\chapter{Stability Domain Enhancement in a Radially Interconnected System with Squirrel Cage Induction Generator - Wind Turbines}\label{Chap4}

In this chapter, stability domain enhancement in the modified IEEE First Benchmark Models (FBM) with a Squirrel Cage Induction Generator - Wind Turbine (SCIG-WT) has been explored. A Static Var Compensator (SVC) has been used in place of a fixed shunt capacitor to provide the needed reactive power for excitation at the terminals of the SCIG. The upper limit of percentage series compensation in practical systems is fixed at a value around $50\%$ in order to avoid instabilities due to SSR. However, using an SVC with a Subsynchronous Damping Controller (SSDC) at the terminals of the SCIG in the place of a fixed shunt capacitor bank, the upper limit of percentage series compensation is enhanced to a value greater than $90\%$. 

% An improved design of a voltage regulator with a subsynchronous damping controller (SSDC) for the SVC has been proposed which improves the stability domain with respect to series compensation. Eigenvalue analysis and time domain simulations reveal the effectiveness of the controller in enhancement of the stability domain of the system permitting a higher level of series compensation, thus enabling a higher amount of power transfer to the grid. 

\section{Design and mathematical model of SVC voltage regulator}

\begin{figure}
    \centering 
    \captionsetup{justification=centering}
    \includegraphics[scale = 1.5]{figure/Chapter 4 Figures/SVC_Block.pdf}
    \caption{Block diagram of voltage regulator for the SCIG}
    \label{fig:Vreg}
\end{figure}

The block diagram of voltage regulator for the SVC is presented in Fig. \ref{fig:Vreg}. The terminal voltage magnitude of the generator ($V_t$) is sensed through a transducer which has a time constant of $T_1$ seconds and the measured value of voltage magnitude ($V_m$) is compared with the reference voltage ($V_{ref}$). The voltage error is processed by a PI controller having proportional and integral gains $K_{P1}$ and $K_{I1}$ respectively. The output of the PI controller dictates the reactive power reference ($Q_{ref}$) for the shunt capacitor. This is compared with the actual value of reactive power injected by the shunt capacitor ($Q_{sh} = V_m^2B_{sh}$) at the generator terminals which is measured ($Q_m$) by a transducer with time constant $T_2$ seconds. 

The reactive power error is processed by a second PI controller with proportional and integral gains $K_{P2}$ and $K_{I2}$ respectively. The susceptance of the SVC ($B_{sh}$) is determined by the second PI controller based on the reactive power error signal. A supplementary signal $-V_{sup}$ which is the output of a Subsynchronous Damping Controller (SSDC) is added to the signals $V_{ref}$ and $-V_m$ at the first summing block of the voltage regulator which is used to control critical bifurcations in the system mainly due to SSR. The structure of the SSDC will be explained further in Section \ref{SSDC_Block}. The relationship between the input and output signal of the voltage measurement block in the laplace domain is given by equation \eqref{Measurement_Laplace}.

\begin{align}
    \frac{V_m(s)}{V_t(s)} = \frac{1}{1 + sT_1}\label{Measurement_Laplace}
\end{align}
where $s$ denotes a Laplace transform operation. Cross-multiplying and taking the inverse Laplace transform, the equation \eqref{Vmeas} which describes the dynamics of the measurement block in the time domain is obtained.

\begin{align}
    T_1\frac{dV_m}{dt} &= -V_m + V_t\label{Vmeas}
\end{align}
Using the same technique, equation \eqref{Qmeas} which describes the dynamics of the reactive power measurement transducer in the time domain is obtained.
\begin{align}
     T_2\frac{dQ_m}{dt} &= -Q_m + Q_{sh}\label{Qmeas}
\end{align}

The input-output relationship of the first PI controller in the Laplace domain is expressed using equation \eqref{Qref_Laplace}. 

\begin{align}
    \frac{Q_{ref}(s)}{V_{ref}(s) - V_{m}(s) - V_{sup}(s)} = K_{P1} + \frac{K_{I1}}{s}\label{Qref_Laplace}
\end{align}
Cross-multiplying the denominator of the left hand side in equation \eqref{Qref_Laplace} and introducing an intermediate state variable $W_1$, equation \eqref{Qref_Laplace_2} which defines the reactive power reference in the Laplace domain is obtained.

\begin{align}
    Q_{ref}(s) = K_{P1}(V_{ref}(s) - V_{m}(s) - V_{sup}(s)) + W_1(s)\label{Qref_Laplace_2}
\end{align}
where 
\begin{align}
    W_1(s) = \frac{K_{I1}}{s}(V_{ref}(s) - V_{m}(s) - V_{sup}(s))\label{W1_Laplace}
\end{align}

Finally, obtaining the inverse Laplace transform of equations \eqref{Qref_Laplace_2} and \eqref{W1_Laplace}, equations \eqref{I1} and \eqref{Qref} which describe the dynamics of the first PI controller are obtained. 

\begin{align}
    \frac{dW_1}{dt} &= K_{I1}(V_{ref} - V_{m} - V_{sup})\label{I1}\\
    Q_{ref} &= K_{P1}(V_{ref} - V_m - V_{sup}) + W_1\label{Qref}
\end{align}

