robust fault diagnosis and load torque estimation in ... fault diagnosis and load torque estimation...

International Journal on Electrical Engineering and Informatics - Volume 10, Number 1, March 2018

Robust Fault Diagnosis and Load Torque Estimation in Electrical Drives

Using Analytical Redundancy Relations and Sliding Mode Observer

Slimane Medjmadj1, Demba Diallo2, and Mohammed Mostefai3

1Laboratoire d’Automatique de Setif (LAS), Univ. Bordj BouArreridj 34000, Algeria

2Laboratoire Génie Electrique et Électronique de Paris (GeePs), CNRS, CentraleSupélec, Univ.

Paris-Sud, Univ. Pierre et Marie Curie, 91192 Gif Sur Yvette, France. 3Laboratoire d’Automatique de Setif (LAS), Univ. Setif 19000, Algeria

[email protected]

Abstract: Electrical drives are more and more used in transportation applications and

industrial process because of their effective cost and high efficiency. They have become

critical components and therefore any fault or failure affecting them may seriously endanger

the users and/or induce huge financial costs. As a consequence, early detection and isolation

of these faults is recommended if not mandatory to enhance the safety, the availability, the

reliability and improve the maintenance and the operational efficiency. Analytical

Redundancy Relations (ARR) are used to compute to structured residuals on which the

diagnosis is based. To smooth the residuals and enhance the fault detection capability, a

sliding mode observer is developed to compute the derivatives. The analysis of the residuals

reveals that they are Gaussian signals with a change in the mean. The comparison between

the two-sigma and the two-sided CUSUM methods has shown that the latter is more efficient

to compute the thresholds. Therefore with the two-sided CUSUM, the rotor resistance fault

is detected with no false alarm. Moreover it is proved that the fault diagnosis is robust to the

load torque variations. Beside, thanks to the residual, a load torque estimator is also

developed. Extensive simulation results prove the validity of our approach.

Index Terms: Electrical drives, Redundancy Relations, Sliding mode, Gaussian distribution,

Fault detection and isolation (FDI).

1. Introduction

The induction machine has ever been the horsepower of electrical machines in the industry,

because of its low cost and structural robustness. However, its intrinsic nonlinearity has slowed

down the penetration of induction drive in the industry or transportation applications.

Fortunately, vector control has now become a reliable and widespread method to drive induction

machine. Therefore, induction motor drive is a serious candidate in electric or hybrid vehicle

powertrain, in electric aircraft, wind turbine, etc. Unfortunately this electrical component can be

affected with different faults [1]. These different faults may yield drastic consequences. The

main problems are related to increasing costs, safety degradation and final product quality

degradation. The detection of the faults allows avoiding unexpected factory stops and saving a

great deal of money [2, 3].

There are mainly three families of methods to diagnose and isolate faults in electrical drives [4]:

• Process-history data-based methods [5] such as signal processing based on chemical analysis,

temperature [6], partial discharge [7] or vibration signal measurement.

• Knowledge based methods such as Artificial Neural Networks, genetic algorithms [8], fuzzy

logic [9].

• Model-based methods such as observers [10, 11], Parity space [12].

When a model of the actuator is available, even if non-linear, model-based methods are very

effective. Figure 1 displays the corresponding flowchart.

Received: February 17th, 2017. Accepted: March 18th, 2018

DOI: 10.15676/ijeei.2018.10.1.13

186

Figure 1. Model-based fault diagnosis

The fault detection and isolation (FDI) [13, 14] block, as it can be seen in Figure 2, is the

main component of the diagnosis procedure. It is based on the computation of the residuals and

the thresholds and also the signature analysis to make a decision.

Figure 2. Schematic diagram of the IM drive and the FDI

The aim of this paper is to investigate the design of a FDI for an induction motor drive with

a faulty rotor yielding an abrupt modification of the rotor resistance. As the model is known and

accurate enough, a Parity space approach is used to determine analytical redundancy relations

(ARR), which are robust against parameters uncertainties. The analytical redundancy approach

requires that the residual generator performs some kind of validation of the nominal relationships

of the system, using the actual input and measured output. If a fault occurs, significant

modifications of the residuals are expected. The key issue is to measure those modifications to

make a decision on the process condition.

