

\chapter{Stability Domain Analysis of Power Systems with Induction Motor Loads supplied through Weak Interconnections}\label{Chap6}

In this chapter, the dynamic modeling and stability domain analysis of a system consisting of an induction motor load supplied supplied through a series compensated feeder has been presented. The topology of the network is affected by the presence of dynamic loads such as induction motors which impacts the stability of the system. Induction motors are utilized in a wide variety of industrial applications like steel rolling mills, compressors, elevators, etc due to it's rugged construction, high efficiency and low maintenance (\cite{Undrill2003}, \cite{8039036}). These motor loads are supplied using soft starters to minimize the effects like high starting currents in the system leading to higher voltage drops. However, as the motor picks up speed, the starter is bypassed and full line voltage appears across the motor terminals under steady state. In this chapter, the stability of the system with induction motor loads supplied through a series compensated network is studied when the motor is perturbed from it's steady state operating point. 

It is very important to understand the impact of these loads on the performance of the power system especially if the interconnection is weak. The steady state stable operating region on the torque-slip characteristics of the induction motor is limited by the impedance of the network in addition to the motor parameters. Series capacitors can be used to enhance the stable operating range of the motor by compensating a part of the reactance of the feeder, thus enabling the motor to operate at a higher slip to supply greater load torques. However, series compensation impacts the dynamic stability of the system when the motor is perturbed from its steady state operation leading to instabilities. The main goal of this work is to reveal bifurcations which arise in these systems as a result of the weak interconnection between the source and load. The impact of X/R ratio of the feeder and induction motor loading on the stability domain with respect to series compensation has been examined through eigenvalue analysis. 

\section{Mathematical modeling of system}\label{MathModel_Chapter6}

\begin{figure}[h]
    \centering
    \includegraphics[scale = 1.5]{figure/Figures IM Grid connection/IM_load_GridConnected_SLD.pdf}
    \caption{Single line diagram of system}
    \label{fig:IM_sld1}
\end{figure}

The single line diagram of system under study is shown in Fig. \ref{fig:IM_sld1}. The induction motor load is fed from a substation directly through a series compensated feeder. The induction motor also has a shunt capacitor at its terminals to provide reactive power compensation. The dynamic equations of the $d$ and $q$ components of the network currents are given by:

\begin{align}
    \frac{X_{net}}{\omega_s}\frac{d}{dt}\{I_{n}\} &= -[Z_{N}]\{I_{n}\} - \{V_{c}\} - \{V_{s}\} + \{V_{g}\}\label{line_current_IM}
\end{align}

where $[Z_{N}] = [Z_{T1} + Z_G + Z_{T2}]$, $X_{net} = X_{t1} + X_g + X_{t2}$  and 

\begin{align}
    [Z_{T1}] = \begin{bmatrix}R_{t1} & -X_{t1}\\X_{t1} & R_{t1}\end{bmatrix},
    [Z_{G}] = \begin{bmatrix}R_g & -X_g\\X_g & R_g\end{bmatrix},
    [Z_{T2}] = \begin{bmatrix}R_{t2} & -X_{t2}\\X_{t2} & R_{t2}\end{bmatrix}\nonumber
\end{align}.

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

\begin{align}
    \{I_{n}\} &= [I_{dn} \hspace{0.35cm} I_{qn}]^{T}\\
    \{V_{s}\} &= [V_{ds} \hspace{0.35cm} V_{qs}]^{T}\\
    \{V_{c}\} &= [V_{dc} \hspace{0.35cm} V_{qc}]^{T}\\
    \{V_{g}\} &= [V_{dg} \hspace{0.35cm} V_{qg}]^{T}
\end{align}

The dynamics of the voltage across the series capacitor is modeled using the equation given below:

\begin{align}
    \frac{1}{\omega_s}\frac{d}{dt}\{V_{c}\} = [Z_{se}]\{I_{n}\} + [N_1]\{V_{c}\}\label{Vse_gridconnected_IM}
\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_{s}\} = [Z_{sh}]\{I_{n} - I_{s}\} + [N_1]\{V_{s}\}\label{Vsh_gridconnected_IM}
\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.

