Preprints of the 21st IFAC World Congress (Virtual)

Berlin, Germany, July 12-17, 2020

Speed and torque estimation of variable

frequency drives with effective values of
stator currents
Y. Kouhi ∗ J. Müller ∗ S. Leonow ∗ M. Mönnigmann ∗
∗
Automatic Control and Systems Theory, Ruhr-Universität Bochum,
Universitätsstraße 150, 44801 Bochum, (e-mail: {yashar.kouhi,
jens.mueller-r55, sebastian.leonow, martin.moennigmann}@rub.de).

Abstract: We introduce a new method for torque and speed estimation of induction motors
under voltage/frequency (V/f) open-loop control. In contrast to existing approaches that need
the phase current, the proposed algorithm only requires the effective value (root mean square)
of the stator current and the synchronous frequency, which are usually available from variable
frequency inverter (VFI) at no additional cost. Our approach is particularly useful for inverter-
fed motor-pumps in which the load varies slowly. We demonstrate the proposed algorithm is able
to estimate the pump torque, speed, differential pressure and flow rate in a hydraulic process
with a progressive cavity pump.

Keywords: Inverter drives; induction motors; torque motors; speed motors; parameter
estimation; positive displacement pumps

of these approaches require high resolution measurements

1. INTRODUCTION of the voltage and current of at least one phase of the
Variable frequency inverter drives are ubiquitous today. motor, however.
Their use is getting more and more common even in ba- It is our main contribution to introduce a new method
sic and fundamental domains such as hydraulic processes for the computation of the speed and the torque based
(Ahonen et al. (2013)). A task of the variable frequency on the motor model, the effective value of the current, the
inverter (VFI) in these applications is to vary the motor synchronous speed, and the magnitude of the voltage. The
speed in order to achieve the desired flow rate of the necessary data are usually provided by the VFI and thus
pump. A mechanism for the determination of the flow rate no extra sensor is needed. As an application, we employ
of the process is required to meet this control objective. this approach for the estimation of the torque and speed
Moreover, in a pumping process often the monitoring of of a progressive cavity pump driven by an induction motor
other signals such as pressure, speed, torque, and efficiency and use the information of the characteristic curves of
of the pump are much desired. Additional measurement the pump to determine the differential pressure and the
device for these signals, even if the installation is physically flow rate of the hydraulic process. It is worth mentioning
possible, increases the cost significantly. Consequently, es- that using the effective value of the stator current and
timation of these quantities based on already existing sig- pump characteristic curves for flow rate estimation of a
nals is of great interest. As many VFIs provide information centrifugal pumping process has already been investigated
on motor operation, it is natural to engage these data for by Leonow and Mönnigmann (2013). However, in their
estimation algorithms. approach, a model is created from the direct effect of the
Speed and torque estimations of an induction motor have flow rate on the motor current, and the motor model and
been widely studied in the context of electric drives, often parameters do not explicitly appear in the estimation.
for achieving the sensorless control of the motor speed. The In Section 2 we present the motor model and propose our
most notable algorithms for speed estimation are those algorithm for the estimation of speed and torque of the
that use the equivalent circuit of the induction motor for induction motor. In Section 3 we introduce the process
finding the slip between the synchronous speed and the ro- control system and explain how the differential pressure
tor speed in steady state (Sen (2007)); direct computation and the flow rate of the pump can be estimated. Moreover,
of the speed through the estimated rotor flux (Kanmachi we verify our algorithm by its implementation on a real
and Taahashi (1993)); observer based estimation such as test bench. Finally, in Section 4 we give a brief conclusion.
model reference adaptive systems (Kubota et al. (1993));
and anisotropy methods in which the speed is computed 2. COMPUTATION OF SPEED AND TORQUE OF
by capturing the spatial harmonics of the rotor position AN INDUCTION MOTOR IN STEADY STATE
on the stator current (Ha and Sul (1999), Holtz (2002)).
Once the speed is estimated, the torque estimation can be The stator model of a three phase induction motor in the
accomplished with the steady state model (Sen (2007)) or rotational coordinate system (d-q) attached to the rotor
the dynamic model of the motor (Kubota et al. (1993)). All flux vector is described by the following equations:

