## 1 Introduction
### Brief State of the Art  
The power flow is an indispensable method in electrical grids that determines the state variables for a given operation scenario [16]. This method is used for protection design [17] and as part of optimization models, such as the optimal selection of conductors [18], location of distributed generators and batteries [19], and reconfiguration [20], among others. The power flow is also part of operation models, such as the secondary [21; 22] and tertiary control [23]. The latter is particularly relevant, since most tertiary controls are based on optimal power algorithms. The difference between the power flow (PF) and optimal power flow (OPF) is that the latter aims to optimize the operation according to a measure of performance, which is known as the objective function, while considering technical constraints. These constraints include the system model represented by the power flow equations [24]. Therefore, the PF is a part of the set of constraints of the OPF. The specialized literature has proposed a broad type of power flow method for monopolar DC networks; such methods are classified into the derivative-based and derivative-free categories. Successive approximation, the triangular method, and the backward-forward sweep algorithm are examples of derivative-free methods, whereas hyperbolic approximation, product approximation, and the Newton-Raphson methods are examples of derivative-based methods (see [25] for a complete review of these methods). As their names indicate, the main difference between derivative-based and derivative-free methods is the requirement of a Jacobian (i.e., a derivative) of the set of algebraic functions. Derivative-based methods usually have a quadratic convergence inherited from [PERSON]'s algorithm, whereas derivative-free methods have linear convergence. Thus, derivative-based methods may converge in a lower number of iterations. However, building the Jacobian and solving the resulting linear systems may be computationally cumbersome; hence, the calculation time may be longer for derivative-based methods [16]. Moreover, some methods are only applicable for radial networks (e.g., the triangular method), while others (most of them derivative-based) are applicable for both radial and meshed grids. These generic power flow methods have only been studied for monopolar DC networks, except for the successive and triangular approximations, which were analyzed in [26], and the Newton-Raphson method, which was analyzed in [27] for bipolar grids. The former proposal only applied to radial systems and required a large number of iterations for convergence. In contrast, the latter proposal was very detailed, including models of loads, droop controls, and voltage controls, making the power flow model non-general and hardly applicable for other studies, such as those of optimal planning, where a simple power flow formulation with constant power loads at its peak magnitude is enough for the analysis. The major challenge in studying the power flow in bipolar LVDC networks is the power injection model that is commonly used to represent the grid. In [28], the problem was reformulated into a current injection equivalent that considered detailed models of loads and droops controls for distributed generators. It was a derivative-based algorithm that required the inverse of the Jacobian matrix in each iteration. Therefore, it may be computationally demanding for large test systems. In [29], the problem was also formulated as an equivalent current injection mode with a locational marginal price to reduce congestion between the two poles. Meanwhile, in [30], a methodology was proposed to reduce the voltage imbalance by redistributing the load to reduce power losses; this method assumed a grounded neutral wire in all of the nodes, which was a noticeable oversimplification of natural systems. These methodologies focused on the operative problem and can hardly be generalized to cover other studies concerning bipolar LVDC networks.