## Apps.PeriodicBoundaryConditions History

March 30, 2021, at 10:19 PM by 10.35.117.248 -
Changed line 37 from:
{$0 \le u \ge 5$}
to:
{$0 \le u \le 5$}
March 10, 2021, at 03:21 PM by 10.35.117.248 -
Changed lines 218-219 from:
* Safdarnejad, S.M., Hedengren, J.D., Baxter, L.L.,
Plant-level dynamic optimization of Cryogenic Carbon Capture with conventional and renewable power sources, Applied Energy, Volume 149, 2015, Pages 354-366, ISSN 0306-2619, https://doi.org/10.1016/j.apenergy.2015.03.100 [[https://www.sciencedirect.com/science/article/pii/S030626191500402X|Article]]
to:
* Safdarnejad, S.M., Hedengren, J.D., Baxter, L.L., Plant-level dynamic optimization of Cryogenic Carbon Capture with conventional and renewable power sources, Applied Energy, Volume 149, 2015, Pages 354-366, ISSN 0306-2619, DOI: 10.1016/j.apenergy.2015.03.100. [[https://www.sciencedirect.com/science/article/pii/S030626191500402X|Article]]
March 10, 2021, at 03:21 PM by 10.35.117.248 -

!!!! Reference

* Safdarnejad, S.M., Hedengren, J.D., Baxter, L.L.,
Plant-level dynamic optimization of Cryogenic Carbon Capture with conventional and renewable power sources, Applied Energy, Volume 149, 2015, Pages 354-366, ISSN 0306-2619, https://doi.org/10.1016/j.apenergy.2015.03.100 [[https://www.sciencedirect.com/science/article/pii/S030626191500402X|Article]]
March 10, 2021, at 03:18 PM by 10.35.117.248 -
Changed lines 27-28 from:
The following example illustrates the use of the boundary condition. Scripts in MATLAB and Python are available below to recreate this solution along with the model equations in APMonitor. Both MATLAB and Python scripts produce equivalent results.
to:
An example illustrates the use of periodic boundary conditions.

{$\min_u \left(x-3\right)^2$}

{$\frac{dx}{dt}+x=cos(t)+u$}

{$x(0)=x(8)=1$}

{$u(0)=u(8)=1$}

{$0 \le u \ge 5$}

%width=550px%Attach:periodic_conditions
.png

Scripts in MATLAB and Python are available below to recreate this solution along with the model equations in APMonitor. Both MATLAB and Python scripts produce equivalent results.
March 10, 2021, at 03:02 PM by 10.35.117.248 -
Changed lines 53-54 from:
to:
Changed line 198 from:
to:
March 10, 2021, at 03:01 PM by 10.35.117.248 -
Changed lines 29-34 from:
[[Attach:periodic_example.zip|Periodic Example Script Files (periodic_example.zip)]]

APMonitor Model

(:toggle hide apmcode button show="Show APMonitor Model File":)
(:div id=apmcode
:)
to:
(:toggle hide gkcode button show="Show GEKKO Python Source":)
(:div id=gkcode:)
from gekko import GEKKO
import numpy as np
m = GEKKO()
m.time = np.linspace(0,8,81)
t = m.Param(m.time)
u = m.MV(1,lb=0,ub=5); u.STATUS=1
x = m.Var(1)
m.periodic(u)
m.periodic(x)
m.Minimize((x-3)**2)
m.Equation(x.dt()+x==m.cos(t)+u)
m.options.IMODE = 6
m.solve()

import matplotlib.pyplot as plt
plt.plot(m.time,u,m.time,x)
plt.legend(['u','x'])
plt.show()
(:sourceend:)
(:divend:)

(:toggle hide apmcode button show="Show APMonitor Model File":)
(:div id=apmcode:)
(:source lang=python:)
Changed line 198 from:
[[Attach:periodic_energy_storage.zip|Periodic Energy Storage in MATLAB/Python (periodic_energy_storage.zip)]]
to:
March 10, 2021, at 02:49 PM by 10.35.117.248 -
Changed lines 7-8 from:
In the APMonitor software, boundary conditions are added for select variables with the use of a periodic object declaration.
to:
Boundary conditions are added for select variables with the use of a periodic object declaration.

'''APMonitor Model'''

Changed lines 17-19 from:
In Python Gekko, there is a ''periodic'' function.
to:
In Python Gekko, there is a ''periodic'' function to add the APMonitor periodic condition.

'''Python Gekko'''
March 10, 2021, at 02:48 PM by 10.35.117.248 -
Changed lines 7-21 from:
In the APMonitor software, boundary conditions are added for select variables with the use of a periodic object declaration. Linking this periodic object to a variable in the model enforces the periodic condition by adding an additional equation that the end point must be equal to the beginning point in the horizon.
to:
In the APMonitor software, boundary conditions are added for select variables with the use of a periodic object declaration.

