Real-time stator resistance estimation for electrical drives -a control perspective

(1)

IN

DEGREE PROJECT ELECTRICAL ENGINEERING, SECOND CYCLE, 30 CREDITS

STOCKHOLM SWEDEN 2020,

Real-time stator resistance

estimation for electrical drives -

a control perspective

FREDRIC SUNESSON

KTH ROYAL INSTITUTE OF TECHNOLOGY

SCHOOL OF ELECTRICAL ENGINEERING AND COMPUTER SCIENCE

(2)

(3)

Real-time stator resistance

estimation for electrical

drives - a control perspective

FREDRIC SUNESSON

Master in Electric Power Engineering Date: September 17, 2020

Supervisor: Gustaf Falk Olson

Company Supervisor: Arpit Gupta & Torbjörn Larsson Examiner: Luca Peretti

School of Electrical Engineering and Computer Science Host company: Volvo cars

Swedish title: Uppskattning av statorresistansen i realtid för elektriska drivsystem - ett kontrollperspektiv

(4)

(5)

iii

Acknowledgements

This thesis was carried out in cooperation with Volvo cars. I hope that this work will be usable for Volvo cars and that it contributes to improve their electric motors.

First of all, I would like to thank Arpit Gupta and Torbjörn Larsson, my su- pervisors at Volvo cars. I am grateful for all help, feedback and inputs needed to improve and carry out this thesis.To Giovanni Zanuso for the possibility of using his drive for experiments, but also valuable inputs during the thesis.

Finally, I would like to thank Gustaf Falk Olson my supervisor at KTH for all weekly meetings with great feedback and inputs.

(6)

iv

Abstract

The automotive industry is in the middle of a rapid transition towards electrical machines. This transition leads to a higher interest in electrical machine management. In this thesis a stator resistance estimation scheme for field oriented controlled motors is proposed and tested in both simulations and experiments.

The proposed scheme uses a DC-current injection in the stationary reference frame with injection in one of the axes. Ohm’s law is used to estimate the stator resistance. The DC-component of the current is found by a zero-crossover detection scheme which calculates the average current over one electric period. The DC-current values is then eliminated from the feedback currents to the field oriented control to minimise disturbances in these controls. The injection is controlled by simple PI-regulators in both axes of the stationary frame to compensate for the cross-coupling between the α and β axis currents.

Simulations shows that the accuracy of the estimated resistance improves and the speed of resistance estimation increases when the injection control is applied in both axis. However a continuous implementation of Ohm’s law is highly oscillatory. To prevent this an accumulator is introduced which aver- ages the estimated resistance over several electrical periods. The trade off using this accumulator is that the speed of detection decreases to achieve higher accuracy. Torque oscillations are inevitable when a DC-current component is injected in the motor. But torque oscillations are solely dependent on the injected DC-current level. The experiments show promising initial results in both DC-current estimation and injection control. However future test are required to further improve the accuracy of the resistance estimation.

(7)

v

Sammanfattning

Fordonsindustrin är mitt i en omsträllning mot elektriska motorer. Detta leder till ett ökat intresse i kontroll av elektriska motorer. I detta arbete föreslås en statorresistans estimering för elektriska motorer med fältorienterings reglering (FOC).

Estimeringen testas både i simuleringar och experiment. Den föreslagna estimeringen utgår från att likström (DC) injiceras i en stationär referensram. Med hjälp av Ohms lag estimeras statorresistansen. Estimeringen av likströmmen kommer från en nollövergångs detekterings algoritm som räknar ut medelvär- det över en elektrisk period. Likströmmen elimineras därefter från feedbacken till övriga regulatorer för att minimera störningar i övriga regulatorer.

Injektionen kontrolleras av enklar PI-regulatorer i både α och β för att kom- pensera för tvärkopplingen mellan dessa. Simuleringarna visar att noggrann- heten ökar och estimerings tiden minskar när likströmmen kontrolleras i både e α och β. Att kontinuerligt implementera Ohms lag är ostabilt. För att stabilisera estimeringen inför en ackumulator som räknar ut medelvärdet för estimering över flera elektriska perioder. Avvägning står dock mellan högre noggrann- het i estimeringen och längre tid för estimeringen. Vridmomentssvägnignar är oundvikliga när likström injiceras i motorn. Dock är vridmomentssvägnig- narna endast beroende på hur stör likström som injiceras. Experimenten visar lovande initiala resultat i både likströms estimeringen och injicerings regle- ringen. Fler tester är dock nödvändiga för att ytterligare förbättra estimeringen.

(8)

List of Figures

2.1 Cross-section of a three phase motors stator windings . . . 4

2.2 The T-model of an induction motor . . . 5

2.3 The inverse Γ model of an induction motor . . . 7

2.4 A basic control scheme of an electric drive . . . 10

2.5 The proposed control scheme with stator resistance estimation 17 3.1 The estimated alpha and beta DC currents with no load and at start up of the motor. . . 22

3.2 The alpha and beta currents with no load when the motor is started. . . 22

3.3 The estimated alpha DC currents with no load, at nominal speed and a fictional step at 2.5 s. . . 23

3.4 The estimated alpha DC currents with no load, at steady state nominal speed and a decrease in speed after 20 seconds. . . 24

3.5 Estimated DC current, reference voltage in the α-frame and the estimated stator resistance . . . 25

3.6 Influence of a resistance estimation in the α-axis in the dq currents seen by the current controller . . . 26

3.7 αβ- frame AC-currents at the start of injection . . . 26

3.8 Motor Torque with the DC-estimation with DC current of 1.12 A started at 5 s . . . 27

3.9 Estimated DC-currents in the alpha and beta frame . . . 29

3.10 Estimated DC-current, voltage and stator resistance. . . 29

3.11 Currents in the dq-frame seen from the current controller . . . 30

3.12 Improved stator resistance estimation . . . 31

3.13 Improved stator resistance estimation with a maximum time of 0.1s and 2 s . . . 32

3.14 Effects of speed change in the improved DC-estimation and estimated DC-current. . . 33

viii

(11)

