Intermediate Variables
Main.Intermediates History
Hide minor edits - Show changes to output
Changed line 1 from:
(:title Intermediates:)
to:
(:title Intermediate Variables:)
Changed line 9 from:
An example of intermediates is with the equations ''z=sqrt(x+y)'' and ''w=(x+y)'^2^''' that both share a common ''x+y'' term. In '''APMonitor''' the ''Intermediates'' keyword is used.
to:
An example of intermediates is with the equations ''z=sqrt(x+y)'' and ''w=(x+y)'^2^''' that both share a common ''x+y'' term. In '''APMonitor''' the ''Intermediates'' keyword defines this special type of variable.
Changed line 9 from:
An example of intermediates is with the equations ''z=sqrt(x+y)'' and ''w=(x+y)'^2^''' that both share a common ''x+y'' term.
to:
An example of intermediates is with the equations ''z=sqrt(x+y)'' and ''w=(x+y)'^2^''' that both share a common ''x+y'' term. In '''APMonitor''' the ''Intermediates'' keyword is used.
Changed lines 1-5 from:
to:
(:title Intermediates:)
(:keywords model reduction, differential, algebraic, modeling language, intermediate, calculation, explicit, speed:)
(:description Intermediate variables in APMonitor and Python Gekko reduce the model size and provide explicit equation calculation.:)
Intermediate variables are used to decrease the complexity of the model. These variables store temporary calculations with results that are not reported in the final solution reports. In many models, the temporary variables outnumber the regular variables. This model reduction often aides the solver in finding a solution by reducing the problem size.
(:keywords model reduction, differential, algebraic, modeling language, intermediate, calculation, explicit, speed:)
(:description Intermediate variables in APMonitor and Python Gekko reduce the model size and provide explicit equation calculation.:)
Intermediate variables are used to decrease the complexity of the model. These variables store temporary calculations with results that are not reported in the final solution reports. In many models, the temporary variables outnumber the regular variables. This model reduction often aides the solver in finding a solution by reducing the problem size.
Changed lines 5-6 from:
to:
An example of intermediates is with the equations ''z=sqrt(x+y)'' and ''w=(x+y)'^2^''' that both share a common ''x+y'' term.
Changed lines 8-16 from:
to:
# APMonitor
Intermediates
i = x+y
End Intermediates
Equations
z=sqrt(i)
w=i^2
End Equations
Intermediates
i = x+y
End Intermediates
Equations
z=sqrt(i)
w=i^2
End Equations
Changed lines 19-20 from:
to:
In [[https://gekko.readthedocs.io/en/latest/|Python Gekko]], intermediates are defined with the Intermediate function.
Added line 22:
# Python Gekko
Changed line 5 from:
In Python Gekko, intermediates are defined with the Intermediate function.
to:
In [[https://gekko.readthedocs.io/en/latest/|Python Gekko]], intermediates are defined with the Intermediate function.
Changed line 5 from:
In Python Gekko, intermediates are defined with with the Intermediate function.
to:
In Python Gekko, intermediates are defined with the Intermediate function.
Added lines 4-17:
In Python Gekko, intermediates are defined with with the Intermediate function.
(:source lang=python:)
y = m.Intermediate(x)
(:sourceend:)
An example of intermediates is with the equations ''z=sqrt(x+y)'' and ''w=(x+y)'^2^''' that both share a common ''x+y'' term.
(:source lang=python:)
i = m.Intermediate(x+y)
m.Equations([z==m.sqrt(i),
w==i**2])
(:sourceend:)
Changed line 1 from:
Intermediate variables are useful to decrease the complexity of the model. These variables store temporary calculations with results that are not reported in the final solution reports. In many models, the temporary variables outnumber the regular variables by many factors. This model reduction often aides the solver in finding a solution by reducing the problem size.
to:
Intermediate variables are useful to decrease the complexity of the model. These variables store temporary calculations with results that are not reported in the final solution reports. In many models, the temporary variables outnumber the regular variables. This model reduction often aides the solver in finding a solution by reducing the problem size.
