Direct Current Transmission Heating Problem - Revision history

FelixMueller at 22:09, 27 June 2016

2016-06-27T22:09:46Z

← Older revision		Revision as of 22:09, 27 June 2016
Line 123:		Line 123:
	-->		-->
	[[Category: MIOCP]]		[[Category: MIOCP]]
			[[Category:ODE model]]

FelixMueller: /* Mathematical formulation */

2016-04-10T09:30:35Z

Mathematical formulation

← Older revision		Revision as of 09:30, 10 April 2016
Line 56:		Line 56:
	\begin{array}{llcll}		\begin{array}{llcll}
	\displaystyle \min_{T, I, u} & \int_{t_0}^{t_f} \sum_{a \in A} P_{1,a}(t) \,\mathrm{d} t \\[1.5ex]		\displaystyle \min_{T, I, u} & \int_{t_0}^{t_f} \sum_{a \in A} P_{1,a}(t) \,\mathrm{d} t \\[1.5ex]
	\mbox{s.t.} & \dot{T}_a~~(t)~~ & = & \beta_{1,a} \cdot e^{\alpha_a T_a~~(t)~~} \cdot I_a~~(t)~~^2 - \beta_{2,a} \cdot T_a~~(t)~~ \qquad & \forall ~~t \in [t_0, t_f],~~ a \in A, \\		\mbox{s.t.} & \dot{T}_a & = & \beta_{1,a} \cdot e^{\alpha_a T_a} \cdot I_a^2 - \beta_{2,a} \cdot T_a \qquad & \forall a \in A, \\
	& d_v & = & \sum_{a \in \delta^{in}(v)} (U_a \cdot I_{a}~~(t)~~ - P_1^a~~(t)~~) - \sum_{a \in \delta^{out}(v)} (U_a \cdot I_a~~(t)~~) \qquad & \forall ~~t \in [t_0, t_f],~~ v \in V \setminus P, \\		& d_v & = & \sum_{a \in \delta^{in}(v)} (U_a \cdot I_{a} - P_1^a) - \sum_{a \in \delta^{out}(v)} (U_a \cdot I_a) \qquad & \forall v \in V \setminus P, \\
	& -u_v~~(t)~~ & = & \sum_{a \in \delta^{in}(v)} (U_a \cdot I_{a}~~(t)~~ - P_1^a~~(t)~~) - \sum_{a \in \delta^{out}(v)} (U_a \cdot I_a~~(t)~~) \qquad & \forall ~~t \in [t_0, t_f],~~ v \in P, \\[1.5ex]		& -u_v & = & \sum_{a \in \delta^{in}(v)} (U_a \cdot I_{a} - P_1^a) - \sum_{a \in \delta^{out}(v)} (U_a \cdot I_a) \qquad & \forall v \in P, \\[1.5ex]
	& T_a(t_0) &=& 0.0 \qquad & \forall a \in A, \\		& T_a(t_0) &=& 0.0 \qquad & \forall a \in A, \\
	& I_a(t) &\in& [0, I_{max, a}] \qquad & \forall a \in A, t \in [t_0, t_f], \\		& I_a(t) &\in& [0, I_{max, a}] \qquad & \forall a \in A, t \in [t_0, t_f], \\

FelixMueller at 17:34, 27 January 2016

2016-01-27T17:34:47Z

← Older revision		Revision as of 17:34, 27 January 2016
Line 122:		Line 122:
	* [[:Category: Casadi \| Casadi code]] at [[Direct Current Transmission Heating Problem (Casadi)]]		* [[:Category: Casadi \| Casadi code]] at [[Direct Current Transmission Heating Problem (Casadi)]]
	-->		-->
			[[Category: MIOCP]]

MirkoHahn: MirkoHahn moved page User:Direct Current Transmission Heating Problem to Direct Current Transmission Heating Problem over redirect

2015-12-09T06:34:33Z

MirkoHahn moved page User:Direct Current Transmission Heating Problem to Direct Current Transmission Heating Problem over redirect

← Older revision	Revision as of 06:34, 9 December 2015
(No difference)

MirkoHahn: MirkoHahn moved page Direct Current Transmission Heating Problem to User:Direct Current Transmission Heating Problem: Moved to User due to model flaws

2015-12-09T06:31:51Z

MirkoHahn moved page Direct Current Transmission Heating Problem to User:Direct Current Transmission Heating Problem: Moved to User due to model flaws

← Older revision	Revision as of 06:31, 9 December 2015
(No difference)

MirkoHahn at 18:05, 8 December 2015

2015-12-08T18:05:18Z

← Older revision		Revision as of 18:05, 8 December 2015
Line 110:		Line 110:
	=== Delayed heating ===		=== Delayed heating ===
	In this scenario, we consider a simple network consisting of two producers producing up to <math>1 \;MW</math> each and a single consumer with a demand of <math>1.25 \;MW</math>. Both producers have direct power links of type C and length <math>1 \;km</math> to the consumer. The time horizon is defined as <math>(t_0, t_f) = (0, 300)</math>. A JSON description of the network can be found under [[Direct Current Transmission Heating Problem: Delayed heating (Network)]].		In this scenario, we consider a simple network consisting of two producers producing up to <math>1 \;MW</math> each and a single consumer with a demand of <math>1.25 \;MW</math>. Both producers have direct power links of type C and length <math>1 \;km</math> to the consumer. The time horizon is defined as <math>(t_0, t_f) = (0, 300)</math>. A JSON description of the network can be found under [[Direct Current Transmission Heating Problem: Delayed heating (Network)]].
			<!--
	== Reference Solutions ==		== Reference Solutions ==

