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A partial outer convexification approach to control transmission lines

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Abstract

In this paper we derive an efficient optimization approach to calculate optimal controls of electric transmission lines. These controls consist of time-dependent inflows and switches that temporarily disable single arcs or whole subgrids to reallocate the flow inside the system. The aim is then to find the best energy input in terms of boundary controls in combination with the optimal configuration of switches, where the dynamics is driven by a coupled system of hyperbolic differential equations. We use a well-known three-step optimization approach based on the idea of partial outer convexification, for which we establish that the analytical requirements for its application hold for each fixed spatial discretization of the underlying partial differential equation, provided that combinatorial constraints are only pointwise in time. A comparison with a direct solver yields very promising results, also for problems with from an application viewpoint important switch up-time and down-time constraints, which are not pointwise in time and thus not fully covered by theory.

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Acknowledgements

SG is supported by the BMBF Project ENets (05M18VMA). AP gratefully acknowledges support by the European Research Council within the ERC Advanced Grant MOBOCON (291 458) and by the German Federal Ministry for Education and Research under Grants 05M2016-MOPhaPro and 05M2018-MOReNet.

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Authors and Affiliations

  1. Department of Mathematics, University of Mannheim, 68131, Mannheim, Germany

    S. Göttlich & C. Teuber

  2. Interdisciplinary Center for Scientific Computing, Heidelberg University, Im Neuenheimer Feld 205, 69120, Heidelberg, Germany

    A. Potschka

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  1. S. Göttlich
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Correspondence to S. Göttlich.

A Details of the Proof of Lemma 3.3

A Details of the Proof of Lemma 3.3

To ensure a clearer presentation of the proof of Lemma 3.3, we provide a detailed derivation for (27) in the following:

$$\begin{aligned}&\Vert [\Phi (t_e,\xi ) - \Phi (t_e,\eta )]\hat{\alpha }\Vert _X\\&=\sum _{r \in A} \sum _{d=0}^{\tilde{l}_r} \left| \sum _{c \in \Omega } \left( \frac{\lambda _r^+}{\Delta x} (\Theta ^+_c(\xi _r(\cdot ,t_e),x_d,\varvec{u}(t_e)) - \Theta _c^+(\eta _r(\cdot ,t_e),x_d,\varvec{u}(t_e)) - \xi ^+_r(x_d,t_e) + \eta ^+_r(x_d,t_e))\right. \right. \\&\left. \left. \qquad -\,b^r_{11} \xi ^+_r(x_d,t_e) + b^r_{11} \eta _r^+(x_d,t_e) - b^r_{12} \xi ^-_r(x_d,t_e) + b^r_{12} \eta ^-_r(x_d,t_e)\right) \hat{\alpha }_c \right| \Delta x\\&\qquad +\,\sum _{r \in A} \sum _{d=0}^{\tilde{l}_r} \left| \sum _{c \in \Omega } \left( \frac{\lambda _r^+}{\Delta x} (\Theta ^-_c(\xi _r(\cdot ,t_e),x_d,\varvec{u}(t_e)) - \Theta _c^-(\eta _r(\cdot ,t_e),x_d,\varvec{u}(t_e)) - \xi ^-_r(x_d,t_e) + \eta ^-_r(x_d,t_e))\right. \right. \\&\left. \left. \qquad -\,b^r_{12} \xi ^+_r(x_d,t_e) + b^r_{12} \eta _r^+(x_d,t_e) - b^r_{11} \xi ^-_r(x_d,t_e) + b^r_{11} \eta ^-_r(x_d,t_e)\right) \hat{\alpha }_c \right| \Delta x \end{aligned}$$

Due to \(\hat{\alpha }_c \le 1\), we get

$$\begin{aligned}&\le \sum _{r \in A} \sum _{d=0}^{\tilde{l}_r} \left| \sum _{c \in \Omega } \frac{\lambda _r^+}{\Delta x} \Big (\Theta ^+_c(\xi _r(\cdot ,t_e),x_d,\varvec{u}(t_e)) - \Theta _c^+(\eta _r(\cdot ,t_e),x_d,\varvec{u}(t_e)) - \xi ^+_r(x_d,t_e) + \eta ^+_r(x_d,t_e)\Big ) \right| \Delta x\\&\qquad +\, \sum _{r \in A} \sum _{d=0}^{\tilde{l}_r} \left| \sum _{c \in \Omega } \frac{\lambda _r^+}{\Delta x} \left( \Theta ^-_c(\xi _r(\cdot ,t_e),x_d,\varvec{u}(t_e)) - \Theta _c^-(\eta _r(\cdot ,t_e),x_d,\varvec{u}(t_e)) - \xi ^-_r(x_d,t_e) + \eta ^-_r(x_d,t_e)\right) \right| \Delta x\\&\qquad +\, \left( \frac{\lambda ^+}{\Delta x}+ b_{11} + b_{12}\right) n_{oc} \Vert \eta - \xi \Vert _X \\&\le \left( 2 \frac{\lambda ^+}{\Delta x} + b_{11} + b_{12}\right) n_{oc} \Vert \eta -\xi \Vert _X =: L_{oc} \Vert \eta -\xi \Vert _X, \end{aligned}$$

