Uncertainty-Aware Capacity Allocation

in Flow-Based Market Coupling

Richard Weinholda,1,∗, Robert Miethb,1

aFaculty VII Economics and Management, TU Berlin, 10623 Berlin, GermanybDepartment of Electrical and Computer Engineering, Tandon School of Engineering,

New York University, New York, NY 10012 USA

Abstract

The effective allocation of cross-border trading capacities is one of the cen-tral challenges in implementation of a pan-European internal energy mar-ket. flow-based market coupling (FBMC) has shown promising results for toachieve better price convergence between market areas, while, at the sametime, improving congestion management effectiveness by explicitly internal-izing power flows on critical network elements in the capacity allocationroutine. However, the question of FBMC effectiveness for a future powersystem with a very high share of intermittent renewable generation is oftenoverlooked in the current literature. This paper provides a comprehensivesummary on FBMC modeling assumptions, discusses implications of externalpolicy considerations and explicitly discusses the impact of high-shares of in-termittent generation on the effectiveness of FBMC as a method of capacityallocation and congestion management in zonal electricity markets. We pro-pose to use an RES uncertainty model and probabilistic security margins onthe FBMC parameterization to effectively assess the impact of forecast errorsin renewable dominant power systems. Numerical experiments on the well-studied IEEE 118 bus test system demonstrate the mechanics of the studiedFBMC simulation. Our data and implementation are published through theopen-source Power Market Tool (POMATO).

Keywords: Flow-based market coupling, zonal electricity markets, optimalpower flow, chance constraints, flow reliability margins

∗Corresponding authorEmail addresses: riw@wip.tu-berlin.de (Richard Weinhold),

robert.mieth@nyu.edu (Robert Mieth)1R. Weinhold and R. Mieth contributed equally to this work. The order of authorship

has been chosen based on the amount of RAM in the authors’ workstations.

Preprint submitted to Elsevier September 14, 2021

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Table 1: Nomenclature.

A. Sets

T Set of time steps within the model horizon.G Set of generators.R Subset R ⊂ G of intermittent generators.N Set of network nodes.CNEC Set of critical network elements and contingencies.Z Set of bidding zones.

B. Variables

Gt Active power generation at t ∈ T indexed by Gg,t, g ∈ GCt Curtailment t ∈ T .It Active nodal power injection at t ∈ T .NPt Zonal net-positions at t ∈ T .EXt Bilateral exchange at t ∈ T indexed by EXt,z,z′ , z, z

′ ∈ ZGredt Active power redispatch at t ∈ T .

αt Generator response at t ∈ T indexed by αg,t, g ∈ GTj,t Standard deviation of the flow t ∈ T on j ∈ CNEC.

C. Parameters

dt Nodal load at t ∈ T .rt Available intermittent generation at t ∈ Tm Mapping of generators/load (subscript) to nodes/zones (su-

perscript).g Upper generation limit.

f Line capacity of CNEC indexed as f j, j ∈ CNEC

f refj,t Reference flow on j ∈ CNEC at t ∈ T .ntc Net-transfer capacity indexed by ntcz,zz, z, z

′ ∈ Zgda Day-ahead scheduled generation at t ∈ T .cda Day-ahead scheduled curtailment at t ∈ T .ωt Real-time deviation on intermittent generation rt at t ∈ T .Ω Uncertainty space ω ∈ Ωε Risk-levelΦ Cumulative distribution function of the standard normal dis-

tributionΣt Variance-Covariance matrix of ω at t ∈ T .

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1. Introduction

The development of the European internal energy market (IEM) for elec-tricity aims to establish a free pan-European trading platform, under a pol-icy triangle of security of supply, affordability and sustainability (EuropeanCommission, 1997). The achievement of coupling the vast majority of Eu-rope’s electricity consumption in a single market is the result of over 20 yearsof continuous policy iteration for more effective market designs and proce-dures (Glachant, 2010). The core of market coupling is defined by capacityallocation and congestion management (CACM) routines, i.e., methods todetermine exchange capacities between market areas such that trade is asunconstrained as possible, while ensuring secure system operation.

While development of efficient methods for CACM where motivated inthe early stages of the IEM by the scarcity of transmission capacity inheritedfrom pre-liberalized market structures (Meeus and Belmans, 2008), currentdevelopments are driven by the system’s transformation towards a sustain-able energy system (Directorate General for Energy, 2019). Methods to de-rive available exchange capacities that are currently employed are the nettransfer capacity (NTC) – and available transfer capacity (ATC) methodsand flow-based market coupling (FBMC), which is the designated IEM tar-get model. FBMC differs from NTC and ATC in how transmission capacityis allocated to markets. While NTC and ATC derive bounds for bilateraltransactions, FBMC derives bounds on the net-position of all involved bid-ding zones, thereby capturing all transactions simultaneously. Hence, FBMCpromises more efficient capacity allocation and better transparency. How-ever, FBMC was inaugurated in 2015 as part of EC Directive 15/1222 (Eu-ropean Commission, 2015), but was envisioned by the responsible marketparties early in the development of the IEM (ETSO, 2001). Therefore, whileFBMC improves (Rte et al., 2015), it was not envisioned to realise policytargets of high shares of renewable energy sources (RES).

As part of the Clean Energy Package (Directorate General for Energy,2019) and the accompanying update to the regulation on CACM, EuropeanCommission (2019a) explicitly addresses specifics in the capacity allocationprocess. Among others, the regulation requires the transmission system op-erators (TSOs) to allocate a minimum of physical line capacity to the market.This kind of engagement with specifics of the process illustrates the regula-tory willingness to closer engage with market design to bring the process inline with sustainability targets. Therefore, FBMC, as an important part ofthe IEM, should become an active element of the transformation process.

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2. Related Literature

Initial publications on FBMC stem from conceptional documentations ofthe involved TSOs and power exchanges (PXs) (ETSO, 2001; ETSO andEuroPEX, 2004). As part of a “dry-run”, meaning theoretical operation, in2008 (Amprion et al., 2011) and a parallel run in 2013 (Rte et al., 2015)the process was evaluated and its efficiency verified. Additionally, FBMC isdescribed in a continuously updated Documentation by the involved TSOsthat is currently available as version 5 (50Hertz et al., 2020). Complemen-tary descriptions and research of the involved parameters was published indifferent articles from TSO work-groups (Schavemaker et al., 2008; Aguadoet al., 2012; Marien et al., 2013).

Over the past years, several academic publications provided various mod-eling approaches to the different components of FBMC. Byers and Hug (2020)provided insights into hte fundamental modeling process and numericallyhighlights the importance of its parametrization, Schonheit et al. (2020b);Finck et al. (2018) evaluated the impact of different generation shift keys, acore parameter that describes how generators will participate in cross-borderexchange, Schonheit et al. (2021a) described the relation of bidding zone con-figuration on the selection of network elements that are part of the capacitycalculation.

