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Article

Flow Allocation in Meshed AC-DC Electricity Grids

Fabian Hofmann 1,†,‡ , Markus Schlott 1,‡ , Alexander Kies 1,* and Horst Stöcker 1, *

1Frankfurt Institute for Advanced Studies (FIAS)

*Correspondence: hofmann@ﬁas.uni-frankfurt.de

† This paper is an extended version of our paper published in Sdewes 2019.

Abstract:

In power systems, ﬂow allocation (FA) methods allow to allocate usage and costs of the

transmission grid to each single market participant. Based on predeﬁned assumptions, the power ﬂow

is split into isolated generator speciﬁc or producer speciﬁc sub-ﬂows. Two prominent FA methods,

Marginal Participation (MP) and Equivalent Bilateral Exchanges (EBE), build upon the linearized

power ﬂow and thus on the Power Transfer Distribution Factors (PTDF). Despite their intuitive and

computationally efﬁcient concept, they are restricted to networks with passive transmission elements

only. As soon as a signiﬁcant number of controllable transmission elements, such as High-voltage direct

current (HVDC) lines, operate in the system, they loose their applicability. This work reformulates the

two methods in terms of Virtual Injection Patters (VIP) which allows to efﬁciently introduce a shift

parameter

q

, tuning contributions of net sources and net sinks in the network. Major properties and

differences of the methods are pointed out. Finally, it is shown how the MA and EBE algorithm can be

applied to generic meshed AC-DC electricity grids: Introducing a pseudo-impedance

¯ω

which reﬂects

the operational state of controllable elements, allows to extend the PTDF matrix under the assumption

of knowing the current system’s ﬂow. Basic properties from graph theory are used for solving the

pseudo-impedance dependent on the position in the network. This directly enables e.g. HVDC lines

to be considered in the MP and EBE algorithm. The extended methods are applied to a low-carbon

European network model (PyPSA-EUR) with a spatial resolution of N=181 and an 18% transmission

expansion. The allocations of VIP and MP, show that countries with high wind potentials proﬁt

most from the transmission grid expansion. Based on the average usage of transmission system

expansion a method of distributing operational and capital expenditures is proposed. Further it is

shown, how injections from renewable resources strongly drive country-to-country allocations and

thus cross-border electricity ﬂows.

Keywords:

Power System Analysis; Flow Allocation; Transmission Cost Allocation; European

Electricity Grid

1. Introduction

The shift from conventional to renewable power sources requires high investments not only on

the generation side but also on the transmission and storage side of a power system. Due to the

dominant dependence on ﬂuctuating wind and solar power potentials, energy has to be shifted in

space and time. For large networks, as the European power system, both elements will play a key

role. Spatial balancing, via a solid transmission grid, will allow power to cover long distances from

wind farms far from load centers. In contrast, temporal balancing allows more self-sufﬁcient areas

which locally produce and store (most likely solar) power. This raises the question of who uses and

proﬁts to what extent from the transmission grid and its upcoming expansions. Flow allocation (FA)

methods allow to efﬁciently quantify the transmission usage per market-participant by decomposing

the network ﬂow into sub-ﬂows driven by isolated power injections. On one hand this opens the

opportunity to distribute transmission costs based on the effective transmission usage of each single

generator and consumer, as broadly reviewed in [

1

]. On the other hand it helps to understand the

operational state of the system and to determine cost-extensive and cost-reducing actions, which helps

to draw up incentives or cost-schemes for an efﬁcient system transformation.

Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 29 January 2020 doi:10.20944/preprints202001.0352.v1

© 2020 by the author(s). Distributed under a Creative Commons CC BY license.

2 of 14

There exist multiple ﬂow allocation methods all differently approaching the determination of

isolated sub-ﬂows. The most prominent candidates happen to be:

(a)

Average Participation, also named Flow Tracing, ﬁrstly presented in [

2

] and used in various

application cases as in [

3

]. It follows the principle of proportional sharing when tracing a power

ﬂow from source to sink

(b) Z

-bus transmission allocation presented in [

4

] which is equivalent to the Power Divider method

[

5

] and very related to the formulation in [

6

]. It derives the contributions of electricity current

injections to the branch currents based on the full AC power ﬂow equations.

