Impedance-Model-Based MIMO Analysis of Power Synchronization Control

Transcription

1 Impedance-Model-Baed MIMO Analyi of Power Synchronization Control Javad Khazaei, Member, IEEE, Zhixin Miao, Senior Member, IEEE, and Lakhan Piyainghe Abtract Thi paper preent impedance-model-baed tability analye of power ynchronization control (PSC) and two type of vector control ued in voltage ource converter-baed high voltage direct current (VSC-HVDC) ytem. The impedance model of a VSC with PSC i firt derived. Stability analyi i then carried out uing multi-input multi-output (MIMO) ytem analyi. A a comparion, power and angle tranfer function-baed analyi i alo conducted. The impedance modelbaed tability analyi reult are validated by the time-domain imulation. Impedance model of two type of vector control are alo derived for comparion. Effect of hort circuit ratio (SCR), power tranfer level, high-pa filter, and the PSC gain are demontrated through analyi and real-time digital imulation in RT-LAB. Index Term power ynchronization control, vector control, weak ac ytem, Nyquit tability criterion, impedance model. I. INTRODUCTION Conventional line current commutating converter (LCC)- HVDC tranmiion ytem cannot perform properly if the interconnected ac ytem i not trong enough. The trength of an ac ytem i commonly defined by hort circuit ratio (SCR), which depend on the HVDC nominal power and the trength of the ac ytem. Normally, ytem with SCR value of le than.5 are conidered a weak ac ytem 6. In contrat to the traditional HVDC ytem, VSC-HVDC can be connected to very weak ac ytem without any reactive compenation 7. Vector current control i the mot popular control cheme for the VSC-HVDC ytem 8, 9. Vector control of the VSC-HVDC can control the converter in weak ac ytem with independent active and reactive power upport to the grid,. However, there are ome barrier regarding the application of vector control epecially when a VSC i connected to a very weak ac ytem. Studie have hown that vector current control of a VSC-HVDC cannot tranfer power level of more than. p.u once connected to a grid with the SCR level of,. Analytical tudie indicated that the limiting factor of vector control can be the interaction between current control and grid inductance 6, and/or phaelocked-loop (PLL) dynamic 6,, 5. It i mentioned in 6, that low frequency reonance may occur due to the interaction of vector current control and weak ac ytem. Additionally, the PLL dynamic will caue problem when the converter i ynchronized with the weak grid. To overcome J.Khazei i with the chool of cience, engineering, and technology (SSET) at Penn State Harriburg, and Z. Miao, and L. Piyainghe are with Department of Electrical Engineering at Univerity of South Florida, Tampa, FL ( jxk79@pu.edu; zmiao@uf.edu; lakhan@mail.uf.edu). the weak grid iue, 5 applied gain cheduling technique to deign the outer loop power/voltage a an MIMO control ytem, which reulted in an increae in the power tranfer level. Another olution i the power ynchronization control (PSC). Compared to the gain cheduling control where the controller gain varie, PSC erve a a claic type controller with a fixed tructure and fixed parameter. The PSC ha alo been hown a a uperior alternative for vector control of VSC- HVDC ytem in connection to weak ac grid, 6 9. Unlike the PLL that ynchronize the converter to the point of common coupling (PCC), the PSC directly ynchronize the converter to the grid through a power control loop. Application of the PSC control method ha been tudied in a few paper. In the firt paper, the power ynchronization i introduced a an alternative to the conventional vector control in weak ac ytem. In 6, the PSC i ued to interconnect two very weak ac ytem. The effect of different parameter on the tability of the ytem i alo invetigated. The application of the PSC for offhore wind farm i introduced in 7. Stability limitation of variou control loop in HVDC ytem enhanced with PSC are tudied in 8. It i found that alternating voltage control make ytem more table compared to reactive power control. Moreover, the impact of the converter with the PSC control on ub-ynchronou reonance (SSR) damping i tudied in 9, which how that the PSC can greatly improve the damping for SSR mode. In the original PSC paper, the power veru angle tranfer function i ued to carry out the tability analyi. The power veru angle tranfer function conider a ingle input ingle output (SISO) ytem, with the effect of voltage on active power ignored. Impedance modeling i a popular approach for converter and grid interaction analyi 6,. The