A Facts Device: Distributed Power-Flow Controller (DPFC)

Transcription

1 A Facts Devce: Dstrbuted Power-Flow Controller (DPFC) Guda Pryanka 1, K. Jaghannath 2, D. Kumara Swamy 3 1, 2, 3 Dept. of EEE, SVS Insttute of Technology, Hanamkonda, T.S, Inda Emal address: ujjwalak32@gmal.com Abstract Ths paper presents a new component wthn the flexble Ac-transmsson system (FACTS) famly, called dstrbuted Powerflow controller (DPFC). The DPFC s derved from the unfed Power-flow controller (UPFC). The DPFC can be consdered as A UPFC wth an elmnated common dc lnk. The actve power exchange between the shunt and seres converters, whch s through the common dc lnk n the UPFC, s now through the transmsson Lnes at the thrd-harmonc frequency. The DPFC employs the dstrbuted FACTS (D-FACTS) concept, whch s to use multple Small-sze sngle-phase converters nstead of the one large-sze Three-phase seres Converter n the UPFC. The large number of Seres converters provdes redundancy, thereby ncreasng the system Relablty. As the D-FACTS converters are sngle-phase and Floatng wth respect to the ground, there s no hgh-voltage solaton requred between the phases. Accordngly, the cost of the DPFC system s lower than the UPFC. The DPFC has the same Control capablty as the UPFC, whch comprses the adjustment of the lne mpedance, the transmsson angle, and the bus voltage. The prncple and analyss of the DPFC are presented n ths paper and the correspondng expermental results that are carred out on a scaled prototype are also shown. Keywords Bdrectona, converter, demonstrates, hgh control capablty seres control. I. INTRODUCTION The growng demand and the agng of network smoke t desrable to control the power flow n power-transmsson systems fast and relably. The flexble ac-transmsson system (FACTS) that s defned by IEEE as a power-electronc based system and other statc equpment that provde control of one or more ac-transmsson system parameters to enhance controllablty and ncrease power-transfer capablty, and can be utlzed for power-flow control. Currently, the unfed power-flow controller (UPFC) s the most powerful FACTS devce, whch can smultaneously control all the parameters of the system: the lne mpedance, the transmsson angle, and bus voltage. Fg. 1. Smplfed representaton of a UPFC. The UPFC s the combnaton of a statc synchronous compensator (STATCOM) and a statc synchronous seres compensator (SSSC), whch s coupled va a common dc lnk, to allow bdrectonal flow of actve power between the seres output termnals of the SSSC and the shunt output termnals of the STATCOM. The converter n seres wth the lne provdes the man functon of the UPFC by njectng a four-quadrant voltage wth controllable magntude and phase. The njected voltage essentally acts as a synchronous ac-voltage source, whch s used to vary the transmsson angle and lne mpedance, thereby ndependently controllng the actve and reactve power flow through the lne. The seres voltage results n actve and reactve power njecton or absorpton between the seres converter and the transmsson lne. Ths reactve power s generated nternally by the seres converter (see e.g., SSSC), and the actve power s suppled by the shunt converter that s back-to-back connected. The shunt converter controls the voltage of the dc capactor by absorbng or generatng actve power from the bus; therefore, t acts as a synchronous source n parallel wth the system. Smlar to the STATCOM, the shunt converter can also provde reactve compensaton for the bus. The components of the UPFC handle the voltages and currents wth hgh ratng; therefore, the total cost of the system s hgh. Due to the common dc-lnk nterconnecton, a falure that happens at one converter wll nfluence the whole system. To acheve the requred relablty for power systems, bypass crcuts and redundant backups (backup transformer, etc.) are needed, whch on other hand, ncrease the cost. Accordngly, the UPFC has not been commercally used, even though; t has the most advanced control capabltes. Ths paper ntroduces a new concept, called dstrbuted power-flow controller (DPFC) that s derved from the UPFC. The same as the UPFC, the DPFC s able to control all system parameters. The DPFC elmnates the common dc lnk between the shunt and seres converters. The actve power exchange between the shunt and the seres converter s through the transmsson lne at the thrd-harmonc frequency. The seres converter of the DPFC employs the dstrbuted FACTS D-FACTS) concept. Comparng wth the UPFC, the DPFC have two major advantages: 1) Low cost because of the low-voltage solaton and the low component ratng of the seres converter and 2) Hgh relablty because of the redundancy of the seres converters. Ths paper begns wth presentng the prncple of the DPFC, followed by ts steady-state analyss. After a short ntroducton of the DPFC control, the paper ends wth the expermental results of the DPFC 95

