Abstract
Five-phase induction machines (5pIMs) are a viable alternative for a wide range of industrial applications. The penetration of drives with 5pIM into industrial applications is slow because of the complexity of the proposed control algorithms. In standard control of induction machines, the stator current is not controlled directly and options for current limitation are restricted. This paper hence addresses a novel current limitation technique for the control algorithm of 5pIM. In addition, a simple speed sensorless control algorithm based on the slip compensation is introduced. A combination of both algorithms is suitable for low-cost medium-precision drive applications. Experimental results, realized on Texas Instruments F28335 DSP, illustrating the performance of novel current limitation technique and sensorless control, are included.
1. Introduction
Five-phase induction machines (5pIMs) have some advantages and drawbacks over three-phase induction machines (3pIMs). One of the biggest advantages is lower current per phase for the same phase voltage and power rating. This results in less stress of semiconductor components, which is favorable for high-voltage machines [1]. Further advantages are lower torque pulsations, less harmonic content, and higher reliability. Because of the higher number of phases, the biggest drawbacks of 5pIM are higher switching losses and more complex control algorithms compared to conventional 3pIM. The application class for 5pIM contains electric and hybrid cars, aircraft and spaceship applications, marine propulsion systems, and high-power traction drives [2, 3]. 5pIMs are especially suitable for safety-critical applications requiring a high degree of redundancy. The degrees of freedom available in 5pIM which do not exist in 3pIM have opened up many new research areas, such as different stator winding connection issues [4] or wound-rotor direct drives [5]. The possibility of redundancy in 5pIM has resulted in improvement of fault-tolerant capabilities, such as open switch fault detection [6, 7], open-phase fault detection and post-fault operation [8, 9], or rotor-bar fault analysis [10].
For many applications where the simplicity of control and total drive cost is an aim, control is preferable [11]. It requires a smaller number of PI controllers than field-oriented control; it employs very simple control structures that result in lower demands on microprocessor resources. It does not require precise values of machine parameters [12]. For safety-critical drives with 5pIM requiring some level of redundancy, control may be used in the case of failure. A switchover of control structure from vector control to simple open-loop control is beneficial when there is a faulty state of the drive or speed sensor. Several control algorithms exist for control, with differences in the type of converter used, the type of variables controlled, and the open-loop vs. closed-loop type of control algorithm. A -based algorithm that generates a trapezoidal induction waveform in the air gap was proposed in [13] where open-loop control is used to impose a fundamental and third-harmonic component to the stator phases to increase the power capability of the machine. Injection of the 5th order harmonic in the sinusoidal references for the 5p-VSI to increase the fundamental output voltage of sinusoidal PWM has been investigated in [14]. It was shown that the linear modulation range was increased and analogy with three-phase sinusoidal PWM with third-harmonic injection was evidenced. Sensorless control with MRAS speed estimator was proposed in [15]. By employing proportional-resonant controllers, disturbance-free operation for open-circuit faults was achieved. The algorithm of 5pIM [15] was compared with a sensorless field-oriented control algorithm, and it provided the results in [16]. It was found that both algorithms deal with an open-phase fault with similar performance. However, field-oriented control is better for dynamic and steady-state performance, but the precise determination of machine parameters highly affected its performance.
The research question barely discussed in the literature is the limitation of drive current in control. Limitation of maximum drive current is an important but often overlooked aspect of control design, and it is unavoidable for industrial implementation. In control, the stator current is not controlled directly and options for current limitation are restricted. Some form of current limiting technique must be implemented in each -controlled drive system to protect the motor and frequency converter against overcurrents. For 3pIM, this problem has been discussed in [17]. The authors proposed a programmable ramp function generator (RFG) and two proportional-integral (PI) controllers. The programmable RFG provides current limiting during transients, and PI controllers limit the stator current in the steady state. The proposed solution has shown a satisfactory performance, but the synthesis of the PI controller has not been included. Another solution was proposed in [18], where an alternative cascade control structure with PID current controller was used for closed-loop control. Industrial drives employ unique structures for stator current limitation in control with one PI controller and programmable RFG [19] or with two PI controllers [20]. A commonly used name for this class of algorithms is controller.
