Shafts

The shaft strategy is plugged onto a machine with the shaft keyword, for example shaft = "Shaft4Mass". It carries the rotor motion \(\delta, \omega\), exchanging the air-gap power \(P_e\) with the electromagnetic model and the mechanical power \(p_m\) with the governor.

shaft =	Class	Description
"Shaft4Mass"	Shaft4Mass	Four-mass torsional shaft: HP – IP – LP – GEN in a linear chain.
"Shaft5Mass"	Shaft5Mass	Five-mass torsional shaft: HP – IP – LP – GEN – EXC in a linear chain.
"SingleMass"	SingleMass	Single rigid rotor mass: the classic swing equation.

The torsional shaft as a chain of rotor masses coupled by stiffness \(K_{ij}\); the single-mass default keeps only the GEN mass.

class hermess.devices.shaft.SingleMass

Single rigid rotor mass: the classic swing equation. The framework default.

Selected on a machine line with shaft = "SingleMass".

Model.

\[\begin{split}\dot{\delta} &= \omega_b \, (\omega - \omega_{ref}) \\ 2 H \, \dot{\omega} &= p_m - P_e - D \, (\omega - \omega_{ref}) - f \, \omega\end{split}\]

with \(\omega_b = 2 \pi f_n\) the base angular frequency, \(\omega_{ref}\) the per-machine reference frequency (1 p.u.), \(p_m\) the governor’s mechanical power and \(P_e\) the electromagnetic air-gap power. The shaft reuses the machine’s own inertia \(H\), damping \(D\) and rotor friction \(f\), so it declares no parameters of its own.

Symbols.

Code	Symbol	Meaning	Default
delta	\(\delta\)	rotor angle wrt the synchronous frame (state) [rad]
omega	\(\omega\)	absolute per-unit rotor speed, 1.0 at synchronism (state) [p.u.]
(machine)	\(H, D, f\)	inertia [s], damping and rotor friction of the host machine

class hermess.devices.shaft.Shaft4Mass

Four-mass torsional shaft: HP – IP – LP – GEN in a linear chain.

Selected on a machine line with shaft = "Shaft4Mass".

Model. Each mass \(i\) carries an angle and an absolute per-unit speed; adjacent masses are coupled by elastic shaft sections (see MultiMassShaft for the general form):

\[\begin{split}\dot{\delta}_i &= \omega_b \, (\omega_i - \omega_{ref}) \\ 2 H_i \, \dot{\omega}_i &= T_{m,i} - T_{e,i} - \sum_{j \sim i} K_{ij} (\delta_i - \delta_j) - D_i (\omega_i - \omega_{ref}) - \sum_{j \sim i} D_{m,ij} (\omega_i - \omega_j)\end{split}\]

The turbine masses HP, IP and LP receive the fractions \(F_{hp}, F_{ip}, F_{lp}\) of the governor’s mechanical power (\(T_{m,i} = F_i \, p_m\), fractions summing to 1); the generator mass alone carries the air-gap power (\(T_{e,gen} = P_e\)), the machine’s own \(H, D\) and the rotor friction \(f \omega\). The generator mass keeps the canonical state names delta / omega, so nothing downstream changes with respect to SingleMass.

Symbols.

Code	Symbol	Meaning	Default
H_hp / H_ip / H_lp	\(H_i\)	turbine mass inertia constants [s]	0.5 / 1 / 4
D_hp / D_ip / D_lp	\(D_i\)	turbine mass self damping	0
K_hp_ip / K_ip_lp / K_lp_gen	\(K_{ij}\)	shaft section stiffness [p.u. torque/rad]	30 / 40 / 60
Dm_hp_ip / Dm_ip_lp / Dm_lp_gen	\(D_{m,ij}\)	shaft section mutual damping	0
F_hp / F_ip / F_lp	\(F_i\)	mechanical-power fractions (sum to 1)	0.3 / 0.3 / 0.4
delta / omega	\(\delta, \omega\)	generator-mass angle [rad] and absolute speed [p.u.] (states)
delta_hp / omega_hp	\(\delta_{hp}, \omega_{hp}\)	HP mass angle and speed (states)
delta_ip / omega_ip	\(\delta_{ip}, \omega_{ip}\)	IP mass angle and speed (states)
delta_lp / omega_lp	\(\delta_{lp}, \omega_{lp}\)	LP mass angle and speed (states)

class hermess.devices.shaft.Shaft5Mass

Five-mass torsional shaft: HP – IP – LP – GEN – EXC in a linear chain.

Selected on a machine line with shaft = "Shaft5Mass".

Model. As Shaft4Mass, with an additional exciter mass EXC on the far side of the generator; for each mass \(i\) of the chain (see MultiMassShaft for the general form):

\[\begin{split}\dot{\delta}_i &= \omega_b \, (\omega_i - \omega_{ref}) \\ 2 H_i \, \dot{\omega}_i &= T_{m,i} - T_{e,i} - \sum_{j \sim i} K_{ij} (\delta_i - \delta_j) - D_i (\omega_i - \omega_{ref}) - \sum_{j \sim i} D_{m,ij} (\omega_i - \omega_j)\end{split}\]

EXC carries no mechanical and no electrical power; it is a passive inertia coupled through the GEN–EXC shaft section, relevant for the higher torsional modes.

Symbols. As Shaft4Mass, plus:

Code	Symbol	Meaning	Default
H_hp / H_ip / H_lp / H_exc	\(H_i\)	non-generator mass inertia constants [s]	0.5 / 1 / 4 / 0.1
D_hp / D_ip / D_lp / D_exc	\(D_i\)	non-generator mass self damping	0
K_hp_ip / K_ip_lp / K_lp_gen / K_gen_exc	\(K_{ij}\)	shaft section stiffness [p.u. torque/rad]	30 / 40 / 60 / 20
Dm_hp_ip / Dm_ip_lp / Dm_lp_gen / Dm_gen_exc	\(D_{m,ij}\)	shaft section mutual damping	0
F_hp / F_ip / F_lp	\(F_i\)	mechanical-power fractions (sum to 1)	0.3 / 0.3 / 0.4
delta / omega	\(\delta, \omega\)	generator-mass angle [rad] and absolute speed [p.u.] (states)
delta_hp / omega_hp	\(\delta_{hp}, \omega_{hp}\)	HP mass angle and speed (states)
delta_ip / omega_ip	\(\delta_{ip}, \omega_{ip}\)	IP mass angle and speed (states)
delta_lp / omega_lp	\(\delta_{lp}, \omega_{lp}\)	LP mass angle and speed (states)
delta_exc / omega_exc	\(\delta_{exc}, \omega_{exc}\)	EXC mass angle and speed (states)