Phase synchronization analysis of bridge oscillators between clustered networks

Abstract

Recent works aim to establish necessary and sufficient conditions to guarantee phase synchronization between clusters of oscillators, usually assuming knowledge of the intra-cluster connections, that is, connections among oscillators of the same cluster. In this context, this paper takes a different approach in studying the stability of the synchronous manifold between clusters. By focusing on the inter-cluster relations between the bridge oscillators, a simplified problem is considered where intra-cluster effects are described as perturbations. Based on Lyapunov’s direct method, a framework is put forward to derive sufficient conditions for the ultimately boundedness of the phase difference between the bridge oscillators. This analysis does not rely on full information on the adjacency matrix describing the specific connections among oscillators within each cluster, an information that is not always available. The established theoretical conditions are compared to numerical simulations in two examples: (i) two interconnected clusters of Kuramoto oscillators, and (ii) a benchmark model of a power grid. Results indicate that the method is effective and that its conservativeness depends on the available network information. This framework can be generalized to different networks and oscillators.

Appendices

A Proof of Theorem

Proof of Theorem 1

Consider Lemma 1. A change of coordinates $\theta _0 = \theta - \bar{\theta }$, with $\bar{\theta } = \sin ^{-1}(\varDelta \omega /2K)$, shifts ${\theta }_0$ to the origin, yielding

$$\begin{aligned} \dot{\theta }_0= & {} \varDelta \omega - 2K \sin (\theta _0 + \bar{\theta }) \nonumber \\= & {} \varDelta \omega - \left( \varDelta \omega \cos \theta _0 + \sqrt{4K^2 - \varDelta \omega ^2}\sin \theta _0\right) , \end{aligned}$$

(21)

where $\left| \frac{\varDelta \omega }{2K}\right| \le \frac{\sqrt{2}}{2} <1$ by assumption.³

Let $V(\theta _0) = \frac{1}{2}\theta _0^2,~\theta _0 \in D_2$, be a positive definite candidate Lyapunov function for system (21), where $D_2 = \{\theta _0 \in \mathbb {R}: |\theta _0|<r<\frac{\pi }{2}\}$. It can be shown that $\dot{V}(\theta _0)$ is definite negative if $\left| \frac{\varDelta \omega }{2K}\right| \le \frac{\sqrt{2}}{2}$ and $\theta _0 \in D_2$. Hence, the candidate function is a Lyapunov function, establishing sufficient conditions for the local asymptotic stability of the origin.

The quadratic Lyapunov function $V(\theta _0)$ satisfies conditions (4) and (6) for $c_1\le \frac{1}{2} \le c_2$ and $c_4\ge 1$. Condition (5) is satisfied as follows:

$$\begin{aligned} -c_3|\theta _0|^2&\ge \theta _0\dot{\theta }_0 = |\theta _0|\hbox {sign}(\theta _0)\dot{\theta }_0, \nonumber \\ -c_3|\theta _0|&\ge \left( \varDelta \omega (1 {-} \cos \theta _0) {-} \sqrt{4K^2 {-} \varDelta \omega ^2}\sin \theta _0\right) \hbox {sign}(\theta _0), \nonumber \\ c_3|\theta _0|&\le \hbox {sign}(\theta _0) h(\theta _0) , \end{aligned}$$

(22)

where $h(x) = |\varDelta \omega |(1 - \cos x) + \sqrt{4K^2 -\varDelta \omega ^2}\sin x$, assuming—without loss of generality—that $\varDelta \omega \le 0$. Since $\left| \frac{\varDelta \omega }{2K}\right| \le \frac{\sqrt{2}}{2}$, we have that $c_3 \le -h(-r)/r$ is the slope of the lower bound of $\hbox {sign}(\theta _0) h(\theta _0)$, for all $|\theta _0(t)|<r$. Figure 8 illustrates this derivation.

Fig. 8

Illustration of equation (22) derivation. The sinusoidal function h(x), where $h(0)=0$, is a sum of both functions depicted in gray dotted lines in interval $x\in \left[ -\pi ,\pi \right]$. The absolute value of the straight line of slope $c_3=-h(-r)/r$ is the lower bound of $\hbox {sign}(x)h(x)$. The region of interest $|x|<r<\frac{\pi }{2}$ is highlighted

By satisfying conditions (4)–(6), system (21) is proven to be exponentially stable. Following Lemma 1, for a perturbation satisfying

$$\begin{aligned} |g(t,\theta )| \le g_\mathrm{max} < \delta = -h(-r) \cdot \epsilon , \quad \forall \theta \in D_1 , \end{aligned}$$

(23)

the solution $\theta (t)$ is bounded by (14), where

$$\begin{aligned} \alpha = -\frac{(1-\epsilon )\cdot h(-r)}{r}, \quad b =-\frac{r\delta }{h(-r)\cdot \epsilon } . \end{aligned}$$

(24)