impedance model of the PSC ha been derived in 9 for a converter connected to a erie compenated tranmiion line. The main focu of 9 i to tudy the effect of power ynchronization control on ubynchronou reonance caued by the interaction between the ynchronou generator and the converter. The analyi in 9 relie on the frequency repone of the impedance matrice. MIMO ytem analyi technique were not adopted in the aforementioned erence. The main contribution of our paper i to analyze the tability and robutne of the PSC for a VSC-HVDC interconnected to a weak ac grid uing impedance model. The MIMO ytem impedance i derived and ytem tability i then evaluated uing ingular value robutne analyi. Two type of vector control are alo conidered for comparion. The impedance model of converter with vector control ha been developed

2 P,Q P,Q Weak AC Grid DC Link Converter Converter V V R g Lg V g AC Filter AC Filter DC Link AC/DC DC/AC i abc i abc Strong AC Grid L g V g Vector Control V V P P PI, PSC v V c d v dq abc V e j c abc v q i a i b i c H() H() abc/dq t PWM Pule Carrier P P Q Q PI PI v abc i d i q Inner Control PLL v d v q PLL dq abc v abc PWM Pule Carrier Fig. : Back-to-back VSC-HVDC connected to a weak ac ytem. Two different control for the rectifier are alo preented. in the literature, a hown in 6,. However, outer control loop have not been conidered. In thi tudy, comprehenive impedance model of vector control will be derived for comparion purpoe. Nyquit theory and the ingular value of the return matrix are applied for the tability and robutne analyi, repectively. The eigen loci of the open-loop ytem and ingular value plot of the return difference matrix are employed in the MIMO tability analyi. The circuit and control tructure of the ytem i hown in Fig.. The PSC i implemented on the rectifier ide of the VSC-HVDC ytem (Converter ). For comparion, the impedance model for vector control of VSC-HVDC are alo derived. Two type of vector control (with and without power outer loop) will be examined. The ret of the paper i organized a follow. Section II decribe the mall-ignal model ued to derive the impedance model. Section III preent the power/angle tranfer function. Section IV preent MIMO ytem-baed tability analyi and validation reult by imulation. Section V conclude the paper. Vg R g PSC Voltage Control L g V Z conv V c R dq AC Filter abc t L v abc Vc PWM Converter AC/DC Carrier Fig. : Simplified model of the ytem with the PSC. The high-pa filter notated a H() in Fig. are omitted in thi model. R = Ω, L =. H, R g =. Ω, L g =.5 H for SCR =, L g =.88 H for SCR = and L g =.8 H for SCR =. II. SMALL-SIGNAL MODEL OF THE SYSTEM A. Circuit Dynamic A the main contribution of thi reearch i weak ac grid, the converter to be tudied i the rectifier (Converter ) that i connected to the weak ac grid. Fig. illutrate a imple repreentation of the ytem in Fig. with the rectifier converter, the ac grid and the filter conidered. The high-pa filter notated in Fig. a H() are not included. Z g () = R g L g i the total impedance of the weak ac grid and it tranformer, Z conv () i the impedance viewed from the point of common coupling (PCC). V c i the converter input voltage and V g i the ac grid voltage. The ac filter i repreented by R and L. A it can be oberved, the converter i equipped with the PSC and an alternating voltage control. The main dynamic related to the inductor in the dq-erence frame (rotating at a ynchronou peed ω ) i expreed a: V V c = L di dt jω LI RI. () where, V, V c, and I are complex vector for the PCC voltage, the converter voltage, and the line current i abc in the dqerence frame. V = v d jv q, V c = v cd jv cq, and I = i d ji q. Further, the converter input voltage i defined uing it magnitude V c and the angle relative to the dq-erence frame θ baed on the PSC control in Fig. : V c = V c e jθ = V c co(θ) jv c in(θ) () where θ i the output from the power control. Since V c = V c e jθ, theore it mall-ignal expreion i V c = jv c e jθ θ e jθ V c = jv c θ e jθ V c. () where the ubcript repreent initial condition. In Laplace domain, we now have: I = V V c R ( jω )L = V jv c θ e jθ V c. () R ( jω )L

