--- title: "Swing equation and swing curve" subject: "Power System Analysis II" chapter: 6 key: "semester_5/power_system_analysis_ii:swing-equation-swing-curve" source: https://www.bctengineer.com/q/semester_5/power_system_analysis_ii/swing-equation-swing-curve site: BCT Engineering --- # Swing equation and swing curve Power System Analysis II · Chapter 6: Power System Stability - Asked in: 2074 Asoj, 8 marks - Video: [Swing Equation Of Synchronous Machine | Swing Curve | Power System Stability | Transient Stability](https://www.youtube.com/watch?v=DvN9fQD9OUQ) — Learn2Earn 4Life - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/power_system_analysis_ii/swing-equation-swing-curve - Whole chapter: https://www.bctengineer.com/md/semester_5/power_system_analysis_ii/chapter-6 ## Question Derive swing equation of a synchronous machine to be applicable in the study of power system stability. What is meant by swing curve? What information is supplied by the swing curve? ## Short answer Swing equation: (2H/ωs)d2δ/dt2 = Pm - Pe in per unit, derived from Newton's law; for an infinite bus, Pe = (EV/X) sinδ. A swing curve plots δ versus t and shows stability (δ peaks then decreases) or loss of synchronism (δ keeps increasing), plus maximum rotor angle and stability margin. ## Answer The **swing equation** relates the rotor angle $\delta$ of a synchronous machine to the difference between mechanical input and electrical output power; it is the basic equation of power system stability studies. #### 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. #### Swing curve A **swing curve** is the plot of rotor angle $\delta$ against time $t$, obtained by solving the swing equation (usually numerically, e.g. point-by-point or Runge-Kutta method) for a given disturbance and clearing time. ```text delta ^ unstable (fault cleared late) | / | _/ | .--. _/ stable: delta peaks | / \ _/ and swings back | / .-. \/ .-. |/ / \___/ \__ +--+-------------------> t d0 tc ``` #### Information supplied by the swing curve 1. **Stable or unstable:** if $\delta$ reaches a maximum and then decreases (oscillates about a new value), the system is stable; if $\delta$ keeps increasing, the machine loses synchronism. 2. **Maximum rotor swing** $\delta_{max}$ and the margin from the critical angle. 3. **Critical clearing time:** by plotting swing curves for different clearing times, the largest time that still gives a stable curve is found; this sets the required speed of relays and breakers. 4. **Frequency and damping of oscillations** of the rotor after the disturbance. 5. In multi-machine systems, **relative angles** between machines show which machines swing together and which separate. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. The swing equation derivation starts from which basic relation? A. Accelerating torque = moment of inertia x angular acceleration B. Terminal voltage = internal emf - drop across Xd C. Power = voltage x current x cos(phi) D. Synchronous speed = 120f/P Answer: A. Newton's law of rotation, J d2(theta)/dt2 = Tm - Te, is the first step of the derivation. (Key point: Starting law) 2. Why is H introduced in the swing equation? A. To account for damper winding losses B. To express M = J*w_sm as 2HS/w_sm using stored kinetic energy per MVA C. To convert power from pu to watts D. To make the equation linear Answer: B. H = stored KE / S, so M = J w_sm = 2HS/w_sm, which lets the equation be written in pu. (Key point: Role of inertia constant H) 3. What is a swing curve? A. Plot of Pe against delta B. Plot of terminal voltage against time C. Plot of rotor angle delta against time from solving the swing equation D. Plot of speed against torque Answer: C. A swing curve is delta(t), obtained by solving the swing equation for a given disturbance and clearing time. (Key point: Swing curve meaning) 4. How is the critical clearing time found using swing curves? A. From the peak value of Pe B. From the frequency of oscillation only C. From the final steady value of delta D. By plotting curves for several clearing times and taking the largest time that still gives a stable curve Answer: D. The longest clearing time that still gives a curve where delta peaks and returns is the critical clearing time. (Key point: Critical clearing time from curves) ## Related in Power System Analysis II - [Swing equation of synchronous machine](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Aswing-equation-derivation) — Power System Analysis II, chapter 6 - [Stability classification and swing equation](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Astability-classification-swing) — Power System Analysis II, chapter 6 - [Stability definitions and swing equation from dynamics](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Astability-define-swing-from-dynamics) — Power System Analysis II, chapter 6 - [Swing equation and inertia constant](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Aswing-equation-inertia-constant) — Power System Analysis II, chapter 6 - [Rotor angle and swing equation, infinite bus](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Arotor-angle-swing-equation) — Power System Analysis II, chapter 6 - [Equivalent swing equation, coherent machines](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Aequivalent-swing-equation-coherent) — Power System Analysis II, chapter 6 ## Related in other subjects - [Power of salient pole machine, power angle curve](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines%3Asalient-pole-power-angle-characteristics) — Electrical Machines, chapter 6 - [Leading kVAR supplied by synchronous motor](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machines_ii%3Apf-correction-leading-kvar) — Electric Machines II, chapter 2 - [How to study stability; test a cubic](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Astability-study-method-cubic) — Digital Control System, chapter 3 - [Power supplied to a string by a rod](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Astring-wave-average-power-rod) — Engineering Physics, chapter 2