Large- and Small-Signal Evaluation of Average Models for Multi-Pulse Diode Rectifiers

Transcription

1 6 IEEE COMPEL Workshop, Rensselaer Polytechnic Institute, Troy, N, USA, July 16-1, 6 Large- and Small-Signal Evaluation of Average Models for Multi-Pulse Diode Rectifiers Sebastian Rosado, Rolando Burgos, Fred Wang, and Dushan Boroyevich Center for Power Electronics Systems The Bradley Department of Electrical and Computer Engineering Virginia Polytechnic Institute and State University Blacksburg, VA 461 USA Abstract This paper analyses different average models of multi-pulse diode rectifiers. In the analysis a and a average model are compared to a detailed model. The main goal is to establish the validity of the model. First the different models are discussed. Then the small and large signal evaluations of the models are presented. The input impedance freuency response of the models is used as basis for the small-signal evaluation. This parameter is important in the small-signal stability of the system. The largesignal evaluation is done by means of time domain simulations carried on with the different models. The results obtained indicate that the average model can provide a good tool for the analysis of circuits using multi-pulse rectifiers. I. INTRODUCTION Multi-pulse diode rectifiers are usually used as front-end converters in aerospace power systems. Compared to other type of converters for the same applications multi-pulse rectifiers are of reduced dimensions, weight, and cost [1], []. In addition, increased power rating of the aircraft power system, as well as the variety of loads and their reuirements are placing increased reuirements on the system and components; this reuires careful analysis. Part of this analysis deals with the stability of the system. For simulation based stability studies detailed models can be appropriate. On the other side for analytical studies, and when it is necessary to use small-signal linearized models it is useful to count with good average models. Two main approaches have been developed for the average modeling of line commutated converters, namely input-output transfer function [3] and closed-form mathematical expressions [4]. Although the former is very simple and straight forward to implement, its dependence on input-output lookup tables makes it cumbersome to use since any parameter variation normally reuires numerous simulations to rebuild these tables [5], [6]. The mathematical approach on the contrary uses explicit euations to relate the input and output variables of the converters, however the complexity of these models is such that especially so for multi-pulse converters numerous simplifying assumptions are reuired, which end up affecting the accuracy of the model regarding its dynamic and freuency responses. Regardless of the approach This work made use of ERC Shared Facilities supported by the National Science Foundation under Award Number EEC /6/$ X/6/$. 6 IEEE. 8 taken, the main issue at hand with this type of converter is that its operation depends on the ac system impedance and thus any model developed should capture this into its operation. This is naturally not practical, reason why these average models are developed taking into consideration only line and generator impedances [7]. In this paper an evaluation is realized with previously developed average models based on the mathematical description of the converter eliminating the need of multiple simulations as in the input-output transfer function modeling approach. The models analyzed range in complexity from: commutation representation to improved AC and DC dynamics. The models are subjected to transient simulations (and compared to a model), and smallsignal models derived numerically from them are used to calculate the d- frame input impedance. The results obtained fully characterize the accuracy of the models studied, showing which models are more propitious for time or freuency domain analysis. Insight is given as well into the physical meaning of the impedance calculations both in d- frame and from the output DC port. All simulations and freuency domain analysis are presented in Matlab/Simulink and Synopsys Saber. II. MULTI-PULSE RECTIFIER MODELS A. Multi-pulse Rectifier Operation Multi-pulse diode rectifiers are composed of an autotransformer and one or more sets of six-pulse diode bridges. The number of bridges is given by the number of phases in the secondary side of the autotransformer, with a minimum of three phases corresponding to one diode bridge. In order to reduce the output voltage ripple the number of phases in the secondary side is increased by an appropriate interconnection of windings. Although the analysis to be presented can be applied to rectifiers of different number of pulses, the discussion concentrates on an eighteen pulse rectifier like the one in Figure 1. The autotransformer in Figure 1 has a three-phase primary and a nine-phase secondary. The nine-phases are eually spaced in the line freuency period being the electrical angle between phases of forty degrees. The output can also be seen as three sets of three-phases eually spaced and each one Authorized licensed use limited to: Dahono Pekik. Downloaded on March 6, at :14 from IEEE Xplore. Restrictions apply.

