Modeling and current control of a Double Salient Permanent Magnet Generator (DSPMG) H. Chen, N. Aït-Ahmed, M. Machmoum, M.E. Zaim, E.Schaeffer IREENA- Institut de Recherche en Energie Electrique de Nantes Atlantique 37 Boulevard de l'Université, BP 406, 44602 Saint-Nazaire Cedex, France Tel.: +33 / (2) - 40172608 Fax: +33 / (2) - 40172618 E-Mail:
[email protected] URL: http://web.polytech.univ-nantes.fr/92391/0/fiche___laboratoire/
Keywords «Non-standard electrical machine», «Electrical machine», «sliding mode regulator», «modeling», «simulation»
Abstract This paper deals with the modeling and current control of non-conventional electrical machine dedicated to marine tidal current applications. The dynamic modeling of a Double Salient Permanent Magnet Generator, based on the analysis of the flux and inductances characteristics calculated by finite element method is deduced. A simple current control on dq reference frame is tested and simulation results are presented to validate the proposed model and to discuss the appropriate control strategy.
Introduction The marine currents have always a very low velocity that is less than 5 m/s [1]. That means if a conventional generator such as Doubly-Fed Induction Generator is used, a gearbox will be a necessary device which causes a maintenance problem in the future [2]. Right now, several studies have been made about low speed electrical machines for the marine current generators and shown that generally Permanent Magnet generators, with high poles number are used [3][4]. This paper deals with the modeling and the current control of a Double Salient Permanent Magnet Generator (DSPMG). Firstly, the general structure of the DSPMG is presented. Secondly, a dynamic analytical model is developed. Then, the behavior of the DSPMG is analyzed. Finally, a simple internal current control loop is tested and simulation results are given and discussed.
Structure of the DSPMG The work realized in our laboratory is focused on a special low speed permanent generator which is a three phase variable reluctance device with permanent magnets. This machine proposed in [3][5] employs concentrated stator windings. The DSPMG under study is presented on Fig. 1. The field excitation is provided by non-rotating Permanent Magnets (PM) located at the stator, the rotor includes 64 teeth and the stator 48 teeth distributed on 12 slots. Comparing with classic machine, this machine has very interesting performances. Indeed, the active mass torque is 12 N*m/kg; the volume torque is 45.5 K N*m/m3 [3].
Fig. 1. Structure of DSPMG
Fig. 2. Inductance variations of DSPMG
The machine model is based on the flux characteristics calculated by finite element method without taken into account the magnetic saturation effect. The self-inductances characteristics are shown on Fig. 2. In order to simplify the model, we neglect the 2nd and higher order Fourier terms. So the expression of the 3-phase self-inductances (La, Lb and Lc), mutual-inductances (Mab, Mbc and Mca) and PM flux-linkage components (φma, φmb and φmc) for DSPMG can be written as: cos cos
(1)
cos cos cos
(2)
cos cos cos
(3)
cos Ω
With
where , Ω and represent respectively the electrical angle, the mechanical speed and the teeth number in rotor respectively. For the PM-flux-linkage expressions, the constant components which magnetize the DSPMG do not exist in the conventional Permanent Magnet Synchronous Generator. From expressions (1), (2) and (3), if L1 & M1 and φ0 are equated to 0, this model will be simplified and will correspond to Permanent Magnet Smooth Synchronous Generator (PMSSG) model. It should be also noted that these inductances are also not like Pole Salient Permanent Magnet Generator (PSPMG). For PSPMG, the variation of the inductances due to the position is twice of the PM flux-linkage frequency. The basic equation for electromagnetic torque [6] [7] is given by the following relation: Γ
(4)
Where p is the number of pole pairs. [Ls] is the stator inductance matrix; [is] and [φs] are current and PM flux-linkage vectors.
The first term of the torque expression represents the reluctant torque (or salient torque); it is linked to the rotor geometry. The second term is the “hybrid” torque and corresponds to the interaction between the stator currents and the rotor flux.
Dynamic model for the DSPMG Here, a new dynamic model of DSPMG for vector-control is derived in three steps: firstly, based on the analysis above, the basic characteristics of DSPMG can be expressed in stator reference frame; secondly, the d-q axes of the DSPMG are defined, as shown on Fig.3; thirdly, the transformation of PM flux-linkage and voltage equations from stator reference to rotor reference are performed due to Concordia and Park transformations.
