I. Introduction

Electrical distribution systems are responsible for supplying electricity to all residential, commercial, and industrial users at medium- and low-voltage levels ^[1]. These networks are typically operated with alternating current (AC) and are built with a radial topology in order to simplify protection coordination schemes and reduce investment costs ^[2].

It is necessary for power systems to have high quality, safety, and efficiency indices. However, the increased demand and environmental constraints, as well as the liberalization of electricity markets, have caused them to operate increasingly close to their stability limits ^[3]. With the recent advances in power electronics, energy generation systems utilizing non-conventional renewable energy sources, in addition to developing efficient energy storage technologies, are transitioning from passive setups to active networks that interact with the user in an intelligent way ^[4]. On the other hand, given the development of energy conversion systems, it is currently possible to implement distribution systems that exclusively use direct current (DC) ^[5] and hybrid schemes that leverage the best traits of AC and DC technologies ^[6], providing a better performance in terms of energy losses and voltage profiles ^[7]. This is due to the fact that the concepts of reactive power and frequency are non-existent in these networks ^[8], which is why their control schemes are easier to design, as their only task is voltage regulation at the DC busbar ^[9].

In the case of DC operation technologies, distribution networks can be built with two typical topologies. The first is associated with monopolar networks, i.e., those with a positive pole and a neutral return conductor ^[10]. The second corresponds to DC networks with a bipolar topology, i.e., those with a positive and a negative pole as well as a neutral return conductor ^[11]. The advantage of a bipolar configuration is that, theoretically, it can transfer twice the power of a monopolar configuration and allows for the connection of loads requiring a voltage level equivalent to twice the power difference between the positive or negative pole and the neutral one ^[12].

To analyze DC distribution networks with bipolar or monopolar topologies, it is necessary to employ power flow tools, as they allow determining the operating state of a network under steady-state conditions ^[7]. However, although there are many reports on the power flow of monopolar networks in the literature ^[13], those on bipolar networks are scarce, and most focus on analyzing the optimal power flow ^[14]. Some works on bipolar networks are presented below.

The authors of ^[11] present a multi-objective optimization methodology for phase balancing in bipolar DC distribution systems. However, they fail to consider the presence of constant-power loads and focus solely on constant impedance and current models, which simplifies the power flow problem and allows obtaining mixed-integer linear programming models.

In ^[14], the authors analyze the effect of asymmetric loads on network congestion, proposing an optimal power flow to obtain marginal prices per node via an equivalent linear formulation.

The authors of ^[15] present a recursive optimal power flow for bipolar DC networks with multiple asymmetric loads. The results of the model are validated in systems with 6 and 33 nodes. The proposed mathematical formulation is developed as a function of the current injection method, and the results are compared against simulations of these networks in the PSCAS/EMTDC.

The work by ^[16] presents a simulation methodology using PSCAD/EMTDC to solve the power flow problem in bipolar networks, for which a three-node test system is employed. However, the authors do not analyze the model’s complexity or a convergence test under load variations. They only alter the constant-power terminals for controlled current sources and leave the complications of the solution to a power system analysis tool.

The study by ^[17] presents the application of a quasi-Newton method for solving the power flow problem in bipolar networks. This method is also known as the Broyden method or the secant, and it ensures a linear convergence to the power flow solution. It is classified as a derivative-free method. The authors validate this methodology in test systems composed of 21, 33, and 85 nodes.

The work by ^[18] presents a generalized methodology for solving the power flow problem in unbalanced bipolar networks, which is based on applying the Newton-Raphson method. A six-node test system is employed to validate the proposed methodology.

In ^[19], the authors present a methodology for studying the optimal power flow in bipolar networks, which considers a linear approximation of the power flow equations. Numerical results in test systems with 21 and 85 nodes demonstrate that the linear approximation of the power flow is effective insofar as distributed energy resources are considered in bipolar networks.

Based on the above-presented review of the state of the art regarding power flow for DC networks with a bipolar topology, this article makes the following contributions: (i) in most cases, calculating voltage values requires solving a system of simultaneous nonlinear equations ^[20], which is why the successive approximations method is applied for bipolar networks with either a solidly grounded or a floating neutral wire and monopolar and bipolar loads (this is an extension of the method presented by ^[21]); and (ii) this work presents the development of a graphic interface for implementing the successive approximations method in bipolar distribution networks, as a tool of interest for researchers and students in the electrical sector. This is a novel tool for electrical networks of a bipolar nature. It is important to mention that, in the specialized literature, there is no availability of open-access software for this type of application. This was identified as a research opportunity to which this article seeks to contribute significantly.