FDI

Inputs

Residuals

Comparison to thresholds

Signature analysis

Model Process

Ω

Inverter

VDC

Defaults

Failure Decision

I.M

Control

FDI

Slimane Medjmadj, et al.

187

The analysis of the residuals (namely comparison to constant or varying thresholds and fault

signature analysis) is usually based on the statistics of the residuals, and a decision rule is then

applied to determine fault occurrence [15] as displayed in Figure 3.

Figure 3. Diagram of the FDI

ARRs are computed with the system inputs, outputs and their derivatives. Differentiation of

output signals is an old and well-known problem. The calculation of successive derivatives of

the output signals with the classical derivative or polynomial approximation [16, 17] produces

the amplification of the noises. Because of this approximation, the computed residuals are highly

noisy, which makes the diagnosis more difficult with the risk of false alarms or non- detection.

In this work, we propose to improve the computation of the successive derivatives with a sliding

mode observer.

Beside the fault detection and diagnosis, the residuals are used to estimate the load torque.

This is a non-measured variable but its knowledge can be useful in the monitoring but also in the

improvement of position or speed estimation [18-19]

This paper is organized as follows: section II is devoted to the description of the principle of

ARRs computation to form the residuals. In section III, two methods of threshold computation

dedicated to Gaussian signals are presented. In section IV the ARRs are computed for the

induction motor drive and the structured residuals are extracted. In the same section, the sliding

mode observer is introduced to calculate the successive derivatives. Section V presents the

simulation results for the load torque estimator and the rotor fault diagnosis. Finally a conclusion

closes the paper.

2. Robust residual Generation for FDI

A. Analytical Redundancy

The fundamental problem of robust fault detection consists in designing a residual generator

which yields residuals or fault indicators that allow a reliable decision between the two

hypotheses: non-faulty or faulty system, despite the disturbances and model uncertainties.

Consider the following system:

,,

,,

zxhy

zxfx (1)

Where n

x is the state vector, m

z is the input vector, y is the measurement vector

and l

is some constant parameter vector. f and h are assumed to be polynomial functions

in their arguments and ttttv

tuz ,,, Where u represents control inputs, v the unknown

inputs, the faults and the noise. Considering Dj successive time derivatives of the jth output:

,,jD

zxjD

jGjD

jy (2)

System Outputs: y

System Inputs: u

Failure

Decision Residuals

Ri

⋮ ⋮

Decision Control

Unit

Estimation of the

Load Torque

Rotor Resistance

Fault Detection

and Isolation

𝑢ሷ , 𝑦ሷ

𝑢ሶ , 𝑦ሶ Differential

Sliding Mode

Observers

Analytical

Redundancy

Relations

Techniques


188

Where jDzresp

jDjy . stands for zrespiy . and its time derivatives up to order js .

The concatenation of (2) for pJj ,. . .,1 gives a system of j jD 1 equations:

,, zxSGy (3)

Where JjjDD , and for simplicity y stands for

pDy

Dy

Dy ...,

2,

1 . ARR are

input/output relations obtained by eliminating the unknown state x in (3). ARR existence

conditions depend on the rank of the Jacobian matrix [20]

x

zxS

G

,,.

Using symbolic computation algorithms a set of polynomial ARR is found:

0,,,, zywzxDGy (4)

B. ARR Decomposition

Since each component of ,, zyw is polynomial in its arguments, the following

decomposition holds [16, 21]:

,,,,,,,,,),,,,,( vuyswvuyd

wvuyw

(5)

Where : ,,,, vuyd

w , is the deterministic component of w and ,,,,, vuysw the

stochastic one are at least of one degree in some components of ,v and . Thus the residual

is equal to zero in the absence of unknown inputs, the faults and the noise. Each of these terms

can also be decomposed as follows:

,,,,,,,,,,,,, vuydnr

wuydr

wuycwvuyd

w (6)

,,,,,,,,,,,, vuysnrwuysrwvuysw (7)

Where the lower scripts c, r and nr stand respectively for the computational, robust and non-

robust to perturbations components. They verify the following properties:

0,0,,,,

0,0,0,,

0,0,,

vuysw

uydnr

w

uydr

w

(8)

The robust ARR exist if all the terms indexed nr are null. Their decomposition gives:

- the computation form is: uycwcomp

,

- the evaluation form is: ,,,,, uysrwuydr

weval

The faults are not detectable if 0,, uydr

w despite that 0 .

