Optimal Reconfiguration of Bipolar DC Networks Using Differential Evolution

Abstract

The search for more efficient power grids has led to the concept of microgrids, based on the integration of new-generation technologies and energy storage systems. These devices inherently operate in DC, making DC microgrids a potential solution for improving power system operation. In particular, bipolar DC microgrids offer more flexibility due to their two voltage levels. However, more complex tools, such as optimal power flow (OPF) analysis, are required to analyze these systems. In line with these requirements, this paper proposes an OPF for bipolar DC microgrid reconfiguration aimed at minimizing power losses, considering dispersed generation (DG) and asymmetrical loads. This is a mixed-integer nonlinear optimization problem in which integer variables are associated with the switch statuses, and continuous variables are associated with the nodal voltages in each pole. The problem is formulated based on current injections and is solved by a hybridization of the differential evolution algorithm (to handle the integer variables) and the interior point method-based OPF (to minimize power losses). The results show a reduction in power losses of approximately 48.22% (33-bus microgrid without DG), 2.87% (33-bus microgrid with DG), 50.90% (69-bus microgrid without DG), and 50.50% (69-bus microgrid with DG) compared to the base case.

1. Introduction

The implementation of microgrids (MGs) is encouraged worldwide due to their technical, economic, and environmental benefits, especially for remote communities where the main power grid is unavailable [1]. The key components of MGs include dispersed generation (DG) units and battery energy storage systems (BESSs), with power conditioning devices that can be AC/DC or DC/DC converters [2].

In recent years, a new paradigm has emerged in electrical distribution networks. Moving away from traditional AC networks, particularly in the context of micro- and nano-grids, DC networks have become a superior solution for various reasons [3]. The compatibility of DC distribution networks with photovoltaic (PV) systems suggests an extensive use of DC microgrids in future developments, as only one conversion stage is required [4]. In addition, DC networks offer advantages in using energy storage systems, enhanced reliability of electrical networks, and support for high-energy efficiency loads [5]. These attributes make DC networks particularly appealing within modern smart power distribution systems.

About configurations, two main types of DC microgrids have been utilized, tested, and studied: unipolar and bipolar. The unipolar configuration, the simpler of the two, consists of only two wires, with all generators, loads, and storage systems connected to the same poles. The bipolar configuration is more complex, featuring three wires (a positive pole, a neutral pole, and a negative pole). In this setup, generators, loads, and storage systems can be connected in various ways across all three poles. This flexibility also allows connections at different voltages, such as between the positive and negative poles or between one pole and the neutral pole [6].

A comparison of both configurations reveals that bipolar DC microgrids offer several advantages. They provide more voltage levels (two instead of one), improved efficiency, and a higher-quality power supply. Another key benefit is enhanced reliability: in the event of a fault in one wire, the load can still be supplied by the other two healthy lines [6,7].

Despite the significant advantages of bipolar DC microgrids, ensuring their full potential requires minimizing electrical losses. This optimization is essential to improve overall performance and sustainability, making it a vital component of modern power distribution systems [8]. One of the most effective measures to enhance performance is the microgrid network reconfiguration process, which involves altering the functional connections between its elements [9].

1.1. Optimal Power Flow for Bipolar DC Grid Analysis

With regard to optimal power flow (OPF) tools for unbalanced bipolar DC microgrids, a few studies have been prosented in the literature. In [10], the OPF solves the optimal economic dispatch by minimizing the total cost through convex optimization based on linear programming modeling. In [11], the current injection method and nonconvex quadratic programming models are used to determine the set point of the output power and the balanced voltage level of the DGs, considering three objective functions: generation cost, voltage unbalance, and network loss. Recursive quadratic programming with conic constraints is employed in [12] to minimize the expected grid power losses for a particular load scenario. In [13], a recursive quadratic programming with linear constraints is used to determine the set point of the output power of dispersed generators to minimize power losses. In [14], an OPF based on the branch flow model for bipolar DC grids with asymmetric loading is proposed to minimize power losses and voltage unbalance using the weighted sum approach solved by a nonlinear programming optimization model.

To minimize voltage unbalance in the bipolar DC microgrid, OPF formulations have also been proposed in the literature. In [15], a multiobjective OPF is developed for power loss and voltage unbalance minimization using the weighted sum approach. Integer decision variables represent the load configuration in the microgrid. In [16], an OPF formulation is used to adjust the decentralized control for bipolar DC networks with z-inverters. In [17], an OPF is proposed for optimal pole swapping.

1.2. Reconfiguration in Microgrids

Reducing power losses in electric distribution systems is a crucial aspect of power system operations, as it directly impacts economic efficiency and energy cost reduction [18].

One effective method to minimize these technical losses is network reconfiguration, which involves selecting the optimal combination of open and closed switches to improve specific performance criteria while preserving a radial network structure [9].

The reconfiguration process in a microgrid scenario presents distinct challenges due to the intermittent nature of the DG, significantly complicating the process. It introduces new complexities with respect to protection and stability [19].

Several studies have addressed the reconfiguration of microgrids under various conditions and constraints. For instance, ref. [20] proposes a Boolean model for microgrid reconfiguration during catastrophic failures of large power systems. In [21], a real-time intelligent reconfiguration algorithm is introduced, which the authors claim can be easily adapted for industrial microgrid applications.