The contribution of this paper is stated as a novel controller introduced for the operation with 5pIM drives. The proposed controller is less complex than those presented for 3pIM but equally effective. Simulation and experimental verification will show its effectiveness for the current limitation. Furthermore, the design and stability analysis of the proposed controller is provided in the paper together with the slip compensation algorithm, which results in a simple sensorless control structure useful for medium-precision 5pIM drives.
2. Theory of Control
Algorithms of control are based on a steady-state model of the machine. A phasor method is usually used for the analysis of steady-state models. If the airgap flux is maintained at a constant value, the stator voltage per phase is expressed as [21]where is the vector of stator voltage, is the vector of stator current, is the stator phase resistance, is the stator leakage reactance, is the electrical speed, and is the vector of air gap flux. If the electrical speed is large enough (i.e., not at low frequencies), the following holds:where is the air gap voltage. Then, the stator phase voltage can be approximated with the air gap voltage:
This means that in the region where equation (3) holds, if the phase voltage is adjusted as follows:we will obtain constant air gap flux. Note that equation (3) is generally applicable for induction machines of several kilowatts and above and for frequencies above 20% of the nominal frequency. Equation (4) says that if we increase the frequency proportionally with the amplitude of the phase voltage, the air gap flux will remain roughly constant. This is a well-known definition of control [22]. If the air gap flux is constant and the load torque is below the value of the rated machine torque, the following equation can be derived for 5pIM:where is the machine torque, is the number of poles, is the slip angular speed, and is the rotor resistance. From equation (5), the torque of the machine is proportional to the squared air gap flux (that will be maintained constant) and proportional to the slip angular speed. If the air gap flux is constant, the torque can be controlled by the control of slip angular speed . The simplest control system for a 5pIM drive is the open-loop configuration shown in Figure 1. The frequency setpoint , calculated from the speed setpoint, enters the function that calculates the stator voltage . The PWM modulator uses the calculated phase voltages to create PWM signals for a five-phase voltage source inverter (5p-VSI) [23]. The frequency converter, which supplies the motor, produces sinusoidal symmetrical voltages. The inverter increases its output voltage to the maximum output voltage, which depends on the voltage of the DC link. When the maximum output voltage is reached, the inverter only increases the output frequency.
The amplitude of voltage is calculated in the function from the frequency using the following relationship:where is the voltage boost that must be designed to cover the voltage drop on the stator resistance at low frequencies. Commonly, a fixed value up to 25% of the rated voltage or a fixed value equivalent to the full load current is used to calculate [24]. A detailed procedure for calculation of and can be found in [25] considering admissible loading levels in a lower frequency range. The value of the voltage boost varies significantly with the change of stator resistance and total leakage inductance. The value of has a significant impact on total performance, and it needs to be calculated precisely. The configuration shown in Figure 1 is simple, and the speed and current sensors are not required, so it is very cost effective. But it has many drawbacks. Steady-state speed deviation is present, even under the no-load condition. The lack of feedback of the phase currents (or DC link current) means a risk of damaging 5p-VSI due to the possible presence of high currents. To overcome these drawbacks, the configuration in Figure 1 must be equipped with auxiliary circuits. A necessary operating condition is that the total drive current must be defined and limited.
3. Novel Current Limitation Method
Fast current protection must be established for the safe operation of the drive. The curve in Figure 1 might be implemented with different characteristics depending on the drive application class and demands for required energy savings [26]. However, for practical implementation of control, a circuit for limitation of drive total current must be always included in the control structure. It must be remembered that in control, a stator current cannot be controlled directly as in the case of vector control [27], where direct and quadrature current components are controlled and limited independently. This approach is not possible for control, and another method must be used. Here, we propose a solution for 5pIM shown as in Figure 2. It consists of three main components: total stator current calculation, voltage-reduction PI controller, and slip compensation.
3.1. Total Stator Current Calculation
The calculation of total drive current for control in industrial drives uses transformations of phase currents (compare [20] or [19]). Here lie the main differences between control of 3pIM and 5pIM. The latter one uses the different transformation of phase current vectors into orthogonal system and, furthermore, employs a different way of transformation angle calculation needed for transformation from to coordinate system.
