Evaluating the impacts of the external supply risk in a natural gas supply chain: the case of the Italian market

Abstract

A large part of the European natural gas imports originates from unstable regions exposed to the risk of supply failure due to economical and political reasons. This has increased the concerns on the security of supply in the European natural gas market. In this paper, we analyze the security of external supply of the Italian gas market that mainly relies on natural gas imports to cover its internal demand. To this aim, we develop an optimization problem that describes the equilibrium state of a gas supply chain where producers, mid-streamers, and final consumers exchange natural gas and liquefied natural gas. Both long-term contracts (LTCs) and spot pricing systems are considered. Mid-streamers are assumed to be exposed to the external supply risk, which is estimated with indicators that we develop starting from those already existing in the literature. In addition, we investigate different degrees of mid-streamers’ flexibility by comparing a situation where mid-streamers fully satisfy the LTC volume clause (“No FLEX” assumption) to a case where the fulfillment of this volume clause is not compulsory (“FLEX” assumption). Our analysis shows that, in the “No FLEX” case, mid-streamers do not significantly change their supplying choices even when the external supply risk is considered. Under this assumption, they face significant profit losses that, instead, disappear in the “FLEX” case when mid-streamers are more flexible and can modify their supply mix. However, the “FLEX” strategy limits the gas availability in the supply chain leading to a curtailment of the social welfare.

Appendices

Appendix A: Complementarity formulation of the welfare optimization problem under the “no flexibility” assumption

In this appendix, we report the complementarity formulation of the welfare optimization problem presented in Sect. 3.4.1.

$$\begin{aligned}&0 \le -\gamma _{nt}+ {\partial C_{nt}\left( X^{G}_{nt}\right) \over \partial X_{nt}^{G}}+\overline{\gamma }_{nt} \perp X^G_{nt} \ge 0\quad \forall n, \forall t \end{aligned}$$

(42)

$$\begin{aligned}&0 \le - (1-\alpha _n) \cdot \delta _{nt}+(1-\alpha _n)\cdot \overline{\delta }_{nt} +\gamma _{nt}+ {\partial LC_{nt}\left( X^{{ LNG}}_{nt}\right) \over \partial X_{nt}^{{ LNG}}}\perp X^{{ LNG}}_{nt} \ge 0\ \forall n, \forall t\nonumber \\ \end{aligned}$$

(43)

$$\begin{aligned}&0 \le - p^{G}_{nmt} + ptc_{nm}^G+ \gamma _{nt}\perp x^G_{nmt} \ge 0\quad \forall n, \forall m, \forall t \end{aligned}$$

(44)

$$\begin{aligned}&0 \le -p^{SpotG}_{t} + \gamma _{nt}+ptc_{n}^{SpotG}+ \sum _{f=1}^F\varGamma _{fn} \cdot \kappa _{ft} \perp x^{SpotG}_{nt}\ge 0\quad \forall n, \forall t \quad \quad \end{aligned}$$

(45)

$$\begin{aligned}&0 \le -p^{{ LNG}}_{nmt} + stc^{{ LNG}}_{nm}+ \delta _{nt} \perp x^{{ LNG}}_{nmt}\ge 0\quad \forall n, \forall m, \forall t \quad \quad \end{aligned}$$

(46)

$$\begin{aligned}&0 \le -p^{Spot{ LNG}}_{nmt} + stc^{{ LNG}}_{nm}+ \delta _{nt} \perp x^{Spot{ LNG}}_{nmt}\ge 0\quad \forall n, \forall m, \forall t \quad \quad \end{aligned}$$

(47)

$$\begin{aligned}&0 \le -p_{st}+ dc_{ms} +\lambda _{mt} \perp z_{mst}\ge 0 \quad \forall m, \forall s, \forall t \end{aligned}$$

