High Performance Direct Torque Control Of Im Using Space Vector Modulation
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HIGH PERFORMANCE DIRECT TORQUE CONTROL OF INDUCTION MOTOR USING SPACE VECTOR MODULATION Jagdish G. Chaudhari Lecturer, Electrical Department, GHRCE, Nagpur (India). email: [email protected]
Sandeep K. Mude Lecturer, Electrical Department, KDKCE, Nagpur (India). email: [email protected]
Abstract This Paper describes sinusoidal pulse width modulation of the voltage impressed to the stator of an induction motor for direct control of torque and stator flux. Instantaneous voltage vectors applied by an inverter have redundancy characteristics which provide some flexibility for selecting the inverter switching modes. By Using this switching freedom, control is achieved according to the following properties; high speed torque control, regulation of the primary flux, minimization of the inverter switching frequency. This utilizes a constant hysteresis band for both developed torque and stator flux and indirectly maintains the switched stator voltage waveforms averaged over a switching period to sinusoidal as in SPWM technique. It improves the dynamic performance of the machine compared to the conventional speed control of induction motor drives. A simulation programme has been developed to verify the results. The inverter duty cycle can then be calculated using the space vector PWM technique. The proposed method is very promising for rapid torque control which is quite different from FOC (Field Orientation Control). Keywords: Space Vector, Torque Control, Estimators.
1. Introduction High dynamic performance of induction motor drives is indispensable in many applications of today’s automatically controlled machines. Induction motor control has attracted much attention recently in the power electronics field. Fieldoriented control has been developed, enabling an ac motor to attain dynamic responses as rapid as for dc motor [2]. The principle of field-oriented control is based on Fleming’s law, which describes the interaction force between fluxes and currents. Many papers have reported the problems associated with compensating various parameters. The current-controlled inverters typically used in the field-oriented drive system develop output waveforms which do not compare favorably with those of the voltage-controlled inverter. The current controlled inverter often causes increased motor harmonic losses and acoustic noise during steady-state operation [5]. This paper proposes new control schemes based on the principle of direct torque control, which can be considered a basic law of torque generation in the induction motor. It makes
1-4244-0038-4 2006 IEEE CCECE/CCGEI, Ottawa, May 2006
Prakash G. Gabhane M.Tech.(IPS), Student, GHRCE, Nagpur (India). email: [email protected]
possible both fast torque response and high-efficiency control at the same time.
2. Need for Direct Torque & Flux Control Inverter fed induction ,motors are increasingly being used in general applications varying the input voltage to the motor with frequency on open loop is one of the very simple and popular methods of speed control. In this method V/F is held constant. In steady state operation, the machine air gap flux is approximately related to V/F. As the frequency approaches zero near zero speed, the magnitude of the stator voltage also tends to zero and this low voltage is absorbed by the stator resistance. Therefore at low speed of operation the stator resistance drop is compensated by injecting an auxiliary voltage so that rated air gap flux and full load torque becomes available up to zero speed. At steady state operation, if the load torque is increased, the slip will increase within the stability limit and a balance will be maintained between the developed torque and the load torque. However, if the supply voltage to the inverter (which is obtained by rectifying the input AC supply) fluctuates, the air gap flux will vary. Also, increase in the stator resistance with temperature results in the variation of air gap flux. Hence, in constant V/F control scheme the air gap flux may drift and as a result the torque sensitivity with slip frequency (or stator current) will vary. If the correct V/F ratio is not maintained the flux may be weak (or may saturate). If the air gap flux decreases, slip frequency (ωs1) will increase for the same torque demand. Also response of the machine detoriates, hence a speed control scheme with independent control of torque and flux loop is desirable. DTC is one such method of speed control.
3. Machine Model Induction motor is modeled by its voltage equations in stator co-ordinates for both stator and rotor as follows:
1090
Vs 0
= =
dψ s + Rs ⋅ is dt dψ r − jω ⋅ ψ r dt
(1) +
Rr ⋅ ir
(2)
where,
σ = 1−
where,
ªV d º Vs = « » ¬V q ¼ ªψ sd º ψs = « » ¬ψ sq ¼ ª i sd º is = « » ¬ i sq ¼ ªψ rd º ψr = « » ¬ψ rq ¼ ª i rd º ir = « » ¬ i rq ¼
The torque expression can also be written in terms of these two flux signals as following:
is the stator voltage
is the stator flux linkage
Te
3 P Lo ψr × ψs 2 2 σLsLr =
is the rotor flux linkage
3 P Lo ψr ψs sin δ 2 2 σLsLr
(8)
where, į is the space angle between these two flux vectors.
