## 5 Simulation Results and Discussion
### Discussions  
In this paper, only simulation results are presented to validate the FDI performance. The proposed scheme is designed for online operation and is independent from control strategies. Some essential issues are discussed with respect to the lack of experimental results.  
* The component parameters of simulation model come from the real laboratory prototype with rated power 2.5 kW. Its controller parameters are designed on the simulation file and can guarantee the control performance. The real waveforms and power characteristics are the same as those of simulation results. The observer design is a dual problem of controller design. Thus, the parameters designed in SIMULINK environment can be applied to the real experiments.
* The threshold selection is the most challenging problem in implementing the proposed algorithm. In real application, the mechanical torque and measurement noise are different from the simulation configuration. This will be further introduced into the observer and error dynamics. These effects can be modeled as generalized unknown disturbances. The upper bound of the disturbances in real application is slightly different from simulation scenarios. However, this does not affect the performance since the upper bounds of disturbances and faults hold for real applications.  
\begin{table}
\begin{tabular}{l l l l l l l} \hline \hline
**FD Scheme** & **Measurements** & **Fault Types** & **Isolability** & **System Model** & **Detection Variables** \\ \hline Bank of observers [28] & 1 voltage, 3 currents, & **Type c** & Single & IM model in \(\alpha-\beta\) & Estimation errors of rotor flux and speed \\ EKF [29] & 1 voltage, 2 currents, & **Type c** & Single & IM model in \(\alpha-\beta\) & Estimation errors of phase currents \\ Adaptive observer [30] & 1 voltage, 2 currents, & **Type c** & Single & IM model in \(d-q\) & Fault inference based on current errors \\ Bank of observers [8] & 1 voltage, 2 currents, & **Type a, Type b,** & 2 faults & IM model in \(\alpha-\beta\) & Geometric residuals \\ TS fuzzy observers [9] & 2 currents, 1 speed & **Type b** & 2 faults & DFIG model in & Estimation errors of the states \\ Integrated filters [10] & \(\mathcal{H}_{-}/\mathcal{H}_{\infty}\) & 2 currents, 1 speed, & **Type a, Type b,** & 2 faults & DFIG model in & Generalized likelihood ratio of residuals \\ TVKF [2] & 2 currents, 1 speed & **Type a, Type b** & 3 faults & PMSG model in & Generalized likelihood ratio of residuals \\ Sliding observer [31] & mode & 3 currents, 1 speed, & **Type c** & 2 faults & PMSG model in & Evaluation of estimation errors \\ This method & 3 currents, 1 speed, & **Type a, Type b** & 3 faults & PMSG model in & Evaluation of the fault estimates \\  & 1 position & **Type c** & & \(\alpha-\beta\) & & \\ \hline \hline \end{tabular}
\end{table}
Table 2: Comparison of a model-based current sensor FDI scheme.  
The harmonics is another issue for current sensor fault diagnosis. The influences of harmonics on system behavior need to be further discussed with respect to system parameter and dynamics variation. However, few results have been presented for dealing with this problem, even for the controller designs in [20; 21; 22]. Recalling the FDI schemes in Table 2, only the method in [2] utilizes the harmonic model of PMSG to diagnose additive and multiplicative faults in current sensors. The state space model and output equation are linear combinations of each order harmonic in frequency domain, which indicates that the residuals can be modeled as the combination of finite harmonics. The proposed FDI takes the time domain behaviors of residuals into consideration. The average value of each fault estimate is calculated with a sliding window. Current sensor faults are evaluated via the threshold function defined in Equation (32). From this perspective, the harmonics will not affect the residual evaluation in time domain analysis.  
Figure 7: Simulation results on simultaneous isolation of phase a and phase c fault.