\pagenumbering{arabic}
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\chapter{INTRODUCTION}\label{chapter:introduction} 
Economical growth in the recent years has led to a huge increase in the demand of electric power from the existing power system. New transmission infrastructure maybe needed between generating stations and load centers as existing lines are likely to become overloaded due to the increase in the amount of power transmitted to meet the system demand. With the ever increasing growth in the population, getting the right of way to install new transmission lines is a huge problem in addition to this being a very expensive solution. A low cost alternative would be to increase the existing power transmission capacity of the network by compensating a percentage of reactance of the lines using series capacitors which has been done for decades \cite{seriescap}, \cite{Miske_seriescap}, \cite{miller2014review}.  The transient stability limit of the system is improved due to reduction of voltage fluctuations and angular swing of the rotor of synchronous machines \cite{Kimbark_seriescap}. 

In addition to an increasing demand for energy, there is a huge need for clean forms of energy resulting in the deployment of more renewable energy sources from namely wind and solar. The installed capacity of off-shore wind farms is expected to reach $100$ GW in the European Union by the year 2030 \cite{soares2020current}. Wind turbines are placed at remote locations especially off-shore wind plants resulting in longer cables of high impedance. This greatly limits the power which can be transmitted to the grid due to higher voltage drops and increasing power losses \cite{ackermann2012wind}. Due to the constraints of the power transmission network, wind turbines are unable to operate at higher wind speeds resulting in under-utilization of the capacity of the electrical generators. This has been illustrated in Fig. \ref{fig:SCIG_SteadyState} for a Type A wind farm connected to the grid through a feeder of high impedance.    

\begin{figure}
\centering
  \includegraphics[scale = 0.85]{figure/Steady_State_WT_SCIG_Introduction.pdf}
  \caption{Steady state torque versus slip of SCIG and wind turbine characteristics}
  \label{fig:SCIG_SteadyState}
\end{figure}

In Fig. \ref{fig:SCIG_SteadyState}, characteristics of the wind turbine along with the characteristics of a Squirrel Cage Induction Generator (SCIG) connected to the grid with and without series capacitor compensation are shown. The wind turbine characteristics show the torque developed by the wind turbine with respect to the slip for a high wind velocity of $17$ m/s. It can be seen from the figure that the system without series compensation, operation of the wind turbine at this speed as the wind turbine characteristics do not intersect with the characteristics of the SCIG. However, operation of the system was possible when the system is compensated using series capacitors. 

Power systems which are weak and operating under heavily stressed conditions are vulnerable to bifurcations which result in a change in the structural and stability properties of a dynamical system. Bifurcations are classified into local and global types and the existence of both types in power systems have been shown in a number of published works. Local bifurcations can be detected by analyzing eigenvalues of the Jacobian matrix obtained by linearizing the non-linear dynamical equations of the system. Most common local bifurcations in power systems and their characteristics have been summarized in Table \ref{bifurcation_types}.  


\begin{table}
  \centering
  \begin{tabular}{p{6cm} p{8.5cm}}
    \toprule
    % \multicolumn{2}{c}{} \\[.5\normalbaselineskip]
    \begin{center} 
\textbf{BIFURCATION} 
\end{center} & \begin{center} 
\textbf{CHARACTERISTICS} 
\end{center} \\
    \midrule
    Saddle-node bifurcation  & Power-flow Jacobian becomes singular\\
     & Characterized by a zero eigenvalue\\
     & Results in voltage instability and collapse \cite{Canizares_saddlenode}, \cite{Vournas_AVR}, \cite{ajjarapu2007computational}, \cite{rajagopalan1992dynamic}\\
     \hline
    Hopf bifurcation & Characterized by a pair of complex eigenvalues crossing over to the right half of the complex plane resulting in a limit cycle \cite{mithulananthan2000indices}, \cite{Jonathan_Hopf}\\
    & Require higher order models to be detected \cite{canizares1996bifurcation}, \cite{rosehart1999bifurcation}, \cite{Rajagopalan_Sauer_Hopf}\\
    & Classified into subcritical or supercritical based on the stability of oscillations \cite{abed1984nonlinear}, \cite{Chin_Hopf}, \cite{pai1995structural}\\
    \hline
    Singularity induced bifurcation & Characteristic of systems modeled using differential and algebraic equations\\
    & Results in an eigenvalue of the reduced system matrix diverging to infinity \cite{Ayasun_SIB}, \cite{Marszalek_SIB}\\
    \hline
    Limit induced bifurcation & Occurs when certain system variables reach their limit resulting in a sudden change in stability\\
    & Characterized by discrete changes in eigenvalue trajectories when limits are reached \cite{Dobson_LIB}, \cite{Chen_LIB}, \cite{Canizares_LIB}\\
    \hline
    Node-Focus bifurcation & Occurs when a pair of complex eigenvalues transform into 2 real eigenvalues moving in opposite directions or 2 real eigenvalues collide and transform into a pair of complex eigenvalues \cite{multiparameter_NFB}\\
   & May not cause a change in the stability but may be a indicate the occurrence of a critical bifurcation like a Hopf bifurcation \cite{Canizares_LIB}, \cite{Ajjarapu_NFB}, \cite{Dobson_NFB}\\
    \bottomrule
  \end{tabular}
  \caption{Local bifurcations in power systems and their characteristics}
  \label{bifurcation_types}
\end{table}

