As traditional control schemes, open-loop Hysteresis and closed-loop pulse-width-modulation (PWM) have been used for the switched reluctance motor (SRM) current controller. The Hysteresis controller induces large unpleasant audible noises because it needs to vary the switching frequency to maintain constant Hysteresis current band. In contract, the PWM controller is very quiet but difficult to design proper gains and control bandwidth due to the nonlinear nature of the SRM. In this thesis, the ac small signal modeling technique is proposed for linearization of the SRM model such that a conventional PI controller can be designed accordingly for the PWM current controller. With the linearized SRM model, the duty-cycle to output transfer function can be derived, and the controller can be designed with sufficient stability margins. The proposed PWM controller has been simulated to compare the performance against the conventional Hysteresis controller based system. It was found that through the frequency spectrum analysis, the noise spectra in audible range disappeared with the fixed switching frequency PWM controller, but was pronounced with the conventional Hysteresis controller. A hardware prototype is then implemented with digital signal processor to verify the quiet nature of the PWM controller when running at 20 kHz switching frequency. The experimental results also indicate a stable current loop operation. / Master of Science

It can be expected that digital controllers will be increasingly used in low voltage, high-current and high frequency voltage regulator modules (VRMs) where conventional analog controllers are currently preferred because of the cost and performace reasons. However, there are still remaining two significant challenges for the spread of the digital control techniques: quantization effects and the delay effects. Quantization effects might introduce the limit cycle oscillations (LCOs) to the converter, which will generate the stability issues. Actually, LCOs can not be totally eliminated theoretically. One way to reduce the possibilities of LCOs is to employ a high resolution Digital Pulse-Width-Modulator (DPWM). However, designing such a DPWM which can meet the requirements of VRMs application requires ultra-high system clock frequency, up to several GHz. Such high frequency is impractical due to huge power consumption. Hybrid DPWM might be an alternative solution but will occupy large silicon area. Single phase digital constant on-time modulation method is another good candidate to improve the DPWM resolution without adding too much cost. However, directly extending this method to multi-phase application, which is the prevalent structure in VRMs application, will introduce some issues. With more phases in parallel, the duty cycle resolution will drop more. To solove the mentioned issue, this work proposed a multi-phase digital constant on-time modulation method. The proposed method will control the control voltage to alternate between two adjacent values, or dither, within one switching period. The outcome is that the phase duty cycle's resolution is improved and independent on phase number. Compared with conventional constant frequency modulation method, the proposed method can achieve about 10 times higher duty cycle resolution for the VRM application. The effectiveness of the proposed method is verified by the simulation as well as the experiment results. Delay effect is another concern for the digital controlled VRMs. There exist several types of delays in the digital feedback loop, including the ADC conversion delay, digital compensator calculation delay, DPWM delay as well as some propagation delays. Usually these delays are inside the digital controller and it is hard to know the exact values. There are several papers talking about the small signal models of the digital voltage mode control. These models are valid only if all the delay terms are known exactly since each delay is considered separately. Actually, this process is not easy. Moreover, there is no literature talking about the complete small signal model of the digital VRMs. But in reallity, different implementations of the sampling process will give different impacts to the loop. This work proposed the small signal signal models of digital VRMs. The analysis is based on the assumptions that DPWM is a double-edge modulation and the sampling instants are aligned with the middle of one phase's off time. At first, the conversion and calculation delay is neglected. The focus of the modeling is on the small signal model of the current sampling methods and the DPWM delay. This model is valid for those digital controllers which have fast ADC and fast calculation capabilities. It is shown that even with a "fast" controller, the current sampling and DPWM might introduce some delay to the loop. After that, the conversion and calculation delay are considered into the modeling. Two time periods, T1ff and T1rr, are employed to describe the total delay effects in the control loop. It is observed that the total delay in the loop is integral times of sampling periods, which is never reported by any other literatures. Therefore, the proposed model only includes one delay term and the value of this delay can be found through a pre-determined lookup table. Finally, the complete small signal model of the digital VRMs considering the conversion and calculation delay is proposed. This model is helpful for the researchers to find the delay effects in their control loop based on the range of the total physical delay in the controller. With the derived small signal mondels of digital VRMs, the design guildeline for AVP control are presented. The digital active-droop control is employed and it borrows the concept of constant output impedance control from the analog world. Two design examples are provided for the verification. / Master of Science

