Comparative Analysis of Lattice-based All-Pass Filter and Second Order Generalized Integrator as Orthogonal System Generator of a PLL

POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 19 |NUMBER: 1 |2021 |MARCH Compa a i e Analysis o La ice-based All-Pass Fil e and Second O de Gene alized In eg a o as O hogonal Sys em Gene a o o a PLL Luciano Emilio BELANDRIA1, Nancy Alejand a AGUDELO1, Joan BERGAS-JANE 2 1Depa men o Elec onic Enginee ing, Na ional Expe imen al Uni e si y o Tachi a, Uni e sidad, 5001 San C is obal, Venezuela 2Cen e o Technological Inno a ion in S a ic Con e e s and D i es, Depa men o Elec ical Enginee ing, Poly echnic Uni e si y o Ca alonia, Diagonal 647, 08028 Ba celona, Spain lb[email p o ec ed]e, [email p o ec ed], joan.gab iel.be [email protected] DOI: 10.15598/aeee. 19i1.4002 A icle his o y: Recei ed No 13, 2020; Re ised Jan 31, 2021; Accep ed Feb 10, 2021; Published Ma 31, 2021. This is an open access a icle unde he BY-CC license. Abs ac . This pape p esen s a s eady-s a e compa i- son o wo me hods ha gene a e an o hogonal ol age sys em o a single-phase Phase-Locked Loop (PLL) s uc u e: a widely accep ed one based on a Second O de Gene alized In eg a o (SOGI) and a new one based on a All-Pass Fil e (APF) wi h La ice s uc u e. Bo h me hods a e e y a ac i e because o hei simple digi al implemen a ion, low compu a- ional load and good pe o mance unde ha monically dis o ed g id condi ions and a iable equency, so hey a e a good al e na i e o o he known me hods. The pape de i es and analyzes he ull s a e space mod- els o he wo me hods. I is shown ha hese wo me hods a e equi alen in he mos common ope a ion condi ions o dis ibu ed ene gy esou ces, al hough he APF s uc u e is clea ly be e han he SOGI one because i main ains i s o hogonal gene a ion abili y o any highe no ch equencies and any lowe sampling equencies. The compa a i e analysis we e alida ed by simula ion using MATLAB/Simulink and expe imen al esul s using a ixed-poin DSP. Keywo ds All-Pass Fil e , O hogonal Signal Gene a o , Phase-Locked Loop, Single-Phase PLL, No ch Fil e , Second O de Gene alized In eg a o . 1. In oduc ion The use, de elopmen and deploymen o Dis ibu ed Ene gy Resou ces (DERs), especially enewable e- sou ces, has inc eased d ama ically in he las decade. This is changing he pa adigm o elec ic gene a ion [1], [2], [3], and [4]. Single-phase g id-connec ed in- e e s a e ound in many DERs, such as pho o ol aic in e e s and ene gy s o age de ices [5] and [6]. In e e sys em con ol mus ensu e ha he powe gene a ion sys em is synch onized wi h he g id, and ha phase angle jumps a e de ec ed o eliable powe deli e y [7] and [8]. Mo eo e , i mus ensu e ha g id- connec ed sys em pe o mance complies wi h ope a ion equi emen s unde he mos common dis o ions, such as line ha monics, no ches, ol age dips, ises and alls, and equency a ia ions. Phase, equency and ampli ude cha ac e ize he single-phase g id ol age signal and knowledge o hese pa ame e s is undamen al in he design o g id- connec ed in e e sys ems [9], like he one in Fig. 1. In o de o mee he new equi emen s o ne wo k codes and op imize in e e pe o mance, a PLL is used o apid synch oniza ion wi h he g id. The main ask o he PLL is o accu a ely de ec he ac ual ol age phase angle a he Poin o Com- mon Coupling (PCC), e en in he p esence o ol age ha monics and unbalance [10] and [11]. This s uc u e, o example, should be used o p o ide any uni y powe ac o ope a ion which in ol es synch oniza ion o he ©2021 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 1 