\chapter{Dynamic analysis of Squirrel Cage Induction Generator - Wind Turbine with a Non-Radial Interconnection to the Grid}\label{Chap5}

In this chapter, the modeling and analysis of an SCIG - WT in the modified IEEE Second Benchmark consisting of an induction generator coupled to the grid through a double circuit line is presented. The previous chapter dealt with the analysis of bifurcations in SCIG-WTs coupled to the grid through a series compensated radial interconnection. The goal of this chapter is to examine the impact of network topology along with other parameters on the stability domain with respect to series compensation in a weak grid. Research presented in \cite{moharana2012subsynchronous} claims that a double-cage induction generator based wind farm may not experience instability due to subsynchronous resonance in the presence of a non-radial interconnection to the grid. In this chapter, it has been shown through eigenvalue calculations and time domain simulations that instability via Hopf bifurcations due to subsynchronous resonance are inevitable in SCIG-WTs even in the absence of a radial interconnection.

\section{Modeling of network}\label{Modeling_DC_SCIG}

\begin{figure}
    \centering
    \includegraphics[scale = 0.95]{figure/Chapter 5 Figures/IEEE_SBM_SLD.pdf}
    \caption{Single line diagram of modified IEEE Second Benchmark Model}
    \label{fig:sld_sbm}
\end{figure}  


The single line diagram of the modified IEEE second benchmark system is shown in Fig. \ref{fig:sld_sbm}. It consists of a SCIG-WT connected to the grid through a step-up transformer followed by a double circuit line which is then coupled to the grid through a short line. Reactive power needed by the SCIG is provided by shunt capacitor banks at the stator terminals. The equations used to model the dynamics of the induction generator have been derived in Chapter \ref{Chapter2}. The matrices given below are used to develop the dynamic model of the network:

\begin{align}
    [Z_{T}] = \begin{bmatrix}R_t & -X_t\\X_t & R_t\end{bmatrix},
    [Z_{G}] = \begin{bmatrix}R_g & -X_g\\X_g & R_g\end{bmatrix},
    [Z_{1}] = \begin{bmatrix}R_1 & -X_1\\X_1 & R_1\end{bmatrix},
    [Z_{2}] = \begin{bmatrix}R_2 & -X_2\\X_2 & R_2\end{bmatrix}\nonumber
\end{align}

The following reactances are defined: $X_{n1} = X_t + X_g + X_1$, $X_{n2} = X_t + X_g$, $X_{n3} = X_t + X_g + X_2$. Also, define matrices: $[Z_{N1}] = [Z_T + Z_1 + Z_G]$, $[Z_{N2}] = [Z_T + Z_G]$ and $[Z_{N3}] = [Z_T + Z_2 + Z_G]$. Using the quantities defined, the currents in the electrical network are modeled using the following differential equations:

\begin{align}
    \frac{X_{n1}}{\omega_s}[U]\frac{d}{dt}\{I_{n1}\} + \frac{X_{n2}}{\omega_s}[U]\frac{d}{dt}\{I_{n2}\} &= -[Z_{N1}]\{I_{n1}\} - [Z_{N2}]\{I_{n2}\} - \{V_{c}\} - \{V_{g}\} + \{V_{s}\}\label{DC_line1}\\
    \frac{X_{n2}}{\omega_s}[U]\frac{d}{dt}\{I_{n1}\} + \frac{X_{n3}}{\omega_s}[U]\frac{d}{dt}\{I_{n2}\} &= -[Z_{N2}]\{I_{n1}\} - [Z_{N3}]\{I_{n2}\} - \{V_{g}\} + \{V_{s}\}\label{DC_line2}
\end{align}

The elements of vectors $\{I_{n1}\}$, $\{I_{n2}\}$, $\{V_{s}\}$ and $\{V_{c}\}$ are given by:

\begin{align}
    \{I_{n1}\} &= [I_{d1} \hspace{0.35cm} I_{q1}]^{T}\\
    \{I_{n2}\} &= [I_{d2} \hspace{0.35cm} I_{q2}]^{T}\\
    \{V_{s}\} &= [V_{ds} \hspace{0.35cm} V_{qs}]^{T}\\
    \{V_{c}\} &= [V_{dc} \hspace{0.35cm} V_{qc}]^{T}
\end{align}

The voltage across the series capacitor in line 1 is modeled using equation given below:

\begin{align}
    \frac{1}{\omega_s}[U]\frac{d}{dt}\{V_{c}\} = [Z_{se}]\{I_{n1}\} + [N_1]\{V_{c}\}\label{series_cap_DC}
\end{align}

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

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

\begin{align}
    \frac{1}{\omega_s}[U]\frac{d}{dt}\{V_{s}\} = [Z_{sh}]\{I_{s} - I_{n1} - I_{n2}\} + [N_1]\{V_{s}\}\label{shunt_cap_DC}
\end{align}

where $[Z_{sh}] = X_{sh}[U]$.

