TVE-MFE 18004
Examensarbete 30 hp Juni 2018
Field Current Control for the Damping of Rotor Oscillations and for the Alternative Start of Synchronous Machines
Further Innovative Applications of Field
Current Active Control besides UMP-Compensation
Roberto Felicetti
Masterprogram i förnybar elgenerering
Master Programme in Renewable Electricity Production
Teknisk- naturvetenskaplig fakultet UTH-enheten
Besöksadress:
Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0 Postadress:
Box 536 751 21 Uppsala Telefon:
018 – 471 30 03 Telefax:
018 – 471 30 00 Hemsida:
http://www.teknat.uu.se/student
Abstract
Field Current Control for the Damping of Rotor Oscillations and for the Alternative Start of Syn-
chronous Machines __________________________________________
Roberto Felicetti
The possibility to save energy in synchronous machines operation by dismissing d-axis damping bars and surrogating them with active excita- tion current control in sectored field winding is proved. In particular a way to recover the energy of rotor oscillations during power regulation is shown by means of a study-case generator whereas a self-starting machine is analytically and numerically designed in view of its next con- struction and test. Principal design requirements and limits for both ap- plications are presented and discussed.
TVE-MFE 18004
Examinator: Juan de Santiago Ämnesgranskare: Urban Lundin Handledare: Johan Abrahamsson
iii
A Rita e Clara
Contents
List of symbols ... vi
1 Introduction ... 1
1.1 Project aims ... 3
1.2 Project background ... 3
1.3 Outline, constraints and limits ... 6
2 Theory... 7
2.1 Synchronous machines ... 7
2.1.1 Types of synchronous machines ... 7
2.1.2 Synchronous machine models ... 8
2.2 Active current control ... 13
2.3 Rotor oscillations and energy recovery ... 16
2.3.1 Rotor hunting and its description ... 16
2.3.2 Energy repartition during the hunting ... 19
2.3.3 Rotor oscillations damping by active current forming ... 21
2.4 Motor/Generator alternative start ... 25
2.4.1 Stator-to-rotor voltage transformation ratio... 25
2.4.2 Starting torque generation ... 28
3 Method ... 32
3.1 Study-case synchronous generator ... 32
3.2 Motor design and parameters ... 33
3.2.1 Motor design ... 33
3.2.2 Motor parameters ... 39
3.2.3 Four sectors rotor winding arrangement... 40
3.3 Simulink models ... 42
3.3.1 Simulink model for rotor oscillations active damping ... 43
3.3.2 Simulink model for the alternative starting ... 45
4 Results and discussion ... 46
4.1 Rotor oscillations and energy recovery ... 46
4.2 Motor/Generator alternative starting ... 54
5 Conclusions ... 62
6 Future work ... 66
v
7 Bibliography ... 67
Appendix ... 69
A.1 ... 69
A.2 ... 71
A.3 ... 73
A.4 ... 77
A.5 ... 79
A.6 ... 82
List of symbols
A Electric load or linear current density [A/m]
B1max Air gap induction fundamental amplitude [T]
c Number of parallel current paths per phase [-]
CD Derivative controller gain [Vs2] CI Integrative controller gain [V]
CP Proportional controller gain [Vs]
cosιR Rated power factor [-]
cosιOE Power factor in over-excitation [-]
D Damping factor [kgm2/s]
Di Machine bore diameter [m]
E Single phase RMS electromotive force (EMF) [V]
EOE RMS electromotive force in over-excitation [V]
f0 Grid frequency [Hz]
H Machine time constant [s]
I RMS phase current [A]
I Moment of inertia [kgm2]
iA...B...C Instantaneous armature phase current [A]
iad Instantaneous d-axis damping bars current [A]
iaq Instantaneous q-axis damping bars current [A]
Id d-axis armature current [A]
id Instantaneous d-axis armature current [A]
if,0 Excitation current for rated voltage at open circuit [A]
if,R Rated excitation current [A]
if,sc Excitation current for rated short circuit current [A]
Iq q-axis armature current [A]
iq Instantaneous q-axis armature current [A]
iref Reference current [A]
Isc,R Short circuit current at rated excitation current [A]
Isc,0 Short circuit current at rated armature voltage [A]
Iswitchmax Absolute maximum current for the switch [A]
i0 Instantaneous homopolar current [A]
JR Rated RMS current density [A/m2]
K Rotor oscillations natural angular frequency [rad/s]
kad Armature to d-axis damping bars transformation ratio [-]
kaq Armature to q-axis damping bars transformation ratio [-]
kd Winding distribution factor [-]
vii
Kd d-axis field form factor [-]kf Induction form factor [-]
kfw Armature to field winding transformation ratio [-]
kp Winding pitch-factor [-]
Kq q-axis field form factor [-]
kR Skin-effect correction factor for DC-resistance [-]
kw Winding factor [-]
kΕ Flux form factor [-]
Lad d-axis damping bars self inductance [H]
Laq q-axis damping bars self inductance [H]
Ld d-axis armature inductance [H]
Lq q-axis armature inductance [H]
lFe Machine equivalent magnetic axial length [m]
Lf Field winding self inductance [H]
Lfρ Field winding stray inductance [H]
Lf,ad Field winding to d-axis damping bars mutual inductance [H]
Lf,aq Field winding to q-axis damping bars mutual inductance [H]
Lm Synchronous magnetization inductance [H]
Lmavg Single phase armature average magnetization inductance [H]
Lmd Single phase d-axis armature magnetization inductance [H]
Lmq Single phase q-axis armature magnetization inductance [H]
Ls Synchronous armature inductance [H]
LXX Armature single phase self inductance of phase X [H]
LXY Armature mutual inductance between phases X and Y [H]
Lρ Armature stray inductance [H]
