Parameter estimation and reliable fault detection of electric motors

Dusan PROGOVAC,Le Yi WANG,George YIN

1.Delphi Corporation,3000 University Drive,Auburn Hills,MI 48326,U.S.A.;

2.Department of Electrical and Computer Engineering,Wayne State University,Detroit,MI 48202,U.S.A.;

3.Department of Mathematics,Wayne State University,Detroit,MI 48202,U.S.A.

Parameter estimation and reliable fault detection of electric motors

Dusan PROGOVAC1,Le Yi WANG2?,George YIN3

1.Delphi Corporation,3000 University Drive,Auburn Hills,MI 48326,U.S.A.;

2.Department of Electrical and Computer Engineering,Wayne State University,Detroit,MI 48202,U.S.A.;

3.Department of Mathematics,Wayne State University,Detroit,MI 48202,U.S.A.

Accurate model identification and fault detection are necessary for reliable motor control.Motor-characterizing parameters experience substantial changes due to aging,motor operating conditions,and faults.Consequently,motor parameters must be estimated accurately and reliably during operation.Based on enhanced model structures of electric motors that accommodate both normal and faulty modes,this paper introduces bias-corrected least-squares(LS)estimation algorithms that incorporate functions for correcting estimation bias,forgetting factors for capturing sudden faults,and recursive structures for efficient real-time implementation.Permanent magnet motors are used as a benchmark type for concrete algorithm development and evaluation.Algorithms are presented,their properties are established,and their accuracy and robustness are evaluated by simulation case studies under both normal operations and inter-turn winding faults.Implementation issues from different motor control schemes are also discussed.

Electric machine;Parameter estimation;Fault detection;Brushless direct current(BLDC)motor;Bias correction;Forgetting factor

1 Introduction

This paper introduces new methods for accurate parameter estimation and reliable fault detection of inverter powered electric motors.Brushless direct current(BLDC)and permanent magnet alternate current(PMAC)motors are used as a benchmark platform to develop our methods.Electric motors are essential parts of electric and hybrid vehicle powertrains[1-3]and other diversified industrial applications[4].Accurate model identification and fault detection are necessary for reliable motor control[5].Motor-characterizing parameters experience substantial changes or sudden jumps due to aging,motor operating conditions,or faults[6,7].Consequently,motor parameters must be estimated ac-curately during operation,leading to a system identification problem[8,9].

This paper employs 3-phase motors as a platform to develop algorithms for identifying motor parameters during normal operations and detecting stator winding faults.To facilitate this study,an enhanced model of 3-phase permanent magnet(PM)motors is developed that accommodates both normal and faulty operating conditions.Due to high measurement noise,motor parameter estimation is a challenging problem.Both motor inputs and outputs are corrupted by noise,leading to an errors-in-variables identification(EIV)problem[10].An EIV structure is known to introduce identification bias[11].Motor faults entail sudden jumps in motor dynamics.To diagnose the faults promptly,identification algorithms must achieve a good balance between fast fault detection(which prefers a short data window),and noise attenuation(which is achieved by averaging,preferably over a large data window).Also,motor controller frameworks are pre-designed and must be accommodated in system identification.

This paper introduces an enhanced least-squares(LS)estimation algorithm that incorporates a function for correcting estimation bias,a forgetting factor for capturing sudden faults,and a recursive structure for efficient real-time implementation.Algorithms are presented,their properties are established,and their accuracy and robustness are evaluated by simulation case studies under both normal operations and inter-turn winding faults.One contribution of this paper is the development of new bias correction algorithms with forgetting factors in a recursive structure.Traditionally,bias correction in an EIV problem was treated by modified correction terms,instrumental methods,or prediction error methods[10-12].The issues of estimation bias and its correction for battery model identification and state-of-charge estimation were discussed in our earlier papers[13,14],without consideration of forgetting factors and the corresponding recursive algorithms.

The rest of the paper is organized into the following sections.Section 2 establishes enhanced model structures for three-phase balanced PM motors in normal and faulty conditions.Identification algorithms are introduced in Section 3.In Section 4,bias-corrected LS algorithms are presented and their bias correction capabilities are established.Section 5 discusses practical aspects of motor estimation,involving different motor control schemes.Reliability of parameter estimation under these schemes is studied.For fast diagnosis of faults,system identification must balance speed and accuracy.Section 6 introduces forgetting factors into our bias correction algorithms.Recursive algorithms are derived.Section 7 concentrates on inter-turn fault diagnosis.Basic algorithms are introduced and evaluated by case studies.Section 8 highlights the main findings of this paper and points out some worthy open problems.Some preliminary ideas of this paper were reported in[15].

