Geometry of power flows in tree networks
Geometry of power flows in tree networks
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DOI：
10.1109/pesgm.2012.6344803
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发表时间：
2012-04
期刊：
影响因子：
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通讯作者：
J. Lavaei;David Tse;Baosen Zhang
中科院分区：
文献类型：
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作者：
J. Lavaei;David Tse;Baosen Zhang
研究了树形网络中的潮流问题及其与优化的关系。结果表明，由于网络的树状拓扑结构，一般的最优潮流问题大大简化。我们的方法是观察电网的注入区域。注入区域只是满足网络和运行约束的母线功率注入的所有向量的集合。感兴趣的几何对象是注入区的帕累托最优点集，因为它们是递增函数最小化的解。我们将注入区域视为高维潮流区域的线性变换，高维潮流区域是所有可行潮流的集合，每条线路的每个方向对应一个。我们证明，如果电压幅值是固定的，那么注入区域就是两个节点潮流区域的乘积，每个节点对应于网络中的一条线路。利用这种分解，我们证明了在实际情况下，在每条直线的角度差不太大的情况下，通过取凸壳，注入区域的Pareto最优点集保持不变。因此，最优潮流问题可以凸化并有效地求解。这一结果改进了以前的工作，因为它不对活动的母线功率约束做出任何假设。对于变电压幅值的情况，我们也得到了一些部分结果。
We investigate the problem of power flow and its relationship to optimization in tree networks. We show that due to the tree topology of the network, the general optimal power flow problem simplifies greatly. Our approach is to look at the injection region of the power network. The injection region is simply the set of all vectors of bus power injections that satisfy the network and operation constraints. The geometrical object of interest is the set of Pareto-optimal points of the injection region, since they are the solutions to the minimization of increasing functions. We view the injection region as a linear transformation of the higher dimensional power flow region, which is the set of all feasible power flows, one for each direction of each line. We show that if the voltage magnitudes are fixed, then the injection region becomes a product of two-bus power flow regions, one for each line in the network. Using this decomposition, we show that under the practical condition that the angle difference across each line is not too large, the set of Pareto-optimal points of the injection region remains unchanged by taking the convex hull. Therefore, the optimal power flow problem can be convexified and efficiently solved. This result improves upon earlier works since it does not make any assumptions about the active bus power constraints. We also obtain some partial results for the variable voltage magnitude case.