# A Recursive Conic Approximation for Solving the Optimal Power Flow Problem in Bipolar Direct Current Grids

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## Abstract

**:**

## 1. Introduction

- i.
- A convex approximation for the power balance constraint associated with constant power loads using a conic representation of the hyperbolic relation between voltages and currents, and a linear approximation for nodes with dispersed generation sources.
- ii.
- An iterative convex solution methodology to minimize the error introduced by the linear approximation of the hyperbolic relation between voltage and currents in the dispersed generation sources via recursive convex programming.

## 2. Optimal Power Flow Modeling

#### 2.1. Objective Function Formulation

#### 2.2. Model Constraints

- i.
- If neutral wire is assumed to be floating, then the variable ${I}_{d,k}^{\mathrm{ground}}$ must be set as zero for all the nodes.
- ii.
- If the neutral wire is assumed to be solidly grounded at all nodes of the network, then ${I}_{d,k}^{\mathrm{ground}}$ is left as a free variable, and the voltage variable regarding the neutral pole (i.e., ${V}_{k}^{o}$) must be set as zero for all nodes of the network.

#### 2.3. Model Interpretation

**Remark**

**1.**

## 3. Iterative Conic Solution Approach

#### 3.1. A Conic Approximation for Constant Power Loads

**Remark**

**2.**

#### 3.2. A Linear Approximation for Generation Sources

**Remark**

**4.**

#### 3.3. Approximate Convex Model and Iterative Conic Solution

**Remark**

**5.**

## 4. Test Feeder Characteristics

#### 4.1. Bipolar DC Configuration

#### 4.2. Branch and Loading Information

#### 4.3. Dispersed Generation Data

## 5. Computational Results

- i.
- A comparative analysis with three different power flow algorithms, two of them derivative-free and another one based on Taylor series expansion.
- ii.
- A comparison between the solution of the OPF problem with three combinatorial optimization methods and the proposed ICS approach.

#### 5.1. Power Flow Solution

- i.
- The derivative-free power flow approaches (i.e., SAPF and SAPF methods) require the same number of iterations in each simulation case. This implies that these methods are numerically equivalent for power flow studies with linear convergence. The HAPF approach for the solidly grounded case exhibited a linear convergence with the same number of iterations as the SAPF and SAPF methods; and in the case of the floating neutral wire, the convergence was quadratic, requiring four iterations, in contrast with the ten iterations taken by the SAPF and SAPF methods.
- ii.
- The main difference regarding the connections of the neutral wire (i.e., the solidly grounded and the floating cases) was the total grid power loss. As expected, with the floating neutral wire, the losses value was $95.4237$ kW, which was reduced to $91.2701$ kW. i.e., a variation of about $4.1536$ kW in favor of the solidly grounded connection. This is an expected behavior, as the current in the neutral wire is directly drained to the earth when it is solidly grounded, which helps to reduce power losses, in contrast with the neutral floating wire, where the presence of asymmetric loads produces neutral circulating currents.
- iii.
- As for processing times, the ICS takes about 5 s to solve the power flow problem, which is a higher value in comparison with the specialized power flow methods, in which only milliseconds are spent. However, it is worth highlighting that the ICS for model (34) involves an optimization problem with infinite feasible solutions and only one global optimal solution. In contrast, the specialized power flow methods can solve the problem without employing combinatorial optimization approach in a master–slave connection.

#### 5.2. OPF Solution

- i.
- The best combinatorial optimization method is the VSA, as demonstrated in [34]. This algorithm found a solution of $22.986$ kW, which is near the optimal solution reached with the ICS ($22.985$ kW). The main difference between both approaches lies in their standard deviation: the VSA reported about $4.23\times {10}^{-6}$, whereas that of the proposed ICS was less than $1\times {10}^{-16}$. These values confirm two things. (i) It is impossible to ensure 100% repeatability with the VSA approach, since the standard deviation is a measure associated with the dispersion between solutions. Even if these are in a closed ball, it is possible to obtain an answer out of it, as is the case of the maximum solution ($22.988$ kW). (ii) Due to the convex nature of the solution space in model (34), the proposed ICS always reaches the exact numerical solution, thereby confirming the standard deviation’s negligible value.
- ii.
- The BHO and SCA got stuck in locally optimal solutions, with values of $23.054$ and $23.066$ kW. However, these solutions can be regarded as acceptable for the power flow solution, since both were less than 0.10 kW away from the optimal one (i.e., solution found with the ICS). Nevertheless, the main problem with these solutions lies in the high variability between their minimum and maximum values, which can be observed in their standard deviation.
- iii.
- Concerning the processing times, it is noted that all of the OPF algorithms in Table 4 required simulation times of between 8 and 13.5 s. However, each one of the combinatorial optimizers requires multiple evaluations in order to determine their average behavior, which means that, after 100 consecutive evaluations, the processing times of the ICS were effectively 100 times higher. This implies that the proposed methodology is the most effective approach, given the fact that no statistical analysis is needed.