The rest of this document is structured as follows. The section titled Materials and methods presents the mathematical development of the successive approximations method for bipolar grid applications considering the presence of monopolar and bipolar loads as well as the possibility of having a neutral wire that is either solidly grounded in all nodes or floating. In the Graphic interface design section, the main characteristics of the MATLAB application development tool are presented. The section titled Results and discussion presents the bipolar configuration of the 21-test feeder used, as well as data on its nodal power consumption and line resistance parameters and the results obtained with the successive approximations method for bipolar networks through its implementation in the developed graphic interface, as well as the corresponding analyses and discussions. Finally, the section titled Conclusions describes the main findings of this research, as well as potential research lines.

II. Materials and Methods

This section presents the general methodology regarding the power flow problem for bipolar DC networks, which is based on the successive approximation’s method proposed in ^[22]. In addition, it presents the design of a graphic interface in the MATLAB environment for solving the power flow problem in unbalanced radial or meshed networks.

A. Power Flow Problem Formulation

In this subsection, matrices will be denoted by capital and italic letters, subscripts will represent nodes, and superscripts will represent iterations.

The power flow problem for bipolar DC networks employs the grid conductance representation ^[23]. The relationship between voltage and current is considered using the nodal admittance matrix.

(1)

Where:

I 𝑔,𝑝𝑛 is the vector containing all currents injected in the positive and negative poles of the slack nodes.

𝐼 𝑑,𝑝𝑛 is the vector containing all demanded currents in the grid’s positive and negative poles.

𝐺

𝐺 𝑔𝑔,𝑝𝑛 is the component of the matrix 𝐺 𝑝𝑛 that relates the slack nodes.

𝐺 𝑔𝑑,𝑝𝑛 is the component of the matrix 𝐺 𝑝𝑛 that relates the slack and demand nodes.

𝐺 𝑑𝑔,𝑝𝑛 is the transpose of the matrix 𝐺 𝑔𝑑,𝑝𝑛 .

𝐺 𝑑𝑑,𝑝𝑛 is the component of the matrix 𝐺 𝑝𝑛 that relates the demand nodes.

𝑉 𝑔,𝑝𝑛 is the vector containing all the voltage outputs of the slack nodes.

𝑉 𝑑,𝑝𝑛 is the vector containing all the voltages in the demand nodes.

Note that the second row of equation (1) expresses the relationship between the demand currents and the voltage vectors. This implies that the equation is not linear, as these currents are a function of the demanded voltages and their powers. Therefore, an iterative method is required to solve the power flow problem ^[5].

To apply the successive approximations method, the second row of equation (1) is reorganized to obtain the voltages of the demand nodes, which was demonstrated in ^[24] and is shown in equation (2).

(2)

The general formula for analyzing the power flow of bipolar DC networks with a solidly grounded neutral wire and unbalanced loads can be obtained by adding an iterative counter to the demand voltage vector based on Tellegen’s second theorem ^[25]. As the demand current vector is a function of the demand voltage vector, the formula for solving the power flow problem is as follows ^[24]:

(3)

The iterative process of the successive approximations method ends when the difference between the voltage magnitudes of the current and previous iteration is lower than the pre-established error. This stopping criterion is defined in equation (4).

(4)

where ɛ is the tolerance error.

One of the great advantages of using equation (3) as an iterative formula is that the inverse of the matrix 𝐺 𝑑𝑑,𝑝𝑛 must only be calculated once, as it does not change throughout the process ^[26].

The main difficulty when it comes to solving the power flow problem corresponds to the hyperbolic relationship between voltages and constant power loads ^[27]. Equations (5) and (6) present the currents of the loads connected between the positive pole and the neutral wire, and those connected between the negative pole and the neutral wire, respectively.

(5)

(6)

Where:

𝐼 𝑑𝑘,𝑝 𝑡 is the current demanded by the positive pole at node k.

𝐼 𝑑𝑘,𝑛 𝑡 is the current demanded by the negative pole at node k.

𝑃 𝑑𝑘,𝑝 is the power consumed at node k by the load connected to the positive pole.

𝑃 𝑑𝑘,𝑛 is the power consumed at node k by the load connected to the negative pole.

𝑉 𝑑𝑘,𝑝 𝑡 is the voltage level of the positive pole at node k.

