Multi-Machine Modelling of Two Series Connected 5-phase Synchronous Machines: Effect of Harmonics on Control E. Semail1, E. Levi2, A. Bouscayrol1, X. Kestelyn1 1
L2EP LILLE ENSAM, 8 Bd Louis XIV,
59 046 Lille, France E-Mail:
[email protected] URL: http://www.univ-lille1.fr/l2ep
2
LIVERPOOL JOHN MOORES UNIVERSITY
School of Engineering, Byrom St Liverpool L3 3AF, UK E-Mail:
[email protected] URL: http://www.eng.livjm.ac.uk/Research/Gr oups/EMAD/home.html
Keywords Multiphase drive, Multi-machine system, Converter machine interactions, Harmonics, Modelling
Abstract This paper deals with the series connection of two 5-phase synchronous machines supplied by a single 5-leg voltage source inverter. Using a specific modelling tool, design constraints on both machines are defined in order to control them independently. The effects of harmonics of back-electromotive force are then examined and improved control is deduced.
Introduction Industrial economic constraints lead to systems where one supply is used for several machines. In traction systems, two DC motors are connected in series in order to use only one chopper. Independent torque control can be achieved using field current control [1]. Parallel connections of AC machines are often used in railway traction applications [1]. For 3-phase AC machines supplied by a 3-leg inverter, series connection cannot yield such independent control. Independent control with series connection however becomes possible for n-phase induction or synchronous machines with n≥5. In [3]-[5], under the assumption of sinusoidal magneto-motive force and back-electromotive force (back-EMF) waveforms, concept of independent control of (n-1)/2 series connected induction machines supplied by an n-leg voltage source inverter (VSI) has been described. In [6] series connections of synchronous and induction machines are also examined. This paper deals with a series connection of two 5-phase permanent magnet synchronous machines (PMSM) supplied by a single 5-leg VSI (Fig. 1). Using a Multi-Machine vectorial characterization developed for a 5-phase synchronous machine [7], the influence of back-EMF harmonics on the PWM vector control is examined. The rules for design of a PMSM, suitable for application in this type of series connection, are deduced from this analysis. First, the Multi-Machine modelling of the drive is performed. The studied two-motor drive is equivalent to four fictitious 2-phase machines that are mechanically and electrically coupled. Each fictitious machine is characterised by a family of harmonics. It is thus possible to deduce conditions that enable carrying out independent control for series connected double-motor drives. Next, rotor flux oriented control is implemented in order to examine effects of harmonics. Simulation results are provided for two types of machines, namely with sinusoidal and trapezoidal back-EMF. Results confirm predicted disturbances induced by harmonics. Finally, an improved control scheme, deduced from Multi-Machine modelling, is proposed to smooth the effects of harmonics.
connection
5-phase machine 1
5-leg VSI 1a
i1a
1b VDC
i1b
1c
i1c
1d
i1d
1e
1'a
2a
1'b
2b
1'c
2c
1'd
2d
1'e
2e
5-phase machine 2
i1e
i2a i2b i2c i2d i2e
N v2eN
Fig. 1: Electrical scheme of the studied drive.
Multi-Machine Modelling of the Drive Multi-Machine vectorial modelling of a single 5-phase synchronous machine In [8], a 5-phase PMSM (Fig. 2) has been shown to be equivalent to a set of two 2-phase machines, called, respectively, Main Machine (MM) and Secondary Machine (SM). Each equivalent machine is characterized by its V DC inductance (resp. Lm and Ls), resistance (resp. Rm and Rs), and back-EMF (resp. em and es ). Moreover, each machine is associated with a harmonic family (Table I).
5-leg VSI
5-phase machine a b c d e
ia ib N
ic id ie νeΝ
Thus the Main Machine (resp. the Secondary Machine) produces Tm (resp. Ts) torque mainly Fig. 2: Electric scheme of a single 5-phase machine. thanks to the first harmonic (resp. third harmonic) of the back-EMF.
