# Optimized Power Flow Control to Minimize Congestion in a Modern Power System

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## Abstract

**:**

## 1. Introduction

## 2. Methods and Materials

#### 2.1. Choice of Power Flow Control Device

#### 2.2. Power Injection Model of UPFC

#### 2.3. Interpretation of the Power Injection Model

#### 2.4. Calculation of the Reactance in the Power Injection Model

#### 2.5. Operational Limits of Unified Power Flow Controllers

#### 2.6. Optimization of UPFC Configuration

#### 2.7. Objective Function

#### 2.8. Configuration of Unified Power Flow Controllers

**${O}_{+}$**. In the optimization

**${O}_{\mathrm{sys}}$**, the number of lines that can be considered is further limited to congested lines. In this case, the number of applicable lines is denoted as ${n}_{\mathrm{pos}}^{\mathrm{sys}}\le {n}_{\mathrm{pos}}$ and it holds that ${n}_{\mathrm{upfcs}}$ is randomly chosen from $\{1,2,\dots ,{n}_{\mathrm{pos}}^{\mathrm{sys}}\}$.

#### 2.9. Metropolis Algorithm and Optimization

- (1)
- Insert a configuration C into the distribution system and evaluate $F\left(C\right)$.
- (2)
- Create trial configuration ${C}^{\prime}$ according to Section 2.8; evaluate $F\left({C}^{\prime}\right)$ and $\Delta F:=F\left({C}^{\prime}\right)-F\left(C\right)$.
- (3)
- ${C}^{\prime}$ replaces C with probability $\mathrm{exp}(-\Delta F/(kT\left)\right)$.

#### 2.10. Parallel Tempering

- For each ${C}_{k}$ at ${T}_{k}$, the Metropolis algorithm is executed$${n}_{\mathrm{s}}=l/{m}_{\mathrm{f}}$$$$\begin{array}{c}\hfill l=4{n}_{\mathrm{upfcs}}+1\hfill \\ \hfill {m}_{\mathrm{f}}={p}_{\mathrm{upfcs}}+{n}_{\mathrm{upfcs}}({p}_{S}+{p}_{r}+{p}_{\gamma})\hfill \end{array}$$Here, l concerns the four columns of the length ${n}_{\mathrm{upfcs}}$ of each configuration and the additional degree of freedom ${n}_{\mathrm{upfcs}}$.
- A value k is chosen randomly $K-1$ times from $\{1,2,\dots ,K-1\}$. Each time, configuration ${C}_{k}$, assigned to ${T}_{k}$, is swapped with ${C}_{k+1}$, assigned to ${T}_{k+1}$, with probability$${p}_{\mathrm{swap}}({C}_{k},{C}_{k+1})=\mathrm{min}(1,\mathrm{exp}[{\Delta}_{k,k+1}({C}_{k},{C}_{k+1})\left]\right)$$$${\Delta}_{k,k+1}({C}_{k},{C}_{k+1})=\left(\frac{1}{{T}_{k}}-\frac{1}{{T}_{k+1}}\right)\left(F\left({C}_{k}\right)-F\left({C}_{k+1}\right)\right).$$If a swap is accepted, ${C}_{k}$ is assigned to ${T}_{k+1}$ and ${C}_{k+1}$ to ${T}_{k}$; otherwise, the configurations are not affected.

#### 2.11. Grid Model

## 3. Results

#### 3.1. Validation of Power Injection Model

#### 3.2. Convergence of Parallel Tempering

#### 3.3. Simple-to-Solve Optimization Problem

#### 3.4. Sensitivity of Simulation Results on UPFC Costs

#### 3.5. Convergence Optimization Approaches with the Greedy Algorithm

#### 3.6. Curtailment with the Optimal UPFC Configuration

#### 3.7. Profitability of the Optimal UPFC Configuration

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Table A1.**Settings of the Metropolis algorithm as part of $\mathrm{PT}\left({O}_{+}\right)$; ${\Delta}_{S}$ in $\mathrm{MVA}$.

