A modified sequence domain impedance definition and its equivalence to the dq-domain impedance definition for the stability analysis of AC power electronic systems

Representations of AC power systems by frequency dependent impedance equivalents is an emerging technique in the dynamic analysis of power systems including power electronic converters. The technique has been applied for decades in DC-power systems, and it was recently adopted to map the impedances in AC systems. Most of the work on AC systems can be categorized in two approaches. One is the analysis of the system in the \textit{dq}-domain, whereas the other applies harmonic linearization in the phase domain through symmetric components. Impedance models based on analytical calculations, numerical simulation and experimental studies have been previously developed and verified in both domains independently. The authors of previous studies discuss the advantages and disadvantages of each domain separately, but neither a rigorous comparison nor an attempt to bridge them has been conducted. The present paper attempts to close this gap by deriving the mathematical formulation that shows the equivalence between the \textit{dq}-domain and the sequence domain impedances. A modified form of the sequence domain impedance matrix is proposed, and with this definition the stability estimates obtained with the Generalized Nyquist Criterion (GNC) become equivalent in both domains. The second contribution of the paper is the definition of a \textit{Mirror Frequency Decoupled} (MFD) system. The analysis of MFD systems is less complex than that of non-MFD systems because the positive and negative sequences are decoupled. This paper shows that if a system is incorrectly assumed to be MFD, this will lead to an erroneous or ambiguous estimation of the equivalent impedance

Dynamic Phasor Modeling of Type 3 Wind Farm including Multi-mass and LVRT Effects

The proportion of power attributable to wind generation has grown significantly in the last two decades. System impact studies such as load flow studies and short circuit studies, are important for planning before integration of any new wind generation into the existing power grid. Short circuit modelling is central in these planning studies to determine protective relay settings, protection coordination, and equipment ratings. Numerous factors, such as low voltage situations, power electronic switching, control actions, sub-synchronous oscillations, etc., influence the response of wind farms to short circuit conditions, and that makes short circuit modelling of wind farms an interesting, complex, and challenging task. Power electronics-based converters are very common in wind power plants, enabling the plant to operate at a wide range of wind speeds and provide reactive power support without disconnection from the grid during low voltage scenarios. This has led to the growth of Type 3 (with rotor side converter) and Type 4 (with stator side full converter) wind generators, in which power electronics-based converters and controls are an integral part. The power electronics in these generators are proprietary in nature, which makes it difficult to obtain the necessary information from the manufacturer to model them accurately in planning studies for conditions such as those found during faults or low voltage ride through (LVRT) periods. The use of power electronic controllers also has led to phenomena such as sub-synchronous control interactions in series compensated Type 3 wind farms, which are characterized by non-fundamental frequency oscillations. The above factors have led to the need to develop generic models for wind farms that can be used in studies by planners and protection engineers. The current practice for short circuit modelling of wind farms in the power industry is to utilize transient stability programs based on either simplified electromechanical fundamental frequency models or detailed electromagnetic time domain models. The fundamental frequency models are incapable of representing the majority of critical wind generator fault characteristics, such as during power electronic switching conditions and sub-synchronous interactions. The detailed time domain models, though accurate, demand high levels of computation and modelling expertise. A simple yet accurate modelling methodology for wind generators that does not require resorting to fundamental frequency based simplifications or time domain type simulations is the basis for this research work. This research work develops an average value model and a dynamic phasor model of a Type 3 DFIG wind farm. The average value model replaces the switches and associated phenomena by equivalent current and voltage sources. The dynamic phasor model is based on generalized averaging theory, where the system variables are represented as time varying Fourier coefficients known as dynamic phasors. The two types models provide a generic type model and achieve a middle ground between conventional electromechanical models and the cumbersome electromagnetic time domain models. The dynamic phasor model enables the user to consider each harmonic component individually; this selective view of the components of the system response is not achievable in conventional electromagnetic transient simulations. Only the appropriate dynamic phasors are selected for the required fault behaviour to be represented, providing greater computational efficiency than detailed time domain simulations. A detailed electromagnetic transient (EMT) simulation model is also developed in this thesis using a real-time digital simulator (RTDS). The results obtained with the average value model and the dynamic phasor model are validated with an accurate electromagnetic simulation model and some state-of-the-art industrial schemes: a voltage behind transient reactance model, an analytical expression model, and a voltage dependent current source model. The proposed RTDS models include the effect of change of flux during faulted conditions in the wind generator during abnormal system conditions instead of incorrectly assuming it is a constant. This was not investigated in previous studies carried out in the real-time simulations laboratory at the University of Saskatchewan or in various publications reported in the literature. The most commonly used LVRT topologies, such as rotor side crowbar circuit, DC-link protection scheme, and series dynamic braking resistance (SDBR) in rotor and stator circuits, are investigated in the short circuit studies. The RTDS model developed uses a multi-mass (three-mass) model of the mechanical drive train instead of a simple single-mass model to represent torsional dynamics. The single mass model considers the blade inertia, the turbine hub, and the generator as a single lumped mass and so cannot reproduce the torsional behaviour. The root cause of sub-synchronous frequencies in Type 3 wind generators is not well understood by system planners and protection engineers. Some literature reports it is self excitation while others report it is due to sub-synchronous control interactions. One publication in the stability literature reports on a small signal analysis study aimed at finding the root cause of the problem, and a similar type of analysis was performed in this thesis. A linearized model was developed, which includes the generator model, a three mass drive train, rotor side converter, and the grid side converter represented as a constant voltage source. The linear model analysis showed that the sub-synchronous oscillations are due to control interactions between the rotor side controller of the Type 3 wind power plant and the series capacitor in the transmission line. The rotor side controls were tuned to obtain a stable response at higher levels of compensation. A real-time simulation model of a 450 MW Type 3 wind farm consisting of 150 units transmitting power via 345 kV transmission line was developed on the RTDS. The dynamic phasor method is shown to be accurate for representing faults at the point of interconnection of the wind farm to the grid for balanced and unbalanced faults as well as for different sub- synchronous oscillation frequencies