3. Decision-Making System

The role of the decision-making system is to process the residuals and to make a decision

concerning the condition of the system; the group of residuals allows isolating certain faults. The

residuals are null when there is no fault but are different if a fault appears. However, due to

noises and model uncertainties, it is mandatory to include thresholds, judiciously computed so

as to avoid false alarms and non- detection.


189

A. Constant Threshold (The 2σ rule)

One robust solution is to compute the thresholds from the mean value and the standard deviation

of the residuals as stated in (9) [22, 23].

mi1

n. R

i(k)

k1

n

i1

n. R

ik mi

2

k1

n

12

Approximately 95% of the values of the residual Ri are in the 2σ interval:

iSimiRiSim .2.2 (10)

The fault can be detected by comparing the residual to the thresholds.

B. Two-sided CUSUM Algorithm

Let us now investigate further the situation discussed in [24, 25] where the signal is Gaussian

with a mean value change. The cumulative sum (CUSUM) algorithm is one of the most used

methods to detect the abrupt change of residuals. This method is founded on the comparison at

every moment of the difference between the value of the report/ratio of probability and its current

minimal value of a residue.

Let us consider a residue r(t), where the mean value after change is either 1 or 1

. It is

relevant to use two CUSUM algorithms together; the first to compute the parameters 0 and v

in healthy case, and the second for detecting the moment of the fault with the computation of

two thresholds named

kg and

kg . The expressions are given in (11) through (15).

The following algorithm makes it possible to determine the breakdown instant of appearance

[26]:

v

01 (11)

v

01 (12)

201

vkrgg kk

(13)

201

vkrgg kk

(14)

kk ggh max,maxmax

(15)

Where h represents the threshold of the residual r(t). The time at which the change is detected

ta is as follows:

hghgkt kka :min (16)

4. Application

A. System Modelling

The system considered in this paper is an induction motor drive, with the state vector defined

as:

tsqi

sdirqrd

txxxxxx ,,,,5,

4,

3,

2,

1

(17)


190

and the output vector txxx

tyyyy )5,

4,

1()

3,

2,

1( . With the rotor angular speed Ω, the

stator currents in the (d,q) reference frame isd , isq and finally the rotor fluxes rd ,rq . The control

inputs are represented by the stator voltage vsd and vsq. The outputs are Ω, isd and isq. Under usual

hypotheses the equations are [16, 17]:

sL

sqvxxpxxx

sL

sdv

xxpxxx

xxpxxMx

xxpxxMx

J

lT

xJ

Bxxxx

JrL

pMx

2135

1

5

3124

1

4

21353

31242

143521

(18)

Rs , Ls are the stator resistance and inductance, Rr , Lr are the rotor resistance and inductance,

M is the mutual inductance between stator and rotor, p is the number of pole pairs, B is the

damping coefficient, J is the moment of inertia and Tl is the load torque (an unknown input).

rL

rR

sL

sR

sL

rL

M

sL

rL

M

rL

rR

sL

sR

1

1,,

²1,,

B. ARR Design

Deriving the speed and currents at 1st order, the rank of the Jacobian matrix

31

21

11

GxJ

GxJ

GxJ is equal

to 5.

There exists one ARR that is not robust against the load torque variations [20, 27]:

0112

12

23122

32112

1 l

TpypBypJypyyypAsLpyyypAsLpR

(19)

This residual can be used to estimate the load torque.

1121

2

31222131ˆ yByJ

yp

yypyAsLpyypyAsLp

lT

(20)

Deriving the speed at 1st order and the currents at 2nd order, the rank of the Jacobian matrix

22

22

32

22

11

11

Gl

TJ

Gl

TJ

GxJ

GxJ

Gl

TJGxJ is equal to 6. There exist two ARRs that are robust against Tl [20, 27].


191

022

112111211231221

1

2

12

2

pypyAAyppypyAAyypyApyR

(21)

022

111211212121321

2

2

12

3

pypyAAypypypAAyypyApyR

(22)

With:

sL

sqvyyA

sL

sdv

yyA.

3.32;.

2.21

C. Structured Residuals for Fault Isolation

The structured residuals are designed so that each residual is sensitive to only one fault. The

detection of a fault depends on the most sensitive residual and its isolation on the less sensitive

one. To isolate the fault source a set of residuals with different responses for each fault is needed.