Copyright lies with the authors 8865

Preprints of the 21st IFAC World Congress (Virtual)
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dψsd
usd = Rs isd + − ωs ψsq , (1)
dt
dψsq
usq = Rs isq + + ωs ψsd , (2) 5
dt
ψsd = Ls isd + Lm ird , (3)
ψsq = Ls isq + Lm irq , (4)

isq
0
where isd , isq and ird , irq are the stator and the rotor
currents in d and q axes, respectively. Likewise, usd ,
usq and ψsd , ψsq describe the stator voltages and the
electromagnetic fluxes in the mentioned coordinate frame.
-5
The index s always refers to the stator variables and the
index r refers to the rotor variables. Rs and Rr represent
the resistances, whereas Ls , Lr , and Lm represent the
-5 0 5
stator, rotor, and mutual inductances, respectively. The
isd
parameter
L2
σ =1− m . (5) Fig. 1. Curves that specify the currents in d and q axes.
Ls Lr
is called leakage factor. Neglecting the motor saturation these two variables results depending on the shaft torque.
and assuming that the leakage and mutual inductances An estimation of the actual motor speed and shaft torque
are constant, the parameter σ is also constant. are therefore of obvious interest. We present our estimation
In steady state, the rotor flux coordinate system rotates algorithm for this purpose in what follows.
with the synchronous rotational speed ωs . In this represen-
tation the d-axis current (isd ) builds up the electromag- 2.1 Estimation of motor speed in V/f control
netic flux ψrd , and the q-axis current (isq ) is proportional
to the electromagnetic torque of the motor. Moreover, the Most VFIs provide the effective value of one phase of
following relations hold in steady state: stator current Ieff as a measurement signal. In our speed
dψsd dψsq Lm estimation algorithm the synchronous rotational speed ωs
= 0, = 0, ird = 0, irq = − isq , (6)
dt dt Lr serves as the input, Ieff is the measured signal, and the
(see, e.g., Quang and Dittrich (2015)). Applying the sim- actual motor speed ωm represents the estimation signal.
plification (6) to (1) and (2), and inserting (3) and (4) in The first step for speed estimation concerns computation
(1) yield of the currents isd and isq . Since the current of each phase
usd = Rs isd − ωs σLs isq , (7) of the motor in stator-fixed coordinate system is almost
sinusoidal in steady state (the effect of higher harmonics of
usq = Rs isq + ωs Ls isd . (8) the current caused by the vector modulation is neglected),
The electromagnetic torque Te and the shaft torque Tm the magnitude of the current vector is = [isd >
are related by the momentum equations according to √ isq ] equals
the maximum current of one phase, which is 2Ieff . Hence,
Te − Tm − F ωm = J ω̇m , (9) the mathematical expressions for the magnitude of the Ieff
where F and J are the friction coefficient and the moment and |us | read:
of inertia, respectively, and ωm represents the mechanical
q
speed in (9). Applying the steady state condition ω̇m = 0 |us | = u2sq + u2sd , (11)
√ q
results in 2 2
Tm = Te − F ωm . (10) Ieff = isq + i2sd , (12)
2
Combining (7), (8), and (11) results in the following
Many inverter driven motors operate under the volt- equation for the magnitude of the stator voltage:
age/frequency (V/f) open-loop control to regulate the (Rs isd −ωs σLs isq )2 + (Rs isq +ωs Ls isd )2 = |us |2 . (13)
motor speed. The principle of the V/f control relies on
keeping the ratio of the magnitude of voltage vector to Equations (12) and (13) describe a circle and an ellipse in
frequency, i.e., |us |/|ωs |, constant. The rotor electromag- isd and isq , respectively, which are illustrated in Figure 1.
netic flux ψrd will then be almost constant and therefore Since the motor currents are positive in normal operation,
the motor can deliver the nominal torque in the entire we require the intersection point in the first quadrant
basic speed region. From a control systems perspective, we marked by the black point in Figure 1. The currents isd
consider the motor under control as a single-input-single- and isq can be computed as follows. Expanding (13) to
output system, where the synchronous speed represents
the control input, the motor speed represents the system |us |2 = Rs2 (i2sd + i2sq ) + 2ωs Rs Ls (1 − σ)isd isq +
(14)
output, and the shaft torque is a disturbance. Note that + ωs2 L2s (σ 2 i2sq + i2sd ),
as the magnitude of the voltage is set proportionally to
the synchronous frequency by the inverter, this variable and substituting
q
is neither control input nor output. We emphasize that isd = 2 − i2
2Ieff (15)
sq
under V/f control the motor speed is not exactly equal
to the synchronous speed, and a slip (up to 3%) between results in

