IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 21, NO. 4, OCTOBER 2006 2071 Steady-State Solution of a Voltage-Source Converter with Full Closed-Loop Control K. L. Lian and P. W. Lehn, Member, IEEE Abstract—An iterative method based on a hybrid time/frequency-domain approach is proposed in this paper to solve for the steady state of a pulse width modulated voltage-source converter (VSC) with a -frame closed-loop controller. The method solves the linear controller equations in the frequency domain, while solving the VSC equations using time domain techniques. The model is validated against time domain simulation results. It is shown that the hybrid time/frequency-domain approach is both highly efficient and accurate, and it provides a viable alterative to brute-force time domain simulation for large signal harmonic analysis of the VSC. Index Terms—Broyden’s method, closed-loop control, harmonics, Jacobian, steady-state analysis, voltage-source converter (VSC). I. INTRODUCTION EREGULATION of electric utilities, together with increasing consumer loads have placed a level of unprecedented stress on power systems. Power electronic equipment is increasingly being installed to meet the evolving needs of modern power systems. At the generation level, back-to-back converter circuits are needed as interfaces for distributed energy sources; at the transmission level, converter based flexible ac transmission systems (FACTS) controllers are employed to improve system stability and at the distribution level converter based Custom Power controllers are used to improve power quality. While many benefits may be realized from large scale introduction of power electronics into the power grid, these come at the expense of generating harmonic distortion. It is therefore essential to predict the harmonics generated by voltage-source converters (VSCs) and to understand how these harmonics will interact with the power system. Brute-force time-domain simulation can provide accurate harmonic analysis results if the simulation step is chosen appropriately. However, the disadvantage of brute-force time domain simulation is that it needs to go through an initialization transient before reaching steady state where Fourier analysis can be performed. Simulation times are especially long when analyzing pulse-width-modulated (PWM) VSCs because the simulation time step must be sufficiently small to capture high frequency switching dynamics while the time interval D Manuscript received August 18, 2005; revised November 18, 2005. Paper no. TPWRD-00482–2005. The authors are with the Energy System Group of University of Toronto, Toronto, ON M5S 3G4, Canada (e-mail: liank@ecf.utoronto.ca; lehn@ecf.utoronto.ca). Digital Object Identifier 10.1109/TPWRD.2006.877081 of simulation must be sufficiently large to provide the correct fundamental frequency solution. Alternative methods have been proposed to rapidly calculate the steady state operating point of various power electronic circuits. These methods can be classified into three categories: fast time domain methods [1]–[10], frequency-domain methods [11]–[16], and hybrid time/frequency-domain methods [17]. To date only frequency-domain methods have been employed for steady-state analysis of the closed-loop three-phase converter circuits. Wood employed linearization about a known operating point to obtain the harmonic domain admittance matrix of the controlled thyristor bridge [15]. He later employed a similar technique to analyze a STATCOM with simple firing angle control [16]. The main limitation of this approach is that it assumes the system harmonics do not influence the operating point of the converter. For the thyristor bridge, this limitation is overcome in [14], where iteration is employed to find the steady-state operating point. However, the iterative approach presented in [14] cannot be directly applied to the VSC circuits because the controllers of VSCs are typically operated in the synchronous reference frame [18]. To date, no steady state model has yet been developed for a conventional PWM VSC with a -frame closed loop controller. This paper presents an iterative large signal method for steady state analysis of the PWM VSC with closed loop -frame curcontrol. The converter controls consist of a rent controller together with a dc bus voltage controller. The proposed approach employs hybrid time/frequency-domain modelling [17] in which the VSC is modeled using a fast time domain approach and controllers are modeled in the frequency domain. In contrast to frequency-domain techniques, the proposed approach is not limited by the Gibbs phenomenon [19], thus far fewer harmonics need be calculated to accurately determine the steady state of the VSC. The simulation results of the hybrid method are compared with those of PSCAD/EMTDC to demonstrate the validity of