mintOC - User contributions [en]

Egerstedt standard problem

2019-09-19T15:09:18Z

ChristophHansknecht: Fixed type, error in relaxed objective function value

{{Dimensions
|nd = 1
|nx = 3
|nw = 3
|nc = 1
|nre = 3
}}The '''Egerstedt standard problem''' is the problem is of an academic nature and was proposed by Egerestedt to highlight the features of an Hybrid System algorithm in 2006 <bib id="Egerstedt2006" />. It has been used since then in many MIOCP research studies (e.g. <bib id="Jung2013" />) for benchmarking of MIOCP algorithms.

== Mathematical formulation ==

The mixed-integer optimal control problem after partial outer convexification is given by


<math>
\begin{array}{llclr}
\displaystyle \min_{x, \omega} & x_3(t_f) \\[1.5ex]
\mbox{s.t.}
& \dot{x}_1 & = & -x_1\omega_1 + (x_1+x_2)\omega_2+(x_1-x_2)\omega_3, \\
& \dot{x}_2 & = & (x_1+2x_2)\omega_1+(x_1-2x_2)\omega_2+(x_1+x_2)\omega_3, \\
& \dot{x}_3 & = & x_1^2+x_2^2, \\[1.5ex]
& x(0) &=& (0.5, 0.5, 0)^T, \\
& x_2(t) & \geq & 0.4, \\
& 1 &=& \sum\limits_{i=1}^3\omega_i(t), \\
& \omega(t) &\in& \{0, 1\},
\end{array}
</math>

for <math>t \in [t_0, t_f]=[0,1] </math>.

== Reference Solutions ==

If the problem is relaxed, i.e., we demand that <math>w(t)</math> be in the continuous interval <math>[0, 1]</math> instead of the binary choice <math>\{0,1\}</math>, the optimal solution can be determined by using a direct method such as collocation or Bock's direct multiple shooting method.

The optimal objective value of the relaxed problem with <math> n_t=6000, \, n_u=40 </math> is <math>x_3(t_f)=0.995906234</math>. The objective value of the binary controls obtained by Combinatorial Integral Approimation (CIA) is <math>x_3(t_f) =3.20831942</math>. The binary control solution was evaluated in the MIOCP by using a Merit function with additional Lagrange term <math> 100 \max\limits_{t\in[0,1]}\{0,0.4-x_2(t)\} </math>.

<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
Image:EgerstedtRelaxed 6000 150 1.png| Optimal relaxed controls and states determined by an direct approach with ampl_mintoc (Radau collocation) and <math>n_t=6000, \, n_u=40</math>.
Image:EgerstedtCIA 6000 150 1.png| Optimal binary controls and states determined by an direct approach (Radau collocation) with ampl_mintoc and <math>n_t=6000, \, n_u=40</math>. The relaxed controls were approximated by Combinatorial Integral Approximation.
</gallery>

== Source Code ==

Model description is available in
* [[:Category:AMPL | AMPL code]] at [[Egerstedt standard problem (AMPL)]]

== References ==
<biblist />


[[Category:MIOCP]]
[[Category:ODE model]]
[[Category:Tracking objective]]
[[Category:Sensitivity-seeking arcs]]



We present numerical results for a benchmark MIOCP from a previous study [157] with the
addition of switching constraints. In its original form, the problem was:

After partial outer convexification with respect to the integer control v, the binary
convexified counterpart problem reads

Continuously Stirred Tank Reactor problem

2019-08-28T23:06:25Z

ChristophHansknecht: /* Parameters */

{{Dimensions
|nd = 1
|nx = 4
|nu = 2
|nre = 2
}}The Continuously Stirred Tank Reactor problem considers a chemical reaction that produces cyclopenthenol while using up cyclepentadiene "by an acid-catalyzed electrophilic hydration in aqueous solution", an exothermal reaction which needs to be cooled. This problem can e.g. be found in <bib id="Diehl2001" />.

