Practice Problems: Stability Analysis

Question 1:
A power system engineer is analyzing a 500 MVA, 20 kV synchronous generator connected to an infinite bus through a transmission line. The generator has a transient reactance of 0.25 pu and the transmission line reactance is 0.35 pu on the generator base. The generator is delivering 400 MW at 0.9 power factor lagging when a three-phase fault occurs at the generator terminals. Calculate the critical clearing angle if the pre-fault power angle is 30°.
(a) 82.5°
(b) 89.3°
(c) 94.7°
(d) 101.2°

Question 2:
A control systems engineer is evaluating the stability of a two-area power system. Area 1 has a total generation capacity of 2000 MW with a frequency bias factor of 1200 MW/Hz. Area 2 has 1500 MW capacity with a frequency bias factor of 900 MW/Hz. A sudden load increase of 150 MW occurs in Area 1. The system frequency before the disturbance was 60 Hz. What is the steady-state frequency deviation?
(a) -0.071 Hz
(b) -0.095 Hz
(c) -0.125 Hz
(d) -0.143 Hz

Question 3:
A transmission engineer is analyzing a 230 kV, 200 km transmission line connecting a 300 MVA generator to an infinite bus. The line has a series reactance of 0.4 Ω/km. The generator internal voltage is 1.1 pu and infinite bus voltage is 1.0 pu. The generator is operating at a power angle of 25°. Calculate the maximum power transfer capability in MW.
(a) 445 MW
(b) 528 MW
(c) 612 MW
(d) 687 MW

Question 4:
A protection engineer is studying the transient stability of a generator-transformer unit feeding into a grid. The generator has H = 4.5 MJ/MVA, rated at 150 MVA. The mechanical input power is 1.0 pu and the initial electrical output is 0.95 pu at δ = 28°. A fault reduces the electrical power to zero. How long can the fault persist before the rotor angle reaches 90° (in cycles at 60 Hz)?
(a) 7.2 cycles
(b) 8.9 cycles
(c) 10.3 cycles
(d) 12.1 cycles

Question 5:
A utility engineer is analyzing a SMIB (Single Machine Infinite Bus) system where a 400 MVA generator with X_d' = 0.30 pu is connected to an infinite bus through two parallel transmission lines each with reactance 0.25 pu. The generator delivers 320 MW at unity power factor. One transmission line is suddenly tripped. Determine the power angle immediately after tripping (assuming constant internal voltage).
(a) 32.8°
(b) 36.5°
(c) 41.2°
(d) 47.9°

Question 6:
A grid operator is evaluating small-signal stability of a power system where a generator operates at P = 0.75 pu on a base of 500 MVA. The synchronizing power coefficient is calculated as 1.25 pu power/rad. The system frequency is 60 Hz and the inertia constant H = 5.0 MJ/MVA. Calculate the natural frequency of oscillation in Hz.
(a) 0.97 Hz
(b) 1.23 Hz
(c) 1.55 Hz
(d) 1.89 Hz

Question 7:
A consulting engineer analyzes a 345 kV interconnection where a 600 MVA generator (H = 3.5 MJ/MVA) operates at 450 MW output. The maximum transferable power is 900 MW. A three-phase fault occurs at t = 0, reducing output power to 0.2 pu. The fault is cleared at t = 0.15 s. Using equal area criterion, determine if the system remains stable (calculate the deceleration area available).
(a) Unstable, area deficit = 0.12 rad-pu
(b) Stable, area margin = 0.08 rad-pu
(c) Stable, area margin = 0.15 rad-pu
(d) Unstable, area deficit = 0.22 rad-pu

Question 8:
A system planner evaluates voltage stability at a load bus receiving power through a transmission line with impedance Z = 5 + j40 Ω. The sending end voltage is maintained at 138 kV. The load consumes 75 MW at 0.85 power factor lagging. Calculate the voltage stability margin as a percentage of critical loading.
(a) 28%
(b) 35%
(c) 42%
(d) 51%

Question 9:
A renewable energy engineer connects a wind farm (equivalent to 200 MVA generation) to a grid through a 50 km, 138 kV line (X = 0.35 Ω/km, R = 0.08 Ω/km). The wind farm operates at 150 MW, 0.92 pf lagging. Grid voltage is 1.0 pu. Calculate the critical clearing time for a three-phase fault at the wind farm terminals if H = 3.0 MJ/MVA and initial power angle is 22°.
(a) 0.18 s
(b) 0.24 s
(c) 0.31 s
(d) 0.38 s

Question 10:
A substation engineer analyzes dynamic stability of a 750 MVA generator (X_d = 1.8 pu, X_d' = 0.35 pu, X_d" = 0.22 pu, T_d0' = 6.0 s, H = 4.2 MJ/MVA) connected to an infinite bus (X_line = 0.15 pu). The generator experiences a 10% step increase in mechanical power from initial loading of 0.80 pu. Calculate the maximum power angle swing (first peak).
(a) 41.2°
(b) 46.8°
(c) 52.3°
(d) 58.7°

