The mentioned faults all affect the hydraulic part of the pump. The performance of the hydraulic part of the pump is modelled by fH andfT. These functions describe the pressure and the torque produced by the pump respectively. Moreover the flow measurement is part of the hydraulic description of the pump. Introducing the faults described above, the description of the hydraulics of the pump becomes,
Hp=fH(Q, ωr)−KfQ2−Cchfc−Cdhfd
Tp=fT(Q, ωr) + ∆Bωr−Cctfc−Cdtfd
y3=Q−Kl
pHp.
In this fault model Kf ∈ R+ represents clogging,∆B ∈ R+ represents rub impact, Kl ∈ R+represents increased leakage flow,fc ∈ R+ represents cavitation, andfd ∈ R+ represents dry runnning. The first three signals model the faults accurately, while the last two terms are linear approximations.
c1 : L0s˙isd=−(Rs+Rr0)isd+A1+vsd
c2 : L0si˙sq =−(Rs+Rr0)isq+A2+vsq
c3 : L0m˙imd=−A1+R0risd
c4 : L0mi˙mq =−A2+R0risq
c5 : Jω˙r=Te−Bωr−Tp
c6 : x˙Q =fA(xQ, Hp,uQ) c7 : A1=R0rimd+zpωrL0mimq
c8 : A2=R0rimq−zpωrL0mimd
c9 : Te=32zpL0m(imdisq−imqisd) c10 : Hp=−ah2Q2p+ah1Qpωr+ah0ωr2 c11 : Tp=−at2Q2p+at1Qpωr+at0ω2r c12 : Qp=hA(xQ)
c13 : y1=isd
c14 : y2=isq
c15 : y3=Hp
c16 : y4=Qp ,
(6.31)
where the constraintsc7andc8are included to make the constraintsc1toc4independent.
Hereby Assumption 6.1.3 is fulfilled globally (Blanke et al., 2003).
In (6.31) the constraints c1 toc4 and c7 toc8 describe the electrical part of the induction motor,c5 describes the mechanical dynamics of the pump, andc6describes the dynamics of the hydraulic applications.c9,c10andc11describe the torque generated by the induction motor, the pressure difference generated by the centrifugal pump, and the load torque of the centrifugal pump respectively. c12 describes the volume flow through the pump and finallyc13toc16describe the measurements on the system.
Two extra constraints can be deduced by differentating c7 andc8 with respect to time. Doing this the following additional constraints are obtained,
cd7 : A˙1=R0ri˙md+zpω˙rL0mimq+zpωrL0m˙imq
cd8 : A˙2=R0ri˙mq−zpω˙rL0mimd−zpωrL0m˙imd. (6.32) Beside the constrains presented above, there is a differential constraint for each variable
˙
xd, meaning that a constraint on the form, di: dxdi
dt = ˙xdi, (6.33)
exists for each element inx˙d. In this expressionidenotes theithelement inx˙d. From the constraints presented in (6.31) and (6.32) the set of variables are identified.
These are given by,
Z= ˙xd∪xd∪xa∪u∪y
whereX = ˙xd ∪xd ∪xa are the unknown variables, which must be matched, and K=u∪yare the known variables.xd,xa,u, andyare given by,
xd=£
isd isq imd imq ωr A1 A2 xTQ¤T xc=£
Hp Qp Tp dT¤T
u=£
vsd vsq
¤T
y=£
y1 y2 y3 y4
¤T .
6.4.2 Cascade Connected Systems
From the model of the system presented in Section 6.3 it is seen that the system is non-linear. Therefore, traditionally a nonlinear transformation is required to enable a design of a residual observer (Garcia and Frank, 1997). This transformation is not easy to find due to the nonlinear nature of the system. Moreover, developing Analytical Redundancy Relations (ARR’s), using for example Groebner basis (Cox et al., 1997) for variable eliminations, have not given any usable result. This is also due to the nonlinear nature of the system. Therefore, the system is divided into two cascade-connected subsystems by using structural consideration on the system (Blanke et al., 2003; Izadi-Zamanabadi and Staroswiecki, 2000).
Using SA it is shown that it is in fact possible to make this split. By analysing the relation describing the electrical part of the induction motor, it is seen from Table 6.4 that these constraints form an over-constraint system, see Definition 6.1.7. Therefore from Theorem 6.1.1 the system is structural observable, meaning that the set of constrainsCe
form an observable subsystem for almost all parameters. This set is given by,
Ce={d1, d2, dd7, dd8, c1, c2, c3, c4, c7, c8, c9, cd7, cd8, c13, c14}, (6.34) Ceis defined as the first subsystem. The remaining relations are defined as the second subsystem, meaning that the constrains of this subsystem are given by,
Cm={d5, d6, c5, c6, c10, c11, c12, c15, c16}. (6.35) The connecting variables between the two subsystems are in this work defined as the estimates ofωrandTe, and will in the following be denotedωˆrandTˆerespectively.
