--- title: "Steady state stability and synchronizing coefficient" subject: "Power System Analysis II" chapter: 6 key: "semester_5/power_system_analysis_ii:steady-state-stability-sync-coefficient" source: https://www.bctengineer.com/q/semester_5/power_system_analysis_ii/steady-state-stability-sync-coefficient site: BCT Engineering --- # Steady state stability and synchronizing coefficient Power System Analysis II · Chapter 6: Power System Stability - Asked in: 2079 Bhadra, 2+4+2 marks - Video: [Synchronizing power coefficient and its importance in stability](https://www.youtube.com/watch?v=tBjoJ7s-B-M) — Infinity knowledge - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/power_system_analysis_ii/steady-state-stability-sync-coefficient - Whole chapter: https://www.bctengineer.com/md/semester_5/power_system_analysis_ii/chapter-6 ## Question What is stability of power system? State and explain the steady state stability and also signify the importance of synchronizing power coefficient. ## Short answer Stability is the ability to remain in synchronism under normal and disturbed conditions. Steady state stability is for small gradual load changes; for a generator-infinite bus, P_e = EV/X sin delta and the limit is EV/X at delta = 90 degrees. Synchronizing power coefficient P_s = dP_e/delta = (EV/X) cos delta; it must be positive for stability and gives stiffness. ## Answer **Stability of a power system** is its ability to remain in synchronism (all machines running at the same electrical speed) under normal operation and to return to a stable operating state after a disturbance. #### Steady state stability Steady state stability is the ability of the system to remain in synchronism when subjected to **small and gradual** changes in load. For a generator connected to an infinite bus through reactance $X$: $$ P_e = \frac{EV}{X}\sin\delta = P_{max}\sin\delta $$ ```text Pe ^ Pmax | .-''-. | Pm .' | '. |----a'----|----b'.-- | /| | \ | / |stable unstable +-+--+-----+--------+--> d 0 d0 90 180 ``` - At point $a$ ($\delta_0 < 90^\circ$): if load increases slightly, $\delta$ increases and $P_e$ also increases, matching the load. The operation is **stable**. - At point $b$ ($\delta > 90^\circ$): an increase in $\delta$ reduces $P_e$, the rotor accelerates further and synchronism is lost. - If $P_m$ is raised slowly, $\delta$ increases until $\delta = 90^\circ$, where $P_e = P_{max}$. This maximum is the **steady state stability limit**: $$ P_{ss} = \frac{EV}{X} $$ Ways to raise it: increase $E$ (excitation) or $V$, reduce $X$ (parallel lines, series compensation). #### Synchronizing power coefficient **Synchronizing power coefficient** $P_s$ is the rate of change of electrical power output with rotor angle at the operating point: $$ P_s = \left.\frac{\partial P_e}{\partial\delta}\right|_{\delta_0} = \frac{EV}{X}\cos\delta_0 = P_{max}\cos\delta_0\ \ \text{(pu power/rad)} $$ It gives the change in power that tends to pull the rotor back to synchronism for a small change in rotor angle: $\Delta P_e = P_s\,\Delta\delta$. It is also called the stiffness of the coupling between the machine and the system. **Importance:** - **Stability criterion:** steady state stability requires $P_s > 0$, i.e. $0 < \delta_0 < 90^\circ$. At $\delta_0 = 90^\circ$, $P_s = 0$ (limit). - **Stability margin:** a larger $P_s$ means a stiffer system and a larger margin. - **Natural frequency of oscillation:** $f_n = \frac{1}{2\pi}\sqrt{P_s\pi f/H}$, used to study rotor oscillations and damping. - Machines operated at light load (small $\delta_0$) have large $P_s$ and are more stable. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Why is operation at delta > 90° unstable? A. Pe increases with delta, so the rotor slows B. An increase in delta reduces Pe, so the rotor accelerates further and synchronism is lost C. Pm falls D. E falls Answer: B. Beyond 90°, dPe/ddelta is negative, so a disturbance grows. (Key point: Stable and unstable regions) 2. The synchronizing power coefficient at operating angle delta_0 is: A. (EV/X) sin(delta_0) B. (EV/X) tan(delta_0) C. (EV/X) cos(delta_0) D. EV/X cos(2 delta_0) Answer: C. Ps = dPe/ddelta = Pmax cos(delta_0). (Key point: Ps expression) 3. A larger Ps means: A. A weaker machine B. A stiffer coupling and larger stability margin C. Instability D. Lower oscillation frequency only Answer: B. Ps acts as the stiffness of the electrical coupling; lightly loaded machines (small delta_0) have large Ps. (Key point: Importance of Ps) 4. Which of these raises the steady state stability limit? A. Increasing X B. Lowering E C. Lowering V D. Reducing X, e.g. with parallel lines or series compensation Answer: D. The limit is EV/X, so increasing E or V or decreasing X raises it. (Key point: Raising the limit) ## Related in Power System Analysis II - [Synchronizing power coefficient and application](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Asynchronizing-power-coefficient) — Power System Analysis II, chapter 6 - [Positive synchronizing power coefficient](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Asynchronizing-coefficient-positive) — Power System Analysis II, chapter 6 - [Steady state vs transient stability; enhancement](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Asteady-transient-stability-enhancement) — Power System Analysis II, chapter 6, asked 2× - [Effect of excitation on steady state stability](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Aexcitation-effect-steady-state) — Power System Analysis II, chapter 6 - [Maximum steady state power with Vt held](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Amax-steady-state-power-terminal-voltage) — Power System Analysis II, chapter 6 - [Stability limits and enhancement techniques](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Asteady-transient-stability-limits-enhancement) — Power System Analysis II, chapter 6, asked 2× ## Related in other subjects - [Equilibrium points and steady-state stability](https://www.bctengineer.com/question.md?key=semester_7%2Futilization_of_electrical_energy%3Aequilibrium-points-steady-state-stability) — Utilization of Electrical Energy, chapter 2 - [Steady/unsteady system; mass conservation for CV](https://www.bctengineer.com/question.md?key=semester_2%2Ffundamental_of_thermodynamics_and_heat_transfer%3Asteady-unsteady-mass-conservation-cv) — Fundamental of Thermodynamics & Heat Transfer, chapter 4, asked 2× - [Short circuit ratio and system stability](https://www.bctengineer.com/question.md?key=semester_8%2Fpower_plant_design%3Ascr-significance-stability) — Power Plant Design, chapter 4, asked 3× - [LTI system: linearity, stability, TI, causality](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Alti-properties-four) — Signals and Systems, chapter 5, asked 2×