# Pre-Selection of the Optimal Sitting of Phase-Shifting Transformers Based on an Optimization Problem Solved within a Coordinated Cross-Border Congestion Management Process

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## Abstract

**:**

## 1. Introduction

- costly actions—shifting the operating points of generating units located in affected areas, reducing demand with DSR (demand side response) or, if other means fail, shedding load (which results in non-zero Energy Not Served (ENS)),
- non-costly actions—switching taps of phase shifting transformers (PSTs) or topology switching (turning selected power system elements on or off by TSOs).

- redispatch—to characterize the shift in generation of thermal units,
- RES curtailment—to denote the reduction of the infeed of renewable energy sources.

## 2. Literature Review

## 3. Theoretical Description—Coordinated Cross-Border Congestion Management Model

#### 3.1. Theoretical Description of the Congestion Management Model

- N-0 state—the case when no outage occurs in the system,
- N-1 state—the case representing the grid with an outage of one transmission line, which is referred to as the critical outage (CO).

- Candidate CBCOs were identified with COs selected out of the XB connections and their nearest neighbors—intra-zonal lines terminating at the border stations.
- The LODFs for all candidate CBCO pairs were calculated.
- The final CBCOs were identified as the ones for which the absolute value LODF was greater than the threshold selected as 5%.

- ${T}_{i}^{+}$ is the power shift up of thermal generator i;
- ${T}_{i}^{-}$ is the power shift down of thermal generator i;
- ${R}_{i}^{-}$ is the curtailed power of RES generator i;
- ${E}_{i}^{+}$ is the variable representing the energy curtailment of the demand or Energy Not Served per demand in bus i;
- ${S}_{i}$ is the variable representing the tap setting of PST i.

- ${N}_{T}$ is the number of thermal generators in the system;
- ${N}_{E}$ is the number of loads in the system;
- ${N}_{R}$ is the number of RES generators in the system;
- ${C}_{{T}_{i}}^{+}$ is the cost of regulating up thermal generator i;
- ${B}_{{T}_{i}}^{-}$ is the revenue from regulating down thermal generator i;
- ${C}_{\mathrm{curt}}$ is the penalty cost for curtailment of RES;
- ${C}_{\mathrm{VOLL}}$ is the penalty cost for the energy curtailment.

- ${F}_{\mathrm{CBCO}}^{max}$ is the power capacity limit for transmission line CB;
- ${F}_{\mathrm{CBCO}}^{0}$ is the initial power flow over line CB in outage state of CO;
- ${T}_{i}^{0}$ is the initial generation point in power of thermal generator i;
- ${T}_{i}^{max}$, ${T}_{i}^{min}$ are, respectively, the maximal and minimal operating power limit of thermal generator i;
- ${R}_{i}^{0}$ is the initial generation point in power of RES generator i;
- ${N}_{S}$ is the number of PSTs in the system;
- ${S}_{i}^{0}$ is the initial tap setting of PST i;
- ${S}_{i}^{max}$, ${S}_{i}^{min}$ are, respectively, the maximal and minimal tap setting of PST i;
- $\mathrm{Balance}\left(\mathbf{V}\right)=0$ is a set of nodal balance equations;
- $\Delta {F}_{\mathrm{CBCO}}\left(\mathbf{V}\right)$ is the set of the flow equations.