2 Enhanced PM motor models under normal operation and faulty conditions

This section describes models of surface mounted PM motors.For working principles,types,mechanisms,and control systems of PM motors,we refer the reader to[4,6]for details.Exploration on modeling and diagnosis of surface mounted PM machines can be found in[5,7,16-18].In this paper,we introduce an enhanced model for PM motors in normal and faulty conditions.The three-phase balanced stator windings under normal operating conditions are illustrated in Fig.1.Under a balanced construction,all phases have the same parameters and are symmetric.

Fig.1 Three phase stator winding.

We start with models of healthy stator windings;see Fig.1,in which the windings are assumed to be sinusoidally distributed.1It should be emphasized that the model structures are also valid under other types of flux linkages and back electromotive force(EMF),such as trapezoidal types.Since the stator windings are balanced,without loss of generality,we use phase a as a generic phase.The state equation for healthy stator windings is

wherevais the phase-a winding terminal voltage(V),iais the phase-a current(A),Ris the phase-a resistance(Ω),and λais the total phase-a flux linkage(Wb).Under the assumption of magnetic linearity and infinite permeability of iron,the flux linkage is related to the phase current and magnetic coupling byHere,Lis the phase-a inductance(H),Mis the stator phase crossinductance(H),λMis the stator/rotor magnetic coupling flux linkage(Wb),fis the electric angular speed of the rotor(Hz),and typically δa=0(rad)(δb=-2π/3 and δc=2π/3).

Assume that there is no saliency,i.e.,the air gap between the rotor and the stator is constant.Then,the stator inductance is constant and does not depend on the relative rotor position.It follows from(1)that

This can be written compactly as

whereIis the identity matrix,and

with

Within this frame,when the stator is subject to a winding fault,model(3)is perturbed.We will use the phase-a fault as a benchmark case in our derivations,see Fig.2.Detection algorithms for phase-b and phase-c faults are similar.

Suppose that the original number of turns of phase a isNafor whichNasturns are shorted.Denote μ=Nas/Na,the ratio of faulty turns.It is noted that the fault introduces a fault currentifthrough the bypass branch of resistanceRfin Fig.2.Fault diagnosis is built on the following enhanced model which captures inter-turn faults with bypass resistance.Here,we assume that the healthy motor model has been identified with model parametersR,L,M,λMestimated.Fault detection aims to identify additional parameters that represent inter-turn faults.From Fig.2,such parameters include μ andRf.

Fig.2 Three phase stator windings with a bypass fault in phase a.

Following the same principles as before,under a fault of 0<μ<1 in phase a with resistanceRf,the model(3)is perturbed to

wherevfis the voltage cross the faulty turns.By eliminatingvf,we obtain

The first equation implies

which,after substituting into the fourth equation,leads to

It is interesting to note that this relationship betweenvfandvais independent of the valueL.

Now,using(6)to eliminateifin the first three equations in(4)results in

These can be compactly expressed as

whereHandgare defined before,andG1=(R,0,0)′,G2=(L,M,M)′.

Next,we discretize(7)for implementation of algorithms on a computer.Suppose that the sampling interval is τ.LetThen,(7)is discretized to

where we have

This paper investigates motor parameter estimation under normal operating conditions and fault detection.Simulation models will be used to schedule a fault appearance.Starting with a normal operation,a fault is then simulated in phase a at a certain time.Our enhanced model is then used to represent the voltagecurrent profiles after the fault.These will be covered in the subsequent sections.

3 Identification algorithms

3.1 Regression models for system identification

The healthy motor model(3)contains four parametersR,L,Mand λM.For system identification,we rewrite(3)in the form of

It is noted that the dimensions areθ∈R4.Also,although physically it is more convenient to view the phase voltages as the input and the currents as the output for the motor models,for system identification we follow(11)to viewvkas the output andikas the input.As a result,in the sebsequent discussions,output noises will refer to voltage measurement noises and input noises will be current measurement noises.

3.2 Algorithms

Due to measurement errors and disturbances,observations are corrupted by noisesCurrent measurement noises introduce a perturbation on the regressorConsequently,the regression relationship that utilizes measured values isk.Here,ekis due to noises on the voltage and δkis induced from the current measurement noises.

Assumption 1The joint vector sequence{[εk,ek]}is stationary and strongly ergodic(in the sense of convergence with probability one(w.p.1))such thatand that both{[εk,ek]}andare ergodic.That is,0,w.p.1 aswhereSis a nonnegative definite matrix w.p.1 asN→∞.Here,E(.)denotes the expectation.