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 19 |NUMBER: 1 |2021 |MARCH DC Powe Sou ce PLL In e e Fil e T ans o me G id PWM PCC ac iac q , , V RMS Con olle Fig. 1: G id-connec ed powe con e sion sys ems. in e e ou pu cu en wi h he g id ol age and o gi e a clean sinusoidal cu en e e ence, among o he unc ions. Fo h ee-phase applica ions, some o he ecen and mos popula g id synch oniza ion echniques o de- ec ing he phase and equency o he mains ol age signal using PLL a e p esen ed in [12], whe e a com- ple e s udy was ca ied ou on he con ol s a egies o Dis ibu ed Gene a ion Powe Sys ems (DGPSs) unde ideal and non-ideal ne wo k condi ions. These ech- niques a e: basic s uc u e adi ional Synch onous Re e ence F ame PLL (SRF-PLL), Enhanced PLL (EPLL), Dual Second O de Gene alized In eg a o PLL (DSOGI-PLL), Mo ing A e age Fil e (MAF) and Decoupled Double Synch onous Re e ence (DDSRF). Indeed, he single-phase s uc u e o PLLs limi s he use o some well-known h ee-phase con ol s a egies [13] and [14]. In single-phase sys ems, less is known abou he ne wo k ope a ing condi ions han in h ee- phase sys ems. Tha is why he mos ad anced me h- ods used o o e come his limi a ion mus c ea e an o hogonal ol age sys em [15] and hen exploi he exis ing h ee-phase con ol me hods. In his line o esea ch, se e al ad anced PLL echniques, ha e been p oposed o single-phase applica ions [16], [17], [18], and [19]. dq Vol age Moni o ing F equency and Phase Es ima o O hogonal Sys em Gene a o a b V V VV V V a b RMS q d q Fig. 2: PLL using he pa k ans o ma ion. Gene al s uc u e o a single-phase PLL algo i hm based on O hogonal Signal Gene a o (OSG), also called Quad a u e Signal Gene a o (QSG), o g id synch oniza ion is p esen ed in [15], [16], [20], [21], [22], [23], and [24]. This s uc u e can use he Pa k T ans- o ma ion, as shown in Fig. 2 o he a c- angen unc- ion depic ed in Fig. 3. The main di e ence be ween OSG-based single-phase PLLs lies in he way o hogo- nal ol age sys ems a e gene a ed. Vol age Moni o ing F equency and Phase Es ima o O hogonal Sys em Gene a o V V V V a b RMS q          b a  an 1_ + q Fig. 3: PLL using he a c angen unc ion. The OSG-based single-phase PLL s uc u es, Fig. 2 and Fig. 3, ound in he s a e-o - he-a me hods a e basically o med by wo blocks. In he i s , an o hog- onal sys em in phase wi h he abo e signal is gene a ed om a single e e ence sinusoidal signal. The second block uses ei he a eedback loop h ough he αβ o dq ans o ma ion o he a c- angen unc ion o de e - mine he phase angle o he e e ence signal. The me hods o OSG o a single-phase PLL mus be easy o apply in p ac ice. Mo eo e , he OSG mus deli e he il e ed ou pu wi hou any delay, because o i s esonance a he undamen al equency, no o be a ec ed by equency changes. The mos common me hods used in he li e a u e o gene a e he o hog- onal ol age a e p esen ed in [15], [19], [20], [21], [25], [26], [27], [28], [29], [30], [31], and [33]. One using a block o a T anspo Delay unc ion is shown in Fig. 4. I in oduces a 90 deg ee phase shi wi h espec o he inpu signal [21] and [28]. Ano he me hod uses he Hilbe T ans o ma ion [29], depic ed in Fig. 5, and he In e se Pa k T ans o ma ion [19], [30], and [31], shown in Fig. 6. Howe e , hese me hods ha e one o mo e o he ollowing de iciencies: equency dependence, high complexi y, non-linea i y, and poo o no il a ion. V V V a b -1 T/4 delay Fig. 4: T anspo delay unc ion. To imp o e he de iciencies men ioned in he OSG me hods, an imp o ed A e age Fil e (AF) is p oposed in [32], wi h a simple s uc u e and implemen a ion, which, in eg a ed in o