Fig. 3. d-q axes of the DSPMG The three phase voltage and flux equations of a DSPMG can be expressed as follows (in abc reference frame): (5.a) 0 0 0 (
0 0
0 , , )
(5.b)
(5.c)
Where ema, emb and emc are the EMF induced on stator windings due to the magnet. According to Fig. 3, we define the d-axis when the PM flux-linkage of phase A is the maximum. As a result, the q-axis is at the position which displaces the d-axis by 90° in electrical degrees anticlockwise. Since the number of teeth in the rotor is 64, the displacement between the d- and q-axis in mechanical degrees will be 1.40625°. Based on equations (1)-(5) and by applying successively Concordia and Park transformations to each stator variables, the following odq model can be deduced. Flux equations: √3 Voltage equations:
√
(6.a) 0
v v v
e e e
i i i
2
with:
√
i i i cos 3 ;
;
(
);
R R
0
2ω
R
0
L
ω
ω
ω
cos 3 ;
sin 3
0
0
(6.b)
and 2ω
L 0
L
and torque expression: ΓDSPMG
Γ ⁄2
Γ
Γ
(6.c)
where Γ , Γ and Γ are defined as follows: Γ
Γ
Γ
To compare DSPMG model to models of classical machines (PMSSG and PSPMG), expressions of corresponding inductances and torque are summarized in TABLE I.
TABLE I: Inductance and torque expressions PMSSG (φ0=0) 2
Γ
PSPMG (L1=M1; φ0=0) 2 3 2 3 2
0
0
0
0
Γ
Γ
DSPMG (p=Nr) 2
1 √2
Γ
cos 3
2
Γ
2
cos 3
2 (
) sin 3
Γ ⁄2
Γ
We can notice that the inductances in the d-q reference frame for DSPMG are different from those of conventional generators. For conventional generators, Ld & Lq are constant, while for DSPMG, they have sinusoidal components with 3 times frequency of EMF. In addition, the mutual inductances are ) or having sinusoidal-variation with 3 times frequency of EMF ( ). either constant ( The torque equation can be easily analyzed as follows: for the PMSSG, the torque (Γ ) is always from PM; for the PSPMG, the torque (Γ Γ ) is not only from the PM, but also from the reluctance. While for DSPMG, both PM and reluctance can create the torque, and there is also a third term Γ produced principally by the mutual inductance between d and q axes [8].
Behavior Analysis of DSPMG To analyze the performances of the studied DSPMG, we consider on the following sinusoidal stator currents, given by equation (7), in phase with EMF wave:
sin(
)
sin(
)
sin(
)
(7)
Im is the amplitude of the current and The obtained dq stator voltages are given by: )
(
2
3
2
(8.a) 3
(8.b)
and the corresponding stator voltage expression, in phase a, is: )
(
(
2
)
2
(8.c)
According to the parameters given in Appendix A, the stator currents, voltages and the electromagnetic torque variations are presented on Fig. 4 and Fig. 5. Two cases are studied: with and without taking into account the mutual term M1. 2200
600 va vb vc
400
2000 1800 1600 Torque (N*m)
Voltage (V)
200
0
1400 1200 1000
-200
800 600 400
-400
200 -600 0
0.005
0.01
0.015
0.02 0.025 Time (s)
0.03
0.035
0 0
0.04
0.005
0.01
a) Voltage in abc reference frame
0.015
0.02 0.025 Time (s)
0.03
0.035
0.04
b) Torque
Fig. 4. Voltages and electromagnetic torque variations (case 1: M1 ≠ 0) 600
2200
va vb vc
400
2000 1800 1600 Torque (N*m)
Voltage (V)
200
0
-200
1400 1200 1000 800 600 400
-400
200
-600 0
0.005
0.01
0.015
0.02 0.025 Time (s)
0.03
0.035
0.04
0 0
0.005
0.01
0.015
0.02 0.025 Time (s)
a) Voltage in abc reference frame b) Torque Fig. 5. Voltages and electromagnetic torque variations (case 2: M1=0)
0.03
0.035
0.04
In case 1, it can be noticed that the obtained abc stator voltages are not exactly sinusoidal. This is due to the last harmonic term having twice pulsation, according to equation (8.c). The torque also presents high ripple due particularly to the third torque component Γ . However, if we neglect the effect of M1 (case 2), the obtained abc stator voltages are quasi sinusoidal and the torque ripple is highly decreased from 30.94% to 10.41% (Fig. 5).
Current Control based on SMC approach In order to control the DSPMG, a state space model can be deduced from equations (6.a) and (6.b). It is given by equation (9): (9) ;
Where:
;
;
;
;
The developed non-linear control law is established using Lyapunov’s Second theorem of stability [9]. For designing the SMC, there are two important points: firstly, how to choose the sliding surface; secondly, how to get the control laws. In our study a Reaching law based on Sliding Mode Control (SMC) scheme has been applied to control the inner loop current performance. The purpose of our system is to well trace the current, according to [10][11] the sliding surface is developed as: (10) Where: Iref is the desired current,
,
_
_
_
Satisfying the reaching condition of sliding mode, the control law is designed such the time taken for reaching the sliding mode is very fast. According to [11], a reaching law approach can be written as follow: ( )
|
(11.a) 0 and are chosen as per the reaching time criterion given as:
0,
Where the constants |
(11.b)
Signum function is defined as: ( )
1; 1;
0 0
So, in our case, the dynamic SMC is designed as the following equation:
(12) The structure of the SMC control scheme of the current control Loop is presented on Fig. 6 and is simulated on MATLAB/SIMULINK by considering the sinusoidal current form given by equation (7).