\section{Linearization of mathematical model}

The dynamic equations of the network presented in Section \ref{Modeling_DC_SCIG} are linearized and presented in this section. Linearized versions of equations \eqref{DC_line1} and \eqref{DC_line2} are given below:

\begin{align}
    \frac{X_{n1}}{\omega_s}[U]\frac{d}{dt}\{\Delta {I}_{n1}\} + \frac{X_{n2}}{\omega_s}[U]\frac{d}{dt}\{\Delta I_{n2}\} &= -[Z_{N1}]\{\Delta I_{n1}\} - [Z_{N2}]\{\Delta I_{n2}\} - \{\Delta V_{c}\}  + \{\Delta V_{s}\}\label{DC_line1_linearized}\\
    \frac{X_{n2}}{\omega_s}[U]\frac{d}{dt}\{\Delta I_{n1}\} + \frac{X_{n3}}{\omega_s}[U]\frac{d}{dt}\{\Delta I_{n2}\} &= -[Z_{N2}]\{\Delta I_{n1}\} - [Z_{N3}]\{\Delta I_{n2}\} + \{\Delta V_{s}\}\label{DC_line2_linearized}
\end{align}

 Since the voltage of the grid is assumed to be fixed under equilibrium conditions, $\{\Delta V_{dqg}\} = 0$ and hence, it does not appear in the linearized equations \eqref{DC_line1_linearized} and \eqref{DC_line2_linearized}. Linearized expressions for equations \eqref{series_cap_DC} and \eqref{shunt_cap_DC} can be found below:

\begin{align}
    \frac{1}{\omega_s}[U]p\{\Delta V_{c}\} &= [Z_{se}]\{\Delta I_{n1}\} + [N_1]\{\Delta V_{c}\}\label{series_cap_DC_linearized}\\
     \frac{1}{\omega_s}[U]p\{\Delta V_{s}\} &= [Z_{sh}]\{\Delta I_{s} - \Delta I_{n1} - \Delta I_{n2}\} + [N_1]\{\Delta V_{s}\}\label{shunt_cap_DC_linearized}
\end{align}

The various vectors used in equations \eqref{DC_line1_linearized} to \eqref{shunt_cap_DC_linearized} are defined below:

\begin{align}
    \{\Delta I_{n1}\} &= [\Delta I_{d1} \hspace{0.35cm} \Delta I_{q1}]^{T}\\
    \{\Delta I_{n2}\} &= [\Delta I_{d2} \hspace{0.35cm} \Delta I_{q2}]^{T}\\
    \{\Delta V_{s}\} &= [\Delta V_{ds} \hspace{0.35cm} \Delta V_{qs}]^{T}\\
    \{\Delta V_{c}\} &= [\Delta V_{dc} \hspace{0.35cm} \Delta V_{qc}]^{T}
\end{align}

\begin{table}[h]
     \centering
       \caption{Subset of parameters of the electrical network (all parameters are specified in per unit on a common base).}
    \label{table:NWparameters_SBM}
    {\begin{tabular}{ cccccc } 
     \toprule
    \,\,\,\,\,\,\,\,\,\,\,\,\,\boldmath{$R_t$}\,\,\,\,\,\,\,\,\,\,\,\,\, & \,\,\,\,\,\,\,\,\,\,\,\boldmath{$X_t$}\,\,\,\,\,\,\,\,\,\,\, & \,\,\,\,\,\,\,\,\,\,\,\boldmath{$R_g$}\,\,\,\,\,\,\,\,\,\,\, & \,\,\,\,\,\,\,\,\,\,\,\boldmath{$X_g$}\,\,\,\,\,\,\,\,\,\,\, & \,\,\,\,\,\,\,\,\,\,\,\boldmath{$X_{sh}$}\,\,\,\,\,\,\,\,\,\,\, & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\boldmath{$V_g$}\,\,\,\,\,\,\,\,\,\,\,\,\, \\ 
     \midrule
    $0$ & $0.05$ & $0$ & $0.05$ & $1.2$ & $1$\\
     \bottomrule
     \end{tabular}}
\end{table}

\section{Eigenvalue analysis of IEEE SBM with SCIG-WT}

The linearized model of the system under study consists of equations \eqref{Stator_IG_linearized}, \eqref{rotor_IG_linearized}, \eqref{swing_IG_linearized}, \eqref{DC_line1_linearized}, \eqref{DC_line2_linearized}, \eqref{series_cap_DC_linearized} and \eqref{shunt_cap_DC_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_{d1}, \Delta I_{q1}, \Delta I_{d2}, \Delta I_{q2}, \Delta V_{ds}, \Delta V_{qs}, \Delta V_{dc}, \Delta V_{qc}, \Delta s_g\}^T
\end{align}