OSG, a enua es he nega i e e - ec o ol age ha monics and unbalances on o hogonal ©2021 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 2 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 19 |NUMBER: 1 |2021 |MARCH V V V a b -1 Hilbe T ans o m Fig. 5: Hilbe ans o ma ion. V V b a b dq LPFLPF V q Vd q V a Fig. 6: In e se pa k ans o ma ion. d-q signals o PLL. P o iding as phase de ec ion and as dynamic esponse wi hou using he second o de il e s, which educe he dynamic esponse ime and he de ec ion o he synch oniza ion uni . In [34], a F equency Lock Loop (FLL) wi h a Gen- e alized In eg a o (GI) was p oposed. In he GI-FLL, he GI pa is a OSG, while he FLL pa uses he sig- nals gene a ed o unknown equency es ima ion pu - poses. The GI wo ks as an adap i e bandpass il e using coo dina e ans o ma ion, which allows o im- p o e he dynamic uning ange, wi h an excellen bal- ance be ween he con e gence speed and he maximum accep able peak es ima ion e o , bu wi h some com- pu a ional cos addi ional o ha needed o accu acy and he FLL has o add he no maliza ion o he gain ha he GI does no ha e. The echnique using he g id ol age demodula ion o ob ain a OSG is p oposed in [35], in a Demodu- la ion Type PLL (DT-PLL) wi h imp o ed DC o se ejec ion capaci y, o he adap i e es ima ion o he phase angle and he equency o a single-phase sys- em, which a oids he use o any low-pass il e . The demodula ion has good dynamic pe o mance and dis- u bance ejec ion abili y. Howe e , due o he p es- ence o igonome ic quan i ies in he es ima o dy- namics, small-signal modelling-based pa ame e uning can be complica ed o DT-PLL. Mo eo e , eal- ime implemen a ion o igonome ic unc ions is compu a- ionally expensi e. A single-phase PLL s uc u e based on SOGI which o e comes he abo e p oblems and a oidance o il e - ing delays due o i s esonance a he undamen al e- quency was p esen ed in [15], [20], [21], [36], [37], [38], [39], [40], and [41]. Thus, he way in which he wo sig- nals a e gene a ed is imp o ed. In [42], [43], and [44] a h ee-phase PLL s uc u e based on a Double Second O de Gene alized In eg a o (DSOGI) is p esen ed. The SOGI s uc u e has also been applied o o he aspec s o powe elec onic con ol, especially in he cu en con ol loop [37] and [38], de ec ion o ha mon- ics [39] and ac i e an i-islanding me hods [22]. Howe e , as al eady indica ed in [15], [37], and [38], he SOGI was designed in he con inuous ime and quad a u e phase delay and ampli ude ipples a e p esen in i s disc e e applica ion. Mo eo e , i is di - icul o apply his s uc u e on a ixed-poin DSP o FPGA due o hei limi ed p ecision and sensi i i y o coe icien ounding. A second-o de APF wi h La ice s uc u e was p o- posed in [45], [46], [47], [48], [49], and [50]. This il e gene a es o hogonal signals necessa y o he PLL, wi h good noise il e ing capabili y bu ampli udes di - e en om ha o he single-phase inpu signal. He e a APF wi h La ice s uc u e is p oposed as OSG which mee s all equi emen s and o e comes all he d aw- backs o he me hods used in single-phase PLL while main aining any uni y gain wi h espec o he inpu . This pape pe o ms a compa a i e s eady- s a e analysis o he s uc u es based on APF and SOGI as pa o a PLL. These s uc u es a e widely used o il e ing he powe supply signal o ensu e he bes possible synch oniza- ion sys em e e ences o use in single-phase con e e s ope a ing in highly dis u bed en i onmen s. Solu ions o he disc e e applica ion o he wo s uc- u es a e also p o ided. Simula ions and a ixed-poin DSP implemen a ion alida e he e ec i eness o one o ano he s uc u e. The es o he pape is o ganized as ol- lows. Sec ion 2. desc ibes he la ice APF as OSG wi h i s main diag ams and equa ions. In Sec. 3. , he SOGI algo i hm is p esen ed. Sec- ion 4. p o ides disc e e ime simula ion esul s o bo h OSGs using as inpu a 50 Hz no malized sinu- soidal signal in bo h il e s. In Sec. 5. , he expe i- men al esul s o a ixed-poin DSP implemen a ion o bo h OSGs unde he same simula ion condi ions a e discussed. Finally, Sec. 6. d aws he conclusions. 