This principle is illustrated by the incidence matrix (Table 1). Each column represents a fault

signature.

For example:

- R4, doesn’t depend on Rr. Therefore it allows the isolation of a rotor resistance fault.

- R6 is robust against Ω and sensitive to Rs, Rr, Ia, Ib and Ic.

Table 1. Incidence Matrix

Rr Rs Ω Or θ Ia Ib Ic

R1(Rr,Rs,Ω,Ia,Ib,Ic) 1 1 1 1 1 1

R2(Rr,Rs,Ω,Ia,Ib,Ic) 1 1 1 1 1 1

R3(Rr,Rs,Ω,Ia,Ib,Ic) 1 1 1 1 1 1

R4(Rs,Ω,Ia,Ib,Ic) 0 1 1 1 1 1

R5(Rr,Ω,Ia,Ib,Ic) 1 0 1 1 1 1

R6(Rr,Rs,Ia,Ib,Ic) 1 1 0 1 1 1

R7(Rr,Rs,Ω,Ib,Ic) 1 1 1 0 1 1

R8(Rr,Rs,Ω,Ia,Ic) 1 1 1 1 0 1

R9(Rr,Rs,Ω,Ia,Ib) 1 1 1 1 1 0

For these residuals the tests of exceeding the thresholds (Ki) are applied separately as follows:

RNi

iKtiRif

iKtiRifti ....1

1

0

(23)

Where 1 ....NR

is the vector of faults signatures.

The response of the residual set is characteristic to each fault and is so called signature or

code of the fault. This operation allows uniform representation of the information.

In [20], it is shown that the structured residual robust to the rotor resistance fault is given by the

determinant of the Sylvester matrix:

,,det,,Re 32324 RRSylvRRsR

(24)


192

0.30.2

1.30.31.20.2

2.31.30.32.21.20.2

3.32.31.33.22.21.2

3.32.33.22.2

3.33.2

32

0000

00

00

0000

,,

SS

SSSS

SSSSSS

SSSSSS

SSSS

SS

RRSylv

Where

1

2

11112

3

10.2

21121211

2

11.2

312112.2

223.2

²

11²

11

1

SpySyypSpyS

yyyyypSypSpyS

yypSpySS

L

VyyS

s

sd

2

2

12111

3

10.3

31131112

2

11.3

211122.3

333.3

²

11²

11

1

SpySyypSpyS

yyyyypSypSpyS

yypSpySS

L

VyyS

s

sq

D. Estimation of the Derivatives Via Sliding Mode Technique

The sliding mode technique is used at first in control theory, but in the recent years sliding

mode is used to design observers for many applications [28, 29].

For this system, we notice that the three analytical redundancy relations found previously

((19), (21) and (22)) are linked to the control inputs, unknown inputs, parameters, outputs and

their derivatives. Let the output signal g(t), considered as the input of the observer, be a

measurable locally bounded function defined on [0, ∞] and let it consist of a base signal having

a derivative with the Lipschitz’s constant (c >0) and a noise. In order to differentiate the unknown

base signal, consider the following auxiliary equation.

(25)

Applying the second order sliding mode algorithm to keep 0)( tg , we obtain:

)(1

)(2/1

)(1

tgSign

tgSigntg

(26)

)(tgS is the sliding surface. Where and λ are >0 and )(t is the output of the

differentiator (observer) which is the estimator of the derivative of g(t). According to [28, 30,

31], the following are the sufficient conditions for the convergence of )(t to ).(tg

c

cc

c

42

(27)


193

The main drawback of this observer is the well-known chattering phenomenon due to the

sign function. To reduce the effects of this phenomenon, we can replace the sign(x) function by 2

.arctan qx

; the coefficient q is a parameter to adjust the slope of the arc tangent function.

5. Simulation results and Discussions

A field oriented voltage vector control is used to drive the induction motor. The parameters

of the induction motor are listed in Table 2.

