Q4Power System Engineering

Question

2. (a) Derive the swing equation of synchronous machine. [8]

(b) Find the expression for angular momentum (M) in terms of machine MVA rating (G), machine inertia constant (H) and system frequency (f). Also find out the expression for per unit angular momentum M (pu). [8]

Answer 1

(a) Derivation of the Swing Equation

Consider a synchronous generator rotor with moment of inertia J (kg-m^2), rotating at synchronous mechanical angular speed omega_sm. Let theta_m be the actual mechanical angular position of the rotor at any instant, measured with respect to a stationary reference axis. During normal synchronous operation, theta_m increases uniformly with time: theta_m = omega_sm*t + delta_m, where delta_m is the rotor's mechanical angular displacement relative to a synchronously rotating reference frame (the rotor angle).

Applying Newton's second law for rotational motion, the net accelerating torque on the rotor equals the moment of inertia times the angular acceleration:

J \frac{d ^{2} θ _{m}}{d t ^{2}} = T_{a} = T_{m} - T_{e}

where Tm is the mechanical (shaft) input torque from the prime mover, Te is the electrical (air-gap) output torque, and Ta is the net accelerating torque. Since theta_m = omega_sm*t + delta_m, and omega_sm is constant, differentiating twice with respect to time gives d^2(theta_m)/dt^2 = d^2(delta_m)/dt^2 (the constant-speed term contributes zero to the second derivative), so:

J \frac{d ^{2} δ _{m}}{d t ^{2}} = T_{m} - T_{e}

Multiplying both sides by omega_sm converts torque to power (P=Tomega) and introduces the angular momentum M=Jomega_sm (in MJ-s per mechanical radian, when powers are in MW), giving the swing equation in terms of mechanical angle:

M \frac{d ^{2} δ _{m}}{d t ^{2}} = P_{m} - P_{e} = P_{a}

Converting from mechanical angle delta_m to electrical angle delta (related by delta = (P/2)*delta_m for a machine with P poles, since electrical angle advances P/2 times faster than mechanical angle) and expressing per unit angular momentum, the swing equation is most commonly written in the standard normalized form using the inertia constant H:

\frac{2 H}{ω _{s}} \frac{d ^{2} δ}{d t ^{2}} = P_{m} - P_{e} (per unit), ω_{s} = 2 π f

This is the swing equation, a second-order nonlinear differential equation (nonlinear because Pe is typically a sinusoidal function of delta, Pe=Pmax*sin(delta), for a simple machine-to-infinite-bus system) governing the rotor angle delta's dynamic response to any imbalance between mechanical input power Pm and electrical output power Pe - it is the fundamental equation underlying all power system transient stability analysis, including the equal area criterion (a graphical/energy-based technique for solving this equation approximately without needing full numerical time-domain integration) examined in detail elsewhere in this examination.

(b) Angular Momentum M in Terms of G, H, and f

The inertia constant H is defined as the ratio of the kinetic energy stored in the rotor at synchronous speed (KE, in MJ) to the machine's MVA rating G: H=KE/G, so KE=G*H (MJ).

The angular momentum M is defined (in absolute, non-per-unit terms) as M=Jomega_sm, and since KE=(1/2)Jomega_sm^2=(1/2)Momega_sm, we have M=2KE/omega_sm. Expressing omega_sm in terms of electrical frequency f (for angular momentum expressed per electrical radian, omega_s=2pif, since electrical angular frequency is what is relevant when delta is measured in electrical radians/degrees, as is standard in swing equation analysis):

M = \frac{2 K E}{ω _{s}} = \frac{2 G H}{2 π f} = \frac{G H}{π f} MJ-s/elec. radian, M = \frac{G H}{180 f} MJ-s/elec. degree

These are the standard expressions for angular momentum M in terms of the machine's MVA rating G, inertia constant H, and system electrical frequency f, matching the specific numerical relation used in part (b) of the previous question of this examination.

For the per unit angular momentum, M is expressed on a chosen system MVA base (Gbase, commonly the same as the machine's own rating G, or a common system base if multiple machines with different individual ratings are being represented on one common base):

M_{p u} = \frac{M}{G _{ba se}} = \frac{G H}{π f G _{ba se}} (elec. radian base), M_{p u} = \frac{2 H}{ω _{s}} when G = G_{ba se}

When the machine's own MVA rating is used as the base (G=Gbase), the per unit angular momentum simplifies to the widely used and recognizable form M(pu)=2H/omega_s=H/(pi*f) (radian base) or 2H/(360f) (degree base) - this is precisely the coefficient appearing directly in front of the second-derivative term in the standard per unit swing equation shown in part (a) above, confirming the internal consistency between the swing equation's standard normalized form and the angular momentum expressions derived here.

It is worth noting that the swing equation derived here in part (a), while presented for a single machine connected to an infinite bus (the simplest and most commonly analyzed case), generalizes directly to a multi-machine power system by writing one such swing equation for each individual generator in the system, each referencing that specific machine's own inertia constant Hi and its own electrical power output Pei (which, in a multi-machine system, depends on that machine's rotor angle relative to every other machine's rotor angle through the full network's power-flow relationships, rather than simply relative to a single fixed infinite-bus reference) - multi-machine transient stability analysis therefore requires simultaneously solving this full set of coupled swing equations, one per machine, typically via numerical time-domain simulation rather than the simpler equal-area-criterion graphical technique that works cleanly only for the reduced single-machine-infinite-bus representation.

It is also worth relating the per unit angular momentum expression derived in part (b) back to its direct role in numerical transient stability simulation software: because M(pu)=2H/omega_s (or the equivalent degree-based form) appears directly as the coefficient of the second-derivative term in the standard per unit swing equation, and because H values for different machines typically fall within a comparatively narrow, well-characterized range (roughly 2-9 MJ/MVA for most conventional turbo- and hydro-alternators, as noted elsewhere in this examination) regardless of the machine's absolute physical size, expressing angular momentum in this normalized per unit form (rather than in absolute, machine-specific units) is precisely what allows a single, standardized numerical stability-simulation algorithm to be applied uniformly across generators of very different physical sizes and MVA ratings within the same simulated multi-machine power system model.

In summary, the swing equation derivation and the angular momentum expressions in terms of G, H, and f together provide the complete mathematical and parametric foundation required to set up and numerically solve any power system transient stability problem, including the equal area criterion applications examined in detail elsewhere in this examination.

These general expressions for M in terms of G, H, and f, and their per unit normalized forms, are precisely the machine-parameter inputs required by virtually every commercial power system stability simulation software package used in industry practice today.

Mastering these derivations equips a student to move confidently from the underlying physics of rotor dynamics through to the standardized, machine-independent per unit formulation used throughout modern power system stability software.

This derivation chain, from Newton's second law through to the normalized per unit swing equation coefficient, represents one of the most frequently examined and practically essential derivations in the entire power system stability curriculum.

Q4Power System Engineering

Question

Answer

(a) Derivation of the Swing Equation

(b) Angular Momentum M in Terms of G, H, and f