Mad d-axis damping bars to armature mutual inductance [H]
Maq q-axis damping bars to armature mutual inductance [H]
Mf Field winding to armature mutual inductance [H]
Nf Number of field winding turns per pole [-]
Nm Armature equivalent number of turns per pole per phase [-]
ns Number of conductors per slot [-]
Nsectors Number of rotor winding sectors [-]
N0 Stator to rotor voltage transformation ratio at rest [-]
p Pole pairs [-]
P Active power [W]
PEM Electromagnetic synchronous power [W]
POE Mechanical power in over-excitation [W]
PR Rated mechanical power [W]
q Number of slots per pole per phase [-]
Q Armature slots number [-]
Q Reactive power [VAR]
R Armature phase resistance [ς]
Rad d-axis damping bars resistance [ς]
Radd Rotor winding additional phase resistance [ς]
Raq q-axis damping bars resistance [ς]
Rf Field winding resistance [ς]
Rr Rotor windings phase resistance [ς]
R75AC Armature AC phase resistance at 75 °C [ς]
R75DC Armature DC phase resistance at 75 °C [ς]
ℑ
d Magnetization reluctance of the d-axis magnetic path [H-1] s Rotor slip [-]SCu Copper wire or bar cross section [m2] Ssector Apparent power per rotor sector [VA]
SR Rated power [VA]
SCR Short Circuit Ratio [-]
TEM Electromagnetic torque [Nm]
Tm Mechanical torque [Nm]
Tsyn Synchronizing torque [Nm]
Td’ Short-circuit d-axis transient time constant [s]
Td’’ Short-circuit d-axis subtransient time constant [s]
Td0’ Short-circuit d-axis transient time constant at open armature [s]
Tq’’ Short-circuit q-axis subtransient time constant [s]
UR Single-phase armature rated voltage [V]
U DC-link voltage [V]
V Single-phase RMS voltage [V]
Vd d-axis armature voltage [V]
vd Instantaneous d-axis armature voltage [V]
Vf,0 Rotor winding open circuit voltage at rest [V]
vf Instantaneous field winding voltage [V]
Vq q-axis armature voltage [V]
vq Instantaneous q-axis armature voltage [V]
Vswitchmax Absolute maximum voltage for the switch [V]
w0 Magnetization energy per machine unit length [W/m]
Xd d-axis synchronous reactance [ς]
Xd’ d-axis transient reactance at grid frequency [ς]
Xd’’ d-axis subtransient reactance at grid frequency [ς]
Xq q-axis synchronous reactance [ς]
Xq’ q-axis transient reactance at grid frequency [ς]
Xq’’ q-axis subtransient reactance at grid frequency [ς]
Pole enclosure [-]
χ Pole span factor [-]
0 Armature current phase constant [rad]
α Oscillation decay constant [s-1]
φ Armature MMF-wave angular mech. position from the d-axis [rad]
φFe Iron density at 25° C [kg/m3] χ Air gap [m]
χ’ Magnetic equivalent air gap [m]
ix
χavg Average airgap [m]χm Tip air gap [m]
χ0 Load angle at t=0 [rad]
Χχ Load angle variation with respect toχ0[rad]
ΧLm Armature magnetizing inductance maximal excursion [H]
ΧP Slip power [W]
ΧTdam Electromagnetic damping torque [Nm]
ΧTm Mechanical torque variation [Nm]
ΧWuse Oscillations maximal theoretical recoverable kinetic energy [W]
δ Relative power regulation [-]
ψ Damping coefficient [-]
γ Efficiency [-]
π Electrical angular position of the d-axis [rad]
Πr+ Progressive rotor MMF wave [A]
Πr+ Regressive rotor MMF wave [A]
λ0 Vacuum permeability [H/m]
θ Short-pitching coefficient [-]
θCu Copper resistivity [ςm]
σ Pole pitch [m]
ι0 Field current phase constant [rad]
βf Relative electromagnetic synchronizing power [-]
βr Relative reluctance synchronizing power [-]
Ξad d-axis damping bars linked flux [Wb]
Ξaq q-axis damping bars linked flux [Wb]
Ξd d-axis armature linked flux [Wb]
Ξq q-axis armature linked flux [Wb]
Ξf Field winding linked flux [Wb]
Ξ0 Armature homopolar linked flux [Wb]
ϖ Electrical angular speed or frequency [rad/s]
ϖ0 Grid angular frequency [rad/s]
ς Rotor mechanical angular speed [rad/s]
1 Introduction
Continuous improvements in construction and management of synchronous machines make it nowadays possible to achieve very high efficiencies (over 98%) for both generators and motors [6]. Nevertheless research for achiev- ing further marginal improvements is still ongoing since their impact on the maintenance and duty costs reduction is anything but negligible, especially for big rate machines.
One example about this effort in heavy vertical axis hydropower units re- search is the partial unloading of the Mitchell’s bearing obtained by means of an electromagnetic thrust bearing [7]. This component allows reducing the friction between bearing body and pads which results in lower energy waste and favorable downsizing of oil cooling apparatus. Another notable example is the compensation of the Unbalanced Magnetic Pull (UMP) [1] performed by the system schematically represented in figure 1.
Figure 1. A digitally controlled drive performs the regulation of the excitation cur- rent in three magnetically independent rotor sectors [1].
It pursues both aims of reducing maintenance costs by eliminating mechani- cal stress on frames and guidance bearings and to reduce additional losses due to magnetic unbalance [8].
The active current control in magnetically independent rotor pole groups plays a key-role in this driving strategy. Figure 2 shows how a radial force ΧF can be generated and displaced around the rotor by controlling suitable currents1 in each sector.
1 The presented UMP compensating mechanism refers here to the dynamic unbalance. The system in figure 1 is also capable to compensate a static unbalance by forcing different levels of current bias in the three rotor sectors.