3 id = i q I() R L ω L ω L R L (R L) (ω L) }{{ } G ω Lv cd (R L)v cq ω L in θ (R L) co θ vd (R L)v cd ω Lv cq ω L co θ (R L) in θ v q (R L) (ω L) }{{ θ (R L) } (ω L) V c (5) V () G G Separating into dq component, we now have the mallignal model of the circuit dynamic in (5). Theore, the mall-ignal repreentation of V can be repreented by: B. Control The key tructure of the PSC i illutrated in Fig.. The idea behind the PSC come from the analogy of the power and angle relationhip in ynchronou machine. In the PSC, ynchronization take place by controlling the active power through the converter. The PSC output i then ued a the erence angle for the VSC PWM unit. By applying thi method, the PLL i not needed. Referring to Fig., the difference between the erence active power of the converter and the meaured power i ent to an integrator controller ( Ki,PSC ), and the output of which preent the ynchronization angle (θ). θ() = K i,psc (P () P ()) (6) The complex power from the PCC bu to the converter i notated a S : S = V I = (v d jv q )(i d ji q ). The mall-ignal equation of active and reactive power are a follow: P = v d i d i d v d v q i q i q v q, Q = v q i d i d v q v d i q i q v d. (7) Theore, the mall-ignal repreentation of the power ynchronization loop i derived by obtaining the mall-ignal model of (6) and replacing P uing (7): T i d θ = Ki,PSC i q } {{ } G v d v q Ki,PSC v d T v q } {{ } G i d i q Ki,PSC P (8) C. AC Voltage Control Loop A illutrated in Fig., the AC voltage controller regulate the magnitude of the rectifier voltage to maintain the grid ide voltage. A imple PI controller i ued to control the magnitude of the voltage, and the output of which i employed to generate the erence converter voltage magnitude V c. Dynamic equation for the alternating voltage controller can be expreed by 9: V c = ( k pv v DP ) (V V ). (9) where V i the erence magnitude of the PCC voltage, and V i the meaured magnitude of PCC voltage, vd v q. V = V (v d v d v q v q ) () Applying the mall-ignal analyi to (9) and replacing V in (9) by () lead to: ( V c = k pv k ) ( iv V k pv k ) vd T iv V vd v d V }{{ v q } G Dq () Parameter J Pq of the PSC are included in the Table III of the Appendix. - Power D. ynchronization Impedance Model of a VSC with PSC Control control Finally, the block diagram conidering the circuit dynamic in (5), PSC in (8), and voltage control in () i preented in Fig.. V () DP DV K i,psc k v pv G G G Dq DV c G G G I() Fig. : Block diagram coniting the circuit dynamic, PSC, and voltage control. The relationhip between the input and output i a follow: I() = (I G G ) (G F G G G ) V () Y conv (I G G ) K i,psc T G ) P () G (kpv kiv V } {{ } I c where I i a identity matrix. Thi notation hould be differentiated from I(), the current vector Laplace form. Y conv i the admittance model of the converter and the impedance model Z conv i the invere of the admittance matrix: Z conv = Yconv. It can be een that the impedance model i alo a matrix. Equation () decribe a Norton equivalent of the converter with the RL filter. The entire ytem in Fig. can now be expreed a a circuit in Fig..

4 Grid V g Zg () I () Y conv Converter Fig. : Impedance model of a converter connected to a grid. III. TRANSFER FUNCTION-BASED ANALYSIS Given the power control loop, it i eay to eek the relationhip between the complex power S c = P c jq c injected to the converter veru the voltage magnitude V c and the angle θ. The following derivation i baed on mall-ignal model of the complex power in the dq erence frame. Our derivation i baed on the complex power definition and line impedance model. Compared to the derivation in, the following derivation i traightforward and eay to follow. If R i ignored, then P = P c. In the dq-erence frame, the complex vector of the converter voltage and the grid voltage are a follow: V c = V c θ, V g = V g () where V g i aumed to be contant, while V c and θ are controlled through the PSC voltage and power control. The electromagnetic dynamic of the line will not be neglected. A a reult, in Laplace domain, the current and voltage relationhip in the dq erence frame i I() = V g() V c () R ( jω ) L. () where R and L are the total reitance and inductance between the grid and the converter. Note that the line impedance model in abc erence frame i R L, while in a ynchronou erence frame it become R ( jω ) L. The mall-ignal model of () i alo derived a follow. Auming that the grid voltage i contant, then V g =. Theore, I() = jv c θ e jθ V c R ( jω ) L. (5) Since the complex power to the converter can be expreed a S c = V c I, it expreion in Laplace domain will be: S c () = V c I () I V c () (6) where the ubcript notate the initial condition. Subtituting I() uing (5) and ubtituting V c uing (), we have S c = jv c R( jω js c ) L V c R( jω )L I e jθ I c T θ() V c () where S c i the initial complex power and S c = V c I. (7) Separating the real and imaginary part, we can find P c = J P θ θ J P V V c. The power to angle tranfer function can be found a: J P θ = Q c ω LV c ( R L) (ω L) (8) where