      
\chapter{Dynamic analysis of Squirrel Cage Induction Generator - Wind Turbine with a Radial Interconnection to the Grid}\label{Chap3}

In this chapter, the stability of SCIG-WTs connected to the grid through a radially connected series compensated line is investigated through eigenvalue analysis and time domain simulations.  Previous research in this area claims that instability in the system due to the induction-generator effect (IGE) of SSR happens only at very high levels of series compensation ($80\%$ compensation) \cite{moharana2011subsynchronous}, \cite{Basepaper}. Although this claim might be true in some cases, it may not be true in cases where the system is weak, such as interconnections involving long lines or feeders of high impedance. Eigenvalue analysis are presented for the linearized system model to analyze bifurcations leading to instabilities in the system. The influence of generator slip as well as the strength of the grid reflected as a function of the X/R ratio of the network has been considered while investigating stability domains with respect to series compensation. 
Time domain simulation of the non-linear dynamic model of the system has been performed to observe system behavior when subjected to a momentary disturbance in the grid voltage.
% In this chapter, it has been shown that even for realistic values of series compensation of $60\%$, the system is unstable due to IGE from SSR when the interconnection to the grid is sufficiently weak.}

\section{Mathematical modeling of network}\label{section:FC_MathModel}

The single line diagram of the system under study is shown in Fig.\ref{fig:sld_fbm}. The terminal of the generator is connected to the grid through a series compensated line to enhance power transfer capability of the network. Reactive power support for the generator is supplied through shunt capacitors connected at the stator terminals.
\begin{figure}
    \centering
    % \includegraphics[width=4in,height=1.55in]{figure/Chapter 3 Figures/IEEE_FBM_SCIG_SLD.pdf}
    \includegraphics[scale = 1]{figure/Chapter 3 Figures/IEEE_FBM_SCIG_SLD.pdf}
    \caption{Single line diagram of modified IEEE First Benchmark Model}
    \label{fig:sld_fbm}
\end{figure}  
The dynamics of the fixed shunt capacitor connected at the stator terminals for reactive power support is described by the equation shown below:

\begin{align}
    \frac{1}{\omega_s}[U]\frac{d}{dt}\{V_{s}\} = [N_1]V_{s} +  [Z_{sh}]\{I_{s}-I_{n}\}\label{Shunt_dyn}
\end{align}

where 

\begin{align}
[U] &=     \begin{bmatrix}
           1 && 0\\
           0 && 1
          \end{bmatrix}\\
[N_1] &= \begin{bmatrix}
        0  & 1\\
    -1 & 0
    \end{bmatrix}\\
[Z_{sh}] &= X_{sh}[U]    
\end{align}

and the elements of the vectors $\{I_{s}\}$, $\{I_{n}\}$ and $\{V_{s}\}$ are given by: 

\begin{align}
    \{I_{s}\} &= [I_{ds} \hspace{0.35cm} I_{qs}]^{T}\\
    \{I_{n}\} &= [I_{dn} \hspace{0.35cm} I_{qn}]^{T}\\
    \{V_{s}\} &= [V_{ds} \hspace{0.35cm} V_{qs}]^{T}
\end{align}
In the above equations, $I_{dn}$ and $I_{qn}$ are $d$ and $q$ components of the network currents. The dynamics of the voltage across the series capacitor along the $d$ and $q$ axis is expressed using the following differential equation:

\begin{align}
    \frac{1}{\omega_s}[U]p\{V_{c}\} = [N_1]\{V_{c}\} +  [Z_{se}]\{I_{n}\}\label{Series_dyn}
\end{align}
where 
\begin{align}
    [Z_{se}] &= X_c[U]
\end{align}

and $\{V_{c}\} = [V_{dc} \hspace{0.35cm} V_{qc}]^{T}$.

