The method of harmonic synchronous reference frames

The frequency selective active filtering was implemented using the method of harmonic syn-chronous reference frames. Previously, the method has been used in (Bojrup et al., 1999;

Mattavelli, 2001; Ponnaluri and Brickwedde, 2001; Yano et al., 1997). The method identifies selected harmonic sequences by transforming them into the harmonic synchronous reference frames. The method is described as a block diagram in Fig. 5.6. The transformation of the supply current vectori_Sto the 5^thnegative sequence harmonic synchronous reference frame is calculated as

i⁵⁻_Sd = Re{(s⁵⁻)^∗i_S}=s⁵⁻_α i_Sα+s⁵⁻_β i_Sβ (5.1) i⁵⁻_Sq = Im{(s⁵⁻)^∗i_S}=s⁵⁻_α iSβ−s⁵⁻_β iSα , (5.2)

5.5 Frequency-domain active filtering with DTC controlled line converter 91

Figure 5.6: Calculation of the negative sequence 5^th harmonicαβ-frame current reference using the method of harmonic synchronous reference frame.

wheres⁵⁻ is a unity vector that is calculated in the control software. In the case of the 5^th negative sequence, the unity vector is rotated with the angular frequency 5ωsto the negative direction. Therefore, it is calculated as

s⁵⁻ = cos(−5ωst) +jsin(−5ωst). (5.3) The grid angular frequencyωshas to be estimated continuously because the actual grid fre-quency slightly deviates from the nominal value. The estimation is implemented by mea-suring the angular frequency of the converter virtual flux linkage vectorψ

1. The angle of the converter virtual flux linkage vector is calculated in every millisecond. The angular frequency is estimated as

ωs,est=∆arg{ψ

1}

∆t , (5.4)

where∆arg{ψ₁}is the change of the virtual converter flux linkage angle between the con-secutive evaluations and∆tis 1 ms. Because the converter virtual flux linkage may contain harmonics the estimate is low-pass filtered to find out the average value that corresponds to the fundamental frequency.

In the harmonic synchronous reference frame the harmonic component of the analyzed sig-nal that matches the frequency and the sequence of the synchronous frame appears as a DC signal. Thus, it may be extracted by using a low-pass filter. The fundamental frequency and the other harmonics of the analyzed signal are shifted in the frequency because of the co-ordinate transformation. In the case of the harmonic synchronous frame corresponding to the negative 5^thor the positive 7^thharmonic sequence the fundamental frequency appears in the 6^thharmonic frequency, which is 300 Hz in a 50 Hz grid. This is the dominating component appearing in the signal transformed to the harmonic synchronous frame and the low-pass fil-ter should be designed to efficiently attenuate this signal component. The characfil-teristics of the low-pass filter are largely determining the potential dynamic performance of the harmonic detection, and further also the dynamic properties of the closed-loop system controlling the harmonic. Therefore, not only the frequency domain but also the time domain characteristics of the low-pass filter are important. The fundamental frequency should be preferably atten-uated at least by 60 dB. The cascade of simple discrete filters, wheren_c is the number of cascaded filter stages is written as

G_lpf(z⁻¹) =

a

1−(1−a)z^{−1 n}c

(5.5)

The parameteraand the order of the resulting filtern_care chosen to provide sufficient atten-uation for 300 Hz and to have adequate time domain characteristics. With the sample time T_s =200 µs and selectionsa =0.008 andn_c =2, an attenuation of 67 dB at 300 Hz and 96 ms for the 90% rise time are obtained. With the negative 11^th(11–) and the positive 13^th (13+) harmonic sequences the fundamental frequency is appearing as 600 Hz component and therefore the low-pass filters for these harmonics could be tuned to provide a faster rise time.