In [22], a risk-constrained reconfiguration approach is presented, where the system topology is derived by solving a chance-constrained optimization problem that considers loss-of-load constraints and the ampacity limits of distribution lines.

A multiobjective optimization problem, formulated using the non-dominated sorting genetic algorithm II (NSGA-II), is proposed in [23] to obtain radial topologies for microgrids within a Pareto front, incorporating the operational constraints and philosophy of isolated microgrid systems.

In [24], a stochastic model for day-ahead management of microgrids is introduced, aiming to maximize benefits while accounting for load demand and wind power generation uncertainties.

An extensive review of microgrid reconfiguration is provided in [19], categorizing research based on objective functions, constraints, optimization methods, and models of uncertainty.

Finally, ref. [25] presents a novel method for post-disturbance microgrid formation based on radiality constraints, achieving greater resilience through enhanced feasibility and optimality.

Despite extensive research on microgrid reconfiguration, several gaps and limitations remain. Notably, there is a lack of studies specifically focusing on DC bipolar microgrids.

1.3. Main Contributions

The reconfiguration of microgrids to minimize power losses is an area of significant research interest, as evidenced by the reviewed literature. However, existing studies often address only specific aspects or operate under simplified conditions. This paper advances the field by proposing a novel optimal power flow (OPF) framework tailored for bipolar DC microgrid reconfiguration. Unlike previous approaches, our method comprehensively addresses dispersed generation (DG) and asymmetrical loads within a unified optimization model. This is a mixed-integer nonlinear optimization problem, in which integer variables are associated with the switch statuses and continuous variables are associated with the nodal voltages in each pole. The problem is formulated on the basis of current injections and is solved by a hybridization of the differential evolution (DE) algorithm [26] (to handle integer variables) and the OPF is based on the interior point method (IPM) [27] (to minimize power losses).

1.4. Paper Organization

This paper is divided into six sections, including the Introduction. Section 2 discusses the basic elements of a bipolar DC microgrid. The proposed approach and the framework used to solve it are presented in Section 3. Section 4 presents the systems used in this work, with 33 and 69 buses. Section 5 shows the results obtained by the proposed approach. General conclusions are drawn in Section 6. Appendix A provides an illustrative example of a small-scale bipolar DC microgrid.

2. DC BipolarMicrogrids

Figure 1 shows a generic representation of a bipolar DC microgrid with various configurations of loads and generation sources. The main goal of a power flow tool for a bipolar DC microgrid is to calculate the nodal voltages (nodes s, j, and i) for each pole (positive p, negative n, and neutral o). In this case, the power flow solves a set of nonlinear equations, illustrated in the Appendix A for the sake of clarity.

3. Proposed Approach

The main goal of this section is to present the proposed approach and the framework for the solution aiming at reconfiguring the bipolar DC microgrid for power loss minimization.

3.1. Mathematical Formulation

The mathematical formulation of the proposed approach is presented in Equation (1), where x represents the continuous optimization variables composed of nodal voltages and y represents the integer variables associated with the status of branches of the bipolar DC microgrid ($y=0$ for an open status and $y=1$ for a closed status). In addition, $f(x,y)$ is the objective function, h represents the set of equality constraints, and $x_{min}/x_{max}$ are the bounds of the continuous variables.

$$\begin{array}{cc}min& f(x,y)\\ & h(x,y)=0\\ \mathrm{st}& x_{min}\le x\le x_{max}\\ & y=0\mathrm{or}y=1\end{array}$$

(1)

The objective function $f(x,y)$ is defined in Equation (2), given by the sum of the power losses in all branches (and their poles).

$$f(x,y)=\sum _{jk\in {\mathsf{\Omega }}_B}C{H}_{jk}\left(R_{jk}^p{I}_{jk}^{p 2}+R_{jk}^o{I}_{jk}^{o 2}+R_{jk}^n{I}_{jk}^{n 2}\right)$$

(2)

where ${\mathsf{\Omega }}_B$ is the set of all branches; $C{H}_{jk}$ is the binary variable representing the status of the branch/switch ($C{H}_{jk}=0/1=\mathrm{off}/\mathrm{on}$)-variable y; $R_{jk}^p$, $R_{jk}^o$, and $R_{jk}^n$ are the resistances of the branch $j-k$ for the positive (p), neutral (o), and negative (n) poles. Finally, the branch currents are $I_{jk}^p$, $I_{jk}^o$, and $I_{jk}^n$.

The first set of equality constraints $h(x,y)$ is presented in Equation (3) and ensures the current balance at each node j and pole of the microgrid.

$$\begin{array}{cc}\hfill & I_{g,j}^p+I_{dg,j}^p-I_{d,j}^p-I_{d,j}^{p-n}-\sum _{k=1}^{nb}C{H}_{jk}·I_{jk}^p=0\hfill \\ \hfill & I_{g,j}^o+I_{dg,j}^o-I_{d,j}^o-I_{d,j}^{ground}-\sum _{k=1}^{nb}C{H}_{jk}·I_{jk}^o=0\hfill \\ \hfill & I_{g,j}^n+I_{dg,j}^n-I_{d,j}^n+I_{d,j}^{p-n}-\sum _{k=1}^{nb}C{H}_{jk}·I_{jk}^n=0\hfill \end{array}$$

