Dynamics of dc bus networks and their stabilization by decentralized delayed feedback

The present paper deals with the dynamics of bus networks, which consist of several identical dc bus systems connected by resistors. It is analytically guaranteed that the stability of a stand-alone dc bus system is equivalent to that of the networks, independent of the number of bus systems and the network topology. In addition, we show that a decentralized delayed-feedback control can stabilize an unstable operating point embedded within the networks. Moreover, this stabilization does not depend on the number of bus systems or the network topology. A systematic procedure for designing the controller is presented. Finally, the validity of the analytical results is confirmed through numerical examples.

Figure 1

Schematic diagram of a dc bus network.

Figure 2

Five different bus networks: Network 0 (no connection), Network 1 (ring network on weak coupling), Network 2 (ring network with two shortcuts on strong coupling), Network 3 (chain network on weak coupling), and Network 4 (network 3 with bus system 4 on strong coupling).

Figure 3

Time-series data for $x_n$ and $u_n^{\left(\varepsilon \right)}(n=1,...,5)$ in a bus network for which the topology and coupling strength are sequentially switched from Network 0 to Network 4: $\varepsilon =0.1$ for Networks 1 and 3 and $\varepsilon =1.0$ for Networks 2 and 4.

Figure 4

Parameter region of stable, spatially uniform equilibrium state ${\mathbf{X}}_{+}^{*}$ with parameter mismatch $p=0.2$ $\left(20\%\right)$. Gray region: $(\overline{a},\overline{b})$, where a bus network switched as shown in Fig. 3 does not diverge. HP, Hopf bifurcation; SN, saddle-node bifurcation; DZ, double-zero bifurcation.

Figure 5

Schematic diagram of a bus system network: system 1 has a high CPL $a_1>a_{\mathrm{HP}}$ and the other systems have low CPLs $a_2=a_3=a_4=a_5<a_{\mathrm{HP}}$. All of the bus systems have $b_1=\cdots =b_5<1$.

Figure 6

Time-series data of states $x_n(n=1,...,5)$ and parameter $a_1$ for the bus system network shown in Fig. 5 $\left(a_2=\cdots =a_5=0.1<a_{\mathrm{HP}}=5/36,b_1=\cdots =b_5=0.2<1,{\varepsilon }_1=\cdots ={\varepsilon }_5=0.05\right)$. All of the states are randomly disturbed with small amplitude $0.03$ at instances when $a_1$ jumps to the next higher level. Bus system 1 is connected only to system 2.

Figure 7

Critical parameter value ${\widehat{a}}_1$ as a function of coupling strength $\varepsilon$ and the degree of system 1, $d_1$. The parameters and the numerical procedure are the same as in Fig. 6.

Figure 8

Schematic diagram of a dc bus system with delayed feedback

Figure 9

Boundary curves of $T$ plotted with respect to $k$ for $g(s,T,0)$ to be stable $\left(a=0.17,b=0.20\right)$. Thin black (Bold red) line: lower (upper) stable intervals of $T$ in Eq. (21).

Figure 10

Functions $f_1\left(\omega \right)$ and $f_2\left(\omega \right)$ plotted with respect to $\omega >0$ for designed $(k,T)=(0.10,5.00)$ $\left(a=0.17,b=0.20\right)$. There exists no $\omega$ such that both $f_1\left(\omega \right)=0$ and $f_2\left(\omega \right)<0$ hold.

Figure 11

Time-series data of $x_n,u_n^{\left(\varepsilon \right)},u_n^{\left(k\right)}(n=1,...,5)$ of the switched bus network $\left(a=0.17,b=0.20\right)$ with decentralized delayed-feedback controller (14) $\left(k=0.10,T=5.00\right)$. The situation is the same as in Fig. 3, except that $a:0.10\to 0.17$ and $k:0\to 0.10$.

Figure 12

Time-series data of $x_n,u_n^{\left(\varepsilon \right)},u_n^{\left(k\right)}(n=1,...,5)$ of the switched bus network $(\overline{a}=0.17,\overline{b}=0.20,\overline{\varepsilon }=0.1$ or $1.0)$ with parameter mismatch $p=0.1$ $(10\%$ error). The situation is the same as in Fig. 11, except for parameter mismatch.

Figure 13

Time-series data of $x_n,a_1$, and $u_n^{\left(k\right)}(n=1,...,5)$ of the controlled bus network $\left(k=0.10,T=5.00\right)$, with a slow time-varying CPL $a_1=0.17+0.15sin2\pi (\tau -100)/400$, shown in Fig. 5. Bus system 1 is connected to bus systems 2 and 3.