In this chapter the FDI problem is introduced. Two different approached are considered, these are the signal-based fault detection approach and the model-based fault detection approach. It is argued that the signal-based approach has its advantages when a model of the system is not available. However, robustness properties are difficult to establish.
For the model-based approach it is argued that the advantages are in its ability to obtain robustness from theoretical consideration, and the drawback is the need for reasonable good models.
This introduction is followed by a state of the art analysis of Fault Detection and Identification (FDI) in centrifugal pump applications. Centrifugal pumps are mostly driven by induction motors, therefore state of the art in the area of FDI in induction motors is also considered. Here, it is stated that the most common faults in induction motors are bearing faults. However, from internal data at Grundfos it is known that stator burnouts are one of the main reason for faults in submersible pumps. Both signal-based and model-based approaches have been used for detecting stator faults. The signal-based approaches are mostly concerned in finding fault signatures in the stator current.
The model-based approaches are mostly based on steady state impedance models of the machine. This basically means that robustness with respect to dynamic changes in the motor speed and motor current is not considered.
From the state of the art analysis of FDI in centrifugal pumps it is seen that differ-ent cdiffer-entrifugal pump faults are considered, and that differdiffer-ent methods are used for their detection. However, the model-based approach is fare less used than signal-based meth-ods. This might be due to the nonlinear nature of the centrifugal pump model. It is well known that frequency converters, making it possible to optimize the operating point of the pump, are used more and more often as drives for centrifugal pumps. However, this means that the detection algorithms should not only be robust with respect to changes in the hydraulic resistance, i.e. the flow through the pump, but also to speed changes. This has not be considered in any of the presented papers.
Model of the Centrifugal Pump
In this chapter the mathematical model of the centrifugal pump is presented. The chapter starts by describing the mechanical construction of a standard centrifugal pump. Here it is argued that the model of the pump can be divided into three subparts,
• The induction motor driving the pump.
• The hydraulics of the pump.
• The mechanical parts of the pump.
The induction motor is modelled using a so-called dq-model of the motor dynamics.
This type of model is extensively described in the literature. The description presented here is based on (Krause et al., 1994; Kazmierkowski, 1994; Novotny and Lipo, 1996).
The steady state performance of the hydraulics of the centrifugal pump is extensively described in the literature too, (Sayers, 1990; Stepanoff, 1957) and others. Here this steady state description is extended to cover the dynamics of the centrifugal pump as well as the steady state operation, making it particular suitable in model based FDI algorithms. The same approach is in (Gravdahl and Egeland, 1999) used for modelling a centrifugal compressor, but here the dynamics are neglected. Dynamics of centrifugal pumps are treated in for example (Bóka and Halász, 2002).
In this work the model is derived using the control volume approach (Roberson and Crowe, 1993). The derived model expresses the theoretical performance of the impeller.
To obtain a model describing the performance of a real pump extra pressure losses are added to the theoretical model (Sayers, 1990; Stepanoff, 1957). The obtained model describes the performance of a single impeller. However, it is shown that the same model structure also describes the performance of a multi stage pump.
The mechanical part of the pump is modelled using simple considerations based on Newton’s second law. The frictions losses in the bearing and seals are modelled by a simple linear friction term, as the friction losses are very small compared to the torque necessary the drive the pump, and therefore are not important in the model.
The first section of this chapter contains a description of the mechanical construction of the centrifugal pump. The second section describes the induction motor model, and the third section contains the derivation of the model modelling the hydraulics of the centrifugal pump. The fourth section presents the mechanical model, and in the fifth section each of the submodels, derived in the previous sections, are composed into the final nonlinear state space model of the centrifugal pump. Finally, concluding remarks end the chapter.
3.1 The Construction of the Centrifugal Pump
In this section the mechanical components of the centrifugal pump are described. This is done in order to give an overview over the construction of the pump. This description is included to help the reader to follow the model derivations presented in the following parts of this chapter.
In Fig. 3.1 a CR5-10 Grundfos centrifugal pump is shown. This centrifugal pump contains the same set of components as almost all other centrifugal pumps, and is in this section used as an example of a standard centrifugal pump. In Fig. 3.1 the pump is sliced revealing the inside of the pump. The CR5-10 pump is a multistage centrifugal pump, meaning that the pressure is increased using a set of identical impellers, see Fig.
3.1. The impellers are the rotating part of the pump, which increase the pressure by the utilization of the centrifugal force induced by the rotation. This effect is formalized in section 3.3.
The pump is driven by an 1.5 [KW] induction motor, which is connected to the pump by a shaft connection, see Fig. 3.1. This is a typical way to drive centrifugal pumps in the rang from 50 [W] up till several hundreds [KW]. The pumps considered in this thesis have the same structure as the one shown in Fig. 3.1.
A signal flow diagram of such a centrifugal pump is shown in Fig. 3.2. Here the pump is divided into four subsystems. These subsystems are,
• The electrical part of the induction motor. This part converts electrical energy into mechanical energy.
• The mechanical part of the induction motor and the pump. This part connects the impeller to the rotor of the induction motor.
• The hydraulic part of the pump. This part converts mechanical energy into hy-draulic energy.
• The hydraulic application. This part absorbs the hydraulic energy delivered by the pump.
The first three of these are parts of the centrifugal pump itself, and the last part is the application in which the pump is placed. As the topic of this thesis is FDI on centrifugal pumps only the first three parts are considered in the following.
Electrical motor
Shaft connection
Centrifugal pump
Impellers
Inlet Outlet
Figure 3.1: A multistage centrifugal pump driven by an induction motor.