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An initial investigation of S and SH with angle resolved photoelectron spectroscopy using synchrotron radiation
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TEC.2017.2735904, IEEE Transactions on Energy Conversion
Power Engineering Letters___________________ Parametric Dynamic Phasor Modeling of Thyristor-Controlled Rectifier Systems Including Harmonics for Various Operating Modes Yingwei Huang, Seyyedmilad Ebrahimi, Navid Amiri, Zhenyu Shan, and Juri Jatskevich Abstract—A parametric dynamic phasor (PDP) model of diode rectifiers has been recently proposed, which uses a computer-aided approach to effectively represent the rectifier dynamics including ac-side harmonics that change with system conditions. This letter extends the PDP methodology to thyristor-controlled rectifier systems where the dc-link voltage is regulated via the thyristor firing control. Steady-state and large-signal transient studies demonstrate that the new/extended model is accurate even for prediction of harmonics for various operating modes. Index Terms—Machine-converter systems, parametric model, thyristor-controlled rectifiers, dynamic phasor, harmonics.
respectively [4]. Without loss of generality, the thyristor firing control via the terminal voltages vabcs and a firing angle α (i.e. Case II) is considered for consistency with prior publication [4].
I. INTRODUCTION
T
HE traditional detailed switch-level models of machine-converter systems render expensive computations and limited scalability for modeling mid- and large-scale ac-dc power systems, where it is challenging to represent the harmonic dynamics that vary frequently with system operating modes. The so-called dynamic phasors (DPs) [1] can simulate the desired (and neglect insignificant) frequency components of the system’s state variables, resulting in flexible modeling accuracy and (orders of magnitude) higher numerical efficiency. The DP models may be derived using analytical or parametric approaches [2]–[3]. Specifically, a recent parametric DP (PDP) model of diode rectifiers [3] relates the ac-dc DP dynamics (including desired ac-side harmonics) using a set of numerically-constructed algebraic functions, in addition to avoiding the fixed-shape switch functions [2] that are complicated and only valid for a particular operating mode. This PDP model is shown to be accurate for various loading conditions, while providing significant numerical advantages. Due to limited space, it is expected that the reader has prior knowledge of [3] and the references therein. This letter extends the PDP methodology [3] to the thyristorcontrolled rectifier systems by including the thyristor firing control. The extended PDP model is valid for harmonic prediction in various operating modes and transients including the firing angle control dynamics, which make it suitable for small- and large-signal system-level studies of many energy conversion systems and industrial applications. II. PDP MODELING OF THYRISTOR-CONTROLLED RECTIFIERS In this letter, the benchmark thyristor-controlled rectifier system depicted in Fig. 1 [4] is considered, where the rectifier can be fed from: i) a rotating machine, or ii) a distribution feeder/transformer (represented by its Thevenin equivalent). Accordingly, the thyristor firing pulses may be generated based upon the sensed rotor position or filtered terminal voltages,
Fig. 1. Benchmark configuration of thyristor-controlled rectifier systems.
Fig. 2. Implementation of the PDP model of thyristor-controlled rectifiers.
A. Relating AC-DC DP Dynamics In DP theory [1], the DPs of system variable u(t) [u = {v, i}] are defined as the complex-form Fourier coefficients k (t) of the time-domain waveform over a sliding interval Ts as t 1 (1) u k (t ) u ( )e jk s d . T t Ts As in [3], the rectifier switching cell in Fig. 1 can be represented by algebraic functions depicted in Fig. 2. Therein, assuming a balance system, the ac-dc voltage and current DPs are related through parametric functions wv,k(·), wi(·), and k(·), where kK, and K={1, 5, 7, …} denotes the order of desired DPs. These parametric functions are defined in (2)-(4), which are determined by the present system operating mode using two inputs: i) the dynamic impedance zd defined in (5) that indicates the loading condition; and ii) the thyristor firing angle α. (2) wv,k ( zd , ) vas k vdc 0 , k K wi ( zd , ) idc
ias
0
1
,
k ( zd , ) ang( vas k ) ang( ias 1 ) , k K zd vdc
0
ias
1
.
