A Study on Speed Control Method of Propulsion Motor for Small Ships with AFE Converter

1. Direct torque control[DTC]

The values that calculate torque and stator magnetic flux by measuring voltage and current in induction motor and compare torque and magnetic flux command every cycle and the errors of torque and magnetic flux determined by the control accuracy of torque and magnetic flux respectively are inputs to the Hysteresis controller. The output of the hysteresis controller is used as the input of the lookup table of the voltage vector, and the desired voltage vector is found here and out-putted.

Direct torque control controls torque and magnetic flux independently, so you can get fast response with accuracy.

The stator flux can be expressed by Eq.(1) by the voltage equation of the induction motor on the three-phase stationary coordinate system.

The value of stator flux in Eq.(1) is obtained as Eq.(2).

Since the stator resistance is very small in Eq.(2), assuming that the voltage drop R_si_s is very small compared to the first term and neglected, the stator flux is proportional to the magnitude of the stator voltage and the direction can be thought of as the same vector.

The relationship is expressed by the Eq.(3).

Here, Δt is the sampling period.

That is, the magnetic flux of the stator moves in the same direction as the stator voltage vector $$ \overrightarrow{V_s}$$. If $$ \overrightarrow{V_s}$$ is an effective vector, λ_s moves at a constant rate with the output voltage vector of the inverter, and at a very small speed if it is an ineffective vector. Therefore, the rotational speed of the magnetic flux is controlled by the ratio of the effective and the ineffective vector of the inverter output voltage.

<Table 1> shows the voltage vectors selected when the stator flux is in the North area. nth region. The vector of the magnetic flux has the direction of selecting the magnetic flux and torque of the motor according to the eight voltage vectors that the inverter can output and the output voltage vector of the inverter selected at the current position. The switching vector is selected considering the direction as well as the magnitude of the magnetic flux.

<Table 1>

Selected vector and means.

[Fig. 1] shows the concept of direct torque control, showing the switching vector and the flux vector. The moving direction of the magnetic flux rotated by the selection of the switching vector in the hysteresis band is shown. The magnetic flux rotates, creating the trajectory of a pulsating circle.

[Fig. 1]

Conception of direct torque control.

Equation(4) relates to the hysteresis band of the magnetic flux.

The choice of flux vector is also related to the magnitude and direction of rotation within the hysteresis width.

First, if the magnetic flux rotates in the clockwise direction, $$ \overrightarrow{V_6}$$ is selected when the magnetic flux is in contact with the lower line of the hysteresis width, and $$ \overrightarrow{V_5}$$ is selected when the magnetic flux is in contact with the upper line of the hysteresis band.

If you are rotating counterclockwise, select $$ \overrightarrow{V_2}$$ and $$ \overrightarrow{V_3}$$. Once the magnetic flux is determined, the torque is determined by equation(5).

When the value of torque T_e reaches the torque command value $$ T_e^{\mathrm{*}}$$, T_e is preferably reduced as late as possible to reduce the switching frequency of the inverter.

It is better to use the ineffective vector among the switching vectors of the inverter, and the torque is divided into two cases according to the rotation direction.

$$ T_e^{\mathrm{*}}-\Delta T_e\le T_e\le T_e^{\mathrm{*}}$$ when λ_s rotates clockwise

$$ T_e^{\mathrm{*}}\le T_e\le T_e^{\mathrm{*}}+\Delta T_e$$ when λ_s rotates counterclockwise

Assuming that λ_s rotates clockwise, an ineffective vector is selected to stop λ_s and reduce T_e when T_s reaches $$ T_e^{\mathrm{*}}$$.

On the contrary, when T_e approaches $$ T_e^{\mathrm{*}}-\Delta T_e$$, an effective vector is selected to rotate the magnetic flux clockwise.

To calculate the optimal voltage vector in the optimal switching voltage vector lookup table, we need to know the stator flux position. This position can be obtained from equations(6a), (6b) and (6c) from the α-axis and β-axis values of the stator linkage flux in the stationary reference frame.