\section{Motivation}

Series capacitors have been utilized to improve the power transfer capability of the existing network for a long time. However, series capacitors result in adverse effects on the stability of power systems when the compensation exceeds a critical value. Instability is due to subsynchronous resonance (SSR) which result in self-excited oscillations (\cite{Kilgore_IGE}, \cite{Wagner_IGE}, \cite{Concordia_IGE}) commonly known as the Induction Generator Effect (IGE) or Torsional Interactions (TI) which may result in shaft fatigues and even shaft failure \cite{Mohave_SSR}, \cite{Ballance_SSR}, \cite{padiyar2012analysis}. 

Series compensation has known to cause subsynchronous oscillations in wind farms as well \cite{mulawarman2011detection}, \cite{adams2012ercot}. SSR is a well-documented phenomenon in Type - C wind turbines which make use of a Doubly Fed Induction Generators (DFIGs). Detailed modeling and analysis of SSR in DFIG based wind farms has been documented in \cite{Lingling_DFIG}, \cite{Huakum_DFIG}, \cite{Mohammadpour_DFIG}, \cite{mohammadpour2015analysis}. Also, mitigation of SSR in DFIG based wind farms has been studied in \cite{Mohammadpour_DFIG_Mit}, \cite{mohammadpour2015_DFIG_Mit}, \cite{Leon_DFIG_Mit}, \cite{Ulas_DFIG_Mit}. However, DFIG based wind turbines may be expensive due to the power electronic converters employed on the rotor side and the grid side. A low cost alternative to DFIG based wind turbines is the Type - A wind turbines based on Squirrel Cage Induction Generators (SCIGs) which are still in use in developing countries. SCIG based wind turbines operate at fixed speed and are usually coupled to the grid. Therefore, these operate in a narrow slip range. SCIGs require reactive power for its operation which is usually supplied by shunt capacitor banks at the stator terminals. 

It was shown earlier in Fig. \ref{fig:SCIG_SteadyState} that operation at high wind speeds may not be possible if the interconnection to the grid is weak unless series compensation is utilized. However, as a result of series compensation, SCIG based wind turbines may also experience SSR and this has been analyzed in \cite{SCIG_1}, \cite{SCIG_2}, \cite{SCIG_3} and \cite{SCIG_4}.  In \cite{Basepaper}, a detailed eigenvalue analysis is presented for a Type-1 wind turbine to study the influences of series compensation and mechanical parameter variations. It has been shown that the SSR due to Induction Generator Effect (IGE) occurs at very high levels of series compensation. However, in this study, IGE is observed for realistic smaller series compensation levels close to 62\% for a weak interconnection. In references \cite{SCIG_1} and \cite{SCIG_2}, the mitigation of SSR is achieved using a supplementary damping controller which was analyzed through time domain simulations. However, the detailed eigenvalue analysis has not been presented.

Traditionally, studies on SSR are performed on systems consisting of a generator or a set of generators feeding power to an infinite bus through a series compensated network by applying bifurcation theory to the IEEE First Benchmark Model (\cite{nayfeh1998bifurcation}, \cite{nayfeh1998application}, \cite{varghese1998}, \cite{zhu1996}). Bifurcation theory has been applied to study SSR in the IEEE Second Benchmark Model in \cite{widyan2010effect} where the influence on exciter gain on SSR has been analyzed. However, the time constant of the exciter is assumed to be fixed, and its influence on SSR has not been addressed. Global bifurcations have been studied in the IEEE First Benchmark Model due to variations in damping and mechanical power of the synchronous generator (\cite{ji1996hard}, \cite{rout2017control}).  Few studies have also been performed on induction motor loads fed through a series compensated feeder from a substation which is modeled as an infinite bus (\cite{mahmoud1980effects}, \cite{limebeer1981subsynchronous}, \cite{ojo1989electro}). 
In these references that have been stated, there is an assumption of a stiff voltage source namely the infinite bus which maybe equivalent to a substation in real time. However, the impact of generator voltage controller parameters on loads supplied from the substation has not been studied so far.