2 II. DPFC OPERATING PRINCIPLE Wthn the DPFC, the transmsson lne presents a common connecton between the AC ports of the shunt and the seres converters. Therefore, t s possble to exchange actve power through the AC ports. The method s based on power theory of non-snusodal components. Accordng to the Fourer analyss, non-snusodal voltage and current can be expressed as the sum of snusodal functons n dfferent frequences wth dfferent ampltudes. The actve power resultng from ths non-snusodal voltage and current s defned as the mean value of the product of voltage and current. Snce the ntegrals of all the cross product of terms wth dfferent frequences are zero, the actve power can be expressed by: P V I cos 1 Where VI and I are the voltage and current at the th harmonc frequency respectvely, and ϕ s the correspondng angle between the voltage and current. Shows that the actve powers at dfferent frequences are ndependent from each other and the voltage or current at one frequency has no nfluence on the actve power at other frequences. The ndependence of the actve power at dfferent frequences gves the possblty that a converter wthout a power source can generate actve power at one frequency and absorb ths power from other frequences. By applyng ths method to the DPFC, the shunt converter can absorb actve power from the grd at the fundamental frequency and nject the power back at a harmonc frequency. Ths harmonc actve power flows through a transmsson lne equpped wth seres converters. Accordng to the amount of requred actve power at the fundamental frequency, the DPFC seres converters generate a voltage at the harmonc frequency, there by absorbng the actve power from harmonc components. Neglectng losses, the actve power generated at the fundamental frequency s equal to the power absorbed at the harmonc frequency. For a better understandng, Fgure 2 ndcates how the actve power s exchanged between the shunt and the seres converters n the DPFC system. The hghpass flter wthn the DPFC blocks the fundamental frequency components and allows the harmonc components to pass, thereby provdng a return path for the harmonc components. The shunt and seres converters, the hgh pass flter and the ground form a closed loop for the harmonc current. selected for actve power exchange n the DPFC. In a threephase System, the 3rd harmonc n each phase s dentcal, whch means they are zero-sequence components. Because the zero-sequence harmonc can be naturally blocked by Y trans- formers and these are wdely ncorporated n power systems (as a means of changng voltage), there s no extra flter requred to prevent harmonc leakage. As ntroduced above, a hgh-pass flter s requred to make a closed loop for the harmonc current and the cutoff frequency of ths flter s approxmately the fundamental frequency. Because the voltage solaton s hgh and the harmonc frequency s close to the cutoff frequency, the flter wll be costly. By usng the zero-sequence harmonc, the costly flter can be replaced by a cable that connects the neutral pont of the Y transformer on the rght sde n fgure 2 wth the ground. Because the -wndng appears open-crcut to the 3rd harmonc current, all harmonc current wll flow through the Y- wndng and concentrate to the groundng cable as shown n fgure 3. Therefore, the large hgh-pass flter s elmnated. Fg. 3. Utlze grounded Y transformer to flter zero-sequence harmonc. Another advantage of usng the 3rd harmonc to exchange actve power s that the groundng of the Y transformers can be used to route the harmonc current n a meshed network. If the network requres the harmonc current to flow through a specfc branch, the neutral pont of the Y transformer n that branch, at the sde opposte to the shunt converter, wll be grounded and vce versa. Fgure 4 shows a smple example of routng the harmonc current by usng the groundng of the Y transformer. Because the floatng neutral pont s located on the transformer of the lne wthout the seres converter, t s an open-crcut for 3rd harmonc components and therefore no 3rd harmonc current wll flow through ths lne. Fg. 2. Actve power exchange between DPFC converters. III. USING THIRD HARMONIC COMPONENTS Due to the unque features of 3rd harmonc frequency components n a three-phase system, the 3rd harmonc s Fg. 4. Route the harmonc current by usng the groundng of the Y transformer. The harmonc at the frequences lke 3rd, 6th, 9th... are all zero-sequence and all can be used to exchange actve power n the DPFC. However, the 3rd harmonc s selected, because t s the lowest frequency among all zero-sequence harmoncs. 96

