# Optimal Power Flow Solution for Bipolar DC Networks Using a Recursive Quadratic Approximation

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## Abstract

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## 1. Introduction

- i.
- A detailed formulation of the OPF problem for bipolar unbalanced distribution networks is presented in this research. In addition, a general procedure to become the hyperbolic relationships between voltages and powers in the demand nodes is introduced to become the exact nonlinear programming (NLP) model that represents the OPF problem into a convex approximation.
- ii.
- A recursive quadratic approximation is proposed to minimize/eliminate the error in the OPF problem’s solution by applying Taylor’s series expansion. This recursive quadratic approximation redefines the linearizing point of the approximated quadratic OPF model iteratively until the desired convergence is reached.

## 2. OPF Formulation for Bipolar DC Networks

#### 2.1. Objective Function

#### 2.2. Set of Constraints

- i.
- ii.
- iii.
- iv.
- v.

## 3. Recursive Quadratic Approximation of the OPF Problem

#### 3.1. Approximation for Nodes with Constant Power Loads

#### 3.2. Approximation for Nodes with Dispersed Generators

#### 3.3. Proposed Recursive Quadratic Approximation

#### 3.4. Test Feeders

#### 3.5. Bipolar DC 21-Bus System

- i.
- The substation bus is located at bus 1, and it operates with a voltage value of $\pm 1$ kV in the positive and negative poles, and it is considered that its neutral pole is solidly grounded.
- ii.
- The total power consumption in the positive pole is 554 kW, and the total power consumption in the negative pole is 445 kW, while the bipolar load sums 405 kW. This grid has a radial grid configuration.

#### 3.6. Bipolar DC 33-Bus System

- i.
- The substation bus is located at bus 1, and it operates with a voltage value of $\pm 12.66$ kV in the positive and negative poles, and it is considered that its neutral pole is solidly grounded.
- ii.
- The total power consumption in the positive pole is 2615 kW, and the total power consumption in the negative pole is 2185 kW, while the bipolar loads sum 2350 kW.

## 4. Computational Results

#### 4.1. Results for the Bipolar DC 21-Bus System

#### 4.1.1. Power Flow Solution

- i.
- As expected, all the power flow approaches, including the proposed RQA find the same value for the total grid power loss with a value of $95.4237$ kW for the neutral solidly-ly grounded operation scenario, and $91.2701$ kW in the neutral floating case. In addition, the SAPF and the TBPF take the same number of iterations in both cases, which is an expected result since both methods are from the same family of solution methods, i.e., derivative-free methods from the family of graph-based theory approaches.
- ii.
- The difference between power losses when considering the neutral wire operating with or without grounded connections is about $4.1536$ kW. This confirms, as expected that the best possible scenario for electrical networks is when the neutral wire is solidly grounded since it minimizes energy losses in this wire while allowing a local voltage reference for each load.
- iii.
- The HAPF approach showed a notable difference regarding the number of iterations when both simulation scenarios are compared. This is because for the neutral wire to be solidly grounded, it only takes 4 iterations. In contrast, when the neutral wire is floating, it takes 13 iterations, which in the first case it has quadratic convergence, whereas in the second case, it has linear convergence.

#### 4.1.2. OPF Solution

- i.
- The proposed RQA finds the best solution to the OPF problem with a value of $22.985$ kW, followed only by a similar value for the VSA. However, the proposed approach has the same numerical solution at each evaluation (convexity of the solution space) with a standard deviation lower than $1\times {10}^{-16}$. However, the VSA approach does not necessarily finds find the same objective function value (see its maximum value). This is an expected behavior for metaheuristics since their random nature makes it impossible to ensure 100% of convergence in nonlinear non-convex optimization problems, as in the case of the exact NLP that represents the studied OPF problem.
- ii.
- The SCA and the BHO approaches have stuck in locally optimal solutions which are explainable in their random nature and their simple evolution rules (less sophisticated than the VSA approach); however, both can be considered adequate approximations to the OPF solution for problems where high precision is not relevant.
- iii.
- Regarding processing times in Table 6, the best metaheuristic optimizer (the VSA approach) takes about 8.3176 s to solve the OPF problem, while the worst approach in terms of processing times corresponds to the BHO method with 13.1513 s. Nevertheless, the proposed RQA takes about 7.7901 s, maintaining a similar time when it was used for solving the power flow problem. Additionally, the proposed model has the main advantage that no statistical studies are required to evaluate its efficiency. The convexity of the solution space in this approximation ensures the 100% of solution repeatability. This does not occur in the case of the combinatorial optimization methods.