the proposed method. II. VSC IN A CLOSED LOOP Fig. 1 shows a general grid connected VSC circuit, where the dc current injection, may represent either a source or a load current. Fig. 1 also shows a typical VSC controller [20]. referCurrent regulators are controlled in the synchronous ence frame to exploit the tracking properties of proportional-integral (PI) regulators. The -axis current determines the active power exchanged with the grid. A dc voltage regulator assigns the -axis reference current. The -axis current determines the reactive power exchanged with the grid. Its reference value may be set to meet grid requirements. Without loss of generality, this 0885-8977/$20.00 © 2006 IEEE 2072 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 21, NO. 4, OCTOBER 2006 Fig. 1. Schematic diagram of a VSC with its internal controllers. paper considers the VSC operated as a STATCOM, where the . dc current injection As shown in Fig. 1, the paper will denote time domain variables using lower case. Upper case variable names will indicate quantities that are in the frequency domain. III. HYBRID METHODS Semlyen and Medina [17] developed hybrid analysis methods based on the realization that nonlinear and time varying components are best modeled using an iterative fast time domain approach while linear components are best described in the frequency domain. Thus the VSC, as a time-varying component, is best modeled in the time domain. In contrast, the PI voltage and current controllers are linear elements most easily specified by their transfer functions. This reasoning naturally leads to the solution flow diagram shown in Fig. 2. However, implementation of such a solution approach is not feasible because the PI controllers have a pole at zero frequency, resulting in singularity and “blow-up” of the solution. Thus the key to solving the steady state solution of the VSC using hybrid techniques lies in resolving the singularity introduced by the PI feedback controllers. The calculation flow diagram of Fig. 3 shows how the system is subdivided into three solution blocks. The first block consists of the pattern generator and the VSC, and is called the “VSC block.” Though not shown in Fig. 3, the VSC block may also contain feedforward or auxiliary controllers, provided that they do not include integral terms. The second block is the ac Fig. 2. Simple solution flow diagram based on the hybrid method. This algorithm will not converge due to singularity in the controller transfer functions at j! = 0. “current control block” while the third is the dc bus “voltage control block.” The calculation paths are chosen to pass backwards through these control blocks. This way integrators are replaced by differentiators, and the singularity at zero frequency is eliminated. Solution of the system proceeds as follows (see Fig. 3). The and (Fig. 1) and active modulating signal harmonics current reference are initialized, and supplied to the blocks has that require them. The initial active current reference two calculation paths. It is supplied to a voltage controller block . It is also supplied to the current confor prediction of troller blocks, where it contributes to the prediction of and . Similarly, , and are supplied to both the VSC block, to calculate the ac current and dc voltage harmonics , , and ) and the current control block, to ( contribute to the prediction of , and . LIAN AND LEHN: STEADY-STATE SOLUTION OF A VSC WITH CLOSED-LOOP CONTROL 2073 Since an infinite bus is assumed, one can assume the forcing functions are perfect sinusoids, which can be modeled as harmonic oscillators [21]. Therefore, (1) can be written as (2) (2) where Fig. 3. Proposed calculation flow diagram for simulating a closed-loop VSC. Initial values (white arrows) supplied to blocks produce predictions (black thick arrows). Calculation paths and mismatch equations are denoted by black thin arrows and question marks, respectively. If all of the current and voltage harmonics predicted by the VSC block are equal to those predicted by the control blocks, then the initial modulating signal harmonics and active current reference harmonics are correct (i.e., the loop is effectively closed); otherwise, the modulating signal harmonics and active current reference harmonics must be iterated until convergence is achieved. and , , model bus voltages , , , respectively. Equation (2) is homogenous and involves only the computation of the exponential matrix over each switching interval. To further reduce the computation time, the VSC can be directly modeled in the alpha–beta reference frame so that only one oscillator is needed [21]. Consequently the size of the state transition matrix of the homogeneous equation is reduced by half IV. MODELS This section briefly describes the fast time domain model of the VSC, and the frequency-domain model of the