A Generalized Nash–Cournot Model for the Northwestern European Natural Gas Markets with a Fuel Substitution Demand Function: The GaMMES Model

Abstract

This article presents a dynamic Generalized Nash–Cournot model to describe the evolution of the natural gas markets. The major players along the gas chain are depicted including: producers, consumers, storage and pipeline operators, as well as intermediate local traders. Our economic structure description takes into account market power and the demand representation tries to capture the possible fuel substitution that can be made between the consumption of oil, coal, and natural gas in the overall fossil energy consumption. We also take into account long-term contracts in an endogenous way, which makes the model a Generalized Nash Equilibrium problem. We discuss some means to solve such problems. Our model has been applied to represent the European natural gas market and forecast, until 2030, after a calibration process, consumption, prices, production, and natural gas dependence. A comparison between our model, a more standard one that does not take into account energy substitution, and the European Commission natural gas forecasts is carried out to analyze our results. Finally, in order to illustrate the possible use of fuel substitution, we studied the evolution of the natural gas price as compared to the coal and oil prices.

Appendix A

This appendix presents the KKT conditions derived from our model. Once the KKT conditions are written, we get the Mixed Complementarity Problem (MCP) given below.

The producers KKT conditions

$$ \begin{array}{rll} \forall t,\ m,\ f,\ p,\ i,\ {\kern-20pt}0 \leq zp^t_{mfpi} {\kern-24pt}&\,\bot \;\; \delta^t \eta_{pi}-\gamma^t_{mfp} {\kern40pt} \leq 0{\kern-22pt} \\ & - \epsilon 2^t_{mfpi}-\eta p^t_{pi} \\ & -\sum\limits_n M2_{in}\alpha p^t_{mpn} \end{array} $$

(21a)

$$ \begin{array}{rll} \forall t,\ m,\ f,\ p,\ d,\ 0 \leq x^t_{mfpd} \, \bot \;\;& \delta^t p^t_{md}\big(x^t_{mfpd}+\overline{x^t_{mfpd}}\,\big) & \leq 0 \\ & +\delta^{t}\frac{\partial p^{t}_{md}}{\partial x^t_{mfpd}} & \\ & \times\big(x^t_{mfpd}+\overline{x^t_{mfpd}}\,\big)x^{t}_{mfpd} & \\ & -\gamma^t_{mfp}-\epsilon 1^t_{mfpd} & \\ & -\sum\limits_n M3_{dn} \alpha p^t_{mpn} & \\ \end{array} $$

(21b)

$$ \begin{array}{rll} \forall t,\ m,\ f,\ p, 0 \leq q^t_{mfp} \, \bot \;\;& -\sum\limits_{t'\geq t} \delta^{t'} \frac{\partial Pc_f}{\partial q} & \leq 0 \\ & \times\left(\,\sum\limits_{t''\leq t'} \sum\limits_m q^{t''}_{mfp},Rf_f\right) & \\ & + \sum\limits_{t'> t} \delta^{t'} \frac{\partial Pc_f}{\partial q} & \\ & \times\left(\,\sum\limits_{t''< t'} \sum\limits_m q^{t''}_{mfp},Rf_f\right) & \\ & - \sum\limits_{t' \geq t} \phi^{t'}_f- \chi^t_{mf} +\gamma^t_{mfp}& \\ &-(-1)^m \big(\vartheta1^t_{f}- \vartheta2^t_{f}\big) & \\ &-\epsilon 3^t_{mfp} +\sum\limits_n M1_{fn} \alpha p^t_{mpn} & \\[4pt] \end{array} $$

(21c)

$$ \begin{array}{rll} \forall t,\ f,\ p, 0 \leq ip^t_{fp} \, \bot \;\; & -\delta^{t} Ip_f-\epsilon 4^t_{fp} & \leq 0 \\ & +\sum\limits_m \sum\limits_{t'\geq t+delay_p} & \\ & \times\chi^{t'}_{mf}(1-dep_f)^{t'-t} & \\ & -\iota p^t_f + Lf_f \sum\limits_{t'\geq t+delay_p} & \\ & \times\iota p^{t'}_{f}(1-dep_f)^{t'-t} & \\ \end{array} $$

(21d)

$$ \forall t,\ p,\ i, 0 \leq up_{pi} \, \bot \;\; \sum\limits_t \eta p^t_{pi}- \eta_{pi} \;\, \leq 0 $$