The inflow into the tank contains only cyclopentadiene (substance <math> A </math>) with temperature <math> \theta_0 </math> and the flow rate <math> \dot{V} </math> can be controlled. The outflow rate is the same as the inflow rate to keep the liquid level in the tank constant.
"The outflow contains a remainder of cyclopentadiene, the wanted product cyclepentenol (substance <math> B </math>) and two unwated by-products, cyclopentanediol (substance <math> C </math>) and dicyclopentadiene (substance <math> D </math>) with concentrations <math> c_A, c_B, c_C, c_D </math>." The latter two are not tracked in the problem as the substances are not of use.
The reaction scheme is given as:

<math>
\begin{array}{ccccc}
A & \overset{k_1}\rightarrow & B \overset{k_2}\rightarrow & C\\
2A & \overset{k_3}\rightarrow & D
\end{array}
</math>

where the reaction rates <math> k_i </math> are a function of the reactor temperature <math> \theta </math> via an Arrhenius law
<math> k_i(\theta) = k_{i0} \cdot \exp ( \frac{E_i}{\theta / ^\circ C + 273.15} ), \quad i=1,2,3. </math>

"The temperature <math> \theta_K </math> in the cooling jacket is held down by an external heat exchanger whose heat removal rate <math> \dot{Q}_K </math> can be controlled."

== Mathematical formulation ==
The problem is given by


<math>
\begin{array}{llcl}
\displaystyle \max_{\dot{V}, \dot{Q}_K} & c_B & & \text{ at the end of reaction} \\[1.5ex]
\mbox{s.t.}
& \dot{c_A} & = & \frac{\dot{V}}{V_R} (c_{A0} - c_A) - k_1 c_A - k_3 c_A^2, \\[0.5cm]
& \dot{c_B} & = & -\frac{\dot{V}}{V_R} c_B + k_1 c_A - k_2 c_B, \\[0.5cm]
& \dot{\theta} & = & \frac{\dot{V}}{V_R} ( \theta_0 - \theta) + \frac{k_w A_R}{\rho C_p V_R} (\theta_K - \theta) - \frac{1}{\rho C_p} (k_1 c_A H_1 + k_2 c_B H_2 + k_3 c_A^2 H_3), \\[0.5cm]
& \dot{\theta_K} & = & \frac{1}{m_K C_{PK}} ( \dot{Q}_K + k_w A_R (\theta - \theta_K)),\\[0.7cm]
& c_A(0) & = & c_{A0},\\
& c_B(0) & = & 0.
\end{array}
</math>

where the various values are given in the Parameters section.

== Parameters ==
These fixed values are used within the model.
{| class="wikitable"
|+Parameters
|-
|Name
|Symbol
|Value
|Unit
|-
|Arrhenius coefficient
|<math>k_{10}</math>
|<math> 1.287 \cdot 10^{12} </math>
|<math> h^{-1} </math>
|-
|Arrhenius coefficient
|<math>k_{20}</math>
|<math> 1.287 \cdot 10^{12} </math>
|<math> h^{-1} </math>
|-
|Arrhenius coefficient
|<math>k_{30}</math>
|<math> 9.043 \cdot 10^{9} </math>
|<math> h^{-1} </math>
|-
|Arrhenius coefficient
|<math>E_1</math>
|<math> -9758.3</math>
|[-]
|-
|Arrhenius coefficient
|<math>E_2</math>
|<math>-9758.3</math>
|[-]
|-
|Arrhenius coefficient
|<math>E_3</math>
|<math>-8560</math>
|[-]
|-
|Reaction enthalpy
|<math>H_1</math>
|<math>4.2</math>
|<math> \frac{kJ}{mol} </math>
|-
|Reaction enthalpy
|<math>H_2</math>
|<math>-11.0</math>
|<math> \frac{kJ}{mol} </math>
|-
|Reaction enthalpy
|<math>H_3</math>
|<math>-41.85</math>
|<math> \frac{kJ}{mol} </math>
|-
|Solution density
|<math>\rho</math>
|<math>0.9342</math>
|<math> \frac{kg}{l} </math>
|-
|Capacity of aqueous solution
|<math>C_p</math>
|<math>3.01</math>
|<math> \frac{kJ}{kg \cdot K} </math>
|-
|Heat transfer coefficient for cooling jacket
|<math>k_w</math>
|<math>4032</math>
|<math> \frac{kJ}{h \cdot m^2 \cdot K} </math>
|-
|Reactor surface area
|<math>A_R</math>
|<math>0.215</math>
|<math> m^2 </math>
|-
|Reactor volume
|<math>V_R</math>
|<math>10</math>
|<math> l </math>
|-
|Coolant mass
|<math>m_K</math>
|<math>5</math>
|<math> kg </math>
|-
|Capacity of coolant solution
|<math>C_{PK}</math>
|<math>2.0</math>
|<math> \frac{kJ}{kg \cdot K} </math>
|-
|Starting concentration of subs. <math> A </math>
|<math>c_{A0}</math>
|<math>5.1</math>
|<math> \frac{mol}{l} </math>
|-
|Inflow temperature
|<math>\theta_0</math>
|<math>104.9</math>
|<math> ^\circ C </math>
|}