\section{Linearization of mathematical model}

Linearized versions of dynamic equations \eqref{line_current_IM}, \eqref{Vse_gridconnected_IM} and \eqref{Vsh_gridconnected_IM} are presented below:

\begin{align}
    \frac{1}{\omega_s}[U]p\{\Delta V_{s}\} &= [N_1]\Delta V_{s} +  [Z_{sh}]\{\Delta I_{n}-\Delta I_{s}\}\label{shunt_IM_linearized}\\
    \frac{1}{\omega_s}[U]p\{\Delta V_{c}\} &= [N_1]\Delta V_{c} +  [Z_{se}]\{\Delta I_{n}\}\label{series_IM_linearized}\\
     \frac{X_{net}}{\omega_s}[U]p\{\Delta I_{n}\} &= -[Z_{nw}]\{\Delta I_{n}\} - \{\Delta V_{c}\} - \{\Delta V_{s}\} + \{\Delta V_{g}\}\label{Line_IM_linearized}
\end{align}

In equation \eqref{Line_IM_linearized}, $\{\Delta V_{g}\} = 0$ since the voltage of the substation is assumed to be fixed under equilibrium conditions. 
\section{Eigenvalue analysis of system}

The complete linearized system model comprises of equations \eqref{shunt_IM_linearized}, \eqref{series_IM_linearized}, \eqref{Line_IM_linearized} along with \eqref{Stator_IM_linearization}, \eqref{rotor_IM_linearization} and \eqref{swing_IM_linearized}. The state variables of the system under study are listed 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_m\}^T
\end{align}

In this section, eigenvalues of the linearized system are presented for various operating conditions and system parameters as the percentage series compensation $K_c = X_c/X_{net}$ is varied from $1\%$ to $98\%$ where network reactance $X_{net} = X_{t1} + X_g + X_{t2} $ and $X_c$ is the reactance of the series capacitor. 

\subsection{Eigenvalues for base case}

\begin{figure}[t!]
    \centering
    \includegraphics[scale = 0.675]{figure/Chapter 6 Figures/Eigenvalues_BaseCase.pdf}
    \caption{Eigenvalue trajectories for base case}
    \label{fig:evas_basecase}
\end{figure}

\begin{table}[t]
    %\centering
    \caption{Eigenvalues of induction motor load for base case}
    \label{tab:evals_slip}
    {\begin{tabular}{ccc}
    \toprule
   \boldmath{$\lambda_i$} & \boldmath{$-\sigma \pm j\omega$}    &  \textbf{Participating States} \\
        \midrule
     \,\,\,\,\,\,\,\,$\lambda_1$ \,\,\,\,\,\,\,\,& \,\,\,\,\,\,\,\,\,\,\,\, $-31.904 \pm j2390.8$\,\,\,\,\,\,\,\,\,\,\,\,  & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$I_{dn}$,$I_{qn}$,$I_{ds}$,$I_{qs}$,$I_{dr}$,$I_{qr}$,$V_{ds}$,$V_{qs}$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \\
        \midrule 
    $\lambda_2$ & $-38.527 \pm j1636.7$ & $I_{dn}$,$I_{qn}$,$I_{ds}$,$I_{qs}$,$I_{dr}$,$I_{qr}$,$V_{ds}$,$V_{qs}$\\
        \midrule
    $\lambda_3$ & $-18.02 \pm j583.89$ & $I_{dn}$,$I_{qn}$,$I_{ds}$,$I_{qs}$, $I_{dr}$,$I_{qr}$,$V_{dc}$,$V_{qc}$\\
       \midrule
    $\lambda_4$ & $0.3674 \pm j168.60$ & $I_{dn}$,$I_{qn}$,$I_{ds}$,$I_{qs}$,$I_{dr}$,$I_{qr}$,$V_{dc}$,$V_{qc}$\\
       \midrule
    $\lambda_5$ & $-32.065$ & $I_{ds}$,$I_{qs}$,$I_{dr}$,$I_{qr}$\\
       \midrule
    $\lambda_6$ & $-18.042 \pm j18.618$ & $I_{ds}$,$I_{qs}$,$I_{dr}$,$I_{qr}$,$s_m$\\
      \bottomrule
    \end{tabular}}
\end{table}