(21e)

$$ \forall t,\ f, 0 \leq \phi^t_f \, \bot \;\; \sum\limits_p\sum\limits_{t'\leq t} \sum\limits_m q^{t'}_{mfp} - Rf_f \;\, \leq 0 $$

(21f)

$$ \begin{array}{rll} \forall t,\ m,\ f,\ \;\, 0 \leq \chi^t_{mf} \;\, \bot \;\, & \sum\limits_p q^t_{mfp} - Kf_f (1-dep_f)^{t} &\;\, \leq 0 \\ & - \sum\limits_p \sum\limits_{t'\leq t-delay_p} & \\ & \times ip^{t'}_{fp}(1-dep_f)^{t-t'} & \\[5pt] \end{array} $$

(22a)

$$ \begin{array}{rll} \forall t,\ m,\ f,\ p,\ \;\, 0 \leq \gamma^t_{mfp} \;\, \bot \;\, & -q^t_{mfp}+\sum\limits_i zp^t_{mfpi}& \;\, \leq 0 \\[-3pt] & +\sum\limits_d x^t_{mfpd} & \\[5pt] \end{array} $$

(22b)

$$ \forall t,\ f,\ \;\, 0 \leq \vartheta1^t_{f} \;\, \bot \;\, \sum\limits_m \sum\limits_p (-1)^m q^t_{mfp} -f\/l_f \;\, \leq 0 $$

(22c)

$$ \forall t,\ f, \, 0 \leq \vartheta2^t_{f} \;\, \bot \;\, -\sum\limits_m \sum\limits_p (-1)^m q^t_{mfp} -f\/l_f \;\, \leq 0 $$

(22d)

$$ \begin{array}{rll} \forall t,\ f,\ \;\, 0 \leq \iota p^t_{f} \;\, \bot \;\, & \sum\limits_p ip^t_{fp} &\;\, \leq 0 \\[-3pt] & - Lf_f Kf_f (1-dep_f)^{t} & \\[-3pt] & -Lf_f \sum\limits_p \sum\limits_{t'\leq t-delay_p} & \\[-3pt] & \times ip^{t'}_{fp}(1-dep_f)^{t-t'} & \\[5pt] \end{array} $$

(22e)

$$ \forall t,\ f, m,\ p,\ d,\ \;\, 0 \leq \epsilon 1^t_{mfpd} \;\, \bot \;\, x^t_{mfpd}-O_{fp} H \;\, \leq 0 $$

(22f)

$$ \forall t,\ m, f,\ p,\ i, \, 0 \leq \epsilon 2^t_{mfpi} \;\, \bot \;\, zp^t_{mfpi}-O_{fp} H \;\, \leq 0 $$

(22g)

$$ \forall t,\ m, f,\ p,\ \;\, 0 \leq \epsilon 3^t_{mfp} \;\, \bot \;\, q^t_{mfp}-O_{fp} H \;\, \leq 0 $$

(22h)

$$ \forall t,\ f,\ p,\ \;\, 0 \leq \epsilon 4^t_{fp} \;\, \bot \;\, ip^t_{fp}-O_{fp} H \;\, \leq 0 $$

(22i)

$$ \begin{array}{rll} \forall t,\ m,\ p,\ n, \; \mbox{free }\;\, \alpha p^t_{mpn} \;\, & \sum\limits_{a} M_6(a,n)fp^t_{mpa} (1-loss_a)&\!\!\!\!=0 \\ &-\sum\limits_{a} M5_{an}fp^t_{mpa} +\sum\limits_f M1_{fn} q^t_{mpf} & \\ & -\sum\limits_d \sum\limits_f M3_{dn} x^t_{mfpd} & \\ & -\sum\limits_i \sum\limits_f M2_{in} zp^t_{mfpi} & \end{array} $$

(22j)

$$ \forall t,\ p,\ i, \; \mbox{free }\;\, \eta p^t_{pi} \;\, up_{pi}-\sum\limits_{f,m} zp^t_{mfpi} \;\, = 0 $$

(23a)

$$ \forall \ p,\ i,\ \;\, \mbox{free } \;\, \eta_{pi} \;\, ui_{pi}-up_{pi} \;\, =0 $$

(23b)