The differential equation which describes the dynamics of the $d$ and $q$ components of the network currents is given below:

\begin{align}
    \frac{X_n}{\omega_s}[U]p\{I_{n}\} = [Z_{nw}]\{I_{n}\} - \{V_{c}\} - \{V_{g}\} + \{V_{s}\}\label{Line_dyn}
\end{align}

where $X_n = X_t + X_g$ and $\{V_{g}\} = [V_{dg} \hspace{0.35cm} V_{qg}]^{T}$ where $V_{dg}$ and $V_{qg}$ are the $d$ and $q$ components of the grid voltage. The elements of matrix $[Z_{nw}]$ are given below:

\begin{align}
   [Z_{nw}] = \begin{bmatrix}
    R_{n} & -X_n\\
    X_{n} & R_n
    \end{bmatrix}
\end{align}

In this study, the resistance of the step-up transformer is assumed to be negligible, i.e $R_t = 0$ pu. Hence, $R_n = R_g$.

% The complete dynamic model of the system is comprised of equations \eqref{Stator_IG}, \eqref{rotor_IG}, \eqref{IG_Te}, \eqref{Shunt_dyn}, \eqref{Series_dyn} and \eqref{Line_dyn}. 
% The state variables of the system under study are listed below:

% \begin{align}
%     X = \{I_{ds}, I_{qs}, I_{dr}, I_{qr}, I_{dn}, I_{qn}, V_{ds}, V_{qs}, V_{dc}, V_{qc}, s_g\}^T
% \end{align}

\section{Linearization of mathematical model}

The dynamic equations of the network presented in Section \ref{section:FC_MathModel} are linearized and presented below:

\begin{align}
    \frac{1}{\omega_s}[U]p\{\Delta V_{s}\} &= [N_1]\{\Delta V_{s}\} +  [Z_{sh}]\{\Delta I_{s}-\Delta I_{n}\}\label{shunt_dyn_linearized}\\
    \frac{1}{\omega_s}[U]p\{\Delta V_{c}\} &= [N_1]\{\Delta V_{c}\} +  [Z_{se}]\{\Delta I_{n}\}\label{series_dyn_linearized}\\
     \frac{X_n}{\omega_s}[U]p\{\Delta I_{n}\} &= [Z_{nw}]\{\Delta I_{n}\} - \{\Delta V_{c}\} - \{\Delta V_{g}\} + \{\Delta V_{s}\}\label{Line_dyn_linearized}
\end{align}

In equation \eqref{Line_dyn_linearized}, $\{\Delta V_{dqg}\} = 0$ since the voltage of the grid is assumed to be fixed under equilibrium conditions. 

\section{Eigenvalue analysis}

The linearized mathematical model of the system consists of equations \eqref{Stator_IG_linearized}, \eqref{rotor_IG_linearized}, \eqref{swing_IG_linearized}, \eqref{shunt_dyn_linearized}, \eqref{series_dyn_linearized} and \eqref{Line_dyn_linearized}, . The complete list of linearized state variables of the system is given below:

\begin{align}
     \Delta X = \{\Delta I_{ds}, \Delta I_{qs}, \Delta I_{dr}, \Delta I_{qr}, \Delta I_{dn}, \Delta I_{qn}, \Delta V_{ds}, \Delta V_{qs}, \Delta V_{dc}, \Delta V_{qc}, \Delta s_g\}^T
\end{align}

These equations are cast in state space form $[M]\Delta \Dot{X} = [A] \Delta X$. The Jacobian matrix of the system is obtained using $[J] = [M]^{-1}[A]$. Eigenvalues of the Jacobian matrix give very useful information regarding stability of the system and computing participation factors using the left and right eigenvectors reveal the contribution of individual system states. Presence of local bifurcations in the system could be detected by analyzing the eigenvalues of the system as certain parameters are varied. 