Q c i the initial reactive power injected into the converter and Q c = Im(S c ). The above tranfer function indicate that there are 6 Hz reonance due to the line electromagnetic dynamic. Thi phenomenon ha been mentioned in and a high-pa filter ha been implemented to increae the damping for thi ocillating mode. The control idea i to increae the total reitance by introducing a virtual reitance via a high-pa filter. The reulting power angle tranfer function become: J P θ = Q c ω LV c ( R H() L) (ω L) (9) where H() i the tranfer function of the high-pa filter. The virtual reitance will provide damping at 6 Hz. A high pa filter will fulfill thi tak: H() =.5. The implementation of the filter hould be in the dq erence frame with a contant rotating peed ω a hown in Fig.. It can be een that the PSC i coupled with the AC grid and a high-pa filter i neceary. On the other hand, the vector control i equipped with a feed forward compenation (, Chap, pg. 5). The feedforward compenation decouple the converter operation from the grid and enhance the diturbance rejection capability. Theore, there i no need for the high pa filter in the vector control baed converter. Dq - J Pq Power ynchronization control DP Fig. 5: The cloed-loop ytem that conit of the PSC control and the plant model. V () G Remark: If we ignore the voltage control impact, we may have a cloed-loop ytem howng in Fig. 5 with the PSC DP control and the plant model Ki,PSC repreented by (9). Dq The Gopen- loop tranfer function i Ki,PSC J P θ. Note that tability analyi in i baed on thi ytem. G G DV kiv kpv DV c G IV. MIMO STABILITY ANALYSIS AND VALIDATION THROUGH TIME-DOMAIN SIMULATION According to Fig., the current can be derived a follow. I() = (Z g Z conv ) (V g () Z conv I c ()) = (Z g Z conv ) Y convy conv (V g () Z conv I c ()) = Y conv Z g Y conv Z conv Y conv (V g () Z conv I c ()) = I Y conv Z g (Y conv V g () I c ()) () where I i the identity matrix. For the above ytem, two aumption are placed. (i) The grid voltage V g () i table; I()

5 5 With PI Power Loop Without PI Power Loop Re( Z Conv ) ( degree ) Re( Z Conv ) 6 8 Frequency ( Hz ) ( degree ) 6 8 Frequency ( Hz ) Re( Z Conv ) ( degree ) 8 6 Frequency ( Hz ) Fig. 6: Comparion between real part of converter impedance, Z conv (jω) firt diagonal component for different controller of rectifier ide converter. (ii) the current i table when the grid impedance Z g i zero, i.e., Y conv V g () I c () i table. The firt aumption i valid for the real-world cenario a long a the grid voltage i within the limit. The econd aumption i valid a long a the inverter converter admittance Y conv i table and current order i table. For properly deigned converter, the econd aumption i alo true. Theore, for the current I() to be table, we only need to examine the denominator: I Y conv ()Z g (). In order to claim that the ytem i table, the zero or root of the characteritic function det(i Y conv ()Z g ()) = hould be located in the left half plane (LHP). A both the grid and converter impedance are matrice, the circuit analyi problem become a multi-input multi-output (MIMO) ytem tability problem. The MIMO ytem tability criterion i given in. The characteritic function, det(i Y conv Z g ) = hould have no zero in the RHP. Such tability criterion can be examined by checking the eigen loci or the Nyquit plot of the eigenvalue of Y conv Z g. If the eigen loci do not encircle (, ), then the ytem i table. Thi technique ha been extenively ued in the impedance-baed tability analyi of the power electronic converter,. On the other hand, the minimum ingular value in the frequency domain of the return difference matrix (I Y conv Z g ) provide an index of gain margin and phae margin 5. The ingular value plot have been adopted in the author previou publication 6 to conduct impedancebaed tability analyi. The minimum ingular value indicate how cloe the return difference matrix i to ingularity or det(i Y conv Z g ) at which frequencie. The greater the minimum ingular value, the more robut the ytem. Reonance frequency can be identified a the frequency where the ingular value i the minimum. Reult of the tability analyi will be validated by the high-fidelity model baed imulation. The topology of the imulation model ha been illutrated in Fig. in detail. For the real-time imulation of the propoed ytem, RT-LAB i ued. The detailed RT-LAB model include pule width modulation (PWM) witching detail and dc ytem dynamic which can lect the nonlinear and dicrete behavior of the model. In addition to the PSC, two type of vector control are deigned for comparion. The firt