\begin{table}[t]
    %\centering
    \caption{Participation factors of $\lambda_{cr}$ $50\%$ compensation}
    \label{table:IM_load_BaseCase}
    {\begin{tabular}{ cccccccc } 
     \toprule
 \,\,\,\,\,\,\boldmath{$I_{ds}$}\,\,\,\,\,\, &\,\,\,\,\,\, \boldmath{$I_{qs}$}\,\,\,\,\,\,& \,\,\,\,\,\,\boldmath{$I_{dr}$} \,\,\,\,\,\,&\,\,\,\,\,\, \boldmath{$I_{qr}$} \,\,\,\,\,\, &\,\,\,\,\,\, \boldmath{$I_{dn}$}\,\,\,\,\,\,&\,\,\,\,\,\,\boldmath{$I_{qn}$} \,\,\,\,\,\,&\,\,\,\,\,\, \boldmath{$V_{dc}$} \,\,\,\,\,\,&\,\,\,\, \boldmath{$V_{qc}$} \,\,\,\,\\ 
 \midrule
0.246 & 0.241 & 0.242 & 0.237 & 0.007 & 0.006 & 0.011 & 0.011 \\
 \bottomrule
\end{tabular}}
\end{table}

% \begin{table}
% \caption{Participation factors of $\lambda_{cr}$ $50\%$ compensation}
% \centering
% \begin{tabular}{ |c|c|c|c|c|c|c|c| } 
%  \hline
%  $I_{ds}$ & $I_{qs}$ & $I_{dr}$ & $I_{qr}$ & $I_{dn}$ & $I_{qn}$ & $V_{dc}$ & $V_{qc}$ \\ 
%  \hline
% 0.246 & 0.241 & 0.242 & 0.237 & 0.007 & 0.006 & 0.011 & 0.011 \\ 
% \hline
% \end{tabular}
% \label{table:IM_load_BaseCase}
% \end{table}

For the system under study, eigenvalues are calculated for a slip of $1\%$ for the induction motor and for a feeder of $X/R$ ratio of $15$ and are plotted on the complex plane as the percentage series compensation is increased from $1\%$ to $98\%$ which can be seen in Fig. \ref{fig:evas_basecase}. From the figure, it is observed that there are 5 oscillatory modes ($\lambda_1$, $\lambda_2$, $\lambda_3$, $\lambda_4$ and $\lambda_6$) characterized by complex conjugate eigenvalues and one non-oscillatory mode ($\lambda_5$) characterized by a real eigenvalue. Among the oscillatory modes, $3$ modes ($\lambda_1$, $\lambda_2$, $\lambda_3$) have frequencies greater than $60$ Hz and $2$ modes ($\lambda_4$, $\lambda_6$)  less than $60$ Hz. For the entire range of percentage series compensation considered, modes $\lambda_1$, $\lambda_2$, $\lambda_3$, $\lambda_5$ and $\lambda_6$ are stable. However, mode $\lambda_4$ starts moving towards the right half of the complex plane as the percentage series compensation increases. The system loses stability due to a Hopf bifurcation which occurs as a result of $\lambda_4$ crossing over from the left to the right half of the complex plane when the percentage series compensation is between $48\%$ and $49\%$. This makes $\lambda_4$ the critical mode which will be referred to as $\lambda_{cr}$. Table \ref{tab:evals_slip} lists the eigenvalues of the system at $50\%$ series compensation and the participation factors for $\lambda_{cr}$ have been listed in Table \ref{table:IM_load_BaseCase}. Observing the participation factors corresponding to the critical mode, it is obvious that the instability in the system is due to SSR as there is participation from the stator and rotor currents of the motor and the series capacitor voltage.

\subsection{Eigenvalues at different X/R ratios of the radial feeder}

\begin{figure}[t!]
    \centering
    \includegraphics[scale = 0.65]{figure/Chapter 6 Figures/Eigenvalues_XRratios.pdf}
    \caption{Eigenvalue trajectories at different X/R ratios}
    \label{fig:evals_strength}
\end{figure}

\begin{figure}[t!]
    \centering
    \includegraphics[scale = 1.2]{figure/Chapter 6 Figures/Hopf_Points.pdf}
  \caption{Hopf bifurcation points at different X/R ratios}
  \label{fig:hopf_points}
\end{figure}