(c)

Marginal Participation (MP) presented in [

7

] and (d) Equivalent Bilateral Exchanges (EBE)

method [

8

] which are based on the linearized power ﬂow equations and extensively explained

later in this paper.

(e)

With-And-Without transits loss allocation presented in [

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.

Non ﬂow-based cost allocation includes another pallette of methods, such as a ’Aumann-Shapley’

method [

10

] which is based on Game Theory or an exogenous approach [

11

] which proposes to

introduce a peer-to-peer market design into the optimal power ﬂow (OPF) calculation. Originally, the

FA methods focus on determining the ﬂow shares on branches. However the work in [

12

] shows that

the FA can be alternatively represented through Virtual Injection Patterns (VIP), that are peer-to-peer

allocations between sources and sinks. Thus, every market generator is associated to a speciﬁc set

of supplied loads and, vice versa, loads are allocated to a speciﬁc set of power suppliers from which

they retrieve their share. The artiﬁcial peer-to-peer transactions can then be used to e.g determine

nodal electricity prices or the CO

2

-intensity of the consumed power, as done in a recent study [

13

].

As all FA methods come along with strengths and weaknesses, it turns out that most of them are

restricted to pure AC or pure passive DC transmission networks only. This applies to non-linear FA as

well as MP and EBE, which rely on the linear Power Transfer Distribution Factors (PTDF). It makes

them inappropriate for large networks, likewise the European one, which consists of multiple AC

subnetworks, i.e. synchronous zones operating at a speciﬁc utility frequency. So far the PTDF-based

allocation is not applicable over borders of these zones and only conceptional propositions have been

made to tackle this issue [

14

]. Note that also FA on distribution network level are inappropriate for

the MP and EBE algorithm, as the characteristic of high resistance-reactance ratios render the linear

approximation of the power ﬂow invalid. This restricts the scope of application for the MP and EBE

algorithms considerably. However, in the following it is shown, how the two algorithms can be applied

to meshed AC-DC transmission networks by incorporating controllable elements as HVDC lines into

the PTDF matrix.

The paper is structured as follows: Sec. 2recalls the MP and EBE algorithms, focusing on both VIP

and ﬂow share formulation. Sec. 3describes the extension of the ﬂow allocation methods for generic

AC-DC networks, realized through the introduction of the so-called pseudo-reactance

¯ω

. In Sec. 4the

extended FA is performed for a highly renewable European power system and Sec. 5summarizes and

concludes the paper.

Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 29 January 2020 doi:10.20944/preprints202001.0352.v1

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2. PTDF Based Flow Allocation Methods

The Marginal Participation (MP) and Equivalent Bilateral Exchanges (EBE) algorithms are both

based on the linear power ﬂow approximation. In order to recall and extent them, let the following

quantities be deﬁned throughout this work

Nodal active power p∈RNPTDF matrix H∈RL×N

Active power ﬂow f∈RLCycle matrix C∈RL×C

Transmission reactance x∈RLVirtual Injection Pattern ˜

P∈RN×N

Transmission resistance r∈RLVirtual Flow Pattern ˜

F∈RL×N

Transmission admittance y∈CLPeer-to-Peer Allocations A∈RN×N

Incidence matrix K∈RN×L

where N denotes the number of buses in the system, L the number of branches (transmission

lines) and C the number of cycles in the network. Note that the equality C = L - N + 1 holds as shown

by Ronellenﬁtsch et al. [

15

,

16

]. The linear power ﬂow approximation assumes that all bus voltage

magnitudes are equal,

ˆvi=ˆvj

, and the series resistances are small compared to series reactances,

rlxl

. Further, voltage angle differences across a line are assumed to be small and no shunt

admittances (at buses or series) to ground are present. These assumptions are usually appropriate for

transmission systems, where high voltages and small resistances lead to a power ﬂow which is mostly