(3)

where:

• $I_{g,j}^p$, $I_{g,j}^o$, and $I_{g,j}^n$ are the current injections at node j for all poles (p, o, and n) by the slack node (if the node is not the slack one, these current injections are null).
• $I_{dg,j}^p$, $I_{dg,j}^o$, and $I_{dg,j}^n$ are the current injections at node j for all poles (p, o, and n) by the dispersed generation sources (if the node j does not contain a dispersed unit, these current injections are null). Equation (4) shows how to calculate these currents.
• $I_{d,j}^p$, $I_{d,j}^o$, and $I_{d,j}^n$ are the current consumption’s at node j for all poles (p, o, and n) as defined in Equation (5). If node j does not have power loads, these currents are null.
• $I_{d,j}^{p-n}$ is the current consumption of bipolar loads connected between positive and negative poles at node j. Equation (5) provides an expression to calculate this current.
• $I_{jk}^p$, $I_{jk}^o$, and $I_{jk}^n$ are the branch currents between nodes j and k (branch $j-k$), calculated as given in Equation (6). These currents are null in two situations. The first one is in the absence of a connection between the nodes (nodes s and i in Figure 1, for instance). The second one is when nodes are interconnected but the switch is open ($C{H}_{jk}=0$).
• $I_{d,j}^{ground}$ is the current drained to the ground if the neutral wire is solidly grounded (as in the slack node in Figure 1). If the neutral wire is assumed to be floating, this current is null.
• ${\sum }_{k=1}^{nb}C{H}_{jk}·I_{jk}^p$, ${\sum }_{k=1}^{nb}C{H}_{jk}·I_{jk}^o$, and ${\sum }_{k=1}^{nb}C{H}_{jk}·I_{jk}^n$ denote the current injections in each system node j (poles p, o, and n), with $nb$ being the number of nodes.

$$\begin{array}{cc}\hfill & I_{dg,j}^p={\displaystyle \frac{P_{dg,j}^p}{V_j^p-V_j^o}}\hfill \\ \hfill & I_{dg,j}^n={\displaystyle \frac{P_{dg,j}^n}{V_j^n-V_j^o}}\hfill \\ \hfill & I_{dg,j}^o={\displaystyle \frac{P_{dg,j}^p}{V_j^o-V_j^p}}+{\displaystyle \frac{P_{dg,j}^n}{V_j^o-V_j^n}}\hfill \end{array}$$

(4)

where $V_j^p$, $V_j^o$, and $V_j^n$ are the voltages at node j and poles positive (p), neutral (o), and negative (n); $P_{dg,j}^p$ and $P_{dg,j}^n$ are constant injections of power from dispersed generators (as illustrated in Figure 1).

$$\begin{array}{cc}\hfill & I_{d,j}^p={\displaystyle \frac{P_{d,j}^p}{V_j^p-V_j^o}}\hfill \\ \hfill & I_{d,j}^n={\displaystyle \frac{P_{d,j}^n}{V_j^n-V_j^o}}\hfill \\ \hfill & I_{d,j}^o={\displaystyle \frac{P_{d,j}^p}{V_j^o-V_j^p}}+{\displaystyle \frac{P_{d,j}^n}{V_j^o-V_j^n}}\hfill \\ \hfill & I_{d,j}^{p-n}={\displaystyle \frac{P_{d,j}^{p-n}}{V_j^p-V_j^n}}\hfill \end{array}$$

(5)

where $P_{d,j}^p$ and $P_{d,j}^n$ are the constant monopolar power loads at node j and the poles p and n; $P_{d,j}^{p-n}$ is the bipolar constant power load between the poles p and n at node j.

$$\begin{array}{c}{I}_{jk}^p={\displaystyle \frac{V_j^p-V_k^p}{R_{jk}^p}}\\ I_{jk}^o={\displaystyle \frac{V_j^o-V_k^o}{R_{jk}^o}}\\ I_{jk}^n={\displaystyle \frac{V_j^n-V_k^n}{R_{jk}^n}}\end{array}$$

(6)

The second set of equality constraints $h(x,y)$ is presented in Equation (7) and ensures the reference voltage at the slack node (node 1).

$$\begin{array}{cc}\hfill & V_1^p=V_{dc}\hfill \\ \hfill & V_1^o=0\hfill \\ \hfill & V_1^n=-V_{dc}\hfill \end{array}$$

(7)

where $V_{dc}$ is the nominal voltage.

Equation (8) presents the constraints $min/max$ for the nodal voltages.

$$\begin{array}{cc}\hfill & V_{j,min}^p\le V_j^p\le V_{j,\max }^p\hfill \\ \hfill & V_{j,min}^n\le V_j^n\le V_{j,\max }^n\hfill \end{array}$$

(8)

It is important to emphasize that if $C{H}_{jk}$ is set to zero (open switch), the disconnection is carried out at all poles simultaneously. In addition, radiality constraints are considered to ensure that no meshed operation occurs. Finally, constraints are also considered to prevent islanded operation [19,28]. These constraints are considered during the implementation of the proposed framework to solve the optimal power flow proposed in this section.