These publications generally limit the scope to specifics of the processand evaluate the relation between parameter choices. However, the process’sability to accommodate policy decisions and impact of integration of largeshares of RES is mostly absent. In Schonheit et al. (2021c) the authorsdescribe the TSOs degree of freedom to influence the parametrization andits impact on results. The analysis also includes a scenario with moderateincrease in RES capacities. Similarly Matthes et al. (2019) analysis FBMCin for the target year 2025 that results in higher shares of RES, but theimplications on FBMC are not discussed.

In this paper we contribute to the body of literature on FBMC in threeways:

• We show how the concept of FBMC allows the inclusion of policy rel-evant considerations.

• We evaluate the capacity allocation methods of NTC/ATC and FBMCin regards to higher shares of RES.

• To alleviate the shortcoming of perfect foresight in models with highRES shares and no explicit representation of a real-time market, wepropose a chance constrained formulation to robustify against networkoverloads caused by real-time deviations.

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3. FBMC Concept and Simulation

Expected point-of-dispatch, GSKs,

FRMs, FAVs, CNECs

IN

flow-based parameters(zonal PTDF, RAM)O

UT

Cross-zonal exchangecapacity allocation

minRAMCNECs

flow-basedparametersIN DA bids

scheduledwelfare-maximizingDA market clearing

OU

TDA generation

schedules

IN

Congestion management

OU

T

Point-of-dispatch

continuouswelfare-maximizing

intraday market clearing

Generation schedules

IntradaybidsI /

O

Generationschedules

Real-timeconditions

D -

2D

- 1

D -

0

Bas

ecas

eM

arke

t C

lear

ing

Co

ng

esti

on

Man

agem

ent

Roles TSOsRegulator Market

Figure 1: FBMC process overview.

FBMC is a three-stage processdesigned to allocate commercial ex-change capacity between adjacentelectricity markets (i.e., cross-borderexchange). It is coordinated bythe local TSOs and PXs and aimsto accommodate inter-zonal electric-ity trading with respect to availabletransmission system capacity andphysical power flows. Fig. 1 providesan overview of the three FBMCstages and shows their timing, in-volved stakeholders, parameters andresults. For any given day D, theFBMC process starts two days in ad-vance at D − 2, with the calculationof the so called basecase. The base-case is informed by TSO-generatedforecasts on the expected point-of-dispatch and provides “a best esti-mate of the state of the [...] sys-tem for day D” (50Hertz et al., 2020,p. 26). Based on these forecasts,on additional predefined policies andon regulatory constrains (see discus-sion below), the TSOs calculate theso called flow-based parameters thatare used to constrain the commer-cial exchange in the following marketclearing stage. Generation and loadbids collected from all bidding areasthat are part of the Central Western Europe (CWE) region are cleared inthe day-ahead (DA) and intraday markets at D− 1 and D− 0, respectively.Lastly, during the re-dispatch stage, the TSOs may require changes to thegenerators’ final point-of-dispatch to resolve any network congestion or otherthreats to system security due to real-time conditions (e.g., load and RESinjections).

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3.1. Preliminaries

The FBMC process and the definition of the flow-based parameters relieson a model of the physical transmission system, which we formalize as follows.Consider an interconnected transmission network with n nodes and l lines.Further, let each node i be defined by its net power injection Ii collected incolumn vector I = [ii]

ni=1 ∈ Rn. If Ii > 0, then node i is a net generator, if

Ii < 0, then node i is a net load. For each vector x there exists exactlyone corresponding vector f = [fj]

lj=1 ∈ Rl that collects the flows along

each line j and is defined by the physics of power flow in a network. Themaximum allowable flow (capacity) of each line j is given by f j collected

in vector [f j]lj=1. Power flow physics in high-voltage transmission networks

allow a linear approximation of the relationship between f and I using apower transfer distribution factor matrix PTDF ∈ Rl×n such that

f = PTDF ·I. (1)

See, e.g., Weinhold and Mieth (2020a) for a more detailed derivation. Fur-ther, assume that the nodes of the network are grouped in z compact marketzones such that each zone k collects nodes Nk and

⋃zk=1Nk = 1, ..., n and⋂z

k=1Nk = ∅. The set of all zones is denoted as Z and the sum of the injec-tions of all nodes in one zone is called the net-position npk of zone k collectedin vector np ∈ Rz = [npk]

zk=1. If npk > 0, then zone k is a net exporter and

if npk < 0, then zone k is a net importer.

3.2. Flow-based formalism

The effectiveness of FBMC to enable least-cost, yet physically feasibleday-ahead market outcomes across interconnected market zones hinges onthe precise definition of the flow-based parameters, which specify (i) howcross-border power exchange affects power flow on transmission lines and (ii)how much capacity on each line is available to accommodate flow caused bycross-border exchange. Each specification (i) and (ii) is given by the followingrespective parameter:

(i) Zonal PTDF; Zonal PTDFz ∈ Rl′×z maps net-position vector np tothe flow on a selection of l′ ≤ l lines in the network, the so called criticalnetwork elements (CNE).

(ii) Remaining available margin (RAM); For each critical network el-ement (CNE), vector RAM ∈ Rl′ defines the absolute capacity that isavailable for cross-border trading in the day-ahead market.

Although these parameters are generated, exchanged and published followingspecific rules requested by regulation and laid out in the documentation of

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the FBMC process in 50Hertz et al. (2020), they depend on the exact choiceof the underlying forecast and meta-parameters that are either chosen at thediscretion of the TSOs or given by policy requirements.

First, the exact definition of PTDFz depends on the chosen CNEs, addi-tional contingency scenarios and so called generation shift key (GSK).

• CNE/CNEC: “A CNE is considered to be significantly impacted byCWE cross-border trade, if its maximum CWE zone-to-zone PTDF islarger than a threshold value that is currently set at 5%.”

(50Hertz et al., 2020, p. 19f)Network elements are considered CNE only if TSOs determine thattheir flow is significantly driven by cross-zonal exchange. This selec-tion prevents heavily loaded lines that are largely insensitive to changesin the zones’ net-positions to limit inter-zonal exchange capacity allo-cation. Further, for each network element the TSOs identify a set ofcontingencies (C), i.e., unplanned outages of network elements, thathighly impact the flow on said CNE. The resulting set of critical net-work elements and contingencies (CNECs), the full PTDF matrix andadditional linear outage factors, which capture flow shifts during theselected contingencies, form the basis for PTDFz computation.

