1 Introduction

With the development of wind power system towards to high power, permanent magnet synchronous motor (PMSM) based on direct-drive wind power system is receiving more and more attention^[1-3]. Compared to the vector control of PMSM, direct torque control has faster torque response speed, which makes it more suitable for wind power system^[4-5].

The observation of stator flux and electromagnetic torque is the basis of direct torque control^[6-9]. The commonly used stator flux observation method mainly includes the voltage model based stator flux observer and the current model based stator flux observer.

The traditional voltage model based stator flux observer is less sensitive to motor parameters. However, the pure integral operation contained in this stator flux observer is affected by the integral initial values, integral drift and the DC offset. Meanwhile, although the conventional current model based stator flux observer is robust to the pure integral operation, it is sensitive to motor parameters. As the generator often operates in the high speed region of wind power system, the voltage model based stator flux observer is more appropriate. In Refs. ^[10-15], different improved voltage models based stator flux observers were proposed and researched. In Ref. ^[10], a low-pass filter was adopted instead of the pure integral operation to eliminate the effect of integral initial values and integral drift on stator flux observation precision, but the DC offset inhibition ability is limited. In Ref. ^[11], a cascading low-pass filter is adopted to achieve the stator flux observation and the observer’s dynamic response is improved through the regulation of the order of the low-pass filter and phase-amplitude compensation module. However, its DC offset inhibition ability is still limited. Different to Refs. ^[10-15], a high-pass filter is used to replace the pure integral operation and an improved stator flux observation method is achieved in Ref. ^[12]. In this observer, the DC offset is eliminated. However, the low-order harmonics, such as 3th, 5th, 7th and so on, contained in the stator voltage and current will affect the stator flux observation precision. Band-pass filter was used to take the place of the pure integral operation in order to improve stator flux observation precision in Refs. ^[13-14]. However, the design procedure is too complicated. In Ref. ^[15], a new stator flux observer was designed based on the fact that the electromotive force is orthogonal to the stator flux, but another two proportion integral coefficients are needed. A neural network based stator flux observer was proposed in Ref. ^[16]. But this method cannot be used in engineering practice. In Ref. ^[17], a stator flux observation algorithm based on the sliding mode observer is proposed. Although the sliding mode observer is robust to the external disturbance, its DC offset inhibition ability is limited as the methods proposed in Refs. ^[10-11]. In Refs. ^[18-20], a voltage and current model based stator flux observer is researched. But the effects of the motor inductance parameters remain to be further studied.

In recent years, Zhao et al. designed a new stator flux observer in Ref. ^[21] based on the second-order generalized integrator (SOGI), which is widely applied to the grid inverter control system. However, the method is sensitive to DC offset too.

In this paper, the method proposed in Ref. ^[21] and influenced by DC offset is analyzed and revealed, then, in order to eliminate the effect of DC offset in the observer proposed in Ref. ^[21], an improved SOGI based stator flux observer is presented and then, system simulation and experimental results of stator flux and torque observation based on the SOGI and the improved SOGI are shown and compared, at last, the conclusion is given.

2 The Stator Flux Observation Method of PMSM Based on SOGI

The pure integral operation contained in this stator flux observer based on voltage model is affected by the integral initial values and integral drift, therefore, an improved voltage model based stator flux observer is proposed in Ref. ^[21] based on SOGI. However, this method is sensitive to DC offset and it will be analyzed as follows.

Fig. 1 Block diagram of SOGI

The transfer function of SOGI meets:

$\frac{{{u}_{0}}\left( s \right)}{u\left( s \right)}=\frac{k{{\omega }_{e}}s}{{{s}^{2}}+k{{\omega }_{e}}s+{{\omega }_{e}}^{2}}$

(1)

$\frac{q\left( s \right)}{u\left( s \right)}=\frac{k{{\omega }_{e}}^{2}}{{{s}^{2}}+k{{\omega }_{e}}s+{{\omega }_{e}}^{2}}$

(2)

When s=jω_e, Eqs.(1) and (2) can be simplified as:

$\left\{ \begin{array}{*{35}{l}} \frac{{{u}_{0}}\left( s \right)}{u\left( s \right)}=1 \\ \frac{q\left( s \right)}{u\left( s \right)}=\frac{1}{j} \\ \end{array} \right.$

(3)

It can be seen from Eq.(3) that when the frequency of input signal u(s) of SOGI is ω_e, the output signal u₀(s) is the same with u(s) and the output signal q(s) is orthogonal to the input signal u(s). Due to this feature, SOGI is widely used in the control system of single phase grid-connected inverters.

Meanwhile, the transfer function of ψ₀ and u can be expressed as:

$\frac{{{\psi }_{0}}\left( s \right)}{u\left( s \right)}=\frac{k{{\omega }_{e}}}{{{s}^{2}}+k{{\omega }_{e}}s+{{\omega }_{e}}^{2}}$

(4)

When s=jω_e, Eq.(4) can be obtained as:

$\frac{{{\psi }_{0}}\left( s \right)}{u\left( s \right)}=\frac{1}{j{{\omega }_{e}}}$

(5)

It is apparent that the signal ψ₀(s) is the integration of input signal u(s) and it is independent of the integral initial values and integral drift. So the SOGI can be used to observe the stator flux of PMSM. If the input signal u(s) is the stator back electromotive force of PMSM, the output signal ψ₀(s) is the observed stator flux. This is the stator flux observer proposed in Ref. ^[21] by Zhao. However, this method is sensitive to DC offset and it will be analyzed in the following sections.