The independent traders’ KKT conditions

$$ \begin{array}{rll} \forall t,\ m,\ p,\ i, \, 0 \leq zi^t_{mpi} \;\, \bot \;\, & -\delta^t \eta_{pi} - \eta i^t_{pi} &\;\, \leq 0 \\[-3pt] & +\psi^t_{mi} & \\[-3pt] & +\sum\limits_n M2_{in} \alpha i^t_{min} & \\[-3pt] & +(1-min_{pi})\upsilon^t_{mpi} & \\[3pt] \end{array} $$

(24a)

$$ \begin{array}{rll} \forall t,\ m,\ i,\ d, \, 0 \leq y^t_{mid} \;\, \bot\;\, & \delta^t p^t_{md}\big(y^t_{mfpd}+\overline{y^t_{mfpd}}\,\big) & \leq 0 \\[-3pt] & \delta^{t} \frac{\partial p^{t}_{md}}{\partial y^t_{mid}}\big(y^t_{mfpd}+\overline{y^t_{mfpd}}\,\big)y^{t}_{mid}& \\[-3pt] & -\psi^t_{mi}-\sum\limits_{n} M3_{dn} \alpha i^t_{min} & \\[3pt] \end{array} $$

(24b)

$$ \forall t,\ i,\ s,\ \;\, 0 \leq r^t_{is} \;\, \bot \;\, -\delta^t Rc_s +\mu^t_{is} - \beta s^t_s \;\, \leq 0 $$

(24c)

$$ \begin{array}{rll} \forall t,\ i,\ s,\ \;\, 0 \leq in^t_{is} \;\, \bot \;\, & -\delta^t (Ic_s+ Wc_s) & \leq 0 \\[-3pt] & -\mu^t_{is}-\sum\limits_m (-1)^{m} \psi^t_{mi} & \\[-3pt] & -\sum\limits_{n} M4_{sn} \alpha i^t_{min} (-1)^m & \\[3pt] \end{array} $$

(24d)

$$ \forall t,\ p,\ i,\ \;\, 0 \leq ui_{pi} \;\, \bot \;\, \sum\limits_t \eta i^t_{pi} + \eta_{pi} \;\, \leq 0 $$

(24e)

$$ \forall t,\ m,\ i,\ {\kern-10pt} \;\,\mbox{free} \;\,\psi^t_{mi}{\kern-10pt} {\kern-20pt}\sum\limits_p zi^t_{mpi}-\sum\limits_d y^t_{mid}+(-1)^m\sum\limits_s in^t_{is} =0 $$

(25a)

$$ \forall t,\ i,\ s,\ \;\, 0 \leq \mu^t_{is} \;\, \bot \;\, in^t_{is}-r^t_{is} \;\, \leq 0 $$

(25b)

$$ \begin{array}{rll} \forall t,\ m,\ i,\ n,\ \;\,\mbox{free} \;\,\alpha i^t_{min} &\sum\limits_{a} M6_{an}f\/i^t_{mia}(1-loss_a) &=0 \\[-3pt] &-\sum\limits_{a} M5_{an}f\/i^t_{mia} & \\[-3pt] &-\sum\limits_d M3_{dn} y^t_{mid} & \\[-3pt] & +\sum\limits_p M2_{in} zi^t_{mpi} & \\[-3pt] & -(-1)^m \sum\limits_s M4_{sn} in^t_{is} & \end{array} $$

(25c)

$$ \forall t,\ p,\ i,\ \;\, \mbox{free } \;\, \eta i^t_{pi} \;\, ui_{pi}-\sum\limits_m zi^t_{mpi} \;\, =0 $$

(25d)

$$ \forall \ p,\ i,\ \;\, \mbox{free } \;\, \eta_{pi} \;\, ui_{pi}-up_{pi} \;\, =0 $$

(25e)

$$ \forall t,\ m,\ p,\ i,\ \;\, 0 \leq \upsilon^t_{mpi} \;\, -zi^t_{mpi} + min_{pi} \sum\limits_m zi^t_{mpi} \;\, \leq 0 $$

(25f)

$$ \forall t,\ s, \, 0 \leq \beta s^t_s \;\, \bot \;\, \sum\limits_i r^t_{is}-Ks_s-\sum\limits_{t'\leq t-delay_s} is^{t'}_s \;\, \leq 0 $$

(25g)