\section{Objectives and scope of this thesis}

This dissertation deals with analyzing stability domains and bifurcations of series compensated power systems with weak interconnections and various network configurations. The main emphasis of this work is to explore how the stability domain of series compensated systems are affected due to varying grid strengths. The following are the objectives and scope of this thesis:

\begin{enumerate}[label={[\arabic*]}]
    \item To develop dynamic models of weak power grids of various topologies.
    \item To perform eigenvalue analysis on the linearized system models to identify local bifurcations due to parameter variations which may affect the stability of the system. 
    \item To design a voltage regulator for an SVC to enhance the stability domain of the system with respect to series compensation in a radially interconnected weak system. 
    \item To present the results of time domain simulations which show the evolution of various system states with respect to time as the system is subject to different types of disturbances and also help identify global bifurcations. 
\end{enumerate}

This dissertation is organized as follows: Dynamic models of synchronous and induction machines and wind turbines have been presented in Chapter 2. Chapter 3 deals with the dynamic analysis of an SCIG-WT using fixed shunt capacitors at the stator terminals with a series compensated radial interconnection to the grid. Analysis of a radially interconnected, series compensated system with an SCIG-WT using an SVC at the stator terminals has been presented in Chapter 4. Enhancement of stability domains with respect to series compensation by means of a new damping controller which mitigates Hopf bifurcations causing instability due to SSR has been studied in Chapter 5 for the radially interconnected, series compensated system with an SCIG-WT and SVC. The dynamic performance of induction motor loads supplied through series compensated networks has been studied in Chapters 6 and 7 where the influence of network and generator control parameters on the stability domain with respect to series compensation is analyzed.

\section{Publications}

A portion of the results presented in this dissertation have been published in $2$ journals, the details of which are listed below. Parts of results presented in Chapters 3, 4 and 5 in this dissertation have been published in [2]. Chapter 7 describes the results presented in [1]. 

\begin{enumerate}[label={[\arabic*]}]
    \item J. Devadason, P.S. Moses, M.A.S. Masoum, "Multiparameter Stability Analysis of Systems with Induction Motor Loads, Weak Interconnections and Series Compensation," WSEAS Transactions on Circuits and Systems, vol. 20, pp. 128-138, 2021. 
    \item  J. Devadason, P.S. Moses, M.A.S. Masoum, "Stability Domain Analysis and Enhancement of Squirrel Cage Induction Generator Wind Turbines in Weak Grids," Energies, MDPI, Open Access Journal, vol. 14(16), pages 1-17, August 2021.
\end{enumerate}

Other results not directly related to this dissertation that have been published during the PhD program can be found in papers listed below:

\begin{enumerate}[label={[\arabic*]}]
    \item J. Devadason, P.S. Moses and W. Fei, "Bifurcation Analysis of Weak Electrical Grids Considering Different Load Representations," 2019 IEEE 7th International Conference on Smart Energy Grid Engineering (SEGE), 2019. 
    \item  J. Devadason and P.S. Moses, "Analysis of Free Acceleration and Steady State Characteristics of Line-Started Induction Motors with Shunt Capacitor Compensation," 2019 IEEE Power \& Energy Society General Meeting (PESGM), 2019.
    \item J. Devadason and P.S. Moses, "Small Signal Stability of a Series Compensated Microgrid Operating in Islanded Mode," 2020 Intermountain Engineering, Technology and Computing (IETC), 2020.
    \item D. Glover, J. Devadason and P.S. Moses, "Multi-Solar PV Allocation for Optimal Sizing and Placement on Distribution Feeders," 2020 International Conference on Smart Grids and Energy Systems (SGES), 2020.
    \item H. B. Sambo, J. Devadason and P.S. Moses, "Interactions of Rooftop Solar Photovoltaic Systems with Symmetrical and Unsymmetrical Faults in Distribution Feeders," 2021 IEEE Kansas Power and Energy Conference (KPEC), 2021.
\end{enumerate}






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