(3) (4) (5)
B. Constructing Parametric Functions The procedure of finding the algebraic relations between ac and dc DPs of the thyristor rectifier are as follows:
0885-8969 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TEC.2017.2735904, IEEE Transactions on Energy Conversion
1) Select the set of desired harmonics, i.e. K, based on the properties of non-linear components in the system; 2) Run detailed simulations of the subject system for a range of operating conditions, which can be achieved using a myriad of steady-state points at different zd and α, or several largesignal transients that span various modes (Section III-D, [3]); 3) Extract desired Fourier coefficients (DPs) from the timedomain waveforms of system’s ac/dc variables, establish the parametric functions at each operating point using (2)-(4); 4) Store these functions in two-dimensional look-up tables, use proper curve fitting or inter/extrapolation if needed. III. COMPUTER STUDIES The system of Fig. 1 with system parameters identical to [3] and the ac filter same as Eq.(11), [4] is considered for study, wherein the dominant 1st, 5th, and 7th order DPs for ac variables and the 0th order DP for dc subsystem are considered. The detailed model implemented using PLECS Blockset and the analytical DP (ADP) model [2] are used for comparison. For the PDP model, the parametric functions (2)-(4) are constructed following the steps in Section II-B, and depicted in Fig. 3. Therein, 2520 solution points have been used, which are more tightly-spaced in the region where the parametric functions become nonlinear.
Fig. 3. Parametric functions as numerically calculated from detailed simulations: (a) wv,1(zd,α); (b) wi(zd,α); (c) 1(zd,α); (d) wv,5(zd,α); (e) 5(zd,α); (f) wv,7(zd,α); and (g) 7(zd,α).
A. Steady-State Response To consider a wide range of steady state operating conditions, the computed total harmonic distortion (THD) (see as Eq. (22), [3]) of ac current ias for different thyristor firing angles with a dc load resistance Rl=25 is shown in Fig. 4. As shown in Fig. 4, the ADP model gives inaccurate THD results at conditions with α>40°, which is due to the fixed-shape switch functions ([3, see Fig. 4]) that are different from the actual current waveforms. In contrast, the THD results are well predicted by the PDP model for the entire firing angle range, which verifies its accuracy in steady state conditions.
Fig. 4. Steady-state THD % of ias for different thyristor firing angles.
B. Large-Signal Transient Response To emulate various large-signal transients, the system (assumed initially in steady-state with Rl =150 and α=10°) is subjected to several load and firing angle changes: i) at t=0.05s, step change of Rl; ii) at t=0.15s, ramp change of Rl; iii) at t=0.35s, step change of α; and iv) at t=0.45s, ramp change of α. In addition, to illustrate the close-loop dynamic responses of the subject models, two thyristor firing control events are included: v) at t=0.65s, activation of the dc voltage controller; and vi) at t=0.75s, step change of the reference voltage. The details of these large-signal transients are also shown in Fig. 5 (a)(b). The close-loop thyristor firing controller is depicted in Fig. 8. The resulting system response of dc variables is depicted in Fig. 5 (c)(d), which shows noticeable error with the ADP model while excellent agreement between the PDP and detailed model results. This is further confirmed with the magnified views of ac variables vas and ias shown in Fig. 6. Specifically, it is noted in Fig. 6 (a) and (d) that as the system enters discontinuous conduction mode (DCM) due to increase of Rl or α, the ADP model fails to reproduce the distorted current waveforms (dips between double peaks), while the PDP model yields accurate waveforms as system operating condition evolves [see lines a) and c)]. It is also seen in Fig. 6 (a) and (c) that during the step-change transients, the ADP model can result in some dynamic errors due to the reduced-order representation of the ac network ([3, see Section IV-E]), which however do not occur with the PDP model. Moreover, Fig. 7 shows magnified views of system response to the activation/change of the dc voltage controller, which again demonstrate significant error with the ADP model [see line b)] and verify the consistency between the PDP and detailed models in close-loop dynamic responses. C. Frequency-Domain Study Finally, to demonstrate capability of the proposed model in the frequency-domain, a small-signal analysis is performed around the same initial steady-state operating point as in Section III-B. The open-loop transfer function F(s) is considered as vˆ ( s) , (6) F ( s) dc ˆ ( s)
where vˆdc ( s) is the change in the output dc voltage due to the small-signal perturbation in the input firing angle ˆ ( s) . The transfer functions as predicted by the subject models are depicted in Fig. 9, where a frequency-sweep technique has been used for the detailed model. As shown in Fig. 9, the ADP model shows inaccurate results for frequency higher than 20 Hz, while the transfer functions evaluated by the detailed and PDP models are in good agreement, especially in the range from 1 to 200 Hz. In general, due to the Fourier transformation of system variables [i.e. averaging of frequency components, as shown in (1)], the DP models should yield accurate frequency-domain response up to about 1/3 of the system switching frequency.
0885-8969 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TEC.2017.2735904, IEEE Transactions on Energy Conversion
Fig. 5. System response to the large-signal transients: (a) profile of Rl; (b) profile of α; (c) resulting dc voltage vdc; and (d) resulting dc current idc.
