# Flow Allocation in Meshed AC-DC Electricity Grids

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## Abstract

**:**

## 1. Introduction

- (a)
- (b)
- Z-bus transmission allocation presented by Conejo et al. [4], which is equivalent to the Power Divider method [5] and strongly related to the formulation by Chang and Lu [6]. It derives the contributions of electricity current injections to the branch currents based on the full AC power flow equations.
- (c)
- (d)
- With-And-Without transit loss allocation presented by Hadush et al. [9], which builds the underlying loss allocation for the Inter-Transmission System Operators Compensation (ITC) mechanism. In contrast to the other methods, it does not determine source-sink relations but calculates losses within regions or countries caused by cross-border flows.

## 2. PTDF-Based Flow Allocation Methods

- Nodal active power $\mathbf{p}\in {\mathbb{R}}^{N}$
- Active power flow $\underline{\mathbf{f}}\in {\mathbb{R}}^{L}$
- Transmission reactance $\underline{\mathbf{x}}\in {\mathbb{R}}^{L}$
- Transmission resistance $\underline{\mathbf{r}}\in {\mathbb{R}}^{L}$
- Transmission admittance $\underline{\mathbf{y}}\in {\mathbb{C}}^{L}$
- Incidence matrix $\mathbf{K}\in {\mathbb{R}}^{N\times L}$
- PTDF matrix $\mathbf{H}\in {\mathbb{R}}^{L\times N}$
- Cycle matrix $\mathbf{C}\in {\mathbb{R}}^{L\times C}$
- Virtual Injection Pattern $\tilde{\mathbf{P}}\in {\mathbb{R}}^{N\times N}$
- Virtual Flow Pattern $\tilde{\mathbf{F}}\in {\mathbb{R}}^{L\times N}$
- Peer-to-Peer Allocations $\mathbf{A}\in {\mathbb{R}}^{N\times N}$

#### 2.1. Equivalent Bilateral Exchanges

#### 2.2. Marginal Participation

## 3. Including Controllable Elements in the PTDF Formulation

#### 3.1. Controllable Elements in Cycles (Case 1)

#### 3.2. Controllable Elements in Tree Networks (Case 2)

## 4. Flow Allocation across European Synchronous Zones

- a flow on a line stays within the bounds of today’s line capacities, or
- a flow exceeds the original transmission capacity and thus makes use of the 18% transmission expansion.

## 5. Summary and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Country Code Lookup

Country Code | Country Name |

AL | Albania |

AT | Austria |

BA | Bosnia and Herzegovina |

BE | Belgium |

BG | Bulgaria |

CH | Switzerland |

CZ | Czechia |

DE | Germany |

DK | Denmark |

EE | Estonia |

ES | Spain |

FI | Finland |

FR | France |

GB | United Kingdom |

GR | Greece |

HR | Croatia |

HU | Hungary |

IE | Ireland |

IT | Italy |

LT | Lithuania |

LU | Luxembourg |

LV | Latvia |

ME | Montenegro |

MK | North Macedonia |

NL | Netherlands |

NO | Norway |

PL | Poland |

PT | Portugal |

RO | Romania |

RS | Serbia |

SE | Sweden |

SI | Slovenia |

SK | Slovakia |

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**Figure 2.**Example of a pure tree network with both AC and DC lines (Case 2). In the PTDF matrix, the pseudo-impedance values for the DC lines are trivially equal to 1 for the given flow pattern.

**Figure 3.**The different synchronous zones of the European power system, as indicated by the different colors. Whereas the Continental European grid is the largest subnetwork, Ireland, the United Kingdom, Scandinavia (with only parts of Denmark), and the Baltic region have their own synchronous zones. These areas are interconnected via DC lines (

**dark green**).

**Figure 4.**Highly renewable PyPSA-EUR network with 181 nodes, 325 AC lines, and 48 controllable DC links. Two scenarios are investigated, one without network expansion and one with an 18% expansion relative to today’s total transmission volume.

**Figure 5.**Average interconnecting flow between the 12 strongest exchanging countries. Remaining countries are grouped into “Other.” A lookup table for the full country names is provided in Appendix A. These aggregated source-sink relations count for both MP and EBEs. The flow allocation leads to broad connections between countries that are geographically far apart, and neighboring countries reveal the strongest interconnections. (

**a**) shows the full allocation, and (

**b**) allocates the flow induced by wind power only. This allocation accounts to 69% of the full cross border flow. Countries along the North Sea coast where most of the wind production is situated, dominate the allocation. Prominent differences to the total flow allocation can be found in Spain, which mainly exports solar power. (

**a**) Full peer-to-peer allocation. (

**b**) Peer-to-peer allocation induced by wind power.

**Figure 6.**Country-wise flow allocation using EBEs (

**a**) and MP (

**b**). The flow allocation per country is split into two parts, one for flows that make use of transmission expansion and one for flows that stay within the original capacity bounds. The lookup table for the full country names is given in Appendix A. Both methods (Figure 6a) state that Great Britain is the strongest user of the transmission grid, despite not being the strongest trader of power in the renewable network simulation, which can be seen in Figure 5.

**Figure 7.**Total allocated flow of an exemplary snapshot in the network for both allocation methods, MP and EBEs, as a function of the shift parameter q. Whereas $q=0$ and $q=1$ result in the same allocation, the MP algorithm allocates more counter-flows as q further approximates to $q=0.5$ (the standard MP setup), which is due to a strong increase of counter flows.

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Hofmann, F.; Schlott, M.; Kies, A.; Stöcker, H.
Flow Allocation in Meshed AC-DC Electricity Grids. *Energies* **2020**, *13*, 1233.
https://doi.org/10.3390/en13051233

**AMA Style**

Hofmann F, Schlott M, Kies A, Stöcker H.
Flow Allocation in Meshed AC-DC Electricity Grids. *Energies*. 2020; 13(5):1233.
https://doi.org/10.3390/en13051233

**Chicago/Turabian Style**

Hofmann, Fabian, Markus Schlott, Alexander Kies, and Horst Stöcker.
2020. "Flow Allocation in Meshed AC-DC Electricity Grids" *Energies* 13, no. 5: 1233.
https://doi.org/10.3390/en13051233