Note that the noises are zero mean,but we do not need the sequences{εk}and{ek}to be independent or uncorrelated.A sufficient condition to ensure the ergodicity in the above assumption is that the underlying sequence is a stationary φ-mixing sequence,which is a sequence whose remote past and distant future are asymptotically independent.The well-known results[19,p.488]then yield that[εk,ek]and{[εk,ek][εk,ek]′}are strongly ergodic.

AfterNobservations,denote

We illustrate our basic algorithms with the following example.

Example 2A PM motor has the following true parameters:R=2.8750Ω,L=0.0064H,M=-0.0021H,λM=0.1750Wb.This model is simulated on a Matlab platform.The sampling frequency is 100kHz or equivalently the sampling interval is τ=0.01ms.The applied voltage profiles are balanced three-phase sinusoid waveforms of peak value 500V and frequency 60Hz.The simulation is run for a total 2000 sampling points.The output(voltage)is corrupted by noise,which is a Gaussian i.i.d.(independent and identically distributed)process of zero mean and standard deviation σv=20V.The LS algorithm(12)is applied.Fig.3 demonstrates the parameter estimation error trajectories.The error is defined as‖θN-θ‖where‖.‖is the Euclidean norm.In this case,estimation is quite accurate.

We demonstrate in Section 4 that if the input is also subject to measurement noise,this algorithm will introduce identification bias,namely,parameter estimates will converge to values different from the true value.

Fig.3 Estimation error trajectories with output noise only.

4 Identification bias and correction

4.1 Errors-in-variables identification and estimation bias

ProofThis follows from

This completes the proof.

Fig.4 Impact of input measurement noise on estimation bias.

4.2 Bias correction by modified LS algorithms

Identification bias can be corrected if Σ andBare known.Algorithm(12)is now modified to

Theorem 5Under the assumptions of Theorem 3,the estimates in(13)satisfy θN→ θ,w.p.1 asN→ ∞.

ProofBy the strong law of large numbers,asN→∞,

This completes the proof.

The modified LS algorithm(13)can be recursified for real-time computational efficiency.The following recursive algorithm was introduced in[13].

Theorem 6[13] The estimates θNin(13)can be updated recursively as

Example 7Continuing the study from Example 4,we note that when the input noise exists and bias correction method is not applied,at the exit point(N=2000)the norm of the estimation error is 2.486(a sample result in simulation).Now we apply our bias correction algorithm,the estimation error at the exit point is reduced to 0.0061.

5 Case studies on parameter estimation and implementation considerations

Practical motors involve certain physical system structures,nonlinearities,auxiliary driving circuits,and time delays.This section includes some more realistic simulation studies that accommodate further motor details.

Two types of stator construction are common for PM machines:sinusoidal winding distribution for permanent magnet synchronous machines(PMSM)and concentrated winding for BLDC motors.In the first case,the back EMF is sinusoidal.The back EMF under concentrated winding is trapezoidal.One important difference between these two types is that synchronous machines have continuous currents through all windings(180-degree current leads).In contrast,BLDC machines will have 'square' currents with 120-degree leads.Consequently,for each winding there is a time interval when there is no current through a particular winding.

Typical PM configurations include the six-step controlled PM motor shown in Fig.5,the filed-oriented control(FOC),and the self-controlled system.In the six-step motor,its inverter has six signal levels and requires the lowest closed-loop bandwidth.Sensor delay is a critical parameter because it results in model mismatch.The FOC motor directly controls the stator rotating magnetic field on the rotating frame to provide maximal torque generation and to ensure smoothness of rotor movements.The self-controlled operation is a simplified FOC that employs a stator-based coordinate frame.It is simple in construction,but requires high bandwidths,generates more noise,and is less smooth in rotor movements than the other two types[18].

We now present simulation studies for stator winding parameter identification under normal operating conditions.The motor is a six-step controlled motor with sinusoid state winding.The motor true parameters are the same as in Example 2.In this case,R,L,Mand λMare to be estimated under a closed-loop configuration.The model sampling time is 0.1ms.A total of 10000 data points are used in this study.Due to PWM control circuits,the driving voltages' profiles are no longer sinusoid waveforms.The phase current waveforms are also quite different.These are shown in Fig.6.

Fig.5 Six-step controlled PM motor.

Fig.6 Phase voltage and current profiles.