Table I: Harmonic Characterisation of Fictitious Machines Fictitious machine
Families of odd harmonics
Main Machine (MM)
1, 9, 11, …, 5h ± 1
Secondary Machine (SM)
3, 7, 13, …, 5h ± 2
This equivalence is based on a generalized Concordia transformation characterized by the [C5] matrix:
[C5 ] =
⎡ ⎢ ⎢ ⎢ ⎢ 2⎢ ⎢ 5⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣
1 2 1 2 1 2 1 2 1 2
1
0
1
2π 5 4π cos 5 6π cos 5 8π cos 5
2π 5 4π sin 5 6π sin 5 8π sin 5
4π 5 8π cos 5 12π cos 5 16π cos 5
cos
sin
cos
⎤ ⎥ ⎥ 4π ⎥ sin 5 ⎥ 8π ⎥ sin ⎥ 5 ⎥ 12π ⎥ sin 5 ⎥ ⎥ 16π ⎥ sin 5 ⎥⎦ 0
(1)
Relationships between actual phase variables (denoted with subscripts a, b, c, d and e) and values of fictitious machines are then defined by: i mach = 0 i mα
[
i mβ
i sα
i sβ
] = [C ] [i
[
vmβ
v sα
v sβ
] = [C ] [v
v mach = 0 vmα
t
t
5
t
a
t
5
aN
ib vbN
ic vcN
id
ie ] v dN
t
veN ]
(2) t
(3)
Currents and voltages obtained using this transformation can be decomposed into two subsystems associated with the main and secondary machines:
[ [
⎧⎪vm = vmα ⎨ ⎪⎩ im = imα
] i β] vmβ
[ [
⎧⎪vs = vsα ⎨ ⎪⎩ is = isα
t
t
m
] iβ] v sβ
t
t
(4)
s
As the machine is supplied by a VSI, vmach = vvsi and consequently:
v m = v mvsi
and
v s = v svsi .
(5)
In order to get a synthetic view, an Energetic Macroscopic Representation (EMR, [1]) is given in Fig. 3. In this representation, the following characteristic equations of the machines are represented by pictograms whose meanings are given in Appendix: v m = R m i m + Lm v s = Rs i s + Ls
( )
d im + em dt
( )
d i s + es dt
em . im = Tm Ω and es . is = Ts Ω
Ttot = Tm + Ts d ⎛1 2⎞ ⎜ J Ω ⎟ = Ttot Ω − Tload Ω dt ⎝ 2 ⎠
(6)
(7) (8) (9) (10)
In EMR, the mechanical coupling (two interleaved orange triangles) simply means that the total torque is the sum of the two 2-phase machine torque contributions (MM and SM). The electrical coupling (two interleaved orange squares) takes into account the generalized Concordia transformation. This synthetic graphic representation is used to deduce systematically the control structure of the 5phase machine. It then appears that it is sufficient to implement two usual vector control algorithms already developed for 3-phase machines [7].
Multi-Machine vectorial modelling of two 5-phase series connected machines Notations are defined according to Fig. 1 and subscripts 1 and 2 are added to previous vectors in order to distinguish between the two synchronous machines. The special connection between the two machines can be expressed by:
⎧⎪v 2 = [v 2 aN v 2bN v 2 cN ⎨ ⎪⎩i 2 = [i 2 a i 2b i 2 c i 2 d
with
⎡1 ⎢0 ⎢ L5 = ⎢0 ⎢ ⎢0 ⎢⎣0
[ ]
v 2 dN
v 2 eN ] = [L5 ][v1'aN t
i 2 e ] = [L5 ][i1a t
i1b
v1'bN
i1c
v1'cN
v1'eN ] = [L5 ] v1' t
v1'dN
i1e ] = [L5 ] i1
(11)
t
i1d
0 0 0 0⎤ 0 0 1 0⎥⎥ 1 0 0 0⎥ ⎥ 0 0 0 1⎥ 0 1 0 0⎥⎦
(12)
Using the relations (2), (11) and the property [C 5 ] −1 = [C 5 ] t , we get a relation between the values of the fictitious machines:
[0
i mα 2
with
i mβ 2
i sα 2
⎡1 ⎢0 ⎢ L5 [C 5 ] = ⎢0 ⎢ ⎢0 ⎢⎣0
[C ] [ ] t
5
i sβ 2
] = [C ] [L ][C ] [0 t
t
5
5
i mα 1
5
i mβ 1
i sα 1
i sβ 1
]
t
(13)
0 0⎤ 0 1 0⎥⎥ 0 0 1⎥ ⎥ 0 0 0⎥ − 1 0 0⎥⎦
0
0
0 0 1 0
(14)
Taking into account (14), the relation (13) is then equivalent to:
[ [
⎧⎪ i s1 = i sα 1 i sβ 1 ⎨ ⎪⎩i m1 = i mα 1 i mβ 1
] = [i ] = [i t
mα 2
t
sα 2
]
i mβ 2 − i sβ 2
t
]
(15)
t
With vectorial notation, we have then:
⎧⎪ i s = i s1 = i m 2 1 * ( ) ⎨ ⎪⎩i m = i m1 = i s 2
(16)
5-leg VSI
fictitious machines
shaft
(6,7,8) DC bus
(10)
mechanical coupling
transformation
(9)
(1) vmvsi
vvsi
VDC ivsi
imach
svsi
load
Tmm
im
MM
im
em
Ω
vsvsi
is
Tsm SM
is
es
Ttot
Ω
Ω Tload
Ω
Fig. 3: Multi-machine energetic macroscopic representation of a 5-phase machine.