\begin{figure}[t]
\centering
  \includegraphics[scale = 0.6]{figure/Chapter 3 Figures/Torque_Slip_FC.pdf}
  \caption{Steady state characteristics of SCIG and wind turbine for $60\%$ series compensation}
  \label{fig:FC_SteadyState}
\end{figure}

\begin{table}[t]
    %\centering
    \caption{Eigenvalues of SCIG-WT for $60\%$ series compensation and $s_g = 1\%$.}%This table has not been referred to within the text of the manuscript. Please cite Table 1 before it appeared.
    \label{tab:FC_Evals}
    {\begin{tabular}{ccc}
    \toprule
   \boldmath{$\lambda_i$} & \boldmath{$-\sigma \pm j\omega$}    &  \textbf{Participating States} \\
     \midrule
  \,\,\,\,\,\,\,\,  $\lambda_1$ \,\,\,\,\,\,\,\, &\,\,\,\,\,\,\,\, $-27.323\pm j1466.5$\,\,\,\,\,\,\,\,  & \,\,\,\,\,\,\,\,\,\,\,\,\, $I_{ds}$,$I_{qs}$,$I_{dr}$,$I_{qr}$,$I_{dn}$,$I_{qn}$,$V_{ds}$,$V_{qs}$,$V_{dc}$,$V_{qc}$\,\,\,\,\,\,\,\,\,\,\,\,\, \\
    \midrule 
   $\lambda_2$ & $-38.79 \pm j712.55$ & $I_{ds}$,$I_{qs}$,$I_{dr}$,$I_{qr}$,$I_{dn}$,$I_{qn}$,$V_{ds}$,$V_{qs}$\\
       \midrule
   $\lambda_3$ & $-12.548 \pm j627$ & $I_{ds}$,$I_{qs}$,$I_{dr}$,$I_{qr}$,$I_{dn}$,$I_{qn}$,$V_{ds}$,$V_{qs}$,$V_{dc}$,$V_{qc}$\\
       \midrule
   $\lambda_4$ & $0.584 \pm j125.45$ & $I_{ds}$,$I_{qs}$,$I_{dr}$,$I_{qr}$,$I_{dn}$,$I_{qn}$,$V_{dc}$,$V_{qc}$\\
       \midrule
   $\lambda_5$ & $-9.4524 \pm j21.922$ & $I_{ds}$,$I_{qs}$,$I_{dr}$,$I_{qr}$,$I_{dn}$,$I_{qn}$,$V_{ds}$,$V_{qs}$,$V_{qc}$,$s_g$\\
       \midrule
   $\lambda_6$ & $-15.979$ & $I_{ds}$,$I_{qs}$,$I_{dr}$,$I_{qr}$,$s_g$\\
       \bottomrule
    \end{tabular}}
\end{table}

Eigenvalues of the system are computed for a wind speed $V_w = 13.5$ m/s which corresponds to a rotor slip $s_g = -0.010724$ pu obtained from the intersection of the steady state characteristics of the induction generator and the wind turbine which is shown in Fig. \ref{fig:FC_SteadyState}. In Table \ref{tab:FC_Evals}, eigenvalues of the system are listed for a series compensation ($K_c = X_c/X_n$) of $60\%$ when the $X/R$ ratio ($X_n/R_n$) of the network is assumed to be $30$. There are $5$ oscillatory modes ($\lambda_1$ to $\lambda_5$) and $1$ non-oscillatory mode ($\lambda_6$).

Among the oscillatory modes, $\lambda_1$, $\lambda_2$ and $\lambda_3$ have frequencies greater than $60$ Hz and are designated as the supersynchronous modes which are found to have negative real parts and hence, are stable. Modes $\lambda_4$ and $\lambda_5$ have frequency less than $60$ Hz which are the subsynchronous modes among which $\lambda_4$ has a positive real part making it unstable. This mode is a result of the participation of stator and rotor currents of the SCIG, line currents and the series capacitor voltage which shows that the system is unstable due to SSR.