4. Direct Torque Control is the rotor current
Rs is the stator resistance Rr is the rotor resistance Equation (1) is for stator voltage and (2) for rotor voltage. As the rotor winding is short circuited, the rotor voltage is zero in magnitude [3]. The induced emf is compensated by the resistive drop and the rotational voltage. Again, stator and rotor flux linkages are related to the stator and rotor currents as follows:
ψ s = L sis + L oir ψ r = L o is + L rir
(3) (4)
Ls is the stator self inductance Lr is the rotor self inductance Lo is the mutual inductance
These two phase variables are obtained from three phase values using the following matrix:
ª fa º ª1 − 1 / 2 − 1 / 2 º « » fb «0 3 / 2 − 3 / 2»¼ « » ¬ «¬ fc »¼
3 P ψ s × is 2 2
Lo ψr Lr
(
T em =
(5)
(6)
Considering stator flux vector and rotor flux vector to be independent state variable, equation (3) and (4) can be modified as following:
σ L sis = ψ s −
Induction motor torque control has traditionally been achieved using Field Oriented Control (FOC). This involves the transformation of stator currents into asynchronously rotating d-q reference frame that is typically aligned to the rotor flux. In this reference frame, the torque and flux producing components of the stator current are decoupled. A PI controller is then used to regulate the output voltage to achieve the reputed stator current and therefore torque. This PI controller limits the transient response of the torque controller. Direct Torque Control (DTC) uses an induction motor model to achieve a desired output torque. By using only current and voltage measurements, it is possible to estimate the instantaneous stator flux and output torque [1]. An induction motor model is then used to predict the voltage required to drive the flux and torque to demanded values within a fixed time period. This calculated voltage is then synthesized using space vector modulation (SVM). The stator flux vector, ψs , and the torque produced by the motor, Tem , can be estimated using (9) and (10) respectively. These only require knowledge of the previously applied voltage vector, measured stator current, and stator resistance.
)
ψ s = V s − r s I s dt
Torque developed within the machine can also be expressed as a cross product of stator flux and current as following:
Te =
=
is the stator current
ωr is the rotor speed in electrical radian/sec.
ª fd º 2 « fq » = ¬ ¼ 3
Lo 2 ψr L sL r
(
3 P ψ s × Is 2 2
(9)
)
(10)
Once the current stator flux magnitude and output torque are known, the change required in order to reach the demand values by the end of the current switching period can be determined. An equivalent circuit of the induction motor in a stationary d-q reference frame is shown in Fig. 1. over a short time period, the change in torque is related to the change in current and from the equivalent circuit equation (11) can be obtained. The voltage E can also be determined by using the stator flux and current vectors.
Δ Is =
(7)
1091
V − E Δt L ′s
(11)
Fig. 1. Equivalent circuit of an induction motor in a d-q reference frame. Fig. 3. Switching voltage space vectors.
By combining (10) and (11), an expression for the change in torque can be obtained as shown in (12). Equation (9) can also be rewritten as an expression for the change in the stator flux, as shown in (13).
( )
(
3 P Δt ψs × V − E 2 2 L ′s Δψs = V s − rs I s Δt = V ⋅ Δt ΔTem =
(
))
Table 1: Optimal vector selection table (2 level)
Δλ
(12)
↑
(13)
These two equations can be solved to find the smallest voltage vector, V , required to drive both the torque and flux to the demand values. The required stator voltage can be calculated by adding on the voltage drop across the stator resistance calculated using the current measured from the last cycle. As shown in Fig. 2, the voltage required to drive the error in the torque and flux to zero is calculated directly. The calculated voltage is then synthesized using Space Vector Modulation [4]. If the inverter is not capable of generating the required voltage then the voltage vector which will drive the torque and flux towards the demand value is chosen and held for the complete cycle.