The same method is used to obtain equations \eqref{I2} and \eqref{Bsh} which are used to describe the dynamics of the second PI controller which processes the reactive power error signal.

\begin{align}
    \frac{dW_2}{dt} &= K_{I2}(Q_{ref} - Q_{m})\label{I2}\\
    B_{sh} &= K_{P2}(Q_{ref} - Q_{m}) + W_2\label{Bsh}
\end{align}

The dynamic equations of the series capacitor and line currents are the same as that of \eqref{Series_dyn} and \eqref{Line_dyn}. However, in the equation describing the dynamics of the shunt capacitor shown in \eqref{Shunt_dyn}, $X_{sh} = B_{sh}^{-1}$ where $B_{sh}$ is defined by equation \eqref{Bsh}.

\begin{figure}
    \centering
    \captionsetup{justification=centering}
    \includegraphics[scale = 1]{figure/Chapter 4 Figures/SSDC_Block.pdf}
    \caption{Block diagram of the proposed SSDC}
    \label{fig:SSDCblock}
\end{figure}

\section{Design and mathematical modeling of SSDC}\label{SSDC_Block}

The design of the subsynchronous damping controller (SSDC) consists of a washout block in series with a compensator which provides a phase lead. The structure of the SSDC is very similar to that of a power system stabilizer used in synchronous generators to eliminate negative damping introduced by high gain exciters (\cite{sauer1998power,kundur2007power}). The signals $X_{sig}$ in Fig. \ref{fig:SSDCblock} considered in this paper are the generator slip ({$s_g$}) and the real power at the stator terminals of the generator ($P_e = V_{ds}I_{ds} + V_{qs}I_{qs}$). The relationship between the input and output signals of the washout block in the Laplace domain is given by the following equation:

\begin{align}
    \frac{sT_w}{1 + sT_w} = \frac{V_{int}(s)}{X_{sig}(s)}
\end{align}

Cross-multiplying and taking the inverse Laplace transform and collecting the derivative terms on the left, equation \eqref{Washout} is obtained. 

\begin{align}
    T_w(\frac{dX_{sig}}{dt} - \frac{dV_{int}}{dt}) &= -X_{sig}\label{Washout} 
\end{align}
Similarly, the dynamic equation corresponding to the lead-compensator block in the time domain is obtained which is given below: 

\begin{align}
    T_{d1}\frac{dV_{sup}}{dt} - K_dT_{d2}\frac{dV_{int}}{dt} &= K_dV_{int} - V_{sup}
\end{align}

\section{Eigenvalue analysis of the system with SVC}\label{bifurcations_SVC}

Eigenvalues of the system are computed for the Jacobian matrix $[J]$ for the linearized system which has been cast in the form $\Delta \Dot{X} = [J]\Delta X$ with the fixed capacitor replaced by the SVC with the voltage regulator described in the previous section. In this section, the SSDC is assumed to be absent. The vector of linearized state variables for this system is given by equation \eqref{StateVariables_SVC} and the parameters of the voltage regulator of the SVC can be found in Table \ref{tab:parameters_SVC}.

\begin{align}
     \Delta X = \{\Delta I_{ds}, \Delta I_{qs}, \Delta I_{dr}, \Delta I_{qr}, \Delta I_{dn}, \Delta I_{qn}, \Delta V_{ds}, \Delta V_{qs}, \Delta V_{dc}, \Delta V_{qc}, \Delta s_g,\nonumber\\\Delta V_m, \Delta Q_m, \Delta W_1, \Delta W_2\}^T\label{StateVariables_SVC}
\end{align}

\begin{table}[h]
   % \centering
    \caption{Parameters of SVC voltage regulator considered in Section \ref{bifurcations_SVC}.}
    \label{tab:parameters_SVC}
    {\begin{tabular}{ccccccc}
        \toprule
       \,\,\,\,\,\,\boldmath{$V_{ref}$} \,\,\,\,\,\,& \,\,\,\,\,\,\,\boldmath{$T_{m1}$ }\,\,\,\,\,\,\,& \,\,\,\,\,\,\,\boldmath{$T_{m2}$} \,\,\,\,\,\,\,&  \,\,\,\,\,\,\,\boldmath{$K_{p1}$ }\,\,\,\,\,\,\,& \,\,\,\,\,\,\,\boldmath{$K_{p2}$ }\,\,\,\,\,\,\,&  \,\,\,\,\,\,\,\boldmath{$K_{i1}$} \,\,\,\,\,\,\,&  \,\,\,\,\,\,\,\boldmath{$K_{i2}$}\,\,\,\,\,\,\, \\ 
        \midrule
        1.05 pu & 1 ms & 1 ms & 0.1 & 0 & 40 & 40\\
        \bottomrule
    \end{tabular}}
\end{table}

\subsection{Eigenvalues for different wind speeds}

\begin{figure}
    \centering
    \includegraphics[scale = 0.65]{figure/Chapter 4 Figures/Eigenvalues_SVC_Vw.pdf}
    \caption{Eigenvalue trajectories for $V_w = 14$ m/s and $V_w = 18$ m/s}
    \label{fig:evals_SVC_Vw}
\end{figure}

