Analysis of Semidefinite Programming Relaxation of Optimal Power Flow for Cyclic Networks
Analysis of Semidefinite Programming Relaxation of Optimal Power Flow for Cyclic Networks
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DOI：
10.1109/cdc.2018.8618935
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发表时间：
2018-12
期刊：
影响因子：
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通讯作者：
H. Mahboubi;J. Lavaei
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文献类型：
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作者：
H. Mahboubi;J. Lavaei
最优潮流（OPF）问题确定了电力网络的最优运行点，该点最小化受物理和网络约束的特定目标函数。近年来，OPF问题的凸松弛问题引起了广泛的关注，研究表明，半定规划（SDP）松弛可以使不同类型的非凸OPF问题达到全局或近全局最优。虽然已知SDP松弛在各种条件下对径向网络都是精确的，但求解循环网络的最优潮流问题还需要进一步的研究。在本文中，我们提出了充分条件下的SDP松弛是特殊的，但重要的循环网络的准确。更确切地说，当目标函数是一个线性函数的有功功率，我们证明了SDP松弛是准确的奇数周期在一定条件下。此外，在不同的技术条件下，还证明了大小为3和4的简单循环的SDP松弛的精确性。在有耗和无耗两种情况下，证明了弱循环网络秩1或秩2 SDP解的存在性。此外，当目标函数是无功功率的增函数时，我们证明了在一定条件下SDP松弛是精确的。这一结果证明了为什么无功功率的总和作为一个低秩促进项的OPF。本文的研究结果为循环网络的不同构建块的SDP行为提供了直观的认识。
The optimal power flow (OPF) problem determines an optimal operating point of the power network that minimizes a certain objective function subject to physical and network constraints. There has been a great deal of attention in convex relaxation of the OPF problem in recent years, and it has been shown that a semidefinite programming (SDP) relaxation can solve different classes of the nonconvex OPF problem to global or near-global optimality. Although it is known that the SDP relaxation is exact for radial networks under various conditions, solving the OPF problem for cyclic networks needs further research. In this paper, we propose sufficient conditions under which the SDP relaxation is exact for special but important cyclic networks. More precisely, when the objective function is a linear function of active powers, we show that the SDP relaxation is exact for odd cycles under certain conditions. Also, the exactness of the SDP relaxation for simple cycles of size 3 and 4 is proved under different technical conditions. The existence of rank-1 or -2 SDP solutions for weakly-cyclic networks is also proved in both lossy and lossless cases. In addition, when the objective function is an increasing function of reactive powers, we prove that the SDP relaxation is exact under certain conditions. This result justifies why the sum of reactive powers acts as a low-rank-promoting term for OPF. The findings of this paper provide intuition into the behavior of SDPs for different building blocks of cyclic networks.