% \begin{figure}[t]
% \centering
% \begin{minipage}{.5\textwidth}
%   \centering
%   \includegraphics[scale = 0.5]{figure/Figures IM Grid connection/Evals_Grid_strength.pdf}
%   \captionof{figure}{\centering Eigenvalues for different grid strengths}
%   \label{fig:evals_strength}
% \end{minipage}%
% \begin{minipage}{.5\textwidth}
%   \centering
%   \includegraphics[scale = 0.5]{figure/Figures IM Grid connection/Hopf_Bifurcation_XR_points.pdf}
%   \captionof{figure}{\centering Hopf bifurcation points at different $X/R$ ratios}
%   \label{fig:hopf_points}
% \end{minipage}
% % \caption*{Dynamic torque-speed characteristics - response to a frequency disturbance}
% \end{figure}

\begin{table}[t!]
\caption{Critical eigenvalues for different X/R ratios at $1\%$ slip}
\label{table:IMload_XR_lightload}
% \centering
{\begin{tabular}{cccc}
\toprule
     \textbf{X/R ratio} & \boldmath{$K_{cr}$} &  \boldmath{$\lambda_{cr} = -\sigma \pm j\omega$} & \textbf{Participating states} \\
    \midrule
    \,\,\,\,\,\,\,\,$10$\,\,\,\,\,\,\,\, & \,\,\,\,\,\,\,\,$54\%$\,\,\,\,\,\,\,\, & \,\,\,\,\,\,\,\,$0.130 \pm j170.2$\,\,\,\,\,\,\,\, & \,\,\,\,\,\,\,\,\,\,\,\,\multirow{3}{*}{$I_{dn}$,$I_{qn}$,$I_{ds}$,$I_{qs}$,$I_{dr}$,$I_{qr}$,$V_{dc}$,$V_{qc}$}\,\,\,\,\,\,\,\,\,\,\,\, \\\cline{1-3}
    %  \hline
    $20$ & $46\%$ & $0.094 \pm j170.26$ & \\\cline{1-3}
    $30$ & $42\%$ & $0.072 \pm j170.29$ & \\
    \bottomrule
\end{tabular}}
\end{table}

% \begin{table}[h]
% \caption{Critical eigenvalues for different X/R ratios at $1\%$ slip}
% \centering
% \begin{tabular}{|c|c|c|c|}
% \hline
%     $X/R$ ratio & $K_c$ & $\lambda_{cr}$ & PARTICIPATING STATES \\
%     \hline
%     $10$ & $54\%$ & $0.130 \pm j170.2$ &  \multirow{3}{*}{$\Delta I_{dn}$,$\Delta I_{qn}$,$\Delta I_{ds}$,$\Delta I_{qs}$,$\Delta I_{dr}$,$\Delta I_{qr}$,$\Delta V_{dc}$,$\Delta V_{qc}$} \\ \cline{1-3}
     
%     $20$ & $46\%$ & $0.094 \pm j170.26$ & \\ \cline{1-3}
    
%     $30$ & $42\%$ & $0.072 \pm j170.29$ &  \\
%     \hline
% \end{tabular}
% \label{table:IMload_XR_lightload}
% \end{table}

\begin{table}[t!]
\caption{Critical eigenvalues for different X/R ratios at $3\%$ slip}
\label{table:IMload_XR_heavyload}
% \centering
{\begin{tabular}{cccc}
\toprule
     \textbf{X/R ratio} & \boldmath{$K_{cr}$} &  \boldmath{$\lambda_{cr} = -\sigma \pm j\omega$} & \textbf{Participating states} \\
    \midrule
   \,\,\,\,\,\,\,\,$10$\,\,\,\,\,\,\,\, & \,\,\,\,\,\,\,\,$52\%$\,\,\,\,\,\,\,\, & \,\,\,\,\,\,\,\,$0.153 \pm j174.19$\,\,\,\,\,\,\,\, & \,\,\,\,\,\,\,\,\multirow{3}{*}{$I_{dn}$,$I_{qn}$,$I_{ds}$,$I_{qs}$,$I_{dr}$,$I_{qr}$,$V_{dc}$,$V_{qc}$}\,\,\,\,\,\,\,\, \\\cline{1-3}
    %  \hline
    $20$ & $44\%$ & $0.017 \pm j174.9$ &  \\\cline{1-3}
    $30$ & $41\%$ & $0.247 \pm j172.91$ &   \\
    \bottomrule
\end{tabular}}
\end{table}