*
(1) with A = Ax x − Ay y for A = Ax x + Ay y .
The series connection between machines 1 and 2 can be interpreted with the proposed vectorial formalism: machines MM2 and SM1 are connected in series, and machines SM2 and MM1 are also connected in series. For the voltages, similar transformation applied to voltage coordinates leads to:
⎧ v1's = v 2 m * ⎨ ⎩v1'm = v 2 s
(17)
Using relationships (5), (6), (7), (16) and (17), we obtain: * di di v mvsi = Rm1 i m1 + Lm1 m1 + e m1 + v1'm = ( Rm1 + R s 2 ) i m1 + (Lm1 + Ls 2 ) m1 + e m1 + e s 2 dt dt d i s1 d i s1 v svsi = R s1 i s1 + Ls1 + e s1 + v1' s = ( R s1 + Rm 2 ) i s1 + (Lm 2 + Ls1 ) + e s1 + e m 2 dt dt
(18) (19)
For the m-component of VSI voltage vmvsi , the series connection of the two machines implies that: (20) Rm= Rs1+Rm2 • Rm2 is added to Rs1 to give a global resistance: (21) Lm= Ls2 +Lm1 • Ls2 is added to Lm1 to give a global inductance: * * (22) • e m1 is added to es 2 to give a global back-EMF noted em : e m = e m1 + e s 2 For the s-component of VSI voltage v svsi , the series connections of the two machines implies that: (23) Rs=Rm1+Rs2 • Rs2 is added to Rm1 to give a global resistance: (24) Ls= Lm2 +Ls1 • Lm2 is added to Ls1 to give a global inductance: (25) • es1 is added to em 2 to give a global back-EMF noted es : e s = e s1 + e m 2 It is then possible to establish an Energetic Macroscopic Representation of the studied system (Fig. 4).
5-leg VSI DC bus
global windings
fictitious machines
shaft
(20-25)
(6, 7)
(10)
transformation
series connections
(1)
(18,19)
mechanical coupling (9)
Tm1
im1
MM1 vmvsi vvsi
VDC ivsi
imach
svsi
im
Ts1
is1
SM1 em
es1
vsvsi
is
is 2
is
es
es 2
im
Ttot1
Ω1
em1
load
Ω1
Ω1
Ω1
Tload1
Ts2 SM2
Ω2 Tm2
im 2
MM2 em 2
Fig. 4: Multi-machine equivalent representation of the studied drive.
Ω2
Ttot2
Ω2
Ω2 Tload2
The first conclusion is that it is not possible to control independently four 2-phase machines with only a 5-leg VSI, which has only four degrees of freedom. As fictitious machines are characterized by harmonic families (Table I), it appears that: • the first current harmonic of machine MM1 is combined with the third back-EMF (26) harmonic of machine SM2 since these machines are connected in series; • the third back-EMF harmonic of machine SM1 interacts with the first current (27) harmonic of machine MM2. These types of interactions (26) and (27) produce pulsating torque components whose frequencies depend on the speeds of the machines M1 and M2. If the speeds are the same, the lower frequency of the pulsating torque is relative to the second harmonic. In general, the control to achieve constant torque is complex unless the back-EMF for SM1 and SM2 machines are zero. This will be the case if the real machines have sinusoidal back-EMF but it is a strong constraint for the designer of the machine.