𝑉 𝑑𝑘,𝑛 𝑡 is the voltage level of the negative pole at node k.

Now, if the loads are connected between the positive and negative poles or vice versa, the currents are calculated as follows:

(8)

(9)

Where:

𝑃 𝑑𝑘,𝑝𝑛 is the constant power load connected between the poles.

B. Extension to Systems that are not Solidly Grounded

When designing bipolar DC networks, especially low-voltage ones, there is the possibility that the neutral wire is not connected to the ground in all nodes. The only node that must be grounded is the network supply node ^[11], implying that the above-presented formulation requires considering said effect in the neutral conductor. To this effect, a new subscript is added to the nomenclature in equation (3), indicating that the neutral wire is not solidly grounded, as shown below:

(10)

Where:

𝑉 𝑑,𝑝𝑜𝑛 𝑡+1 is the vector containing all the voltages of the demand nodes in the following order: positive, neutral, and negative.

𝐼 𝑑,𝑝𝑜𝑛 is the vector containing all the currents demanded in the positive and negative poles of the grid.

𝑉 𝑔,𝑝𝑜𝑛 is the vector containing all the voltage outputs of the generators in the following order: positive, neutral, and negative.

𝐺 𝑑𝑑,𝑝𝑜𝑛 is the component of the conductance matrix that relates the demand nodes, including the conductance of the neutral conductor.

𝐺 𝑑𝑔,𝑝𝑛 is the component of the conductance matrix that relates the slack and demand nodes, including the conductance of the neutral conductor.

The most significant effect of including this in the power flow problem is related to the demand current. Therefore, if a load is connected between a pole and the neutral conductor, the demand currents are calculated as follows:

(11)

(12)

(13)

Where:

𝑃 𝑑𝑘,𝑝𝑜 is the constant power load connected between the positive pole and the neutral conductor.

𝑃 𝑑𝑘,𝑛𝑜 is the constant power load connected between the negative pole and the neutral conductor.

𝑉 𝑑𝑘,𝑜 𝑡 is the voltage of the neutral conductor at node k in iteration t.

C. Graphic Interface

This section presents the implementation of the graphic interface using the mathematical model shown in the previous section. This was done using the MATLAB 2021b software and the application development tool called AppDesigner. In the MATLAB environment, to access AppDesigner, the command appdesigner must be entered in the command window, as shown in Figure 1.

Fig. 1
Opening AppDesigner

Next, the Blank app option is selected, which will open the workspace along with its design and code views, through which the graphic interface and the program code must be adjusted, respectively ^[28].

The design view is divided into three windows: calculations, graphs, and system. The first of these includes the text editing boxes where the users will enter the program’s input data, as shown in Figure 2.

Fig. 2
Calculations window (part 1).

For data with multiple inputs, the use of tables is advised, as this facilitates data capture and a better display for the user, which is why this paper uses two tables for entering the load and line data. To facilitate data entry, an option that allows entering information from an Excel file (.xlsx) was added. Two images were added (the logos of the university), as well as two text boxes, one to name the program and its authors and one to show the results after execution. Finally, a button was added to be pressed when all the input data were ready. Figure 3 shows all the above.

Fig. 3
Calculations window (part 2)

In the graphs window, an axis is added, which allows drawing the entered data on the Cartesian plane, as shown in Figure 4.

Fig. 4
Graphs window

This window shows the voltage profiles at the end of the program’s execution. To the system window, an axis is also added, albeit without a ruler and with a transparent background. Thus, this axis does not look like a graph in the Cartesian plane. This window plots an approach to the system’s topology using graph theory ^[29].

It is in the code view that all the entered data are captured. Here, the boxes and tables containing the data are named and stored in global variables for use anywhere in the code and at any time during execution.

When pressing the aforementioned Calculate button, the code that calculates the power flow begins its execution. Once this process is complete, the app sends the results obtained to the text area, along with a graph showing the voltage profiles.

After adjusting the functional part of the program, operation details are added in order to make the interface more user-friendly, e.g., the addition of colors for a better look, as shown in Figure 5.

Fig. 5
Final graphic interface

III. Results and Discussion

This section presents the test system considered in this work, the numerical validations carried out for its radial and meshed configurations, and an analysis and a discussion of the results obtained.

