Frequency Control

Power Balance

Primary Control

- cinetics of the rotating masses

								dv
linear motion:	F	=  m · a  =	m ·	v	;	v	=
								dt
rotation:	Tq	=	J ·	ω	;	ω	=	2 π f

T_q : mechanical shaft torque   [ Nm = Ws ]

J : total moment of inertia of the rotating masses   [ kg m² ]

ω	: angular acceleration   [ 1 / s2 ] ;     ω : angular velocity   [ 1 / s ]

-- torque balance of rotating turbine/generator shaft

		dω		1
ω	=		=		· ( TqT − TqG )
		dt		J

T_qT : turbine torque (accelerates)

T_qG : generator torque (retards)

-- mechanical power balance

with   P  =  ω · T_q

ω	ω	=   J -1 · ( ω TqT − ω TqG )   =   J -1 · ( PT − PG )   =   J -1 ·  Pacc

 P_T(t)  −  P_G(t)  =  P_acc(t)   :   accelerating power ( =  dE_kin / dt   with  E_kin  =  J · ω² / 2 )

ω		ω		P0		Pacc(t)		1		Pacc(t)
−	·	−	=		·		=		·
ω0		ω0		ω02 J		P0		TA		P0

 T_A  =  ω₀² J / P₀  ≈  8 ... 20 s  :     accelerating time constant

- electrical power balance

 P_G(t)  =  P_L(t)  +  ΔP

 ΔP :   disturbance of power balance (load increase, generator outage or tie line outage) at t = t₀

 ΔP_G(t)  =  P_G(t)  −  P₀  =  P_L(t)  −  P₀  +  ΔP

 ΔP_G(t)  =  ΔP_L(t)  +  ΔP

- turbine controller (frequency or speed control)

dΔPT		1	· ( −ΔPT(t)  −  KTC · ( f (t)  −  f0 )  )	simplified turbine model (first order delay)
	=
dt		TT

 ΔP_T(t)  =  P_T(t)  −  P₀   ;     Δf (t)  =  f (t)  −  f₀

final state

dΔP_T / dt  =  0   =>   ΔP_T  =  − K_TC · Δf

ΔPT		KTC · f0		Δf		Δf
	=   −		·		=   −   kTC  ·
P0		P0		f0		f0

- frequency dependence of load

P_L(t)  =  P₀  +  K_L · Δf (t)

ΔP_L(t)  =  P_L(t) − P₀  =  K_L · Δf (t)

ΔPL		KL · f0		Δf				Δf
	=		·		=	kL	·
P0		P0		f0				f0

- states of primary control

-- initial state (before disturbance)

 ΔP  =  0  for  t  <  t₀

 =>   P_T(t)  =  P_G(t)  =  P_L(t)  =  P₀   ;     ΔP_T(t)  =  ΔP_G(t)  =  ΔP_L(t)  =  0

-- disturbance at t = t₀

 ΔP  ≠  0  for  t  ≥  t₀

-- immedeately after disturbance at t  =  t₀

 Δf  =  0  ;    ΔP_T  =  0  ;    ΔP_L  =  0   but   ΔP_G  =  ΔP  =  − P_acc

ω		ω		1		Pacc(t0)		1		ΔP
−	·	−	=		·		=   −		·		initial slope only depends on TA and ΔP
ω0		ω0		TA		P0		TA		P0

-- final state (after primary control)

ω	=  0   =>   Pacc  =  ΔPT  −  ΔPG  =  0

 − K_TC · Δf  −  K_L · Δf  −  ΔP  =  0   =>   Δf  =  −  ΔP / ( K_TC  +  K_L )  =  −  ΔP / K_S

 K_S  =  K_TC  +  K_L

	Δf		ΔP		P0		ΔP		1
		=   −		·		=   −		·
	f0		P0		f0  ·  KS		P0		kS

Secondary Control

 ΔP_SC(t)  =  − K_SC · ∫ Δf (t) dt

dΔPT		1	· ( −ΔPT(t)  −  KTC · Δf (t)  −  KSC · ∫ Δf (t) dt  )	turbine with secondary control
	=
dt		TT

- final state

 Δf  =  0   =>   frequency recovered to normal ;   primary control relieved