Eigenvalues of the system with an SVC at the SCIG terminal in the IEEE - FBM have been plotted in Fig. \ref{fig:evals_SVC_Vw} for wind speeds of $14$ and $18\hspace{0.1cm}m/s$ as the series compensation $K_c$ is varied from $0.001$ up to $0.98$. Initially, $X/R$ ratio of the system is assumed to be $20$. These wind speeds are considered here as the slip at which the induction generator operates is close to $1\%$ and $3\%$ for wind speeds of $14$ and $18$ m/s respectively and the performance can be compared with that of the system compensated with a fixed shunt capacitor as described in \ref{FC_IG_Slips}. From the eigevalue plot, it is observed that there are $2$ kinds of bifurcations which occur. One is the node-focus bifurcation which does not result in a change in the stability of the system and hence is non-critical. However, Hopf bifurcations are also observed in the system due to SSR making it highly critical as the stability of the system is lost after the bifurcation occurs. 

When the  eigenvalues of the system for lower wind speed of $14$ m/s are analyzed, $6$ complex conjugate eigenvalue pairs (stable foci) and $3$ stable nodes are observed. Among the $6$ complex conjugate pairs, $3$ are low frequency modes: $\lambda_1 = -4.2745 \pm j9.4714$, $\lambda_2 = -8.2775 \pm j14.712$ and $\lambda_3 = -18.917 \pm j376.28$. The $3$ stable nodes are $\lambda_4 = -1021.3$, $\lambda_5 = -925.66$ and $\lambda_6 = -46.858$. As~$K_c$ is increased, $\lambda_1$ moves further into the left half of the complex plane and the frequency of oscillation decreases; at $K_c = 0.559$, $\lambda_1 = -9.4476 \pm j0.52247$, which is a stable focus, changes into 2 stable nodes: $\lambda_{11} = -9.6715$ and $\lambda_{12} = -9.2474$, which is the result of a node-focus bifurcation. Eigenvalue $\lambda_2$ moves further into the left half of the complex plane as $K_c$ increases. When $K_c$ changes from $0.703$ to $0.704$, $\lambda_3 = -0.0259 \pm j103.85$ crosses over to the right half of the complex plane and changes to $\lambda_2 = 0.0057 \pm j103.66$ as a result of a Hopf bifurcation due to SSR. When $K_c$ increases, $\lambda_4$ and $\lambda_5$ move toward each other; when $K_c$ changes from $0.938$ to $0.939$, $\lambda_4 = -979.87$ and $\lambda_5 = -968.33$ collide and transform into a pair of complex-conjugate eigenvalues $\lambda_{45} = -974.14 \pm j4.0936$ as the result of a node-focus bifurcation.
It should be noted that the system is already unstable, due to the earlier Hopf bifurcation. Additionally, $\lambda_6$ starts moving along the real axis toward the right as $K_c$ increases, but it still remains in the left half of the complex plane.

When the wind speed is 18 m/s, the~operating slip of the SCIG is $3.051\%$. For a small value of compensation $K_c = 0.001$, $7$ complex conjugate pairs of eigenvalues and one real eigenvalue all on the left half of the complex plane are observed. Among~the complex conjugate pairs of eigenvalues, $4$ of them are of low frequency: $\lambda_1 = -974.04\pm j14.893$, $\lambda_2 = -12.633 \pm j17.108$, $\lambda_3 = -0.4911 \pm j7.3045$ and $\lambda_4 = -18.911\pm j376.28$, which follow different pathways on the complex plane as $K_c$ is varied. As~the series compensation is increased, $\lambda_1$ starts moving toward the right half of the complex plane, and its frequency of oscillation decreases as well. When $K_c$ changes from $0.074$ to $0.075$, a~node-focus bifurcation results from $\lambda_1$ splitting into $2$ real eigenvalues: $\lambda_{11} = -975.15$, $\lambda_{12} = -972.57$, which then move in opposite directions as $K_c$ is increased. However, these eigenvalues remain in the left half of the complex plane as $K_c$ is increased to $0.95$. As $K_c$ changes from $0.697$ to $0.698$, a second node-focus bifurcation takes place where $\lambda_3 = -7.3577 \pm j0.51245$ transforms into $2$ real eigenvalues: $\lambda_{31} = -7.6109$, $\lambda_{32} = -7.136$, and move in opposite directions. An increase in $K_c$ also causes $\lambda_4$ to move toward the right half of the complex plane; a Hopf bifurcation occurs in the system when $K_c$ changes from $0.742$ to $0.743$ as $\lambda_4$, which is now $-0.0118 \pm j97.353$, changes to $0.02075 \pm j97.166$, and the system becomes unstable due to SSR. The stability domain is slightly larger in the system with SVC as the critical percentage series compensation $K_{cr}$ changes from $0.655$ corresponding to the system without SVC to $0.704$ for the system which uses an SVC for reactive power support at the SCIG terminals at a slip of approximately $1\%$. Similarly, $K_{cr}$ increases from $0.681$ to $0.743$ for a slip close to $3\%$.

