14th International Power Electronics and Motion Control Conference, EPE-PEMC 2010
Induction Motor Torque Control in Field Weakening Regime by Voltage Angle Control Petar Matić*, Aleksandar Rakić†, Slobodan N. Vukosavić† *
Faculty of Electrical Engineering, Banjaluka, RS, Bosnia and Herzegovina, e-mail:
[email protected] † Faculty of Electrical Engineering, Belgrade, Serbia, e-mail:
[email protected];
[email protected]
Abstract — This paper presents a novel algorithm for the induction motor torque control in field weakening region. Proposed method insures maximum DC bus utilization and offers DTC performance through the stator voltage angle control. The algorithm is simple, without the outer flux loop nor the inner current loop. Dynamic response is preserved over wide speed range by means of gain-scheduling. The paper comprises the implementation details and experimental results. Keywords — Induction Motor, Vector Control, Direct Torque Control, Field Weakening.
I. INTRODUCTION Decoupled torque and flux control of an induction machine (IM) is commonly achieved through the Field Oriented Control (FOC) or the Direct Torque Control (DTC). In the former, the stator current component aligned with the rotor flux controls the flux amplitude, while the perpendicular component controls the torque. The later method is based on setting the stator voltage vector according to the torque and flux references and it does not have an inner current loop [1-3]. In both cases stator or rotor flux is kept constant below the nominal speed and decreased proportionally to the rotor speed above the nominal speed, in the field weakening region. The FOC control schemes require current controller and coordinate transformations. They can be based on rotor flux or stator flux reference frames. In most cases, the stator current controller is located in synchronous dq reference frame and it requires both direct and inverse Park transforms. The benefit of first DTC schemes [2, 3] was sought in relieving the controller from numerical tasks of performing the current control and coordinate transformations. With the numerical throughput of present digital signal processors (DSPs), these problems are of a lesser importance. Yet, the absence of an explicit current controller results in a better DC bus utilization, which is the primary subject discussed in this paper. For the proper operation of the current loops within FOC schemes, a minimum voltage margin has to be preserved both in the constant flux and the field weakening regime. Therefore the FOC scheme results in flux amplitudes lower than attainable with the given DC bus voltage. Consequently, the peak torque capability of the drive is decreased while the power losses are increased. For the IM drives experiencing stepwise torque changes in the field weakening region, the flux has to be further reduced and the voltage margin further increased in order to avoid saturation of the current loops [4-11]. Additional flux reduction contributes to further performance loss of FOC
978-1-4244-7855-2/10/$26.00 ©2010 IEEE
drives. Therefore, DTC schemes have the potential of improving the IM drive performance in the field weakening mode, and it is believed that this potential has not been fully exploited. A large number of proposed DTC algorithms, tend to regulate either stator or rotor flux and use stator voltage equations in a stationary reference frame without the inner current regulators. In order to regulate rotor flux, these structures must include coordinate transformations, which are not necessary when stator flux is controlled [23, 26, 28]. Most of the DTC concepts are based on some kind of IM model inversion [23-30] and therefore they are parameter sensitive. In this paper, a novel DTC structure is proposed, capable of operating in the field weakening mode with uncompromised flux, dynamic response, efficiency and peak torque capability. Instead of using dead-beat control [22, 23, 26, 27, 30] or coordinate transformations [24, 25, 26, 28,29], proposed structure is simple, with the torque controller having proportional and integral actions. Controller gains are variable in the flux weakening region and they change through the gain scheduling scheme. This structure is robust and insensitive to parameter variations, unlike common DTC schemes based on model inversion . In Section II, mathematical model of induction motor operating in the field weakening regime is derived. It is assumed that the torque controller makes the full utilization of the DC bus voltage. Hence, the 3-phase inverter is assumed