LIST OF FIGURES ix

3.15 Resistance estimation results with a rapid change in resistance 34 3.16 Resistance estimation results with a slow change in resistance . 34 3.17 Experimental AC-currents and estimated DC-current in both

α & β frame . . . 35

3.18 The low-frequency oscillations in α and β DC-currents estimated 36 3.19 Mechanical measured speed . . . 37

3.20 Experimental DC-injection currents in both α & β frame . . . 38

3.21 Measured phase currents of the motor when injection is started after 2.2 seconds . . . 38

3.22 Reference voltages for the resistance estimation . . . 39

3.23 Experimental online stator resistance estimation . . . 39

3.24 Post-process resistance estimation with averaging DC-values over 60, 600 and 6000 samples . . . 40

A.1 Simulink model . . . 45

A.2 The inverse gamma model of the induction motor in Simulink 46 A.3 The Speed controller in Simulink . . . 47

A.4 The current controller in Simulink . . . 48

A.5 The current model in Simulink . . . 49

A.6 The resistance estimation block in Simulink . . . 50

(12)

List of Tables

2.1 Induction machine data . . . 19

2.3 Simulink control parameters . . . 19

2.2 Base values used for simulations . . . 20

2.4 Experimental Control parameters . . . 20

3.1 Different injected currents and corresponding Torque ripple. At no load and nominal speed. . . 27

3.2 Torque ripple at different loads and speeds. . . 28

x

(13)

Abbreviations

FOC Field-oriented control DC Direct Current

AC Alternating Current IM Induction Machine MMF Magnetomotive force PWM Pulse-width modulation SVM Space vector modulation

FPGA Field-Programmable Gate Array SoC System-on-a-Chip

VSC Voltage Source Converter RPM Revolutions Per Minute

xi

(14)

List of Symbols

Symbol Description Unit

a_c Current controller bandwidth rad/s

a_s Speed controller bandwidth rad/s

b Viscous damping constant kgm²/s

b_a Active viscous damping constant kgm²/s

E¯^s Back EMF space vector in a stationary reference frame V

f_s Sampling frequency Hz

F(s) Controller transfer function -

G(s) Complex system transfer function -

G_cl(s) Closed loop complex system transfer function -

n_p Number of pole pais -

i_p p-axis current where, p=α, β, d, q A

¯i^s_s Stator-current space vector in a stationary reference frame A

¯i^s_r Rotor-current space vector in a stationary reference frame A

¯i^r_r Rotor-current space vector in a rotating reference frame A

J Inertia kgm²

L_lr Rotor leakage inductance H

L_ls Stator leakage inductance H

L_m Mututal stator and rotor magnetizing inductance H

L_M Transformed magnetizing inductance H

L_σ Transformed total leakage inductance H

P_e Electric power tranfered to the shaft W

Rr Rotor winding resistance Ω

R_R Transformed rotor resistance Ω

R_s Stator resistance Ω

R_s,est Estimated Stator resistance Ω

s Slip %

T_s Sampling period time s

T_{P W M} Inverter period time s

xii

(15)

LIST OF TABLES xiii

u_p p-axis voltage where, p=α, β, d, q V

¯

u^s_s Stator-voltage space vector in a stationary reference frame V

¯

u^s_r Rotor-voltage space vector in a stationary reference frame V

¯

u^r_r Rotor-voltage space vector in a rotating reference frame V

τe Electrical torque Nm

ψ¯_a^s Air gap flux Vs

ψ¯_s^s Stator-flux space vector in a stationary reference frame Vs ψ¯_r^s Rotor-flux space vector in a stationary reference frame Vs ψ¯_r^s Transformed rotor-flux space vector in a stationary reference frame Vs ψ¯_r^r Rotor-flux space vector in a rotating reference frame Vs

θ₁ Flux angle rad

ω₁ Electrical angular frequency rad/s

ωr Electrical rotor speed rad/s

ω_m Mechanical motor speed rad/s

(16)

(17)

Chapter 1 Introduction

1.1 Background

In recent years the realization of climate change has grown tremendously within the society. The UN Paris Agreement from 2015 states that all countries should aim for a reduction of green house gas emissions. According to the Paris Agreement the global warming is to be held within 2^◦ C. The transport sector stands today for about 15% of the global green house gas emissions [1]. The automotive industry is standing before one of the grandest changes in the his- tory of the industry. The conventional Internal combustion engine, which has been the predominant technique used for the passed century is being replaced.

The alternative that stands out is the battery powered electric motors. An example is Volvo Cars which in 2017 stated that they are committed to electrify all models sold after 2019[2]. This is fueled both by public opinion and the will to be up to date with regulations supposed by governments.

But the expanding usage of electric vehicles and motors also bring new challenges for developers. The control and management of the motor is a crucial part for an efficient and robust performing motor. One of the largest contributor to faults in an electric motor is overheating and proper thermal management of the motor could increase the lifetime of the motor significantly [3]. The thermal management of a motor could be performed in numerous ways and the most obvious is with built in temperature sensors. A drawback of thermal sensors is that it is associated with an extra cost both economically and space wise. An alternative to external thermal sensors is to estimate the stator resistance which is proportional of the temperature. Hence the temperature of the motor can be deduced by the resistance estimation. Since the temperature of

1

(18)

2 CHAPTER 1. INTRODUCTION

the motor varies with motor mode, load condition etc. it has to be performed continuous to ensure reliable results. Another positive effect of the resistance estimation is that it can be used to update the control algorithm in the FOC to enhance performance of the drive.

1.2 Aim

With a DC current injection in the stator windings the resistance can easily be found by applying Ohm´s law. But there are two challenges involved; First:

how should the DC-injection be tuned to keep the torque ripple and interference with the current and speed control at a minimum. Second: How are the DC-values obtained. The aim of this thesis is therefore to create an algorithm to estimate the stator resistance online using a current injection technique. The algorithm should be that general that it could be applicable to multiple AC- drives.