${\mathit{T}}_{\mathit{k}}$ | ${\mathbf{\Delta}}_{\mathbf{U}}$ | ${\mathbf{\Delta}}_{\mathit{S}}$ | ${\mathbf{\Delta}}_{\mathit{r}}$ | ${\mathbf{\Delta}}_{\mathit{\gamma}}$ | ${\mathit{p}}_{\mathbf{U}}$ | ${\mathit{p}}_{\mathit{S}}$ | ${\mathit{p}}_{\mathit{r}}$ | ${\mathit{p}}_{\mathit{\gamma}}$ |
---|---|---|---|---|---|---|---|---|

9.8 | 1 | 3 | 0.001 | 1 | 0.45 | 0.55 | 0.65 | 0.65 |

11.3 | 1 | 3 | 0.002 | 4 | 0.45 | 0.55 | 0.65 | 0.65 |

13.4 | 1 | 4 | 0.002 | 5 | 0.50 | 0.60 | 0.60 | 0.60 |

14.5 | 1 | 4 | 0.003 | 7 | 0.53 | 0.63 | 0.63 | 0.63 |

16.53 | 2 | 5 | 0.003 | 7 | 0.52 | 0.62 | 0.72 | 0.72 |

19 | 2 | 6 | 0.003 | 7 | 0.58 | 0.68 | 0.78 | 0.78 |

23 | 2 | 6 | 0.007 | 10 | 0.65 | 0.75 | 0.85 | 0.85 |

28.5 | 2 | 12 | 0.007 | 15 | 0.70 | 0.80 | 0.90 | 0.90 |

45.5 | 3 | 17 | 0.020 | 19 | 0.85 | 0.95 | 0.95 | 0.95 |

650 | 13 | 65 | 0.030 | 60 | 1 | 1 | 1 | 1 |

${\mathit{T}}_{\mathit{k}}$ | ${\mathbf{acc}}_{\mathbf{swap}}$ | ${\mathbf{acc}}_{\mathbf{MA}}$ |
---|---|---|

${T}_{1}$ | 61.55 | 55.1 |

${T}_{2}$ | 53.43 | 49.71 |

${T}_{3}$ | 77.41 | 50.85 |

${T}_{4}$ | 65.48 | 49.44 |

${T}_{5}$ | 67.31 | 44.46 |

${T}_{6}$ | 58.36 | 47.39 |

${T}_{7}$ | 59.76 | 47.96 |

${T}_{8}$ | 60.35 | 54.28 |

${T}_{9}$ | 55.45 | 51.95 |

${T}_{10}$ | - | 53.52 |

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**Figure 1.**Basic setup of UPFC (adapted from [16]).

**Figure 2.**The PIM of UPFC on a line (adapted from [16]).

**Figure 4.**Flow charts of the Metropolis algorithm and the Parallel Tempering approach (prob. = probability, swc = sweep counter, MA = Metropolis algorithm).

**Figure 5.**Validation of PFC by the PIM of a UPFC on two different lines: (

**a**) Line with positive active power ${P}_{\mathrm{i}}$ at the initial bus and $\delta =4.2\xb0$ (in the reference case); (

**b**) Line with ${P}_{\mathrm{i}}<0$ and $\delta =-2.3\xb0$ (in the reference case).

**Figure 7.**Convergence behavior of the greedy algorithm optimization with (

**a**) $\mathrm{GrA}\left({O}_{+}\right)$ considering costs, (

**b**) with $\mathrm{GrA}\left({O}_{\mathrm{sys}}\right)$, and (

**c**) with $\mathrm{GrA}\left({O}_{+}\right)$ neglecting costs.

**Figure 8.**The reference case (above) and with the optimal UPFCs (beneath). Geographical information based on OpenEnergy project [37].

**Table 1.**Overview of optimization variants. For ${n}_{\mathrm{pos}}$ and ${n}_{\mathrm{pos}}^{\mathrm{sys}}$ refer to Section 2.8.

Variant | ${\mathit{O}}_{\mathbf{sys}}$ | ${\mathit{O}}_{+}$ |
---|---|---|

Degrees of freedom | Number of UPFCs ${n}_{\mathrm{upfcs}}$ | Number of UPFCs ${n}_{\mathrm{upfcs}}$ |

Sizing $\left|{\tilde{S}}_{\mathrm{se}}\right|$ | Sizing $\left|{\tilde{S}}_{\mathrm{se}}\right|$ | |

Placement: Only congested lines ${n}_{\mathrm{pos}}^{\mathrm{sys}}$ | Placement: All ${n}_{\mathrm{pos}}$ lines | |

Control parameters r, $\gamma $ | Control parameters r, $\gamma $ | |

Constraints | $r={r}_{\mathrm{max}}$ | $r\in [0,{r}_{\mathrm{max}}]$ |

$\delta +\gamma \in \{90\xb0,270\xb0\}$ | $\gamma \in [0\xb0,360\xb0]$ | |

${n}_{\mathrm{upfcs}}\le {n}_{\mathrm{pos}}^{\mathrm{sys}}$ | ${n}_{\mathrm{upfcs}}\le {n}_{\mathrm{pos}}$ | |

