Equation

Basin of Capture of a Synchronous generator

$$\left\{ \begin{array}{lll}\dot{x}_{1} & = & x_{2}\\\dot{x}_{2} & = & -0.5x_{2}-\sin(x_{1}+0.412)+\sin(0.412)\end{array}\right. $$

Target set: $x_{1}^{2}+x_{2}^{2}\leq (0.4)^{2}$ (Willems, J. (1971). Direct method for transient stability studies in power system analysis. IEEE Transactions on Automatic Control, 16(4), 332-341.)

Search space: $[-6, 6]\times[-6,6]$

Figure

Code

from pyinvariant import *

# Define the search space
space = IntervalVector([[-6.0, 6.0],[-6.0,6.0]])

# Create the grpah structure
smartSubPaving = SmartSubPaving(space)

# Create the Domain
dom_outer = Domain(smartSubPaving, FULL_WALL)
dom_outer.set_border_path_in(False)
dom_outer.set_border_path_out(False)
f_sep = Function("x[2]", "(x[0])^2+(x[1])^2-(0.42)^2")
s_outer = SepFwdBwd(f_sep, LEQ) # possible options : LT, LEQ, EQ, GEQ, GT
dom_outer.set_sep_output(s_outer);

dom_inner = Domain(smartSubPaving, FULL_DOOR)
dom_inner.set_border_path_in(True)
dom_inner.set_border_path_out(True)
s_inner = SepFwdBwd(f_sep, GEQ) # possible options : LT, LEQ, EQ, GEQ, GT
dom_inner.set_sep_input(s_inner);

# Create the Dynamics
f = Function("x[2]", "(x[1], -0.5*x[1]-sin(x[0]+0.412)+sin(0.412))")
dyn = DynamicsFunction(f, BWD)

# Create the two Maze associated with the Domain and the dynamics
maze_inner = Maze(dom_inner, dyn)
maze_outer = Maze(dom_outer, dyn)

# Contract the system
for i in range(15):
	print(i)
	smartSubPaving.bisect()
	maze_outer.contract()
	maze_inner.contract()

# Visualization
visu = VibesMaze("Synchronous generator Basin of Capture", maze_outer, maze_inner)
visu.setProperties(0,0,512,512)
visu.show()
visu.drawCircle(0.0, 0.0, 0.4, "black[red]");