% \begin{table}[h]
% \caption{Critical eigenvalues for different X/R ratios at $3\%$ slip}
% \centering
% \begin{tabular}{|c|c|c|c|}
% \hline
%     $X/R$ ratio & $K_c$ & $\lambda_{cr}$ & PARTICIPATING STATES \\
%     \hline
%     $10$ & $52\%$ & $0.153 \pm j174.19$ &  \multirow{3}{*}{$\Delta I_{dn}$,$\Delta I_{qn}$,$\Delta I_{ds}$,$\Delta I_{qs}$,$\Delta I_{dr}$,$\Delta I_{qr}$,$\Delta V_{dc}$,$\Delta V_{qc}$} \\ \cline{1-3}
     
%     $20$ & $44\%$ & $0.017 \pm j174.9$ & \\ \cline{1-3}
    
%     $30$ & $41\%$ & $0.247 \pm j172.91$ &  \\
%     \hline
% \end{tabular}
% \label{table:IMload_XR_heavyload}
% \end{table}

An increase in $X/R$ ratio for a given vaue of resistance reduces the short circuit capacity of the system due to an increase in the impedance of the network as a result of which the strength of the interconnection decreases. Presence of a series capacitor improves the short circuit ratio by increasing the short circuit capacity by reducing the net reactance of the system thereby resulting in an increase in the system strength while simultaneously extending the steady state stable operating range of the motor. However, the upper limit of percentage series compensation is determined by the Hopf bifurcation point which is influenced by the $X/R$ ratio of the feeder. Eigenvalues of the linearized system are calculated and plotted on the complex plane which is shown in Fig. \ref{fig:evals_strength} when the series compensation is varied for $X/R$ ratios of $10$, $20$ and $30$ with the induction motor operating at a slip of $1\%$. The system with an $X/R$ ratio of 30 becomes unstable at a smaller value of $K_c$.  This means the stability domain of the system with respect to series compensation shrinks as the system becomes weaker due to high values of line impedance. Critical eigenvalues of the system are listed in Table \ref{table:IMload_XR_lightload}. When eigenvalue analysis was conducted for a higher slip of $3\%$, the critical series compensation at which the system becomes unstable is slightly lower for the same $X/R$ ratios considered and this can be observed from Table \ref{table:IMload_XR_heavyload} and from Fig.\ref{fig:hopf_points}. 

\section{Time domain simulation results}
The non-linear dynamic equations of the system (\eqref{line_current_IM}, \eqref{Vse_gridconnected_IM}, \eqref{Vsh_gridconnected_IM}, \eqref{Stator_IM}, \eqref{rotor_IM} and \eqref{swing_IM}) are programmed in SIMULINK.   responses of the system are plotted for a disturbance in the grid frequency injected in the same way as described in Chapter \ref{Chap4} using the exponential function $f(t) = f_o - exp(-\tau t)\sin(\omega_n t)$ where $f_o = 60$ Hz, $\tau = 1.95$, $f_n = 1.2$ Hz and $\omega_n = 2\pi f_n$ rad/s. The series compensation of the system is set at $50\%$ and a slip of $1\%$ is considered for the induction motor. The time responses of the system to a step change in the grid voltage was very similar in shape and hence are not shown here. 

\begin{figure}[t!]
    \centering
    \includegraphics[scale = 0.5]{figure/Chapter 6 Figures/StatorVoltage_XRratio.pdf}
  \caption{Stator voltage response to a disturbance in grid frequency for (a) X/R ratio of 20 (b) X/R ratio of 30}
  \label{fig:IM_Vs_XR}
\end{figure}

\begin{figure}[t!]
    \centering
    \includegraphics[scale = 0.5]{figure/Chapter 6 Figures/LineCurrent_XRratio.pdf}
  \caption{Line current response to a disturbance in grid frequency for (a) X/R ratio of 20 (b) X/R ratio of 30}
  \label{fig:IM_IL_XR}
\end{figure}

% \begin{figure}
% \centering
% \begin{minipage}{.5\textwidth}
%   \centering
%   \includegraphics[scale = 0.51]{figure/Figures IM Grid connection/StatorVoltage_XR20.pdf}
%   \captionof*{figure}{\centering X/R ratio of 20}
%   \label{fig:IM_Vs_XR20}
% \end{minipage}%
% \begin{minipage}{.5\textwidth}
%   \centering
%   \includegraphics[scale = 0.51]{figure/Figures IM Grid connection/StatorVoltage_XR30.pdf}
%   \captionof*{figure}{\centering X/R ratio of 30}
%   \label{fig:IM_Vs_XR30}
% \end{minipage}
%  \caption{Stator voltage response to a disturbance in grid frequency}
% \end{figure}

