mintOC - User contributions [en]

Template:Current News

2019-04-23T12:13:57Z

AndreasPotschka: News flash for Transmission Lines

{| style="background:#EEEEFF;"

{{News|2019/3/14| Added multiple control problems with [[:Category: Gekko | Python GEKKO]] implementation}}
{{News|2018/9/12| Added [[Control of Transmission Lines]] problem}}
{{News|2016/7/26 | Added [[Industrial robot]] problem}}
{{News|2016/7/24 | Added [[Continuously Stirred Tank Reactor problem | CSTR problem]]}}
{{News|2016/6/30 | Added [[Electric Car]] problem}}
{{News|2016/5/5| Added multiple control problems with [[:Category: AMPL/TACO | AMPL with TACO]] implementation}}
{{News|2016/2/23|Added [[Control of Heat Equation with Actuator Placement | Actuator Placement]] control problem and [[:Category:Elliptic | Elliptic]], [[:Category:Parabolic | Parabolic]] and [[:Category:Hyperbolic | Hyperbolic]] categories}}

|}

<noinclude>
== Older news ==

{| style="background:#eaecd0;"

{{News|2016/1/10|Added [[Van der Pol Oscillator]], [[Batch reactor]], [[Cushioned Oscillation]] and [[Goddart's rocket problem]] control problems}}
{{News|2015/11/10|Moved mintoc.de to a new server}}
{{News|2012/09/01|Added the [[Lotka Experimental Design]] and a [[:Category:Optimum Experimental Design | category for experimental design problems]]}}
{{News|2011/09/29|Added the [[:Category:AMPL/TACO | first set of AMPL optimal control problems]] using the TACO toolkit}}
{{News|2010/11/21|Added New York [[Subway ride]] control problem}}
{{News|2010/11/18|Extended description of [[:Category:Problem characterization | problem characterization]]}}
{{News|2010/11/18|Description of benchmark library as [http://mathopt.de/PUBLICATIONS/Sager2011b.pdf pdf file] preprint}}
{{News|2010/08/16|Added [[Bang-bang approximation of a traveling wave]] 1D PDE example}}
{{News|2010/02/11|Launch of EU project [http://embocon.org embocon.org]}}
{{News|2009/11/20|Added [[External Links]] page}}
{{News|2009/10/26|Revision and Correction of several [[:Category:Optimica | optimica]] models}}
{{News|2009/08/26|Knitro solution for [[F-8 aircraft (AMPL)]]}}
{{News|2009/08/11|[[F-8 aircraft]] new local minimum found with [[:Category:Optimica | optimica]]/ipopt}}
{{News|2009/07/31|New category [[:Category:Optimica | optimica]] introduced}}
{{News|2009/07/07|[[:Category:AMPL]] revised page with discretized MINLPs in AMPL format}}

|}

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</noinclude>

Control of Transmission Lines

2018-09-13T07:38:15Z

AndreasPotschka: /* Reference Solution */

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 ==

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

<math>
\partial_t \xi_r(x,t) + \Lambda_r \xi_r(x,t) + B_r \xi_r(x,t) = 0,
</math>

with a diagonal 2x2-matrix <math>\Lambda_r</math> and a symmetric matrix <math>B_r</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

<math>
\partial_t \boldsymbol{\xi} + \boldsymbol{\Lambda} \partial_x \boldsymbol{\xi} + \mathbf{B} \boldsymbol{\xi} = 0
</math>

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 matrices <math>\mathbf{\Lambda}^\pm</math> of size <math>m \times m</math> according to

<math>
\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).
</math>

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

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

<math>
\displaystyle \min_{\boldsymbol{v} \in \mathcal{V}, \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,
</math>

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 ==

A detailed account of the network structures and parameter settings can be found in <bib id="Goettlich2018" /> and in the source code below.

== Discretization ==

The mixed-integer variables <math>\boldsymbol{v}(t)</math> are transcribed via Partial Outer Convexification and the dynamics are discretized using Finite Volumes with upwind fluxes for the characteristic variables and explicit first-order time-stepping.