The widespread integration of power converters in applications such as microgrids and data centers has introduced significant stability challenges. This dissertation presents a novel approach to modeling and comprehensive stability analysis for both single-phase and three-phase converters, addressing vital gaps in the existing literature. The first part of the dissertation (Chapters 2 to 4) focuses on single-phase power supply units, proposing an impedance model and a loop gain model based on dq-frame analysis. These models have been validated through extensive experimental testing, demonstrating their effectiveness in stability analysis across a range of system configurations, including single-phase, three-phase three-wire, and three-phase four-wire systems. The second part (Chapters 5 and 6) examines three-phase converters used for integrating renewable energy into microgrids. It introduces a grid-forming control, followed by a detailed investigation into its impedance modeling and stability assessment. This part specifically tackles the challenges posed by the appearance of right-half-plane poles in stability analysis, proposing a new stability margin index to address these issues. The efficacy of these research findings is further substantiated by the development and implementation of a Power-Hardware-in-the-Loop testbed, providing practical validation. Overall, this dissertation has enhanced the modeling, understanding, and management of stability issues in power electronics systems, offering valuable insights and methodologies that are likely to influence future research and development in the field. / Doctor of Philosophy / Power electronics play a crucial role in many of today's advanced technologies, including Renewable Energy (like wind and solar power), Electric Vehicles, Cloud Computing, and Artificial Intelligence. In renewable energy, power electronics are key for converting energy sources for efficient grid integration. Electric vehicles rely on power converter systems for charging their batteries and driving their motors. Similarly, in Cloud Computing and Artificial Intelligence, power electronics ensure that the computers and servers in data centers have a steady and reliable power supply for operation. However, using these advanced power electronics on a large scale, like in wind farms or data centers, can lead to challenges, including many reported system instability issues. These issues highlight the importance of a thorough analysis and understanding of the behavior and interaction of power electronics systems. In addressing these challenges, power electronics converters, conceptualized as a blend of circuits and control systems, demand comprehensive modeling from the ground up. Such modeling is essential to understanding their behavior, ranging from individual components to the entire system. This is key to establishing a clear connection between intricate design details and overall system performance. With power electronics systems becoming more complex and the continual emergence of new technologies, there remains a significant array of unanswered questions, especially in the domain of stability analysis for AC power electronics systems. This dissertation delves into two prominent modeling methods for stability analysis: impedance modeling and loop gain modeling. By exploring and addressing specific gaps identified in prior research, this work aims to contribute to a more profound understanding and enhanced application of these critical methods. The research presented in this dissertation is methodically divided into two main sections. The first section, including Chapter 2 to Chapter 4 is dedicated to exploring single-phase converter power supply units (PSUs) systems. This section introduces innovative models for analyzing their stability, applicable to single-phase PSUs in various system configurations, including both single-phase and three-phase setups. This modeling approach is a significant step forward in understanding and enhancing the stability of single-phase PSU loads. The second section, including Chapter 5 and Chapter 6, delves into the analysis of three-phase converters used in integrating renewable energy sources into microgrids. A notable feature of these converters is their grid-forming control mechanism, which includes a new frequency and power droop control loop. This part also explores modeling the impact of these converters on microgrid stability. Moreover, the issue of right-half-plane (RHP) poles in impedance analysis- a complex problem that can affect stability analysis is addressed. It proposes innovative methods for measuring stability in such conditions. In conclusion, this research made advancements in the modeling for stability analysis of power converter systems. For single-phase converters, the developed impedance model and loop gain model, based on dq-frame analysis, have been proven to be accurate. These models are versatile for stability analysis in various AC systems with single-phase PSU loads. In the study of three-phase converters, the grid-forming converter was successfully designed to support the grid as a distributed energy resource interface. This design contributes positively to microgrid stability. Furthermore, to address the presence of RHP poles in stability analysis, a new stability margin index was defined to better understand and manage these challenges. These findings represent important steps forward in the field of power electronics and contribute valuable insights for future research and development.