3 The relatonshp between the exchanged actve power at the th harmonc frequency P and the voltages generated by the converters s expressed by the well known the power flow equaton and gven as: Vsh, Vse, P sn( sh, se, ) X Vse, Where X s the lne mpedance at th frequency, V sh, and are the voltage magntudes of the th harmonc of the shunt and seres converters, and ( sh, se, ) s the angle dfference between the two voltages. As shown, the mpedance of the lne lmts the actve power exchange capacty. To exchange the same amount of actve power, the lne wth hgh mpedance requres hgher voltages. Because the transmsson lne mpedance s mostly nductve and proportonal to frequency, hgh transmsson frequences wll cause hgh mpedance and result n hgh voltage wthn converters. Consequently, the zero-sequence harmonc wth the lowest frequency - the 3rd harmonc - has been selected IV. DPFC CONTROL To control multple converters, a DPFC conssts of three types of controllers: central control, shunt control and seres control, as shown n fgure 5. Fg. 5. DPFC control block dagram. The shunt and seres control are localzed controllers and are responsble for mantanng ther own converters parameters. The central control takes care of the DPFC functons at the power system level. The functon of each controller s lsted: Central control: The central control generates the reference sgnals for both the shunt and seres converters of the DPFC. Its control functon depends on the specfcs of the DPFC applcaton at the power system level, such as power flow control, low frequency power oscllaton dampng and balancng of asymmetrcal components. Accordng to the system requrements, the central control gves correspondng voltage reference sgnals for the seres converters and reactve current sgnal for the shunt converter. All the reference sgnals generated by the central control concern the fundamental frequency components. Seres control: Each seres converter has ts own seres control. The controller s used to mantan the capactor DC voltage of ts own converter, by usng 3 rd harmonc frequency components, n addton to generatng seres voltage at the fundamental frequency as requred by the central control. Shunt control: The objectve of the shunt control s to nject a constant 3 rd harmonc current nto the lne to supply actve power for the seres converters. At the same tme, t mantans the capactor DC voltage of the shunt converter at a constant value by absorbng actve power from the grd at the fundamental frequency and njectng the requred reactve current at the fundamental frequency nto the grd. The detaled schematcs and desgns of the DPFC control wll be ntroduced n followng chapters. V. VARIATION OF THE SHUNT CONVERTER In the DPFC, the shunt converter should be a relatvely large three-phase converter that generates the voltage at the fundamental and 3rd harmonc frequency smultaneously. A conventonal choce would be a three-leg, three-wre converter. However, the converter s an open crcut for the 3rd harmonc components and s therefore ncapable of generatng a 3rd harmonc component. Because of ths, the shunt converter n a DPFC wll requre a dfferent type of 3- phase converter. There are several 3-phase converter topologes that can generate 3rd harmonc frequency components, such as mult-leg, mult-wre converters or three sngle-phase converters [July 99]. These solutons normally ntroduce more components, thereby ncreasng total cost. A new topology for the DPFC shunt converter s proposed. The topology utlzes the exstng Y-_ transformer to nject the 3rd harmonc current nto the grd. A sngle- phase converter s connected between the transformer s neutral pont and the ground, and njects a 3rd harmonc current nto the neutral pont of the transformer. Ths current evenly spreads nto the 3-phase lne through the transformer. The converter can be powered by an addtonal back-to-back converter connected to the low-voltage sde of the transformer. The crcut scheme of ths topology s shown n fgure 6. For a symmetrcal system, the voltage potental at the neutral pont and fundamental frequency s zero. Accordngly, the sngle-phase converter only handles the 3rd harmonc voltages, whch are much lower than the voltage at the fundamental frequency. As the sngle-phase converter s only used to provde actve power for the seres converter, the voltage and power ratng are small. In addton, the snglephase converter uses the already present Y transformer as a grd connecton. The sngle-phase converter s powered by another converter through a common DC lnk. In the case of the system wth a STATCOM, the sngle-phase converter can be drectly connected back-to-back to the DC sde of the STATCOM, as shown n fgure 6. Fg. 6. DPFC shunt converter confguraton. 97