3.2. Binary Variable Treatment

In this paper, binary variables associated with the switch status are treated by the sigmoid function defined in Equation (9). This strategy allows for the use of continuous optimization methods to solve the optimal power flow discussed in Section 3.1. Figure 2 shows the sigmoid function: using a continuous optimization variable $z_{jk}$, it is possible to calculate the sigmoid value $sig\left(z_{jk}\right)$. Using the condition posed in Equation (10), it is possible to deal with the variable $C{H}_{jk}$ (binary in nature). Finally, Equation (11) constrains the variable $z_{jk}$.

$$sig\left(z_{jk}\right)={\displaystyle \frac{1}{1+e^{-10\left(z_{jk}-0.5\right)}}}$$

(9)

$$C{H}_{jk}=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}sig\left(z_{jk}\right)\ge 0.5\hfill \\ 0,\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

(10)

$$\begin{array}{cc}\hfill & 0\le z_{jk}\le 1\hfill \end{array}$$

(11)

3.3. Individuals’ Structures

The approach proposed in Section 3.1 is solved using a framework based on differential evolution and optimal power flow. Individuals of differential evolution follow the structure defined in Equation (12).

$${ind}_k=\left[\begin{array}{ccccccc}{z}_1\hfill & z_2\hfill & \cdots \hfill & z_p\hfill & \cdots \hfill & z_{nbr-1}\hfill & z_{nbr}\hfill \end{array}\right]$$

(12)

where $nbr$ is the number of branches of the bipolar DC microgrid; ${ind}_k$ denotes the $k^{th}$ individual; and $z_p$ denotes the $p^{th}$ branch.

3.4. Fitness Function Calculation

For each individual $in{d}_k$ in Equation (12), a fitness function $fi{t}_k$ is calculated following the procedure depicted in Figure 3. In step 01, the continuous individual’s values in Equation (12) are read. Step 02 applies the sigmoid function as discussed in Section 3.2. Once the status of the switches is known (on/off), using graph theory [19,28] it is possible to verify two situations in step 03: radiality and connection. The radiality constraint ensures that no mesh operation occurs. The connection constraint ensures that no microgrid nodes (or regions) operate isolated. If the system is not radial and connected in step 04, the penalty term is different from zero ($penalt={10}^4$) in step 07. Otherwise, the penalty term is set to zero ($penalt=0$) and the optimal power flow in Section 3.1 is solved in step 05. In this case, the binary values ($C{H}_{jk}$, y variables) are known and fixed at 0 or 1. As a result, the OPF becomes continuous in nature and is solved by the interior point method [27]. If the OPF converges, the algorithm proceeds to step 08. Otherwise, a penalty term is calculated in step 07 ($penalt={10}^4$). Once the minimum power losses or penalty term are calculated, the fitness function $fi{t}_k$ is given by Equation (13). Finally, step 09 returns the fitness function $fi{t}_k$.

$$fi{t}_k=P_{loss}+penalt$$

(13)

where $P_{loss}$ is the minimal power losses calculated by the OPF. If the OPF is not evaluated due to step 04, $P_{loss}$ can be set to zero, since $penalt$ is set to ${10}^4$.

In the proposed method, power loss minimization is addressed directly through the fitness function (Equation (13)) used in the differential evolution algorithm. Specifically, for each potential configuration generated by the DE algorithm, the interior point method-based OPF computes the total power losses across the microgrid. The DE algorithm then iteratively adjusts the configurations, seeking to minimize these losses. By incorporating power losses as a critical component of the fitness function, the DE algorithm effectively prioritizes configurations that result in lower losses, thereby optimizing the overall performance of the DC microgrid.

3.5. Proposed Framework for Optimal Reconfiguration

The proposed approach is solved by the differential evolution algorithm [26]. It is a population-based metaheuristic that iteratively improves a specific solution to an optimization problem using an evolutionary process. Figure 4 shows the DE algorithm. In step 01, parameters such as population size, mutation scaling factor, and crossover rate are defined according to [29]. In step 02, feasible solutions are generated considering radiality and connection constraints. Step 03 evaluates each solution (individual), as discussed in Section 3.4 and depicted in Figure 3. If the maximum number of generations (iterations) is reached in step 04, the results are returned in step 05. Otherwise, in step 06, for each individual ${ind}_k$, three other individuals must be randomly selected. It is imperative that all selected individuals are different from each other and from the original individual, to meet the criteria outlined by Equation (14).

$$in{d}_{r1}\ne in{d}_{r2}\ne in{d}_{r3}\ne in{d}_k$$

(14)

Once the three individuals ($in{d}_{r1}$, $in{d}_{r2}$ and $in{d}_{r3}$) are randomly selected, the mutant vector is calculated according to Equation (15).

$$m=in{d}_{r1}+\zeta ·\left(in{d}_{r2}-in{d}_{r3}\right)$$

(15)

where $\zeta$ is a constant within the range [0–2] that controls the amplitude of the differential variation $\left(in{d}_{r2}-in{d}_{r3}\right)$, also known as the mutation scaling factor. In this study, a value of $\zeta =0.5$ was adopted to simulate the conventional methodology.

In step 07 the mutant vector is then applied individually to each individual gene ($z_1,z_2,\cdots ,z_{nbr}$) in an operation known as crossover. The probability of mutation is controlled by the crossover rate $cr$. In this work, $cr=0.9$ is used. The crossover process is described by Equation (16).

$${ind}_k^{\prime }\left(z\right)=\left\{\begin{array}{cc}m\left(z\right)& \mathrm{if}\mathrm{rand}\le cr\hfill \\ {ind}_k\left(z\right)& \mathrm{otherwise}\hfill \end{array}\right.$$

(16)

where ${ind}_k\left(z\right)$ denotes the $k^{th}$ component of ${ind}_k\left(z\right)$.

