Hopf bifurcations in a power System and damping oscillations

Bu çalışmada, Senkronaltı Rezonans (SSR) oluşan bir elektrik güç sistemindeki Hopf çatallanmaları incelenmekte ve kararsız burulma salınımlarının sönümlendirilmesi için tasarlanan bir kontrolör sunulmaktadır. SSR araştırmaları için geliştirilen IEEE İkinci Gösterge Modelinin birinci sistemi kullanılmıştır. Senkron makinenin amortisör sargıları doğrusal olmayan modele dahil edilmiştir. Seri kompanzasyon kapasitörü tesis edilmiş olan enerji iletim hatlarına bağlı senkron makineler, potansiyel olarak senkronaltı elektrik modunun, türbin-generatör şaft sisteminin burulma salınım modları ile etkileşimine maruz kalabilirler. Modellenen elektrik güç sisteminde meydana gelen Hopf çatallanmalarının hangi tip olduğu, birinci Lyapunov katsayılarının analitik olarak hesaplanması ile belirlenmiştir. Sabit ikaz uygulanan modelde kritik-altı Hopf çatallanması meydana gelmektedir. Diğer yandan, Otomatik Gerilim Düzenleyicisinin (AVR) eklenmesi ile modelde kritik-üstü Hopf çatallanması oluşmaktadır. Ayrıca, SSR sonucu ortaya çıkan kararsız burulma salınımlarının sönümlendirilmesi için, zaman gecikmeli geri besleme kontrol teorisine dayanan bir kontrolör tasarlanmıştır. Kontrol girdisi olarak sadece senkron makine rotorunun açısal hızını kullanan kontrolörün zaman gecikme ve kazanç parametreleri için uygun değerler, sistemin dinamik cevabını değerlendiren bir performans endeksi hesaplanarak belirlenmiştir. Tasarlanan kontrolörün etkili sonuçlar verdiği benzetimler yardımı ile gösterilmiştir. Kontrolör etkinliği değerlendirilirken, AVR ve kontrolör çıkış sınırlayıcıları da modele dahil edilmiş ve seri kapasitör kompanzasyonunun pratik işletme değerleri için kontrolörün etkili olduğu görülmüştür.  Anahtar Kelimeler: Senkronaltı Rezonans, hopf çatallanması, lyapunov katsayıları, zaman gecikmeli geri besleme kontrolü.Series capacitor compensation of AC transmission lines as a means of increasing load carrying capacity and enhancing transient stability has a widespread use in power systems, particularly in the North America. On the other hand, potential danger of interaction between torsional oscillation modes of the turbine generator shaft system and the subsynchronous electrical mode may arise in electric power systems consisting of turbine-generators con-nected to transmission lines with series compensation capacitors. This phenomenon is called Subsynchronous Resonance (SSR). Unless adequate measures are implemented, unstable torsional mode oscillations due to SSR can lead to catastrophic turbine-generator shaft failures, as occured at Mohave power plant in 1970 and 1971. Since then, considerable effort to the analysis of SSR phenomenon has been devoted by researchers and industry professionals. In this study, Hopf bifurcations in a power system susceptible to SSR is investigted and a novel controller based on the delayed feedback control theory to stabilize unstable torsional oscillations caused by SSR is presented. Bifurcation theory is employed for the analysis of torsional oscillations in a power system which consists of a synchronous generator connected to an infinite busbar through two parallel transmission lines one of which is equipped with a series compensation capacitor. The first system of the IEEE Second Benchmark Model for Subsynchro-nous Resonance studies has been used. Damper windings of the synchronous generator are included in the nonlinear model. The state equations representing the dynamics of the electrical system, mechanical system and the excitation system are obtained separately and then combined into one set of ordinary differential equations. Occurrence of Hopf bifurcations in the model at certain values of the series compensation factor has been verified by tracing the eigenvalues of the Jacobian matrix evaluated at equilibrium. Loss of stability occurs in the first and second torsional oscillation modes through Hopf bifurcation due to SSR. Instead of using the Floquet multipliers method, the first Lyapunov coefficient has been computed analytically to determine the type of Hopf bifurcations (supercritical or subcritical), thereby stability condition of the limit cycles emanating from the Hopf bifurcation points. It is found that subcritical Hopf bifurcation occurs in the model without an Automatic Voltage Regulator (AVR). On the other hand, in the model with AVR, supercritical Hopf bifurcation occurs. Time domain simulations in MATLAB-Simulink are presented to demonstrate the validity of analytic findings.The proposed controller is based on the delayed feedback control theory. The Time Delay Auto-Synchronization (TDAS) controller requires the measurement of the synchronous generator rotor angular speed as the only input. The difference between the value of the controller input signal in -time unit in the past and its current value is multiplied by a gain to obtain an output signal which is combined into the AVR as the stabilizing signal.The effective performance of the controller in providing sufficient damping for the unstable torsional oscillations depends on the correct setting of time delay and gain parameters of the controller. For this purpose, an optimization performance index (OPI), which evaluates damping performance of the controller in time-domain dynamic responses of the model, has been defined.Time-domain simulations in MATLAB-Simulink were carried out to evaluate the effectiveness of the TDAS controller in providing additional damping for the unstable torsional oscillations at various series compensation levels, provided that the time delay and gain parameters are correctly set. Performance of the TDAS controller has been investigated in the presence of AVR limiters in order to obtain a realistic assessment. The TDAS controller output limiter is also included so that the impact on the AVR's voltage regulating performance remains limited. Time domain simulations are presented to demonstrate the effectiveness of the proposed controller even in the presence of the limiters within the practical operating ranges of series capacitor compensation.     Keywords: Subsynchronous Resonance, hopf bifurcation, lyapunov coefficients, delayed feedback control