#### 5.3. Complementary Analysis

- i
- The minimum voltage in the benchmark case (without dispersed generation) occurred at the positive pole, i.e., a value of $0.8883$ pu at node 17, which implies that the voltage regulation of the bipolar DC network was about $11.17\%$. This is a significant result, as it demonstrates that, without dispersed generation, the 21-bus network does not fulfill the voltage regulation condition for distribution networks (typically $\pm 10\%$). However, this only occurs for the positive pole because it is the most loaded pole. In the case of the negative pole, the minimum voltage was $-0.9098$, which implies that the voltage regulation for this pole is below the permitted limits.
- ii.
- The behavior of the neutral pole shows that, once the dispersed generation is introduced into the distribution network, it helps balance the voltage behavior of the systems. Note that the maximum voltage in the neutral wire was $0.02434$ pu, i.e., 24.34 V at node 17 in the benchmark case. In contrast, when the dispersed generation was optimally dispatched, the maximum voltage deviation in the neutral wire was about $0.0139$ pu, i.e., 13.90 V at node 12.
- iii.
- The presence of dispersed generation in bipolar DC networks has important effects on the performance of the voltage profile. Note that node 17, for the positive and negative poles, has a magnitude of 1.0 pu, i.e., an ideal voltage profile due to the injection of power at this node with the dispersed sources. In addition, the voltage regulation for this system with dispersed sources was $3.32\%$. The minimum voltage at node 12 had a magnitude of $0.9668\phantom{\rule{3.33333pt}{0ex}}\mathrm{pu}$ in the negative pole.

## 6. Conclusions and Future Work

- i.
- The proposed ICS reached an equal solution for the power flow problem when compared to three different specialized power flow approaches (SAPF, TBPF, and HAPF) for both simulation cases associated with the neutral wire’s connection, i.e., the solidly grounded and floating cases.
- ii.
- The OPF solution showed that the ICS found the global optimal solution for the 21-bus grid, with a value of $22.985$ kW. The VSA approach, with a near-optimal solution, only followed the ICS. However, the proposed method always reached an equal optimal solution due to the convex nature of the solution space. In contrast, the VSA and the other comparison methods (BHO and SCA) can get stuck in locally optimal solutions, as evidenced by the statistical analysis.
- iii.
- The voltage profile analysis showed that, for the benchmark case, when the neutral wire is floating, the voltage regulation in the test feeder is about $11.17\%$. However, when the dispersed generation is optimally dispatched, the voltage regulation of the 21-bus grid is improved, with a final value of $3.32\%$. This implies that dispersed generators allow for an improvement of about $7.85\%$.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Schematic representation of monopolar and bipolar DC networks: (

**a**) monopolar configuration and (

**b**) bipolar configuration.

**Figure 2.**Proposed iterative convex solution for the approximated OPF model defined in Equation (34).

**Figure 4.**Voltage profile performance before and after optimally dispatching dispersed generation sources.

**Figure 5.**Convergence behavior of the proposed ICS methodology for the power flow and optimal power flow problems.

Node j | Node k | ${\mathit{R}}_{\mathbf{jk}}$ ($\mathsf{\Omega}$) | ${\mathit{P}}_{\mathit{d},\mathit{k}}^{\mathit{p}}$ | ${\mathit{P}}_{\mathit{d},\mathit{k}}^{\mathit{n}}$ | ${\mathit{P}}_{\mathit{d},\mathit{k}}^{\mathit{p}-\mathit{n}}$ |
---|---|---|---|---|---|