To understand further the impact of current measurement noises,we compare two cases:1)Only voltage(output in system identification)measurements have noises;2)both voltage and current measurements are subject to noises.The least-squares algorithm(12)is applied.In the first case,only the output(voltage)is corrupted by noise,which is a sequence of Gaussian i.i.d.random variables with zero mean and standard deviation σv=50V.Estimates are shown in Fig.7.The top plot shows that when no input noise exists,the LS algorithm generates highly accurate estimates.When an input noise is added to the current measurements,which is a Gaussian i.i.d.sequence with zero mean and standard deviation σi=10A,the bottom plot illustrates that the parameter estimation has a bias,which is about 2.1,or a relative estimation error 72%.This is a persistent bias that does not decrease with an increase in data size.

Fig.7 Impact of input measurement noise on estimation bias.

The bias correction algorithm(13)is then applied.The estimated parameter values at the exit point are listed in Table 1.The norm of the estimation error is 0.011,or a relative error 0.3826%.

Table 1 Estimates from bias-corrected LS algorithm.

6 Fast tracking and forgetting factors

whereWN=diag{λN,λN-1,...,λ,1}.When λ=1,it is reduced to the un-weighted LS algorithm.When λ is close to 0,only most recent data are used in estimating parameters.λ is called a 'forgetting factor'.There is a key trade-off in selecting λ.If λ is close to 1,then historical data remain heavily weighted.Consequently,fault detection will be slow.On the other hand,if λ is small,the fault detection will be faster,but noise attenuation capability will be compromised,which follows from the laws of large numbers[19].

LetThen,(14)can be written as

whose solution can be obtained by the LS result withYNreplaced byQNYNand ΦNbyQNΦN,as

When both input and output noises are taken into consideration,(15)becomes

However,when input noises cause bias in LS estimates,(16)will be subject to bias as well.We note that,which implies.As a result,On the other hand,

We should point out that since 0<λ<1,the factorin(13)is no longer needed here.We now derive a recursive algorithm for(17).LetandBy stationarity,these quantities do not depend onN.

Theorem 8Given a forgetting factor 0<λ≤1,the bias-corrected LS estimate θNwith forgetting factor λ in(17)can be updated recursively as

ProofFrom(17),On the other hand,

Let.Then,

By the matrix inversion lemma

Moreover,

Define.Then,

Finally,

This completes the proof.

7 Diagnosis of stator winding faults

7.1 Inter-turn fault

Fig.8 shows rotor speed trajectories for a six-step controlled motor when an inter-turn fault happens in phase a att=2s with fault bypass resistanceRf=100Ω.It is noted that when a fault happens the closed-loop regulation has difficulty in maintaining the required rotor speed if the leakage insulation is close to a short circuit.We will present a new detection algorithm which can detect such a fault with accuracy.

Fig.8 Six-step controlled motor with a winding fault at t=2s,Rf=100 Ω.

7.2 Estimation of κ

The estimation algorithms under normal operating conditions provide nominal values of balanced stator winding parameters.In this section,we concentrate on fault detection.Fault detection methods for multi-phase electrical motors can take advantage of balanced phase designs.Since all phases are symmetric,faults will alter parameter values and create an imbalanced condition between any pair of phases that can be used for detecting and isolating faults.Stator winding faults can spread quickly.Without prompt detection and protective actions,the condition can deteriorate rapidly.As aresult,it is extremely important that fault detection is fast,which creates a challenging situation for designing identification algorithms.Rotor speed fluctuations can be affected by both faults and load variations.As a result,fault detection and isolation from rotor speed fluctuations are not reliable in the majority of practical situations.

wherezkand ψkcan be easily derived from(8).For computation ofzkand ψk,we point out that the inverse ofHcan be explicitly computed as

It is apparent that all previous algorithms remain viable,withykreplaced byzkand φkby ψk.As a result,we will not spell out the details here.To distinguish from the previous expressions,we will express the bias correction algorithm as

Note that the correction terms ξ andbare scalars,and the inverse is changed to a division here.

Example 9We first examine the bias from measurement noises.From the regressor expression in(18),the voltage measurement noises will cause estimation bias.We evaluate estimation biases on κ by applying i.i.d.Gaussian measurement noises of zero mean but different variances.σvis the standard deviation of the voltage measurement noise,and σiis the standard deviation of the current measurement noise.Table 2 illustrates estimation errors when noise variances increase.The sampling interval is 10ms,the estimation data length is 10000,μ=0.5(50%inter-turn fault),Rf=10,L=0.0064,M=-0.0021,R=2.8750.Apparently,the estimation biases are quite significant.

Table 2 Estimation errors on κ without bias correction.

Example 10In comparison,if the bias-corrected estimation algorithm(19)is applied,the estimation accuracy can be significantly improved.This is shown in Table 3 under the same simulation conditions.Since the noises are i.i.d.,b=0.Hence,the bias correction is based on ξ.

Table 3 Estimation errors on κ with bias correction.

