CHAPTER 1 Introduction and Literature Review
1.14 Mathematical Modeling
1.14.1 Mathematical Modeling for CP Fault Detection
CPs are majorly made up of the stationary and rotating mechanical components and fluid flow components. Hence, they involve complex fluid-structure interactions. They produce non- stationary signals, and therefore these systems are complicated to accurately model.
Isermann (1984) reviewed fault detection using methods based on modeling and estimation.
Both determinate and indeterminate parameters of the model were used. Every parameter had a certain acceptance limit for the standard value. The model used data from the sensor to monitor essential parameters in the system. An alarm was activated if parameters were said to be outlying their tolerance limits. Later, the site and source of the fault were recognized. Next fault assessment was done. Lastly, a judgment was made whether to consider the fault tolerable or not. The developed technique was applied to a CP energized by a direct current motor. The drawback of this technique was that it indicated that leakage in a CP was not a defect. That is the model was sensitive to the process noise but insensitive to leaks.
In some of the approaches of modeling CP systems, blocks were created to identify the mechanical parts, fluid parts, etc. (Wolfram et al., 2001a; Kallesoe et al., 2004; Kallesoe et al., 2006; Samanipour et al., 2017). These methods involved developing models and parameters for each of the sub-systems and thereby calculating the residuals. The model developed by Wolfram et al. (2001a) consisted of three models for the CP, pipe, and mechanical subsystems, respectively. Separate neural-fuzzy models were used to denote corresponding normal state for each subsystem. To implement the fault detection residuals were calculated. The models were designed in such a way that the individual residuals were only sensitive to the used measurements and the underlying sub-process, and were insensitive
to the faults in measurements or other sub-processes. The faults that were considered are the sensor faults, gap losses, obstruction, cavitation, bearing friction and impeller defects.
Kallesoe et al. (2004) developed a model by dividing the CP system into four sub-systems where each subsystem was sensitive to a subset of faults considered. The developed model was based on the structural analysis (to identify the subsystems), analytical redundancy relation (ARR) and observers design. One of the residual observers was designed for each of the four subsystems. These residuals were further used for the identification/ isolation of the faults. On applying the developed methodology to an industry benchmark condition, it was observed that the algorithm was successful in identifying four out of five faults in the CP.
Where, the five faults considered on the CP were clogging, bearing faults, leakage, cavitation and dry run. The signals that were acquired for the detection were shaft torque, pressure head developed by the CP and the flow through the CP. The method was however sensitive to the change in the operating parameter of the CP.
In another work, Kallesoe et al. (2006) developed a mathematical system model of the CP which needed only two inputs/ variables, and they are the electrical motor measurements (stator voltages, motor current) and the delivery pressure of the CP. The CP model was distributed into two sub-systems, the electrical system, and mechanical/ hydraulic systems.
The faults accounted for were the CP clogging, friction generated because of either rub impact or bearing defects, degradation of CP performance because of cavitation, and dry run. Thus, the considered faults were identified using the ARR. A large residual implied a deviation of the system’s actual behavior and that was expected by the developed model, suggesting a fault
in the system. Though this method detected the CP faults, the transient phases of the system created difficulties for the algorithm.
Samanipour et al. (2017) developed a nonlinear electro-pump model to diagnose cavitation defects in CPs. The model constituted three blocks, including blocks for the induction motor, mechanical CP parts, and hydraulic CP parts. The pressure and torque values were ident ified from the developed model, and their nonconformity to the performance curve of the CP was identified as the residual. The extracted features from the residuals were input to a self- organizing map neural network to identify the state of the system. The low cavitation was detected with 88% accuracy, while the high cavitation was identified with 96% accuracy.
Different other approaches used to model the CP system are given in refs. (Nordmann and Aenis, 2004; Harihara and Parlos, 2006, 2008b, 2008a). Nordmann et al. (2004) constructed a mathematical model of a rotor using the inertia, stiffness, damping, and loading function matrices. The input and output characteristics of the active magnetic bearings (AMB) on the CP were measured. The CP data, which were measured, included the rotor speed, CP flow rate, head developed, radial clearance of the piston seal and the impeller seal. From the AMB, the AMB force, pre-magnetization current, windings per pole pair, air gap, and the cross- sectional area of the pole were used. Using this data, the bearing force was estimated. Twenty- five quantifiable transfer functions created relating the inputs with the force excitation patterns worked as indications to the response of the system. Substantial differences from the model estimated parameters signified the presence of a fault in the system. Recognition of the fault was done by estimating the fault-symptom relations or by generating faults in the model
parameters and relating the faulty model parameters with the actual data. This method was tested on a CP with no fault, and dry-run faults.
A team of researchers developed a sensorless model for CPs using the motor current signature analysis. ‘Sensorless’ here meant that there was no add-on or intrusive sensor. None of CP or motor design attributes were used to develop the model. The developed model was successful in identifying various cavitation states in a CP with very few false alarms. The system model inputs used were different transformed signals processed from the voltages and currents, for example, the voltage level, voltage imbalance, etc. A function was constructed based on the time-varying inputs in the polynomial NARX form. The developed model constituted sampling the motor electrical signals, down-sampling the signals to lower frequency ranges, and later scaling these signals to per-unit values. At a specific CP operating state, the model projected the baseline system response. The residuals were generated by comparing this to that of the actual measured response of the system. When residual was found to be beyond a certain threshold fault was alarmed. Else, the system was considered healthy. Based on various experiments a 5% threshold was used in the study. They applied the aforementioned technique to detect the cavitation (Harihara and Parlos, 2006), successfully, identify varying severity of impeller cracks (Harihara and Parlos, 2008b) and detect motor faults and pump faults (Harihara and Parlos, 2008a). It was observed that the motor line current analysis based model could identify the faults in a shorter span when compared to the model using vibration parameters as an indicator.