The pipeline operator KKT conditions

$$ \begin{array}{rll} \forall t,\ m,\ p,\ a,\ \;\, 0 \leq fp^t_{mpa} \;\, \bot \;\, & -\delta^t \big(Tc_{a}+\tau^t_{ma}\big) - \tau^t_{ma} &\;\, \leq 0 \\[-3.5pt] & +\sum\limits_n M6_{an} \alpha p^t_{mpn} (1-loss_a) &\\[-3.5pt] & -\sum\limits_n M5_{an} \alpha p^t_{mpn} & \\[3pt] \end{array} $$

(26a)

$$ \begin{array}{rll} \forall t,\ m,\ i,\ a,\ \;\, 0 \leq f\/i^t_{mia} \;\, \bot \;\,& -\delta^t \big(Tc_{a}+\tau^t_{ma}\big) - \tau^t_{ma} & \;\, \leq 0 \\[-3.5pt] &+\sum\limits_n M6_{an} \alpha i^t_{min} (1-loss_a) & \\[-3.5pt] & -\sum\limits_n M5_{an} \alpha i^t_{min} & \\[3pt] \end{array} $$

(26b)

$$ \begin{array}{rll} \forall t,\ a,\ \;\, 0 \leq ik^t_{a} \;\, \bot \;\,& -\delta^{t} Ik_{a}+\sum\limits_{t'\geq t+delay_i} \tau^{t'}_{ma} & \;\, \leq 0 \\[-3.5pt] & -\iota a^t_a + La_a \sum\limits_{t'\geq t+delay_i} \iota a^{t'}_{a} & \\[3pt] \end{array} $$

(26c)

$$ \begin{array}{rll} \forall t,\ m,\ a,\ \;\,0 \leq \tau^t_{ma} \;\,\bot\;\, &\sum\limits_p fp^t_{mpa}+ \sum\limits_i f\/i^t_{mia}&\leq 0 \\[-3.5pt] &-Tk_{a}-\sum\limits_{t'\leq t-delay_i} ik^t_{a}& \\[3pt] \end{array} $$

(26d)

$$ \forall t,\ a,\ \;\, 0 \leq \iota a^t_{a} \;\, \bot \;\, ik^t_{a}-Tk_{a}-\sum\limits_{t'\leq t-delay_i} ik^t_{a} \;\, \leq 0 $$

(26e)

$$ \begin{array}{rll} {\kern-6pt}\forall t,\ m,\ p,\ n, \;\mbox{free} \;\,\alpha p^t_{mpn} &\sum\limits_{a} M_6(a,n)fp^t_{mpa} (1\!-\!loss_a) &=0 \\[-3.5pt] &\!-\!\sum\limits_{a} M5_{an}fp^t_{mpa} \!+\!\sum\limits_f M1_{fn} q^t_{mpf} &\\[-3.5pt] &-\sum\limits_d \sum\limits_f M3_{dn} x^t_{mfpd} & \\[-3.5pt] & -\sum\limits_i \sum\limits_f M2_{in} zp^t_{mfpi} & \end{array} $$

(26f)

$$ \begin{array}{rll} \forall t,\ m,\ i,\ n,\ \;\, \mbox{free } \;\, \alpha i^t_{min} \;\, & \sum\limits_{a} M6_{an}f\/i^t_{mia}(1-loss_a) &\;\, =0 \\ &-\sum\limits_{a} M5_{an}f\/i^t_{mia} -\sum\limits_d M3_{dn} y^t_{mid} & \\ & +\sum\limits_p M2_{in} zi^t_{mpi} & \\ & -(-1)^m \sum\limits_s M4_{sn} in^t_{is} & \end{array} $$

(27)

The storage operator KKT conditions

$$ \begin{array}{rll} \forall t,\ s, \;\, 0 \leq is^t_s \;\, \bot \;\,& -\delta^{t} Is_s+\sum\limits_{t'\geq t+delay_s} \beta s^{t'}_s & \;\, \leq 0 \\ & -\iota s^t_s + Ls_s \sum\limits_{t'\geq t+delay_s} \iota s^{t'}_{s} & \\ \end{array} $$

(28a)

$$ \forall t,\ s, \, 0 \leq \beta s^t_s \;\, \bot \;\, \sum\limits_i r^t_{is}-Ks_s-\sum\limits_{t'\leq t-delay_s} is^{t'}_s \;\, \leq 0 $$

(28b)

$$ \forall t,\ s, \, 0 \leq \iota s^t_s \;\, \bot \;\, is^t_{s}-Ls_s Ks_s- Ls_s\sum\limits_{t'\leq t-delay_s} is^{t'}_s \;\, \leq 0 $$

(28c)