The total current calculation in Figure 2 employs a measurement of currents in all machine phases. The phase currents are transformed to the stator coordinates using the following transformation matrix [28]:
The Park transformation is used to transform currents to coordinates. The phase transformation angle needed for transformation from to coordinate system is, unlike field-oriented control, obtained from the precalculated frequency as shown in Figure 2. Finally, the RMS value of total machine current is evaluated as
3.2. Design of Voltage-Reduction PI Controller
As the stator current is not directly controlled in control, the only ways to ensure current limitation are to reduce the reference voltage , to reduce the reference frequency , or to reduce both simultaneously. This can be achieved using several approaches [17–20]. In most cases, two PI controllers are used: one for reduction of voltage and the other for reduction of frequency. Other solutions need a special modification to the RFG that enables it to be frozen when the drive current exceeds its limit. However, no discussion about the stability of the current controller is provided in the available literature. Here, we propose a novel and less complex solution using only one PI controller and a feedback corrective term. Its design and stability will be researched as well.
In Figure 2, the measured feedback current given by equation (8) is compared to the maximum admissible current and enters the voltage-reduction PI controller. It should be noted that the PI controller is employed here for reduction of the stator current by decreasing the stator voltage, i.e., PI controller is not used for the speed control. The output of the PI controller is voltage that decreases the voltage magnitude that comes out from the function. An inverse function is calculated with the aim of the original function equation (6) as
If the motor current reaches the current limit during acceleration, the proposed controller reduces the output voltage to keep the current below the current limit . If the load on the motor is so high during stationary operation that the motor current reaches the current limit, the controller reduces the voltage and speed setpoint until a stable working point is set and the motor current drops into the permissible range. If the load is reduced, the controller increases the speed setpoint until the setpoint is reached and a new stable working point is reached.
A circuit for the synthesis of voltage reduction for the PI controller is shown in Figure 3. is the transfer function of the voltage-reduction PI controller:where is given as the sum of computational delay and delay of the sample-hold mechanism [29]. It is calculated from the PWM carrier frequency as follows:where is the stator gain, is the electrical time constant of the stator, and is the time constant of the current sensing path which is given aswhere is the stator current sampling period and is the time constant of total stator current as shown in Figure 2.
Usually, holds, and a substitute time constant can be used. Then, the parameters of the PI controller can be calculated using the pole-placement method.
3.3. Synthesis of Voltage-Reduction PI Controller
The closed-loop transfer function of the system in Figure 3 including the PI controller is given as
Parameters of the system given by equation (13) can be found in Table 1. The system has 3 poles and one zero. It can be shown that zero in nominator might be cancelled by precompensation with a low-pass filter with the time constant (precompensation filter in Figure 2). By comparing the compensated system equation (13) with the desired closed-loop transfer function,where is damping, is the natural frequency, and is the arbitrary displacement of the 3rd pole’s real axis. Using the following poles:PI controller values can be found as
The final parameters in equation (16) enable the design of a PI controller based on the desired damping and desired natural frequency. However, the real axis displacement of pole depends on the desired values. An improper combination of desired values may cause a shift of pole to unstable area. A stability analysis is therefore necessary. By using Routh stability criterion, we can show that the PI controller parameters have to meet the following conditions:
The performance of the controller is highly dependent on the value of and the placement of . Decrease of damping coefficient value results in more accurate current limitation. On the other hand, it increases oscillations of . For fixed value of , the natural frequency or settling time must be chosen to ensure is located within .
4. Slip Compensation
The torque and the slip of the machine increase during loading. This is observed from equation (5). Without slip compensation in the control structure, a steady-state speed deviation will occur. For control, this deviation is usually treated using two different approaches: a closed-loop structure with speed measurement, as shown in Figure 4(a), and an open-loop structure, for example, the one given in Figure 4(b). All of these methods are part of the large group of slip compensation methods summarized in [30].
(a)
(b)
The reference speed and the actual speed in Figure 4(a) are compared, and the resulting slip speed enters the PI controller. The output of the PI controller is saturated by the maximum and minimum values of the slip frequency , which is then added to the actual frequency, resulting in output frequency for control (compare with Figure 2). This method can achieve reasonable accuracy, but it has two disadvantages. It is costly due to the necessary speed measurement. This measurement can be replaced by some speed estimation algorithm. Nevertheless, it will still require PI controller tuning and additional loops for speed estimation, which makes it more complex. This approach would not be preferable for simple and low-cost control. The proposed solution, shown in Figure 4(b), is based on the following principle.
