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Энергетика. Изв. высш. учеб. заведений и энерг. объединений СНГ. Т. 66, № 3 (2023), с. 205-214 Епе^ейка. Ргос. dS Higher Educ. Inst. ада Power Eng. Assoc. V. 66, No 3 (2023), рр. 205-214 205
https://doi.org/10.21122/1029-7448-2023-66-3-205-214 UDC 62-83-52(075.8)
Reference Input Signals Formation for Induction Motor Control in Traction Drive
O. F. Opeiko1)
1)Belarusian National Technical University (Minsk, Republic of Belarus)
© Белорусский национальный технический университет, 2023 Belarusian National Technical University, 2023
Abstract. The purpose of this work is to build the analytical improved (with resistances estimation) real time computation of the reference inputs for rotor flux and torque in the vector control system of an induction motor of a traction electric drive. The reference inputs must maximize electromagnetic torque in conditions of voltage source instability, particularly in magnetic field weakening mode. The conventional way to control the field weakening mode is to form flux coupling task inversely proportional to the speed or inversely proportional to the square of the speed in second and third zones respectively. Such reference input signals are not able to provide the maximum torque capability over the entire speed range, and the improved torque capability is achieved in different ways. For instance, voltage feedback is useful for the torque capability enhancement in conditions of internal and external perturbations. A wide change in speed with the weakening of the flux reveals the nonlinear properties of an induction electric motor. However, in vector control systems, proportional-integrating (PI) regulators are usually used. Therefore, firstly, linear PI controllers must be robust, and secondly, the reference input signals for flux and torque must guaranty linear, not saturated state of each PI controller. The proposed expressions for calculating reference inputs for induction motor rotor flux and electromagnetic torque as functions of actual rotor speed are the approximate expressions. The estimation of the possible error shows that the error is acceptable. Simulation is performed for the vector control system of an induction motor and taking into account the calculation of the control signal by the microcontroller and the dynamics of the frequency invertor. The simulation of the resulting system validates the effectiveness of the control system using the proposed expressions for the formation of real-time reference input signals for setting the flux and torque.
Keywords: vector control, reference input signals, induction motor, electromagnetic torque, field weakening, simulation
For citation: Opeiko O. F. (2023) Reference Input Signals Formation for Induction Motor Control in Traction Drive. Energetika. Proc. CIS Higher Educ. Inst. and Power Eng. Assoc. 66 (3), 205-214. https://doi.org/10.21122/1029-7448-2023-66-3-205-214
Формирование сигналов задания системы тягового асинхронного электропривода
О. Ф. Опейко1)
''Белорусский национальный технический университет (Минск, Республика Беларусь)
Реферат. Цель данной работы - формирование в реальном времени сигналов задания пото-косцепления и момента в системе векторного управления асинхронным электродвигателем
Адрес для переписки
Опейко Ольга Федоровна
Белорусский национальный технический
просп. Независимости, 65,
220013, г. Минск, Республика Беларусь
Тел.: +375 17 293-95-61
Address for correspondence
Opeiko Olga F.
Belаrusian National Technical University 65, Nezavisimosty Ave., 220013, Minsk, Republic of Belarus Tel.: +375 17 293-95-61 [email protected]
тягового электропривода. Сигналы задания должны максимизировать момент в условиях нестабильности напряжения источника питания, в частности в режиме ослабления магнитного поля. Обычный способ управления режимом ослабления поля заключается в формировании задания потокосцепления обратно пропорционально скорости либо квадрату скорости. Так формируемые сигналы задания не способны обеспечить максимум располагаемого момента во всем диапазоне изменения скорости, и увеличение располагаемого момента достигается различными путями. Например, обратная связь по напряжению используется для увеличения располагаемого момента в условиях внутренних и внешних возмущений. Широкое изменение скорости при ослаблении потокосцепления выявляет нелинейные свойства асинхронного электродвигателя. Однако в системах векторного управления обычно применяются пропорционально-интегрирующие (ПИ) регуляторы. Следовательно, во-первых, линейные ПИ-регуляторы должны быть робастными, во-вторых, сигналы задания для потокосцепления и момента должны гарантировать линейное без насыщения функционирование каждого ПИ-регулятора системы управления. Предложенные выражения для расчета входных сигналов задания для потокосцепления ротора и электромагнитного момента как функции текущего значения скорости ротора являются приближенными выражениями. Оценка возможной погрешности показывает, что погрешность допустима. Имитационное моделирование выполнено для системы векторного управления асинхронным электродвигателем и с учетом вычисления сигнала управления микроконтроллером и динамики преобразователя частоты. Имитационное моделирование системы подтверждает эффективность управления с применением предложенных выражений для формирования в реальном времени сигналов задания потокосцепления и момента.