(:source lang=python:)
Objects
q = periodic
End Objects
(:sourceend:)

In Python Gekko, there is a ''periodic'' function.

(:source lang=python:)
m.periodic(q)
(:sourceend:)

Linking this periodic object to a variable in the model enforces the periodic condition by adding an additional equation that the end point must be equal to the beginning point in the horizon.
March 10, 2021, at 02:46 PM by 10.35.117.248 -
Changed lines 61-62 from:
to:
(:divend:)
(:toggle hide gekkocode button show="Show GEKKO Python Source":)
(:div id=gekkocode:)
(:divend:)
March 10, 2021, at 02:45 PM by 10.35.117.248 -
Changed lines 15-17 from:
(:source lang=matlab:)
to:
(:toggle hide apmcode button show="Show APMonitor Model File":)
(:div id=apmcode:)
(:source lang=python
:)
Changed lines 39-40 from:
vx >= 0 !used for energy storage representation
vy >=
0 !used for energy recovery representation
to:
vx >= 0  # slack variable for energy storage representation
vy >=
0  # slack variable for energy recovery representation
March 10, 2021, at 02:43 PM by 10.35.117.248 -
Changed lines 14-17 from:
->Attach:periodic_model.png

MATLAB Script
->Attach:periodic_script_matlab.png
to:

(:source lang=matlab:)
Objects
q = periodic
End Objects

Connections
s = q.x
End Connections

Constants
eps = 0.7
End Constants

Parameters
d
p
End Parameters

Variables
s >= 0 , = 100
stored
recovery
vx >= 0 !used for energy storage representation
vy >= 0 !used for energy recovery representation
End Variables

Equations
minimize p
p + recovery/eps - stored >= d
p - d = vx- vy
stored = p-d + vy
recovery = d- p + vx
$s = stored - recovery/ eps stored * recovery <= 0 End Equations File *.plt New Trend p s d End File (:sourceend:) March 10, 2021, at 02:41 PM by 10.35.117.248 - Changed line 35 from: http://apmonitor.com/wiki/index.php/Apps/PeriodicBoundaryConditions to: https://apmonitor.com/wiki/index.php/Apps/PeriodicBoundaryConditions March 10, 2021, at 02:41 PM by 10.35.117.248 - Deleted lines 18-21: Python Script ->Attach:periodic_script_python.png ->Attach:periodic_plot.png March 10, 2021, at 02:40 PM by 10.35.117.248 - Changed line 117 from: Attach:periodic_storage2.png to: %width=550px%Attach:periodic_storage2.png March 10, 2021, at 02:40 PM by 10.35.117.248 - Changed lines 29-31 from: [[Attach:periodic_energy_storage.zip|Periodic Energy Storage Script Files (periodic_energy_storage.zip)]] Attach:periodic_storage.png to: (:source lang=python:) #!/usr/bin/env python3 # -*- coding: utf-8 -*- """ Created on Mon Mar 8 21:34:49 2021 Gekko implementation of the simple energy storage model found here: https://www.sciencedirect.com/science/article/abs/pii/S030626191500402X Useful link: http://apmonitor.com/wiki/index.php/Apps/PeriodicBoundaryConditions @author: Nathaniel Gates, John Hedengren """ import numpy as np import matplotlib.pyplot as plt import matplotlib.ticker as mtick from gekko import GEKKO m = GEKKO(remote=False) t = np.linspace(0, 24, 24*3+1) m.time = t m.options.SOLVER = 1 m.options.IMODE = 6 m.options.NODES = 3 m.options.CV_TYPE = 1 m.options.MAX_ITER = 300 p = m.FV() # production p.STATUS = 1 s = m.Var(100, lb=0) # storage inventory store = m.SV() # store energy rate vy = m.SV(lb=0) # store slack variable recover = m.SV() # recover energy rate vx = m.SV(lb=0) # recover slack variable eps = 0.7 d = m.MV(-20*np.sin(np.pi*t/12)+100) m.periodic(s) m.Equations([p + recover/eps - store >= d, p - d == vx - vy, store == p - d + vy, recover == d - p + vx, s.dt() == store - recover/eps, store * recover <= 0]) m.Minimize(p) m.solve(disp=True) #%% Visualize results fig, axes = plt.subplots(4, 1, sharex=True) ax = axes[0] ax.plot(t, store, 'C3-', label='Store Rate') ax.plot(t, recover, 'C0-.', label='Recover Rate') ax = axes[1] ax.plot(t, d, 'k-', label='Electricity Demand') ax.plot(t, p, 'C3--', label='Power Production') ax = axes[2] ax.plot(t, s, 'C2-', label='Energy Inventory') ax = axes[3] ax.plot(t, vx, 'C2-', label='$S_1$') ax.plot(t, vy, 'C3--', label='$S_2\$')
ax.set_xlabel('Time (hr)')

for ax in axes:
ax.legend(bbox_to_anchor=(1.01, 0.5), \
loc='center left', frameon=False)
ax.grid()
ax.set_xlim(0, 24)
loc = mtick.MultipleLocator(base=6)
ax.xaxis.set_major_locator(loc)

plt.tight_layout()
plt.show()
(:sourceend:)