The purpose of this investigation is to identify appropriate location of capacitor banks and sources of reactive power by studying power system stability in the vicinity of system equilibrium states. The locations for reactive power sources are determined by identifying those modes of the system that participate most in the system behavior in general and in dictating the final state of the system after experiencing faults or disturbances. To identify the relevant modes of the system that participate most in the system dynamic, we shall make use of modal and participation analysis for different system conditions. We also apply modal and participation analysis to a system in order to identify the components of greatest impact that result in the most efficient system control. The ideas developed in this study are used to analyze and identify weak boundaries of the IEEE 39- Bus system that contribute to the system’s instability.

O presente trabalho propõe a utilização conjunta dos conceitos de modelagem politópica e estabilidade quadrática para avaliação da robustez de desempenho de estabilizadores de sistemas de potência (ou PSSs, do inglês, Power System Stabilizers). Controladores de amortecimento do tipo PSS têm sido amplamente utilizados em sistemas elétricos de potência desde o final da década de 60. A maioria destes estabilizadores que hoje estão em operação foi projetada segundo uma abordagem clássica, que envolve a linearização das equações do sistema em torno de um ponto de equilíbrio e controle através de um compensador de avanço-atraso de fase. Este procedimento de projeto é bastante difundido devido à facilidade do uso de tais técnicas e ao baixo custo de implementação. No entanto, uma das principais desvantagens inerentes a essa abordagem vem justamente da linearização, pois a validade do controle projetado fica restrita a uma vizinhança do ponto de operação no qual o sistema foi linearizado. Sendo assim, não há garantia formal de desempenho satisfatório do controlador, uma vez que as condições operativas do sistema variam normalmente ao longo do dia. Mesmo que o desempenho seja verificado, após o projeto, para pontos de operação diferentes daquele no qual foi feito a linearização (procedimento que é tipicamente empregado em estudos de estabilidade a pequenas perturbações), o mesmo estará garantido formalmente apenas nas proximidades dos pontos verificados. A presente pesquisa busca o preenchimento desta lacuna referente à falta de garantia formal de desempenho em condições não nominais de operação. Com o intuito de garantir formalmente a robustez de desempenho dos controladores, utilizou-se o conceito de estabilidade quadrática associado a uma modelagem politópica do sistema de potência para verificação do fator de amortecimento mínimo dentre todos os modos de oscilação do sistema (o qual é usualmente adotado em sistemas de potência como critério de desempenho ou, equivalentemente, como indicador de margem de estabilidade a pequenas perturbações). A modelagem politópica é usada como alternativa para a obtenção de um modelo de sistema dinâmico que leva em conta as incertezas referentes ao ponto de operação. Neste tipo de modelagem, ao invés de se considerar apenas um ponto de operação nominal, leva-se em conta um conjunto particular de pontos de operação típicos do sistema (os quais comporão os vértices de um conjunto convexo, chamado de politopo). Posteriormente, com base no conceito de estabilidade quadrática, pode-se garantir que um controlador projetado para garantir um desempenho mínimo aos vértices de um politopo estenderá tal garantia também a qualquer ponto de operação que tiver uma descrição linearizada pertencente a este politopo. Os resultados obtidos demonstram que a associação desses dois conceitos fornece uma alternativa viável e vantajosa para a avaliação da robustez de estabilidade e desempenho em sistemas de potência. O procedimento proposto pode ser usado de maneira complementar ao cálculo de autovalores tipicamente empregado na indústria, estendendo a garantia formal de robustez a um conjunto mais amplo de pontos de operação. / The present work proposes the joint use of polytopic modeling and quadratic stability concepts to evaluate the performance robustness of power systems stabilizers (or PSSs). PSS-type damping controllers have been widely used in electric power systems since the end of 6th decade of this century. The majority of these stabilizers, which are in operation nowadays, were designed according to a classical control approach. This method involves linearization of the system equations around an equilibrium point and control through a lead-lag phase compensator. This procedure has a widespread application in power systems due to the simplicity of the technique and the low implementation cost. However, one of the main disadvantages inherent to this method lies exactly in the linearization, since the validity of the designed control is restricted to a neighborhood of the operation point in which the linearization has been done. Since the system operating condition changes throughout the day, we cannot have a formal guarantee of a satisfactory controller performance. Even if the controller performance is checked for different operating points after the design, the performance will be formally guaranteed only in the neighborhoods of the verified points. The present research aims to fill this gap associated to the lack of a formal performance guarantee in an off-nominal operation condition. With the objective of formally guaranteeing the controller performance, the concept of quadratic stability, associated to a polytopic modeling of the system, was used to check the minimum damping factor among all system modes of oscillation (which is usually adopted in power systems as a performance criteria or, equivalently, as an index of small-signal stability margin). The polytopic modeling is used as an alternative to obtain the dynamic system model that accounts for the uncertainty in the operating point. In this type of modeling, instead of considering only one nominal operating point, a particular set of typical system operating points is chosen (which will compose the vertices of a convex set, called polytope). Later, based on the quadratic stability concept, it is possible to guarantee that a controller designed to achieve a minimum performance index at the vertices of the polytopic set will extend this property to any operation point belonging to this set. The obtained results show that the association of these two concepts provides a viable and advantageous alternative for the evaluation of the stability and performance robustness in power systems. The proposed procedure can be used as a complement to the eigenvalue calculation used in the industry, extending the formal robustness guarantee to a broader set of operating points.

