Resonance damping of LCL filters via input admittance frequency shaping

© 2016 IEEE. Pe sonal use o his ma e ial is pe mi ed. Pe mission om IEEE mus be ob ained o all o he uses, in any cu en o u u e media, including ep in ing/ epublishing his ma e ial o ad e ising o p omo ional pu poses, c ea ing new collec i e wo ks, o esale o edis ibu ion o se e s o lis s, o euse o any copy igh ed componen o his wo k in o he wo ks Resonance damping o LCL fil e s ia inpu admi ance equency shaping Jo ge Pé ez, San iago Cób eces, Daniel Piza o, F ancisco Ja ie Rod íguez Sánchez Depa amen o de Elec ónica Uni e sidad de Alcalá Email: jo [email p o ec ed] Robe G iñó Ins . o Indus ial and Con ol Enginee ing.(IOC) Uni e si a Poli ècnica de Ca alunya (UPC) Email: [email p o ec ed] Abs ac —This pape p esen s a no el ac i e damping ech- nique o cu en -con olled g id-connec ed powe con e e s h ough LCL fil e s. Based on H∞syn hesis algo i hms, a g id cu en con olle is ob ained so ha he g id-connec ed powe con e ed-based applica ion admi ance esembles a gi en equency e e ence. By defining a low esis i e admi ance as he e e ence, conside ed applica ion esonance is e ec i ely damped, educing g id cu en oscilla ions unde g id ol age a ia ions and a oiding hei associa ed s abili y p oblems. P esen ed g id cu en con olle senses only he PCC g id ol age and cu en , and is expe imen ally es ed in bo h ime and equency domains. Addi ionally, he e ec i eness o p esen ed damping me hod is p o ed unde di e en g id impedance scena ios. I. INTRODUCTION Pulsewid h modula ed (PWM) powe elec onic con e e s a e usually connec ed o he g id h ough inpu fil e s in o de o ensu e low THD sinusoidally shaped g id cu en s [1]. Rega ding g id cu en con ol applica ions, one o he mos common opologies is he LCL fil e [2], which has be e fil e ing capabili y han simple opologies as, o example, he L fil e , bu also inc eases con ol complexi y [3]. Tha is due o he p esence o wo complex conjuga e poles in he con olled plan , which esonan ly inc eases i s open-loop gain (i.e., i s admi ance alue). This esonance will gene a e cu en oscilla ions (a he esonan equency) unde changes in he sys em ope a ing poin (e.g., PCC ol age pe u ba ions). These oscilla ions can be mo e o less du able o e ime and high in magni ude depending on how damped he esonance is. Mo eo e , hey can become uns able in p esence o non-dissipa i e closed- loop sys ems [4] and weak g ids [5]. Damping echniques o esonan sys ems ha e ecei ed significan a en ion by he specialized li e a u e. They can be so ed in wo big g oups; ac i e damping [2], [3], [5]–[10], whe e he sys em con olle is modified o damp he esonance, and passi e damping [1], [11]–[14], whe e passi e elemen s, commonly esis ances, a e added o he fil e o displace he esonan poles. Passi e damping is a simple solu ion bu comes a he cos o ex a powe loss and educ ion o he high- equency a enua ion capabili y [13]. Ac i e damping echniques o e come passi e damping d awbacks, bu i s e - ec i eness, howe e , is limi ed by he swi ching equency, VSC PCC LCL il e + - + - + - Powe con olle Cu en con olle Fig. 1. One-line equi alen diag am o he conside ed sys em. In blue he measu ed a iables, in g een he cu en con olle and in ed he inpu admi ance o be shaped. and addi ional passi e componen s may s ill be needed o damp high- equency esonance [11]. This pape p oposes a new ac i e damping echnique o g id-connec ed powe elec onic con e e s h ough LCL fil e s by di ec ly shaping, in he equency domain, he closed- loop inpu admi ance o he