EP1564411B1 - Method for detecting operation errors of a pump aggregate - Google Patents

Abstract

In a process to detect a fault during operation of an electric motor-powered pump, a monitoring has two electric wires linking the electric motor to a power monitoring system, and a further sensor monitoring a hydraulic parameter in the pump. The electrical and hydraulic values are logged and processed in an algorithm by a data processing system, the results being compared with pre-stored values. Also claimed is a commensurate pump monitoring system.

Description

The invention relates to a method for determining errors in the operation of a pump unit, in particular a centrifugal pump unit according to the features specified in the preamble of claim 1.

Out EP-A-1 286 056 It belongs to the state of the art to detect in a pump unit cavitation in the pump by means of sensors which record the pump pressure and the flow rate. The data recorded by the sensors are fed to a classification system which determines through a neural network whether and to what extent cavitation exists.

Out EP-A-0 321 295 It belongs to the state of the art to detect by means of a variety of sensors hydraulic system data and mechanical data of the pump motor to detect critical operating conditions of the pump and turn off the pump if necessary in time when the pump z. B. runs dry or promotes closed valves.

In the meantime, pump units are therefore considered to be the state of the art for providing a large number of sensors, on the one hand in order to detect operating conditions, and on the other hand also to determine fault conditions of the system and / or the pump unit. The disadvantage here, however, is that the sensors required in this context not only consuming and expensive, but is often prone to failure. Gats to create, which is executable with the least possible sensor technology and a device for carrying out the method.

This object is achieved according to the invention by the features specified in claim 1. Advantageous embodiments of the method according to the invention will become apparent from the dependent claims, the following description and the figures.

The basic idea of the present invention is to detect characteristic data which are generally available anyway or at least with little effort to determine the motor's electrical variables and at least one variable hydraulic variable of the pump for the electric motor and the hydraulic-mechanical pump and to evaluate them by mathematical linkage. In the simplest form, this is done by comparison with predetermined values, wherein both the comparison and the result is done automatically by means of electronic data processing, which thus determines whether an error in the operation of the pump is present or not.

The method according to the invention requires a minimum of sensor technology and, in the case of modern pumps, which are typically frequency-controlled, which in any case have digital data processing, can generally be implemented by software. It is particularly advantageous that the electrical power of the motor determining variables, namely typically the voltage applied to the motor and the motor current, anyway within the frequency converter electronics are available, so that for detecting a hydraulic variable, eg the pressure only one Pressure sensor is required, which incidentally in modern pumps also often Standard features. The predetermined values required for comparison can be stored in digital form in corresponding memory components of the engine electronics.

According to the invention, it is provided that, on the one hand, the two electrical quantities of the motor determining the electrical power of the motor, preferably the voltage applied to the motor and the current supplying the motor, are mathematically linked to obtain at least one comparison value and, on the other hand, the at least one variable hydraulic variable the pump and a further determining the performance of the pump mechanical or hydraulic variable to obtain at least one further comparison value are mathematically linked, in which case it is determined by comparison with predetermined values based on the result of the mathematical link, whether an error exists or not. The mathematical combination is carried out for the motor-side data by appropriate for the electrical and / or magnetic relationships in the engine determining equations whereas equations are used for the pump, which describe the hydraulic and / or mechanical system. The values resulting from the respective links are compared either directly or with predetermined values stored in the memory electronics, after which the electronic data processing automatically determines whether an error exists or not. In the direct comparison, the error magnitude is calculated as a deviation between a quantity derived from the engine model, e.g. B. T _e or ω and a corresponding from the mechanical-hydraulic model resulting size determined. The method according to the invention has the advantage that it requires little storage space for the given values.

It can not only be determined with the inventive method whether an error exists, but it can also be advantageous even the error can be specified, ie it can be determined which error it is.

As a hydraulic variable to be detected, the pressure or differential pressure generated by the pump is advantageously used, since this size can be detected on the aggregate side and the provision of such a pressure sensor in numerous pump designs today is state of the art.

As an alternative or in addition to detecting the pressure, the amount delivered by the pump can advantageously also be used as a hydraulic variable. The detection of the flow rate can also be done on the aggregate side, also for this are less complex and long-term stable measuring systems at your disposal.

Since the absolute pressure detection of the pressure generated by the pump always represents a differential pressure measurement against the outside atmosphere, it is often cheaper to detect the differential pressure formed between the suction and discharge side of the pump instead of the absolute pressure, which is also much cheaper to further process as a hydraulic variable of the pump.

$${\mathit{Lʹ}}_s=\frac{{\mathit{di}}_{\mathit{sq}}}{\mathit{dt}}=-{\mathit{Rʹ}}_s{i}_{\mathit{sq}}+\frac{L_m}{L_r}\left({\mathit{Rʹ}}_r{\mathit{\psi }}_{\mathit{rq}}+z_p\omega {\mathit{\psi }}_{\mathit{rd}}\right)+v_{\mathit{sq}}$$

$$\frac{{\mathit{d\psi }}_{\mathit{rd}}}{\mathit{dt}}=-{\mathit{Rʹ}}_r{\mathit{\psi }}_{\mathit{rd}}-z_p\omega {\mathit{\psi }}_{\mathit{rq}}+{\mathit{Rʹ}}_r{L}_m{\mathit{i}}_{\mathit{sd}}$$

$$\frac{{\mathit{d\psi }}_{\mathit{rq}}}{\mathit{dt}}=-{\mathit{Rʹ}}_r{\mathit{\psi }}_{\mathit{rq}}+z_p\omega {\mathit{\psi }}_{\mathit{rd}}+{\mathit{Rʹ}}_r{L}_m{\mathit{i}}_{\mathit{sq}}$$

$$T_e=z_p\frac{3}{2}\frac{L_m}{L_r}\left({\mathit{\psi }}_{\mathit{rd}}{\mathit{i}}_{\mathit{sq}}-{\mathit{\psi }}_{\mathit{rq}}{\mathit{i}}_{\mathit{sd}}\right)$$