where we exploit that

$$\begin{aligned}&\sum _{r \in A} \sum _{d=0}^{\tilde{l}_r} \left| \sum _{c \in \Omega } \frac{\lambda _r^+}{\Delta x} \Big (\Theta ^+_c(\xi _r(\cdot ,t_e),x_d,\varvec{u}(t_e)) - \Theta _c^+(\eta _r(\cdot ,t_e),x_d,\varvec{u}(t_e)) - \xi ^+_r(x_d,t_e) + \eta ^+_r(x_d,t_e)\Big ) \right| \Delta x\\&= \sum _{r \in A} \sum _{d=1}^{\tilde{l}_r} \left| \sum _{c \in \Omega } \frac{\lambda _r^+}{\Delta x} \Big ( \xi ^+_r(x_{d-1},t_e) - \eta ^+_r(x_{d-1},t_e) \Big ) \right| \Delta x\\&\quad +\, \sum _{r \in A \setminus A_Q} \left| \sum _{c \in \Omega } \frac{\lambda _r^+}{\Delta x} \left( \sum _{k \in \delta ^-_{\alpha (r)}} ~^c\!d^{+}_{rk} \frac{\lambda _k^+}{\lambda _r^+} \xi ^+_k\left( x_{\tilde{l}_k},t_e\right) - \sum _{k \in \delta ^-_{\alpha (r)}} ~^c\!d^{+}_{rk} \frac{\lambda _k^+}{\lambda _r^+} \eta ^+_k\left( x_{\tilde{l}_k},t_e\right) \right) \right| \Delta x\\&\quad +\, \sum _{r \in A_Q} \left| \sum _{c \in \Omega } \frac{\lambda _r^+}{\Delta x} \left( \underbrace{u_r(t_e) - u_r(t_e)}_{=0} \right) \right| \Delta x\\&= \sum _{r \in A} \sum _{d=0}^{\tilde{l}_r-1} \left| \sum _{c \in \Omega } \frac{\lambda _r^+}{\Delta x} \left( \xi ^+_r(x_d,t_e) - \eta ^+_r(x_d,t_e) \right) \right| \Delta x\\&\quad +\, \sum _{r \in A } \left| \sum _{c \in \Omega } \frac{\lambda _r^+}{\Delta x} \left( \sum _{k \in A} ~^c\!d^{+}_{rk} \frac{\lambda _k^+}{\lambda _r^+} \xi ^+_k(x_{\tilde{l}_k},t_e) - \sum _{k \in A} ~^c\!d^{+}_{rk} \frac{\lambda _k^+}{\lambda _r^+} \eta ^+_k(x_{\tilde{l}_k},t_e) \right) \right| \Delta x\\&\le \sum _{r \in A} \sum _{d=0}^{\tilde{l}_r-1} \left| \sum _{c \in \Omega } \frac{\lambda _r^+}{\Delta x} \left( \xi ^+_r(x_d,t_e) - \eta ^+_r(x_d,t_e) \right) \right| \Delta x\\&\quad +\,\sum _{r \in A } \sum _{c \in \Omega } \frac{\lambda _r^+}{\Delta x} \left( \sum _{k \in A} ~^c\!d^{+}_{rk} \frac{\lambda _k^+}{\lambda _r^+} \left| \xi ^+_k(x_{\tilde{l}_k},t_e) - \eta ^+_k(x_{\tilde{l}_k},t_e) \right| \right) \Delta x\\&= \sum _{r \in A} \sum _{d=0}^{\tilde{l}_r-1} \left| \sum _{c \in \Omega } \frac{\lambda _r^+}{\Delta x} \left( \xi ^+_r(x_d,t_e) - \eta ^+_r(x_d,t_e) \right) \right| \Delta x\\&\quad +\, \sum _{k \in A } \sum _{c \in \Omega } \underbrace{\sum _{r \in A} ~^c\!d^{+}_{rk}}_{=1} \frac{\lambda _r^+}{\Delta x} \frac{\lambda _k^+}{\lambda _r^+} \left| \xi ^+_k(x_{\tilde{l}_k},t_e) - \eta ^+_k\left( x_{\tilde{l}_k},t_e\right) \right| \Delta x\\&= \sum _{r \in A} \sum _{d=0}^{\tilde{l}_r-1} \left| \sum _{c \in \Omega } \frac{\lambda _r^+}{\Delta x} \left( \xi ^+_r(x_d,t_e) - \eta ^+_r(x_d,t_e) \right) \right| \Delta x\\&\quad +\, \sum _{k \in A } \sum _{c \in \Omega } \frac{\lambda _k^+}{\Delta x} \left| \xi ^+_k\left( x_{\tilde{l}_k},t_e\right) - \eta ^+_k\left( x_{\tilde{l}_k},t_e\right) \right| \Delta x\\&\le \sum _{r \in A} \sum _{d=0}^{\tilde{l}_r} \sum _{c \in \Omega } \frac{\lambda _r^+}{\Delta x} \left| \xi ^+_r(x_d,t_e) - \eta ^+_r(x_d,t_e) \right| \Delta x \end{aligned}$$

The same computations can be done for the remaining summand.

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Göttlich, S., Potschka, A. & Teuber, C. A partial outer convexification approach to control transmission lines. Comput Optim Appl 72, 431–456 (2019). https://doi.org/10.1007/s10589-018-0047-6

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