A. Test System

To demonstrate the effectiveness of this proposal and validate the data, a medium-voltage asymmetric bipolar DC distribution network was considered, which is composed of 21 nodes, 22 lines (meshed system), and 21 constant power loads. This feeder was originally used as a balanced network, as presented in ^[23]. However, in this case, two lines are added, with the configuration shown in Figure 6. Note that, in order to ensure open access to the work presented in this document, the developed tool can be downloaded at Power flow for bipolar networks.

Fig. 6
Nodal connection scheme for the 21-node test feeder.

The 21-node test system’s slack node operates at ±1000V. The rest of the electrical parameters can be found in Table 1.

Table 1

Electrical parameters of the 21-node test system.

For the studied case, two scenarios are evaluated. In the first, the neutral wire is considered to be solidly grounded in all nodes, and in the second, it is considered to be floating.

Two lines are added to the original study case. This information can be found in Table 2. One line goes from node 7 to node 9, and the other one is between nodes 11 and 16 to avoid exceeding a 5% regulation in all nodes. Not surpassing this limit is advisable, which is not possible without these two lines. This will be confirmed in the computational validation.

Table 2

Electrical parameters of the additional lines

B. Numerical validations and discussion

Considering the above-presented system, the original one was simulated. The results are shown in Table 3. In these numerical results, it can be observed that, in the scenario with the neutral wire grounded in all nodes, some nodes exceed a 10% regulation in the positive pole, specifically nodes 16, 17, and 18. The node with the lowest voltage is no. 17, with a 10.99% regulation. Compared to its counterpart, the negative pole exhibits the lowest voltage at node 18, with a 9.14 regulation. This is due to the fact that the load is unbalanced. The total load reported for the positive pole is 554 kW, while that of the negative pole is 445 kW, which represents a difference of over 100 kW.

Table 3

Voltage for the two simulation cases in the original system

In the simulation where the neutral wire is only grounded at the slack node, it is observed that the neutral conductor of nodes 16 and 17 exceeds 20 V in magnitude, which may have negative consequences for the sensitivity of some devices. It should also be emphasized that voltage regulation slightly decreases compared to the previous case, but this is not a significant fact.

To demonstrate the effectiveness of the application for solving the power flow of unbalanced bipolar DC networks, we included a scenario with two additional lines (Table 2), aiming to reduce voltage regulation at critical nodes and use a meshed network to confirm that our approach works with both topologies.

Table 4 shows the results of the new simulation case. Accordingly, it can be stated that the power losses for the solidly grounded scenario is 75.1832 kW, 3.5540 kW less than those of the case with the floating neutral wire. This is associated with the currents flowing through the neutral wire, as a result of load imbalances between the positive and negative poles.

Table 4

Voltage for the two simulation cases in the modified system

On the other hand, voltage regulation decreases significantly with respect to the original system, i.e., from 10.99% in the radial network to 7.47% in the meshed one. In the positive pole, the voltage goes from 890.1027 V to 925.313 V. This may be due to the fact that adding lines changed the current distribution, and the most remote nodes do not perceive such a drastic voltage difference.

Figure 7 shows the voltage profiles for the modified study case with the solidly grounded neutral wire, and Figure 8 presents the voltage profile for the modified scenario with the floating neutral conductor.

Fig. 7
Voltage profiles for the modified study case (solidly grounded neutral wire).

Fig. 8
Voltage profiles for the modified study case (floating neutral wire)

V. Conclusions

This work analyzed solidly grounded and non-grounded bipolar DC networks, to which the same successive approximations method was applied, demonstrating its effectiveness in both cases. In addition, this method can be used in both radial and meshed grids. The implementation of a graphic interface facilitates its use; as it is user-friendly, this interface can be used by anyone with basic knowledge of the subject.

A graphical analysis of voltage profiles allows quickly determining key factors in a bipolar DC network, such as the system’s electrical center (i.e., the node with the highest voltage), the nodes in which the highest load is concentrated, and, in some cases and with the help of a system topology scheme, the lines with the greatest losses.

As possible research works, the following could be conducted in the future: (i) implementing new power flow methods for bipolar grids based on Taylor approximations and the triangular method; (ii) studying the phase balancing problem by combining the successive approximations method and a metaheuristic algorithm of a binary nature; and (iii) proposing a convex reformulation to study the optimal power flow problem in bipolar networks with a high penetration of distributed energy resources.

Acknowledgments

This research work was carried out within the framework of the academic production graduation modality to obtain the degree of Electrical Engineer from the curricular project in Electrical Engineering of Universidad Distrital Francisco José de Caldas.

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Notes

Author notes