type ha an outer active power and voltage control loop, an inner current control, and a PLL a hown in Fig.. The econd type ha no outer feedback loop. Intead, the current erence are computed directly from power erence. Thi type of the vector control ha been analyzed in and the detailed vector control deign and impedance derivation are preented in Appendice B and C. To verify the trength of the ytem in analyi and imulation, three different SCR value are conidered; SCR = reemble a very weak ac ytem, SCR = for a normal ac ytem, and SCR = which i a trong ytem. The bae value to calculate the SCR are included below: V bae (LL) = kv, S bae = MVA, P HVDC = MVA, L f =. H, L g =.5 H for SCR =, L g =.88 H for SCR = and L g =.8 H for SCR =. A. Impedance of the Converter for Different Control In the firt cae, impedance of the converter are derived for three eparate control. Fig. 6 how the D diagonal (dd) component of Z conv () with the three controller, where the gray urface i the urface where the real part of the impedance i zero. It i oberved that with the PSC, the converter impedance i poitive for the frequency range of more than 5 Hz for low power angle (or low power tranfer level). It i alo noted that a the power angle (or power level) increae, the poitive impedance occur at a higher frequency. For intance, for the power angle of 8 degree, the poitive impedance happen at 7 Hz. Theore, a the power level increae, the impendence become more negative at the fundamental frequency. Compared to the PSC, the vector control ha negative impedance for frequency range of le than Hz at low power angle. With an increae in the power angle (or power level), the impedance will be more negative. Thi how that the PSC provide a poitive impedance or more damping within the frequency range where the vector control provide a negative impedance. B. Stability under Different SCR Scenario The firt cae tudy i deigned to compare the PSC and two other type of vector control for ytem with three different SCR. The Nyquit plot for different controller are illutrated

6 6 Imaginary λ, λ λ SCR= SCR=.5. Hz SCR=.5 SCR= 6. Hz SCR=.5 6. Hz.5 λ (a) Real Imaginary SCR= 8. Hz λ,λ λ SCR= SCR= λ SCR= 6.6 Hz SCR= (b) Real Imaginary λ 7.7 Hz λ,λ SCR= SCR= SCR= SCR= SCR= λ 5. Hz (c) Real Fig. 7: Comparion of eigen loci of Y conv (jω)z g (jω) for different SCR value and controller of the rectifier ide converter. The power tranfer level i MW. The high pa filter i included for the PSC. (a) PSC; (b) vector control with PI power loop; (c) vector control without PI power loop Singular Value Singular Value Singular Value SCR= SCR= SCR= Singular Value (ab) 5 Singular Value (ab) 5 PI Power Loop Singular Value (ab) 5 Simple Power Loop Frequency (Hz) Frequency (Hz) Frequency (Hz) Fig. 8: Comparion of ingular value plot of I Y conv ()Z g () for the different SCR value and controller of the rectifier ide converter. The power tranfer level i MW. in Fig. 7 and the ingular plot for the return matrix are hown in Fig. 8. Fig. 7 how that for SCR = none of the Nyquit plot encircle (-,) in a clockwie direction. All the control method reult in table operation. When the SCR decreae, all of the controller performance deteriorate due to the reduction in the gain and phae margin a illutrated in Fig. 7. By decreaing the SCR to, the vector control encircle the point (, ) in a clockwie direction, indicating intability. However, the PSC till provide a robut performance. The ingular value plot in Fig. 8 how that the PSC with the high pa filter ha the minimum ingular value greater than for variou SCR cenario. Thi indicate a robut tability margin and immunity toward the SCR. On the other hand, the vector control have the minimum ingular value of le than. When the SCR reduce, thi value alo reduce, which indicate that the vector control are prone to intability when the grid i weak. The ingular value plot indicate that the reonance frequency in the dq-erence frame i 6 Hz for PSC. The eigen loci plot alo indicate that the reonance frequency i around 6 Hz for one locu. Simulation reult for the vector control (with outer loop) V (p.u) Q (p.u) Vector Control SCR= Time (ec) Fig. 9: Simulation reult (real power, PCC voltage, and reactive power) for a tep change in the real power when the vector control i applied and the SCR i. are hown in Fig. 9 and Fig.. For the ake of implicity, only two SCR value are included. The vector control aim to control the active power and terminal voltage at the rectifier tation. The tep erence power change i applied to change the erence power from.8 p.u to p.u at the time 5 ec.