conside ed applica ion. A model- e e ence app oach is adop ed, whe e designe defines he desi ed admi ance a he esonan equency and an H∞syn- hesis algo i hm ob ain he (sub)op imal con olle ha makes he closed-loop inpu admi ance ma ch he gi en e e ence [17]. The sys em esonance will be e ec i ely damped, hen, by defining a pu ely esis i e admi ance as he e e ence. The es o he pape is o ganized as ollows. Sec ion II models he plan o he conside ed applica ion and in oduce he esonan beha iou o he LCL fil e . Sec ion III de elop he p oposed sys em con ol, s a ing wi h he con olle ob- jec i es and ending up wi h i s design. Sec ion IV discuss he esul s o he p oposed con olle , expe imen ally es ed in bo h ime and equency domains. Addi ionally, i shows how obus he con olle is, ega ding bo h s and-alone s abili y and damping capabili y unde di e en g id condi ions. The pape ends wi h a b ie discussion o i s conclusions. II. DYNAMIC MODELLING This pape conside s a g id cu en con ol o a shun con e e connec ed o he g id h ough an LCL fil e (see Fig. 1). The sys em is modelled in s a iona y (i.e., αβ) e e ence ame, which allows e ec i e con olle ope a ion e en unde unbalanced condi ions. Addi ionally, as αβ axes a e uncoupled, he o iginal MIMO con ol p oblem is educed o he con ol o wo iden ical SISO p oblems. Fo he sake o no a ion simplici y, only one o he con olled channels is conside ed in his pape o bo h plan modelling and con olle design. G id cu en idynamic o conside ed sys em is ep esen ed in he Laplace domain as ollows: I(s)=G(s)·U(s)+Gd(s)·Vs(s),(1) whe e I(s),U(s)and Vs(s)a e he g id injec ed cu en , he VSC a e age ou pu ol age and PCC ol age, espec- i ely. T ans e unc ions G(s)and Gd(s)a e he open-loop command- o-ou pu and inpu open loop admi ance, espec- i ely, which will ollow he nex dynamic exp essions i an ideal PCC ol age (i.e., Lg=0) is conside ed: G(s)=−Zc(s) ZL1(s)·Zc(s)+ZL2(s)·(ZL1(s)+Zc(s)),(2) Gd(s)= ZL1(s)+Zc(s) ZL1(s)·Zc(s)+ZL2(s)·(ZL1(s)+Zc(s)),(3) whe e Zc(s)=1/(sC),ZL1(s)=sL1+R1and ZL2(s)= sL2+R2a e he capaci o , con e e -side coil and g id-side coil impedances, espec i ely. Bo h G(s)and Gd(s)ha e a pai o complex conjuga e poles which will p oduce an inc ease o hei espec i e gains a he esonance equency ω es =(L1+L2)/(L1·L2·C) ad/s. O main impo ance in he s abili y o he conside ed applica ion is he esonance in Gd(s)( e e o Fig. 7 o see i s equency domain ep esen a ion), as i may cause high cu en oscilla ions and e en sys em ins abili y unde PCC ol age changes. Tha p oblem is inc eased i he sys em is connec ed o a weak g id (i.e., wi h high g id impedance) [5]. The esonance mus be p ope ly damped in o de o assu e he co ec and obus ope a ion o he sys em. III. SYSTEM CONTROL A. Con ol objec i es The main objec i e o his pape is he design o a g id cu en con olle ha , in addi ion o ack a gi en e e ence i∗, can e ec i ely damp he sys em esonance. In o de o do so, his pape p oposes he equency shaping o wo closed loop ans e unc ion, acking ans e unc ion T(s)and closed-loop admi ance Y(s), which ela e he g id cu en e e ence i∗and he PCC ol age s o he ob ained g id cu en , espec i ely. Fig. 1 shows, in g een, he in eg a ion o he p oposed con olle Kin he sys em. I has h ee inpu s: he measu ed PCC ol age s, he sensed g id cu en iand i s co esponding e e ence i∗. E en hough he con olle is ob ained as a MIMO sys em in he syn hesis p ocess, is in e es ing o di ide i s ans e ma ix in ows: K(s)=[KsK e Ki]T. Con- olle ac ua ion udynamic can be exp essed in he Laplace domain as ollows: U(s)=Ks(s)·Vs(s)+K e (s)·I∗(s)+Ki(s)·I(s).