Figure 2. Compensating pull generation mechanism graphically explained For the dynamic unbalance compensation it is beneficial that the air gap induction adjustments react promptly and unweakened to the commanded control currents, especially when the machine is a fast rotating one.
Unfortunately many authors [2][3][4] had proved that the magnetizing in- ductances strongly decrease during transient conditions due to the reaction of damping circuits, field winding and eddy currents.
It is therefore advisable to dismiss the damping bars when it comes to a sec- tor-wise excitation current control. Nevertheless damping bars are part of the current state of art for synchronous generators and motors. Traxler-Samek, Lugand and Schwery [5] recall the reasons for that by pointing out all per- formances they ensure:
a) damping of torque oscillations
b) reduction of parasitic air gap magnetic field harmonics c) suppression of negative-sequence field
d) protection of the excitation winding at fault e) transient stability
f) asynchronous start.
However, the field current active control per rotor sectors proposed in [1]
discloses new ways to perform the duties enlisted above, overcoming more- over the intrinsic energy wasting nature of damping circuits.
In the present work two beneficial applications of the field current active control technique are presented for alternatively achieving points a) and f).
1.1 Project aims
This thesis aims to prove how the field current active control can surrogate and even improve at least two duties performed by damping bars in synchro- nous machines: damping of rotor oscillations and asynchronous start.
About the first application, by means of a 36-poles, 175 MVA study-case hydropower generator, a way to damp the rotor oscillations by recovering their related kinetic energy is theoretically proved and simulated.
A self-starting 4-poles, 35 kW synchronous motor without damping bars is analytically and numerically designed in view of its next construction in order to test the second application. At the same time a starting strategy by field current active forming is studied and its validity checked by simulation.
Principal design requirements, actual limits and future perspectives of both applications are presented and discussed in the following.
1.2 Project background
The damping bars are particularly interested by losses when a slip between the rotating armature Magneto Motive Force (MMF) and the machine spin- ning rotor arises2.
By that mechanism the kinetic energy of rotor oscillations is intentionally dissipated after few mechanical periods. In this kind of praxis, the more fre- quent the power adjustments producing the oscillations the grater the amount of wasted energy. An alternative but conservative oscillations damp- ing technique would be then of advantage, especially for big synchronous generators undergoing a large amount of power regulations per day and for synchronous motors performing e.g. S3 and S6 duties according to the inter- national standard IEC 60034-1 (or equivalent German norm VDE 0530) [10]. This mechanism, which pretends to be conservative, must necessarily be reversible. So if the UMP-compensation system is able to detect perturba- tions in the excitation current [9] being compensated in turn by perturbations impressed on the field current [1], the rotor oscillations, which induce har- monics in the field current [3], must be necessarily compensated by control- ling a suitable current in the rotor winding.
R1)3The excitation system shown in figure 1, which has revealed to be effec- tive for solving the UMP issue is therefore suitable to perform the conserva- tive rotor oscillations damping by actively controlling the field current.
In the asynchronous behavior of synchronous motor and generators the loss- es produced in the damping bars are intentionally used for speeding up the rotor close to the synchronism.
2 It is known that also at synchronism air gap induction harmonics and negative sequence armature currents provoke some losses in the damping bars but they are disregarded here.
3 R stands for remark
From the theory of the asynchronous machine [4] it is known that the elec- tromagnetic torque is proportional to the rotor copper losses according to the equation:
s R 3 pI s
p P
T
r0 2 r 0
Cu r
em
< ϖ
< ϖ
. ( 1.1 )In a wound rotor asynchronous machines a constant torque all over the start is classically achieved by connecting a three phase additional resistive load Raddto the rotor winding, the value of which changes continuously or step- wise, so that:
∋ ( const s
s R R
r add∗ <
. ( 1.2 )
The rotor active power balance in this kind of drive is highlighted in figure 3 by means of the equivalent single-phase circuit of an asynchronous machine.
Figure 3. Asynchronous speed control by slip power dissipation: the red boxes represent the dissipated energy, the green ones the kinetic energy.
Figure 4 offers correspondingly a graphical idea of the energy balance relat- ed to the start of an asynchronous machine at constant torque by help of con- tinuously varying additional rotor resistances: in a) at constant frequency; in b) by means of three subsequent frequency and armature voltage steps4.
Figure 4. Asynchronous start: a) at constant frequency, b) with three frequency steps
4 The magnetizing flux constancy is ensured by two conditions: 1) Volt/Hertz constant ratio at the armature supply; 2) prevalent resistive nature of the rotor circuit (R >> sϖL). For high slip the second requirement urges the reduction of the supply frequency and voltage. This perfor- mance would require the usage of an expensive cycloconverter.
Figure 5 shows the replacement of the additional external resistance by an external inverter connected to the rotor single phase circuit. It can be ob- served that for the sake of torque generation it is absolutely irrelevant if the voltage dropΧVadd in the rotor winding is produced by the insertion of an additional resistance Radd or by an active component capable of generating it.
Figure 5.Asynchronous speed control by slip power recovery
However, this last driving strategy presents the advantage of recovering that part of slip power which is irremediably lost when the additional resistance is used. Its beneficial effect can be observed in figure 6.
Figure 6.Asynchronous start at constantly variable frequency
The rotor current is held constant as far as the back Electro Motive Force (EMF) Er enables it and so are torque and rotor losses (red area). The rest and major part of the slip power is recovered by the inverter and given back to the DC-link (blue area).
All theoretical considerations presented in this paragraph about a three phase rotor winding can be valuably applied to the excitation winding of a syn- chronous machine even though it behaves as a single phase asynchronous rotor winding all over the start. The inverter proposed by [1] for actively supplying the excitation circuit can conveniently be used as an active rectifi- er which enables the energy to flow from the rotor to the DC link.
R2) The excitation system shown in figure 1 is therefore suitable to perform the asynchronous start of a synchronous machine by partially recovering its slip power via the field current active control.
