*5.1. Voltage Sensor Fault Detection*

Multiple voltage sensor faults were injected at different cell capacities in simulation. One specific case will be shown as an example. At a cell capacity of 16.47 Ah, a bias fault of +0.5 V was added to the voltage sensor at the time 30,000 s. The diagnostic results are plotted in Figure 9. Figure 9a,c,e show the deviation between the filtered and unfiltered data. As can be seen, the error increases significantly at the fault injection time. Figure 9b,d,f show the corresponding CUSUM values for the errors. Both the CUSUM values for *R*1 and C1 exceed the threshold at 30,003 s, which is 3 s after the voltage sensor fault occurs. The CUSUM value for *R*0 takes longer to respond to the fault, which is expected for voltage sensor faults, and also helps to achieve correct fault isolation. The detected voltage sensor fault signal is plotted in Figure 9g. Table 3 presents results for detection time of the voltage sensor faults of different fault sizes and cell capacities at an injection time of 30,000 s.

**Table 3.** Summary of detection time for voltage sensor faults at different sizes and cell capacities.


**Figure 9.** Errors and diagnostic results in the case of voltage sensor fault. (**a**) Error from *R*0; (**b**) CUSUM control chart for *R*0; (**c**) Error from *R*1; (**d**) CUSUM control chart for *R*1; (**e**) Error from *C*1; (**f**) CUSUM control chart for *C*1; (**g**) Isolated voltage sensor fault FU signal.

### *5.2. Current Sensor Fault Detection*

Similar to the simulation done for voltage sensor fault diagnosis validation, current sensor faults of various sizes were injected at different available cell capacities. The case that will be shown as an example is at a cell capacity of 16.47 Ah, where a gain fault of +10% was injected at the time 30,000 s. The diagnostic results are plotted in Figure 10. The errors were also found to increase at the time of fault injection, as seen in Figure 10a,c,e. Figure 10b,d,f show that the CUSUM values all exceed their respective thresholds after the fault occurs. The CUSUM for the error of R0 is the fastest to exceed the threshold, at 30,165 s; while the CUSUM values for *R*1 and *C*1 exceed their thresholds afterward. This indicates a current sensor fault, according to the proposed FDI scheme. Figure 10g shows the detected and isolated current sensor fault signal. The detection time for current sensor faults suffers from a delay, as the CUSUM values take longer to pass their thresholds. Lowering these thresholds

should give faster detection time, but risks giving false detection, which is a common trade-off in practice [3]. Table 4 summarizes the results for detection time for the current sensor at an injection time of 30,000 s, with different fault sizes and at different cell capacities.

**Figure 10.** Errors and diagnostic results in the case of current sensor fault. (**a**) Error from *R*0; (**b**) CUSUM control chart for *R*0; (**c**) Error from *R*1; (**d**) CUSUM control chart for *R*1; (**e**) Error from *C*1; (**f**) CUSUM control chart for *C*1; (**g**) Isolated current sensor fault FI signal.


**Table 4.** Summary of detection time for current sensor faults at di fferent sizes and cell capacities.

For both voltage and current sensors, more simulations were conducted at di fferent injection times, sizes, and capacities to test the validity and e ffectiveness of the proposed FDI scheme; but it is impossible to show all the results individually, so a summary will be presented. The injection times were set at 10,000 s, 20,000 s, and 30,000 s. It should be noted that faults were not added at the beginning of the runs, due to the proposed FDI scheme's aforementioned inability to detect faults during the converging period of the RLS algorithm, which typically lasts an hour at the start of the battery operation. The considered faults for the voltage sensor are [±0.1 V; ±0.5 V; ±10%], while the considered faults for the current sensor are [±4 A; ±7 A; ±10%]. Approximately 200 runs were simulated. Table 5 shows the results for maximum, minimum, and mean detection time (DT—time from fault occurrence to correct detection of fault), false detection rate (FDR—fraction of tests where fault is detected, but there is no fault) and missed detection rate (MDR—fraction of tests where fault is not detected, but there is a fault). The isolation time depends on the fault size; the larger the fault, the faster the isolation time. It is thus concluded that faults can be detected within a reasonable time using the proposed FDI scheme, with no false detection or missed detection.

**Table 5.** Summary of the performance evaluation metrics.