Finally, in step 08 the individual ${ind}_k$ should be replaced by the mutant individual ${ind}_k^{\prime }$ if ${ind}_k^{\prime }$ produces a better objective function value. For further information on evolutionary operators, the reader is referred to [26,29].

The proposed framework for optimal reconfiguration is formulated as a mixed-integer optimization problem, involving discrete variables (e.g., switch statuses) and continuous variables (e.g., nodal voltages). Differential evolution handles the discrete variables, while the interior point method-based OPF optimizes the continuous variables.

To ensure binary switch statuses, the DE algorithm incorporates the sigmoid function, as outlined in Section 3.2. DE generates continuous candidate solutions, which are passed through the sigmoid function to determine the probability of a switch being open or closed. If the sigmoid output exceeds 0.5, the switch is considered closed (binary value 1); otherwise, it is open (binary value 0), according to Equation (10). This approach ensures binary status while allowing DE to explore a continuous search space.

Once the switch statuses are determined, they are fed into the IPM, which optimizes the continuous variables for that configuration (nodal voltages). The DE and IPM are tightly coupled: DE iteratively proposes new configurations, and for each, the IPM calculates associated power losses. This iterative process continues until DE reaches the predefined maximum number of generations, 50 generations in this study.

The fitness function detailed in Section 3.4 ensures effective interaction between DE and IPM. For each candidate solution, DE utilizes IPM to calculate electrical losses. This process guarantees that every solution is evaluated, allowing for iterative refinement of the microgrid configuration while satisfying the system’s constraints.

4. Bipolar Test Feeders and General Parameters

4.1. System Description and Data

The simulations were conducted on two DC bipolar microgrids. The first is a 33-bus bipolar DC microgrid, comprising 33 nodes and adapted from the original IEEE 33-bus grid proposed in [30]. The same assumptions were recently used in [13]. The second bipolar DC microgrid consists of a 69-bus configuration, adapted from the IEEE 69-bus grid proposed in [31,32]. The power basis is set at 1 MVA and the voltage basis is ±12.66 kV. Detailed information on the total power consumption by monopolar and bipolar loads is provided in

Table 1. Total power consumption (kW).

System	Monopolar Positive Pole	Monopolar Negative Pole	Bipolar
33-bus	2615	2185	2350
69-bus	1485.4	1215.7	1101.1

.

Regarding the 33-bus microgrid:

• The one-line diagram is presented in Figure 5, with the open branches in the base case represented by dashed lines.
• System parameters are shown in

  Table 2. Parametric information regarding the 33-bus grid (closed switches in the base case). Adapted from [30].

  Node j	Node k	${\mathit{R}}_{\mathit{jk}}$ ($\mathbf{\Omega }$)	${\mathit{P}}_{\mathit{d},\mathit{k}}^{\mathit{p}}$ (kW)	${\mathit{P}}_{\mathit{d},\mathit{k}}^{\mathit{n}}$ (kW)	${\mathit{P}}_{\mathit{d},\mathit{k}}^{\mathit{p}-\mathit{n}}$ (kW)	Switch
  1	2	0.0922	100	150	0	$S_1$
  2	3	0.4930	90	75	0	$S_2$
  3	4	0.3660	120	100	0	$S_3$
  4	5	0.3811	60	90	0	$S_4$
  5	6	0.8190	60	0	200	$S_5$
  6	7	0.1872	100	50	150	$S_6$
  7	8	1.7114	100	0	0	$S_7$
  8	9	1.0300	60	70	100	$S_8$
  9	10	1.0400	60	80	25	$S_9$
  10	11	0.1966	45	0	0	$S_{10}$
  11	12	0.3744	60	90	0	$S_{11}$
  12	13	1.4680	60	60	100	$S_{12}$
  13	14	0.5416	120	100	200	$S_{13}$
  14	15	0.5910	60	30	50	$S_{14}$
  15	16	0.7463	110	0	350	$S_{15}$
  16	17	1.2890	60	90	0	$S_{16}$
  17	18	0.7320	90	45	0	$S_{17}$
  2	19	0.1640	90	150	0	$S_{18}$
  19	20	1.5042	150	50	115	$S_{19}$
  20	21	0.4095	0	90	0	$S_{20}$
  21	22	0.7089	0	90	145	$S_{21}$
  3	23	0.4512	90	110	35	$S_{22}$
  23	24	0.8980	120	0	40	$S_{23}$
  24	25	0.8960	150	100	100	$S_{24}$
  6	26	0.2030	60	80	0	$S_{25}$
  26	27	0.2842	60	0	225	$S_{26}$
  27	28	1.0590	0	0	130	$S_{27}$
  28	29	0.8042	120	75	65	$S_{28}$
  29	30	0.5075	100	100	0	$S_{29}$
  30	31	0.9744	50	150	125	$S_{30}$
  31	32	0.3105	175	100	75	$S_{31}$
  32	33	0.3410	95	60	120	$S_{32}$

  and

  Table 3. Parametric information regarding the 33-bus grid (open switches in the base case). Adapted from [30].

