This article establishes sufficient conditions for the uniqueness of AC power flow solutions via the monotonic relationship between real power flow and the phase angle difference. More specifically, we prove that the P-Θ power flow problem has at most one solution for any acyclic or GSP graph. In addition, for arbitrary cyclic power networks, we show that multiple distinct solutions cannot exist under the assumption that angle differences across the lines are bounded by some limit related to the maximal girth of the network. In these cases, a vector of voltage phase angles can be uniquely determined (up to an absolute phase shift) given a vector of real power injections within the realizable range. The implication of this result for the classical power flow analysis is that under the conditions specified above, the problem has a unique physically realizable solution if the phasor voltage magnitudes are fixed. We also introduce a series-parallel operator and show that this operator obtains a reduced and easier-to-analyze model for the power system without changing the uniqueness of power flow solutions.

中文翻译：

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使用单调性和网络拓扑的潮流解决方案的独特性


本文通过有功潮流与相角差之间的单调关系，为交流潮流解的唯一性建立了充分条件。更具体地说，我们证明 P-θ 潮流问题对于任何非循环图或 GSP 图至多有一个解。此外，对于任意循环电力网络，我们表明，在假设线路上的角度差受到与网络最大周长相关的某些限制的假设下，不可能存在多个不同的解决方案。在这些情况下，给定可实现范围内的有功功率注入向量，可以唯一地确定电压相角向量（直到绝对相移）。该结果对于经典潮流分析的含义是，在上述指定的条件下，如果相量电压幅值固定，则该问题具有唯一的物理上可实现的解决方案。我们还引入了串并联算子，并表明该算子在不改变潮流解决方案的唯一性的情况下获得了简化且更易于分析的电力系统模型。