An Off-line Maximum Torque Control Strategy of Wound Rotor Synchronous Machine with Nonlinear Parameters

Abstract

Belt-driven Starter Generator (BSG) differs from other mild hybrid systems as the crankshaft of vehicle are not run off. Motor permits a low-cost method of adding mild hybrid capabilities such as start-stop, power assist, and mild levels of regenerative braking. Wound rotor synchronous motor (WRSM) could be adopted in BSG system for HEV e-Assisted application instead of the interior permanent magnet synchronous motor (IPMSM). In practice, adequate torque is indispensable for starter assist system, and energy conversion should be taken into account for the HEV or EV as well. Particularly, flux weakening control is possible to realize by adjusting both direct axis components of current and field current in WRSM. Accordingly, this paper present an off-line current acquisition algorithm that can reasonably combine the stator and field current to acquire the maximum torque, meanwhile the energy conversion is taken into consideration by losses. Besides, on account of inductance influence by non-uniform air gap around rotor, nonlinear inductances and armature flux linkage against current variation are proposed to guarantee the results closer to reality. A computer-aided method for proposed algorithm are present and results are given in form of the Look-up table (LUT). The experiment shows the validity of algorithm.

1. Introduction

BSG is a kind of mild hybrid system with simple structure and light-weight as shown in Fig. 1. It provides more power to traction system, and also be able to electrically start and assist the engine via belt.

Fig. 1.BSG system structure

Recently, WRSM could be a challenger to displace the IPMSM that is frequently used in BSG system due to plentiful merits such as high starting toque, wide constant power range, freedom field current control via external circuit substitute for permanent magnet, and controllable power factor. Inversely, there are some drawbacks as well. For instance, the exciting current should be injected via slip ring-brush system, and external circuit makes the entire circuit more complex, shrinks the life cycle and increases the maintenance cost. Moreover, electromagnetic interference and electromagnetic compatibility are occurred due to the outer circuit [1]. Despite these drawbacks, it is still worthy to do research on WRSM, because of its merits and the mentioned problems could be solved along with more research proceed on. Literature [2] investigated the performances between the WRSM and IPMSM under the same configuration, where discloses that the former has the preferable efficiency in the high speed region, the poor efficiency in the low speed region than the latter.

WRSM has three control variables: d-, q-axis components of current and field current. It allows the machine operating at an arbitrary power factor even unity power [3]. Currently, widely used maximum torque control are solving partial differential and quadruplicate equation for acquiring the reference current where the inductances are constant [4, 5]. However, the inductances and armature flux linkage are impacted by non-uniform air gap around rotor in WRSM, which lead to the calculated current inaccuracy, and adversely affect its overall efficiency. In addition, the maximum torque control in the Flux-weakening (FW) region has to take into account of the controllable field current. As a consequence, this study presents an off-line control strategy to acquire the reference current for maximum torque control by nonlinear inductances with consideration of losses. The algorithm is able to obtain the optimal combination of stator current and field current by means of computer-aided method. The calculated currents refer to torque and flux command are stored in order to force the flux linkage against the terminal voltage variation [5]. Therefore, the desired current could respond rapidly in case that the battery capacity is insufficient.

The proposed control strategy was deduced in MATLAB with 6kW prototype WRSM. The strategy was demonstrated by simulation and experiment.

 

2. Mathematical model

2.1 Nomenclature

vds, vqs, vfr d-, q-axis components of terminal voltage and rotor terminal voltage; vods, voqs d-, q-axis voltage components of iron loss; V0 Terminal voltage at rated speed; iods, ioqs d-, q-axis components of armature current with consideration of iron loss; idr, iqr, ifr d-, q-axis component of damper winding current and field excitation current; icds, icqs d-, q-axis current components of iron loss; iods, ioqs d-, q-axis current components of torque generation; Rs, Rfr Armature and rotor resistance; Rdr, Rqr d-, q-axis components of damper winding resistance; Rc Equivalent iron loss resistance; Lds, Lqs d-, q- axis components of stator inductance; Lmd, Lmq Magnetizing inductance; Ldr, Lqr d-, q-axis components of damper winding; Ldr, Lqr d-, q-axis components of damper winding; Ldr, Lqr d-, q-axis components of damper winding inductance; Lls, Llfr Stator and rotor leakage inductance; Lldr, Llqr d-, q-axis components of leakage inductance of damper winding; ψds, ψqs d-, q-axis components of flux linkage in stator side; ψdr, ψqr d-, q-axis components of flux linkage in rotor side; ψf Excited flux linkage; ωe Electrical angular velocity; ωm Mechanical angular velocity; Pn Motor pole pairs; s=d/dt Differential operator; Pc Copper loss; PcS, PcR Stator and rotor copper loss; D Outer radius of rotor; L Stack length of motor; M Mass of rotor; Kf Friction factor (1~3); n Motor speed.

