Maximum Torque per Ampere Control of Permanent Magnet Assisted Synchronous Reluctance Motor: An Experimental Study

In recent times, permanent magnet assisted synchronous reluctance motors (PMaSRM) have been considered as suitable traction motors for electric vehicle applications. In this type of machine, where the share of reluctance torque is more significant than the excitation torque, it is more appropriate to use a control strategy that can fully utilize the reluctance torque. This paper deals with a new structure of permanent magnet-assisted synchronous reluctance motors that was designed and manufactured in a previous study. This paper suggests applying, in a first study, a constant parameter maximum torque per ampere (MTPA) strategy to make a contribution towards the control of such structure that is becoming increasingly attractive in the field of electric transportation. This method is usually used to control interior permanent magnet synchronous motors to minimize the copper losses of the system. Before implementing and simulating this method, the mathematical models of the suggested motor and the inverter are given. An experimental study is conducted on a small-scale 1 kW prototype PMaSRM using a MicrolabBox Dspace to test and examine the proposed control. Simulation and experimental results are presented in this article in order to verify the validity of the developed control strategy.


Introduction
Electrified transportation, especially electric vehicles, are going to play a big role towards achieving the net-zero emission by 2050. Currently, most electric vehicles use permanent magnet synchronous motors since they can provide high power and torque density together with good efficiency. However, the rare earth magnets used in those motors are becoming increasingly expressive over the last few years. Moreover, this is also not sustainable. To overcome this issue, machine designers are taking many approaches to develop motors with a reduced amount of permanent magnets without compromising their required performances.
In recent times, permanent magnet assisted synchronous reluctance motors (PMaSRM) have been considered as suitable traction motors for electric vehicle applications. In this type of machine, where the share of reluctance torque is more significant than the excitation torque, it is more appropriate to use a control strategy that can fully utilize the reluctance torque. This paper deals with a new structure of permanent magnet-assisted synchronous reluctance motors that was designed and manufactured in a previous study. This paper suggests applying, in a first study, a constant parameter maximum torque per ampere (MTPA) strategy to make a contribution towards the control of such structure that is becoming increasingly attractive in the field of electric transportation. This method is usually used to control interior permanent magnet synchronous motors to minimize the copper losses of the system. Before implementing and simulating this method, the mathematical models of the suggested motor and the inverter are given. An experimental study is conducted on a small-scale 1 kW prototype PMaSRM using a MicrolabBox Dspace to test and examine the proposed control. Simulation and experimental results are presented in this article in order to verify the validity of the developed control strategy.
Among the various alternative solutions available in the literature, permanent magnet assisted synchronous reluctance motors (PMaSRM) appear to be a very suitable choice for electric vehicle propulsion applications. The electromagnetic torque of this machine has two components; one is called permanent magnet torque generated by the interaction between the rotor magnetic field and the stator's windings magnetomotive force (or current), while the other is the reluctance torque. Because the d-and q-axis stator current contribute to torque generation, there are different possible combinations of these currents for the desired torque [1] [2]. To improve the system's efficiency and reduce motor losses, various control strategies can be adopted, such as unit power factor control [3], = 0 control, and maximum torque per Ampere Control (MTPA) [4]. The maximum torque per-ampere is a control technique that has been subject to much research as it focuses on producing the required torque using the minimum current magnitude, which helps to reduce copper losses [5]. The review of the literature reveals that in the majority of existing studies, MTPA strategy is applied to control permanent interior magnet synchronous machine (IPMSM) in order to exploit the double torque generation in the most effective way. The work in [6] [7] presents results of the MTPA control when using nominal values for motor parameters. However, parameter variations can be a considerable challenge when implementing the technique. That's why online motor parameter estimation has been suggested in reference [8]- [10] to overcome this issue.
Unlike the IPMSM, the permanent magnet assisted synchronous motor is a motor where the share of reluctance torque is more significant compared to the PM electric torque. To make a contribution towards the control of a structure of motor that is becoming increasingly attractive, this paper suggests the use of the MTPA control technique. The structure and the design of the motor are proposed in [11]. In this first article, only the constant parameter MTPA technique is considered. Mathematical modeling, control design, and experimental results are presented to verify the validity of the developed method.
The rest of this paper is organized as follows: Section II describes the considered motor and also presents the mathematical model of the motor. Details of the developed MTPA control strategy are given in Section III. Simulation results are given in Section IV, while experimental results are provided in Section V. Finally, Section VI concludes this paper

