Extended Permanent Magnet Synchronous Motors Speed Range Based on the Active and Reactive Power Control of Inverters

Abstract

The paper proposes a new speed control method to improve control quality and expand the Permanent Magnet Synchronous Motors speed range. The Permanent Magnet Synchronous Motors (PMSM) speed range enlarging is based on the newly proposed power control principle between two voltage sources instead of winding current control as the conventional Field Oriented Control method. The power management between the inverter and PMSM motor allows the Flux-Weakening obstacle to be overcome entirely, leading to a significant extension of the motor speed to a constant power range. Based on motor power control, a new control method is proposed and allows for efficiently reducing current and torque ripple caused by the imbalance between the power supply of the inverter and the power required through the desired stator current. The proposed method permits for not only an enhanced PMSM speed range, but also a robust stability in PMSM speed control. The simulation results have demonstrated the efficiency and stability of the proposed control method.

1. Introduction

Permanent Magnet Synchronous Motors (PMSM) are increasingly widely used in civil and industrial applications because of their high efficiency, high power density, low maintenance cost, and lower permanent magnet price. The vector control method’s popularity, including Field Oriented Control (FOC) [1,2,3,4,5,6] and Direct Torque Control (DTC) [7,8,9,10], stems from its advantages of simplicity in the control structure, ease of operation, reliability, and high efficiency [11,12].

There are two main parts of the operating speed range of PMSM operation, including constant torque range and constant power range, as mentioned in Figure 1. The speed control principle in the constant torque region (in which the PMSM rotor speed is below the rated speed) is suitable for the vector control method when the operating voltage is lower than the inverter output voltage limit. When the PMSM speed exceeds the rated speed, the Back Electromotive Force (B-EMF) generated in the winding, caused by a permanent magnet flux on the rotating rotor in the lumen of the stator windings, can increase beyond the limit of the voltage of the inverters. For motors that do not use permanent magnets, such as induction motors, reducing the current rotor windings can reduce the BEMF voltage [13,14,15]. Unfortunately, this way is not possible with rotor motors that use permanent magnets. Due to the limit of output voltage, the inverter cannot increase the voltage to stabilize the stator current control.

The input DC voltage is proportional to the maximum amplitude of the output voltage of the inverter. Therefore, in some studies, the expansion of the motor operating speed is achieved by increasing the inverter input voltage [16,17]. However, the step-up voltage method encounters the main limitation of expanding the power drive room for DC–DC voltage converter circuits, and only changes in a narrow speed range. In addition, inverters are designed to operate within a specific voltage limit. Increasing DC voltage to increase motor speed increases inverter costs in applications requiring a high-speed operation.

To ensure that the active power is transmitted to the motor, the reactive power is drawn out of the inverter to ensure that a low voltage amplitude source can push power into a higher voltage source. Then, the d-axis current increases the negative amplitude during entering the FW region, which manifests reactive power inflowing from the PMSM to the inverter.

On the other hand, during operation in the FW zone, the inverter voltage cannot exceed the inverter output voltage limit. However, controlling the stator current using current controllers will cause the estimated controller output voltage to exceed a given boundary. This trouble leads to an increased stator current ripple, resulting in increasing electromagnetic torque ripple and reducing the efficiency in reference velocity tracking.

Several improvements have been used in PMSM speed control in FW to improve control efficiency and reduce current ripple during voltage limit violation in the FW zone. These methods include the Auto-tuning method [18,19,20], Fuzzy Logic Speed Control [21,22,23,24,25], Improved UDE-Based Flux-Weakening Control [26], and Single Current Regulator Control [27,28,29,30].

When entering the FW zone, the d-axis current is adjusted to increase negatively based on the PI controller, where input is the deviation between the maximum voltage amplitude supplied by the inverter and the voltage at the current controller output. As the voltage fluctuation is more extensive in the FW region, the d-axis current ripple also increases. The fuzzy logic speed control method reduces the d-axis current ripple desired by replacing a PI controller with a Fuzzy PI controller. The Fuzzy PI current controller’s output voltage is less violated than the PI current controller, leading to a reduced ripple of the d-axis current component. However, this violation’s nature persists and adversely affects the control quality when the current controller output voltage estimated value still exceeds the limit voltage value of the inverter.

