The first topic of the work presented in this chapter is realization of over-constrained subsystems identified using Structural Analysis (SA). It is well known that there is a straightforward connection between Analytical Redundant Relations (ARR’s) and mini-mal over-constrained subsystem. Unfortunately, the obtained ARR’s are in general func-tions of the derivatives of the measurements, which are difficult to calculate when the measurements are corrupted by noise.
To overcome this problem a new method for rewriting a subsystem identified using SA to a state space description is developed. The obtained state space description has the property, that the only unknown variables are the states of the system. The state space
0 2 4 6 8 10
−1.5
−1
−0.5 0 0.5 1 1.5
Robustness
time [sec]
residuals
r1 r2 r3
0 2 4 6 8 10
0 0.5 1 1.5 2 2.5 3
time [sec]
Decisions
D1 D2 D3 Valve opening step 1 Valve opening step 2
(a) Robustness test.
0 2 4 6 8 10 12 14 16 18
−1.5
−1
−0.5 0 0.5 1 1.5
Clogging
time [sec]
residuals
r1 r2 r3
0 2 4 6 8 10 12 14 16 18
0 0.5 1 1.5 2 2.5 3
time [sec]
Decisions
D1 D2 D3
(b) Detection of the faultKfclogging.
Figure 6.6: Test results from test of the developed algorithms on the test setup. The top figures shows the obtained residuals and the bottom figures shows decision signals obtained from CUSUM algorithms.
0 2 4 6 8 10
−2
−1 0 1 2
Leakage Flow
time [sec]
residuals
r1 r2 r3
0 2 4 6 8 10
0 0.5 1 1.5 2 2.5 3
time [sec]
Decisions
D1 D2 D3
(a) Detection of the faultKlleakage flow.
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−2
−1 0 1 2
Rub Impact
time [sec]
residuals
r1 r2 r3
0 2 4 6 8 10 12 14 16
0 0.5 1 1.5 2 2.5 3
time [sec]
Decisions
D1 D2 D3
(b) Detection of the fault∆Brub impact.
Figure 6.7: Test results from test of the developed algorithms on the test setup. The top figures shows the obtained residuals and the bottom figures shows decision signals obtained from CUSUM algorithms.
0 2 4 6 8 10 12 14 16 18
−5 0 5
Cavitation
time [sec]
residuals
r1 r2 r3
0 2 4 6 8 10 12 14 16 18
0 0.5 1 1.5 2 2.5 3
time [sec]
Decisions
D1 D2 D3 End of cavitation
(a) Detection of the faultfccavitation.
0 2 4 6 8 10 12 14 16 18
−15
−10
−5 0 5 10 15
Dry Running
time [sec]
residuals
r1 r2 r3
0 2 4 6 8 10 12 14 16 18
0 0.5 1 1.5 2 2.5 3
time [sec]
Decisions
D1 D2 D3 Air valve opened Air valve closed
(b) Detection of the faultfddry running.
Figure 6.8: Test results from test of the developed algorithms on the test setup. The top figures shows the obtained residuals and the bottom figures shows decision signals obtained from CUSUM algorithms.
description can then be used for derivation of residual observers. In the design of these observers decoupling of unknown inputs has not to be considered, as the only unknown variables are the states of the system. The method is tested on a satellite model, showing that a residual observer can be obtained using this approach. Moreover the method is used to develop residual observer for the centrifugal pump application, which is the second topic of the chapter.
The second topic of the chapter is fault detection and isolation in a centrifugal pump placed in an arbitrary hydraulic application. An algorithm, which is capable of detection and isolation of five faults in a centrifugal pump, is developed. The proposed alogrithm is independent of the application in which the pump is placed. This makes the algorithm robust and usable in a wide range of applications, such as submersible application, waste water application, and heating application.
Tests have shown that it is possible to distinguish between four of the five faults un-der consiun-deration, using three chosen residuals. But it is also shown that the algorithm is sensitive to the operating point. This is partly due to dependency between the operating point and the parameters in the model and partly due to flow sensor problems at zero flow. Even though the algorithm has a small inherent dependency of the operating point, it still performs considerably better than algorithms built on a linearized model, when the operating point is changed.
FDI on the Centrifugal Pump: A Steady State Solution
In this chapter Structural Analysis (SA) is utilized to derive Analytical Redundant Re-lations (ARR’s), which can be used for fault detection. The concept of SA is described in Section 6.1 in the previous chapter. The obtained ARR’s are based on a steady state model of the pump. Here a steady state model denotes a model describing the pump under constant speed, pressure and flow conditions. The developed detection algorithm is only using electrical measurements on the motor, and the pressure and flow measure-ments on the centrifugal pump. The electrical measuremeasure-ments are in this case Root Mean Square (RMS) measurements of the voltage and current, the voltage frequency, and the electrical angle between the voltage and current. These measurements become constant when the system is running under steady state conditions, and can therefore be sampled at any given sample rate, provided the steady state conditions. Therefore, by using only these measurements the microprocessor load of the derived algorithm can be chosen freely. Moreover, the steady state measurements of the electrical signals are sometimes made available by modern motor protection units. All in all this makes the algorithm suitable for implementation in cost sensitive products.
