4.3 Pump Applications
4.3.5 Sensor Configuration Analysis
eq1 ∼ Increased flow eq2 ∼ Decreased flow eq3 ∼ Not defined
The connection between these effects, and the faults and disturbing events in the system is given by the following logical equation,
eq1
eq2
eq3
←
0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0
µf
d
¶
(4.20) wheref is the fault vector anddis disturbing event vector, both defined in (4.17).
Effects on the impeller eye pressure measurement
It is well known that the inlet pressure has large impact on the pump performance, as cavitation will occur if this pressure becomes too low. If cavitation does occure it will destroy the pump over time. Therefore, by measuring the pressure at the impeller eye it might be possible to detect decreases in this pressure and thereby detect the possibilities for cavitation. Moreover measuring the pressure at the impeller eye pressure noise due to cavitation might be measurable. The effects in the impeller eye pressure are found in componentc7Inlet of the Pump andc5Hydraulic part of the centrifugal pump, where the mean pressure is coming fromc7and the pressure noise due to cavitation is coming fromc5. The measurable effects are,
eeh1 ∼ Noise like pressure oscillations eeh2 ∼ Impeller eye pressure not defined eeh3 ∼ Impeller eye pressure too low
The connection between these effects and the faults and disturbing events in the system is given by the following logical equation,
eeh1
eeh2 eeh3
←
0 0 1 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0
µf
d
¶
(4.21)
connections between faults and effects were established for each of the sensors in the previous section.
The robustness of the different sensor configurations could be analysed using Theo-rem 4.2.2, but by looking at the description of the disturbing events in Section 4.3.3 the following mutually exclusive expressions of the disturbing events are recognized,
de2= 1 → de3= 0 de3= 1 → de2= 0 de4= 1 → de5= 0 de5= 1 → de4= 0 dh2= 1 → dh1= 0 dh1= 1→ dh2= 0
asde2,de4anddh2are increased voltage, frequency and flow respectively, andde3,de5
anddh1are decreased voltage, frequency and flow. These dependencies are taken into account using Corollary 4.2.1. When the robustness properties are establised the pos-sibilities for fault identification of the detectable fault can be analysed, using Theorem 4.2.3.
The results are presented in two logical vectors Rr and Ri, where Rr contains the results of the robustness analysis, andRi contains the results of the identificability analysis. Such that,
if the property holds forfjthenrj := 1,elserj := 0
wherefjis thejthcomponent in the fault vectorfandrjis thejthcomponent in either RrorRidependent on which property is analysed. The fault vectorf is given by,
f =¡
fe1 fh1 fh2 fh3 fm1 fin1 fin2
¢
where eachfjis described in Section 4.3.3. Three sensor configurations are considered in the following, these are,
1. Sensors measuring the electrical quantities, e.i. the voltage and current measure-ments.
2. Sensors measuring the electrical quantities and the pressure difference across the pump.
3. Sensors measuring the electrical quantities, the pressure difference and the volume flow.
4. Sensors measuring the electrical quantities, the pressure difference, the volume flow and the impeller eye pressure.
The results of the analysis of these different sensor configurations are presented in the following.
Electrical measurements
Firstly, only the effects in the electrical measurements are considered, e.i. the effects measurable using the current and voltage sensors. The connection from the faults and disturbing events, to their effect on the electrical quantities is given by (4.18). This equation is on the form,
e←£
Af Ad
¤·
·f d
¸
meaning that Theorem 4.2.2 and Corollary 4.2.1 can be used to establish the logical robust fault detection possibilities. The result of this analysis is shown below,
Rr=¡
0 0 0 0 0 0 0¢
. (4.22)
FromRr it is seen that it is not possible to distinguish any of the faults from possible logical combinations of disturbing events. This means that non of these effects can be used in a robust signal-based fault detection scheme.
Electrical and pressure difference measurements
Secondly, consider the effects on the electrical measurements and the pressure difference measurement. These are given by (4.18) and (4.19). The result of this analysis is shown below,
Rr=¡
0 1 0 0 1 1 0¢
. (4.23)
Here it is seen that 3 of the 6 faults are detectable using signal-based methods.
Normally it is impossible to measure the high frequency components in the pressure signal using standard pressure sensors. To analyse the detection properties under this assumption, the high frequency pressure component eop,h5is removed from the analy-sis. Moreover it is assumed that the pressure value is always available, meaning that the effect eop,h6is also removed. The result of this test is shown below,
Rr=¡
0 1 0 0 1 1 0¢
. (4.24)
By comparing this result with the result from the analysis including the high frequency components, it is seen that all the fault information, not corrupted by disturbing events, is contained in the low frequency parts of the pressure signal.
Electrical, pressure difference and flow measurements
In the third analysis the electrical measurements are combined with both a pressure measurement containing high frequency components and a flow measurement. These are given by (4.18), (4.19) and (4.20) repectively. The result of this test is shown below,
Rr=¡
0 1 0 0 1 1 0¢
. (4.25)
Comparing these results, with the results of the analysis where only the electrical mea-surements and the pressure sensor are considered, it is seen that no addition information is added using the flow sensor. This is in fact true when only logical combinations are considered. But, as it will be shown in Chapter 6, The flow sensor can be used for dis-turbance decoubling, when model-based methods are used. Hereby, it becomes possible to detect the faultfh3corresponding to increased leakage flow inside the pump.
Electrical, pressure difference, flow and impeller eye pressure measurements As the last analysis a pressure sensor measuring the impeller eye pressure is added. This means that effect in the electrical quantities, the pressure difference, the volume flow, and the impeller eye pressure are assumed known. These are given by (4.18), (4.19), (4.20) and (4.21) respectively. The result of this analysis is shown below,
Rr=¡
0 1 1 0 1 1 1¢
. (4.26)
Here it is seen that only the faultsfe1andfh1corresponding to inter-turn short circuit and leakage flow are undetectable. To check the possibilities for identification of the the 5 detectable faults, the effects not corrupted by disturbing events are analysed using Theorem 4.2.3. The logical expression of the faults is given by the following expression.
eem,i4
eem,v1
edh3
edh4
edh6
eq3
eeh2
eeh3
←
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1
·¡ f¢
.
In this expression the effects, which can be corrupted by disturbing events, are removed.
The result of the detectability analysis is shown below, Ri=¡
x 0 0 x 0 1 0¢
(4.27) wherexcorresponds to the fault not robust detectable. Here it is seen that only one fault is distinguishable from the other faults, when all the effects corrupted by disturbing events are removed.