 
 \chapter{Dynamic Interactions of Synchronous Generators
with Induction Motor Loads in Series Compensated Systems}\label{chap7}

Dynamic modeling and stability domain analysis of a system consisting of a synchronous generator supplying an induction motor load through a series compensated weak network has been carried out in this chapter. In the previous chapter, the induction motor load is supplied from a stiff voltage source at the substation. However, the assumption of a stiff voltage source is eliminated by considering the complete dynamic model of the generator and it's control system. The impact of $X/R$ ratio of the feeder and generation control system parameters on the stability domain with respect to series compensation has been examined through eigenvalue calculations and time domain simulations to reveal new bifurcations in the system.

% \textcolor{blue}{From the studies conducted, it was observed that the stability domain of the system with respect to series compensation depends on the grid strength in addition to the excitation system parameters. Eigenvalue analysis shows that there is a strong correlation between the exciter gain, time constants of the measurement transducer and exciter, and the series compensation level. The main contribution of this chapter is to reveal new bifurcations which arise in these systems which has been studied through eigenvalue analysis and time domain simulations for various combinations of system parameters.} 

\section{Mathematical modeling of system}

\begin{figure}[h]
    \centering
    \includegraphics[scale = 1.2]{figure/Figures_IM_DG/SLD_IM_DG.pdf}
    \caption{Single line diagram of system under study}
    \label{fig:IM_DG_SLD}
\end{figure}

The single line diagram of the system under study is shown in Fig. \ref{fig:IM_DG_SLD}. It consists of a Synchronous Generator (SG) supplying power to a substation (SS) through a transformer and a short line. The substation is connected to an induction motor load through a series compensated long feeder. Traditionally, studies related to subsynchronous resonance have been performed in system consisting of a generator connected to an infinite bus  through a series compensated line. Here, this study is extended to study the effect of series compensation on an induction motor load supplied through a series compensated feeder from the substation. This is a more realistic study as the effect of series compensation on the network beyond the substation is studied in very little detail in the past. The influence of large dynamic loads on power system stability has been studied in \cite{mahmud2013effects}. Although the exciter gain and direct axis time constant of the induction motor has been found to impact stability of the system, the influence of time constants of the measurement transducer and the exciter has not been modeled. In this paper, the exciter and measurement transducers have been modeled using first order transfer functions in order to study the influence of these time constants on the stability domain of the system with respect to series compensation. 

\subsection{Modeling of network}
The dynamics of the terminal voltage of the synchronous machine is given by the following equation:

\begin{align}
   \{V_{t}\}  &= \frac{1}{\omega_s}X_{nw}[U]\frac{d}{dt}\{I_{n}\} + [Z_{nw}]\{I_{n}\} + \{V_{c}\} + \{V_{m}\}\label{Vt_SG}
\end{align}

where $[Z_{nw}] = [Z_{T1} + Z_{L1} + Z_{T2} + Z_{L2} + Z_{T3}]$, $X_{nw} = X_{t1} + X_{n1} + X_{t2} + X_{n2} + X_{t3}$  and 

\begin{align}
    [Z_{T1}] = \begin{bmatrix}R_{t1} & -X_{t1}\\X_{t1} & R_{t1}\end{bmatrix},
    [Z_{L1}] = \begin{bmatrix}R_{n1} & -X_{n1}\\X_{n1} & R_{n1}\end{bmatrix},
    [Z_{T2}] = \begin{bmatrix}R_{t2} & -X_{t2}\\X_{t2} & R_{t2}\end{bmatrix},\nonumber
\end{align}

\begin{align}
    [Z_{L2}] = \begin{bmatrix}R_{n2} & -X_{n2}\\X_{n2} & R_{n2}\end{bmatrix},
    [Z_{T3}] = \begin{bmatrix}R_{t3} & -X_{t3}\\X_{t3} & R_{t3}\end{bmatrix},
    % [X_{nw}] = \begin{bmatrix}
    % 1 & 0\\0 & 1
    % \end{bmatrix}X_{net}\nonumber
\end{align}

The elements of vectors $\{I_{n}\}$, $\{V_{m}\}$, $\{V_{c}\}$ and $\{V_{t}\}$ are given by:

\begin{align}
    \{I_{n}\} &= [I_{dn} \hspace{0.35cm} I_{qn}]^{T}\\
    \{V_{m}\} &= [V_{dm} \hspace{0.35cm} V_{qm}]^{T}\\
    \{V_{c}\} &= [V_{dc} \hspace{0.35cm} V_{qc}]^{T}\\
    \{V_{t}\} &= [V_{dt} \hspace{0.35cm} V_{qt}]^{T}
\end{align}

The voltage across the series capacitor is modeled using equation \eqref{Vse_Chap7} given below:

\begin{align}
    \frac{1}{\omega_s}\frac{d}{dt}\{V_{c}\} = [Z_{se}]\{I_{n}\} + [N_1]\{V_{c}\}\label{Vse_Chap7}
\end{align}

where $[Z_{se}] = X_c[U]$.

The dynamics of the shunt capacitor voltage at the terminals of the induction motor are described by the following equation:

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

where $[Z_{sh}] = X_{sh}[U]$ and $\{I_{s}\} = [I_{ds} \hspace{0.35cm} I_{qs}]^{T}$ are the $d$ and $q$ components of the stator current of the induction motor.

\subsection{Integration of SG and Network Models}
The dynamic model of the synchronous generator is expressed in a reference frame fixed on it's rotor to eliminate time varying inductances. However, when the synchronous machine is interfaced with the rest of the network, the terminal voltage of the generator in the rotor reference frame ($V_d$, $V_q$) can be expressed in a synchronously rotating reference frame ($V_{dt}$, $V_{qt}$) using equation \eqref{transform1} shown below:

% The terminal voltage of the synchronous machine ($V_d$, $V_q$) in the rotor reference frame is transformed to a synchronously rotating reference frame ($V_D$, $V_Q$) as given by:

\begin{equation}
    \begin{bmatrix}V_{dt} \\ V_{qt}\end{bmatrix} = \underbrace{\begin{bmatrix}
    \sin\delta & \cos\delta\\
   -\cos\delta & \sin\delta
    \end{bmatrix}}_{\text{$T_r$}}
    \begin{bmatrix}V_d \\ V_q\end{bmatrix}\label{transform1}
\end{equation}

where $[T_r]$ is the transformation matrix. The inverse transformation is accomplished through matrix $[Q] = [T_r]^{-1}$. In this system, the synchronous machine is directly connected to the network which means the stator currents of the generator are the same as those of the network currents. The transformation of the generator stator currents in the rotor reference frame to the synchronously rotating reference frame is achieved using equation \eqref{transform2}. 