According to the invention, an electrical motor model and for the mathematical combination of the mechanical-hydraulic pump size a mechanical-hydraulic pump / motor model is used for the mathematical link for determining the electrical power of the motor. Here, as the electric motor model, it is preferable to use one defined by the equations (1) to (5) or (6) to (9) or (10) to (14). $${\mathit{L\text{'}}}_s=\frac{{\mathit{di}}_{\mathit{sd}}}{\mathit{di}}=-{\mathit{R\; \text{'}}}_s{i}_{\mathit{sd}}+\frac{L_m}{L_r}\left({\mathit{R\; \text{'}}}_r{\mathit{\psi }}_{\mathit{rd}}+z_p\omega {\mathit{\psi }}_{\mathit{rd}}\right)+v_{\mathit{sd}}$$

$${\mathit{L\text{'}}}_s=\frac{{\mathit{di}}_{\mathit{sq}}}{\mathit{dt}}=-{\mathit{R\; \text{'}}}_s{i}_{\mathit{sq}}+\frac{L_m}{L_r}\left({\mathit{R\; \text{'}}}_r{\mathit{\psi }}_{\mathit{rq}}+z_p\omega {\mathit{\psi }}_{\mathit{rd}}\right)+v_{\mathit{sq}}$$

$$\frac{{\mathit{d\psi }}_{\mathit{rd}}}{\mathit{dt}}=-{\mathit{R\; \text{'}}}_r{\mathit{\psi }}_{\mathit{rd}}-z_p\omega {\mathit{\psi }}_{\mathit{rq}}+{\mathit{R\; \text{'}}}_r{L}_m{\mathit{i}}_{\mathit{sd}}$$

$$\frac{{\mathit{d\psi }}_{\mathit{rq}}}{\mathit{dt}}=-{\mathit{R\; \text{'}}}_r{\mathit{\psi }}_{\mathit{rq}}+z_p\omega {\mathit{\psi }}_{\mathit{rd}}+{\mathit{R\; \text{'}}}_r{L}_m{\mathit{i}}_{\mathit{sq}}$$

$$T_e=z_p\frac{3}{2}\frac{L_m}{L_r}\left({\mathit{\psi }}_{\mathit{rd}}{\mathit{i}}_{\mathit{sq}}-{\mathit{\psi }}_{\mathit{rq}}{\mathit{i}}_{\mathit{sd}}\right)$$

$$\mathit{\omega }\mathit{=}{\mathit{\omega }}_{\mathit{s}}\mathit{-}\mathit{s}{\mathit{\omega }}_{\mathit{s}}$$

$$I_r=\frac{V_s}{Z_r\left(s\right)}$$

$${\mathit{T}}_{\mathit{e}}^{}=\frac{3R_r{I}_r^2}{s}$$

Equations (1) to (5) represent an electric dynamic motor model for an asynchronous motor. $$V_s=Z_s\left(s\right)I_s$$

$$\mathit{\omega }\mathit{=}{\mathit{\omega }}_{\mathit{s}}\mathit{-}\mathit{s}{\mathit{\omega }}_{\mathit{s}}$$

$$I_r=\frac{V_s}{Z_r\left(s\right)}$$

$${\mathit{T}}_{\mathit{e}}^{}=\frac{3R_r{I}_r^2}{s}$$

$$L_s\frac{{\mathit{di}}_{\mathit{sq}}}{\mathit{dt}}=-{\mathit{R}}_s{\mathit{i}}_{\mathit{sq}}-z_p{\mathit{\omega }L}_s{\mathit{\psi }}_{\mathit{rd}}+v_{\mathit{sq}}$$

$$\frac{{\mathit{d\psi }}_{\mathit{rd}}}{\mathit{dt}}=-z_p\omega {\mathit{\psi }}_{\mathit{rq}}$$

$$\frac{{\mathit{d\psi }}_{\mathit{rq}}}{\mathit{dt}}=z_p\omega {\mathit{\psi }}_{\mathit{rd}}$$

$$T_e=z_p\frac{3}{2}\left({\mathit{\psi }}_{\mathit{rd}}{\mathit{i}}_{\mathit{sq}}-{\mathit{\psi }}_{\mathit{rq}}{\mathit{i}}_{\mathit{sd}}\right)$$

Equations (6) to (9) also represent an electric static motor model for an asynchronous motor. $$L_s\frac{{\mathit{di}}_{\mathit{sd}}}{\mathit{dt}}=-{\mathit{R}}_s{\mathit{i}}_{\mathit{sd}}+z_p{\mathit{\omega }L}_s{\mathit{\psi }}_{\mathit{rq}}+v_{\mathit{sd}}$$

$$L_s\frac{{\mathit{di}}_{\mathit{sq}}}{\mathit{dt}}=-{\mathit{R}}_s{\mathit{i}}_{\mathit{sq}}-z_p{\mathit{\omega }L}_s{\mathit{\psi }}_{\mathit{rd}}+v_{\mathit{sq}}$$

$$\frac{{\mathit{d\psi }}_{\mathit{rd}}}{\mathit{dt}}=-z_p\omega {\mathit{\psi }}_{\mathit{rq}}$$

$$\frac{{\mathit{d\psi }}_{\mathit{rq}}}{\mathit{dt}}=z_p\omega {\mathit{\psi }}_{\mathit{rd}}$$

$$T_e=z_p\frac{3}{2}\left({\mathit{\psi }}_{\mathit{rd}}{\mathit{i}}_{\mathit{sq}}-{\mathit{\psi }}_{\mathit{rq}}{\mathit{i}}_{\mathit{sd}}\right)$$

Equations (10) to (14) represent an electric dynamic motor model for a permanent magnet motor.

i_sd

den Motorstrom in Richtung d

i_sq

den Motorstrom in Richtung q

ψ_rd

den magnetischen Fluss des Rotors in d-Richtung

ψ_rq

den magnetischen Fluss des Rotors in q-Richtung

T_e

das Motormoment

ν_sd

die Versorgungsspannung des Motors in d-Richtung

ν_sq

die Versorgungsspannung des Motors in q-Richtung

ω

die Winkeigeschmdndigkeit des Rotors und Laufrades

R'_s

den Ersatzwiderstand der Statorwicklung

R'_r

den Ersatzwiderstand der Rotorwicklung

L_m

den induktiven Kopplungswiderstand zwischen Stator-und Rotorwicklung

L'_s

den induktiven Ersatzwiderstand der Statorwicklung

L_r

den induktiven Widerstand der Motorwicklung

_p

die Polpaarzahl

I_s

den Phasenstrom

V_s

die Phasenspannung

ω_s

die Frequenz der Versorgungsspannung

ω

die tatsächliche Rotor und Laufraddrehzahl

s

den Motorschlupf

Z_s (s)

die Statorimpedanz

Z_r (s)

