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A bond graph representation of switching devices known for a long time has been a modulated transformer with a modulus b(t)∈{0,1}∀t≥0 in conjunction with a resistor R:Ron accounting for the ON-resistance of a switch considered non-ideal. Besides other representations, this simple model has been used in bond graphs for simulation of the dynamic behaviour of hybrid systems. A previous article of the author has proposed to use the transformer–resistor pair in bond graphs for fault diagnosis in hybrid systems. Advantages are a unique bond graph for all system modes, the application of the unmodified standard Sequential Causality Assignment Procedure, fixed computational causalities and the derivation of analytical redundancy relations incorporating ‘Boolean’ transformer moduli so that they hold for all system modes. Switches temporarily connect and disconnect model parts. As a result, some independent storage elements may temporarily become dependent, so that the number of state variables is not time-invariant. This article addresses this problem in the context of modelling and simulation of fault scenarios in hybrid systems. In order to keep time-invariant preferred integral causality at storage ports, residual sinks previously introduced by the author are used. When two storage elements become dependent at a switching time instance ts, a residual sink is activated. It enforces that the outputs of two dependent storage elements become immediately equal by imposing the conjugate3 power variable of appropriate value on their inputs. The approach is illustrated by the bond graph modelling and simulation of some fault scenarios in a standard three-phase switched power inverter supplying power into an RL-load in a delta configuration. A well-developed approach to model-based fault detection and isolation is to evaluate the residual of analytical redundancy relations. In this article, analytical redundancy relation residuals have been computed numerically by coupling a bond graph of the faulty system to one of the non-faulty systems by means of residual sinks. The presented approach is not confined to power electronic systems but can be used for hybrid systems in other domains as well. In further work, the RL-load may be replaced by a bond graph model of an alternating current motor in order to study the effect of switch failures in the power inverter on to the dynamic behaviour of the motor.
Hybrid system models exploit the modelling abstraction that fast state transitions take place instantaneously so that they encompass discrete events and the continuous time behaviour for the while of a system mode. If a system is in a certain mode, e.g. two rigid bodies stick together, then residuals of analytical redundancy relations (ARRs) within certain small bounds indicate that the system is healthy. An unobserved mode change, however, invalidates the current model for the dynamic behaviour. As a result, ARR residuals may exceed current thresholds indicating faults in system components that have not happened. The paper shows that ARR residuals derived from a bond graph cannot only serve as fault indicators but may also be used for bond graph model-based system mode identification. ARR residuals are numerically computed in an off-line simulation by coupling a bond graph of the faulty system to a non-faulty system bond graph through residual sinks. In real-time simulation, the faulty system model is to be replaced by measurements from the real system. As parameter values are uncertain, it is important to determine adaptive ARR thresholds that, given uncertain parameters, allow to decide whether the dynamic behaviour in a current system mode is the one of the healthy system so that false alarms or overlooking of true faults can be avoided. The paper shows how incremental bond graphs can be used to determine adaptive mode-dependent ARR thresholds for switched linear time-invariant systems with uncertain parameters in order to support robust fault detection. Bond graph-based hybrid system mode identification as well as the determination of adaptive fault thresholds is illustrated by application to a power electronic system easy to survey. Some simulation results have been analytically validated.