driven by active power injection

p

. They allow to map the latter linearly to the active power ﬂow

f

through the PTDF matrix

f=H p (1)

where the PTDF matrix is deﬁned as

H=diag yKTKdiag yKT+(2)

with the series admittance

y=x−1

and

()+

denoting the generalized inverse. We recall that the PTDF

matrix can be provided with a slack, that is one or more buses which ’absorb’ total power imbalances

of

p

. In formulation

(2)

the slack is distributed over all nodes in the system, however it is possible to

modify it by adding a column vector kto the PTDF matrix

H→H+k(3)

without touching the result of the linear power ﬂow equation

(1)

for a balanced injection pattern.

Corresponding to equation (1), the nodal power balance is expressed by

p=K f (4)

As pointed out earlier, the FA methods can be perceived from two different angles. One is looking at

the impact of generators or consumers on the network ﬂow they cause in the network respectively. The

Virtual Flow Pattern (VFP) matrix

˜

F

is a L

×

N matrix of which the

n

th column denotes the sub-ﬂow

induced by bus

n

. The other is looking at the peer-to-peer transactions given by the Virtual Injection

Pattern matrix

˜

P

of size N

×

N of which the

n

th column denotes the effective, balanced injection

pattern of bus

n

. The two quantities contain the same information and are, similarly to equations

(1)

and (4), linked through

˜

F=H˜

P(5)

Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 29 January 2020 doi:10.20944/preprints202001.0352.v1

4 of 14

and

˜

P=K˜

F(6)

Further the sum of their columns equals the original ﬂow

f=˜

F 1N(7)

and the original injection pattern respectively

p=˜

P 1N(8)

where

1N

represents a vector of ones of length N. Further let

A

denotes a

N×N

matrix with

peer-to-peer transactions

m→n

given by the entry

Amn

. For a given VIP matrix, those values

are straightforwardly obtained by

˜

Pmn −˜

Pnm

, that is taking all power in the injection pattern of

n

coming from

m

to

n

and all the negative power in the injection pattern of

n

coming from

m

. In matrix

notation this leads to

A=˜

P−˜

PT+(9)

where

()+

denotes to restrict to positive entries only. The latter should be taken for granted as only

positive source to sink relations are considered. Thus, for non-zero values

Amn

, bus

m

is always a net

producer and bus na net consumer.

Equivalent Bilateral Exchanges

The deﬁnition of the EBE algorithm refers to peer-to-peer exchanges between net producers

(sources) and net consumers (sinks) in the network [

8

]. It assumes that a source provides all sinks in the

network which are proportionally scaled down while all other sources are ignored. Correspondingly,

it assumes that power ﬂowing into a sink comes from all sources which are proportionally scaled

down while all other sinks are ignored. In order to allow weighting the net producers differently

from net consumers, let 0

≤q≤

1 be a shift parameter which allows tuning their contributions (the

lower, the stronger net consumers are taken into account). Further, let

p+/−

denote injections by

only sources/sinks and let

γ=pT

+1(N)−1

denote the inverse of the total positive injected power.

Therefore the VIP is given by

˜

Pebe =qP++γp−pT

++ (1−q)P−−γp+pT

−(10)

The ﬁrst term represents injection patterns of net producers only which deliver power to sinks. The

second term comprises injection patterns of all net consumers. The corresponding VFP matrix ˜

Febe is

given by

˜

Febe(q) = H˜

Pebe(q)(11)

If

q

is set to 0.5, 50% of the ﬂow is allocated to net producers and 50% to net consumers. Inserting

(10)

into (11) allows to write the VFP in form of

˜

Febe =H P (12)

with a given slack of

k=γf−

for net producers and

k=−γf+

for net consumers in the PTDF matrix

(see eq. (3)). It is taken as granted that the separate ﬂow patterns induced by the market participants

5 of 14

can be of completely different nature than the original ﬂow. Therefore the FA method may leads to

counter-ﬂows which are counter aligned to the network ﬂow f.