All the rotor variables are referred to the stator.

2.2 WRSM model

The performance of synchronous machine strongly depends on the vector control strategy. Equivalent circuits of WRSM based on d-q model are shown in Fig. 2.

Fig. 2.Dynamic equivalent circuits of machine in the synchronously rotating coordinate

Based on them, the WRSM mathematical model was described by Eq. (1) with considering d-, q-axis components of damper winding.

It reveals that the electromechanical dynamics of WRSM are nonlinear Multiple Input Multiple Output system，which is very complicated for computation. Hence, in order to intuitively analyze the system, the steady state model is derived by equating all the time derivatives to zero and the power transferred to the rotor winding is zero, and all the power across the air gap is converted to mechanical power. Then the equations in steady state without considering damper winding were rewritten as follow.

In Fig. 2, it is observed that armature current is consists of iron loss components and torque generated components. The torque generated current can be expressed as follows.

Substitute (7), (8) into (5), (6), the equivalent iron loss currents are obtained:

The torque can be calculated as following:

Where ψa is the phase flux linkage established by field current. Notice that magnetizing flux is involved in the excited flux linkage. Hence, the phase flux linkage at load condition is used to replace the excited flux linkage ψf.

2.3 Loss model

In order to consider the efficiency, loss model is necessary. The dominant losses in WRSM mainly consist of copper losses, iron losses, and mechanical losses.

As seen from equivalent circuit, copper losses are caused by stator and rotor winding resistance in the form of heat dissipation. Hereby, copper losses are formulated as:

Iron losses are composed of the eddy current loss and hysteresis loss in core. Literature [6] provides an iron losses calculation method that only can be used for sinusoidal variation. Due to the harmonic components in the magnetic flux density, the iron losses are one of the most difficult phenomenon to model accurately. Hereby, the aforementioned iron loss resistance is utilized to calculate the iron losses refer to practical test. It is an operating frequency function as shown in Eq. (14), where the iron losses are proportional to the square of frequency, and flux linkage can be controlled by field current in the WRSM. Iron losses can be modeled as a function of frequency and field current with finite element analysis method by (15). Nevertheless, terminal voltage and total iron losses should calculate iron loss resistance at every operating condition, which results in the enormous and complicated computation since the conditions vary according to the current and its phase angle. Thus, the back electro-motive force (back-EMF) in Fig. 3 and total iron loss at no-load are used to calculate the iron loss resistance.

Fig. 3.Back-EMF versus field current

Mechanical losses are not involved in the equivalent circuit, whereas these exist during the machine running and should be taken into account in order to estimate the accurate efficiency. In WRSM, it mainly consists of draft loss and frictional loss on the rotor resistance and internal friction among mechanical components, respectively. It is hardly possible to estimate mechanical losses precisely. Normally the mechanical losses are less than 1~2% of total losses. Hence, a simplified model was adopted on the basis of experiment in this study. The mechanical losses are related to motor speed for the synchronous motor [7]. Herewith mechanical losses are modeled as a function of motor speed according to the test and expressed as:

The motor efficiency is expressed by:

 

3. Algorithm realization

Generally, BSG motor operates in constant torque region and constant power region according to operating speed. In this section, an off-line algorithm is analyzed according to the operating region.

