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Steady State Analysis ​

The SteadyStateAnalysis is used to give the solution of the model at steady state.

Method Overview ​

The Dyad compiler uses the physical description of the system to generate a system of differential-algebraic equations:

x′=f(x,y,p,t)0=g(x,y,p,t)

where x are the differential variables, y are the algebraic variables, p are the parameters, and t is the independent variable (time). The steady state is defined as the value when the final time is taken to be infinity, i.e. (x(∞),y(∞)). If the system is stable, non-oscillatory, and non-chaotic, then this is equivalent to point where x′=0, or the values (x,y) s.t.

0=f(x,y,p,t)0=g(x,y,p,t)

Example Definition ​

For this example we will run a steady state analysis on the basic Hello World Newton's Law of Cooling Thermal model from the Getting Started tutorial:

dyad
# A simple lumped thermal model
component Hello
  # Ambient temperature
  parameter T_inf::Temperature = 300
  # Initial temperature
  parameter T0::Temperature = 320
  # Convective heat transfer coefficient
  parameter h::CoefficientOfHeatTransfer = 0.7
  # Surface area
  parameter A::Area = 1.0
  # Mass of thermal capacitance
  parameter m::Mass = 0.1
  # Specific Heat
  parameter c_p::SpecificHeatCapacity = 1.2
  variable T::Temperature
relations
  # Specify initial conditions
  initial T = T0
  # Newton's law of cooling/heating
  m*c_p*der(T) = h*A*(T_inf-T)
metadata {"Dyad": {"tests": {"case1": {"stop": 10, "expect": {"initial": {"T": 320}}}}}}
end

We can set a few run-specific parameters as follows:

dyad
analysis ThermalSteadyState
  extends DyadInterface.SteadyStateAnalysis(abstol=1e-8, reltol=1e-8)
  model = Hello(T_inf=T_inf, h=h)
  parameter T_inf::Temperature = 300
  parameter h::CoefficientOfHeatTransfer = 0.7
metadata {"Dyad": {"using": "DyadInterface: AbstractSteadyStateAnalysisSpec, SteadyStateAnalysisSpec, SteadyStateAnalysis"}}
end

Then in the Julia REPL, we can run the generated analysis via:

julia
result = ThermalSteadyState()
Steady State Analysis Solution for SteadyStateAnalysis
retcode: Success
u: 1-element Vector{Float64}:
 299.99999999999994

We can get the steady state values by examining the SimulationSolutionTable artifact which gives us a Julia DataFrame of the solution values:

julia
using DyadInterface: artifacts
artifacts(result, :SimulationSolutionTable)
1×1 DataFrame
RowT(t)
Float64
1300.0

Analysis Arguments ​

The following arguments define a SteadyStateAnalysis:

Required Arguments ​

  • model: the Dyad model that the analysis is being applied to.

Optional Arguments ​

  • alg::String: chooses the solver algorithm for the solution process. The default is "auto". The choices are:

    • "auto" - uses the NonlinearSolve.jl automated polyalgorithm.

  • abstol::Real: chooses the absolute tolerance for the solver. Defaults to 1e-8.

  • reltol::Real: chooses the relative tolerance for the solver. Defaults to 1e-8.

Experimental Arguments ​

  • IfLifting::Boolean: sets the iflifting compiler pass in the ModelingToolkit code generation. Currently defaults to false, but will default to true after the feature is made non-experimental.

Artifacts ​

A SteadyStateAnalysis returns the following artifacts:

Customizable Plot ​

No customizable plot is given.

Standard Plots ​

  • SimulationSolutionTable: Returns a DataFrame of the solution values at the steady state.

Further Reading ​