\begin{table}[t!]
    %\centering
    \caption{Eigenvalues of modified IEEE-SBM with IG1 for $60\%$ series compensation.}
    \label{tab:evals_SBM_basecase_1}
    {\begin{tabular}{ccc}
    \toprule
   \boldmath{$\lambda_i$} & \boldmath{$-\sigma \pm j\omega$}    &  \textbf{Participating States} \\
        \midrule
     \,\,\,\,\,\,\,\,$\lambda_1$ \,\,\,\,\,\,\,\,& \,\,\,\, $-30.963\pm j1835.9$\,\,\,\,  & $I_{ds}$, $I_{qs}$, $I_{dr}$, $I_{qr}$, $I_{d1}$, $I_{q1}$, $I_{d2}$, $I_{q2}$, $V_{ds}$, $V_{qs}$\\
        \midrule 
     $\lambda_2$ & $-39.772 \pm j1082$ & $I_{ds}$, $I_{qs}$, $I_{dr}$, $I_{qr}$, $I_{d1}$, $I_{q1}$, $I_{d2}$, $I_{q2}$, $V_{ds}$, $V_{qs}$\\
        \midrule
     $\lambda_3$ & $-12.999 \pm j658.23$ & $I_{ds}$, $I_{qs}$, $I_{dr}$, $I_{qr}$, $I_{d1}$, $I_{q1}$, $I_{d2}$, $I_{q2}$, $V_{ds}$, $V_{qs}$, $V_{dc}$, $V_{qc}$\\
       \midrule
    $\lambda_4$ & $-24.067 \pm j376.67$ & $I_{ds}$, $I_{qs}$, $I_{dr}$, $I_{qr}$, $I_{d2}$, $I_{q2}$\\
       \midrule
    $\lambda_5$ & $-7.5794 \pm j94.595$ & $I_{ds}$, $I_{qs}$, $I_{dr}$, $I_{qr}$, $I_{d1}$, $I_{q1}$, $I_{d2}$, $I_{q2}$, $V_{dc}$, $V_{qc}$, $s_g$\\
       \midrule
    $\lambda_6$ & $-5.8228 \pm j17.499$ & $I_{ds}$, $I_{qs}$, $I_{dr}$, $I_{qr}$, $I_{d1}$, $I_{q1}$, $V_{qc}$, $s_g$\\
       \midrule
    $\lambda_7$ & $-11.297$ & $I_{ds}$, $I_{qs}$, $I_{dr}$, $I_{qr}$\\
      \bottomrule
    \end{tabular}}
\end{table}

\begin{table}[t!]
    %\centering
    \caption{Eigenvalues of modified IEEE-SBM with IG2 for $60\%$ series compensation.}
    \label{tab:evals_SBM_basecase_2}
    {\begin{tabular}{ccc}
    \toprule
   \boldmath{$\lambda_i$} & \boldmath{$-\sigma \pm j\omega$}    &  \textbf{Participating States} \\
        \midrule
   \,\,\,\,\,\, $\lambda_1$\,\,\,\,\,\, & \,\, $-36.005\pm j1557.7$\,\, & $I_{ds}$, $I_{qs}$, $I_{dr}$, $I_{qr}$, $I_{d1}$, $I_{q1}$, $I_{d2}$, $I_{q2}$, $V_{ds}$, $V_{qs}$, $V_{dc}$, $V_{dc}$\\
    \midrule
   $\lambda_2$ & $-60.742 \pm j802.89$ & $I_{ds}$, $I_{qs}$, $I_{dr}$, $I_{qr}$, $I_{d1}$, $I_{q1}$, $I_{d2}$, $I_{q2}$, $V_{ds}$, $V_{qs}$\\
       \midrule
   $\lambda_3$ & $-14.089 \pm j650.26$ & $I_{ds}$, $I_{qs}$, $I_{dr}$, $I_{qr}$, $I_{d1}$, $I_{q1}$, $I_{d2}$, $I_{q2}$, $V_{ds}$, $V_{qs}$, $V_{dc}$, $V_{qc}$\\
       \midrule
   $\lambda_4$ & $-21.495 \pm j375.84$ & $I_{ds}$,$I_{qs}$,$I_{dr}$,$I_{qr}$,$I_{d2}$,$I_{q2}$\\
       \midrule
   $\lambda_5$ & \boldmath{$3.081 \pm j105.54$} & $I_{ds}$, $I_{qs}$, $I_{dr}$, $I_{qr}$, $I_{d1}$, $I_{q1}$, $I_{d2}$, $I_{q2}$, $V_{ds}$, $V_{qs}$, $V_{dc}$, $V_{dc}$, $s_g$\\
      \midrule
   $\lambda_6$ & $-23.434 \pm j25.302$ & $I_{ds}$, $I_{qs}$, $I_{dr}$, $I_{qr}$, $I_{d1}$, $I_{q1}$, $V_{qc}$, $s_g$\\
       \midrule
   $\lambda_7$ & $-39.378$ & $I_{ds}$, $I_{qs}$, $I_{dr}$, $I_{qr}$, $I_{d1}$, $I_{q1}$, $V_{dc}$, $V_{qc}$, $s_g$\\
       \bottomrule
    \end{tabular}}
\end{table}