1 | 2 | 0.053 | 70 | 100 | 0 |

1 | 3 | 0.054 | 0 | 0 | 0 |

3 | 4 | 0.054 | 36 | 40 | 120 |

4 | 5 | 0.063 | 4 | 0 | 0 |

4 | 6 | 0.051 | 36 | 0 | 0 |

3 | 7 | 0.037 | 0 | 0 | 0 |

7 | 8 | 0.079 | 32 | 50 | 0 |

7 | 9 | 0.072 | 80 | 0 | 100 |

3 | 10 | 0.053 | 0 | 10 | 0 |

10 | 11 | 0.038 | 45 | 30 | 0 |

11 | 12 | 0.079 | 68 | 70 | 0 |

11 | 13 | 0.078 | 10 | 0 | 75 |

10 | 14 | 0.083 | 0 | 0 | 0 |

14 | 15 | 0.065 | 22 | 30 | 0 |

15 | 16 | 0.064 | 23 | 10 | 0 |

16 | 17 | 0.074 | 43 | 0 | 60 |

16 | 18 | 0.081 | 34 | 60 | 0 |

14 | 19 | 0.078 | 9 | 15 | 0 |

19 | 20 | 0.084 | 21 | 10 | 50 |

19 | 21 | 0.082 | 21 | 20 | 0 |

Node | Connection | Capacity (kW) |
---|---|---|

3 | p | 300 |

3 | n | 100 |

11 | p | 400 |

17 | p | 200 |

17 | n | 300 |

Neutral Wire Solidly Grounded | |||
---|---|---|---|

Method | Losses (pu) | Iterations | Time (ms) |

SAPF | 0.954237 | 13 | 0.5275 |

TBPF | 0.954237 | 13 | 0.8340 |

HAPF | 0.954237 | 13 | 1.5542 |

ICS | 0.954237 | 2 | — |

Neutral Wire Floating | |||

Method | Losses (pu) | Iterations | Time (ms) |

SAPF | 0.912701 | 10 | 0.4911 |

TBPF | 0.912701 | 10 | 0.7672 |

HAPF | 0.912701 | 4 | 1.0212 |

ICS | 0.912701 | 2 | — |

Method | Ref. | Evaluations | Iterations | Pop. Size |
---|---|---|---|---|

Black-hole optimizer (BHO) | [32] | |||

Sine-cosine algorithm (SCA) | [33] | 100 | 1000 | 20 |

Vortex-search algorithm (VSA) | [34] |

**Table 5.**Comparative analysis between combinatorial optimizers and the ICS methodology (all values in pu).

Method | Min. | Mean | Max. | Std. Dev. | Time (s) |
---|---|---|---|---|---|

SCA | 0.23054 | 0.25305 | 0.29703 | $1.39\times {10}^{-2}$ | 6.7870 |

BHO | 0.23066 | 0.23183 | 0.23329 | $5.90\times {10}^{-4}$ | 13.1513 |

VSA | 0.22986 | 0.22986 | 0.22988 | $4.23\times {10}^{-6}$ | 8.3176 |

ICS | 0.22985 | 0.22985 | 0.22985 | <$1\times {10}^{-16}$ | 11.6125 |

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## Share and Cite

**MDPI and ACS Style**

Montoya, O.D.; Grisales-Noreña, L.F.; Hernández, J.C.
A Recursive Conic Approximation for Solving the Optimal Power Flow Problem in Bipolar Direct Current Grids. *Energies* **2023**, *16*, 1729.
https://doi.org/10.3390/en16041729

**AMA Style**

Montoya OD, Grisales-Noreña LF, Hernández JC.
A Recursive Conic Approximation for Solving the Optimal Power Flow Problem in Bipolar Direct Current Grids. *Energies*. 2023; 16(4):1729.
https://doi.org/10.3390/en16041729

**Chicago/Turabian Style**

Montoya, Oscar Danilo, Luis Fernando Grisales-Noreña, and Jesús C. Hernández.
2023. "A Recursive Conic Approximation for Solving the Optimal Power Flow Problem in Bipolar Direct Current Grids" *Energies* 16, no. 4: 1729.
https://doi.org/10.3390/en16041729