4 VI. DPFC ADVANTAGES The DPFC can be consdered as a UPFC that employs the DFACTS concept and the concept of exchangng power through harmonc. Therefore, the DPFC nherts all the advantages of the UPFC and the D-FACTS, whch are as follows. 1) Hgh control capablty. The DPFC can smultaneously control all the parameters of the power system: the lne mpedance, the transmsson angle, and the bus voltage. The elmnaton of the common dc lnk enables separated nstallaton of the DPFC converters. The shunt and seres converters can be placed at the most effectvely locaton. Due to the hgh control capablty, the DPFC can also be used to mprove the power qualty and system stablty, such as lowfrequency power oscllaton dampng, voltage sag restoraton, or balancng asymmetry. 2) Hgh relablty. The redundancy of the seres converter gves an mproved relablty. In addton, the shunt and seres converters are ndependent, and the falure at one place wll not nfluence the other converters. When a falure occurs n the seres converter, the converter wll be short-crcuted by bypass protecton, thereby havng lttle nfluence to the network. In the case of the shunt converter falure, the shunt converter wll trp and the seres converter wll stop provdng actve compensaton and wll act as the D-FACTS controller. 3) Low cost. There s no phase-to-phase voltage solaton requred by the seres converter. Also, the power ratng of each converter s small and can be easly produced n seres producton lnes. However, as the DPFC njects extra current at the thrd harmonc frequency nto the transmsson lne, addtonal losses n the transmsson lne and transformer should be aware of VII. ANALYSIS OF THE DPFC In ths secton, the steady-state behavor of the DPFC s analyzed, and the control capablty of the DPFC s expressed n the parameters of the network and the DPFC. To smplfy the DPFC, the converters are replaced by controllable voltage sources n seres wth mpedance. Snce each converter generates the voltage at two dfferent frequences, t s represented by two seres-connected controllable voltage sources, one at the fundamental frequency and the other at the thrd-harmonc frequency. Assumng that the converters and the transmsson lne are lossless, the total actve power generated by the two frequency voltage sources wll be zero. The multple seres converters are smplfed as one large converter wth the voltage, whch s equal to the sum of the voltages for all seres converter, as shown n fgure 7. The DPFC s placed n a two-bus system wth the sendngend and the recevng-end voltages Vs and Vr, respectvely. The transmsson lne s represented by an nductance L wth the lne current I. The voltage njected by all the DPFC seres converters s Vse,1 and Vse,3 at the fundamental and the thrdharmonc frequency, respectvely. The shunt converter s connected to the sendng bus through the nductor Lsh and generates the voltage Vsh,1 and Vsh,3 ; the current njected by the shunt converter s Ish. The actve and reactve power flow at the recevng end s Pr and Qr, respectvely. Ths representaton conssts of both the fundamental and thrdharmonc frequency components. Based on the superposton theorem, the crcut n fgure 7 can be further smplfed by beng splt nto two crcuts at dfferent frequences. The two crcuts are solated from each other, and the lnk between these crcuts s the actve power balance of each converter, as shown n fgure 8. Fg. 8. DPFC equvalent crcut. (a) Fundamental frequency. (b) Thrd harmonc frequency. The power-flow control capablty of the DPFC can be llustrated by the actve power Pr and reactve power Qr receved at the recevng end. Because the DPFC crcut at the fundamental frequency behaves the same as the UPFC, the actve and reactve power flow can be expressed as follows (1) 2 2 V V se,1 Pr Pr 0 Qr Qr 0 X1 Where Pr0, Qr0, and θ are the actve, reactve power flow, and the transmsson angle of the uncompensated system, Xse,1 = ωlse s the lne mpedance at fundamental frequency, and V s the voltage magntude at both ends. In the PQ-plane, the locus of the power flow wthout the DPFC compensaton f (Pr0, Qr0) s a crcle wth the radus of V 2/ X1 around the center defned by coordnates P = 0 and Q = V 2/ X1. Each pont of ths crcle gves the Pr0 and Qr0 values of the uncompensated system at the correspondng transmsson angle θ. The boundary of the attanable control range for Pr and Qr s obtaned from a complete rotaton of the voltage Vse,1 wth ts maxmum magntude. Fgure 9 shows the control range of the DPFC wth the transmsson angle θ. 2 Fg. 7. DPFC smplfed representaton. Fg. 9. DPFC actve and reactve power control range wth the transmsson angle θ. 98