#### 3.2. Modified DC Power Flow Formulation

#### 3.3. PSDF and PTDF Formulation

## 4. Theoretical Description—Methods for Pre-Selecting the Candidates of the PST Investments

#### 4.1. Multiplier Indicator (MI)

- A possibility of changing the phase angle of each branch in the model was added—each branch is allowed to introduce a phase shift ${L}_{i}$ like a PST. In particular, the new power flow equations now take the following form:$$\begin{array}{cc}\hfill {\overline{\Delta F}}_{\mathrm{CBCO}}\left({\mathbf{V}}_{\mathbf{L}}\right)& =\Delta {F}_{\mathrm{CBCO}}\left({\mathbf{V}}^{\U0001f7c9}\right)+\hfill \\ \hfill & +\sum _{i=1}^{{N}_{L}}{\mathrm{PSDF}}_{\mathrm{CBCO}}^{{L}_{i}}\xb7{A}_{{L}_{i}}\xb7{L}_{i},\hfill \end{array}$$
- ${\mathrm{PSDF}}_{\mathrm{CBCO}}^{{L}_{i}}$ is the PSDF coefficient of flow over CBCO with respect to phase angle ${L}_{i}$ of branch i (acting like a PST),
- ${A}_{{L}_{i}}$ is the angle per tap sensitivity assigned to branch i—the change of the phase angle resulting from shifting one tap (a reference value of ${A}_{{L}_{i}}$, based on the values for existing PSTs, is used for the “candidates”)
- ${N}_{L}$ is the number of branches in the grid,
- ${S}_{i}^{\U0001f7c9}$ are the optimal tap settings of existing PSTs, obtained in the cross-border congestion management optimal solution for the given grid scenario,
- ${\mathbf{V}}^{\U0001f7c9}=\{{\mathbf{T}}^{+}-{\mathbf{T}}^{-},{\mathbf{R}}^{-},{\mathbf{E}}^{+},{\mathbf{S}}^{\U0001f7c9}\}$ is the set of vector variables from Equation (2) with the optimal PST tap settings,
- ${\mathbf{V}}_{\mathbf{L}}=\{{\mathbf{T}}^{+}-{\mathbf{T}}^{-},{\mathbf{R}}^{-},{\mathbf{E}}^{+},\mathbf{L}\}$ is the modified set of vector variables.

- The new phase shift ${L}_{i}$ is set to a parameter ${\alpha}_{i}$, by additional constraints:$${\forall}_{i\in \{1,\dots ,{N}_{L}\}}{L}_{i}={\alpha}_{i}.$$
- The $\mathbf{L}$ variables are defined as continuous, which makes the optimization problem continuous (not MILP) and thus the Lagrange multipliers of each constraint can be obtained.

- ${\mu}_{\mathrm{CBCO}}^{+}$ and ${\mu}_{\mathrm{CBCO}}^{-}$—assigned to constraints limiting power flows for CBCOs (in both directions),
- ${\lambda}_{i}$—assigned to constraints setting the phase angles equal to ${\alpha}_{i}$.

#### 4.2. Congestion Factor (CF)

- ${\mathrm{TSDF}}_{\mathrm{CBCO}}^{{L}_{i}}={\mathrm{PSDF}}_{\mathrm{CBCO}}^{{L}_{i}}\xb7{A}_{{L}_{i}}$ is the so-called Tap-Shift Distribution Factor (TSDF) joining the PSDF and the angle per tap sensitivity of the branch i,
- ${\mathrm{CV}}_{\mathrm{CBCO}}^{\mathrm{scen}}$ is the congestion volume for a CBCO in grid scenario $\mathrm{scen}$, which is the value of power flow over the CB line that exceeds its thermal limit, defined by Equation (20) with:
- –
- ${\mathbf{V}}_{\mathbf{0}}^{\U0001f7c9}=\{\mathbf{0},\mathbf{0},\mathbf{0},{\mathbf{S}}^{\U0001f7c9}\}$—the set of vector variables in the optimal point including zero-valued vectors,
- –
- ${\mathbf{S}}^{\U0001f7c9}$—the vector of optimal tap settings of the existing PSTs for the scenario $\mathrm{scen}$.

#### 4.3. Combined Methodology for Pre-Selecting the PST Candidates

## 5. EU-SysFlex Scenarios

- with high detailed resolution: Poland, Germany, Czech Republic, Slovakia,
- with medium detailed resolution: Austria, Hungary, Ukraine,
- an equivalent representation of other European countries connected to the synchronous grid.

## 6. Results

#### 6.1. Congestion Management Model

#### 6.2. Long-Term Analysis—Candidate Selection for PST Investments

#### 6.2.1. Multiplier Indicator Method

#### 6.2.2. Congestion Factor Method

#### 6.2.3. Summary of the Validation Results

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Data Consideration for the Generation Cost

## Appendix B. Possible Extensions of the Multiplier INdicator

## Appendix C. Possible Extensions of the Congestion Factor

## Appendix D. Combination of PST Candidates—Horizontal Congestion Factor

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**Figure 1.**Congestions on critical branch–critical outages (CBCOs) per borders. The width of the orange line is proportional to the product of average congestion severity (when the congestion is non-zero) and the total number of congested CBCOs per border across 24 grid scenarios. Source: own calculations.

**Figure 2.**Top 6 locations according to the top 9 MI values (cf. Table 3). The volume of the circle at location is proportional to the MI aggregated for branches in the same location. Source: own calculations.