\subsection{Eigenvalues for different generator slips}\label{FC_IG_Slips}

\begin{figure}[t]
\centering
  \includegraphics[scale = 0.6]{figure/Chapter 3 Figures/Eigenvalues_SCIG_FC.pdf}
  \caption{Eigenvalues for low and high generator slips}
  \label{fig:FC_Vw}
\end{figure}

The eigenvalues of the system under study when the degree of compensation $K_c$ is varied from $0.005$ to $0.98$ are calculated and plotted on the complex plane for 2 different generator slips $s_g = 1\%$ and $s_g = 3\%$ which correspond to low and high values wind speeds respectively as shown in Fig. \ref{fig:FC_Vw}. The $X/R$ ratio of the network is assumed to be $30$. The variations in torque input from the wind is assumed to be slow and hence is neglected in this analysis ($\Delta T_w \approx 0$). This greatly simplifies the computations as the operating slip $s_g$ does not have to be calculated for each level of series compensation. %The system consists of $5$ pairs of complex conjugate eigenvalues and $1$ real eigenvalue. Among the complex conjugate eigenvalues, there are $3$ pairs which have frequency above $60\hspace{0.1cm}Hz$ (Supersynchronous modes) and $2$ pairs have a frequency below $60\hspace{0.1cm}Hz$ (Subsynchronous modes).

It was observed from Fig. \ref{fig:FC_Vw} that the supersynchronous modes are all stable for the entire range of series compensation considered for $s_g$ of $1\%$ and $3\%$. The damping of the subsynchronous mode decreases with an increase in $K_c$ and when $K_c$ is between $0.575$ and $0.58$, a Hopf bifurcation occurs due to subsynchronous resonance when $s_g$ was $1\%$. At a higher slip $s_g$ of $3\%$, the system is unstable for a compensation level $K_c$ from $0.5\%$ due to the presence of a real eigenvalue on the right half of the complex plane resulting in the system being unstable. The equilibrium is called a `saddle-focus' characterized by 3 leading eigenvalues given by $\lambda_{1,2} = -8.8894 \pm j16.957$ and $\lambda_{3} = 3.5009$. Participation factors for these leading eigenvalues are presented in Table \ref{table:1}. 

% \begin{table}
%     \caption{Participation factors of leading eigenvalues for $s_g = 3\%$ at a compensation of $0.5\%$}
%     \centering
%     \begin{tabular}{ |c|c|c|c|c|c|  } 
%      \hline
%      $\lambda = -\sigma \pm j\omega$ & $I_{ds}$ & $I_{qs}$ & $I_{qs}$ & $I_{qs}$ & $s_g$\\ 
%      \hline
%      \hline
%      3.7758 & 0.114 & 0.297 & 0.13 & 0.366 & 0.091\\
%      \hline
%      $-8.8295 \pm j17.017$ & 0.247 & 0.183 & 0.308 & 0.225 & 0.034\\
%      \hline
%      \end{tabular}
%     \label{table:1}
% \end{table}

\begin{table}[t!]
     %\centering
       \caption{Participation factors of critical eigenvalue pair for $s_g = 1\%$ and $60\%$ series~compensation.}
    \label{table:participation_FC_sg_1}
    {\begin{tabular}{ ccccccccc } 
     \toprule
    \boldmath{$\lambda = -\sigma \pm j\omega$} & \boldmath{$I_{ds}$} & \boldmath{$I_{qs}$} & \boldmath{$I_{dr}$ }& \boldmath{$I_{qr}$} & \boldmath{$I_{d}$} & \boldmath{$I_{q}$} &\,\,\,  \boldmath{$V_{dc}$}\,\,\, & \boldmath{$V_{qc}$}\\ 
     \midrule
  \,\, \,\,\,\,\,\, $0.584 \pm j125.38$\,\,\,\,\,\, \,\,& 0.246 & 0.24 & 0.243 & 0.237 & 0.007 & 0.007 & 0.009 & 0.009\\
     \bottomrule
     \end{tabular}}
\end{table}


