Multi-objective Predictive Control of 3L-NPC Inverter Fed Sensorless PMSM Drives for Electrical Car Applications

ABSTRACT


INTRODUCTION
The permanent magnet synchronous machine is now well established in electric vehicle drive systems, electric rail and aerospace traction and naval propulsion. Its increasing use in electric motors is due to its high power density, high efficiency and high reliability [14]. However, control of torque ripples and current harmonics is necessary to ensure proper system operation. Many [1] in 1986, the classical version of which is characterized by simplicity, fast dynamics and robustness [16]. Among these algorithms, the finite control set (FCS) algorithm based on prediction using the machine or system model takes an important place for optimal and multi-objective controls. Like any model-based control method, the FS-MPC algorithm is multi-objective since several sub-objectives can be defined for the same cost function, such as torque ripple control, stator flux control, stator current harmonic distortion control, speed control or inverter neutral point voltage control. The implementation of several sub-objectives on the same cost function creates a real computational load problem [5,7]. The use of the three-level NPC inverter allows for a considerable reduction in current harmonics compared to the conventional inverter and also offers ease of balancing the voltages across the capacitors. However, the large number of iterations to obtain the 27 control voltage vectors increases considerably the search time for the optimal solution. Numerous works have dealt with multi-objective three-level NPC inverter controls with the target being the computational load [2][3][4][5][6][7] or the minimization of torque ripples [21,22] by reducing the number of voltage vectors at the prediction stage [2][3], the weighting factors, the choice of which is decisive for the reliability of the control [6,8], and the operation of the machine without angular position or speed sensors [7,14].
To reduce the computational load, some authors prefer to reduce the number of prediction vectors. This approach impacts the number of iterations as well as the computation time of the optimal vector. Thus, [4] proposes to take into account the stator flux but also to reduce the prediction vectors. This results in a reduction of the computational complexity by the rate of 49% compared to the conventional method. In [5] the control algorithm is based on 20 fixed duty cycle modulated voltage space vectors. In [2][3] a new technique is proposed based on reducing the number of voltage vectors from 27 to 17 at the prediction stage.
Others prefer a strategic choice of the weighting factors allowing to refine the various sub-objectives of the order. The work of [6] proposes a method that consists in making an appropriate choice of the weighting factors by applying the principle of coefficient variation done online. The new cost function is constructed to target torque ripple and switching loss reduction with improved low-speed operation. It produces a reduction in the range of 29.85% to 33.33% in torque ripple and common mode voltage respectively. To further emphasize the weighting factors, [7] proposes the use of speed and current error weighting coefficients in the cost function. The resulting method performs well under steep transient conditions with a short prediction horizon and low computational load. The DC bus voltage balance is achieved by regulating the time allowed to each inverter string while they are connected to the neutral point. In this paper, [8] proposes an optimal selection of the weighting factors of the cost function based on artificial intelligence. The authors demonstrate in this scientific contribution that the proposed method can be applied not only to cost functions with multiple conflicting objectives but also to control objectives that cannot be unified without imposing an additional computational burden. On the other hand, [9] proposes a predictive torque control algorithm based on voltage vector selection that eradicates weighting factors in the cost function. This was achieved by defining only a cost function taking into account the flow error to select the optimal voltage vector in order to avoid the constant adjustment of the weighting factors. The optimization is done in two steps but the switching frequency remains variable and high. To overcome the issue of high and variable switching frequency inherent in most control algorithms, [10] proposes an algorithm that improves robustness and performance of FCS-MPC by providing "dynamic error correction" and "modified revised prediction" with reduced switching frequency.
In addition to the cost function complexity problem due to the likely large number of the control subobjectives, which results in a long computation time for the optimal solutions, the problem of saving space for the implementation of the control system and the overall financial cost of the system remains for motorization systems. The latter is generally related to the use of encoders to read the angular position of the machine rotor in order to deduce the speed of rotation. The use of electrical speed and angle observers in the motor thus bypasses the need to obtain the rotor position via the mechanical sensor, which can reduce the reliability of the system, increase the sensitivity of the system to external disturbances and increase the financial cost of the control system [14].
In this paper, the proposed method consists of reducing the voltage vectors at the prediction stage from 27 to 19 with the objective of controlling the current, electromagnetic torque, motor speed and neutral point voltage of the 3-level NPC inverter. To avoid computational redundancy that could increase the time to find the optimal solution, the proportionality between the q-axis component of the current in the d-q frame and the electromagnetic torque of the surface-magnetized synchronous motor is exploited to control both variables. To reduce the interactions between external disturbances and the system, a sliding mode observer is used to obtain the estimated motor speed and rotor angular position. The algorithm is built on three essential steps in order to make the calculation digestible and simulated under Matlab/Simulink. A comparison of the results obtained is made with those of [3], [10], [11], [14], [19], [21] and [22]. This paper is constituted as follows: In sections 2, the research Method are presented. Sections 3 and 4 are devoted to the presentation and discussion of the results obtained under Matlab/Simulink and conclusion.