2. La ice-Based APF In he app oach p oposed in [49], no uni y gain has been conside ed while, in his pape , a new s uc u e o OSG based on APF, as illus a ed in Fig. 7, is p oposed cha ac e ized by uni y gain [50]. In he sys em, he ou pu signals x1(n)and x2(n) ha e a −90 deg ee and 0 phase shi , espec i ely, ©2021 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 3 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 19 |NUMBER: 1 |2021 |MARCH -1 -1 -cos 1 -sin 1 sin 1 u(n) x (n) 1 sin -1 2 -sin 2 -cos 1 sin 2 x (n) 2 -sin -1 2 y(n) Fig. 7: APF as OSG wi h uni y gain. wi h espec o he inpu signal wi h uni y gain, a e de ined as: x1(n) = cos θ1(1−sin θ2)z−1 1 + sin θ1(1 + sin θ2)z−1+ sin θ2z−2u(n), (1) x2(n) = sin θ1(sin θ2−1)z−1+ (sin θ2−1)z−2 1 + sin θ1(1 + sin θ2)z−1+ sin θ2z−2u(n). (2) The space-s a e equa ion o he APF wi h La ice s uc u e can be ob ained om he sys em in Fig. 7, is de ined as:   x1(n+ 1) x2(n+ 1) y(n)  = = −sin θ1cos θ1sin θ2cos θ1(1 −sin θ2) −cos θ1−sin θ1sin θ2sin θ1(sin θ2−1) 0−(1 + sin θ2) sin θ2  ·  x1(n) x2(n) u(n)  , (3) whe e θ1is associa ed wi h he no ch equency ω0, a which he APF o e s a phase shi o π adians, and θ2is associa ed wi h he 3 dB a enua ion Bandwid h BW o he il e . They a e de ined as: θ1=ω0 s−π 2,(4) θ2= a csin     1− an BW 2 1 + an BW 2     ,(5) BW =2πB s ,(6) whe e sand Bco espond o he sampling equency and bandwid h in Hz, espec i ely. Independen adjus men o no ch equency and bandwid h is a de- si able a ibu e. Adjus ing he bandwid h o any sampling equency om Eq. (6), ei he di ec ly o adap i ely, allows he APF o ejec e y low equen- cies, e en ejec ing he DC o se , wi hou adding an- o he ype o il e and wi hou in e e ing wi h he uning equency. This ea u e o he APF can comply wi h hose o he OSG wi h DC o se ejec ion ca- pabili y p oposed in [33], wi h only he adjus men o a single pa ame e . The s uc u e in Fig. 7, is heo e ically s able and nume ically well beha ed in ime- a ying en i onmen s [49]. Each o a ion angle ωk(k = 1, 2) is di ec ly con olled, so ha θ1and θ2a e con e ed in o he adjus able pa ame e s o adap i e pe o mance. Elemen s Aand Bo he s uc u e a e ex ac ed om Eq. (3): A=−sin θ1cos θ1sin θ2 −cos θ1−sin θ1sin θ2,(7) B=cos θ1(1−sin θ2) −sin θ1(1−sin θ2).(8) Figu e 8, shows he Bode diag am o he APF as no malized OSG o a uning equency o 50 Hz. Magni ude (dB) Phase (deg) F equency (Hz) -80 -60 -40 -20 0 100101102103 -180 -90 0 90 180 x (n) x (n) 1 2 Fig. 8: Bode diag am o la ice-based APF as no malized OSG. 3. The SOGI A SOGI is equi alen o wo P opo ional-In eg al (PI) con olle s in synch onous e e ence ames compensa - ing he sequences o posi i e and nega i e [15], [20], [37], [38], [39], and [40]. SOGI is p oposed o ob ain a ze o s eady s a e e o using sinusoidal e e ences wo king on he αβ s a ion- a y e e ence ame. This me hod has been included in ha monic elimina ion algo i hms (because ha mon- ics ac in a e y na ow band a ound hei esonance equency); g id sequence de ec ion and quad a u e signal gene a ion algo i hms; algo i hms o con e e synch oniza ion o he powe supply; and o mul i- equency