Single Current Regulator Control approach allows elimination of the output voltage violation when the voltage amplitude is kept at the rated value, as shown in [29]. However, the voltage angle change is based on a current error, without considering the inverter power balance and the load power caused by the wave. Results are demonstrated in [29], on the proposed solutions of Single Current Regulator Control method proposed in [27,31]. Research [29] can only minimize power imbalance by limiting voltage angle variation, but does not wholly address the power imbalance’s harmful effects.

Another creative direction in minimizing current and torque ripple in the FW region aims to optimize the current controller parameters. The PSO algorithm is used in [32] to find suitable values when working conditions were changed. A Fuzzy-PI controller is used instead of the PI controller in the more refined determination of the d-axis reference current when the motor operates in the FW region. These innovative methods minimize output torque ripple, while reducing the reference current’s effect on the control voltage. However, these methods increase the complexity, and require an amount of computation in the speed control process of PMSM.

The novelty and contributions of this paper are tri-fold. Firstly, this paper proposes the new PMSM speed control based on balancing the output inverter energy and the motor’s load energy. The proposed approach differs from the known solution because it does not use rated speed to switch from a constant torque region to an FW region. A change in reactive power leads to a change in rotor speed. Second, the energy transformation to the machine is done by sending energy between two voltage sources, the inverter output voltage and the BEMF of the motor. Eventually, a detailed analysis of the power transmission principle between the two voltage sources is discussed in the following section.

Over the entire operating speed range of the PMSM, the increase (or decrease) of the rotor speed is accomplished by increasing (or decreasing) the active power of the inverter, which enables the roaming to operate without the need to switch different control methods between the two operating zones. Power control will minimize current and torque ripple, and the power supplied to the PMSM will also better monitored. This advantage is especially significant in mobile systems with batteries. The new control method allows efficient reduction of the electric current and torque ripples in the motor through direct control of the inverter output voltage.

The remainder of this paper is structured as such: Part 2 presents the PMSM math model and the principle of transferring power from the inverter to the PMSM; Part 3 introduces the principle of controlling a PMSM motor’s speed based on the active power control; Part 4 shows the results obtained after the simulation modeling for the proposed algorithm with the Direct Current Calculation and Vector Current Control algorithms used in the comparison; Part 5 includes the conclusion.

2. Speed Control of PMSM Drive Systems

2.1. PMSM Model and Its Limitations

According to [33], Equations (1) and (2) show the relationship between stator voltage, stator current, and rotor speed. The dq coordinate system is performed through the Park transformation, converting from the abc three-dimensional space to the dq two-dimensional space with almost constant values.

$${\mathrm{v}}_{\mathrm{d}}{=\mathrm{R}}_{\mathrm{S}}\times {\mathrm{i}}_{\mathrm{d}}{+\mathrm{L}}_{\mathrm{d}}\times \frac{{\mathrm{i}}_{\mathrm{d}}}{\mathrm{dt}}-\mathrm{Pp}\times {\mathsf{\omega }}_{\mathrm{m}}\times {\mathrm{L}}_{\mathrm{q}}\times {\mathrm{i}}_{\mathrm{q}}$$

(1)

$${\mathrm{v}}_{\mathrm{q}}={\mathrm{R}}_{\mathrm{S}}{\mathrm{i}}_{\mathrm{q}}+{\mathrm{L}}_{\mathrm{q}}\frac{{\mathrm{i}}_{\mathrm{q}}}{\mathrm{d}\mathrm{t}}+\mathrm{P}\mathrm{p}.{\mathsf{\omega }}_{\mathrm{m}}.{\mathrm{L}}_{\mathrm{d}}.{\mathrm{i}}_{\mathrm{d}}+\mathrm{P}\mathrm{p}.{\mathsf{\omega }}_{\mathrm{m}}.{\mathsf{\lambda }}_{\mathrm{p}\mathrm{m}}$$

(2)

where v_d, v_q, i_d and i_q are the components voltage and current on the dq-axis, respectively. L_d and L_q are the stator coil inductance components in the dq coordinate system, λ_pm is the flux linkage of the permanent magnet on the rotor, R_S is the coil resistance, ω is the rotor speed, and Pp is the number of pole pairs.

