Dynamic Optimization

%width=100px%Attach:solve_odes.png One of the most important factors in efficient and reliable solution of dynamic systems is the model formulation. Changes in model formulation are not intended to change the equations, only to put them into a form that allows solvers to more easily find an accurate solution. The ordinary differential equation {`\frac{dx}{dt}+x=12`} can be solved analytically with separate & integrate or [[https://apmonitor.com/pdc/index.php/Main/FirstOrderSystems|LaPlace transforms]]. The ODE can also be solved numerically with Excel or Python.

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(:title Formulation Strategies:)

%width=200px%Attach:solve_odes.png One of the most important factors in efficient and reliable solution of dynamic systems is the model formulation. Changes in model formulation are not intended to change the equations, only to put them into a form that allows solvers to more easily find an accurate solution. The ordinary differential equation {`\frac{dx}{dt}+x=12`} can be solved analytically with separate & integrate or [[https://apmonitor.com/pdc/index.php/Main/FirstOrderSystems|LaPlace transforms]]. The ODE can also be solved numerically with Excel or Python.

%width=200px%Attach:solve_odes.png One of the most important factors in efficient and reliable solution of dynamic systems is the model formulation. Changes in model formulation are not intended to change the equations, only to put them into a form that allows solvers to more easily find an accurate solution. The ordinary differential equation {`\frac{dx}{dt}+x=12`} can be solved analytically with separate & integrate or LaPlace transforms. The ODE can also be solved numerically with Excel or Python.

%width=100px%Attach:solve_odes.png One of the most important factors in efficient and reliable solution of dynamic systems is the model formulation. Changes in model formulation are not intended to change the equations, only to put them into a form that allows solvers to more easily find an accurate solution. The ordinary differential equation {`\frac{dx}{dt}+x=12`} can be solved analytically with separate & integrate or LaPlace transforms. The ODE can also be solved numerically with Excel or Python.

One of the most important factors in efficient and reliable solution of dynamic systems is the model formulation. Changes in model formulation are not intended to change the equations, only to put them into a form that allows solvers to more easily find an accurate solution. The ordinary differential equation {`\frac{dx}{dt}+x=12`} can be solved analytically with separate & integrate or LaPlace transforms. The ODE can also be solved numerically with Excel or Python.

One of the most important factors in efficient and reliable solution of dynamic systems is the model formulation. Changes in model formulation are not intended to change the equations, only to put them into a form that allows solvers to more easily find an accurate solution. The ordinary differential equation {`\frac{dx}{dt}+x=12`} can be solved analytically with separate & integrate or LaPlace transforms. It can also be solved numerically.

One of the most important factors in efficient and reliable solution of dynamic systems is the model formulation. Changes in model formulation are not intended to change the equations, only to put them into a form that allows solvers to more easily find an accurate solution. The ordinary differential equation {`\frac{dx}{dt}+x=12`} can be solved analytically with separate & integrate or LaPlace transforms. It can also be solved numerically with approximate methods.

One of the most important factors in efficient and reliable solution of dynamic systems is the model formulation. Changes in model formulation are not intended to change the equations, only to put them into a form that allows solvers to more easily find an accurate solution. In each section below are some of the key strategies related to model creation and formulation. The discussion begins with a basic introduction to the APMonitor Modeling Language and Gekko Optimization Suite.

One of the most important factors in efficient and reliable solution of dynamic systems is the model formulation. Changes in model formulation are not intended to change the equations, only to put them into a form that allows solvers to more easily find an accurate solution. In each section below are some of the key strategies related to model creation and formulation. The discussion begins with a basic introduction to the APMonitor Modeling Language.

[[https://gekko.readthedocs.io/en/latest/|GEKKO]] is an object-oriented Python library to facilitate local execution of APMonitor. It is a Python package for machine learning and optimization of mixed-integer and differential algebraic equations. It is coupled with large-scale solvers for linear, quadratic, nonlinear, and mixed integer programming (LP, QP, NLP, MILP, MINLP). Modes of operation include parameter regression, data reconciliation, real-time optimization, dynamic simulation, and nonlinear predictive control.