↓
ΔTem ↑ 0 ↓ ↑ 0 ↓
Sectors 1 V2 V0 V6 V3 V7 V5
2 V3 V7 V1 V4 V0 V6
3 V4 V0 V2 V5 V7 V1
4 V5 V7 V3 V6 V0 V2
5 V6 V0 V4 V1 V7 V3
6 V1 V7 V5 V2 V0 V4
6. Discussion of Result Direct torque control for induction motor and a power converter suitable for such application such as induction motor control are presented in this paper. Need for direct torque control and principle of DTC is explained in great details. For complete control of motor [6], software program is done in Matlab 7.0.
Fig. 2. DTC using space vector modulation block diagram.
Fig. 4. Plot of direct axis flux and quadrature axis flux.
5. Space Vector Concept In a three phase inverter, it is well known that the three phase inverter can produce eight output states. Switching state [1 0 0] means, upper switch in phase ‘a’ is closed and upper switch in phase ‘b’ and ‘c’ are open. Thus eight output states of inverter represents eight space vectors, two vectors V0[0 0 0] and V7[1 1 1] are null and remaining six are of equal magnitude and arranged 600 apart in space diagram as shown in fig(3). The table 1 shows the optimum voltage switching vector look-up table.
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Fig. 5. Plot of flux magnitude with respect to time.
Finally from the results, the control variable, the stator flux and the torque are controlled directly with the help of two hysterisis controllers. These control variables are estimated from the stator quantities.
7. Conclusions Direct torque control system for a three-phase induction motor is described; the motor is operated continuously from zero to full speed. It has been proved that the instantaneous primary flux can be calculated by means of computer software and therefore, the data acquisition is simplified significantly. It is shown that by selecting a space non-zero voltage vector and its time width, the amplitude of electromagnetic torque also can be controlled and that by selecting a space zero voltage vector and its time width, the amplitude of electromagnetic torque also can be controlled. Therefore, flux and torque control can be achieved separately.
Fig. 6. Plot of torque with respect to time.
Acknowledgements The authors would like to express their appreciation to Dr. S. G. Tarnekar, Professor and Head, Electrical Engineering Department, Dr. B. N. Jajoo, Principal, G. H. Raisoni college of Engineering, Nagpur University, Nagpur, Maharashtra (India) and greatful to Dr. M. V. Aware, Assistant Professor in Electrical Engineering Department, Visvesvaraya National Institute of Technology, Nagpur (Deemed University).
Fig. 7. Plot of d-q axis current with respect to time.
References
Fig. 8. Plot of direct axis voltage with respect to time.
The result obtained and all these results i.e. Fig. (4). Plot of direct axis flux with respect to quadrature axis flux; Fig. (5). Plot of flux magnitude with respect to time; Fig. (6). Plot of torque with respect to time; Fig. (7). Plot of d-q axis current with respect to time and Fig. (8). Plot of direct axis voltage with respect to time, shows noticeable reduction in torque ripples and suitable only for low to medium power applications due to its higher switching frequency. From Fig. (6) it is observed that the torque in the DTC schemes is not as steady as expected, the reason for this is the use of imperfect current sensors. The current waveforms are too glitchy. From Fig. (4) we find that the DTC scheme offers a much more circular path.
[1] W. Leonhard, “Field-orientation for controlling acmachines – principle and application: A tutorial,” 3rd Int. Conf. on Power Electronics and Variable Speed Drives, pp.227-282, July 1988. [2] I. Takahashi and T. Noguchi, “A new quick response and high efficiency control strategy of an induction motor,” Conf. Rec. IEEE-IAS Annual Meeting, pp.496-502, Oct. 1985. [3] I. Takahashi and Y. Ohmori, “High-performance direct torque control of an induction motor,” IEEE Trans. Industry Applications, vol.25, no.2, pp.257-264, Mar. /April 1989. [4] T. G. Habetler and D. M. Divan, “Control strategies for direct torque control using discrete pulse modulation,” Conf. Rec. IEEE-IAS, pp.514 -522, 1989. [5] Thomas G. Habetler, “Direct torque control for induction machines using vector modulation,” IEEE Trans. Industry Applications, vol.28, no.5, Sep. /Oct. 1992, pp.1045-1054. [6] Marian P. Kazmierkocoski and Anderzei B. Kasprowicz, “Improved direct torque and flux vector control of PWM inverter-fed induction motor drives,” IEEE Trans. Industrial Electronics, vol.42, No.4, Aug.1991, pp.344349.