• GSKs: “A GSK aims to deliver the best forecast of the impact onCritical Network Elements of a net-position change [and] is calculatedaccording to the reported available market driven power plant potentialof each TSO divided by the sum of market driven power plant potentialin the bidding zone. ”. (50Hertz et al., 2020, p. 38ff)Since line flows f depend on the injection at every node, the computa-tion of PTDFz requires an estimation of how changes in net-position npare distributed to changes in net injections I. Assuming that the differ-ence between forecasted net-position in the basecase and the realizednet-position in the market phase is small, the set of generators thatwill serve this difference by “shifting” their production levels can beanticipated. The resulting nodal contribution to net-position changesis formally captured in vector GSK ∈ Rn×z that maps a change ∆npin net-positions to a change in nodal injections ∆I such that

∆I = GSK ∆np, (2)

which further provides a clearer definition of PTDFz as

PTDFz = PTDF ·GSK . (3)

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Note that this approach implicitly attributes all possible changes tonet-position np to changes of dispatchable generators and is ignorantto np changes caused by forecast errors of load and RES.

Second, the final RAM value can be corrected by additional flow reliabilitymargins (FRMs) and final adjustment values (FAVs).

• FRM: “[F]or each Critical Network Element, a Flow Reliability Margin(FRM) has to be defined, that quantifies at least how [...] uncertaintyimpacts the flow on the Critical Network Element.”

(50Hertz et al., 2020, p. 47)flow reliability margins (FRMs) are static RAM reductions that are in-formed by past observations on how the flow on each CNE changedbetween the basecase forecast and the point-of-dispatch realization.Therefore, FRM captures forecast uncertainty of load and RES injec-tions.

• FAV: “With the Final Adjustment Value (FAV), operational skills andexperience that cannot be introduced into the Flow Based-system canfind a way into the Flow Based-approach by increasing or decreasing theremaining available margin (RAM) on a CNE for very specific reasons[...] to eliminate the risk of overload on the particular CNE.”

(50Hertz et al., 2020, p. 25)FAVs decrease or increase RAM based on the operational experienceof the TSOs. It may reflect remedial actions at the point of dispatchor other complex system security considerations.

Finally, the available commercial exchange capacity between market zonesis defined by constraining the zonal net-positions relative to the basecase. Forthe basecase, the TSOs forecast the expected flow f bc on all CNEs and theexpected net-positions npbc. The net-positions that can be realized duringthe market stage (npda) can only differ from npbc if CNE limits f (correctedby FRMs and final adjustment values (FAVs)) are maintained:

PTDFz(npda − npbc) ≤ f − (FRM + FAV)− f bc. (4a)

Note that the explicit introduction of npbc in (4a) is necessary to ensure thevalidity of the GSKs and, thus, PTDFz. See also Schonheit et al. (2020b) fora broader discussion. Eq. (4a) can be rewritten as follows:

PTDFz ·npda ≤ f − (FRM + FAV)− f bc + PTDFz ·npbc (4b)

⇔ PTDFz ·npda ≤ f − (FRM + FAV)− f ref (4c)

⇔ PTDFz ·npda ≤ RAM . (4d)

8

In Eq. (4c), f ref = f bc−PTDFz ·npbc denotes the reference flow that capturesa residual between the parameter choices made in PTDFz and the forecastedf bc. While f ref can be assumed small, it is not necessarily zero. Eq. (4d)yields the desired limit on market-based net-positions npda subject to theflow-based parameters PTDFz and RAM (50Hertz et al., 2020, p. 60). Thespace of all possible net-positions that fulfill (4d) is called flow-based domain.

Irregardless of the formal RAM definition, regulations, e.g., the EuropeanCommission (2019a, Art. 16), prescribe specific conditions that define a mini-mal percentage of CNE capacity that must be made available for cross-borderexchange and that ensures the feasibility of long-term traded capacities.

• minRAM: “CNECs with a RAM of less than the minRAM factor mul-tiplied by Fmax at zero-balance are assigned an AMR value (adjustmentfor minRAM) in order to increase the RAM.”

(50Hertz et al., 2020, p. 64)The minRAM criterion defines a lower bound for the RAM based onthe CNEs capacity and is applied after FRMs and FAVs:

RAM = max(minRAM ·f, f − (FRM + FAV )− f ref ). (5)

• Long-term allocations: “the long-term-allocated capacities of theyearly and monthly auctions have to be included in the initial FlowBased-domain” (50Hertz et al., 2020, p. 66)This requirement ensures that trades on energy futures and bilateraldelivery contracts outside of the day-ahead or intraday market clearingstage have to be feasible within the flow-based domain.

FRMs, FAVs, minimum remaining available margin (minRAM) and long-term allocations either enlarge the flow-based domain to enable higher priceconvergence (minRAM), shrink the flow-based domain to accommodate se-curity margins (FRM), or go both ways (FAV).

3.3. Flow based discussion

It is clear, that the resulting flow-based parameters do not only reflectformal definitions, but also methods to account for uncertainty or imperfec-tions, e.g., arising from zonal aggregation and required forecasts. Further,their specific configuration depends on policy considerations on the desiredlevel of restrictions on commercial cross-border exchange. Regulation statesa clear goal of achieving higher price convergence (European Commission,2019b). As a result, solely cross zonal tie lines are encouraged to be nomi-nated as CNEs (ACER and CEER, 2018, p.8) and the a minRAM of 70% willbe required by 2025. This indicates that large trading domains are desirable

9

and that potentially higher cost for congestion management (e.g., real-timeredispatch) fall into the responsibility of local TSOs.

In previous academic studies, derivation and application of the flow-basedparameters is mostly understood as a strictly formal process. Studies gen-erally focus on the formal dimension in their numerical experiments by de-scribing the relation of a specific parametrization policy to a chosen metric,e.g., system cost or welfare, with the goal to provide a better understandingof parameter choices. Current literature on flow-based parameter policiesin relation to system cost exist for GSKs (Voswinkel et al., 2019), min-RAMs (Schonheit et al., 2021b), commercial exchange and uncertainty inthe basecase parametrization (Byers and Hug, 2020) and selection of CNECs(Schonheit et al., 2021c). All contribute to better understanding the relationbetween the parameters, however comparability remains difficult since it re-quires similar definitions on how flow-based domains should be used. Moststudies do not explicitly discuss which overall target the capacity allocationstrives for.

Since the current regulation explicitly requires a parametrization to pro-vide higher capacities to the markets with the goal to ensure the integrationof higher shares of RES (European Commission, 2019a) it is important tomake these considerations part of the modeling and academic process. With-out such considerations the effectiveness of FBMC to accommodate highershares of RES cannot be definitively answered. With this paper we aim tocontribute to this discussion by providing a transparent parametrization ofthe FBMC process and the underlying market simulations an by numeri-cally show the effects of higher shares of RES. We explicitly discuss differentconsideration regarding the permissiveness of day-ahead trading domains byminRAM and CNEC selection and the effect on total system cost and conges-tion management. In addition we provide a sensible way to include risk-awaresecurity margins FRMs in the modeling process. The permissive capacity al-location in systems with high shares of intermittent generation raises thequestion of operability. Thus we include process-considerations regardingexpected deviations from scheduled generation to make the system more ro-bust.

