## 3 Open-Circuit Fault Diagnosis Strategy
### Fault Location  
As previously analyzed, both single-switch and double-switch faults within the same phase are indicated by \(R_{x}\) exceeding \(T_{x}\). Therefore, the fault detection variables can only detect and isolate faulty branches, rather than pinpoint the specific faulty switches. This paper performs initial location using variables \(W_{x}\) and \(w_{x}\), and incorporates \(\eta_{x}\) to precisely identify faulty switches.  
First, taking into account the distinct characteristics of the observed and actual currents, the initial location variables are introduced to distinguish between positive- and negative-level switches. Their specific representations are as follows:  
\[W_{x}=\frac{\int_{t-T}^{t}i_{x}dt}{\int_{t-T}^{t}|i_{x}|dt},w_{x}=\frac{\int_{t -T}^{t}\dot{i}_{x}dt}{\int_{t-T}^{t}|\dot{i}_{x}|dt} \tag{24}\]  
According to the preceding analysis of switch OC faults, \(W_{x}=w_{x}\) under normal inverter operation, \(W_{x}<w_{x}\) when the positive-level switches (\(T_{x1}\) and \(T_{x3}\)) fault, \(W_{x}>w_{x}\) during the negative-level switches (\(T_{x2}\) and \(T_{x4}\)) fail. To accurately determine the location of the faulty switches, the fault location variable \(\eta_{x}\) is introduced as follows:  
\[\eta_{x}=\frac{\left(\frac{1}{T}\int_{t-T}^{t}|i_{x}|^{2}dt\right)^{1/2}}{ \frac{1}{T}\int_{t-T}^{t}|i_{x}|dt} \tag{25}\]  
Under normal or steady-state operation, the location variable \(\eta_{x}\) remains stabilized at approximately \(1.11\). In T\({}^{2}\)3L inverters, faults in the bridge-arm switches and midpoint switches affect the output current to different degrees.A fault in midpoint switch (\(T_{x2}\) or \(T_{x3}\)) causes a slight current attenuation, resulting in \(\eta_{x}\in[1.11,\eta_{1}]\),where \(\eta_{1}\) denotes the maximum value of \(\eta_{x}\) under this condition. A fault in bridge-arm switch (\(T_{x1}\) or \(T_{x4}\)) leads to significant current attenuation, corresponding to \(\eta_{x}\in[1.11,\eta_{2}]\) with \(\eta_{2}>\eta_{1}\). Due to partial overlap between the characteristic ranges of the two fault types, a classification threshold \(\delta_{1}=1.002\times\eta_{1}\) is designed to enhance identification accuracy. Midpoint switch faults are detected when \(\eta_{x}\in[1.11,\delta_{1})\), whereas \(\eta_{x}\geq\delta_{1}\) indicates the bridge-arm switch faults. To improve robustness under practical disturbances such as noise, the identification interval for midpoint switch faults is narrowed to \(\eta_{x}\in[1.12,\delta_{1})\).  
The above method is applicable to scenarios with partial double-switch faults in the same phase. The distinctive topology of the T\({}^{2}\)3L inverter results in the complete disappearance of the forward current when both positive-level switches (\(T_{x1}\) and \(T_{x3}\)) fail simultaneously. This phenomenon leads to a pronounced increase in \(\eta_{x}\). In this case, the positioning variable satisfies \(\eta_{x}\in[1.11,\eta_{3}]\), where \(\eta_{3}>\eta_{2}\). Similarly, when both \(T_{x2}\) and \(T_{x4}\) experience simultaneous faults, \(\eta_{x}\) also falls within the range \(\eta_{x}\in[1.11,\eta_{3}]\). Therefore, a threshold value of \(\delta_{2}=1.002\times\eta_{2}\) is defined to differentiate this case from other fault conditions. When \(\eta_{x}\geq\delta_{2}\), it indicates either concurrent faults of \(T_{x1}\&T_{x3}\) or concurrent faults of \(T_{x2}\&T_{x4}\). Based on the preceding analysis, the fault location criteria for the T\({}^{2}\)3L inverter are defined in Equation (26):  
\[F_{xi}=\begin{pmatrix}T_{x1},&W_{x}<w_{x}\ \&\ \eta_{x}\geq\delta_{1}\\ T_{x4},&W_{x}>w_{x}\ \&\ \eta_{x}\geq\delta_{1}\\ T_{x2},&W_{x}>w_{x}\ \&\ \eta_{x}\in[1.12,\delta_{1})\\ T_{x3},&W_{x}<w_{x}\ \&\ \eta_{x}\in[1.12,\delta_{1})\\ T_{x1}\ \&\ \ T_{x3},&W_{x}<w_{x}\ \&\ \eta_{x}\geq\delta_{2}\\ T_{x2}\ \&\ \ T_{x4},&W_{x}>w_{x}\ \&\ \eta_{x}\geq\delta_{2}\end{pmatrix} \tag{26}\]  
where, \(F_{xi}\) represents the faulty switches.