Fig. 7. Magnified views of system close-loop dynamic response to: (a) activation of the dc voltage controller; and (b) step change in reference voltage.
Fig. 8. The close-loop thyristor firing controller for dc voltage regulation: KP = 1 0.1, KI = 10, and H (s) , where a 45 106 , and b 14 103 . a 2 s bs 1
Fig. 9. Transfer function from the input firing angle to the output dc voltage as predicted by the subject models. IV. CONCLUSIONS
Fig. 6. Magnified views of ac variables vas and ias response to: (a) step change of Rl ; (b) ramp change of Rl ; (c) step change of α; (d) ramp change of α.
An extended PDP model of thyristor-controlled rectifier systems is presented in this letter. The case studies validate the effectiveness of the PDP model in predicting system dynamics including harmonics for various operating modes. V. REFERENCES [1]
When the disturbance frequency approaches the switching frequency or goes beyond, it would be expectant to observe deviation of results between the detailed and DP models, since the switching patterns of the detailed model may start changing and the basic assumptions of the generalized averaging are no longer valid. However, since the input is the firing angle command, the lower frequency dynamics are of more importance for studying stability and designing controllers for the system.
[2]
[3]
[4]
S. R. Sanders, J. M. Noworolski, X. Z. Liu, and G. C. Verghese, "Generalized averaging method for power conversion circuits," IEEE Trans. Power Electron., vol. 6, pp. 251–259, Apr. 1991 C. Liu, A. Bose, and P. Tian, “Modeling and analysis of HVDC converter by three-phase dynamic phasor,” IEEE Trans. Power Del., vol. 29, no.1, pp. 3-12, Feb. 2014. Y. Huang, L. Dong, S. Ebrahimi, N. Amiri, and J. Jatskevich, “Dynamic phasor modeling of line-commutated rectifiers with harmonics using analytical and parameter approaches,” IEEE Trans. Energy Convers., vol. 32, no.2, pp. 534-547, Jun. 2017. H. Atighechi, S. Chiniforoosh, K. Tabarraee, and J. Jatskevich, “Average-value modeling of synchronous-machine-fed thyristorcontrolled-rectifier systems,” IEEE Trans. Energy Convers., vol. 30, no.2, pp. 487-497, Jun. 2015.
0885-8969 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
Power Engineering Letters___________________ Parametric Dynamic Phasor Modeling of Thyristor-Controlled Rectifier Systems Including Harmonics for Various Operating Modes Yingwei Huang, Seyyedmilad Ebrahimi, Navid Amiri, Zhenyu Shan, and Juri Jatskevich Abstract—A parametric dynamic phasor (PDP) model of diode rectifiers has been recently proposed, which uses a computer-aided approach to effectively represent the rectifier dynamics including ac-side harmonics that change with system conditions. This letter extends the PDP methodology to thyristor-controlled rectifier systems where the dc-link voltage is regulated via the thyristor firing control. Steady-state and large-signal transient studies demonstrate that the new/extended model is accurate even for prediction of harmonics for various operating modes. Index Terms—Machine-converter systems, parametric model, thyristor-controlled rectifiers, dynamic phasor, harmonics.
respectively [4]. Without loss of generality, the thyristor firing control via the terminal voltages vabcs and a firing angle α (i.e. Case II) is considered for consistency with prior publication [4].
I. INTRODUCTION
T
HE traditional detailed switch-level models of machine-converter systems render expensive computations and limited scalability for modeling mid- and large-scale ac-dc power systems, where it is challenging to represent the harmonic dynamics that vary frequently with system operating modes. The so-called dynamic phasors (DPs) [1] can simulate the desired (and neglect insignificant) frequency components of the system’s state variables, resulting in flexible modeling accuracy and (orders of magnitude) higher numerical efficiency. The DP models may be derived using analytical or parametric approaches [2]–[3]. Specifically, a recent parametric DP (PDP) model of diode rectifiers [3] relates the ac-dc DP dynamics (including desired ac-side harmonics) using a set of numerically-constructed algebraic functions, in addition to avoiding the fixed-shape switch functions [2] that are complicated and only valid for a particular operating mode. This PDP model is shown to be accurate for various loading conditions, while providing significant numerical advantages. Due to limited space, it is expected that the reader has prior knowledge of [3] and the references therein. This letter extends the PDP methodology [3] to the thyristorcontrolled rectifier systems by including the thyristor firing control. The extended PDP model is valid for harmonic prediction in various operating modes and transients including the firing angle control dynamics, which make it suitable for small- and large-signal system-level studies of many energy conversion systems and industrial applications. II. PDP MODELING OF THYRISTOR-CONTROLLED RECTIFIERS In this letter, the benchmark thyristor-controlled rectifier system depicted in Fig. 1 [4] is considered, where the rectifier can be fed from: i) a rotating machine, or ii) a distribution feeder/transformer (represented by its Thevenin equivalent). Accordingly, the thyristor firing pulses may be generated based upon the sensed rotor position or filtered terminal voltages,
Fig. 1. Benchmark configuration of thyristor-controlled rectifier systems.
