Thyristor-based excitation system state-space modeling for synchronous generator applications

Thyristor-based rectifier bridges

The principal references used for the basic understanding of the thyristor rectifier bridges comes from Mohan (2003), Kundur (1994) and Erickson R. (2004), as they all touch the subject, but Erickson R. (2004) goes more into detail. All of them advance the theory of the commutation equivalent resistance due to line voltage notches caused by the line inductance effects. The commutation equivalent resistance has also been used for modeling purposes by Chaĳarurnudomrung K.,K. N. A. (2010). The switching process is also thoroughly explained in Kundur (1994), Mohan (2003) and Erickson R. (2004), which gives the basic knowledge needed to make a thyristor rectifier bridge firing board. In Chaĳarurnudomrung K. (2010), the rectifier bridge is modeled as an averaged equivalent transformer model by doing the Fourier analysis on the square-shaped line current curves. The result is called the switching function, and can be used to obtain all the steady-state equations of the rectifier bridge given by Mohan (2003), Kundur (1994) and Erickson R. (2004).

The Park Transform

The Park transform is introduced in Kundur (1994) and Krause (2002), but the transform matrices that they use are not the same. For this research, the Park transform matrix used is the one that comes from Krause (2002). Since grids are not always perfectly balanced, properties of the Park transform under unbalanced conditions must be studied. The thesis Dupré (2019) does it, and explains how two sinusoidal signals of same amplitude shifted by 90 degrees can be injected in the inverse transform to obtain a clear harmonics-free negative sequence on the three phase side of the transform. The theory of how the Park transform can be used to transfer differential equations into qd-frame is given in both Kundur (1994) and Krause (2002). Since a phase-angle reference is required for the Park transform, a synchronization device has to be used such as a PLL.

Synchronization devices

The book Best (2003) gives basic understanding of phase-locked loops, and is a good entry-level source on the subject. The author actually explains how and why mathematical models are made for such devices. The performance characteristics needed for a PLL to recover the phase angle properly are given, and the trade-off of the bandwidth versus the response time is explained. Also, the book has a chapter on how to incorporate filters into the loop and stabilize the devices through control problems solving. However, the phase detector used in the book are not what was used in the research simply because the qd-frame phase detector fits better in the body of the research, introducing the need for qd-frame signal analysis.

Two PLL devices are to be designed in the research. One of them uses the usual abc-frame signals and is the Unified Three-Phase Signal Processor (UTSP), first introduced in Karimi (2008), then improved and simplified in Karimi H.,Y. S. (2019). The UTSP is a PLL that is actually able to decouple the sequences of a three-phase signal, and give their angles and amplitudes separately. The other one uses Park transform as a phase detector, meaning that the signals going in the filter are in qd-frame. The article Karimi H. (2012) uses a combination of Clarke and Park transforms as a phase detector since the PLL is to be used for single-phase synchronization. The synchronization device of this research needs to be for three-phase circuits applications, so the Clarke transform was removed. However, the stability analysis method is clearly explained, and thus the filter can be adapted for lower frequency harmonics applications. The line inductance is also assumed to be of high value in the physical applications of the excitation system model, since the excitation transformer is design to attenuate short-circuits, and has high leakage inductance values, therefore notches are always present on the three-phase signals entering the rectifier bridge’s synchronization system.

To properly design a synchronization system, the input signals have to be correctly characterized to design the filter accordingly, which must be designed before the loop controller can stabilize the whole thing and set the performance characteristics. The book Corinthios (2009) gives all the knowledge needed for Fourier analysis as well as filter design, with the mathematical background needed to understand exactly how the Fourier analysis can be conducted on the bridge input phase voltages. The article Graham (1993) gives the method for a Fourier analysis done of the input bridge voltage. The Fourier analysis is done on a trapezoidal-shaped line currents (in presence of line voltage notches), then the equivalent impedance seen by the bridge is used to calculate the voltage drop in the equivalent inductance to find the input bridge voltage frequency characterstics. The results given by the Fourier analysis had to be transferred into qd-frame. The only article found on the Park transform’s frequency domain properties Zhang B.,Yi S. (2000), but the results are not proven in any way, and the method’s basic hypothesis are not exactly clearly given. Still, this article gave away the idea that some work could be done on the properties of the Park transform when used on signals containing harmonics.