4. Model Formulation

4.1. Market Simulation

In addition to computing flow-based parameters, modeling and studyingthe three-step FBMC process as shown in Fig. 1 requires a simulation ofbasecase, market clearing and congestion management processes. We modelall of these steps as a multi-period economic dispatch (ED) problem, where

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each step is constrained by a specific set of network or transport constraints.The ED is given as:

min∑t∈T

c(Gt) + p(eTCt) (6a)

s.t. 0 ≤ Gt ≤ g ∀t ∈ T (6b)

0 ≤ Ct ≤ rt ∀t ∈ T (6c)

mnGt + mn(rt − Ct)− dt = It ∀t ∈ T (6d)

mzGt + mz(rt − Ct)−mzdt = NPt ∀t ∈ T (6e)

NPt,z =∑z′∈Z

EXt,z,z′ −EXt,z′,z ∀t ∈ T ,∀z ∈ Z (6f)

eT It = 0 ∀t ∈ T (6g)

where t indicates the market clearing time steps (e.g., hour or 15 minutes)and T is the set of times steps. Objective function (6a) minimizes systemcost given by the cost of generation c(Gt) and the cost of curtailing RESpCt, where c(·) is a generator cost function model, Gt is the vector of gen-erator production levels, Ct is the vector of RES curtailment, p is a scalarcurtailment penalty and e is the vector of ones in appropriate dimensions.Constraint (6b) enforces limits g on generator outputs Gt and constraint (6c)limits curtailment Ct to available capacity rt. Finally, Eqs. (6d) and (6e)balance power on a nodal and zonal resolution. Nodal energy balance (6d)defines nodal power injections It in terms of nodal load and generation bymapping generator and RES injections to each node via map mn. Similarly,the zonal energy balance defines the zonal net-position NPt as the differencebetween zonal load and generation mapping resources and loads into eachzone via map mz. Eq. (6f) defines auxiliary variable EXt,z,z‘ ≥ 0, which de-notes the bilateral exchange from zone z to zone z′.2 Eq. (6g) enforces systembalance. Note that all decision variable of the model are written in capitalletters, while are parameters are given as lower case symbols.

Nodal power injections or zonal net-positions of ED (6) can be subjectto limitations given by the transmission system capacity and chosen powerflow model. Hence, FBMC-based market clearing can be modeled by using

2Note, when constraints apply to net-positions only, the EX remains unconstrained.Therefore, a small penalty factor for EX is included in the objective function to discourageexcessive exchange but does not distort model results with less than 0.5% of the objectivevalue.

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ED (6) and additionally enforcing

NPt ∈ F z := x : PTDFzt x ≤ RAMt ∀t ∈ T , (7)

where F z is the flow based domain as derived in Section 3 above. Not thatthe zonal PTDF may be different for each time step, indicated by indext. Alternatively, nodal market clearing, i.e., an economic dispatch that is isconstrained by all network transmission lines, can be modeled by constraining(6) with

It ∈ Fn := x : PTDFn x ≤ f ∀t ∈ T . (8)

Nodal market clearing limits the cross-zonal exchange only implicitly by tak-ing into account the transmission capacity of the whole network. On theother hand, we can constrain cross-zonal exchange EXt,z,z‘ directly usingstatic bilateral NTCs:

EXt ∈ Fntc := x : 0 ≤ x ≤ ntc ∀t ∈ T . (9)

Notably, the approach in (9) does not include a physical power flow model.All FBMC steps can be modeled through (6) in combination with either

(7), (8) or (9) as summarized in the FBMC column of Table 2. Notably, thebasecase is a nodal market clearing, following the intuition that the basecaseshould resemble D-0 as well as possible. The day-ahead market is clearedzonally with flow-based parameters PTDFzt and RAM derived from the base-case results as per (4). D-0 congestion management again relies on a nodalnetwork representation (8) and requires additional constraints that impose

Table 2: Model configuration for FBMC, Nodal and NTC market clearing.

FBMC NTC Nodal

D-2:Basecase

(6a) s.t.(6b)–(6g), (8)

– –

D-1:Market Clearing

(6a) s.t.(6b)–(6g), (7)

(6a) s.t.(6b)–(6g), (9) (6a) s.t.

(6b)–(6g), (8)D-0:CongestionManagement

(6a)+(10a) s.t.(6b)–(6g), (8),(10b), (10c)

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cost for deviating from the market clearing results:

C(Gred) = cred∑t∈T

|Gredt | (10a)

Gt − gdat = Gredt ∀t ∈ T (10b)

Ct ≥ cda ∀t ∈ T , (10c)

where gdat and cdat are the decisions on Gt and Ct from the previous marketclearing stage. Note that we do not explicitly model an intraday market stageand, for now, assume that load and RES injections do not change betweenthe market stage and the real-time point of dispatch.

For reference, zonal market clearing using static bilateral NTCs and anodal market clearing are modeled and their resulting formulations are item-ized in Table 2 as well. The NTC market clearing is modeled in two steps,because it does not require a basecase computation. The necessary con-gestion management step is the same as for FBMC. The nodal market isa one-shot optimization of (6) subject to nodal power flow constraints (8).Notably, the nodal market does not require a congestion management stage,because all

4.2. Probabilistic FRMs via Chance Constraints

The formulations of the previous section model FBMC under perfect fore-sight of load and RES injections. The intention to provide efficient commer-cial exchange capacities to the market in combination with high shares ofintermittent renewable generation poses the question of operability and howthe forecasting characteristics of the basecase can be used to robustify resultsagainst RES uncertainty. As outlined in Section 3 above, the FBMC conceptrecognizes the existence of forecast uncertainties in the basecase by intro-ducing FRMs, which are tuned based on historical data and TSO-definedrisk levels (50Hertz et al., 2020, Fig. 4-2). Specifically, TSOs use histori-cal data to estimate the (1− ε)-percentile of the absolute deviation betweenforecasted basecase flows f bc, corrected by changes in the market schedule,and the realized real-time flows. By setting FRMs to at least the value ofthis (1 − ε)-percentile, TSOs ensure that lines are not overloaded due RESor load forecast errors with a probability of (1− ε). Thus, ε defines the risklevel and is usually chosen small (e.g., 1− 5%).

To date, studies on FBMC have not considered FRMs in terms of an un-certainty model and risk-threshold, but rather employ fixed security marginsthat are applied uniformly to all CNEs, see e.g., Schonheit et al. (2021b,2020a); Wyrwoll et al. (2019). While such simplifications may be motivated

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by a lack of historical data to simulate the FRM computation process pre-scribed by regulation, ignoring the specific impact of real-time control actionscaused by intermittent renewable injections may obstruct a clear assessmentof the effectiveness of FBMC in high RES systems. As an alternative, wepropose to model risk-aware FRMs that explicitly internalize an RES uncer-tainty model and the impact of real-time generator control actions on eachCNE. To this end, instead of creating an empirical uncertainty model of theflow forecast error on each CNE, we rely on a parameterized RES forecasterror distribution and control participation factors. This approach leveragesresults on chance-constrained optimal power flow proposed by Bienstock et al.(2014).