die Rotorimpedanz

R_r

den Ersatzwiderstand der Rotorwicklung

R_s

den Ersatzwiderstand der Statorwicklung

L_s

Den induktiven Widerstand der Statorwicklung,

wobei d uns q zwei senkrecht zueinander stehende Richtungen senkrecht zur Motorwelle sind,Represent in equations (1) to (14)

i _sd

the motor current in direction d

i _sq

the motor current in direction q

ψ _rd

the magnetic flux of the rotor in d-direction

_{q rq}

the magnetic flux of the rotor in q-direction

T _e

the engine torque

ν _sd

the supply voltage of the motor in d-direction

ν _sq

the supply voltage of the motor in q-direction

ω

the angularity of the rotor and impeller

R _'s

the equivalent resistance of the stator winding

R ' _r

the equivalent resistance of the rotor winding

L _m

the inductive coupling resistance between stator and rotor winding

L ' _s

the inductive equivalent resistance of the stator winding

L _r

the inductive resistance of the motor winding

_p

the pole pair number

I _s

the phase current

V _s

the phase voltage

ω _s

the frequency of the supply voltage

ω

the actual rotor and impeller speed

s

the engine slip

Z _s ( s )

the stator impedance

Z _r ( s )

the rotor impedance

R _r

the equivalent resistance of the rotor winding

R _s

the equivalent resistance of the stator winding

L _s

The inductive resistance of the stator winding,

where d and q are two perpendicular directions perpendicular to the motor shaft,

For the mechanical-hydraulic pump / motor model, Equation (15) and at least one of Equations (16) and (17) are advantageously used.

und mindestens eine der Gleichungen $$H_p=-a_{h2}{Q}^2+a_{h1}\mathit{Q\omega }+a_{h0}{\omega }^2$$

$$T_p=-a_{t2}{Q}^2+a_{t1}\mathit{Q\omega }+a_{t0}{\omega }^2$$

in deren
$\frac{\mathit{d\omega }}{\mathit{dt}}$

  die zeitliche Ableitung der Winkelgeschwindigkeit des Rotors,
T_p   das Pumpendrehmoment,
J  das Massenträgheitsmoment von Rotor, Laufrad und im Laufrad gebundener Förderflüssigkeit,
B  die Reibungskonstante,
Q  der Förderstrom der Pumpe,
H_p   der von der Pumpe erzeugten Differenzdruck,
a_h2, a_h1 , a _h0  die Parameter, die den Zusammenhang zwischen Drehzahl des Laufrades, Förderstrom und Differenzdruck beschreiben und
a _t2, a _t1 a _t0  die Parameter, die den Zusammenhang zwischen Drehzahl des Laufrades, Förderstrom und Massenträgheitsmoment beschreibenEquation (15) represents the mechanical relationships between motor and pump, whereas equations (1b) and (17) describe the mechanical-hydraulic relationships in the pump. These equations are: $$\mathit{J}\frac{\mathit{dw}}{\mathit{dt}}\mathit{=}{\mathit{T}}_{\mathit{e}}\mathit{-}\mathit{B\omega }\mathit{-}{\mathit{T}}_{\mathit{P}}$$

and at least one of the equations $$H_p=-a_{H2}{\mathrm{<figure-callout\; id="2"\; label="Q"\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}">Q</figure-callout>}}^2+a_{H1}\mathit{Q\omega }+a_{H0}{\omega }^2$$

$$T_p=-a_{t2}{\mathrm{<figure-callout\; id="2"\; label="Q"\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}">Q</figure-callout>}}^2+a_{t1}\mathit{Q\omega }+a_{t0}{\mathrm{<figure-callout\; id="2"\; label="\omega "\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}">\omega </figure-callout>}}^2$$

in theirs
$\frac{\mathit{dw}}{\mathit{dt}}$

the time derivative of the angular velocity of the rotor,
T _{p is} the pump torque,
J the mass moment of inertia of rotor, impeller and in the impeller bound delivery liquid,
B the friction constant,
Q is the flow rate of the pump,
H _{p is} the differential pressure generated by the pump
a _h2 , a _h1 , a _{h 0 are} the parameters describing the relationship between the speed of the impeller, the flow rate and the differential pressure, and
a _{t 2} , a _{t 1} a _{t 0 are} the parameters that describe the relationship between the speed of the impeller, the flow rate and the mass moment of inertia

Claim 8 defines, by way of example, how mathematical links are made to determine whether an error exists or not. On the storage of predetermined values can be completely omitted here in principle. The basic idea of this, concrete method is, on the one hand with the aid of the engine model, to determine the engine torque resulting from the electrical variables on the motor shaft and the rotational speed, the latter also being able to be measured. With the aid of equations (16) and / or (17) a relationship between pressure and flow rate on the one hand and between power / torque and flow rate on the other hand is determined. It is then advantageously checked with equation (15) whether the variables calculated with the aid of the motor model and those with the aid of the pump model after the onset of the measured hydraulic variable calculated quantities or not, whereby in the case of a mismatch an error is registered. It is thus compared, as it were, whether or not the drive variables resulting from the electric motor model coincide with those resulting from the hydraulic-mechanical pump model. If this is the case, the pump set works without errors, otherwise there is an error that can be specified even further.

In order to give the system a certain tolerance, it may be useful to set a tolerance _band by variance of at least one of the variables a _h0 to a _h2 , a _t0 to a _t2 , B and J in order to register an error only if it is is also operationally relevant.

In order to be able to specify the type of error more precisely, it is expedient in addition to the two electrical variables to determine two hydraulic variables, preferably by measuring, and to insert the determined values into the equations according to claim 7, so that then four error variables r ₁ to r ₄ result , Based on the combination of these error variables, the type of error is then determined on the basis of predetermined limit value combinations. This too is done automatically by electronic data processing.

In an alternative development of the method according to the invention, in addition to the two electrical variables, two hydraulic variables can preferably be determined by measuring and the determined values are compared with predetermined values to determine the type of error, wherein in each case the predetermined values define an area in three-dimensional space and it is determined whether or not the quantities determined lie on these areas (r * ₁ to r * ₄ ) and, based on the combination of the values, the type of error on the basis of given values Limit value combinations are determined. The type of error can then be determined, for example, from the following table: Error type defect size r ₁ , r ₁ * r ₂ , r ₂ * r ₃ , r ₃ * r ₄ r ₄ * comparison area Increased friction due to mechanical defects 1 0 1 1 Reduced promotion / lack of pressure 0 1 1 1 Defective intake / lack of delivery 1 1 0 1 Fjord failure 1 1 1 1

With the aid of the method according to the invention, it is thus possible not only to determine or not to determine the fault-free operating state of the pump set with a minimum of sensor, but moreover to specify it in detail in the event of a fault, so that a corresponding error signal is generated in the pump set which indicates the type of error. This signal may optionally be transmitted to remote locations where the function of the pump set is to be monitored.