% When the wind speed of the system is $14$ m/s and $K_c$ changes from $0.703$ to $0.704$, the critical mode $\lambda_{cr} = -0.0259 \pm j103.85$ crosses over to the right half of the complex plane and changes to $0.0057 \pm j103.66$ as a result of a Hopf bifurcation due to SSR. Similarly, for a wind speed of $18$ m/s, the system loses stability when the subsynchronous mode $\lambda_{cr}$ changes from $-0.0118 \pm j97.353$ to $0.02075 \pm j97.166$ as $K_c$ changes from $0.742$ to $0.743$. The stability domain is slightly larger in the system with SVC as the critical percentage series compensation $K_{cr}$ changes from $0.655$ corresponding to the system without SVC to $0.704$ for the system which uses an SVC for reactive power support at the SCIG terminals at a slip of $1\%$. Similarly, $K_{cr}$ increases from $0.681$ to $0.743$ for a slip of $3\%$.  

\subsection{Eigenvalues for various grid strengths}\label{Evals_SVC_XR}

\begin{figure}[t!]
    \centering
    \includegraphics[scale = 0.65]{figure/Chapter 4 Figures/Eigenvalues_SVC_XR.pdf}
    \caption{Eigenvalue trajectories for various grid strengths}
    \label{fig:evals_SVC_XR}
\end{figure}

\begin{table}[t!]
\caption{Critical series compensation for system with and without SVC}
\label{tab:Kcr_XRratio}
\centering
{\begin{tabular}{ccc}
\toprule
     \textbf{X/R ratio} & \textbf{\boldmath{$K_{cr}$} without SVC} &  \textbf{\boldmath{$K_{cr}$} with SVC} \\
    \midrule
    \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$10$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$0.809$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$0.858$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\\
    \midrule
     15 & 0.717 & 0.768 \\
     \midrule
     20 & 0.655 & 0.704\\
     \midrule
     25 & 0.611 & 0.658\\
     \midrule
     30 & 0.579 & 0.623\\
     \bottomrule
\end{tabular}}
\end{table}

% \begin{table}[h]
%     \centering
%     \begin{tabular}{|c|c|c|}
%     \hline
%     $X/R$ ratio & $K_{cr}$ without SVC  & $K_{cr}$ with SVC \\
%     \hline
%     10 & 0.809 & 0.858\\
%     \hline
%     15 & 0.717 & 0.768 \\
%     \hline 
%     20 & 0.655 & 0.704\\
%     \hline 
%     25 & 0.611 & 0.658\\
%     \hline
%     30 & 0.579 & 0.623\\
%     \hline
%     \end{tabular}
%     \caption{Critical series compensation for system with and without SVC}
%     \label{tab:Kcr_XRratio}
% \end{table}

Eigenvalues are computed for various levels of series compensation for $3$ different grid strength which is indicated by the $X/R$ ratio of the line and are traced on the complex plane as given in Fig. \ref{fig:evals_SVC_XR}. The system is analyzed for a lower wind speed of $14\hspace{0.1cm}m/s$ because it is observed from the previous section that subsynchronous oscillations occurs at low values of series compensation $K_c$ when the power generated is low. The system with $X/R = 10$ has a node-focus bifurcation as $K_c$ changes from $0.563$ to $0.564$ as the stable focus $\lambda = -9.5344 \pm j0.4132$ transforms into $2$ stable nodes each moving in the opposite direction as $K_c$ is increased. When $K_c$ changes from $0.857$ to $0.858$, the complex conjugate eigenvalue pair $\lambda_{cr} = -0.0528 \pm j73.976$ crosses over to the right half of the complex plane and becomes $\lambda_{cr} = 0.0069 \pm j73.81$ as a result of a Hopf bifurcation which gives rise to unstable oscillations at subsynchronous frequency. 

Similar behavior is encountered in the system for higher $X/R$ ratios but at lower values of $K_c$. The node-focus bifurcation occurs when $K_c$ changes from $0.558$ to $0.559$ and $0.557$ to $0.558$ for $X/R$ ratios of $20$ and $30$ respectively. The system undergoes a Hopf bifurcation when $K_c$ is close to $0.704$ and $0.623$ with $\lambda_{cr} = 0.0057 \pm j103.66$ and $\lambda_{cr} = 0.0085 \pm j120.02$ for $X/R$ ratios of $20$ and $30$ respectively. Table \ref{tab:Kcr_XRratio} shows the critical value of series compensation for the systems with and without SVC for various $X/R$ ratios. The system with SVC has a slightly larger stability domain than the system without SVC. Also, as the strength of the grid decreases, the compensation level at which Hopf bifurcation occurs becomes smaller making it crucial to implement a damping controller in order to be able to mitigate this phenomenon causing instability. 