One feature of WRSM which differs from IPMSM is d-axis inductance larger than q-axis, therefore it is able to control in arbitrary quadrant. The proposed current vector operating in the first and second quadrant is shown in Fig. 4. In the low speed region, motor operates at the point A in the first quadrant where both d-, q-axis component of currents are positive, and maximum field current is maintained to ensure the torque maximization. As speed increased, the maximum terminal voltage is reached due to output capability of inverter. Thereby flux weakening control is executed to extend the speed on account of terminal voltage limitation, which is implemented by adjusting both d-axis component of current and field current. The motor operating point goes towards the second quadrant along the negative direct axis from point A to B to maximize the torque. Then the total flux is decreased along the B to the centre M as the speed increasing. Note the point M is controllable by means of field current control [3].

Fig. 4.Current vector of proposed control strateg

Despite of there are corresponding equation for each control algorithm [8], but the characteristic analysis is much difficult to be dealt by them for WRSM due to the nonlinear inductances, that impacted by several controllable variables such as excitation current, armature current and phase angle. This paper suggests a computer aided method which uses the iteration loop to calculate the motor characteristics and the reference currents for motor controlling. Before the base speed, the loop condition is the maximum torque and current due to maximum torque per ampere (MTPA) control. After base speed, the maximum power is the loop condition. Once loop condition is satisfied, the calculated data is stored. The procedure of the algorithm implementation is shown in Fig. 5.

Fig. 5.The flowchart of the proposed algorithm

The currents are calculated by an offline program for deriving the look-up tables. Four pre-prepared data such as d-, q-axis components of inductance, iron loss and back-EMF are employed as program called files. Maximum field current is utilized to maximize the flux for providing the maximum torque before the motor rated speed. In the flux weakening region, the algorithm extracts the maximum torque by means of comparing the torque generated by identical d-axis component current with the reduced field current gradually, and the torque generated by identical field current with the varying d-axis component current. On the other hand, the different combination of the stator current in each angle and field current are computed to seek which combination could produces the maximum torque with nonlinear parameters. Terminal voltage suppression is the execution condition and Eqs. (13) ~ (17) used to the losses calculation. The space vector modulation is adopted, where the maximum output voltage is Vdc-link / , the total magnetic flux linkage is calculated as follow.

Eventually, the reference current related to the torque command and total flux command are stored as LUT automatically.

 

4. Simulation

In particular, saliency of WRSM varies owing to the nonlinear characteristics that are induced by magnetic saturation. Consequently, the nonlinear parameters are employed by Finite Element Analysis Method, which aims to obtain more realistic characteristic.

Tri-linear interpolation method is utilized to build these parameters for reference current computation as follows.

(xd, yd, zd) are desired point, x0 and x1 represent the point below and above x in the cubic space lattice and similarly for y0, y1, z0 and z1. For finding the 3-D point, firstly interpolated along the x-axis and obtained:

Where v represents the function value, then interpolated along the y-axis and obtained:

Finally, interpolate along the z-axis and obtained:

where vi is the desired value of underlying 3-D function v at the points in arrays. The result of the tri-linear interpolation is not related to the interpolating calculation’s order which obeys the commutative law of tensor product. In this paper, x, y, z are Ia, β and ifr, respectively. vi is the desired point such as Ld, Lq, and ψa.

Fig. 6 show the calculated d-, q-axis components of inductance and phase flux linkage profiles. It is evident that Lq has not much change in total current distribution, whereas the Ld behaves significant nonlinearity with saturation.

Fig. 6.Nonlinear inductances and phase flux linkage

Fig. 7 show the calculated d-, q-axis components and field current profiles. Fig. 7 (a) shows the filed current is much utilized for flux weakening control with considering losses at the high speed and lower torque requirement. Otherwise, the negative d-axis component current is more utilized and the field current keep constant control in order to sustain the higher torque at the high speed. The variations can be seen from Fig. 7 (a) and (c). Fig. 7 (b) shows the variation of q-axis component current according to the variation of the d- axis component current in the constant stator current.

Fig. 7.Current mapping according to Flux & Torque

 

5. Test Bench and Experiment

6kW prototype machine was utilized to verify the aforesaid algorithm in the drive system as shown in Fig. 8. Field current is regulated by H-bridge circuit via a slip ring, which has the rapid response than the traditional chopper circuit [2]. Table 1 provides the specification of the machine and driver system. Several performance evaluations with proposed drive system were tested.