Modeling of the excitation system

In Krause (2002) and Kundur (1994), the excitation systems are explained, setting the hypothesis used for the excitation system model creation. Those references actually tell us that a transformer is in the excitation loop, as well as a harmonic filter, telling us the exact topology of the excitation circuit used to build the mathematical model. Since the synchronous machine modeling is always in qd-frame, we will use the Park transform to model the whole excitation system. The book Balabanian N. (1969) gives strong techniques for state-space modeling of complex systems called circuit tearing. The technique tells us that the separate subsystems can be modeled and validated separately, before being put together in a functioning global model. Also, the state-space representation modeling method is thoroughly explained in Balabanian N. (1969), as well as the meaning of such mathematical knowledge on a physical system. The state-space model of the excitation system can be used to extract eigenvalues (or closed-loop poles) of the system, much needed for the control system design of the excitation system. It can also be used to conduct transient analysis on the system, which is how the model will be validated.

Table des matières

INTRODUCTION
CHAPTER 1 LITERATURE REVIEW
CHAPTER 2 THYRISTOR-BASED SIX-PULSE RECTIFIER BRIDGE THEORY
2.1 Three-phase rectifier bridge ideal switching behavior
2.2 Expression of the switching function of the rectifier bridge
2.3 Steady-state equations equations in abc reference frame
2.4 Line inductance effects on the rectifier bridge
2.5 Rectifier bridge input phase voltages and line currents Fourier analysis
2.5.1 Conclusions on thyristor-based six-pulse rectifier bridge theory
CHAPTER 3 THE PARK’S TRANSFORM – SYNCHRONOUS REFERENCE FRAME
3.1 Definition of the Park’s transform matrix
3.2 Expressing parameters into the Park’s referential frame
3.2.1 Observed parameters under unbalanced three-phase signal conditions
3.2.2 Observed parameters for a three-phase positive-sequence signal with harmonics
3.2.3 Observed parameters for a harmonics-polluted three-phase negativesequence signal
3.2.4 Inverse transform properties
3.3 Power expressed into Park’s referential
3.4 Conclusions on synchronous reference frame transform
CHAPTER 4 SYNCHRONIZATION DEVICES
4.1 Synchronization systems input voltage input signals
4.1.1 Input signals characterization of the UTSP
4.1.2 Input signals characterization of the qd-frame PLL
4.2 qd0-PLL model derivation and design
4.2.1 Mathematical model derivation of the qd0-PLL
4.2.2 Root-locus drawing of the qd0-PLL
4.2.2.1 Root-locus drawing of the qd0-PLL without filter
4.2.2.2 Root-locus drawing of the qd0-PLL with first order low-pass filter
4.2.2.3 Root-locus drawing of the qd0-PLL with second order low-pass filter
4.2.3 Qd0-PLL design
4.3 UTSP model derivation and design
4.3.1 UTSP design
4.4 UTSP and qd0-PLL comparison
4.5 Conclusions on proposed PLLs as synchronization devices for excitation systems
CHAPTER 5 EXCITATION SYSTEM STATE-SPACE MODELING USING PARK’S TRANSFORM
5.1 Transformer state-space model expressed in per units
5.2 Transformer state-space model validation with simulation tools
5.3 Line voltage notch filter state-space model
5.4 Line voltage notch filter state-space model validation with simulation tools
5.5 Rectifier bridge state-space model
5.6 Rectifier bridge state-space model validation with simulation tools
5.7 Complete excitation system state-space model representation
5.8 Conclusions on the state-space model of the excitation system
CHAPTER 6 VALIDATION OF THE COMPLETE EXCITATION SYSTEM STATE-SPACE MODEL
6.1 Experimental state-space model validation method
6.2 Firing angle set-point and perturbation measurement
6.2.1 Firing angle set-point measurement method
6.2.2 Firing angle perturbation amplitude measurement method
6.3 Experiments at low perturbation amplitude
6.3.1 Simulations model performances under discontinuous conduction mode with low perturbation amplitude
6.3.2 Simulation models performances for high frequency perturbations with low perturbation amplitude
6.3.3 Simulation models performances during continuous conduction and low perturbation frequencies with low perturbation amplitudes
6.4 Discussion on experimental results
6.4.1 Effects of the on-state resistance of the thyristors on the output voltage
6.4.2 Effects of the voltage input imbalance on the DC-side voltage and currents
6.4.3 Effects of the firing angle command update maximum frequency of the rectifier bridge
6.5 Discussion on the state-space model validity
6.5.1 Synchronization system effects on the rectifier bridge dynamics
6.5.2 Firing angle perturbation measurement method
6.5.3 Rectifier bridge dynamics with added impedance on the secondary side of the transformer
6.5.4 Simulation errors and instability
6.6 Conclusions on the state-space model validity
CHAPTER 7 DISCUSSION ON THE FIRST ORDER APPROXIMATION
7.1 Effects of the transformer mutual inductance
7.2 Effects of the transformer leakage inductances
7.3 Effects of the transformer winding resistances
7.4 Conclusions on the first-order approximation
CONCLUSION