Electric Power System (Model of Dobson and Chiang)

exd15
Variables and parameters:

V: magnitude of the load voltage

$\delta$: phase angle of V

$\delta_m$: generator voltage phase angle

$\omega$: rotor speed

Q1: reactive power demand (bifurcation parameter: $\lambda:=Q_1$)

Model equations:

\begin{displaymath}\dot \delta_m =&\ \omega\cr \end{displaymath}


\begin{displaymath}M\dot\omega =&-d_m\omega +P_m-E_mVY_m\sin(\delta_m-\delta)\cr \end{displaymath}


\begin{displaymath}K_{q\omega}\dot\delta =&-K_{qv2}V^2-K_{qv}V+Q(\delta_m,\delta,V)-Q_0-Q_1\cr \end{displaymath}


\begin{displaymath}TK_{q\omega}K_{pv}\dot V =&\ K_{p\omega}K_{qv2}V^2+(K_{p\omega}K_{qv}-K_{q\omega}K_{pv})V\cr \end{displaymath}


\begin{displaymath}& +K_{q\omega}(P(\delta_m,\delta, V)-P_0-P_1) -K_{p\omega}(Q(\delta_m,\delta,V)-Q_0-Q_1)\cr\end{displaymath}

with variables and constants

\begin{displaymath}P(\delta_m,\delta,V) =& -E_0VY_0\sin(\delta)+E_mVY_m\sin(\delta_m-\delta)\cr \end{displaymath}


\begin{displaymath}Q(\delta_m,\delta,V) =&\ E_0VY_0\cos(\delta)+E_mVY_m\cos(\delta_m-\delta)-(Y_0+Y_m)V^2\cr\end{displaymath}


\begin{displaymath}&M=0.01464,\ Q_0=0.3,\ E_0=1.0,\ E_m=1.05,\ Y_0=3.33,\ Y_m=5.0,\cr \end{displaymath}


\begin{displaymath}&K_{p\omega}=0.4,\ K_{pv}=0.3,\ K_{q\omega}=-0.03,\ K_{qv}=-2.8,\ K_{qv2}=2.1,\cr \end{displaymath}


\begin{displaymath}&T=8.5,\ P_0=0.6,\ P_1=0.0,\ C=12.0,\ P_m=1.0,\ d_m=0.05.\cr \end{displaymath}


Comments

Figure 1
Bifurcation diagram V versus load $\lambda$ (detail: $\lambda\geq 2.41$, all stationary states and some periodic states; for the latter ones $V_{\min}$ is depicted).

Figure 2
Bifurcation diagram V and $V_{\min}$ versus load $\lambda$. detail: $2.555\leq\lambda\leq 2.568$, all stationary states and some periodic states (S: stationary, P: periodic, s: stable, u: unstable, HB: Hopf bifurcation, PD: periodic doubling)

Figure 3
Phase diagram (projection). Horizontal axis: $\delta_m$; vertical axis: V. ( $\lambda=2.55878$)

Figure 4
Phase diagram (projection). Horizontal axis: $\delta_m$; vertical axis: V. ( $\lambda=2.5649$)

Figure 5
Phase diagram (projection). Horizontal axis: $\delta_m$; vertical axis: V. ( $\lambda=2.566525$)

Figure 6
Phase diagram, periodic orbit, unstable $\lambda=2.5628$, T=4.6376, projection: horizontal axis $\omega$, vertical axis V.

Figure 7
Phase diagram, two periodic orbits for $\lambda=2.56016$, one stable (T=3.802) and one unstable (T=1.913) projection: horizontal axis $\omega$, vertical axis V.

Figure 8
Phase diagram, unstable periodic orbit, $\lambda=2.5602$, T=7.62, projection: horizontal axis $\omega$, vertical axis V.

Figure 9
Phase diagram, $\lambda=2.5603$, $0\leq t\leq 100$. projection: horizontal axis $\omega$, vertical axis V.

Figure 10
$V(t),\ \lambda=2.55988,\ 0\leq t\leq 1.873$. Periodic orbit: ``simple'' period.

Figure 11
$V(t),\ \lambda=2.56016,\ 0\leq t\leq 3.802$. Periodic orbit: ``double'' period.

Figure 12
$V(t),\ \lambda=2.5602,\ 0\leq t\leq 7.62$. Periodic orbit: ``fourfold'' period.

Figure 13
$V(t),\ \lambda=2.5602$

Figure 14
$V(t),\ \lambda=2.5603$

Figure 15
V(t), with $\lambda$ varying linearly from $\lambda=2.5583$ to $\lambda =2.572$. Collapse of voltage.

Figure 16
Phase diagram corresponding to previous figure. Projection to $(\omega,V)$-plane, $0\leq t\leq 68.67$, with $\lambda$ varying linearly from $\lambda=2.5583$ to $\lambda=2.572$.

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