We model the uncertain injection from RES generators as rt(ω) = rt+ωt,where ωt is a zero-mean random vector that captures the forecast error ofrenewable generation rt. We assume that the distribution can be modeledas a normal distribution such that ωt ∼ N(0,Σt), where Σt denotes thecovariance matrix of ωt. Following the empirical results of Dvorkin et al.(2016), we will rely on the assumption that ωt is normally distributed for theremainder of this paper, but note that the proposed approach can be modifiedto accommodate different distributional assumptions, see e.g., Roald et al.(2015); Dvorkin (2020). Next, we realize that, because the basecase andday-ahead markets are cleared based on forecast rt, error ωt creates a systemimbalance and assume that the systems reaction to this imbalance can beanticipated. Similar to how GSKs capture the estimated distribution of ∆npto all generators, we introduce vector of control participation factors αt thatmaps the reaction of generators to imbalance ωt as:

Gt(ωt) = Gt − αt(eTωt). (11)

Since ωt and, thus, Gt(ωt) are random variables, we first formulate thezonal ED as a probabilistic problem

min E[C(G(ω))] (12a)

s.t.

P[0 ≤ Gg,t(ω) ≤ gg] ≥ 1− ε ∀t ∈ T , ∀g ∈ G (12b)

P[PTDFzj,t ·NPt(w) ≤ f j − frefj,t ] ≥ 1− ε ∀t ∈ T , ∀j ∈ CNEC (12c)

mzgGt(ω) + mz

rrt(ω)− Ct − dt = NPt(w) ∀t ∈ T , ∀ω ∈ Ω. (12d)

Objective (12a) minimizes the expected system cost. Constraints (12b) and(12c), ensure that the probability a generator can not fulfill its requiredresponse αt(e

Tωt) or of a CNEC being overloaded at least (1− ε). These socalled chance constraints resemble the value-at-risk, a risk metric commonly

14

used in the finance industry (Bienstock et al., 2014). Lastly, Eq. (12d) ensuresthat the system is balanced for all possible outcomes of ω ∈ Ω.

Problem (12) can not be solved directly, but allows a computationallytractable deterministic reformulation. First, recall that ωt is zero mean,i.e., E[ωt] = 0. For a linear cost function model c(G) we therefore getE[c(G(ωt))] = c(G(ωt)). See, e.g., Mieth (2021) for the derivations forquadratic cost functions. Next, chance-constraints (12b) and (12c) can bereformulated by realizing that for any random vector x ∼ N(µ,Σ) it holdsthat (Bienstock et al., 2014):

P[x ≤ x] ≥ (1− ε) ⇔ µ+ Φ−1(1− ε)σ(x) ≤ x, (13)

where Φ is the cumulative distribution function of the standard normal dis-tribution and σ(x) is the standard deviation of x. Using (13) to reformulate(12b) and (12c) we get:

E[Gg,t(ωt)] = E[Gg,t − αg,t(eTωt)] = Gg,t (14)

σ(Gg,t(ωt)) =√

Var[αg,t(eTωt)] =√α2g,t(e

TΣte) = αg,tSt (15)

E[PTDFzj,t ·NPt(w)] = E[PTDFzj,t(mzgGt(ω) +mz

rrt(ω)−mzddt)]

= PTDFzj,t(mzgGt +mz

rrt −mzddt)

(16)

σ[PTDFzj,t ·NPt(w)] =√

Var[PTDFzj,t(mzgGt(ωt) +mz

rrt(ωt)−mzddt)]

=√

[PTDFj,t(mzr −mz

gαet)]Σt[PTDFj,t(mz

r −mzgαe

t)]T

= ‖PTDFz(mzr −mz

gαeT )Σ

1/2t ‖2,

(17)

where we define S2t = eTΣte and ‖·‖2 denotes the 2-norm. For more a more

detailed explanation we refer to Bienstock et al. (2014) or Mieth (2021).Lastly, (12d) holds for all ωt, if the system is balanced in expectation andthe sum of all control actions is exactly equal to the system imbalance, i.e.,:

eTαt(eTω) = eTωt ⇔ eTαt = 1. (18)

15

Thus, the deterministic reformulation of (12) is given as:

min C(G(ω)) (19a)

s.t. Gt + zεαs ≤ g ∀t ∈ T (19b)

−Gt + zεαs ≥ 0 ∀t ∈ T (19c)

PTDFzj,t NPt ≤ f j − frefj,t − zεTj,t ∀t ∈ T , ∀j ∈ CNEC (19d)

‖PTDFzj,t(mzr −mz

gαeT )Σ1/2‖2 ≤ Tj,t ∀t ∈ T , ∀j ∈ CNEC (19e)

mzgGt + mz

r rt −mzddt = NPt ∀t ∈ T (19f)

eTαt = 1 ∀t ∈ T (19g)

where we use zε = Φ−1(1 − ε) for a more concise notation and introduceauxiliary variable Tj,t to denote the standard deviation of the flow acrossCNEC j. Constraint (19e) is a second-order conic constraint, which is convexand can be solved efficiently by modern off-the-shelf solvers. The variable αcan be chosen as parameter, similarly to the GSK, or optimized as a decisionvariable in (19). In this paper we use the latter approach, thus allowing foran optimized generator response. Additionally, the risk-level is defined withε = 5% and Σ is calculated with an assumed standard deviation of 10% · r,that is consistent with other publications (Mieth, 2021; Dvorkin et al., 2016,p.29).

We interpret term zεTj,t in Eq. (19d) as the endogenous FRM that re-duces capacity for each CNEC based on the RES uncertainty model. Fig. 2shows the resulting impact on the flow-based domain for a single time stepfor exchange from Zone 1 to Zone 2 on the x-axis and Zone 2 to Zone 3 onthe y-axis. See Fig. 3 in Section 5 below for an illustration of the zones. Eachline in the figure corresponds to a row of the zonal power transmission dis-tribution factor (PTDF) matrix and colored light grey for CNECs and darkgrey for CNE. The The FRMs (dashed blue) reduce the flow-based domainby the blue area. Subsequently, the market outcome (red dot) will also moveinward, leading to lower commercial exchange. However, Fig. 2 shows, thatthe necessary margin to achieve (1− ε) security differs among CNEs, whichindicates that a fixed FRM proxy margin would over- or underestimate theRAM on some CNE.

16

-3000 0 500

-1000

0

500

-500

-2000 -1000

CNE Constraints FRM

CNEC Constraints Market Outcome

Figure 2: Flow-based domain with FRMs for exchange Zone 1 - Zone 2 (x-axis) and Zone2 - Zone 3 (y-axis).

17

5. Case Study

5.1. Data Set

Zone 1

Zone 2

Zone 3

Figure 3: Topology of the IEEE 118 bus case, zones indicated by color. Changes intopology from the original data are indicated with filled nodes.