[[Attach:periodic_energy_storage.zip|Periodic Energy Storage in MATLAB/Python (periodic_energy_storage.zip)]]

Attach:periodic_storage2.png
May 28, 2014, at 10:49 PM by 107.188.175.164 -

!!!! Periodic Energy Storage
May 28, 2014, at 10:49 PM by 107.188.175.164 -

Attach:periodic_storage.png
May 28, 2014, at 10:48 PM by 107.188.175.164 -
Changed line 27 from:
[[Attach:periodic_storage.zip|Periodic Energy Storage Script Files (periodic_storage.zip)]]
to:
[[Attach:periodic_energy_storage.zip|Periodic Energy Storage Script Files (periodic_energy_storage.zip)]]
May 28, 2014, at 10:48 PM by 107.188.175.164 -
Changed lines 9-10 from:
The following example illustrates the use of the boundary condition. Scripts in MATLAB and Python are available below to recreate this solution.
to:
The following example illustrates the use of the boundary condition. Scripts in MATLAB and Python are available below to recreate this solution along with the model equations in APMonitor. Both MATLAB and Python scripts produce equivalent results.
Changed lines 13-16 from:
* MATLAB Script
to:
APMonitor Model
->Attach:periodic_model.png

MATLAB Script
Changed line 19 from:
* Python Script
to:
Python Script

->Attach:periodic_plot.png
----

A further example demonstrates a more complicated model for energy storage and retrieval. In this case, energy is stored during the first hours of the day when demand is lower. The power generation runs at a constant level while the energy storage is able to follow the cyclical demand. Energy storage is set to a periodic boundary condition to ensure that the beginning and end of the day have at least 100 units of stored energy. Scripts are available in both MATLAB and Python.

[[Attach:periodic_storage.zip|Periodic Energy Storage Script Files (periodic_storage.zip)]]
May 28, 2014, at 10:42 PM by 107.188.175.164 -
Changed lines 13-18 from:
(:table border=1 width=100%:)
(:cell:) MATLAB
(:cell:)
Python
(:cellnr:) Attach:periodic_script_matlab.png
(:cell:) Attach:periodic_script_python.png
(:tableend:)
to:
* MATLAB Script
->Attach
:periodic_script_matlab.png

*
Python Script
->
Attach:periodic_script_python.png
May 28, 2014, at 10:36 PM by 107.188.175.164 -
Changed lines 16-17 from:
(:cellnr:)
(:cell
:) Attach:periodic_script_matlab.png
to:
(:cellnr:) Attach:periodic_script_matlab.png
May 28, 2014, at 10:35 PM by 107.188.175.164 -
(:title Periodic Boundary Conditions:)
(:keywords periodic, Circadian rhythm, differential, algebraic, modeling language, numerical, boundary condition:)
(:description Solve dynamic estimation and optimization problems with periodic boundary conditions.:)

Periodic boundary conditions arise in any situation where the end point must be equal to the beginning point. This type of boundary condition is typical where something is repeating many times but the optimization or simulation only needs to take place over one cycle of that sequence. An examples of a repeating process is the body's natural [[https://en.wikipedia.org/wiki/Circadian_rhythm|Circadian rhythm]] or a power plant that produces power to follow daily demand cycles. Examples of periodic boundary conditions in natural cycles or in manufacturing processes give importance to these conditions in numerical simulation.

In the APMonitor software, boundary conditions are added for select variables with the use of a periodic object declaration. Linking this periodic object to a variable in the model enforces the periodic condition by adding an additional equation that the end point must be equal to the beginning point in the horizon.

The following example illustrates the use of the boundary condition. Scripts in MATLAB and Python are available below to recreate this solution.

[[Attach:periodic_example.zip|Periodic Example Script Files (periodic_example.zip)]]

(:table border=1 width=100%:)
(:cell:) MATLAB
(:cell:) Python
(:cellnr:)
(:cell:) Attach:periodic_script_matlab.png
(:cell:) Attach:periodic_script_python.png
(:tableend:)