Analysis of a dc bus system with a nonlinear constant power load and its delayed feedback control

This paper tackles a destabilizing problem of a direct-current (dc) bus system with constant power loads, which can be considered a fundamental problem of dc power grid networks. The present paper clarifies scenarios of the destabilization and applies the well-known delayed-feedback control to the stabilization of the destabilized bus system on the basis of nonlinear science. Further, we propose a systematic procedure for designing the delayed feedback controller. This controller can converge the bus voltage exactly on an unstable operating point without accurate information and can track it using tiny control energy even when a system parameter, such as the power consumption of the load, is slowly varied. These features demonstrate that delayed feedback control can be considered a strong candidate for solving the destabilizing problem.

Figure 1

Conceptual diagram of a dc bus system (CPL: constant power load; PV: photovoltaic cell).

Figure 2

Simplified dc bus system.

Figure 3

Nullclines of system (3) (bold line: $\dot{x}=0$; dotted line: $\dot{y}=0$).

Figure 4

Bifurcation curves of system (3): (a) ${\mathit{p}}_{+}$ and (b) ${\mathit{p}}_{-}$. SN: saddle-node bifurcation; HP: Hopf bifurcation; HO: Homoclinic bifurcation; DZ: double-zero bifurcation; BW: breakaway.

Figure 5

Phase portraits of system (3) ($b=0.40$). These portraits correspond to the arrowed broken line A in Fig. 4. D: saddle; $S_{\mathrm{f}}$: stable focus; $U_{\mathrm{f}}$: unstable focus; $U_{\mathrm{n}}$: unstable node; ULC: unstable limit cycle.

Figure 6

Phase portraits of system (3) ($b=1.20$). These portraits correspond to the arrowed broken line B in Fig. 4. $S_{\mathrm{n}}$: stable node.

Figure 7

Simplified dc bus system with delayed feedback control.

Figure 8

Sketch of ${\psi }_{1,2},{\omega }_{1,2},T$ introduced in Theorem 1: $g(s,T)$ is stable within the gray intervals.

Figure 9

Stabilizable parameter space. Inequalities (18) hold in the gray region.

Figure 10

Stability regions in controller parameter space, $k$ vs $T$ ($a=0.17$, $b=0.20$). Thin black line: $({\psi }_1+2\pi l)/{\omega }_1$; bold red line: $({\psi }_2+2\pi l)/{\omega }_2$.

Figure 11

Time-series data of state variables $x$ and $y$, parameter $a$, and control signal $u$ for $b=0.20$. Controller (12) with $k=0.10$ and $T=5.0$ is applied to system (3) at $\tau =150$.

Figure 12

Trajectory of $(x,y)$ on the phase plane ($b=0.2$): (a) $a=0.10\to 0.17$ at $\tau =150$, (b) $a=0.10$, and (c) $a=0.17\to 0.17+0.01sin0.05\pi (\tau -250)$ at $\tau =250$. Open circle: initial state $\left(x\right(0),y(0\left)\right)$; red (gray) point: fixed point ${\mathit{p}}_{+}$D.