Problem 1 -

a. Consider an isolated generating station with a local load and with:

• Governor time constant and gain T_G = 0, K_G = 1,
• Turbine time constant and gain T_T = 1, K_T = 1
• System time constant T_P = 20
• Total damping D^~ = D + D_L = 0.01 and system gain K_P = 1/0.01 = 100
• Regulation R = 2.5%
• ΔP_ref = 0

 • ΔP_L = u(t), that is, ΔP_L =

Determine and sketch Δω(t).

b. Suppose the generator is connected to a large system through a tie-line with stiffness (synchronizing coefficient) K₁ω₀ = T = 10. Assume the large system is approximated by an infinite bus. Making reasonable approximations, and under the above conditions solve for Δω(t), sketch it and compare your results with the above. Assume exciter keeps the voltage constant, Δ|E'|= 0.

You may employ MATLAB or MAPLE to sketch your results.

Problem 2 -

Consider the Single-Machine-Infinite-Bus system with no local load and lossless line connecting the generator to infinite bus, i.e., Z = JX and Y = 0. Assume zero stator resistance, r = 0, and machine parameters  X^~_d = X_d + X, X^~'_d = X'_d + X, X^~_q = X_q + X.

1. Derive the following formula for the generated power and calculate the constants K1 and K2 using it.

P = (|E'||V_∞|/X^~'d)sin(δ) + |V_∞|²/2((1/X^~_q) - (1/X^~'_d))sin(2δ)

2. Derive the following formula for the generated power and calculate the constants K1 and K2 using it.

P = (|E||V_∞|/X^~d)sin(δ) + |V_∞|²/2((1/X^~_q) - (1/X^~_d))sin(2δ)

3. Are the results from (2) and (3) the same? If not which one is correct? Calculate the correct results from both (2) and (3).

Assume the following numerical values;

a. Operating point; V_∞ = 1∠0, E = 2∠60, f = 60 Hz

b. Machine parameters; T'_do = 3sec. , M = 2H = 6 sec.

c. Reactance's: X_d = 1.0, X'_d = 0.2, X_q = 0.4, X = 1.0.

4. Calculate the voltage behind transient reactance E', and the terminal voltage and its components, V, V_d,  V_q.

5. Calculate the numerical values of the constants K1 and K2.

6. Using mechanical loop equation (shaft dynamics) find the approximate frequency of oscillation of the machine verses infinite bus.

Problem 3 -

You are to analyze, and design a PSS to damp, the low frequency oscillation of a generator connected to the grid. A "Single Machine Infinite Bus" model has been identified for the generator. The operating points of interest are identified and evaluated, i.e., |V|⁰, V⁰_d, V⁰_q, |I|⁰, I⁰_d, I⁰_q, |E'|⁰, δ⁰, |V_∞| are known. Furthermore, the six constants K1 to K6 as well as the model for the variation of the current components with the variation of Δδ and Δ|E'| have been calculated, i.e., we have available

To overcome saturation effects, the current, in addition to the voltage, is fed back to the exciter. Therefore, following model for the exciter has also been identified.

How do you propose to incorporate this exciter into the Heffron-Phillips model? Extend the Heffron-Phillips model to accommodate this type of exciter and draw the block diagram of the extended H-P model.

 Problem 4 -

The following Figure shows a single diagram of three phase, 60-Hz synchronous generator, connected through a transformer and parallel transmission lines to an infinite bus (SMIB). All reactances are given in per-unit on a common system base in Table I. The infinite bus receives 1.0 per unit real power at 0.95 p.f. lagging in steady state operation when both transmission lines are in service and there is no fault in the system. Now, suppose that a permanent single-line to ground fault with fault resistance R_f = 1(pu) occurs at the beginning of the second transmission line L₂. The line differential protection assert the opening bit to the circuit breaker CB then remain open.

Calculate the critical clearing angle. In this problem H = 3.0(pu - sec), P_m = 1.0(pu) and ω_p.u. = 1.0 in the swing equation.

Note: the curve of swing equations before, during and after fault, required equations are required.