(48)

$$\begin{aligned}&0 \le -(1-\beta _m) \cdot \eta _{mt}+{\partial RC_{m}\left( Y^{{ LNG}}_{mt}\right) \over \partial Y^{{ LNG}}_{mt}}+(1-\beta _m) \cdot \overline{\eta }_{mt} \perp Y^{{ LNG}}_{mt} \ge 0 \quad \forall m, \forall t \end{aligned}$$

(49)

$$\begin{aligned}&0 \le -\lambda _{mt} -\psi _{nm}^G+\varPi _{nm}^{G} \cdot p^{G}_{nmt}+ \sum _{f=1}^F\varGamma _{fn} \cdot \kappa _{ft} \perp y^G_{nmt} \ge 0 \quad \forall n, \forall m, \forall t \end{aligned}$$

(50)

$$\begin{aligned}&0 \le -\lambda _{mt}+p^{SpotG}_{t} \perp y^{SpotG}_{mt} \ge 0 \quad \forall m, \forall t \end{aligned}$$

(51)

$$\begin{aligned}&0 \le - \lambda _{mt} -\psi _{nm}^{{ LNG}}+\varPi _{nm}^{{ LNG}}\cdot p^{{ LNG}}_{nmt} +\eta _m\perp y^{{ LNG}}_{nmt} \ge 0 \quad \forall n, \forall m, \forall t \quad \quad \end{aligned}$$

(52)

$$\begin{aligned}&0 \le - \lambda _{mt}+p^{{ LNG}}_{nmt} +\eta _m \perp y^{Spot{ LNG}}_{nmt} \ge 0 \quad \forall m, \forall t \quad \quad \end{aligned}$$

(53)

$$\begin{aligned}&0 \le -p^{SpotG}_{t}+\lambda _{mt} \perp q^{SpotG}_{mt} \ge 0 \quad \forall m, \forall t \end{aligned}$$

(54)

$$\begin{aligned}&0 \le -\mu _{m} + {\partial I_{m1}\left( i_{m1}\right) \over \partial i_{m1}} +\nu _{m}+\lambda _{m1}\perp i_{m1} \ge 0 \quad \forall m, t=1 \end{aligned}$$

(55)

$$\begin{aligned}&0 \le -\lambda _{m2} +\mu _{m}+\sigma _m+\phi _m\perp w_{m2} \ge 0 \quad \forall m, \forall t=2 \end{aligned}$$

(56)

$$\begin{aligned}&0 \le \bar{X}_{n}-X^G_{nt} \perp \overline{\gamma }_{nt} \ge 0 \quad \forall n, \forall t \end{aligned}$$

(57)

$$\begin{aligned}&0 \le \bar{L}_{n}- (1-\alpha _n) \cdot X^{{ LNG}}_{nt} \perp \overline{\delta }_{nt} \ge 0 \quad \forall n, \forall t \end{aligned}$$

(58)

$$\begin{aligned}&0 \le \bar{R}_{m}- (1-\beta _m) \cdot Y^{{ LNG}}_{mt} \perp \overline{\eta }_{mt} \ge 0 \quad \forall m, \quad \forall t\end{aligned}$$

(59)

$$\begin{aligned}&0 \le \sum _{n=1}^N y_{nmt}^{G}+ \sum _{n=1}^N y^{{ LNG}}_{nmt}+ y^{SpotG}_{mt}+ \sum _{n=1}^N y^{Spot{ LNG}}_{nmt}+\nonumber \\&-i_{mt} -\sum _{s=1}^S z_{mst}- q_{mt}^{SpotG}\perp \lambda _{mt} \ge 0 \quad \forall m, t=1 \end{aligned}$$

(60)

$$\begin{aligned}&0 \le \sum _{n=1}^N y_{nmt}^{G}+ \sum _{n=1}^N y^{{ LNG}}_{nmt}+ y^{SpotG}_{mt}+ \sum _{n=1}^N y^{Spot{ LNG}}_{nmt}+w_{mt} +\nonumber \\&-\sum _{s=1}^S z_{mst}- q_{mt}^{SpotG} \perp \lambda _{mt} \ge 0 \quad \forall m, \quad t=2 \end{aligned}$$