Changed line 15 from:
%blue%A%red%P%black%Monitor is designed to encourage the user to construct well-posed models for numerical solution. One limitation that may be encountered is the 100 variable limit in each equation. Excessive use of intermediate variables may lead to the violation of this limit. If this limit is encountered, the user can remediate this problem by converting an intermediate variable to a regular implicit variable.
to:
%blue%A%red%P%black%Monitor and [[https://gekko.readthedocs.io/en/latest/|Python Gekko]] are designed to encourage the user to construct well-posed models for numerical solution. Excessive use of intermediate variables may lead to the violation of this limit. If this limit is encountered, the user can remediate this problem by converting an intermediate variable to a regular implicit variable.
Changed line 3 from:
Intermediate variables are declared in the ''Intermediates ... End Intermediates'' section of the model file. The intermediate variables may be defined in one section or in multiple declarations throughout the model. Intermediate variables are parsed sequentially, from top to bottom. To avoid inadvertent overwrites, intermediate variable can be defined once. In the case of intermediate variables, the order of declaration is critical. If an intermediate is used before the definition, an error reports that there is an uninitialized value.
to:
Intermediate variables are declared in the ''Intermediates ... End Intermediates'' section of the model file or with ''m.Intermediates()'' in Python Gekko. The intermediate variables may be defined in one section or in multiple declarations throughout the model. Intermediate variables are parsed sequentially, from top to bottom. To avoid inadvertent overwrites, intermediate variable can be defined once. In the case of intermediate variables, the order of declaration is critical. If an intermediate is used before the definition, an error reports that there is an uninitialized value.
Changed lines 3-4 from:
Intermediate variables are declared in the ''Intermediates ... End Intermediates'' section of the model file. The intermediate variables may be defined in one section or in multiple declarations throughout the model. Intermediate variables are parsed sequentially, from top to bottom. To avoid inadvertent overwrites, intermediate variable can be defined once. In the case of intermediate variables, the order of declaration is critical. If a variable is used before it's definition, it will contain a default value of 1.
to:
Intermediate variables are declared in the ''Intermediates ... End Intermediates'' section of the model file. The intermediate variables may be defined in one section or in multiple declarations throughout the model. Intermediate variables are parsed sequentially, from top to bottom. To avoid inadvertent overwrites, intermediate variable can be defined once. In the case of intermediate variables, the order of declaration is critical. If an intermediate is used before the definition, an error reports that there is an uninitialized value.
Changed line 7 from:
The intermediate variables are processed before the implicit equation residuals, every time the solver requests model information. As opposed to implicitly calculated variables, the explicit variables are calculated once and substituted into other explicit or implicit equations.
to:
The intermediate variables are processed before the implicit equation residuals, every time the solver requests model information. As opposed to implicitly calculated variables, the intermediates are calculated repeatedly and substituted into other intermediate or implicit equations.
Changed lines 21-22 from:
(:table border=1 width=100% align=left bgcolor=#EEEEEE cellspacing=0:)
(:cellnr:)
(:cellnr
to:
(:source lang=python:)
Changed lines 67-69 from:
to:
(:sourceend:)
(:source lang=python:)
(:source lang=python:)
Changed lines 88-89 from:
to:
(:sourceend:)
Changed lines 92-93 from:
!!!! Example 4
to:
!!!! Example Problem
Changed line 106 from:
to:
Attach:download.png [[https://apmonitor.com/do/index.php/Main/DynamicOptimizationBenchmarks|Dynamic Optimization with Intermediates]] (see Problem #4)
Changed line 105 from:
!!!! Solution to [[https://apmonitor.com/do/index.php/Main/DynamicOptimizationBenchmarks|Dynamic Optimization with Intermediates]]
to:
!!!! [[https://apmonitor.com/do/index.php/Main/DynamicOptimizationBenchmarks|Dynamic Optimization with Intermediates]]
Added lines 91-173:
!!!! Example 4
* Batch reactor with consecutive reactions A->B->C
{$\max_{T(t)} x_2 \left( t_f \right)$}
{$\mathrm{subject \; to}$}
{$\frac{dx_1}{dt}=-k_1 \, x_1^2$}
{$\frac{dx_2}{dt}=k_1 \, x_1^2 - k_2 \, x_2$}
{$k_1 = 4000 \, \exp{\left(-\frac{2500}{T}\right)}$}
{$k_2 = 6.2e5 \, \exp{\left(-\frac{5000}{T}\right)}$}
{$x(0) = [1 \; 0]^T$}
{$298 \le T \le 398$}
{$t_f=1$}
!!!! Solution to [[https://apmonitor.com/do/index.php/Main/DynamicOptimizationBenchmarks|Dynamic Optimization with Intermediates]]
(:html:)
<iframe width="560" height="315" src="https://www.youtube.com/embed/yFprG0iJQUE" frameborder="0" allowfullscreen></iframe>
(:htmlend:)
(:toggle hide gekko4 button show="Show GEKKO (Python) Code":)
(:div id=gekko4:)
(:source lang=python:)
import numpy as np
import matplotlib.pyplot as plt
from gekko import GEKKO
m = GEKKO()
nt = 101
m.time = np.linspace(0,1,nt)
# Parameters
T = m.MV(value=362,ub=398,lb=298)
T.STATUS = 1
T.DCOST = 0
# Variables
x1 = m.Var(value=1)
x2 = m.Var(value=0)
p = np.zeros(nt)
p[-1] = 1.0
final = m.Param(value=p)
# Intermediates
k1 = m.Intermediate(4000*m.exp(-2500/T))
k2 = m.Intermediate(6.2e5*m.exp(-5000/T))
# Equations
m.Equation(x1.dt()==-k1*x1**2)
m.Equation(x2.dt()==k1*x1**2 - k2*x2)
# Objective Function
m.Obj(-x2*final)
m.options.IMODE = 6
m.solve()
print('Objective: ' + str(x2[-1]))
plt.figure(1)
plt.subplot(2,1,1)
plt.plot(m.time,x1.value,'k:',LineWidth=2,label=r'$x_1$')
plt.plot(m.time,x2.value,'b-',LineWidth=2,label=r'$x_2$')
plt.ylabel('Value')
plt.legend(loc='best')
plt.subplot(2,1,2)
plt.plot(m.time,T.value,'r--',LineWidth=2,label=r'$T$')
plt.legend(loc='best')
plt.xlabel('Time')
plt.ylabel('Value')
plt.show()
(:sourceend:)
(:divend:)
(:html:)
<iframe width="560" height="315" src="https://www.youtube.com/embed/gptRJ5x7Ybs" frameborder="0" allow="autoplay; encrypted-media" allowfullscreen></iframe>
(:htmlend:)
Changed lines 38-39 from:
(y+2/x)^(x*z) * (log(tanh(sqrt(y-x+x^2))+3))^2 = 2+sinh(y)+acos(x+y)+asin(x/y)
to:
(y+2/x)^(x*z) * (log(tanh(sqrt(y-x+x^2))+3))^2 = &
2+sinh(y)+acos(x+y)+asin(x/y)
2+sinh(y)+acos(x+y)+asin(x/y)
Deleted lines 0-1:
Changed line 21 from:
(:table border=1 width=50% align=left bgcolor=#EEEEEE cellspacing=0:)
to:
(:table border=1 width=100% align=left bgcolor=#EEEEEE cellspacing=0:)
Added lines 14-17:
!!! Limitations
%blue%A%red%P%black%Monitor is designed to encourage the user to construct well-posed models for numerical solution. One limitation that may be encountered is the 100 variable limit in each equation. Excessive use of intermediate variables may lead to the violation of this limit. If this limit is encountered, the user can remediate this problem by converting an intermediate variable to a regular implicit variable.
Added lines 1-85:
!! Intermediate Variables and Equations
Intermediate variables are useful to decrease the complexity of the model. These variables store temporary calculations with results that are not reported in the final solution reports. In many models, the temporary variables outnumber the regular variables by many factors. This model reduction often aides the solver in finding a solution by reducing the problem size.
Intermediate variables are declared in the ''Intermediates ... End Intermediates'' section of the model file. The intermediate variables may be defined in one section or in multiple declarations throughout the model. Intermediate variables are parsed sequentially, from top to bottom. To avoid inadvertent overwrites, intermediate variable can be defined once. In the case of intermediate variables, the order of declaration is critical. If a variable is used before it's definition, it will contain a default value of 1.