to provide the maximum voltage for the given operating conditions over the whole field weakening range. In order to get an insight into the torque dynamics, a small signal operating point model is developed. This model is used to obtain the transfer function for small input disturbance (Section III). Said transfer function is used for creating the gain scheduling algorithm in Section IV, where the complete torque controller is detailed and explained. Performances of the proposed method are first investigated through a series of computer simulations (Section V). Experimental results, included in Section VI clearly demonstrate the advantages of the proposed scheme. II. MATHEMATICAL MODEL OF INDUCTION MOTOR IN FIELD WEAKENING Mathematical model of induction machine is given in equations (1) through (7), dΨs u s = Rs i s + + jω s Ψ s (1) dt dΨr 0 = Rr i r + + j (ω s − ω m )Ψ r (2) dt
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Ψ s = Ls i s + M i r Ψ r = Lr i r + M i s
(4) 3 Te = P(Ψ s × i s ) (5) 2 ωm = Pω (6) dω J = Te − TL (7) dt where Ψ s = Ψsd + jΨsq and Ψ r = Ψrd + jΨrq are the d – and q – axes stator and rotor flux vectors, respectively, u s = u sd + ju sq and i s = i sd + ji sq are the d – and q – axes stator voltage and current vectors, i r = i rd + ji rq is rotor current vector. R s and R r are stator and rotor resistance, Ls and Lr are stator and rotor self – inductance, M is mutual inductance, Te and TL are the motor and load torque, P is number of pole – pairs, ω s ,
ω m , and ω are respectively synchronous, electrical, and rotor speeds, J is motor inertia and j is imaginary unit. Core losses and saturation are neglected. In the flux weakening regime, the stator voltage is limited due to a limited DC bus voltage. Ideally, the peak value of the line-to-line voltage can reach the bus voltage. Namely, for the bus voltage of UDC, the RMS value of the line-to-line voltage cannot exceed 0,707 ×UDC. In terms of dq voltage components, the voltages ud and uq are limited by us =
2 u sd
+
2 u sq
≤ U MAX ,
(8)
where UMAX represents the peak phase voltage available and is equal to U DC / 3 . Due to voltage constraint (8), the torque producing component of the stator current and the flux producing component are coupled [26-27]. With a limited voltage, a torque rise increases the active component of the stator current and, hence, the voltage drop across series impedances. In turn, this diminishes the back electromotive force and reduces the flux. Therefore, the steady state values of the torque and flux are coupled in the field weakening regime. In addition, dynamic coupling between the torque and flux loops may lead to unfavorable oscillations. Therefore, one of the goals of the proposed torque controller is to suppress the torque – flux dynamic coupling and to provide a smooth, well damped response. In order to fully utilize DC bus voltage, possibility of using maximum available voltage (9) in the whole flux weakening region is investigated. Synchronous reference frame is aligned with stator voltage as shown in Fig.1. u s = U MAX , (9)
u sd = U MAX , u sq = 0 .
(10) (11)
q
b
(3)
us ϑ
d a
c
Fig. 1. Synchronous reference frame orientation
The only one control variable in model (1-7) is synchronous frequency ω s related to the angle ϑ from Fig. 1. as: dϑ ωs = . (12) dt Assuming that rotor speed stays constant during the torque transients (i.e. mechanical transients are much slower than electrical ones), equations (1-5) can be normalized and rewritten in form of state-space model as follows:
⎡ ⎤ dΨd 1 k = ωb ⎢U − ' Ψd + ωe Ψq + r' ΨD ⎥ , dt Ts Ts ⎢⎣ ⎥⎦ ⎡ ⎤ dΨq 1 k = ωb ⎢− ωe Ψd − ' Ψq + r' ΨQ ⎥ , dt Ts Ts ⎣⎢ ⎦⎥ ⎤ ⎡k 1 dΨD = ωb ⎢ s' Ψd − ' ΨD + (ωe − ω r )ΨQ ⎥ , dt Tr ⎢⎣ Tr ⎥⎦
(13) (14) (15)
⎤ ⎡k 1 = ωb ⎢ s' Ψq − (ωe − ω r )ΨD − ' ΨQ ⎥ , (16) dt Tr ⎣⎢ Tr ⎦⎥ 1 te = Ψ Ψ − ΨQ Ψd . (17) σls D q In model (13-17) U is stator voltage amplitude in perunit [p.u] Ψd , Ψq , ΨD and ΨQ are stator and rotor dΨQ
(
)
fluxes in [p.u], k s = M / Ls , k r = M / Lr are stator and rotor
coupling
Tr' = ωbσLr / Rr
coefficients,
Ts' = ωbσLs / Rs ,
are stator and rotor transient time
constants in [p.u], ω e and ω r are synchronous and rotor speed in [p.u], σ = 1 − M 2 / Ls Lr is leakage coefficient,
ls is stator inductance in [p.u], te is motor torque in [p.u] and ωb is base speed. State space model (13-17) can be expressed as: x = f1 ( x, u ) , y = f 2 (x ) , where state and input vectors x and u are
[
x = Ψd
Ψq
ΨD
ΨQ
]T ,
(18) (19) (20)
u = [ωe ωr ]T , (21) with ω e as control variable, ω r as disturbance and y = t e as output.