When analyzing the PMSM in the stationary state, the current variation is zero; the voltage Equations are rewritten in (3) and (4).

$${\mathrm{v}}_{\mathrm{d}}={\mathrm{R}}_{\mathrm{S}}\times {\mathrm{i}}_{\mathrm{d}}-\mathrm{P}\mathrm{p}\times {\mathsf{\omega }}_{\mathrm{m}}\times {\mathrm{L}}_{\mathrm{q}}\times {\mathrm{i}}_{\mathrm{q}}$$

(3)

$${\mathrm{v}}_{\mathrm{q}}={\mathrm{R}}_{\mathrm{S}}\times {\mathrm{i}}_{\mathrm{q}}+\mathrm{P}\mathrm{p}\times {\mathsf{\omega }}_{\mathrm{m}}\times {\mathrm{L}}_{\mathrm{d}}\times {\mathrm{i}}_{\mathrm{d}}+\mathrm{P}\mathrm{p}\times {\mathsf{\omega }}_{\mathrm{m}}\times {\mathsf{\lambda }}_{\mathrm{p}\mathrm{m}}$$

(4)

Electromagnetic torque is determined by (5)

$${\mathrm{T}}_{\mathrm{e}}=\frac{3}{2}\times \mathrm{P}\mathrm{p}\left({\mathsf{\lambda }}_{\mathrm{p}\mathrm{m}}\times {\mathrm{i}}_{\mathrm{q}}+\left({\mathrm{L}}_{\mathrm{d}}-{\mathrm{L}}_{\mathrm{q}}\right)\times {\mathrm{i}}_{\mathrm{q}}\times {\mathrm{i}}_{\mathrm{d}}\right)$$

(5)

According to Newton’s Second Law for Rotation, the rotational acceleration of the rotor is determined by (6).

$$\frac{\mathrm{d}{\mathsf{\omega }}_{\mathrm{m}}}{\mathrm{d}\mathrm{t}}=\frac{1}{\mathrm{J}}\left({\mathrm{T}}_{\mathrm{e}}-{\mathrm{T}}_{\mathrm{L}}-\mathrm{F}{\mathsf{\omega }}_{\mathrm{m}}\right)$$

(6)

where:

• J: The inertia of motor and load combined referred to the motor shaft.
• F: The combined viscous friction coefficient of rotor and load.
• θ: Rotor angular position.
• T_L: Shaft load torque.
• θm: Angular velocity of the rotor (mechanical speed)

During operation, the stator voltage and current are limited below their rated values. The voltage limit is determined based on the motor insulation capacity and the maximum voltage amplitude generated by the inverter, determined by the dc bus voltage. Current limits are usually related to the motor heat dissipation, and the manufacturer provides its value. The Equations described the limits of voltage and current are expressed through (7) and (8), respectively:

$${\mathrm{v}}_{\mathrm{d}}^2+{\mathrm{v}}_{\mathrm{q}}^2={\mathrm{v}}_{\mathrm{s}}^2\le {\mathrm{v}}_{\mathrm{s},\max }^2$$

(7)

$${\mathrm{i}}_{\mathrm{d}}^2+{\mathrm{i}}_{\mathrm{q}}^2={\mathrm{i}}_{\mathrm{s}}^2\le {\mathrm{i}}_{\mathrm{s},\max }^2$$

(8)

where v_s represents the vector amplitude of the stator voltage, v_s,max is the maximum output voltage of the inverter, i_s is the vector amplitude of the stator current, and i_s,max denotes the maximum output current of the inverter.