(4) Subs i u ing exp essions (4), (2) and (3) in (1) gi es he closed loop g id cu en dynamic in he Laplace domain: I=(1−GKi)−1GK e    T(s) I∗+(1−GKi)−1(Gd+GKs)   Y(s) Vs, (5) whe e (s and-alone) sys em s abili y depends only on he sys em open-loop ans e unc ion L=−GKi. The con olle design ollows a model- e e ence app oach; ha is, a con olle K(s)is ob ained in o de o make he closed-loop ans e unc ions Y(s)and T(s) esemble wo gi en model e e ences, Y e (s)and T e (s). By defining a plain low admi ance e e ence a he sys em esonance equency, oscilla ions can be e ec i ely damped and, as a esul , de i ed s abili y p oblems a e a oided. As i is shown in Fig. 1, g id cu en e e ence i∗is gene a ed by an ou e loop powe con olle , whose objec i e may be, o example, main ain he DC-bus ol age DC equal o a gi en e e ence ∗ DC e en unde changes o he load RL (i.e., he sys em ac s as an ac i e ec ifie ) [15]. I s design is ou o he scope o his wo k, which is cen ed in he inne cu en con olle design. B. Con olle design Con olle K(s)is ob ained h ough an H∞syn hesis algo i hm, whose inpu is he gene alized plan P[16]. I is jus a i ual MIMO plan wi h he s uc u e showed below: z z z =Pw w w u u u,(6) whe e w w wis called he exogenous inpu s ec o o he sys em, z z zis he so-called ou pu e o signals, u u uis he ac ua ion ec o ha will be compu ed o he con olle and is he measu emen s ou pu ec o . H∞syn hesis p ocess will compu e a (sub)op imal con- olle K, which minimizes he infini y no m1o he closed- loop sys em N ha esul s om he eedback in e connec ion o Pand K, and ela es exogenous inpu ec o w w wand e o ec o z z z=Nw w w: min KN(K)∞=min K z z z2 w w w2 ≤γ(7) Fig. 2 shows he conside ed i ual plan P(in ed) used o ob aining he con olle K(in blue) and he esul ing closed- loop sys em N.Pis o med by he open-loop plan s Gand Gd(in o ange) and a se o added elemen s only necessa y o he syn hesis p ocess (in pu ple) which a e explained below. T e (s)and Y e (s)a e he desi ed ( e e ence model) ack- ing and admi ance ans e unc ions. Thei ou pu s, i and iy, 1The infini y no m o a MIMO sys em H(s)in he equency domain is defined as H(s)∞ supω¯σ(H(jω)), whe e ¯σ(H(jω)) is he maximum singula alue o H(jω). Fo SISO sys ems ha is simplified o H(s)∞  supω|H(jω)| s * i i i y e y i e w u z N * i s i u d G G e T e Y u W W y W P K i s * i u s Fig. 2. S uc u e used o he H∞syn hesis. In ed he gene alized plan P(s). W apped inside o i , in o ange he open loop plan G(s)and open loop admi ance Gd(s), and in pu ple he e e ences and weigh s added o he syn hesis. Finally, he ob ained con olle is ep esen ed in g een. ma k he e e ence cu en s due o acking and admi ance con ol e ec s, espec i ely. G id cu en iis sub ac ed om bo h e e ence cu en s, gi ing wo di e en e o s: e o acking and ey o admi ance con ol. Bo h con olle objec- i es (i.e., admi ance and acking shaping) can be achie ed by means o minimizing hese wo e o s. Bu , as bo h e o s depend on he g id cu en , hey can no minimized a he same equencies (i.e., ican no be equal o bo h i and iy a he same equency). Addi ionally, con ol bandwid h mus be limi ed by means o con ol e o uminimiza ion. To handle his ade-o , h ee equency-weigh s (W (s),Wy(s) and Wu(s) espec i ely) mul iplies each signals. Ou pu and inpu signal ec o s o gene alized plan Pdefined in Fig. 2 a e hen: z z z=⎡ ⎣ W ·e Wy·ey Wu·u ⎤ ⎦ =⎡ ⎣ s i∗ i ⎤ ⎦w w w= s i∗u u u=u(8) As syn hesized con olle Kshould minimize z z z, inc easing one weigh a a gi en equency (while se ing he o he wo ela i ely low) should