Given that

p=p++p−

, the peer-to-peer relations are straightforwardly obtained by ﬁrst,

reformulating (10) to

˜

Pebe =P−−γp+pT

−+q|P|+γp−pT

++γp+pT

−

| {z }

symmetric

(13)

and second, inserting the new expression into

(9)

(

|·|

denotes the absolute value). Note, that the

symmetric term cancels out, which makes Aindependent of q, and ﬁnally boils it down to

Aebe =−γp+pT

−(14)

which one-on-one reﬂects the deﬁnition of the EBE ﬂow allocation.

Marginal Participation

The MP algorithm, in contrast to EBE, comes from a sensitivity analyzing perspective which

directly deﬁnes the VFP matrix. As originally proposed, it measures each line’s active power ﬂow

sensitivity against changes in the power balances of the buses. The sensitivity characteristics are then

multiplied with the nodal power imbalances, which gives the sub-ﬂows induced by the buses. As

the work in [

14

] describes in detail, the choice of the slack

k

is used for tuning contributions of net

producers and net consumers. However, aiming at introducing the shift parameter qcorrectly and in

an generalized way, we propose straightforwardly to deﬁne the VIP matrix as

˜

Pmp =˜

Pebe +sP+γp−pT

−−γp+pT

+(15)

where

s

is set to

s=1

2−q−1

2

. Here, we used the full VIP pattern for the EBE method and added a

term which only takes effect for 0

<q<

1. This makes

˜

Pmp

and

˜

Pebe

the same for full allocation to

consumers (q=0) or producers (q=1). The standard 50%-50% split leads to

˜

Pmp(q=0.5) = P−1

2γ|p|pT(16)

which, when inserting into (5), leads to an effective slack of

kmp(q=0.5) = 1

2γ(f−−f+)(17)

which reﬂects the standard setup of the MP algorithm as originally proposed. Overall, the newly

introduced formulation given in eq. (15) generalizes the algorithm for the shift parameter

q

and

matches the EBE method for full consumer/producer contributions. Again note that the

˜

FMP

may

contain counter-ﬂows which are not aligned with

f

. Especially in the realm around

q=

0.5 this is

even more likely for the MP method, as ﬂows are also allocated from one net producer to another net

producer. It shall be noted that the second term in eq. (15) is symmetric, which makes the peer-to-peer

allocation again independent of the shift parameter q, which leads to

Amp =−γp+p−=Aebe (18)

So, it can the be concluded that the peer-to-peer relations for both EBE and MP are the same,

whereas the ﬂow allocations and virtual injections patterns differ for mixed producer-consumer

contributions.

6 of 14

3. Including Controllable Elements in the PTDF Formulation

Despite representing the two methods in form of Virtual Injection Patterns, eqs. (10) and (15),

the ﬂow allocation in form of

(5)

is still restricted to the scope of the PTDF matrix, namely to

AC-subnetworks or pure passive DC networks. Up to now, proposals for incorporating HVDC lines

within the MP or EBE [

14

,

17

] are rather of conceptional nature and do not derive all mathematical

details. In the following, it is shown how the PTDF matrix can be reformulated and extended by

introducing a ﬂow-dependent pseudo-impedance

¯ω(f)

for controllable DC lines, which here shall

represent all controllable branch elements. The resulting extended PTDF matrix then solves Equation

(1) for mixed AC-DC networks.

In a network of N nodes and L lines, let there be

LAC

AC lines,

LPDC

passive DC lines and

LCDC

controllable DC lines. As stated in [

15

], a graph can always be decomposed into cycles and trees. Due

to different physical laws, the two cases are treated separately, in terms of a DC line being part of a

cycle in the network (Case 1) or being part of a tree in the network (Case 2).

3.1. Controllable Elements in Cycles (Case 1)

Let

CAC

denote the cycle matrix for all pure AC line cycles and

fAC ∈RLAC

be the ﬂow on all AC

lines, then, according to the linear approximation, the Kirchhoff Voltage Law states that the ﬂows in

every closed cycle weighted by the reactants sum up to zero, namely

CAC diag (x)fAC =0. (19)

As a counterpart, let

CPDC

be the cycle matrix for passive DC lines and

fDC

passive

the ﬂow on those lines.

