Modeling of FACTS Devices Based on SPWM VSCs - CIEP

Fig. 3. Single-phase equivalent circuit.discontinuities produced during the commutation process areavoided if a smoother pulse train obtained with the Fourierseries replaces in (3).The switching instants are obtained with the solution <strong>of</strong>the nonlinear algebraic equations set describing the sawtoothwaveform and . In principle, the computation <strong>of</strong> the switchinginstants requires the application <strong>of</strong> an iterative process for thesolution <strong>of</strong> the nonlinear algebraic equations. However, canbe approximated through a first order Taylor series, to allowthe original nonlinear algebraic system to be transformed into alinear algebraic system. Let’s be the approximated switchinginstants. The equations defining the straight lines with positiveslope for the triangular signal <strong>of</strong> Fig. 4(a) are given byThe equations defining the straight lines with negative slopefor the triangular signal are given by(4)(5)Fig. 4. Transforming a desired continuous signal v(a) SPWM process. (b) Switching function (S).into a SPWM signal.where is the fundamental period <strong>of</strong> and is the positiveslope, calculated as(6)achieve this result, a sawtooth wave (tri) is drawn. Now, conductiontakes place whenever lies above the triangular waveform(tri), and conduction ceases whenever it lies below. The resultingpulse train contains the signal . The SPWM processis illustrated by Fig. 4(a). The corresponding switching functionis shown in Fig. 4(b). This switching function correspondsonly to one phase; however, it can be extended to the other twophases.In practice, is much shorter than that shown in Fig. 4(a).Consequently, the change in during one carrier period isconsiderably less than the indicated in Fig. 4(a). In other words,a higher carrier frequency directly improves the reproduction<strong>of</strong> the original signal ; the carrier frequency is defined as.The frequency-modulation radio is defined as ,where is the frequency <strong>of</strong> . Higher implies a highswitching frequency. As we have mentioned before, there areseveral simulation problems related to the switching process.In addition, the numerical error increases with the increase inthe switching frequency.In the next section, the two proposed models for the fast andefficient simulation <strong>of</strong> VSC power converters are detailed.III. SINUSOIDAL PULSE-WIDTH MODULATIONCONVERTER REPRESENTATIONA. Fourier ModelIt can be observed that the switching function shown inFig. 4(b) contains discontinuities at each switching instant. TheThe control signalis given byits first-order Taylor approximation aroundFor instance, the switching instant for iscomputed throughwith, thenis(7)(8)(9)(10)For , with , we obtainIn general, for(11)(12)

Fig. 5. Time-domain comparison between the discontinuous switching functionS and the Fourier approach for different harmonic orders.where(13)Equation (12) is the general expression for the approximatedcomputation <strong>of</strong> the switching instants. Once has been calculated,the pulse train is automatically obtained. The proposedmodel can be resumed by using an approximation <strong>of</strong> in terms<strong>of</strong> the Fourier series, e.g., , beingwhereis the highest harmonic order, and(14)(15)(16)Fig. 6. Spectrum comparison between the discontinuous switching function Sand the Fourier approach for different harmonic orders.harmonic orders is presented. In Fig. 5(a) a time-domain comparisonbetween and for a maximum harmonic order <strong>of</strong> 19(1140 Hz) is shown, in Fig. 5(b) for a maximum harmonic order<strong>of</strong> 113 (6780 Hz), and in Fig. 5(c) for a maximum harmonicorder <strong>of</strong> 300 (18 kHz), respectively. Fig. 6 shows the harmonicdomain comparison; in Fig. 6(a) a harmonic domain comparisonbetween and for a maximum harmonic order <strong>of</strong> 19(1140 Hz) is shown, in Fig. 6(b) for a maximum harmonic order<strong>of</strong> 113 (6780 Hz), and in Fig. 6(c) for a maximum harmonicorder <strong>of</strong> 300 (18 kHz), respectively. The obtained responses arein close agreement.It can be noticed from Fig. 6 that for a Fourier series approximationwith a maximum harmonic order <strong>of</strong> , the switchingfunction contains up to the harmonic <strong>of</strong> , where all theharmonics are included. If higher precision is required, an iterativeprocess can be implemented with (12) to increase theprecision, so that(17)For the balanced case, is calculated for one phase, andfor the other phases it is obtained with the appropriate phaseangle if is multiple <strong>of</strong> three. In addition, if is odd theswitching function have half-wave symmetry, consequently, alleven-numbered harmonics vanish. In addition(18)Using half-wave symmetry property, we only need to computethe half <strong>of</strong> the switching times with (18).It is clear that with this Fourier series approximation, the harmonicnumber for the analysis can be selected. The harmonicnumber introduced in the analysis increases the computationaleffort for each integration step; however, this aspect is compensatedwith the use <strong>of</strong> larger integration steps, since the discontinuitiesno longer exist.In Fig. 5, a time-domain comparison between the discontinuousswitching function and the Fourier approach for different(19)for , (19) corresponds to (12). It can be noticed that (19) isthe iterative Newton method. Expression (19) can be also representedas(20)The harmonic spectrum <strong>of</strong> (14) is finite; for this reason,there are not aliasing problems if the selected integration stepis chosen according to the Nyquist criterion. For example, forequal to 51 with a fundamental frequency <strong>of</strong> 60 Hz, theintegration step must be at leasts. A smaller integrationstep would be necessary if numerical stability problemsare detected, since this integration step is in general large.B. Smooth Switching FunctionA numerical error is introduced during the numerical integration<strong>of</strong> electric systems including power-electronic switchescoming from the discontinuities <strong>of</strong> the ideal switch model.

