action Characteristics
Since there are two trends in the change of overload protection action time, in order to more clearly describe the change trend of the overload protection action characteristics of a low-voltage circuit breaker and its impact on the service state, the linear degradation model was used to analyze the degradation of the overload protection action time, as shown in Figure 5. In the figure, \(\mu_{0}\) represents the design parameter of the circuit breaker overload protection action time. The actual overload protection action time can be described using a normal distribution with a mean value of \(\mu_{0}\). \(T\) and \(T^{\prime}\) in the figure represent the failure time, corresponding to the occurrence of a malfunction fault or refusal fault of the circuit breaker overload protection, i.e., the average lifetime of the thermal release of the low-voltage circuit breaker. Based on the above assumptions, the low-voltage circuit breaker overload protection service state transfer process is as follows:  
Figure 5: Circuit breaker action time change process. In Figure 6, the transfer probability \(\lambda_{i}\Delta t\) is the state transfer probability between the states of low-voltage circuit breaker overload protection at \(\Delta t\) time, which is satisfied by the service state transfer probability:  
\[ \begin{cases}p_{ij}(\Delta t)=P[X(t+\Delta t)=j|X(t)=i]\\ p_{ii}(\Delta t)=P[X(t+\Delta t)=i|X(t)=i]\\ p_{ii}(\Delta t)+\sum\limits_{j\
eq i}p_{ij}(\Delta t)=1\end{cases} \quad i,j=1,2,3,4,5 \tag{9}\]  
where \(p_{ij}(\Delta t)\) is the state \(i\) of the circuit breaker in \(t\) moments, in \(t+\Delta t\) moments there is a probability of state transfer to state \(j\), and \(p_{ii}(\Delta t)\) is the state \(i\) of the circuit breaker in \(t\) moments, in \(t+\Delta t\) moments there is a probability of maintaining state \(i\). According to Equation (9), the transfer probability is written in matrix form and denoted as:  
\[\mathbf{P}(\Delta t)= \begin{pmatrix}1-\lambda_{1}\Delta t&\lambda_{1}\Delta t &0&0&0\\ 0&1-\lambda_{2}\Delta t&\lambda_{2}\Delta t&0&0\\ 0&0&1-\lambda_{3}\Delta t&\lambda_{3}\Delta t&0\\ 0&0&0&1-\lambda_{4}\Delta t&\lambda_{4}\Delta t\\ 0&0&0&0&1\end{pmatrix}  \tag{10}\]  
Continuous time chi-squared Markov chains have an important feature [36]:  
\[\left\{ \begin{array}{l}q_{ij}=\lim\limits_{\Delta t\to 0}\frac{p_{ij}( \Delta t)}{\Delta t},i\
eq j\\ q_{ii}=\lim\limits_{\Delta t\to 0}\frac{1-p_{ii}(\Delta t)}{\Delta t} \end{array} \right. \tag{11}\]  
Then, the corresponding transfer density matrix \(\mathbf{M}\) is:  
\[\mathbf{M}=\lim\limits_{\Delta t\to 0}\frac{\mathbf{P}(\Delta t)-\mathbf{I}}{ \Delta t}= \begin{pmatrix}-\lambda_{1}&\lambda_{1}&0&0&0\\ 0&-\lambda_{2}&\lambda_{2}&0&0\\ 0&0&-\lambda_{3}&\lambda_{3}&0\\ 0&0&0&-\lambda_{4}&\lambda_{4}\\ 0&0&0&0&0\end{pmatrix}  \tag{12}\]  
where \(\mathbf{I}\) is the unit matrix. According to the full probability formula, the state probability of \(t+\Delta t\) moments in any state \(j\) is [13]:  
\[p_{j}(t+\Delta t)=\sum\limits_{i=1}^{5}p_{i}(t)p_{ij}(\Delta t) \tag{13}\]  
The matrix form is:  
\[\mathbf{P}(t+\Delta t)=\mathbf{P}(t)\mathbf{P}(\Delta t) \tag{14}\]  
where the probability of the state i at time t pi(t) = P(X(t) = i), \(i=1\sim 5\), the state probabilities \(\mathbf{P}(t)\) = (p\({}_{1}\)(t), p\({}_{2}\)(t), p\({}_{3}\)(t), p\({}_{4}\)(t), p\({}_{5}\)(t)), P(t + \(\Delta t\)) = (p\({}_{1}\)(t + \(\Delta t\)), p\({}_{2}\)(t + \(\Delta t\)), p\({}_{3}\)(t + \(\Delta t\)), p\({}_{4}\)(t + \(\Delta t\)), p\({}_{5}\)(t + \(\Delta t\))). Figure 6: Markov-process-based state transfer diagram for circuit breaker overload protection. By the definition of differentiation, let converge go to 0:  
\[\lim_{\Delta t\to 0}\frac{\mathbf{P}(t+\Delta t)-\mathbf{P}(t)}{\Delta t}=\frac{d \mathbf{P}(t)}{dt} \tag{15}\]  
Solve Equations (14) and (15) together and the probability of each state at the initial moment is \(\mathbf{P}(0)=(1,0,0,0,0)\), then the state probability of the state at any moment is:  
\[\left\{