1.3 Method

This thesis consists of three main steps. Firstly a literature review where the current state of the art in online resistance estimation by DC current injection is examined and a supposed algorithm is introduced. Secondly simulations are used to verify the results. The final step is to verify the algorithm by experiments on an induction motor.

(19)

Chapter 2 Theory

In this chapter the basic principles of the Induction motor (IM). The theoretical background of the FOC is described, the resistance estimation theory and the technical description of the test motor.

2.1 The basics of AC-machines

Alternating current machines are basically designed to convert electric energy to mechanical energy. The two fundamental parts of AC machines is the stationary stator windings which are carrying AC currents which introduces a rotating magnetic field in the machine. This rotating magnetic field then produces a force on the rotor which produces torque to the shaft of the motor.

The difference between the two main motor types synchronous motor and induction motor is the magnetization of the rotor. In an induction machine the magnetization of the rotor is fed from the stator current, i.e. the stator current magnetize the stator and the rotor indirectly. In a synchronous motor the magnetization of the rotor is directly by the permanent magnets.

2.1.1 State space representation

A common procedure in the control and analysis of drive systems are to rep- resent the state variables, current, voltage and flux as state space vectors. A cross section of a three phase stator is displayed in Figure 2.1. Where “a”

indicates that the current is flowing into the paper and “a⁰” indicates that the current is flowing in the positive direction out of the paper. The axis of a shows the direction where the MMF given from the current in phase "a" flow through has the maximum positive value. Since the currents which flows in the

3

(20)

4 CHAPTER 2. THEORY

𝑎

𝑎′

𝑏′

𝑏 𝑐′

𝑐

𝑎-axis

𝑐-axis 𝑏-axis

Figure 2.1: Cross-section of a three phase motors stator windings different windings are displaced by the angle 120 degrees, the MMF will be displaced by 120 degrees as well. Since these vectors only have an amplitude in the plane perpendicular to the shaft of the machine it is possible to describe these by complex numbers. This means that the three-phase machine can be analysed as a two-phase system with alpha and beta components of the state variables[4]. If the real axis is aligned with the a-axis the transform from the three-phase system to the two phase has the following matrix.

i_α iβ

=

2

3 −¹₃ −¹₃ 0 ^√¹

3 −^√¹

3

=



 i_a i_b i_c



 (2.1)

Even though the complexity of the system now has been reduced one ob- stacle remains. Since the vectors described above are fixed at the stator, they will be viewed by the rotor as fluctuating. Since the mechanical work of the machine is bound to the shaft connected to the rotor a rotating reference frame is convenient. This rotating reference frame is rotating with the speed of the airgap flux induced by the stator currents and have the same amplitude as the stator currents but different argument. This is called the synchronous reference frame. If θ is the angle between the α axis and the real synchronous frame axis d. the transform has the following form.

i_d i_q

=

cos(θ) sin(θ)

−sin(θ) cos(θ)

i_α i_β

(2.2)

(21)

CHAPTER 2. THEORY 5

If the machine is in steady state, the currents in the synchronous reference frame will be seen as DC-components. This also allows for the control system to work much more easily since just PI-control is enough to handle DC- components.

2.1.2 Induction machine basics

The induction machine is the most used machine in industrial appliances. His- torically the main usage has been constant speed operations. At the present variable speed drive of an induction motor by means of VSC is state of the art.

T-model

The normally used model of the induction motor is the T model shown in Figure 2.2.

𝑅_𝑠

𝐿_𝑚

𝑅_𝑟 ҧ𝑖𝑠

𝑠

+

− ത 𝑢_𝑠^𝑠

𝐿𝑠𝑙

𝑗𝜔𝑟𝜓ത𝑟𝑠

ҧ𝑖𝑟 𝑠

ҧ𝑖𝑚𝑠

𝐿𝑟𝑙

+

−

Figure 2.2: The T-model of an induction motor

It is in the T-model assumed that the stator resistance is equal in all three phases. The law of induction states that the stator voltage that is not dissipating in the stator resistance will build up flux in the stator windings.

¯

u^s_s = R_s¯i^s_s+∂ ¯ψ_s^s

∂t (2.3)

The rotor has a similar equation if viewed from a rotational reference frame, where the rotor is rotating with the speed of the rotor. When viewed from the

(22)

6 CHAPTER 2. THEORY

rotor there will be no induced voltage from the rotation hence the rotor flux dynamics is identical to the stator flux dynamics. The difference is that the rotor of an induction machine is short circuited. Hence ¯u^r_r = 0.

0 = R_r¯i^r_r+∂ ¯ψ^r_r

∂t (2.4)

If this equation is transformed to the stator fixed reference frame the following relation is found.

0 = jω_rψ¯^s_r− R_r¯i^s_r− ∂ ¯ψ_r^s

∂t (2.5)

The equations for the induction machine can there for be found to be

∂ ¯ψ_s^s

∂t =¯u^s_s− R_s¯i^s_s (2.6)

∂ ¯ψ_r^s

∂t =jω_rψ¯_r^s− R_r¯i^s_r (2.7) For the stator and rotor respectively. Assuming linear relations, the airgap flux can be represented as.

ψ¯_a^s = L_m(¯i^s_s+ ¯i^s_r) (2.8) Where Lmis the magnetization inductance. The total stator and rotor flux can be expressed as,

ψ¯_s^s=L_m(¯i^s_s+ ¯i^s_r) + L_sl¯i^s_s (2.9) ψ¯_r^s=L_m(¯i^s_s+ ¯i^s_r) + L_rl¯i^s_r (2.10)

Combining all these equation yields.

¯

u^s_s =R_s¯i^s_s+ L_sl¯i^s_s+ L_m∂(¯i^s_s+ ¯i^s_r)

∂t (2.11)

jω_rψ¯^s_r =R_r¯i^s_r+ L_rl¯i^s_r+ L_m∂(¯i^s_s+ ¯i^s_r)

∂t (2.12)

This is known as the T equivalent model for the induction machine. Even though the T-model represents the physical motor correctly it is inefficient to use in dynamic and control design purposes since it is over parametrized[5].