% \begin{figure}
% \centering
% \begin{minipage}{.5\textwidth}
%   \centering
%   \includegraphics[scale = 0.51]{figure/Figures IM Grid connection/LineCurrent_XR20.pdf}
%   \captionof*{figure}{\centering X/R ratio of 20}
%   \label{fig:IM_Il_XR20}
% \end{minipage}%
% \begin{minipage}{.5\textwidth}
%   \centering
%   \includegraphics[scale = 0.51]{figure/Figures IM Grid connection/LineCurrent_XR30.pdf}
%   \captionof*{figure}{\centering X/R ratio of 30}
%   \label{fig:IM_Il_XR30}
% \end{minipage}
%  \caption{Line current response to a disturbance in grid frequency}
%  \label{fig:IM_IL}
% \end{figure}

\begin{figure}[t!]
    \centering
    \includegraphics[scale = 0.55]{figure/Chapter 6 Figures/Rotor_Speed_XRratio.pdf}
  \caption{Induction motor speed response to a disturbance in grid frequency}
  \label{fig:IM_speed_XR}
\end{figure}

\begin{figure}[t!]
    \centering
    \includegraphics[scale = 0.47]{figure/Chapter 6 Figures/PhasePlane_XRratio.pdf}
  \caption{Phase plane trajectory of motor speed in response to a disturbance in grid frequency}
  \label{IM_PP_XR}
\end{figure}

\begin{figure}[t!]
    \centering
    \includegraphics[scale = 0.5]{figure/Chapter 6 Figures/Periodic_Solutions_PV_QV.pdf}
  \caption{Periodic solutions of P,Q and V for a grid frequency disturbance}
  \label{LimitCycle_s1}
\end{figure}

% \begin{figure}[t!]
% \centering
% \begin{minipage}{.5\textwidth}
%   \centering
%   \includegraphics[scale = 0.55]{figure/Figures IM Grid connection/Phase_Plane_IM_XR20.pdf}
%   \captionof*{figure}{\centering X/R ratio of 20}
%   \label{fig:IM_PP_XR20}
% \end{minipage}%
% \begin{minipage}{.5\textwidth}
%   \centering
%   \includegraphics[scale = 0.55]{figure/Figures IM Grid connection/Phase_Plane_XR30.pdf}
%   \captionof*{figure}{\centering X/R ratio of 30}
%   \label{fig:IM_PP_XR30}
% \end{minipage}
%  \caption{Phase plane trajectory response to a disturbance in grid frequency}
% \label{IM_PP_XR}
% \end{figure}


% \begin{figure}[t!]
% \centering
% \begin{minipage}{.5\textwidth}
%   \centering
%   \includegraphics[scale = 0.6]{figure/Figures IM Grid connection/PV_Characteristics_XR.pdf}
%   \captionof*{figure}{\centering P-V characteristics}
%   \label{fig:LC_PV_1}
% \end{minipage}%
% \begin{minipage}{.5\textwidth}
%   \centering
%   \includegraphics[scale = 0.6]{figure/Figures IM Grid connection/QV_Characteristics_XR.pdf}
%   \captionof*{figure}{\centering Q-V characteristics}
%   \label{fig:LC_QV_1}
% \end{minipage}
% \caption{Periodic solutions of P,Q and V for a grid frequency disturbance}
% \label{LimitCycle_s1}
% \end{figure}

The rotor speed of the induction motor is shown in Fig. \ref{fig:IM_speed_XR} from which it can be observed that when $X/R$ ratio is 20, the rotor speed oscillates around $0.2$ pu. However, as the grid becomes weaker with a higher $X/R$ ratio of 30, the rotor speed of the induction motor oscillates at a negative speed of around $-0.2$ pu. The phase-plane trajectories for the $X/R$ ratios considered are shown in Fig. \ref{IM_PP_XR}.  Oscillations in electric torque produced by the motor attain dangerously high values as the motor stalls as a result of subsynchronous oscillations. 

