Analysis of Turbine Generator Steady-State Reactances for Load Conditions

SHIN�ICHI WAKUI, KAZUMASA IDE, MIYOSHI TAKAHASHI, GUSTAVO MONTOYA,

FUTOSHI HIYAMA, and YASUOMI YAGIHitachi Ltd., Japan

SUMMARY

This paper describes variations of turbine generator

steady-state reactances for load conditions. When the tur-

bine generator is of small size, magnetic saturation of the

stator and rotor core is a problem. Therefore, it is important

to understand the variations of the reactances for load

conditions. The reactances of a 592-MVA turbine generator

are calculated with d- and q-axis equations considering

magnetic saturation and two-dimensional numerical mag-

netic field analysis. The results are as follows.

(1) The equivalent synchronous reactances consid-

ering cross-magnetizing are smaller than the original syn-

chronous reactances for load conditions.

(2) Numerical values of the d- and q-axis mutual

reactance and the field and q-axis mutual reactance are at

their maximum when the d-axis linkage flux is nearly equal

to the q-axis linkage flux under constant voltage.

(3) The d-axis synchronous reactance drops with

increasing reactive power under constant voltage and active

power, but the q-axis synchronous reactance increases.

(4) The variation of the equivalent d-axis synchro-

nous reactance with increasing reactive power is similar to

that of the d-axis synchronous reactance, but the variation

of the equivalent q-axis synchronous reactance is different

from that of the q-axis synchronous reactance. © 1998

Scripta Technica. Electr Eng Jpn, 123(3): 73�83, 1998

Key words: Turbine generator; d- and q-axis equa-

tions; magnetic field analysis; magnetic saturation; reac-

tance.

1. Introduction

In the market for rotating machines in recent years,

the demand for lower prices is increasing, in addition to

demands regarding the performance of the machines, and

low-cost implementation by reducing machine size be-

comes necessary. One of the problems in such reduction is

the pronounced magnetic saturation of the rotor and stator

cores due to the high magnetic field implementation, and

compensation of the accuracy drop in the characteristic

calculations by nonlinear magnetic characteristics becomes

necessary.

In the general method, the characteristics of a syn-

chronous machine under load conditions are calculated by

first calculating the no-load and three-phase short-circuit

characteristics and then using these and the numerical val-

ues statistically processed by Potier reactances and so forth

[1]. The generator performance must be determined origi-

nally based on the characteristics under load conditions;

however, it my not necessarily be the case that the electrical

characteristics under load conditions can be accurately

grasped as long as a method close to such an estimation is

adopted.

The characteristic values which show the electrical

performance under load conditions are the reactances.

These reactances are used in the analysis of stability and so

forth for large-capacity synchronous machines to be oper-

ated in power systems; however, in general, they are used

by dividing them into two values, the saturated value and

the unsaturated value. The saturated value is calculated for

a state corresponding to the time when a no-load rated

voltage occurs; however, under various load states, no

account is taken of when the flux level and the direction of

the main flux inside the generator change, and thus the

reactance values change.

Analytical techniques for performing power system

analysis considering the effect of saturation in the generator

reactances have been proposed [2, 3]. However, the power

system voltage is fixed, and for the general operation pat-

terns of generators whose power factor and output change,

no examples have been confirmed in which the reactance

values under various load states are calculated with allow-

CCC0424-7760/98/030073-11

© 1998 Scripta Technica

Electrical Engineering in Japan, Vol. 123, No. 3, 1998Translated from Denki Gakkai Ronbunshi, Vol. 117-D, No. 3, March 1997, pp. 364�372

73

ance for magnetic coupling of both the d axis and the field

with the q axis at saturation.

Therefore, it may be necessary to consider generator

performance under load conditions and to reconstruct the

design guidelines of the generator in order to enhance

magnetic loading and reduce machine size. When attention

is paid to these characteristics under load conditions, it is

necessary to determine quantitatively the variations of the

reactances of the generator under various load conditions.

