Find the corresponding equilibrium value for the airspeed

Question

Problem 1

A simplified model of a glider is given by

Y = -cos(y)g/v + ng/v

V = -sin(y)g = -k₁n²g/v² - k₂ V²g

Where Y is the flight path angle in radians, v is the airspeed in m/s  n= L/mg is the load factor; L is the lift in Newtons, m is the mass in kg, and g = 9.81m/s² is the acceleration of gravity. k₁ = 60 and k₂ = 5 • 10^-5 are constant for the glider.

1) Given that y = -0.3176rad find the necessary load factor to maintain equilibrium.

2) Find the corresponding equilibrium value for the airspeed. There are several solutions for the airspeed, we take the higher value.

3) Let the state vector be x = [y.v)T, let the input be n and the output of interest be a, derive the linearized system for the operating point of the previous questions. Determine the numerical values of matrices A, B, C.

4) Write code to simulate the linear and the nonlinear systems. Run the code and compare, show and discuss your results.

Problem 2

For a generation unit attached to an infinite bus system (SMIB), the swing equation can be written as

Mδ = P_{m - EV/X} sinδ - Dω

where
• M= 2H/w_o, H is the normalized inertia constant, M is the generation inertia, and coo is the synchronous speed.
• D is the generator damping coefficient.
• E is the sending end voltage and V is the receiving end voltage.
• X is the total reactance of the equivalent transmission link between the two ends.
• δ is the rotor angle and w is the rotor speed. The relationship between them is

δ= w

• P_m, is the mechanical energy. It represents the input.

1) Write the system under standard state space form. The state vector is x = [δ, w)^T'. The output is w.

2) Find the two operating points in terms of P_m and other variables.

3) Find the linearized system near each of the operating points

4) For the stable operating point, simulate and compare between the linear and the nonlinear systems when the mechanical power is constant.

5) For the stable operating point, simulate and compare between the linear and the nonlinear systems when the mechanical power has the profile shown in figure 1.

Suggested numerical values X = 0.5, D = 0.138, V = 1, B = 1.35, fo = 60, H = 10, Pm = 0.6. Feel free to pick appropriate values for the initial conditions.

Question 3:

Our goal is to simulate the dynamics of a multi-machine system. The swing equation for the il h machine is given by

1) Write the state equations for a three-machine system where machine 3 is taken as a reference. The state vector in this case is

2) Simulate the system and plot the time evolution of δ₂₃ and δ₁₃. Is the system stable? Assume that all G_ij = 0

E₁ = 1.0566; E₂ = 1.0502; E_s = 1.0170

P_m1 = 0.8566; P_m2 = 0.502; P_m3 = 0.70

B₁₂ = 1.5113; B₂₃ = 1.088; Hsi = 1.226

For the values of H, you can take: H₁ = 23.64; H₂ = 6.40; H₃ = 3.01; or, to simplify things, take H₁ = H₂ = H₃. Pick appropriate values for the other variables (such as coo). Suggested initial conditions for the relative angles: δ₁₂₀ = -17.4598°; δ₂₃₀ = 6.5563°; δ₃₁₀ = 10.9035°.

For the method to converge you need to use appropriate values for the initial conditions of ω₁ - ω₃ and ω₂ - ω₃.