  Node j	Node k	${\mathit{R}}_{\mathit{jk}}$ ($\mathbf{\Omega }$)	Switch
  21	8	2	$S_{33}$
  9	15	2	$S_{34}$
  22	12	2	$S_{35}$
  18	33	0.5000	$S_{36}$
  25	29	0.5000	$S_{37}$

  . Table 2 provides the data of closed switches, and Table 3 presents the data of open switches.
• Two simulation scenarios are taken into consideration. The first scenario does not consider dispersed generators in the system, which is associated with higher losses. The second scenario considers dispersed generators allocated at different poles and nodes of the bipolar DC microgrid, as discussed in [13]. The dispersed generator’s parameters are presented in

  Table 4. Power of dispersed units (kW) in DC 33-bus grid [13].

  Node	Generation Positive Pole (kW)	Generation Negative Pole (kW)
  10	555.9692	-
  12	-	500.8079
  15	835.0393	623.0057
  30	1013.3334	-
  32	-	803.9153

  , adjusted in [13] to minimize power losses in the base case (line configuration in Figure 5).

Regarding the 69-bus microgrid, Figure 6 shows the one-line diagram, with the open switches in the base case represented by dashed lines.

Table 5. Parametric information regarding the 69-bus grid (closed switches in the base case)—part 1. Adapted from [30].

Node j	Node k	${\mathit{R}}_{\mathit{jk}}$ ($\mathbf{\Omega }$)	${\mathit{P}}_{\mathit{d},\mathit{k}}^{\mathit{p}}$ (kW)	${\mathit{P}}_{\mathit{d},\mathit{k}}^{\mathit{n}}$ (kW)	${\mathit{P}}_{\mathit{d},\mathit{k}}^{\mathit{p}-\mathit{n}}$ (kW)	Switch
1	2	0.0005	0	0	0	$S_1$
2	3	0.0005	0	0	0	$S_2$
3	4	0.0015	0	0	0	$S_3$
4	5	0.0251	0	0	0	$S_4$
5	6	0.3660	1.3	1.3	0	$S_5$
6	7	0.3811	20.2	20.2	0	$S_6$
7	8	0.0922	37.5	37.5	0	$S_7$
8	9	0.0493	30	0	0	$S_8$
9	10	0.8190	0	28	0	$S_9$
10	11	0.1872	0	0	145	$S_{10}$
11	12	0.7114	0	0	145	$S_{11}$
12	13	1.0300	8	0	0	$S_{12}$
13	14	1.0440	0	8	0	$S_{13}$
14	15	1.0580	0	0	0	$S_{14}$
15	16	0.1966	0	0	45.5	$S_{15}$
16	17	0.3744	60	0	0	$S_{16}$
17	18	0.0047	0	60	0	$S_{17}$
18	19	0.3276	0	0	0	$S_{18}$
19	20	0.2106	1	0	0	$S_{19}$
20	21	0.3416	57	57	0	$S_{20}$
21	22	0.0140	0	5.3	0	$S_{21}$
22	23	0.1591	0	0	0	$S_{22}$
23	24	0.3463	0	0	28	$S_{23}$
24	25	0.7488	0	0	0	$S_{24}$
25	26	0.3089	14	0	0	$S_{25}$
26	27	0.1732	0	14	0	$S_{26}$
3	28	0.0044	26	0	0	$S_{27}$
28	29	0.0640	0	26	0	$S_{28}$
29	30	0.3978	0	0	0	$S_{29}$
30	31	0.0702	0	0	0	$S_{30}$
31	32	0.3510	0	0	0	$S_{31}$
32	33	0.8390	14	0	0	$S_{32}$
33	34	1.7081	0	19.5	0	$S_{33}$
34	35	1.4740	6	0	0	$S_{34}$

,

Table 6. Parametric information regarding the 69-bus grid (closed switches in the base case)—part 2. Adapted from [30].

Node j	Node k	${\mathit{R}}_{\mathit{jk}}$ ($\mathbf{\Omega }$)	${\mathit{P}}_{\mathit{d},\mathit{k}}^{\mathit{p}}$ (kW)	${\mathit{P}}_{\mathit{d},\mathit{k}}^{\mathit{n}}$ (kW)	${\mathit{P}}_{\mathit{d},\mathit{k}}^{\mathit{p}-\mathit{n}}$ (kW)	Switch
3	36	0.0044	26	0	0	$S_{35}$
36	37	0.0640	0	26	0	$S_{36}$
37	38	0.1053	0	0	0	$S_{37}$
38	39	0.0304	24	0	0	$S_{38}$
39	40	0.0018	0	24	0	$S_{39}$
40	41	0.7283	1.2	0	0	$S_{40}$
41	42	0.3100	0	0	0	$S_{41}$
42	43	0.0410	6	0	0	$S_{42}$
43	44	0.0092	0	0	0	$S_{43}$
44	45	0.1089	39.22	0	0	$S_{44}$
45	46	0.0009	0	39.22	0	$S_{45}$
4	47	0.0034	0	0	0	$S_{46}$
47	48	0.0851	79	0	0	$S_{47}$
48	49	0.2898	384.7	0	0	$S_{48}$
49	50	0.0822	0	384.7	0	$S_{49}$
8	51	0.0928	40.5	0	0	$S_{50}$
51	52	0.3319	3.6	0	0	$S_{52}$
9	53	0.1740	0	4.35	0	$S_{53}$
53	54	0.2030	0	0	26.4	$S_{54}$
55	56	0.2813	0	0	0	$S_{55}$
56	57	1.5900	0	0	0	$S_{56}$
57	58	0.7837	0	0	0	$S_{57}$
58	59	0.3042	0	0	100	$S_{58}$
59	60	0.3861	0	0	0	$S_{59}$
60	61	0.5075	414.67	414.67	414.67	$S_{60}$
61	62	0.0974	32	0	0	$S_{61}$
62	63	0.1450	0	0	0	$S_{62}$
63	64	0.7105	113.5	0	113.5	$S_{63}$
64	65	1.0410	0	0	59	$S_{64}$
11	66	0.2012	18	0	0	$S_{65}$
66	67	0.0047	0	18	0	$S_{66}$
12	68	0.7394	28	0	0	$S_{67}$
68	69	0.0047	0	28	0	$S_{68}$