2 connected to a diode bridge rectifier. Each of this six-pulse rectifiers conduct one third of the total output power. Figure 1: Eighteen pulse diode rectifier circuit schematic In modeling these circuits usually it is assumed that the leakage inductances have the same value in all nine secondary phases. In order for the circuit to achieve good performance it is desirable that this leakage inductance is of reduced value. Nonlinearities due to the iron core of the transformer are also neglected. The circuit is usually connected to an LC filter in the DC side, this filter can be included as part of the model. The basic input-output voltage relationships are given by: didc [ V] [ S] = V abc abc dc = Rdcidc + Ldc + Vload (1-a) dt [ i] [ S] idc abc = (1-b) abc Where [S] abc represents the action of the multi-pulse rectifier, L dc is the inductance connected at the output and R dc its resistance; V load is the voltage at the load. Switching models of multi-pulse rectifiers can be implemented in a circuit simulator for accurate computer simulations [8]. For control and stability studies average models are preferred. Some average models providing good simulation fidelity have been proposed [],[1]. The next paragraphs briefly review and compare those previous models. Additionally, some characteristics allowing for simplification are discussed. B. Full-order Model The methodology to obtain the close form mathematical model of a diode rectifiers have been exposed in [11] for three-phase, six-pulse rectifiers. The circuit analysis of a multi-pulse rectifier is based on the same methodology. When the DC current is assumed constant during the averaging period, the static model is obtained. The dynamic model allowed for a DC current change during one period, but still considers it constant for the AC current calculation. Otherwise, the model presented in [] also accounted for the DC current change in the AC current calculation producing an improvement in the model fidelity [1]. In this type of models the circuit operation is divided in two stages: conduction and commutation. During conduction only one of each the positive and negative terminal diodes are conducting. In a different way, during commutation, the DC current is transferred from one branch of the bridge to another at either the positive or negative side of the bridge. The model operation is represented in the block diagram in Figure. The first step is to calculate the voltage V d from the applied d- voltage components. Subtracting the voltage drop at the circuit parasitic components and the one created by the commutation, the DC voltage can be calculated. For these calculations it is good to refer the parameters to the secondary side by affecting the magnitudes and parameters by the proper transfer ratio. 18 µ + sin µ Vdc = Vpk sin ( R Ron ) + + ωlc () µ ωlc + ( R+ Ron) idc Von In () L c is the inductance seen by the commutation circuit and is composed by the autotransformer leakage plus any inductance in the primary circuit; whereas R represents the parasitic resistance of the transformer and primary circuit. R on and V on are the diode voltage drop and forward resistance. The calculation in the previous euation reuires the knowledge of µ, the commutation angle. This angle can be calculated by means of the following expression: ω LI c dc µ = arccos1 (3) Vpk sin Figure Block diagram of the operation of the model in [] The commutation takes place during the time corresponding to µ/ω. At that time the current evolution is dictated by the commutation circuit. Knowing the commutation angle µ, the line current is calculated as the sum of its commutation and conduction components. As mentioned above the accuracy can be improved by accounting for the DC current change as proposed in []. The methodology and details about the derivation of the expressions calculating the actual values of all the circuit magnitudes can be followed in the mentioned references. C. Simplified Model The analysis to be presented in this section assumes small harmonic content in the input AC magnitudes and small ripple in the output DC voltage and current when compared to the di dc dt Authorized licensed use limited to: Dahono Pekik. Downloaded on March 6, at :14 from IEEE Xplore. Restrictions apply.