\subsection{Eigenvalues for different measurement time delays}

\begin{figure}
    \centering
    \includegraphics[width = 6in, height = 2.75in]{figure/Chapter 4 Figures/Eigenvalues_SVC_Tdelay_1.pdf}
    \caption{Eigenvalue trajectories for various measurement delays}
    \label{fig:evals_SVC_TimeDelay}
\end{figure}

In control systems, the reference value of a quantity is compared against the actual value which is measured using a transducer. Measurement transducers have finite time constants which are usually ignored. However, these time constants have an effect on the stability domain of the system with respect to series compensation which would be studied in this section. 
Time delays in measurement of voltage magnitude and reactive power have been modeled using a first order transfer function. Eigenvalues  of the system are traced when $K_c$ is varied from $0.001$ to $0.95$ and are plotted on the complex plane for $3$ different measurement time delays in the voltage regulator. The system is analyzed for higher wind speed $V_w = 18\hspace{0.1cm}$m/s because certain unique bifurcations occur in this system for this wind speed. When the measurement time delay is $1\hspace{0.1cm}$ms, the system undergoes a node-focus bifurcation at $K_c=0.075$ and $K_c=0.704$ and a Hopf bifurcation due to SSR at $K_c=0.743$. With a measurement delay of $10\hspace{0.1cm}$ms, the system undergoes a Hopf bifurcation at $K_c=0.679$ and a node-focus bifurcation at $K_c=0.691$ as a stable focus transforms into $2$ stable nodes but the system is already unstable at $K_c = 0.679$. 

As the measurement time delay is increased to $100\hspace{0.1cm}$ms, for $K_c = 0.001$, there are $2$ pairs of complex conjugate eigenvalues $\lambda_1 = 1.4651 \pm j19.368$ and $\lambda_2 = 0.2911 \pm j9.7375$ on the right half of the complex plane which results in the system being unstable. When $K_c$ increases, these two pairs move towards the left half of the complex plane and when $K_c=0.038$, $\lambda_1 = 1.1153 \pm j19.34$ and $\lambda_2 = -0.0075 \pm j10.014$. Though the real part of $\lambda_2$ is negative, the system still is unstable as $\lambda_1$ has a positive real part. When $K_c$ approaches $0.172$, $\lambda_1$ changes to $-0.0068 \pm j19.208$ which means all the eigenvalues of the system have negative real parts indicating a stable system. Since the system changes from an unstable state to a stable state because of the movement of a complex conjugate pair of eigenvalues to the left half of the complex plane from the right, the system undergoes a reverse Hopf bifurcation. 

The reason for this phenomenon can be explained as follows: the system is unstable for small values of series compensation due to increased time delays in measurement. When the series compensation is increased, the system becomes more and more stable because increasing the series capacitive reactance decreases the effective line reactance which decreases the time constant of the electrical circuit thus compensating for measurement delays. Hence, the system can be stabilized by increasing the series compensation above a certain threshold when instability due to large measurement delays are experienced. The upper limit of $K_c$ is determined by the Hopf bifurcation which occurs due to SSR at $K_c=0.683$ where $\lambda_{cr} = 0.0223 \pm j108.57$. Finally, a node-focus bifurcation takes place at $K_c=0.865$ where a stable focus changes into $2$ stable nodes but the system has already become unstable due to SSR. 

\section{Eigenvalues of the system with SVC and SSDC}
In this section, the~performance of the designed damping controllers based on slip signal (SSDC-A) and real power (SSDC-B) is analyzed through computing eigenvalue trajectories as $K_c$ is varied from $0.001$ to $0.95$ for various $X/R$ ratios, wind speeds and measurement time delays. The controller gains of the SVC and SSDC are tuned for the various scenarios considered to obtain a stable system characterized by all eigenvalues on the left half of the complex plane up to a series compensation of $95\%$. The parameters of the network and wind speeds are the same as those considered in Section \ref{bifurcations_SVC}. 

\subsection{Eigenvalues for low and high wind speeds}\label{SSDC_Vw}

\begin{figure}
    \centering
    % \captionsetup{justification=centering}
    \includegraphics[width = 6in, height = 2.75in]{figure/Chapter 4 Figures/Eigenvalues_SSDC_Vw.pdf}
    \caption{Eigenvalues for low and high wind speeds with (a) SSDC - A (b) SSDC - B}
    \label{fig:SSDC_Vw}
\end{figure}

Eigenvalue trajectories of the system for an X/R ratio of 20 with SSDC-A are computed for wind speeds $V_w$ of 14 and 18 m/s and are presented in Figure \ref{fig:SSDC_Vw} on the left. The subsynchronous mode starts moving toward the right half of the complex plane but even for a compensation of $K_c = 0.95$, the corresponding eigenvalue pairs are $\lambda_{sub} = -3.2013 \pm j79.178$ and $\lambda_{sub} = -8.2223 \pm j72.016$ for wind speeds of $14$ and $18\hspace{0.1cm}$m/s, respectively, which have negative real parts. The eigenvalue closest to the imaginary axis is $\lambda = -0.00276$, which corresponds to $V_w = 18\hspace{0.1cm}$m/s. 