and

Table 7. Parametric information regarding the 69-bus grid (open switches in the base case). Adapted from [30].

Node j	Node k	${\mathit{R}}_{\mathit{jk}}$ ($\mathbf{\Omega }$)	Switch
11	43	0.5	$S_{69}$
13	21	0.5	$S_{70}$
15	46	1	$S_{71}$
50	59	2	$S_{72}$
27	65	1	$S_{73}$

present the system parameters. Two scenarios are considered: (i) system without dispersed generators and (ii) system considering generators as shown in

Table 8. Power of dispersed units (kW) in DC 69-bus grid.

Node	Generation Positive Pole (kW)	Generation Negative Pole (kW)
11	500	-
20	-	500
61	500	500

.

4.2. Limits and Optimization Parameters

The proposed approach is carried out considering the voltage limits defined in Equation (17). Since the differential evolution algorithm is stochastic in nature, multiple executions are necessary to obtain a reliable estimate of the optimal solution. In this study, the algorithm was executed 10 times, with a population of 25 individuals over 50 generations (resulting in 25 × 50 = 1250 power flow calculations). The choice of 10 executions was made to strike a balance between computational efficiency and statistical robustness. By running the algorithm multiple times, we can better assess the consistency and stability of the results, ensuring that the solutions are not unduly influenced by random variations inherent in the stochastic process.The simulations are conducted using the MATLAB platform and an Intel Core i7 2.10 GHz computer with 32 GB of RAM and a Windows 11 64-bit operating system.

$$\begin{array}{c}0.9\le V_j^p\le 1.1pu\\ -1.1\le V_j^n\le -0.9pu\end{array}$$

(17)

5. Results and Discussion

Figure 7 confirms the importance of several simulations of the differential evolution algorithm, since power losses (solution) vary for a set of executions, regardless of the system or scenario. The reason is the random nature of the differential evolution algorithm. In addition, it can be seen that the dispersed generation allocation reduces the power losses (in scenario 02), as is well known in the literature.

Table 9. Computational burden (minutes).

33-Bus (Scenario 01)	33-Bus (Scenario 02)	69-Bus (Scenario 01)	69-Bus (Scenario 02)
6.14	0.47	4.29	0.90

presents the average computational burden for each of the 10 solutions. An interesting observation is that the system scale does not influence the computational burden: the 69-bus grid required less computational time than the 33-bus grid in Scenario 01, without dispersed units. Furthermore, the presence of dispersed generators significantly reduced the computational burden in scenario 02 (for both microgrids). The reason is that the algorithm starts from an operating condition close to the optimal value.

It is important to note two distinct behaviors observed in the second scenario of both simulations. For the 33-bus microgrid, the dispersed generation configuration defined in [13], designed to minimize power losses, was employed. The reconfiguration process achieved a slight reduction in power losses (2.87%—from 28.4942 kW to 27.6775 kW) compared to the system without generators. The modest reductions are attributed to the base case that already incorporates dispersed generators with power outputs optimized for minimal losses, leaving limited potential for further loss reduction. In contrast, the second scenario of the 69-bus microgrid employed a random configuration of distributed generators, resulting in a significant reduction of 50.50% in power losses after the reconfiguration process (from 21.6746 kW to 10.7298 kW).

To better understand Figure 7,

Table 10. Quartile analysis for 33-bus and 69-bus microgrids (power losses in kW).

	33-Bus (Scenario 01)	33-Bus (Scenario 02)	69-Bus (Scenario 01)	69-Bus (Scenario 02)
Quartile 1	183.9619	28.0374	34.6140	11.1281
Quartile 2	199.7020	28.8781	35.3977	11.4715
Quartile 3	206.5961	29.8194	37.4434	12.3247

provides a quartile analysis based on the data obtained from 10 simulations. Quartiles are statistical measures that divide a data set into four equal parts, each representing 25% of the data. They are particularly useful for understanding the distribution and spread of data. Taking scenario 01 of the 33-bus microgrid as an example, the first quartile (Q1) is 183.96 kW, which means that 25% of the data points are less than or equal to this value. The second quartile (Q2), also known as the median, is 199.70 kW, indicating that 50% of the data points are less than or equal to this value, and 50% are greater. The third quartile (Q3) is 206.60 kW, which means that 75% of the data fall below this point. These quartiles help to understand the distribution of the data, providing insight into potential skewness or the presence of outliers. In this example, the quartiles suggest that the majority of the data points are clustered between 183.96 kW and 206.60 kW, with the median at 199.70 kW representing the central tendency of the data.