== Reference Solution ==

We consider an extended tree network with 2 producers, 5 consumers and real-world demand data.
After partial outer convexification (POC) and discretization, Ipopt delivers an NLP solution with objective value 2.804 and the following relaxed POC multipliers:
[[File:translines_relaxed_control.png|500px|Relaxed POC multipliers for switched controls]]

Sum-Up Rounding with SOS1-constraint delivers the following integer feasible POC multipliers:
[[File:translines_switched_control.png|500px|POC multipliers for switched controls after Sum-Up-SOS1-Rounding]]

Reoptimization with fixed Sum-Up Rounding decisions delivers an objective value of 3.152 and the following controls:
[[File:translines_cont_control.png|500px|Continuous controls]]

The next figure compares the consumer demands (red) with the obtained power delivery (blue).
[[File:translines_demand_delivery.png|500px|Demand and delivery]]

==Source Code==

The C++ code for the results in the paper is not publicly available, but a more user-friendly Python/CasADi-Version (without dwell-time constraints) is available on [https://github.com/apotschka/poc-transmission-lines GitHub/poc-transmission-lines].

==References==
<biblist />


[[Category:MIOCP]]
[[Category:PDE model]]
[[Category:Hyperbolic]]
[[Category:Tracking objective]]
[[Category:Mesh-dependent integer variables]]
[[Category:Energy Networks]]
[[Category:Mesh-independent integer variables]]
[[Category:Casadi]]

Control of Transmission Lines

2018-09-13T07:37:15Z

AndreasPotschka: /* Reference Solution */

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 ==

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

<math>
\partial_t \xi_r(x,t) + \Lambda_r \xi_r(x,t) + B_r \xi_r(x,t) = 0,
</math>

with a diagonal 2x2-matrix <math>\Lambda_r</math> and a symmetric matrix <math>B_r</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

<math>
\partial_t \boldsymbol{\xi} + \boldsymbol{\Lambda} \partial_x \boldsymbol{\xi} + \mathbf{B} \boldsymbol{\xi} = 0
</math>

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 matrices <math>\mathbf{\Lambda}^\pm</math> of size <math>m \times m</math> according to

<math>
\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).
</math>

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

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

<math>
\displaystyle \min_{\boldsymbol{v} \in \mathcal{V}, \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,
</math>

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 ==

A detailed account of the network structures and parameter settings can be found in <bib id="Goettlich2018" /> and in the source code below.

== Discretization ==

The mixed-integer variables <math>\boldsymbol{v}(t)</math> are transcribed via Partial Outer Convexification and the dynamics are discretized using Finite Volumes with upwind fluxes for the characteristic variables and explicit first-order time-stepping.

== Reference Solution ==

We consider an extended tree network with 2 producers, 5 consumers and real-world demand data.
After partial outer convexification (POC) and discretization, Ipopt delivers an NLP solution with objective value 2.804 and the following relaxed POC multipliers:
[[File:translines_relaxed_control.png|500px|Relaxed POC multipliers for switched controls]]

Sum-Up Rounding with SOS1-constraint delivers the following integer feasible POC multipliers:
[[File:translines_switched_control.png|500px|POC multipliers for switched controls after Sum-Up-SOS1-Rounding]]

Reoptimization with fixed Sum-Up Rounding decisions delivers an objective value of 3.152 and the following controls:
[[File:translines_cont_control.png|500px|Continuous controls]]

The next figure compares the consumer demands with the obtained power delivery.
[[File:translines_demand_delivery.png|500px|Demand and delivery]]

==Source Code==

The C++ code for the results in the paper is not publicly available, but a more user-friendly Python/CasADi-Version (without dwell-time constraints) is available on [https://github.com/apotschka/poc-transmission-lines GitHub/poc-transmission-lines].

==References==
<biblist />


[[Category:MIOCP]]
[[Category:PDE model]]
[[Category:Hyperbolic]]
[[Category:Tracking objective]]
[[Category:Mesh-dependent integer variables]]
[[Category:Energy Networks]]
[[Category:Mesh-independent integer variables]]
[[Category:Casadi]]

Control of Transmission Lines

2018-09-13T07:32:32Z

AndreasPotschka:

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 ==

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

<math>
\partial_t \xi_r(x,t) + \Lambda_r \xi_r(x,t) + B_r \xi_r(x,t) = 0,
</math>

with a diagonal 2x2-matrix <math>\Lambda_r</math> and a symmetric matrix <math>B_r</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

<math>
\partial_t \boldsymbol{\xi} + \boldsymbol{\Lambda} \partial_x \boldsymbol{\xi} + \mathbf{B} \boldsymbol{\xi} = 0
</math>

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 matrices <math>\mathbf{\Lambda}^\pm</math> of size <math>m \times m</math> according to

<math>
\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).
</math>

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

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

<math>
\displaystyle \min_{\boldsymbol{v} \in \mathcal{V}, \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,
</math>

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 ==

A detailed account of the network structures and parameter settings can be found in <bib id="Goettlich2018" /> and in the source code below.