1093
Sandeep K. Mude Lecturer, Electrical Department, KDKCE, Nagpur (India). email: [email protected]
Abstract This Paper describes sinusoidal pulse width modulation of the voltage impressed to the stator of an induction motor for direct control of torque and stator flux. Instantaneous voltage vectors applied by an inverter have redundancy characteristics which provide some flexibility for selecting the inverter switching modes. By Using this switching freedom, control is achieved according to the following properties; high speed torque control, regulation of the primary flux, minimization of the inverter switching frequency. This utilizes a constant hysteresis band for both developed torque and stator flux and indirectly maintains the switched stator voltage waveforms averaged over a switching period to sinusoidal as in SPWM technique. It improves the dynamic performance of the machine compared to the conventional speed control of induction motor drives. A simulation programme has been developed to verify the results. The inverter duty cycle can then be calculated using the space vector PWM technique. The proposed method is very promising for rapid torque control which is quite different from FOC (Field Orientation Control). Keywords: Space Vector, Torque Control, Estimators.
1. Introduction High dynamic performance of induction motor drives is indispensable in many applications of today’s automatically controlled machines. Induction motor control has attracted much attention recently in the power electronics field. Fieldoriented control has been developed, enabling an ac motor to attain dynamic responses as rapid as for dc motor [2]. The principle of field-oriented control is based on Fleming’s law, which describes the interaction force between fluxes and currents. Many papers have reported the problems associated with compensating various parameters. The current-controlled inverters typically used in the field-oriented drive system develop output waveforms which do not compare favorably with those of the voltage-controlled inverter. The current controlled inverter often causes increased motor harmonic losses and acoustic noise during steady-state operation [5]. This paper proposes new control schemes based on the principle of direct torque control, which can be considered a basic law of torque generation in the induction motor. It makes
1-4244-0038-4 2006 IEEE CCECE/CCGEI, Ottawa, May 2006
Prakash G. Gabhane M.Tech.(IPS), Student, GHRCE, Nagpur (India). email: [email protected]
possible both fast torque response and high-efficiency control at the same time.
2. Need for Direct Torque & Flux Control Inverter fed induction ,motors are increasingly being used in general applications varying the input voltage to the motor with frequency on open loop is one of the very simple and popular methods of speed control. In this method V/F is held constant. In steady state operation, the machine air gap flux is approximately related to V/F. As the frequency approaches zero near zero speed, the magnitude of the stator voltage also tends to zero and this low voltage is absorbed by the stator resistance. Therefore at low speed of operation the stator resistance drop is compensated by injecting an auxiliary voltage so that rated air gap flux and full load torque becomes available up to zero speed. At steady state operation, if the load torque is increased, the slip will increase within the stability limit and a balance will be maintained between the developed torque and the load torque. However, if the supply voltage to the inverter (which is obtained by rectifying the input AC supply) fluctuates, the air gap flux will vary. Also, increase in the stator resistance with temperature results in the variation of air gap flux. Hence, in constant V/F control scheme the air gap flux may drift and as a result the torque sensitivity with slip frequency (or stator current) will vary. If the correct V/F ratio is not maintained the flux may be weak (or may saturate). If the air gap flux decreases, slip frequency (ωs1) will increase for the same torque demand. Also response of the machine detoriates, hence a speed control scheme with independent control of torque and flux loop is desirable. DTC is one such method of speed control.
3. Machine Model Induction motor is modeled by its voltage equations in stator co-ordinates for both stator and rotor as follows:
1090
Vs 0
= =
dψ s + Rs ⋅ is dt dψ r − jω ⋅ ψ r dt
(1) +
Rr ⋅ ir
(2)
where,
σ = 1−
where,
ªV d º Vs = « » ¬V q ¼ ªψ sd º ψs = « » ¬ψ sq ¼ ª i sd º is = « » ¬ i sq ¼ ªψ rd º ψr = « » ¬ψ rq ¼ ª i rd º ir = « » ¬ i rq ¼
The torque expression can also be written in terms of these two flux signals as following:
is the stator voltage
is the stator flux linkage
Te
3 P Lo ψr × ψs 2 2 σLsLr =
is the rotor flux linkage
3 P Lo ψr ψs sin δ 2 2 σLsLr
(8)
where, į is the space angle between these two flux vectors.