Then Ohm’s law states that ﬂows in a closed cycle weighted by underlying resistance sum up to zero

CPDC diag (r)fDC

passive =0. (20)

In a network with pure AC subnetworks and pure passive DC subnetworks,

x

and

r

are not overlapping.

Equation

(19)

and

(20)

can thus be combined by using

z=hr xiT

,

z∈RL−LCDC

and collecting the

ﬂows on all passive branches in fpassive =hfAC fDC

passiveiT, which leads to

"CAC

CPDC#diag (z)fpassive =0. (21)

Note that ﬂows on controllable DC lines are not considered in equation

(21)

as those are not given

by physical laws and not a function of

(C,r,p)

. In order to include controllable DC lines, let

Cmixed

denote the cycle matrix with all cycles in which (I) controllable DC lines with nonzero ﬂow appear and

(II) no controllable DC line with zero ﬂow. Note that (II) guarantees that topologically irrelevant cycles

are excluded as controllable DC with zero ﬂow are not affecting the total ﬂow pattern. We introduce a

pseudo-impedance ¯ω∈RLCDC for controllable DC lines, which fulﬁlls equation

Cdiag "z

¯ω#!f=0, (22)

where

C

denotes all cycles in the AC/DC super grid in the form

C=hCAC CPDC CmixediT

and

f

the

full network ﬂow in the form

f=hfpassive fcontrol.iT

. In order to solve equation

(22)

for

¯ω

, we only

7 of 14

consider mixed cycles, as all other cycles are not affected by

¯ω

. We then split the mixed cycle matrix

into two parts, according to Cmixed =hCmixed

passive Cmixed

control.i

0=Cmixed diag "z

¯ω#!f(23)

=hCmixed

passive Cmixed

control.idiag "z

¯ω#!"fpassive

fcontrol.#(24)

=Cmixed

control. diag (¯ω)fcontrol. +Cmixed

passive diag (z)fpassive (25)

=Cmixed

control. diag (fcontrol.)¯ω+Cmixed

passive diag (z)fpassive (26)

In the last step it was used that for any equally shaped vectors

a

and

b

, the relation

diag (a)b=

diag (b)ais valid. Isolating ¯ωin Equation (26) leads to

¯ω=−Cmixed

control. diag (fcontrol.)+Cmixed

passive diag (z)fpassive (27)

Figuratively, the pseudo-impedance stands for the reciprocal contribution of a DC line to the current

ﬂow

f

within the considered cycle. It indicates the impedance a controllable DC line would have, if it

were a passive AC line. Hence, the higher the ﬂow on a DC line the lower its pseudo-impedance.

In Figure 1we give a short example of DC lines embedded in cycles in a network with N = 4, L

AC

= 2

and L

CDC

= 3. The pseudo-impedance in

(28)

directly results from the ﬁrst four quantities, namely the

Incidence Matrix, ﬂow, injection and reactance. The arrow sizes in Figure 1are proportionally set to

their ﬂow.

Figure 1.

Example for pure cycle network with both AC and controllable DC lines (

Case 1

). When

creating a PTDF matrix for such a ﬂow pattern the pseudo-impedance values for the DC lines are given

by equation (27). Relevant corresponding network quantities are given in equation (28).

8 of 14

K=

01101

1−1 0 0 0

−1 0 −1 1 0

000−1−1

,p=

10

−7

−7

4

,"fAC

fDC#=

−3

4

5

−5

1

,x="0.5

0.5#,"z

¯ω#=

2

2

10

10

52

(28)

3.2. Controllable Elements in Tree Networks (Case 2)

If the controllable DC lines are not embedded in cycles, one can consider them as topologically

being a part of a tree network. For such a tree network with L

<

N, the Incidence matrix is non-singular.

Thus, equation

(1)

is straightforwardly derived from Equation

(4)

as

H=K+

is well-deﬁned. Further,

extracting values for the pseudo-impedance becomes trivial as

diag yKTKdiag yKT+=K+(29)

is solved by yDC

control. =1 where y=hypassive ycontrol.iTand thus ¯ω=1.