#### 4.2. Results for the Bipolar DC 33-Bus System

- i.
- The maximum reduction regarding power loss reduction is reached when the DGs in both poles are optimally dispatched. This reduction is higher than 90% concerning the benchmark case. In this scenario, the usage of the DGs capacities was about $77.5594\%$ and $62.6904\%$ in the positive and negative poles, respectively.
- ii.
- In the case of power dispatch only one pole, the best reduction of power losses is reached in the positive pole, with a value of $37.3828\%$, while for the negative pole, this reduction reaches a value of $8.6662\%$. Two possible facts explain this: (i) the positive pole is the most charged (more power loss when compared with the negative pole), and (ii) the location of the DGs may be better than the DGs in the negative pole.
- iii.
- Power loss reductions in Table 7 confirm that the OPF problem in bipolar DC networks is indeed a complex optimization problem where the superposition method is not applicable since the model is nonlinear. This is demonstrated by the combination of the generations in the positive and negative poles is different from the solution when all the distributed generators are optimally dispatched.

## 5. Conclusions and Future Works

- i.
- The proposed RQA can efficiently solve the power flow problem in bipolar DC networks by finding the same numerical result in power losses compared with specialized power flow methods as the cases of the SAPF, TBPF, and the HAPF approaches. When the neutral wire is solidly grounded, the total grid power loss was $91.2701$ kW, which increased to $95.4237$ kW in the case of a non-grounded connection.
- ii.
- Comparative analysis with combinatorial optimizers showed that the proposed RQA efficiently solved the OPF problem in the case of the neutral wire floating. Numerical comparisons with the BHO, the SCA, and the VSA demonstrated that the RQA found the optimal solution with a final value of $22.985$ kW with a standard deviation lower than $1\times {10}^{-16}$. Only the VSA approach found a similar objective function value. However, its standard deviation is about $4.23\times {10}^{-6}$, which means that statistical analysis is required to confirm its effectiveness, which is not the case for the proposed RQA owing to the convex nature of the solution space.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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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 | Location | Capacity (kW) |
---|---|---|