Fig. 6. Structure of the current control loop The sliding mode parameters were taken as ε=0.001, k=100. Simulations show that the proposed sliding mode controller provides high-performance dynamic characteristics. The results of the simulated control scheme are presented on Fig. 7. 50
0.1 id ref id SMC
0.06
40
0.04
35
0.02
30
0 -0.02
25 20
-0.04
15
-0.06
10
-0.08
5
-0.1 0
0.02
0.04
0.06
0.08
iq ref iq M1=0 SMC iq M1≠ 0 SMC
45
Current (A)
Current (A)
0.08
0 0
0.1
0.02
0.04
Time (s)
0.06
0.08
0.1
Time (s)
a) Current in d axis
b) Current in q axis
40 error ia error ib error ic
30
Current Error(A)
20 10 0 -10 -20 -30 -40 0
0.02
0.04
0.06
0.08
0.1
Time (s)
c) Current error in abc phases Fig. 7. SMC current control results The controller parameters were then varied and their effect on the settling time of the speed response is presented in Fig. 8 and Fig. 9. It can be concluded that with increasing the value of the parameter K, the settling time decreases; with increasing the value of ε, it causes more chattering in steady state as shown in Fig. 9.b.
50 iq ref iq k=100 iq k=200 iq k=300
45 40
Current (A)
35 30 25 20 15 10 5 0 0
0.02
0.04
0.06
0.08
0.1
Time (s)
Fig. 8. Influence of the variation of k (ε is fixed, 0.001) 48.9898
50 iq ref iq ε =0.001 iq ε =0.01 iq ε =0.1
45 40
48.9898 48.9898
30
Current (A)
Current (A)
35
25 20
48.9898 48.9898 48.9898
15
48.9898
10
48.9898
5 0 0
iq ref iq ε =0.001 iq ε =0.01 iq ε =0.1
48.9898
0.02
0.04
0.06
0.08
0.1
48.9898 4
4.0002
Time (s)
4.0004 4.0006 Time (s)
4.0008
4.001
a) Transient state b) Steady state Fig. 9. Influence of the varation of ε (k is fixed, 100) Robustness tests have also been done for this machine using SMC controller, we have changed the values of self- and mutual- inductances, and get almost the same figures like Fig.7. The studied SMC controller is compared with a Proportional Integral (PI) based controller, whose parameters were computed by a poles compensation method. The simulation results are showed in Fig. 10. 60
0.1
iq ref iq M1=0 PI iq M1≠ 0 PI
id ref id PI
0.08
50 0.06
40 Current (A)
Current (A)
0.04 0.02 0 -0.02
30
20 -0.04 -0.06
10
-0.08 -0.1 0
0.02
0.04
0.06 Time (s)
a) Current in d axis
0.08
0.1
0 0
0.02
0.04
0.06 Time (s)
b) Current in q axis
0.08
0.1
40
40 error ia error ib error ic
30
20 Current Error (A)
Current Error (A)
20 10 0 -10
10 0 -10
-20
-20
-30
-30
-40 0
error ia error ib error ic
30
0.02
0.04
0.06
0.08
0.1
-40 0
0.02
Time (s)
0.04
0.06
0.08
0.1
Time (s)
c) Current error with M1=0
d) Current error with M1≠0
Fig. 10. PI current control results
Conclusion In this paper, a precise Park model for Double Salient Permanent Magnet Generator is proposed. An additional torque component exists in comparison with conventional machines due the mutual between d and q axis. A robust sliding mode current regulator is used to control the nonlinear system. Results show that our assumption for the sinusoidal current supply is not the best choice. To reduce the torque ripple in the future, one way consists on redesigning the machine and minimizing the effect of mutual inductance. The second alternative, in order to get a constant torque from the DSPMG for every mechanical speed, will be to analyze the performances under other specific waveforms of the stator currents.
Appendix A Parameters of DSPMG (with 40 turns in every slot): Nr=64 Rs=0.08837 Ω L0=0.0255 H M1=0.0025 H φ0=1.455 Wb φ1=0.4805 Wb
L1=0.0025 H Im=-40 A
M0=-0.0124 H ω=50*π*2 rad/s
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[10] Gao W. and Hung C.: Variable structure control of nonlinear systems: A new approach, IEEE Transaction on Industrial Electronics, Vol.40, N°1, February 1993 [11] Zhang S. Z., Ma X.L.: A PMSM Sliding-mode Control System Based-on Exponential Reaching Law, International Conference on Computational Aspects of Social Networks, CASoN 2010, Sep. 2628, 2010 Taiyuan, China, pp. 412-414