The numerical experiments build on the Pena et al. (2017) version of theIEEE-118 bus case, which augments the well known case study by additionalgeneration technologies, zonal configuration and hourly load and RES in-jection timeseries for a full year. In Pena et al. (2017) the authors kept theoriginal topology from the original IEEE-118 bus network, but line capacitiesare generously (4x-5x) scaled with installed capacity.

18

biomass

natural gas

oil

coal

water ror_ts

sun

wind

5

10

15

20

med

ium re

s

original

0

high

res

25Zone 1

Cap

acity

[GW

]

5

10

med

ium re

s

original

0

high

res

12.5

Zone 3

med

ium re

s

original

0

5

high

res

7.5

2.5

Zone 2

Figure 4: Installed capacities for the three scenarios original, medium res and high res.

The original data in Pena et al. (2017) is complemented in this studywith two scenarios that further increase the share of RES generation in thetotal generation over the model horizon from 26% in the original data to50% (medium res) and 70% (high res). The resulting installed capacitiesare itemized in Fig. 4 for the three scenarios original, medium res and highres. To better reflect the scarcity of transmission capacity, all line capacitiesare scaled down by 30%. Additionally, the zonal configuration was slightlyadjusted so all zones have shared borders. This is indicated in Fig. 3 bynodes that are filled solid black, which where allocated to “Zone 2” (red)and are now allocated to “Zone 3” (green).

We analyze the effectiveness of FBMC by comparing system cost, whichare composed of generation cost at the market clearing stage and additionalcongestion management (redispatch) in D-0. See also (10). We set the cost forredispatch to 30$ per MWh and curtailment cost to 5$ per MWh. Note thatcongestion management does not just revert a zonal solution to an optimalnodal solution, but tries to achieve a network-feasible solution with minimaldeviations from the zonal solution.

19

5.2. Zonal benchmarks

Generation Costs Curtailment CostsRedispatch Costs

Cos

ts [

mio

$]

Nod

al

NTC

-50

NTC

-100

NTC

-200

NTC

-500

NTC

-1000

Uniform

high res medium res original

1200

1400

1600

1800

2000

2200

Nod

al

NTC

-50

NTC

-100

NTC

-200

NTC

-500

NTC

-1000

Uniform

Nod

al

NTC

-50

NTC

-100

NTC

-200

NTC

-500

NTC

-1000

Uniform

Figure 5: Cost composition of zonal NTCs market clearing and the nodal reference.

Fig. 5 shows the benchmark reference results for the three scenarios fornodal market clearing and zonal market clearing subject to NTCs. See alsoTable 2. System cost are evaluated after congestion management and includecost for curtailment and redispatch. The Nodal solution on the left repre-sents the economic optimum that does not require redispatch. Zonal marketclearing is shown with increasing NTCs from left to right. The “Uniformpricing“ case describes market clearing without any commercial exchangeconstraints.

The results show the expected pattern:

• Generation cost decrease with higher shares of cheap generation fromRES.

• The nodal dispatch is the least cost solution, with no redispatch re-quired. Additionally, due to cost for congestion management, zonalsolutions often show increased generation cost.

• For zonal market clearing, more exchange capacities lead to lower gen-eration cost. However, cost for congestion management can outweighthese savings.

Notably, an NTCs of 500 MW leads to the lowest cost dispatch for zonalmarket clearing. In the following, this results will be used as a benchmarkfor the FBMC results.

20

5.3. Flow-based parametrization

As described in Section 3.1, the flow-based parameters are used to solvethe day-ahead stage in FBMC. They are composed of the zonal PTDF andRAM values and calculated from basecase generation schedules gbc, powerflows f bc and net-positions npbc. For our numerical experiments we aim tostay as close as possible to the reference given by the documented processby 50Hertz et al. (2020). We model the basecase as nodal pricing with fullnetwork representation (nodal). The zonal PTDF is composed of all cross-border lines and internal lines with a zone-to-zone PTDF value larger than5%. Contingencies are included based on a 20% line-to-line sensitivity in caseof an outage using so called load-outage distribution factors as per Jiachunet al. (2009), i.e. lines are considered contingencies that distribute 20% ofline loading to the CNE. For all scenarios we use a so called Pro-Rata ap-proach to calculate GSKs, i.e., changes in net-position are distributed basedon the online dispatchable generation capacity at each time step (50Hertzet al., 2020; Dierstein, 2017). Consistent with current practises (Amprion,2019), a 20% minRAM is employed. This also ensures feasibility of (4c) asthe feasible region (7) is cases where a suboptimal GSK leads to negativeRAM values. The resulting FBMC configurations is denoted as FBMC inthe following results section. To illustrate the the impact of less restric-tive flow-based parameterization a second parametrization is evaluated thatonly considers cross-border lines as CNEs and enforces a minRAM of 70%,denoted as FBMC+.

All scenarios and market configurations where solved using the openPower Market Tool (POMATO) (Weinhold and Mieth, 2020b) written inPython and Julia. The calculation where done on standard PC hardwarewith a Ryzen 7 and 32GB of memory using the Gurobi solver (Gurobi Op-timization LLC, 2018) for the deterministic configurations and the Moseksolver (MOSEK ApS, 2021) for configurations using chance constraints.

21

5.4. Deterministic FBMC

Basecase

FBM

C D

-1

FBM

C D

-0

NTC

-500

1000

1500

2000

Cos

ts [

mio

$]

Generation Costs Curtailment CostsRedispatch Costs

high res medium res original

Basecase

FBM

C D

-1

FBM

C D

-0

NTC

-500

Basecase

FBM

C D

-1

FBM

C D

-0

NTC

-500

Figure 6: Cost composition of FBMC and NTC market clearing. For each scenario:Basecase, D-1 zonal market clearing and D-0 redispatch and NTC reference.

Fig. 6 shows the FBMC results for the three scenarios. The total sys-tem cost in D-0 are also depicted in Table 3. The three market steps aredepicted as Basecase (D-2), FBMC D-1 for the day-ahead stage and FBMCD-0 for congestion management. NTC-500 is included as the least cost zonalreference. For each scenario the basecase represents the expected market out-come, from which the flow-based parameters are derived and used to clearthe market stage. The composition of this market result shows differentcharacteristics for the three scenarios. For the original data, the cost arehigher compared to the nodal Basecase. The flow-based parameters overlyconstrains zonal exchange and therefore lead to sub-optimal market result,whereas for the high res scenario shows lower cost in the market result. Theadditional cost for congestion management depend on the systems abilityto accommodate the market result. Here the FBMC and NTC solutionsare very close, with FBMC generally resulting in higher generation cost,but lower cost for congestion management. Similarly to the reference re-sults from Section 5, the NTC and FBMC solutions illustrate the trade-offbetween capacity allocation and congestion management, where FBMC ismore restrictive and generally leads to less congestion management at highercost in generation.