Since a large part of this thesis is evolving around the control the T-equivalent does not fit this master thesis scope. The inverse Γ model is a better choice.

(23)

CHAPTER 2. THEORY 7

Inverse Γ model

The inverse gamma equivalent is a simplified version of the T-equivalent and shown in Figure 2.3.

𝑅_𝑠

𝐿_𝑀

𝑅𝑅

ҧ𝑖𝑠 𝑠

+

− ത 𝑢𝑠𝑠

𝐿_𝜎

𝑗𝜔_𝑟𝜓ത_𝑅^𝑠 ҧ𝑖𝑅 𝑠

ҧ𝑖𝑀 𝑠

Figure 2.3: The inverse Γ model of an induction motor

Since the inductor currents are linearly dependent they can be simplified and only one leakage inductance is sufficient. This is accomplished by adding a transformation factor f when new rotor variables are determined.

ψ¯_R^s =f ¯ψ_r^s (2.13)

¯i^s_R =¯i^s_R

b (2.14)

If f = ^L_L^m_r the flux equations (2.9) and (2.10) simplifies to ψ¯^s_s =L_s¯i^s_s+L²_m

Lr

¯i^s_R (2.15)

ψ¯^s_R=L²_m

L_r(¯i^s_R+ ¯i^s_s) (2.16) Two new parameters can now be introduced,

L_M = L²_m

L_r (2.17)

L_σ = L_s− L_M = L_sl+ L_rl (2.18)

(24)

8 CHAPTER 2. THEORY

The rotor resistance can be transformed into RR= (^L_L^m

r)²R_r. which gives:

¯

u^s_s=R_s¯i^s_s+ L_σ∂¯i^s_s

∂t + L_M∂¯i^s_M

∂t (2.19)

jω_rψ¯_R^s =R_R¯i^s_R+ L_M∂¯i^s_M

∂t (2.20)

This gives the dynamic equivalent for the machine in both stator and rotor currents. But since the rotor current is hard to sensor for most control purposes only the stator current is used. So the rotor current is eliminated in the following way.

¯i^s_R= ψ¯_R^s LM

− ¯i^s_s (2.21)

Which gives the final dynamical inverse Γ model for the induction machine.

¯

u^s_s =R_s¯i^s_s− L_σ∂¯i^s_s

∂t − ∂ ¯ψ_R^s

∂t (2.22)

∂ ¯ψ^s_R

∂t =R_R¯i^s_s− (R_R LM

− jω_r) ¯ψ^s_R (2.23)

Torque production

The complex instantaneous power in a three phase system are defined as, S =¯ 3

2u¯^s(¯i^s)∗ (2.24)

where ¯z^∗ = x − jy is the complex conjugate for the complex number z = x + jy. The complex conjugate ¯i^s makes the expression reference frame in- variant[5]. The active power can then be defined as

P = 3

2Re(¯u^s(¯i^s)∗) = 3

2Re(¯u¯i) (2.25)

If core losses are neglected the power that does not dissipate in the stator and rotor resistances goes to the rotor EMF and is converted to mechanical power.

The power to the shaft can then be defined as, P_e= −3

2Re(jω_rψ¯_R¯i^∗_R) (2.26) Since Re(j ¯z) = Re(j(x + jy)) = −Im(¯z)), the ¯iRis defined as (2.21) in the direction opposite to the rotor EMF the power to the shaft can be defined as

(25)

CHAPTER 2. THEORY 9

P_e = 3

2ω_rIm( ¯ψ_R^∗¯i_s) (2.27) Since power is the product of torque and angular speed, using ωr = n_pω_m therefore the following torque expression can be defined,

τ_e = P_e

ω_m = 3n_p

2 Im( ¯ψ_R^∗¯i_s) (2.28) if this is split in the d and q axis withψ¯_R = ψ_d+ jψ_q it can be written as,

τ_e= 3n_p

2 (ψ_di_q− ψ_qi_d) (2.29) This relation also holds for the stator flux withψ¯_R = ¯ψ_s− L_σi¯_sthe following relation holds

τ_e = 3np

2 Im( ¯ψ_s^∗¯i_s) (2.30) If perfect field orientation (all parameters are perfectly known) is assumed the torque expression simplifies to

τ_e = 3n_p

2 ψ_Ri_q (2.31)

Current model

Induction motors unlike the synchronous machine does not rotate synchronously.

If load is applied the rotor always lags the electrical rotation of the stator. This lag is called slip and is defined as

s = ω₁− ω_r

ω₁ (2.32)

The flux angle θ1 provided will because of the slip not be the same as the positioning angle of the rotor. Measurements of the flux angle in an induction motor could be sensored using Hall elements placed in the airgap. But these are very expensive and should be avoided [5]. An estimation of the flux angle is done by the following relation

ω1− ωr = R_Ri_q

ψ_R (2.33)

Assuming perfect field orientation, the references for the q-axis current iq,ref and the reference flux the following relations is used to find the flux angle and stator electrical frequency.

ω₁ = R_Ri_q,ref

ψ_R,ref (2.34)

θ˙₁ = ω₁ (2.35)

(26)

10 CHAPTER 2. THEORY

2.2 Control Theory

2.2.1 Control basics

A standard control scheme of an electric drive is shown in Figure 2.4

Current Controller Speed

Controller M 𝜔_𝑚

𝜔_𝑚^𝑟𝑒𝑓+ +

− −

𝑖𝛼𝛽

𝑢_𝑑𝑞^𝑟𝑒𝑓

αβ dq

dq αβ

Inverter 𝑢_𝛼𝛽^𝑟𝑒𝑓 𝑖_𝛼𝛽^𝑟𝑒𝑓

Figure 2.4: A basic control scheme of an electric drive

As seen in the figure, the input reference to the model is basically the desired speed ωm which forms the outer loop of the control system. The inner loop is the current controller which takes it’s input from the speed controller and takes it´s feedback from the inverter. The inverter allows for variable amplitude and frequency to be applied to the drive. In both controllers PI- regulators are used. The current controller could basically be made in the stator fixed reference frame. The problem with this scheme is that the PI-controllers then would have to work with sinusoidal references. A PI-controller with sinusoidal input are incapable of giving a zero steady-state error. For this reason the synchronous reference frame is used in a pair of two for both the idand iq

currents. Regarding the tuning of both controllers it is important that the current controller’s bandwidth is smaller than that of the speed controller. Since the current dynamics are faster than the mechanical dynamics it implies that the current controller should be faster.