== Reference solution ==

"The result of a steady state optimization of the yield <math> = \frac{c_B |_S}{c_{A0}} </math> with respect to the design parameter <math> \theta_0 </math> (feed temperature) and the two controls yields the steady stae and controls"
<math> c_A = 2.1402 \frac{mol}{l}, c_B = 1.0903\frac{mol}{l}, \theta = 114.19^\circ C, \theta_K = 112.91^\circ C </math> and <math> \frac{\dot{V}}{V_R} = 14.19 h^{-1}, \dot{Q}_K = -1113.5 \frac{kJ}{h} </math>.

== Source Code ==

Model descriptions are available in

* [[:Category:AMPL/TACO | AMPL/TACO code]] at [[Continuously Stirred Tank Reactor (TACO)]]


[[Category:MIOCP]]
[[Category:ODE model]]
[[Category:Chemical engineering]]

Continuously Stirred Tank Reactor problem

2019-08-28T23:05:13Z

ChristophHansknecht: Fixed incorrect parameter

{{Dimensions
|nd = 1
|nx = 4
|nu = 2
|nre = 2
}}The Continuously Stirred Tank Reactor problem considers a chemical reaction that produces cyclopenthenol while using up cyclepentadiene "by an acid-catalyzed electrophilic hydration in aqueous solution", an exothermal reaction which needs to be cooled. This problem can e.g. be found in <bib id="Diehl2001" />.

The inflow into the tank contains only cyclopentadiene (substance <math> A </math>) with temperature <math> \theta_0 </math> and the flow rate <math> \dot{V} </math> can be controlled. The outflow rate is the same as the inflow rate to keep the liquid level in the tank constant.
"The outflow contains a remainder of cyclopentadiene, the wanted product cyclepentenol (substance <math> B </math>) and two unwated by-products, cyclopentanediol (substance <math> C </math>) and dicyclopentadiene (substance <math> D </math>) with concentrations <math> c_A, c_B, c_C, c_D </math>." The latter two are not tracked in the problem as the substances are not of use.
The reaction scheme is given as:

<math>
\begin{array}{ccccc}
A & \overset{k_1}\rightarrow & B \overset{k_2}\rightarrow & C\\
2A & \overset{k_3}\rightarrow & D
\end{array}
</math>

where the reaction rates <math> k_i </math> are a function of the reactor temperature <math> \theta </math> via an Arrhenius law
<math> k_i(\theta) = k_{i0} \cdot \exp ( \frac{E_i}{\theta / ^\circ C + 273.15} ), \quad i=1,2,3. </math>

"The temperature <math> \theta_K </math> in the cooling jacket is held down by an external heat exchanger whose heat removal rate <math> \dot{Q}_K </math> can be controlled."