Line 121:		Line 121:

	* [[:Category: Casadi \| Casadi code]] at [[Direct Current Transmission Heating Problem (Casadi)]]		* [[:Category: Casadi \| Casadi code]] at [[Direct Current Transmission Heating Problem (Casadi)]]
			-->

MirkoHahn: /* Mathematical formulation */

2015-12-08T17:21:02Z

Mathematical formulation

← Older revision		Revision as of 17:21, 8 December 2015
Line 58:		Line 58:
	\mbox{s.t.} & \dot{T}_a(t) & = & \beta_{1,a} \cdot e^{\alpha_a T_a(t)} \cdot I_a(t)^2 - \beta_{2,a} \cdot T_a(t) \qquad & \forall t \in [t_0, t_f], a \in A, \\		\mbox{s.t.} & \dot{T}_a(t) & = & \beta_{1,a} \cdot e^{\alpha_a T_a(t)} \cdot I_a(t)^2 - \beta_{2,a} \cdot T_a(t) \qquad & \forall t \in [t_0, t_f], a \in A, \\
	& d_v & = & \sum_{a \in \delta^{in}(v)} (U_a \cdot I_{a}(t) - P_1^a(t)) - \sum_{a \in \delta^{out}(v)} (U_a \cdot I_a(t)) \qquad & \forall t \in [t_0, t_f], v \in V \setminus P, \\		& d_v & = & \sum_{a \in \delta^{in}(v)} (U_a \cdot I_{a}(t) - P_1^a(t)) - \sum_{a \in \delta^{out}(v)} (U_a \cdot I_a(t)) \qquad & \forall t \in [t_0, t_f], v \in V \setminus P, \\
	& u_v(t) & = & \sum_{a \in \delta^{in}(v)} (U_a \cdot I_{a}(t) - P_1^a(t)) - \sum_{a \in \delta^{out}(v)} (U_a \cdot I_a(t)) \qquad & \forall t \in [t_0, t_f], v \in P, \\[1.5ex]		& -u_v(t) & = & \sum_{a \in \delta^{in}(v)} (U_a \cdot I_{a}(t) - P_1^a(t)) - \sum_{a \in \delta^{out}(v)} (U_a \cdot I_a(t)) \qquad & \forall t \in [t_0, t_f], v \in P, \\[1.5ex]
	& T_a(t_0) &=& 0.0 \qquad & \forall a \in A, \\		& T_a(t_0) &=& 0.0 \qquad & \forall a \in A, \\
	& I_a(t) &\in& [0, I_{max, a}] \qquad & \forall a \in A, t \in [t_0, t_f], \\		& I_a(t) &\in& [0, I_{max, a}] \qquad & \forall a \in A, t \in [t_0, t_f], \\

MirkoHahn: /* Parameters */

2015-12-08T16:28:53Z

Parameters

← Older revision		Revision as of 16:28, 8 December 2015
Line 109:		Line 109:

	=== Delayed heating ===		=== Delayed heating ===
	In this scenario, we consider a simple network consisting of two producers producing <math>1 \;MW</math> each and a single consumer with a demand of <math>1.25 \;MW</math>. Both producers have direct power links of type C and length <math>1 \;km</math> to the consumer. The time horizon is defined by <math>(t_0, t_f) = (0, 300)</math>. A JSON description of the network can be found under [[Direct Current Transmission Heating Problem: Delayed heating (Network)]].		In this scenario, we consider a simple network consisting of two producers producing up to <math>1 \;MW</math> each and a single consumer with a demand of <math>1.25 \;MW</math>. Both producers have direct power links of type C and length <math>1 \;km</math> to the consumer. The time horizon is defined as <math>(t_0, t_f) = (0, 300)</math>. A JSON description of the network can be found under [[Direct Current Transmission Heating Problem: Delayed heating (Network)]].

	== Reference Solutions ==		== Reference Solutions ==

MirkoHahn: /* Mathematical formulation */

2015-12-08T16:27:36Z

Mathematical formulation

← Older revision		Revision as of 16:27, 8 December 2015
Line 67:		Line 67:
	</p>		</p>

	The differential state <math>T_a</math> describes the temperature of power line <math>a</math> while the controls <math>I_a</math> and <math>u_v</math> represent the current in a power line <math>a \in A</math> and the power produced in a production node <math>v \in P</math> respectively		The differential state <math>T_a</math> describes the temperature of power line <math>a</math> while the controls <math>I_a</math> and <math>u_v</math> represent the current in a power line <math>a \in A</math> and the power produced in a production node <math>v \in P</math> respectively.

	== Parameters ==		== Parameters ==

MirkoHahn: /* Mathematical formulation */

2015-12-08T16:27:16Z

Mathematical formulation

← Older revision		Revision as of 16:27, 8 December 2015
Line 67:		Line 67:
	</p>		</p>

	The differential state <math>T_a</math> describes the temperature of power line <math>a</math> while the controls <math>I_a</math> and <math>u_v</math> represent the current in a power line <math>a \in A</math> and the power produced in a production node <math>v \in P</math>		The differential state <math>T_a</math> describes the temperature of power line <math>a</math> while the controls <math>I_a</math> and <math>u_v</math> represent the current in a power line <math>a \in A</math> and the power produced in a production node <math>v \in P</math> respectively

	== Parameters ==		== Parameters ==