[[https://apmonitor.com/wiki/index.php/Main/SlackVariables|Slack variables]] are defined to transform an inequality expression into an equality expression with an added slack variable. The general expression for inequality constraints in the DAE expression is ''g(dx/dt,x,p)>0''. An equivalent mathematical expression of the general inequality is ''g(dx/dt,x,p)=s'' and ''s>0''. This form is desirable with solvers such as interior point methods where the initial guess must satisfy all inequality constraints or be on the inside of the feasible region. In the slack variable form, an initial guess value greater than zero for the new slack variable ''s'' satisfies this requirement. The APMonitor Modeling Language automatically transforms all inequality constraints into equivalent equality constraints with added slack variables.

* [[https://apmonitor.com/online/view_pass.php?f=slack.apm | Click to Solve a Slack Variable Optimization Problem]]

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Attach:download.png [[Attach:simulate_reservoirs.zip|Reservoir Simulation in MATLAB and Python]]

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There is an additional optional designation of parameters as either fixed values (''FVs'') or manipulated variables (''MVs''). Variables can be optionally designated as state variables (SVs) or controlled variables (''CVs''). The terminology of ''FV, MV, SV,'' and ''CV'' is from the process systems engineering community. In this community the ''MVs'' are designated as the inputs that are potentially changed by the controller and ''CVs'' are model outputs that are driven to target conditions. The terms ''FVs'' refer to either measured or unmeasured disturbances to the system and ''SVs'' are simply designated for viewing purposes as variables of importance. These parameter and variable classifications are specified in MATLAB or Python scripts (See [[https://apmonitor.com/wiki/index.php/Main/INFO|additional details on FV, MV, SV, and CV classification]]).

 V'_flow_in1_' = 0.13 (July-Mar), 0.21 (Apr-June)

vin = [0.13,0.13,0.13,0.13,0.21,0.21,0.21,\

vin = [0.13,0.13,0.13,0.21,0.21,0.21,0.13,\

 Outflow River Rates (km'^3^'/yr)

!!!!Solution

 V'_evap1_' = 0.03
 V'_evap2_' = 0.05
 V'_evap3_' = 0.02
 V'_evap4_' = 0.00

Simulate the height of the reservoirs over the course of a year, starting in January, with higher inlet flowrates in the spring due to melting snow. Use a [[https://youtu.be/6Cc693cmqCc|mass balance]] to describe the change in volume and height of each body of water. This is a simple simulation model that assumes no active control. In actual practice, water outlet flows are actively managed to maintain reservoir levels for recreation, provide sufficient river flow rates, limit river flow rates to avoid flooding, and serve agricultural and community needs. Utah Lake and the Great Salt Lake also have additional inlet sources such as the Payson River (Utah Lake) and the Weber River and Bear River (Great Salt Lake) that are not considered in this simulation.

Simulate the height of the reservoirs over the course of a year, starting in January, with higher inlet flowrates in the spring due to melting snow. This is a simple simulation model that assumes no active control. In actual practice, water outlet flows are actively managed to maintain reservoir levels for recreation, provide sufficient river flow rates, limit river flow rates to avoid flooding, and serve agricultural and community needs. Utah Lake and the Great Salt Lake also have additional inlet sources such as the Payson River (Utah Lake) and the Weber River and Bear River (Great Salt Lake) that are not considered in this simulation.

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'''Objective:''' Formulate a dynamic model with model quantities such as constants, parameters, and variables and model expressions such as intermediates and equations. Use time-varying inputs, initial conditions, and mass balance equations to specify the problem inputs and dynamics. Create a MATLAB or Python script to simulate and display the results.

Simulate the height of the reservoirs over the course of a year, starting in January, with higher inlet flowrates in the spring due to melting snow. This is a simple simulation model that assumes no active control. In actual practice, water outlet flows are actively managed to maintain reservoir levels for recreation, provide sufficient river flow rates, limit river flow rates to avoid flooding, and serve agricultural and community needs.

Suppose that there is a spillway from each upstream body of water to the lower body of water with a flow that is proportional to the square root of the reservoir height. There is no outflow from the Great Salt Lake except due to evaporation. Develop a simplified dynamic model of the height change in each reservoir from water mass balances. Below are relevant constants to use for outflow correlations and initial holdups. Below are constants such as area (km'^2^') and usage requirements (km'^3^'/yr), inlet and outlet flow correlations (km'^3^'/yr), and initial conditions for the volumes (km'^3^'). 