Similarly, for the system with SSDC-B, eigenvalue trajectories are computed for wind speeds of 14 and 18 m/s and are plotted in Figure \ref{fig:SSDC_Vw} to the right. It is observed that the damping of the subsynchronous mode starts reducing as the amount of series compensation $K_c$ is increased from $0.001$ up to $0.793$; beyond that, the damping starts increasing as the compensation is increased. A similar phenomenon is observed for {$V_w = 18\hspace{0.1cm}$m/s}, except that the damping of the subsynchronous mode decreases for $K_c$ up to $0.725$; beyond this, the damping increases. However, all the eigenvalues of the system for both low and high wind speeds lie on the left half of the complex plane, which shows that the Hopf bifurcation is eliminated for the entire range of series compensation considered. The parameters of the SVC voltage regulator and SSDC-A and B are listed in Table \ref{tab:parameters_SVC_1}.

\begin{table}
    %\centering
    \caption{SVC control gains and SSDC considered in Section~\ref{SSDC_Vw}.}
    \label{tab:parameters_SVC_1}
    {\begin{tabular}{ccccccccc}
        \toprule
        \,\,\, \boldmath{$SSDC$}\,\,\, & \,\,\boldmath{$K_{p1}$}\,\,  &\,\,\boldmath{$K_{p2}$} \,\, &\,\,\boldmath{ $K_{i1}$} \,\, &\,\,  \boldmath{$K_{i2}$}\,\, & \,\, \boldmath{$K_{d}$ }\,\, &\,\,\,\,  \boldmath{$T_{d1}$}\,\,\,\,  &\,\,\,\, \boldmath{$T_{d2}$} \,\,\,\, &\,\,\,\, \boldmath{$T_{w}$}\,\,\,\, \\
        \midrule
        A & 0.5 & 0.1 & 1 & 1 & 450 & 10 ms & 150 ms & 10 s\\
        \midrule
        B & 0.1 & 0 & 100 & 100 & 0.1 & 10 ms & 100 ms & 10 s\\
        \bottomrule
    \end{tabular}}
\end{table}


\subsection{Eigenvalues for varying grid strengths}\label{SSDC_XR}

\begin{figure}
    \centering
     \captionsetup{justification=centering}
    \includegraphics[width = 6in, height = 2.75in]{figure/Chapter 4 Figures/Eigenvalues_SSDC_XR.pdf}
    \caption{Eigenvalues for various grid strengths with (a) SSDC - A (b) SSDC - B}
    \label{fig:SSDC_XR}
\end{figure}

Eigenvalues of the system with SSDC-A and SSDC-B are traced on the complex plane for varying grid strengths, which is a function of the $X/R$ ratio. It was shown in Section \ref{Evals_SVC_XR} that as the strength of the grid decreases, the percentage series compensation at which the system becomes unstable due to SSR becomes smaller. In Figure \ref{fig:SSDC_XR}, it is shown that with SSDC-A and SSDC-B, the eigenvalues of the system are all found to lie on the left half of the complex plane, thus eliminating the Hopf bifurcation due to SSR. In SSDC-B, with controller gains $K_{p1},K_{p2} = 0.1$ and $K_{i1},K_{i2} = 50$ and a stabilizer gain $K_{SS}$ of $0.3$, all the modes are found to be well damped for $X/R$ ratios to about $15$. Beyond that, the stabilizer gain is increased to $0.7$ and all the modes are found to be well damped for $X/R = \{10,20,30\}$, and those are shown in Figure \ref{fig:SSDC_XR}b.  The values of control gains of the SVC and SSDC are presented in Table \ref{tab:parameters_SVC_2}. Re-tuning of the stabilizer and PI controllers in the voltage regulator are not needed with SSDC-A for all the $X/R$ ratios considered here.

\begin{table}
    %\centering
    \caption{SVC control gains and SSDC parameters considered in Section~\ref{SSDC_XR}.}
    \label{tab:parameters_SVC_2}
    {\begin{tabular}{ccccccccc}
        \toprule
      \,\,\, \boldmath{$SSDC$}\,\,\, & \,\,\boldmath{$K_{p1}$}\,\,  &\,\,\boldmath{$K_{p2}$} \,\, &\,\,\boldmath{ $K_{i1}$} \,\, &\,\,  \boldmath{$K_{i2}$}\,\, & \,\, \boldmath{$K_{d}$ }\,\, &\,\,\,\,  \boldmath{$T_{d1}$}\,\,\,\,  &\,\,\,\, \boldmath{$T_{d2}$} \,\,\,\, &\,\,\,\, \boldmath{$T_{w}$}\,\,\,\, \\
        \midrule
        A & 0.55 & 0.1 & 1 & 1 & 450 & 10 ms & 150 ms & 10 s\\
        \midrule
        B & 0.1 & 0 & 50 & 50 & 0.7 & 10 ms & 100 ms & 10 s\\
        \bottomrule
    \end{tabular}}
\end{table}

\subsubsection{Eigenvalue trajectories for varying measurement time delays in the SVC}\label{SSDC_Tdelay}

\begin{figure}
    \centering
     \captionsetup{justification=centering}
    \includegraphics[width = 6in, height = 2.75in]{figure/Chapter 4 Figures/Eigenvalues_SSDC_Tdelay.pdf}
    \caption{Eigenvalues for different measurement delays with (a) SSDC - A (b) SSDC - B}
    \label{fig:SSDC_Tdelay}
\end{figure}

