# Structural Methodologies for Distributed Fault Detection and Isolation
## 5 Case Study  
The ADAPT-Lite system is designed to emulate the operation of generic spacecraft electrical power distribution systems [30]. The system has five subsystems: (1) the battery, (2) the Direct Current (DC) electric load, (3) the inverter, (4) the Alternating Current (AC) resistive electric load, and (5) the electric fan as a second inductive load for the AC system (see Figure 11). Seven measurements are made on the system: \(y_{E240}\), \(y_{E242}\), and \(y_{E281}\) represent DC voltage measurements in the system; \(y_{IT240}\) represents the battery current; \(y_{E265}\) represents the inverter AC output voltage; \(y_{IT267}\) is the inverter AC output current; and \(y_{ST516}\) is the fan rotational speed. Six faults are considered in the system: \(f_{E240}\) and \(f_{E242}\) are sensor faults in \(y_{E240}\), and \(y_{E242}\), respectively; \(f_{dc}\) represents a fault in the DC load; \(f_{INV}\) models inverter faults; \(f_{ac}\) represents a fault in the AC load; and \(f_{fan}\) is a fan fault. The ADAPT-Lite system has several Circuit Breakers (CB236, CB262, CB266, and CB280), and relays (EY244, EY260, EY281, EY272, and EY275) and, therefore, operates as a hybrid system with multiple modes (configurations). In previous work [28], we discussed structural diagnosis for hybrid systems. In this paper, we focus on distributed diagnosis. Therefore, we assume all the circuit breakers and relays are on and there is no mode change in the system. The set of equations in each subsystem is derived as follows. _Subsystem 1 (battery)_: The set of equations:  
\[ \begin{split} e_{a1}:\dot{v}_{0}&=\frac{1}{C_{0}} (-i_{B})\qquad\qquad e_{a5}:v_{B}=v_{0}-v_{s}\qquad\quad e_{a9}:y_{E240}=v_{1}+ f_{E240}\\ e_{a2}:v_{0}&=\int\dot{v}_{0}dt\qquad\qquad e_{a6}:v_{ 1}=v_{B}\qquad\qquad\quad e_{a10}:y_{E242}=v_{2}+f_{E242}\\ e_{a3}:\dot{v}_{s}&=\frac{1}{C_{s}}(i_{B}R_{s}-v _{s})\qquad e_{a7}:v_{2}=v_{1}\qquad\qquad e_{a11}:y_{IT240}=i_{B},\\ e_{a4}:v_{s}&=\int\dot{v}_{s}dt\qquad\qquad e_{a8}:v_{ 3}=v_{2}\end{split}  \tag{13}\]  
where \(V_{a1}=\{\dot{v}_{0},v_{0},i_{B},\dot{v}_{s},v_{s},v_{B},v_{1},v_{2},v_{3}\}\) is the set of unknown variables in this subsystem, the set of measurements is \(M_{a1}=\{y_{E240},y_{E242},y_{IT240}\}\), \(F_{a1}=\{f_{E240},f_{E242}\}\) represents subsystem faults, and \(C_{0}\), \(C_{s}\) and \(R_{s}\) are the component parameters in the subsystem. The battery is directly connected to the second subsystem (DC load). _Subsystem 2 (DC load)_: The DC load is modeled by an electric resistance, \(R_{dc}\). The set of equations for this subsystem is:  
\[ \begin{split} e_{a12}:v_{3}&=v_{dc}\qquad\qquad e_{ a14}:i_{dc}=f_{dc}\frac{v_{dc}}{R_{dc}}\\ e_{a13}:i_{B}&=i_{dc}+i_{inv}\qquad e_{a15}:y_{E281}=v_{ dc},\end{split}  \tag{14}\]  
Figure 11: ADAPT-Lite subsystems [30]. CB, Circuit Breaker. where \(V_{a2}=\{v_{3},v_{dc},i_{B},i_{dc},i_{inv}\}\) are unknown variables, \(M_{a2}=\{y_{E281}\}\) are measurements, \(F_{a2}=\{f_{dc}\}\) are faults, and \(R_{dc}\) is a component parameter in the subsystem. Subsystems 1 and 2 are first order connected, and their shared variables are \(V_{a1}\cap V_{2}=\{v_{3},i_{B}\}\). _Subsystem 3 (inverter)_: The inverter converts DC power to AC. When there is no fault in the subsystem and the input voltage, \(v_{in}\), is above 18 V, the output voltage, \(v_{rms}\), stays at 120 V. \(R_{inv}\) represents the internal resistance in the inverter, and \(e\) is the inverter efficiency coefficient. The set of equation for the subsystem is:  
\[ \begin{split} e_{a16}:v_{in}&=v_{dc}\qquad\qquad \qquad\qquad\qquad e_{a18}:i_{inv}=\frac{v_{rms}.i_{rms}}{e.v_{in}}+\frac{v_{inv }}{R_{inv}}\\ e_{a17}:v_{rms}&=120.(v_{in}>18).f_{INV}\qquad e_{a 19}:y_{E265}=v_{rms},\end{split}  \tag{15}\]  
where \(V_{a3}=\{v_{in},v_{dc},v_{rms},i_{inv},i_{rms}\}\) are unknown variables, \(M_{a3}=\{y_{E265}\}\) is a measurement, \(F_{a3}=\{f_{INV}\}\) is a fault, and \(\{e,R_{inv}\}\) are parameters of the subsystem.