\begin{table}[t!]
    %\centering
      \caption{Participation factors of leading eigenvalues for $s_g = 3\%$ at a compensation of $0.5\%$.}
    \label{table:1}
    {\begin{tabular}{ cccccc} 
     \toprule
    \,\,\,\,\,\,\,\,\,\,\,\boldmath{$\lambda = -\sigma \pm j\omega$}\,\,\, \,\,\,\,\,\,\,\, & \,\,\,\,\,\,\,\,\boldmath{$I_{ds}$}\,\,\,\,\,\,\,\, &\,\,\,\,\,\,\,\, \boldmath{$I_{qs}$} \,\,\,\,\,\,\,\,&\,\,\,\,\,\,\,\, \boldmath{$I_{qs}$} \,\,\,\,\,\,\,\,&\,\,\,\,\,\,\,\, \boldmath{$I_{qs}$} \,\,\,\,\,\,\,\,&\,\,\,\,\, \boldmath{$s_g$}\,\,\,\,\,\, \\ 
     \midrule
     3.5009 & 0.103 & 0.307 & 0.119 & 0.376 & 0.093\\
     \midrule
     $-8.8894 \pm j16.957$ & 0.250 & 0.181 & 0.310 & 0.223 & 0.033\\
     \bottomrule
     \end{tabular}}
\end{table}


As the compensation increased, the unstable eigenvalue moves to the left half of the complex plane and when $K_c > 31.5\%$, $\lambda_{cr}$ mode crosses over from the right half to the left half of the complex plane resulting in a bifurcation making the system stable. The system remains stable as the compensation level increases to $60\%$ beyond which a Hopf bifurcation results making the system unstable due to subsynchronous resonance. The critical series compensation level at which instability due to Hopf bifurcation is observed is slightly larger for a higher operating slip of the induction generator.  

\subsection{Eigenvalues for different grid strengths}

\begin{figure}[t!]
  \centering
  \includegraphics[scale = 0.6]{figure/Chapter 3 Figures/Eigenvalues_FC_XRratio.pdf}
  \caption{Eigenvalues for different grid strengths}
  \label{fig:FC_XR}
\end{figure}

\begin{figure}[t!]
  \centering
  \includegraphics[scale = 0.6]{figure/Chapter 3 Figures/Stability_Boundary_FC.pdf}
  \caption{Stability boundary with respect to X/R ratio of network.}
  \label{fig:Stab_boundary}
\end{figure}

% \begin{table}[h]
% \caption{Critical eigenvalues for different X/R ratios}
% \centering
% \begin{tabular}{|c|c|c|c|}
% \hline
%     $X/R$ ratio & $K_{cr}$ & $\lambda_{cr}$ & Dominant states \\
%     \hline
%     $15$ & $72\%$ & $0.146 \pm j101.19$ & \multirow{3}{*}{$I_{ds}$,$I_{qs}$,$I_{dr}$,$I_{qr}$,$I_{dL}$,$I_{qL}$,$V_{dc}$,$V_{qc}$} \\\cline{1-3}
%     %  \hline
%     $20$ & $65.5\%$ & $0.009 \pm j114.05$ &  \\\cline{1-3}
%     $25$ & $61.5\%$ & $0.122 \pm j122.25$ &  \\
%     \hline
% \end{tabular}
% \label{table:FC_XRratio}
% \end{table}

\begin{table}[t!]
\caption{Critical eigenvalues for different X/R ratios}
\label{table:FC_XRratio}
% \centering
{\begin{tabular}{cccc}
\toprule
     \textbf{X/R ratio} & \boldmath{$K_{cr}$} &  \boldmath{$\lambda_{cr} = -\sigma \pm j\omega$} & \textbf{Participating states} \\
    \midrule
    \,\,\,\,\,\,\,\,$15$\,\,\,\,\,\,\,\, & \,\,\,\,\,\,\,\,$72\%$\,\,\,\,\,\,\,\, & \,\,\,\,\,\,\,\,$0.146 \pm j101.19$\,\,\,\,\,\,\,\, & \,\,\,\,\,\,\,\,\multirow{3}{*}{$I_{ds}$,$I_{qs}$,$I_{dr}$,$I_{qr}$,$I_{dn}$,$I_{qn}$,$V_{dc}$,$V_{qc}$}\,\,\,\,\,\,\,\, \\\cline{1-3}
    %  \hline
    $20$ & $65.5\%$ & $0.009 \pm j114.05$ &  \\\cline{1-3}
    $25$ & $61.5\%$ & $0.122 \pm j122.25$ &  \\
    \bottomrule
\end{tabular}}
\end{table}