 V'_evap' = 1e-5 * Area, for fresh water
 V'_evap' = 0.5e-5 * Area, for salt water

In Utah, water flows into the (1) [[https://en.wikipedia.org/wiki/Jordanelle_Reservoir|Jordanelle reservoir]], to the (2) [[https://en.wikipedia.org/wiki/Deer_Creek_Dam_and_Reservoir|Deer Creek reservoir]], to (3) [[https://en.wikipedia.org/wiki/Utah_Lake|Utah Lake]], and finally to the (4) [[https://en.wikipedia.org/wiki/Great_Salt_Lake|Great Salt Lake]]. Suppose that there is a spillway from each upstream body of water to the lower body of water with a flow that is proportional to the square root of the reservoir height. There is no outflow from the Great Salt Lake except due to evaporation. Develop a simplified dynamic model of the height change in each reservoir from water mass balances. Below are relevant constants to use for outflow correlations and initial holdups. Below are constants such as area (km'^2^') and usage requirements (km'^3^'/yr), inlet and outlet flow correlations (km'^3^'/yr), and initial conditions for the volumes (km'^3^').

 Outflow Rates (km'^3^'/yr)
 V'_flow_out1_' = 0.1 sqrt(h'_1_')
 V'_flow_out2_' = 0.05 sqrt(h'_2_')
 V'_flow_out3_' = 0.2 sqrt(h'_3_')

 V'_evap1_' = 20
V'_evap2_' = 100
V'_evap3_' = 30
V'_evap4_' = 0

In Utah, water flows into the (1) [[https://en.wikipedia.org/wiki/Jordanelle_Reservoir|Jordanelle reservoir]], to the (2) [[https://en.wikipedia.org/wiki/Deer_Creek_Dam_and_Reservoir|Deer Creek reservoir]], to (3) [[https://en.wikipedia.org/wiki/Utah_Lake|Utah Lake]], and finally to the (4) [[https://en.wikipedia.org/wiki/Great_Salt_Lake|Great Salt Lake]]. Suppose that there is a spillway from each upstream body of water to the lower body of water with a flow that is proportional to the square root of the reservoir height. There is no outflow from the Great Salt Lake except due to evaporation. Develop a simplified dynamic model of the height change in each reservoir from water mass balances. Below are relevant constants to use for outflow correlations and initial holdups. Lookup estimates for the height (km), area (km'^2^'), and initial volumes (km'^3^'). Simulate the height of the reservoirs over the course of a year with higher inlet flowrates in the spring due to melting snow.

 V'_flow_in1_' = variable
 V'_flow_in2_' = V'_flow_out1_'
 V'_flow_in3_' = V'_flow_out2_'
 V'_flow_in4_' = V'_flow_out3_'

This is a simple simulation model that assumes no active control. In actual practice, water outlet flows are actively managed to maintain reservoir levels for recreation, provide sufficient river flow rates, limit river flow rates to avoid flooding, and serve agricultural and community needs.

In Utah, water flows into the (1) [[https://en.wikipedia.org/wiki/Jordanelle_Reservoir|Jordanelle reservoir]], to the (2) [[https://en.wikipedia.org/wiki/Deer_Creek_Dam_and_Reservoir|Deer Creek reservoir]], to (3) [[https://en.wikipedia.org/wiki/Utah_Lake|Utah Lake]], and finally to the (4) [[https://en.wikipedia.org/wiki/Great_Salt_Lake|Great Salt Lake]]. Suppose that there is a spillway from each upstream body of water to the lower body of water with a flow that is proportional to the square root of the reservoir height. There is no outflow from the Great Salt Lake except due to evaporation. Develop a simplified dynamic model of the height change in each reservoir from water mass balances. Below are relevant constants to use for outflow correlations and initial holdups. Lookup estimates for the height (m), area (m'^2^'), and initial volumes (m'^3^'). Simulate the height of the reservoirs over the course of a year with higher inlet flowrates in the spring due to melting snow.