Eigenvalues of the system with SSDC for $3$ values of measurement delays (1, 10 and 100 ms) are plotted on the complex plane and are presented in Figure \ref{fig:SSDC_Tdelay}. The~parameters of SSDC-A and -B and that of the SVC with SSDC-A are same as those presented in Table~\ref{tab:parameters_SVC_1}. For~the system with SSDC-A, all the modes are damped and no re-tuning of the PI controls of the voltage controller is needed. However, with SSDC-B, the~controller gains of the SVC have to be re-tuned in order to obtain a stable system up to $95\%$ series compensation for each of the time delays considered here; those values are presented in Table \ref{tab:parameters_SVC_3}. Hence, it is shown that both SSDC-A and SSDC-B are able to mitigate subsynchronous oscillations as well as the unstable low frequency modes,  which appear for a time delay of $100$ ms in the system without the damping controller. 

\begin{table}[t!]
    %\centering
    \caption{SVC control gains with SSDC-B considered in Section \ref{SSDC_Tdelay}.}
    \label{tab:parameters_SVC_3}
    {\begin{tabular}{ccccccccc}
        \toprule
        \,\,\,\,\,\,\,\,\,\,\,\, \boldmath{$T_{m1}$, $T_{m2}$}\,\,\,\,\,\,\,\,\,\,\,\, &\,\,\,\,\,\,\,\,\,\,\,\, \boldmath{$K_{p1}$}\,\,\,\,\,\,\,\,\,\,\,\, &\,\,\,\,\,\,\,\,\,\,\,\, \boldmath{$K_{p2}$}\,\,\,\,\,\,\,\,\,\,\,\, &\,\,\,\,\,\,\,\,\,\,\,\, \boldmath{$K_{i1}$} \,\,\,\,\,\,\,\,\,\,\,\,& \,\,\,\,\,\,\,\,\,\,\,\boldmath{$K_{i2}$ }\,\,\,\,\,\,\,\,\,\,\, \\

        \midrule
        1 ms & 0.1 & 0 & 200 & 100 \\
        \midrule
        10 ms & 0.1 & 0.2 & 100 & 100\\
        \midrule
        100 ms &  0.1 & 0.3 & 300 & 10\\
        \bottomrule
    \end{tabular}}
\end{table}

\section{Time domain simulations}

\begin{figure}[t!]
    \centering
    % \captionsetup{justification=centering}
    \includegraphics[scale = 0.6]{figure/Chapter 4 Figures/Frequency_Disturbance.pdf}
    \caption{Disturbance simulation in grid frequency}
    \label{fig:frequency}
\end{figure}

The dynamic responses of the system are presented in this section to disturbances in grid voltage and frequency. The disturbance with respect to the grid voltage involves a reduction in the voltage magnitude by $15\%$  for about $0.1$ s and then the voltage is restored to $1$ pu. In real time, this may occur due to a three phase fault at a bus resulting in a  reduction in all the surrounding bus voltages and the fault is assumed to be cleared in 6 cycles ($0.1$ seconds). A frequency disturbance is simulated using a mathematical function given by 
\begin{align}
f(t) = f_o - exp(-\tau t)\sin(\omega_n t)
\end{align}
where $\tau$ is the damping term and $\omega_n$ is the frequency of the disturbance. Here, $\tau = 2.5$ and $\omega_n = 2\hspace{0.1cm}$rad/s which is of the order of frequency involving electromechanical oscillations in the grid.  This mimics the frequency changes in the grid due to the response of speed governors to disturbances in the grid. As the frequency of the is perturbed, the $d$ and $q$ components of the grid voltage change accordingly which are shown in Fig. \ref{fig:frequency}.

\begin{figure}[t!]
    \centering
    \captionsetup{justification=centering}
    \includegraphics[scale = 0.55]{figure/Chapter 4 Figures/SVC_FC_StatorVoltage.pdf}
    \caption{Stator voltage magnitude response to a (a) Grid voltage disturbance, and (b) Grid frequency disturbance}
    \label{fig:SVC_FC_Vs}
\end{figure}

\begin{figure}[t!]
    \centering
    \captionsetup{justification=centering}
    \includegraphics[scale = 0.55]{figure/Chapter 4 Figures/SVC_FC_Linecurrent.pdf}
    \caption{Line current magnitude response to a (a) Grid voltage disturbance, and (b) Grid frequency disturbance}
    \label{fig:SVC_FC_IL}
\end{figure}

\begin{figure}[t!]
    \centering
    \captionsetup{justification=centering}
     \includegraphics[scale = 0.55]{figure/Chapter 4 Figures/SVC_FC_RotorSpeed.pdf}
    \caption{Rotor speed response to a (a) Grid voltage disturbance, and (b) Grid frequency disturbance}
    \label{fig:SVC_FC_wg}
\end{figure}