The surfaces formed in the three-dimensional space on the basis of predetermined values are typically space-curved surfaces whose values have previously been determined by the factory based on the respective unit or aggregate type and stored in the digital data memory on the aggregate side. The aforementioned comparison surfaces r * ₁ to r * _{4 are arranged} in a three-dimensional space which at r * ₁ from the torque, the flow and the rotor speed, at r * ₂ from the head, the flow rate and the rotor speed, for r * ₃ from the torque, the head and the rotor speed as well are formed for r * ₄ from the torque, the delivery head and the flow rate.

The variables defined in the table by the comparison surfaces r * ₁ to r * ₄ indicate the respective operating state, wherein the number 0 means that the respective value lies within the area defined by the predetermined values and 1 outside. For example, the error combination defined in the table due to increased friction due to mechanical defects can mean bearing damage or an otherwise caused increased frictional resistance between the rotating parts and the stationary parts of the aggregate. The error combination marked under the generic term reduced promotion / missing pressure can be caused, for example, by errors or wear on the pump impeller or an obstacle in the pump inlet or outlet. Defined under the generic term defect in the intake / missing flow error combination can be caused for example by defect of the ring seal at the suction of the pump. The error combination under the generic term Förderausfall can have a variety of causes and should be further specified if necessary. This delivery failure may be caused by a blocked shaft or jammed pump impeller, shaft breakage, pump impeller loosening, cavitation due to excessively low pressure at the pump inlet, and dry running.

The operating states designated in the table by the variables R ₁ to R ₄ are based on mathematical calculations of fault parameters r ₁ to r ₄ according to the equations (19) to (22), wherein the corresponding error amount becomes zero when a correct operation is present and the value 1 in case of error. The table is to be understood in terms of the type of error in a similar manner as described above. Figuratively speaking, each represents the mistake sizes r ₁ to r _{4 are} spaced from the corresponding areas r * ₁ to r * ₄ . However, the error quantities do not necessarily correspond to the areas r * ₁ to r * ₄ . The error quantities r ₁ to r ₄ correspond to the equations (19) to (22) and correspond to the areas r * ₁ to r * ₄ in the FIGS. 7 to 10 ,

In order to further differentiate the nature of the error, it is provided in a further development of the invention that, when a fault is detected, the pump unit is actuated with a different speed, in order then to be able to narrow down the detected error more precisely on the basis of the resulting measuring results.

Preferably, the mechanical-hydraulic pump / motor model comprises not only the pump unit itself, but also beyond at least parts of the acted upon by the pump hydraulic system, so that errors of this hydraulic system can be determined.

in der

K_J

die Konstante ist, die Massenträgheit der Flüssigkeitssäule im Rohrsystern beschreibt,

K_V

die Konstante, die die flowabhängigen Druckverluste im Ventil beschreibt und

K_L

die Konstante ist, die die flowobhängigen Druckverluste Im Rohrsystem beschreibt,

H_p

den Differenzdruck der Pumpe,

P_out

den Druck am verbraucherseitigen Ende der Anlage,

P_in

den Zulaufdruck,

Z_out

das statische Druckniveau am verbraucherseitigen Ende der Anlage,

Z_ln

das statische Druckniveau am Pumpeneingang,

p

die Dichte des Fördermedium

g

die Gravitationskonstante

Sind.In this case, the hydraulic system is advantageously defined by equation (18), which represents the change in the delivery flow over time. $$K_J\frac{\mathit{dQ}}{\mathit{d\tau }}=H_p-\left({\mathit{p}}_{\mathit{out}}\mathit{+}{\mathit{\rho gz}}_{\mathit{out}}\mathit{-}{\mathit{p}}_{\mathit{in}}\mathit{-}{\mathit{\rho gz}}_{\mathit{in}}\right)\left(K_v+K_l\right){\mathrm{<figure-callout\; id="2"\; label="Q"\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}">Q</figure-callout>}}^2$$

in the

K _J

the constant is the mass inertia of the liquid column in the tube system,

K _V

the constant which describes the flow-dependent pressure losses in the valve and

K _L

is the constant that describes the flow-dependent pressure losses in the pipe system,

H _p

the differential pressure of the pump,

P _out

the pressure at the consumer end of the system,

P _in

the inlet pressure,

Z _out

the static pressure level at the consumer end of the system,

Z _ln

the static pressure level at the pump inlet,

p

the density of the fluid

G

the gravitational constant

Are.

$$\{r_2=q_2(-a_{h2}{Q}^2+a_{h1}\mathit{Q\omega }+a_{h0}{\omega }^2-H_p)$$

$$\{\begin{array}{ll}\mathit{Qʹ}& =\frac{a_{h1}\omega +\sqrt{a_{h1}^2{\omega }^2-4a_{h2}\left(H_p+a_{h0}{\omega }^2\right)}}{2a_{h2}}\\ J\frac{{\mathit{d}\hat{\mathit{\omega }}}_1}{\mathit{dt}}& =-B{\hat{\omega }}_3-\left(-a_{t2}{Q}^{\prime 2}+a_{t1}\mathit{Q}\mathit{\prime }\mathit{\omega }+a_{t0}{\omega }^2\right)+T_e+k_3\left(\omega -{\hat{\omega }}_3\right)\\ r_3& =q_3(\omega -{\hat{\omega }}_3)\end{array}$$

$$\{\begin{array}{ll}\mathit{\omega ʹ}& =\frac{a_{h1}\omega +\sqrt{a_{h1}^2{\omega }^2-4a_{h2}\left(H_p+a_{h0}{\omega }^2\right)}}{2a_{h2}}\\ J\frac{d{\hat{\mathit{\omega }}}_4}{\mathit{dt}}& =-B{\hat{\omega }}_4-\left(-a_{t2}{Q}^2+a_{t1}\mathit{Q\omega ʹ}+a_{t0}{\mathrm{\omega ʹ}}^2\right)+T_{\mathit{e}}+k_4\left(\mathrm{\omega ʹ}-{\hat{\omega }}_4\right)\\ r_4& =q_4(\mathrm{\omega ʹ}-{\hat{\omega }}_4)\end{array}$$

in denen

k ₁,k ₃,k ₄

Konstanten,

q ₁,q ₂,q ₃,q ₄

Konstanten,

Q'

Die berechnete Fördermenge auf Basis von aktueller Drehzahl und gemessenem Druck,

™₁

die berechnete Rotordrehzahl auf Grundlage der mechanisch-hydraülischen Gleichungen (15) und (17),

™ ₃

die berechnete Rotordrehzahl auf Grundlage der Gleichungen (15), (16) und (17).