 V'_evap3_' = 0.005 A'_4_'

 V'_flow_in2_' = m'_out1_'
 V'flow_in3' = m'_out2_'
 V'flow_in4' = m'_out3_'
 V'^flow_out1^' = 0.1 sqrt(h'_1_')
 V'^flow_out2^' = 0.05 sqrt(h'_2_')
 V'^flow_out3^' = 0.2 sqrt(h'_3_')
 V'^flow_out4^' = 0
 V'^evap1^' = 0.01 A'_1_'
 V'^evap2^' = 0.01 A'_2_'
 V'^evap3^' = 0.01 A'_3_'
 V'^evap3^' = 0.005 A'_4_'

 V'flow_in1' = variable
 V'flow_in2' = m'_out1'
 V'flow_in3' = m'_out2'
 V'flow_in4' = m'_out3'

In Utah, water flows into the (1) [[https://en.wikipedia.org/wiki/Jordanelle_Reservoir|Jordanelle reservoir]], to the (2) [[https://en.wikipedia.org/wiki/Deer_Creek_Dam_and_Reservoir|Deer Creek reservoir]], to (3) [[https://en.wikipedia.org/wiki/Utah_Lake|Utah Lake]], and finally to the (4) [[https://en.wikipedia.org/wiki/Great_Salt_Lake|Great Salt Lake]]. Suppose that there is a spillway from each upstream body of water to the lower body of water with a flow that is proportional to the square root of the reservoir height. There is no outflow from the Great Salt Lake except due to evaporation. Develop a simplified dynamic model of the height change in each reservoir from water mass balances. Below are relevant constants to use for outflow correlations and initial holdups. Lookup estimates for the height (m) and area (m^_2_^).

m^_out1^ = 0.1 h^_1_^
 m^_out2^ = 0.05 h^_2_^
 m^_out3^ = 0.2 h^_3_^
 m^_out4^ = 0
 m^_evap1^ = 0.01 A^_1_^
 m^_evap2^ = 0.01 A^_2_^
 m^_evap3^ = 0.01 A^_3_^
m^_evap3^ = 0.005 A^_4_^

!!!!Solution

Slack variables are defined to transform an inequality expression into an equality expression with an added slack variable. The general expression for inequality constraints in the DAE expression is ''g(dx/dt,x,p)>0''. An equivalent mathematical expression of the general inequality is ''g(dx/dt,x,p)=s'' and ''s>0''. This form is desirable with solvers such as interior point methods where the initial guess must satisfy all inequality constraints or be on the inside of the feasible region. In the slack variable form, an initial guess value greater than zero for the new slack variable ''s'' satisfies this requirement. The APMonitor Modeling Language automatically transforms all inequality constraints into equivalent equality constraints with added slack variables.

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There is an additional optional designation of parameters as either fixed values (FVs) or manipulated variables (MVs). Variables can be optionally designated as state variables (SVs) or controlled variables (CVs). The terminology of ''FV, MV, SV,'' and ''CV'' is from the process systems engineering community. In this community the ''MVs'' are designated as the inputs that are potentially changed by the controller and ''CVs'' are model outputs that are driven to target conditions. The terms ''FVs'' refer to either measured or unmeasured disturbances to the system and ''SVs'' are simply designated for viewing purposes as variables of importance. These parameter and variable classifications are specified in MATLAB or Python scripts (See [[https://apmonitor.com/wiki/index.php/Main/INFO|additional details on FV, MV, SV, and CV classification]]).

Collections of constants, parameters, variables, intermediates, equations, objects, and connections constitute a model. The model file is created and stored in a text file with extension ''apm''. Several text editors are available that support syntax highlighting such as Notepad++ and gEdit (---see installation instructions). Below is an example model that demonstrates the use of many of these sections to create a 5th order differential equation model.

Certain functions such as ''abs'', ''if..then'', ''min'', ''max'', ''signum'', and discontinuous functions can be included in models but need to be posed in a way to allow efficient solution by solvers that perform better with continuous first and second derivatives. There are alternative methods to reformulate the problems. Two popular approaches are as [[https://apmonitor.com/wiki/index.php/Apps/MpecExamples|MPECs (Mathematical Programs with Equilibrium Constraints)]] or with [[https://apmonitor.com/me575/index.php/Main/LogicalConditions|binary variables that switch on or off certain elements of the equations]].