**Figure 3.**Top 7 locations according to the top 8 CF values (cf. Table 4). The volume of the circle at location is proportional to the CF aggregated for branches in the same location. Source: own calculations.

**Table 1.**Average congestion severity (when the congestion is non-zero) and total number of congested CBCOs per border across 24 grid scenarios. Source: own calculations.

Border | Average Congestion [MW] | Number of Congestions |
---|---|---|

DE-AT | 35.61 | 25 |

PL-CZ | 129.16 | 26 |

CZ-SK | 48.48 | 35 |

SK-UA | 75.26 | 12 |

PL-DE | 140.06 | 24 |

DE-CZ | 99.14 | 5 |

SK-HU | 42.71 | 6 |

**Table 2.**Comparison of congestion management costs aggregated over 24 grid scenarios, maximal congestion management costs across 24 grid scenarios, and total redispatch volumes across 24 grid scenarios with and without the use of phase-shifting transformers (PSTs). Source: own calculations.

PST Usage | Total Congestion Management Cost [EUR] | Maximal Congestion Management Cost [EUR] | Total Volume of Redispatching [MW] |
---|---|---|---|

Used | 12,579.28 | 2561.32 | 3573.09 |

Not used | 134,023.58 | 58,598.78 | 16,435.82 |

**Table 3.**Top 9 (cut-off at multiplier indicator (MI) = 600) branches according to the multiplier indicators and congestion management costs, aggregated over 24 grid scenarios, obtained after placing a new PST at a given branch. (The letters A–I in parentheses serve as identifiers of locations to be compared with congestion factor (CF) results in Table 4 below). Source: own calculations.

Branch Location | MI | Congestion Management Costs [EUR] |
---|---|---|

line (A) on DE-AT border | 3045.15 | 394 |

DE station (B) next to DE-AT border | 1285.90 | 808 |

DE transformer (C) next to DE-AT border | 916.84 | 490 |

DE transformer (D) next to DE-AT border | 866.77 | 641 |

line (E) on DE-AT border | 695.62 | 5391 |

line (F) on DE-AT border | 690.63 | 5422 |

DE line (G) next to DE-AT border | 651.30 | 2007 |

line (H) on PL-CZ border | 629.70 | 9407 |

DE line (I) next to DE-AT border | 610.58 | 2335 |

**Table 4.**Top 8 (cut-off at CF = 5000) branches according to the congestion factors and congestion management costs, aggregated over 24 grid scenarios, obtained after placing a new PST at a given branch. (The letters in parentheses serve as identifiers of locations to be compared with MI results in Table 3 above). Source: own calculations.

Branch Location | CF | Congestion Management Costs [EUR] |
---|---|---|

line (A) on DE-AT border | 21,441.05 | 394 |

DE station (B) next to DE-AT border | 11,078.74 | 808 |

line (H) on PL-CZ border | 7436.30 | 9407 |

line (J) on CZ-SK border | 6343.77 | 9569 |

CZ station (K) next to PL-CZ border | 6142.07 | 10,474 |

PL station (L) next to PL-CZ border | 5908.53 | 9454 |

DE transformer (C) next to DE-AT border | 5319.99 | 490 |

line (M) on DE-CZ border | 5163.14 | 3546 |

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**MDPI and ACS Style**

Urresti-Padrón, E.; Jakubek, M.; Jaworski, W.; Kłos, M.
Pre-Selection of the Optimal Sitting of Phase-Shifting Transformers Based on an Optimization Problem Solved within a Coordinated Cross-Border Congestion Management Process. *Energies* **2020**, *13*, 3748.
https://doi.org/10.3390/en13143748

**AMA Style**

Urresti-Padrón E, Jakubek M, Jaworski W, Kłos M.
Pre-Selection of the Optimal Sitting of Phase-Shifting Transformers Based on an Optimization Problem Solved within a Coordinated Cross-Border Congestion Management Process. *Energies*. 2020; 13(14):3748.
https://doi.org/10.3390/en13143748

**Chicago/Turabian Style**

Urresti-Padrón, Endika, Marcin Jakubek, Wojciech Jaworski, and Michał Kłos.
2020. "Pre-Selection of the Optimal Sitting of Phase-Shifting Transformers Based on an Optimization Problem Solved within a Coordinated Cross-Border Congestion Management Process" *Energies* 13, no. 14: 3748.
https://doi.org/10.3390/en13143748