Optimization of coasting points in a mass rail transit system

Bir raylı taşıma sisteminde enerji tüketimi birçok farklı parametreye bağlıdır. Ancak, bir raylı sistemde tren rotası boyunca hız profilini en uygun hale getirmek, enerji tüketimini azaltmada en etkili yöntemlerden biridir. Makalede trenler için boşta gitme noktalarının optimize edilebilmesi için yeni geliştirilmiş bir yöntem sunulmaktadır. Çalışmada önerilen yaklaşım GA (Genetik Algoritma), YSA (Yapay Sinir Ağları), ve çok hatlı ve çok trenli sistem simulasyon yazılımının birlikte kullanımıyla gerçekleştirilmiş olan gerçekçi sistem modellemesini içermektedir. Simulasyon yazılımı araçların regeneratif frenleme yapabilme ve düşük gerilimdeki performansını da modellemektedir. Simulasyon yazılımı, YSA için eğitim ve test verilerinin oluşturulmasında kullanılmıştır. Bu veriler YSA’ların eğitiminde ve bu eğitilmiş YSA’lar ise değişik boşta gitme konumları için yolculuk süresi ve enerji tüketimini tahmin etmede kullanılmıştır. Optimizasyon kısmında hedef yolculuk süresi, hedef enerji tüketimi ve ağırlık faktörleri içeren bir uygunluk fonksiyonu sunulmuş ve YSA’lar bu uygunluk fonksiyonunu hesaplayan optimizasyon prosedürünün hızını arttırmada kullanılmıştır. Uygunluk fonksiyonunu minimize eden en uygun boşta gitme noktalarının belirlenmesinde GA araştırma yöntemi kullanılmıştır. Gerek GA’nın gerekse kullanılan uygunluk fonksiyonunun değişik parametreleri için optimizasyon çalışmalarının tekrar edilmesi işlemi mevcut literatürde bulunan yöntemlerle çok fazla zaman almaktadır. Çalışmada önerilen yöntemde eğitilmiş olan YSA’ların kullanılması ile parametre değişiklikleri için tekrarlanan optimizasyon çalışmlarında simülatörün kullanılmasına gerek kalmamakta, dolayısı ile yeni parametreler için optimizasyon sonuçları çok hızlı elde edilebilmektedir. Anahtar Kelimeler: Raylı sistem, enerji verimliliği, boşta gitme, yapay sinir ağı, optimizasyon.Energy consumption of a rail transit system depends on many parameters such as train weight, maximum speed, power supply system voltage level, and operation concepts. One of the most effective methods of reducing energy consumption in a rail transit system is optimizing the speed profile of the trains along the route: Trains consume the maximum energy during flat-out mode operation where they speed up to the maximum speed and keep that speed until it reaches to the braking point which is determined by deceleration rate. This type of operation gives the shortest journey time. A small trade-off from this journey time gives high saving in energy consumption.  This subject poses an optimization problem which could be very complicated. In this study, a new efficient method will be presented for optimization of the coasting points for trains in a global manner. The new method suggest using Artificial Neural Networks (ANN) together with classical approach of simulator tool and genetic algorithms.Trains run along the line according to a timetable. Timetables define the travelling time for every train from every station to station. Timetables always include some slack time for an unexpected time loss which could be caused by faulty equipment, but mostly by passengers. Slack times and station dwell times are very important for providing a punctual service. Delays disturb the punctual operation as well as reducing energy efficiency by consuming the slack times which can be used in normal operation conditions for energy efficient driving. Simulation software is used for creating training and test data for ANNs. These data are used for training of ANNs. Test data are used to validate the outputs of trained ANNs. The trained ANNs are then used for estimating energy consumption and travel time for new sets of coasting points. Finally, the outputs of ANNs are optimized to find optimal train coasting points. For this purpose, a fitness function with target travel time, energy consumption and weighting factors is proposed. Genetic algorithms were used for this search purposes. An interesting observation is that the use of ANNs increases the speed of optimization, and gives researchers the ability to test different optimization with differing genetic algorithm parameters. Proposed method is used for optimizing coasting points for minimum energy consumption for a given travel time of first 10 km section of Istanbul Aksaray-Airport metro line, where trains operate every 150 seconds. The section covers 9 passenger stations, which means 8 coasting points for each line. It has been demonstrated that an 16 input ANN can be trained with acceptable error margins for such a system .The optimization method finds the near optimum points for different target travel times and weight factors of the fitness function. It was found for the given line configurations that the energy saving potential with coasting schemes for the same amount of time increase is far less in the multi-train case, where trains regenerate and feed each other. In the 3 station case, 4.81% increase in travel time with optimum coasting points compared to the flat-out case creates 30.85% energy saving, whereas in the 5 station case, 4.65% increase in travel time with optimal coasting scheme creates only 18.25% energy saving. Using ANN and GA in combination speeds up optimization, and bigger line segments with more stations can also be optimized with this proposed method. A simple way of application of the proposed method is using a Driver Information System (DIS). Aksaray - Airport LRT system has been recently equipped with such a system. It is planned to find the optimum coasting points for different headways, and enter these values into the DIS as time and location dependent values. The DIS will give a warning to the driver for start of coast at these pre-defined locations. It should be noted, however, that some operational parameters are changing dynamically in real life. The train weights and the station waiting times, for instance, affect the energy consumption and the travel time. Authors are aware of staggering difficulties in finding optimum coasting points online for such operational variants. Nevertheless, the paper reveals the advantages of using ANN, and its possible application to optimization of coasting points for the case of multiple stations and multiple lines, and hopefully paves the road for future research in this direction. Keywords: Mass rail transit, energy efficiency, coasting, artificial neural networks, optimization.