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 Preprints of the 21st IFAC World Congress (Virtual)
Berlin, Germany, July 12-17, 2020


|us |2 − 2(Rs2 + ωs2 L2s )Ieff
2
= (1 − σ)ωs Ls 2Rs isq × parallel to the mutual inductance (see, e.g., Sen (2007)).
q  This resistance is defined by
× 2Ieff 2 − i2 − ω L (1 + σ)i2 .
sq s s sq (16) 3 (ωs |ψm |)2
RFe = , (24)
2 PFe
It proves to be convenient to introduce the auxiliary
variables where ψm is the mutual flux linkage. ψm can be expressed
in terms of the stator and rotor currents with the formula
a1 = 2Rs , a2 = ωs Ls (1 + σ), >
ψm = Lm [isd + ird isq + irq ] . (25)
|us |2 − 2(Rs2 + ωs2 L2s )Ieff
2 (17)
a3 = , Subsequently, substituting the rotor currents from (25) by
ωs Ls (1 − σ) the ones in (6), the magnitude of mutual flux linkage can
which permit expressing (16) as be determined from the
q q stator currents
a3 = a1 isq 2Ieff2 − i2 − a i2 .
sq 2 sq (18) |ψm | = Lm (isd + ird )2 + (isq + irq )2
s
Rearranging (18) yields L2rσ 2
= Lm i2sd + i , (26)
(a3 + a2 i2sq )2 = a21 i2sq (2Ieff
2
− i2sq ) L2r sq
a23 + 2a3 a2 i2sq + a22 i4sq = a21 i2sq (2Ieff
2
− i2sq ) where Lrσ is the rotor leakage inductance. It is an es-
tablished method for the characterization of the iron re-
a23 + 2 a3 a2 − a21 Ieff2
i2sq + (a21 + a22 )i4sq = 0


p sistance to run a no-load test and to use the measured
2 −a3 a2 + a21 Ieff2
± a1 −2a3 a2 Ieff 2 + a2 I 4 − a2
1 eff 3 values of stator voltages and currents of the three phases
isq = 2 2 , and the power factor (Sen (2007)). However, since only
a1 + a2
(19) the signals provided by the inverter are available in our
case, we do not have a direct access to the mentioned
where we used a21 + a22 > 0 in the last step, which holds signal as a function of time. Thus, we consider a simple
since a1 > 0 and a2 > 0 by definition. Furthermore, (18) approximation for modeling the iron resistance suggested
yields a3 + a2 i2sq > 0, or equivalently, by Quang and Dittrich (2015). The iron resistance can be
i2sq > −a3 /a2 . (20) stated by an expression proportional to the synchronous
speed
This implies the solution in the last line of (19) with the ωs
positive sign applies. Hence, RFe = RFe,n , (27)
s ωs,n
p
2 +a
−a3 a2 + a21 Ieff 2 2 4
1 −2a3 a2 Ieff + a1 Ieff − a3
2 where RFe,n is the iron resistance at the nominal frequency
isq = , (21) ωs,n . We briefly explain how to approximate the parameter
a21 + a22
s RFe,n from the nominal data of the motor. To this end, we
first compute the nominal currents isd,n and isq,n with
p
2 −a
a3 a2 + (a21 + 2a22 )Ieff 2 2 4 2
1 −2a3 a2 Ieff + a1 Ieff − a3
isd = , the algorithm from Section 2.1 and the nominal motor
a21 + a22
data (current, voltage, and synchronous frequency). The
(22)
nominal mutual flux ψm,n then results with (26). Next,
where isd is calculated with (15). The speed of a motor recalling (6), the iron losses for the nominal operating
with the number of pole pairs zp can be calculated from the point equal
difference between synchronous speed and the slip speed PFe,n = Pelec,n − Pm,n − PRs ,n − PRr ,n − Pfr,n
given by the following equation 2
    = 3Un Ieff,n cos(ϕn ) − Pm,n − 3Rs Ieff,n +
1 Rr Lm 1 Rr isq
ωm = ωs − isq = ωs − , (23) 3 L 2 2
zp Lr ψrd zp Lr isd − Rr m i − Pfr,n . (28)
2 L2r sq,n
(for the derivation of (23) see, e.g., Melkebeek (2018),
pp. 678). We assume the nominal data of the effective value of the
voltage of one phase Un , effective value of stator current
2.2 Computation of the motor torque Ieff,n , mechanical power Pm,n , the power factor cos(ϕn ),
and the friction losses Pfr,n to be available from the motor
Assuming the currents isq and isd and the speed ωm are manufacturer. Thus, by substituting the expression for the
available, the mechanical power Pm of the motor and the iron losses (28) into (24), the iron resistance RFe,n can be
shaft torque Tm can be computed. We first model the determined. Combining (24), (26), and (27) the iron losses
3 (ωs |ψm |)2 L2 2
 