As to the attempts to calculate the field current under

any load condition, Chari and others have proposed a

method using a two-dimensional magnetic field analysis

and the d�q axis equations [4, 5]. However, in that method,

since the aforementioned magnetic couplings between the

d and q axes and between the field and q axis which are

typical of saturation have not been considered, it is difficult

to allow for these effects. In order to solve this problem, the

authors and others have previously proposed a method for

calculating the field current and reactances under any load

condition while taking into account the magnetic couplings

which are typical of saturation; they reported that a good

analytical accuracy can be obtained even in a wide region

of saturation, compared to the case of not considering the

magnetic couplings which are typical of saturation [6].

In this paper, by applying the previously proposed

calculation method for reactances under any load condition,

the variations of the reactances under various load states are

quantitatively analyzed, investigated, and consolidated,

taking into account the magnetic couplings between the d

and q axes and between the field and q axis at saturation.

First, we will describe reactance calculations by the ex-

tended d�q axis equations with allowance for magnetic

saturation and the equivalent transformation of these reac-

tances into the reactances in the conventional d�q axis

equations without allowance for magnetic saturation. Then,

using a 592-MVA turbine generator, the variations of the

reactances are calculated for the case when the voltage and

power factor are constant and the output is varied, and for

the case when the voltage and output are constant and the

power factor is varied; and the results of transforming them

into the aforementioned equivalent reactances are shown.

As a result, the following become clear quantitatively: the

variations of the reactances expressing the magnetic cou-

plings which are typical of saturation when the load

changes, as well as the differences between the values of

the reactances defined by the d�q axis equations with

allowance for magnetic saturation and the values of the

equivalently transformed reactances defined by the conven-

tional d�q axis equations.

We will describe the above results in this paper.

2. Calculation Method for Field Current and

Reactances

In this section, we will describe the method for cal-

culating the field current and reactances under any load

condition by using the d�q axis equations with allowance

for magnetic saturation and a two-dimensional magnetic

field analysis, and will explain the method for equivalently

transforming the reactances thus obtained into the reac-

tances in the conventional d�q axis equations without al-

lowance for magnetic saturation.

2.1 Calculation method for field current and

reactances considering magnetic saturation

If the d-axis armature linkage flux fd, the q-axis

armature linkage flux fq, and the field linkage flux ffconsidering magnetic saturation are expressed in the per-

unit system, we obtain the following equations [6]:

where xd is the d-axis synchronous reactance, xq is the q-axis

synchronous reactance, xf is the field winding self-reac-

tance, xad is the d-axis armature reaction reactance, xdq is

the mutual reactance between the d and q axes, xfq is the

mutual reactance between the field and q axis, id is the

d-axis armature current, iq is the q-axis armature current,

and if is the field current. In Eqs. (1) to (3), the respective

reactances will be changed by the magnetic saturation.

Moreover, based on Eqs. (1) to (3), the d-axis arma-

ture voltage vd and q-axis armature voltage vq can be ex-

pressed by

where r is the armature winding resistance.

Figure 1 shows the vector diagram of the synchro-

nous generator. Figure 1(a) is the case without allowance

for magnetic saturation and Fig.1(b) is the case considering

magnetic saturation. They are drawn based on Eqs. (4) and

(5). In this figure, ef as well as ed and eq are, respectively,

the voltage induced by the field current and the voltage

drops produced by the d- and q-axis armature currents; efand ed are vectors in the q-axis direction and eq is a vector

in the d-axis direction in the case without allowance for

(1)

(2)

(3)

(4)

(5)

74

magnetic saturation. These relations will no longer hold

when magnetic saturation is considered.

Figure 2 shows a flow chart of the calculations of the

field current and reactances, and illustrates the method of

calculating the field current and reactances under a specific

load state by using d�q axis equations (1) to (3) and

two-dimensional sectional magnetic field analysis. As pre-

sented, these calculations are based on the following flow

[6].

(1) The initial values of the armature voltage, arma-

ture current, power factor, and reactances corresponding to

the load state to be calculated are input.

(2) By using Eqs. (1) to (3), the d�q axis armature

currents and field current, as well as the d�q axis armature

linkage fluxes and field linkage flux, are calculated.

(3) By using these d�q axis armature currents and

field current, the two-dimensional sectional magnetic field

is analyzed, the d�q axis armature linkage fluxes and field

linkage flux are determined, and the reactances are calcu-

lated [6]. Here, the d�q axis armature linkage fluxes are

calculated based on the concept presented in the appendix.

