Main.Intermediates History

November 27, 2021, at 06:56 PM by 10.35.117.248 -
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(:title Intermediates:)

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(:title Intermediate Variables:)

November 27, 2021, at 06:55 PM by 10.35.117.248 -
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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.

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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.

November 27, 2021, at 06:54 PM by 10.35.117.248 -
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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.

November 27, 2021, at 06:53 PM by 10.35.117.248 -
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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.

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(: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.

November 27, 2021, at 06:50 PM by 10.35.117.248 -
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In Python Gekko, intermediates are defined with the Intermediate function.

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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.

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y = m.Intermediate(x)

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1. APMonitor

Intermediates

i = x+y

End Intermediates

Equations

z=sqrt(i)
w=i^2

End Equations

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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:

In Python Gekko, intermediates are defined with the Intermediate function.

1. Python Gekko
November 27, 2021, at 06:48 PM by 10.35.117.248 -
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In Python Gekko, intermediates are defined with the Intermediate function.

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In Python Gekko, intermediates are defined with the Intermediate function.

November 27, 2021, at 06:47 PM by 10.35.117.248 -
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In Python Gekko, intermediates are defined with with the Intermediate function.

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In Python Gekko, intermediates are defined with the Intermediate function.

November 27, 2021, at 06:46 PM by 10.35.117.248 -

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:)

October 30, 2019, at 03:10 PM by 136.36.211.159 -
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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.

October 30, 2019, at 02:40 PM by 136.36.211.159 -
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APMonitor 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:

APMonitor and 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.

October 30, 2019, at 02:39 PM by 136.36.211.159 -
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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.

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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.

May 31, 2019, at 11:39 PM by 10.37.66.43 -
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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.

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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.

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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.

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(:source lang=python:)

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(:cellnr:)

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(:sourceend:)

(:source lang=python:)

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(:tableend:)

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(:sourceend:)

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Example Problem

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Dynamic Optimization with Intermediates

to: Dynamic Optimization with Intermediates (see Problem #4)
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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 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)

1. Parameters

T = m.MV(value=362,ub=398,lb=298) T.STATUS = 1 T.DCOST = 0

1. Variables

x1 = m.Var(value=1) x2 = m.Var(value=0)

p = np.zeros(nt) p[-1] = 1.0 final = m.Param(value=p)

1. Intermediates

k1 = m.Intermediate(4000*m.exp(-2500/T)) k2 = m.Intermediate(6.2e5*m.exp(-5000/T))

1. Equations

m.Equation(x1.dt()==-k1*x1**2) m.Equation(x2.dt()==k1*x1**2 - k2*x2)

1. 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:)

June 16, 2015, at 06:48 PM by 45.56.3.184 -
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(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)
June 16, 2015, at 06:47 PM by 45.56.3.184 -
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Intermediate Variables and Equations

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(:table border=1 width=100% align=left bgcolor=#EEEEEE cellspacing=0:)

October 02, 2008, at 09:22 PM by 158.35.225.228 -
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Example

September 26, 2008, at 07:38 PM by 158.35.225.231 -

Limitations

APMonitor 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.

September 26, 2008, at 07:17 PM by 158.35.225.231 -

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:)