When the PMSM motor speed exceeds the rated speed, it is possible to ignore the voltage drop on the stator coil since they are insignificant in comparison to the back EMF on the stator winding. On the other hand, in the steady-state, the flux linkage is considered constant, resulting in the derivative of the current to zero $(\mathrm{d}{\mathrm{i}}_{\mathrm{d}}/\mathrm{d}\mathrm{t}=0;\mathrm{d}{\mathrm{i}}_{\mathrm{q}}/\mathrm{d}\mathrm{t}=0)$. Therefore, the stator voltage Equations in (1) and (2) are rewritten as follows:

$${\mathrm{v}}_{\mathrm{d}}=-{\mathsf{\omega }}_{\mathrm{e}}{\mathrm{L}}_{\mathrm{q}}{\mathrm{i}}_{\mathrm{q}}$$

(9)

$${\mathrm{v}}_{\mathrm{q}}={\mathsf{\omega }}_{\mathrm{e}}{\mathrm{L}}_{\mathrm{d}}{\mathrm{i}}_{\mathrm{d}}+{\mathsf{\omega }}_{\mathrm{e}}{\mathsf{\lambda }}_{\mathrm{p}\mathrm{m}}$$

(10)

Substitute (9) and (10) into (7), the stator voltage limit in the constant power region is described by (11).

$${\left({\mathrm{L}}_{\mathrm{q}}{\mathrm{i}}_{\mathrm{q}}\right)}^2+{\left({\mathrm{L}}_{\mathrm{d}}{\mathrm{i}}_{\mathrm{d}}+{\mathsf{\lambda }}_{\mathrm{p}\mathrm{m}}\right)}^2\le \frac{{\mathrm{v}}_{\mathrm{s},\max }^2}{{\mathsf{\omega }}_{\mathrm{e}}^2}$$

(11)

The voltage and current restriction relationships of the IPMSM in the whole region are shown in Figure 2, which indicates both of the voltage-limiting circle and current-limiting circle.

2.2. Principle of Transferring Power from the Inverter to the Motor

Consider the static state of the PMSM working with speed ω₁ and electromagnetic torque T_e1. the electromagnetic torque must balance with the load torque (friction torque included) to keep the speed constant. At this time, the motor provides the power P₁ (P₁ = ω₁ × T_e1). Ignore the energy conversion losses and the stator winding losses, and the PMSM power consumption is equal to the inverter’s electrical power.

The inverter needs to provide a power P₂ (P₂ = ω₂ × T_e2) in converting this operating point to a new one with velocity ω₂ and electromagnetic torque T_e2. Thus, the working condition change is to change the power supplied to the motor from the inverter.

Changing the rotor speed cannot be instantaneous but takes a short time with a specific route. During this process, the inverter’s power can change with the principle that if the power increases, the speed will increase, and vice versa.

When the rotor rotates once, the permanent magnet’s magnetic field will sweep Pp times the number of coils on it. Thus, the BEMF generated when the permanent magnet’s magnetic field turns with the speed ω_m induced on the stator coil is determined by Faraday’s induction law (12).

$$\mathrm{E}=\mathrm{P}\mathrm{p}\times {\mathsf{\omega }}_{\mathrm{m}}\times {\mathsf{\lambda }}_{\mathrm{p}\mathrm{m}}$$

(12)

With the synchronous motor in the stationary state, the magnetic field speed equals the rotor speed, ω_e = ω_m × Pp. The Equation (12) is rewritten according to the stator magnetic field rate, as shown in (8).

$$\mathrm{E}={\mathsf{\omega }}_{\mathrm{e}}\times {\mathsf{\lambda }}_{\mathrm{p}\mathrm{m}}$$

(13)

When two alternating current sources have the same rotating speed of the magnetic field, based on recommendations in [33], Figure 3 represented the equivalent circuit of these two voltage sources.