Besides, in cases where the q-axis armature current be-

comes zero, such as in three-phase short circuit, a suffi-

ciently smaller current than the others is input so that the

reactances can be calculated.

Fig. 1. Vector diagram of synchronous generators.

Fig. 2. Flow chart of field current and reactance calculation.

75

(4) The calculated reactances are corrected by con-

sidering the effect of the air ducts and end steps on the stator

side and the group slots on the rotor side as well as the

leakage reactances of the armature winding end portion [6].

(5) From the three-dimensional corrected reac-

tances, the d- and q-axis armature linkage fluxes and the

field linkage flux are calculated and updated.

(6) Whether or not the updated values of the linkage

fluxes agree with the values before update is judged. If they

agree, it is judged that the solutions of the currents, fluxes,

and reactances have converged; and if not, the updated

values of the reactances are replaced with the values before

update and the calculations of (2) to (5) are repeated.

2.2 Equivalent reactances

Equations (1) to (3) are the equations for the armature

linkage fluxes between the d and q axes and between the

field and q axis considering magnetic coupling; however,

they differ from the forms of the d- and q-axis equations of

the synchronous machines generally used. For this reason,

if the equations for the armature linkage fluxes are ex-

pressed by considering the effect of magnetic couplings

xdq between the d and q axes and coupling xfq between the

field and q axis in d- and q-axis synchronous reactances

xd and xq, we obtain

where xl is the armature leakage reactance and xaq is the

q-axis armature reaction reactance.

Here, xdeq and xqeq will be called the equivalent d-axis

synchronous reactance and equivalent q-axis synchronous

reactance, respectively; and xadeq and xaqeq will be called the

equivalent d-axis armature reaction reactance and equiva-

lent q-axis armature reaction reactance, respectively. These

reactances are defined in order to express the armature

linkage fluxes equal to the conventional equations without

the coupling of the d axis to the field and to the q axis.

However, as is also clear from the derivation of Eqs. (6) and

(7), these equivalent reactances can be compared with the

equations for the conventional linkage fluxes, but cannot be

used in the equation for the field linkage flux.

3. Variations of Reactances under Load Conditions

In this section, using a 592-MVA turbine generator,

we will first analyze the field current and reactances by the

analytical method shown in the preceding section, compare

them with the measured results, and confirm the calculation

accuracy of the proposed analytical technique under vari-

ous operating states. Then, we will calculate and investigate

the variations of the reactances for the case when the voltage

and power factor are constant and the output is varied, as

well as for the case when the voltage and output are constant

and the power factor is varied.

3.1 Generator under analysis

Table 1 shows the main specifications of the gener-

ator under consideration. The generator under considera-

tion is a 592-MVA turbine generator and is a

standardized-design machine of this class.

Figure 3 shows the three-phase short-circuit and no-

load saturation curves. The calculated values and measured

values of the three-phase short-circuit curve agree well. For

the no-load saturation curve, the calculated values of the

field current are slightly larger than the measured values;

however, a good calculation result is obtained even in the

region above the rated voltage where the magnetic satura-

tion is large. The reason why the calculated field currents

at no load are somewhat larger than the measured values

my be presumed to be that although the field flux linking

the end portion of the armature winding has occurred under

the state of no armature reaction, it is not considered in the

analysis.

Table 2 shows the analysis cases under load condi-

tions. Cases 1 to 4 are inductive loads near the rated power

factor 0.85, where the output is set at about 40, 60, 80, and

100% of the rated load, respectively. Cases 5 and 6 represent

a noninductive load with an output set at about 100% and

a capacitive load with an output set at about 100%, respec-

tively.

Figure 4 shows the calculated field currents under

load conditions. Figure 4(a) shows Cases 1 to 4 of Table 2,

namely, the field currents when the power factor is constant

at 0.85 p.u. and the load is increased. Figure 4(b) shows

Cases 5 and 6 of Table 2, namely, the field currents when

(6)

(7)

76

the output is constant at about 100% and the power factor

is varied. It is seen that the calculated field currents under

load conditions agree well with the measured values, with

little error, regardless of the magnitude of the load.