\begin{table}[t]
    %\centering
    \caption{Participation factors of $\lambda_{cr}$ with IG2 at $60\%$ compensation.}
    \label{table:sbm_part_1}
    {\begin{tabular}{ cccccccc } 
     \toprule
 \,\,\,\,\,\,\boldmath{$I_{ds}$}\,\,\,\,\,\, &\,\,\,\,\,\, \boldmath{$I_{qs}$}\,\,\,\,\,\,& \,\,\,\,\,\,\boldmath{$I_{dr}$} \,\,\,\,\,\,&\,\,\,\,\,\, \boldmath{$I_{qr}$} \,\,\,\,\,\, &\,\,\,\,\,\, \boldmath{$I_{d1}$}\,\,\,\,\,\,&\,\,\,\,\,\,\boldmath{$I_{q1}$} \,\,\,\,\,\,&\,\,\,\,\,\, \boldmath{$V_{dc}$} \,\,\,\,\,\,&\,\,\,\, \boldmath{$V_{qc}$} \,\,\,\,\,\\ 
 \midrule
0.229 & 0.212 & 0.227 & 0.208 & 0.026 & 0.026 & 0.032 & 0.032 \\ 
 \bottomrule
\end{tabular}}
\end{table}

The linearized equations were cast in the form $[M_c][\Delta \Dot{X} = [A_c]\Delta X$ and the Jacobian matrix $[J]$ is computed as $[J] = [Mc]^{-1}[A_c]$. Eigenvalues of the Jacobian matrix are computed and plotted on the complex plane to observe the movement of eigenvalues of the system when the degree of series compensation $K_c$ given by equation \eqref{Kc_SBM} is chosen as the bifurcation parameter.
\begin{equation}
    K_c = \frac{X_c}{X_t + X_1 + X_g}\label{Kc_SBM}
\end{equation}
Two different induction generators are considered in this study. The first generator will be denoted by IG1 which has the same parameters as that of the SCIG considered in Chapter \ref{Chap3}. A second induction generator (which will be referred to as IG2) with a different set of parameters is also considered in this analysis. Parameters of IG1 and IG2 are listed in Tables \ref{table:IG1} and \ref{table:IG2} respectively in \ref{appendixA}. Parameters of the electrical network except that of the double circuit line can be found in Table \ref{table:NWparameters_SBM}. Eigenvalues of the system under study with IG1 and IG2 for an $X/R$ ratio of 15 for lines 1 and 2 in the double circuit ($X_1/R_1 = X_2/R_2 = 15$) with $R_1 = R_2 = 0.04$ pu and a wind speed of $10$ m/s have been listed in Tables \ref{tab:evals_SBM_basecase_1} and \ref{tab:evals_SBM_basecase_2} respectively. The series compensation considered is $60\%$ which is realistic for a weak interconnection. From Table \ref{tab:evals_SBM_basecase_1}, it is obvious that the system is stable with IG1 as all eigenvalues have negative real parts. However, in Table \ref{tab:evals_SBM_basecase_2}, $\lambda_5$ is found to have a real part which is positive resulting in an unstable system. This makes $\lambda_5$ the critical mode $\lambda_{cr}$ and the participation factors corresponding to the dominant states for $\lambda_{cr}$ are listed in Table \ref{table:sbm_part_1}.


Among the 13 eigenvalues of the system, there are 6 complex conjugate pairs ($\lambda_1$ to $\lambda_6$) and 1 real eigenvalue ($\lambda_7$). The frequencies of $\lambda_1$ to $\lambda_3$ are greater than system frequency among which $\lambda_1$ and $\lambda_2$ correspond to the modes which arise due to the interaction of induction generator currents with the shunt capacitor voltage. The other eigenvalue pair which has a frequency greater than the system frequency ($\lambda_3$) corresponds to the modes produced due to interactions of generator currents with the series capacitor voltage. There is one mode which has a frequency very close to the system frequency ($\lambda_4$) which has highest participation from the generator currents and currents flowing in line 2 (uncompensated line) of the double circuit. There are 2 modes in the subsynchronous frequency range ($\lambda_5$,$\lambda_6$) among which $\lambda_5$ is the critical mode as it becomes unstable as the compensation exceeds a certain limit. The other subsynchronous mode ($\lambda_6$) is dominated by the d-axis components of the generator currents and slip  while the real eigenvalue ($\lambda_7$) has highest participation from the q-axis components of the generator currents and slip of the induction generator. 

\subsection{Eigenvalues of the system for low and high wind speeds}

\begin{table}[h]
\caption{Critical eigenvalues for different wind speeds}
\label{table:Vw_SBM_1}
% \centering
{\begin{tabular}{cccc}
\toprule
     \boldmath{$V_w$} & \boldmath{$K_{cr}$} &  \boldmath{$\lambda_{cr} = -\sigma \pm j\omega$} & \textbf{Participating States} \\
    \midrule
    \,\,\,\,\,\,\,\,$9$\,\,\,\,\,\,\,\, &\,\,\,\,\,\,\,\, $82\%$ \,\,\,\,\,\,\,\,& \,\,\,\,\,\,\,\,$0.413 \pm j45.958$ \,\,\,\,\,\,\,\,&\,\,\,\,\,\,\,\, \multirow{3}{*}{$I_{ds}$,$I_{qs}$,$I_{dr}$,$I_{qr}$,$I_{d1}$,$I_{q1}$,$V_{dc}$,$V_{qc}$,$s_g$} \,\,\,\,\,\,\,\, \\\cline{1-3}
    %  \hline
    $10$ & $82\%$ & $0.3079 \pm j45.832$ &  \\\cline{1-3}
    $11$ & $82\%$ & $0.1695 \pm j45.690$ &  \\\cline{1-3}
    $12$ & $83\%$ & $0.8255 \pm j43.600$ &  \\
    \bottomrule
\end{tabular}}
\end{table}

