A double fed induction generator (DFIG) is a popular wind turbine system due to its high energy efficiency, reduced mechanical stress on the wind turbine, and relatively low power rating of the connected power electronics converter of low costs [1-4]. The vector control of the rotor current is used to control the active and reactive power independently and stably using a rotating reference frame fixed on the gap flux [5-7]. Therefore, precise measurement of the rotor currents is very important in the vector control [8, 9].
The rotor- and grid-side currents are measured from the current sensors of the converter through low pass filters and A/D converters. In this process, current measurement errors can be generated due to the non-linearity of the current sensors, quantization error of A/D converters, and the thermal drift of the analog electric elements. These errors cause the ripples in the produced power of the DFIG [10, 17]. The errors can be classified as offset errors, scaling errors and non-linearity errors.
In the rotor side control, the ripples by offset and scaling errors have one and two times component of the fundamental rotor current frequency, respectively. Then the ripples of the rotor side lead to the fluctuations of the stator side. The ripple frequency by the offset error is same as the difference between the grid frequency and the slip frequency, and it by the scaling error is same as the difference between the grid frequency and two times of the slip frequency .
In , a method was proposed in order to solve the problem of the offset and scaling errors in the DFIG system. The method obtains the two errors directly from the phase currents of the rotor side, and compensates the two errors on the current measurement path. It can improve the steady state performance of the DFIG by eliminating the effects of the offset and scaling error. However, this method cannot guarantee the complete compensation in the transient state because the phase currents of the stationary reference frame are used as inputs for the calculation of the errors.
This paper analyzes the error components of the offset and the scaling on the basis of the synchronous reference frame of the DFIG, and proposes a compensation method to reduce the effect of the errors. The proposed method adopts the synchronous d-axis current of the rotor as the input signal for compensation due to the several advantages. This method can be simply implemented by calculating the several integral and subtracting operations. Also the method is well operated not only under the steady state but also under the transient state, and robust to the variation of the machine parameters. The validity of the proposed method is verified through the experiments.
2. Description of DFIG [1-7]
Fig. 1 illustrates the configuration of a DFIG system. As shown in Fig. 1, the stator magnetizing current is obtained from the grid.
Fig. 1.Configuration of DFIG system
The voltage equations and the stator flux of the synchronous reference frame are given in (1-3) and (4).
where ωe is the synchronous angular speed, Ls is the selfof stator and Lm is inductance the magnetizing inductance.
The voltage equations and the rotor flux of the synchronous reference frame are given in (5-7) and (8).
where ωr is the angular speed of the rotor.
The produced active and the reactive powers of DFIG are given by (9) and (10).
By using (1-3), and (4), the active and reactive power equations of DFIG can be rewritten as follows:
As known in (11) and (12), the synchronous d- and q-axis currents of the rotor are directly proportional to the reactive and active powers of a DFIG, respectively. In this paper, two rotor current sensors are used for the vector control of back-to-back converter of a DFIG.
3. Effect of Measurement Error of Rotor Current [19, 20]
Fig. 2 shows the measurement path of the rotor currents for DFIG control. Because of the non-linearity of the hall sensors, the thermal drift of the analog elements, quantization error of the A/D converters, and unbalance of each element, the errors are generated from current measurement path and inevitable even if the control system is well designed and constructed.
3.1 Effect of offset error
Fig. 3 shows the block diagram of the calculating process about the influence of rotor phase current offset error.
The offset errors may be caused by an imbalance of current sensors and the measurement path or other problems as mentioned above. The offset errors are calculated as:
Fig. 2.Path of rotor phase current measurement
Fig. 3.Effect of rotor phase current offset error
where iar, ibr are ideal a- and b-phase rotor currents and ΔIar , ΔIbr are offsets of a- and b-phase, respectively.
From (13), the synchronous d-q axis ripple currents of the rotor are expressed as:
where sωe is the slip angular speed of the rotor, Iedr_offset and Ieqr_offset are the ripple components of the rotor currents due to the offset error, respectively. As known from (14) and (15), the synchronous d- and q- axis currents of the rotor have the fundamental components of the slip frequency.
The synchronous d- and q- axis current ripples by the offset in the stator can be derived in (16) and (17), respectively.
