<tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$N -1$ security is an important criterion for the development and planning of distribution networks. The existing <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$N-1$ contingency analysis methods of distribution networks are primarily based on the unit of components

and the faults of each component are analyzed and verified one by one. The calculation process of these methods is complicated

and the positioning effect of the weak links needed for distribution network planning is insufficient. For this reason

this paper proposes a method of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$N-1$ contingency parallel analysis for main transformers based on the incidence matrix. First

the main transformer incidence matrix is established

and the load transfer process is classified into three types

according to different transfer modes

and the contact matrices are then established respectively. Second

considering the capacity constraint of main transformers and tie-lines

the corresponding capacity matrices are established. The maximum transfer capability (MTC) of the distribution network in the case of each main transformer fault is calculated by contact matrices and capacity matrices

and the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$N-1$ contingency analysis result is obtained by comparing MTC with the load that needs transferring. Meanwhile

the transfer margin of the main transformers and tie-lines after transferring can be obtained in the process of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$N-1$ contingency analysis

so as to find any weak links. An example is given to verify the correctness and effectiveness of the proposed method. In this paper

the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$N-1$ contingency analysis results of all main transformers in the distribution network are expressed analytically through matrices operations

and the weak links can then be intuitively identified. The cumbersome operation of checking each component is avoided

and the computational efficiency is improved.