When simulated for a longer period of time, the system was found to enter a state of undamped oscillations which are bounded in magnitude which is the result of the emergence of a limit cycle as a consequence of the Hopf bifurcation. The size of the limit cycle is found to be a function of the $X/R$ ratio of the feeder which is shown in Fig. \ref{LimitCycle_s1}. The limit cycle was the smallest for the system with $X/R$ ratio of 30 which may be due to a higher voltage drop owing to a greater value of feeder reactance. 

The responses of line currents of the system for X/R ratios of 20 and 30 are shown in Fig. \ref{fig:IM_IL_XR}. For both loading conditions, an current magnitude of almost $16$ pu has been observed close to $1$ second. For a smaller X/R ratio of 20, the line current undergoes oscillations between 0 and $9$ pu past $1$ second. A double frequency component of the current is seen in the line current waveform during the entire simulation. When the simulation was repeated for a higher X/R ratio of 30, the line current approaches a value of almost $14.2$ pu close to $1.45$ seconds. The oscillations start decreasing in magnitude till around $6.8$ seconds. Between $6.8$ and $10.9$ seconds, the oscillations appear to have disappeared, however, beyond $10.96$ seconds, they reappear and start increasing in magnitude gradually. Beyond $15$ seconds, the oscillations are bounded in magnitude between $1.5$ and $4$ pu and this can be attributed to a limit cycle. Also, it should be noted that the double frequency component in the line current exists only till about $6$ seconds beyond which it disappears. 

\begin{figure}[t]
    \centering
    \includegraphics[scale = 0.65]{figure/Chapter 6 Figures/PhasePlane_LoadStep.pdf}
    \caption{Rotor speed and electrical torque response to a load step change}
    \label{fig:LC_Te_speed}
\end{figure}   

\begin{figure}[t]
    \centering
    \includegraphics[scale = 0.47]{figure/Chapter 6 Figures/PeriodicSolution_LoadStep.pdf}
    \caption{Limit cycles with respect to real and reactive power and motor terminal voltage for a load step change}
    \label{fig:LC_power}
\end{figure}   

The system was also subjected to a step change in the load torque for an X/R ratio of 20, operating slip of $1\%$ and series compensation of $60\%$ and the dynamic responses of the variables were observed. The system variables followed the same wave shape as those corresponding to a frequency disturbance in the system with X/R ratio of 20 for a load step change of $20\%$ on the induction motor. However, when the load torque disturbance was increased to $40\%$, the time responses of the variables are very similar to the case when the system with $X/R$ ratio of 30 was subjected to a frequency disturbance. The rotor speed, torque with respect to time and the limit cycle with respect to rotor speed and torque are shown in Fig. \ref{fig:LC_Te_speed}. The limit cycles with respect to real power and voltage magnitude and reactive power and voltage magnitude are shown in Fig. \ref{fig:LC_power}. The reason for the reversal of induction motor may be explained as follows: when the system is subjected to a disturbance, there are self-excited oscillations of increasing magnitudes as a result of induction generator effect of SSR. This results in voltages and currents at subsynchronous frequencies induced in the stator winding of the induction motor in addition to the applied voltage at supply frequency. The resultant emfs induced in the stator winding may result in an opposite phase sequence resulting in a magnetic field rotating in the opposite direction and the induction motor follows the direction of the resultant magnetic field and starts rotating in the opposite direction.

\section{Conclusions}
Eigenvalue analysis of an induction motor load fed from the substation through a series compensated weak network exhibits Hopf bifurcations which can be attributed to subsynchronous resonance. As the load on the induction motor increases, the operating slip increases. This has little influence on the percentage series compensation level at which the Hopf bifurcation occurs. However, for decreasing grid strengths due to increase in the impedance of the feeder, the stability domain with respect to series compensation decreases as SSR occurs at lower levels of percentage series compensation. Size of limit cycles generated as a result of Hopf bifurcation due to SSR depends on the grid strength. Limit cycles are smallest when the interconnection is the weakest. Motor stalling is observed as a result of SSR. However, the direction of rotation of the motor during the limit cycle is dependent on the grid strength as well as the magnitude of disturbance. There is a possibility of reversal in the direction of rotation for lower grid strengths and disturbances of higher magnitudes. In this analysis, the substation voltage is assumed to be at a constant value which in practice is possible only if there is a voltage regulating device like a STATCOM or SVC installed. Hence, to mitigate harmful oscillations in the system due to Hopf bifurcations, a supplementary damping controller is needed.  