RESEARCH METHOD 2.1. Mathematical model of the surface-mounted PMSM
The mathematical model of the terminal voltage of the permanent magnet synchronous motor in the d-q frame is given by equation (1).
In the d-q frame, the current, electrical speed and electromagnetic torque are given by equations (2), (3) and (4) For the surface-mounted PMSM subject to the Maximum Torque Per Ampere, L d = L q = L and the electromagnetic torque is proportional to the q-axis component of the d-q reference current (

3L-NPC inverter
The 3L-NPC inverter is a distributed neutral inverter topology in which loopback diodes are used to make the connection to the neutral reference point. The 3L-NPC inverters, are more popular in motor drive applications, compared to other multi-level inverter topologies [11,20]. This topology has been studied for motor drives in several works [2][3][4][16][17][18][19]. Figure 1 below shows this topology.
Some of these switching states can produce the same voltage vector amplitude but with opposite effects on capacitor charging and discharging, which significantly affect motor performance [3].

Park and Clarke Transformations
Cost function v c2 (k)

Prediction
Speed ω e and θ e electrical angle observer The proposed three-step multi-objective FS-MPC method as shown in figure 3 consists of the following parts: estimation of speed and electrical angle by a sliding mode observer, selection of the optimal voltage vector, prediction and minimization of the cost function. The stator currents and DC bus capacitor voltages are measured while the motor speed and motor electrical angle are observed. The selection of the optimal voltage vector is performed by taking into account exclusively the 19 non-redundant vectors of the voltage space consisting of a zero vector, the 6 short vectors, the 6 medium vectors and the 6 long vectors. The control variables considered in the proposed method are current, motor speed, neutral point voltage under the constraint of overcurrent protection.

Speed and electrical angle observer
The speed and angle observer bypasses the obtaining of the rotor position via the mechanical sensor that may influence the reliability of the system due to external disturbances [14]. The model of the machine in the α-β frame of reference developed from the back-electromotive forces e α and e β is defined by (4): The conventional sliding mode observer applied to the surface-mounted PMSM is described by the relation (5).
Where î denotes the estimated values, and k SMO the constant gain of the observer. The conventional SMO uses the signum function which causes the proliferation of disturbances in the system [13]. Thus the sigmoid function is preferred because of its moderate interactions with the disturbances. This sigmoid function is defined by the relation (6) (7) below is obtained by substituting the signum function for the sigmoid function in equation (6).
The SMO described by equation (7) By introducing the second-order identity matrix into (10), the discrete equation of the sliding-mode observer is deduced by (10).
The electrical angular position of the rotor is estimated using the negative arc tangent method based on the back-EMF. The observed electrical angular measurement of the rotor is obtained as follows: Figure 4 shows the voltage vector distribution of the used 3-level NPC inverter.  (11) where v α and v β are the real and imaginary parts respectively if the voltage vector whose expression is given by (12 The voltage V an , V bn and V cn are given by table 1.

Prediction
Since the control variables of the proposed method are current, motor speed, electromagnetic torque and inverter neutral point voltage, by applying Euler's formula to equations (1), (2) and (3) above, equations (13), (14) and (15) are deduced. [ The discrete equations for the voltages across capacitors C 1 and C 2 and their currents are given by (16) respectively: Where ci V represent the voltage across capacitor {1, 2}, c1 i the current through capacitor C 1 and c2 i the current through capacitor C 2 . The predicted voltage at the neutral point of the inverter can therefore be deduced by (17): Keeping this voltage close to 0 is essential for balanced operation of the inverter.

