Small-signal models of synchronous generators, FACTS devices and the power system
4.2 Small-signal models of synchronous generators
4.2.2 Generator modelling assumptions
Sec. 4.2 Generator modelling assumptions 93 Test procedures that are used to identify synchronous generator models for dynamic analy- sis commonly employ the Operational Parameter representation of the generator. As ex- plained in Section 4.2.12 this representation comprises three d-axis transfer-functions and one q-axis transfer-function to completely characterise the machine. The test procedures identify the gains and time constants of these transfer-functions. These transfer-function constants are referred to as the “standard parameters” such as , , , , etc. In order to employ the coupled-circuit formulation of the em equations when only the standard parameters are provided it is necessary to transform the standard parameters to the coupled- circuit parameters as outlined in Section 4.2.14. A troublesome aspect of using the standard parameters is that over the years two alternative and inconsistent definitions of the param- eters have evolved. The ‘Exact’ definitions correspond to the exact roots of the above trans- fer-functions. In the ‘Classical’ definitions the d-axis standard parameters are related to the parameters of the equivalent circuit of the machine by the classical relationships which are based on the assumptions that (i) during the transient period the damper winding resistances are infinite; (ii) during the subtransient period the resistance of the field winding is zero and the resistances of the second damper winding is infinite; (iii) finally, during the sub-subtran- sient period the resistances of the field and first damper winding are assumed to be zero. In the q-axis, analogous assumptions are made to arrive at the classical definitions of the q-axis standard parameters in terms of the coupled-circuit parameters. It is important to know if the generator standard parameters that are provided conform to the ‘Exact’ or ‘Classical’
definitions and if necessary to transform them appropriately to suit the requirements of the simulation model in use. This is particularly important for the q-axis parameters.
The em equations in some widely-used simulation packages are formulated directly in terms of the classically-defined standard parameters. This is referred to as the Classical Parameter formulation in this book. It is emphasised that the Classical Parameter formulation is exactly equivalent to the coupled-circuit formulation provided: (i) that the unequal coupling between the d-axis rotor windings is neglected, and (ii) that the same method for representing magnetic satu- ration is employed in the two models.
94 Generators, FACTS devices & system models
• In the stator voltage equations the transformer voltage terms, and , are omitted because, in the bandwidth of interest, they are negligible in comparison with the speed voltage terms, and [7].
• For the purpose of calculating the d- and q-axis stator voltages the perturbation of the per-unit rotor-speed from the per-unit synchronous speed is assumed to be negligible and thus, for this purpose, . Rotor angle and speed perturbations are, necessarily, represented in the rotor equations of motion.
• A consequence of the above assumption is that the per-unit power transferred across the airgap is independent of perturbations in the rotor speed. Thus, it is also necessary to assume that the mechanical power developed by the turbine is independent of per- turbations in rotor speed.
• The generator equations are expressed in per-unit form in which the base quantities are summarized in Section 4.2.3.2. In particular, the Lad-base reciprocal per-unit sys- tem is chosen for the rotor windings [8, 9, 10].
• As recommended in IEEE Std. 421.5 [11] a non-reciprocal per-unit system is assumed to be employed for the representation of generator excitation systems. As explained in Section 4.2.7 it is therefore necessary to appropriately adjust the scaling of the field current and voltage at the interface between the generator and excitation system mod- els.
4.2.3 Electromagnetic model in terms of the per-unit coupled-circuit parameters The development of the per-unit generator equations in the rotating dq coordinate system from the ideal coupled-circuit representation of the synchronous machine in the stationary abc coordinate system is provided in reference [12]. In this section the per-unit coupled-cir- cuit formulation of the electromagnetic equations of the machine are listed. These equations are then linearized about the initial steady-state operating point of the machine in Section 4.2.11. The Operational Parameter and Classical Parameter formulations for the em equations are summarized in Sections 4.2.12 and 4.2.13 respectively.
The third to eighth-order coupled-circuit machine models require data in the form of the d- and q- axis equivalent circuit winding resistances and leakage and mutual inductances.