The current ripples of the 3-phase reference frame can be an be obtained in (18, 19) and (20).
where , and are the offset errors of 3-phase current. As known from (18, 19) and (20), the frequency component of the stator has (1–s) due to the offset effect of the rotor side.
3.2 Effect of scaling error
Fig. 4 shows the block diagram of the calculating process about the influence of rotor phase current scaling error.
If the rotor currents contain scaling errors without offsets, they can be expressed as follows:
where, I is the real value of the phase current without offset and scaling errors, Ka and Kb denote the scale factor of a-, and b- phase currents, respectively.
From (21), the synchronous d-q axis ripple currents of the rotor can be derived as:
The synchronous d- and q- axis current ripples by the scaling error in the stator can be derived in (24) and (25), respectively.
The current ripples of the 3-phase reference frame can be obtained in (26, 27) and (28).
Fig. 4.Effect of rotor phase current scaling error
where , and are the scale errors of 3-phase current. As known from (26), (27) and (28), the frequency component of the stator has (1 – 2s) due to the scaling error of the rotor side.
4. Detection and Compensation of Current Errors
4.1 Input signal adoption for compensation
Among previous equations, the synchronous d- and qaxis currents of the rotor and stator can be selected for the input signal of the compensator. The q-axis of the stator current is not suitable for the input signal of the proposed compensator because the synchronous q-axis current of the stator is always fluctuating according to the intensity of the wind. The offset errors of the rotor and stator are given in (14) and (16), respectively. The scaling errors of the rotor and stator are given in (22) and (24), respectively. As known from (14, 16, 22) and (24), the amplitude of the stator current is lower than that of the rotor current. Moreover, the synchronous d-axis current of the rotor is nearly zero or constant for the vector control. Therefore, the input signal of the proposed compensator uses the synchronous d-axis current of the rotor in this paper as shown in Table 1.
4.2 Compensation of current measurement errors
The offset and scaling errors ( ΔIar , ΔIar , Ka – Kb) can be obtained from the selected (14) and (22) according to the specific rotor slip angle as shown in Table 2 and 3.
Due to the proportional integral (PI) current regulator of the synchronous d-and q-axis, DC components of (22) and (23) are compensated automatically, respectively. Therefore, the front sine term can be only considered regardless of the rear DC term in (22) and (23).
Table 1.Input signal adoption for compensation
This compensation method using the Table 2 and 3 is very simple but difficult to bring the reliable error values due to the effects of inaccurate sampling point and the switching noise of IGBT.
However, the proposed compensation method can have high accuracy and reliability due to the use of the average values through continuous integrals.
In order to compensate the measurement errors, two parts of the integrating operation are needed. The first is the integral of detecting the offset error, and the second is detecting the scaling error.
The first parts are divided into two steps as shown in Figs. 5, and Fig. 6. The first step is the integral of the synchronous d-axis current of the rotor according to the slip angle θsl from 1/2π to 3/2π as shown in the sector I of Fig. 5. As shown in (29), the calculation makes the cancellation of the sine term of (14), and can obtain the offset-a, ΔIar .
Table 2.Detection of offset error on a specific slip angle
Table 3.Detection of scale error on a specific slip angle
Fig. 5.Detection of offset-a (ΔIar)
After acquiring the offset-a ( ΔIar ), the second step is the integral of the current of (14) from 0 to π according to the slip angle as shown in the sector II of Fig. 6. In this integral calculation, the cosine term of (14) is removed, and the offset-b ( ΔIbr ) can be obtained as shown in (30).
After that, the scaling error of the rotor currents can be obtained from the second part. As shown in Fig. 7, the error is acquired by integrating the current (18) from π/3 to 5π/6 as shown in (31).
Fig. 6.Detection of offset-b (ΔIbr)
Fig. 7.Detection of scale error (Ka , Kb)
Fig. 8 shows the block diagram of the proposed compensation scheme. The synchronous d-axis current of the rotor side is used for an input signal of the integrators as shown in Fig. 8. The offset regulator consists of two steps. One is for compensating the a-phase offset error, and the other is for the b-phase offset error. These two have integral type regulators (Koffset/s). These integral regulators force ε1 and ε2 to be zero as shown in Fig. 8. The controller for compensating the scaling error also has the same regulator of the offset error, and forces ε3 to be zero. These integral controllers have a memory function to store the compensated values of the offset and scaling, respectively. The proposed algorithm automatically compensates the offset and scaling errors by using the integral type regulators as shown in Fig. 8.