The low-pass filtered space-vector componentsi⁵⁻_Sd,f andi⁵⁻_Sq,f represent the detected har-monic current space-vector in the harhar-monic synchronous frame. Error signals in both direc-tions are calculated by subtracting the detected harmonic components from the corresponding references. Typically, the harmonic supply current referencesi⁵⁻_Sd,refandi⁵⁻_Sq,refare zeros, but not necessarily, if some predetermined harmonic current is needed. The error signals are fed to the PI-controllers, which produce converter harmonic current references in the harmonic synchronous frame,i⁵⁻_1d,refandi⁵⁻_1q,ref. These references are then transformed to the station-aryαβ-frame with equations

i⁵⁻_1α,ref = Re{s⁵⁻i⁵⁻_1,ref}=s⁵⁻_α i⁵⁻_1d,ref−s⁵⁻_β i⁵⁻_1q,ref (5.6) i⁵⁻_1β,ref = Im{s⁵⁻i⁵⁻_1,ref}=s⁵⁻_α i⁵⁻_1q,ref+s⁵⁻_β i⁵⁻_1d,ref . (5.7) The transformation from theαβ-frame to the fundamental wave dq-frame is calculated cor-responding to Eqs. (4.50) and (4.51)

i^5−(dq_1d,ref¹⁺⁾ = ψ1αi⁵⁻_1α,ref+ψ1βi⁵⁻_1β,ref ψ1

(5.8) i^5−(dq_1q,ref¹⁺⁾ = ψ1αi⁵⁻_1β,ref−ψ1βi⁵⁻_1α,ref

ψ₁ . (5.9)

where the superscript (dq¹⁺) is used to explicitly indicate that the harmonic component is expressed in the fundamental frequency positive sequence oriented synchronous reference frame. The total current references are calculated by summing the fundamental wave refer-ence components and all harmonic referrefer-ence components. With four harmonic sequrefer-ences the total current references are calculated as

i1d,ref,tot = i1d,ref+i^5−(dq_1d,ref¹⁺⁾+i^7+(dq_1d,ref¹⁺⁾+i^11−(dq_1d,ref ¹⁺⁾+i^13+(dq_1d,ref ¹⁺⁾ (5.10) i1q,ref,tot = i1q,ref+i^5−(dq_1q,ref¹⁺⁾+i^7+(dq_1q,ref¹⁺⁾+i^11−(dq_1q,ref ¹⁺⁾+i^13+(dq_1q,ref ¹⁺⁾ . (5.11) These current references are used in the converter virtual flux linkage oriented current vector system shown in Fig. 4.8 on page 59.

The control of four harmonic sequences, 5–, 7+, 11– and 13+, is implemented to the digital signal processor. The harmonic current reference calculations are implemented in a 200 µs control loop. The 200 µs time level is realized by splitting up the 100 µs time level so that every other cycle new references are calculated to the sequences 5– and 7+ and every other cycle to the sequences 11– and 13+. The current vector control is in 100 µs time level. The calculation of the unit vectors (e.g. (5.3)) is divided in 100 µs and 1 ms time levels. In the 1 ms time level the angle which the unit vectors^νshould rotate in 100 µs is calculated with the estimated grid frequencyωs,estas

∆φs^ν =ωs,est∆t , (5.12)

5.5 Frequency-domain active filtering with DTC controlled line converter 93

where∆tis 100 µs. Also, the cosine and sine of the∆φ_sν are calculated in the 1 ms time level. In the 100 µs time level the corresponding unit vectors^ν is in every round rotated for the angle∆φ_sν as

"

s^ν_{α, n+1} s^ν_{β, n+1}

#

=

"

cos(∆φs^ν) −sin(∆φs^ν) sin(∆φs^ν) cos(∆φs^ν)

# "

s^ν_{α, n} s^ν_{β, n}

#

, (5.13)

wherenis the time index. In (5.13) the rotation direction is positive (i.e. counter-clockwise).

The unit vectors are synchronized to the converter virtual flux linkage vector by observing the zero-crossings of the unit vector components and the converter virtual flux linkage vector components. If the unit vector appears to have rotated too much or too little, the rotation angle∆φs^ν is adjusted by a small offset. The synchronization eliminates slow drifting of the harmonic d- and q-axes with respect to the fundamental d- and q-axes. The drifting, however, is a slow phenomenon and not a problem if zero harmonic references are used. But, if some predetermined harmonic current is injected the synchronism is required to maintain a fixed relation between the harmonic and the fundamental co-ordinate systems, and, thus, between the harmonic and the fundamental waveforms. Instead of the virtual converter flux linkage the harmonic synchronous co-ordinates may also be synchronized to the line voltage or the virtual line flux linkage.