== Discretization ==

The mixed-integer variables <math>\boldsymbol{v}(t)</math> are transcribed via Partial Outer Convexification and the dynamics are discretized using Finite Volumes with upwind fluxes for the characteristic variables and explicit first-order time-stepping.

== Reference Solution ==

We consider an extended tree network with 2 producers, 4 consumers and real-world demand data.
After partial outer convexification (POC) and discretization, Ipopt delivers an NLP solution with objective value 2.804 and the following relaxed POC multipliers:
[[File:translines_relaxed_control.png|500px|Relaxed POC multipliers for switched controls]]

Sum-Up Rounding with SOS1-constraint delivers the following integer feasible POC multipliers:
[[File:translines_switched_control.png|500px|POC multipliers for switched controls after Sum-Up-SOS1-Rounding]]

Reoptimization with fixed Sum-Up Rounding decisions delivers an objective value of 3.152 and the following controls:
[[File:translines_cont_control.png|500px|Continuous controls]]

The next figure compares the consumer demands with the obtained power delivery.
[[File:translines_demand_delivery.png|500px|Demand and delivery]]

==Source Code==

The C++ code for the results in the paper is not publicly available, but a more user-friendly Python/CasADi-Version (without dwell-time constraints) is available on [https://github.com/apotschka/poc-transmission-lines GitHub/poc-transmission-lines].

==References==
<biblist />


[[Category:MIOCP]]
[[Category:PDE model]]
[[Category:Hyperbolic]]
[[Category:Tracking objective]]
[[Category:Mesh-dependent integer variables]]
[[Category:Energy Networks]]
[[Category:Mesh-independent integer variables]]
[[Category:Casadi]]

File:Translines cont control.png

2018-09-12T14:46:04Z

AndreasPotschka: Add description

Control of Transmission Lines: Continuous controls after SUR-SOS1.

Control of Transmission Lines

2018-09-12T14:44:06Z

AndreasPotschka: /* Reference Solution */

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 ==

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

<math>
\partial_t \xi_r(x,t) + \Lambda_r \xi_r(x,t) + B_r \xi_r(x,t) = 0,
</math>

with a diagonal 2x2-matrix <math>\Lambda_r</math> and a symmetric matrix <math>B_r</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

<math>
\partial_t \boldsymbol{\xi} + \boldsymbol{\Lambda} \partial_x \boldsymbol{\xi} + \mathbf{B} \boldsymbol{\xi} = 0
</math>

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

<math>
\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).
</math>

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

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

<math>
\displaystyle \min_{\boldsymbol{v} \in \mathcal{V}, \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,
</math>

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 ==

A detailed account of the network structures and parameter settings can be found in <bib id="Goettlich2018" /> and in the source code below.

== Discretization ==

The mixed-integer variables <math>\boldsymbol{v}(t)</math> are transcribed via Partial Outer Convexification and the dynamics are discretized using Finite Volumes with upwind fluxes for the characteristic variables and explicit first-order time-stepping.

== Reference Solution ==

We consider an extended tree network with 2 producers, 4 consumers and real-world demand data.
After partial outer convexification (POC) and discretization, Ipopt delivers an NLP solution with objective value 2.804 and the following relaxed POC multipliers:
[[File:translines_relaxed_control.png|500px|Relaxed POC multipliers for switched controls]]

Sum-Up Rounding with SOS1-constraint delivers the following integer feasible POC multipliers:
[[File:translines_switched_control.png|500px|POC multipliers for switched controls after Sum-Up-SOS1-Rounding]]

Reoptimization with fixed Sum-Up Rounding decisions delivers an objective value of 3.152 and the following controls:
[[File:translines_cont_control.png|500px|Continuous controls]]

The next figure compares the consumer demands with the obtained power delivery.
[[File:translines_demand_delivery.png|500px|Demand and delivery]]

==Source Code==

The C++ code for the results in the paper is not publicly available, but a more user-friendly Python/CasADi-Version (without dwell-time constraints) is available on [https://github.com/apotschka/poc-transmission-lines GitHub/poc-transmission-lines].