\begin{table}[h]
\caption{Critical eigenvalues for different wind speeds}
\label{table:Vw_SBM_2}
% \centering
{\begin{tabular}{cccc}
\toprule
     \boldmath{$V_w$} & \boldmath{$K_{cr}$} &  \boldmath{$\lambda_{cr} = -\sigma \pm j\omega$} & \textbf{Participating States} \\
    \midrule
    \,\,\,\,\,\,\,\,$9$\,\,\,\,\,\,\,\, &\,\,\,\,\,\,\,\, $52\%$ \,\,\,\,\,\,\,\,& \,\,\,\,\,\,\,\,$0.2996 \pm j123.37$ \,\,\,\,\,\,\,\,& \multirow{3}{*}{$I_{ds}$,$I_{qs}$,$I_{dr}$,$I_{qr}$,$I_{d1}$,$I_{q1}$,$I_{d2}$,$I_{q2}$,$V_{dc}$,$V_{qc}$,$s_g$}  \\\cline{1-3}
    %  \hline
    $10$ & $52\%$ & $0.1147 \pm j123.26$ &  \\\cline{1-3}
    $11$ & $53\%$ & $0.2095 \pm j120.82$ &  \\\cline{1-3}
    $12$ & $54\%$ & $0.2212 \pm j118.37$ &  \\
    \bottomrule
\end{tabular}}
\end{table}

Critical eigenvalues of the system at wind speeds from $9$ to $12$ m/s for the system with IG1 and IG2 are listed in Tables \ref{table:Vw_SBM_1} and \ref{table:Vw_SBM_2} respectively. The $X/R$ ratio of the system for both lines was chosen to be $15$. 
 The system with IG1 loses stability when compensation exceeds $81\%$ for $V_w = 9, 10, 11$ m/s and $82\%$ for $V_w = 12$ m/s. However, for $V_w = 9, 10$ m/s, the system with IG2 loses stability when the compensation exceeds $51\%$ which is much smaller than the critical value of series compensation compensation of the system with IG1 which means the stability domain of the system with IG2 is smaller. The dominant states in the critical modes are the stator and rotor currents of the induction generator. Since the critical mode arises due to the interaction of generator currents, network currents and the series capacitor voltage, the reason for instability is identified to be SSR. Hence, from the eigenvalue analysis for low and high wind speeds, it is inferred that the critical value of series compensation depends on the parameters of the generator and in the system with non-radial interconnection to the grid whereas wind speed does not have much influence over the critical value of series compensation. 

\subsection{Eigenvalue analysis for low, medium and high X/R ratios}

Eigenvalues are computed for the system with IG1 and IG2 as the series compensation is varied from $1\%$ to $95\%$ and are plotted on the complex plane as shown in Fig. \ref{fig:evas_sb} for various $X/R$ ratios. The parameters of the network except for the double circuit line are listed in Table \ref{table:NWparameters_SBM}. The resistance of lines 1 and 2 is assumed to be equal to $0.04$ pu and the wind speed considered here is $10\hspace{0.1cm}$m/s. Here, it is assumed that the $X/R$ ratios of lines 1 and 2 are equal ($X_1/R_1 = X_2/R_2$). It can be seen from Fig. \ref{fig:evas_sb} that for an $X/R$ ratio of 5  which signifies a relatively strong grid, the system with IG1 loses stability at $90\%$ series compensation whereas the system with IG2 remains completely stable as the percentage series compensation is increased from $1\%$ to $95\%$. When the eigenvalue trajectory of the system is computed for $X/R$ ratio of 10 , the system with IG1 loses stability through a Hopf bifurcation when the compensation exceeds $85\%$. Similarly, as the $X/R$ ratio of the system was changed to $15$, the system with IG1 loses stability as the percentage series compensation becomes greater than $82\%$.

When the same analysis was performed for the system with IG2, the system loses stability when the compensation percentage exceeds $65\%$ and $52\%$ for $X/R$ ratios of $10$ and $15$ respectively. Although the stability domain for the system with IG2 was slightly larger than that of the system with IG1 for an $X/R$ ratio of 5 for lines 1 and 2, the system with IG1 has a larger stability domain as the $X/R$ ratio was increased to $10$ and then to $15$. It has been observed that in the system with an SCIG-WT with IG1 using non-radial interconnection to the grid, a Hopf bifurcation due to SSR still occurs although the stability domain with the non-radial interconnection is larger compared to the system using a radial interconnection. It has also been observed that the stability domain with respect to series compensation depends on the generator parameters as well since the system with IG2 has a smaller stability domain compared to that of the system with IG1 when connected to the same network for $X/R$ ratios of $10$ and $15$.