Fig. 2. Implementation of the PDP model of thyristor-controlled rectifiers.
A. Relating AC-DC DP Dynamics In DP theory [1], the DPs of system variable u(t) [u = {v, i}] are defined as the complex-form Fourier coefficients k (t) of the time-domain waveform over a sliding interval Ts as t 1 (1) u k (t ) u ( )e jk s d . T t Ts As in [3], the rectifier switching cell in Fig. 1 can be represented by algebraic functions depicted in Fig. 2. Therein, assuming a balance system, the ac-dc voltage and current DPs are related through parametric functions wv,k(·), wi(·), and k(·), where kK, and K={1, 5, 7, …} denotes the order of desired DPs. These parametric functions are defined in (2)-(4), which are determined by the present system operating mode using two inputs: i) the dynamic impedance zd defined in (5) that indicates the loading condition; and ii) the thyristor firing angle α. (2) wv,k ( zd , ) vas k vdc 0 , k K wi ( zd , ) idc
ias
0
1
,
k ( zd , ) ang( vas k ) ang( ias 1 ) , k K zd vdc
0
ias
1
.
(3) (4) (5)
B. Constructing Parametric Functions The procedure of finding the algebraic relations between ac and dc DPs of the thyristor rectifier are as follows:
0885-8969 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TEC.2017.2735904, IEEE Transactions on Energy Conversion
1) Select the set of desired harmonics, i.e. K, based on the properties of non-linear components in the system; 2) Run detailed simulations of the subject system for a range of operating conditions, which can be achieved using a myriad of steady-state points at different zd and α, or several largesignal transients that span various modes (Section III-D, [3]); 3) Extract desired Fourier coefficients (DPs) from the timedomain waveforms of system’s ac/dc variables, establish the parametric functions at each operating point using (2)-(4); 4) Store these functions in two-dimensional look-up tables, use proper curve fitting or inter/extrapolation if needed. III. COMPUTER STUDIES The system of Fig. 1 with system parameters identical to [3] and the ac filter same as Eq.(11), [4] is considered for study, wherein the dominant 1st, 5th, and 7th order DPs for ac variables and the 0th order DP for dc subsystem are considered. The detailed model implemented using PLECS Blockset and the analytical DP (ADP) model [2] are used for comparison. For the PDP model, the parametric functions (2)-(4) are constructed following the steps in Section II-B, and depicted in Fig. 3. Therein, 2520 solution points have been used, which are more tightly-spaced in the region where the parametric functions become nonlinear.
Fig. 3. Parametric functions as numerically calculated from detailed simulations: (a) wv,1(zd,α); (b) wi(zd,α); (c) 1(zd,α); (d) wv,5(zd,α); (e) 5(zd,α); (f) wv,7(zd,α); and (g) 7(zd,α).
A. Steady-State Response To consider a wide range of steady state operating conditions, the computed total harmonic distortion (THD) (see as Eq. (22), [3]) of ac current ias for different thyristor firing angles with a dc load resistance Rl=25 is shown in Fig. 4. As shown in Fig. 4, the ADP model gives inaccurate THD results at conditions with α>40°, which is due to the fixed-shape switch functions ([3, see Fig. 4]) that are different from the actual current waveforms. In contrast, the THD results are well predicted by the PDP model for the entire firing angle range, which verifies its accuracy in steady state conditions.
Fig. 4. Steady-state THD % of ias for different thyristor firing angles.