(61)

$$\begin{aligned}&0 \le i_{m1}-w_{m2} \perp \mu _{m}\ge 0 \quad \forall m \end{aligned}$$

(62)

$$\begin{aligned}&0 \le \bar{I}_{m}- i_{m1} \perp \nu _{m}\ge 0 \quad \forall m \end{aligned}$$

(63)

$$\begin{aligned}&0 \le \bar{W}_{m}- w_{m2} \perp \sigma _{m} \ge 0 \quad \forall m\end{aligned}$$

(64)

$$\begin{aligned}&0 \le {{WG}_{m} }- \theta _2 \cdot w_{m2} \perp \phi _{m} \ge 0 \quad \forall m \end{aligned}$$

(65)

$$\begin{aligned}&0 \le \sum _t \theta _t \cdot y_{nmt}^{G} -\tau _{nm} \perp \psi _{mn}^{G}\ge 0 \quad \forall n, \forall m \end{aligned}$$

(66)

$$\begin{aligned}&0 \le \sum _t \theta _t \cdot y_{nmt}^{{ LNG}} -\xi _{nm}\perp \psi _{mn}^{{ LNG}} \ge 0 \quad \forall n, \forall m \end{aligned}$$

(67)

$$\begin{aligned}&0 \le \varUpsilon _{ft}-\left( \sum _{n=1}^N\sum _{m=1}^M \varGamma _{fn} \cdot y^{G}_{nmt}+ \sum _{n=1}^N \varGamma _{fn} \cdot x^{SpotG}_{nt} \right) \perp \kappa _{ft}\ge 0 \quad \forall t \end{aligned}$$

(68)

$$\begin{aligned}&0 \le p_{st}-a_{st}+b_{st} \cdot d_{st} \perp d_{st} \ge 0 \quad \forall s, \forall t\end{aligned}$$

(69)

$$\begin{aligned}&X^G_{nt}-\left( \sum _{m=1}^M x^G_{nmt}+x^{SpotG}_{nt}+X^{{ LNG}}_{nt} \right) = 0 \quad \forall n, \forall t \qquad (\gamma _{nt}: \mathrm{free}) \end{aligned}$$

(70)

$$\begin{aligned}&(1-\alpha _n) \cdot X^{{ LNG}}_{nt}-\left( \sum _{m=1}^M x^{{ LNG}}_{nmt}+\sum _{m=1}^M x^{Spot{ LNG}}_{nmt} \right) = 0 \quad \forall n, \forall t \qquad (\delta _{nt}: \mathrm{free})\end{aligned}$$

(71)

$$\begin{aligned}&(1-\beta _m) \cdot Y^{{ LNG}}_{mt} -\left( \sum _{n=1}^N y^{{ LNG}}_{nmt}+\sum _{n=1}^N y^{Spot{ LNG}}_{nmt}\right) = 0 \ \forall m, \forall t \ ({\eta }_{mt}: \mathrm{free})\end{aligned}$$

(72)

$$\begin{aligned}&x_{nmt}^{G}- \varPi _{nm}^{G}\cdot y_{nmt}^{G} = 0 \quad \forall n, \forall m, \forall t \ (p^{G}_{nmt}: \mathrm{free}) \end{aligned}$$

(73)

$$\begin{aligned}&\sum _{n=1}^N x_{nt}^{SpotG}+q_{mt}^{SpotG}-\sum _{m=1}^M y_{mt}^{SpotG} = 0 \quad \forall t \quad (p^{SpotG}_t: \mathrm{free}) \end{aligned}$$

(74)

$$\begin{aligned}&{x}^{{ LNG}}_{mnt}- \varPi _{nm}^{{ LNG}}\cdot y_{mnt}^{{ LNG}}= 0 \quad \forall n, \forall m, \quad \forall t \qquad (p_{mnt}^{{ LNG}}: \mathrm{free}) \end{aligned}$$