The quartiles of the four analyses indicate that the data are consistently concentrated, with relatively small variations between the first and third quartile values in each analysis. The median, in all analyses, is close to the center of the data, suggesting little dispersion and a distribution clustered around specific central values. In particular, the analyses of the 33-bus (scenario 01) and 69-bus (scenarios 01 and 02) microgrids show an especially narrow distribution, highlighting limited variability. Overall, this suggests that the data have low variability and tend to concentrate around central values, which could be indicative of consistency or homogeneity in the results provided by the DE algorithm.

The results provided by the proposed approach are presented in

Table 11. Results obtained by the proposed approach.

Bipolar DC Microgrid	Optimal Solution Open Switches (Losses)	Base Case Open Switches (Losses)
33-bus (scenario 01)	$S_7$, $S_{11}$, $S_{14}$, $S_{16}$, $S_{27}$ 178.3846 kW	$S_{33}$, $S_{34}$, $S_{35}$, $S_{36}$, $S_{37}$ 344.4797 kW
33-bus (scenario 02) [13]	$S_{13}$, $S_{21}$, $S_{26}$, $S_{33}$, $S_{36}$ 27.6775 kW	$S_{33}$, $S_{34}$, $S_{35}$, $S_{36}$, $S_{37}$ 28.4942 kW
69-bus (scenario 01)	$S_{10}$, $S_{14}$, $S_{56}$, $S_{62}$, $S_{70}$ 33.9455 kW	$S_{69}$, $S_{70}$, $S_{71}$, $S_{72}$, $S_{73}$ 69.1413 kW
69-bus (scenario 02)	$S_{10}$, $S_{14}$, $S_{17}$, $S_{40}$, $S_{55}$ 10.7298 kW	$S_{69}$, $S_{70}$, $S_{71}$, $S_{72}$, $S_{73}$ 21.6746 kW

. The optimal solution for each system is associated with the best solution obtained in the set of 10 simulations, as shown in Figure 7. The first observation is the power loss minimization achieved through the reconfiguration process: 48.22% (33-bus, scenario 01), 2.87% (33-bus, scenario 02), 50.90% (69-bus, scenario 01), and 50.05% (69-bus, scenario 02). Regarding the 33-bus bipolar microgrid, it is clear that dispersed generators contribute to a reduction of 84.48% in the optimal solution (from 178.3846 kW to 27.6775 kW) and 91.73% in the base case (from 344.4797 kW to 28.4942 kW). A similar observation is seen for the 69-bus microgrid.

Finally, Figure 8, Figure 9, Figure 10 and Figure 11 show the voltage profiles for the positive, negative, and neutral poles for both bipolar DC grids in both scenarios. It can be seen that the voltage limits are respected.

6. Conclusions

This paper presented an optimal power flow for reconfiguration of a bipolar DC microgrid to minimize power losses, considering asymmetric loads and dispersed generators. The problem was formulated as a mixed-integer nonlinear optimization problem in which integer variables represent the switch statuses and continuous ones stand for the nodal voltages. The solution was carried out using the differential evolution algorithm (considering the binary variables as continuous through the sigmoid function) and an optimal power flow expressed by the injection current method for loss minimization, solved by the interior point method. The following points summarize the observations:

• Results for microgrids with 33 and 69 buses confirmed the benefits of the reconfiguration process for loss minimization. Reductions compared to the base case were approximately 48.22% (33-bus microgrid without DG), 2.87% (33-bus microgrid with DG), 50.90% (69-bus microgrid without DG), and 50.50% (69-bus microgrid with DG).
• For the 33-bus microgrid with DG, the reductions were smaller since this simulation considered a base case with dispersed generators whose power was also optimized for power loss minimization [13]. In this case, the gains from the reduction of losses are minimal. However, for the 69-bus with DG, the reduction was around 50.50% since the DG placement has not been carried out by an optimization approach aiming at minimizing power losses.
• The computational burden is feasible. An interesting observation for the 33-bus microgrid is that the scenario with DG converged in 0.47 min, compared to 6.14 min for the scenario without DG. This indicates better convergence of the interior point method when voltages are close to optimal values in the base case.
• The differential evolution algorithm is stochastic in nature and its performance is considered suitable when taking into account a set of simulations. However, there is room for further investigation and improvement, such as the application of other metaheuristics.

Furthermore, the proposed approach demonstrates the potential for improving the efficiency of bipolar DC microgrids, particularly in scenarios with distributed generation. However, the general usability of the method extends beyond the specific case studied here; it can be adapted for other types of microgrids or larger distribution systems with appropriate modifications.

Despite its effectiveness, the approach has certain limitations. The computational complexity of the mixed-integer nonlinear problem can increase with the size of the network, potentially affecting real-time application. Additionally, the reliance on precise load and generation data may limit the method’s robustness in highly dynamic environments.

Future research will focus on improving the scalability of the optimization process, integrating advanced forecasting techniques for load and generation profiles, and exploring other optimization methods to further improve computational efficiency.