## 4 Simulations
### Key Factors Affecting the Malfunction Probability  
(1) False trip probability for inconsistent delays  
In this simulation, the false trip probability follows the braking ratio criterion (\(I_{cd}\geq kI_{res}\)) with \(I_{res}\) being calculated as \(I_{res}=\left|\dot{I}_{n}\right|+\left|\dot{I}_{n}\right|.\) From Equation (13), it can be found that the key factors affecting the false trip probability are the differential current's standard deviation \(\sigma\) and the inconsistent delay \(t_{cd}\).  
Figure 12 shows the changes of the false trip probability with \(t_{cd}\) ranging from 0 ms to 8 ms and \(\sigma\) ranging from 0 to 1.0.  
From Figure 12, we can conclude that when the standard deviation of differential current is equal to 0, the false trip probability experiences a step change at an inconsistent delay of 6.88 ms. Hence, the observation that the maximum inconsistent delay is 6.88 ms is supported by China industry's practical application which indicates the maximum inconsistent delay of differential protections ranges from 5.9 ms to 8.2 ms [25].  
Furthermore, with the increase of \(\sigma\), the probability changes with respect to inconsistent delay becomes slower. Especially, for \(t_{cd}=t_{\max}\), the probability changes dramatically at \(\sigma=0\), while it remains almost constant at \(\sigma=1\). This means that a larger \(\sigma\) indicates less differential protection sensitivity.  
(2) Refuse-operation probability for consistent delays  
According to (13), the refuse-operation probability is mainly dependent on the distribution of the consistent delay (the normal distribution standard deviation of consistent delay \(\sigma_{1}\), the expected value of consistent delay \(t_{d}\)) and the threshold of the consistent delay \(t_{\max}\).  
According to the real industrial experience in China, we selected the threshold of consistent delay \(t_{\max}\) as 12 ms. Figure 13 shows the changes of the refuse-operation probability with the expected consistent delay \(t\) ranging from 0 ms to 20 ms and the standard deviation of consistent delay \(\sigma_{1}\) ranging from 0 ms to 15 ms.  
Figure 12: Key factors affecting false trip probability.  
In Figure 13, when the normal distribution standard deviation of differential current is equal to 0, the refuse-operation probability experiences a step change at a consistent delay of 12 ms. This is in agreement with the fact that the threshold of consistent delay \(t_{\text{max}}\) was selected as 12 ms. Hence, with the increase of \(\sigma_{1}\), the probability changes with respect to consistent delay becomes slower. This means that a larger \(\sigma_{1}\) indicates less differential protection sensitivity. We selected consistent delay \(t_{d}\) as 10 ms and \(\sigma_{1}\) as 1 ms, then the refuse-operation probability change with the threshold ranging from 0 to 20 ms was obtained as shown in Figure 14.  
Figure 14 shows that the larger the threshold is, the smaller the probability of refuse-operation is. When the consistent delay is equal to the threshold, the probability of refuse-operation is equal to 0.5.  
(3) Refuse-operation probability for bit errors  
According to (16), the threshold of bit error rate \(w_{\text{max}}\) and the real time bit error rate \(w_{1}\) are two relevant factors to the refuse-operation probability. We selected the real time bit error rate \(w_{1}\) as \(5\times 10^{-4}\), then the relationship between \(w_{\text{max}}\) and refuse-operation probability was obtained as shown in Figure 15.  
Figure 14: Refuse-operation probability with threshold.  
Figure 13: Refuse-operation probability considering consistent delay distribution.  
From Figure 15, we find that the larger the threshold is, the smaller the refuse-operation probability is. When the real time bit error rate is equal to the threshold, the probability of refuse-operation is equal to 0.5.