iron losses in the induction motor to achieve a better 3 ωs,n 2
accuracy in the estimation of the mechanical power and PFe = = Lm ωs i2sd + rσ i . (29)
2 RFe 2 RFe,n L2r sq
accordingly the torque. Iron losses PFe appear in the form result as a function of the frequency ωs .
of eddy-current losses and hysteresis losses. Since the rotor
frequency is much smaller than the stator frequency at Once the iron losses are known, we can compute the air
the frequencies considered here, the iron losses in rotor gap power according to the formula
can be neglected. The iron losses significantly increase in Pag = Pelec − PRs − PFe , (30)
inverter driven motors due to the harmonic components
of the voltages and currents caused by the vector space where Pelec and PRs indicate the electric power and
modulation. These losses are usually modeled by adding ohmic losses of the stator, respectively. In the rotor flux
a resistance RFe in the equivalent circuit of the motor coordinate system, they are given by

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Preprints of the 21st IFAC World Congress (Virtual)
Berlin, Germany, July 12-17, 2020

Pelec = 3
us is = 3
(usd isd + usq isq ), (31) ωs
2 2 ωm/ωp ∆p
VFI Ieff estimation PCP
PR s = 3
2 Rs (i2sd + i2sq ). (32) Tm/Tp Q
V/f-curve |us| algorithm char. curve
where the currents isd and isq results from (21) and (22)
and the voltages usd and usq result from (7) and (8). In
summary, ωs RFe,n motor parameters
L2 ωs,n 2 L2 2
  
Pag = 23 (1 − σ)Ls isq isd − m isd + rσ i ωs .
RFe,n L2r sq
(33) plate data

The friction losses are proportional to the square of the

2
motor speed ωm according to Pfr = F ωm . Using the
following expression for the mechanical power Fig. 2. Block diagram for the estimation.
Pm = Pag (1 − s) − Pfr , (34) Figure 2 shows a block diagram of the quantities and their
we can compute the shaft torque as role in the estimation algorithm as applied to the PCP.
Pm 1 The PCP is driven by an induction motor. We assume
Tm = = (Pag (1 − s) − Pfr ) (35) the synchronous frequency is set by the operator of the
ωm ωm
pump as the input and the effective value of the current
L2 ωs,n 2 L2rσ 2
  