$\left|{\tilde{S}}_{\mathrm{se}}\right|\in \left\{1,2,\dots ,100\right\}\mathrm{MVA}$ | $\left|{\tilde{S}}_{\mathrm{se}}\right|\in \left\{1,2,\dots ,100\right\}\mathrm{MVA}$ |

Variables | Probability p | Variables Change with Increment $\mathbf{\Delta}$ |
---|---|---|

${n}_{\mathrm{upfcs}}$ | ${p}_{\mathrm{upfcs}}$ | ${n}_{\mathrm{upfcs}}^{\prime}={n}_{\mathrm{upfcs}}+{\Delta}_{\mathrm{upfcs}}H$ |

${S}_{i}$ | ${p}_{S}$ | ${S}_{i}^{\prime}={S}_{i}+{\Delta}_{S}H$ |

${r}_{i}$ | ${p}_{\mathrm{r}}$ | ${r}_{i}^{\prime}={r}_{i}+{\Delta}_{\mathrm{r}}H$ |

${\gamma}_{i}$ | ${p}_{\gamma}$ | ${\gamma}_{i}^{\prime}={\gamma}_{i}+{\Delta}_{\gamma}H$ |

Parameter | Change |
---|---|

${\Delta}_{\mathrm{upfcs}}$ | 1 UPFC |

${\Delta}_{S}$ | 5 $\mathrm{MVA}$ |

${\Delta}_{r}$ | 0.01 |

${\Delta}_{\gamma}$ | 30° |

${p}_{\mathrm{upfcs}}$ | 0.3 |

${p}_{S}$ | 0.4 |

${p}_{r}$ | 0.6 |

${p}_{\gamma}$ | 0.8 |

**Table 4.**The four best results from simulations where costs are considered (${F}_{\mathrm{REF}}=642.65$).

UPFC Config. | $\mathbf{PT}\left({\mathit{O}}_{+}\right)$ | $\mathbf{GrA}\left({\mathit{O}}_{+}\right)$ | $\mathbf{GrA}\left({\mathit{O}}_{\mathbf{sys}}\right)$ |
---|---|---|---|

best | $F=53.73$ | $F=47.98$ | $F=23.94$ |

${F}_{\mathrm{vio}}=9.93$ | ${F}_{\mathrm{vio}}=8.36$ | ${F}_{\mathrm{vio}}=8.73$ | |

second best | $F=55.77$ | $F=51.00$ | $F=24.31$ |

${F}_{\mathrm{vio}}=17.62$ | ${F}_{\mathrm{vio}}=7.73$ | ${F}_{\mathrm{vio}}=8.71$ | |

third best | $F=62.74$ | $F=62.12$ | $F=24.67$ |

${F}_{\mathrm{vio}}=10.83$ | ${F}_{\mathrm{vio}}=16.76$ | ${F}_{\mathrm{vio}}=8.72$ | |

fourth best | $F=63.92$ | $F=64.38$ | $F=24.89$ |

${F}_{\mathrm{vio}}=19.02$ | ${F}_{\mathrm{vio}}=21.69$ | ${F}_{\mathrm{vio}}=8.74$ |

Line Loading | Reference Case | With UPFC Installation |
---|---|---|

>100% | 14 lines | 1 lines |

80–100% | 9 lines | 19 lines |

70–80% | 6 lines | 9 lines |

0–70% | 112 lines | 112 lines |

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## Share and Cite

**MDPI and ACS Style**

Bodenstein, M.; Liere-Netheler, I.; Schuldt, F.; von Maydell, K.; Hartmann, A.K.; Agert, C.
Optimized Power Flow Control to Minimize Congestion in a Modern Power System. *Energies* **2023**, *16*, 4594.
https://doi.org/10.3390/en16124594

**AMA Style**

Bodenstein M, Liere-Netheler I, Schuldt F, von Maydell K, Hartmann AK, Agert C.
Optimized Power Flow Control to Minimize Congestion in a Modern Power System. *Energies*. 2023; 16(12):4594.
https://doi.org/10.3390/en16124594

**Chicago/Turabian Style**

Bodenstein, Max, Ingo Liere-Netheler, Frank Schuldt, Karsten von Maydell, Alexander K. Hartmann, and Carsten Agert.
2023. "Optimized Power Flow Control to Minimize Congestion in a Modern Power System" *Energies* 16, no. 12: 4594.
https://doi.org/10.3390/en16124594