controller. A. VSC Model (3) where The essence of the fast time domain method is to determine and thereby yield the perithe correct initial system states odic solution without going through system transients. In [3] an “augmented matrix method” is introduced to determine the corby augmenting forcing functions to the state marect state trices of linear switched circuits. The differential equation of a VSC is given in (1) and (1) The correct initial conditions can be solved by partitioning the system matrix of (3) and performing a partial inversion [4]. In addition, further augmentation of the system matrices may be carried out to permit the harmonics of the state variables to be analytically solved [3], [4]. Consequently, the augmented matrix method can determine system harmonics without applying the discrete Fourier transform (DFT) to simulation data sets—thus avoiding limitations imposed by the Nyquist theorem. where B. Current Controller Model , switching functions. in Fig. 1 and , , are the Fig. 4 shows the current controller of a VSC [18] in the syngives the steady chronous reference frame, where state response of the PI current controller at harmonic frequen, and , the ac current harmonics at the cies. For given 2074 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 21, NO. 4, OCTOBER 2006 Fig. 5. Block diagram of the dc voltage controller in the frequency domain. Complete expressions for , , , , , , are listed in the Appendix. C. Voltage-Controller Model Fig. 4. Block diagram of the current controllers in the frequency domain. current controllers, and found as in (4) and can be back calculated Fig. 5 shows the outer dc bus voltage controller that assigns , where gives the active current reference the steady-state response of the PI voltage controller at harmonic frequencies. , the dc voltage harmonics are back calculated to Given be (8) (8) For each harmonic of interest, an equation in the form of (8) is required, yielding the matrix relation (9) (9) Complete expressions for in the Appendix. , , and are listed V. NUMERICAL ANALYSIS (4) Note that (4) must be evaluated at each harmonic of interest. For multiple harmonics of interest one equation of the form of (4) is required for each harmonic. In general, we may write this in compact form as (5) (5) To form a mismatch equation between solutions of the current controller block and the VSC block, (5) must be transformed frame. This can be done by multiplying both sides into the of (5) by the connection matrix, C [22], [23] (6) are to be iniHowever, since the modulating signals tialized in the -frame as indicated in Fig. 3, a more convenient form of the equation is given by (7) (7) The steady-state solution of the closed-loop VSC is found by using a hybrid time/frequency-domain approach based on a Newton-type algorithm. A. Flowchart Fig. 6 shows the overall flow diagram of the proposed large signal analysis of a closed-loop VSC. Note that the algorithm initializes the modulating voltage signals in terms of positiveand negative-sequence quantities ( , and ), rather than , and as mentioned in the previous sections to simplify initialization and improve convergence under unbalanced operation [23]. frame comSequence components are linearly related to ponents through transform (10) (10) and , together with , are passed to the where VSC and controller blocks. In the VSC block, the modulating signal harmonics are converted to the abc components and fed to a PWM pattern generator. Any desirable modulating strategy may be implemented by the pattern generator. LIAN AND LEHN: STEADY-STATE SOLUTION OF A VSC WITH CLOSED-LOOP CONTROL 2075 Fig. 7. Simplified open-loop VSC model with the assumption of the switching frequency being infinite. The mismatch equations set the stage for a Newton-type it, and erative method, which starts with an initial seed value by (12) generates the sequence, (12) where . The convergence and efficiency are dependent on the initial , and the Jacobian, . Sections V-B and V-C will values, be devoted to the discussions on initialization and finding the Jacobian matrix. B. Initialization Proper initialization is critical for Newton’s method. Poor initialization is not only detrimental to the speed of the simulation but can, in extreme cases, even lead to overall convergence problems. A good seed value must therefore be found for the first iteration. Given that the dominant harmonic produced by a VSC is the positive sequence fundamental component, a simple phasor solution, based on Fig. 7 is used for initialization. The simplified model assumes infinite switching frequency and neglects both control action and converter losses. The phasor analysis yields phase A modulation index and firing angle associated with the fundamental frequency , positive-sequence component for a given value of , , and . All other harmonic