7 7 V (p.u) Q (p.u) Vector Control SCR= Time (ec) Fig. : Simulation reult (real power, PCC voltage, and reactive power) for a tep change in the real power when the vector control i applied and the SCR i. V(p.u) Q (p.u) SCR= Time (ec) Fig. : Simulation reult (real power, PCC voltage, and reactive power) for a tep change in the real power when the PSC i applied and the SCR i. It i oberved that when the SCR i, the vector control can follow the erence active power tep change. In contrat to the trong ytem, when the ytem i weak (SCR i ), the vector control fail to upport the power tranfer to.8 p.u. Simulation reult are in agreement with the tability analyi illutrated in Fig. 7 and 8. V (p.u) Q (p.u) SCR= Time (ec) Fig. : Simulation reult (real power, PCC voltage, and reactive power) for a tep change in the real power when the PSC i applied and the SCR i. P (MW) P (MW) P (MW) 5 5 Without Outer Loop (b) (a) With Outer Loop P=5 MW P= MW P=8 MW (c) Time (ec) Fig. : Simulation reult for different active power level for three different control approache. (a) Vector control without outer power loop. (b) Vector control with outer power loop. (c) Power ynchronization control. A three-phae fault i applied at t = 5 at the inverter ac ide and cleared after one cycle. The SCR for thi cae i et to. Fig. - preent the imulation reult for the PSC in trong and weak ac grid. Same a the previou cae, an active power change i applied to change the power tranfer from.8 p.u. to p.u. at 5 econd. It i hown that for both cae, the ytem i table and the PSC can uccefully tranfer the amount of active power which i needed. Compared to the vector control, the PSC can tranfer p.u. even in a very weak ac ytem connection. Multiple power tranfer level for three eparate controller are preented in Fig.. The firt figure i for the vector control with imple power controller. Compared to a imple power controller, the vector control with PI outer loop provide a better reult for power level of 8 MW and MW. However, the vector control fail to tranfer 5 MW in thi cae. In contrat, the PSC can tranfer all three active power level. C. Effect of the High-Pa Filter The effect of the high-pa filter i examined in thi ubection. The tudied filter will move the 6 Hz reonant pole to the left hand plane (LHP) and provide more tability margin to the PSC controller. The reult of the root loci (on the open K loop tranfer function J i,psc P θ ) comparion for two cae (with and without filter) are hown in Fig.. It i oberved that by adding the filter, two reonant 6 Hz pole will move to the left in the -plane, which indicate more damping. Simulation reult in Fig. 5 validate the analyi reult. The ytem i imulated without the filter at the beginning and the filter i activated at 5 econd. It i oberved that after activating the filter, 6 Hz ocillation caued by reonant pole will be damped well.

8 8 Imaginary Axi 5 Without Filter With Filter 5 Real Axi Imaginary Axi 5 5 With Filter No Filter Real Axi Fig. : Effect of filter on root locu curve of the ytem when SCR i (a).85.8 P ( p.u ) P ( p.u ) P ( p.u ) (a).....5k.6 PSC =9 (b) K PSC = K PSC = (c) Time ( ec ) Fig. 7: Real-time imulation reult for different PSC gain loop when the SCR =. (a): PSC loop gain i 9, (b): PSC loop gain i, (c) PSC loop gain i 5. The dynamic are initiated by a tep change in the power order P (b) Time (ec) Fig. 5: Real-time imulation reult for the effect of filter when SCR i hown in different cale. (a) Effect of filter during the operation. (b) Zoom in verion of the firt ubplot. D. Effect of the PSC Loop Gain A large PSC integral gain enure a fat repone. However, a large gain can reult in intability. Thi iue wa mentioned in without preenting any analyi and imulation. In thi ubection, the root loci of the open loop tranfer function (J P θ K i,psc ) are ued to indicate the range of,psc. Singular Value (ab) K PSC =5 Singular Value 6 8 Frequency (Hz) No Filter With Filter Singular Value (ab) K PSC = Singular Value Frequency (Hz) Fig. 8: Effect of high-pa filter and large gain. Imaginary Axi 5 Without Filter Sytem: F_new Gain: 9 Pole:.9 7i Damping:.6 Overhoot (%): Frequency (rad/ec): 8 the gain i increaed from 5 to, the minimum ingular value become le than. The MIMO ytem analyi reult corroborate with the analyi reult concluded from the power and angle tranfer function analyi and imulation Real Axi Fig. 6: Root loci of PSC for SCR equal to, with the high pa filter. Imaginary Axi 5 With Filter Reult of root locu analyi of the ytem with PSC control are illutrated in Fig. 6. It i oberved that by increaing the PSC gain to 9, the 6 Hz reonance pole 5 will move to the right hand ide of the -plane and make the ytem untable Real Axi Simulation reult for different PSC gain are preented in Fig. 7. When the PSC gain i et to 9, large ocillation are oberved and power command can not be followed. However, with the gain a low a 5, the ytem will be table. The developed impedance model i alo ued to invetigate the effect of the filter and a large PSC gain. Fig. 8 how the effect of the filter and large gain on ingular value plot. The finding verify that the high-pa filter application help to improve the minimum ingular value. On the other hand, when V. CONCLUSION Stability analyi of the VSC-HVDC ytem connected to a very weak ac ytem ha been preented in thi tudy. Three different control for VSC are conidered including a power ynchronization control, a vector control with PI outer loop, and a vector control with imple power control. The impedance-baed analyi i implemented to derive the converter input impedance conidering the detailed control. The derived impedance model i then ued to invetigate the interaction between the converter and the weak ac grid. The MIMO ytem analyi i utilized to evaluate the ytem tability under everal control trategie and operating condition. The cae tudie how that the PSC i capable to handle higher power tranfer compared to the vector control. Moreover, the 6 Hz reonance tability of PSC i greatly improved by a high-pa filter. The PSC gain ha a limit. The overall reult of the imulation and analyi demontrate the uperior performance of PSC in term of power tranfer when the ac grid i weak.

