The liberalization of the European gas markets started in the 1990s and = lead to the current situation in which European transmission system operato= rs (TSOs) typically operate under the so-called Entry-Exit system [1, 2, 3]= . The timing of this system is as follows: The TSOs have to publish so-call= ed technical capacities at every entry or exit point of the their network. = Afterward, gas traders can book capacities that are bounded above by the co= rresponding technical capacity. The booking is a capacity right that ensure= s that the trader can inject (as an entry customer) or withdraw (as an exit= customer) balanced amounts of gas up to the booked capacity. The latter pr= ocess is called nomination and the TSOs have to be able to transport all po= ssible nomination situation as they are (via the bookings) conformal to the= published technical capacities.

The current entry-exit system can be addressed by mathematical modeling =
in various ways. From the perspective of the evaluation of this market desi=
gn one is faced with multilevel models that are made up of the following le=
vels:

a) Computation of technical capacities by the TSO

b) Bookin=
g by gas traders

c) Nomination by gas traders

d) Transport by the TSO=

For the ease of presentation we refrained from discussing secondary=
intra-day markets [4, 5].

The first mathematical challenge is the r=
obustness the TSO has to address when computing the technical capacities in=
level a): All balanced nominations that are restricted by the bookings tha=
t themselves have to be in line with the TSO's technical capacities have to=
transportable by the TSO. Feasibility of transport depends on the physical=
model of gas flow and of the chosen models of technical entities of gas tr=
ansport networks like compressor and control valve stations, filters, measu=
rement devices, etc. The former is typically modeled by systems of nonlinea=
r and hyperbolic partial differential equations (the Euler equations, cf., =
e.g., [6]) on a graph. The latter are mainly modeled by algebraic but highl=
y nonlinear discrete-continuous equality and inequality systems [7, 8]. Ass=
uming the TSO's goal of cost-minimal transport of nominations, the levels a=
) and d) alone lead to adjustable robust mixed-integer nonlinear optimizati=
on problems that are subject to hyperbolic partial differential equations o=
n a graph.

Since the acting agents (TSO and gas traders) in this mar=
ket game typically have different objectives one is readily confronted with=
multilevel optimization or complementarity problems in levels b) and c) th=
at are intermediate levels in the overall equilibrium problem.

Although the mathematical model described so far is extremely challengin= g and far beyond the border of what can be solved with the current state of= mathematical theory and algorithmic technology, there are still a lot of p= ossible extensions of this setting. One possible extension is the considera= tion of uncertainty in the given setting: cf., e.g. [9, 10]. Typically, the= exact gas demand is unknown before booking and nomination. Thus, both stoc= hastic and robust optimization techniques may be employed to address this i= ssue.

**References**

[1] Dir 1998/30/EC Directive 98/30/EC of the European Parliament and of =
the Council of 22 June 1998 concerning common rules for the internal market=
in natural gas (OJ L 204 pp. 1=E2=80=9312).

[2] Dir 2003/55/EC Dire=
ctive 2003/55/EC of the European Parliament and of the Council of 26 June 2=
003 concerning common rules for the internal market in natural gas and repe=
aling Directive 98/30/EC (OJ L 176 pp. 57=E2=80=9378).

[3] Dir 2009/=
73/EC Directive 2009/73/EC of the European Parliament and of the Council co=
ncerning common rules for the internal market in natural gas and repealing =
Directive 2003/55/EC (OJ L 211 pp. 36=E2=80=9354).

[4] Keyaerts, N. =
and W. D=E2=80=99haeseleer (2014). "Forum shopping for ex-post gas-balancin=
g services." In: Energy Policy 67, pp. 209=E2=80=93221. doi: 10.1016/j.enpo=
l.2013.11.062.

[5] Keyaerts, N., M. Hallack, J.-M. Glachant, and W. =
D=E2=80=99haeseleer (2011). "Gas market distorting effects of imbalanced ga=
s balancing rules: Inefficient regulation of pipeline flexibility." In: Ene=
rgy Policy 39.2, pp. 865=E2=80=93876. doi: 10.1016/j.enpol. 2010.11.006.

[6] Brouwer, J., I. Gasser, and M. Herty (2011). "Gas Pipeline Models =
Revisited: Model Hierarchies, Nonisothermal Models, and Simulations of Netw=
orks." In: Multiscale Modeling & Simulation 9.2, pp. 601=E2=80=93623. d=
oi: 10.1137/100813580.

[7] Fu=CC=88genschuh, A., B. Gei=C3=9Fler, R.=
Gollmer, A. Morsi, M. E. Pfetsch, J. Ro=CC=88vekamp, M. Schmidt, K. Spreck=
elsen, and M. C. Steinbach (2015). "Physical and technical fundamentals of =
gas networks." In: Evaluating Gas Network Capacities. Ed. by T. Koch, B. Hi=
ller, M. E. Pfetsch, and L. Schewe. SIAM-MOS series on Optimization. SIAM. =
Chap. 2, pp. 17=E2=80=9343. doi: 10.1137/1.9781611973693.ch2.

[8] Sc=
hmidt, M., M. C. Steinbach, and B. M. Willert (2016). "High detail stationa=
ry optimization models for gas networks: validation and results." In: Optim=
ization and Engineering 17.2, pp. 437=E2=80=93472. doi: 10.1007/s11081-015-=
9300-3.

[9] Zhuang, J. and S. A. Gabriel (2008). =E2=80=9CA compleme=
ntarity model for solving stochastic natural gas market equilibria.=E2=80=
=9D In: Energy Economics 30.1, pp. 113=E2=80=93147. doi: 10.1016/j.eneco.20=
06.09.004.

[10] Gabriel, S. A., J. Zhuang, and R. Egging (2009). =E2=
=80=9CSolving stochastic complementarity problems in energy market modeling=
using scenario reduction.=E2=80=9D In: European Journal of Operational Res=
earch 197.3, pp. 1028=E2=80=931040. doi: 10.1016/j.ejor. 2007.12.046.**[11] V. Grimm, L. Schewe, M. Schmidt, and G. Zo=CC=88ttl. A Multilevel Mo=
del of the European Entry-Exit Gas Market. Tech. rep. Friedrich-Alexander-U=
niversita=CC=88t Erlangen-Nu=CC=88rnberg, 2017. URL: http://w=
ww.optimization-online.org/ DB_HTML/2017/05/6002.html.**

**Contributor:**

Dr Martin Schmidt, Friedrich-Alexander-Universit=C3=A4t