Question 11:
A relay engineer is coordinating protection for a generating station where three 200 MVA generators (each with H = 5.5 MJ/MVA) operate in parallel. The combined output is 480 MW feeding into a network through a transformer and line (total X = 0.18 pu on 600 MVA base). A bolted three-phase fault occurs on the high-voltage bus. The critical clearing angle is determined to be 98°. Calculate the critical clearing time.
(a) 0.22 s
(b) 0.28 s
(c) 0.35 s
(d) 0.41 s

Question 12:
A planning engineer evaluates a 500 kV transmission corridor where power transfer between two areas must not exceed stability limits. Area A has equivalent generation with E = 1.15 pu, Area B voltage is 1.0 pu. The transfer reactance is 0.42 pu. For a critical clearing angle of 105° and initial loading at 70% of maximum power, determine the initial power angle.
(a) 27.3°
(b) 31.8°
(c) 36.5°
(d) 44.2°

Question 13:
A SCADA engineer monitors a two-machine system where Machine 1 (H₁ = 4.0 MJ/MVA, 300 MVA) and Machine 2 (H₂ = 5.5 MJ/MVA, 400 MVA) swing together against the rest of the system. Calculate the equivalent inertia constant for the two machines on a 500 MVA common base.
(a) 4.12 MJ/MVA
(b) 4.78 MJ/MVA
(c) 5.24 MJ/MVA
(d) 5.91 MJ/MVA

Question 14:
A distribution engineer analyzes a microgrid where a 5 MVA diesel generator (droop = 4%) connects to a 3 MVA battery storage system (droop = 3%) supplying a common load. Initially, the diesel supplies 3.5 MW and battery supplies 2.0 MW at 60 Hz. A load increase of 1.0 MW occurs. Calculate the new frequency.
(a) 59.32 Hz
(b) 59.47 Hz
(c) 59.68 Hz
(d) 59.81 Hz

Question 15:
A commissioning engineer tests a 400 MVA synchronous condenser (X_d = 1.5 pu, X_q = 1.0 pu) installed for voltage support at a 230 kV bus. The condenser operates at zero real power with terminal voltage at 1.05 pu and draws 80 MVAR (capacitive). Calculate the power angle δ.
(a) 0°
(b) 12.3°
(c) 18.7°
(d) 24.5°

Question 16:
A transmission planning engineer designs a series compensation scheme for a 400 km, 500 kV line (X_L = 0.35 Ω/km) connecting a 1000 MVA generator (X_d' = 0.28 pu) to a load center. To improve transient stability, 40% series compensation is applied. Calculate the new steady-state stability limit if E = 1.2 pu and V = 1.0 pu.
(a) 1850 MW
(b) 2180 MW
(c) 2640 MW
(d) 3120 MW

Question 17:
A stability engineer performs modal analysis on a multi-machine power system. The A-matrix eigenvalue analysis reveals a complex eigenvalue pair: λ = -0.35 ± j7.85. Calculate the damping ratio and frequency of this oscillatory mode.
(a) ζ = 0.044, f = 1.25 Hz
(b) ζ = 0.058, f = 1.42 Hz
(c) ζ = 0.071, f = 1.68 Hz
(d) ζ = 0.085, f = 1.95 Hz

Question 18:
A project engineer evaluates adding a power system stabilizer (PSS) to a 350 MVA generator currently experiencing 0.5 Hz oscillations with 2.5% damping. The PSS design adds phase compensation of 45° at the oscillation frequency and gain of 15 pu. Estimate the improved damping ratio after PSS installation.
(a) 6.8%
(b) 8.2%
(c) 9.5%
(d) 11.3%

Question 19:
A forensic engineer investigates a cascading outage where a 600 MVA generator (H = 3.8 MJ/MVA, X_d' = 0.32 pu) was operating at 480 MW, δ = 35° when a nearby fault occurred. The fault was cleared in 0.18 s by opening one of two parallel lines, changing total transfer reactance from 0.25 pu to 0.45 pu. Using equal area criterion, determine if the generator remained stable.
(a) Stable with 15° margin
(b) Stable with 8° margin
(c) Unstable by 6°
(d) Unstable by 12°

Question 20:
An operations engineer monitors a 765 kV interconnection where the tie-line power flow is limited by transient stability to 2400 MW. The sending-end equivalent has E = 1.18 pu (on 3000 MVA base) and the transfer reactance is 0.38 pu. A proposed FACTS device (STATCOM) can regulate the receiving-end voltage from 0.98 pu to 1.05 pu. Calculate the percentage increase in stability limit with voltage regulation.
(a) 5.2%
(b) 7.1%
(c) 9.4%
(d) 11.8%