## 1 Introduction  
Over the last decades, a number of power system blackouts caused by natural disasters or human factors have occurred, which led to a series of serious effects on the functionality and reliability of the networks, economic growth, and social stability [1,2]. The failure of a few critical nodes or lines can make the system very vulnerable to attacks, which may be the major reason causing the blackouts [3, 4, 5, 6]. Thus, it is very important to identify the critical nodes and lines so that the system reliability and efficiency can be improved by monitoring and protecting them. In the past decades, studying the vulnerability of power systems has been a hot issue and many worthy results have been obtained [3]. Most of them assess the vulnerability of the power system by studying the system stability including transient [7], voltage [8,9], and frequency [10] stability, _etc._  
It is well recognized that the complex network theory is very suitable to solve large-scale practical problems. Based on the graph theory, the subject system can be studied from the point of view of structure and dynamical functions of an array of nodes and lines, in which the degree of dependence on the dimensionality of the system can be obviously reduced. Recent developments in this field have opened up a new way to power system research. In recent years, there has been an increasing interest in the study of assessing the vulnerability of the power system based on the complex network theory [3,11, 12, 13, 14, 15, 16, 17, 18]. For example, [13] analyzed the structural vulnerability of the North American power grid and concluded that the nodes with high degree play an important role in power systems. Considering the influence of physical topological structure, the concept of betweenness is proposed to identify important nodes or lines in [14, 15, 16, 17]. Additionally, a novel betweenness index was defined in [11], in which the reactance was defined as the weight of the lines. However, it is not necessary that the power only flows through the shortest path in a power system. Then, the idea of using the maximum flow theory to analyze the vulnerability of power systems was first proposed in [18]. [PERSON] _et al._ have defined a centrality index to identify the critical lines in power systems with improved results in [3], which considered the maximum flow from the generator nodes to the load nodes based on the Ford-Fulkerson theorem. It should be noted that, although lots of worthy results have been obtained, there still exist some problems that were not well handled. Firstly, few of the existing literatures take the system's controllability into account. However, if the power systems cannot offer full control over the networks, when several important nodes are in fault, some serious phenomenon may occur, e.g., the phenomenon of voltage collapse, which has been well discussed in [19,20]. Secondly, in order to calculate the indexes such as the betweenness index [17] and the centrality index [3], weight of each link (transmission line) needs to be known. However, for most real networks, the weight of each link is either unknown or known only approximately and time-dependently. These problems are the main motivation of our study. Network controllability is a vast area of research with a long history [21, 22, 23, 24, 25, 26, 27, 28]. The structural controllability was first presented and studied in [26], which proved that a physical system is controllable if the system is structurally controllable. Furthermore, in [29], [PERSON] _et al._ extended the results in [26] and proposed a network controllability model to solve the computable problems of the network controllability. It concluded that the minimum number of driver nodes can be calculated by the maximum matching method which was introduced in [30], and the structural controllability problem can be translated into an equivalent geometrical problem.