Again, we show a small example for a tree network of N = 6, L

CDC

= 4 and L

AC

= 1 with given ﬂows

and injections (30) and topology shown in Figure 2.

K=

01000

10000

−1−1 1 0 0

0 0 −1 1 1

000−1 0

0000−1

,p=

8

−7

0

0

−7

6

,"fAC

fDC#=

−7

8

1

7

−6

,x=h0.5i,"z

¯ω#=

2

1

1

1

1

(30)

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 1 for the given ﬂow pattern.

The extended impedance in the form of

hz¯ωiT

can now be inserted in eq. (3) by replacing

y

by

its inverse. The resulting PTDF matrix now solves eq. (1) for a meshed AC-DC network.

9 of 14

4. Flow Allocation across European Synchronous Zones

In the following the formalism presented in sections 2and 3is applied to a highly

renewable European network model in order to extract general sink-source relations and

transmission ﬂow behavior. We show how cross-border ﬂows are mainly driven by wind power

and transmission system usage derived from the MP algorithm is allocated to the countries.

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, United Kingdom,

Scandinavia (with only parts of Denmark) and the Baltic region have

their own synchronous zone. These are interconnected via DC lines

(dark green).

The European power system

comprises ﬁve synchronous zones of

heterogeneous sizes. As displayed in

Figure 3these are the synchronous

grid of Continental Europe (blue),

representing the largest with 24

countries, the North of Europe

(grey), the Baltic States (pink), United

Kingdom (brown) and Ireland (light

green). They are interconnected by

HVDC lines in dark green.

Each synchronous zone

distributes power through AC

lines and, in nominal operational

state, levels out all load within the

subnetwork. The power ﬂow on

the passive AC lines are determined

by the Kirchhoff Current Law and

Kirchhoff Voltage Law and are in

direct relation to the nodal power

injection. Therefore, when the

line loading of passive branches is

getting close to the capacity limits,

Transmission System Operators (TSO)

have to regulate and reduce the

critical power ﬂows by redispatching.

However, with upcoming HVDC

projects realized within the Ten Year

Network Development Plan (TYNDP)

[

18

] more controllable elements will

allow to distribute power more

efﬁciently.

The openly available power system model PyPSA-EUR, presented in [

19

] and available at [

20

],

suits due to it’s realistic topology representing a realistic European meshed AC-DC network. The

model itself is based on reﬁned data of the European transmission system containing all substations

and AC lines at and above 220 kV, all HVDC lines as well as most of today’s conventional generators.

It is accessible via an automated software pipeline, which allows to examine different scenarios, e.g.

by varying transmission network expansion limits, CO

2

caps or coupling of the heat, transport and

electricity sector as done by Brown et al. [

21

]. In order to represent a highly-renewable future scenario,

the network is clustered, simpliﬁed and linearly cost-optimized allowing generator expansion and

18% total transmission capacity expansion. Further, the CO

2

cap is set to 5% of the 1990’s emission

level. Available generation technologies are onshore and offshore wind, solar PV, natural gas and

Run-of-River.

Available storage technologies are pumped-hydro-storage (PHS), hydro dams, batteries and

hydrogen storage. Note that hydro dams (hydro) do not have the ability to store power from the

10 of 14

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 18% expansion

relative to today’s total transmission volume.

electricity grid, but are supplied by natural water inﬂow. All generator and storage types are allowed

to be expanded except for PHS and hydro. Dispatch and expansion were calculated using the linear

power ﬂow approximation, neglecting line losses, and minimizing the total system costs consisting

of capital and operational expenditures of the different network components. The resulting network,

shown in Figure 4, comprises N=181 buses and L=373 lines, of which 48 are controllable DC lines.

The left hand side shows power generation and original transmission capacities, the right side shows

capacity distribution of storages and transmission expansion. The energy production is strongly

relying on wind power, which produces 40% of the yearly total in offshore regions and 18% in onshore

regions. Solar power on the other hand accounts for 23% of the total energy production. The rest is

covered be hydro power (10%), Open Cycle Gas Turbines (5%) and Run-of-River (4%). The average

electricity price lays at 58 e/MWh.