(27)

CHAPTER 2. THEORY 11

2.2.2 Current control

To obtain the current control parameters the transfer function of the electrical equation is needed. Since the control will be conducted in the synchronous dq- frame the equation has to be transferred into synchronous coordinates. Equa- tion (2.19) is considered with the back- EMF ^{∂ ¯}^ψ

s R

∂t noted asE¯^s,

¯

u^s_s= R¯i^s_s− L_σ∂¯i^s_s

∂t − ¯E^s (2.36)

Since the rotating frame dq is oriented with the rotor flux the derivativeE¯^sis perpendicular to the dq frame. The back EMF can therefore be written as,

E¯^s = jEe^jθ (2.37)

Assuming perfect field orientation (2.36) is transferred into the synchronous reference frame,

Lσ

∂¯i

∂t = ¯u − (R + jω1Lσ)¯i − ¯E (2.38) If is is assumed that the input voltage is set to E and the system is in steady state the complex output power will be:

S =¯ 3 2

E¯i¯ ^∗ (2.39)

which if divided into active and reactive power will yield,

P =3/2E i_q (2.40)

Q =3/2E i_d (2.41)

Hence the d-axis will be controlling the reactive power called the flux producing current and q-axis will control the active power generation called the torque producing current.

Applying the Laplace transform to equation (2.38) the following transfer function is obtained:

G(s) = 1

(s + jω₁)L_σ + R_s (2.42)

The term jω1 is the cross-coupling between the two currents and can be seen if equation 2.38 is split in the d-and q axis,

L_σ∂ ¯i_d

∂t = ¯u_d− R_si¯_d+ ω₁L_σi¯_d− ¯E (2.43) L_σ∂ ¯i_q

∂t = ¯u_q− R_si¯_q− ω₁L_σi¯_q− ¯E (2.44)

(28)

The first part in control design is to cancel this cross-coupling. This is done by adding a decoupler to an inner-feedback loop. To this loop a active resistance R_acan also be added to speed up the dynamics of the controller. The dynamics can then be seen as

L_σ∂¯i

∂t = ¯u − (R_s+ R_a)¯i − ¯E (2.45) This system has the transfer function

G(s) = 1

L_σs + R_s+ R_a (2.46)

The PI controllers which can be written as:

F(s) = k_p+k_i

s (2.47)

where kp is the proportional gain and ki is the integrating gain of the PI controller. The closed loop transfer function is now,

Gcl(s) = F(s)G(s)

1 + F(s)G(s) (2.48)

This yields that the PI-controller parameters should be,

kp =acLσ (2.49)

k_i =(R_s+ R_a) (2.50)

where a^c is the closed-loop-system bandwidth. Since PI controllers work of the error between the reference and the feedback the controller can be described as,

e =iref − i (2.51)

dI

dt =e (2.52)

v_ref =k_pe + k_iI + (jω₁L_σ− R_a)i + E (2.53) Where E is the back-EMF which in this case is seen as a load disturbance which is subtracted from the input value of the controller.

2.2.3 Speed controller

The mechanical dynamics for a three phase motor is, Jdω_m

dt = 3n_p

2 ψ_Ri_q− τ_l (2.54)

(29)

where J is the total inertia of the rotor and the mechanical load, τe is the electrical torque produced by the motor and τ^l is the total load torque. The load torque can be separated into a viscous part and an externally applied load torque τL.

τ_l = bω_m+ τ_L (2.55)

As stated earlier the current which is responsible for the torque production is the q-axis current. Hence the output from the speed controller should be the q-current reference into the current controller. To speed up the controller a inner feedback loop is introduced with an active viscous damping, ba. This is done to move the pole from −b/J to −αswhere αsis the desired closed-loop bandwidth. The active viscous damping is defined as,

b_a= α_sJ − b

ψ_R (2.56)

The transfer function of the new dynamics with the inner feedback loop can now be determined to be,

G_s(s) = ψ_R/J

s + αs (2.57)

The transfer function of the controller can therefore be determined to be F_s(s) = α_s

s G⁻¹_s (s) (2.58)

Where the PI controller parameters can be determined to be, kps = α_sJ

ψ_R (2.59)

kis = α²_sJ

ψ_R (2.60)

2.2.4 Controller saturation and anti-windup

The PI-controllers in both the current and speed control are limited by the physical system. This should be implemented in the control algorithm. This must be integrated into the control. It is done by saturating the output to some maximum allowed value, in the speed controller the maximum allowed current in the q-frame is used and in the current controller the maximal allowed voltage is used. Hence the output x is altered to,

x_sat = sat(x, x_max) (2.61)

(30)

Another problem which needs to be considered is the integrator windup. This occurs if there is a large step in the input reference. The integrator then starts to accumulates the error and when the the input gets close to the reference the integrator has been powered up so that the controlled variable overshoots the reference. The input then settles at the reference after the wind up has been removed by an accumulation of a negative control error. If this occurs at the saturation limit of the controller it could cause potential failure of the machine.

If the current controller is used as an example the error of the integrator in (2.51) is modified to eawgiving,

e =iref − i (2.62)

dI

dt =e_aw (2.63)

v_ref =k_pe + k_iI + (jω₁L − R_a)i + E (2.64) The modified integration error is now calculated backwards to ensure that the saturation region never gets reached. If v^sat,ref is the saturated value of the current controller output and vref is the non saturated value the modified error can be written as,

e_aw = e + 1

k_p(u_sat,ref − u_ref) (2.65) which ensures that the controlled variable never reaches the saturated region.