1.3 Outline, constraints and limits
Active current control applications envisaged in the previous paragraph are duly examined from a theoretical point of view in chapter 2. Chapter 3 ex- plains how the Simulink models have been conceived and built, which made it possible to perform the simulation work on both generator and motor. Fur- thermore, it illustrates the design steps for a 35 kW synchronous motor.
The obtained results for the simulations are presented and discussed in chap- ter 4 . In chapter 5 some conclusions are drawn whereas chapter 6 deals with suggestions for future work.
Coming to the constraints posed from the beginning to the present Master Thesis, rated power lower than 100 kW and pole pairs equal to 2 were the only requirements to be fulfilled for the motor design. They descend from the need to keep the handling of future experimental work as easy and inex- pensive as possible, considering all voltage- and power-level limitations to be faced in a laboratory test-setting.
The intrinsic limits of the present work are essentially two:
1) iron saturation effects are not included in the Simulink models;
2) contributions of eddy currents, magnetic hysteresis and mechanical friction to the damping of rotor oscillations are neglected.
The simplicity in the model gained by that way pays a cost to the accuracy of the simulated currents profiles but ensures at the same time less calculation effort.
Anyway not having accounted for those second order effects, far from changing the nature of the electromagnetic phenomena under examination, has made them even more clear and understandable.
2 Theory
2.1 Synchronous machines
2.1.1 Types of synchronous machines
A classical synchronous machine (SM) is a two-windings5 alternating current machine in which a first winding called armature is connected to a supply line operating at constant frequency f0 and a second one, called field wind- ing, provides the machine’s excitation being supplied by direct current.
While the armature of SM shows more or less the same kind of structure - which usually means in slots distributed short-pitched three-phase lap- or wave- double layer winding with more parallel strands - the field winding execution is strongly dependent on the chosen rotor type. Synchronous ma- chine can be essentially sorted out according to their rotor structure in:
∂ round or cylindrical rotor
∂ salient pole rotor
The first sort shows a smooth rotor surface like the one depicted in figure 7 and it is typical for critical steam high speed turbogenerators.
Figure 7.Cylindrical rotor structure6
The second type encompasses all those rotor wheel arrangements which produce an angular varying radial air gap height due either to rotor saliencies (figure 8) or to claw poles (figure 9).
Hydropower and back-up generators as well as synchronous motor are al- most exclusively built with salient pole rotors even though turbogenerators which are driven by subcritical low speed steam turbines (nuclear and geo- thermal power groups) are often manufactured in the same way.
Claw poles rotor arrangements, especially in the brushless execution, are used when the machine must work under harsh service conditions for long
5 Reluctance and permanent magnet synchronous machines are not included in the study
6 Figure from [3]
time, without possibility of any maintenance but still granting its duty (mines and railways cars motors and generators [3], car and trucks alternators [11]).
Figure 8.Rotor with saliencies7
Figure 9.Claw pole rotor8
2.1.2 Synchronous machine models
Synchronous machine models are used for describing and foreseeing mo- tors/generators steady state or transient behaviors. A simple inversion of the armature current sign convention transforms a generator model into a motor one and vice versa. This is why only the generator current convention9 has been considered in this review.
7 Figure from [3]
8 Figure form [3]
9 Armature positive current comes out from the phase terminal
The most simple model which fits for cylindrical rotor unsaturated machines in symmetrical steady state behavior is Behn-Eschenburg’s single phase equivalent circuit shown in figure 10.
Figure 10.Behn-Eschenburg’s equivalent circuit for cylindrical rotor machine It shows the internal motional electromotive force E proportional to the exci- tation current if and it takes in account the effects on the output voltage regu- lation due to load current I by means of the synchronous armature reactance ϖLs. The synchronous stator inductance Ls encompasses the equivalent mag- netizing inductance Lm seen from the armature winding during a symmetric load and the armature stray inductance Lρ:
∗
ρ< L L 2
L
s3
m . ( 2.1 )The symbolic equation related to this model is:
∋ R j L ( I E
V < , ∗ ϖ
s√
( 2.2 )and it can be represented in phasorial way as shown in figure 11:
Figure 11. Behn-Eschenburg’s model phasors diagram.
As soon as an anisotropic rotor is used the magnetization inductance is not constant anymore but, according to the relative angular position between rotor and chosen armature phase axis φ, it fluctuates periodically around an average value Lmavgand between two extreme values separated byΧLm. The armature self-inductance as well as the mutual inductances between phases become all functions of the relative rotor to stator angular position:
∋ (
φ < ∗Χ cos2pφ∗Lρ 2L L
LAA avgm m , ( 2.3 )
∋ (
⌡
∑ φ ∗ ο
∗ Χ ,
<
φ 3
p 4 2 2 cos L L
2
L
AB1
avgm m , ( 2.4 )∋ (
⌡
∑ φ ∗ ο
∗ Χ ,
<
φ 3
p 2 2 2 cos L L
2
L
AC1
avgm m . ( 2.5 )When the rotor saliency is aligned with the phase axis the magnetizing in- ductance presents its maximal value Lmd:
2 L L
Ldm avgm Χ m
∗
< . ( 2.6 )
Lays the rotor saliency orthogonal to the phase axis then it assumes its smallest value Lmq:
2 L L
Lqm avgm Χ m ,
< . ( 2.7 )
In order to have a machine model dealing with constant parameters the Blondel’s two-axis armature reaction theory is needed. Figures 12 and 13 show the application of Kirchhoff’s Voltage Law (KVL) to the EMFs gener- ated respectively by the magnetic flux linkages acting along the d- and the q- axis.