According to Figure 1, the transparent power transmitted to the BEMF terminal is determined according to (14) with the current determined according to (15):

$$\mathrm{S}=3\times \mathrm{E}\times {\mathrm{I}}_{\mathrm{S}}$$

(14)

$${\mathrm{I}}_{\mathrm{S}}=\frac{\left|{\mathrm{U}}_{\mathrm{S}}\right|\angle \mathsf{\delta }-\left|\mathrm{E}\right|\angle 0^0}{\left|{\mathrm{Z}}_{\mathrm{S}}\right|\angle \mathsf{\beta }}$$

(15)

From Equations (14) and (15), we get

$$\mathrm{S}=3\frac{\left|{\mathrm{U}}_{\mathrm{S}}\right|\times \left|\mathrm{E}\right|}{\left|{\mathrm{Z}}_{\mathrm{S}}\right|}\angle \left(\mathsf{\beta }-\mathsf{\delta }\right)-3\frac{{\left|\mathrm{E}\right|}^2}{\left|{\mathrm{Z}}_{\mathrm{S}}\right|}\angle \mathsf{\beta }$$

(16)

The active and reactive power is derived from (16) and determined according to (17) and (18) respectively.

$$\mathrm{P}=3\frac{\left|{\mathrm{U}}_{\mathrm{S}}\right|\times \left|\mathrm{E}\right|}{\left|{\mathrm{Z}}_{\mathrm{S}}\right|}\cos \left(\mathsf{\beta }-\mathsf{\delta }\right)-3\frac{{\left|\mathrm{E}\right|}^2}{\left|{\mathrm{Z}}_{\mathrm{S}}\right|}\cos \mathsf{\beta }$$

(17)

$$\mathrm{Q}=3\frac{\left|{\mathrm{U}}_{\mathrm{S}}\right|\times \left|\mathrm{E}\right|}{\left|{\mathrm{Z}}_{\mathrm{S}}\right|}\sin \left(\mathsf{\beta }-\mathsf{\delta }\right)-3\frac{{\left|\mathrm{E}\right|}^2}{\left|{\mathrm{Z}}_{\mathrm{S}}\right|}\sin \mathsf{\beta }$$

(18)

If resistance is neglected, then Zs = X_L and β = 90^°. At this time, Equations (17) and (18) are rewritten as (19) and (20), respectively, as follows.

$$\mathrm{P}=3\frac{\left|{\mathrm{U}}_{\mathrm{S}}\right|\times \left|\mathrm{E}\right|}{{\mathrm{X}}_{\mathrm{L}}}\sin \mathsf{\delta }$$

(19)

$$\mathrm{Q}=3\frac{\left|\mathrm{E}\right|}{{\mathrm{X}}_{\mathrm{L}}}\left(\left|{\mathrm{U}}_{\mathrm{S}}\right|\cos \mathsf{\delta }-\left|\mathrm{E}\right|\right)$$

(20)

The active power P transmitted between the two voltage sources is determined by the Equation (21) where δ is an angular deviation between the phase of inverters voltage vector and phase of back electromotive force vector in the stator winding, as shown in Figure 4. In order to stabilize PMSM operations under electric motor drive mode, δ is kept within (0; π/2).

$$\mathrm{P}=3\frac{\left|{\mathrm{U}}_{\mathrm{S}}\right|\times \left|\mathrm{E}\right|\times \sin \mathsf{\delta }}{{\mathrm{X}}_{\mathrm{L}}}$$

(21)

The two voltage sources’ impedance value is kept constant in the stationary state, with a constant rotor velocity according to (23), with a new inductance determined according to (10). The amplitude of BEMF is determined based on (13).

$$\mathrm{L}={\mathrm{L}}_{\mathrm{d}}={\mathrm{L}}_{\mathrm{q}}$$

(22)

$$({\mathrm{X}}_{\mathrm{L}}={\mathsf{\omega }}_{\mathrm{e}}\times \mathrm{L})$$

(23)

Combined with (13) and (23), the power applied to the motor from the inverter is determined according to (24).

$$\mathrm{P}=3\frac{\left|{\mathrm{U}}_{\mathrm{S}}\right|\times {\mathsf{\lambda }}_{\mathrm{p}\mathrm{m}}\times \sin \mathsf{\delta }}{\mathrm{L}}$$

(24)