==References==
<biblist />


[[Category:MIOCP]]
[[Category:PDE model]]
[[Category:Hyperbolic]]
[[Category:Tracking objective]]
[[Category:Mesh-dependent integer variables]]
[[Category:Energy Networks]]
[[Category:Mesh-independent integer variables]]
[[Category:Casadi]]

Control of Transmission Lines

2018-09-12T14:43:30Z

AndreasPotschka: /* Reference Solution */

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 ==

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

<math>
\partial_t \xi_r(x,t) + \Lambda_r \xi_r(x,t) + B_r \xi_r(x,t) = 0,
</math>

with a diagonal 2x2-matrix <math>\Lambda_r</math> and a symmetric matrix <math>B_r</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

<math>
\partial_t \boldsymbol{\xi} + \boldsymbol{\Lambda} \partial_x \boldsymbol{\xi} + \mathbf{B} \boldsymbol{\xi} = 0
</math>

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

<math>
\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).
</math>

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

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

<math>
\displaystyle \min_{\boldsymbol{v} \in \mathcal{V}, \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,
</math>

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 ==

A detailed account of the network structures and parameter settings can be found in <bib id="Goettlich2018" /> and in the source code below.

== Discretization ==

The mixed-integer variables <math>\boldsymbol{v}(t)</math> are transcribed via Partial Outer Convexification and the dynamics are discretized using Finite Volumes with upwind fluxes for the characteristic variables and explicit first-order time-stepping.

== Reference Solution ==

We consider an extended tree network with 2 producers, 4 consumers and real-world demand data.
After partial outer convexification (POC) and discretization, Ipopt delivers an NLP solution with objective value 2.804 and the following relaxed POC multipliers:

[[File:translines_relaxed_control.png|800px|Relaxed POC multipliers for switched controls]]

Sum-Up Rounding with SOS1-constraint delivers the following integer feasible POC multipliers:

[[File:translines_switched_control.png|800px|POC multipliers for switched controls after Sum-Up-SOS1-Rounding]]

Reoptimization with fixed Sum-Up Rounding decisions delivers an objective value of 3.152 and the following controls:

[[File:translines_cont_control.png|800px|Continuous controls]]

The next figure compares the consumer demands with the obtained power delivery.

[[File:translines_demand_delivery.png|800px|Demand and delivery]]

==Source Code==

The C++ code for the results in the paper is not publicly available, but a more user-friendly Python/CasADi-Version (without dwell-time constraints) is available on [https://github.com/apotschka/poc-transmission-lines GitHub/poc-transmission-lines].

==References==
<biblist />


[[Category:MIOCP]]
[[Category:PDE model]]
[[Category:Hyperbolic]]
[[Category:Tracking objective]]
[[Category:Mesh-dependent integer variables]]
[[Category:Energy Networks]]
[[Category:Mesh-independent integer variables]]
[[Category:Casadi]]

File:Translines relaxed control.png

2018-09-12T14:37:29Z

AndreasPotschka: Control of Transmission Lines: Optimal relaxed POC multipliers

Control of Transmission Lines: Optimal relaxed POC multipliers

Control of Transmission Lines

2018-09-12T14:35:33Z

AndreasPotschka: Some results figures.

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 ==

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

<math>
\partial_t \xi_r(x,t) + \Lambda_r \xi_r(x,t) + B_r \xi_r(x,t) = 0,
</math>

with a diagonal 2x2-matrix <math>\Lambda_r</math> and a symmetric matrix <math>B_r</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

<math>
\partial_t \boldsymbol{\xi} + \boldsymbol{\Lambda} \partial_x \boldsymbol{\xi} + \mathbf{B} \boldsymbol{\xi} = 0
</math>

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

<math>
\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).
</math>

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

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

<math>
\displaystyle \min_{\boldsymbol{v} \in \mathcal{V}, \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,
</math>

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 ==

A detailed account of the network structures and parameter settings can be found in <bib id="Goettlich2018" /> and in the source code below.

== Discretization ==

The mixed-integer variables <math>\boldsymbol{v}(t)</math> are transcribed via Partial Outer Convexification and the dynamics are discretized using Finite Volumes with upwind fluxes for the characteristic variables and explicit first-order time-stepping.

== Reference Solution ==

[[File:translines_switched_control.png|800px|POC multipliers for switched controls]]
[[File:translines_cont_control.png|800px|Continuous controls]]
[[File:translines_demand_delivery.png|800px|Demand and delivery]]

==Source Code==

The C++ code for the results in the paper is not publicly available, but a more user-friendly Python/CasADi-Version (without dwell-time constraints) is available on [https://github.com/apotschka/poc-transmission-lines GitHub/poc-transmission-lines].