% Hence, the stator currents of the synchronous generator in the rotor reference frame has to be transformed synchronously rotating reference frame Similarly, the network currents ($I_d$, $I_q$) in the rotor reference frame are transformed to the synchronously rotating reference frame ($I_{dL}$, $I_{qL}$) using the following equation:

\begin{equation}
    \begin{bmatrix}I_{dn} \\ I_{qn}\end{bmatrix} = \underbrace{\begin{bmatrix}
    \sin\delta & \cos\delta\\
   -\cos\delta & \sin\delta
    \end{bmatrix}}_{\text{$T_r$}}
    \begin{bmatrix}I_d \\ I_q\end{bmatrix}\label{transform2}
\end{equation}

Using equations \eqref{transform1}, \eqref{transform2} and \eqref{Vt_SG} in \eqref{Fdq_stator_final}, the final equation used to interface the synchronous generator with electrical network and load is obtained which is given by equation \eqref{IM_SG_integ}.

\begin{equation}
    \frac{1}{\omega_s}[X_{sg1}Q - QX_{nw}]\frac{d}{dt}I_{n} + \frac{1}{\omega_s}[N_1^2]\frac{d}{dt}E'_{sg} + \frac{1}{\omega_s}[X_{sg1}Q_{d}I_{n} ]\frac{d}{dt}\delta = [Z_{sg}Q + QZ_{nw}]I_{n} + \nonumber\\
    \omega[N_1^2]E'_{sg} + [Q]V_{c} + [Q]V_{m}\label{IM_SG_integ}
\end{equation}
where, in \eqref{IM_SG_integ}, the matrix $[Q_d]$ is the derivative of matrix $[Q]$ with respect to the rotor position $\delta$. The elements of matrices $[Q]$ and $[Q_d]$ are given below:

\begin{equation}
    [Q] = \begin{bmatrix}
    \sin\delta & -\cos\delta\\
    \cos\delta & \sin\delta
    \end{bmatrix}\label{matrixQ}
\end{equation}

\begin{equation}
    [Q_d] = \begin{bmatrix}
    \cos\delta & \sin\delta\\
-\sin\delta & \cos\delta
    \end{bmatrix}\label{matrixQd}
\end{equation}

\begin{figure}[t!]
    \centering
    \includegraphics[scale = 1.1]{figure/Figures_IM_DG/Exciter_Simple.pdf}
    \caption{Block diagram of AVR}
    \label{fig:AVR_simple}
\end{figure}

\section{Modeling of AVR and Governor}\label{AVR_Governor}
The block diagram of the Automatic Voltage Regulator (AVR) of the SG is shown in Fig. \ref{fig:AVR_simple}. The terminal voltage magnitude ($V_t$) of the SG is measured by a transducer of time constant $T_x$ seconds and is compared with the reference voltage $V_{ref}$. The AVR is represented using a first order transfer function consisting of a gain $K_a$ and time constant $T_a$ seconds. The speed governor dynamics are modeled using a first order differential equation which determines the mechanical torque input ($T_m$) to the rotor of the SG. The differential equations which govern the dynamics of the AVR and the speed governor are given below:

\begin{align}
    T_x\frac{dV_{x}}{dt} &= -V_x + V_t\label{eqn:Vx}\\
    T_a\frac{dE_{fd}}{dt} &= -E_{fd} + K_a(V_{ref} - V_x)\label{eqn:Efd}\\
    T_{g}\frac{dT_{m}}{dt} &= -T_{m} + P_c - (1/R_d)(\omega - 1)\label{eqn:Tm}
\end{align}

In equations \eqref{eqn:Vx} to \eqref{eqn:Tm}, $V_x$ is the output of the measurement transducer, $E_{fd}$ is the input voltage applied to the field winding on the rotor of the SG, $P_c$ is the control input which is fixed and $R_d$ is the droop of the speed governor. The values of $T_g$ and $R_d$ are assumed to be $50$ ms and $5\%$ respectively. 

\section{Eigenvalue analysis}\label{S3}
The complete state-space model of the system is developed and is linearized and cast in the form $\Delta \Dot{X} = [J]\Delta X$ where $X$ is the vector of system state variables and $\Delta X$ is the linearized state variable vector. The state variables considered in this system are the $d$ and $q$ axis components of stator and rotor currents, terminal voltage and slip of the induction motor, field and damper winding emfs, mechanical torque, rotor angle and speed of the synchronous generator, $d$ and $q$ axis components of the line current and series capacitor voltage and variables $V_x$ and $E_{fd}$ associated with the AVR as explained in Section \ref{AVR_Governor}. 
The eigenvalues of the linearized mathematical model of the system are calculated and analyzed for various values of system parameters and the results are presented in the following sections:

\subsection{Eigenvalues of base case}

\begin{figure}[t!]
    \centering
    \includegraphics[scale = 0.68]{figure/Chapter 7 Figures/Eigenvalues_BaseCase_SG_IM.pdf}
    \caption{Eigenvalue trajectory for base case}
    \label{fig:evals_basecase}
\end{figure}

Eigenvalues of the system under study are computed for the following operating conditions: the induction motor slip is at $2\%$ and the $X/R$ ratio of the line is assumed to be $20$, exciter of the SG has a gain and time constant of $500$ and $25$ ms respectively and the measurement transducer time constant of the AVR is $10$ ms. The series compensation level of the system ($K_c = X_c/X_{nw}$) is varied from $1\%$ to $95\%$.

