We propose an adaptive control strategy for the simultaneous estimation of topology and synchronization in complex dynamical networks with unknown, time-varying topology. Our approach transforms the problem of time-varying topology estimation into a problem of estimating the time-varying weights of a complete graph, utilizing an edge-agreement framework. We introduce two auxiliary networks: one that satisfies the persistent excitation condition to facilitate topology estimation, while the other, a uniform-$\delta$ persistently exciting network, ensures the boundedness of both weight estimation and synchronization errors, assuming bounded time-varying weights and their derivatives. A relevant numerical example shows the efficiency of our methods.

There is an increasing need for monitoring and controlling uncertainties brought by distributed energy resources in distribution grids. For such goal, accurate three-phase topology is the basis for correlating and exterminating measurements in unbalanced distribution networks. Unfortunately, such topology knowledge is often unavailable due to limited investment, especially for secondary distribution grids. Also, the bus phase connectivity information is inaccurate due to human errors or outdated records. For this challenge, we utilize smart meter data at different phases for an information-theoretic approach to learn the structures. Specifically, we convert the system of three unbalanced phasors into symmetrical components, namely the positive, negative, and zero sequences. Then, we prove that Chow-Liu algorithm can find the optimal topology by utilizing power flow equation and the conditional independence relationships implied by the radial three-phase structure of distribution grids with the presence of incorrect bus phase labels. At last, by utilizing Carson's equation, we prove that the bus phase connection can be correctly identified using voltage measurements. For validation, we extensively simulate on IEEE $37$- and $123$-bus systems using real data from PG\&E, ADRES Project, and Pecan Street. We observe that the algorithm is highly accurate for finding three-phase topology in distribution grids even with strong load unbalancing condition and DERs. This ensures close monitoring and controlling DERs in distribution grids.