The gains (Koffset and Kscale) of the compensators can be set between 0 and 1, respectively. The smaller the gain is given, the slower the response and more accurate the compensating performance can be obtained. The final compensating equations are achieved as in (32) from the output (Offset_a, Offset_b, Scale_a,b) of the integral regulators as shown in Fig. 8.
Fig. 8.Block diagram of proposed compensation scheme
5. Experimental Results
Fig. 9 shows the experimental setup. Experimental tests were performed in a 3.3-kW DFIG test rig. The DFIG is coupled to the permanent magnet synchronous motor (PMSM) of 3-kW controlled by a variable speed driver providing torque and speed regulation. The rotor of DFIG and PMSM are controlled separately by microprocessor. The switching frequency of both converters is 5 kHz. Space vector PWM is used for switching pulse generation with a sampling frequency of 10 kHz. The parameters of the experimental system are shown in Table 4.
Fig. 10 shows the stator phase currents, the synchronous d-axis rotor current and their FFT results under the following conditions; ΔIar = 0.5, ΔIbr = 0.5, Ka = 0.99, and Kb=1.05. The rotor is rotating at 50Hz and the synchronous speed of this induction generator is 60 Hz. Hence, the slip frequency is 10 Hz. Fig. 10(b) is the d-axis current of the rotor side and its FFT result. The d-axis current has one and two times ripples of the slip frequency, fsl, by the offset and scaling error, respectively. As the result, the a- and bphase current of the stator side in Fig. 10(a) have (1 – s) and (1 – 2s) times ripples of the synchronous frequency, fe, respectively as shown in Figs. 10(c) and (d).
Fig. 9.Configuration of experimental system
Table 4.The Experimental Parameters
Fig. 11 shows the synchronous d-axis rotor current and its FFT result before the compensation of two errors. The experimental results were obtained under these error conditions; ΔIar = 0.1, ΔIbr = 0.3, Ka = 0.99, and Kb = 1.05. The conditions are the maximum tolerable errors of the worst case based on their datasheets of the current sensors, low pass filter, matching circuit, and A/D converter. As shown in FFT result of Fig.11, the synchronous d-axis current has one and two times of the slip frequency of the rotor. If the FFT result is compared with the previous waveform, the two times ripple by scaling error has the same amplitude and the one times ripple by the offset error is decreased due to the reduced offset.
Fig. 12 shows three output values of the compensator in the block diagram of Fig 8. At first, the offset-a and -b values are calculated by the two integrators, respectively. After that, the third integrator obtains the scaling error by integrating the two times ripple component until it is nearly zero. In this paper, Koffset = 0.1 and Kscale = 0.05 are chosen for stable and precise operation.
Fig. 10.Stator phase currents, synchronous d-axis rotor current and their FFT result
Fig. 11.Synchronous d-axis rotor current before compensating operation
Fig. 12.Characteristics of compensation
Fig. 13.D-axis current after compensating offset error
Fig. 14.D-axis current after compensating scaling error
Fig. 13 shows the compensation result removed the offset error of Fig. 11. As shown in the FFT result, the ripple of the slip frequency is eliminated, and the synchronous d-axis current has only two times of the slip frequency of the rotor. Then the scaling error compensator becomes active. As the result, the ripple components by both the offset and scaling errors are nearly compensated in the synchronous d-axis current of the rotor as shown in Fig.14. Therefore, power quality is improved and the vibrations of the generator by low frequency ripples are reduced because the stator phase currents do not have the low frequency ripples by offset and scaling errors.
This paper proposed a compensation method to solve the offset and scaling problem by the measurement errors of the current sensors in a DFIG. The principal feature of the proposed method is using the synchronous d-axis current of the rotor as an input signal of the compensator. Therefore, the proposed method has the several attractive features: robustness with regard to the variation of the machine variables, application to the steady and transient states, easy implementation. The feasibility and effectiveness of the proposed compensating method were verified through experimentation.