This gives the final control algorithm for the current controller to be, dI

dt =e + 1

k_p(u_sat,ref − u_ref) (2.66)

u_ref =k_pe + k_iI + (jω₁L − R_a)i + E (2.67)

u_sat,ref =sat(u_ref, u_max) (2.68)

2.3 Discretization and delays

Digital implementation of the control for a motor can not run in continuous time. There needs to be time for sampling of measurements, analysis, and calculation of the switching times which will be sent to the inverter. The sampling frequency, fs, is the number of samples a discrete system takes during one second. In electrical machine control systems sampling frequencies is in the range of thousands of Hertz. In order to have well performing control system the sampling time of the measurements must have the exact same time

(31)

period as the inverter i.e.

Ts= TP W M (2.69)

If these two do not match the inverter will not be able to function since the current values of the inverter legs will differ from period to period. Over one switching period the following happens: The currents are measured by sensors, they are digitised and the control algorithm calculates the switching periods of the next cycle, at the end of the cycle the calculated references are transferred to the inverter and new current measurements are made. This implies that there will be a delay of one sampling period for each control reference. Since the controls are implemented in a rotating reference frame, this will lead to a rotation of the frame by one sampling period, hence this rotation needs to be added to the reference to be able to have a well performing drive.

The delay using a synchronous sampling is 1.5T^s[5].

There is a requirement on the current control to be slower than the sampling time. The current control bandwidth must be below 4% of the angular sampling frequency[5].

α_c<0.04ω_T ω_T =2πf_T (2.70)

Both switches for each phase in the inverter cannot be closed at the same time since it would cause a short circuit. Because of this at a switching both switches are open during a short period of time. This gives uprising switching losses which needs to be compensated.Hence for a certain reference voltage an extra voltage is required to compensate for this dead time. The main problem is that the losses changes at different current values. It also complicates further by the fact that it is different for different converters.

2.4 Current injection

The interest of current injection in drive systems for stator resistance estimation has increased in latter years. The main drawback is that an injection of a DC current in a running drive system will lead to a disturbance in the torque and speed of the machine, ripples will occur. There are mainly to schools of how this could be mitigated. There could be a current injection together with an injection in the second harmonic which would basically reduce the ripple to zero[6]. The problem with this approach is described in [7] that since the injection has to take place in the dq-reference frame. If the machine is running in speed sensorless mode the estimation of the flux has to be obtained. This

(32)

on the other hand depends on the stator resistance. This could according to [7]

lead to a cause and effect problem. Because of this the selected form of injection in this thesis is a single DC-current injection in the stator fixed reference frame αβ. The DC currents has to be small enough to not cause unnecessary large torque ripples, but also large enough to be able to get reliable results.

If both the stator current and the voltage is known in the alpha frame Ohm’s law is used to get the stator resistance in the following manner,

R_s,est = u_sα,DC

i_s_α_,DC . (2.71)

This may seem simple but comes with challenges. First the DC-components needs to be extracted since the α and β is a stationary frame the current form will be sinusoidal. There are multiple approaches that could be used to find the DC-component of the current, one is by using low-pass filters that filters all but the DC-component. The main draw back of low-pass filters is that the cut- off frequency needs to bee tuned which decreases the estimations execution time.[7]. An alternative to this presented by Matić et al. [3]. Their approach is to average the AC signal over one AC-period. Since it is known that the signal contains harmonics at frequencies equal to a multiple of the fundamental.

These harmonics would be eliminated using the proposed method. Since the θ₁ is modulus, hence it is resest at every electrical period this step could be used as the zero crossover detection. The result of the estimated DC-current will then be used in the next AC-period to control the injected DC voltage.

The second challenge is to mitigate the cross-coupling which arises from the current controller. Since the current controller imposes a cross coupling cancellation in the dq-reference frame cross coupling arises in the αβ-frame [7].

Hence a injection in one of the frames will lead to a reaction in the other. This is mitigated by controlling both the α and β currents in the resistance estimation scheme. If the current injection is supposed to be performed in the α-frame the current in the β-frame is controlled to zero[7].

The third challenge is that the original FOC will see the DC injection as disturbances and will try to reject them. Zanuso, Peretti, and Sandulescu [7] presents a solution to this problem. If the injection is controlled by PI- controllers the currents which are obtained by the DC-estimation is withdrawn from the current which is feed back to the current controller to avoid unnecessary rejection from the current controller. The bandwidth of the injection controllers are also discussed. The resistance estimation should not be as fast as the current controller to avoid interference. But since the dynamics are faster than the mechanical system it should be faster than the speed control loop. The control scheme for the DC-current injection can be seen in Figure

(33)

2.5

Using these approaches the resistance estimation could be conducted continu- ously during the DC estimation, hence one sample of the DC-reference voltage and the estimated DC-current value. Zanuso, Peretti, and Sandulescu [7] uses a similar approach but adds that that the sampling time does not always match the AC period time. Their solution is to measure DC-components over several AC-periods until the least minimum multiple of both the sampling time and the AC-period time could be found. If γ is defined as:

γ = T_c

T_s (2.72)

where Tcis the electrical period the goal is to find the smallest possible integer of γ

Current Controller Speed

Controller M 𝜔_𝑚

𝜔_𝑚^𝑟𝑒𝑓 + +

− −

𝑖_𝛼𝛽 𝑢_𝑑𝑞^𝑟𝑒𝑓

αβ dq

dq αβ

Inverter 𝑢_𝛼𝛽^𝑟𝑒𝑓 𝑖_𝛼𝛽^𝑟𝑒𝑓

+

− 𝑖_{𝛼𝛽 𝐷𝐶}^𝑟𝑒𝑓

𝑖_{𝛼𝛽 𝐷𝐶} DC Current

Controller

−+ 𝑢_{𝛼𝛽,𝐷𝐶}^𝑟𝑒𝑓

++

Resistance estimation 𝑖_𝐷𝐶

𝑢_𝐷𝐶^𝑟𝑒𝑓

Figure 2.5: The proposed control scheme with stator resistance estimation The torque ripples that occurs in the speed and torque arises from the stationary MFF that is imposed in the stator. This will cause two more torque components than in normal operation. The first one will be because of the in- teraction between the stationary MMF and the induced rotating field, this will pulsate with synchronous speed. Secondly a constant braking torque component will be added since the stationary field in the stator will be seen from the rotor as a rotating field in the opposite direction to the rotor speed, this field

(34)

will have the angular frequency of the the angular rotor speed. Matić et al. [3]

finds a reduced torque expression to be, τ_e= 3n_p

2 (i_di_q− I_DCi_dsin(ω_vt) −

√2 2ωvTR

(I_DC)²) (2.73)

where T^R= ^L_R^M

R is the rotor time constant and ω^v is the angular frequency of the voltage.