™ ₄

die berechnete Rotordrehzahl auf Grundlage der Gleichungen (15), (16) und (17).

ω'

die berechnete Rotordrehzahl aufgrund gemessenen Förderdrucks und gemessener Fördermenge

r ₁ - r ₄

Fehlergrößen und

r ₁* - r ₄*

durch drei Variable bestimmte Flächen sind, die einen fehlerfreien Betrieb der Pumpe repräsentieren.

The error quantities r ₁ to r ₄ are advantageously defined by the equations (19) to (22): $$\{\begin{array}{ll}J\frac{d{\hat{\mathit{\omega }}}_1}{\mathit{dt}}& =-B{\hat{\omega }}_1-\left(-a_{t2}{Q}^2+a_{t1}\mathit{Q\omega }+a_{t0}{\omega }^2\right)+T_e+k_e\left(\omega -{\hat{\omega }}_1\right)\\ r_1& =q_1(\omega -{\hat{\omega }}_1)\end{array}$$

$$\{r_2=q_2(-a_{H2}{Q}^2+a_{H1}\mathit{Q\omega }+a_{H0}{\omega }^2-H_p)$$

$$\{\begin{array}{ll}\mathit{Q\; \text{'}}& =\frac{a_{H1}\omega +\sqrt{a_{H1}^2{\omega }^2-4a_{H2}\left(H_p+a_{H0}{\omega }^2\right)}}{2a_{H2}}\\ J\frac{{\mathit{d}\hat{\mathit{\omega }}}_1}{\mathit{dt}}& =-B{\hat{\omega }}_3-\left(-a_{t2}{Q}^{\text{'}2}+a_{t1}\mathit{Q}\mathit{\text{'}}\mathit{\omega }+a_{t0}{\omega }^2\right)+T_e+k_3\left(\omega -{\hat{\omega }}_3\right)\\ r_3& =q_3(\omega -{\hat{\omega }}_3)\end{array}$$

$$\{\begin{array}{ll}\mathit{\omega \text{'}}& =\frac{a_{H1}\omega +\sqrt{a_{H1}^2{\omega }^2-4a_{H2}\left(H_p+a_{H0}{\omega }^2\right)}}{2a_{H2}}\\ J\frac{d{\hat{\mathit{\omega }}}_4}{\mathit{dt}}& =-B{\hat{\omega }}_4-\left(-a_{t2}{Q}^2+a_{t1}\mathit{Q\omega \text{'}}+a_{t0}{\mathrm{\omega \text{'}}}^2\right)+T_{\mathit{e}}+k_4\left(\mathrm{\omega \text{'}}-{\hat{\omega }}_4\right)\\ r_4& =q_4(\mathrm{\omega \text{'}}-{\hat{\omega }}_4)\end{array}$$

in which

k ₁ , k ₃ , k ₄

constant,

q ₁ , q ₂ , q ₃ , q ₄

constant,

Q '

The calculated flow rate based on actual speed and pressure,

™ ₁

the calculated rotor speed based on the mechanical-hydraulic equations (15) and (17),

™ ₃

the calculated rotor speed based on equations (15), (16) and (17).

™ ₄

the calculated rotor speed based on equations (15), (16) and (17).

ω '

the calculated rotor speed based on measured discharge pressure and measured flow rate

r ₁ - r ₄

Error sizes and

r ₁ * - r ₄ *

by three variables are certain areas that represent a faultless operation of the pump.

In order to carry out the method according to the invention for determining the fault in operating states of a centrifugal pump assembly there are means for detecting two electrical variables determining the motor power and means for detecting at least one variable hydraulic size of the pump and means for detecting at least one further mechanical or mechanical performance determining the power of the pump provide hydraulic size and an electronic evaluation device which determines a fault condition of the pump unit based on the detected variables. In the simplest form, therefore, a sensor system for detecting the voltage applied to the motor supply voltage and the supply current and for detecting the pressure applied by the pump, preferably differential pressure and the delivery rate or the rotational speed is provided here. In addition, an evaluation device is to be provided, which may be in the form of digital data processing, for example a microprocessor, in which the method according to the invention is implemented by software. In order to be able to carry out the comparison between detected or calculated values and specified values (eg recorded and stored by the factory), an electronic memory is also to be provided. In modern frequency converter controlled pump units all the aforementioned hardware requirements are already in place, so that is only to ensure sufficient dimensioning of the electronic data processing system, in particular the storage means and the evaluation device. All components, with the exception of the sensors required for the detection of hydraulic variables, are preferably an integral part of the motor and / or pump electronics, so that constructively so far no further provisions for carrying out the method according to the invention are to be made. Another embodiment may be a separate module provided in a panel or control panel, in the same way as a motor protection switch, but with the monitoring and diagnostic features as described above.

The embodiments described here relate to centrifugal pumps, as can be seen from the mechanical-hydraulic pump model. Such pumps may be, for example, industrial pumps, submersible pumps for sewage or water supply and heating circulation pumps. Particularly advantageous is a diagnostic system according to the invention in canned pumps, since through early fault detection, the looping through of the can and thus outlet of the pumped liquid, eg. B. in the living area, preventively prevented. In the application of the invention in Verdrängerpumpenbereich the mechanical-hydraulic pump model must be adjusted according to the different physical relationships. The same applies to the use of other engine types for the electric motor model.

Moreover, according to the invention means are provided for generating and transmitting at least one error message to a display unit arranged on the pump unit or elsewhere, be it in the form of one or more indicator lights or a display with an alphanumeric display. In this case, the transmission can be wireless, for example via infrared or radio but also wired, preferably in digital form.