3 fundamental freuency component. Under these assumptions all the power is carried on by the fundamental freuency. If additionally the rectification process is assumed lossless the power at both AC and DC sides must be the same. The power at the AC side is given by: PAC = 3Vt It cosϕ (4) where V t and I t are the rms of the line-ground voltage and the line current and ϕ the angle between the two. Otherwise, the power in the DC side is, PDC = VdcIdc (5) Neglecting the magnetizing current component, the current input and output must be the same. On the other side, the voltage experiences a drop due to the commutation process. Therefore, from (4) and (5) the following relationship can be established V dc = Gdc vd + v cosϕ (6) Conversely, the currents in the AC side can be calculated form the DC side magnitudes. vd cosϕ+ vsinϕ id = Gdc Idc (7-a) vd + v vd sinϕ+ v cosϕ id = Gdc Idc (7-b) vd + v The factor G dc must consider the conversion factor of the abc d matrix transformation and the transformer ratio of the autotransformer. For the matrix transformation of the nine-phase circuit used (in the Appendix), the following relationship apply. Vpk = Vt = vd + v (8) 3 The phasor diagram in Figure 3 represents the fundamental components of the AC magnitudes and their d- decomposition. Given the euations (6) and (7) it also can be seen as a representation of the average values of the DC magnitudes, affected by a scale factor. 3I t 3V t Figure 3: Phasor diagram for the AC side of the rectifier Considering the euation (6) the voltage drop created during the commutation is given by drop dc d ( 1 cos ) V = G v + v ϕ () Additionally, the DC voltage euation () under lossless conditions reduces the euation (1): 18 Vdc = Vpk sin ωlc i (1) dc In this euation the first term on the right represents the ideal DC voltage in case the commutation effect is negligible. Moreover, the second term corresponds to the voltage drop produced by the commutation. Vdrop = ω Lc idc (11) From euations () and (11), and considering the peak component relationship in (8), the following expression provides the angle between the voltage and current fundamental components. ω Lc I dc ϕ = arccos1 (1) Gdc 3V pk The euations (1-a), (6), (7), and (1) provide the complete description of the multi-pulse rectifier model. These euations can be represented in the circuit schematic of Figure 4. Figure 4: Simplified euivalent circuit of the multi-pulse rectifier D. Discussion on the coefficients and angles In the calculation of V dc in (6) it was assumed that the output voltage corresponds to a rectifier of infinite number of pulses. In such case the output voltage corresponds to V pk. However, a look at euation (1) shows that in the 18-pulse rectifier, the ideal output voltage is; Vdc = Vpk sin The error introduced by the approximation is small, about 1.%. In fact, the true ratio can be accounted for in the calculation of G dc. However, the influence on the output magnitudes is not significant and can be neglected considering the other assumptions in the model. Euation (1) provides the phase shift angle between the fundamental components of the voltage and current in the DC circuit. This shift is introduced by the leakage reactance of the autotransformer and the inductance in the primary circuit, which are generically considered in L c. This inductance is also responsible for the commutation time interval given by the commutation angle µ. A look at euations (3) and (1) show the close relation between these two angles. Figure 5 shows the 1 Authorized licensed use limited to: Dahono Pekik. Downloaded on March 6, at :14 from IEEE Xplore. Restrictions apply.