Ключевые слова: векторное управление, ориентированное по потокосцеплению ротора, сигналы задания, асинхронный электродвигатель, электромагнитный момент, ослабление поля, имитационное моделирование
Для цитирования: Опейко, О. Ф. Формирование сигналов задания системы тягового асинхронного электропривода / О. Ф. Опейко // Энергетика. Изв. высш. учеб. заведений и энерг. объединений СНГ. 2023. Т. 66, № 3. С. 205-214. https://doi.org/10.21122/1029-7448-2023-66-3-205-214
Introduction
The control of induction motor in the field weakening region became an actual problem with the vector control development [1-8]. The early implementations of the flux weakening control of rotor or stator field-oriented control and direct torque control use the simple inverse of the rotor speed method. As it was shown in [5] and [6], such a control cannot maintain maximum torque capability over the entire speed range. The improved torque capability is achieved when the stator flux reference is varied as the inverse of the rotor speed [5-8]. As usual, the torque is maximized with omission of the stator resistance [5, 8]. The inverter voltage limit and current limit are considered simultaneously in the (d, q)-axis plane of the stator current to obtain an optimal rotor flux command [6]. Three sub regions of the speed can be recognised [5, 6].
The papers [7-24] present ways to improve the torque maximization when the field is weakening. Closed-loop methods use the proportional integral (PI) controllers to adjust the flux level when the voltage commands from current regulators are beyond the inverter voltage limit [15, 16]. The advantage of the closed loop methods is the relative independence of machine parameters. However, the additional PI controllers make the controller more complicated and tuning more difficult. In [8, 11] the control schemes are presented and analysed
for the field-weakening region. These control schemes fully utilize the maximum available voltage and current and can produce the maximum possible torque in the entire field-weakening region. The field weakening is mainly actual for the traction drives and others wide speed range applications. For a traction drive, especially for the electric vehicles, efficiency of induction motors applications depends hardly on the field weakening control [14, 15, 25-28]. So, in the papers [5-24, 29] the field weakening control synthesis comprises two directions. Firstly, the reference values are designed [5, 6, 29]. Next, the feedback control is developed for the robustness improvement. Mainly, for torque maximisation, the voltage loop is in usage [7-22]. In [29] the artificial neural network is applied for flux and speed control. As it was shown in [14, 15] when extracting the possible output torque for deep flux-weakening operation, the torque ripples are typical. The band-stop filter is proposed in [14], but that means the stability margin reducing, and also the system becomes more complicate. Hence, the reason is to provide near maximum torque control avoiding torque ripples.
As usual, in vector control for induction motors PI controllers are in employment. That means the hidden assumption of plant linearity or linearization. However, in field weakening region the induction motor nonlinearity is revealing. But the linear control still effective, when the reference values are correctly constructed, such all regulators still not saturated.
In this work, the aim is to build the analytical improved (with resistances estimation) real time computation of the references for rotor flux and torque in field weakening region to maximise torque value for induction motor in the vector control traction drive system. The reference inputs must maximize electromagnetic torque in conditions of voltage source instability, particularly in field weakening region. The electromagnetic parameters of induction motor are assumed to be known from the on line identification [30-32].
Induction motor control in the traction drive
In the traction drive, the flux and torque control are useful. The input uC of references block determines acceleration or deceleration of traction motor. The car driver's command uC must provide the reference rotor flux and current i* computation, as shown in the control system block diagram (Fig. 1).
-K>
¥
-
PI1 C 1dq
'dq
PI2
Abs(Udq)
Udq
T §
*
u
U
C
Fig. 1. The induction motor control system
That denotes the torque reference as M * = kM ¥*i*, but instead of M * the
current component i* is in employment as a reference for current control. The control structure uses plants variables in the synchronous frame (d, q). The PI1 is the flux controller, its output iq is the magnetizing current reference value. Block C corrects current reference value i*0 depending on i* through the conditions for the magnitude of the stator current vector: if i*2 + i*0 < I2m, then i* = i*0, otherwise i* = (I2m - i*2 ) . The flux PI controller PI1 has two inputs to estimate the voltage source u* [7-16] and flux ¥* reference values. The magnitude uf = Abs (udq ) = (u2d + u* ) is needed for the comparisons
with source voltage u*. The block T offers coordinate and phase direct and inverse transforms, and the block Plant contains the induction motor with frequency inverter as voltage source.