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III. TORQUE DYNAMICS IN FIELD WEAKENING A. Linearization of Induction Motor Model Dynamic of torque response in field weakening with voltage constraint (9) satisfied is given by non-linear model (18-19). In order to analyze torque dynamic response, this model will be linearized around the stationary operating point in which values are denoted 0
with superscript “ 0 ”. Stationary input vector u is:
[
]
[
Ψq0
T
u 0 = ωe0 ω r0 . (22) Vector of state variables in operating working point is: 0
x =
Ψd0
],
T ΨQ0
ΨD0
B. Static Characteristics in Field Weakening In order to analyze steady states in field weakening as a function of load, motor slip is introduced as s = ωe − ω r . (35) Motor torque (29), and stator and rotor flux modulus (24-28) are plotted in Figs. 2-4 as functions of synchronous speed and steady state slip s 0 . These plots are obtained for test motor whose physical parameters are given in the Appendix. As speed increases, torque and fluxes decrease due to voltage limit. 3 s0 = 0 s0 = 0.05
(23)
2.5
and can be found from (13-17) using (22) by letting d / dt = 0 as:
2
Ψq0 = −
)
(
)
⎧ 0 '⎡ '2 0 0 2⎤ UTs' ⎪ωe Ts ⎢1 + Tr ωe − ω r ⎥ ⎣ ⎦
(24)
⎫ +⎪ ⎬, ⎪ ⎭
(25)
{
{
(
[
(
) (
)}
(26)
)
(
0
-0.5 1
2
⎡ Ψq0 ⎢ ∂f − Ψ0 B = 1 = ωb ⎢ 0d ⎢Ψ ∂u ⎢ Q ⎢⎣ ΨD0
C=
[
∂f 2 k = r − ΨQ0 ∂ x σls
1 Ts'
0
0 kS TR'
−
(
1 − ' TR
ωe0
− ω r0
0 ⎤ ⎥ 0 ⎥ , ΨQ0 ⎥ ⎥ ΨD0 ⎥⎦
ΨD0
2.4
2.6
0.9
2.8
3
⎤ ⎥ ⎥ kR ⎥ ⎥ Ts' ⎥ ,(32) ωe0 − ω r0 ⎥ ⎥ 1 ⎥ − ' ⎥ TR ⎥⎦
s0 = 0.1 s0 = 0.15
(28)
(29)
( )
2.2
s0 = 0.05
+⎫ ⎪ ⎬. ⎪ ⎭
0.8
0.7
0.6
0.5
0.4
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
ωe0 [p.u]
Fig. 3. Stationary Stator Flux vs. Synchronous Speed 1 s0 = 0 s0 = 0.05
0.9
s0 = 0.1
)
s0 = 0.15
0.8 0.7
Ψr0 [p.u]
−
2
s0 = 0
0
Ts'
1.8
1
)]
kR
1.6
ωe0 [p.u]
Linearized operating point model of (18-19) is in form: x = A x + Bu , (30) y = Cx , (31) where
ωe0
1.4
(27)
)
⎡ 1 ⎢ ' ⎢ Ts ⎢− ω 0 ⎢ e ∂f A = 1 = ωb ⎢ ∂x ⎢ kS ⎢ TR' ⎢ ⎢ 0 ⎢⎣
1.2
Fig. 2. Stationary Motor Torque vs. Synchronous Speed
)}
(
1
0.5
⎨ D ⎪ ' 0 0 ⎩+ k s k r Tr ωe − ω r UTs' k s ΨD0 = 1 − k s k r + ωe0Ts'Tr' ωe0 − ω r0 , D UT ' k ΨQ0 = − s s ωe0Ts' + Tr' ωe0 − ωr0 , D where ⎧(1 − k k )2 + ω 0T ' + k k T ' ω 0 − ω 0 s r e s s r r e r ⎪ D=⎨ '2 0 0 2⎡ 2 2 0 2 '2 ⎤ ⎪+ Tr ωe − ω r ⎢1 − k s k r + ωe Ts ⎥ ⎣ ⎦ ⎩ 1 Te0 = Ψ 0 Ψ 0 − ΨQ0 Ψd0 . σls D q
(
1.5
e
(
⎧1 − k k + T '2 ω 0 − ω 0 2 ⎫ , ⎨ s r r e r ⎬ ⎩ ⎭
T 0 [p.u]
UTs' D
s0 = 0.15
Ψs0 [p.u]
Ψd0 =
s0 = 0.1
0.6 0.5 0.4
(33)
0.3 0.2 0.1 1
Ψq0
]
− Ψd0 .
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
ωe0 [p.u]
(34)
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Fig. 4. Stationary Rotor Flux vs. Synchronous Speed
C. Motor Dynamics in Field Weakening From linearized model (30-31) transfer function and static gain G0 can be found as
G( p ) = C ( pI − A)−1 B ,
(37)
−1
(38) G0 = −CA B , where p is complex variable. Analytic solution of transfer function (37) is rather complicated. Static gain and the positions of the open-loop poles are numerically calculated for the test motor and plotted with solid lines in Fig 5 and Fig. 6. 30 Approx, s0 = 0
( ξ s , ξ r ) and natural frequencies ( ωns , ωnr ) are:
Approx, s0 = 0.15
10
0
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
,
( ) ω (1 + s ω T ) ,
ξ s = 1/ ωb 1 + ωe2Ts'2 , 2
b
2 '2 e r
1 + ω e2 , Ts'2
(41)
(42) (43) (44)
1 (45) + s 2ω e2 , Tr'2 where s stands for motor slip (35). Approximated static gain (41) is also shown in Fig. 5, plotted with dash lines. Difference between approximated and exact values of gain is neglible. It can be concluded that the static gain approximation decreases in inverse proportion to synchronous speed squared. The positions of the open-loop poles in approximated transfer function (40) are shown in Fig. 6. It can be observed that approximated motor poles are very close to the exact ones. The analysis above showed that motor dynamics in field weakening can be fully represented by simple fourth-order transfer function given by (40).