If the drive output voltage and frequency are held constant when the motor load is increased, the shaft speed will decrease due to additional required torque, which results in a greater amount of slip. This will result in an increase in current that can be detected by the current transducers. Slip compensation automatically compensates for a decrease in the speed by applying additional output frequency to the existing output frequency. The final output frequency is “shifted” to a higher value, resulting in an increase in actual speed. Note that the amount of slip does not decrease, and the motor shaft still slips back. However, the speed deviation is partially reduced. The following relationship between measured currents and slip frequency was found [30]:
The calculated slip frequency in Figure 4(b) is saturated by the minimum and maximum values of the slip and then multiplied by the calculated nominal slip value in Hz to obtain the relative slip. The relative slip is then multiplied by adjusting the function that turns off the slip compensation for 0 to 6% of the nominal frequency, and the slip compensation is increased linearly up to 10% of nominal frequency. For higher frequencies, the full-slip compensation is applied. Adjusting the function reduces the slip compensation for lower frequencies as instabilities may occur in that region [31]. The final corrective term is then added to the output frequency as seen in Figure 2. This method does not require information about the actual speed nor any tuning procedure, and it is very simple. However, a knowledge of the rotor resistance and inductance is required, but these parameters are often automatically calculated during the initial commissioning of the drive.
5. Simulation Results
5.1. Current Limitation and Slip Compensation
Simulation results for the proposed algorithm are shown in Figures 5 and 6. The simulation was provided for the star-connected model of 5pIM, and machine parameters are given in Table 2. Values V/Hz and V were used for control, based on guidelines given in [25].
(a)
(b)
(c)
(d)
(a)
(b)
(c)
Figure 5 shows a step reference change from zero to the nominal speed 2830 rpm and loading by 50% of nominal load torque at s. The machine was unloaded at . The maximum drive current was set to 2A (approx. 120% of the nominal motor current). The performance of the current limitation is shown in (B) in Figure 5(a), where it can be observed that the current limitation works correctly during the acceleration phase. The performance of the slip compensation is shown in (A) in Figure 5(a), where detailed speed drops for enabled and disabled slip compensation are shown for comparison. The performance of the slip compensator is also shown in Figure 5(c), where the upper figure shows responses of drive frequencies when the slip compensation was enabled. Enabled slip compensation produces a corrective frequency signal fSLIP_CORR that corresponds to the estimated value of actual slip. This signal is added to the reference frequency fREF what causes the shift of fOUT frequency. The frequency value is given as (compare with Figure 2)
The amount of slip does not actually decrease, but it is simply shifted. For disabled compensation in Figure 5(d), the slip correction has zero value, and consequently, there is a constant steady-state speed error even for unloaded drive.
Different current limits are shown in Figure 6. It shows a step reference change from zero to the 50% of nominal speed. Then, two values of load torque were applied: at s and of nominal torque at s. Three different current limits were tested. It can be observed that the current limitation works correctly during the acceleration phase and for higher load torque values. In both cases, the total drive current does not exceed its preset maximum value. For the lower value of load torque, the current limitation does not actuate as it is not necessary. Details of corrective term being active in the current limitation are included in Figure 6 at the bottom. It can be observed that for the higher load torque value, the value of rises up and the actual speed of the drive falls down up to the instant when the drive suddenly becomes unloaded at s. After that, the actual drive speed quickly recovers to its setpoint. It can be concluded that the drive current limitation works correctly under different conditions.