In the synchronous frame (d, q) the induction motor dynamics can be described with state vector xT = (¥ d, id, iq ) and control vector uTdq = (ud, uq ),
where ¥ = (¥ d, ¥ q ) is the rotor flux vector with ¥ d = 0, and idiq = (id, iq ) is the stator current vector. The equations are as follow:
¥ d = -a¥ d + aLidid ;
ddd=(- ^d+ud+kd a¥ d )-+®0iq ; (1)
diq t _ \ -1
= (-Riq + uq - k2Ppo¥d )) - ®0id ;
Jo = M-Mc ; M = kM ¥diq. (2)
In these expressions the stator and rotor resistances R1, Rd, Q; the stator and rotor inductances L1, Ld, H and the mutual inductance L12, H are employed as follow: a = R2/L2, k1 = Lj2/L1, k2 = L12/L2, Re = R1 + k2R2. Then, the leakage factor is c = 1 - k1k2, and the inductance Le = cL1 characterizes the flux leakage. The rotor angular speed is o , rad/s, and O = oopp, where pp is the pair of pole number. The expression (2) for electromagnetic torque M contains the coefficient kM = (3/2) k2pp, J is the moment of inertia, and MC is the load torque. It is assumed, that parameters L12, R1, L1, T2 = L2/R2 are online identified [30-32].
Maximum torque conditions
The reference values for flux and current components for torque maximization must be evaluated for the steady state conditions in (d, q) frame. The steady state equations are produced from (1), (2) as follow:
0 = -a^ d +aL12id;
0 = (-RJd + Ud + k2a^ d ) +®0iq; (3)
0 = (-RJq + Uq - k2Ppd )Le^ - C0id ;
M = Mc; M = kM; C = C - ® = aijid . (4)
The first harmonic magnitudes U, I of voltage and current are defined from the
2 2 2 2 2 2 expressions U = Ud + u I = id + iq. From (3) and with respect to the expression
a>s = a i/id for the slip frequency in the steady state, the current can be expressed
as 12 = i2d (1 + c2Ja2). So, the stator voltage magnitude is U2 =i2d f , ®0).
With a,j = RjLj, aj = RjLl, the function f (ras, ra0) can be written in form
f (,ra0) = 2L ((1 + c(/a2)(a2rad +aj2) + ra) - a2C +2 (1 -a)axras ra0/a), or as follow:
f (c, ®0 ) = 2 L (A2®2 + A1®s + A ). (5)
The rotor flux angular speed ©0 is unmeasured and can be calculated from the rotor speed c , as follow: c0 = + C . The variables Ai (i = 0, 1, 2) depend on elec-
( 2 2 2 \ _2 1
ax +0 ®0 )a , АJ = 2ra0aia- x
x(1 -a), A0 = ®o +ai2. The torque (4), in the form M = kMLl2idctiJ a permits to express the condition CM/ 5®s = kML12 (id + 2id did/ dccs )^a = 0 of torque maximum, depending on sliding frequency , in the form id + c(id )Ccos =0. But the torque maximization must be done with voltage restriction U2 = id f (cs c0),
where CU2/cccS = 0. The nonlinear algebraic equations can be represented as follow:
id + C(id2 )/dcs =0;
c(id ) , . 2 Cf(c,®0) _
-cc^y (, ®G )+id——-= c = c0-c = aiq/id. (6)
The function f derivative is Cf (®s, ®g )/5©s = 2A2®s + A1). We must submit C (id )/c©s = -id2 from the first equation (6) to the second one with respect to (5) and then divide by L2 id2 > 0. Hence, for the maximizing sliding frequency c*, we have the equation -(A2©¡2 + Ax®* + Ag ) + (2A2®S + A) = 0. It
is equal to A2o>12 - Ag = 0. So, the optimal value c* as a function of the synchronous frequency ©0 becomes as follow:
„ (а,2 +Юп |
^ = a2/V 2°^- (7)
(2 2 2\' ах + о ю° I
As a result, the function f (©:, c° ) = f * (co°) takes a form
f * ) = L (а? + C + ai®° (1 - c^^^af+C^yja^Tc2^2)) 2. (8)
With respect to the expression (5) = a iqlid the optimal references for ¥ = Ll2id and iq take a form
* = Ll2U .* = ra*_* = ras U (Q)
d = VW) ' ^ a ld~ «fM ■ (9)
The expressions (7)-(9) contain the variable q0, which is not measurable. The rotor speed ra is measured by sensor or estimated. In case of submitting the value of ra = rapn instead of ra0, the error occurs. The absolute error may be
evaluated from (7) with respect to a(®S2 ) = 2®* A®*; a(®2 ) = 2®0A®0, and takes a form
Аю„ =
Л
« УА (c2) a 2 a2 (1 -c2 )c°
d м (2 +c2c2 )2 ю:*
Аю°. (10)
