Modeling and simulation of gas networks coupled to power grids

Abstract

A mathematical framework for the coupling of gas networks to electric grids is presented to describe in particular the transition from gas to power. The dynamics of the gas flow are given by the isentropic Euler equations, while the power flow equations are used to model the power grid. We derive pressure laws for the gas flow that allow for the well-posedness of the coupling and a rigorous treatment of solutions. For simulation purposes, we apply appropriate numerical methods and show in an experimental study how gas-to-power might influence the dynamics of the gas and power network, respectively.

Appendix: A Proof of Lemma 3

The following two technical results are the key ingredients to prove Lemma 3.

Lemma 13

Let $g \in C^1(\mathbb {R}^+, \mathbb {R}^+)$be a non-negative function, $g\ge 0$, and let G be given by $G(\rho ) = \int _\rho ^{\rho _l}g(s)\mathrm{d}s$. Then There holds

• 1. If $\rho ^2g(\rho ) \overset{\rho \rightarrow 0}{\rightarrow } 0$, then $\rho G(\rho ) \overset{\rho \rightarrow 0}{\rightarrow } 0$.

• 2. If $\rho G(\rho ) \overset{\rho \rightarrow 0}{\rightarrow } 0$, then $\liminf _{\rho \rightarrow 0}\rho ^2g(\rho ) =0$.

Proof

• 1. By assumption $\rho ^2g(\rho ) \overset{\rho \rightarrow 0}{\rightarrow } 0$. For $m \in \mathbb {N}$ choose $\rho _m >0$ such that $g(\rho )\le \tfrac{1}{m\rho ^2}$ for $\rho <\rho _m$. Now choose $\rho _{m,0}<\rho _m$ so small that

  $$\begin{aligned} \rho _{m,0} \left( \int _{\rho _m}^{\rho _l}g(s)\mathrm{d}s -\frac{1}{m\rho _m}\right) \le \frac{1}{m} . \end{aligned}$$

  (69)

  Then, for $\rho <\rho _{m,0}$, there holds

  $$\begin{aligned} \rho \int _{\rho }^{\rho _l}g(s)\mathrm{d}s&\le \rho \int _{\rho _m}^{\rho _l}g(s) \mathrm{d}s + \frac{1}{m}\rho \int _{\rho }^{\rho _m}\frac{1}{s^2}\mathrm{d}s\nonumber \\&= \rho \left( \int _{\rho _m}^{\rho _l}g(s) \mathrm{d}s-\frac{1}{m\rho _m}\right) +\frac{1}{m}\nonumber \\&\le \frac{2}{m}. \end{aligned}$$

  (70)

  Therefore $\lim _{\rho \rightarrow 0}\rho \int _{\rho }^{\rho _l}g(s)\mathrm{d}s \le \tfrac{2}{m}$ for all $m \in \mathbb {N}$. As $g\ge 0$, we also have $0\le \lim _{\rho \rightarrow 0}\rho \int _{\rho }^{\rho _l}g(s)\mathrm{d}s$. Summarizing, we get $\lim _{\rho \rightarrow 0}\rho G(\rho ) = \lim _{\rho \rightarrow 0}\rho \int _{\rho }^{\rho _l}g(s)\mathrm{d}s = 0$.

• 2. We prove by contradiction: Assume there are $\rho _0>0$ and $a >0$ such that $\rho ^2g(\rho ) \ge a$ for $\rho < \rho _0$. Then $g(\rho )\ge \tfrac{a}{2}\tfrac{1}{\rho ^2}$ for such $\rho$. Therefore

  $$\begin{aligned} \rho \int _\rho ^{\rho _l}g(s)\mathrm{d}s \ge \rho \int _{\rho _0}^{\rho _l}g(s) \mathrm{d}s + \rho \frac{a}{2}\int _{\rho }^{\rho _0} \frac{1}{s^2} \mathrm{d}s \rightarrow 0 + \frac{a}{2} \rho \left[ -\frac{1}{s} \right] _\rho ^{\rho _0} \rightarrow \frac{a}{2}, \end{aligned}$$

  (71)

  which contradicts $\rho G(\rho ) \overset{\rho \rightarrow 0}{\longrightarrow } 0$. $\square$

Lemma 14

Let $g \in C^1(\mathbb {R}^+, \mathbb {R}^+)$, $g\ge 0$ and $\liminf _{\rho \rightarrow 0} \rho ^2g(\rho ) = 0$. Let also $\left( \rho ^2g(\rho ) \right) ^\prime \ge 0$. Then, $\limsup _{\rho \rightarrow 0} \rho ^2 g(\rho ) = \liminf _{\rho \rightarrow 0} \rho ^2 g(\rho ) = 0$.

Proof

We prove by contradiction. Let $\limsup _{\rho \rightarrow 0} \rho ^2 g(\rho ) > \liminf \rho ^2 g(\rho )$. Then it is easily seen that $\liminf _{\rho \rightarrow 0} \left( \rho ^2g(\rho ) \right) ^\prime = -\infty < 0$ resulting in a contradiction. $\square$