CORRELACIONES ESTRATÉGICAS

Physics-based models can be used for FDD purposes using the technique of analytical redundancy; the process of comparing the outputs of a system with the output of a mathematical model in order deviations from a healthy condition. Using the input and output signals of a process, comparisons are made in order to highlight faults in the actual process. This requires that the process is modelled with a reasonable degree of accuracy.

Obtaining accurate models can be achieved through utilising in-depth and verified mathematical relationships or determining certain physical characteristics of the system using identification techniques [55]. The comparison of the mathematical model of the system with the actual system generates ‘residuals’ which allow for the diagnosis of faults [56]. In theory, when faults are present the residual values will be large and when the system is healthy the residuals will be small. The three popular ways to generate residuals are [57]:

1. Parameter Estimation. Generates residuals that are the difference of healthy model parameters and model parameters estimated from system input/output relationships.

2. Observers. Observers estimate the output of the system. Residuals are generated from the difference between measured and estimated signals. 3. Parity Relations. The residuals are generated by direct comparison of the

model and system outputs.

The direct-quadrature (DQ) induction motor model in DQ [21] or DQ0 [9] form finds widespread use in the field of induction motor modelling [9], [21], [58]–[70] . In addition to the DQ transform the Concordia transform also results in a two-

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axis model which can then be used for simulation purposes [71], [72]. The model is obtained by transforming the three-phase system into a two-phase system using a transform similar to the Park transform (see previous section). The DQ model is relatively simple to implement in modern software [9] and compared to some of the more complex motor models has a minor computational burden.

The simple DQ0 model in its most basic form has no ability to simulate faulty machines since it assumes sinusoidal balanced winding distributions in the stator and sinusoidal balanced equivalent windings for the rotor. The rotor cage is simulated as a three-phase set of windings since the magnet-motive force (MMF) distribution of a rotor cage approximates a sinusoidal distribution. Several extensions to the basic DQ0 model introduce the capability to simulate motors in unbalanced modes of operation i.e. operation in the presence of faults, and also simulate machines taking into account the non-uniform distribution of conductors in the stator and rotor which makes the DQ0 model suitable for use in a model-based IM FDD system.

The DQ0 balanced rotor equivalent windings can be replaced by a detailed model of the rotor including equivalent circuits for each of the individual rotor loops (two adjacent bars and a section of each end-ring) [21]. The inductance and resistance terms for a given number of loops can then be modified to simulate damage or complete bar breakages. This method has also been applied to an ABC (3 axis) reference frame model [73]. An additional feature of the detailed rotor loop model is the ability to locate the angular position of the rotor fault with respect to some arbitrary reference angle [74]. In research terms this is an interesting development but offers little practical use since the

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absolute position of the rotor will not be tracked over time nor will it be recorded when the machine is assembled, therefore knowledge of the location of the rotor fault is largely irrelevant.

If the detailed rotor cage model is coupled with a winding function approach (WFA) model of the stator winding then the DQ0 model gains the ability to simulate rotor slot harmonics (RSH) [59]. The rotor slot harmonics are generated as the rotor bars pass the individual winding coils. The WFA model replaces the sinusoidal stator winding distributions with a ‘winding function’ which provides information on the density of conductors as a function of angular position, these parameters then dictate the air-gap field strength. For this kind of model a detailed knowledge of the machine construction is required including information regarding the air-gap width, number of stator coils, number of turns per coil and the number of rotor bars. This information is not commonly available and can only be obtained direct from the manufacturer by special request which makes techniques requiring a large number of detailed machine parameters such as this prohibitive. The DQ0 model including RSH does however provide very useful information since the RSH are useful for two key purposes; speed estimation and eccentric air-gap fault detection [75].

Winding function methods calculate machine inductances based on the stator winding layout, rotor cage dimensions and electrical parameters. The inductances are also a function of the air-gap permeance. Furthermore, the air- gap permeance is a function of air-gap width and thus if the air-gap width is provided as a function of stator and rotor angular positions then the conditions for both static and dynamic eccentricity faults can be modelled [76], [77].

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Stator winding turn-to-turn faults can be modelled by including parameters in the model which allow modification of the stator inductance and resistance matrices; this creates a set of electrically unbalanced stator windings which simulate the stator turn-to-turn winding fault [9]. The most commonly applied method used to extend the simple IM to a stator fault model is to include parameters which define the number or percentage of shorted turns in one winding. The magnitude of this parameter then modifies the inductance and resistance terms for that particular winding. Some stator fault models exist which allow for localization of the fault by including an angle parameter estimating the mechanical angle between the inter-turn short and the stator winding a-axis [74]. The modifications that are made to the stator inductance and resistance matrices are applied to the ABC 3-axis model then, depending on whether the 2-axis or 3-axis model is to be simulated, the matrices are inputted directly into the simulation [78] or are converted to 2 axes (DQ0 model) by applying the mathematical transform [9].

The final major fault zone, the shaft bearings require additional information on the dimensions of the machine. These parameters are crucial for effective modelling, namely; contact angle, inner race diameter, outer race diameter and number of rolling elements. Once these details have been included in the model, periodic frequencies relating to the impacts between bearing contact surfaces and a bearing fault (inner race, outer race, cage or rolling element) can be predicted. This information can then be used to simulate the periodic changes in load torque and air-gap width (and thus air-gap permeance) which allows the effect of bearing fault on the motor line currents to be modelled [79]. More advanced bearing models have been developed which allow modelling of bearing dynamics under high frequency resonant conditions by including the

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effects of the stiffness of the bearing supports, components and the shaft, masses of the individual components and also measured vibration responses [80]. Clearly several additional parameters are required for inclusion of bearing subsystems within a motor model adding to the complexity and creating an issue as to how the values for these parameters are obtained or estimated.

The preceding paragraphs highlight the common approaches used to create and simulate physical models of the induction motor in healthy configuration and also in the presence of rotor, air-gap eccentricity, stator and bearing faults. In addition there is a description of the techniques that utilise these models to provide useful ‘features’ for use in a FDD system, namely, parameter estimations, observers, or parity relations. If enough information can be obtained on a given induction motor the modelling approach provides a powerful method of detecting and diagnosing a fault easily since the variation in specific parameters and states can be linked directly to specific faults.