5 To ensure the seres converters to nject a 360 rotatable voltage, an actve and reactve power at the fundamental frequency s requred. The reactve power s provded by the seres converter locally and the actve power s suppled by the shunt converter. Ths actve power requrement s gven by * X1 Pse,1 Re( Vse,1I1 ) s 2 r Sr0 sn( r0 r ) V r Where ϕ r 0 s the power angle at the recevng end of the uncompensated system, whch equals tan 1 (Pr0/Qr0) and ϕr s the power angle at recevng end wth the DPFC compensaton. The lne mpedance X1 and the voltage magntude Vr are constant; therefore, the requred actve power s proportonal to sr Sr0 sn( r0 r ), whch s two tmes the area of the trangle that s formed by the two vectors S r0 and Sr. Fgure 10 llustrates the relatonshp between Pse,1 and the power flow at the recevng end at a certan power angle θ. Fg. 10. Relatonshp between Pse,1 and the power flow at the recevng end. Consequently, the requred actve power by the seres converter can be wrtten as follows: P CA se,1 ( O, r0, r) / r Where the coeffcent C X V and A ( O, r0, r) s the area of the trangle (0, Sr0, Sr). The angle dfference ϕr0 ϕr can be postve or negatve, and the sgn gves the drecton of the actve power through the DPFC seres converters. The postve sgn means that the DPFC seres converters generate actve power at the fundamental frequency and vse versa. The actve power requrement vares wth the controlled power flow, and the actve power requrement has ts maxmum when the vector Sr Sr0 s perpendcular to the vector Sr0. The relatonshp between the power flow control range and the maxmum actve power requrement can be represented by X1 Sr0 Pse,1,max S 2 r, c V Where Sr,c s the control range of the DPFC. Each converter n the DPFC generates two frequency voltages At the same tme. Accordngly, the voltage ratng of the each Converter should be the sum of the maxmum voltage of the two frequences component V V V se,max se,1,max se,3,max Durng the operaton, the actve power requrement of the seres converter vares wth the voltage njected at the fundamental frequency. When the requrement s low, the seres voltage at the thrd-harmonc frequency wll be smaller than Vse,3,max V se,3,max r. Ths potental voltage that s between Vse,3 and can be used to control the power flow at the fundamental frequency, thereby ncreasng the power-flow control regon of the DPFC. When Src, s perpendcular to the uncompensated power S r0, the seres converters requre maxmum actve power, and the radus of the DPFC control regon s gven by Vr Vse,1,max Src, X1 If Sr, c s n the same lne as Sr0, the seres converters only provde the reactve compensaton and the boundary of the DPFC control regon wll extend to Vr ( Vse,1,max Vse,3,max ) Src, X1 It shows that the control regon of the DPFC can be extended to a shape that s smlar as an ellpse,. Fg. 12. DPFC power-flow control range. Fg. 11. Maxmum actve power requrement of the seres converters. To obtan the same control capablty as the UPFC, the ratng of the DPFC converter at the fundamental frequency should be the same as the one for the UPFC. Because the voltages and currents at the thrd-harmonc frequency have to be added, the ratng of the DPFC converter s slghtly larger than the UPFC. The ncreased ratng s related wth the actve power exchanged at the thrd-harmonc frequency. For a transmsson lne, the lne mpedance X1 s normally around 0.05 p.u. (per unt). Assumng the bus voltages V and 99

6 uncompensated power flow Sr 0 s 1 p.u., and then, from (7), we can see that to control 1-p.u. power flow, the exchanged actve power s around 0.05 p.u. Even wth ths extra voltage and current at the thrdharmonc frequency, the cost of the DPFC s stll much lower than the UPFC, for the followng reasons: 1) the UPFC converter handles the lne-to-lne voltage solaton that s much larger than voltage njected by the seres converter; 2) no land requrement for the seres converter; and 3) the actve and passve components for the DPFC converter are lowvoltage components (less than 1kV and 60 A), whch s much cheaper than the hgh-voltage components n the UPFC. VIII. LABORATORY RESULTS An expermental setup has been bult to verfy the prncple and control of the DPFC. One shunt converter and sx sngle phase seres converters are bult and tested n a scaled network, as two solated buses wth phase dfference are connected by the lne. Wthn the expermental setup, the shunt converter s a sngle-phase nverter that s connected between the neutral pont of the Y Δ transformer and the ground. The nverter s powered by a constant-voltage source. The specfcatons of the DPFC expermental setup are lsted n the wthn the setup, multple seres converters are controlled by a central controller. The central controller gves the reference voltage sgnals for all seres converters. The voltages and Currents wthn the setup are measured by an osclloscope and processed n computer by usng the MATLAB. The photograph of the DPFC expermental setup s llustrated n Fg. 18.To verfy the DPFC prncple, two stuatons are demonstrated: the DPFC behavor n steady state and the step response. In steady state, the seres converter s controlled to nsert a voltage vector wth both d- and q- component, whch s Vse d, ref = 0.3 V and Vse, q, ref = 0.1 V. one operaton pont of the DPFC setup. For clarty, only the waveforms n one phase are shown. The voltage njected by DPFC operaton n steady state lne. DPFC