de ec ion. The ans e unc ion o a SOGI o a single sinu- soidal signal is [37] and [38]: G(s) = 2s s2+ω2 0 ,(9) whe e ω0is he esonance equency and s he Laplace ope a o . The in eg a o ou pu con ains no ©2021 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 4 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 19 |NUMBER: 1 |2021 |MARCH only he in eg a ed inpu bu also an insigni ican ad- di ional componen . In o de o use hem o gene a e quad a u e signals, he o iginal opology is modi ied, as shown in Fig. 9. The esul ing ma hema ical ex- p ession is: G(s) = x(s) y(s) =ω0s s2+ω2 0 .(10) (s) y(s) _ + w 1 s 1 s x(s) 0 w 0 Fig. 9: Con inuous- ime SOGI. The new ans e unc ion has wo poles a ±jω0and a ze o a he o igin, jus as Eq. (9). The only di e ence be ween he wo exp essions is in he gain, which is no signi ican as a as he inal pe o mance is conce ned. Howe e , he unc ion leads o a mo e gene al s uc u e ha can be used o bo h powe con e e s con ol and synch oniza ion asks. The SOGI s uc u e o gene a ion o o hogonal sig- nals, also known as OSG-SOGI, is ou lined in Fig. 10. As can be seen, he basic elemen is a SOGI [15], [20], [37], [38], [39], and [40]. The con inuous- ime ans e unc ions a e: ’( s) ( s)=Ksω0s s2+Ksω0s+ω2 0 ,(11) q ’( s) ( s)=Ksω2 0 s2+Ksω0s+ω2 0 .(12) ’(s) _ + w 1 s 1 s x(s) 0 w 0 _ + K (s) s q ’(s) SOGI Fig. 10: Con inuous- ime OSG-SOGI. Disc e e- ime implemen a ion o he SOGI can be ac- complished by disc e izing he con inuous- ime ans e unc ions o by using disc e e in eg a o s [15], [37], and [38]. Figu e 11 shows he disc e e- ime SOGI s uc u e. I s ou pu beha es like ha o he con inuous- ime SOGI in Fig. 9 upon applica ion o a s ep signal o am- pli ude 1 a inpu x(n), ha is a sinusoidal signal o pulsa ion ω0and ampli ude 1. The choice o his SOGI is based on he use o disc e e Eule Backwa d In eg a- o wi h compu a ional delay added in se ies wi h he eedback gain, modeling he inhe en delay caused by he p og amming p ocess. Mo eo e , i s s uc u e is mo e simila o ha o he classical PI con olle . ++ ++ x(n) y(n) _ + Z-1 Z-1 (n) Z-1 w T 0 S w T 0 S Eule Backwa d In eg a o Eule Backwa d In eg a o Fig. 11: SOGI based on Eule Backwa d In eg a o and com- pu a ional delay. The ans e unc ion o he disc e e- ime SOGI is: G(z) = ω0Ts−ω0Tsz−1 1+(ω2 0T2 sz−1−2) + z−2.(13) Figu e 12 displays he SOGI based on disc e e Eu- le Backwa d In eg a o wi hin an OSG-SOGI, which ensu es ha signals ’ and q ’ a e quad a u e signals a all ope a ing equencies. The OSG-SOGI allows independen adjus men o he no ch equency ω0and o he 3 dB a enua ion bandwid h BW, conside ing he sampling pe iod Ts, acco ding o: K =ω0Ts,(14) Ks=BW ω0 √0.98.(15) ++ ++ u(n) ’(n) _ + Z-1 Z-1 q ’(n) Z-1 Eule Backwa d In eg a o _ + K s Z-1 Eule Backwa d In eg a o SOGI K K Fig. 12: Disc e e- ime OSG-SOGI s uc u e. ©2021 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 5 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 19 |NUMBER: 1 |2021 |MARCH The OSG-SOGI can also be exp essed by s a e equa- ion o m based on he inal ci cui o Fig. 13, which is equi alen o ha o Fig. 12, whe e x1(n)and x2(n)a e he signals co esponding o he o hogonal sys em and a e deno ed as in he APF, wi h a −90 and 0 deg ee phase shi , espec i ely, and uni y gain wi h espec o he inpu signal o powe u(n)a ins an n. + + u(n) ’(n) _ + Z-1 q ’(n) _ + K s Z-1 + + x (n) 1 x (n) 2 K K Fig. 13: De ini i e OSG-SOGI s uc u e. This o hogonal sys em is gi en by: x1(n) = =KsK2 z−1 1+(KsK −2 + K2 )z−1+ (1−KsK )z−2u(n), (16) x2(n) = =KsK z−1−KsK z−2 1+(KsK −2 + K2 )z−1+ (1−KsK )z−2u(n). (17) Figu e 14 o e s he Bode diag am o he OSG-SOGI, o a uning equency o 50 Hz. Conside ing ha ou - pu s x1(n+ 1) and x2(n+ 1) a e he same answe s q ’(n) and ’(n), he s a e equa ion is:   x1(n+ 1) x2(n+ 1) y(n)  = =  1−K2 K (1 −KsK )KsK2 −K 1−KsK KsK 0 1 0  ·  x1(n) x2(n) u(n)  . (18) Elemen s Aand Bo he s uc u e can be ex ac ed om Eq. (18): A=1−K2 K (1 −KsK ) −K 1−KsK ,(19) B=KsK2 KsK .