Though not commonly used the 3d3q-c1 model has been included because there is evidence in the literature that this number of rotor circuits may be required to adequately represent some machines. Test methods have already been developed to identify machine models with this number of rotor windings, for example [13, 14, 15,.16] Figure 4.2 shows the stator and rotor winding flux linkages in terms of the winding mutual and leakage inductances and the winding currents. The unequal mutual coupling between the d-axis rotor windings is repre-
pd
pq
d
q
0
= 0
Sec. 4.2 Vector and matrix nomenclature 95 sented by the inductances and which are referred to as the Canay inductances. The shielding of the field winding by the damper windings has been identified in the literature as important in correctly predicting the field voltage and current [13, 17, 18]. Unequal mutual coupling between the q-axis rotor windings is not represented since these windings are not directly observable. The d- and q-axis equivalent circuits for the 3d3q-c1 model are shown in Figure 4.3.
Figure 4.2 Per-unit d-axis flux linkage distribution showing the unequal mutual coupling between the rotor windings as described by Canay [17, 18].
Importantly, in the following analysis, once the equations for the eighth-order model are de- fined, the lower-order coupled-circuit models are readily derived. All lower-order coupled-cir- cuit models are formed by deleting the equations and variables associated with those damper windings to be omitted in the formulation of the simpler model.
Two equivalent approaches to the representation of magnetic saturation are accommodated in the formulation of the model. The specific details of the non-linear saturation functions and their linearization are provided in Section 4.2.8.
4.2.3.1 Notes on vector and matrix nomenclature
The nomenclature in the following table applies to matrices and vectors that are extensively employed in the formulation of the models.
Lc1 Lc2
efd ifd
i1d
i2d
id
pd
fl
f1
f12
ad
Flux linkage contributions:
d-axis rotor winding flux linkages:
f 1d 2d
d-axis stator winding flux linkages:
d:
fl = Lfdifd
1l = L1di1d
2l = L2di2d
l = Ll–id
f1 =Lc1ifd+i1d
f12 = Lc2ifd+i1d+i2d
ad = Ladifd+i1d+i2d–id
fd = ad+fl+f1+f12 Lad+Lfd+Lc1+Lc2
ifd+Lad+Lc1+Lc2i1d+Lad+Lc2i2d–Ladid
=
1d= ad+1l+f1+f12 Lad+Lc1+Lc2
ifd+Lad+L1d+Lc1+Lc2i1d+Lad+Lc2i2d–Ladid
=
2d= ad+2l+f12 Lad+Lc2
ifd+Lad+Lc2i1d+Lad+L2d+Lc2i2d–Ladid
=
d = ad+l
Ladifd+i1d+i2d–Lad+Llid
=
1l
2l
l
96 Generators, FACTS devices & system models Table 4.1 Vector and matrix nomenclature
4.2.3.2 Summary of the generator per-unit system
The principal base quantities for the machine are normally (kV), (MVA) and (Hz) which are respectively values for the stator RMS line-to-line voltage, the stator three- phase apparent power and the stator frequency. Usually, but not necessarily, the generator rated values of these quantities are chosen. Additionally, the base value of time is chosen to be one-second. Finally, the relationship between mechanical and electrical angles requires knowledge of the number of rotor pole pairs, .
Symbol Meaning
Denotes a column vector
Denotes a matrix or, depending on the context, a vector.
, Denotes vector and matrix transposition respectively Null matrix or vector. The null matrix has zero rows and/
or zero columns. The possibility of a non-zero number of rows or columns ensures dimensional consistency of matrix equations. The dimension is to be inferred from the context.
The identity matrix. The dimension is to be inferred from the context and in some situations may be the scalar iden- tity (i.e. 1) or the null or empty matrix.
The zero matrix or vector. The dimension is to be inferred from the context and in some situations may be the scalar zero (i.e. 0) or the null matrix.
Denotes a column vector whose entries are all ones. The dimension is to be inferred from the context.
• If is a nx1 vector then is a n x n matrix of all ones.
• is the sum of all the elements in
Denotes block diagonalization which is the result of appending the mx x nx matrix and the my x ny matrix
to create the (mx + my) x (nx + ny) matrix as shown. If and are scalars or diagonal matrices the result is a diagonal matrix.
x˜ X x˜
T XT
Ø
I
0
u = 1 1 1 T u uTu
uTx
˜ x
˜
X Y
D X 0
= 0 Y
X Y
X Y
Vusb Susb fusb
tb
npp
Sec. 4.2 Summary of generator per-unit system 97
Figure 4.3 The d- (top) and q-axis (bottom) equivalent circuits for the 3d3q-c1 generator model represented by three rotor windings in each axis and unequal mutual coupling be- tween the d-axis rotor windings. (Note: inductance & flux-linkage values are scaled by
because the base value of time is one second).