During operation, L and λ_pm have slight changes due to magnetic saturation and magnet temperature. However, in comparison with U_S and δ, this change is not significant and can be ignored. Therefore, we can consider that L and λ_pm are constants in the control process. We can calculate transmitted power based on (25) where constant k = 3 × λ_pm/L.

$$\mathrm{P}=\mathrm{k}\times \left|{\mathrm{U}}_{\mathrm{S}}\right|\times \sin \mathsf{\delta }$$

(25)

During operation, the inverter provides active power P to the motor. Reactive power Q does not participate in the transfer of energy from the stator to the rotor, but only serves as the power supply for the magnetization of the magnetic circuits. The reactive power Q only plays a specific role in the switching support, so the goal is to minimize the reactive power Q supplied to the BEMF source.

The goal in motor control is zero reactive power QR at the BEMF end. Thus, the reactive power Q at the stator head only compensates for the magnetic circuit’s loss. The reactive power Q obtained at the BEMF input is determined according to (20).

For Q = 0, then U_S must satisfy (26).

$${\mathrm{U}}_{\mathrm{S}}=\frac{\mathrm{E}}{\cos \mathsf{\delta }}$$

(26)

When considering the internal resistance of the stator winding, the voltage drop across the winding is determined as follows:

$$\Delta \mathrm{U}={\mathrm{I}}_{\mathrm{s}}\mathrm{Z}=\left({\mathrm{I}}_{\mathrm{d}}+\mathrm{j}{\mathrm{I}}_{\mathrm{q}}\right)\left({\mathrm{R}}_{\mathrm{S}}+\mathrm{j}{\mathrm{X}}_{\mathrm{L}}\right)=\left({\mathrm{I}}_{\mathrm{d}}{\mathrm{R}}_{\mathrm{S}}-{\mathrm{I}}_{\mathrm{q}}{\mathrm{X}}_{\mathrm{L}}\right)+\mathrm{j}\left({\mathrm{I}}_{\mathrm{q}}{\mathrm{R}}_{\mathrm{S}}+{\mathrm{I}}_{\mathrm{d}}{\mathrm{X}}_{\mathrm{L}}\right)$$

(27)

Due to $\left({\mathrm{I}}_{\mathrm{q}}{\mathrm{R}}_{\mathrm{S}}+{\mathrm{I}}_{\mathrm{d}}{\mathrm{X}}_{\mathrm{L}}\right)\gg \left({\mathrm{I}}_{\mathrm{d}}{\mathrm{R}}_{\mathrm{S}}-{\mathrm{I}}_{\mathrm{q}}{\mathrm{X}}_{\mathrm{L}}\right)$, it leads to the voltage drop magnitude which is calculated by (28).

$$\left|\Delta \mathrm{U}\right|\simeq \left({\mathrm{I}}_{\mathrm{q}}{\mathrm{R}}_{\mathrm{S}}+{\mathrm{I}}_{\mathrm{d}}{\mathrm{X}}_{\mathrm{L}}\right)$$

(28)

Combined (26) and (28), the inverter voltage amplitude is calculated according to (29).

$$\left|{\mathrm{U}}_{\mathrm{S}}\right|=\left|\frac{\mathrm{E}}{\cos \mathsf{\delta }}\right|+\left|\left({\mathrm{I}}_{\mathrm{q}}{\mathrm{R}}_{\mathrm{S}}+{\mathrm{I}}_{\mathrm{d}}{\mathsf{\omega }}_{\mathrm{e}}\mathrm{L}\right)\right|$$

(29)

Thus, the active power P to change the working point from the point with power P₁ to the operation point with power P₂ is determined by adjusting the inverter voltage range. The angle δ is calibrated to the new working point, so that the reactive power supplied to the BEMF is zero.

3. The Principle of Controlling the Speed of a PMSM Motor Is Based on the Power Control PQ

The control diagram of the PMSM motor is shown in Figure 5 below, based on the control principle analyzed above.