B. Large-Signal Transient Response To emulate various large-signal transients, the system (assumed initially in steady-state with Rl =150 and α=10°) is subjected to several load and firing angle changes: i) at t=0.05s, step change of Rl; ii) at t=0.15s, ramp change of Rl; iii) at t=0.35s, step change of α; and iv) at t=0.45s, ramp change of α. In addition, to illustrate the close-loop dynamic responses of the subject models, two thyristor firing control events are included: v) at t=0.65s, activation of the dc voltage controller; and vi) at t=0.75s, step change of the reference voltage. The details of these large-signal transients are also shown in Fig. 5 (a)(b). The close-loop thyristor firing controller is depicted in Fig. 8. The resulting system response of dc variables is depicted in Fig. 5 (c)(d), which shows noticeable error with the ADP model while excellent agreement between the PDP and detailed model results. This is further confirmed with the magnified views of ac variables vas and ias shown in Fig. 6. Specifically, it is noted in Fig. 6 (a) and (d) that as the system enters discontinuous conduction mode (DCM) due to increase of Rl or α, the ADP model fails to reproduce the distorted current waveforms (dips between double peaks), while the PDP model yields accurate waveforms as system operating condition evolves [see lines a) and c)]. It is also seen in Fig. 6 (a) and (c) that during the step-change transients, the ADP model can result in some dynamic errors due to the reduced-order representation of the ac network ([3, see Section IV-E]), which however do not occur with the PDP model. Moreover, Fig. 7 shows magnified views of system response to the activation/change of the dc voltage controller, which again demonstrate significant error with the ADP model [see line b)] and verify the consistency between the PDP and detailed models in close-loop dynamic responses. C. Frequency-Domain Study Finally, to demonstrate capability of the proposed model in the frequency-domain, a small-signal analysis is performed around the same initial steady-state operating point as in Section III-B. The open-loop transfer function F(s) is considered as vˆ ( s) , (6) F ( s) dc ˆ ( s)
where vˆdc ( s) is the change in the output dc voltage due to the small-signal perturbation in the input firing angle ˆ ( s) . The transfer functions as predicted by the subject models are depicted in Fig. 9, where a frequency-sweep technique has been used for the detailed model. As shown in Fig. 9, the ADP model shows inaccurate results for frequency higher than 20 Hz, while the transfer functions evaluated by the detailed and PDP models are in good agreement, especially in the range from 1 to 200 Hz. In general, due to the Fourier transformation of system variables [i.e. averaging of frequency components, as shown in (1)], the DP models should yield accurate frequency-domain response up to about 1/3 of the system switching frequency.
0885-8969 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TEC.2017.2735904, IEEE Transactions on Energy Conversion
Fig. 5. System response to the large-signal transients: (a) profile of Rl; (b) profile of α; (c) resulting dc voltage vdc; and (d) resulting dc current idc.
Fig. 7. Magnified views of system close-loop dynamic response to: (a) activation of the dc voltage controller; and (b) step change in reference voltage.
Fig. 8. The close-loop thyristor firing controller for dc voltage regulation: KP = 1 0.1, KI = 10, and H (s) , where a 45 106 , and b 14 103 . a 2 s bs 1
Fig. 9. Transfer function from the input firing angle to the output dc voltage as predicted by the subject models. IV. CONCLUSIONS
Fig. 6. Magnified views of ac variables vas and ias response to: (a) step change of Rl ; (b) ramp change of Rl ; (c) step change of α; (d) ramp change of α.
An extended PDP model of thyristor-controlled rectifier systems is presented in this letter. The case studies validate the effectiveness of the PDP model in predicting system dynamics including harmonics for various operating modes. V. REFERENCES [1]
When the disturbance frequency approaches the switching frequency or goes beyond, it would be expectant to observe deviation of results between the detailed and DP models, since the switching patterns of the detailed model may start changing and the basic assumptions of the generalized averaging are no longer valid. However, since the input is the firing angle command, the lower frequency dynamics are of more importance for studying stability and designing controllers for the system.
[2]
[3]
[4]
S. R. Sanders, J. M. Noworolski, X. Z. Liu, and G. C. Verghese, "Generalized averaging method for power conversion circuits," IEEE Trans. Power Electron., vol. 6, pp. 251–259, Apr. 1991 C. Liu, A. Bose, and P. Tian, “Modeling and analysis of HVDC converter by three-phase dynamic phasor,” IEEE Trans. Power Del., vol. 29, no.1, pp. 3-12, Feb. 2014. Y. Huang, L. Dong, S. Ebrahimi, N. Amiri, and J. Jatskevich, “Dynamic phasor modeling of line-commutated rectifiers with harmonics using analytical and parameter approaches,” IEEE Trans. Energy Convers., vol. 32, no.2, pp. 534-547, Jun. 2017. H. Atighechi, S. Chiniforoosh, K. Tabarraee, and J. Jatskevich, “Average-value modeling of synchronous-machine-fed thyristorcontrolled-rectifier systems,” IEEE Trans. Energy Convers., vol. 30, no.2, pp. 487-497, Jun. 2015.
0885-8969 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.