Difference between revisions of "Control of Transmission Lines"

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(Mathematical formulation.)
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This problem was provided by Göttlich, Potschka, and Teuber <bib id="Goettlich2018" />. It is governed by a 2x2-system of conservation laws based on the telegraph equations for single transmission lines, which are then connected to form a network. The objective is to minimize the quadratic deviation of the load delivered to customer nodes from their demand by continuously controlling the power inflow to the network at the energy producer nodes and by discrete but time-varying switches at the coupling nodes inside the network.
  
 
== Mathematical formulation ==
 
== Mathematical formulation ==
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The dynamics on the <math>r</math>-th transmission line with spatial variable <math>x \in [0, l_r]</math>, temporal variable <math>t \in [0, T]</math>, and state variable <math>\xi_r(x,t) = (\xi^+_r(x, t), \xi^-_r(x, t))</math> containing the characteristic variables for right-traveling and left-traveling components are given by the hyperbolic PDE system
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<p>
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<math>
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\partial_t \xi_r(x,t) + \Lambda \xi_r(x,t) + B \xi_r(x,t) = 0,
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</math>
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</p>
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with a diagonal 2x2-matrix <math>\Lambda</math> and a symmetric matrix <math>B</math>. We combine all <math>m</math> single line states to a large state vector <math>\boldsymbol{\xi}(x,t)</math> to obtain the system
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<p>
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<math>
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\partial_t \boldsymbol{\xi} + \boldsymbol{\Lambda} \partial_x \boldsymbol{\xi} + \mathbf{B} \boldsymbol{\xi} = 0
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</math>
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</p>
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and formulate the coupling between the lines and the continuously controlled power inflow <math>\boldsymbol{u}(t)</math> as boundary conditions involving distribution matrices <math>\mathbf{D}^\pm(v)</math>, which depend on a discrete switching signal <math>\boldsymbol{v}(t)</math>, and constant distribution matrices <math>\mathbf{\Lambda}^\pm</math> of size <math>m \times m</math> according to
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<p>
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<math>
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\begin{pmatrix}
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\boldsymbol{\Lambda}^+ & 0\\
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0 & \mathbf{D}^-(\boldsymbol{v}(t))
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\end{pmatrix} \boldsymbol{\xi}(0,t) =
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\begin{pmatrix}
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\mathbf{D}^+(\boldsymbol{v}(t)) & 0\\
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0 & \boldsymbol{\Lambda}^-
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\end{pmatrix} \boldsymbol{\xi}(l,t) +
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\begin{pmatrix}
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\boldsymbol{\Lambda}^+ & 0\\
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0 & 0
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\end{pmatrix} \boldsymbol{u}(t).
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</math>
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</p>
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The continuous control <math>\boldsymbol{u}(t)</math> is subject to simple bounds.
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The objective is to track the given demands <math>Q_s(t)</math> of consumers, which can be formulated as
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<p>
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<math>
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  \displaystyle \min_{\boldsymbol{u} \in \mathcal{U}} \frac{1}{2}\sum_{s \in V_S} \int_0^{T} \left( Q_s(t) - \sum_{r \in \delta_{s}} \xi_r^+(l_r,t) \right)^2 dt,
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</math>
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</p>
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where <math>V_S</math> is the set of consumer nodes and <math>\delta_s</math> is the set of all lines adjacent to vertex <math>s</math>.
  
 
== Parameters ==
 
== Parameters ==
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==References==
 
==References==
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<biblist />
  
 
<!--List of all categories this page is part of. List characterization of solution behavior, model properties, ore presence of implementation details (e.g., AMPL for AMPL model) here -->
 
<!--List of all categories this page is part of. List characterization of solution behavior, model properties, ore presence of implementation details (e.g., AMPL for AMPL model) here -->

Revision as of 14:57, 12 September 2018

This problem was provided by Göttlich, Potschka, and Teuber [Goettlich2018]Author: G{\"o}ttlich, Simone; Potschka, Andreas; Teuber, Claus
Institution: University of Mannheim
Note: Optimization Online 6312
Title: A partial outer convexification approach to control transmission lines
Url: http://www.optimization-online.org/DB_HTML/2017/11/6312.html
Year: 2018
Link to Google Scholar. It is governed by a 2x2-system of conservation laws based on the telegraph equations for single transmission lines, which are then connected to form a network. The objective is to minimize the quadratic deviation of the load delivered to customer nodes from their demand by continuously controlling the power inflow to the network at the energy producer nodes and by discrete but time-varying switches at the coupling nodes inside the network.

Mathematical formulation

The dynamics on the r-th transmission line with spatial variable x \in [0, l_r], temporal variable t \in [0, T], and state variable \xi_r(x,t) = (\xi^+_r(x, t), \xi^-_r(x, t)) containing the characteristic variables for right-traveling and left-traveling components are given by the hyperbolic PDE system

\partial_t \xi_r(x,t) + \Lambda \xi_r(x,t) + B \xi_r(x,t) = 0,

with a diagonal 2x2-matrix \Lambda and a symmetric matrix B. We combine all m single line states to a large state vector \boldsymbol{\xi}(x,t) to obtain the system

\partial_t \boldsymbol{\xi} + \boldsymbol{\Lambda} \partial_x \boldsymbol{\xi} + \mathbf{B} \boldsymbol{\xi} = 0

and formulate the coupling between the lines and the continuously controlled power inflow \boldsymbol{u}(t) as boundary conditions involving distribution matrices \mathbf{D}^\pm(v), which depend on a discrete switching signal \boldsymbol{v}(t), and constant distribution matrices \mathbf{\Lambda}^\pm of size m \times m according to

\begin{pmatrix} \boldsymbol{\Lambda}^+ & 0\\ 0 & \mathbf{D}^-(\boldsymbol{v}(t)) \end{pmatrix} \boldsymbol{\xi}(0,t) = \begin{pmatrix} \mathbf{D}^+(\boldsymbol{v}(t)) & 0\\ 0 & \boldsymbol{\Lambda}^- \end{pmatrix} \boldsymbol{\xi}(l,t) + \begin{pmatrix} \boldsymbol{\Lambda}^+ & 0\\ 0 & 0 \end{pmatrix} \boldsymbol{u}(t).

The continuous control \boldsymbol{u}(t) is subject to simple bounds.

The objective is to track the given demands Q_s(t) of consumers, which can be formulated as

\displaystyle \min_{\boldsymbol{u} \in \mathcal{U}} \frac{1}{2}\sum_{s \in V_S} \int_0^{T} \left( Q_s(t) - \sum_{r \in \delta_{s}} \xi_r^+(l_r,t) \right)^2 dt,

where V_S is the set of consumer nodes and \delta_s is the set of all lines adjacent to vertex s.

Parameters

Discretization

Reference Solution

Source Code

References

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