Figure 12.Blondel’s equivalent circuit for the d-axis armature reaction The inductance Ld represented in figure 12 is the synchronous direct axis inductance:
∗
ρ< L L 2
L
d3
dm . ( 2.8 )Figure 13.Blondel’s equivalent circuit for the q-axis armature reaction
The inductance Lq represented in figure 13 is the synchronous quadrature axis inductance:
∗
ρ< L L 2
L
q3
qm . ( 2.9 )The steady state equations related to the circuital models above are:
d q q
d Vsin L I RI
V < χ<,ϖ , , ( 2.10 )
q d
d
q Vcos L I E RI
V < χ<,ϖ ∗ , . ( 2.11 )
They can be represented in phasorial way as shown in figure 14:
Figure 14. Blondel’s model phasors diagram
As soon as the phase currents, the excitation current or the rotor speed un- dergo some transients, the previous models are not useful anymore. The reasons for that are mainly that they do not give explicit and separated ac- count for two kinds of electromotive forces:
∂ the transformational ones, which are related to all possible magnetic linkages among circuits on the armature and on the rotor, under assump- tion of their relative immobility;
∂ the motional ones, which are produced by the relative movement be- tween rotor and stator.
Clarke et al. [12] and Park [13] have proposed two different changes of ref- erence frame called respectively α0-frame and dq0-frame in order to per- form, by means of only two orthogonal phases, the same armature MMF provided by a set of time varying currents in three 120° spaced phases.
These are shown in figure 15.
Moreover, Park’s transformation shows the same advantage of the Blondel’s two axis armature reaction theory since it results in circuital models with constant parameters.
Machine inductances in Park’s model are not functions of the electric angle π=pφ anymore. This makes it possible to approach transient problems by solving differential equations with constant coefficients.
Figure 15. Clarke’s (a) and Park’s (b) reference frames
The electric differential equations for a synchronous machine with damping circuits on both d- and q-axis are:
, Ξ <
, Ξ <
, Ξ <
∗ Ξ <
∗ Ξ ϖ , Ξ <
∗ Ξ ϖ
∗ Ξ <
. i dt R
d
i dt R
d
i R dt v
d
Ri dt v
d
Ri dt v
d
Ri dt v
d
aq aq aq
ad ad ad
f f f f
0 0 0
q d q q
d q d d
( 2.12 )
The following electromechanical differential equation is to be added:
∋ (
, Ξ , Ξ ϖ <
d q q d
m
p i i
2 T 3 I p dt
d
. ( 2.13 )The several linked fluxes shown in the former equations are expressed as linear combinations of the currents by means of constant inductance coeffi- cients:
√
∗
√
∗
√
∗
√ ,
√ ,
√ ,
<
Ξ
√
∗
√
∗
√
∗
√ ,
√ ,
√ ,
<
Ξ
√
∗
√
∗
√
∗
√ ,
√ ,
√ ,
<
Ξ
√
∗
√
∗
√
∗
√ ,
√ ,
√ ,
<
Ξ
√
∗
√
∗
√
∗
√ ,
√ ,
√ ,
<
Ξ
√
∗
∗
∗
√ ,
√ , ,
<
Ξ
. i L i 0 i 0 i 0 i M i 0
i 0 i L i L i 0 i 0 i M
i 0 i L i L i 0 i 0 i M
i 0 i 0 i 0 i L i 0 i 0
i M i 0 i 0 i 0 i L i 0
i 0 i M i M i 0 i 0 i L
aq aq ad f 0 q aq d aq
aq ad ad f f , ad 0 q d ad ad
aq ad ad , f f f 0 q d f f
aq ad f 0 0 q d 0
aq aq ad f 0 q q d q
aq ad ad f f 0 q d d d
( 2.14 )
In [2] it is shown how to obtain circuital representations of (2.12) and (2.14) after having referred all rotor parameters to the armature side by means of the following stator-to-rotor transformation ratios:
0 d
f
fw L L
M 2 k 3
< , ;
0 d
ad
ad
L L
M 2 k 3
< ,
;0 q
aq
aq L L
M 2 k 3
< , . ( 2.15 ) All parameters which have been referred to the armature side are represented with a prime sign in the Park’s circuits of figures 16 and 17.
Figure 16.Park’s equivalent circuit for the d-axis
Figure 17.Park’s equivalent circuit for the q-axis
2.2 Active current control
From the analysis of linear networks in time domain it is known that the current response i(t) to a step-wise voltage of amplitude U applied over the time interval t-t0 to an inductive-resistive load is equal to:
∋ (
⌡
∑ ,
∗
<
,, LR
t t
0
0
e R 1 ) U t ( i t
i
. ( 2.16 )If the duration of the voltage step is way shorter than the inductor time con- stant the increment of current over the interval t-t0can be expressed by:
∋
0(
0
t t
L ) U t ( i ) t ( i
i < , ? ,
Χ
. ( 2.17 )Is the impressed voltage U positive the current rises otherwise is U negative the current drops (two points control). In some cases it is also possible to force the impressed voltage to zero (three points control), circumstance which makes the current stationary. In figure 18 a current control by means of a H-bridge is represented, which makes use of the presented strategy.
Figure 18.Two points current control by hysteresis comparator (bang-bang control) A time varying reference current iref is compared with the phase current sensed by a measurement device (green dot) in a Schmitt’s trigger. The ob- tained digital error signal is used to drive the switches in the bridge. As long as the phase current is smaller than the actual current value plus a given tol- eranceΧI the red wired switches are switched on and the phase current keeps growing. As soon as the phase current overcomes the upper tolerance limit the trigger output changes state and forces the blue wired switches to turn on. The voltage on the load becomes then negative and the load current starts dropping until it reaches the lowest tolerance limit. At that point the trigger output changes state again and the process continues forth and forth. By this way the actual value of the phase current is forced to follow the reference one, being trapped in the tolerance band (bang-bang strategy).
A further very widespread current control technique is based on the Pulse Width Modulation (PWM). In figure 19 it can be seen how the departureδ of the actual current i from the reference one iref (colored in blue in the figure) produces a control voltage vδ (shaded brown) by means of a PID controller.