The two FA methods are applied on the whole simulation year, which consists of 2920 time steps

representing a three hour time-resolution. We choose the standard formulation with

q=

0.5, where the

difference between MP and EBE are expected to be the largest. As the analysis of the three dimensional

data

˜

P(t)

and

˜

F(t)

requires detailed reporting, we want to restrict on international power trafﬁcs for

the following.

Figure 5(a) shows the average source-sink allocation, given by

AMP =AEBE

, aggregated to

countries. Colors of the outer circle indicate the overall exchange of the considered country. Colors

of the inner circle and inbound connections represent either exports of the country (same color as

outer circle) or imports from other countries (corresponding colors of other countries in the outer

circle). Self-assigned allocations are connections with the country itself, as one can clearly see for e.g.

Germany. The allocated cross border ﬂows (CBF) are dominated by large exporters and importers

in the system. Therefore, only 12 countries with the largest cross border exchange are represented

separately, whereas the remaining countries are aggregated in ’Other’. In the cost-optimized setup

Germany, France and United Kingdom are the strongest exporters and importers. However, there are

countries having a much higher export-import ratio, as for example Greece or Netherlands. Note

that by deﬁnition of the peer-to-peer relations for both methods eq. (18) does not takes geographical

distance into account thus leads to large-distance exchanges, e.g. Germany-Spain or Finland-Italy.

However, neighboring countries disclose the strongest interconnections. On the one hand this applies

particularly to countries along the North Sea coast where transmission expansion allows strong

interactions. On the other hand, optimizing the cost of capacity expansion and disposition may lead to

more installation and generation in regions near load centers. The FA methods now allow to further

breakdown the cross border ﬂow allocation. Figure 5(b) shows the same setup but only includes power

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(a) Full peer-to-peer allocation (b) Peer-to-peer allocation induced by wind power

Figure 5.

Average interconnecting ﬂow between 12 strongest exchanging countries, remaining countries

are grouped into ’Other’. These aggregated source-sink relations count for both MP and EBE. On one

hand the ﬂow allocation leads to broad connections between countries which are geographically far

apart, on the other hand neighboring countries reveal the strongest interconnections. Where (a) shows

the full allocation, ﬁgure (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

transfers induced by wind power injection. Remarkably, the country-to-country allocation hardly

changes as the overall CBF allocation is mainly driven by countries with strong wind production.

Indeed, CBF induced by wind power covers 69% of the total CBF.

As the peer-to-peer relations are dominated by strong exporters of power, we want to have a look

at the transmission grid usage, which is given by

˜

F

. Especially, the usage of the transmission expansion

might be of current interest in regard to cost allocation of grid expansion projects. Therefore, we split

the ﬂow finto two categories:

•a ﬂow on a line stays within the bounds of today’s line capacities, or

•

a ﬂow exceeds the original transmission capacity, thus makes use of the 18% transmission

expansion

In ﬁgs. 6(a) and 6(b) the average induced transmission for each country is shown. Each bar is split

according to the two categories. First of all it is remarkable how similar EBE and MP correlate on

aggregated country level. Despite strong absolute differences, the relative proportions are almost the

same. The strongest transmission grid users are Great Britain, France, Germany and Spain, followed by

Norway after a large gap. More or less similarly are the usages of transmission expansion distributed.

So, even though Great Britain is not the strongest power exporter or importer, it has the highest average

share of ﬂows in the system. This indicates that due to its topological situation and injection behavior,

its power exports and imports are, in average, of longer spatial distance, which pushes the usage of

the original and expanded transmission grid.

The transmission grid usage strongly anti-correlates with the amount of storage capacity. Thus

countries like Italy which have high solar power shares and strong battery storage capacities, have

proportionally dependence on the transmission grid. Thus, for a simpliﬁed approach of allocating

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(a) Equivalent Bilateral Exchanges

(b) Marginal Participation

Figure 6.