\begin{table}[t]
    %\centering
    \caption{Eigenvalues for base case}
    \centering
    \label{table:IMDG_table1}
    {\begin{tabular}{ccc}
    \toprule
   \boldmath{$\lambda_i$} & \boldmath{$-\sigma \pm j\omega$}    &  \textbf{Participating States} \\
        \midrule
     \,\,\,\,\,\,\,\,$\lambda_1$ \,\,\,\,\,\,\,\,& \,\,\,\,\,\,\,\,$-40.188 \pm j2086.8$\,\,\,\,\,\,\,\, & \,\,\,\,\,\,\,\,\,\, ${I}_{dn}$, ${I}_{qn}$,  ${I}_{ds}$, ${I}_{qs}$, ${I}_{dr}$, ${I}_{qr}$, ${V}_{dm}$, ${V}_{qm}$\,\,\,\,\,\,\,\,\,\, \\
        \midrule 
    $\lambda_2$ & $-51.454 \pm j1330$ & ${I}_{dn}$, ${I}_{qn}$, ${I}_{ds}$, ${I}_{qs}$, ${I}_{dr}$, ${I}_{qr}$, ${V}_{dm}$, ${V}_{qm}$\\
        \midrule
    $\lambda_3$ & $-7.4056 \pm j557.92$ & ${I}_{dn}$, ${I}_{qn}$, ${I}_{ds}$, ${I}_{qs}$, ${I}_{dr}$, ${I}_{qr}$, ${V}_{dc}$, ${V}_{qc}$ \\
       \midrule
    $\lambda_4$ & $-0.2358 \pm j194.8$ & ${I}_{dn}$, ${I}_{qn}$, ${I}_{ds}$, ${I}_{qs}$, ${I}_{dr}$, ${I}_{qr}$, ${V}_{dc}$, ${V}_{qc}$\\ 
       \midrule
    $\lambda_5$ & $-7.1786 \pm j28.759$ & $I_{ds}$, $I_{qs}$, $I_{dr}$, $I_{qr}$, $E'_{q}$, $V_{x}$, $E_{fd}$\\ 
       \midrule
    $\lambda_6$ & $-10.157 \pm j20.166$ & $I_{ds}$, $I_{qs}$, $I_{dr}$, $I_{qr}$, $E'_{q}$, $s_{m}$\\ 
       \midrule
    $\lambda_7$ & $-106.99$ & $I_{ds}$, $I_{qs}$, $I_{dr}$, $I_{qr}$, $E'_{q}$, $V_{x}$, $E_{fd}$\\ 
        \midrule
    $\lambda_8$ & $-24.485$ & $I_{ds}$, $I_{qs}$, $I_{dr}$, $I_{qr}$, $E_{fd}$\\
        \midrule
    $\lambda_9$ & $-17.112$ & $I_{ds}$, $I_{qs}$, $I_{dr}$, $I_{qr}$, $T_m$, $\omega$\\
        \midrule
    $\lambda_{10}$ & $-2.7822$ & $T_m$, $\omega$\\ 
        \midrule
     $\lambda_{11}$ & $-1.4834$ & $I_{ds}$, $I_{qs}$, $I_{dr}$, $I_{qr}$, $E'_d$\\
       \bottomrule
    \end{tabular}}
\end{table}

% \begin{table*}
% \caption{Eigenvalues for base case}
% \centering
% \begin{tabular}{|c|c|c|}
% \hline
%       $\lambda_i$ & $-\sigma \pm j\omega$    &  Dominant states \\
%     \hline
%     $\lambda_1$ & $-40.188 \pm j2086.8$ &  ${I}_{dn}$, ${I}_{qn}$, ${I}_{qL}$, ${I}_{ds}$, ${I}_{qs}$, ${I}_{dr}$, ${I}_{qr}$, ${V}_{dm}$, ${V}_{qm}$ \\ 
%      \hline
%     $\lambda_2$ & $-51.454 \pm j1330$ & ${I}_{dL}$, ${I}_{dL}$, ${I}_{qL}$, ${I}_{ds}$, ${I}_{qs}$, ${I}_{dr}$, ${I}_{qr}$, ${V}_{dm}$, ${V}_{qm}$\\ 
%      \hline
%     $\lambda_3$ & $-7.4056 \pm j557.92$ & ${I}_{dL}$, ${I}_{dL}$, ${I}_{qL}$, ${I}_{ds}$, ${I}_{qs}$, ${I}_{dr}$, ${I}_{qr}$, ${V}_{dc}$, ${V}_{qc}$ \\
%      \hline
%     $\lambda_4$ & $-0.2358 \pm j194.8$ & ${I}_{dL}$, ${I}_{dL}$, ${I}_{qL}$, ${I}_{ds}$, ${I}_{qs}$, ${I}_{dr}$, ${I}_{qr}$, ${V}_{dc}$, ${V}_{qc}$\\ 
%      \hline
%     $\lambda_5$ & $-7.1786 \pm j28.759$ & $I_{ds}$, $I_{qs}$, $I_{dr}$, $I_{qr}$, $E'_{q}$, $V_{x}$, $E_{fd}$\\ 
%      \hline
%     $\lambda_6$ & $-10.157 \pm j20.166$ & $I_{ds}$, $I_{qs}$, $I_{dr}$, $I_{qr}$, $E'_{q}$, $s_{m}$\\ 
%      \hline
%     $\lambda_7$ & $-106.99$ & $I_{ds}$, $I_{qs}$, $I_{dr}$, $I_{qr}$, $E'_{q}$, $V_{x}$, $E_{fd}$\\ 
%      \hline
%     $\lambda_8$ & $-24.485$ & $I_{ds}$, $I_{qs}$, $I_{dr}$, $I_{qr}$, $E_{fd}$\\ 
%      \hline
%     $\lambda_9$ & $-17.112$ & $I_{ds}$, $I_{qs}$, $I_{dr}$, $I_{qr}$, $T_m$, $\omega$\\ 
%      \hline
%     $\lambda_{10}$ & $-2.7822$ & $T_m$, $\omega$\\ 
%      \hline
%      $\lambda_{11}$ & $-1.4834$ & $I_{ds}$, $I_{qs}$, $I_{dr}$, $I_{qr}$, $E'_d$\\ \hline
% \end{tabular}
% \label{table:IMDG_table1}
% \end{table*}

\begin{table}[t]
\caption{Critical eigenvalues for different X/R ratios}
\label{table:IMDG_Strength}
\centering 
{\begin{tabular}{cccc}
\toprule
     \textbf{X/R ratio} & \boldmath{$K_{cr}$} &  \boldmath{$\lambda_{cr} = -\sigma \pm j\omega$} & \textbf{Participating states} \\
    \midrule
    \,\,\,\,$10$\,\,\,\, & \,\,\,\,\,\,\,\,$74\%$\,\,\,\,\,\,\,\, & \,\,\,\,\,\,$0.0522 \pm j192.07$\,\,\,\,\,\, & \,\,\,\,\multirow{2}{*}{$I_{dn}$,$I_{qn}$,$I_{ds}$,$I_{qs}$,$I_{dr}$,$I_{qr}$,$V_{dc}$,$V_{qc}$}\,\,\,\, \\\cline{1-3}
    %  \hline
    $30$ & $56\%$ & $0.0792 \pm j189.43$ &  \\
    \bottomrule
\end{tabular}}
\end{table}

% \begin{table*}
% \caption{Critical eigenvalues for different X/R ratios}
% \centering
% \begin{tabular}{|c|c|c|c|}
% \hline
%     $X/R$ ratio & $K_c$ & $\lambda_{cr}$ & Dominant states \\
%     \hline
%     $10$ & $74\%$ & $0.0522 \pm j192.07$ 
%     & \multirow{2}{*}{$\Delta I_{dL}$,$\Delta I_{qL}$,$\Delta I_{ds}$,$\Delta I_{qs}$,$\Delta I_{dr}$,$\Delta I_{qr}$,$\Delta V_{dc}$,$\Delta V_{qc}$} \\\cline{1-3}
     