A simplified procedure is based on Fig. 1 shown. In an electric motor model 1, the variable electrical power-determining variables flow, in particular the voltage V _abc and the current i _abc . The product of these quantities defines the electrical power consumed by the motor. From this motor model, such as given by the equations (1) to (5) or (6) to (9) or (10) to (14), the torque T _e on the shaft of the motor and the rotational speed ω derivable from the engine, as they result arithmetically on the basis of the engine model. This power-dependent electrical Sizes of the motor are linked to the determined mechanical delivery height H (pressure) in a pump model 2, for example according to equations (16) and (17), in which case the result is compared with predetermined operating values determined on the basis of defined operating points. If these input variables agree with the specified values, the pump set operates without errors. On the other hand, if the difference is greater than a predetermined amount, then an error signal r is generated which signals a malfunction of the pump.

1. 1. erhöhte Reibung aufgrund mechanischer Defekte,
2. 2. reduzierte Förderung/fehlender Druck,
3. 3. Defekt im Ansaugbereich/fehfende Fördermenge und
4. 4. Förderausfall.

In the embodiment according to the invention Fig. 2 become the same as at Fig. 1 the input voltage v _abc and the motor current i _{abc are used} as input values for the motor model 1 to determine the torque Te present at the motor shaft and the rotational speed of the shaft ω. These derived from the engine model 1 values and the sensory variables of the head H (pressure) and the flow rate Q are mathematically linked together in a mechanical-hydraulic pump model 3, for example, by the equations (19) to (22) developed. In this case, four error quantities r ₁ to r _{4 are} generated, whereby a fault-free operation is present if they all assume the value zero and thus the operating points in the in the FIGS. 7 to 10 shown in detail surfaces r * ₁ to r * ₄ lie. These areas shown there are defined from a variety of operating points in the proper operation of the pump unit and factory-generated and stored digitally in the memory module of the evaluation electronics. Alternatively or additionally, it is determined whether or not the error variables r ₁ to r ₄ determined on the basis of the mechanical-hydraulic pump model are zero or not, according to this result an evaluation takes place in accordance with the above-described table. Depending on whether an error size is present or not, when an error occurs in total four faulty operating states of the pump aggregate are detected, namely those falling under the abovementioned preambles:

1. 1. increased friction due to mechanical defects,
2. 2. reduced promotion / lack of pressure,
3. 3. Defective in the intake area / decreasing flow rate and
4. 4. Delivery failure.

With the method according to the invention, not only the pump unit itself, but also parts of the system can be monitored, in which the pump unit is arranged. The system is structured as in Fig. 3 is shown in detail. Again, an electric motor model is provided, the input variables V _abc and i _abc and the example, a static motor model according to the equations (6) to (9) is based, as it is well known and based on Fig. 5 is shown. The output variable of this static engine model is the engine torque T _e , which in turn flows via the equation (15) input into the mechanical part of the pump model 3 a. The hydraulic part of the pump model 3b is defined by equations (16) and (17), via which the hydraulic part of the plant 4 is coupled. The hydraulic part of the plant is defined by equation (18) and based on Fig. 4 schematically shown, in which P _in the pressure inlet of the pump, H _p the differential pressure of the pump, Q the flow rate, P _out the pressure at the consumer end of the system and V ₁ the flow losses within the. Represent pump. Z _out is the static pressure level at the consumer end of the system and Z _at the pump inlet.

Fig. 3 clarifies the relationship between the engine model, the mechanical part of the pump model, the hydraulic part of the pump model and the hydraulic part of the plant. While in the hydraulic parts of the pump model 3b and the hydraulic part of the plant Inlet and out flow, and go in the hydraulic part of the pump model 3b, the speed ω _r , which also enters the engine model 1. The torque determined from the hydraulic part of the pump model 3b in turn enters the mechanical part of the pump model 3a for determining the rotational speed.

The equations described above for the mathematical description of pump and motor are to be understood as examples only and can be replaced, if necessary, by other suitable equations, as they are known from the relevant specialist literature. The above with these models detectable error in the operation of a pump set or differentiation by types of errors can be further diversified by appropriate error algorithms.

To ensure that even small production tolerances or measurement errors do not lead to the emission of error signals, it is expedient to select the parameters a _h and a _t given in equations (16) and (17) not constant, but in each case a lower or upper limit value set to produce a certain bandwidth, as in Fig. 6 are shown. In the left-hand curve shown there, the power is plotted against the flow rate and in the right-hand curve the delivery height is plotted against the flow rate.

LIST OF REFERENCE NUMBERS

1 -

Electric engine model

2 -

Simplified pump model

3 -

Extended pump model

3a -

Mechanical part of the pump model

3b -

Hydraulic part of the pump model

4 -

Hydraulic part of the plant

Claims (16)

1. A method for determining faults on operation of a pump assembly, with which at least two electrical variables of the motor which determine the electrical power of the motor, and at least one changing hydraulic variable of the pump, as well as at least one further mechanical or hydraulic variable which determines the power of the pump, are acquired, and on the one hand, the two electrical variables of the motor which determine the electrical power of the motor, are mathematically linked for achieving at least one comparison value, and on the other hand the at least one changing hydraulic variable of the pump, as well as at least one further mechanical or hydraulic variable determining the power of the pump are mathematically linked for achieving at least one comparison value, wherein a mathematical, electrical motor model (1) is used in combination with a mathematical, mechanical-hydraulic pump model / motor model (3) for the mathematical linking, and one determines whether a fault is present or not by way of the results of the mathematical linkings by comparison with predefined values.
2. A method according to claim 1, characterised in that the voltage prevailing at the motor and the current feeding the motor are acquired as the electrical variables determining the electrical power of the motor.
3. A method according to one of the preceding claims, characterised in that if the presence of a fault is determined, one then further determines as to which fault it is a case of.
4. A method according to one of the preceding claims, characterised in that the acquired hydraulic variable is the pressure produced by the pump.
5. A method according to one of the preceding claims, characterised in that the acquired hydraulic variable is the delivery quantity of the pump.
6. A method according to one of the preceding claims, characterised in that the acquired hydraulic variable is the differential pressure between the suction side and the pressure side of the pump.
7. A method according to one of the preceding claims, characterised in that the electrical motor model (1) is formed by the following equations $${\mathit{Lʹ}}_s=\frac{{\mathit{di}}_{\mathit{sd}}}{\mathit{dt}}=-{\mathit{Rʹ}}_s{i}_{\mathit{sd}}+\frac{L_m}{L_r}\left({\mathit{Rʹ}}_r{\mathit{\psi }}_{\mathit{rd}}+z_p\omega {\mathit{\psi }}_{\mathit{rd}}\right)+v_{\mathit{sd}}$$

