When the shunt elements in the $\Pi$ circuit line model are assumed zero, it has been proved that branch flow models are equivalent to bus injection models and that the second-order cone relaxation of optimal power flow problems on a radial network is exact under certain conditions. In this note we propose a branch flow model that includes nonzero line shunts and prove that the equivalence and the exactness of relaxation continue to hold under essentially the same conditions as for zero shunt elements.

中文翻译：

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关于具有分流器的支流模型的说明

当分流元件在 $\Pi$假设电路线模型为零，证明了支流模型等价于母线注入模型，径向网络上最优潮流问题的二阶锥弛豫在一定条件下是准确的。在本说明中，我们提出了一个包含非零线路分流器的分支流模型，并证明在与零分流器元件基本相同的条件下，松弛的等效性和精确性继续保持。