(20) Magni ude (dB)Phase (deg) F equency (Hz) -80 -60 -40 -20 0 100101102103 -180 -135 -90 -45 0 45 90 135 180 q (n) (n) Fig. 14: Bode diag am o he OSG-SOGI. 4. Beha io and Compa ison o OSGs The s uc u es o he la ice-based APF in Fig. 7 and he OSG-SOGI in Fig. 13 can be easily implemen ed as ma ix sys em consis ing o elemen s Aand Bonly, hus we will ha e he equa ion o s a e o he o m: x1(n+ 1) x2(n+ 1)=A B·x(n) u(n).(21) Bo h s uc u es ha e he abili y o gene a e, in phase wi h he inpu signal, an o hogonal sys em. Figu e 15 shows he implemen a ion o he la ice-based APF and OSG-SOGI s uc u es in s a e equa ion. Two pa ame- e s iden i ied as x1(n)and x2(n), which cons i u e he o hogonal sys em, a e obse ed. + A B u(n) x (n) 1 x (n) 2 + Z-1 x (n) Fig. 15: OSG in s a e equa ion. Bo h s uc u es a e simula ed unde he same con- di ions o compa e hem as Band-Pass Fil e s (BPF) which gene a e a dis u bance- ee o hogonal sys em wi h one signal image o he undamen al inpu signal and ano he one delayed 90 deg ees wi h espec o he same inpu signal. The wo il e s a e e alua ed using any uni y ampli ude sinusoidal signal a he equency o 50 Hz as he undamen al inpu signal. Is se s a sam- pling equency o 20 kHz, 50 Hz uning equency and a low bandwid h o 4 Hz, na ow enough o many ap- plica ions whe e a e y selec i e magni ude esponse is equi ed, wi h a high quali y ac o . The simula ion is implemen ed in MATLAB/Simulink. The s a e equa- ©2021 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 6 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 19 |NUMBER: 1 |2021 |MARCH ion om Eq. (3) o he APF wi h he alues es ima ed unde hese condi ions is: x1(n+ 1) x2(n+ 1)= =0.9998766 0.0156876 0.0000197 −0.0157073 0.9986209 0.0012557·  x1(n) x2(n) u(n)  . (22) And he s a e equa ion om Eq. (18) o he OSG- SOGI wi h he alues es ima ed: x1(n+ 1) x2(n+ 1)= =0.9997532 0.0156884 0.0000195 −0.0157080 0.9987560 0.0012440·  x1(n) x2(n) u(n)  . (23) Figu e 16 gi es he equency esponse o ou pu s x2(n)/u(n), o Eq. (2) and Eq. (17), co esponds o he APF and he OSG-SOGI. Fo equencies o e 10 Hz bo h s uc u es main ain a s ong simila i y in bo h magni ude and phase. Bu o equencies below 10 Hz hey al eady begin o di e en ia e, he APF begins o inc ease i s phase o alues g ea e han 90 deg ees. Fo equencies below 1 Hz he di e ence in magni ude begins o be mo e no iceable. Magni ude (dB)Phase (deg) F equency (Hz) -80 -60 -40 -20 0 100101102103104 -180 -90 0 90 180 OSG-SOGI APF La ice Fig. 16: Bode diag am o x2(n)/u(n)in La ice-based APF and OSG-SOGI. As shown in Eq. (22) and Eq. (23), he alues o he coe icien s o bo h il e s a e nea ly iden ical un- de hese ope a ing condi ions, he di e ences be ween he coe icien s a e less han 0.0001233. Hence, hei equency esponses should ha e a e y simila beha - io . The equency esponses a e ob ained o x2(n)/u(n), in he APF and in he OSG-SOGI o a ange o undamen al equencies, om 100 Hz o 10 kHz, wi h a bandwid h o 4 Hz, a sampling equency o 20 kHz, o bo h il e s. In Fig. 17, one can obse e ha bo h OSG beha e like a BPF, bu o undamen al equencies g ea e han 400 Hz, he magni ude esponses begin o ha e di e ences, al hough hey main ain hei uning. F om equencies g ea e han 3 kHz, he SOGI begins o lose i s uning, bo h in i s magni ude and phase esponse. -150 -100 -50 0 Magni ude (dB) 101102103104 -180 -90 0 90 180 Phase (deg) F equency (Hz) OSG-SOGI APF La ice Fig. 17: Bode diag am o x2(n)/u(n)in La ice-based APF and OSG-SOGI o a ious uning equencies. 