%     $30$ & $56\%$ & $0.0792 \pm j189.43$ &  \\
%     \hline
% \end{tabular}
% \label{table:IMDG_Strength}
% \end{table*}

The eigenvalues of the system along with the corresponding dominant state variables for each mode are presented in Table \ref{table:IMDG_table1}. for a series compensation of $60\%$. There are $6$ oscillatory modes ($\lambda_1$ to $\lambda_6$) and $5$ non-oscillatory modes ($\lambda_7$ to $\lambda_{11}$). The system starts out to be stable as the series compensation is at $1\%$ and remains stable up to $62\%$. As the compensation is increased beyond $62\%$, the system becomes unstable due to SSR as the oscillatory mode $\lambda_4$ crosses over to the right half of the complex plane resulting in an unstable system. The same analysis when repeated for a lower $X/R$ ratio of the system ($X/R = 10$) revealed a Hopf bifurcation due to SSR at a higher value of critical series compensation ($K_{cr} = 74\%$, $\lambda_{cr} = 0.0522 \pm j192.07$) and for a higher $X/R$ ratio of $30$, the critical value of series compensation is $K_{cr} = 56\%$ with $\lambda_{cr} = 0.0792 \pm j189.43$.

\subsection{Influence of AVR parameters on stability domain}

The influence of exciter parameters on the stability domain of the system with respect to series compensation level is explored in this section through eigenvalue analysis. The parameters under study are the gain of the exciter and the time constants of the AVR and measurement transducer.

\subsubsection{Influence of exciter gain}

\begin{figure}[t!]
    \centering
    \includegraphics[scale = 0.68]{figure/Chapter 7 Figures/Eigenvalues_Ka_2000.pdf}
    \caption{Eigenvalue trajectory for $K_a = 2000$}
    \label{fig:evals_Ka}
\end{figure}

\begin{figure}[t!]
    \centering
    \includegraphics[scale = 1.1]{figure/Chapter 7 Figures/Hopf_Points_ExcGain.pdf}
    \caption{Hopf bifurcation points for varying exciter gains}
    \label{fig:hopf_excgain}
\end{figure}

\begin{table}[t]
\caption{Critical eigenvalues for AVR gain of $2000$}
\label{table:IMDG_table2}
\centering 
{\begin{tabular}{ccc}
\toprule
     \boldmath{$K_{cr}$} &  \boldmath{$\lambda_{cr} = -\sigma \pm j\omega$} & \textbf{Participating states} \\
    \midrule
    \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$15\%$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, & \,\,\,\,\,\,\,\,\,\,\,\,\,\,$0.9597 \pm j75.013$\,\,\,\,\,\,\,\,\,\,\,\,\,\, & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$I_{ds}$,$I_{qs}$,$I_{dr}$,$I_{qr}$,$E'_{q}$,$V_{x}$,$E_{fd}$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \\
    %  \hline
    \midrule
    $50\%$ & $0.094 \pm j170.26$ & $I_{dn}$,$I_{qn}$,$I_{ds}$,$I_{qs}$,$I_{dr}$,$I_{qr}$,$V_{dc}$,$V_{qc}$ \\
    \bottomrule
\end{tabular}}
\end{table}

% \begin{table*}
% \caption{Critical eigenvalues for AVR gain of $2500$}
% \centering
% \begin{tabular}{|c|c|c|}
% \hline
%      $K_c$ & $\lambda_{cr}$ & Dominant states\\
%     \hline
%     $15\%$ & $0.9597 \pm j75.013$ &  $\Delta I_{ds}$,$\Delta I_{qs}$,$\Delta I_{dr}$,$\Delta I_{qr}$,$\Delta E'_{q}$,$\Delta V_{x}$,$\Delta E_{fd}$ \\ 
%      \hline
%     $50\%$ & $0.094 \pm j170.26$ & $\Delta I_{dL}$,$\Delta I_{qL}$,$\Delta I_{ds}$,$\Delta I_{qs}$,$\Delta I_{dr}$,$\Delta I_{qr}$,$\Delta V_{dc}$,$\Delta V_{qc}$\\ 
%     \hline
% \end{tabular}
% \label{table:IMDG_table2}
% \end{table*}

Eigenvalues of the system are computed for the system under study for the following operating conditions: the induction motor slip is at $2\%$ and the $X/R$ ratio of the feeder ($X_{n2}/R_{n2}$) is assumed to be $25$, exciter of the SG has a gain and time constant of $2000$ and $25$ ms respectively and the measurement transducer time constant of the AVR of the SG is $10$ ms.
The eigenvalue trajectory of the system are presented in Fig. \ref{fig:evals_Ka} as the degree of series compensation $K_c$ is varied from $1\%$ to $98\%$. The non-oscillatory modes are stable for the entire range of series compensation considered. When $K_c$ is at $1\%$, there is an oscillatory mode on the right half of the complex plane ($\lambda_{c1} = 2.2759 \pm j76.585$) making the system unstable. As the compensation increases, $\lambda_{c1}$ moves toward the left half of the complex plane and when $K_c$ exceeds $24\%$, $\lambda_{c1}$ moves into the left half of the complex plane which results in a stable system as a consequence of a reverse Hopf bifurcation. As $K_c$ further increases, the system becomes unstable once again as another oscillatory mode moves to the right half of the complex plane when $K_c$ exceeds $49\%$. The critical eigenvalues at this point are $\lambda_{c2} = 0.1471 \pm j205.38$. Instability in this case is a result of a Hopf bifurcation occurring due to SSR. Critical eigenvalues and participation factors of the system for $2$ different series compensation levels of $15\%$ and $50\%$ are presented in Table \ref{table:IMDG_table1}. 

The eigenvalues of the system were further computed for various exciter gains for $X/R$ ratios of $10$ and $30$. In Fig. \ref{fig:hopf_excgain}, the variation of Hopf bifurcation points with respect to exciter gain and series compensation levels are shown. For an AVR gain of $100$, the system with $X/R$ ratios of $10$ and $30$ loses stability through a Hopf bifurcation due to SSR when the series compensation level is $80\%$ and $60\%$ respectively. As the AVR gain increases, the percentage series compensation at which SSR occurs decreases resulting in smaller stability domains. 