Table 3 shows the calculated transient reactances of

the 592-MVA turbine generator. The measured value is the

transient reactance xdg when a three-phase sudden short-cir-

cuit test is conducted from a no-load operating state at a

terminal voltage V = 0.3 p.u.; it corresponds to the unsatu-

rated value. The calculated value of xdg is calculated as the

sum of the armature leakage reactance xl and the field

leakage reactance xfl under the operating state Ia = 0.3 p.u.,

terminal voltage V = 0.3 p.u., and power factor 1.00 p.u.; it

corresponds to the unsaturated value. The calculated value

agrees with the measured value with an error of about 5%.

It is clear from the above results that the proposed

calculation method can calculate the field current and so

forth with good accuracy regardless of the operating states;

and it provides data with which the reactances under various

operating states can be calculated with good accuracy.

3.2 Variations of reactances at constant power

factor

Table 4 shows the reactance calculations at constant

power factor. Cases 1 to 3 are the calculation cases when

the armature current is varied while the specific power

factors, namely, a lag power factor of 0.85 p.u., a lead power

factor of 0.90 p.u., and 1.00 p.u. are set constant.

Figure 5 shows the variations of the synchronous

reactances and leakage reactances when the power factors

are set constant. Figure 5(a) plots the d-axis synchronous

reactance xd, the q-axis synchronous reactance xq, and the

leakage reactance xl; it is seen that xq becomes smaller as

the load becomes larger regardless of the power factor. On

the other hand, xd is about constant at a leading power factor

of 0.90 p.u. and at 1.00 p.u., where the flux level is low,

regardless of the load, and decreases as the load becomes

Fig. 3. Three-phase short-circuit and no-load saturation

curves.

Table 1. Specifications of 592-MVA turbine generator

Item Stator Rotor

Capacity (MVA) 592

Armature voltage (V) 20

Armature current (A) 17,090

Power factor (p.u.) 0.85 (lagging)

Number of poles 2

Frequency (Hz) 50

Number of rotations (rpm) 3000

Axial length of core (mm) 5800

Connection method 2Y �

Number of slots 48 32

Outer diameter of core (mm) 2466 1079.5

Cooling scheme water cooling direct hydrogen cooling

77

larger at a lagging power factor of 0.85. The decreased

reactances are caused by the fact that when the load be-

comes larger, since a voltage drop due to the leakage

reactance will occur, the fluxes of the rotor and stator cores

become large, and the cores will be further saturated. The

trend of xd becoming smaller is more pronounced when the

armature current lags farther behind the terminal voltage;

in contrast, the trend of xq becoming smaller is more pro-

nounced when the armature current is increasingly leading

the terminal voltage.

This can be understood from the fact that when the

power factor is increasingly leading, since the ratio of the

field magnetomotive force to the total magnetomotive force

becomes smaller due to the magnetizing action, and the

effect of the armature reaction affecting the fluxes inside

the generator becomes relatively larger, the fluxes inside the

generator incline toward the q-axis direction as the power

factor becomes increasingly leading. Namely, in a cylindri-

cal rotor such as a turbine generator, it is generally regarded

that xd @ xq, and this is shown numerically by the fact that

it holds when the power factor leads to an increasing extent.

Moreover, when the load becomes larger, since the synchro-

nous reactances xd and xq become smaller and the inherent

steady-state stability will be enhanced [7], synchronous

operation can be continued stably as the load becomes

larger. On the other hand, the armature leakage reactance

xl tends to increase slightly as the load becomes larger, in

opposition to the armature reaction reactances xd and xq, and

a difference due to the power factor is scarcely seen.

Figure 5(b) shows the equivalent d-axis synchronous

reactance xdeq and equivalent q-axis synchronous reactance

xqeq; xdeq and xqeq are calculated using Eqs. (6) and (7).

xdeq and xqeq decrease, regardless of the calculation cases,

due to magnetic saturation as the armature current becomes

larger. Moreover, for the same armature current value, xdeqand xqeq decrease in order of lagging power factor 0.85 p.u.

where the magnetic saturation is large, power factor 1.00

p.u., and leading power factor 0.90 p.u.; and when

xdeq, xqeq, and xd, xq are compared, xdeq and xqeq are smaller

than xd, xq, respectively, and the difference due to the dif-

ference of power factors also becomes smaller.

