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Electric Power System (Model of Dobson and Chiang)

Original source of the model: I. Dobson, H.-D. Chiang: Towards a Theory of Voltage Collapse in Electric Power Systems. Systems and Control Letters 13 (1989) 253-262.

The example exhibits a turning point, a Hopf bifurcation, and a period doubling sequence. All of the bifurcations are essential ingredients of a risk analysis for electric power systems. Figures 15 and 16 show a simulation for varying $\lambda$ (the reactive power demand). The simulation illustrates a collapse of the voltage V of a generator.

For the other figures, the parameter is kept (quasi-)stationary. Figure 1 and 2 show the bifurcation scenario, the other figures show phase portraits and time history for a selection of parameter values. The development of a blue-sky catastrophe is illustrated in Figures 3-5. Figures 10-14 report on the growing complexity of the orbits when $\lambda$ is passing through the range of period doubling. This leads to chaos, which is illustrated in Figure 9.

For further reference on the risk aspect, see R. Seydel: Assessing Voltage Collapse. Proceedings of a Conference on Risk Analysis. Paris 1998.

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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