Docs: Reduced Order Machine Models

Fri, 31 Jul 2026 00:00:00 +0000

The full dq0 machine keeps every rotor winding as a state. Reduced order models keep the rotor flux linkages that matter on the timescale of interest and represent the rest algebraically, which removes the fastest states and lets the machine be stepped at the same rate as the network.

This page derives the equations. Nothing here depends on how they are arranged in software.

Per unit system and operational parameters

All quantities are in the reciprocal per unit system referred to the direct axis mutual inductance [Kundur1994]. With rated apparent power $S_n$ and rated line to line voltage $V_n$ as the stator base, the base peak phase voltage, base current and base impedance are

$$V_{base} = \frac{\sqrt{2}}{\sqrt{3}} V_n, \qquad I_{base} = \frac{S_n}{\frac{3}{2} V_{base}}, \qquad Z_{base} = \frac{V_n}{S_n / V_n}.$$

The machine is described by operational parameters rather than by winding data: the synchronous inductances $L_d$ and $L_q$, the transient inductances $L_d’$ and $L_q’$, the subtransient inductances $L_d’’$ and $L_q’’$, the corresponding open circuit time constants $T_{d0}’$, $T_{q0}’$, $T_{d0}’’$, $T_{q0}’’$, and the inertia constant $H$. A sixth parameter $T_{aa}$, the armature to field coupling time constant, distinguishes the two sixth order variants.

Which states each order retains

The state variables are the voltages behind the transient and subtransient reactances, $E_d’$, $E_q’$, $E_d’’$ and $E_q’’$, together with the two mechanical states. They are defined from the terminal quantities by

$$E_d' = V_d - L_q' I_q, \qquad E_q' = V_q + L_d' I_d,$$

and analogously for the subtransient pair with $L_d’’$ and $L_q’’$.

Order	Electrical states	Physical content
3	$E_q'$	Field winding only; the q axis rotor is neglected.
4	$E_d’$, $E_q'$	Field winding and one q axis damper.
5	$E_q’$, $E_d’’$, $E_q’'$	Adds both subtransient windings, no q axis transient state.
6a	$E_d’$, $E_q’$, $E_d’’$, $E_q’'$	Full transient and subtransient set, with $T_{aa} \neq 0$.
6b	$E_d’$, $E_q’$, $E_d’’$, $E_q’'$	Same states with $T_{aa} = 0$.

Every order carries the same two mechanical states, so the third order model has five states in total and the sixth order models have eight.

Voltage behind reactance form

Written directly, the stator equations couple the machine currents to the network currents, and the machine inductances appear in the axis frame while the network is solved in the phase frame. The voltage behind reactance form removes that coupling: the machine is expressed as an internal voltage in series with a reactance that is constant in the axis frame, so the only quantity that changes between steps is the internal voltage.

The internal voltage is not a free variable. Applying the trapezoidal rule to the rotor flux equations over one step $\Delta t$ gives it as a recursion in quantities already known at the start of the step. For the transient states,

$$E_{h,d} = A_d' \, I_q + B_d' \, E_d', \qquad E_{h,q} = A_q' \, I_d + B_q' \, E_q' + D_q' \left( E_{f}[k-1] + E_{f}[k] \right),$$

where $E_f$ is the field voltage supplied by the excitation system. The coefficients follow from the trapezoidal integration and depend only on the parameters and the step size:

$$A_d' = \frac{\Delta t \, Z_d'}{2 T_{q0}' + \Delta t}, \qquad B_d' = \frac{2 T_{q0}' - \Delta t}{2 T_{q0}' + \Delta t},$$

$$A_q' = \frac{-\Delta t \, Z_q'}{2 T_{d0}' + \Delta t}, \qquad B_q' = \frac{2 T_{d0}' - \Delta t}{2 T_{d0}' + \Delta t}, \qquad D_q' = \frac{\Delta t \left( 1 - T_f \right)}{2 T_{d0}' + \Delta t}.$$

The reactance differences are $Z_q’ = L_d - L_d’ - Y_d$ and $Z_d’ = L_q - L_q’ - Y_q$. For the orders without subtransient states $Y_d = Y_q = 0$ and $T_f = 0$. For the sixth order variant with armature coupling,

$$Y_d = \frac{T_{d0}''}{T_{d0}'} \frac{L_d''}{L_d'} \left( L_d - L_d' \right), \qquad Y_q = \frac{T_{q0}''}{T_{q0}'} \frac{L_q''}{L_q'} \left( L_q - L_q' \right), \qquad T_f = \frac{T_{aa}}{T_{d0}'}.$$

The subtransient states obey a recursion of the same shape with $T_{d0}’’$ and $T_{q0}’’$ in place of the transient time constants, and with the transient history entering as a forcing term.

Two properties of this form matter. The coefficients are computed once for a fixed step size, since they contain no state. And because $B_d’$ and $B_q’$ are the trapezoidal amplification factors $(2T - \Delta t)/(2T + \Delta t)$, they lie strictly inside the unit interval for any positive time constant, so the flux recursion is unconditionally stable regardless of step size.

Mechanical equations

The rotor obeys the swing equation in per unit,

$$2H \frac{d \omega_r}{dt} = T_m - T_e, \qquad \frac{d \theta}{dt} = \omega_{base} \, \omega_r,$$

with the electrical torque taken from the axis frame quantities as $T_e = V_d I_d + V_q I_q$. The load angle follows from the initial operating point as the phase of $V + j L_q I$, which is the standard construction of the q axis position from terminal conditions.

Solution schemes

The recursion above evaluates the internal voltage from quantities at the previous step, so the machine and the network are solved once per step in sequence. Two refinements exist for cases where that single pass is not accurate enough.

The predictor corrector method takes the single pass result as a prediction, re-evaluates the flux recursion using the corrected terminal quantities, and repeats until the change between successive passes falls below a tolerance. It converges to the solution of the implicit trapezoidal step rather than to its explicit approximation, at the cost of repeated network solutions.

The two stage predictor method splits the step differently: it advances the machine state on a predicted terminal voltage, then applies a single correction derived from the network solution, without iterating to convergence. It costs one extra network solve per step and removes most of the one step delay error.

Both are schemes for solving the same equations. They do not change the model order or the retained states.

Validity

The reduced order models assume the stator transients are fast enough to be neglected, so the stator is treated as algebraic. This is the assumption that makes the model valid for electromechanical studies and invalid where stator dynamics matter, such as during the first cycles of a close-in fault or for subsynchronous interaction. Neglecting the q axis rotor entirely, as the third order model does, additionally removes damping that is physically present, so a third order machine oscillates more than the same machine represented at fourth order.

References

[Kundur1994] P. Kundur, Power System Stability and Control. New York: McGraw-Hill, 1994. Source of the reciprocal per unit system and of the operational parameter definitions used throughout this page.

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