Orientador: Percival Bueno de Araujo / Banca: Laurence Duarte Colvara / Banca: Gideon Villar Leandro / Resumo: O principal objetivo deste trabalho é a utilização do modelo de injeção de potência do dispositivo FACTS TCSC (Thyris-tor Controlled Series Capacitor) na análise da estabilidade a pequenas perturbações de sistemas elétricos de potência. Para atingir este objetivo é deduzido o modelo de injeção de potência do TCSC, cujo equacionamento é adicionado ao Modelo de Sensibilidade de Po-tência (MSP), utilizado para representar o sistema elétrico de potência (SEP). Para o amortecimento das oscilações eletromecânicas de baixa frequência do SEP são utilizados dois modelos para os controladores, um que considera somente um ganho proporcional e outro comumente chamado na literatura de controlador suplementar de amortecimento (POD - Power Oscillation Damping) que contém também blocos de avanço- atraso de fase. Ambos os controladores devem atuar em conjunto com o TCSC para fornecer amortecimento ao SEP. Neste trabalho o sinal de entrada para os dois controladores é a variação da potência ativa na linha de transmissão de instalação do TCSC e seus parâmetros são ajustados de duas formas: pelo método dos resíduos e utilizando o toolbox rltool (SISO) do software MATLAB. Para a validação do equacionamento desenvolvido foram realizadas simulações em um sistema de potência simétrico, de duas áreas, composto de 4 geradores e 10 barras / Abstract: This work presents a power injection model for the Thyristor Controlled Series Compensator (TCSC), a Flexible AC Transmission Systems (FACTS) device, for small signal stability analysis in the electric power systems. To achieve such goal, the TCSC injection power model equations are summed up to the Power Sensitivity Model (PSM) which is used to represent the electric power system (EPS). For the low frequency electromechanical oscillation damping, two models are used to represent the controller: 1. A proportional control and; 2. A supplementary control known as Power Oscillation Damping (POD), which also comprehends lead-lag blocks. Both controllers, previously cited, must work together with the TCSC to damp oscillations in the EPS. In this work, the input signal for both controllers is the real power flow variation in the transmission line where the TCSC is placed and the controllers parameters are adjusted using the residues method and the Matlab toolbox rltool (SISO). Several simulations in a symmetrical, two areas power system, composed of four generators and ten busses, are provided in way to validate the power injection model and are discussed in this work / Mestre

O presente trabalho propõe a utilização conjunta dos conceitos de modelagem politópica e estabilidade quadrática para avaliação da robustez de desempenho de estabilizadores de sistemas de potência (ou PSSs, do inglês, Power System Stabilizers). Controladores de amortecimento do tipo PSS têm sido amplamente utilizados em sistemas elétricos de potência desde o final da década de 60. A maioria destes estabilizadores que hoje estão em operação foi projetada segundo uma abordagem clássica, que envolve a linearização das equações do sistema em torno de um ponto de equilíbrio e controle através de um compensador de avanço-atraso de fase. Este procedimento de projeto é bastante difundido devido à facilidade do uso de tais técnicas e ao baixo custo de implementação. No entanto, uma das principais desvantagens inerentes a essa abordagem vem justamente da linearização, pois a validade do controle projetado fica restrita a uma vizinhança do ponto de operação no qual o sistema foi linearizado. Sendo assim, não há garantia formal de desempenho satisfatório do controlador, uma vez que as condições operativas do sistema variam normalmente ao longo do dia. Mesmo que o desempenho seja verificado, após o projeto, para pontos de operação diferentes daquele no qual foi feito a linearização (procedimento que é tipicamente empregado em estudos de estabilidade a pequenas perturbações), o mesmo estará garantido formalmente apenas nas proximidades dos pontos verificados. A presente pesquisa busca o preenchimento desta lacuna referente à falta de garantia formal de desempenho em condições não nominais de operação. Com o intuito de garantir formalmente