Fault-induced delayed voltage recovery in a long inhomogeneous power-distribution feeder

We analyze the dynamics of a distribution circuit loaded with many induction motors and subjected to sudden changes in voltage at the beginning of the circuit. As opposed to earlier work by Duclut et al. [Phys. Rev. E 87, 062802 (2013)], the motors are disordered, i.e., the mechanical torque applied to the motors varies in a random manner along the circuit. In spite of the disorder, many of the qualitative features of a homogeneous circuit persist, e.g., long-range motor-motor interactions mediated by circuit voltage and electrical power flows result in coexistence of the spatially extended and propagating normal and stalled phases. We also observed a new phenomenon absent in the case without inhomogeneity or disorder. Specifically, the transition front between the normal and stalled phases becomes somewhat random, even when the front is moving very slowly or is even stationary. Motors within the blurred domain appear in a normal or stalled state depending on the local configuration of the disorder. We quantify the effects of the disorder and discuss the statistics of distribution dynamics, e.g., the front position and width, total active and reactive consumption of the feeder, and maximum clearing time.

Figure 1

Electric and mechanical torques as functions of the mechanical frequency $\omega /{\omega }_0$ for a range of motor terminal voltages $v$, reference mechanical torque $T_0=0.32$, and $\alpha =0.1$. For the $v=0.86$ electrical torque curve (red triangles), there are three equilibrium solutions indicated by intersections with the mechanical torque curve (black solid line). The solution with the highest $\omega /{\omega }_0$ is the “normal” stable solution with the induction motor rotating near the grid frequency ${\omega }_0$. The “stalled state” with $\omega /{\omega }_0\simeq 0$ is also stable while the intermediate solution is unstable. For $v>v_c^{+}=0.9$ (light blue line with plus signs), there is only one solution corresponding to the normal state. For $v<v_c^{-}=0.83$ (green dashed line), there is only one solution corresponding to the stalled state. The points $(a,b,c,d)$ correspond to the same labels in Fig. 2.

Figure 2

Hysteretic behavior of an induction motor stable steady states in Fig. 1 as the voltage $v$ at its terminals is varied. The dashed red (solid black) curves indicate the path of equilibrium states as the voltage $v$ is decreased (increased) starting from the high-voltage normal (low-voltage stalled) state. The vertical lines at the spinodal voltages $v_c^{\pm }$ indicated the abrupt hysteretic transitions between states. $v_c^{\pm }$ correspond to the same labels in the legend of Fig. 1. The points ($a,b,c,d$) correspond to the states where the motor must make transitions from normal to stalled ($c\to a$) and from stalled to normal ($b\to d$).

Figure 3

Steady-state profile along the feeder for all motors in the “normal” state for a sample of disorder drawn from the $\Delta =0.0157$ and $z_d=0.015$ for $v_0=1$. Notice, that while the disorder is clearly seen in active and reactive density consumptions $p$ and $q$, its effect on $\omega$, the flows $\rho$ and $\varphi$, and the voltage $v$ is significantly reduced.

Figure 4

Typical sequence of snapshots of the dynamics for “Case A—Stalling front” for $\Delta =0.0157$ and $z_d=0.005$. The color of the lines is the same as in Fig. 3. As the stalling front passes from $z=L$ to lower $z$, the disorder in ${\tau }_0$ interacts with the state transition in Fig. 2 to amplify the disorder in ${\tau }_0$.

Figure 5

Gaussian fits to the probability density distribution functions over $\rho \left(0\right)$, $\varphi \left(0\right)$, and $z_f$ after reaching steady state for Case A—Stalling front dynamics. Statistics are collected over 100 samples, each disorder ensemble parametrized by $\Delta$ and $z_d$. The vertical dashed line indicates the position of the front for no disorder, i.e., $\Delta =0$ and $z_d=0$.

Figure 6

Probability distribution functions over $\rho \left(0\right)$, $\varphi \left(0\right)$, and $z_f$ observed for the $\Delta =0.0079$ and $z_d=0.005$ ensemble after reaching steady state following Case A—Stalling front dynamics for different values of $v_0$.

Figure 7

Typical sequence of snapshots of the dynamics for “Case B—Restoration front” for $\Delta =0.0157$ and $z_d=0.005$. The color of the lines is the same as in Fig. 3. As the restoration front passes from $z=0$ into the circuit, the disorder in ${\tau }_0$ interacts with the state transition in Fig. 2 to amplify the disorder in ${\tau }_0$.

Figure 8

Gaussian fits to the probability density distribution function over $z_f$ after reaching steady state for Case B—Restoration front dynamics. Statistics are collected over 100 samples, each disorder ensemble parametrized by $\Delta$ and $z_d$. The vertical dashed line indicates the position of the front for no disorder, i.e., $\Delta =0$ and $z_d=0$.

Figure 9

Histograms and respective Gaussian fits for the probability density distribution function of the maximum clearing time, ${\tau }_{\text{cl}}$, measured under experiments C for four different ensembles and for the case of no disorder (vertical red lines).

Figure 10

Histograms and respective probability density distribution functions for the width of the burry regions, $z_w$, shown for the final state of the respective ensembles (see the legend) in experiments A and B.