Fig. 21.Transient solution comparison for the UPFC.Fig. 19. Steady-state solution comparison for the SSSC.Fig. 22. Steady-state solution comparison for the UPFC.Fig. 20. Harmonic content comparisons. (a) Harmonic content in the dcvoltage. (b) Magnitude <strong>of</strong> selected harmonics in phase A <strong>of</strong> the terminalvoltage. (c) Magnitude <strong>of</strong> selected harmonics in the phase A <strong>of</strong> the seriescurrent.Simulink PSCAD/EMTDC, and the two proposed models. Anintegration step <strong>of</strong> 0.1 s has been used in Simulink, 100 sfor the Fourier approach method, 30 s for the hyperbolictangent model, and 0.1 s in PSCAD/EMTDC. Observe thatthe proposed models closely agree with the solution obtainedwith Simulink and PSCAD/EMTDC.A comparison between the periodic steady-state solutionsis shown in Fig. 19; the capacitor voltage is shown inFig. 19(a), the terminal voltage at node 1 in Fig. 19(b),and the phase A <strong>of</strong> the series current in Fig. 19(c), respectively.Observe that all solutions are again in very closeagreement.Fig. 20(a) shows the harmonic content for , in Fig. 20(b)for , and in Fig. 20(c) for . It can be seen that the proposedmodels are in very good agreement with the solution obtainedwith Simulink and PSCAD/EMTDC.3) Unified Power Flow Controller (UPFC): The UPFC includesthe shunt control described in [15], and the series controlproposed in [16]. The initial condition is zero, except for thedc capacitor voltage, which is 2 p.u. The series controller regulatesthe real ( p.u.) and reactive ( p.u.)power flows by adjusting the injected series voltage. The shuntconverter regulates the dc-side capacitor voltage ( p.u.)and the sending end voltage ( p.u.). The modulationindex is . The maximum harmonic order in the Fourierapproach is 69 (4140 Hz).Fig. 21 shows the transient solution comparison for the capacitorvoltage <strong>of</strong> the UPFC operating in open-loop control.For this operating mode, the initial condition has been assumedzero. This solution has been computed using Simulink, PSCAD/EMTDC and the two proposed models. An integration step <strong>of</strong>0.1 s has been used in Simulink and PSCAD/EMTDC, 80 sfor the Fourier approach method, and 30 s for the hyperbolictangent model.A comparison between the periodic steady-state solution obtainedwith Simulink, the Fourier model, the hyperbolic tangentmodel, and PSCAD/EMTDC is shown in Fig. 22. Fig. 22(a)shows the capacitor voltage , Fig. 22(b) shows the voltage<strong>of</strong> phase A in bus 1, and Fig. 22(c) shows the series current<strong>of</strong> phase A . The slight difference between the solutionobtained with PSCAD/EMTDC and the other solutions is becausethe UPFC implemented in PSCAD/EMTDC is operatingin open-loop control, meanwhile, the other solutions are operatingin closed-loop control; however, all the solutions are ingood agreement.