a robustez de desempenho dos controladores, utilizou-se o conceito de estabilidade quadrática associado a uma modelagem politópica do sistema de potência para verificação do fator de amortecimento mínimo dentre todos os modos de oscilação do sistema (o qual é usualmente adotado em sistemas de potência como critério de desempenho ou, equivalentemente, como indicador de margem de estabilidade a pequenas perturbações). A modelagem politópica é usada como alternativa para a obtenção de um modelo de sistema dinâmico que leva em conta as incertezas referentes ao ponto de operação. Neste tipo de modelagem, ao invés de se considerar apenas um ponto de operação nominal, leva-se em conta um conjunto particular de pontos de operação típicos do sistema (os quais comporão os vértices de um conjunto convexo, chamado de politopo). Posteriormente, com base no conceito de estabilidade quadrática, pode-se garantir que um controlador projetado para garantir um desempenho mínimo aos vértices de um politopo estenderá tal garantia também a qualquer ponto de operação que tiver uma descrição linearizada pertencente a este politopo. Os resultados obtidos demonstram que a associação desses dois conceitos fornece uma alternativa viável e vantajosa para a avaliação da robustez de estabilidade e desempenho em sistemas de potência. O procedimento proposto pode ser usado de maneira complementar ao cálculo de autovalores tipicamente empregado na indústria, estendendo a garantia formal de robustez a um conjunto mais amplo de pontos de operação. / The present work proposes the joint use of polytopic modeling and quadratic stability concepts to evaluate the performance robustness of power systems stabilizers (or PSSs). PSS-type damping controllers have been widely used in electric power systems since the end of 6th decade of this century. The majority of these stabilizers, which are in operation nowadays, were designed according to a classical control approach. This method involves linearization of the system equations around an equilibrium point and control through a lead-lag phase compensator. This procedure has a widespread application in power systems due to the simplicity of the technique and the low implementation cost. However, one of the main disadvantages inherent to this method lies exactly in the linearization, since the validity of the designed control is restricted to a neighborhood of the operation point in which the linearization has been done. Since the system operating condition changes throughout the day, we cannot have a formal guarantee of a satisfactory controller performance. Even if the controller performance is checked for different operating points after the design, the performance will be formally guaranteed only in the neighborhoods of the verified points. The present research aims to fill this gap associated to the lack of a formal performance guarantee in an off-nominal operation condition. With the objective of formally guaranteeing the controller performance, the concept of quadratic stability, associated to a polytopic modeling of the system, was used to check the minimum damping factor among all system modes of oscillation (which is usually adopted in power systems as a performance criteria or, equivalently, as an index of small-signal stability margin). The polytopic modeling is used as an alternative to obtain the dynamic system model that accounts for the uncertainty in the operating point. In this type of modeling, instead of considering only one nominal operating point, a particular set of typical system operating points is chosen (which will compose the vertices of a convex set, called polytope). Later, based on the quadratic stability concept, it is possible to guarantee that a controller designed to achieve a minimum performance index at the vertices of the polytopic set will extend this property to any operation point belonging to this set. The obtained results show that the association of these two concepts provides a viable and advantageous alternative for the evaluation of the stability and performance robustness in power systems. The proposed procedure can be used as a complement to the eigenvalue calculation used in the industry, extending the formal robustness guarantee to a broader set of operating points.