ω nr = ωb
5
3
ωe0 [p.u]
Fig. 5. Stationary Gain vs. Synchronous Speed
As motor speed increases, static gain of the linearized model decreases, and even becomes negative for some loads. This effect can be neglected since it occurs at high loads, i.e. when motor slip is larger than break-down slip. 1000
IV. DIRECT TORQUE CONTROL IN FIELD WEAKENING
800
In order to formulate control law for direct torque control in field weakening, torque dynamics of (40-45) is further analyzed. Natural frequency of stator poles (44) is much larger than frequency of rotor poles (45), meaning that rotor dynamics is dominant in torque transients. Motor torque can be regulated by simple proportional – integral torque regulator (PI), parameterized in the form:
600 400 200 Imag
(σl s )2 ω e2
Approx, s0 = 0.1
15
0 -200 -400 -600 -800 -1000 -76.5
U 2 k sTr
G0 App =
ω ns = ωb
Approx, s0 = 0.05 Exact, s0 = 0.1 Exact, s0 = 0.15
ΔTe/Δωe
, (40) 2 ⎞⎛ 2 ⎞ ⎛ ξ 2 p ξ p 2 p p s r ⎜1 + + 2 ⎟ + 2 ⎟⎜1 + ⎟⎜ ⎜ ω ns ω ns ω nr ω nr ⎟⎠ ⎠⎝ ⎝ where static gain G0 App , stator and rotor damping ratios
Exact, s0 = 0.05
20
-5 1
G0 App
G App ( p ) =
ξ r = 1/
Exact, s0 = 0 25
approximated motor transfer function is expressed as
-76
-75.5
-75
-74.5
-74
-73.5
-73
-72.5
-72
Real
Fig. 6. The positions of the Open-loop Poles (Blue lines – Exact, Green Lines – Approximated)
As it can be seen from Fig. 5 and Fig. 6, static gain is highly dependent on synchronous speed and the motor dynamics is defined by two pairs of complex-conjugate poles. In order to solve (37) analytically, an approximation is introduced, i.e., rotor leakage flux is neglected: Lr ≈ M . (39) Using approximation (39), rotor coupling coefficient becomes k r = 1 . Introducing (39) into model (30-31),
⎛ 1 1⎞ K PI ( p ) = K C ⎜⎜ + ⎟⎟ , (45) ⎝ ωC p ⎠ where ωC is desired closed-loop bandwidth and K C is the regulator gain needed to achieve it. Closed-loop bandwidth ωC is selected as a half rotor natural frequency for the non-loaded motor, regarded as the worst case (with the lowest phase margin). Adequate tuning of PI regulator parameters is: ω (46) ωC = b ' , 2Tr
ωC
ωb (σl s )2
ωe2 . (47) G0 App 2k sTr'2U 2 Block diagram of proposed direct torque control method is shown in Fig. 7. KC =
T4-111
=
T*
+ -
T
Gain Scheduled PI
s
E
ωe
+
1 ϑ p
Ε
+ ωm
Ua
Uα
UMA X cos
Space Vector Modulator
Uβ
UMA X sin
Ub Uc
Induction Motor
ia Torque & Speed Estimator
ib
Fig. 7. Block Diagram of the Proposed Torque Controller in Field Weakening Regime.
