When a fault hits a transmission line, every generator in the grid suddenly speeds up or slows down. Whether they settle back into step — or swing out of synchronism and trigger a blackout — is the question of power system stability. Stability, the swing equation and the equal-area criterion are core topics for PGCB, BPDB, NWPGCL, EGCB, BCS (EEE) written exams and grid-operation vivas. This post explains them with an animated equal-area plot and swing curve, worked numbers and MCQs.
- Stability = ability of a power system to return to a steady operating state after a disturbance.
- Classes: rotor-angle (steady-state/small-signal and transient), voltage and frequency stability.
- Power transfer: P = (EV/X) sin δ; steady-state limit Pmax = EV/X at δ = 90°.
- Swing equation: (2H/ωs) d²δ/dt² = Pm − Pe.
- Equal-area criterion: the system is transiently stable if the decelerating area can equal the accelerating area (A2 = A1). Clear faults before the critical clearing angle/time.
1. What Is Power System Stability?
All synchronous generators connected to a grid rotate in step (synchronism) at a speed corresponding to 50 Hz. Power system stability is the ability of the system, for a given operating condition, to regain a state of equilibrium after being subjected to a disturbance — such as a short circuit, loss of a line, loss of a large generator or a sudden load change — with system variables staying within limits so that practically the entire system remains intact. Major grid disturbances, such as the nationwide blackouts in Bangladesh in November 2014 and October 2022, show why stability studies matter.
2. Classification
| Type | Concerned with | Typical disturbance |
|---|---|---|
| Steady-state / small-signal stability | Ability to remain in synchronism after small, gradual changes | Slow load changes, small oscillations |
| Transient stability | Ability to remain in synchronism after a large, sudden disturbance (first few seconds) | Three-phase fault, line tripping, loss of generation |
| Voltage stability | Ability to maintain acceptable bus voltages | Heavy reactive load, lack of reactive support |
| Frequency stability | Ability to keep frequency near 50 Hz after generation–load imbalance | Sudden loss of a large plant or interconnection |
3. Power-Angle Curve and Steady-State Limit
Pe = (E V / X) sin δ- E = generator internal EMF, V = infinite-bus voltage, X = total reactance between them, δ = load (power) angle.
- Maximum power Pmax = EV/X occurs at δ = 90° — the steady-state stability limit.
- Stable operation requires dP/dδ > 0 (synchronising power coefficient positive), i.e. δ < 90°; in practice δ is kept around 30–40° to leave a margin.
- Pmax can be raised by increasing E (excitation) or V, or reducing X (parallel lines, series capacitors, higher voltage).
4. Swing Equation and Inertia Constant
The rotor dynamics are governed by Newton’s law for rotation. In per-unit form:
(2H / ωs) · d²δ/dt² = Pm − Pe = Pa or M d²δ/dt² = Pa, M = H/(πf)- Pm = mechanical input (turbine), Pe = electrical output, Pa = accelerating power.
- Inertia constant H = stored kinetic energy at synchronous speed ÷ machine MVA rating (MJ/MVA, i.e. seconds). Typical values: thermal units 4–9 s, hydro units 2–4 s.
- If Pm > Pe the rotor accelerates (δ increases); if Pe > Pm it decelerates.
5. Equal-Area Criterion (Animation)
For a single machine connected to an infinite bus, transient stability can be judged without solving the swing equation:
- Before the fault the machine runs at δ0 where Pm = Pe on the pre-fault curve.
- During the fault the transfer reactance rises, the power curve drops, Pe < Pm, and the rotor accelerates. The energy gained is the accelerating area A1.
- After clearing the post-fault curve applies; now Pe > Pm, and the rotor decelerates, giving up energy — the decelerating area A2.
- The rotor stops advancing (and swings back) when A2 = A1. If the available decelerating area is smaller than A1, δ keeps increasing beyond the unstable equilibrium and synchronism is lost.
6. Critical Clearing Angle and Time
The critical clearing angle δcr is the largest rotor angle at which the fault can be cleared and the system still remains stable (A1 = maximum available A2). For pre-fault, faulted and post-fault maxima P1, P2, P3:
cos δcr = [ Pm(δmax − δ0) + P3 cos δmax − P2 cos δ0 ] / (P3 − P2), δmax = π − sin⁻¹(Pm/P3)The corresponding time is the critical clearing time tcr. Protection and breakers must clear the fault faster than tcr. For a fault with P2 = 0 (fault at the generator terminals): tcr = √[2H(δcr − δ0)/(πf Pm)].
7. Methods to Improve Stability
- Faster fault clearing: high-speed relays and breakers (2–3 cycle); auto-reclosing (single-pole) for transient faults.
- Reduce transfer reactance: parallel lines, bundled conductors, higher transmission voltage, series capacitor compensation.