The base values of the rotor winding currents and flux-linkages are determined such that (i) the per-unit mutual inductances between all pairs of windings are reciprocal; (ii) the mutual inductances between all d-axis rotor windings and the stator are equal to the per-unit d-axis mutual inductance ; (iii) the mutual inductances between all q-axis rotor windings and the stator are equal to . Furthermore, the base values of rotor-winding voltages are cho- sen such that the form of the rotor winding voltage equations in SI units and in per-unit are identical. This choice of base values for the rotor quantities is equivalent to that recommend- ed by Rankin in 1945 [8, 9] and is referred to as the “Lad-base reciprocal per-unit system”
[10]. It has gained very wide, if not universal, acceptance in the power system analysis field.
On the above basis for the generator per-unit system are derived the base values for the me- chanical, stator winding and rotor winding quantities in Table 4.2.
Lad
b ---
Ll
b --- rs rq
id
ifd+i1d+i2d
p ad
b ---
p d
b ---
vd
r2d L2d
b ---
i2d ifd+i1d
Lc2
b ---
Lc1
b ---
L1d
b ---
r1d i1d
Lfd
b ---
rfd
ifd
efd rfd Lad
u
---Efd
=
Laq
b ---
Ll
b --- rs rd
iq
p aq
b ---
p q
b ---
vq
r3q L3q
b ---
i3q L2q
b ---
r2q i2q
L1q
b ---
i1q r1q
1b
Lad Laq
98 Generators, FACTS devices & system models Table 4.2 Base values for generator quantities (Note: a bar above a quantity (e.g. )
means the SI value and the subscript ‘b’ denotes the base value of a quantity.).
Base
Quantity SI Units Description
Principal base quantities from which all other base quantities are derived kV (rms,
ph-ph)
Arbitrary choice, but usually rated RMS phase-to-phase stator voltage (sometimes referred to as VBASE).
MVA Arbitrary choice, but usually three-phase MVA rating of the machine (sometimes referred to as MBASE).
Hz
Arbitrary choice, but usually rated generator frequency.
(This is not necessarily the same as nominal frequency of the system to which the generator is connected. For example, when a generator rated at 60 Hz is connected to a 50 Hz system or vice-versa).
Number of pole pairs.
s Base value of time is chosen to be 1 second.
Derived base quantities
s-1
Base value of the time differential operator:
where s-1. (elec)
rad/s Base electrical frequency: . V(peak,
ph-n)
Stator base voltage: peak value of phase to neutral voltage .
VA Machine three-phase VA (apparent power) base: .
Joules Base energy: .
A (peak,
line) Stator base current: peak value of line current .
Stator base resistance / impedance: .
H Stator base inductance: .
Wb-turns Stator base flux-linkages: .
(mech)
rad/s Base mechanical rotor speed: .
Nm Base mechanical torque: .
Nm Base electrical torque: .
x
Vusb Susb
fusb
npp tb
pb p
t̅
d d
tbt d
d 1
tb ----dt
d pbp
= = = = pb 1
tb ---- 1
= =
b b = 2fusb
vsb
vsb = 2 3 Vusb103
Sb Sb = Susb106
Ub Ub = Sbtb = Sb1
isb isb = 2 3 Sbvsb
Zsb Zsb = vsbisb
Lsb Lsb = Zsbb = sbisb
sb sb = Lsbisb = vsbb
mb mb = bnpp
Tmb Tmb = Sbmb
Tb Tm = TmbTmb
Sec. 4.2 Parameter and variable definitions 99
4.2.3.3 Parameter and variable definitions
The parameters and variables used in the formulation of the model are listed in Tables 4.3 and 4.4 respectively together with their base values as defined in Table 4.2.
d-axis rotor quantities
(per-unit unsaturated d-axis mutual inductance is used in the following.) A Base field current in the reciprocal per-unit system of units:
.
V Base field voltage in the reciprocal per-unit system: . A Base field current in the non-reciprocal per-unit system:
.
V
Base field voltage in the non-reciprocal per-unit system:
, is the field resistance in at the specified tem- perature.
A Base current of the d-axis damper windings:
, . V Base voltage of the d-axis damper windings:
, .