Here v_αs, v_βs, i_αs, i_βs are the measured stator currents and voltages.

Therefore, the torque is as shown in equation(7).

[Fig. 2(a)] shows the flux hysteresis controller. Since the stator flux always has a (+) value, the comparator output consists of two stages (1, 0). That is, Δλ_s = 1 when the stator flux error is larger than the positive hysteresis width H_F , and Δλ_s = 0 when it is smaller than the negative hysteresis width –H_F .

[Fig. 2]

Characteristics of torque and flux hysteresis comparator.

[Fig. 2(b)] shows the torque component hysteresis comparator used in the direct torque control inverter. In general, unlike the magnetic flux component hysteresis comparator, the output ΔT_e is thre e steps (-1.0.1) so that the torque command in both the forward and the reverse direction can be followed.

<Table 2> shows the optimal switching voltage vector lookup table based on the hysteresis comparator characteristics of torque and magnetic flux.

<Table 2>

Optimum switching voltage vector look-up table

2. Direct Vector Control

This method, called magnetic flux feedback control, directly detects the air gap magnetic flux close a feedback loop on this signal by a Hall sensor or a sensing coil, or uses a magnetic flux model composed of motor terminal voltage, current and speed after that, magnitude and position of the rotor magnetic flux vector is obtained. In the direct measurement method, it is difficult to install mechanically because the Hall element or the detection coil must be installed in the stator slot, and the design of the filter is complicated because the harmonics generated by the slot vary with the motor speed. Therefore, it is common to use the magnetic flux vector estimated indirectly from the motor constant, voltage, current, speed, and the like.

3. Indirect Vector Control

This method, called slip frequency or feed-forward control, calculates slip command angular velocity using the magnetic flux current, torque current and motor constant on the synchronous rotational reference frame, and adds the rotor speed to the integral value, after that estimates and controls the field angle.

It is a kind of predictive control. This method is sensitive to changes in motor parameters and requires encoders and resolvers for rotor speed information, but it is more widely applied because no flux sensor or model is required. To illustrate this control method, a vector diagram of the stator current vector decomposed into the current components of each reference axis is shown in [Fig. 3].

[Fig. 3]

Vector diagram for indirect vector control.

The α - β axis is fixed to the stator, the d - q axis rotates at the synchronous angular velocity ω_e, and the rotor flux axis coinciding with the d axis rotates while maintaining a slip angle(θ_sl) with respect to the rotor axis(Refer to <Fig. 1>). Therefore, it can be seen that the magnetic flux current i_ds and the torque current i_qs of the stator separately control the rotor flux and the torque. In addition, the desired magnetic flux angle θ_e can be obtained by adding the rotational magnetic angle θ_r and θ_sl obtained from the magnetic flux setting value, it used for the coordinate transformation.

In the case of a squirrel cage induction motor, when the voltage equation of the rotor is converted into a synchronous rotational coordinate system, equations(14a) and (14b) are obtained.

Here, ω_sl ( = ω_e - ω_r) is the slip angular velocity.

And, the d - q axis rotor current component obtained from the stator and rotor linkage fluxes is expressed by equations(15a) and (15b).

Substituting equations(15a) and (15b) into equations(14a) and (14b) and eliminating the rotor current values results in equations(16a) and (16b).

When perfect vector control is performed in induction motor control, the q-axis component of the rotor linkage flux becomes zero and the d-axis component is also constant, so the torque is controlled only by the q-axis current component. Therefore, the following equation holds.

Substituting equation(17) into equations(16a) and (16b) yields equations(18) and (19).

In addition, in the case of constant flux control, dλ_dr / dt = 0, so that equation(20b) is obtained from equation(14b).

Substituting the above equation into equations(18), (19) and (14), the following magnetic flux, slip and torque equations can be obtained.

Therefore, Eqs.(20) to (23) are the motor models when the indirect vector control theory is applied to the induction motor and the magnetic flux angle is the following equation which is an integral value obtained by adding the motor angular velocity and the slip command angular velocity of Equation(23).