The results from the more permissive FBMC+ configuration, which en-forces a 70% minRAM and only considers cross-border CNEs, further il-lustrates this point. Table 3 shows the cost decomposition and Table 4

22

Table 3: System cost including generation and congestion management (CM) for eachscenario.

FBMC FBMC+ NTC-500 Nodal

original

Generation 2091.45 2083.82 2091.01 2075.16Curtailment 0.03 0.08 0.01 0Redispatch 48.76 89.28 62.89 0

total CM 48.79 89.36 62.9 0total 2140.24 2173.18 2153.92 2075.16

medium res

Generation 1498.85 1499.01 1496.28 1461.68Curtailment 7.8 9.97 8.55 3.06Redispatch 110 115.91 110.61 0

total CM 117.8 125.88 119.17 3.06total 1616.66 1624.89 1615.44 1464.74

high res

Generation 1256.1 1239.05 1245.6 1176.48Curtailment 42.1 42.73 41.57 31.81Redispatch 234.89 243.57 241.63 0

total CM 276.99 286.29 283.2 31.81total 1533.09 1525.35 1528.8 1208.29

23

show the respective redispatch (R), curtailment (C) and combined conges-tion management (sum) volumes for each scenario and include the FBMC+

configuration. Here, the FBMC+ proves less restrictive than the NTC andFBMC configurations. With the original data, the relaxed flow-based pa-rameters FBMC+ lead to overall higher cost, due to increased congestionmanagement. For higher shares of intermittent renewable generation, largerexchange capacities become more efficient and, while still with the highercongestion management volumes, provide the lowest cost for zonal marketclearing.

Table 4: Quantities of redispatch (R) and curtailment (C) in TWh for each scenario.

original medium res high resC R C+R C R C+R C R C+R

FBMC 0.01 1.63 1.64 1.56 3.67 5.23 8.42 7.83 16.25FBMC+ 0.02 2.98 3 1.99 3.86 5.85 8.55 8.12 16.67NTC-500 0 2.1 2.1 1.71 3.69 5.4 8.31 8.05 16.36Nodal 0 0 0 0.61 0 0.61 6.36 0 6.36

5.5. Probabilistic FRMs

The high res scenario reaches a share of intermittent renewable generationof 60 %. The dispatch of these share proves challenging when accounting forexpected deviation between the day-ahead market stage and real-time. InSection 4.2 we propose a chance-constrained formulation for FRMs, thatreduce the flow-based domain based on an assumed distribution of forecasterrors ωt. The results in the previous section illustrate the trade-off betweenpermissive capacity allocation and increased congestion management, whichcan be desirable depending of the associated cost. However, the cost forcongestion management will change, and presumably increase, if the real-time availability is subject to forecast errors. For the numerical experimentwe take a closer look at the FBMC+ scenario, as it provides the larges tradingcapacities and would be subject the largest impact of forecast errors.

Including FRMs, denoted as FBMC+ CC will, in a deterministic case,lead to lower capacities allocated to the market and extension to higher costin congestion management compared to the FBMC+ scenario without FRMs.To evaluate the impact of forecast errors that occur in real time, congestionmanagement is run multiple times for both FBMC+ CC and FBMC+ subjectto randomized real-time deviations ω that follow a normal distribution witha relative standard deviation of 10% from expected in-feed. The generator

24

response α, that is an endogenous result from the chance constraint formula-tion, is used for both scenarios to calculate the generator response real-timedeviations. The numeric results are obtained from 20 full-year runs, withhourly independent real-time deviations.

Table 5 shows that, indeed, the resulting expected cost for congestionmanagement are indeed lower in the FBMC+ CC, illustrating its higher ro-bustness against real-time deviations. Here, we see the same values as inTable 3 with no deviations at real-time ω = 0 with FBMC+ CC resultingin higher cost for congestion management. With real-time deviations ω > 0cost for congestion management are overall lower in the FBMC+ CC dueto reduced exchange margins in response of expected deviations and the setgenerator’s response.

Table 5: Cost for redispatch (R) and curtailment (C) in relation to forecast error ω.

ω = 0 ω > 0C R C+R C R C+R

FBMC+ 42.73 243.57 286.29 41.42 248.41 289.83FBMC+ CC 44.43 246.6 291.03 42.09 245.16 287.25

The result is visualized in Fig. 7 that visualizes the range of hourly costfor congestion management in for randomized different real-time deviations.The blue band shows the range of cost for FBMC+ and the red band show costfor FBMC+ CC that includes the FRMs. The solid lines are the hourly costwithout real-time deviations. The figure visualizes that, while often aligned,the FBMC+ CC case provides a tighter band that on average provides lowercost.

25

Jun 16 Jun 18 Jun 20 Jun 22 Jun 24 Jun 26

20

40

60

Hou

rly

Cos

ts [

k $]

Figure 7: Range of hourly cost for congestion management for randomised real-time devi-ations ω for FBMC+ (blue) and FBMC+ CC (red)

5.6. Conclusion

In this paper we illustrate considerations necessary to model FBMC, inparticular parameter choices that are policy relevant and consider high sharesof intermittent renewable generation. The numerical experiments, conductedwith the open power system model POMATO, highlight the effectivenessof FBMC in comparison to zonal market configurations using NTCs andquantifies the trade-off between permissive capacity allocation and increasedcongestion management.

Results on the extended IEEE 118 bus network of Pena et al. (2017)show, that for high shares of intermittent generation more permissive com-mercial exchange capacities provide the most effective configurations. Here,the additional cost for congestion management do not overcompensate lowercost in the market clearing stage. For smaller RES shares, more restrictivedomains prove efficient.

With high shares of intermittent generation, the system becomes moresusceptible to real-time deviation or forecast errors. We propose a chanceconstraint formulation that builds on the forecasting characteristics of the D-2 basecase and results in risk aware security FRMs in the day-ahead marketclearing stage. Initially, the reduced commercial exchange capacity provideshigher system cost but proves more robust against real-time deviations thanthe deterministic counter part. The proposed formulation is a suitable ex-tension to modeling FBMC.

26

Aknowledements

The authors gratefully acknowledge the support by the German Fed-eral Ministry for Economic Affairs and Energy (BMWi) in the project“Modellierung (De-)Zentraler Energiewenden: Wechselwirkungen, Koordina-tion und Losungsansatze aus systemorientierter Perspektive” (MODEZEEN,03EI1019B)

References

50Hertz, Amprion, APG, Creos, Elia, Rte, TenneT, TransnetBW, 2020. Doc-umentation of the CWE FB MC Solution - Version 5.0. Technical Report.

ACER, CEER, 2018. Annual Report on the Results of Monitoring the Inter-nal Electricity and Natural Gas Markets in 2017 - Electricity WholesaleMarkets Volume. Report.

Aguado, M., Bourgeois, R., Bourmaud, J., Van Casteren, J., Ceratto, M.,Jakel, M., Malfiet, B., Mestdag, C., Noury, P., Pool, M., Van Den Reek,W., Rohleder, M., Schavemaker, P., Scolari, S., Weis, O., Wolpert, J., 2012.Flow-Based Market Coupling in the Central Western European Region: Onthe Eve of Implementation. Technical Report C5-204. Cigre.