3 is provided by the inverter. The amplitude of the stator
= zp (1−σ)Ls isq isd − m i + i −F ωm ,
2 RFe,n sd L2r sq voltage is obtained from the V/f table. If the motor
for a motor with number of pole pairs zp , where resistances and inductances are not known, they can be
ωs − zp ωm determined with established algorithms (see, e.g., Quang
s= (36) and Dittrich (2015)). We determined the iron resistance
ωs at nominal speed from the motor data provided by the
is the slip parameter. manufacturer as explained in Section 2.2 and estimated the
Table 1 summarizes the steps needed for the computation speed and the shaft torque of the motor with the algorithm
of the motor speed and torque. summarized in Table 1. The estimated motor speed and
shaft torque are the input signals to the pump block shown
Table 1. Computational steps for estimation of in Figure 2. A gear box with gear ratio γ and efficiency
the motor speed and torque. η ≤ 1 is used in our test setup. It can be accounted for in
ωp and Tp with
estimation algorithm: 1
ωp = ωm , (37)
γ
Required signals and parameters:
Tp = γ η Tm . (38)
• Ieff is the measured signal.
• ωs is the input. We estimate the differential pressure over the pump with
• |us | is computed from the V/f table. the characteristic curves available from the pump manu-
• The iron resistance at nominal speed (e.g., derived facturer. The characteristics that apply in our test case,
from (27)) and the motor parameters are required. specifically, the mechanical power (Pp [kW]) as a function
Computational steps: of the speed (n [rpm]) for a set of given differential pres-
i) Calculate currents isq from (21) and isd from (22).
sures (∆p [bar]) are given by the dashed lines in Figure 3.
ii) Calculate motor speed from (23). Since the relationship between the mechanical power Pp
iii) Calculate motor torque from (35). and the speed n are linear for the PCP according to
Figure 3, the mechanical torque is constant for a constant
differential pressure. The mechanical power also is a linear
3. IMPLEMENTATION AND RESULTS function of the differential pressure for a given speed ac-
cording to the characteristic curves. Therefore, the torque
We apply the proposed algorithm to a test setup with Tp respects a function of the form
a progressive cavity pump (PCP). We summarize some Pp
aspects about the PCP needed for the remainder of the Tp = = Tp,0 + α∆p, (39)
ωp
paper in Section 3.1. The proposed algorithm is applied in
Section 3.2. where Tp,0 refers to the torque at zero differential pressure
and where α can be determined from manufacturer data
3.1 Inverter-driven progressive cavity pump such as the characteristics shown in Figure 3. The expres-
sion (39) can be viewed as a simple form of the torque
Progressive cavity pumps are frequently used in industrial models for a PCP presented by Moraes-Duzat (2000) and
systems for pumping high viscosity and abrasive media Wittrisch and Cholet (2012). The simple form results if
(Wittrisch and Cholet (2012), Paladino et al. (2011)). only friction and hydraulic components of the torque are
They are not understood as well as centrifugal pumps. taken into account and the other torque components, e.g.
We estimate the speed, torque, differential pressure over viscous torque, are neglected. In this case Tp,0 can be
the pump, and the flow rate, because these quantities are considered as the friction torque and the term α∆p repre-
essential for a reliable operation of PCP. sents the hydraulic torque, where the constant parameter

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Berlin, Germany, July 12-17, 2020

12
ar
10 0b d) e) FT
f) g)
8 VFI M Tp , ωp
Q [m3 /h]

ar

Pp [kW]
6b
6 3
PT PT
4 6 bar 2
2 1 a) c) b)
0 bar
0 100 200 300 400 Fig. 4. Components of the pumping process test setup:
n [rpm] a) container; b) PCP; c) control valve; d) VFI; e)
induction motor; f) gear box. Measurement devices:
g) torque and speed transmitter; PT: pressure trans-
Fig. 3. Characteristic curves for the progressive cavity mitter; FT: flow transmitter.
pump. 1
α depends on the geometry of the PCP. However, note that
the starting torque of a PCP is large (see e.g. Wittrisch 8
and Cholet (2012), pp. 27) and the approximation (39) is 7 isq
only valid for the middle range speeds. We experimentally