Fig. 8.WRSM drive system with proposed algorithm

Table 1.Specifications

On account of the real situation, maximum torque was evaluated and test results are shown in Fig. 9 and Fig. 10. The comparison between tested results and designed value in different condition is provided in Table 2. The first test is under the condition of maximum power and torque, and the second is under the condition of maximum power and speed. Water-cooling system is used to maintain the motor temperature. The tested torque is approximate to the simulated data at 20 ℃. At the higher temperature of 65 ℃, the maximum torque have less than 5% declined that can be acceptable.

Fig. 9.Maximum torque comparison

Fig. 10.Maximum torque characteristic in different temperature

Table 2.(Sim is the abbreviation of simulation.)

Fig. 11 is the vector plane of tested current in the flux weakening control region. According to the speed increased, the more negative d-axis current is used till to the −70A, and q-axis current is decreased to about 20A at high speed region.

Fig. 11.Experimental id - iq plane in the FW region

Fig. 12 is the maximum torque test in different field current condition. As seen, the field current have a significant contribution for maximum torque generation at the low speed region, and make a contribution to flux weakening with d-axis component of current at the high speed region.

Fig. 12.Experimental maximum torque characteristic in field current control

Current control is implemented according to the torque and flux command in Fig. 13. The start mode is activated when the load starts, and after the motor started with the required torque and reached to 2500 rpm, torque e-Assist was shut up and current commands become zero.

Fig. 13.2500RPM in Start-up mode

Fig. 14 is the torque assist mode with targeted engine where 2.5:1 pulley ratio is selected. The engine speed is 800 rpm where motor speed is 2000rpm. With the 24 Nm torque started, the maximum field current and phase current were given to follow the engine commands. While the engine speed rise up to the 800 rpm, the torque e-Assist was released promptly, meanwhile currents were released to follow the torque command as well.

Fig. 14.Torque e-Assist mode for BSG system

Fig. 15 is 8kW load motor that is assembled to establish the efficiency and regenerative power map. The coolant temperature was fixed at 25℃. In addition, in order to test the regeneration power, constant field current 2A was employed in rated operating and the coolant temperature was fixed at 65℃. Since the power generating is proceeded during the driving. Fig. 16 shows the tested results, where maximum efficiency and maximum regenerative power were 96% and 5.2 kW in the motoring and generating mode, respectively.

Fig. 15.Test bench with load motor

Fig. 16.Tested efficiency and generation power map

 

6. Conclusion

This study presented an offline strategy to acquire the reference current with taking losses and torque maximization into account for BSG application. It provides a computer-aided method to implement the d-axis component of current and field current distribution reasonably for flux weakening control. With the algorithm, the torque maximization and preferable efficiency could be achieved which have been verified by experiment.

Generally, the constant or online detection parameters are frequently used in the previous studies. However, the motor parameters are influenced by a number of factors such as current and temperature in practice. With this consideration, the nonlinear parameters that could be adopted for algorithm development and determined by stator current, field current and phase angle. It contributes the algorithm to agree better with the reality.

The experiment results demonstrated the obtained maximum torque was in accordance with the simulated data at normal temperature status. However, the temperature effect cannot disregard which cause about 5% torque decline because a number of seasons such as thermal loss.

Due to the BSG characteristic, the motor usually have the contribution in low speed and high torque region. It implies this motor is often used to be starting torque assistant. Test results reveal the WRSM have approximate efficiency with the IPMSM, but the manufacturing cost lower in the absence of permanent magnet, and the performances in each operating region is more suitable for BSG e-Assist application such as the higher torque in the start mode. The payoff of proposed algorithm is practical, not merely theoretical.

Unquestionable, some of the undesirable current harmonics are involved in the alternative current that lead to the converter efficiency reduction. Meanwhile, induce voltage distortion in the alternating current supply that causes the electromagnetic compatibility problems. Therefore, further work is still need to proceed. Sorting these problems have assistance to improve efficiency and compatibility.