# A Novel Low-Complexity Fault Diagnosis Algorithm for Energy Internet in Smart Cities
## 4 Performance Analysis  
### Theoretical Analysis  
In this section, we analyze the performance of the \(SFBT\) in terms of: (I) Detection Accuracy (abbreviated as \(DA\), which is the ratio between number of faulty meters detected as faulty and the total number of faulty meters); (II) False Alarm Rate (abbreviated as \(FAR\), which is the number of fault-free meters detected as faulty to the total number of fault-free meters); (III) False Positive Rate (abbreviated as \(FPR\), which is the ratio of the number of faulty meters detected as fault-free to the total number of faulty meters); and (IV) message complexity. Let \(p_{r}\) be the probability of a meter being faulty, \(p\) be the probability that a faulty meter detected as fault-free, and \(p^{{}^{\prime}}\) denote the accuracy of the \(SFBT\), respectively. Then, the following three theorems hold. **Theorem 1**: _The \(DA\) of the proposed algorithm \(SFBT\) is_  
\[DA_{SFBT}=\left\{ \begin{array}{ll}\frac{|MDS||(1-p)\triangleup+1|}{\pi},&t_{ i}^{r}\leq 3\\ \frac{|MDS|(\frac{\triangleup}{2}|(1-p_{r})+p|+1)}{n},&t_{i}^{r}>3\end{array}  \right.. \tag{21}\]  
The \(SFBT\) detects faulty meters in two phases. In the first phase, there are up to \(|MDS|(1-p_{r})\) fault-free cluster-heads detected along with \(|MDS|p_{r}\) faulty ones. That implies there are at most \(|MDS|(1-p_{r})\ \bigtriangleup\ p_{r}\) faulty cluster-members correctly diagnosed within corresponding clusters. According to the modified sigma rule, it is apparent that, if \(t_{i}^{r}\leq 3\), then the detection accuracy is according to the \(DA\) of the three sigma test given by [PERSON] [38], which is  
\[DA=\left\{ \begin{array}{ll}p_{r}-p,&t_{i}^{r}\leq 3\\ (1-p_{r})+p,&t_{i}^{r}>3\end{array} \right.. \tag{22}\]That suggests we can correctly detect \(|MDS|p_{r}\bigtriangleup\) faulty cluster-members. Therefore, the \(DA\) of \(SFBT\) is deduced as  
\[DA_{SFBT}=\left\{ \begin{array}{ll}\frac{|MDS|[(1-p)\bigtriangleup+1]}{n},&t^{ r}_{i}\leq 3\\ \frac{|MDS|[(1-p_{r})+p]\bigtriangleup+1]}{n},&t^{r}_{i}>3\end{array} \right.,\]  
where \(|MDS|\leq\frac{1+ln(\delta+1)}{\delta+1}n\)[40], \(n\) denotes the number of meters, and the minimum and the maximum degree of graph \(G\) is denoted by \(\delta\) and \(\bigtriangleup\), respectively. Theorem 3.2: _The \(FPR\) of the algorithm \(SFBT\) is_  
\[FPR_{SFBT}=\left\{ \begin{array}{ll}\frac{n(p_{r}+p)-|MDS|[1+(1-p_{r})\bigtriangleup ]}{n},&t^{r}_{i}\leq 3\\ \frac{n(1-p_{r}-p)-|MDS|[1+(1-p_{r})\bigtriangleup]}{n},&t^{r}_{i}>3\end{array}  \right.. \tag{23}\]  
Proof: Similar to the proof of Theorem 3.1, the \(SFBT\) can correctly detect up to \(|MDS|p_{r}\) faulty cluster-heads. With the help of fault-free cluster-heads, there are at most \(|MDS|p_{r}(1-p_{r})\bigtriangleup\) faulty cluster-members detected as faulty ones. According to literature [38], the \(FPR\) of the three sigma test is  
\[FPR=\left\{ \begin{array}{ll}p_{r}+p,&t^{r}_{i}\leq 3\\ 1-(p_{r}+p),&t^{r}_{i}>3\end{array} \right.. \tag{24}\]  
Therefore, we can deduce that the \(FPR\) of the \(SFBT\) is  
\[FPR_{SFBT}=\left\{ \begin{array}{ll}\frac{n(p_{r}+p)-|MDS|[1+(1-p_{r})\bigtriangleup ]}{n},&t^{r}_{i}\leq 3\\ \frac{n(1-p_{r}-p)-|MDS|[1+(1-p_{r})\bigtriangleup]}{n(1-p_{r})},&t^{r}_{i}>3 \end{array} \right..\]  
Theorem 3.3: _The FAR for the algorithm \(SFBT\) is_  
\[FAR_{SFBT}=\left\{ \begin{array}{ll}\frac{np_{r}-|MDS|[(1-p_{r})[1+\bigtriangleup (1-p_{r})]}{n(1-p_{r})},&t^{r}_{i}\leq 3\\ \frac{np-|MDS|[(1-p_{r})[1+\bigtriangleup(1-p_{r})]}{n(1-p_{r})},&t^{r}_{i}>3 \end{array} \right.. \tag{25}\]  
Proof: Similar to the proofs of previous theorems, there are up to \(|MDS|(1-p_{r})\) fault-free cluster-heads correctly detected by the \(SFBT\). That implies the fault states of corresponding cluster-members, which are as many as \(|MDS|(1-p_{r})\bigtriangleup(1-p_{r})\) meters, are easily verified. The \(FAR\) of the three sigma test is given in Reference [38] as follows:  
\[FAR=\left\{ \begin{array}{ll}\frac{p_{r}}{1-p_{r}},&t^{r}_{i}\leq 3\\ \frac{p}{1-p_{r}},&t^{r}_{i}>3\end{array}