components are initialized near zero. C. Jacobian Matrix The Jacobian matrix needed for Newton-type iteration is given by the partial derivative of the mismatch equations . with respect to the input vector Consequently Fig. 6. Proposed flowchart for hybrid method. The switching times produced by the pattern generator are fed to the VSC model. The VSC model outputs cur, and ), and dc voltage harmonics, rent harmonics ( . In the controller blocks, , and are converted -components, , and via (10). These voltages, to are substituted into (7) and (9) to yield together with , and and . Once , , , , , and are obtained, the mismatch equations can be found by (11). (11) (13) Since the VSC block contains two nonlinear systems, the PWM pattern generator and the converter itself, it is difficult to obtain an analytical expression for the Jacobian of the entire . In addition, one needs to re-derive the JaVSC block cobian of the VSC block each time a different type of pattern generator is used. An alternative is to compute the Jacobian numerically [6], [10], [17], [24] based on a sequential perturbation of the input vector calculated at the nominal case. The Jacobian , on the other hand, can be eiof the controller block ther evaluated numerically or obtained analytically. 2076 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 21, NO. 4, OCTOBER 2006 D. Broyden’s (Quasi-Newton) Method The amount of computation required to carry out a NewtonRaphson solution is extensive. As mentioned, the Jacobian of the VSC block is obtained numerically, which requires cycles of simulation for an Jacobian [4]. In addition, the Newton-Raphson method requires evaluation and inversion of the Jacobian matrix each iteration as well as evaluation of the mismatch equations in each iteration. In 1965, Brodyden [25] introduced a Quasi-Newton’s method in which only an approximate Jacobian matrix is needed to start the iteration process. Refinement of the Jacobian matrix is achieved by introducing correction terms in the iteration loop. This avoids re-evaluating the partial derivatives for each column and yields the inverse of the Jacobian matrix by simple matrix-vector multiplications. The total arithmetic calculations , rather than as required by the original required is Newton’s method [26]. A simple algorithm for implementing Broyden’s method [26] can be found in the Appendix. VI. SIMULATION RESULTS A. System Under Study To validate the proposed method, its steady state solutions are compared with those obtained by brute force time domain simulation (PSCAD/EMTDC). The system under study is as shown in Fig. 1. The block diagram of the current and voltage controllers are the same as in Fig. 4 and Fig. 5. The parameters and PWM strategy are extracted from the IEEE benchmark model of the D-STATCOM [18]. However, in order to show the uncharacteristic harmonics more clearly for the unbalanced case, the frequency modulation ratio is reduced from the original value of 27 to 9 (i.e. a switching frequency of 540 Hz), and the value of the dc capacitance is reduced from the original value . A simulation time step of 14 is chosen of 4860 to 1000 for PSCAD/EMTDC to limit integration error. Low order harmonics are expected to be heavily influenced by control action. To demonstrate how harmonics are affected by the controls, two different current controllers are employed. The two current controllers differ by a factor of 10 in their gains. Both have their zero placed to cancel the left-half plane pole of [18]. The transfer the linearized VSC model, which is at functions of the slow and fast current controllers are and ( , respectively. The same slow outer dc voltage controller is employed for both studies. In the following sections the VSC is studied under first balanced and then unbalanced operating conditions. B. Balanced Operation The VSC is assumed to connect to a perfectly balanced threephase system, with system voltages given by TABLE I TOTAL NUMBER OF ITERATIONS AND CPU TIME UNDER BALANCED OPERATION 1) Initialization and Efficiency: As mentioned in Section V-B, under balanced condition, phasor analysis is used to initialize the positive sequence fundamental component quantities; other harmonics are initialized close to zero. For the , , and system under study, resulting in initial values of and equal to , equivalent to an initial positive-sequence 0.8911 and . fundamental component of While the converter model is not inherently bandwidth limited, only a finite number of harmonics can be considered for computation of the controller response. Given that measured current and voltage signals are typically low pass filtered before entering the controllers, the modulating harmonics need only be included up to the cut-off frequency of these filters. If no feedback filters exist, as is the case in Fig. 1, the number of modulating harmonics