ACKNOWLEDGMENTThe authors thank the Universidad Michoacana de SanNicolás de Hidalgo (UMSNH) through the División de Estudiosde Posgrado, Facultad de Ingeniería Eléctrica, for thefacilities that were granted to carry out this investigation. Theauthors would also like to thank Pr<strong>of</strong>. Dr. E. Barcenas fromthe Instituto Tecnológico Superior de Irapuato (ITESI) for hisvaluable suggestions and comments.REFERENCESFig. 23. Harmonic content comparisons. (a) Harmonic content in the dcvoltage. (b) Magnitude <strong>of</strong> selected harmonics in phase A <strong>of</strong> the terminalvoltage. (c) Magnitude <strong>of</strong> selected harmonics in the phase A <strong>of</strong> the seriescurrent.A validation <strong>of</strong> the proposed models is further illustrated inFig. 23, which shows the harmonic spectrum for the capacitorvoltage in Fig. 23(a), the voltage <strong>of</strong> phase A at node 1 inFig. 23(b), and the phase A <strong>of</strong> the series current in Fig. 23(c),respectively; all the solutions are in excellent agreement.V. CONCLUSIONTwo efficient VSC models based on Fourier series approachand hyperbolic tangent have been proposed for the six-pulseconverter. The proposed models have been used for the computation<strong>of</strong> the solution <strong>of</strong> <strong>FACTS</strong> devices connected to a powernetwork.It has been shown that the proposed models can be used tocompute both the transient and the periodic steady-state solution<strong>of</strong> a power network containing VSC-based <strong>FACTS</strong>.The response given by the proposed methods has been successfullyvalidated against the solution obtained with the widelyaccepted digital simulators Simulink and PSCAD/EMTDC, respectively,in all the cases the obtained results were in excellentagreement. In addition, it has been shown through simulationsthat the proposed models allow significant larger integrationsteps, as compared with the ideal switch model approach,e.g., for the conducted studies, more than 800 times the integrationstep needed by Simulink and PSCAD/EMTDC when theFourier approach was used, and at least 300 times when the hyperbolictangent method was applied.The proposed VSCs models have been only implemented in anetwork including <strong>FACTS</strong> devices; however, these models canbe used in any power-electronic device based on SPWM sixpulse converters, or even for multilevel converters based on arrangement<strong>of</strong> six-pulse converters.[1] N. G. Hingorani and L. Gyugyi, Understanding <strong>FACTS</strong>. Piscataway,NJ: IEEE Press, 2000.[2] E. Uzunovic, C. A. Cañizares, and J. Reeve, “Fundamental frequencymodel <strong>of</strong> unified power flow controller,” in Proc. North AmericanPower Symp., Cleveland, OH, Oct. 1998, pp. 294–299.[3] A. 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IEEE, vol. 86, no. 2, pp. 447–457, Feb. 1998.[14] Power System Blockset User’s Guide 2001, TEQSIM Int. Inc..[15] B. Mahyavanshi and G. Radman, “A study <strong>of</strong> interaction betweendynamic load and STATCOM,” presented at the Southeastern Symp.System Theory, Cookeville, TN, Mar. 5–7, 2006.[16] H. Fujita, Y. Watanabe, and H. Akagi, “Transient analysis <strong>of</strong> a unifiedpower flow controller and its applications to design <strong>of</strong> the DC link capacitor,”IEEE Trans. Power Electron., vol. 16, no. 5, pp. 735–740, Sep.2001.Juan Segundo-Ramírez received the B.S. degree in electromechanical engineeringfrom the Instituto Tecnológico de Lázaro Cárdenas, Lázaro Cárdenas,Mexico, in 2001, the M.Sc. degree from the CINVESTAV Guadalajara,Guadalajara, Mexico, in 2004, and is currently pursuing the Ph.D. degreeat the División de Estudios de Posgrado, Facultad de Ingeniería Eléctrica,Universidad Michoacana de San Nicolás de Hidalgo (UMSNH), Morelia,México.His area <strong>of</strong> research is the dynamic and steady-state analysis <strong>of</strong> powersystems.Aurelio Medina (SM’02) received the Ph.D. degree from the University <strong>of</strong>Canterbury, Christchurch, New Zealand, in 1992.He was a Postdoctoral Fellow at the University <strong>of</strong> Canterbury, New Zealand,for one year and at the University <strong>of</strong> Toronto, Toronto, ON, Canada, for twoyears. He is currently a staff member with the Facultad de Ingeniería Eléctrica,Universidad Michoacana de San Nicolás de Hidalgo (UMSNH), Morelia,México, where he is the Head <strong>of</strong> the Division for postgraduate studies. His researchinterests are in the dynamic and steady-state analysis <strong>of</strong> power systems.