• The active power is adjusted through the speed PI controller (PI_S) from the actual speed difference compared to the reference speed.
• The power PI controller is used to change the inverter voltage amplitude, to change the power supplied to the inverter according to the reference power,
• To ensure that reactive power Q is not fed to the BEMF source, the inverter voltage must be kept at a value according to (18) and must not exceed the inverter limit value to optimize motor operation. Inverter voltage must be kept at a determined value; some reactive power must be added (or drawn) from the inverters. The determination of reactive power Q is done via a voltage controller with the input of the error between the reference voltage and the actual voltage at the inverter output.
• The reactive power PI controller corrects the voltage angle δ between U and E, respectively, to change the reactive power to the reference value. The BEMF vector angle is always perpendicular to the direction of the d-axis current component, as shown in Figure 4; the voltage across the inverters’ coordinate system dq is determined according to (19) and (20).

$${\mathrm{v}}_{\mathrm{d}}=-\left|{\mathrm{U}}_{\mathrm{S}}\right|\sin \left(\mathsf{\delta }\right)$$

(30)

$${\mathrm{v}}_{\mathrm{q}}=\left|{\mathrm{U}}_{\mathrm{S}}\right|\cos \left(\mathsf{\delta }\right)$$

(31)

4. Simulation Validation

4.1. Simulation IPMSM Speed Control Model

The PMSM motor speed control model is built on the simulation software Matlab/Simulink used to evaluate the proposed control efficiency.

Table 1. PMSM parameters.

Parameter	Symbol	Value	Unit
Rated power		400	W
Rated phase current	${\mathrm{I}}_{\max }$	2.7	Arms
Rated speed	${\mathsf{\omega }}_{\mathrm{rate}}$	3000	RPM
Rated torque	${\mathrm{T}}_{\mathrm{rate}}$	1.27	N.m
Number of poles pairs	P	4
Stator resistance	$\mathrm{R}$	2.35	Ω
Stator inductance	${\mathrm{L}=\mathrm{L}}_{\mathrm{d}}{=\mathrm{L}}_{\mathrm{q}}$	8.5	mH
Flux linkage	${\mathsf{\lambda }}_{\mathrm{pm}}$	0.0615	Wb
Inertia	J	167 × 10−6	Kg.m2
Viscous friction	F	106.9 × 10−6	N.m.s/rad
DC Bus Voltage	${\mathrm{V}}_{\mathrm{dc}}$	200	V

shows the PMSM motors’ parameters used in the simulation; these parameters belong to the EMJ-04APB22 motor, which Wang et al. used in their study [34]. The Texas Instruments Company recommends the use of the EMJ-04APB22 motor in the quality assessment of controllers. The model is built with three main blocks, the PMSM motor block, the control unit, and the inverter block Figure 6 shows the simulation PMSM model. The Vector Current Control (VCC) method [35] and the Direct Current Calculation (DCC) method [36] were used to evaluate the proposed control method’s effectiveness. The simulation results under different operating conditions are satisfactorily introduced in detail in the following sections.

4.2. Simulation Results

The PMSM motor is used to test the proposed method’s control quality in constant torque and constant power zones at various speeds. The speed variation between two operating zones, as well as within the same operating region of PMSM, is used to evaluate the effect of extending the operating speed range and minimizing the fluctuations in electric torque, current, and associated flux of PMSM in the FW region of the proposed method. Figure 7 shows the speed control results. The obtained results show that the control methods can adhere well to the reference speed.

The under-rated speed enlargement results show that DCC and VCC control methods got lower overshoot than the proposed method. The reason for this is that when controlling the constant torque region, it is much more efficient to change the speed by changing the current when having to indirectly control the speed by holding power inject to the motor. Although the response is slower, the proposed method is more stable, with the speed’s ripple less than the other DCC and VCC ways.

The speed enlargement results in the FW region show that the two DCC and VCC methods also have a lower overshoot than the proposed method. However, these two methods’ speed ripple is much higher, and thus the quality of speed control is also significantly reduced. The reason is that when entering the voltage limit area, the method of current management and voltage correction through the current controller will no longer be as effective as in the constant torque region. Then, the proposed method shows the advantages when the two control zones’ speed control quality is the same and stable.

