Abstract
In this study, the voltage stability margin for direct current (DC) networks in the presence of constant power loads is analyzed using a proposed convex mathematical reformulation. This convex model is developed by employing a second-order cone programming (SOCP) optimization that transforms the non-linear non-convex original formulation by reformulating the power balance constraint. The main advantage of the SOCP model is that the optimal global solution of a problem can be obtained by transforming hyperbolic constraints into norm constraints. Two test systems are considered to validate the proposed SOCP model. Both systems have been reported in specialized literature with 6 and 69 nodes. Three comparative methods are considered: (a) the Newton-Raphson approximation based on the determinants of the Jacobian matrices, (b) semidefinite programming models, and (c) the exact non-linear formulation. All the numerical simulations are conducted using the MATLAB and GAMS software. The effectiveness of the proposed SOCP model in addressing the voltage stability problem in DC grids is verified by comparing the objective function values and processing time.
Highlights
Direct current (DC) electrical networks are promising grids capable of supplying multiple users at different voltage levels from high-voltage DC (HVDC) to low-voltage DC (LVDC) in monopole or bipole configurations [1,2]
Even if recently reported approaches typically use the exact non-linear formulation (GAMS solvers) and heuristic searches (Newton-Raphson) [11,12] because of the non-convexities introduced by the power balance constraints, it is not possible to ensure a global optimum, even if for both test feeders these solutions coincide with convex approaches, i.e., the semidefinite programming model [9] and the newly proposed second-order cone programming (SOCP) model
= ||vi ||2 v j where the pre-multiplication by vi v j is required to transform the hyperbolic relation between voltages into a conic constraint [19]. Please note that this relaxation is possible because all the voltage variables must be positive for satisfactory operation of DC grid, including in extreme cases, such as the voltage stability margin analyzed in this study
Summary
Direct current (DC) electrical networks are promising grids capable of supplying multiple users at different voltage levels from high-voltage DC (HVDC) to low-voltage DC (LVDC) in monopole or bipole configurations [1,2]. The main advantage of this approach is that it guarantees a global optimum and unique solution by transforming the exact non-linear non-convex optimization problem into a convex problem [13,14] This approach has not been previously proposed for the analysis of voltage stability in DC networks. Even if recently reported approaches typically use the exact non-linear formulation (GAMS solvers) and heuristic searches (Newton-Raphson) [11,12] because of the non-convexities introduced by the power balance constraints, it is not possible to ensure a global optimum, even if for both test feeders these solutions coincide with convex approaches, i.e., the semidefinite programming model [9] and the newly proposed SOCP model.
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