That control voltage, which represents the modulating signal, is compared with a triangular wave vc of given frequency called carrier (orange). The output state of the comparator is then used to drive the switches in the bridge which establish the voltage vAB on the load (black). As a result of it the cur- rent in the load (green) reproaches the reference one.
Usually, in order to decrease the harmonic distortion of the controlled cur- rent the frequency of the carrier is chosen way higher than the highest fre- quency the fundamental of iref can assume.
Figure 19.Two levels current control by pulse width modulation (PWM)
2.3 Rotor oscillations and energy recovery
2.3.1 Rotor hunting and its description
It is known from the theory of synchronous machines that for each synchro- nous steady state there is a defined position between the armature magneto- motive force and the rotor, which is represented by the load angle χ10. With reference to figure 20, any change that affects the power generated or ab- sorbed by the machine inevitably modifies the position of the rotor relative to the armature rotating field.
Figure 20.Mechanical angles and speeds in a salient pole synchronous machine This usually results in damped oscillatory perturbations which interest me- chanical rotor angleπ, rotor angular speed ς, electromagnetic torque, arma- ture current and instantaneous electric power at the same time.
Such a phenomenon is known as “hunting” [14]. It can be described in the most general case and under the assumption of small departures “Χ” in the neighborhood of the steady-state load angleχ0by the following differential equation called swing equation:
dt D d T
dt T d p
I
syn 2 m
2
Χ χ < Χ , √ Χ χ , Χ χ
. ( 2.18 )
The variation of the torque at the shaft ΧTmin (2.18) can be regarded as cause of the hunting and the load angle departure Χχ as its effect, whereas the synchronizing torque Tsyn and the damping factor D stand respectively for the conservative and the dissipative torques opposing this kind of behav- ior.
10 The load angle is usually defined as an electrical angle.
The typical pseudo-periodical rotor response, which is also solution of (2.18), results in (see appendix A.2):
∋ 1 e
Ktcos K 1
2t (
max
, , ψ
χ Χ
<
χ
Χ
, ψ.
( 2.19)WhereΧχmax and K are respectively the amplitude of the load angle depar- ture and the pulsation during the free oscillations whereas ψ is the damping coefficient related to the damping factor D.
Before investigating a way to recover the harmonic inertial work of the rotor, otherwise converted in heat by the damping bars, it is relevant to present an estimate of the maximal attainable energy from a given hunting behavior. To that end the analysis of free oscillations for D=0 reveals that the amount of energy exchanged in a period between rotor and power network is the same which decays exponentially in time in the damping circuits, in the field and armature windings and even in the iron core due to the induced losses. A first estimation of it, for oscillations triggered by a step-wise relative power regulationδ, has been found (see appendix A.2) to be
0 r
0 f
0 0
R
useful
cos 2 cos 2
H 2 W S
χ
√ β
√
∗ χ
√ β
ϖ
√ ϖ
? δ
Χ
, ( 2.20 )whereβf and βr are respectively the coefficient of the maximal electromag- netic- and reluctance-related synchronizing powers expressed in p.u.:
d R
f
X ~
S E 3 U √
<
β
, ( 2.21 )
⌡
∑ ,
<
β 1
X ~ X ~ X ~ S
U 2 3
q d
d R
2
r . ( 2.22 )
It must be observed that the reactances used in (2.21) and (2.22) are smaller than the Blondel’s synchronous reactances because of the partial magnetiz- ing flux capture exerted by the damping bars and the field winding when the rotor hunts. The flux is forced to leave the main way and to flow in some measure through the magnetic stray paths. More precisely, according to the frequency of the rotor oscillations those inductances can range between their classical transient and sub-transient values, being closer to the subtransient values the higher said frequency:
' X X ~ ''
X
d′
d′
d , ( 2.23 )' X X ~ ''
X
q′
q′
q . ( 2.24 )Since the pulsation of free oscillations and the maximal load angle departure are given respectively by
0 0 r
0 f
H 2
2 cos 2
K < β √ cos χ ∗ √ β √ χ √ ϖ
( 2.25 )and
0 r
0 f
max β √cosχ ∗2β √cos2χ
< δ χ
Χ , ( 2.26 )
considered that the maximal angular frequency departure is related to (2.25) and (2.26) by
max max
< K Χ χ ϖ
Χ
, ( 2.27 )(2.20) can finally be expressed as:
H 2 S
W R
0 useful ϖmax
ϖ
? Χ
Χ . ( 2.28 )
By using (2.28) on the study-case generator presented in paragraph 3.1 it is possible to find out the potentially recoverable energies respectively for the synchronous, the transient and the subtransient behavior.
Table 1.Electromechanical oscillations parameters for synchronous behavior
βf βr Χχmax K f T ςmax ςmin ΧWmax ΧWuse
[p.u.] [p.u.] [°] [rad/s] [Hz] [s] [rad/s] [rad/s] [kWh] [kWh]
1.804 0.171 2.61 9.19 1.46 0.68 17.476 17.430 0.907 0.453 Table 2.Electromechanical oscillations parameters for transient behavior
βf βr Χχmax K f T ςmax ςmin ΧWmax ΧWuse
[p.u.] [p.u.] [°] [rad/s] [Hz] [s] [rad/s] [rad/s] [kWh] [kWh]
5.723 -1.010 1.27 13.19 2.10 0.48 17.469 17.437 0.632 0.316 Table 3.Electromechanical oscillations parameters for sub-transient behavior
βf βr Χχmax K f T ςmax ςmin ΧWmax ΧWuse
[p.u.] [p.u.] [°] [rad/s] [Hz] [s] [rad/s] [rad/s] [kWh] [kWh]
8.736 -0.359 0.654 18.36 2.92 0.34 17.465 17.442 0.454 0.227
A relevant result can be concluded by observing (2.28):
R3) the maximal attainable recovery of kinetic energy from the rotor oscilla- tions triggered by a single step-wise relative power regulation δ in a given synchronous machine, is proportional to the entity of the relative regulation itself, to the machine rated power and to its own inertial time constant.