2.5 Technical description

The simulations are performed in the software Matlab v.2019b the attached Simulink. All parameters are calculated in Matlab, these are then exported to Simulink which performs the simulation and then the data is exported to Matlab for analysis. The simulation is performed in the per-unit system where the base values displayed in Table The Matlab initiating code is found in Ap- pendix B code and the Simulink models are found in Appendix A. All motor parameters in the simulations are extracted from the experimental motor.

The motor used at the experimental test bench is a 11 kW BEVI (3SEI 160-M4), motor parameters are displayed in Table 2.5.1. Since the available voltage at the test site is limited to 400 V is the motor parameters scaled down to limit the nominal voltage to available levels. Hence the nominal voltage is 400 V and the nominal and the power is reduced to 6250 W. The motor drive is a modified ABB ACS880 15 kW. The drive is based on PWM, using SVM for control of the inverter. The main processor has been replaced by a Field- Programmable Gate Array (FPGA) a ZedBoard Zynq-7000 ARM/FPGA SoC Development Board. The HDL design of the FPGA is written in the software Xlinx Vivado 2017.3. The drive routine is written in C in the software Xil- inx Software Development Kit (XSDK) v.2017.3. The system is powered by a DC-bus which provides a DC-voltage to the VSC. The current sensors which are used in at the experimental test bench is Sensitec CMS3030. The current sensors are implemented in the simulations by adding noise in the current measurements to get a more realistic model. The voltage sensors used at the test bench are PicoTecg Ta043.

2.5.1 Simulation & Experimental parameters

In the simulations the per unit system is used where all parameters are transformed using the base values displayed in Table 2.5.1

(35)

Parameter Notation Value Unit

Nominal Voltage Y-connected U 690 V, RM S Nominal Current Y-connected I 12.1 A, RM S

Supply Frequency f 50 Hz

Nominal Output power P 11 kW

Nominal Speed n_N 1470 rpm

Power factor cos(ϕ) 0.83 −

Efficiency η_N 91.65 %

Number of pole pairs p 2 −

Nominal slip s 2 %

Nominal torque TN 71.46 N m

Stator Resistance R_s 0.742 Ω

Rotor Resistance R_r 0.89 Ω

Leakage inductance L_l 0.05 H

Magnetizing inductance Lm 0.349 H

Moment of inertia J 0.088 kg m²

Table 2.1: Induction machine data

The control parameters used for the simulations where tuned by the meth- ods described in Section 2.2. Except for the injection control which where tuned by trial and error. These are all shown in Table 2.3

Current proportional gain k_p,c 10.9861 V /A Current integral gain ki,c 2413.9 V /As

Active resistance R_a 9.6792 Ω

Speed proportional gain k_p,s 7.4894x10⁻⁴ N ms/rad

Speed integral gain k_i,s 0.2353 N m/rad

Active viscous damping ba 0.4212 kg m²/s Injection proportional gain k_p,DC 17.1074 V /A Injection integral gain k_i,DC 17.9149 V /As

Table 2.3: Simulink control parameters

The experimental motors control was simpler than the Simulink controllers without active resistance and active viscous damping. The current controller are tuned by direct synthesis, all other parameters are tuned by trial and error.

These are all displayed in Table 2.4.

(36)

Base Voltage V_base 690 V

Base Current Ibase 12.1 A

Base angular frequency ω_base 100π rad/s

Base time t_base _ω¹

base s

Base impedance Z_base 57.02 Ω

Base inductance L_base 0.1815 H

Base Flux ϕ_base 2.20 W b

Base Torque τ_base 71.46 N m

Base moment of inertia J_base 0.0014 kg m² Base viscous damping b_base 0.4549 kg m²/s

Table 2.2: Base values used for simulations

Current proportional gain k_p,c 22.3128 V /A Current integral gain k_i,c 1419.5 V /As Speed proportional gain k_p,s 1 N ms/rad

Speed integral gain k_i,s 3 N m/rad

Injection proportional gain k_p,DC 17 V /A Injection integral gain k_i,DC 10 V /As

Table 2.4: Experimental Control parameters

(37)

Chapter 3 Results

In this are first the results from the simulations presented and then the experimental results.

3.1 Simulations

In this section the simulation results are presented for the DC-estimation scheme, control in two different ways, and an improved resistance estimation.

3.1.1 DC-estimation

The first test is done to analyse the accuracy of the DC-current estimation.

This is implemented by simulating the motor model at nominal speed without any external load applied. Hence the DC-level detected should be zero. As shown in Figure 3.1 the DC-level settles within 0.5 seconds after the start of the motor.

21

(38)

22 CHAPTER 3. RESULTS

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

time (s) -6

-4 -2 0

i,DC (A)

Alpha stator current

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

time (s) 0

5 10

i,DC (A)

Beta stator current

Figure 3.1: The estimated alpha and beta DC currents with no load and at start up of the motor.

The first offset can be explained by the AC-currents starting transient. As seen in Figure 3.2.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

time (s) -20

-10 0 10 20

i (A)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

time (s) -20

0 20 40

i (A)

Figure 3.2: The alpha and beta currents with no load when the motor is started.