Cost Function
The cost function will be evaluated for the 19 states to find the best combination in the three stages. The proposed cost function for this method aims to control both the current, the electromagnetic torque, the motor speed and the voltage at the neutral point of the inverter. The proposed cost function will therefore take into account three types of errors: ▪ The pipe errors (g 1 ) which comprise the quadratic form of the d-axis component of the current and voltage at the neutral point. This function is defined by (18).
=  + +  + ▪ The tracking errors (g 2 ) which include the quadratic form of the q-axis component of current and torque. This function is defined by (19).  2  2  2  2  d  q  max  d  q  max  3  m   i  i  I  if i  i  I  g  I  (20)  0 Otherwise are the weighting factors of current i d , neutral point voltage V z , current i q and electromagnetic torque T em respectively. As the motor is surface magnetised and the electromagnetic torque is set proportional to the current on the q-axis of the d-q frame, equation (19) becomes (21).
This property of proportionality between the electromagnetic torque makes it possible to eliminate the calculation redundancy in g 2 of ( )

Simulation results
To verify the effectiveness of the proposed method, simulations were carried out in the Matlab/Simulink virtual laboratory. Table 2 below shows the different parameters used in these simulations. Firstly, to study the steady state performance of the control, tests were carried out at constant speed and low load torque and then at high load torque. The electrical and observed quantities are given in Figure 5.

Discussion
From the results shown in Figure 5, it can be seen that the speed ω e , the electromagnetic torque T em , the current ic1 and the neutral point voltage v z are well regulated during constant speed operation. At low load torque, the electromagnetic torque is characterised by small ripples with margins of ± 0.3 N.m while the inverter neutral point voltage remains very close to 0 with an amplitude of the order of 10 -4 V. These ripples are thus sufficiently attenuated compared to those obtained by [19] which are about 2.5 N.m and those of [21] which are about 0.4 N.m. The proposed method makes it possible to maintain a constant voltage across capacitors C 1 and C 2 with negligible variations compared to 1 V obtained by [3] or 0.208 V obtained by [11]. It also provides a low harmonic content line current with a THD of 1.69% at 60 Hz for heavy load operation. In this case, there is a considerable decrease in THD compared to the work of [21] where a THD of 9.33% at 27 Hz for a fundamental current of 3.027 A was obtained, that of [10] where a THD of 3.21% was obtained at 40 Hz for a fundamental current of 6.3 A and that of [3] where a THD of 2.12% was obtained for an asynchronous machine. This was achieved by using the 3-level NPC inverter whose output voltage is shown in Figure 5.a) The superposition of the observed rotor angle with that measured by a sensor shown in Figure 6 allows the effectiveness of the sliding mode observer of the rotor speed and electrical angle to be verified. Compared to the work of [14] where the average error is 0.1 rad at 120 rpm, a significant decrease in the estimation error is noticed.
When the load torque is varied from 2 to 6 N.m, Figure 7 shows that the electromagnetic torque reacts positively to the load variation with small ripples but with great dynamics without overshoot. The variations of the voltage at the neutral of the inverter remain controlled as well as the variations of the current id which remain in the vicinity of 0. Figure 8 shows the ability of the proposed control system to adapt to the reversal of the motor's direction of rotation. Here, the electromagnetic torque is characterised by very good dynamics. It follows its reference perfectly, without overshoot and with a remarkable reduction in ripples. The speed of rotation obtained follows its reference, without any static error and without probable vibration at the machine level since its undulations are lower than 2 rpm for an average speed of 200 rpm. A significant reaction of the current id during the phase of changing the direction of rotation is nevertheless observed. During operation in each direction of rotation, the voltage at the neutral point of the inverter remains around 0.
The feasibility of the proposed method over the conventional FS-MPC studied in [19] is evaluated with the help of Figure 9, which presents the electromagnetic torque and speed in the low speed regime with engine acceleration. While for the conventional FS-MPC the torque ripples remain high in the low speed regime, our method achieves low torque ripples in this speed regime. These ripples are of the order of ± 0.1 N.m with an overshoot of 10 N.m at the moment of acceleration. This value remains significantly lower than that obtained by [22], which is higher than 12 N.m.

CONCLUSION
This paper proposes a multi-objective torque predictive control approach for surface-magnetized synchronous motor drives fed by a three-level clamped neutral point inverter. It uses the principle of reducing the number of candidate voltage vectors in the search for the optimal solution and the proportionality between the electromagnetic torque and the current component on the q-axis of the d-q frame to reduce the computational load despite its high number of sub-objectives. A sliding mode observer is used to estimate the motor speed and the angular position of the rotor. It achieves torque and speed control as well as a significant reduction in current harmonics.