operaton n steady state: seres converter voltage. DPFC operaton n steady state: bus voltage and current at the Δ sde of the transformer. seres converter, the current through the lne, and the voltage and current at the Δ sde of the transformer are llustrated. The constant thrd-harmonc current njected by the shunt converter evenly dsperses to the three phases and s supermposed on the fundamental current, as shown n The voltage njected by the seres converter also contans two frequency components n. The ampltude of the pulse wdth modulated (PWM) waveform represents the dc-capactor voltage, whch s well mantaned by the thrd-harmonc component n steady state. As shown, the dc voltage has a small oscllaton; however, t does not nfluence the DPFC control. Demonstrates the thrd-harmonc flterng by the Y Δ transformers. There s no thrd-harmonc current or voltage leakng to the Δ sde of the transformer. The DPFC controls the power flow through transmsson lnes by varyng the voltage njected by the seres converter at the fundamental frequency. Illustrate the step response of the expermental setup. A step change of the fundamental reference voltage of the seres converter s made, whch conssts of both actve and reactve varatons, as. As shown, the dc voltage of the seres converter s stablzed before and after the step change. To verfy f the seres converter can nject or absorb actve and reactve power from the grd at the fundamental frequency, the power s calculated From the measured voltage and current n. The measured data n one phase are processed n the computer by Fg. 13.Model of the seres converter control. Fg. 14. Reference voltage for the seres converters (Vdref, Vqref). Fg. 15.Step response of the DPFC: seres converter voltage. Fg. 16.Step response of the DPFC: lne current. 100

7 Fg. 17.Step response of the DPFC: bus voltage and current at the Δ sde of the transformer. Step response of the DPFC: actve and reactve power njected by the seres converter at the fundamental frequency usng MATLAB. To analyze the voltage and current at the fundamental frequency, the measured data that contans harmonc dstorton are fltered by a low-pass dgtal flter wth the 50-Hzcutoff frequency. Because of ths flter, the calculated voltage and current at the fundamental frequency have a 1.5 cycle delay to the actual values, thereby causng a delay of the measured actve and reactve power llustrated the actve and reactve. A comparson s made between the measured power and the calculated power. We can see that the seres converters are able to absorb and nject both actve and reactve power to the grd at the fundamental frequency. Fg. 22. Step response of the DPFC: bus voltage and current at the Δ sde of the transformer. Fg. 18. Step response of the DPFC: actve and reactve power njected by the seres converter at the fundamental frequency. Fg. 19. Reference voltage for the seres converters. Fg. 20. Step response of the DPFC: seres converter voltage. Fg. 21. Step response of the DPFC: lne current. IX. CONCLUSION Ths paper has presented a new concept called DPFC. The DPFC emerges from the UPFC and nherts the control capablty of the UPFC, whch s the smultaneous adjustment of the lne mpedance, the transmsson angle, and the busvoltage magntude. The common dc lnk between the shunt and seres converters, whch s used for exchangng actve power n the UPFC, s elmnated. Ths power s now transmtted through the transmsson lne at the thrd-harmonc frequency. The seres converter of the DPFC employs the D- FACTS concept, whch uses multple small sngle-phase converters nstead of one large-sze converter. The relablty of the DPFC s greatly ncreased because of the redundancy of the seres converters. The total cost of the DPFC s also much lower than the UPFC, because no hgh-voltage solaton s requred at the seres-converter part and the ratng of the components of s low. The DPFC concept has been verfed by an expermental setup. It s proved that the shunt and seres converters n the DPFC can exchange actve power at the thrd-harmonc frequency, and the seres converters are able to nject controllable actve and reactve power at the fundamental frequency. REFERENCES [1] Y.-H. Song and A. Johns, Flexble ac transmsson systems (FACTS) (IEE Power and Energy Seres), London, U.K.: Insttuton of Electrcal Engneers, vol. 30, [2] N. G. Hngoran and L. Gyugy, Understandng FACTS: Concepts and Technology of Flexble AC Transmsson Systems, New York: IEEE Press, [3] L. Gyugy, C. D. Schauder, S. L. Wllams, T. R. Retman, D. R. Torgerson, and A. Edrs, The unfed power flow controller: A new approach to power transmsson control, IEEE Trans. Power Del., vol. 10, no. 2, pp , [4] A. A. Edrs, Proposed terms and defntons for flexble ac transmsson system (facts), IEEE Trans. Power Del., vol. 12, no. 4, pp ,

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