4 evolution of these two angles as function of the DC current for a generic eighteen-pulse rectifier. angle (rad) fiϕ mu µ Idc (pu) Figure 5: V, I phase shift ϕ, and commutation angle µ III. SMALL-SIGNAL ANALSIS In most cases the small-signal behavior of a non-linear circuit can be analyzed using a linearization of the original model around the operation point. In particular, the inputoutput impedances (or admittances) are important to evaluate of the stability of a DC or AC system [1]. Therefore, the input admittance of both average models:, and was evaluated. In d- reference the admittance has four components as indicated by (13). dd d in = d (13) The admittance freuency response of the full order model is shown in Figure 6. The reference frame is aligned with the d- axis of the d- frame Bode Diagram 5/D-Q L-C Filter B (1) From: A TRU D 5/D-Q L-C Filter B (3) dd d components. dc represents the input admittance of the DC circuit connected to the multi-pulse rectifier, including the DC filter. Additionally, the input admittance freuency response is shown in Figure 7. The corresponding dc freuency response is shown in Figure i V VV d d d dd = = cos ϕ + sinϕcosϕ Gdcdc v d Vd + V Vd + V VV id d d = = cos ϕ + sinϕcosϕ Gdcdc v Vd + V Vd + V i V V V d d = = cos ϕ sinϕcosϕ Gdcdc v d Vd + V Vd + V i V V V d = = cos ϕ sinϕcosϕ Gdcdc v Vd + V Vd + V V (14-a,b,c,d) Bode Diagram 5/D-Q L-C Filter B1 (1) 5/D-Q L-C Filter B1 (3) Figure 7: Input admittance of the multi-pulse rectifier model according to euations (14-a,b,c,d) 3 1 dd d Freuency (Hz) Bode Diagram d 5/ATRU Mk.VII (1) To: ATRU D 5/LC Filter () d Freuency (Hz) Figure 6: Input admittance of the multi-pulse rectifier fullorder model obtained from the euivalent circuit The admittance of the model can be obtained analytically from euations (6) and (7). This produces the results of formulas (14-a,b,c,d) for the four admittance Magnitude (db) Phase (deg) Freuency (Hz) Figure 8: Input admittance of DC circuit Authorized licensed use limited to: Dahono Pekik. Downloaded on March 6, at :14 from IEEE Xplore. Restrictions apply.

5 It is possible to observe the similarity between the responses in Figure 8 and the dd component in Figure 7. Additionally, the other three components are largely attenuated. With a proper alignment the multi-pulse rectifier reflects the behavior of the DC circuit on the primary AC side. On the contrary, the response in Figure 6 does not reflect that behavior. Further analysis indicates that the difference is introduced by the perturbation of the commutation angle. To achieve appropriate results the commutation angle µ, or the V,I angle ϕ in the model must be kept constant during the small-perturbation, linearization process. IV. LARGE-SIGNAL ANALSIS This section evaluates the behavior of the multi-pulse rectifier average models beyond the description provided by the small-signal freuency response of the circuit. The evaluation is made through computer simulations. A. Transient response The transient behaviors of three different models: detailed, average, and average, was simulated in order to observe the evolution of the different circuit magnitudes. Figure shows the waveforms for V dc and I dc for a sudden load connection and disconnection. Vdc (V) id (A) i (A) Figure 1: AC current component evolution during a load connection-disconnection transient B. Large-signal stability behavior An important aspect of the large-signal evaluation is related to the model ability to represent the large-signal stability behavior. The circuit in Figure 11 was used for this largesignal evaluation. The DC side of the rectifier connects to a constant power load (CPL) type. In this load the current level is adjusted in order to maintain the power unchanged at any input voltage. For a voltage perturbation, the current response is not instantaneous and has a response of first order type, which corresponds to the bandwidth of the load controller. 3 5 Idc (A) Figure : Transient evolution for V dc and I dc a transient originated by a load connection-disconnection The evolution of the AC current for the same transient than the previous Figure is shown in Figure 1. In these two Figures it is possible to observe the closeness of the response of the three models. There are however slight differences mainly between the two average models and the models originated in the assumptions made during the modeling. 3 Figure 11 Circuit schematic of the circuit used for large-signal stability analysis The circuit is also composed of a synchronous generator with excitation regulation. The DC filter is provided by L and C; whereas R p represents a small resistive load. The sudden connection of the controlled load was simulated. When the load power level is increased the circuit reaches a point where the load connection produces instability. It is important that simulation study can identify this load level producing instability. To compare model performances, the simulation of the circuit in Figure 11 was done for the detailed model and the average model. Figure 1 shows the V dc, I dc evolution for a stable case. Increasing the load by Authorized licensed use limited to: Dahono Pekik. Downloaded on March 6, at :14 from IEEE Xplore. Restrictions apply.