Country-wise ﬂow allocation using Equivalent Bilateral Exchanges (a) and Marginal

Participation (b). The ﬂow allocation per country is split into two parts, one for ﬂows which make use

of transmission expansion and one for ﬂows which stay within the original capacity bounds. Both

methods (ﬁg. 6(a)) state that Great Britain is the strongest user of the transmission grid who however is

not the strongest trader of power in the renewable network simulation, as found Figure 5.

capital and operational expenditures of the transmission grid to countries, one can legitimately propose

to allocate capital expenditures proportional to the use of transmission expansion ﬂow (the upper

parts of the bars) and operational expenditures proportional to the ﬂow allocation within the original

capacity bound (lower parts).

Finally, we close with a short discussion about higher absolute ﬂow contributions in the MP

allocation compared to EBE. As pointed out earlier, with the 50%-50% split the MP algorithm leads to

effective ﬂows from net producer to net producer. This leads to a higher shares of counter ﬂows in

the allocation, which on the other hand have to be balanced according to eq. (7). Figure 7shows the

ratio between the sum of absolute allocated ﬂows and the total transmission as a function of

q

. Indeed,

in the realm 0

<q<

1 the MP method allocates much more ﬂow, peaking at

q

= 0.5 with more than

6 times of the total transmitted power, whereas the EBE method stays steadily at around 3 times of

the total transmission. The two lower lines reﬂect the sum of all allocated counter-ﬂows which, as

to expect, lays 0.5 below the half of the absolute allocation sums. In other words, each counter-ﬂow

cancels out with an aligned ﬂow, and the remaining aligned ﬂows sum up to f.

Note that it can be assumed that this effect scales with the network resolution, as the number of

possible peer-to-peer connections scales with

ON2

. For

q=

0.5,

˜

Pmp

exploits all of these connections,

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Figure 7.

Total allocated ﬂow of an exemplary snapshot in the network for both allocation methods,

Marginal Participation (MP) and Equivalent Bilateral Exchanges (EBE) 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 the more q approximates

q=

0.5 (the standard MP setup), which is due to a strong

increase of counter ﬂows.

which makes the appearance of counter-ﬂows much more likely. However further research needs to be

done to sustain this argument.

5. Summary and Conclusions

A mathematical consistent extension of the Power Transfer Distribution Factors matrix (PTDF)

which incorporates the operational state of controllable elements as high-voltage direct current lines

was presented. By introducing a ﬂow dependent pseudo-impedance vector

¯ω(f)

of the size of

controllable elements in the grid, the PTDF matrix was reformulated for meshed AC-DC networks.

Thereby, it becomes essential to differentiate between the controllable elements being part of a

independent cycle in the network or not, as both cases are affected by a different set of physical

constraints. The extension was propagated to the reformulated and extended Marginal Participation

and Equivalent Bilateral Exchanges algorithm, ﬂow allocation methods which are both based on the

PTDF matrix. Thereby, both algorithms become applicable for meshed AC-DC networks and thus

for the European power system. On the basis of a future scenario model of the European power

system a ﬂow allocation was performed to determine cross border transactions and transmission grid

usage per country. It could be shown that the FA methods can appropriately be used to quantify the

usage of transmission expansion and opens a possible distribution scheme for capital expenditures on

transmission projects.

Author Contributions:

Conceptualization, F.H. and M.S.; methodology, F.H.; software, F.H.; validation, F.H.,

M.S. and A.K.; formal analysis, F.H.; investigation, F.H.; resources, F.H.; data curation, F.H.; writing–original

draft preparation, F.H.; writing–review and editing, F.H, M.S, A.K.; visualization, F.H.; supervision, H.S.; project

administration, H.S.; funding acquisition, A.K., H.S. All authors have read and agreed to the published version of

the manuscript.

Funding:

This research was funded by the by the German Federal Ministry for Economics Affairs and Energy in

the frame of the NetAllok project [22].

Acknowledgments: We thank Mirko Schäfer and Tom Brown for stready support and fruitful discussions.

Conﬂicts of Interest: The authors declare no conﬂict of interest.

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