So far, important vector control relations such as equations(20) to (23) have been derived from the rotor voltage and the rotor linkage flux equation of the vector control of induction motors. But, the stator voltage equation of the vector control induction motor must be known to construct a current controller that determines the performance of the actual vector control system.

This can be obtained by substituting the vector control relation equations (20a) and (20b) into the stator voltage equation and stator linkage flux equation, and the result is as follows.

[Fig. 4] shows the induction motor model applying the indirect vector control theory based on the above results.

[Fig. 4]

The block diagram of indirect vector controlled induction motor

4. Induction Motor Control Using AFE Rectifier

4.1 AFE Rectifier

In small coastal ships such as cruise ships and small working ships, it is difficult to install a DFE rectifier that requires the installation of passive filters and phase shifting transformers to alleviate harmonics due to space constraints. Therefore, AFE rectifier, which is a kind of active filter, can be applied as a good technology(Jeon et al., 2017).

[Fig. 5] shows the AFE rectifier consisting of rectifier stage by connecting diode and IGBT in anti-parallel.

[Fig. 5]

Circuit diagram of AFE rectifier.

Each IGBT element in the rectifier stage is switched via PWM control. In the DC stage, a small capacitor of 1/50 size was installed in place of the large capacitor, which is a disadvantage of the DFE method.(Hur et al., 2018).

4.2 Induction Motor Inverter

[Fig. 6] shows a typical induction motor speed control topology. Each IGBT element in the inverter stage is switched by PWM using the hysteresis current control equally for three speed control methods.

[Fig. 6]

Circuit diagram of conventional converter.

1. Simulation

In this paper, computer simulation is performed using PSIM program in high speed and low speed to verify the characteristics of three speed control methods of induction motor using AFE rectifier. Each speed control method is compared by switching to 100 [μs], which is the same as the inverter applied to small ships.

In the low speed region 300[rpm], step load torque was applied, and in the high speed region 1500[rpm], the load was proportional to the speed square. <Table 3> shows the parameters and system constants of the induction motor used in the simulation.

<Table 3>

Parameters of induction motor and system constants used for computer simulation.

[Fig. 7] shows the PSIM block diagram of the induction motor drive simulation. Hysteresis controller is used to control current.

[Fig. 7]

PSIM diagram of converter system.

2. Induction Motor Simulation with Direct Torque Control

[Fig. 8] shows the speed response characteristics of the induction motor when the speed command is applied from 0[rpm] to 1,500[rpm]. (a) shows the change of motor speed and fine oscillation occurs compared to the vector control method. (b) shows the load change proportional to the speed square, and is similar to the conventional method in the transient state. (c) shows the phase current of the input stage and shows the sine wave similar to the vector control method.

[Fig. 8]

Simulation responses for step change of speed setting (0→1,500[rpm]).

THD is 6.59[%] and slightly higher than vector control. (d) is the motor maximum load current, and the maximum value is shown around 19[A] under the steady state.

[Fig. 9] shows the response characteristics of induction motor when load torque of 20[N·m] is applied during steady state operation at 300[rpm]. (a) shows the change of motor speed and oscillation is bigger than at high speed.

[Fig. 9]

Simulation responses for step change of load torque (300[rpm], 20[N·m]).

(b) shows the torque change before and after applying the load torque. (c) shows the change of shape of input phase current before and after load torque application, and the shape of input stage current is similar to the vector control method after load is stabilized. In addition, the change in current size due to the application of load torque is smaller than the conventional method. (d) shows the motor load current before and after applying the load torque, and after the load is stabilized, the maximum value flows around 24[A].

3. Induction Motor Simulation with Direct Vector Control

[Fig. 10] shows the speed response characteristics of the induction motor when the speed command is applied from 0[rpm] to 1,500[rpm]. (a) shows a change in motor speed, and (b) shows a load change proportional to the speed square.

[Fig. 10]

Simulation responses for step change of speed setting (0→1,500[rpm]).

(c) is a phase current waveform of the input terminal. THD contains 5.14[%], indicating that a good quality current is supplied. (d) shows the motor-side load current, and the maximum current is shown around 19[A] under the steady state.