==References==
<biblist />


[[Category:MIOCP]]
[[Category:PDE model]]
[[Category:Hyperbolic]]
[[Category:Tracking objective]]
[[Category:Mesh-dependent integer variables]]
[[Category:Energy Networks]]
[[Category:Mesh-independent integer variables]]
[[Category:Casadi]]

File:Translines demand delivery.png

2018-09-12T14:29:45Z

AndreasPotschka: Control of transmission lines: Demand and optimal delivery

Control of transmission lines: Demand and optimal delivery

File:Translines switched control.png

2018-09-12T14:28:48Z

AndreasPotschka: Control of transmission lines: Switched controls

Control of transmission lines: Switched controls

File:Translines cont control.png

2018-09-12T14:27:20Z

AndreasPotschka:

Control of Transmission Lines

2018-09-12T14:23:06Z

AndreasPotschka:

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 ==

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

<math>
\partial_t \xi_r(x,t) + \Lambda_r \xi_r(x,t) + B_r \xi_r(x,t) = 0,
</math>

with a diagonal 2x2-matrix <math>\Lambda_r</math> and a symmetric matrix <math>B_r</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

<math>
\partial_t \boldsymbol{\xi} + \boldsymbol{\Lambda} \partial_x \boldsymbol{\xi} + \mathbf{B} \boldsymbol{\xi} = 0
</math>

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

<math>
\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).
</math>

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

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

<math>
\displaystyle \min_{\boldsymbol{v} \in \mathcal{V}, \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,
</math>

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 ==

A detailed account of the network structures and parameter settings can be found in <bib id="Goettlich2018" /> and in the source code below.

== Discretization ==

The mixed-integer variables <math>\boldsymbol{v}(t)</math> are transcribed via Partial Outer Convexification and the dynamics are discretized using Finite Volumes with upwind fluxes for the characteristic variables and explicit first-order time-stepping.

== Reference Solution ==

==Source Code==

The C++ code for the results in the paper is not publicly available, but a more user-friendly Python/CasADi-Version (without dwell-time constraints) is available on [https://github.com/apotschka/poc-transmission-lines GitHub/poc-transmission-lines].

==References==
<biblist />


[[Category:MIOCP]]
[[Category:PDE model]]
[[Category:Hyperbolic]]
[[Category:Tracking objective]]
[[Category:Mesh-dependent integer variables]]
[[Category:Energy Networks]]
[[Category:Mesh-independent integer variables]]
[[Category:Casadi]]

Control of Transmission Lines

2018-09-12T14:11:32Z

AndreasPotschka: Parameters, Discretization, Source Code

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 ==

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

<math>
\partial_t \xi_r(x,t) + \Lambda_r \xi_r(x,t) + B_r \xi_r(x,t) = 0,
</math>

with a diagonal 2x2-matrix <math>\Lambda_r</math> and a symmetric matrix <math>B_r</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

<math>
\partial_t \boldsymbol{\xi} + \boldsymbol{\Lambda} \partial_x \boldsymbol{\xi} + \mathbf{B} \boldsymbol{\xi} = 0
</math>

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

<math>
\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).
</math>

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

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

<math>
\displaystyle \min_{\boldsymbol{v} \in \mathcal{V}, \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,
</math>

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 ==

A detailed account of the network structures and parameter settings can be found in <bib id="Goettlich2018" /> and in the source code below.

== Discretization ==

The mixed-integer variables <math>\boldsymbol{v}(t)</math> are transcribed via Partial Outer Convexification and the dynamics are discretized using Finite Volumes with upwind fluxes for the characteristic variables and explicit first-order time-stepping.

== Reference Solution ==

==Source Code==

The C++ code for the results in the paper are not publicly available, but a more user-friendly Python/CasADi-Version is available on [https://github.com/apotschka/poc-transmission-lines GitHub/poc-transmission-lines].

==References==
<biblist />


[[Category:MIOCP]]
[[Category:PDE model]]
[[Category:Hyperbolic]]
[[Category:Tracking objective]]
[[Category:Mesh-dependent integer variables]]
[[Category:Energy Networks]]
[[Category:Mesh-independent integer variables]]
[[Category:Casadi]]

Control of Transmission Lines

2018-09-12T13:57:42Z

AndreasPotschka: Mathematical formulation.

Control of Transmission Lines

2018-09-12T11:40:58Z

AndreasPotschka: Empty template for Control of Transmission Lines

== Mathematical formulation ==

== Parameters ==

== Discretization ==

== Reference Solution ==

==Source Code==

==References==


[[Category:MIOCP]]
[[Category:PDE model]]
[[Category:Hyperbolic]]
[[Category:Tracking objective]]
[[Category:Mesh-dependent integer variables]]
[[Category:Energy Networks]]
[[Category:Mesh-independent integer variables]]
[[Category:Casadi]]