, , Wb-turns Base flux-linkages of d-axis rotor windings:
, & .
, , Base resistance of the d-axis rotor windings:
, & .
q-axis rotor quantities
(per-unit unsaturated q-axis mutual inductance is used in the following.) A Base current of the q-axis damper windings:
, . V Base voltage of the q-axis damper windings:
, .
Wb-turns Base flux-linkages of the q-axis damper windings:
, .
Base resistance of the q-axis damper windings:
, . Base
Quantity SI Units Description
Lad = LadLsb ifdb
ifdb = LadsbLafd = LadLafdisb
efdb efdb = Sbifdb
Ifdb Ifdb = ifdbLad
Efdb Efdb = rfdIfdb rfd
ikdb
ikdb = LadsbLakd = LadLakdisb k = 1 2 vkdb vkdb = Sbikdb k = 1 2
fdb 1db
2db fdb = vfdbb 1db = v1dbb 2db = v2dbb rfdb r1db
r2db rfdb = vfdbifdb r1db = v1dbi1db r2db = v2dbi2db
Laq = LaqLsb ikqb
ikqb = LaqsbLakq = LadLakqisb k = 13 vkqb vkqb = Sbikqb k = 13
kqb kqb = vkqbb k = 13
rkqb rkqb = vkqbikqb k = 13
100 Generators, FACTS devices & system models Table 4.3 Summary of the parameters in the per-unit coupled-circuit
representation of the 3d3q-c1 synchronous machine model.
Per-unit Parameter Base Value
(see Tab. 4.2) Description
n/a
The base frequency (elec. rad/s) which appears explic- itly in the per-unit equations due to the choice of one second as the base value of time.
Aggregate inertia constant of the generating unit.
Refer to Appendix 4–II.2 for derivation.
, Aggregate incremental mechanical damping torque coefficient of the generating unit. Refer to
Appendix 4–II.2 for derivation.
Stator resistance, assumed identical in the d- and q- axes.
Stator leakage inductance, assumed identical in the d- and q- axes.
, Respectively the d- and q-axis unsaturated airgap
mutual inductance between the corresponding stator and rotor windings.
The operating point dependent values of the d- and q- axis mutual inductances. (Note: may be a varia- ble depending on the method used to represent mag- netic saturation).
, , , , Resistances of the field winding and the first and sec- ond d-axis damper windings respectively.
, ,
See Note (1)
Leakage inductances of the field winding and the first and second d-axis damper windings respectively.
These inductances represent flux that links only their respective windings.
Mutual inductance between the field and first damper winding which represents flux linkages between these windings but which do not link the stator or the sec- ond damper winding. To neglect unequal coupling between the d-axis rotor windings . Mutual inductance between the three d-axis rotor windings which represents flux that links all three d- axis rotor windings but not the stator.
, , , ,
Resistances of the three q-axis damper windings.
b
H 2Ub b2
D Tmbmb
Tbb
rs Zsb
Ll Lsb
Lad
u Laq
u Lsb
L˜adq = Lad Laq T Lsb L
˜adq
rfd r1d r2d rfdb r1db r2db
Lfd L1d L2d
Lc1
Lc1 = Lc2 = 0 Lc2
r1q r2q r3q r1qb r1qb r2qb
Sec. 4.2 Parameter and variable definitions 101
In the literature on models of generators reference is often made to per-unit machine reac- tances (e.g. , , etc.) rather than per-unit machine inductances (e.g. , ). In this book we adopt per-unit machine inductances. It should be noted that in the per-unit system used the values of per-unit reactances and inductances can be used interchangeably.
, , See Note (2)
Leakage inductances of the three q-axis damper wind- ings. Note that unequal coupling between the q-axis rotor windings is not represented since the q-axis is observable only from the stator.
(1) The d-axis rotor-winding per-unit inductance matrix is defined in terms of the corresponding matrix in terms of SI units where the d-axis rotor-
winding leakage inductance matrix is defined in (4.19), (H) and
.
(2) The q-axis rotor-winding per-unit inductance matrix is defined in terms of the corresponding matrix in terms of SI units where the q-axis rotor-wind-
ing leakage inductance matrix is defined in (4.20), (H) and
.
