Small-signal models of synchronous generators, FACTS devices and the power system
4.2 Small-signal models of synchronous generators
4.2.1 Structure of the per-unit linearized synchronous generator models
Before developing the details of the per-unit linearized model of the synchronous generator the overall mathematical structure of the model is described by means of the block diagram in Figure 4.1. The linearized model is formulated in the rotating direct- and quadrature-axis (dq) coordinate system in which the d-axis is aligned with the magnetic north pole of the field winding and the q-axis leads the d-axis by 90 deg. (electrical). The model is linearized about an initial steady-state operating point that is defined by the initial stator terminal quantities which are typically obtained from a power flow solution. The calculation of the generator initial conditions from the specified terminal quantities is described in Section 4.2.9. The two principal components of the linearized generator model in Figure 4.1 are the electro- magnetic (em) equations and the shaft equations of motion. The em equations comprise a set of differential equations that describe the dynamic characteristics of the d- and q-axis ro- tor-winding flux linkages together with algebraic equations for the stator voltage compo- nents.
As described in Section 4.2.10 the connection of the generator stator to the network requires the transformation of the perturbations in the stator voltage and current components in the generator dq coordinate system to the corresponding components in the synchronously ro- tating RI-network coordinate system1.
There are two control inputs to the generator. The first is the perturbation in the field volt- age ( ) developed by the excitation system. As explained in Section 4.2.7 it is important to note that is expressed in the non-reciprocal per-unit system of the generator field.
The field voltage input in this per-unit system must be converted to the reciprocal per-unit system which is employed in the formulation of the generator electromagnetic equations.
Similarly, the field current in the reciprocal per-unit system must be converted to the non- 1. RI refers to the Real and Imaginary components in the network coordinate system.
Efd
Efd
Sec. 4.2 Structure of per-unit linearized generator models 91 reciprocal per-unit system for use in the excitation system model. The second input is the perturbation in the mechanical torque ( ) developed by the turbine / governor system which is expressed in per-unit on the generator base value of mechanical torque. Models for the excitation and turbine / governor systems are not included in this chapter.
The generator models are designated by a code of the form ndmq-c{0,1} in which n and m are the number of d- and q-axis rotor-windings respectively; c1 and c0 are used to indicate, respectively, that unequal mutual coupling between the d-axis rotor windings is represented or neglected. The 1d0q-c0 model comprises three state-variables: the rotor-angle, rotor- speed and d-axis field flux-linkages. This is the basis for the Heffron-Phillips model [3, 4]
that is frequently used for developing concepts for generator controls. It is, however, not recommended for use in power system analysis. The 3d3q-c1 model, the most complex mod- el considered in this work, comprises a field winding, and two damper windings in the d-axis and three q-axis damper windings; unequal mutual coupling between the d-axis rotor wind- ings is represented. This model – with eight state-variables comprising the six rotor-winding flux-linkage variables and rotor angle and rotor speed – is the most complex model encoun- tered in small-signal analysis of large power systems. The most commonly employed models in large scale small-signal stability studies are the fifth and sixth order models 2d1q-c0 and 2d2q-c0 in which unequal mutual coupling effects are neglected.
The formulation of the em equations for the 3d3q-c1 model described in Section 4.2.3 is based on the ideal coupled-circuit representation of the synchronous machine for which the model parameters are the resistances, mutual and leakage inductances of the windings. As explained in Section 4.2.4 the em equations developed for this model are readily modified to represent machine models with fewer damper windings in the respective axes. In particu- lar, the structure of the em equations and their interface with other components in the over- all model of the generator are unaffected by changes in the number of damper windings.
The linearized coupled-circuit formulation of the state- and algebraic equations of the com- plete generator model are given in matrix form in equation (4.117) on page 133 followed in Table 4.9 by a step-by-step procedure for calculating the associated coefficient matrices.
Tm
92 Generators, FACTS devices & system models
Figure 4.1 Structure of the per-unit linearized model of the synchronous generator. (Refer to Tables 4.3 and 4.4 for descriptions of the parameter and variable symbols in this figure).
KI
Network
v˜RI
g
i˜RI
g
R v0
˜dq U v0
˜dq0
+
R 0 i
˜dq U i0
˜dq0
+
px
˜r 0
Ar Bri 0 CvrDvi–I
x
˜r
i
˜dq
v
˜dq bre
0 efd +
=
Electromagnetic Equations (see Note)
p
p
0
0 b 0 0 0
0 D
2H---
– 1
2H---
– 0 0
0 0 –1 CtidqCtvdq
Tg
i
˜dq
v˜dq 0 1 2H---
0
Tm +
=
Rotor Equations of Motion (Sections 4.2.5 & 4.2.6)
i
˜dq v
˜dq
Turbine / Governor
Model
Tm
Input Signals (Includes Generator
Outputs) Excitation
System Model
rfd Lad
u
---
Lad
u
efd
ifd
Efd
Ifd
Conversion between reciprocal (R)
and non-reciprocal (NR) field winding
per-unit systems (Section 4.2.7) Transformation from generator dq to network
RI coordinate system (Section 4.2.10)
Input Signals (Includes Generator
Outputs)
Compute perturbations in stator quantities
(Section 4.3.1)
NR R
dq RI
Vt P Q I
Note: Coupled-circuit formulation of linearized em equations is devel- oped in Section 4.2.11. The Classical Parameter formulation is developed in Section 4.2.13.4. In the Classical Parameter formulation the conversion between the reciprocal and non-reciprocal per-unit systems of the field winding is embedded in the em equations.
KV
i˜RI
n v
˜RI
n
On generator per-unit system
Conversion from generator (g) to network (n) per-unit
system. (Section 4.2.10)
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.