!!! Explicit calculation
The intermediate variables are processed before the implicit equation residuals, every time the solver requests model information. As opposed to implicitly calculated variables, the explicit variables are calculated once and substituted into other explicit or implicit equations.
!!! Clipping
When the intermediate variable is solely a function of parameters (not variables), the value may be clipped. This is accomplished by adding inequalities to the expression, separated by a comma. The inequalities may also be a function of other intermediate or regular variables.
!!! Example
(:table border=1 width=50% align=left bgcolor=#EEEEEE cellspacing=0:)
(:cellnr:)
! Original model
Model example
Parameters
p = 2
End Parameters
Variables
x
y
z
End Variables
Equations
exp(x*p)=y
z = p*$x + x
(y+2/x)^(x*z) * (log(tanh(sqrt(y-x+x^2))+3))^2 = 2+sinh(y)+acos(x+y)+asin(x/y)
End Equations
End Model
! Model simplified with use of intermediate variables
Model example
Parameters
p = 2
End Parameters
Variables
x
y
z
End Variables
Intermediates
exp_result = exp(x*p)
left = (y+2/x)^(x*z) * (log(tanh(sqrt(y-x+x^2))+3))^2
right = 2+sinh(y)+acos(x+y)+asin(x/y)
End Intermediates
Equations
exp_result=y
z = p*$x + x
left = right
End Equations
End Model
(:cellnr:)
! Example with intermediate variable clipping
Model example
Parameters
p = 1
End Parameters
Variables
x
End Variables
Intermediates
pi = p*3.1415, <1.5
End Intermediates
Equations
x = 0.5 * pi
End Equations
End Model
(:tableend:)
Intermediate variables are useful to decrease the complexity of the model. These variables store temporary calculations with results that are not reported in the final solution reports. In many models, the temporary variables outnumber the regular variables by many factors. This model reduction often aides the solver in finding a solution by reducing the problem size.
Intermediate variables are declared in the ''Intermediates ... End Intermediates'' section of the model file. The intermediate variables may be defined in one section or in multiple declarations throughout the model. Intermediate variables are parsed sequentially, from top to bottom. To avoid inadvertent overwrites, intermediate variable can be defined once. In the case of intermediate variables, the order of declaration is critical. If a variable is used before it's definition, it will contain a default value of 1.
!!! Explicit calculation
The intermediate variables are processed before the implicit equation residuals, every time the solver requests model information. As opposed to implicitly calculated variables, the explicit variables are calculated once and substituted into other explicit or implicit equations.
!!! Clipping
When the intermediate variable is solely a function of parameters (not variables), the value may be clipped. This is accomplished by adding inequalities to the expression, separated by a comma. The inequalities may also be a function of other intermediate or regular variables.
!!! Example
(:table border=1 width=50% align=left bgcolor=#EEEEEE cellspacing=0:)
(:cellnr:)
! Original model
Model example
Parameters
p = 2
End Parameters
Variables
x
y
z
End Variables
Equations
exp(x*p)=y
z = p*$x + x
(y+2/x)^(x*z) * (log(tanh(sqrt(y-x+x^2))+3))^2 = 2+sinh(y)+acos(x+y)+asin(x/y)
End Equations
End Model
! Model simplified with use of intermediate variables
Model example
Parameters
p = 2
End Parameters
Variables
x
y
z
End Variables
Intermediates
exp_result = exp(x*p)
left = (y+2/x)^(x*z) * (log(tanh(sqrt(y-x+x^2))+3))^2
right = 2+sinh(y)+acos(x+y)+asin(x/y)
End Intermediates
Equations
exp_result=y
z = p*$x + x
left = right
End Equations
End Model
(:cellnr:)
! Example with intermediate variable clipping
Model example
Parameters
p = 1
End Parameters
Variables
x
End Variables
Intermediates
pi = p*3.1415, <1.5
End Intermediates
Equations
x = 0.5 * pi
End Equations
End Model
(:tableend:)