Short circuit modeling of wind turbine generators

Modeling of wind farms to determine their short circuit contribution in response to faults is a crucial part of system impact studies performed by power utilities. Short circuit calculations are necessary to determine protective relay settings, equipment ratings and to provide data for protection coordination. The plethora of different factors that influence the response of wind farms to short circuits makes short circuit modeling of wind farms an interesting, complex, and challenging task. Low voltage ride through (LVRT) requirements make it necessary for the latest generation of wind generators to be capable of providing reactive power support without disconnecting from the grid during and after voltage sags. If the wind generator must stay connected to the grid, a facility has to be provided to by-pass the high rotor current that occurs during voltage sags and prevent damage of the rotor side power electronic circuits. This is done through crowbar circuits which are of two types, namely active and passive crowbars, based on the power electronic device used in the crowbar triggering circuit. Power electronics-based converters and controls have become an integral part of wind generator systems like the Type 3 doubly fed induction generator based wind generators. The proprietary nature of the design of these power electronics makes it difficult to obtain the necessary information from the manufacturer to model them accurately. Also, the use of power electronic controllers has led to phenomena such as sub-synchronous control interactions (SSCI) in series compensated Type 3 wind farms which are characterized by non-fundamental frequency oscillations. SSCI affects fault current magnitude significantly and is a crucial factor that cannot be ignored while modeling series compensated Type 3 wind farms. These factors have led to disagreement and inconsistencies about which techniques are appropriate for short circuit modeling of wind farms. Fundamental frequency models like voltage behind transient reactance model are incapable of representing the majority of critical wind generator fault characteristics such as sub-synchronous interactions. The Detailed time domain models, though accurate, demand high levels of computation and modeling expertise. Voltage dependent current source modeling based on look up tables are not stand-alone models and provide only a black-box type of solution. The short circuit modeling methodology developed in this research work for representing a series compensated Type 3 wind farm is based on the generalized averaging theory, where the system variables are represented as time varying Fourier coefficients known as dynamic phasors. The modeling technique is also known as dynamic phasor modeling. The Type 3 wind generator has become the most popular type of wind generator, making it an ideal candidate for such a modeling method to be developed. The dynamic phasor model provides a generic model and achieves a middle ground between the conventional electromechanical models and the cumbersome electromagnetic time domain models. The essence of this scheme to model a periodically driven system, such as power converter circuits, is to retain only particular Fourier coefficients based on the behavior of interest of the system under study making it computationally efficient and inclusive of the required frequency components, even if non-fundamental in nature. The capability to model non-fundamental frequency components is critical for representing sub-synchronous interactions. A 450 MW Type 3 wind farm consisting of 150 generator units was modeled using the proposed approach. The method is shown to be highly accurate for representing faults at the point of interconnection of the wind farm to the grid for balanced and unbalanced faults as well as for non-fundamental frequency components present in fault currents during sub-synchronous interactions. Further, the model is shown to be accurate also for different degrees of transmission line compensation and different transformer configurations used in the test system