   

   $${\mathit{Lʹ}}_s=\frac{{\mathit{di}}_{\mathit{sq}}}{\mathit{dt}}=-{\mathit{Rʹ}}_s{i}_{\mathit{sq}}+\frac{L_m}{L_r}\left({\mathit{Rʹ}}_r{\mathit{\psi }}_{\mathit{rq}}+z_p\omega {\mathit{\psi }}_{\mathit{rd}}\right)+v_{\mathit{sq}}$$

   

   $$\frac{{\mathit{d}\mathrm{\psi }}_{\mathit{rd}}}{\mathit{dt}}=-{\mathit{Rʹ}}_r{\mathit{\psi }}_{\mathit{rd}}-z_p\omega {\mathit{\psi }}_{\mathit{rq}}+{\mathit{Rʹ}}_r{L}_m{\mathit{i}}_{\mathit{sd}}$$

   

   $$\frac{{\mathit{d}\mathrm{\psi }}_{\mathit{rq}}}{\mathit{dt}}=-{\mathit{Rʹ}}_r{\mathrm{\psi }}_{\mathit{rq}}+z_p\omega {\mathrm{\psi }}_{\mathit{rd}}+{\mathit{Rʹ}}_r{L}_m{\mathit{i}}_{\mathit{sq}}$$

   

   $$T_e=z_p\frac{3}{2}\frac{L_m}{L_r}\left({\mathrm{\psi }}_{\mathit{rd}}{\mathit{i}}_{\mathit{sq}}-{\mathrm{\psi }}_{\mathit{rq}}{\mathit{i}}_{\mathit{sd}}\right)$$

   

   
   or $$V_s=Z_s\left(s\right)I_s$$

   

   $$\mathit{\omega }\mathit{=}{\mathit{\omega }}_{\mathit{s}}\mathit{-}\mathit{s}{\mathit{\omega }}_{\mathit{s}}$$

   

   $$I_r=\frac{V_s}{Z_r\left(s\right)}$$

   

   $${\mathit{T}}_e=\frac{3R_r{I}_r^2}{s}$$

   

   $$L_s\frac{{\mathit{di}}_{\mathit{sd}}}{\mathit{dt}}=-{\mathit{R}}_s{\mathit{i}}_{\mathit{sd}}+z_p{\mathit{\omega }L}_s{\mathrm{\psi }}_{\mathit{rq}}+v_{\mathit{rd}}$$

   

   $$L_s\frac{{\mathit{di}}_{\mathit{sq}}}{\mathit{dt}}=-{\mathit{R}}_s{\mathit{i}}_{\mathit{sq}}-z_p{\mathit{\omega }L}_s{\mathrm{\psi }}_{\mathit{rd}}+v_{\mathit{sq}}$$

   

   $$\frac{{\mathit{d}\mathrm{\psi }}_{\mathit{rd}}}{\mathit{dt}}=-z_p\omega {\mathrm{\psi }}_{\mathit{rq}}$$

   

   $$\frac{{\mathit{d}\mathrm{\psi }}_{\mathit{rq}}}{\mathit{dt}}=-z_p\omega {\mathrm{\psi }}_{\mathit{rd}}$$

   

   $$T_e=z_p\frac{3}{2}\left({\mathrm{\psi }}_{\mathit{rd}}{\mathit{i}}_{\mathit{sq}}-{\mathrm{\psi }}_{\mathit{rq}}{\mathit{i}}_{\mathit{sd}}\right)$$

   

   
   in which

   i_sd is the motor current in direction d

   i_sq the motor current in direction q

   ψ _rd the magnetic flux of the rotor in the d-direction

   ψ _rq the magnetic flux of the rotor in the q-direction

   T_e the motor moment

   ν _sd the supply voltage of the motor in the d-direction

   ν_sq the supply voltage of the motor in the q-direction

   ω the angular speed of the rotor and impeller

   R'_s the equivalent resistance of the stator winding

   R' _r the equivalent resistance of the rotor winding

   L_m the inductive coupling resistance between the stator and the rotor winding

   L'_s the inductive equivalent resistance of the stator winding

   L_r the inductive resistance of the rotor winding

   z_p the pole pair number

   I_s the phase current

   V_s the phase voltage

   ω _s the frequency of the supply voltage

   ω the actual rotor and impeller rotational speed

   s the motor slip

   Z_s (s) the stator impedance

   Z_r (s) the rotor impedance

   R_r the equivalent resistance of the rotor winding

   R_s the equivalent resistance of the stator winding

   L_s the inductive resistance of the stator winding

   wherein d and q are two directions perpendicular to the motor shaft and perpendicular to one another
   and wherein the mechanical-hydraulic pump/motor model is formed by the equation $$\mathit{J}\frac{\mathit{d\omega }}{\mathit{dt}}\mathit{=}{\mathit{T}}_{\mathit{e}}\mathit{-}\mathit{B\omega }\mathit{-}{\mathit{T}}_{\mathit{P}}$$

   

   
   and at least one of the equations $$H_p=-a_{h2}{Q}^2+a_{h1}\mathit{Q\omega }+a_{h0}{\omega }^2$$

   

   $$T_p=-a_{t2}{Q}^2+a_{t1}\mathit{Q\omega }+a_{t0}{\omega }^2$$

   

   
   in which

   $\frac{\mathit{d\omega }}{\mathit{dt}}$

   

   is the temporal derivative of the angular speed of the rotor,

   T_p the pump torque,

   J the moment of mass inertia of the rotor, impeller and the delivery fluid contained in the impeller,