% \begin{figure}[t!]
%     \centering
%     \captionsetup{justification=centering}
%      \includegraphics[scale = 0.55]{figure/Chapter 4 Figures/SVC_FC_RealPower.pdf}
%     \caption{Real power response to a (a) Grid voltage disturbance, and (b) Grid frequency disturbance}
%     \label{fig:SVC_FC_Pg}
% \end{figure}

The dynamic responses of the system with SVC are plotted in Figs. \ref{fig:SVC_FC_Vs} to \ref{fig:SVC_FC_wg}  for an $X/R$ ratio of $30$ and a wind speed of $14$ m/s and $60\%$ series compensation. The controller parameters of SVC are chosen as $K_{P1} = 0.1$, $K_{P2} = 0$, $K_{I1} = K_{I2} = 40$. In this system, the damping controller is absent. The dynamic responses of the system without SVC described in Chapter 3 are also plotted for comparison. It can be observed from the plots that the system with SVC has a damped response to the disturbance in grid voltage and frequency. However, the system with fixed capacitors at the SCIG terminals is unstable as the dynamic responses show growing oscillations with respect to time. 

\begin{figure}[t!]
    \centering
    \captionsetup{justification=centering}
    \includegraphics[scale = 0.57]{figure/Chapter 4 Figures/Stator_Voltage_Response.pdf}
    \caption{Stator voltage magnitude response to a (a) Grid voltage disturbance, and (b) Grid frequency disturbance}
    \label{fig:SVC_SSDC_Vs}
\end{figure}

\begin{figure}[t!]
    \centering
    \captionsetup{justification=centering}
    \includegraphics[scale = 0.57]{figure/Chapter 4 Figures/Real_power_response.pdf}
    \caption{Real power delivered in response to a (a) Grid voltage disturbance, and (b) Grid frequency disturbance}
    \label{fig:SVC_SSDC_Pg}
\end{figure}

\begin{figure}[t!]
    \centering
    \captionsetup{justification=centering}
    \includegraphics[scale = 0.57]{figure/Chapter 4 Figures/Reactive_power_response.pdf}
    \caption{Reactive power delivered in response to (a) Grid voltage disturbance, and (b) Grid frequency disturbance}
    \label{fig:SVC_SSDC_Qg}
\end{figure}

The dynamic responses of the system with SVC but without the SSDC and the system with the SVC equipped with an SSDC based on the induction generator slip and the real power at the terminals of the SCIG are presented. The system is simulated for a wind speed of $14$ m/s and an $X/R$ ratio of 20 with a series compensation level of $71\%$. Responses of the voltage magnitude, real power and reactive power delivered at the stator terminals of the SCIG show oscillations of increasing amplitude for the case with SVC and without the SSDC due to SSR. The system equipped with the SSDC results in an oscillatory response which is well-damped. 

\begin{figure}[t!]
    \centering
    % \captionsetup{justification=centering}
    \includegraphics[scale = 0.67]{figure/Chapter 4 Figures/Phase_Plane_A_New.pdf}
    \caption{Dynamic torque-speed characteristics in response to a grid voltage disturbance}
    \label{fig:SSDCA_Te_Speed}
\end{figure}

\begin{figure}[t!]
    \centering
    % \captionsetup{justification=centering}
    \includegraphics[scale = 0.67]{figure/Chapter 4 Figures/Phase_Plane_B_New.pdf}
    \caption{Dynamic torque-speed characteristics in response to a grid frequency disturbance}
    \label{fig:SSDCB_Te_Speed}
\end{figure}

Oscillations in the stator and rotor currents result in growing oscillations in the electrical torque produced which causes a mismatch in the mechanical torque input from the wind and the electrical torque output of the generator which is manifested as oscillations of increasing magnitude in the speed of the rotor. However, the system with SSDC returns to the same operating point after a few cycles which can be observed from the phase-plane trajectory with respect to rotor speed shown in Figs. \ref{fig:SSDCA_Te_Speed} and \ref{fig:SSDCB_Te_Speed} when the system is subjected to a disturbance in the grid voltage and grid frequency respectively. 

\section{Conclusion}

An SVC used in the place of a fixed capacitor is able to eliminate the unstable node at small values of series compensation. In addition to a Hopf bifurcation, a node-focus bifurcation is also introduced into the system which may not  affect the stability of the system. Also, the series compensation level at which SSR occurs is increased with the SVC. The Hopf bifurcation point is affected by the strength of the grid in systems with fixed capacitors as well as an SVC equipped with a voltage regulator. As the strength of the grid decreases, the system becomes unstable for smaller values of series compensation due to SSR due to a decrease in the resonant frequency as the $X/R$ ratio of the line increases. However, the Hopf bifurcation point can be shifted in the system with SVC by proper adjustment of controller parameters in the voltage regulator. 
    
With higher time delays, there is a possibility of unstable low frequency oscillations which are eliminated by higher levels of series compensation. Hence, stable operation of a system with higher measurement delays might not be possible without a minimum level of series compensation. A supplementary damping controller is needed for stable operation of the system for high levels of series compensation. The proposed SVC voltage regulator with the supplementary damping controller based on the generator slip signal as well as the real power at the terminals is able to successfully damp subsynchronous oscillations when tuned properly.
 