102103104 -80 -60 -40 -20 0 Magni ud(dB) APF La ice OSG-SOGI (a) Fo a ious uning equencies. Magni ud(dB) 102103104105 -25 -20 -15 -10 -5 0 5 APF La ice OSG-SOGI (b) Fo a ious sampling equencies. 101102103104 -35 -30 -25 -20 -15 -10 -5 0 5 F ecuency(Hz) Magni ud(dB) APF La ice OSG-SOGI 500Hz 14 kHz 17 kHz 20 kHz 23 kHz 29 kHz 11 kHz 8 kHz 5 kHz 2 kHz 26 kHz (c) Fo a ious sampling equencies and a ia ion o he uning equency. Fig. 18: Magni ude o x2(n)/u(n)in he La ice-based APF and OSG-SOGI. ©2021 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 7 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 19 |NUMBER: 1 |2021 |MARCH -3 -2 -1 0 a21 -1 0 1 a22 102103104 -2 0 2x 10 -3 F ecuency(Hz) a23 -6 -4 -2 0 2 a11 0 1 2 3 a12 0 2 4x 10-3 a13 10210 10 F ecuency(Hz) 3 4 APF La ice OSG-SOGI (a) Fo a ious uning equencies. -10 -5 0 5 a11 0 1 2 3 a12 -0.5 0 0.5 1 a13 10210310410 5 F ecuency(Hz) -3 -2 -1 0 a21 -1 0 1 a22 -0.4 -0.2 0 0.2 a23 102103104105 F ecuency(Hz) APF La ice OSG-SOGI (b) Fo a ious sampling equencies. Fig. 19: Beha io o he ma ix pa ame e s o he s a e equa- ion o APF and OSG-SOGI. Figu e 18 displays he esul o he magni ude o x2(n)/u(n)o he la ice APF and he OSG-SOGI wi h a bandwid h o 4 Hz and a sampling equency o 20 kHz a a ious equencies, o a uning equency o 50 Hz wi h sampling equencies anging om 100 Hz o 100 kHz and o a ious sampling equencies and a ia ions o he equency o uning. As can be no- iced, om a uning equency o 600 Hz and a sam- pling equency below 1 kHz, he uning o he SOGI is no accu a e, and so a e i s cha ac e is ics as OSG, which does no happen wi h he APF. The magni ude in dB o x2(n)/u(n)is ob ained by e alua ing he APF and he OSG-SOGI wi h a band- wid h o 4 Hz, depic ed in Fig. 18(a) o a sampling equency o 20 kHz and he a ia ion o he uning equency. In Fig. 18(b) he uning equency is kep a 50 Hz wi h a ia ion o he sampling equency om 100 Hz o 100 kHz and in Fig. 18(c) he esul o he magni ude is shown o se e al sampling equencies (500 Hz, 2 kHz, 5 kHz, 8 kHz, 11 kHz, 14 kHz, 17 kHz, 20 kHz, 23 kHz, 26 kHz, and 29 kHz) and a ia ion o he uning equency. As can be seen, he APF main- ains a cons an magni ude o 0 dB o all he a ia- ions in all he e alua ed anges. On he o he hand, he SOGI loses he uning s a ing om a uning e- quency o 500 Hz and a sampling equency lowe han 1 kHz, and he e o e hence i s cha ac e is ics as OSG. Figu e 19 shows he beha io o he ma ix pa ame- e s o he s a e equa ion o he APF Eq. (3) and he OSG-SOGI Eq. (18) o a ious uning and sampling equencies. The pa ame e s begin o di e mo e sig- ni ican ly om uning equencies g ea e han 500 Hz and sampling equencies below 1 kHz in mos cases. This is demons a ed by he esul s in Fig. 16, Fig. 17, and Fig. 18. Figu e 20, has he magni ude esponse su ace o x2(n)/u(n), as a unc ion o he uning equency and he sampling equency. Obse ing ha o he APF, he esul is a comple ely la su ace wi h a magni- ude o 0 dB cons an o any a ia ion o he equen- cies. This does no occu o he OSG-SOGI, whe e i is obse ed ha he magni ude esponse a ies as he equencies a y, mo ing away om he equi ed pass- band. Being he wo s case when he uning equency inc eases and he sampling equency dec eases. 