5.2. Analysis of Mismatch in Rotor Resistance
The control structure in Figure 2 depends on the measurement of currents which are affected by the change of parameters. The slip compensation heavily depends on the value of rotor resistance . It is found that is difficult to be accurately measured since is not easily accessible. Furthermore, rotor resistance varies significantly with the motor temperature. The normal temperature change of C, which is expected in induction machines with insulation class B, causes a change in rotor and stator resistances of 34% [32]. This must be investigated because the rotor resistance has a strong link with the speed. Therefore, a mismatch in rotor resistance was tested in this section. The nominal value or resistance has been used for the calculation of slip compensation, and then actual rotor resistance was varied for three different values: , of , and of . The results are shown in Figure 7. Two different step changes of reference speed were executed, followed by the step load torque of of nominal torque in s. It can be observed that with the increase of rotor resistance value, the total drive current is decreasing. The mismatch in the rotor resistance value has the highest impact to the load torque rejection capability of the drive. However, it must be emphasised that the machine is operated only as a sensorless drive. Thus, it is very difficult to compensate actual speed drop without any speed sensor, relying only on current measurements that are strongly affected by mismatched parameters.
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(b)
(c)
6. Experimental Results
Experimental tests were performed for the real prototype of 5pIM with the parameters given in Table 2. The control algorithm to generate real-time executable code was prepared in MATLAB/Simulink and then compiled and loaded to the Texas Instruments TMS320F28335 DSP. The modular model approach [33] for rapid control prototyping with two-level architecture was used, and we can find the implementation details in [23]. The experimental setup comprises the 5pIM with distributed winding, a loading induction machine controlled by an industrial converter, and a prototype of five-phase VSI with IGBT transistors. An interrupt service routine for the algorithm was triggered with , and the PWM frequency was fixed to kHz. We set the DC link value to V because of the hardware setup limitations. The experimental setup is shown in Figure 8.
Experimental results for verification of drive current limitation during acceleration are given in Figure 9. At first, the drive is magnetized at time s. The value of total current in this period is given by the applied voltage boost value V. The value of the voltage boost is the same for different settings. At s, the fast ramp s was applied to the reference speed to change from zero to rpm. It can be observed that the output current and the phase currents to are limited to given maximum values during acceleration. The lower the value of the maximum current is, the longer the acceleration transients take.
(a)
(b)
Experimental results for drive current limitation under steady-state loading tests are shown in Figure 10. For both cases, the maximum current limit was set to A. Slip compensation was disabled in Figure 10(a). Therefore, a steady-state speed error is present even for unloaded drive. The slip compensation was re-enabled in Figure 10(b). In both experiments, the drive was first loaded with a step load torque value of Nm (approx. s), then unloaded, and then again loaded with a step load torque value of 2.5 Nm (approx. s). For higher load torque value, maximum current level of A of the drive was exceeded and the actual speed started to decrease. We can observe that the control algorithm keeps the total drive current at its maximum value. If this situation lasts longer, the drive will have to be switched off. In this experiment, the drive was unloaded after two seconds so that the actual speed recovered to its previous reference value. When comparing Figures 10(a) and 10(b), the effect of the slip compensation is visible in detailed speed responses. For disabled slip compensation, the speed dropped from to rpm in Figure 10(a), whereas in Figure 10(b), it dropped only to rpm and then started to recover slowly to its setpoint.
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(b)
7. Conclusion
In this paper, a current limiting technique for sensorless algorithm for five-phase induction machines is presented. Two elements of the control structure have been studied particularly. At first, a novel drive current limitation algorithm was introduced and provided, employing only one PI controller. Existing industrial solutions use at least two PI controllers or one PI controller in combination with a freezing function in the ramp function generator. We overcame this problem by using feedback inverse signal to reduce the reference frequency if an overcurrent condition is detected. Thus, presented solution does not require any modification of the ramp function generator compared to other solutions, and it is less complex. Next, we presented a slip compensation technique for the improvement of steady-state speed control. It was shown that the slip compensation technique developed for three-phase induction machines can successfully be applied also to five-phase induction machines with the differences in the used transformation matrix that have been also discussed in the paper. The proposed combination of control algorithms enables sensorless operation with no speed measurement or phase voltage measurement. It is purely based on phase current measurement, but it requires a knowledge of rotor parameters. We usually obtain these values during the industrial drive’s commissioning procedure. The future work should be focused on the operation of the controller in the regenerative mode and field weakening region.
Data Availability
The datasets used in this article are available from the corresponding author on reasonable request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This study was supported by the Slovak Research and Development Agency under contract nos. APVV-18-0436, APVV-16-0270, and APVV-19-0210 and the Scientific Grant Agency of the Ministry of Education of the Slovak Republic under the project VEGA 1/0493/19.