Per-unit Parameter Base Value
(see Tab. 4.2) Description
L1q L2q L3q
Lrd LaduuT+Llrd b Sb ---
irdbLrdirdb
= =
Lrd
Llrd Lrd
Lffd Lf1d Lf2d Lf1d L11d L12d Lf2d L12d L22d
=
irdb = Difdbi1dbi2db
Lrq LaquuT+Llrq b Sb ---
irdqLrqirqb
= =
Lrq
Llrq Lrq
L11q L12q L13q L12q L22q L23q L13q L23q L33q
=
irqb = Di1qbi2qbi3qb
Xd Xad Ld Lad
102 Generators, FACTS devices & system models Table 4.4 Summary of variables in the per-unit coupled-circuit representation of the
3d3q-c1 synchronous machine.
Variable (in per-unit) Base Value
(see sec. Tab. 4.2) Description
d- and q-axis stator terminal voltage respec- tively.
d- and q-axis stator winding current respec- tively. Direction of positive stator current is from the generator into the network.
d- and q-axis airgap flux linkages respectively.
d- and q-axis stator flux linkages respectively.
d-axis rotor flux linkages in which subscripts
‘fd’ refers to the field winding and ‘1d’ and ‘2d’
refer respectively to the first and second d-axis damper windings.
q-axis rotor flux linkages in which the sub- scripts ‘1q’, ‘2q’ and ‘3q’ refer respectively to the three q-axis damper windings.
d-axis rotor-winding voltages in which is the per-unit field winding voltage in the recip- rocal base system. The damper windings are short-circuit so their voltages are zero.
q-axis rotor winding voltages are identically zero since the damper windings are short-cir- cuit.
d-axis rotor winding currents. is the per- unit field winding current in the reciprocal base system.
q-axis rotor winding currents.
Note: For compactness the d- and q-axis rotor winding variables are aggregated as follows:
, , .
, , Per-unit field current and voltage respectively
in the non-reciprocal base system.
v˜dq = vd vqT vsb
˜idq = id iq T isb
˜adq = ad aq T sb
˜dq = d q T sb
˜rd = fd 1d 2d T fdb 1db 2db T
˜rq = 1q2q 3q T 1qb 2qb 3qb T
v˜rd = efd 0 0 T efdb v1db v2db T
efd
v˜rq = 0 0 0 T v1qb v2qb v3qb T
˜ird = ifd i1d i2d T ifdb i1db i2db T
ifd
˜irq = i1q i2q i3q T i1qb i2qb i3qbT
˜rdq
˜rd
˜rq
= v
˜rdq v˜rd v˜rq
= i
˜rdq
˜ird
˜irq
=
Ifd Efd Ifdb Efdb
Sec. 4.2 Summary of the generator coupled-circuit equations 103
4.2.3.4 Summary of the coupled-circuit formulation of the generator electromagnetic equa- tions
Summarized below are the per-unit coupled-circuit equations describing the electromagnet- ic behaviour of the generator in the rotating dq coordinate system. These equations are de- veloped from first principles in [12].
The d- and q-axis rotor-winding voltage equations are respectively:
and (4.2)
;
The demagnetizing components of the d- and q-axis excitation current that is required to account for the effects of magnetic saturation in the respective axes. This saturation excita- tion current is incorporated in the model only if the second method of saturation modelling in Section 4.2.8.2 is employed.
where As above, but the non-reciprocal per-unit sys- tem is employed. This representation of the demagnetizing effects of magnetic saturation is employed in the Classical Parameter formula- tion of the em equations in Section 4.2.13.
(elec. rad)
Relative rotor angle being the angular position of the d-axis with respect to the synchronously rotating network reference (in elec. rad).
(elec. rad)
Stationary rotor angle being the angular posi- tion of the d-axis with respect to a stationary reference (in elec. rad).
Rotor-speed.
Synchronous speed. Note, if the nominal sys- tem frequency is equal to the generator base frequency then .
Electrical power output.
Electromagnetic (or airgap) torque.
Variable (in per-unit) Base Value
(see sec. Tab. 4.2) Description
˜isdq = isd isq T ifdb i1qb T
I˜sdq = IsdIsqT Isd Lad
uisd
= Isq Laq
uisq
=
ifdb Lad
u
i1qb Laq
u
b
0 b
0 = 1
Pe Sb
Tg Tmb
1
b ---p
fd
1d
2d 1 0 0
efd
rfd 0 0 0 r1d 0
0 0 r2d
ifd i1d i2d –
=