Table 2. Induction Machine Characteristics Parameter Value

Nominal Power Pn 4 kW

Nominal Torque 25 N.m

Number of Pole Pairs p 2

Stator Resistance Rs 1.2 Ω

Stator Inductance Ls 0.1554 H

Rotor Resistance Rr 1.8 Ω

Rotor Inductance Lr 0.1568 H

Mutual Inductance M 0.15 H

Rotor Inertia J 0.07 kg/m²

The sampling period is equal to 1ms and the simulation is performed during 4s. With no loss

of generality, the reference speed of the machine is fixed in all this study at 1200 rev/min that is

equivalent to 125.6rd/s. The inverter is driven with a SPWM technique.

A. Load Torque Estimation

Figure 4.a displays the real and estimated load torque. The estimation error (Figure 4.b) is

almost zero except during the start-up. Except during the initial state, the dynamic performances

of the estimator are very good.

a. Real and estimated load torque

b. Estimation error

Figure 4. The load torque estimator


194

B. Fault Diagnosis

One of the most important purposes of fault diagnosis is to detect unexpected changes (faults)

in a monitored process as early as possible, before they lead to a complete breakdown. For the

diagnosis, the following load torque profile (Figure. 5) is assumed as an example.

Figure 5. Load torque applied

The residuals generated by this method contain successive derivatives of the outputs, which

are sensitive to noise and then can be amplified by using classical derivative. To reduce this

phenomenon, we have used a differential sliding mode observer to estimate those successive

derivatives and the results are better. Figure.6 shows the time evolution of residuals R1, R2 and

R3 in the healthy case with classical derivative.

a. R1

b. R2

c. R3

Figure 6. Residuals in healthy case with classical derivatives

We observe in Figure.6, that residuals R1, R2 and R3 generate false alarms due to peaks that

are caused by derivatives. If we increase the thresholds, we will increase the delay of detection,

and moreover some faults would not be detected.

Figure 7 shows the residuals computed with the sliding mode observer. We can observe

clearly that the residuals are smoother and for the same thresholds, no false alarm is generated.


195

a. R1

b. R2

c. R3

Figure 7. Residuals in healthy case with sliding mode observer

These results confirm that the use of a second order sliding mode observer to estimate the

successive derivatives of each output reduces the occurrence of false alarms.

Many motor faults have an electrical reason. Rotor bar faults are usually associated with high

temperatures, high mechanical loading particularly during the starting time, or any defective

casting or poor jointing during the manufacturing process. Initially, they started as high resistance

causing high temperature and then progress as cracking or small holes in the rotor bars.

In healthy state, rotor resistance varies slowly due to operating conditions. We can assume

therefore that an abrupt variation can be considered as a fault.

However in a closed-loop speed drive, despite the fault, the operation points remain in close

domains as it can be seen in Figure.8. that displays the trajectory in the (d,q) reference frame of

currents for different rotor resistance values (Rr.n and ±25% Rr.n).

Figure 8. (d, q) current pattern in case for different rotor resistance values


196

We can also observe in Figure.9 that the speed regulation is also still effective despite the

variations of the rotor resistance.

Figure 9. Speed regulation

These results show that the drive is still under control but with a faulty rotor. A monitoring

system is therefore necessary to detect the fault as it may lead to efficiency reduction, accelerated

ageing or failure.

The role of the decision-making system is to process the residuals and to make a decision

concerning the condition of the drive. Figure.10 shows the probability density function (pdf) of

the two residuals R2 and R3 under different values of resistance rotor.

a. R2

b. R3

Figure 10. Pdf of the two residuals under different rotor resistance values.

We can notice that the pdf is the sum of three Gaussian distribution with mean value µ and

variance σ². In this case, the probability density function is given by:

2

3

3

2

2

2

2

1

1

2

1

3

2

1

2

2

1

1

111

2

1

yyy

eeeyp

Where : 2

3

2

2

2

1 , 31

, 0321 ,

02 and

31

We can observe from Table 3 a change in the means and variances parameters for the

different rotor resistance faults (±25% Rr.n). Theoretically, µ=0 but the actual value cannot be

zero because of the imbalances due to the SPWM and derivatives computation.


197

Table 3. Characteristics of The Residuals

Residual R2 Residual R3

Rr (Ω) Rr (Ω)

1.35 1.8 2.25 1.35 1.8 2.25

μ -0.014 -0.009 -0.016 0.018 0.01 0.012

σ 2.828 3.336 2.713 3.746 4.432 3.590

max 4.988 5.677 4.760 6.708 7.783 6.274

min -5.051 -5.707 -4.872 -6.434 -7.67 -6.32

To evaluate the method, we suppose that at t = 2s, a rotor resistance fault (25% is added to

the value of Rr,n) appears. This failure has an influence on all variables of the system because of

the control loop that propagates the fault effect.