% \subsubsection{Variations in $K_c$ for equal $X/R$ ratios for lines 1 and 2}

\begin{figure}[t!]
    \centering
    \includegraphics[scale = 0.65]{figure/Chapter 5 Figures/Eigenvalues_XR_IG2.pdf}
    \caption{Eigenvalue trajectories for equal $X/R$ ratios with IG2}
    \label{fig:evas_sb}
\end{figure}

\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!]
\caption{Critical eigenvalues of IG2 for equal $X/R$ ratios}
\label{table:sbm_part_XR_2}
% \centering
{\begin{tabular}{cccc}
\toprule
     \textbf{X/R ratio} & \boldmath{$K_{cr}$} &  \boldmath{$\lambda_{cr} = -\sigma \pm j\omega$} & \textbf{Participating states} \\
    \midrule
    \,\,\,\,\,\,\,\, $5$\,\,\,\,\,\,\,\, & \,\,\,\,\,\,\,\,$-$\,\,\,\,\,\,\,\, & \,\,\,\,\,\,\,\,$-$\,\,\,\,\,\,\,\, & \,\,\,\,\,\,\,\,$-$\,\,\,\,\,\,\,\, \\
    \midrule
    %  \hline
    \,\,\,\,\,\,\,\,$10$\,\,\,\,\,\,\,\, & \,\,\,\,\,\,\,\,$65\%$\,\,\,\,\,\,\,\, & \,\,\,\,\,\,$0.388 \pm j96.347$\,\,\,\,\,\, & \multirow{2}{*}{$I_{ds}$,$I_{qs}$,$I_{dr}$,$I_{qr}$,$I_{d1}$,$I_{q1}$,$I_{d2}$,$I_{q2}$,$V_{dc}$,$V_{qc}$,$s_g$} \\\cline{1-3}
    % \hline
    $15$ & $52\%$ & $0.115 \pm j123.23$ &  \\
    \bottomrule
\end{tabular}}
\end{table}

\section{Time domain simulations}
The non-linear dynamic model of the system described by equations \eqref{Stator_IG}, \eqref{rotor_IG}, \eqref{swing_IG} and \eqref{DC_line1} to \eqref{shunt_cap_DC} is programmed using MATLAB-SIMULINK and the responses of various system variables are observed when the system is subject to disturbances. Two kinds of disturbances are considered here - the first one is a momentary decrease in the grid voltage to $0.85$ pu from 1 pu lasting for $0.1$ seconds and the other disturbance is the loss of the uncompensated line in the double circuit. A wind speed of $10$ m/s is considered and $\frac{X_1}{R_1} = \frac{X_2}{R_2} = 15$ for a line resistance $R_1 = R_2 = 0.04$ pu. The values of other parameters of the network can be found in Table \ref{table:NWparameters_SBM} and the parameters of IG2 listed in Table \ref{table:IG2} are considered for the SCIG. Series compensation $K_c$ (defined in equation \eqref{Kc_SBM}) of $55\%$ is considered. 

\subsection{Response to a disturbance in the grid voltage}

\begin{figure}[t!]
  \centering
  \includegraphics[scale = 0.65]{figure/Chapter 5 Figures/Stator_Voltage_VoltDist.pdf}
  \captionof{figure}{\centering Stator voltage response to a disturbance in grid voltage}
  \label{fig:FC_Vs_DCL}
\end{figure}

\begin{figure}[t!]
  \centering
  \includegraphics[scale = 0.6]{figure/Chapter 5 Figures/Line_Current_VoltDist.pdf}
  \captionof{figure}{\centering Line current response to a disturbance in grid voltage}
  \label{fig:FC_IL_DCL}
\end{figure}

\begin{figure}[t!]
  \centering
   \includegraphics[scale = 0.6]{figure/Chapter 5 Figures/Real_Power_VoltDist.pdf}
  \captionof{figure}{\centering Real power response to a disturbance in grid voltage}
  \label{fig:FC_Pe_DCL}
\end{figure}

\begin{figure}[t!]
  \centering
   \includegraphics[scale = 0.6]{figure/Chapter 5 Figures/Rotor_Speed_VoltDist.pdf}
  \captionof{figure}{\centering Rotor speed response to a disturbance in grid voltage}
  \label{fig:FC_Speed_DCL}
\end{figure}

% \begin{figure}[t!]
% \centering
% \begin{minipage}{.5\textwidth}
%   \centering
%   \includegraphics[scale = 0.325]{figure/Chapter 4 Figures/Stator_Voltage_VoltDist.pdf}
%   \captionof{figure}{\centering Stator voltage response to a disturbance in grid voltage}
%   \label{fig:FC_Vs_DCL}
% \end{minipage}%
% \begin{minipage}{.5\textwidth}
%   \centering
%   \includegraphics[scale = 0.325]{figure/Chapter 4 Figures/Line_Current_VoltDist.pdf}
%   \captionof{figure}{\centering Line current response to a disturbance in grid voltage}
%   \label{fig:FC_IL_DCL}
% \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 = 0.325]{figure/Chapter 4 Figures/Rotor_Speed_VoltDist.pdf}
%   \captionof{figure}{\centering Generator speed response to a disturbance in grid voltage}
%   \label{fig:FC_Speed_DCL}
% \end{minipage}%
% \begin{minipage}{.5\textwidth}
%   \centering
%   \includegraphics[scale = 0.325]{figure/Chapter 4 Figures/Real_Power_VoltDist.pdf}
%   \captionof{figure}{\centering Real power response to a disturbance in grid voltage}
%   \label{fig:FC_Pe_DCL}
% \end{minipage}
% % \caption{Performance of the proposed SSDC in a grid frequency disturbance}
% \end{figure}