   B the friction constant,

   Q the delivery flow of the pump,

   H_p the differential pressure produced by the pump,

   a _h2, a _h1 , a _h0 the parameters which describe the relationship between the rotational speed of the impeller, the delivery flow and the differential pressure and

   a _t2, a _t1, a _t0 the parameters which describe the relation between the rotational speed of the impeller, the delivery flow and the moment of mass inertia.
8. A method according to claim 7, characterised in that the variables a_h0- a_h2 and a_t0 - a_t2 are fixed in the equations (16) and (17), as well as the variables B and J in the equation (15), that a motor moment (T_e) is determined from the electrical motor model (1) according to the equations (1) - (5) or (6) - (9) or (10) - (14), and the rotational speed is either computed according to the equations (1) - (5) or (6) - (9) or (10) - (14) or measured, whereupon with the help of the equations (16) and/or (17), one determines a relationship between pressure and delivery quantity on the one hand and/or between power/moment and delivery quantity on the other hand, whereupon preferably one checks with equation (15) as to whether the variables computed with the help of the motor model (1) agree or not with those variables computed with the help of the pump model (3) after the substitution of the measured hydraulic variables, wherein a fault is registered should there be no agreement.
9. A method according to claim 7, characterised in that a tolerance band is fixed by way of variance of at least one of the variables a_h0- a_h2 and a_t0- a_t2 and B and J.
10. A method according to one of the preceding claims, characterised in that for determining the type of fault, additionally to the two electrical variables, two hydraulic variables are determined, preferably by way of measurement, and the determined values are substituted into the equations according to claim 8, in a manner such that several fault variables (r₁ - r₄) result, wherein by way of the combination of fault variables, one determines the type of fault by way of predefined boundary value combinations.
11. A method according to one of the preceding claims, characterised in that for determining the type of fault, additionally to the two electrical variables, two hydraulic variables are determined, preferably by way of measurement, and the determined values or values derived therefrom are compared to predefined values, wherein the predefined values in each case define a surface, wherein one determines whether the determined variables or those derived therefrom lie on one of these surfaces (r*₁ - r*₄) or not, and by way of the combination of the fault variables, one determines the type of fault by way of predefined boundary value combinations.
12. A method according to one of the preceding claims, characterised in that the evaluation of the fault type is effected by way of the following table: fault type fault variable r ₁ , r ₁* r ₂, r ₂ * r ₃ , r ₃ * r ₄, r ₄ * comparative surface increased friction on account of mechanical defects 1 0 1 1 reduced delivery/ absent pressure 0 1 1 1 defect in suction region/ absent delivery quantity 1 1 0 1 delivery stoppage 1 1 1 1
13. A method according to one of the preceding claims, characterised in that on determining a fault, the pump assembly is activated with a changed rotational speed, in order by way of the measurement results which then set in, to more accurately specify the determined fault.
14. A method according to one of the preceding claims, characterised in that the mechanical-hydraulic pump model / motor model (3) also includes at least parts of the hydraulic system (4) affected by the pump, in a manner such that faults of the hydraulic system (4) may also be determined.
15. A method according to claim 14, characterised in that the hydraulic system (4) is defined by the equation $$K_J\frac{\mathit{dQ}}{\mathit{dt}}=H_p-\left({\mathit{p}}_{\mathit{out}}\mathit{+}{\mathit{\rho gz}}_{\mathit{out}}\mathit{-}{\mathit{p}}_{\mathit{in}}\mathit{-}{\mathit{\rho gz}}_{\mathit{in}}\right)\left(K_v+K_l\right)Q^2$$

    

    
    in which

    K _J is the constant which describes the mass inertia of the fluid column in the pipe system,

    K_V the constant which describes the flow-dependent pressure losses in the valve, and

    K_L the constant which describes the flow-dependent pressure losses in the pipe system,

    H_p the differential pressure of the pump,

    P_out the pressure at the consumer-side end of the installation,

    P_in the supply pressure,

    Z_out the static pressure level at the consumer-side end of the installation,

    Z_in the static pressure level at the pump entry,

    p the density of the delivery medium

    g the gravitational constant
16. A method according to one of the preceding claims, characterised in that the variables r₁ - r₄ are defined by the equations $$\{\begin{array}{ll}J\frac{d{\hat{\mathit{\omega }}}_1}{\mathit{dt}}& =-B{\hat{\omega }}_1-\left(-a_{t2}{Q}^2+a_{t1}\mathit{Q\omega }+a_{t0}{\omega }^2\right)+T_e+k_e\left(\omega -{\hat{\omega }}_1\right)\\ r_1& =q_1(\omega -{\hat{\omega }}_1)\end{array}$$

    

    $$\{r_2=q_2(-a_{h2}{Q}^2+a_{h1}\mathit{Q\omega }+a_{h0}{\omega }^2-H_p)$$

    

    $$\{\begin{array}{ll}\mathit{Qʹ}& =\frac{a_{h1}\omega +\sqrt{a_{h1}^2{\omega }^2-4a_{h2}\left(H_p+a_{h0}{\omega }^2\right)}}{2a_{h2}}\\ J\frac{d{\hat{\mathit{\omega }}}_3}{\mathit{dt}}& =-B{\hat{\omega }}_3-\left(-a_{t2}{\mathit{Qʹ}}^2+a_{t1}\mathit{Qʹ\omega }+a_{t0}{\mathrm{\omega }}^2\right)+T_e+k_3\left(\mathrm{\omega }\mathrm{-}{\hat{\mathrm{\omega }}}_{\mathrm{3}}\right)\\ r_3& =q_3(\mathrm{\omega }\mathrm{-}{\hat{\mathrm{\omega }}}_{\mathrm{3}})\end{array}$$

    

    $$\{\begin{array}{ll}\mathit{\omega ʹ}& =\frac{-a_{h1}{H}_p+\sqrt{a_{h1}^2{H_p}^2-4a_{h2}\left(H_p+a_{h0}{Q}^2\right)}}{2a_{h2}}\\ J\frac{d{\hat{\mathit{\omega }}}_4}{\mathit{dt}}& =-B{\hat{\omega }}_4-\left(-a_{t2}{Q}^2+a_{t1}\mathit{Q\omega ʹ}+a_{t0}{\mathrm{\omega ʹ}}^2\right)+T_{\mathit{e}}+k_4\left(\mathrm{\omega ʹ}-{\hat{\omega }}_4\right)\\ r_4& =q_4(\mathrm{\omega ʹ}-{\hat{\omega }}_4)\end{array}$$

    

    
    in which

    k₁,k₃,k₄ represent constants,

    q ₁,q ₂,q ₃,q ₄ constants,

    Q' the computed delivery quantity on the basis of current rotational speed and measured pressure,

    ^™ ₁ the computed rotor rotational speed on the basis of the mechanical-hydraulic equations (15) and (17),

    ^™ ₃ the computed rotor rotational speed on the basis of the equations (15), (16) and (17),

    ω' the computed rotor rotational speed on the basis of the measured delivery pressure and measured delivery quantity

    r ₁ - r ₄ fault variables, and

    r ₁ *- r ₄ * surfaces determined by three variables, which represent a fault-free operation of the pump.

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