7.3 Fast fault detection with forgetting factor

One critical requirement for fast fault detection is to make the identification algorithms rely more heavily on the recent data.As discussed in Section 6,this can be achieved by employing forgetting factors.To illustrate the impact of forgetting factors on the speed of fault detection,we select different values of λ and show the corresponding trajectories of estimates in tracking κ af-ter a fault occurrence in Fig.9.It is clear that to achieve fast tracking capability,a relatively small λ should be selected.

Fig.9 Estimation of κ under different forgetting factor λ.

7.4 Statistics for ξ and b

The bias correction algorithm(19)relies on the knowledge of ξ andbto devise correction actions.In practical applications,such covariance values may not be availablea priori.As a result,they need to be estimated also.We now present an estimation scheme for ξ andb.For simplicity,we assume that all sensor noises are Gaussian i.i.d.random variables.

The identification equation for κ isZN= ΨNκ.We assume that the three-phase motor model is known.Hence ΨNis known.The measurement equations areSubstituting these equations in ΨN,we obtainThe bias correction term is the limitwhich can be used to estimate ξ.

8 Conclusions

Parameter identification for three-phase motors is a difficult task due to measurement noise.The methodology introduced in this paper enhances the traditional LS algorithms with integrated bias removal and forgetting factors.By relating winding faults to changes in a characterizing variable,fault detection is explored in the identification framework.In addition,by exploiting the symmetry of phases in balanced PM stator windings,we introduce a ratio test to isolate faults with fast response and convergence.We demonstrate that our bias removal algorithms can significantly improve fault detection reliability when measurement noises are present.A related topic is control algorithms for running a three-phase motor when only two phases are functional.Combined with our fault detection algorithms,this joint diagnosis and control strategy can potentially provide robustness in motor operations when faults occur.

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12 November 2013;revised 15 January 2014;accepted 21 February 2014

DOI10.1007/s11768-014-0178-y

?Corresponding author.

E-mail:lywang@wayne.edu.Tel.:+313-577-4715;fax:+313-577-1101.

?2014 South China University of Technology,Academy of Mathematics and Systems Science,CAS,and Springer-Verlag Berlin Heidelberg

Dusan PROGOVACreceived the B.S.degree in Electrical Engineering in 1988 and his M.S.degree in Mathematics in 1987 all from University of Southern California,Los Angeles.Since 1988,he has been working as Senior Engineering Specialist for General Dynamics Land System,Senior Project Engineer for TRW,Project Design Engineer at Ford Motor Company,and Software Engineer at Delphi Corporation.He also worked as Associate Lecturer for Department of Mathematics,University of Wisconsin at Milwaukee.His research interests are in the areas of information complexity,system identification,detection of abrupt changes,fault detection and vehicle powertrain control systems.He presented his papers at several conferences and he has been Program Committee Member for International Conferences.E-mail:dusan.progovac@delphi.com.

Le Yi WANGreceived the Ph.D.degree in Electrical Engineering from McGill University,Montreal,Canada,in1990.Since1990,he has been with Wayne State University,Detroit,Michigan,where he is currently a professor in the Department of Electrical and Computer Engineering.His research interests are in the areas of complexity and information,system identification,robust control,H∞optimization,time-varying systems,adaptive systems,hybrid and nonlinear systems,information processing and learning,as well as medical,automotive,communications,power systems,and computer applications of control methodologies.He was a keynote speaker in several international conferences.He was an associate editor of the IEEE Transactions on Automatic Control and several other journals,and currently is an associate editor of the Journal of System Sciences and Complexity and Control Theory and Technology.He is a Fellow of IEEE.E-mail:lywang@wayne.edu.

George YINjoined Wayne State University in 1987 and became a professor in 1996.Working on stochastic systems,he is Chair of SIAM Activity Group in Control and Systems Theory and is one of the Board of Directors of American Automatic Control Council.He was Co-Chair of SIAM Conference on Control&Its Application,2011,Co-Chair of 1996 AMS-SIAM Summer Seminar and 2003 AMS-IMS-SIAM Summer Research Conference,Coorganizer of 2005 IMA Workshop on Wireless Communications.He chaired the SIAM W.T.and Idalia Reid Prize Committee,the SIAG/Control and Systems Theory Prize Committee,and the SIAM SICON Best Paper Prize Committee.He is an associate editor of Control Theory and Technology,SIAM Journal on Control and Optimization,and on the editorial board of many other journals and book series.He was an associate editor of Automatica and IEEE T-AC.He was President of Wayne State University's Academy of Scholars.He is a Fellow of IEEE.Email:gyin@math.wayne.edu.