Increasing the $X/R$ ratio of the line results in an increase in the line impedance for a given value of line resistance. This decreases the Short Circuit Ratio (SCR) at the point of interconnection (POI) as the Thevenin impedance of the system up to the POI increases as the impedance of the line increases which results in a weaker interconnection to the grid. 
The eigenvalues of the system for a generator slip of $1\%$ are calculated for $X/R$ ratios of $15$, $20$ and $25$ and are plotted in Fig. \ref{fig:FC_XR}. Supersynchronous modes and the non-oscillatory mode is observed to be stable for the entire range of series compensation considered but the subsynchronous mode becomes less damped as the series compensation level increases. Subsynchronous resonance occurs at compensation levels of $72\%$, $65.5\%$ and $61.5\%$ for $X/R$ ratios of $15$, $20$ and $25$ respectively which shows that as the grid becomes weaker, there is a higher possibility of SSR at lower values of series compensation. This is further evident in Fig. \ref{fig:Stab_boundary}, where the critical value of series compensation is plotted as a function of $X/R$ ratio of the network, which gives the stability boundary with respect to series compensation.

\section{Time domain simulations}

In this section, the results of time domain simulations performed on the system under study are presented. The non-linear dynamic model of the system comprised of equations \eqref{Stator_IG}, \eqref{rotor_IG}, \eqref{swing_IG}, \eqref{Shunt_dyn}, \eqref{Series_dyn} and \eqref{Line_dyn} is simulated using SIMULINK and the traces of various system state variables are observed. The series compensation level considered is $60\%$ which is a practical value of series compensation level for an $X/R$ ratio of $30$. 

\begin{figure}[t!]
  \centering
  \includegraphics[scale = 0.5]{figure/Chapter 3 Figures/Stator_Voltage_FC.pdf}
  \captionof{figure}{\centering Stator voltage response to a disturbance in grid voltage}
  \label{fig:FC_Vs}
\end{figure}

\begin{figure}[t!]
  \centering
  \includegraphics[scale = 0.5]{figure/Chapter 3 Figures/Line_Current_FC.pdf}
  \captionof{figure}{\centering Line current response to a disturbance in grid voltage}
  \label{fig:FC_IL}
\end{figure}

\begin{figure}[t!]
  \centering
   \includegraphics[scale = 0.5]{figure/Chapter 3 Figures/Real_Power_FC.pdf}
  \captionof{figure}{\centering Real power response to a disturbance in grid voltage}
  \label{fig:FC_Pt}
\end{figure}

\begin{figure}[t!]
  \centering
   \includegraphics[scale = 0.5]{figure/Chapter 3 Figures/Rotor_Speed_FC.pdf}
  \captionof{figure}{\centering Rotor speed response to a disturbance in grid voltage}
  \label{fig:FC_Wg}
\end{figure}

% \begin{figure}[t!]
% \centering
% \begin{minipage}{.5\textwidth}
%   \centering
%   \includegraphics[scale = 0.3]{figure/Chapter 3 Figures/Stator_Voltage_FC.pdf}
%   \captionof{figure}{\centering Stator voltage response to a disturbance in grid voltage}
%   \label{fig:FC_Vs}
% \end{minipage}%
% \begin{minipage}{.5\textwidth}
%   \centering
%   \includegraphics[scale = 0.3]{figure/Chapter 3 Figures/Line_Current_FC.pdf}
%   \captionof{figure}{\centering Line current response to a disturbance in grid voltage}
%   \label{fig:FC_IL}
% \end{minipage}
% % \caption{Performance of the proposed SSDC in a grid frequency disturbance}
% \end{figure}

% \begin{figure}[t!]
% \centering
% \begin{minipage}{.5\textwidth}
%   \centering
%   \includegraphics[scale = 0.3]{figure/Chapter 3 Figures/Real_Power_FC.pdf}
%   \captionof{figure}{\centering Real power response to a disturbance in grid voltage}
%   \label{fig:FC_Pt}
% \end{minipage}%
% \begin{minipage}{.5\textwidth}
%   \centering
%   \includegraphics[scale = 0.3]{figure/Chapter 3 Figures/Rotor_Speed_FC.pdf}
%   \captionof{figure}{\centering Rotor speed response to a disturbance in grid voltage}
%   \label{fig:FC_Wg}
% \end{minipage}
% % \caption{Performance of the proposed SSDC in a grid frequency disturbance}
% \end{figure}