- Fast excitation systems and power system stabilisers (PSS) to damp oscillations.
- FACTS devices: SVC, STATCOM, TCSC, UPFC; HVDC links for controlled power flow.
- Braking resistors, fast valving of turbines, generator tripping and under-frequency load shedding schemes.
- Higher inertia machines and spinning reserve for frequency stability.
8. Solved Problems
Problem 1. E = 1.2 pu, V = 1.0 pu, X = 0.6 pu. Find the steady-state limit. Pmax = 1.2 × 1.0/0.6 = 2.0 pu.
Problem 2 (the animated case). Pm = 1.0 pu; pre-fault P1 = 2.0, faulted P2 = 0.5, post-fault P3 = 1.5 pu. Find δ0, δmax and δcr.
- δ0 = sin⁻¹(1/2) = 30°
- δmax = 180° − sin⁻¹(1/1.5) = 180° − 41.8° = 138.2° (2.412 rad)
- cos δcr = [1 × (2.412 − 0.524) + 1.5 cos 138.2° − 0.5 cos 30°]/(1.5 − 0.5) = [1.888 − 1.118 − 0.433]/1 = 0.337 ⇒ δcr ≈ 70.3°
- With H = 5 s at 50 Hz, a numerical solution of the swing equation gives δ ≈ 45° at 0.15 s (fault cleared in time — stable, as in the animation) and a critical clearing time of about 0.25 s.
9. Practice MCQs (click to see answer)
1. Steady-state stability limit occurs at load angle: (a) 0° (b) 45° (c) 90° (d) 180°
2. Transient stability deals with: (a) Small, slow changes (b) Large, sudden disturbances (c) Harmonics only (d) DC systems
3. The equal-area criterion is applicable to: (a) Multi-machine systems directly (b) A single machine connected to an infinite bus (or two-machine system) (c) DC networks (d) Distribution feeders
4. The unit of inertia constant H is: (a) kg·m² (b) MJ/MVA (seconds) (c) rad/s (d) MW
5. Transient stability can be improved by: (a) Slower breakers (b) Increasing series reactance (c) Faster fault clearing (d) Reducing excitation
6. Series capacitors improve stability by: (a) Increasing line reactance (b) Reducing net line reactance (c) Increasing resistance (d) Reducing voltage
7. In the swing equation, accelerating power equals: (a) Pm + Pe (b) Pm − Pe (c) Pe/Pm (d) PmPe
8. A power system stabiliser (PSS) acts through the: (a) Turbine governor (b) Excitation system (c) Circuit breaker (d) Transformer tap changer
9. Maximum power transfer between two sources is: (a) EV/X (b) EV·X (c) X/EV (d) E/V
10. Under-frequency load shedding mainly protects: (a) Voltage stability (b) Frequency stability (c) Insulation (d) Harmonic distortion
10. বাংলায় দ্রুত রিভিশন
- পাওয়ার সিস্টেম স্টেবিলিটি: কোনো বিঘ্নের (ফল্ট, লাইন ট্রিপ, জেনারেটর হারানো) পর সিস্টেমের পুনরায় সাম্যাবস্থায় ফেরার ক্ষমতা।
- প্রকার: স্টেডি-স্টেট, ট্রানজিয়েন্ট (রোটর অ্যাঙ্গেল), ভোল্টেজ ও ফ্রিকোয়েন্সি স্টেবিলিটি।
- P = (EV/X) sin δ; সর্বোচ্চ শক্তি δ = ৯০°-এ, Pmax = EV/X।
- সুইং সমীকরণ: (2H/ωs) d²δ/dt² = Pm − Pe; H = ইনারশিয়া ধ্রুবক (MJ/MVA)।
- সম-ক্ষেত্রফল নীতি: ত্বরণ ক্ষেত্রফল A₁ = মন্দন ক্ষেত্রফল A₂ হলে সিস্টেম স্থিতিশীল; ক্রিটিক্যাল ক্লিয়ারিং কোণ/সময়ের আগে ফল্ট দূর করতে হবে।
- উন্নতির উপায়: দ্রুত রিলে ও ব্রেকার, অটো-রিক্লোজিং, সিরিজ ক্যাপাসিটর, প্যারালাল লাইন, দ্রুত এক্সাইটেশন ও PSS, FACTS, HVDC, লোড শেডিং।
Related Posts
- 220 Most Repeated Power Sector MCQs
- Equivalent Circuit of a Transmission Line
- EEE Job Viva Voce Preparation
- EEE Departmental Job Exam Preparation
References: P. Kundur, Power System Stability and Control; Hadi Saadat, Power System Analysis; Nagrath & Kothari, Modern Power System Analysis. Images and animations: original work by Totalecer.
Thank you for visiting my blog.