4. Direct Torque Control is the rotor current
Rs is the stator resistance Rr is the rotor resistance Equation (1) is for stator voltage and (2) for rotor voltage. As the rotor winding is short circuited, the rotor voltage is zero in magnitude [3]. The induced emf is compensated by the resistive drop and the rotational voltage. Again, stator and rotor flux linkages are related to the stator and rotor currents as follows:
ψ s = L sis + L oir ψ r = L o is + L rir
(3) (4)
Ls is the stator self inductance Lr is the rotor self inductance Lo is the mutual inductance
These two phase variables are obtained from three phase values using the following matrix:
ª fa º ª1 − 1 / 2 − 1 / 2 º « » fb «0 3 / 2 − 3 / 2»¼ « » ¬ «¬ fc »¼
3 P ψ s × is 2 2
Lo ψr Lr
(
T em =
(5)
(6)
Considering stator flux vector and rotor flux vector to be independent state variable, equation (3) and (4) can be modified as following:
σ L sis = ψ s −
Induction motor torque control has traditionally been achieved using Field Oriented Control (FOC). This involves the transformation of stator currents into asynchronously rotating d-q reference frame that is typically aligned to the rotor flux. In this reference frame, the torque and flux producing components of the stator current are decoupled. A PI controller is then used to regulate the output voltage to achieve the reputed stator current and therefore torque. This PI controller limits the transient response of the torque controller. Direct Torque Control (DTC) uses an induction motor model to achieve a desired output torque. By using only current and voltage measurements, it is possible to estimate the instantaneous stator flux and output torque [1]. An induction motor model is then used to predict the voltage required to drive the flux and torque to demanded values within a fixed time period. This calculated voltage is then synthesized using space vector modulation (SVM). The stator flux vector, ψs , and the torque produced by the motor, Tem , can be estimated using (9) and (10) respectively. These only require knowledge of the previously applied voltage vector, measured stator current, and stator resistance.
)
ψ s = V s − r s I s dt
Torque developed within the machine can also be expressed as a cross product of stator flux and current as following:
Te =
=
is the stator current
ωr is the rotor speed in electrical radian/sec.
ª fd º 2 « fq » = ¬ ¼ 3
Lo 2 ψr L sL r
(
3 P ψ s × Is 2 2
(9)
)
(10)
Once the current stator flux magnitude and output torque are known, the change required in order to reach the demand values by the end of the current switching period can be determined. An equivalent circuit of the induction motor in a stationary d-q reference frame is shown in Fig. 1. over a short time period, the change in torque is related to the change in current and from the equivalent circuit equation (11) can be obtained. The voltage E can also be determined by using the stator flux and current vectors.
Δ Is =
(7)
1091
V − E Δt L ′s
(11)
Fig. 1. Equivalent circuit of an induction motor in a d-q reference frame. Fig. 3. Switching voltage space vectors.
By combining (10) and (11), an expression for the change in torque can be obtained as shown in (12). Equation (9) can also be rewritten as an expression for the change in the stator flux, as shown in (13).
( )
(
3 P Δt ψs × V − E 2 2 L ′s Δψs = V s − rs I s Δt = V ⋅ Δt ΔTem =
(
))
Table 1: Optimal vector selection table (2 level)
Δλ
(12)
↑
(13)
These two equations can be solved to find the smallest voltage vector, V , required to drive both the torque and flux to the demand values. The required stator voltage can be calculated by adding on the voltage drop across the stator resistance calculated using the current measured from the last cycle. As shown in Fig. 2, the voltage required to drive the error in the torque and flux to zero is calculated directly. The calculated voltage is then synthesized using Space Vector Modulation [4]. If the inverter is not capable of generating the required voltage then the voltage vector which will drive the torque and flux towards the demand value is chosen and held for the complete cycle.