esul in he minimiza ion o i s inpu a ha equency. Following his design c i e ion, and gi en he con olle objec i es s a ed in he p e ious subsec ion, sui able equency-weigh s a e ep esen ed in Fig. 3. As i is shown, acking o cu en e e ence i∗is only desi ed a he g id undamen al equency, 60 Hz in his applica ion, meanwhile admi ance shaping is desi ed, mainly, a supe -synch onous equencies, which is he loca ion o he LCL-fil e esonance (a sub-synch onous admi ance shaping is also conside ed o damp possible low equency g id oscilla ion). Finally, con ol bandwid h is limi ed a high equencies. Laplace exp essions o he selec ed weigh s a e showed below: W (s)=K s2+2ζnω1s+ω2 1 s2+2ζdω1s+ω2 1 ,(9) whe e K is he ini ial acking gain, ω1is he undamen al equency in ad/s and ζdand ζnwill define bo h maximum 102101100101102103104 100 50 0 50 100 Magni ude (dB) Admi ance shaping a ea Admi ance shaping a ea Re e ence acking a ea Con ol s opband F equency (Hz) Fig. 3. Magni ude o he selec ed weigh s in he equency domain. The equency spec um is di ided in a eas acco ding o he con olle objec i e in ha ange. )( sW )( y sW )( usW )( e sT )( e sY )('sG )( dsG )(zG )(sG ()sK ()zK syn hesis ()sP 1 )(()zz' GzG Disc e ize h ough ZOH me hod Add compu a ional delay Make con inuous h ough Tus in app oxima ion Con inuous plan wi h delay and ZOH dynamics sis Disc e ize h ough Tus in me hod Con inuous open loop model C o n i n u o u s o p e n l o o p m o d e l Design speci ica ions Fig. 4. Flux diag am o he inne con olle Ksyn hesis and disc e iza ion p ocess gain peak and wid h o he weigh esonance; Wy(s)=Ky s2+2ζdω1s+ω2 1 s2+2ζnω1s+ω2 1 ·1 (1/ωy)s+1 (10) whe e an ini ial weigh Ky>K is defined, ζdand ζnchange posi ions o define a complemen a y no ch o he acking weigh esonance and an addi ional pole is defined a high equency ωy o delimi he admi ance shaping egion; Wu(s)=Ku (1/ωu1)s+1 (1/ωu2)s+1,(11) whe e a ze o in he equency ωu1will inc ease he ini ial low ac ua ion weigh Ku, and a pole a e y high equency ωu2 is added jus o ulfil H∞syn hesizing me hod equi emen s. As o he e e ence model, a acking e e ence T e =1is se , which will esul in pe ec cu en acking a undamen al equency (i.e., I(jω1)≈I∗(jω1)), meanwhile a low esis i e admi ance o Y e =0.1Ω −1is conside ed, which will esul in a p ope ly damped LCL esonance. Syn hesized con olle K(s)is ob ained in Laplace con inu- ous domain. In o de o be implemen ed in a digi al pla o m, AC p og ammable Powe supply DSP-FPGA Con ol pla o m VSC LCL l e Fig. 5. Pic u e o expe imen al se -up. TABLE I. EXPERIMENTAL SETUP PARAMETERS Sn17.5kVA L13.4mH Vg120 VR128.8mΩ ω12π60 ads−1L21.7mH V∗ DC 700 VR218.6mΩ Tsw 400 μsC18 μF Ts200 μsCDC 4.7mF i mus be p e iously disc e ized. The disc e iza ion p ocess is summed up in Fig. 4. I is impo an o poin ou ha he ob ained con olle , and hen he close-loop admi ance, akes in o accoun he phase lag in oduced by bo h PWM modula ion and compu a ional delay. Fo mo e in o ma ion abou he disc e iza ion p ocess o he admi ance and acking shaping o powe con e e s e e o [17]. IV. RESULTS P oposed algo i hm has been es ed in bo h simula ions and an expe imen al se -up. The la e consis o an AC p og ammable powe supply Pacific Sma Sou ce 345-AMX, emula ing he g id, and a 17.5kVA wo-le el VSC connec ed o i h ough an LCL fil e . Passi e loads RLa e connec ed o he DC-side o es he pla o m unde di e en ope - a ing poin s. Con ol algo i hm is implemen ed on a Texas Ins umen s DSP TMS320DSK6713. Table I sums up se -up pa ame e s, whe e Tsis he sampling pe iod o he con ol algo