With the growth of renewable energy usage and energy storage adoption in recent decades, people have started to reevaluate the possible roles of dc systems in current and future electrical systems. The dc voltage distribution has been applied in various applications, such as data centers and aircraft industry, for high efficiency and power density. However, for some applications such as subsea gas and oil fields, and ocean observatory systems, the dc current distribution is preferred over dc voltage distribution for its low cost and robustness against cable faults. Design and control of dc power distribution systems for different applications is an emerging research area with complex technical challenges. This dissertation solves the technical challenges in analysis, design, modeling, control and protection of series resonant converters (SRCs) for dc current distribution applications. An optimum design that has high efficiency, high reliability, and minimum required control efforts for the SRC with constant input current has been achieved and demonstrated by applying the analysis and design procedures developed in this dissertation. The modeling and analysis presented in this dissertation represents an operating condition that has not been studied in the literature and could be easily extended to other resonant converter topologies. Explicit analytical expressions have been provided for all key transfer functions, including input impedance and control-to-output, offering valuable resources to design feed-back regulation and to evaluate system stability. Based on the control strategies and control design presented in this dissertation, stable and reliable operation of dc current distribution systems with long distance cable has been achieved and demonstrated. The proposed analysis, design procedure, stability evaluation, control strategy and protection techniques in this dissertation can be applied to a wide range of similar scenarios as well, which greatly increases their value.

In this thesis, the investigation focuses on the analysis of the high frequency characteristics and the nonlinearity of the transistors. In view of the III-V semiconductors which have excellent high frequency performance and the advantage for high frequency circuit design, the 0.15£gm InGaAs based pseudomorphic high electron mobility transistors provided by WIN semiconductor Corp. were used in this study. The high frequency measurement was utilized to extract both extrinsic and intrinsic components of the transistors, and further to establish the small signal equivalent model in each bias condition. According to the physical definition of the extracted gm, gds and the relationship with the output current, RF I-V curves could be determined through the integration procedure. The nonlinearity of the transistors can be attributed to the nonlinear input capacitance Cgs and Cgd, and the voltage dependent current source. The high frequency nonlinear models proposed in this thesis were based on classic Angelov model. For the high frequency application, the frequency dependent characteristics of the nonlinear sources would be taken into consideration through the combination of the RF I-V curves and extracted intrinsic components. Thus, the nonlinearities could be able to describe by nonlinear function through the fitting process and model the output performance completely. The accuracy of the models could be confirmed through the comparison between the simulation and the measurement result. Obviously, the high frequency models which include the high frequency effect and the nonlinear characteristics have excellent agreement with the experimental data.

Made available in DSpace on 2014-06-11T19:22:32Z (GMT). No. of bitstreams: 0 Previous issue date: 2010-05-28Bitstream added on 2014-06-13T19:06:57Z : No. of bitstreams: 1 pina_ap_me_ilha.pdf: 797000 bytes, checksum: d243ccbbde4d3e77f76b36fd34886251 (MD5) / Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) / Este trabalho apresenta estudos referentes à modelagem do dispositivo FACTS STATCOM para posterior inclusão nas equações do Modelo de Sensibilidade de Potência multimáquinas. O objetivo final da modelagem é o estudo da estabilidade a pequenas perturbações de sistemas elétricos de potência. São considerados dois modelos para o dispositivo: um primeiro modelo permite apenas a compensação de potência reativa, enquanto que num segundo modelo é possível a compensação tanto de potência ativa como de potência reativa. Também são sugeridos controladores para o dispositivo FACTS STATCOM e, neste trabalho, estes controladores são descritos por blocos de primeira ordem. Com o equacionamento do sistema elétrico realizado, seu modelo é implementado computacionalmente para se efetuar simulações para se avaliar a estabilidade a pequenas perturbações. As simulações estão baseadas na análise no domínio do tempo e no domínio da frequência, utilizando os dois modelos desenvolvidos para o STATCOM. A partir dos resultados obtidos pelas simulações, análises são realizadas, e discutidos os principais aspectos referentes à estabilidade a pequenas perturbações de sistemas elétricos de potência / This work presents studies referred to the modeling of the FACTS STATCOM device to include in multi-machine Power Sensitivity Model equations. The aim is to study electrical system stability under small perturbations. Two models are considered for the device: the first one allows only the reactive power compensation, while the other one allows the reactive or active compensation. Controllers for the FACTS STATCOM device are also suggested, and in this work they are described by first order blocks. As the electrical system equations are finalized, the model is computationally implemented to effectuate simulations and evaluate the stability under small perturbations. The simulations are based on the time and frequency domain using the two models developed for the FACTS STATCOM device. Considering the results obtained by the simulations the analysis are realized and discussed the principal aspects referred to the electrical Power system stability under small perturbations