As the AVR gain was increased to $1600$ for an $X/R$ ratio of $30$, the system was unstable for $K_c$ of $1\%$ to $3\%$ and when $K_c$ is increased beyond $4\%$, the system becomes stable causing a reverse Hopf bifurcation. The system remains stable for values of $K_c$ up to $47\%$ and as $K_c \geq 48\%$, the system loses stability due to SSR.  As the AVR gain is increased to $1800$, the system experiences a reverse Hopf bifurcation (RHB) and a Hopf bifurcation (HB) at $K_c$ values of $16\%$ and $47\%$ respectively. Similar phenomenon was observed for the system with $X/R$ ratio of $10$, however the stability domains were found to be much higher. For example, for an AVR gain of $1800$ in the system with $X/R = 10$, the RHB and HB points correspond to $K_c$ values of $5\%$ and $62\%$ respectively. Hence, from this it can be inferred that for higher AVR gains, value of $K_c$ at which the reverse Hopf bifurcation occurs increases which further reduces the size of the stability domain. Overall, the size of the stability domain with respect to $K_c$ is much smaller for a weaker system especially for high AVR gains.

\subsection{Influence of exciter time constant}

The time constant of the AVR has a significant impact on the stability domain of the system with respect to series compensation which would be described in this section. Eigenvalues of the system were analyzed as the series compensation is varied from $1\%$ to $98\%$ for various values of exciter time constant. This was done for systems with $X/R$ ratios of $10$ and $30$. This analysis was performed for AVR gains of $500$ and $1000$ and the results are presented in Tables \ref{tab:evals_ka_500} and \ref{tab:evals_ka_1000} respectively. Stability domain of the system with respect to $K_c$ is increased as the AVR time constant is increased from $1$ ms to $40$ ms as the Hopf bifurcation point due to SSR is moved from $K_c = 51\%$ to $K_c = 77\%$ for the system with $X/R = 10$. When the time constant is increased to $42$ ms, instability due to a Hopf bifurcation occurs due to a mode corresponding to the exciter at $K_c = 75\%$. As the time constant increases further to $70$ ms, the stability domain shrinks further as the Hopf bifurcation point due to the exciter is shifted to $K_c = 6\%$. When $T_a \geq 80$ ms, the stable operation of the system is not possible for any level of $K_c$ as the system is unstable for $K_c$ starting at $1\%$. 

The same phenomenon occurs for the system with $X/R = 30$, however, the stability domain starts decreasing when $T_a \geq 59$ ms which is higher compared to the previous case with $X/R = 10$. Stable operation of the system is impossible for $T_a \geq 100$ ms which is also higher compared to the previous case. Also, when the time constant $T_a = 59$ ms, as $K_c$ changes from $58\%$ to $59\%$, 2 pairs of complex eigenvalues ($\lambda_{c1} = 0.0277 \pm j184.7$, $\lambda_{c2} = 0.0194 \pm j23.495$ at $K_c = 59\%$) crossover from the left to the right half of the complex plane resulting in a Hopf-Hopf bifurcation. 

% \begin{table}[t!]
% \caption{Critical eigenvalues of IG1 for equal $X/R$ ratios}
% \label{table:sbm_part_XR_1}
% % \centering
% {\begin{tabular}{cccc}
% \toprule
%      \textbf{X/R ratio} & \boldmath{$K_{cr}$} &  \boldmath{$\lambda_{cr} = -\sigma \pm j\omega$} & \textbf{Participating states} \\
%     \midrule
%     \,\,\,\,\,\,\,\, $5$\,\,\,\,\,\,\,\, & \,\,\,\,\,\,\,\,$90\%$\,\,\,\,\,\,\,\, & \,\,\,\,\,\,\,\,$0.325 \pm j26.94$\,\,\,\,\,\,\,\, & $I_{ds}$,$I_{qs}$,$I_{dr}$,$I_{qr}$,$I_{d1}$,$I_{q1}$,$I_{d2}$,$I_{q2}$,$V_{dc}$,$V_{qc}$,$s_g$ \\
%     \midrule
%     %  \hline
%     $10$ & $85\%$ & $0.152 \pm j39.256$ 
%     & \multirow{2}{*}{$I_{ds}$,$I_{qs}$,$I_{dr}$,$I_{qr}$,$I_{d1}$,$I_{q1}$,$V_{dc}$,$V_{qc}$,$s_g$} \\\cline{1-3}
%     %  \hline
%     $15$ & $82\%$ & $0.308 \pm j45.832$ &  \\
%     \bottomrule
% \end{tabular}}
% \end{table}

\begin{table*}[t!]
    \centering
    \begin{tabular}{ *{5}{c} }
    \toprule
\multirow{2}{4em}{$T_a$ (ms)}   & \multicolumn{2}{c}{$X/R = 10$}
            & \multicolumn{2}{c}{$X/R = 30$}\\
                    
    \cline{2-5}
    % \midrule
   &   $K_{cr}$ ($\%$)  &   $\lambda_{cr} = -\sigma \pm j\omega$  &   $K_{cr}$ ($\%$)  &   $\lambda_{cr} = -\sigma \pm j\omega$  \\
    \midrule
    \,\,\,\,\,\,\,\,1\,\,\,\,\,\,\,\,   & \,\,\,\,\,\,\,\,  51\,\,\,\,\,\,\,\,  &   \,\,\,\,\,\,\,\,$0.0203 \pm j224.79$\,\,\,\,\,\,\,\,  & \,\,\,\,\,\,\,\,  38 \,\,\,\,\,\,\,\, &  \,\,\,\,\,\,\,\, $0.0609 \pm j223.53$  \,\,\,\,\,\,\,\, \\
    \midrule
    5   &   57 &   $0.0423 \pm j214.73$  &   43  &   $0.0962 \pm j212.61$   \\
    \midrule
    10   &   65  &   $0.0473 \pm j203.49$  &   49  &   $0.0689 \pm j201.35$   \\
    \midrule
    20   &   72  &   $0.0279 \pm j194.52$  &   54  &   $0.0034 \pm j192.75$   \\
    \midrule
    30   &   75  &   $0.027 \pm j190.88$  &   57  &   $0.0831 \pm j187.81$   \\
    \midrule
    40   &   77  &   $0.0644 \pm j188.5$  &   58  &   $0.0547 \pm j186.24$   \\
    \midrule
    50   &   54  &   $0.0019 \pm j24.798$  &   59  &   $0.0783 \pm j184.65$   \\
    \midrule
    60   &   30  &   $0.0292 \pm j23.87$  &   58  &   $0.0428 \pm j23.408$   \\
    \midrule
    70   &   6  &   $0.0012 \pm j23.116$  &   45  &   $0.0469 \pm j22.703$   \\
    \midrule
    80   &   1  &   $0.4935 \pm j22.09$  &   32  &   $0.0189 \pm j22.111$   \\
    % \hline
    % 90   &   1  &   $0.9694 \pm j21.141$  &   20  &   $0.0179 \pm j21.57$   \\
    % \hline
    % 100   &  1  &   $1.3212 \pm j20.331$  &   9  &   $0.0362 \pm j21.062$   \\
    \bottomrule
\end{tabular}
    \caption{Critical eigenvalues for various AVR time constants at an AVR gain of 500}
    \label{tab:evals_ka_500}
\end{table*}

