## 2 Channel Reliability Modeling for 5G Communications
### Upward Equivalence of Branch Reliability Parameters Considering Switching Action Failures  
Faults in any element on a distribution branch may spread to the upstream area, and whether or not fault spreading occurs depends mainly on the successful operation of the branch's switches. Since the action of the circuit breaker does not require the cooperation of the control center communication and can disconnect the branch in time with a certain probability \(P_{br}\) after detecting the fault current, the equivalent fault rate for the feeder f with the presence of a circuit breaker at the first end can be expressed as  
\[\lambda_{eq,f}=(1-p_{br})\sum\limits_{i=1}^{N_{f}}\lambda_{i} \tag{15}\]  
where \(\lambda_{eq,f}\) is the equivalent failure rate of branch \(f\). \(\lambda_{i}\) is the failure rate of the element on branch \(i\) or the lower branch. \(N_{f}\) is the total number of elements on branch \(f\) or the lower branch.  
When considering physical faults alone, it is often determined that the switches can be reworked using manual methods, and the equivalent outage time of the fault in the area downstream of the switch is the switch action time, or the sum of the switch action time and the manual operation time, which fails to reflect the self-healing process of faults in a highly automated distribution network. Therefore, this paper offers a formula for calculating the equivalent outage time of switch \(r_{eq,f}\).  
\[r_{eq,f}=p_{br}t_{br}+(1-p_{br})\sum\limits_{i=1}^{N_{f}}\lambda_{i}r_{i} \tag{16}\]  
where \(r_{i}\) is the repair time of the upper element or lower branch of branch \(i\). \(t_{br}\) is the switching action time.  
At the same time, whether or not a fault spreads in a component of a branch circuit does not only depend on one of the upstream switches; all of the upstream switches may successfully operate and isolate the fault. Therefore, for branch circuits containing sectionalized switches, isolation of the fault requires the coordination of multiple switches. The branch circuit can be divided into N regions according to the sectionalized switches. For a distribution branch circuit containing sectionalized switches, once a fault occurs, all of its upstream sectionalized switches will try to isolate the fault sequentially under the remote control commands, so the equivalent fault rate of this branch circuit is  
\[\lambda_{eq,f}=\sum\limits_{i=2}^{N}\left[\lambda_{i}\prod\limits_{j=i-1}^{M} \left(1-p_{sw,j}\right)\right] \tag{17}\]  
where \(\lambda_{L,i-1}\) is the failure rate of the _i_-1 st region. \(M\) is the number of sectionalized switches in the branch.  
The repair time for the region downstream of switch \(i\), if the switch successfully isolates the fault, is the sectionalized switch action time \(t_{sw}\); otherwise, the upstream switch will actuate up to the branch circuit breaker or branch node. Therefore, for the region downstream of switch \(i\), the equivalent fault time is  
\[r_{eq,j}=\sum\limits_{i=j}^{M}\left[p_{sw,j}\prod\limits_{k=j}^{i-1}(1-p_{sw,k})\right]t_{sw}+\prod\limits_{i=0}^{M-j}(1-p_{sw,j+i})r_{j} \tag{18}\]  
Included among these is  
\[r_{j}=\frac{1}{\sum\limits_{i=1}^{j}\lambda_{L,i}}(\lambda_{L,i}r_{L,j}+\sum \limits_{k=1}^{j-1}\lambda_{L,k}r_{eq,j-1}) \tag{19}\]  
where \(r_{j}\) is the equivalent outage time of the area downstream of switch \(j\) in the branch circuit. \(r_{L,j}\) is the repair time of switch \(j\).