The obtained algorithm is robust with respect to parameter variations and the oper-ating point of the pump respectively, making it usable in real life applications. In the development of the FDI algorithm, the approach, presented in Section 6.4, of dividing the system into two cascade-connected subsystems is used. This is done to avoid two large residual expressions. The first of these subsystems consists of the electrical part of the induction motor driving the pump, and the second subsystem consists of the me-chanical and hydraulic part of the pump. Theoretical considerations regarding to Struc-tural Analysis (SA) are, among others, found in (Blanke et al., 2003; Izadi-Zamanabadi, 2001).
The faults, which only affect the second subsystem, are detected using Analytical
Redundancy Relations (ARR’s). These relations are obtained through the utilization of structural analysis (Blanke et al., 2003) and the Groebner basis algorithm (Cox et al., 1997). Using this approach the obtained ARR’s are polynomial. This is utilized in the development of a detection algorithm, which is robust with respect to parameter variations. Theoretical results on using the Groebner basis are given in (Staroswiecki and Comtet-Varga, 2001). Moreover, an overview of model-based methods using ARR’s is given in (Staroswiecki, 2000).
This chapter starts by presenting the steady state model of a centrifugal pump placed in an arbitrarily hydraulic application. This is done in Section 7.1. Structural analysis on the derived steady state model is considered in Section 7.2, which includes; the division of the system into two suitable subsystems, derivation of an expression for calculating the connecting variables, and ARR expressions for fault detection. In Section 7.3 a method for robust fault detection using ARR’s is considered. Section 7.4 presents some test results obtained on the industrial test bench, which also was used to obtain the test results presented in Chapter 6. Finally concluding remarks end the chapter.
7.1 Steady State Model of the System
In this section, the model of the centrifugal pump derived in Chapter 3, is reformulated to describe the pump under steady state conditions, only. The section is divided into three subsections, where the first one is concerned with the steady state model of the motor. The second one is concerned with the steady state model of the mechanical and hydraulic parts of the pump, and finally the last one is concerned with fault modelling.
The obtained model will in the following be used in the derivation of a FDI algorithm based on low bandwidth measurements, such as Root Mean Square (RMS) measurement of the electrical quantities.
7.1.1 Steady State Motor Model
The model of the induction motor is described in Section 3.2. In this section both a model of a Y-connected and a∆-connected motor is derived. The obtained models are prestented in (3.9) and (3.10) respectively. Comparing these models it is seen that the following equations describe the motor in both cases,
L0sdidtsdq =−(Rs+R0r)isdq+ (R0r−zpωrJL0m)imdq+vsdq
L0mdimdqdt =R0risdq−(R0r−zpωrJL0m)imdq. (7.1) The variablesvsdq,isdq, andimdqare in general unknown. However, in the Y-connected case the measurable electrical quantities at the terminals of the motor itdq andvtdq
equalsisdq andvsdq respectively. In the∆-connected case these currents and voltages are given by the transformationsitdq = Ciisdq andvsdq = Bvvtdq respectively, see Section 3.2.3.
To obtain a steady state model of the motor, a transformation of the motor states is used. This transformation takes the states described in the stator fixed reference frame xdq and transform these to an arbitrary frame. In this case the arbitrary frame rotates with the frequency of the supply voltage ωe. The new states are denoted xedq. The transformation is given byxdq =T(θe)xedq, whereθeis the angle between the stator fixed frame and the arbitrary rotating frame.Tis given by
T(θe) =
·cos(θe) −sin(θe) sin(θe) cos(θe)
¸
, (7.2)
whereθeis a function of time. Using this transformation to transformvsdq,isdq, and imdq, a new model is obtained with the property that all states are constant during steady state operation. Setting the derivatives of the states equal to zero, the following steady state model of the motor is obtained,
−ωeL0sIsqe =−(Rs+R0r)Isde + (R0rImde −zpωrL0mImqe ) +Vsde ωeL0sIsde =−(Rs+R0r)Isqe + (R0rImqe +zpωrL0mImde ) +Vsqe
−ωeL0mImqe =−(R0rImde −zpωrL0mImqe ) +R0rIsde ωeL0mImde =−(R0rImqe +zpωrL0mImde ) +R0rIsqe .
