A key feature of the Modelica language is its objectoriented nature: components are instances of classes and they can aggregate other components, so that extremely large models can be efficiently designed as "trees of components". However, the structural analysis of Modelica models, a necessary step for generating simulation code, often relies on the flattening of this hierarchical structure, which undermines the scalability of the language and results in widely-used Modelica tools not being able to compile and simulate such large models. In this paper, we propose a novel method for the modular structural analysis of Modelica models. An adaptation of Pryce's Sigma-method for non-square DAE systems, along with a carefully crafted notion of component interface, make it possible to fully exploit the object tree structure of a model. The structural analysis of a component class can be performed once and for all, only requiring the information provided by the interface of its child components. The resulting method alleviates the exponential computation costs that can be yielded by model flattening; hence, its scalability makes it ideally suited for the modeling and simulation of large cyber-physical systems.