2.3.2 Energy repartition during the hunting
In order to determine in what measure the kinetic energy of the rotor oscilla- tions supplies the losses in the armature winding and in the rotor damping circuits11, a closer insight into the electromechanical energy conversion dur- ing a transient is needed.
Park’s model for a salient pole synchronous machine, which has been pre- sented in paragraph 2.1, is the tool used in the following. One of its expected duties is that to establish numerical relationships between the magnitude of the angular deviation of the rotor from its steady state position and the inten- sities of all energy-wasting induced currents during the hunting occurrence.
Since all quantities involved in the oscillating behavior have sinusoidal pseudo periodical nature the accomplishment of this task is conveniently performed in an approximated way by solving a succession of quasi-steady states through the symbolic method.
2.3.2.1 Method of departures
As long as the oscillations interesting the rotor are small all departures which intervene in the machine quantities are also small and undergo the superposi- tion principle12. Park’s equations [2] can be then rewritten in term of super- posed steady state values and their related departures:
⌡
∑ ∗
∗
⌡
∑ ∗
<
∗
⌡
∑ ∗
∗
⌡
∑ ∗
<
∗
⌡
∑ ∗
∗
⌡
∑ ∗
<
∗
⌡
∑ ∗ ,
∗
⌡
∑ ∗ ,
<
∗
⌡
∑ , ,
∗
⌡
∑ , ,
<
∗
. i dt R
Δ dΨ i
dt R Δv dΨ
0
i dt R
Δ dΨ i
dt R Δv dΨ
0
i dt R
Δ dΨ i
dt R Δv dΨ
v
Ri Ψ dt ω
Δ dΨ Ri
Ψ dt ω
Δv dΨ v
Ri Ψ dt ω
Δ dΨ Ri
Ψ dt ω
Δv dΨ v
ad ad ad aq
aq aq aq
ad ad ad ad
ad ad ad
f f f
f f f
f f
q d r q q
d r q q
q
d q d r
d q d r
d d
( 2.29 )
11 It has been here generally referred to rotor damping circuits in order to encompass all those closed galvanic paths present in the rotor which are seat of induced currents during the hunt- ing. They include not only dumping bars and excitation winding but rotor iron core too.
12 It states that the effect caused by two or more causes is the sum of the responses that would have been produced by each cause acting individually. This principle requires and implies system linearity, which is always fulfilled for small amplitude stimuli in systems with C2 transfer functions.
By considering the impressed voltages on the rotor circuits constant and by recalling the initial condition of equilibrium (2.29) reduces to:
Χ
∗ ΧΞ
<
Χ
∗ ΧΞ
<
Χ
∗ ΧΞ
<
Χ , ΧΞ ϖ
∗ Ξ ϖ Χ
∗ ΧΞ
<
Χ
Χ , ΧΞ ϖ , Ξ ϖ Χ , ΧΞ
<
Χ
. i dt R
0 d
i dt R
0 d
i dt R
0 d
i dt R
v d
i dt R
v d
aq aq aq
ad ad ad
f f f
q d r d r q q
d q r q r d d
( 2.30 )
Stapleton [16] and Krause et al. [17], who make extensive use of the small departures method, both suggest the following reasonable simplifications for (2.30):
a) voltages generated owning to changes in speed are negligible (this means voltage terms involvingΧϖ);
b) voltages induced in the armature by rate of change of armature flux linkages are negligible (terms in d/dtΧi);
c) voltage drops on resistances are negligible in both armature and excitation winding (terms in RΧi).
Under those assumptions the simplified system becomes:
Χ
∗ ΧΞ
<
Χ
∗ ΧΞ
<
ΧΞ
?
ΧΞ ϖ
∗
? Χ
ΧΞ ϖ ,
? Χ
. i dt R
0 d
i dt R
0 d
dt 0 d v v
aq aq aq
ad ad ad
f d r q
q r d
( 2.31 )
Tables 1,2 and 3 show that rotor oscillations and all related quantities evolve at very low angular frequency K with a period which is several times multi- ple of the electric one. Since the energy repartition question aim only to de- termine the relative values of the energies involved in the different damping circuits, the absolute amplitudes of the departures do not matter here. The analysis can be then limited to the first half rotor oscillation. By taking in account (2.10) (2.11) (2.14) and (2.31) it is possible to solve the pseudo si- nusoidal currents departures in the frequency domain:
∋ (
∋ (
∗ ,
<
∗
∗ ,
<
, ,
,
<
∗
∗ ,
<
√ ,
,
∗
<
√
. Δi jKL R Δi
jKM 0
Δi jKL R Δi jKL Δi
jKM 0
Δi jKL Δi jKL Δi
jKM 0
Δi M ω Δi M ω Δi
L ω Δδ sinδ Vˆ
Δi M ω Δi
L ω Δδ
cosδ Vˆ
aq aq aq q
aq
ad ad ad f ad f, d
ad
ad ad f, f
f d
f
ad ad 0 f f 0 d
d 0 0
aq aq 0 q
q 0
0 ( 2.32 )
(2.32) can be written in form of matrix as:
.
i i i i i
jKL R jKM
jKL R jKL jKM
jKL jKL
jKM
M M
0 L
M L
0
0 0 0 sin Vˆ
cos Vˆ
aq ad f q d
aq aq aq
ad ad ad , f ad
ad , f f
f
ad 0 f 0 d
0
aq 0 q
0 0
0
Χ Χ Χ Χ Χ
√
∗ ,
∗
∗ ,
∗
∗ ,
ϖ
∗ ϖ
∗ ϖ
,
ϖ , ϖ
∗
<
χ Χ
√
χ ,
χ ( 2.33 )
The first relevant result descending from (2.33) is that
R5) the intensities of current departures are all proportional to the ampli- tude of the oscillations. This fact lets foresee that all currents peaks decay in the same way the rotor oscillations do.