current [A]
6 isd
verified this approximation applies with high accuracy for
the estimation of the torque in the range of 100-400 rpm. Ieff
5
The values Tp,0 = 15.03 and α = 5.97 apply in our test 4
setup. After determining Tp with (38), the differential pres-
sure can be determined with (39). Subsequently, we deter- 3
mine the flow rate with the characteristic curve available
2
from the pump manufacturer. This second characteristic 0 20 40 60 80 100 120
curve, which states the flow rate Q as a function of the time [s]
speed n for fixed differential pressures ∆p, is given in
Figure 3 (solid lines). For our specific test setup, a char-
acteristic curve corresponding to a differential pressure Fig. 5. Estimation of the currents isd and isq .
∆p [bar] ∈ {0, 2, 4, 6} can be approximated by
points. The PCP is driven by an induction motor 2 . The
Q(t) = Qnref ,∆p + knref ,∆p (n(t) − nref ), (40) iron resistance at nominal operating point is approximated
where Qnref ,∆p [m3 /h] ∈ {2.9, 2.72, 2.15, 0.536} is the by (27), where for this case RFe,n = 628 Ω has been
flow rate value corresponding to the reference speed calculated. The motor and pump are connected via a gear
nref [rpm] = 100 and knref∈{0.0283, 0.0283, 0.0285, 0.0298}. box with the transmission ratio γ = 2.94 and the efficiency
The flow rate Q(t) corresponding to a differential pressure η = 0.96. By using the algorithm introduced in Quang
∆p ∈ [∆p1 , ∆p2 ] at a speed n(t), where ∆p1 = 2b ∆p and Dittrich (2015) (pp. 207), we determined the motor
2 c
∆p parameters
and ∆p2 = 2b 2 c + 2 (b·c is the floor operator) can be
computed in the following way. We obtain the variable Rs = Rr = 1.16Ω, Lm = 0.16 H,
Q1 (t) by inserting n(t), Qnref ,∆p1 , and knref ,∆p1 in (40). Ls = Lr = 0.19 H, σ = 0.0812 (43)
Analogously, the variable Q2 (t) is computed at the same from the nominal data. The friction coefficient of the
speed with the differential pressure ∆p2 . Then, the flow shaft is taken from the motor manufacturer data sheet as
rate Q(t) can be calculated by the linear interpolation of F = 7.69 × 10−4 . The VFI is set to V/f open loop mode.
Q1 (t) and Q2 (t) as We record the pump speed, torque, flow rate, and pressure
Q(t) = θ Q1 (t) + (1 − θ)Q2 (t), (41) signals of the pump separately by additional sensors for
verification purposes. Note that these additional sensors
where the parameter 0 ≤ θ ≤ 1 is defined by
are not used for the proposed algorithm. All data is
∆p2 − ∆p recorded with the sampling time 1 ms.
θ= . (42)
2
Starting from the synchronous frequency 10 Hz, we incre-
ment the motor synchronous frequency by steps of 10 Hz
3.2 Results every 30 s. The position of the control valve c) is kept
constant. Figure 5 shows the estimated current signals isd
Figure 2 shows a sketch of the experimental setup. The and isq , and the measured signal Ieff . The flux producing
fluid pumped from the container a) passes through the current isd is almost constant as expected, whereas the
2
PCP b) and the control valve c), and eventually flows back SK-112MH/4 manufactured by Nord GmbH. Nominal data: power
to the container a). The control valve produces a back 4 KW, current 8A in star connection, frequency fn = 50 Hz, effective
pressure, allowing the realization of different operational voltage of one phase with respect to the star connection point
Un = 230 V, speed nn = 1440 rpm, power factor cos ϕn = 0.83,
1 10-6L manufactured by Seepex GmbH with undersize rotor. and number of pole pairs zp = 2.

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400 60
300

Tp [Nm]
40
n [rpm]

200
20
100
0 0
20 40 60 80 100 120 20 40 60 80 100 120
time [s] time [s]
8 10
∆p [bar]

6 8

Q [m3 /h]
4 6
2 4
0 2
20 40 60 80 100 120 20 40 60 80 100 120
time [s] time [s]

Fig. 6. Estimation signals (dashed) and reference measurements (solid) for the PCP process.

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