% \begin{table*}[t!]
%     \centering
%     \begin{tabular}{| *{5}{c|} }
%     \hline
% \multirow{2}{4em}{$T_a$ (ms)}   & \multicolumn{2}{c|}{$X/R = 10$}
%             & \multicolumn{2}{c|}{$X/R = 30$}\\
                    
%     \cline{2-5}
%   &   $K_{cr}$ ($\%$)  &   $\lambda_{cr} = -\sigma \pm j\omega$  &   $K_{cr}$ ($\%$)  &   $\lambda_{cr} = -\sigma \pm j\omega$  \\
%     \hline
%     1   &   51  &   $0.0203 \pm j224.79$  &   38  &   $0.0609 \pm j223.53$   \\
%     \hline
%     5   &   57 &   $0.0423 \pm j214.73$  &   43  &   $0.0962 \pm j212.61$   \\
%     \hline
%     10   &   65  &   $0.0473 \pm j203.49$  &   49  &   $0.0689 \pm j201.35$   \\
%     \hline
%     20   &   72  &   $0.0279 \pm j194.52$  &   54  &   $0.0034 \pm j192.75$   \\
%     \hline
%     30   &   75  &   $0.027 \pm j190.88$  &   57  &   $0.0831 \pm j187.81$   \\
%     \hline
%     40   &   77  &   $0.0644 \pm j188.5$  &   58  &   $0.0547 \pm j186.24$   \\
%     \hline
%     50   &   54  &   $0.0019 \pm j24.798$  &   59  &   $0.0783 \pm j184.65$   \\
%     \hline
%     60   &   30  &   $0.0292 \pm j23.87$  &   58  &   $0.0428 \pm j23.408$   \\
%     \hline
%     70   &   6  &   $0.0012 \pm j23.116$  &   45  &   $0.0469 \pm j22.703$   \\
%     \hline
%     80   &   1  &   $0.4935 \pm j22.09$  &   32  &   $0.0189 \pm j22.111$   \\
%     % \hline
%     % 90   &   1  &   $0.9694 \pm j21.141$  &   20  &   $0.0179 \pm j21.57$   \\
%     % \hline
%     % 100   &  1  &   $1.3212 \pm j20.331$  &   9  &   $0.0362 \pm j21.062$   \\
%     \hline
% \end{tabular}
%     \caption{Critical eigenvalues for various AVR time constants at an AVR gain of 500}
%     \label{tab:evals_ka_500}
% \end{table*}

\begin{table*}[t!]
    \centering
    \begin{tabular}{ *{5}{c} }
    \toprule
\multirow{2}{4em}{$T_a$ (ms)}   & \multicolumn{2}{c}{$X/R = 10$}
            & \multicolumn{2}{c}{$X/R = 30$}\\
                    
    \cline{2-5}
   &   $K_{cr}$ ($\%$)  &   $\lambda_{cr} = -\sigma \pm j\omega$  &   $K_{cr}$ ($\%$)  &   $\lambda_{cr} = -\sigma \pm j\omega$  \\
    \midrule
    \,\,\,\,\,\,\,\,1 \,\,\,\,\,\,\,\,  & \,\,\,\,\,\,\,\,  37\,\,\,\,\,\,\,\, & \,\,\,\,\,\,\,\,  $0.1122 \pm j247.35$\,\,\,\,\,\,\,\,  &  \,\,\,\,\,\,\,\, 27 \,\,\,\,\,\,\,\, &  \,\,\,\,\,\,\,\, $0.1024 \pm j247.53$ \,\,\,\,\,\,\,\,  \\
    \midrule
    10   &   56  &   $0.0463 \pm j215.27$  &   42  &   $0.0250 \pm j213.76$   \\
    \midrule
    20   &   66  &   $0.0725 \pm j201.77$  &   50  &   $0.1073 \pm j199.22$   \\
    \midrule
    30   &   70  &   $0.0172 \pm j196.79$  &   53  &   $0.0520 \pm j194.2$   \\
    \midrule
    40   &   73  &   $0.0694 \pm j193.13$  &   55  &   $0.0609 \pm j190.94$   \\
    \midrule
    50   &   36  &   $0.0147 \pm j36.324$  &   56  &   $0.032 \pm j189.36$   \\
    \bottomrule
\end{tabular}
    \caption{Critical eigenvalues for various AVR time constants at an AVR gain of 1000}
    \label{tab:evals_ka_1000}
\end{table*}

The exciter gain is increased to $1000$ and the same analysis was performed on the systems with $X/R$ ratios of $10$ and $30$ and the results are presented in Table \ref{tab:evals_ka_1000}. The stability domain with respect to $K_c$ is greater for the system with $X/R = 10$ up to a value of $T_a$ close to $40$ ms compared to the system with $X/R = 30$. When $T_a = 50$ ms, the system with $X/R = 10$ has a much smaller stability domain as the system loses stability due to a Hopf bifurcation with the exciter mode causing instability at $K_c = 36\%$. This is much smaller compared to the Hopf bifurcation point which corresponds to $K_c = 56\%$ due to SSR for the system with $X/R = 30$. Also note that as $T_a$ varies from $1$ to $40$ ms, the stability domain increases as the critical value of $K_c$ causing a Hopf bifurcation due to SSR increases for each of the systems with $X/R$ ratios of $10$ and $30$ respectively. Furthermore, the system also undergoes a reverse Hopf bifurcation for $T_a\geq64$ ms at smaller values of $K_c$ in addition to the Hopf bifurcation which occurs due to SSR as $K_c$ is increased. This shows that for it may be impossible to operate a weak system in a stable manner without a minimum level of series compensation.  