i hm and Tsw is he con e e ’s IGBTs swi ching pe iod. A pic u e o he expe imen al se -up is shown in Fig. 5. A. F equency domain esul s Fig. 6 shows heo e ical acking ans e unc ion T(ob- ained ollowing (5)), which is equal o i s e e ence T e a he undamen al equency, as was specified in he design p o- cess. Fig. 7 shows he admi ance shaping esul s. Theo e ical admi ance Yis equal o he gi en e e ence Y e a bo h sub and supe -synch onous equencies as i was specified. F equency (Hz) M agn i u d e (dB) 100101102103 −10 −5 0 5 T e T Fig. 6. Ob ained acking ans e unc ion Tand i s e e ence T e . F equency (Hz) −200 −150 −100 −50 0 Magni ude (dB) Y e GdYYiden Phase (deg) 100101102103 −180 −90 0 90 180 Fig. 7. Admi ance equency esul s. Open-loop admi ance Gdis shown in black. Admi ance e e ence model Y e is shown on blue colo . The closed loop syn hesised admi ance ( heo e ical) Yis shown on pu ple. Red c osses show he expe imen ally measu ed admi ance Yiden . Yiden is an expe imen al admi ance measu emen ob ained by adding a h ee phase con olled sinusoidal signal o he ol age gene a ed by he AC powe supply and analysing, in s eady- s a e, he cu en esponse o he con e e a ha equency. O main impo ance is he esonance equency o Gd, whe e mo e poin s o Yiden a e aken, which p o es he good ac i e damping capabili ies o he p oposed me hod. B. Sys em obus ness Fig. 8 shows he ob ained sys em sensi i i y unc ion S= (1 −GKi)−1. I s infini y no m S∞is a good in e se indica o o he design s and-alone obus ness [16], ha is, how much plan pa ame e s may change un il he designed sys em becomes uns able. A commonly design c i e ion o obus con olle is o syn hesize loops wi h S∞<6dB, which will assu e a gain ma gin bigge han 6dB and a phase ma gin bigge han 30o. This c i e ion is ulfilled, as i can be seen in Fig. 8, which p o es he design obus ness. The e ec s o g id impedance on he p esen ed ac i e damping echnique a e es ed below. Conside ing a non-ideal induc ance Lga he PCC, he g id cu en is modified as ollows: I(s)=G(s)·U(s)+G d(s)·Vg(s),(12) F equency (Hz) M agn i u d e (dB) 100101102103 −80 −60 −40 −20 0 20 S Fig. 8. Ob ained sensi i i y unc ion S. Blue dash-do ed line ma ks he accep ed 6dB sensi i i y gain limi o obus sys ems. Lg(p.u) F equency(Hz) |Y|(-1) Lg(p.u) F equenc y (Hz) Fig. 9. E ec o modi ying he g id induc ance Lgon he ob ained close loop admi ance modulus |Y(s)|. whe e Vg(s)=Vs(s)+(s·Lg)·I(s)and G(s)=−Zc(s) ZL1(s)·Zc(s)+Zg(s)·(ZL1(s)+Zc(s)),(13) G d(s)= ZL1(s)+Zc(s) ZL1(s)·Zc(s)+Zg(s)·(ZL1(s)+Zc(s)) (14) He e Zg(s)=s(Lg+L2)+R2is he new g id-side impedance. Subs i u ing exp essions (4), (13) and (14) in (12) gi es he new closed loop inpu admi ance Y(s)seen om he PCC: Y(s)= I(s) Vs(s)=G(s)·Ks(s)+G d(s) 1−(G(s)·Ki(s)+s·Lg·Gd(s)) (15) Fig. 9 shows closed loop inpu admi ance modulus |Y(s)|in he equency domain unde changes o he g id induc ance Lg∈[0.05,1] p.u 2. P esen ed damping echnique is e ec i e e en unde high g id induc ances (i.e., weak g ids), as i is shown in he minimum modulus inc ease o e he gi en admi ance e e ence o 0.1Ω −1. C. Time domain esul s Fig. 10 shows expe imen al ime domain e olu ion o he sys em a e he in oduc ion o a DC-load o 4.2 kW. Fig. 2Lgis exp essed in pe uni alues o he con e e nominal impedance Zn=( √3Vg)2/Sn. G id ol age G id cu en DC-bus ol . Zoom 1 Zoom 1 Zoom 2 G id ol age G id ol age G id cu en G id cu en DC-bus ol . DC-bus ol . +250V -250V +30A -30A +780V +580V +250V -250V +30A -30A +780V +580V +250V -250V +30A -30A +780V +580V 5ms/di 50ms/di 0ms -5000ms -3165ms-3215ms-3390ms-3890ms Zoom 2 Fig. 10. Connec ion o a 4.2kW DC-load wi h a null eac i e e e ence. Top shows he comple e ansien . Zoom 1 ocuses on he cu en s and DC- ol age e olu ion a e he connec ion. Zoom 2 shows g id cu en s and ol ages in s eady-s a e. G id ol age G id cu en DC-bus ol . Zoom 1 Zoom 1 +250V -250V +60A -60A +720V +680V +250V -250V +60A -60A +720V +680V 20ms/di -790000us -1290000us -1084000ms G id ol age G id cu en DC-bus ol . 