Figure 11 shows the results of fault diagnosis by using the 2σ method on R2 and R3. Clearly,

the fault is not detected because the residuals remain under the thresholds computed with the 2σ

method.

a. R2

b. R3

Figure 11. 2σ algorithm results

Let us now analyse the results of the two-sided CUSUM algorithm. The evolution of

kg and

kg for the two residuals R2 and R3 are presented on Figure 12.

Thresholds computed from R2


198

Thresholds computed from R3

Figure 12. Two-sided CUSUM algorithm results

As it can be seen in Figureure.12, on the contrary of the 2σ method, the fault is detected with

no false alarm in 215 ms. The rate of no-detection corresponds to the number of samples during

the detection time.

Table 4. Results of The Two-Sided Cusum Algorithm Parameter R2 R3

0.1364 0.1976

µo -0.0056 0.0119

Max kg 697.201 924.326

Max kg 698.517 923.851

h 698.517 924.326

Alarm time 2.215 2.220

Rate of no detection 0.107 0.110

Rate of false alarm 0 0

Therefore, if a fault induces in change in the mean, the two-sided CUSUM algorithm is more

efficient than the 2σ one. In order to enhance the fault diagnosis, it is necessary to check the

robustness to load torque variations. Figure.13 represents the load torque variations, from the

nominal torque (25 N.m) to an overload of 40% (Tl=35N.m).

Figure 13. Load torque (real and estimated)

Residual R1 (Figure.14 (a)) can be used to detect an overload (Figure.13), while residuals R2

and R3 are clearly robust against the load torque variations.

v


199

a. R1

b. R2

c. R3

Figure 14. Residuals robustness evaluation

6. Conclusion

This work has presented the application of the analytical redundancy relations to derive

structured residuals for the rotor fault detection in an induction motor drive. To smooth the

residuals, the derivatives are successfully computed with a sliding mode observer. Comparison

with the classical derivative shows the superiority of the method.

Within the Fault Detection and Isolation scheme, the residuals have been analysed and

because they are Gaussian, thresholds are determined with the two-sided CUSUM algorithm.

The results have shown that the rotor resistance abrupt variation is detected with no false alarm.

The robustness of the detection against the load torque variation has also been proved.

Beside, the residuals have also been used to estimate accurately the load torque. The results

are very encouraging and further investigations on the use of residuals for parameter estimation

and other fault detection and diagnosis are being conducted. Experimental validations should

also emphasize the effectiveness of the approach.

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Slimane MEDJMADJ received his engineering, magister and PhD degrees in

electrotechnique from the University of Setif 1, Setif, Algeria in 1996, 2005

and 2015, respectively. He is currently an Assistant Professor in the

Department of Electromechanical, Bordj Bou Arréridj University, Bordj Bou

Arréridj, Algeria. He is a Member of the Laboratory of Control (LAS),

University of Setif. His main research interests include diagnosis and fault

tolerant control of electric drives.

Demba DIALLO Senior Member IEEE SM05, received the M.Sc. and Ph.D.

degrees both in Electrical and Computer Engineering, from the National

Polytechnic Institute of Grenoble, France, in 1990 and 1993.

He’s currently Professor at the University Paris Sud and is with the Group of

Electrical Engineering Paris. His current area of research includes fault

diagnosis, fault tolerant control and energy management. The applications are

related to more electrified transportation sytems (EV and HEV) and

microgrids.

Mohammed MOSTEFAI was born in B.B.A., Algeria, on April 16, 1965. He

received the Engineering Diploma degree in electronics from the University of

Setif, Sétif, Algeria, in 1988, and the Ph.D. degree in automatic and software

engineering from the University of Lille, Lille, France, in 1994. Currently, he

is a Full Professor with the University of Setif. Since February 2006, he has

held several scientific and administrative responsibilities, first as President of

the scientific committee, then as Director of the Automatic Control Laboratory,

and finally as Vice President for Research with the University of Setif. His

research interests include study, modeling, analysis, and control of industrial systems. These are

mainly production systems, ranging from electro-energy systems (electric machines and

networks) up to large automated manufacturing systems.


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