included is chosen by considering 1) the low-pass nature of the power circuit itself and 2) the diminishing amplitude of PWM harmonics with frequency. Based on the open loop analysis [27], [28], and engineering are very small experience, harmonic magnitudes after for the studied system and have negligible effect on the converter switching times. Furthermore, for balanced operation, there is harmonics because only no need to include all of the characteristic harmonics will be nonzero [27]. Therefore, the number of harmonics to be initialized is substantially reduced. Thus, the following harmonics are initialized. , 5, 7, 11, 13, 17, 19, 23, 25, 29, • Modulating signal and 31. , 2, 4, 6, 8, 10, 12, 14, 16, • Active current reference 18, 20, 22, 24, 26, 28, 30, and 32. Table I shows the total number of iterations and CPU time for the hybrid method versus PSCAD/EMTDC (on a 550-MHz, Pentium III, Windows 2000 workstation). As Table I shows, a significant number of iterations and simulation time are greatly saved by the hybrid method. 2) Current and Voltage Harmonics: Once the closed loop modulating signal harmonics are obtained, these signals are passed to the PWM pattern generator to determine the steady state switching times. Applying the switching times to the the open loop augmented VSC model yield the ac current and dc voltage harmonics. Note that although, the modulating (i.e., 31st) harmonics are only specified up to the harmonic, the ac current and dc voltage harmonics may be calculated up to much higher frequencies. The assumption is merely that harmonics above the 31st do not propagate through the control. Once correct switching times are determined, the ac current and dc voltage harmonics may be found to arbitrarily high frequency [4]. Figs. 8 and 9 show the ac current space vector harmonic spectrum due to the slow and fast current controllers LIAN AND LEHN: STEADY-STATE SOLUTION OF A VSC WITH CLOSED-LOOP CONTROL Fig. 8. AC current space vector complex spectrum under balanced operation ). (due to C Fig. 9. AC current space vector complex spectrum under balanced operation ). (due to C under balanced operation. The nonzero ac current harmonics shown in the figures (from left to right) are [ 53 47 41 35 29 23 17 11 5 1 7 13 19 25 31 37 43 49 55]. Figs. 10 and 11 show the dc voltage harmonic spectrum due to the slow and fast current controllers, respectively, under balanced operation. Note that since dc voltage harmonics roll off very rapidly, the magnitude of dc voltage harmonics are plotted using a log scale to allow comparison of small high-frequency components. C. Unbalanced Operation A 10% negative sequence voltage is superimposed on the system voltage to study operation under unbalanced conditions. The system voltage is given by 2077 Fig. 10. DC voltage spectrum under balanced operation (due to C ). Fig. 11. DC voltage spectrum under balanced operation (due to C ). Note that a 10% bus voltage imbalance is selected for visualization purpose only, and that such a high level of imbalance would not ordinarily occur in a power system. 1) Initialization and Efficiency: For unbalanced operation, one expects uncharacteristic harmonics to appear in the spectrum of the modulating signals. The harmonics of interest must be either predicted a priori—a challenging task—or all odd harmonics up to the maximum harmonic of interest must be included. For the sample case, all odd harmonics up to are all considered. Initialization of these harmonics is carried out as follows. , 5, 7, 11, 13, 17, 19, 23, • Modulating signal: for 25, 29, and 31, the initialized values are set to those converged to under balanced operation. Other harmonics are . arbitrarily initialized to • Active current reference: all even harmonics are initialized to those converged to under balanced operation. Table II shows the total number of iterations and CPU time for the hybrid method versus PSCAD/EMTDC. Again, a great 2078 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 21, NO. 4, OCTOBER 2006 TABLE II TOTAL NUMBER OF ITERATIONS AND CPU TIME UNDER UNBALANCED OPERATION Fig. 14. DC voltage spectrum under unbalanced operation (due to C ). Fig. 15. DC voltage spectrum under unbalanced operation (due to C ). Fig. 12. AC current space vector complex spectrum under unbalanced operation (due to C ). using a log scale to allow comparison of small high-frequency components. VII. DISCUSSIONS Fig. 13. AC current space vector complex spectrum under unbalanced operation (due to C ). number of iterations and simulation time are saved by use of the hybrid method. 