The input to the control algorithm is the torque reference, from which the estimated torque is subtracted and the torque error signal is obtained and fed to the torque controller block. Torque controller is gain scheduled proportional-integral controller (45) with saturation. Output of the regulator is machine slip s = ωe − ω m , (48) which is limited to the break-down slip 1 s≤± ' . (49) Tr Saturation is adjusted to prevent windup of the control loop. As it can be seen from (45) and (47), PI regulator gains are operating point sensitive, i.e. gains are scheduled by synchronous speed squared. Gain scheduling mechanism adopts torque controller to obtain projected response for all working points. The output of the torque controller is the slip angular frequency command, which represents an increment to the estimated rotor speed. Reference synchronous speed is obtained by adding reference slip with the estimated speed ω mE . Speed command is integrated and the voltage angle command ϑ (12) is calculated, from which stationary-frame reference voltages are: Uα = U cosϑ , (50) U β = U sin ϑ . (51) Reference voltages are fed to the Space Vector Modulator to generate machine voltages U a , U b and
U c . Two line currents, ia and ib are measured. Gain-scheduling mechanism is providing accurate closed-loop response in the wide-range of field weakening operating regimes by using maximal voltage available. Torque and speed estimator estimates motor torque and rotor speed. A simple open loop flux estimator based on stationary frame stator voltage equations in normalized form is used:
∫ ∫ = ∫ eβ dt =ω ∫ (U β − r iβ )dt ,
ΨαE = eαE dt =ωb (U α − rs iα )dt ,
(52)
ΨβE
(53)
E
b
s
ΨαE
where
ΨβE
and
are estimated stator flux
components, eαE and eβE are stator back emfs, rs is stator resistance in [p.u], and iα = ia , (54) ⎛ ia ⎞ iβ = 3 ⎜ + ib ⎟ ⎝2 ⎠ (55) are stator currents in stationary reference frame calculated from measured currents. Estimated motor torque is: T E = ΨαE iβ − ΨβE iα , (56) and estimated rotor fluxes are: 1 ΨαEr = ΨαE − σl s iα , (57) kr 1 ΨβEr = ΨβE − σl s i β . (58) kr Motor slip in [p.u] is calculated from estimated torque and estimated rotor flux as: rr (58) sE = TE , 2 E E 2 Ψαr + Ψβr where rr is rotor resistance in [p.u]. Rotor speed is calculated from estimated synchronous speed which is obtained from rotor flux angle:
(
)
(
)
(
)
( ) ( )
ΨβEr ⎞ dϑrE d⎛ (59) = ⎜ arctan E ⎟ , dt dt ⎜⎝ Ψαr ⎟⎠ by subtracting estimated slip as: (60) ω mE = ω sE − s E . Proposed estimator is usual in DTC algorithms [23-27] enabling fast and accurate estimation. In practical realization, instead of pure integration (52-53), low pass filter should be used to avoid integrator drift. Back emf’s,
ω sE =
eαE and eβE in field weakening are large enough to
estimate speed accurately. Machine current can be easily limited by limiting machine slip, i.e., by limiting output of the gainscheduled PI regulator (49) to lower values than breakdown slip according to current capability of drive.
