--- title: "Stability definitions and swing equation from dynamics" subject: "Power System Analysis II" chapter: 6 key: "semester_5/power_system_analysis_ii:stability-define-swing-from-dynamics" source: https://www.bctengineer.com/q/semester_5/power_system_analysis_ii/stability-define-swing-from-dynamics site: BCT Engineering --- # Stability definitions and swing equation from dynamics Power System Analysis II · Chapter 6: Power System Stability - Asked in: 2068 Chaitra, 3+5 marks - Video: [Swing Equation derivation | Power system Stability | Power system Analysis](https://www.youtube.com/watch?v=ZpfAQ3G_8DE) — Simple n Easy Learning - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/power_system_analysis_ii/stability-define-swing-from-dynamics - Whole chapter: https://www.bctengineer.com/md/semester_5/power_system_analysis_ii/chapter-6 ## Question Define steady-state and transient stability of a power system. Starting from the basic principle of dynamics, "Accelerating torque for a synchronous generator is given by the product of moment of inertia and the angular acceleration of the rotor" derive the swing equation of the rotor of the synchronous generator. ## Short answer Steady-state stability is the ability to remain in synchronism under small gradual changes, with limit Pmax = EV/X at delta = 90 degrees. Transient stability is the ability to remain in synchronism after large sudden disturbances. From Newton's law, J d2theta/dt2 = Tm - Te, converting to power and electrical angle gives (2H/omega_s) d2delta/dt2 = Pm - Pe, with Pe = Pmax sin delta. ## Answer #### Definitions - **Steady state stability:** the ability of a power system to remain in synchronism when subjected to **small and gradual** changes in load. Its limit for a machine on an infinite bus is $P_{max} = EV/X$ (at $\delta = 90^\circ$). - **Transient stability:** the ability of a power system to remain in synchronism after a **large and sudden** disturbance (fault, switching out a line, loss of a generator). It is checked over the first swing, about 1 s. #### Swing equation from the basic principle of dynamics The principle states: accelerating torque = moment of inertia × angular acceleration. #### Derivation Consider the rotor of a synchronous generator. Let - $J$ = moment of inertia of rotor and turbine (kg·m²) - $\theta_m$ = angular position of the rotor with respect to a stationary axis (mech. rad) - $T_m$ = mechanical (shaft) torque, $T_e$ = electromagnetic torque (N·m) **Step 1: Newton's law of rotation.** Accelerating torque = moment of inertia × angular acceleration: $$ J\frac{d^2\theta_m}{dt^2} = T_a = T_m - T_e $$ **Step 2: Measure angle from a synchronously rotating frame.** The rotor angle $\delta_m$ is the angle of the rotor relative to a reference axis rotating at synchronous speed $\omega_{sm}$: $$ \theta_m = \omega_{sm}t + \delta_m \;\Rightarrow\; \frac{d\theta_m}{dt} = \omega_{sm} + \frac{d\delta_m}{dt}, \quad \frac{d^2\theta_m}{dt^2} = \frac{d^2\delta_m}{dt^2} $$ so $$ J\frac{d^2\delta_m}{dt^2} = T_m - T_e $$ **Step 3: Convert torque to power.** Multiply by the rotor speed $\omega_m$ (power = torque × speed): $$ J\omega_m\frac{d^2\delta_m}{dt^2} = P_m - P_e $$ $M = J\omega_m$ is the **angular momentum** (inertia constant). Since the speed changes very little during a swing, $\omega_m \approx \omega_{sm}$ and $M$ is taken as constant: $$ M\frac{d^2\delta_m}{dt^2} = P_m - P_e $$ **Step 4: Introduce the inertia constant $H$.** $H$ is the stored kinetic energy at synchronous speed per MVA rating $S$: $$ H = \frac{\tfrac12 J\omega_{sm}^2}{S}\ \ \text{(MJ/MVA or s)} \;\Rightarrow\; M = J\omega_{sm} = \frac{2HS}{\omega_{sm}} $$ Substituting and dividing by $S$: $$ \frac{2H}{\omega_{sm}}\frac{d^2\delta_m}{dt^2} = \frac{P_m - P_e}{S} = P_m - P_e\ \ \text{(pu)} $$ **Step 5: Use electrical angle.** For a machine with $p$ poles, $\delta = \frac{p}{2}\delta_m$ and $\omega_s = \frac{p}{2}\omega_{sm}$, so $\delta/\omega_s = \delta_m/\omega_{sm}$: $$ \boxed{\frac{2H}{\omega_s}\frac{d^2\delta}{dt^2} = P_m - P_e} \qquad \text{or} \qquad \frac{H}{\pi f}\frac{d^2\delta}{dt^2} = P_m - P_e $$ with $\delta$ in electrical radians and $\omega_s = 2\pi f$. If $\delta$ is in electrical degrees, $\dfrac{H}{180f}\dfrac{d^2\delta}{dt^2} = P_m - P_e$. For a machine connected to an infinite bus through reactance $X$, $P_e = \dfrac{EV}{X}\sin\delta = P_{max}\sin\delta$, so $$ \frac{H}{\pi f}\frac{d^2\delta}{dt^2} = P_m - P_{max}\sin\delta $$ This is the **swing equation**. It is a second-order non-linear differential equation; its solution $\delta(t)$ describes how the rotor swings after a disturbance. If damping is included, a term $D\frac{d\delta}{dt}$ is added on the left. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Steady state stability refers to the ability to remain in synchronism after: A. A three-phase fault B. Loss of a generator C. Switching out a line D. Small and gradual changes in load Answer: D. Steady state stability concerns small, gradual changes; transient stability concerns large, sudden disturbances. (Key point: Steady state stability) 2. Transient stability is normally judged over: A. The first swing, about 1 s after a large disturbance B. Several minutes after a load change C. The steady-state operating point only D. A full day of load cycle Answer: A. It is the ability to keep synchronism after a large sudden disturbance, checked over about the first second. (Key point: Transient stability) 3. In the derivation, how is the torque equation converted into a power equation? A. By dividing by the speed w_m B. By multiplying by the rotor speed w_m since power = torque x speed C. By differentiating once more D. By setting Te = 0 Answer: B. Multiplying J d2(delta_m)/dt2 = Tm - Te by w_m gives M d2(delta_m)/dt2 = Pm - Pe. 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