Comprehensive STATCOM Control For Distribution And Transmission System Applications

This thesis presents the development of a comprehensive STATCOM controller for load compensation, voltage regulation and voltage balancing in electric power distribution and transmission networks. The behavior of this controller is first validated with published results. Subsequently, the performance of this STATCOM controller is examined in a realistic Hydro One distribution feeder for accomplishing the compensation of both mildly and grossly unbalanced loads, and balancing of network voltages using PSCAD/EMTDC software. The STATCOM voltage control function is utilized for increasing the connectivity of wind plants in the same distribution feeder. The thesis further presents a frequency scanning technique for simple and rapid identification of the potential of subsynchronous resonance in induction generator based wind farms connected to series compensated lines, utilizing MATLAB software. This technique is validated by published eigenvalue analysis results. The voltage control performance of the developed comprehensive STATCOM controller is then demonstrated for different scenarios in the modified IEEE First SSR Benchmark transmission system for mitigating subsynchronous resonance in series compensated wind farms using industry grade PSCAD/EMTDC software

Generalised dq-dynamic phasor modelling of STATCOM connected to a grid for stability analysis

The synchronous dq based small-signal stability using the eigenvalue analysis and impedance methods is widely employed to assess system stability. Generally, the harmonics are ignoredin stability analysis which may lead to inaccuracies in stability predictions, particularly, when the system operates in a harmonic-richenvironment. Typically, the harmonic state-space method (HSS) facilitates stability studies of linear time-periodic (LTP) systems, which considersthe impact of harmonics. The use of the dq-dynamic phasor state space and impedance method offers significant advantages over the HSS counterpart, as it reduces system order, is more suitable for studying control systems, retains mutual coupling of harmonics, and simplifies the stability study under unbalanced conditions. This paper extends dynamic phasor modelling for studying stability of modern power systems that include power converters. It is shown that the proposed method reproduces the typical response of STATCOM at the fundamental frequency as well as at significant low-order harmonics using both eigenvalues and impedance analysis. Quantitative validations of the proposed extended models against synchronous dq small signal models confirm their validity

Frequency-Domain Control Design in Power Systems

The scope of this thesis encompasses two main subjects: fixed-structure data-driven control design on one side, and control design in power systems on the other. The overall goal is to identify challenging and relevant problems in power systems, to express them as rigorous specifications from the viewpoint of control systems, and to solve them by developing and applying advanced methods in robust control. This work aims to combine expertise from both fields to open up a holistic perspective and bridge the gap between control and power systems. First, the derivation of a novel fixed-structure, data-driven frequency-domain control design method for multivariable systems is described. A key feature of the method is that only the frequency response of the plant is required for the design, and no parametric model is required. The designed controllers are fully parametrized in terms of matrix polynomial functions and can take a centralized, decentralized or distributed structure. The controller performance is formulated as H_2 and H_infinity constraints on any loop transfer function. A convex formulation of the optimization problem is derived, and it is shown that the solution converges to a locally optimal solution of the original problem. The versatility of the design method is demonstrated in various simulation examples, as well as in experiments on two electromechanical setups. Next, a frequency-domain modeling approach for power grids is discussed. A model based on dynamic phasors is developed that represents the electromagnetic and electromechanic dynamics of lines, inverters, synchronous machines and constant power loads. It also offers a modular structure that makes it straightforward to combine white-, grey- and blackbox models in a single framework. Then, the control design method and dynamic phasor model are applied in two relevant power systems case studies. First, the design of a decentralized current controller for parallel, grid-connected voltage source inverters in a typical distribution grid is considered. It is shown how performance specifications can be formulated as frequency-domain constraints in order to attenuate the resonances introduced by the output filters and coupling effects, and to provide robustness against model uncertainties and grid topology changes. The controllers for all VSIs are designed in a single step, and stability and performance is guaranteed by design. Furthermore, an approach for plug-and-play control design is presented. The results are validated in numerical simulation as well as in power-hardware-in-the-loop experiments. The second study concerns the design of a distributed controller that combines primary and secondary frequency and voltage control for an islanded, meshed low-voltage grid with any number of voltage source inverters and synchronous generators in a single framework. No assumption on the R/X-ratio of the lines is made, and it is shown how advanced control specifications such as proportional active power sharing, zero frequency steady-state error and decoupling can be formulated as constraints on the norm of weighted sensitivity functions. Furthermore, the communication delays of the distributed controller are considered exactly during the design. The controller is implemented in numerical simulation, and results show significantly improved performance as compared to the classical hierarchical structure