Defining Isde = 0 in the above model, meaning that the rotating reference frame is aligned with the currentIsqe, the final steady state model of the motor is obtained,
−ωeL0sIsqe = (R0rImde −zpωrL0mImqe ) +Vsde
0 =−(Rs+Rr0)Isqe + (R0rImqe +zpωrL0mImde ) +Vsqe
−ωeL0mImqe =−(Rr0Imde −zpωrL0mImqe )
ωeL0mImde =−(Rr0Imqe +zpωrL0mImde ) +R0rIsqe .
(7.3)
The torque expression of the motor is given in (3.11) and repeated here for the sake of convenience,
Te= 3
2zpL0m(imdisq−imqisd) .
Using the transformationT, given in (7.2), a torque expression of the new variables is obtained. The expression becomes
Te= 3 2zpL0m¡
Imde Isqe¢
. (7.4)
Equations (7.3) and (7.4) form the model of the induction motor during steady state op-erations. However, the electrical quantitiesVsde,Vsqe, andIsqe are not directly measurable.
Therefore, a connection between these quantities and the measurable quantities must be established. The measurable quantities are the RMS values of the supply voltageVrms
and currentIrms, the supply frequencyωeand the electrical angle between the voltage
and currentφ. The angleφis in this case defined as the angle between the voltage vec-torVetdq =£
Vtde Vtqe¤T
, and the current vectorIetdq = £
0 Itqe¤T
, where subscriptt denotes quantities available at the terminals of the motor. The Euclidean length of the vectorsVetdqandIetdqis connected to the RMS values of the measurable electrical quan-tities by|Vtdqe |=√
2Vrmsand|Ietdq|=√
2Irms. Using the RMS valuesVrmsandIrms
and the angleφthe voltagesVtde andVtqe, and the currentItqe can be calculated using Vtde =−√
2Vrmssin(φ) Vtqe =√
2Vrmscos(φ) Itqe =√
2Irms. (7.5) In the case of a Y-connected motor the electrical quantities at the motor terminalsVtde, Vtqe, andItqe and the quantitiesVsde,Vsqe, andIsqe in (7.3) are equivalent. However, in the case of a∆-connected motor the transformation matricesBv andCi have to be considered. These matrices are defined in Section 3.2.3. It can be shown that
Bv=√
3T∆ C−1i = 1
√3T∆,
whereT∆ is a rotation matrix. Using these expressions in (7.1) to obtain the steady state model, it is seen that (7.3) can be used to model the steady state operation of the motor when it is connected in a∆-connection if the following scalings of the electrical quantities are used,
Vtde =√
3Vtde Vtqe =√
3Vtqe Isqe = 1
√3Itqe . (7.6)
7.1.2 Steady State Pump Model
The hydraulic and the mechanical parts of the pump are derived in Section 3.4 and Section 3.3 respectively. Moreover, a model of a general hydraulic application is given in Section 6.3.2. In Section 3.4 it is shown that the mechanical part is described by the following differential equation,
Jdωr
dt =Te−Bωr−Tp
whereTeis the torque produced by the motor,ωris the speed of the shaft, andTp the load torque produced by the pump. Under steady state operation the speed is constant, i.e. dωdtr = 0, meaning that the steady state model of the mechanical part becomes,
0 =Te−Bωr−fT(Qp, ωr). (7.7) The hydraulic parts of the pump is considered as the parts involved with the energy transformation from mechanical to hydraulic energy. In the model derived in Section 3.3 this energy transformation is described by two maps given by
Tp=fT(Qp, ωr) Hp=fH(Qp, ωr), (7.8)
whereTpandHp are the load torque and pressure produced by the pump respectively.
The arguments of the mapsfT andfHin (7.8) are the volume flowQpand the speed of the pumpωr. The mapsfT andfH are given by,
fH(Qp, ωr) =ρg¡
−ah2Q2p+ah1Qpωr+ah0ωr2¢ fT(Qp, ωr) =−at2Q2p+at1Qpωr+at0ωr2.
A general model of a hydraulic application of a centrifugal pump is given in Sec-tion 6.3.1. Here it is argued that such a model, in most cases, can be described by the following state space model,
˙
xQ=fA(xQ, Hp,d) Qp=hA(xQ),
wherexQis the state vector of the application model,HpandQpare the pressure and volume flow of the centrifugal pump, anddis a vector containing some unknown input signals to the application. If it is assumed that the derivative of the statesx˙Qequals zero during steady state operation, the steady state model becomes,
0=fA(xQ, Hp,d)
Qp=hA(xQ). (7.9)
7.1.3 The Fault Models
In Section 6.3.2 five faults are included in the centrifugal pump model. These faults are all affecting the hydraulic part of the pump. The faults are,
1. clogging inside the pump,
2. increased friction due to either rub impact or bearing faults, 3. increased leakage flow,
4. performance degradation due to cavitation, 5. dry running.
In the model of the system used here, the hydraulic part of the pump is modelled by the mapsfH,fT and the flow measurement. Introducing the faults described above, the description of the hydraulic part 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.