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V. SIMULATION RESULTS Response, ω = 1 p.u. r
Computer simulations of the proposed algorithm are conducted in order to initially verify expected performances. Torque reference is in the form of the rectangular pulse sequence with the amplitude of nominal motor torque. Closed-loop responses of the torque, rotor flux, machine slip and the phase current are obtained at the rotor speed of 1 p.u., 1.5 p.u., and 2 p.u. and shown in Fig. 8 through Fig. 11. At the speed of 2 p.u., the torque reference cannot be achieved, as it can be observed in Fig. 8. In this situation, motor slip hits the break down limit at 0.24 p.u. in Fig. 10 and the torque reaches the maximal achievable breakdown value.
Response, ω = 1.5 p.u. r
0.3
Response, ω = 2 p.u. r
0.25
ωk [p.u]
0.2
0.15
0.1
0.05
0 0
0.1
0.2
0.3
0.4
0.5 t [s]
0.6
0.7
0.8
0.9
1
Fig. 10. Motor Slip Responses at Various Speeds in Field Weakening Regime
1.6 Reference Response, ω = 1 p.u. r
1.2
Response, ω = 2 p.u.
2
ωr = 1 p.u.
1.4
iα [p.u],
Response, ω = 1.5 p.u. r r
-2 0
1 0.8
0.1
0.2
0.3
0.4
0.5 t [s]
0.6
0.7
0.8
0.9
1
0.1
0.2
0.3
0.4
0.5 t [s]
0.6
0.7
0.8
0.9
1
0.1
0.2
0.3
0.4
0.5 t [s]
0.6
0.7
0.8
0.9
1
iα [p.u],
ωr = 1.5 p.u.
2
e
T [p.u]
0
0.6 0.4
0 -2 0
0.2
ωr = 2 p.u.
2 iα [p.u],
0 -0.2 0
0.1
0.2
0.3
0.4
0.5 t [s]
0.6
0.7
0.8
0.9
1
-2 0
Fig. 8. Torque Reference and Responses at Various Speeds in Field Weakening Regime
1.5
Fig. 11. Phase Current Responses at Various Speeds in Field Weakening Regime
Response, ω = 1 p.u.
VI. EXPERIMENTAL RESULTS
r
Response, ω = 1.5 p.u. r
Response, ω = 2 p.u. r
Ψr [p.u]
1
0.5
0 0
0.1
0.2
0.3
0.4
0.5 t [s]
0.6
0.7
0.8
0.9
0
1
Fig. 9. Rotor Flux Responses at Various Speeds in Field Weakening Regime
Rotor flux transients due to torque change are shown on Fig. 9. On break down limit (when the slip is limited), rotor flux is on its minimum value, and further flux decrease is not permitted. As no current limit was introduced in simulation, obtained current amplitudes in Fig. 11 are larger than nominal.