\begin{figure}[t!]
    \centering
    \includegraphics[scale = 0.65]{figure/Chapter 5 Figures/PhasePlane_VoltDist.pdf}
    \caption{Phase plane trajectory when subjected to a disturbance in grid voltage}
    \label{fig:Phase_Plane_Speed_Vg_sbm}
\end{figure}

% The dynamic torque-speed characteristics of an SCIG-WT for a disturbance in grid voltage have been plotted in Fig. \ref{fig:dyn_Te_speed_Vg_sbm}. It is observed in Fig. \ref{fig:dyn_Te_speed_Vg_sbm}, the rotor speed starts oscillating around the steady state speed at $0.5$ s when the disturbance is encountered. The oscillations start small at $0.5$ s and increase in magnitude as time increases. At around $2.1$ s, the speed of the rotor reduces to a value significantly below the synchronous speed resulting in stalling of the motor and past $2.5$ s, the speed oscillates between $0.55$ and $0.65$ s under steady state. However, the torque oscillations are between $-3.72$ and $5.4$ pu alternating between motoring and generating mode of operation. 

The responses of stator voltage magnitude, line current magnitude, real power at the generator terminals and the rotor speed of the induction generator to a momentary decrease in the grid voltage are shown in Figs. \ref{fig:FC_Vs_DCL} to \ref{fig:FC_Speed_DCL}. The system is observed to be unstable as $K_c$ is greater than the critical value for the given operating conditions which confirms the findings from the eigenvalue analysis. Stator voltage magnitude plotted in Fig. \ref{fig:FC_Vs_DCL} shows a decrease between $1$ and $1.1$ seconds as the disturbance persists. As the disturbance is removed at $1.1$ seconds, the stator voltage gradually starts building up and reaches a high value of almost $2.22$ pu at $3.63$ seconds followed by which the oscillations start reducing in amplitude till $3.8$ seconds. Beyond $3.8$ seconds, there is a slight increase in the magnitude of oscillations lasting for a few cycles as the system transitions to an oscillatory state where the oscillations in the stator voltage are bounded between $0.6$ pu and $1.6$ pu past $4.2$ seconds. This is a result of the limit cycle as a consequence of the Hopf bifurcation due to SSR. These bounded oscillations (between $1.38$ pu and $3.08$ pu) are also observed in the magnitude of line current in Fig. \ref{fig:FC_IL_DCL}. Bounded oscillations between $-2.19$ pu and $2.85$ pu are observed in the real power transmitted from the stator terminals to the grid which implies that power is continuously being transferred back and forth between the stator terminals and the grid when the system enters an oscillatory state.

Rotor speed of the induction generator in Fig. \ref{fig:FC_Speed_DCL} shows oscillations of small amplitude around the equilibrium point ($1.014$ pu) intially. However, between $3.5$ and $3.6$ seconds, the rotor speed of the induction generator reduces rapidly causing it to stall. Beyond $3.6$ seconds, the speed of the generator recovers slightly before entering an oscillatory state where generator speed oscillates between $0.57$ pu and $0.63$ pu beyond $4.2$ seconds. These bounded oscillations in the system states are manifested as a limit cycle in the phase-plane of the rotor speed shown in Fig. \ref{fig:Phase_Plane_Speed_Vg_sbm} and in the characteristics of the voltage plotted with respect to real power at the stator terminals of the SCIG which can be seen in \ref{fig:FC_Vs_DCL}. 

\subsection{Response to a loss of a line in the double circuit}

\begin{figure}[t!]
    \centering
    \includegraphics[scale = 0.6]{figure/Chapter 5 Figures/PhasePlane_LineLoss.pdf}
    \caption{Phase - plane trajectory with respect to rotor speed for the loss of a line}
    \label{fig:PP_line_Loss}
\end{figure}

\begin{figure}[t!]
    \centering
    \includegraphics[scale = 0.6]{figure/Chapter 5 Figures/QV_Characteristics_LineLoss.pdf}
    \caption{Stator voltage response when system experiences loss of a line}
    \label{fig:QV_lineLoss}
\end{figure}

\begin{figure}[t!]
    \centering
    \includegraphics[scale = 0.6]{figure/Chapter 5 Figures/PV_Characteristics_LineLoss.pdf}
    \caption{P-V characteristics when system experiences loss of a line}
    \label{fig:PV_lineLoss}
\end{figure}

\begin{figure}[t!]
    \centering
    \includegraphics[scale = 0.55]{figure/Chapter 5 Figures/DFT_LineLoss.pdf}
    \caption{Instantaneous stator voltage of phase A with the corresponding frequency spectrum for the system with (a) Grid voltage disturbance (b) Loss of uncompensated line}
    \label{fig:DFT_VaL}
\end{figure}

\begin{figure}[t!]
    \centering
    \includegraphics[scale = 0.55]{figure/Chapter 5 Figures/DFT_IaL.pdf}
    \caption{Instantaneous stator voltage of phase A with the corresponding frequency spectrum for the system with (a) Grid voltage disturbance (b) Loss of uncompensated line}
    \label{fig:DFT_IaL}
\end{figure}

