Control of Transmission Lines - Revision history

JonathanHelgert at 07:07, 16 July 2021

2021-07-16T07:07:25Z

@@ Line 48: / Line 48: @@
 == 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.
+The mixed-integer variables <math>\boldsymbol{v}(t)</math> are transcribed via Partial Outer Convexification, i.e. the coupling conditions read
 == Reference Solution ==
@@ Line 67: / Line 91: @@
 ==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].
+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]. A Python/Gurobi-Version (including dwell-time constraints) that solves the problem to global optimality is available on [https://github.com/jhelgert/pyTranslines GitHub/pyTranslines]. The latter also contains a specialized heuristic based on the CIAP decomposition in order to quickly find feasible solutions.
 ==References==

AndreasPotschka: /* Reference Solution */

2018-09-13T07:38:15Z

Reference Solution

← Older revision		Revision as of 07:38, 13 September 2018
Line 62:		Line 62:
	[[File:translines_cont_control.png\|500px\|Continuous controls]]		[[File:translines_cont_control.png\|500px\|Continuous controls]]

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

AndreasPotschka: /* Reference Solution */

2018-09-13T07:37:15Z

Reference Solution

← Older revision		Revision as of 07:37, 13 September 2018
Line 52:		Line 52:
	== Reference Solution ==		== Reference Solution ==

	We consider an extended tree network with 2 producers, 4 consumers and real-world demand data.		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:		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]]		[[File:translines_relaxed_control.png\|500px\|Relaxed POC multipliers for switched controls]]

AndreasPotschka at 07:32, 13 September 2018

2018-09-13T07:32:32Z

← Older revision		Revision as of 07:32, 13 September 2018
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	</math>		</math>
	</p>		</p>
	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		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
	<p>		<p>
	<math>		<math>

AndreasPotschka: /* Reference Solution */

2018-09-12T14:44:06Z

Reference Solution

← Older revision		Revision as of 14:44, 12 September 2018
Line 54:		Line 54:
	We consider an extended tree network with 2 producers, 4 consumers and real-world demand data.		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:		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]]
	[[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:		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]]
	[[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:		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]]
	[[File:translines_cont_control.png\|~~800px~~\|Continuous controls]]

	The next figure compares the consumer demands with the obtained power delivery.		The next figure compares the consumer demands with the obtained power delivery.
			[[File:translines_demand_delivery.png\|500px\|Demand and delivery]]
	[[File:translines_demand_delivery.png\|~~800px~~\|Demand and delivery]]

	==Source Code==		==Source Code==

AndreasPotschka: /* Reference Solution */

2018-09-12T14:43:30Z

Reference Solution

← Older revision		Revision as of 14:43, 12 September 2018
Line 52:		Line 52:
	== Reference Solution ==		== Reference Solution ==

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

AndreasPotschka: Some results figures.

2018-09-12T14:35:33Z

Some results figures.

← Older revision		Revision as of 14:35, 12 September 2018
Line 51:		Line 51:

	== Reference Solution ==		== 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==		==Source Code==

AndreasPotschka at 14:23, 12 September 2018

2018-09-12T14:23:06Z

← Older revision		Revision as of 14:23, 12 September 2018
Line 54:		Line 54:
	==Source Code==		==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].		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==		==References==

AndreasPotschka: Parameters, Discretization, Source Code

2018-09-12T14:11:32Z

Parameters, Discretization, Source Code

← Older revision		Revision as of 14:11, 12 September 2018
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	<p>		<p>
	<math>		<math>
	\partial_t \xi_r(x,t) + \~~Lambda~~ \xi_r(x,t) + B \xi_r(x,t) = 0,		\partial_t \xi_r(x,t) + \Lambda_r \xi_r(x,t) + B_r \xi_r(x,t) = 0,
	</math>		</math>
	</p>		</p>
	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		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
	<p>		<p>
	<math>		<math>
Line 37:		Line 37:
	<p>		<p>
	<math>		<math>
	\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,		\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>		</math>
	</p>		</p>
Line 43:		Line 43:

	== Parameters ==		== 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 ==		== 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 ==		== Reference Solution ==

	==Source Code==		==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==		==References==

AndreasPotschka: Mathematical formulation.

2018-09-12T13:57:42Z

Mathematical formulation.

@@ Line 1: / Line 1: @@
-== 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
 == Parameters ==
@@ Line 12: / Line 51: @@
 ==References==
 <!--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 -->