Fig. 2 shows the bode diagram of the transfer function given in Eq.(4) when the synchronous frequency ω_e is 31.4 rad/s. As can be seen from Fig. 2, when k is increased, the gain of DC offset is magnified, which means that the DC offset inhibiting ability of Eq.(4) is weakened when k is increased. However, even when k is increased to 10, the gain of DC offset is still less than zero, which means that the algorithm shown in Eq.(4) still has the ability to restrain DC deviator.

Similarly to Fig. 2, Fig. 3 is given when ω_e=3.14 rad/s. From Fig. 3, it can be seen obviously that when k is increased to 10, the gain of DC offset is large than zero, i.e. the influence of DC offset is magnified. So it is concluded that the strategy proposed in Ref.^[21] is easily influenced by DC offset. So an improved stator flux observation technique based on SOGI is proposed in this paper and it will be described in the next part.

3 Improved Stator Flux Observation Method Based on SOGI

In the actual motor control system, the sampled voltage and current always contains DC bias. In this situation, when the stator flux observation method proposed in Ref.^[21] is adopted, the obtained stator flux will contain DC offset, therefore, an improved stator flux observation method is proposed based on an improved SOGI as shown in Fig. 4.

Compared to Fig. 1, the improved SOGI adds a branch. This added branch can estimate the DC offset contained in the input signal and the estimated DC offset is subtracted from the input signal. As a result, the output signal is insensitive to DC offset. Thus, the stator flux can be observed accurately based on this improved SOGI. The detailed analysis will be given in the following sections.

The transfer function of the improved SOGI meets:

$\frac{{{u}_{0}}\left( s \right)}{u\left( s \right)}=\frac{{{k}_{0}}{{\omega }_{e}}({{s}^{2}}+{{\omega }_{e}}^{2})}{{{s}^{3}}+({{k}_{0}}+k){{\omega }_{e}}{{s}^{2}}+{{\omega }_{e}}^{2}s+{{k}_{0}}{{\omega }_{e}}^{3}}$

(6)

$\frac{{{\psi }_{0}}\left( s \right)}{u\left( s \right)}=\frac{k{{\omega }_{e}}s}{{{s}^{3}}+({{k}_{0}}+k){{\omega }_{e}}{{s}^{2}}+{{\omega }_{e}}^{2}s+{{k}_{0}}{{\omega }_{e}}^{3}}$

(7)

Choosing ω_e=3.14 rad/s and k₀=1, the bode diagram of the transfer functions given in Eqs.(6) and (7) are shown in Figs. 5 and 6.

From Fig. 5, it is seen that whatever k is, the transfer function of Eq.(6) is an all-pass filter for the DC signal. Besides, it is a band stop filter for the signal whose frequency is ω_e. Because of this property, the DC offset contained in the input signal can be observed by Eq.(6) . Meanwhile, although the gain of DC offset of Eq.(7) as shown in Fig. 6 is enlarged when k is increased, it is still less than zero, which means that the DC offset can be still restrained compared to Fig. 3. Substitute s=0 into Eq.(7) , the result of Eq.(7) is zero, which means that in theory the DC offset can be eliminated by Eq.(7) completely. What’s more, substitute s=jω_e into Eq.(7) , Eq.(8) is obtained as:

$\frac{{{\psi }_{0}}\left( s \right)}{u\left( s \right)}=\frac{1}{j{{\omega }_{e}}}$

(8)

From Eq.(8) , it is known that the improved SOGI can be used to observe the stator flux.

Besides, the design of k₀ is researched. Choosing ω_e=3.14 rad/s and k=1, the bode diagram of the transfer function given in Eq.(7) is shown in Fig. 7 with different k₀.

It is seen from Fig. 7 that if k₀ is reduced, the low-frequency gain of Eq.(7) is increased. Meanwhile, if k₀ is too large, the resonance peak of Eq.(7) is increased. As a result, in order to decrease the low-frequency gain with the resonance peak of Eq.(7) , a compromise value of k₀ must be chosen. In this paper, according to Fig. 7, k₀ is suggested to be chosen between 0.2 and 3.

Finally, the proposed stator flux observer based on improved SOGI is obtained as shown in Fig. 8. In Fig. 8, u_sα and u_sβ are the stator voltage of PMSM in the stationary α-β reference frame, and i_sα and i_sβ are the stator current of PMSM in the stationary α-β reference frame, and R_s is stator resistance, and ψ_sα and ψ_sβ are the observed stator flux.

Based on Fig. 8, the electromagnetic torque can be observed based on Eq.(9) .