There is an additional optional designation of parameters as either fixed values (FVs) or manipulated variables (MVs). Variables can be optionally designated as state variables (SVs) or controlled variables (CVs). The terminology of ''FV, MV, SV,'' and ''CV'' is from the process systems engineering community. In this community the ''MVs'' are designated as the inputs that are potentially changed by the controller and ''CVs'' are model outputs that are driven to target conditions. The terms ''FVs'' refer to either measured or unmeasured disturbances to the system and ''SVs'' are simply designated for viewing purposes as variables of importance. These parameter and variable classifications are specified in MATLAB or Python scripts.

MPECs - max, min, abs, if statements, etc

* '''Equations''' are either equality constraints as ''f(dx/dt,x,p)=0'', inequality constraints as ''g(dx/dt,x,p)>0'', or expression of the objective with statements that begin with keywords ''maximize'' or ''minimize''.

* '''Constants''' are values that never change. Integer values may be defined to give sizes to arrays (See [[https://apmonitor.com/wiki/index.php/Main/Constants|additional details on constants]])..

* '''Constants''' are values that never change. Integer values may be defined to give sizes to arrays.
* '''Parameters''' are values that are nominally fixed at initial values but can be changed with input data, by the user, or can become calculated by the optimizer to minimize an objective function if they are indicated as decision variables.
* '''Variables''' are always calculated values as determined by the set of equations. Some variables are either measured and/or controlled to a desired target value.
* '''Intermediates''' are explicit equations where the variable is set equal to an expression that may include constants, parameters, variables, or other intermediate values that are defined previously. Intermediates are not implicit equations but are explicitly calculated with each model function evaluation.

* '''Objects''' are object-oriented extensions of APMonitor that may include parameters, variables, equations, and objective terms.
* '''Connections''' are simple equality constraints that relate object variables to model parameter or variables.

There is an additional optional designation of parameters as either fixed values (FVs) or manipulated variables (MVs). Variables can be optionally designated as state variables (SVs) or controlled variables (CVs). The terminology of ''FV, MV, SV,'' and ''CV'' is from the process systems engineering community. In this community the ''MVs'' are designated as the inputs that are potentially changed by the controller and ''CVs'' are model outputs that are driven to target conditions. The terms ''FVs'' refer to either measured or unmeasured disturbances to the system and ''SVs'' are simply designated for viewing purposes as variables of importance.

Collections of constants, parameters, variables, intermediates, equations, objects, and connections constitute a model. The model file is created and stored in a text file with extension ''apm''. Several text editors are available that support syntax highlighting such as Notepad++ and gEdit (see installation instructions).

 ! 5th order differential equation model with:

!!!!Slack variables

include slack variable sections from optimization course

Models consist of sections including constants, parameters, variables, intermediates, and equations. All expressions used in the equations section must be created in one of the prior sections. The initialization of individual parameters or variables is sequential in the order that they are listed in the model file. Equations, however, can be listed in any order because equations are solved simultaneously.

* '''Parameters''' are values that are nominally fixed at initial values but can be changed with input data, by the user, or can become calculated by the optimizer to minimize an objective function if they are indicated as degrees of freedom.

Collections of the above elements constitute a model and are stored in a text file with extension ''apm''. Several text editors are available that support syntax highlighting such as Notepad++ and gEdit (see installation instructions).

In the above model comments are designated with ''!, %,'' or ''#''. Another symbol is the dollar sign ''$'' that indicates a differential variable or ''dx'_i_'/dt''. The square brackets with the range of ''[1:n]'' creates 5 separate variables or ''x[1], x[2], x[3], x[4],'' and ''x[5]''. Each variable is initialized with a lower bound of ''0'' and an initial condition of ''u=3''. The variable ''y'' is defined in the intermediates section. This variable is a renaming of x[5] and is used in the objective function. The quantity x[n] could also be used in the objective function instead with the same result. There are no degrees of freedom for this problem.

 ! state space model with:

   x[1:n] = 0, >=0

   y = K * u

In the above model comments are designated with !, %, or #. The other symbol that is the $ that indicates a differential variable.

(--include additional symbols--)

One of the most important factors in efficient and reliable solution of dynamic systems is the model formulation. Changes in model formulation are not intended to change the equations, only to put them into a form that allows solvers to more easily find an accurate solution. In each section below are some of the key strategies related to model creation and formulation. The discussion begins with a basic introduction to

(:title Formulation Strategies for Simulation of Dynamic Systems:)