2) Current and Voltage Harmonics: Figs. 12 and 13 show harmonic spectrum due the ac current space vector to the slow and fast current controllers under unbalanced operation. All odd harmonics are shown. Figs. 14 and 15 show dc voltage harmonic spectrum due to the slow and fast current controllers, respectively. All even harmonics are shown. Note that similar to the case of balanced operation, the magnitude of dc voltage harmonics are plotted As can be noted from Figs. 8–15, excellent agreement exists between the proposed hybrid technique and the brute-force time domain (PSCAD/EMTDC) simulation results. and For balanced or unbalanced operation, both successfully regulate the positive sequence fundamental current to 30 A. However, the ac current spectra for the two controllers differ at other frequencies. This is true even under the case of balanced operation. Comparing Figs. 8 and 9, one sees that harmonics at 47, 13, and 55 differ. Note that suppresses these three does not. These difharmonics to almost zero, while ferences are sufficiently large to produce different 60-Hz operating points. Table III lists phase A modulation index and the firing angle of the fundamental component of the modulating signal produced by PSCAD/EMTDC and those converged to by the hybrid method. As can be noted, the firing angle LIAN AND LEHN: STEADY-STATE SOLUTION OF A VSC WITH CLOSED-LOOP CONTROL TABLE III OPERATION POINTS OF THE FUNDAMENTAL COMPONENT OF PHASE A MODULATION SIGNAL UNDER BALANCED OPERATION issued by the fast controller is about half of that issued by the slow controller. .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . 2079 and between PSCAD/EMTDC and Discrepancies of the hybrid method result from the fact that ideal bidirectional switches are employed in the hybrid method whereas switches in PSCAD/EMTDC are not perfectly ideal. Minor phase discrepancies are also attributable to delays between control and circuit solutions in PSCAD/EMTDC. Under unbalanced operation, the ac current harmonic spectra for the two controllers differ very significantly; as evident from Fig. 12 and Fig. 13. For instance, the negative sequence fundawhile it is mental current component is about 26A for is employed. Table IV shows how the only 13.2A when 2080 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 21, NO. 4, OCTOBER 2006 3) TABLE IV OPERATION POINTS OF THE FUNDAMENTAL COMPONENT OF PHASE A MODULATION SIGNAL UNDER UNBALANCED OPERATION where ; , and 4) 5) set 6) end. ; ; B. Matrices in Section IV differing harmonic distortion levels impact the 60-Hz operating point of the converter. The table lists the modulation indices and firing angles of phase A, B, and C for both the hybrid method and PSCAD/EMTDC simulations. Figs. 10, 11, 14, and 15 show that the dc component of the is successfully regulated to its reference capacitor voltage value of 230 V in all cases. As seen by comparing Figs. 10 and 11, for balanced operation, the ac current controller gain has little impact on dc voltage harmonics. In contrast, for unbalanced operation Figs. 14 and 15 show that a design of the ac current controller has a significant influence on dc voltage harmonics. Particularly noticeable discrepancies exist in the second, fourth, and eighteenth harmonic voltages. VIII. CONCLUSION An efficient hybrid time/frequency-domain method has been presented for steady-state analysis of the VSC with closed-loop control. The method identifies the exact timing of converter switching events, thereby avoiding limitations imposed by the Gibbs phenomenon. Only harmonics that influence the switching times are included in the iterative portion of the solution algorithm, thus minimizing computational burden. Once the timing of the switching events is accurately known, an arbitrary number of VSC harmonics may be calculated analytically. Accuracy of the model is demonstrated through comparison with simulation results obtained from PSCAD/EMTDC. In comparison to simulation, the hybrid time/frequency-domain method obtains the steady-state solution using roughly one-tenth the computation time. The simulation test cases considered also demonstrate the importance of including the effects of control action when calculating the harmonic steady state of the VSC. Control action is shown to play a particularly important role in the case of unbalanced operation, where active power exchange at frequencies other than the positive sequence fundamental can be significant. APPENDIX A. Broyden’s Method Step 1) initialization (i.e., ) to obtain ; Step 2) set Step 3) while 1) begin; 2) evaluate ; and ); In the equation at the bottom of the previous page, , , where , , , , , , , , , , , , , , , , , , , , , , . Note that . REFERENCES [1] T. Aprille and T. Trick, “Steady-state of nonlinear circuits with periodic inputs,” Proc. IEEE, vol. 60, no. 1, pp. 108–114, Jan. 1972. [2] T. Kato and W. Tachibana, “Periodic steady-state analysis of an autonomous power electronic system by a modified shooting method,” IEEE Trans. Power Electron., vol. 13, no. 3, pp. 522–527, May 1998. [3] H. 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