Note that $\delta$ and b are dependent on r, where $-h(-r)$ is a monotonic function for all $r<\frac{\pi }{2}$ and radius r, in turn, can be chosen arbitrarily small. Thus, for a given $\varDelta \omega$ and based on an optimal choice of r, one can define a minimum threshold $K_\mathrm{c}$, where condition $\left| \frac{\varDelta \omega }{2K}\right| \le \frac{\sqrt{2}}{2}$ is satisfied, via

$$\begin{aligned}&g_\mathrm{max}< \delta = -h(-r)\cdot \epsilon \nonumber \\&g_\mathrm{max}< \left( -|\varDelta \omega | + 2K\cos \left( r - \tan ^{-1}\beta \right) \right) \cdot \epsilon \nonumber \\&g_\mathrm{max} < \left( 2K - |\varDelta \omega |\right) \cdot \epsilon \nonumber \\&K_\mathrm{c} > \frac{g_\mathrm{max}+|\varDelta \omega |\epsilon }{2\epsilon } . \end{aligned}$$

(25)

where $r = \tan ^{-1}\beta$ and $\beta =\sqrt{4K^2-\varDelta \omega ^2}/ |\varDelta \omega |$. It is straightforward to derive (15) from (23) and (24). $\square$

Remark 6

In Theorem 1, the choice of r was defined in order to maximize $\delta$, thus extending Theorem 1 applicability to network systems with higher perturbations $g_\mathrm{max}$. Moreover, this guarantees that, for a given $g_\mathrm{max}$, the theoretical estimate of $K_\mathrm{c}$ is minimum. However, this choice of r is not concerned with providing the smallest theoretical estimate of b. Indeed, if it is of interest to obtain a tighter theoretical ultimate bound, the optimization problem in (25) can be reformulated to find the optimal r for a minimum b subject to $g_\mathrm{max} < \delta (r)$.

Fig. 9

Theoretical ultimate bound b versus the coupling strength K, for different values of r and $(\bar{\omega }_1,\bar{\omega }_2) = (1.0, 1.5)$. Numerical results are shown in black solid line. Theoretical results are shown for $r=\tan ^{-1}\beta$ (circle); $r=\frac{\pi }{2}$ (triangle); $r=\frac{\pi }{4}$ (diamond), and $r=\frac{\pi }{8}$ (square). For each r, the threshold $K_\mathrm{c}(r)$ is determined based on (23) and traced in the plot by a dashed line of respective color. (Color figure online)

Figure 9 shows how the choice of r is a trade-off between the estimates of b and $K_\mathrm{c}$. If r is a constant for all values of K, the smaller r, the closer the theoretical estimate of b is to the numerical results and the higher the threshold $K_\mathrm{c}$. In Theorem 1, an adaptive choice of $r=\tan ^{-1}\beta$ is determined in order to acquire a minimum value for $K_\mathrm{c}$—aiming for an increase in applicability of Theorem 1 in exchange for precision of b. Nevertheless, it can be seen numerically in Fig. 9 that its respective $K_\mathrm{c}$ is indeed the smallest, with a satisfactory estimate of b.

B Generalization to multiple bridges

This section shows how the proposed framework could be generalized to deal with network systems with multiple clusters and “bridges”. Consider the network in Fig. 2c as a guideline to study this generalization and assume that all nodes are composed of Kuramoto oscillators as in (10). As illustrated in Fig. 2c, this set of interconnected clusters can be simplified by a set of bridge oscillators plus the perturbation terms. Let $\theta _{ij}\equiv \phi _i-\phi _j$ and $\varDelta \omega _{ij} =\omega _i-\omega _j$, then the stability of the bridge oscillators synchronous manifold can be studied through the following set of equations:

$$\begin{aligned} \dot{\theta }_{23}&= \varDelta \omega _{23} - 2a_{23}\sin \theta _{23} -a_{24}\sin \theta _{24} + g_{23}(t,\theta ) \nonumber \\ \dot{\theta }_{24}&= \varDelta \omega _{24} - 2a_{24}\sin \theta _{24} -a_{23}\sin \theta _{23} + g_{24}(t,\theta ) \nonumber \\ \dot{\theta }_{15}&= \varDelta \omega _{15} - 2a_{15}\sin \theta _{15} +a_{56}\sin \theta _{56} + g_{15}(t,\theta ) \nonumber \\ \dot{\theta }_{56}&= \varDelta \omega _{56} - 2a_{56}\sin \theta _{56} -a_{15}\sin \theta _{15} + g_{56}(t,\theta ) \end{aligned}$$

(26)

where $g_{ij} = \sum _{k\in \mathcal {C}_i}a_{ik}\sin \theta _{ki} -\sum _{k\in \mathcal {C}_j}a_{jk}\sin \theta _{kj}$.

As mentioned earlier, since the study of PS between bridge oscillators can be a first step to achieve full cluster PS, it might be relevant to determine under which conditions all interconnected bridge oscillators PS simultaneously. The proposed perturbation approach could be relevant in this case when the intra-cluster dynamics $g_{ij}(t,\theta )$ are not exactly known, but its bounds are. In this situation, for the network of Fig. 2c, the sufficient conditions for phase synchronization of all bridge oscillators can be derived by proving the conditions for exponential stability of the four-dimensional system (26) and establishing the upper bound that the perturbation term must follow according to Lemma 1.