0 1 2 x 10 4 0 1 2 3 x 10 4 -300 -200 -100 0 F ecuency(Hz) Sampling F ecuency(Hz) Magni ud(dB) -350 -300 -250 -200 -150 -100 -50 0 (a) APF. 0 1 2 x 10 4 0 1 2 3 x 10 4 -300 -200 -100 0 F ecuency(Hz) Sampling F ecuency(Hz) Magni ud(dB) -350 -300 -250 -200 -150 -100 -50 0 (b) OSG-SOGI. Fig. 20: Su ace o he magni ude o x2(n)/u(n) o a ious sampling equencies and a ia ion o he uning e- quency. ©2021 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 8 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 19 |NUMBER: 1 |2021 |MARCH 5. Simula ion and Expe imen al Resul s In o de o alida e he analysis, he APF and he OSG-SOGI a e simula ed by using MATLAB/Simulink p og am and an expe imen al se up was implemen ed. A CDM2480 es pla o m, wi h a TMS320F2812 ixed-poin DSP om Texas Ins umen s as he cen al elemen , sui able o mo ion con ol and powe elec onics applica ions, was used o implemen and es bo h s uc u es as OSG. The ixed poin DSP, wi h a clock equency o 150 MHz, was used o gene a e he inpu signal, he OSG algo i hms and he ou pu signals wi h a 12 bi D/A con e e . All simula ed and expe imen al esul s, we e ob- ained using he s uc u es in Fig. 7 and Fig. 13. The ou pu s x1( )and x2( )o he APF and he OSG-SOGI a e ob ained. The OSG inpu signal and pa ame e s a e he same o he simula ion and he expe imen al pa , whe e he inpu signal is a sinusoid wi h uni y ampli ude and equency equal o he uning equency o he il e . A bandwid h o 4 Hz was se o he de- sign o he OSGs. In he DSP, a ixed-poin Q15 base was used o global calcula ions whe eas a Q30 base was used o calcula ions o he il e s. Vol age(V) 0.5 0.505 0.51 0.515 0.52 0.525 0.53 0.535 0.54 -1 Time(sec) -1 0 1 -1 0 1 -1 0 1 -1 0 1V e V e 90º V e V e 90º Vol age(V)Vol age(V) Vol age(V) 2APF 1APF 2SOGI 1SOGI (a) Simula ion. V e 2APF V e 90º 1APF V e 2SOGI V e 90º 1SOGI (b) DSP Implemen a ion. Fig. 21: x1( )and x2( )o he APF and he OSG-SOGI o a sampling equency o 20 kHz and uning equency o 50 Hz. In he expe imen al esul s ob ained o he OSGs, channels A and B in blue a e x2( )and x1( )o he APF wi h [1 V/di ]; channels C and D in g een a e x2( )and x1( )o he SOGI wi h [1 V/di ]. Co e- sponding e e ence signals (in ed), an o he unda- men al inpu signal on channels A and C and delayed signals 90 deg ees wi h espec o his inpu signal on channels B and D, we e added. The esul s o a equency o 50 Hz o he inpu sig- nal and uning, wi h a sampling equency o 20 kHz, a e p esen ed in Fig. 21, whe e i is e idenced ha bo h OSGs beha e in acco dance wi h he es ablished, bo h in he simula ion and in he DSP implemen a ion. The ou pu s x1( )and x2( ) ollow hei e e ences and e ain hei wa e o m, and main ain he gain o 1, nec- essa y condi ion o bo h OSGs ope a ing as PLLs. -1 0 1 -1 0 1 -1 0 1 0.5 0.5005 0.501 0.5015 0.502 0.5025 0.503 0.5035 0.504 -1 0 1 Time(sec) Vol age(V) V e V e 90º V e V e 90º Vol age(V) Vol age(V) Vol age(V) 2APF 1APF 2SOGI 1SOGI (a) Simula ion. V e 2APF V e 90º 1APF V e 2SOGI V e 90º 1SOGI (b) DSP Implemen a ion. Fig. 22: x1( )and x2( )o he APF and he OSG-SOGI o a sampling equency o 20 kHz and uning equency o 500 Hz. The sampling equency is kep a 20 kHz and he equency o he inpu and uning signal was inc eased o 500 Hz in Fig. 22 and 1000 Hz in Fig. 23. In he simula ion o bo h uning equencies, he signals o he o hogonal sys em co esponding o he APF ha e a good ollow-up o he e e ence signals wi h a gain o 1. The OSG-SOGI, loses i s cha ac e is ics like OSG, wi h an ad ance o 0.0002 seconds, ela i e o he e - e ence signals, which in oduces a phase shi , as well as a dec ease in he gain, which is no eaching 1. ©2021 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 9