Examination of Harmonic Effects on Control in Rotating Reference Frame With only one 5-leg VSI it is not possible to control independently the four 2-phase fictitious machines. The chosen control strategy is to control only the main machines MM1 and MM2. These machines, which are associated with the first harmonics, have greater back-EMF and produce, for given Joule losses, greater torque than SM1 and SM2 machines. As in [7], two standard (the same as for a 3-phase machine) vector control schemes in rotating reference frame have been implemented, one for each of the two machines (MM1 and MM2). To confirm the predictions (pulsating torques) obtained with the Multi-Machine modelling, two different cases have been considered. In the first case, the back-electromotive force of both machines is supposed to be sinusoidal. In the second case, back-EMF of machine no. 2 is still sinusoidal but back-EMF of machine no. 1 has also a third harmonic component.
Independent control with two machines with sinusoidal back-EMF When each machine has a sinusoidal back-EMF, the synthesis of control is easy since only the MM1 and MM2 have back-EMF. Two vector control schemes in two rotor flux oriented frames have been implemented. The algorithm is the same as the one developed for 3-phase machine. To prove that the two real machines can be independently controlled, the speed reference is changed at 0.5s only for machine no. 2 (Fig. 5). In Fig. 6, we observed a modification of the currents in the MM2 machine but also of the SM1 which is in series. In Fig. 7 and Fig. 8, it can be seen that there is nevertheless no impact on the torque of machine no. 1. There are not any pulsating torque components. The two real machines are independently controlled. 100 90
25 Speed machine n°1 red Reference speed machine n°1 blue Speed machine n°2 green Reference speed machine n°2 cyan
80 70
20 15
I 10 (A)
Speed rds/
60 50
5
40
0
30
-5
20
-10
10 0
Currents in MM2 and SM1 two-phase machines
30
-15 0
0.1
0.2
0.3
0.4 0.5 time s
0.6
0.7
0.8
0.9
Fig. 5: Speeds of the two 5-phase machines.
-20
0
0.1
0.2
0.3
0.4 0.5 time s
0.6
0.7
0.8
0.9
Fig. 6: Currents in MM2 and SM1 machines
30
5 Torque machine n°1 red Reference Torque machine n°1 blue Torque machine n°2 green Reference Torque machine n°2 cyan
25
Torque machine n°1 red Reference Torque machine n°1 blue Torque machine n°2 green Reference Torque machine n°2 cyan
4.5 4
20 3.5 Torques Nm
Torques Nm
15 10 5
3 2.5 2 1.5
0
1
-5 -10
0.5 0 0
0.1
0.2
0.3
0.4 0.5 time s
0.6
0.7
0.8
0.9
Fig. 7: Torques of the two 5-phase machines.
0.4
0.42
0.44
0.46
0.48
0.5 time s
0.52
0.54
0.56
0.58
0.6
Fig. 8: Zoom of torques from Fig. 7.
Examination of the case with only one machine with trapezoidal back-EMF In this case, we consider the same control but the back-EMF of machine no. 1 contains a third harmonic with amplitude equal to 30 % of the first harmonic. For the speeds, there are no significant differences (Fig. 9). However, as far as the torques are concerned, the situation is different. Using the EMR representation it is possible to analyse the effect of the third harmonic of back-EMF. In order to achieve the new speed reference, the speed controller changes the reference torque of MM2 at 0.5s. Then, currents in the machine n°2 suddenly increase. The interaction (27) of i 2 m with e1s implies a perturbation in the torque of machine no. 1 (Fig. 10 and Fig. 11). Nevertheless, the speed controller of the machine no. 1 should compensate this effect by modifying the reference torque of the MM1 machine. In Fig. 10 and Fig. 11 we can see that the reference torque of MM1 (in red) is effectively more pulsating than the real torque of the machine n°1 (in blue) itself. To verify this compensation it is sufficient to see the beginning of the simulation (0