3 | p | 300 |

3 | n | 100 |

11 | p | 400 |

17 | p | 200 |

17 | n | 300 |

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 | 0.0922 | 100 | 150 | 0 |

2 | 3 | 0.4930 | 90 | 75 | 0 |

3 | 4 | 0.3660 | 120 | 100 | 0 |

4 | 5 | 0.3811 | 60 | 90 | 0 |

5 | 6 | 0.8190 | 60 | 0 | 200 |

6 | 7 | 0.1872 | 100 | 50 | 150 |

7 | 8 | 1.7114 | 100 | 0 | 0 |

8 | 9 | 1.0300 | 60 | 70 | 100 |

9 | 10 | 1.0400 | 60 | 80 | 25 |

10 | 11 | 0.1966 | 45 | 0 | 0 |

11 | 12 | 0.3744 | 60 | 90 | 0 |

12 | 13 | 1.4680 | 60 | 60 | 100 |

13 | 14 | 0.5416 | 120 | 100 | 200 |

14 | 15 | 0.5910 | 60 | 30 | 50 |

15 | 16 | 0.7463 | 110 | 0 | 350 |

16 | 17 | 1.2890 | 60 | 90 | 0 |

17 | 18 | 0.7320 | 90 | 45 | 0 |

2 | 19 | 0.1640 | 90 | 150 | 0 |

19 | 20 | 1.5042 | 150 | 50 | 115 |

20 | 21 | 0.4095 | 0 | 90 | 0 |

21 | 22 | 0.7089 | 0 | 90 | 145 |

3 | 23 | 0.4512 | 90 | 110 | 35 |

23 | 24 | 0.8980 | 120 | 0 | 40 |

24 | 25 | 0.8960 | 150 | 100 | 100 |

6 | 26 | 0.2030 | 60 | 80 | 0 |

26 | 27 | 0.2842 | 60 | 0 | 225 |

27 | 28 | 1.0590 | 0 | 0 | 130 |

28 | 29 | 0.8042 | 120 | 75 | 65 |

29 | 30 | 0.5075 | 100 | 100 | 0 |

30 | 31 | 0.9744 | 50 | 150 | 125 |

31 | 32 | 0.3105 | 175 | 100 | 75 |

32 | 33 | 0.3410 | 95 | 60 | 120 |

Node | Location | Capacity (kW) |
---|---|---|

10 | p | 800 |

12 | n | 1000 |

15 | p | 950 |

15 | n | 950 |

30 | p | 1350 |

31 | n | 1125 |

**Table 5.**Comparison between power flow methods and the RQA for solidly-ly grounded and neutral floating scenarios in the 21-bus grid.

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 |

RQA | 0.954237 | 4 | — |

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 |

RQA | 0.912701 | 4 | — |

**Table 6.**Evaluation of the different alternatives to solve the OPF problem in bipolar asymmetric DC networks (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 |

SQA | 0.22985 | 0.22985 | 0.22985 | <$1\times {10}^{-16}$ | 7.7901 |

Case | Power Loss (kW) | Gen. Positive Pole (kW) | Gen. Negative Pole (kW) | Reduction (%) |
---|---|---|---|---|

Ben. case | 344.4797 | — | — | — |

Case 1 | 215.7037 | $\left[\begin{array}{c}\left(10\right)\phantom{\rule{0.166667em}{0ex}}327.4197\\ \left(15\right)\phantom{\rule{0.166667em}{0ex}}457.9217\\ \left(30\right)\phantom{\rule{0.166667em}{0ex}}576.6784\end{array}\right]$ | — | 37.3828 |

Case 2 | 314.6265 | — | $\left[\begin{array}{c}\left(12\right)\phantom{\rule{0.166667em}{0ex}}179.3277\\ \left(15\right)\phantom{\rule{0.166667em}{0ex}}151.0976\\ \left(32\right)\phantom{\rule{0.166667em}{0ex}}334.9579\end{array}\right]$ | 8.6662 |

Case 3 | 28.4942 | $\left[\begin{array}{c}\left(10\right)\phantom{\rule{0.166667em}{0ex}}555.9692\\ \left(15\right)\phantom{\rule{0.166667em}{0ex}}835.0393\\ \left(30\right)\phantom{\rule{0.166667em}{0ex}}1013.3334\end{array}\right]$ | $\left[\begin{array}{c}\left(12\right)\phantom{\rule{0.166667em}{0ex}}500.8079\\ \left(15\right)\phantom{\rule{0.166667em}{0ex}}623.0057\\ \left(32\right)\phantom{\rule{0.166667em}{0ex}}803.9153\end{array}\right]$ | 91.7283 |

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## Share and Cite

**MDPI and ACS Style**

Montoya, O.D.; Gil-González, W.; Hernández, J.C.
Optimal Power Flow Solution for Bipolar DC Networks Using a Recursive Quadratic Approximation. *Energies* **2023**, *16*, 589.
https://doi.org/10.3390/en16020589

**AMA Style**

Montoya OD, Gil-González W, Hernández JC.
Optimal Power Flow Solution for Bipolar DC Networks Using a Recursive Quadratic Approximation. *Energies*. 2023; 16(2):589.
https://doi.org/10.3390/en16020589

**Chicago/Turabian Style**

Montoya, Oscar Danilo, Walter Gil-González, and Jesus C. Hernández.
2023. "Optimal Power Flow Solution for Bipolar DC Networks Using a Recursive Quadratic Approximation" *Energies* 16, no. 2: 589.
https://doi.org/10.3390/en16020589