Permanent Magnet assisted Synchronous Reluctance Motor Model PMaSRM Mathematic Model
The PMaSRM motor has three stator windings and a permanent magnet on the rotor that can be illustrated in the frame as shown in Fig. 1. The mathematic model of this motor in the natural frame of reference (ABC frame) in matrix notation is given by the following equations [12]:  The voltage equation:  The flux linkage equation: Where , , Ѱ are the phase voltage vector, the stator current vector and the flux linkage vector, [ ] is the stator resistance matrix, [Ls] is the inductance matrix and [Ѱ m ] is the magnitude of the flux linkage established by the permanent magnet. They are defined as follows: , [ ] = [ ] Using Clark transformation followed by Park transformation, it is possible to get the mathematic model in the frame [13]. Fig. 2 illustrates the definition of -axis and -axis of the permanent magnet assisted synchronous reluctance motors used in this work In this PMaSRM topology, the magnets are placed along the -axis rather than -axis. In the reference frame, the stator voltage equations are expressed as follows: Where Ѱ and Ѱ are respectively the flux linkage along axis and axis, is the number of pole-pairs, is the mechanical angular speed and is the stator resistance. Since the magnets are inserted along the axis, the equations of flux linkage in axis are given by: Where is the -axis inductance, the -axis inductance and Ѱ the flux linkage of permanent magnets. By substituting the flux linkage equations (5) and (6) in (3) and (4), the -axis motor current dynamics can be obtained as: Assuming that the motor parameters are constant (Table 1), the generated electromagnetic torque can be expressed in the reference frame by: Substituting (5) and (6) in (9), the torque equation can be found as: In order to control the motor, it is important to consider its dynamic behavior. The relation between electrical torque and load torque is given by the mechanical equation, which is: Where is the inertia of the motor and coupled load, is the load torque and is the friction coefficient, and Ω is the mechanical angular speed.

Inverter Mathematic Model
The three-phase inverter used to drive the PMaSRM is fed by a DC voltage source and has three-phase legs, each comprised of two switches, as shown in Fig. 3. The switches are controlled by using sinusoidal pulse width modulation (SPWM) [6] control by comparing a sinusoidal wave and a triangular carrier signal that has a frequency of 10kHz. For each leg of the inverter, we define three connection function ( = 1, 2, 3 = { 1 if is on and ′ is off 0 if is off and ′ is on (12) Where and ′ are switches. The line voltages of the inverter can be expressed by: The relationship between line voltages and phase voltages are given by: From (13) and (14), the output phase voltages can be expressed using the connection functions as following [16]

MTPA Control Strategy
The objective of the MTPA control strategy (maximum torque per ampere) is to find, for a given torque, the minimum value of currents, and . To fully understand this principle, the electromagnetic torque versus current angle, according to the different current amplitude, are illustrated in Fig. 4. For each current Amplitude, it can be seen that there is a specific value of the current angle for which the torque reaches its maximum value [17]- [19].

Constant Parameter MTPA Control
For a given torque, there are numerous possible combinations of and . Among the possible solutions, there's one pair that can minimize the current magnitude. Ignoring the variation of the motor parameters, finding the optimal solution requires solving the following optimization problem: It can also be stated as: To obtain the current expressions for the maximum torque, the differentiation of torque is made equal to zero as shown in equation (17) [18] [20] as Keeping the stator current constant and substituting (10) and = √ 2 + 2 into equation (17), the relationship between and can be written as: For any given torque, it is possible to find the optimal and that reduce the copper losses. As mentioned previously, the motor parameters ( , , Ѱ ) are considered constant, and the effect of saturation, as well as temperature influence, are neglected in all the analytical approaches.