\subsection{Influence of time constant of measurement transducer}

\begin{table*}[t!]
\centering
    \begin{tabular}{ *{5}{c} }
    \toprule
\multirow{2}{4em}{$T_x$ (ms)}   & \multicolumn{2}{c}{$X/R = 10$}
            & \multicolumn{2}{c}{$X/R = 30$}\\
                    
    \cline{2-5}
   &   $K_{cr}$ ($\%$)  &   $\lambda_{cr} = -\sigma \pm j\omega$  &   $K_{cr}$ ($\%$)  &   $\lambda_{cr} = -\sigma \pm j\omega$  \\
    \midrule
    \,\,\,\,\,\,\,\,1\,\,\,\,\,\,\,\,   &  \,\,\,\,\,\,\,\, 64 \,\,\,\,\,\,\,\, & \,\,\,\,\,\,\,\,  $0.052 \pm j205.81$ \,\,\,\,\,\,\,\, & \,\,\,\,\,\,\,\,  48\,\,\,\,\,\,\,\,  & \,\,\,\,\,\,\,\,  $0.0699 \pm j204$ \,\,\,\,\,\,\,\,  \\
    \midrule
    5   &   69 &   $0.0862 \pm j198.52$  &   52  &   $0.0973 \pm j196.32$   \\
    \midrule
    10   &   74  &   $0.0522 \pm j192.07$  &   56  &   $0.0792 \pm j189.43$   \\
    \midrule
    15   &   77  &   $0.0866 \pm j188.41$  &   58  &   $0.0713 \pm j186.17$   \\
    \midrule
    20   &   78  &   $0.0426 \pm j187.26$  &   59  &   $0.0583 \pm j184.59$   \\
    \midrule
    25   &   56  &   $0.014 \pm j26.955$  &   60  &   $0.0919 \pm j183.01$   \\
    \midrule
    30   &   30  &   $0.0039 \pm j26.951$  &   60  &   $0.0459 \pm j183.06$   \\
    \midrule
    35   &   1  &   $0.0933 \pm j27.902$  &   47  &   $0.0412 \pm j26.127$   \\
    \bottomrule
\end{tabular}
    \caption{Critical eigenvalues for various measurement time constants at an AVR gain of 500}
    \label{tab:evals_tx_ka_500}
\end{table*}

\begin{table*}[t!]
    \centering
    \begin{tabular}{ *{5}{c} }
    \toprule
\multirow{2}{4em}{$T_x$ (ms)}   & \multicolumn{2}{c}{$X/R = 10$}
            & \multicolumn{2}{c}{$X/R = 30$}\\
                    
    \cline{2-5}
   &   $K_{cr}$ ($\%$)  &   $\lambda_{cr} = -\sigma \pm j\omega$  &   $K_{cr}$ ($\%$)  &   $\lambda_{cr} = -\sigma \pm j\omega$  \\
    \midrule
    \,\,\,\,\,\,\,\,1\,\,\,\,\,\,\,\,   & \,\,\,\,\,\,\,\,  53 \,\,\,\,\,\,\,\, &  \,\,\,\,\,\,\,\, $0.0661 \pm j221.22$ \,\,\,\,\,\,\,\, &  \,\,\,\,\,\,\,\, 39 \,\,\,\,\,\,\,\, &  \,\,\,\,\,\,\,\, $0.004 \pm j221$  \,\,\,\,\,\,\,\, \\
    \midrule
    5   &   61 &   $0.1190 \pm j208.75$  &   46  &   $0.0354 \pm j206.65$   \\
    \midrule
    10   &   68  &   $0.0008 \pm j199.28$  &   52  &   $0.1094 \pm j195.83$   \\
    \midrule
    15   &   73  &   $0.1027 \pm j192.97$  &   55  &   $0.084 \pm j190.8$   \\
    \bottomrule
\end{tabular}
    \caption{Critical eigenvalues for various measurement time constants at an AVR gain of 1000}
    \label{tab:evals_tx_ka_1000}
\end{table*}

Eigenvalue analysis was conducted for the system with $X/R$ ratios of $10$ and $30$ for various values of time constant of the measurement transducer. Just as it was described in the previous section with respect to $T_a$, here, the stability domain increases for both the systems with $X/R$ ratios of $10$ and $30$ as $T_x$ increases up to $20$ ms. Beyond that, as $T_x\geq25$, the system with $X/R = 10$ becomes unstable at lower values of $K_c$ due to the exciter mode. This phenomenon is observed in the system with $X/R = 30$ as well, however, at a higher value of $T_x$ at $35$ ms. Hence, up to a certain value of $T_x$, the stability domain of the stronger system is bigger than that of the weaker system. However, beyond a certain value of $T_x$, the weaker system has a larger stability domain. As the AVR gain is increased to $1000$, the system shows similar behavior. In addition to Hopf bifurcations due to SSR and exciter mode destabilization, reverse Hopf bifurcations occur as well at small values of $K_c$. Hence, systems using AVRs with high gains need a minimum level of $K_c$ in order to have stable operation. Depending on the values of $T_x$ and $T_a$, the upper limit of $K_c$ is determined either by SSR or exciter mode destabilization. 

\section{Time domain simulation results}\label{S4} 
The nonlinear dynamic model of the system is programmed in SIMULINK and the results generated are presented in this section. The responses of the system with respect to time are studied for AVR gain $K_a$ of $500$ and different values of time constants of the AVR control system. The series compensation level for the system is at $50\%$ which is realistic for a weak system with an $X/R$ ratio of $30$. 

\subsection{Exciter time constant $T_a = 10\hspace{0.1cm}$ms}

\begin{figure}[t!]
\centering
\begin{subfigure}{0.5\textwidth}
  \centering
  % include first image
  \includegraphics[scale = 0.37]{figure/Chapter 7 Figures/IM_Speed_Ta_10ms.pdf}  \caption{Induction motor speed}
  \label{fig:speed_Ta_10_full}
\end{subfigure}
\quad
\begin{subfigure}{0.5\textwidth}
  \centering
  % include second image
  \includegraphics[scale = 0.4]{figure/Chapter 7 Figures/IM_Voltage_Ta_10ms.pdf}  
  \caption{Motor terminal voltage}
  \label{fig:voltage_Ta_full}
\end{subfigure}
\caption{Time responses for the system without AVR limits}
\label{Timeresponse_Ta_nolim}
\end{figure}

The responses of induction motor speed and terminal voltage are shown in Fig.\ref{Timeresponse_Ta_nolim} for a disturbance in the form of a $20\%$ increase in the load torque on the induction motor. In this case, the  AVR limits have not been modeled as a result of which, oscillations in the voltage increase with time. This is a consequence of a subcritical Hopf bifurcation as a result of SSR which causes excessive voltages and currents. 

