Non-reciprocal definition of the per-unit field voltage and current

Small-signal models of synchronous generators, FACTS devices and the power system

4.2 Small-signal models of synchronous generators

4.2.7 Non-reciprocal definition of the per-unit field voltage and current

Although the L_ad-base reciprocal per-unit system has a number of advantages from the perspective of representing the generator it is usually the case that a different per-unit system, referred to as the “non-reciprocal” or “unity-slope” per-unit system, is used when representing the excitation system of the generator. Thus, it is necessary to establish the relationship between these two per-unit systems for the purpose of interfacing between the field winding of the generator and excitation system models.

To begin, consider the generator represented by the per-unit equations in the reciprocal per- unit system and neglecting magnetic saturation. Suppose that the generator is on open-circuit and rotating steadily at one per-unit speed. Under this steady-state condition , and the rates of change of all variables in the dq coordinate system are zero. From the d-axis rotor voltage equations (4.2) on page 103 it is deduced that and the d-axis damper winding currents are zero (i.e. ). From the corresponding q-axis equations (4.3) the q-axis damper winding currents are also found to be zero (i.e. ). Given these initial values of the winding currents it is deduced from

p D

2H ---

 

  

– 1

2H---

 

 C_tvdqv

˜^dq

– 1

2H ---

 

 C_tidqi

˜^dq

– 1

2H---

 

  T_m +

p 1

2H---P_m–P_g–D   – ₀

= P_g

p 1

2H---P_m–P_e–D   – ₀–r_si_d²+i_q²

p 1

2H₀

--- P_m–P_e–D₀ 2r_s i_d

0i_d i_q

0i_q

 + 

 – 

₀ = 1 r_s0

p 1

2H---P_m–P_e–D

2H M = 2H

i_d = i_q = 0  = ₀ = 1

e_fd = r_fdi_fd i_1d = i_2d = 0

˜i^rq = 0

116 Generators, FACTS devices & system models the d- and q-axis airgap flux-linkage equations (4.15) on page 105 and the d- and q-axis flux- linkage equations (4.23) on page 106 that and . From the d- and q-axis stator voltage equations (4.24) it follows that and . If the field current in the reciprocal per-unit system is one per-unit then the stator voltage is per-unit (neglecting saturation) and the field voltage is per-unit.

Thus, an equivalent way of defining the base field current in the L_ad-base reciprocal per-unit system is:

The base field current in the reciprocal per-unit system is that field current, in Amperes, which is required to generate per-unit stator voltage on the airgap line when the machine is open-circuit and rotating steadily at one per-unit speed. The base field voltage is the corresponding field voltage in Volts divided by the per-unit field winding resistance¹ at the specified field winding temperature.

The above reciprocal definition of the base field current and voltage is ^notconsistent with the non-reciprocal definition of the base values of the field quantities which is recommend- ed in Annex B of IEEE Std. 421.5 [11] for the modelling of excitation systems. The following definition of the non-reciprocal per-unit system for the field current and voltage is consistent with that given in IEEE Std. 421.5. Note that in the reciprocal per-unit system quantities related to the field current and voltage are denoted by lower case ‘i’ and ‘e’ re- spectively whereas the corresponding quantities in the non-reciprocal per-unit system are denoted by upper-case ‘I’ and ‘E’.

The base field current in the non-reciprocal per-unit system is that field current, in Amperes, which is required to generate 1.0 per-unit stator voltage on the airgap line when the machine is open-circuit and rotating steadily at one per-unit speed. The base field voltage in this per-unit system is the field voltage in Volts, corrected to the specified field winding temperature, required to generate the base field current

1. From Table 4.2, the per-unit field resistance is where is the field resistance in ohms and (ohm) is the base value of field resistance in the reciprocal per-unit system.

_d L_ad

ui_fd

= _q = 0

v_d = _q = 0 v_q _d L_ad

ui_fd

= =

L_ad

e_fd = r_fd

i_fdb

L_ad

e_fdb

r_fd =  r_fd r_fdb r_fd r_fdb = S_bi_fdb²

I_fdb

E_fdb

I_fdb

Sec. 4.2 Definition of non-reciprocal per-unit field voltage and current 117 The above definitions lead to the following mathematical conversions between the per-unit field current (and voltage ) in the non-reciprocal per-unit system and the corresponding value of ( ) in the reciprocal per-unit system:

and . (4.65)

The conversion between the reciprocal and non-reciprocal definitions of the field current is shown graphically in the generator open-circuit characteristic in Figure 4.4 in which three field current scales are shown: (i) Amperes, (ii) per-unit on the reciprocal base system; and (iii) per-unit on the non-reciprocal base system.

Figure 4.4 Generator open-circuit characteristic with the field current scaled in Amperes, and in per-unit according to the reciprocal and non-reciprocal per-unit systems.

The scaling required at the interface between the model of the exciter and the generator field winding is depicted in Figure 4.5. It is assumed that the output from the exciter is the field voltage in per-unit in the non-reciprocal per-unit system and the input to the generator is the per-unit field voltage in the reciprocal system. It is assumed that the generator per-unit field current in the reciprocal system is, from model signal flow perspective, an output signal from the generator which is input to the model of the exciter in per-unit in the non-reciprocal per-unit system.

I_fd E_fd

i_fd e_fd

I_fd L_ad

ui_fd

= E_fd L_ad

r_fd ---e_fd

Ifd (A)

ifd (pu) (reciprocal) Ifd (pu) (non−reciprocal) Ifdb

1/Ladu

1.0

ifdb

= Ladu I

fdb

1.0

Ladu

0 0 1.0 Ladu

Open−circuit stator voltage (pu) and flux−linkages (pu)

Airgap line

O.C.C.

Non−reciprocal Reciprocal

118 Generators, FACTS devices & system models

Figure 4.5 Interface between the generator and exciter model taking account of the conversion between the reciprocal and non-reciprocal per-unit systems in the

respective models.

In the reciprocal per-unit system it follows from (4.2) on page 103 that under steady-state condition . By applying the conversion in (4.65) to this relationship it follows that in the non-reciprocal per-unit system the steady-state value of the field voltage and current are equal (i.e. ).

Some of the reasons why the non-reciprocal per-unit system is preferred [10] are:

• The measured generator open-circuit characteristic (O.C.C.) rarely extends beyond a stator voltage of 1.1 per-unit and never to . Thus, direct graphical determination of the base field current from the measured O.C.C. is straight-forward in the non-reciprocal system whereas supplementary calculation is required to determine the base field current in the reciprocal system.

• The numerical value of per-unit field-voltage in the reciprocal per-unit system is very small whereas, under steady-state conditions, in the non-reciprocal system.

Although IEEE Std. 421.5 recommends the use of the non-reciprocal per-unit system for modelling of the excitation system, it is sometimes the case that vendors or testing contrac- tors provide excitation system model parameters on a different per-unit system. For exam- ple, sometimes the base value of field current is defined as that field current, in Amperes, that is required to produce rated stator voltage when the generator is operating at ^{rated out-} put and frequency. Therefore, it is essential that those who are entering data into simulation programs understand the basis on which model parameters are supplied and, if necessary, adjust parameter values to comply with the per-unit system assumed by the simulation pro- gram being used.

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