50ms/di -1284000us Fig. 11. Response unde g id balanced ol age dip when DC-bus is loaded wi h 4.2kW. All phases all o 60% o i s alue keeping hei phase un ouched. Top iew shows he comple e ansien in g id ol ages, cu en s and DC-bus ol age. Lowe iew ocuses on he dip ini ial edge. 11 shows he esponse o he sys em unde a balanced dip o 60% o he nominal g id ol age alue. Bo h expe imen s show good cu en acking pe o mance. O main impo ance is he cu en esponse unde he g id dip: as can be seen, no oscilla ion a equency ω es can be app ecia ed on i , which p o es again he good ob ained esonance damping. V. CONCLUSION This wo k p esen s a no el ac i e damping echnique o cu en -con olled g id-connec ed powe con e e s h ough esonance fil e s like he LCL. Resonance damping will esul in smalle cu en oscilla ions unde PCC ol age a ia ions and, as a consequence, will imp o e sys em s abili y in con- nec ion o weak g ids. P esen ed ac i e damping echnique is based on equency shaping o he closed-loop inpu admi ance o he conside ed applica ion, and only needs g id cu en and g id ol age mea- su emen s (i.e., no addi ional LCL fil e senso s a e needed). This is achie ed by a model- e e ence app oach, whe e, a e designe specifies a desi ed admi ance e e ence, an H∞ syn hesis algo i hm will ob ain he (sub)-op imal con olle ha shapes he sys em admi ance, in he equency domain, o ollow he gi en e e ence. By defining a esis i e admi ance e e ence he esonance can be e ec i ely damped, and ela ed s abili y p oblems a e a oided. Cu en con ol design o achie e his goal is explained. Expe imen al es o he ob ained con olle a e done and hei esul s a e shown. P oposed me hod shows good cu en acking and esonance damping capabili ies. The la e is p o ed expe imen ally in bo h equency and ime domains. Mo eo e , damping echnique is demons a ed ( heo e ically) e ec i e e en unde ex emely high g id impedance, wi h bo h good phase and gain ma gins. Fu u e wo ks will s udy he esonance damping and s abili y obus ness unde fil e and g id modelling unce ain ies, and how hey a e a ec ed by he addi ion o di e en con ol measu emen s (like fil e capaci o ol age o con e e side cu en ). ACKNOWLEDGEMENT This wo k o he Uni e si y o Alcala g oup was sup- po ed in pa by he spanish esea ch p ojec s CONPOSITE (ENE2014-57760-C2-2-R Minis e io de Economía y Com- pe i i idad) and PRICAM (S2013/ICE-2933 Conseje ía de educación, ju en ud y depo e de la Comunidad de Mad id). The wo k o Robe G iñó was suppo ed in pa by he Spanish Resea ch P ojec DPI2013-41224-P. REFERENCES [1] T. Nussbaume , M. L. Heldwein, and J. W. Kola , “Di e en ial mode inpu fil e design o a h ee-phase buck- ype pwm ec ifie based on modeling o he emc es ecei e ,” Indus ial Elec onics, IEEE T ansac ions on, ol. 53, no. 5, pp. 1649–1661, 2006. [2] J. He and Y. W. Li, “Gene alized closed-loop con ol schemes wi h embedded i ual impedances o ol age sou ce con e e s wi h lc o lcl fil e s,” Powe Elec onics, IEEE T ansac ions on, ol. 27, no. 4, pp. 1850–1861, 2012. [3] S. G. Pa ke , B. P. McG a h, and D. G. 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