The phase - plane trajectory of the rotor speed when the system is subjected to the loss of the uncompensated line in the double circuit at $1$ second is shown in \ref{fig:PP_line_Loss}. The response is similar to that of the previous case where stalling of the SCIG occurs after undergoing a few cycles of oscillations around the equilibrium point. The rotor speed goes slightly below synchronous speed at $1.35$ seconds beyond which it decreases rapidly. After about 3 seconds, the system enters an oscillatory state where the oscillations in the rotor speed are bounded between $0.2$ and $0.4$ pu. On the phase plane, the trajectory starts spiraling outwards from the equilibrium point and is pulled towards the limit cycle shown in Fig. \ref{fig:PP_line_Loss}.

The plot of the stator voltage at the SCIG terminals with respect to time (Fig. \ref{fig:QV_lineLoss}) reveal a much larger overvoltage when the system enters the oscillatory state as the oscillations in the voltage are bounded between $2.34$ pu and $3.05$ pu between $3.5$ and $4$ seconds. In practice, the protection system would be triggered immediately within a few cycles when the voltage oscillations start when the disturbance is encountered resulting in tripping of the SCIG. The dynamic P-V characteristics reveal that the trajectory starting at the equilibrium point diverges away from it, moving towards the limit cycle when the system undergoes the loss of the uncompensated line which is shown in Fig. \ref{fig:PV_lineLoss}. The periodic solutions of the stator voltage with respect to the reactive and real power at the stator terminals are also shown in Figs. \ref{fig:QV_lineLoss} and \ref{fig:PV_lineLoss}.

The instantaneous stator voltage of phase A along with it's frequency spectrum is computed for the case when the system is subjected to a momentary disturbance in the grid voltage as well as when the system is subjected to the loss of the uncompensated line in the double circuit. The waveforms are shown past $5$ seconds which corresponds to the time past which the system enters an oscillatory state due to the limit cycle. From Fig. \ref{fig:DFT_VaL}, $35$ Hz component is observed to be dominant followed by a $60$ Hz component and the maximum amplitude of the voltage is found to be almost $1.7$ pu. However, an instantaneous voltage of approximately $3$ pu can be observed when the system is subjected to the loss of the uncompensated line. The frequency spectrum in this case shows a dominant $18$ Hz component. This is followed by small contributions from $25$ Hz and $60$ Hz components. Hence, in both cases, a subsynchronous frequency component dominates the stator voltage when the system exhibits oscillatory behavior. The frequency spectrum of the line current in phase A shown in Fig. \ref{fig:DFT_IaL} reveals a  dominant $60$ Hz component for both types of disturbances considered followed by a $35$ Hz component when the system undergoes a disturbance in the grid voltage momentarily and an $18$ Hz component when the uncompensated line is lost. The frequency components depend on the type of disturbance which the system is subject to.

\section{Conclusion}
In this chapter, the dynamic modeling, eigenvalue analysis and detailed time domain simulation results for the IEEE second benchmark model has been presented. For systems with non-radial interconnections as this, it was concluded that there is no possibility of subsynchronous resonance in \cite{moharana2012subsynchronous}. However, this cannot be generalized for systems with non-radial interconnections as in this chapter, an SCIG-WT connected to the grid through a double circuit line still experiences instabilities due to subsynchronous resonance. An SCIG-WT with a non-radial interconnection may possess a larger stability domain with respect to series compensation compared to the same SCIG-WT in a radial interconnection to the grid. As the grid becomes weaker  for higher $X/R$ ratios, the system loses stability through a Hopf bifurcation for smaller values of percentage series compensation. The Hopf bifurcation point also depends on the parameters of the induction generator but is not affected very much for different wind speeds. 
% A Hopf bifurcation due to subsynchronous resonance can be avoided by using lines of different $X/R$ ratios in the double circuit. 
Time domain simulations are performed using SIMULINK software considering a momentary reduction in the grid voltage and loss of the uncompensated line in the double circuit. From the results of the time domain simulation, the nature of the Hopf bifurcation is found to be sub-critical as the system significantly diverges away from the equilibrium point. The frequency components in the periodic solutions of various system variables depends on the type of disturbance which the system undergoes. Hence, network topologies have a significant impact on the stability domain of series compensated systems which makes detailed dynamic modeling and stability studies highly important while planning a power system. In the next 2 chapters, the impact of series compensation on dynamic loads namely induction motors will be explored. 