(75)

$$\begin{aligned}&x_{nmt}^{Spot{ LNG}}- y_{nmt}^{Spot{ LNG}} = 0 \quad \forall n, \forall m, \forall t \quad (p^{Spot{ LNG}}_{nmt}: \mathrm{free}) \end{aligned}$$

(76)

$$\begin{aligned}&\sum _{m} {z}_{mst}- d_{st} = 0 \quad \forall s, \forall t \qquad (p_{st}: \mathrm{free}) \end{aligned}$$

(77)

Appendix B: Complementarity formulation of the welfare optimization problem under the “flexibility” assumption

To model the “flexibility” assumption we only replace constraints (31) and (32) in the welfare optimization problem described in Sect. 3.4.1 with the constraints (39) and (40) presented in Sect. 3.4.2. This modification leads to some changes in the KKT conditions presented in “Appendix A”. In particular, conditions (50), (52), (66), and (67) are respectively substituted with the following ones:

$$\begin{aligned}&0 \le -\lambda _{mt} +\psi _{nm}^G+\varPi _{nm}^{G} \cdot p^{G}_{nmt}+ \sum _{f=1}^F\varGamma _{fn} \cdot \kappa _{ft} \perp y^G_{nmt} \ge 0 \quad \forall n, \forall m, \forall t \end{aligned}$$

(78)

$$\begin{aligned}&0 \le - \lambda _{mt} +\psi _{nm}^{{ LNG}}+\varPi _{nm}^{{ LNG}}\cdot p^{{ LNG}}_{nmt} +\eta _m\perp y^{{ LNG}}_{nmt} \ge 0 \quad \forall n, \forall m, \forall t \quad \end{aligned}$$

(79)

$$\begin{aligned}&0 \le \tau _{nm}-\sum _t \theta _t \cdot y_{nmt}^{G} \perp \psi _{mn}^{G}\ge 0 \quad \forall n, \forall m \end{aligned}$$

(80)

$$\begin{aligned}&0 \le \xi _{nm} -\sum _t \theta _t \cdot y_{nmt}^{{ LNG}}\perp \psi _{mn}^{{ LNG}}\ge 0 \quad \forall n, \forall m \end{aligned}$$

(81)

More precisely, since constraints (39) and (40) impose upper bounds on the primal variables $y^G_{nmt}$ and $y^{{ LNG}}_{nmt}$, the associated dual variables $\psi _{mn}^{G}$ and $\psi _{mn}^{{ LNG}}$ enter with a positive sign in the KKT conditions (78) and (79) of these primal variables. The reverse happens in the complementarity formulation of the optimization problem under the “no flexibility” assumption. Since constraints (31) and (32) define lower bounds on variables $y^G_{nmt}$ and $y^{{ LNG}}_{nmt}$ the associated dual variables $\psi _{mn}^{G}$ and $\psi _{mn}^{{ LNG}}$ enter with a negative sign in the KKT conditions (50) and (52) of these primal variables. Finally, all the other KKT conditions are as indicated in “Appendix A”.

Appendix C: Additional results

Fig. 7

Volumes of gas exchanged via LTCs under the “Risk” and “No FLEX” assumptions (mcm/day)

Fig. 8

Volumes of LNG exchanged via LTCs under the “Risk” and “No FLEX” assumptions (mcm/day)

Fig. 9

Volumes of spot gas exchanged under the “Risk” and “No FLEX” assumptions (mcm/day)

Fig. 10

Volumes of spot LNG exchanged under the “Risk” and “No FLEX” assumptions (mcm/day)

Fig. 11

Volumes of natural gas exchanged via LTCs under the “Risk” and “FLEX” assumptions (mcm/day)

Fig. 12

Volumes of spot gas exchanged under the “Risk” and “FLEX” assumptions (mcm/day)

Fig. 13

Volumes of spot LNG exchanged under the “Risk” and “FLEX” assumptions (mcm/day)