The problem of the optimal placement and dimensioning of constant power sources (i.e., distributed generators) in electrical direct current (DC) distribution networks has been addressed in this research from the point of view of convex optimization. The original mixed-integer nonlinear programming (MINLP) model has been transformed into a mixed-integer conic equivalent via second-order cone programming, which produces a MI-SOCP approximation. The main advantage of the proposed MI-SOCP model is the possibility of ensuring global optimum finding using a combination of the branch and bound method to address the integer part of the problem (i.e., the location of the power sources) and the interior-point method to solve the dimensioning problem. Numerical results in the 21- and 69-node test feeders demonstrated its efficiency and robustness compared to an exact MINLP method available in GAMS: in the case of the 69-node test feeders, the exact MINLP solvers are stuck in local optimal solutions, while the proposed MI-SOCP model enables the finding of the global optimal solution. Additional simulations with daily load curves and photovoltaic sources confirmed the effectiveness of the proposed MI-SOCP methodology in locating and sizing distributed generators in DC grids; it also had low processing times since the location of three photovoltaic sources only requires $233.16$s, which is 3.7 times faster than the time required by the SOCP model in the absence of power sources.

中文翻译：

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直流配电网中DG的最佳放置和尺寸确定的混合整数圆锥公式

本研究从凸优化的角度出发，解决了直流（DC）配电网中恒定电源（即分布式发电机）的最佳布置和尺寸确定问题。原始的混合整数非线性规划（MINLP）模型已通过二阶锥规划转换为混合整数圆锥等效形式，从而产生了MI-SOCP近似值。提出的MI-SOCP模型的主要优点是可以通过结合分支定界方法来解决问题的整数部分（即电源的位置）和内点来确保全局最优发现解决尺寸问题的方法。与GAMS中可用的精确MINLP方法相比，在21节点和69节点的测试馈线中的数值结果证明了其效率和鲁棒性：对于69节点的测试馈线，精确的MINLP求解器陷入局部最优解中，而提出的MI-SOCP模型可以找到全局最优解。通过每日负荷曲线和光伏电源的其他模拟，证实了所提出的MI-SOCP方法在直流电网中对分布式发电机进行定位和确定尺寸时的有效性；它的处理时间也很短，因为三个光伏源的位置仅需要 通过每日负荷曲线和光伏电源的其他模拟，证实了所提出的MI-SOCP方法在直流电网中对分布式发电机进行定位和确定尺寸时的有效性；它的处理时间也很短，因为仅需放置三个光伏源即可 通过每日负荷曲线和光伏电源的其他模拟，证实了所提出的MI-SOCP方法在直流电网中对分布式发电机进行定位和确定尺寸时的有效性；它的处理时间也很短，因为三个光伏源的位置仅需要$233。16$s，比没有电源时SOCP模型所需的时间快3.7倍。