\begin{figure}[t!]
\centering
\begin{subfigure}{0.55\textwidth}
  \centering
  % include first image
  \includegraphics[scale = 0.37]{figure/Chapter 7 Figures/IM_Speed_Ta_10ms_Limits.pdf} 
  \caption{Induction motor speed}
  \label{fig:speed_Ta_10_full_limit}
\end{subfigure}
\quad
\begin{subfigure}{0.55\textwidth}
  \centering
  % include second image
  \includegraphics[scale = 0.4]{figure/Chapter 7 Figures/IM_Voltage_Ta_10ms_Limits.pdf} 
  \caption{Motor terminal voltage}
  \label{fig:voltage_Ta_full_limit}
\end{subfigure}
\caption{Time responses for the system with AVR limits}
\label{Timeresponse_Ta_limit}
\end{figure}

\begin{figure*}[t!]
\centering
\includegraphics[scale = 0.48]{figure/Chapter 7 Figures/PhasePlane_Ta_10ms.pdf}
\caption{Phase plane trajectory of induction motor speed for $T_a = 10\hspace{0.1cm}$ms}
\label{PP_Ta_10ms}
\end{figure*}

Limits on the excitation system have been imposed and the simulations were repeated and the time responses are presented in Fig. \ref{Timeresponse_Ta_limit}. Exciter limits impose an upper and lower limit on the field voltage applied to the rotor winding of the synchronous machine resulting in an arrest on the growth of oscillations in the system states. The system now behaves as if the nature of the bifurcation is supercritical resulting in sustained oscillations of fixed amplitude with respect to time. Phase plane trajectory of the induction motor speed  in Fig. \ref{PP_Ta_10ms} reveal an unstable system with growing oscillations if the AVR limits are neglected and a system with oscillations bounded in magnitude if the AVR limits are taken into account.

\subsection{Exciter time constant $T_a = 70\hspace{0.1cm}$ms}

\begin{figure}[t!]
\captionsetup{justification=centering}
\centering
\begin{subfigure}{0.5\textwidth}
\captionsetup{justification=centering}
  \centering
  % include first image
  \includegraphics[scale = 0.1]{figure/Chapter 7 Figures/IM_Speed_Ta_70ms.jpg} 
  \caption{Induction motor speed}
  \label{fig:speed_Ta_70}
\end{subfigure}

\begin{subfigure}{0.5\textwidth}
\captionsetup{justification=centering}
  \centering
  % include second image
  \includegraphics[scale = 0.1]{figure/Chapter 7 Figures/IM_Voltage_70ms.jpg}  
  \caption{Motor terminal voltage}
  \label{fig:voltage_Ta_70ms}
\end{subfigure}
\caption{Time responses for the system without AVR limits}
\label{Timeresponse_Ta_70ms}
\end{figure}

Time traces of induction motor speed and terminal voltage are presented in Fig. \ref{Timeresponse_Ta_70ms} when the AVR limits of the synchronous machine are neglected. The system shows an oscillatory response bounded in magnitude characterizing a supercritical Hopf bifurcation. The amplitude of oscillations appear to continually change within a bound for the quantities presented. However, with the AVR limits included, the oscillations appear to be bounded with fixed amplitude characterizing a supercritical bifurcation in this case as well. These are evident from the time responses of induction motor speed and terminal voltage as shown in Fig. \ref{Timeresponse_Ta_70_limit}.

\begin{figure}[t!]
\centering
\begin{subfigure}{0.5\textwidth}
  \centering
  % include first image
  \includegraphics[scale = 0.1]{figure/Chapter 7 Figures/IM_Speed_Ta_70ms_Limits.jpg} 
  \caption{Induction motor speed}
  \label{fig:speed_Ta_70_limit}
\end{subfigure}

\begin{subfigure}{0.5\textwidth}
  \centering
  % include second image
  \includegraphics[scale = 0.1]{figure/Chapter 7 Figures/IM_Voltage_Ta_70ms_Limts.jpg}  
  \caption{Motor terminal voltage}
  \label{fig:voltage_Ta_70_limit}
\end{subfigure}
\caption{Time responses for the system with AVR limits}
\label{Timeresponse_Ta_70_limit}
\end{figure}

\begin{figure*}[t!]
\centering
\includegraphics[scale = 0.48]{figure/Chapter 7 Figures/Phase_Plane_Ta_70ms.pdf}
\caption{Phase plane trajectory of induction motor speed for $T_a = 70\hspace{0.1cm}$ms}
\label{PP_Ta_70ms}
\end{figure*}

Phase-plane trajectories of induction motor speed when the AVR time constant is set at $70$ ms are presented in Fig. \ref{PP_Ta_70ms}. In the case where the AVR limits are neglected, a supercritical Hopf bifurcation is observed in the system which appears to undergo repeated period doublings. This is evident in the one to the left in Fig. \ref{PP_Ta_70ms}. The amplitude of oscillations although bounded appear to change continuously in a random manner which is the characteristic of 'quasi-chaos'. However, with the AVR limits modeled, the phase-plane trajectory of induction motor speed shows a well-defined limit cycle and the absence of a period doubling bifurcation. This is due to the fact that with AVR limits modeled, the voltage applied to the field winding is allowed to vary only between $\pm3$ pu. This limit is much lower than the minimum amplitude of oscillations in $E_{fd}$ when it varies in a random manner when AVR limits are neglected. This imposes a bound on the variations in the generator terminal voltage which is reflected at the motor terminal voltage and impacts the speed of the motor which varies only within a limit.

\begin{figure}
\centering
\includegraphics[scale = 0.85]{figure/Chapter 7 Figures/PDB_Chaos.pdf}
\caption{Period doublings for various step changes in load torque}
\label{PDB_Chaos}
\end{figure}

Further investigations were carried out with regards to the period-doubling bifurcation observed in the system when AVR limits were neglected. The system was subject to different step increases in the induction motor load torque and phase-plane trajectories of the induction motor speed are plotted and are shown in Fig. \ref{PDB_Chaos}. With the system subject to a $25\%$ increase in the load torque, the number of period doublings are hard to count as the trajectories are so close to each other. As the step change was increased to $30\%$, $2$ period doublings can be observed. The number of period doublings reduced to $1$ when the load torque disturbance was increased to $35\%$ and finally, the period doubling bifurcation is found to be absent when the load torque disturbance magnitude was increased to $45\%$.

\section{Conclusion}
In this chapter, the dynamic analysis of induction motor loads fed through weak interconnections was performed through analyzing eigenvalues of the system and time domain simulations. Control system parameters of the AVR have a significant impact on the stability domain with respect to series compensation. The AVR gain, time constants of AVR and measurement transducers have to be taken into account while determining percentage series compensation to avoid instability due to subsynchronous resonance or exciter mode destabilization. Depending upon the speed of response of the AVR and measurement transducer, the stability domain of the weaker system may be bigger or smaller than that of a relatively stronger system. This shows that a supplementary controller is needed  to operate the system in a stable manner. 