Figure 8a shows the results of the electromagnetic moment of comparative control methods. The simulation results show that all three approaches give a small ripple in the constant moment region. Electromagnetic torque responds to changes in the load torque and the need for the engine’s acceleration or deceleration speed. Figure 8b shows electromagnetic moment results at a constant power range. The electromagnetic moment oscillation of VCC, DCC and the proposed approach is (0.35–0.6) N.m, (0.38–0.5) N.m, (0.42–0.48) N.m, respectively. The result of the flux linkage ripple percent is 26.3%, 13.64%, and 6.67%, respectively. This means that the VCC and DCC methods’ electromagnetic torque has a more significant ripple than the proposed method while entering the FW region. The results demonstrate the proposed method’s ability to minimize the electromagnetic torque ripple. The task of reducing electromagnetic surges plays a crucial role in improving not only the control quality but also the electric drive system’s stability.

The stator winding power loss is determined based on the stator current and stator resistance, as shown in (21). Figure 9 shows the power loss simulation results. The results show that the proposed control method has the same effect of minimizing losses as the other comparative methods. These results show that the proposed control method effectively reduces power loss as the current-based control method, similar to the FOC control method.

$${\mathrm{P}}_{\mathrm{loss}}={\mathrm{I}}_{\mathrm{s}}^2.{\mathrm{R}}_{\mathrm{s}}$$

(32)

Figure 10a shows the flux linkage results. The flux linkage in the control method is greater than that of the flux linkage of the permanent magnet when operating in the constant torque region. The flux linkage decreases when entering the constant power region. Figure 10b shows the flux linkage results at a constant power range. It is important to note that flux linkage oscillation of VCC, DCC and the proposed approach is (0.038–0.045) Wb, (0.04–0.043) Wb, (0.04–0.041) Wb, respectively. This leads to the flux linkage ripple percentage, at 18%, 7.5%, and 2.5%, respectively. Hence, the VCC and DCC methods’ flux linkage value has a more significant ripple than the proposed method.

Figure 11a shows the resulting stator currents controlled by different methods. The similarity of the currents within the constant moment range can be seen immediately. However, when entering the FW region, the quality of the stator current has a big difference. Zooming in on these results in Figure 11b from 3.2 s to 3.5 s shows the difference between the comparative methods. The current fluctuation of the VCC method is the most significant (2.4–3.3) A, with a fluctuation rate of 15.8%. On the contrary the current instability of the proposed method is the smallest (2.6–2.8) A, with a fluctuation rate of 3.7%. Hence, a current ripple of the proposed method is significantly smaller than two other comparative approaches in the FW mode.

Figure 12 shows the voltage simulation results on the stator winding. The recorded results show that the proposed method’s voltage is more stable than the DCC and VCC methods when operating in the FW region. In the area of the constant moment, the ripple of the current and voltage amplitude is equivalent. Moreover, it also shows the superiority of the proposed method in expanding the range of PMSM operating speeds in comparison with the DCC and VCC control methods.

5. Conclusions

The paper proposes an advanced method to expand the PMSM speed range by controlling the inverter power instead of the conventional current control approach. Speed control based on power balance allows flexible control, no longer considering higher or lower rotor speed conditions along with switch control strategies. After comprehensive simulations, it is possible to state that the proposed control method is capable of stable operation from 300 to 5500 rpm, regardless of changing the PMSM speed control method between two separate working stages: the constant power region and constant torque region. By controlling the power injected into the PMSM, the proposed control method allows for reasonable control of the storage systems’ discharge power, necessary in mobile drive applications. The proposed control method contains the same number of control loops, and does not need to provide any additional data compared to the two comparative control methods. In comparing simulation results with the two traditional DCC and VCC control methods, the proposed control method has proven to improve not only control quality and stability, but also to minimize electromagnetic torque and current ripples. The fact is that, during PMSM operation, the algorithm uses PMSM parameters to calculate the voltage to be supplied by the inverter, which leads to the requirement of accurately determining PMSM parameters. These limitations lead to the subsequent development direction of the research, to further improve the control efficiency of the proposed PMSM speed control algorithm.