The relative error 5 s = Ara*J raS depending on 50 = Ara0/ ra0 derives from (10) and (7) and takes a form
5 = (1 -g2) 5 11 s (l + a2®0/aj2)(/©2 +l) 0
The denominator has a minimum on ®2 = a1ila, so, the maximal error is 5Smax =50(l-a2)/(l + a)2. However, in field weakening region, as usual, ©2 >a?/a2, then 5S <0.550(l-a2)/(l + a2). If in expressions (7)-(9) the value ra0 is changed by ® = ©0 -ras, the error Ara0 = ras and relative error 50 = Ara0 = ras/ra0 occur in slip frequency ras computation. But the slip frequency in (8) is growing with frequency q0, and its limit is raS = a/a. Consequently, the relative error for ® = ©0 -ras is as follow:
(l-a2) a
5s < 277—2Y^. (l2)
2 (l + a2) ara0
Hence, with small o, the relative error of calculating c is lower, then a half of the motor slip s = ro*/raG = a/®Ga. This is acceptable, and expressions (7)-(9) after changing ® = ®G -®s instead of ®G takes a form:
*2 2 c =a
(a2 +®2)
(ad +a2©2)'
f* (cc) = L2 (a2 +c2 +a1c(1 -a) + ^(a2 +c2 )/( +a2c2 )); (13)
* *
c * m U
= L12U . ,* =.®L7 =
- I *, , ; V _ ~
_ a a
As it can be seen, these expressions (13) are useful for the computation of references for high-speed operation vector control. The relative error is not exceeding (12).
Simulation results
The simulation of dynamics behavior of the traction drive system is needed for testing the proposed control structure. The simulation results are realized for the vector-controlled induction motor 180 KW, 1500 rpm in traction drive with speed sensor (Fig. 2) and speed sensorless (Fig. 3) in high-speed operating range including field weakening region.
Fig. 2. Starting and speed acceleration with field weakening: a - torque M and speed CO; b - current components id, iq
a
Figure 2 shows the start and acceleration process with field weakening for the vector control system with speed sensor. There are three zone of the operation [5, 6] when field weakening is in usage. The first one is from zero to the nominal speed, that is seen in Fig. 2 for the time t <0.5 s. In the second zone the field is weakening with the maximal stator current in the time interval from t - 0.5 s to t - 1.6 s. In the third zone, t > 1.6 s the flux weakening control provides the maximal voltage with id decreasing so, that idliq = ®sla = const.
In the Fig. 3 the start and acceleration process with field weakening for the speed sensorless vector control is shown. The sensorless system is more critical for the internal and external disturbances and, as it is shown in Fig. 3 a, has the ripples in torque in accelerating. In addition, with the sensorless control, the dynamics of the system becomes slower. Hence, the speed increases less intensive in Fig. 3, then for the system with speed sensor in Fig. 1.
From the comparison of Fig. 2, 3, the torque and acceleration are lover in case of Fig. 3. However, in both cases the dynamics performance is appropriate for the traction applications.
Fig. 3. Speed sensorless system, starting and speed acceleration with field weakening: a - torque M and speed co; b - current components id, iq; c - estimated and actual speed
a
CONCLUSIONS
1. The expressions (14) can be useful for references flux and torque
current iq* real time computation. Such expressions provide correct operating
of the system via the cost criteria, avoiding regulators saturation. The expressions (14) are approximate, then the absolute and relative errors are estimated through (11)-(13).
2. The simulations results demonstrate the efficiency of the real time implementation of the expressions (8), (10) in vector control system in high-speed operating range including field weaking region for the torque control in the system with speed sensor and in speed sensorless system.
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Received: 21 October 2022 Accepted: 27 December 2022 Published online: 31 May 2023