${{T}_{e}}={{p}_{n}}({{\psi }_{s\alpha }}{{i}_{s\beta }}-{{\psi }_{s\beta }}{{i}_{s\alpha }})$

(9)

where T_e is the observed electromagnetic torque and p_n is the pole pairs of PMSM.

4 Simulation Results

The rated parameters of PMSM used in the simulation and experiment are listed as U_n=380 V, f_n=50 Hz, T_n=70 Nm, n_p=2, R_s=0.6 Ω, L_d=24 mH, L_q=24 mH, and f=1.2 Wb.

In the simulation, the PMSM is operating at 600 r/min with i_d=0, i_q =10 A. A DC offset of -10 V is added to u_sα at 0.4 s, meanwhile, another DC offset of 10 V is added to u_sβ at 0.4 s, then, the speed steps down to 300 r/min at 0.8 s. It should be pointed out that the simulation model is established based on Matlab/Simulink. The DC offsets are added to u_sα and u_sβ respectively based on two step signal sources in Matlab/Simulink.

In the first simulation, the method proposed in Ref. ^[21] is tested and the simulation results are given in Fig. 9. As shown in Fig. 9(a), the actual electromagnetic torque is about 35 Nm in the whole simulation procedure. When DC offset is added to the stator voltage at 0.4 s, a DC drift is appeared in the observed stator flux ${{{\hat{\psi }}}_{s\alpha }}$、${{{\hat{\psi }}}_{s\beta }}$ as shown in Fig. 9(b). Meanwhile, a big stator flux observation error is appeared as shown in Fig. 9(c). Due to this DC drift, the estimated toque contains a large fundamental frequency pulsation. When the speed decreases to 300 r/min, the influence of the added DC offset is enlarged obviously as shown in Figs. 9(c) and 9(d).

In the second simulation, the method proposed in this paper is tested and the simulation results are given in Fig. 10. As can be seen from Figs. 10(a) and 10(b), although the stator flux observation errors are enlarged immediately when the DC offsets are added suddenly at 0.4 s, the errors decrease to zero fast within less than 0.1 s. As a result, the observed stator flux and electromagnetic torque is not affected by the added DC offset, which means that the stator flux and torque observation precision is improved. Besides, the added DC offset can be estimated too, as shown in Figs. 10(d) and 10(e). Meanwhile, it should be pointed out that the electro-magnetic torque observation error is large at 0.8 s. This is due to the sudden step of the speed in the simulation. In the practical PMSM control system, the speed step rarely happens due to the large inertia of PMSM. Thus the electromagnetic torque observation error is limited.

In the third simulation, the harmonic suppression characteristic of the proposed method is tested and compared to the conventional method proposed in Ref.^[21]. In this simulation, the PMSM is operating at 300 r/min with i_d=0, i_q =10 A. A harmonic current as shown in Eq.(10) is injected into the sampled current in order to test the harmonic suppression characteristic. The simulation result of the proposed method is shown in Fig. 11(a), and the simulation result of the conventional method is shown in Fig. 11(b).

${{i}_{3}}=3\text{sin}(60\pi t)$

(10)

It is apparent to see from Fig. 11 that when aharmonic current as shown in Eq.(10) is injected into the sampled current, a 3th harmonic is appeared in the stator flux observation error. Both the proposed method and the conventional method have similar stator flux observation error, which means that both of the two methods have similar harmonic suppression ability.

5 Experimental Results

In the experiment, an induction motor controlled by another converter is used as a prime mover to drive the tested PMSM. The control system is developed based on DSP 28335. The sample frequency is 4 kHz and the switching frequency is 2 kHz. The proposed stator flux and electromagnetic torque observation method is achieved in the discrete control system. In the first situation, the speed is controlled as 600 r/min. Two DC offsets of 10 V are added to the stator voltage u_sα and u_sβ respectively at the time marked in Fig. 12. It should be pointed out that in the DC offsets are added to the sampled stator voltage in the DSP, rather than added to actual stator voltage. As can be seen from Fig. 12, the observed stator flux and electromagnetic torque are heavily affected by the intentional added DC offset based on the method in Ref.^[21]. As a comparison, Fig. 13 is given based on the proposed method. As can be seen from Fig. 13, the influence of DC offset is completely eliminated.

In the second situation, the speed is controlled as 900 r/min. The same DC offset is added at the time marked in Fig. 14. Fig. 14 is given based on the method designed in Ref.^[21] and the result based on the proposed method is given in Fig. 15.

Compared Fig. 14 with Fig. 12, it can be seen when the speed is accelerated, the influence of DC offset is decreased. In other words, the lower the speed is, the stronger the influence of DC offset is. In addition, it is apparent that the influence of DC offset is completely eliminated based on the method proposed in this paper as can be seen from Fig. 15.

6 Conclusions

This paper proposes an improved stator flux and electromagnetic torque observation method of PMSM based on an improved SOGI. Compared with the SOGI method, the simulation and experimental results verity that the improved method is robust to the DC offset contained in the stator voltage or current of PMSM and the stator flux and electromagnetic torque observation accuracy is improved effectively.