A disturbance in the grid voltage is applied at $0.5$ s when the voltage magnitude of the grid decreases from $1$ pu to $0.8$ pu and is restored to $1$ pu again at $0.6$ s. In a realistic system, this may be equivalent to a three phase fault occurring at a bus causing a reduction in bus voltages and the fault being cleared in 6 cycles ($0.1$ s) resulting in a restoration of the bus voltages. The stator voltage at the terminals of the SCIG and the line current magnitudes are shown in Figs. \ref{fig:FC_Vs} and \ref{fig:FC_IL} respectively. The SCIG initially operates with a terminal voltage of $1.18$ pu under equilibrium conditions. When the reduction in the grid voltage takes place, there is a momentary decrease in the terminal voltage of the SCIG. Beyond $0.8$ s, the voltage starts increasing up to $9.44$ s. There is a significant spike in the voltage at $9.57$ s where the magnitude almost becomes $3.8$ pu and beyond this, the voltage starts decreasing. The voltage finally settles into an oscillatory state where the magnitude of voltage at the SCIG terminals oscillates between $1.5$ and $2.1$ pu. 

The traces of real power generated by the SCIG and the speed of the induction generator are shown in Figs. \ref{fig:FC_Pt} and \ref{fig:FC_Wg} respectively. The responses of stator voltage and real power generated indicates that the system is unstable due to a Hopf bifurcation resulting from SSR. The rotor speed of the induction generator shows oscillations around the steady state value initially between $0.5$ and $9.4$ s. After this, the rotor speed drops rapidly causing the generator to stall. The reason for this is that there are voltages and currents induced at subsynchronous frequency which increase in magnitude as a result of self excitation due to SSR. This produces a magnetic field rotating at subsynchronous speed in the air gap which increases in strength beyond $9.4$ s, as the magnitude of currents and voltages increase rapidly. Hence, the magnetic field induced by the subsynchronous currents dominates the main field and the rotor speed of the induction machine attempts to catch up with the speed of the magnetic field rotating at subsynchronous speed as a result of which the speed starts decreasing beyond $9.4$ s.

\begin{figure}[t!]
  \centering
  \includegraphics[scale = 0.6]{figure/Chapter 3 Figures/Phase_Plane_FC.pdf}
  \caption{Phase plane trajectory of rotor speed in response to a grid voltage disturbance}
  \label{fig:FC_PP_Speed}
\end{figure}

The phase-plane trajectory of the generator speed along with the limit cycle is shown in Fig. \ref{fig:FC_PP_Speed}. The system appears to have an oscillatory response of fixed amplitude beyond $18$ s as a result of a limit cycle generated due to the Hopf bifurcation. The generator speed shows oscillations between $0.17$ pu and $0.24$ pu which is far away from the equilibrium speed of $1.0103$ pu. Hence, the nature of Hopf bifurcation can be concluded to be subcritical as the trajectory has diverged away from the equilibrium value significantly. 

\section{Conclusion}

In this chapter, the dynamic analysis of an SCIG-WT supplying power to the grid through a weak interconnection has been presented. Eigenvalue analysis and participation factors reveal the main reason for instability in the system is the Hopf bifurcation which occurs due to subsynchronous resonance for smaller values of generator slips as the series compensation exceeds a certain critical percentage. However, for higher generator slip values, there is a possibility of non-oscillatory instability for small values of series compensation which is the characteristic of a saddle-focus equilibrium point. Series compensation has to be above a lower threshold value for the system to operate in a stable manner, however, above a certain upper limit, the system exhibits SSR causing the system to become unstable. From the time domain simulation results, it can be concluded that the nature of the Hopf bifurcation due to SSR is subcritical as the limit cycle generated is far away from the equilibrium point.