↓
ΔTem ↑ 0 ↓ ↑ 0 ↓
Sectors 1 V2 V0 V6 V3 V7 V5
2 V3 V7 V1 V4 V0 V6
3 V4 V0 V2 V5 V7 V1
4 V5 V7 V3 V6 V0 V2
5 V6 V0 V4 V1 V7 V3
6 V1 V7 V5 V2 V0 V4
6. Discussion of Result Direct torque control for induction motor and a power converter suitable for such application such as induction motor control are presented in this paper. Need for direct torque control and principle of DTC is explained in great details. For complete control of motor [6], software program is done in Matlab 7.0.
Fig. 2. DTC using space vector modulation block diagram.
Fig. 4. Plot of direct axis flux and quadrature axis flux.
5. Space Vector Concept In a three phase inverter, it is well known that the three phase inverter can produce eight output states. Switching state [1 0 0] means, upper switch in phase ‘a’ is closed and upper switch in phase ‘b’ and ‘c’ are open. Thus eight output states of inverter represents eight space vectors, two vectors V0[0 0 0] and V7[1 1 1] are null and remaining six are of equal magnitude and arranged 600 apart in space diagram as shown in fig(3). The table 1 shows the optimum voltage switching vector look-up table.
1092
Fig. 5. Plot of flux magnitude with respect to time.
Finally from the results, the control variable, the stator flux and the torque are controlled directly with the help of two hysterisis controllers. These control variables are estimated from the stator quantities.
7. Conclusions Direct torque control system for a three-phase induction motor is described; the motor is operated continuously from zero to full speed. It has been proved that the instantaneous primary flux can be calculated by means of computer software and therefore, the data acquisition is simplified significantly. It is shown that by selecting a space non-zero voltage vector and its time width, the amplitude of electromagnetic torque also can be controlled and that by selecting a space zero voltage vector and its time width, the amplitude of electromagnetic torque also can be controlled. Therefore, flux and torque control can be achieved separately.
Fig. 6. Plot of torque with respect to time.
Acknowledgements The authors would like to express their appreciation to Dr. S. G. Tarnekar, Professor and Head, Electrical Engineering Department, Dr. B. N. Jajoo, Principal, G. H. Raisoni college of Engineering, Nagpur University, Nagpur, Maharashtra (India) and greatful to Dr. M. V. Aware, Assistant Professor in Electrical Engineering Department, Visvesvaraya National Institute of Technology, Nagpur (Deemed University).
Fig. 7. Plot of d-q axis current with respect to time.
References
Fig. 8. Plot of direct axis voltage with respect to time.
The result obtained and all these results i.e. Fig. (4). Plot of direct axis flux with respect to quadrature axis flux; Fig. (5). Plot of flux magnitude with respect to time; Fig. (6). Plot of torque with respect to time; Fig. (7). Plot of d-q axis current with respect to time and Fig. (8). Plot of direct axis voltage with respect to time, shows noticeable reduction in torque ripples and suitable only for low to medium power applications due to its higher switching frequency. From Fig. (6) it is observed that the torque in the DTC schemes is not as steady as expected, the reason for this is the use of imperfect current sensors. The current waveforms are too glitchy. From Fig. (4) we find that the DTC scheme offers a much more circular path.
[1] W. Leonhard, “Field-orientation for controlling acmachines – principle and application: A tutorial,” 3rd Int. Conf. on Power Electronics and Variable Speed Drives, pp.227-282, July 1988. [2] I. Takahashi and T. Noguchi, “A new quick response and high efficiency control strategy of an induction motor,” Conf. Rec. IEEE-IAS Annual Meeting, pp.496-502, Oct. 1985. [3] I. Takahashi and Y. Ohmori, “High-performance direct torque control of an induction motor,” IEEE Trans. Industry Applications, vol.25, no.2, pp.257-264, Mar. /April 1989. [4] T. G. Habetler and D. M. Divan, “Control strategies for direct torque control using discrete pulse modulation,” Conf. Rec. IEEE-IAS, pp.514 -522, 1989. [5] Thomas G. Habetler, “Direct torque control for induction machines using vector modulation,” IEEE Trans. Industry Applications, vol.28, no.5, Sep. /Oct. 1992, pp.1045-1054. [6] Marian P. Kazmierkocoski and Anderzei B. Kasprowicz, “Improved direct torque and flux vector control of PWM inverter-fed induction motor drives,” IEEE Trans. Industrial Electronics, vol.42, No.4, Aug.1991, pp.344349.
1093