Amprion, 2019. Amprion Market Report 2019 - Flow Based Market Coupling:Development of the Market and Grid Situation 2015-2018. Report.

Amprion, APX-ENDEX, Belpex, Creos, Elia, EnBW, EPEX SPOT, RTE,TenneT, 2011. CWE Enhanced Flow-Based MC feasibility report.

Bienstock, D., Chertkov, M., Harnett, S., 2014. Chance-Constrained Opti-mal Power Flow: Risk-Aware Network Control under Uncertainty. SIAMReview 56, 461–495.

Byers, C., Hug, G., 2020. Modeling flow-based market coupling: Base case,redispatch, and unit commitment matter, in: 2020 17th International Con-ference on the European Energy Market, pp. 1–6.

Dierstein, C., 2017. Impact of Generation Shift Key determination on flowbased market coupling, in: 2017 14th International Conference on theEuropean Energy Market, pp. 1–7.

Directorate General for Energy, 2019. Clean Energy for All Europeans. WhitePaper. European Commision.

27

Dvorkin, Y., 2020. A chance-constrained stochastic electricity market. IEEETransactions on Power Systems 35, 2993–3003.

Dvorkin, Y., Lubin, M., Backhaus, S., Chertkov, M., 2016. Uncertainty Setsfor Wind Power Generation. IEEE Transactions on Power Systems 31,3326–3327.

ETSO, 2001. Co-Ordinated Auctioning: A Market-Based Method for Trans-mission Capacity Allocation in Meshed Networks. Technical Report.

ETSO, EuroPEX, 2004. Flow-Based Market Coupling: A Joint ETSO-EuroPEX Proposal for Cross-Border Congestion Management and Inte-gration of Electricity Markets in Europe. Technical Report.

European Commission, 1997. Directive 96/92/EC concerning common rulesfor the internal market in electricity.

European Commission, 2015. Commission Regulation (EU) 2015/1222: Es-tablishing a guideline on capacity allocation and congestion management.

European Commission, 2019a. Commission Regulation (EU) 2019/943 onthe internal market for electricity.

European Commission, 2019b. Directive (EU) 2019/944 on common rulesfor the internal market for electricity.

Finck, R., Ardone, A., Fichtner, W., 2018. Impact of Flow-Based MarketCoupling on Generator Dispatch in CEE Region, in: 2018 15th Interna-tional Conference on the European Energy Market, pp. 1–5.

Glachant, J.M., 2010. The Achievement of the EU Electricity Internal Marketthrough Market Coupling. EUI Working Papers RSCAS 2010/87. FlorenceSchool of Regulation.

Gurobi Optimization LLC, 2018. Gurobi Optimizer Reference Manual.

Jiachun, G., Yong, F., Zuyi, L., Shahidehpour, M., 2009. Direct Calculationof Line Outage Distribution Factors. IEEE Transactions on Power Systems24, 1633–1634.

Marien, A., Luickx, P., Tirez, A., Woitrin, D., 2013. Importance of de-sign parameters on flowbased market coupling implementation, in: 201310th International Conference on the European Energy Market, Stock-holm, Sweden. pp. 1–8.

28

Matthes, B., Spieker, C., Klein, D., Rehtanz, C., 2019. Impact of a MinimumRemaining Available Margin Adjustment in Flow-Based Market Coupling,in: 2019 IEEE Milan PowerTech, pp. 1–6.

Meeus, L., Belmans, R., 2008. Electricity market integration in Europe, in:16th Power Systems Computation Conference 2008, p. 1505.

Mieth, R., 2021. Risk-Aware Control, Dispatch and Coordination in Sustain-able Power Systems. Dissertation. Technische Universitat Berlin. Berlin.

MOSEK ApS, 2021. MOSEK Modeling Cookbook. Documentation Release3.2.3.

Pena, I., Martinez-Anido, C.B., Hodge, B.M., 2017. An extended IEEE 118-bus test system with high renewable penetration. IEEE Transactions onPower Systems 33, 281–289.

Roald, L., Oldewurtel, F., Van Parys, B., Andersson, G., 2015. SecurityConstrained Optimal Power Flow with Distributionally Robust ChanceConstraints. arXiv Preprint 1508.06061. arXiv:1508.06061.

Rte, Amprion, Creos, Elia, TenneT, TransnetBW, 2015. CWE Flow BasedMarket Coupling project: Parallel Run performance report.

Schavemaker, P., Croes, A., Otmani, R., Bourmaud, J.Y., Zimmermann, U.,Wolpert, J., Reyer, F., Weis, O., Druet, C., 2008. Flow-Based Allocationin the Central Western European Region. Technical Report. Cigre.

Schonheit, D., Bruninx, K., Kenis, M., Most, D., 2021a. Improved Selectionof Critical Network Elements for Flow-Based Market Coupling Based onCongestion Patterns. Technical Report. ZBW – Leibniz Information Centrefor Economics.

Schonheit, D., Dierstein, C., Most, D., 2021b. Do minimum trading capacitiesfor the cross-zonal exchange of electricity lead to welfare losses? EnergyPolicy 149, 112030.

Schonheit, D., Kenis, M., Lorenz, L., Most, D., Delarue, E., Bruninx, K.,2021c. Toward a fundamental understanding of flow-based market couplingfor cross-border electricity trading. Advances in Applied Energy 2, 100027.

Schonheit, D., Kenis, M., Lorenz, L., Most, D., Delarue, E., Brunix, K.,2020a. Toward understanding flow-based market coupling: An open-accessmodel. Energy Systems Integration & Modeling Group Working PaperSeries .

29

Schonheit, D., Weinhold, R., Dierstein, C., 2020b. The impact of differentstrategies for generation shift keys (GSKs) on the flow-based market cou-pling domain: A model-based analysis of Central Western Europe. AppliedEnergy 258, 114067.

Voswinkel, S., Felten, B., Felling, T., Weber, C., 2019. Flow-Based MarketCoupling – What Drives Welfare in Europe’s Electricity Market Design?HEMF Working Paper No. 08/2019. University of Duisburg-Essen, Houseof Energy Markets & Finance.

Weinhold, R., Mieth, R., 2020a. Fast Security-Constrained Optimal PowerFlow Through Low-Impact and Redundancy Screening. IEEE Transactionson Power Systems 35, 4574–4584.

Weinhold, R., Mieth, R., 2020b. Power Market Tool (POMATO) for theAnalysis of Zonal Electricity Markets. arXiv Preprint 2011.11594v1.

Wyrwoll, L., Blank, A., Muller, C., Puffer, R., 2019. Determination ofPreloading of Transmission Lines for Flow-Based Market Coupling, in:2019 16th International Conference on the European Energy Market, pp.1–6.

30