[Fig. 11] shows the response characteristics when load torque of 20[N·m] is applied during steady state operation at 300[rpm] at one second after starting.

[Fig. 11]

Simulation responses for step change of load torque (300[rpm], 20[N·m]).

(a) shows the motor speed change, and (b) shows the torque change before and after applying the load torque. (c) shows the change of the phase current shape of the input stage before and after load torque application, (d) shows the motor load current before and after load torque application, and the maximum value is shown around 17[A] in the normal state.

4. Induction Motor Simulation with Indirect Vector Control

[Fig. 12] shows the speed response characteristics of the induction motor when the speed command is applied from 0[rpm] to 1,500[rpm]. (a) shows the change of motor speed and stable control result similar to the direct vector control method. (b) shows the load change proportional to the speed square and the wide load range similar to the direct vector control method in the transient state. (c) shows the input phase current and shows the same shape as the direct vector control method. The THD contains 4.97[%]. (d) shows the motor load current, and the maximum value is around 19[A] under the steady state.

[Fig. 12]

Simulation responses for step change of speed setting (0→1,500[rpm]).

[Fig. 13] shows the response when the load torque of 20[N ·m] is applied during steady state operation at 300[rpm].

[Fig. 13]

Simulation responses for step change of load torque (300[rpm], 20[N·m]).

(a) shows the motor speed change and stable control results under transient conditions similar to the direct vector control method. (b) shows the torque change before and after applying the load torque. (c) shows the change of shape of input phase current before and after applying load torque, and after the load is stabilized, the input current is closer to sine wave than direct vector control. In addition, the change of the current magnitude due to the load torque is larger than the direct vector control method. (d) shows the motor load current before and after applying the load torque, and the maximum value is about 17[A] after the load is stabilized. The results show the effectiveness of the control of system in a steady state as well as in transient stability.

5. Review simulation results

When comparing Fig. 8(a), Fig. 10(a), Fig. 12(a) and Fig. 8(b), Fig. 10(b), Fig. 12(b), respectively, speed command is applied from 0[rpm] to 1,500[rpm], which is a high speed range, it can be seen that stable control is achieved in the direct vector control and indirect vector control methods in the transient area. And in the direct torque control, fine oscillation occurs, but it is confirmed that it is relatively stable.

When comparing Fig. 8(c), Fig. 10(c) and Fig. 12(c), the THD of the direct torque control method is 6.59[%] and higher than THD of the direct vector control method 5.14[%], and the indirect vector control method THD 4.97[%] shows a current waveform close to the sine wave, indicating that the input current is very good quality.

When comparing Fig. 8(d), Fig. 10(d), and Fig. 12(d), the maximum value of the load current in the steady state section is not significantly different before and after 19[A] similarly, in the transient state, it was confirmed that the current is stably controlled in all three methods.

When comparing Fig. 9(a), Fig. 11(a), Fig. 13(a) and Fig. 9(b), Fig. 11(b) and Fig. 13(b), the speed command is given from 0[rpm] to 300[rpm], the low speed range, the load is applied, it can be seen that fine oscillation occurs in the direct vector control method and continuous oscillation occurs in the direct torque control method as compared to the indirect vector control method which is stably controlled.

When comparing Fig. 9(c), Fig. 11(c), and Fig. 13(c), the waveform shows a near sine wave in the steady state section after the load is applied, indicating that the quality of the input current is excellent, and it is confirmed that the magnitude of the current change is large in the indirect vector control method.

When comparing Fig. 9(c), Fig. 11(c), and Fig. 13(c), after applying the load torque, it can be seen that the average values of the load current between the conventional method and the proposed method are about 17[A] in the steady state section, we can see that the difference is not big, and even during transient states, the proposed method is similar to the existing method and it was confirmed that stable control is possible.

As a result of comparing the switching frequency and gain value similar to the actual operation of electric propulsion system for small crafts, the indirect vector control method is superior in input current quality and speed control performance of induction motor for propulsion of small electric propulsion ship comparing with the direct torque control method used in large ships.