It is also seen from the column at the right hand side of Table 6.4 that the faults treated in this chapter are not affecting the relations in the first subsystem. Therefore, it is only necessary to consider the second subsystem when designing residual generators for FDI on the system. Hence, the fault detection algorithm can be divided into two parts as shown in Fig. 6.3.
Using the relations describing the first part, an adaptive observer is designed. This observer observes the variablesωrandTeconnecting the two parts. The design of this observer is considered in Section 6.5. But first SA is used to identify minimal over-constrained subsystems (see Definition 6.1.11) in the system formed by the constraints Cm. The obtained results are presented in the following subsection,
Table 6.4: The structure table of the system.ximplies that the given variable is not cal-culable using the constraint but does appear in the constraint (Izadi-Zamanabadi, 2001).
The symbols
➀
show a matching for the first part of the system.KnownUnknownFaults y1y2y3y4vsdvsquQ˙xQxQTpHpQpTeωr˙ωr˙imd˙imqimdimq˙A1˙A2A1A2˙isd˙isqisdisqKfKl∆Bfcfd c61111 d61x c1011 d5x1 c51111 c9111111 c8111111 c1311 c14111 d31x d41x c9➀1111 cd8➀11111 cd71➀1111 c3➀11 c4➀11 c71➀11 c811➀1 dd1➀x dd2➀x c11➀11 c21➀11 d1➀x d2➀x c141➀ c151➀
First
subsystem Second
subsystem r
vabc iabc
ωr
Te
H
Observer
FDI
Q
Figure 6.3: The division of the fault detection algorithm. In the first subpart the model of the electrical part of the motor is used, and in the second subpart the models of the mechanical and hydraulic parts of the system are used.
6.4.3 Structural Analysis on the Second Subsystem
The SA is in this section used for identification of four minimal over-constrained sub-systems. Each of these subsystem can be utilized for detecting a subset of the faults in the centrifugal pump. The second subsystem is described by the set of constraints Cmin (6.35). Beside these constraints two extra constraints are defined to describe the connecting variablesωˆrandTˆe. These constraints are given by,
c15 : ωˆr=ωr
c16 : Tˆe=Te.
In Table 6.5 the graph of the second subsystem is shown. Here the two extra constraints are added.
Table 6.5: The structural model of the second subsystemCmobtained in Section 6.4.2.
Known Unknown Faults
y3 y4 ωˆr Tˆe uQ x˙Q xQ ω˙r Tp ωr Hp Qp Te Kf Kl ∆B fc fd
c6 1 1 1 1
d6 1 x
c10 1 1
d5 1 x
c5 1 1 1 1
c9 1 1 1 1 1 1
c8 1 1 1 1 1 1
c15 1 1
c13 1 1
c14 1 1 1
c16 1 1
Using the definitions and procedures described in (Blanke et al., 2003; Izadi-Zamanabadi, 2001), four minimal over-constrained subsystems are identified. The set
of constraints contained in each of these system are given by, Cm1={d5, c5, c9, c14, c15, c16} Cm2={c8, c13, c14, c15}
Cm3={d5, c5, c8, c9, c13, c15, c16} Cm4={d5, c5, c8, c9, c13, c14, c16}.
(6.36)
From these four minimal over-constrained subsystems, it is seen that the constraintsd6, c6, andc10 are not contained in any of the matchings. These constraints describe the application in which the pump is placed. Therefore, when these are not used in a match it means that the matching is independent of the application model, and therefore no knowledge about the application is necessary for the algorithm to work.
Looking at the column to the right in table 6.5 the faults affecting each of the sub-systemsCmican be identified. The connections between the faults and the subsystems are shown below,
Cm1:{Kl,∆B, fc, fd} Cm2:{Kf, Kl, fc, fd} Cm3:{Kf,∆B, fc, fd} Cm4:{Kf, Kl,∆B, fc, fd}.
(6.37)
These connections show that the given faults are Structurally monitorable from the given set of constraints, see Theorem 6.1.2 (Blanke et al., 2003).
From the connection in (6.37) it is seen that the faultsfc andfd are indistinguish-able from a structural point of view, meaning that isolation of these faults is impossible using residual generator built on these sets of constraints. Moreover, it is seen that no additional information is added usingCm4. Therefore the set,
{Cm1, Cm2, Cm3},
contains the obtainable information about the faults in the system. The last relationCm4
could be used for validation in a robust fault detection scheme.