The second relevant result is that
R6) the voltage sources which force the currents departures on the d-axis and on the q-axis are respectively proportional to the sine and to the cosine of the steady state load angle. This fact implies that for small load angles the contribution of the d-axis current departures to the damping is less relevant than that of the q-axis ones. For big load angles vice versa the q-axis cur- rents departures play a minor role than the d-axis ones.
This result is very important because it highlights that, at least for small load angles, the d-axis does not play a major role in damping the rotor oscilla- tions. The cause behind this weak damping effect via the d-axis current con- trol at small load angles is explained in the next paragraph. It is relevant to remark here how favorable it is, that the current departures on the d-axis depends on the sinus of the load angle. That means in fact that the active field current regulation for energy recovery gets more and more effective the higher the load angle is. Fortunately the conclusion of paragraph 2.2.1 goes in the same direction since the stronger the power regulation effort the high- er the amount of energy which can be potentially recovered from the oscilla- tions.
2.3.3 Rotor oscillations damping by active current forming
Park’s theory of the synchronous machine [2][3][4][18][19][20] offers a way to express the electromagnetic torque in terms of d- and q-axis quantities:
∋
d q q d(
em
p i i
2
T < 3 ξ , ξ
. ( 2.34 )By specializing the expressions of linked fluxes (2.14) for a machine without damping bars and by substituting them in (2.34), the following expression for the torque is obtained:
∋
q d(
d q f f qem
pM i i
2 i 3 i L L 2 p
T < 3 , ∗
. ( 2.35 )(2.35) shows that
R7) the control over the torque by means of the excitation current, in a ma- chine which performs the excitation flux only along the d-axis, it can be pur- sued exclusively by interacting with the stator quadrature current.
According to this remark and having in mind the damping torque seen in (2.18) the solution of the active damping issue can be stated as it follows:
R8) determining a suitable excitation current departureΧif which is able to produce a damping torque proportional to the speed departure Χϖr by means of its only interaction with the q-axis component of the armature cur- rent.
By putting remark n.8 in analytical formula it is obtained:
i p i 2pM
Tdamp 3 f f q r ϖ,ϖ0
<
ϖ Χ
× Χ
<
Χ . ( 2.36 )
From Blondel’s equation (2.10) it can be deduced that the armature current on the quadrature axis depends essentially from the armature voltage on the d-axis:
0 q 0 q 0
d
q
sin
L V L
i v χ
< ϖ
? ϖ
. ( 2.37 )The excitation field current departure can be obtained instead by integrating the control voltage applied to the excitation field and by neglecting the rotor resistance:
∋ (
σ σ?
Χi L1
〉
t v d0 f f
f . ( 2.38 )
In particular, for compensating the delay introduced in the current control chain by the high excitation field time constant a derivative controller is advisable, which processes the error signal on the rotor speed according to:
∋ (
D∋ (
D∋
0(
f
dt
C d dt t
C d t
v < , δ < , ϖ , ϖ
, ( 2.39 )where CD is a constant to be determined.
The expression for the field departure current becomes then:
∋ ( ∋
0(
f D t
0 f f
f L
d C L v
i < 1 σ σ<, ϖ,ϖ
Χ
〉
. ( 2.40 )By recalling (2.27)
χ Χ
<
ϖ ,
ϖ dt
d
0 ( 2.41 )
and by substituting it in (2.40) and then in (2.36) together with (2.37) the expression for the breaking torque can be rewritten as:
dt sin d L V L pM 2 C 3
T 0
q 0 f
f D
dam
χ χ Χ , ϖ
<
Χ . ( 2.42 )
This kind of damping torque is qualified as viscose or Newtonian, since it always opposes the speed of the oscillations, exactly how a viscose medium does on a pendulum oscillating in it.
By comparing the damping torque term in (2.18) with (2.42) an expression for the damping factor D can be achieved:
0 q 0 f
f
D sin
L V L M 2 p3 C
D χ
< ϖ . ( 2.43 )
(2.43) is the key for performing the conservative electromagnetic dumping of the rotor oscillations:
R9) the strength of the damping mechanism is set by the factor D which can be tuned via the excitation field by means of the derivative controller gain CD.
In order to obtain the fastest settling time of the perturbations the system must be critically damped with damping coefficient ψ equal to one (figure 22). Appendix A.1 shows that this condition is met when the decay constant of the oscillationsα is equal to the rotor natural angular frequency K:
K I D 2
p <
<
α
. ( 2.44 )Figure 21.Normalized indicial response of a 2nd order system for different values of the damping coefficient13
13(Wikipedia https://en.wikipedia.org/wiki/Damping_ratio#/media/File:2nd_Order_Damping_Ratios.svg) – Public domain source
By substituting (2.43) in (2.44) it is possible to obtain the critic value for the constant CD:
0 R f
f q
0 cr
,
D
V sin
S M
L H L K 3 C 8
χ
< ϖ
. ( 2.45 )Some important remarks can be done by observing (2.45):
R10) CD,cr is inversely proportional to the load angle which suggests a high control effort needed for damping the oscillations at low power levels;
R11) CD,cr is directly proportional to q-axis inductance (the synchronous or transient one) which means that strong rotor anisotropy and low resistive damping circuit on the q-axis facilitate the active damping task;
R12) control effort is proportional to the machine angular momentum (HS/ϖ0) and it is therefore inversely proportional to the number of pole- pairs;
R13) high values of rotor field stray inductance would require an unneces- sarily heavier control effort.