The DC estimations dynamical behavior is tested by simulating the motor

(39)

CHAPTER 3. RESULTS 23

as in the previous case. But a virtual DC-current step of 2 A is imposed on the currents feed to the DC-estimation scheme at the time 2.5 seconds. The reaction time is seen in Figure 3.3 to be 40 ms.

2.4 2.42 2.44 2.46 2.48 2.5 2.52 2.54 2.56 2.58 2.6

time (s) -0.5

0 0.5 1 1.5 2 2.5

i,DC (A)

Figure 3.3: The estimated alpha DC currents with no load, at nominal speed and a fictional step at 2.5 s.

To analyse the DC-estimations response to speed changes the speed reference is decreased from 1470 to 1000 when the system is at steady state.

(40)

19.9 19.95 20 20.05 20.1 20.15 20.2 20.25 20.3 20.35 20.4 time (s)

-0.5 0 0.5

i,DC (A)

19.9 19.95 20 20.05 20.1 20.15 20.2 20.25 20.3 20.35 20.4 time (s)

-1 -0.5 0

i,DC (A)

Figure 3.4: The estimated alpha DC currents with no load, at steady state nominal speed and a decrease in speed after 20 seconds.

As seen in Figure 3.4 there is a disturbance in both DC-estimations since it detects the transient current response from the current controllers. This settles within 250 ms for both cases.

(41)

3.1.2 Resistor estimation in α frame

The motor is simulated at steady state, nominal speed and the DC-injection is performed in the alpha-frame without control of the beta frame DC-current.

The current injection is started after 5 seconds and the reference current is 1.12 A (i.e. 10 % of rated current). The results from the DC estimation can be seen in Figure 3.2.

4.9 5 5.1 5.2 5.3 5.4 5.5

time (s) 0

0.5 1

i ,inj (A)

Alpha DC current

i, inj i, inj,ref

4.9 5 5.1 5.2 5.3 5.4 5.5

time (s) 0

10 20

v,DC (V)

DC-voltage

4.9 5 5.1 5.2 5.3 5.4 5.5

time (s) 0

5

Rest ()

Estimated stator resistance

Figure 3.5: Estimated DC current, reference voltage in the α-frame and the estimated stator resistance

The rise time of the DC-current is 140 ms and the results obtained from the resistance estimation is 0.5 Ω with which is oscillating ± 0.14 Ω. The real value of the stator resistance is 0.7416 Ω hence an underestimation of 33 %.

There is also a steady state error in the injected α-current.

(42)

4 4.5 5 5.5 6 6.5 7

time (s) 6.4

6.6 6.8 7

i (A)

d currents

id id, ref

4 4.5 5 5.5 6 6.5 7

time (s) 1

1.2 1.4 1.6

i (A)

q currents

iq iq, ref

Figure 3.6: Influence of a resistance estimation in the α-axis in the dq currents seen by the current controller

4 4.5 5 5.5 6 6.5 7

time (s) -10

-5 0 5 10

i (A)

4 4.5 5 5.5 6 6.5 7

time (s) -10

-5 0 5 10

i (A)

Figure 3.7: αβ- frame AC-currents at the start of injection

As seen in figure 3.6 the dq-current see a disturbance from the DC-injection.

In both frames the initial disturbance is large but settles in the d frame but remains in the q-frame at a substantial size. In Figure 3.7 The α and β frame shows the clear injection in the α frame and a small disturbance in the β frame.

The electric torque of the currents are shown in Figure 3.8

(43)

4 4.5 5 5.5 6 6.5 7

-2 0 2 4 6 8 10 12 14

Figure 3.8: Motor Torque with the DC-estimation with DC current of 1.12 A started at 5 s

The electrical torque ripple caused by the DC-injection are ± 5.9 Nm.

The dependence of torque ripple by the injected current level is displayed in Table 3.1.2.

Injected DC-current [A] Percentage of Nominal current Torque Ripple [Nm]

0.121 1 % 1.1

0.605 5 % 3.2

1.21 10 % 5.9

1.815 15 % 8.9

2.42 20 % 11.85

Table 3.1: Different injected currents and corresponding Torque ripple. At no load and nominal speed.

With an injected current of 1.21 A the torque ripple dependence on speed and load torque is shown in Table 3.1.2.

(44)

Speed [rpm] Load Torque [Nm] Torque Ripple [Nm]

1470 0 5.9

1470 20 5.9

1470 40 5.9

1470 60 5.9

1000 0 5.9

500 0 5.9

100 0 5.9

Table 3.2: Torque ripple at different loads and speeds.

As table 3.1.2 and 3.1.2 shows the torque ripple is only dependent on the amount of injected DC-current.

3.1.3 Resistor estimation in α and β frame

In this scheme the injection is still in the α-axis but the DC-estimation takes place in in the β-frame as well with the cancellation of the DC-offset off the currents feedback to the current controller. The β-axis DC-current is also controlled to zero. As seen in Figure 3.9 the current which is controlled in the alpha frame has a rise time of 125 ms. The current in the beta frame sees a small disturbance at the start of the injection and then settles at 20 mA.

Real-time stator resistance estimation for electrical drives -a control perspective

Real-time stator resistance

estimation for electrical drives -

a control perspective

FREDRIC SUNESSON

Real-time stator resistance

estimation for electrical

drives - a control perspective

FREDRIC SUNESSON

Acknowledgements

Abstract

Sammanfattning

Contents

List of Figures

List of Tables

Abbreviations

List of Symbols

Chapter 1

Introduction

1.1 Background

1.2 Aim

1.3 Method

Chapter 2

Theory

2.1 The basics of AC-machines

2.1.1 State space representation

2.1.2 Induction machine basics

2.2 Control Theory

2.2.1 Control basics

2.2.2 Current control

2.2.3 Speed controller

2.2.4 Controller saturation and anti-windup

2.3 Discretization and delays

2.4 Current injection

2.5 Technical description

2.5.1 Simulation & Experimental parameters

Chapter 3

Results

3.1 Simulations

3.1.1 DC-estimation

3.1.2 Resistor estimation in α frame

3.1.3 Resistor estimation in α and β frame