6 1kW (less than 1%) produces the unstable behavior of Figure 13. As indicated by the simulations, both models agreed in identifying the stability limit in a band of about 8% of the total power. vdc (V), idc (A) vdc (V), idc (A) vdc - idc - 1 vdc idc Figure 1 Response for a stable load connection vdc - idc - vdc - idc time (sec) Figure 13 Response for an unstable load connection V. CONCLUSION The model analysis and comparison presented shows, in general, good agreement among the evaluated average models and the model. Exception is made for the case of the average model freuency response. However, the origin of this difference has been identified. The average model has the advantage of providing good response with much less calculation effort than the average model. From the analysis it is also possible to conclude that the dynamic inherent to the multipulse rectifier is of reduced impact when compared to the dynamics of the AC and DC circuits where it is connected. This is valid at least for the type of power systems of interest. ACKNOWLEDGEMENTS This work was conducted under the CPES project Stability of AC Power Systems sponsored by The Boeing Company. This work made use of Engineering Research Center Shared Facilities supported by the National Science Foundation under NSF Award Number EEC APPENDIX Matrix transformation for a nine-phase system to the d- rotating reference frame T d 1 = 3 3 cosθ + 4 cos θ 8 cosθ + cosθ cosθ 3 cosθ + 3 cosθ 8 cosθ 4 cosθ + sinθ + 4 sinθ 8 sinθ + sin θ sinθ 3 sinθ + 3 sinθ 8 sinθ 4 sinθ + REFERENCES [1] Kaz Furmanczyk and Mark Stefanich, "Demonstration of very high power airborne AC to DC converter," in SAE Power System Conference, Reno, Nevada, Nov. -4, 4. [] F.J.Chivite-Zabalza, A.J. Forsyth, D.R.Trainer, "Analysis and practical evaluation of an 18-pulse rectifier for aerospace applications," in Proceedings of the Second International Conference on Power Electronics, Machines and Drives,4, vol 1. pp [3] I. Jadric, D. Borojevic, M. Jadric, Modeling and control of a synchronous generator with an active DC load, IEEE Trans. on Power Electronics, Vol.15, Issue, pp , March [4] S. Sudhoff, Analysis and Average Value Modeling of Dual Line-Com Converter 6-phase Synchronous Machine, IEEE Trans Energy Conversion, Sep 13 [5] C.T.Tinsley, E.M.Hertz, R. Carie, D.Boroyevich, "Average modeling of a hexagon transformer and rectifier," in IEEE PESC Rec. 3, vol. 4 pp [6] J. Jatskevich, A. Davoudi, S. D. Pekarek, Parametric Average-Value Model of Synchronous Machine-Rectifier Systems, IEEE Transactions on Energy Conversion, Issue, 5, pp. 1-1 [7] S. D. Sudhoff, K. A. Corzine, H. J. Hegner, D. E. Delisle, Transient and Dynamic Average-Value Modeling of Synchronous Machine fed Load-Commutated Converters, IEEE Trans. on Energy Conversion, Vol.11, No.3, Sep 16. [8] K. J. Karimi, A. C. Mong, Modeling nonlinear loads for aerospace power systems, Proc. IECEC, pp , July 4 [] H. Zhu, R. P. Burgos, F. Lacaux, A. Uan-Zo-li, D. K. Lindner, F. Wang, D. Boroyevich, Evaluation of Average Model of Nine-phase Diode Rectifiers with Improved AC and DC Dynamics, APEC 6 [1] H. Zhu, R. P. Burgos, F. Lacaux, A. Uan-Zo-li, D.K. Lindner, F. Wang, D. Boroyevich, Average modeling of three-phase and nine-phase diode rectifiers with improved AC current and DC voltage dynamics, Proc of IECON 5. pp. 6-1, Nov. 5 [11] P. C. Krause, O. Wasynczuk, S. D.Sudhoff, Analysis of Electric Machinery and drive systems, New ork: IEEE Press, [1] M. Belkhayat, Stability Criteria for AC Power Systems with Regulated Loads, PhD. Dissertation, Purdue University, West Lafayette, 17 T 4 Authorized licensed use limited to: Dahono Pekik. Downloaded on March 6, at :14 from IEEE Xplore. Restrictions apply.