2.1. General Representation
A general representation of a bipolar DC network is given in
Figure 1. It consists of a grid with a positive pole
p, a negative pole
n, and a neutral pole
o that may be solidly grounded. In a given node
k, the bipolar grid could contain multiple loads connected between positive and negative poles
or between positive and neutral poles
(analogously, a negative and neutral pole
). Naturally, the system may be unbalanced. To deal with multiple constant power loads, we start with the analysis of the solidly grounded case and then extend the formulation to the non-grounded case.
The bipolar DC grid is represented by an oriented graph
, where
= {1,2,…,
l} is the set of hypernodes and
is the set of hyperbranches. Each hypernode and hyperbranch has three components represented by the set
= {
p,
o,
n}; these contain the positive, neutral, and negative poles. In this manuscript, the subscripts
k and
m are entries of
, and the electrical connections are represented by the nodal conductance matrix
Gbus. Capital letters represent vector variables or constants, while capital bold letters represent matrix variables or constants, and their entries are lower-case letters. The main variables, constants, and indices are presented in the nomenclature below. In the present work, we call the loads connected between a positive and negative pole a C-type connection, and we call the loads connected between a positive and neutral pole an E-type connection (see
Figure 2).
2.2. Solidly Grounded Case
The voltage
is zero for solidly grounded systems. Therefore, the general power flow equation for a multinodal bipolar DC can be represented by (
1) (in the following equations,
are entries of the set
):
where
is the current generation in the power source connected at node
k with an E-type connection in the pole
h;
is the current demanded at node
k for a constant-power terminal with an E-type connection in the pole
h;
is the current demanded at node
k for a constant-power terminal with a C-type connection in the pole
h;
is the component (entry) of the conductance matrix that relates nodes
k and
j and the poles
h and
b, respectively;
is the voltage at node
k for the pole
b.
The definitions of the demanded current for the E-type and C-type connections are presented in Equations (
2) and (
3).
where
is the power consumed at node
k in the pole
h with respect to the neutral wire;
represents the power consumption at node
k between poles
h and
w. Both (
2) and (
3) have hyperbolic form.
A linear approximation of (
2) and (
3) is proposed here. It consists of a Taylor expansion of the hyperbolic representations
and
around the operation point
, which is defined in (
4) and (
5).
Equations (
4) and (
5) are replaced in (
2) and (
3), so the equivalent currents for the E-type and C-type connections are found:
We separate the general power flow Equation (
1) into two terms: The first term is related to the slack node, and the second term is associated with the demand nodes. This separation is presented in Equations (
8) and (
9), respectively.
where
contains the power injections in the slack node for the positive and negative poles (
);
and
are vectors that represent the demanded currents in the E-type and C-type connections. In this definition of the variables,
l represents the number of nodes in the bipolar DC grid;
and
are vectors that contain the voltage output in the slack node and the voltage variables in the demand nodes for poles
, respectively;
,
,
, and
are submatrices of the general conductance matrix
, which relates the slack and demanded nodes.
Equation (
8) is linear, since the variables of the power flow problem are the current injections in the slack node, i.e.,
, and the voltages in the demand nodes
. Hence, its solution is reached when the solution of Equation (
9) is found. However, Equation (
9) needs to be organized, since vectors with currents
and
are dependent of the demanded voltages, as given in (
6) and (
7).
Let us define the following vectors and matrices:
where
is the vector of bipolar constant-power loads,
is a vector that contains monopolar constant-power terminals, and
and
are the identity matrix and an auxiliary matrix that allow the calculation of the voltage differences for bipolar constant-power terminals.
Equations (
10) and (
11) allow the generalization of the currents in (
6) and (
7), as presented below:
where
is a linearized vector with the voltage at the iteration
t. If Equations (
12) and (
13) are replaced in Equation (
9), then a recursive power flow formula is derived for the bipolar power flow problem with the neutral cable solidly grounded, as presented in Equation (
14).
Finally, if we rearrange Equation (
14) to obtain a general expression for
, the following general power flow formula is reached:
where
Note that the recursive power flow Formula (
15) is evaluated from the initial point
by assigning the voltage at each node that is equal to the substation bus until the convergence criterion is met:
where
is the acceptable tolerance. Notice that, in this study, the metric for evaluating the convergence process is the voltage variation between iterations; however, some authors make use of the power loss formula as the metric. This requires two additional calculations, the first of which is that of the current across lines and the second of which is that of the power losses in each line; this is less direct than assuming the voltage variation as the metric of convergence [
27].
2.3. Non-Grounded Neutral Wire
This subsection explores the extension of the power flow formulation for bipolar DC networks where the neutral wire is uniquely grounded in the substation bus. The main difference in this case is that the current per pole depends on the difference between the pole and load voltages. We define the vector of demanded current per node as
:
where
,
, and
are the net current injections at node
k in the positive, neutral, and negative poles, respectively. These currents are calculated as follows:
Equations (
18)–(
20) have the same hyperbolic relation defined for the bipolar current
in Equation (
3). This implies that the currents for the positive, neutral, and negative poles can be approximated as presented in (
7) by using the Taylor series expansion with its linear term.
To obtain the power flow formulation for the non-grounded neutral bipolar DC case, we rewrite the matrix Equation (
9), as presented in Equation (
21).
where
and
are the vectors of demanded currents and voltages in the demand nodes.
is the voltage output in the slack node.
and
are submatrices associated with the bipolar conductance matrix considering the neutral wire. To define the general power flow formula for a neutral non-solidly grounded wire, the following vectors are defined.
in addition, we define the following auxiliary matrices:
matrices
,
, and
, were constructed by the authors in order to vectorize the recursive formula, and they are defined as follows:
Now, the following approximation for the demanded current
is found:
With the approximation of the currents in Equation (
24), it is possible to reach a general formula for bipolar DC grids by combining it with Equation (
21), as presented in Equation (
25).
where
Note that the recursive power flow Formula (
25) is evaluated from the initial point
by assigning the voltage at each node that is equal to the substation bus until the convergence criterion is met, i.e.,