== Mathematical formulation ==
The problem is given by


<math>
\begin{array}{llcl}
\displaystyle \max_{\dot{V}, \dot{Q}_K} & c_B & & \text{ at the end of reaction} \\[1.5ex]
\mbox{s.t.}
& \dot{c_A} & = & \frac{\dot{V}}{V_R} (c_{A0} - c_A) - k_1 c_A - k_3 c_A^2, \\[0.5cm]
& \dot{c_B} & = & -\frac{\dot{V}}{V_R} c_B + k_1 c_A - k_2 c_B, \\[0.5cm]
& \dot{\theta} & = & \frac{\dot{V}}{V_R} ( \theta_0 - \theta) + \frac{k_w A_R}{\rho C_p V_R} (\theta_K - \theta) - \frac{1}{\rho C_p} (k_1 c_A H_1 + k_2 c_B H_2 + k_3 c_A^2 H_3), \\[0.5cm]
& \dot{\theta_K} & = & \frac{1}{m_K C_{PK}} ( \dot{Q}_K + k_w A_R (\theta - \theta_K)),\\[0.7cm]
& c_A(0) & = & c_{A0},\\
& c_B(0) & = & 0.
\end{array}
</math>

where the various values are given in the Parameters section.

== Parameters ==
These fixed values are used within the model.
{| class="wikitable"
|+Parameters
|-
|Name
|Symbol
|Value
|Unit
|-
|Arrhenius coefficient
|<math>k_{10}</math>
|<math> 1.287 \cdot 10^{13} </math>
|<math> h^{-1} </math>
|-
|Arrhenius coefficient
|<math>k_{20}</math>
|<math> 1.287 \cdot 10^{12} </math>
|<math> h^{-1} </math>
|-
|Arrhenius coefficient
|<math>k_{30}</math>
|<math> 9.043 \cdot 10^{9} </math>
|<math> h^{-1} </math>
|-
|Arrhenius coefficient
|<math>E_1</math>
|<math> -9758.3</math>
|[-]
|-
|Arrhenius coefficient
|<math>E_2</math>
|<math>-9758.3</math>
|[-]
|-
|Arrhenius coefficient
|<math>E_3</math>
|<math>-8560</math>
|[-]
|-
|Reaction enthalpy
|<math>H_1</math>
|<math>4.2</math>
|<math> \frac{kJ}{mol} </math>
|-
|Reaction enthalpy
|<math>H_2</math>
|<math>-11.0</math>
|<math> \frac{kJ}{mol} </math>
|-
|Reaction enthalpy
|<math>H_3</math>
|<math>-41.85</math>
|<math> \frac{kJ}{mol} </math>
|-
|Solution density
|<math>\rho</math>
|<math>0.9342</math>
|<math> \frac{kg}{l} </math>
|-
|Capacity of aqueous solution
|<math>C_p</math>
|<math>3.01</math>
|<math> \frac{kJ}{kg \cdot K} </math>
|-
|Heat transfer coefficient for cooling jacket
|<math>k_w</math>
|<math>4032</math>
|<math> \frac{kJ}{h \cdot m^2 \cdot K} </math>
|-
|Reactor surface area
|<math>A_R</math>
|<math>0.215</math>
|<math> m^2 </math>
|-
|Reactor volume
|<math>V_R</math>
|<math>10</math>
|<math> l </math>
|-
|Coolant mass
|<math>m_K</math>
|<math>5</math>
|<math> kg </math>
|-
|Capacity of coolant solution
|<math>C_{PK}</math>
|<math>2.0</math>
|<math> \frac{kJ}{kg \cdot K} </math>
|-
|Starting concentration of subs. <math> A </math>
|<math>c_{A0}</math>
|<math>5.1</math>
|<math> \frac{mol}{l} </math>
|-
|Inflow temperature
|<math>\theta_0</math>
|<math>104.9</math>
|<math> ^\circ C </math>
|}

== Reference solution ==

"The result of a steady state optimization of the yield <math> = \frac{c_B |_S}{c_{A0}} </math> with respect to the design parameter <math> \theta_0 </math> (feed temperature) and the two controls yields the steady stae and controls"
<math> c_A = 2.1402 \frac{mol}{l}, c_B = 1.0903\frac{mol}{l}, \theta = 114.19^\circ C, \theta_K = 112.91^\circ C </math> and <math> \frac{\dot{V}}{V_R} = 14.19 h^{-1}, \dot{Q}_K = -1113.5 \frac{kJ}{h} </math>.

== Source Code ==

Model descriptions are available in

* [[:Category:AMPL/TACO | AMPL/TACO code]] at [[Continuously Stirred Tank Reactor (TACO)]]


[[Category:MIOCP]]
[[Category:ODE model]]
[[Category:Chemical engineering]]