Implementation of MTPA
The block diagram of the PMaSRM drive with MTPA control is illustrated in Fig. 5. To implement the MTPA control method, -axis and -axis stator currents need to be controlled according to their references * , * . In this study, the reference for the d axis is calculated using the following torque equation: Where is the measured q-axis stator current and * is the reference torque generated by the PI controller after comparing the actual speed with the given speed. For the -axis current, the reference is given by the equation (16) [21] [22].
It is to be noted here that It is possible to find from the -axis and -axis current reference the relationship between torque and current through Matlab two-dimensional curve fitting function [5] [19]. The most common approach in the implementation of -axis current and speed control loops is the use of proportional-integral controllers. The conventional PI control process is written in the form of the transfer function as: Where is the proportional action coefficient and is the integral action coefficient. By simply adjusting those two parameters, the drive can quickly respond to errors between the command and the actual value. In this study, the PI controllers have been designed following reference [23], where it is possible to determine the gains in a simple way using motor parameters. All parameters used for the simulation are presented in Table 2.

Simulation results
In this Section, Matlab/Simulink-based simulation study is reported. Parameters of the PMaSRM are given in Table 1. This paper simulates MTPA control of PMaSRM in two parts, a noload acceleration phase and a constant speed phase where a load torque is added.
After applying a reference speed of 500 rpm, the motor speed follows this given value with a slight reduction due to load variation, which is quickly adjusted. Under no-load operation, the motor produces an electromagnetic torque that overcomes the friction, and both -axis andaxis currents are used for overcoming the inertia. Speed, torque, and current waveforms under no-load operation can be seen in Fig. 6 between 0 and 1s.
When a load is introduced (Tr=2.5 Nm), the current and torque values change, as shown in Fig. 6. The system uses both -axis and -axis currents to produce the reference torque with the most effective way to have the lowest stator current resulting in reduced copper losses. It is important to note that for a salient motor, it is more appropriate to use the MTPA control strategy as it can fully utilize the reluctance torque.

Experimental Results
The experimental study was carried out in in France, in Groupe de Recherche en Energie Electrique (GREEN) laboratory. The test bench is given in Fig. 7. To implement the control strategy, MicrolabBox dSPACE was used. It's a compact system that offers the possibility to set up experiments quickly and easily. The MicrolabBox has more than 100 channels with different I/O types, a dual-core processor with 2 GHz, and a programmable FPGA. It is also supported by Real-Time Interface (RTI) and ControlDesk software packages to enable the linkage with Simulink. After compiling the model on Simulink, the RTI sends the generated C code to the MicrolabBox. This code is then converted to PWM pulses in order to control the power converter. Using the analog-to-digital converter (ADC), the converter signals are supplied to the Microlabbox. It's possible to visualize online the evolution of all the obtained signals and measurements and to adjust all the parameters at run time using the Control desk monitoring software. The PMaSRM motor and the connection between the inverter and the MicrolabBox are represented in Fig. 8 and Fig. 9, respectively.  After implementing the MTPA control strategy under a load operating mode (Tr=2.5 Nm) and applying the same speed reference as simulated, the obtained waveforms are illustrated in Fig. 10. The actual speed reaches its reference value of 500 rpm, and the electromagnetic torque is oscillating around an average value of 2.52 Nm. The value of the measured current of the d-and q-axis in the steady-state nearly matches the results obtained in the simulation with a slight difference of 0.2 A that can be noticed. For phase stator current, it is nearly sinusoidal with an amplitude of about 2A. However, a slight deformation is witnessed. This deformation begins mainly in the interchange section between the top half of the current and the bottom half and can be caused by the dead time of MOSFETs. Compensation of this dead-time would be considered in future work.

Conclusion
In this study, a constant parameter MTPA control law was established to control a PMaSRM motor. After giving the mathematical model of the motor and the inverter and explaining the control strategy, the traction system model was simulated using Matlab Simulink. It has been seen from the simulation results that both current components are used with the most effective way for producing torque. To validate the proposed technique, experiments were performed on a PMaSRM previously designed and implemented for testing in a laboratory setup. PMaSRM is a motor that has the capability required for traction systems of EVs and HEVs. However, the control system needs to be improved in order to obtain a high performance under all conditions. In future work, we will improve this control strategy by applying an adaptive control in order to consider the parameter variation due to magnetic saturation and change in temperature.