9 9 ACKNOWLEDGEMENT The author would like to acknowledge the help from Prof. Lingling Fan. The author alo wih to acknowledge anonymou reviewer whoe comment helped improve thi paper. APPENDIX A SYSTEM AND CONTROLLER PARAMETERS Thi ection provide the parameter of the ytem and controller. TABLE I: Sytem Parameter of VSC-HVDC Model Quantity ac ytem line voltage ac ytem frequency bae power dc rated voltage dc cable parameter dc cable length Value kv 6 Hz MW 8 kv.9 Ω/km,.59 mh/km,. µf/km km Separating equation () into dq-axe, the plant model for the current control deign i derived: L di d dt Ri d = v d v d ωli q () u d L di q dt Ri q = v q ωli d () u q The plant model for the current controller i aumed a /(RL) for both d and q axe. The input are u d, u q, while the output are i d and i q. The feedback control are deigned for the dq-axi to track the erence current. In addition, to generate the dq component of the converter voltage, the cro coupling and feed-forward voltage term hould be added to the deign. A implified inner current control bloc illutrated in Fig. 9. The loop gain of ytem i repreented by: TABLE II: Parameter Simplified of Outer Individual Loop VSC Model Switching frequency 6 Hz Grid filter.8 H (for SCR ) Grid filter i.88 d i P H d (for SCR ) k Grid filter p V.5 H (for SCR ) dc capacitor 96 µf P i d, i d k p Simplified Inner Loop Model u d L R Plant Model TABLE III: Parameter of PSC controller, rmp SC =5 PI controller k pv=.5, v = TABLE IV: Parameter of Vector Controller parameter bandwidth (rad/) Current controller k p=5, = 5 Alternating voltage controller k p=., = (SCR=) 5 (SCR =) (SCR=) Outer PI controller k p=., =5 5 APPENDIX B VECTOR CONTROL AND BANDWIDTHS The inner current control for the vector control hould be deigned to be much fater than the outer control loop. The converter voltage in abc frame i notated a v abc and the current i notated a i abc. The voltage at the point of the common coupling (PCC) i notated a v. An RL circuit i conidered between the converter and the PCC. Theore: L d i dt R i = v v. () where. i the pace vector. The dq-erence frame i now utilized. It i aumed that the d-axi i aligned with the pace vector of the PCC voltage, thu: L d(i d ji q ) dt jωl(i d ji q ) R(i d ji q ) = v d jv q v d. () Fig. 9: Simplified block diagram for inner loop control. l() = k p L ( ) k p R. (5) L A it i mentioned in, the plant pole i fairly cloe to the origin. Theore, thi plant pole i canceled by the compenator zero and the loop gain become: l() = k p L The cloed-loop tranfer function can be repreented a: G Inner () = l() l() = τ (6) (7) where τ = L k p and = R τ. The inner loop gain are deigned o that the bandwidth of the inner loop with k p = 5, =, and L =.H i around 5 rad/. Compared to the inner current controller, the outer loop i deigned to be very low to lect the dynamic change. The implified block diagram of the outer control loop i illutrated in Fig.. A the dq-erence frame i aligned with the PCC voltage, the real and reactive power can be expreed a P = V i d and Q = V i q, where, V i the magnitude of PCC voltage. A cloed-loop implified tranfer function i repreented a:

10 P P k p Simplified Outer Loop Model i =i d, Fig. : Simplified block diagram for outer loop control. G outer () = ( ) kp ki V ( ) = k p ki V d ( V V k p V kp ) (8) Thi i a firt-order tranfer function ( to the form of a τ, where τ i the time contant (τ = kp )) and the ytem bandwidth can be found a /τ. In thi tudy, the outer loop gain are deigned o that the bandwidth of the outer loop with k p =. and = 5 i rad/. Thi bandwidth i time lower than the inner control bandwidth. The ac voltage PI controller i deigned baed on Q = V i q, where V i the PCC voltage. Furthermore, the PCC voltage change V i proportional to Q. Hence, the plant model i derived a: V = Q SCR = V SCR i q (9) The cloed-loop ytem tranfer function can be computed a: V V = k p / (k p SCR)/ () with the aumption that V i approximately pu. Theore, the time contant and the bandwidth are a follow: τ = k p SCR () ω bw = /τ () For k p =., = and SCR =, the bandwidth i rad/. For SCR =, the bandwidth i 5 rad/. APPENDIX C DERIVATION OF IMPEDANCE MATRIX FOR VECTOR CONTROL The dynamic of the inner current controller can be expreed a: ( V c c = k p k ) i (Īc Īc ) jωlīc τ V c () where upercript c denote the converter d q frame, V c i the PCC voltage converted to dq frame, and I i the converter dq frame current obtained from I abc. It i noted that a firt order filter i alo included in the deign and τ i the time contant of the filter (.). Rearranging () and eparating it to d q component 6: I c gc () = I c yi () g c () V c y i () () G c() Y i() where I c, I c and V c are vector of d axi and q axi variable in converter d q frame. { gc () = kpki L k p (5) y i () = (L k p)(τ) Now, impedance of the vector controlled converter can be derived if I can be expreed by voltage and current vector in the grid d q frame. The PLL i in charge of converting the component from the converter d q frame to the grid d q frame by ynchronizing the angle. In the next ubection, the effect of outer loop and the PLL will be added to derive the impedance model of the vector controlled converter. Applying the mall ignal analyi to () will reult in: I c gc () = I c yi () } g c () {{ } G c() A. Effect of Outer Loop } y i () {{ } Y i() V c (6) Outer PI outer loop are illutrated in Fig., where the d axi will control the active power and q axi i in charge of the converter input voltage. The primary dynamic equation of the PI outer loop for deriving the erence d q axi current can be expreed a: { I c d, = (P P )F P I I c q, = (V V )F V I (7) Where V i the magnitude of the PCC voltage, V i the erence magnitude of the PCC voltage, F P I i the PI controller tranfer function (k pp kip ) for power control loop and, F V I i the PI controller tranfer function (k pv kiv ) to track the PCC voltage magnitude. Applying the mall ignal analyi to (7): { I c d, = F P I P Iq, c (8) = F V I V The magnitude of the PCC voltage can be derived by: V = Vd V q, and the converter output active power can imply be derived by: P = V d I d V q I q, theore, applying mall ignal analyi: P = I d V d V d I d I q V q V q I q (9) V = V d V d V q V q Vd V q = V d V V d V q V V q ()

11 By replacing the (9) and () into (8): I c FP = I v d F P I v q I c G () FP I I d F P I I q V d V F V I Vq V F V I } {{ } G () V c () The impedance of the vector controlled converter with outer loop and without the PLL i then derived by ubtituting () into (6): B. The PLL Effect I c = G e()g () Y i () V c () I G e ()G () ZRec c Impedance analyi of the PLL ha been performed in 6 in detail. The final equation are adopted here and detailed analyi can be found in 6. The main goal i to convert the Īc and V c from the converter frame to the grid frame component (Ī and V ) by : Q I c V = I G P LL Q V V G P LL Gp P LL V c = V V G () P LL G P LL where G P LL i a imple PI controller a:g P LL = K P LL k P LL i p. For deriving the impedance model of the vector controlled converter with the outer power loop and the PLL effect, () i ubtituted into (), then the reult will be: Z Rec () = Gp P LL Z c Rec G P LL () REFERENCES O. B. Nayak, A. Gole, D. Chapman, and J. 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Athan et al., Robutne reult in linear-quadratic gauian baed multivariable control deign, Automatic Control, IEEE Tranaction on, vol. 6, no., pp. 75 9, L. Piyainghe, Z. Miao, J. Khazaei, and L. Fan, Impedance modelbaed SSR analyi for TCSC compenated type- wind energy delivery ytem, IEEE Tran. Sutain. Energy., vol. 6, no., pp , 5. Javad Khazaei (S M 6) received hi Bachelor degree in Electrical Engineering from Mazandaran Univerity (9) and Mater degree from Urmia Univerity () in Iran. He received hi Ph.D degree at Univerity of South Florida (USF) in Summer 6. He i currently an Aitnat Profeor at Penn State Harriburg. Hi reearch interet include Microgrid modeling, Renewable Energy Integration, and Power Electronic application. Zhixin Miao (S M SM 9) i with the Univerity of South Florida (USF), Tampa. Prior to joining USF in 9, he wa with the Tranmiion Aet Management Department with Midwet ISO, St. Paul, MN, from to 9. Hi reearch interet include power ytem tability, microgrid, and renewable energy. Lakhan Piyainghe i a recent Ph.D. graduate from USF SPS lab. He received hi Bachelor degree in Electrical Engineering in 6 from Univerity of Moratuwa, Sri Lanka. He tarted hi Ph.D. tudy at USF in Fall and hi reearch interet include dynamic modeling and analyi of power electronic ytem and power ytem.