Experimental verification is done on a test setup with two mechanically coupled 750W induction machines. These machines are powered by transistor driven inverters with paralleled DC busses. Four quadrant testing is performed using this test rig, but due to test bench speed limit, the motor has been fed with a reduced voltage, leading to base speed to be derated to 750rpm. The control algorithm is implemented on fixed processor board with PWM period to be 2kHz. Motor was operated using V/f control up to field weakening regime. After reaching base speed, DTC control scheme depicted in Fig. 7 was implemented. Experiments have been carried out to investigate drive performance in field weakening region when command torque is in recentagular shape with shaft speed of 2 p.u. Firstly, torque response on positive reference change (from zero to 50% of rated torque) is shown in Fig. 12. Secondly, torque response on negative reference change (from 50% to zero) is shown in Fig. 13.
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Fig. 12. Torque Step From 0 to 50% Rated Torque, Speed 2 p.u (Upper Trace, Motor Current, 2A/div, Lower Trace, Motor Torque, 2.5Nm/div, Time 0,25s/div)
Fig. 15. Torque Step From -50% Rated Torque to 50% Rated Torque (Upper Trace, Motor Current, 2A/div, Lower Trace, Motor Torque, 2.5Nm/div, Time 0,25s/div)
Finally, motor behavior when torque reference is in form of rectangular shape pulses (nominal torque change pulses from -50% to 50%, i.e., nominal torque transition in field weakening, speed 2 p.u.) is shown in Fig. 16-17.
Fig. 13. Torque Step Reference From 50% Rated Torque Down to Zero, (Upper Trace, Motor Current, 2A/div, Lower Trace, Motor Torque, 2.5Nm/div, Time 0,25s/div)
Torque response when reference change is from positive to negative (50% rated to -50% rated) and negative to positive (-50% rated to 50% rated) when speed is 2 p.u. is shown in Fig 14-15. In this regime, machine changes regime from motoring to generating and vice versa. As can be seen, the motor behaves as shown on simulations, namely torque response is aperiodic.
Fig. 16. Torque Step From -50% to 50% in Recantangular Pulse shape (Upper Trace, Motor Current, 2A/div, Lower Trace, Motor Torque, 2.5Nm/div, Time 0,25s/div)
Fig. 17 shows gain scheduling coefficient (47) in p. u and rotor flux for the torque change shown in Fig. 16.
Fig. 17. Gain Scheduling Coefficient (47) and Rotor Flux for Torque Change Shown in Fig. 16. (Upper Trace Kc in p.u, Lower Trace Rotor Flux, 0,4Wb/div, Time 0,25s/div)
Fig. 14. Torque Step From 50% Rated Torque to -50% Rated torque (Upper Trace, Motor Current, 2A/div, Lower Trace, Motor Torque, 2.5Nm/div, Time 0,25s/div)
Results shown on Fig. 12-15 prove the robustness of the proposed DTC algorithm and show excellent dynamic performance in high speed region.
VII. CONCLUSIONS State of the art field weakening control algorithms are typically based on trajectory calculation from steady state machine models and therefore fail to optimize dynamic response. In addition, FOC solutions suffer from current regulator saturation, which is sometimes prevented by maintaining significant voltage margin. The end result in all these implementations is suboptimal operation. In contrast, the proposed DTC solution offers following benefits:
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A. Flux optimization is accomplished automatically by closed loop regulation, in contrast to other solutions which are based on open loop flux reference calculations. B. Best possible DC bus voltage utilization is enabled by maintaining maximum available voltage modulus and only regulating voltage angle. This is accomplished without rotational transformations. C. Gain scheduling with respect to speed is introduced. The gain scheduling enables optimal aperiodic transient response of the system. Gain scheduling algorithm is derived and analyzed from machine dynamic model to accomplish this goal. APPENDIX Parameters of motor used in simulation and experiment are: 750W , 380V , 50 Hz , 1450rpm , Rs = 10.5Ω ,
Rr = 11Ω , M = 0.557 H , Ls = 0.579H , Lr = 0.579H . ACKNOWLEDGMENT This work was supported by the Ministry of Science and Technology of Republic of Srpska. REFERENCES [1]
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