The reactances xd, xq based on the original definition

are directly dominated by the magnitudes of the d-axis

component id and q-axis component iq of the armature

Table 2. Calculation cases

Item Case 1 Case 2 Case 3 Case 4 Case 5 Case 6

Active power (p.u.) 0.356 0.507 0.677 0.845 0.840 0.797

Reactive power (p.u.) 0.193 (lag) 0.312 (lag) 0.427 (lag) 0.519 (lag) 0.030 (lag) 0.291 (lead)

Armature voltage (p.u.) 1.020 0.995 0.997 1.003 0.995 1.005

Armature current (p.u.) 0.404 0.605 0.802 0.989 0.848 0.854

Power factor (p.u.) 0.880 (lag) 0.851 (lag) 0.846 (lag) 0.852 (lag) 0.999 (lag) 0.940 (lead)

Fig. 4. Calculated field currents for load conditions.

78

current, respectively; however, since xdeq, xqeq are the reac-

tances equivalently expressing the magnetic couplings be-

tween the d and q axes and between the field and q axes,

they are affected by iq and id, respectively, and the difference

due to the power factors also becomes smaller. Moreover,

due to the flux in the d-axis direction produced by iq, the

flux in the d-axis direction produced by the vector sum of

id and if is demagnetized and the magnetic coupling in the

d-axis direction becomes smaller, so that xdeq becomes

smaller than xl. For the q axis, due to the same reason, since

the magnetic coupling in the q-axis direction becomes

smaller, xqeq becomes smaller than xq.

Figure 6 shows the variations of the mutual reactance

xdq between the d and q axes, the mutual reactance xfqbetween the field and q axis, and the d- and q-axis armature

linkage fluxes fd and fq with respect to the armature current

Ia. Figure 6(a) shows the variations of xdq, xfq and fd, fq for

the case of lagging power factor 0.85 p.u. (Case 1); this is

the case when Ia = 1.0 p.u. is the rated load. xdq and xfqroughly agree and become larger with increasing armature

current Ia. fd becomes smaller as the armature current

becomes larger, and in contrast, fq becomes larger.

Figure 6(b) shows the case of a leading power factor

of 0.90 p.u. (Case 2); here xdq and xfq roughly agree and

Table 3. Calculated and measured transient reactances

30% voltage

Armature voltage (p.u.) 0.30

Armature current (p.u.) 0.30

Power factor (p.u.) 1.00

xdg measured value (p.u.) 0.338

xdg calculated value (p.u.) 0.356

Measured value/calculated value (%) 94.94

Table 4. Calculation of reactances under constant power factor

Item Case 1 Case 2 Case 3

Active power (p.u.) 0�0.85 0�0.90 0�1.00

Reactive power (p.u.) 0�0.53 (lag) 0�0.44 (lead) 0

Armature voltage (p.u.) 1.00 1.00 1.00

Armature current (p.u.) 0�1.00 0�1.00 0�1.00

Power factor (p.u.) 0.85 (lag) 0.90 (lead) 1.00

Fig. 5. Variation of synchronous reactances and leakage

reactances under constant power factors.

79

become larger up to an armature current of Ia = 0. 4 p. u. as

Ia increases. However, they decrease when Ia exceeds 0.4

p.u. fd becomes smaller when the armature current be-

comes larger, but in contrast, fq becomes larger; however,

fd becomes smaller than fq when the armature current Iaexceeds 0.5 p.u.

Figure 6(c) shows the case of a power factor of 1.00

(Case 3). In the range where the armature current Ia is

smaller than 0.6 p.u., xdq and xfq become larger with increas-

ing Ia, but slowly become smaller when it exceeds this

value. Moreover, fd becomes smaller as Ia becomes larger,

and, in contrast, fq becomes larger. Besides, fd becomes

smaller than fq when the armature current reaches about 0.7

p.u.

In Fig. 6, it is seen that xdq and xfq have their maximum

values when fd and fq are about equal, and the maximum

value becomes about the same as the armature leakage

reactance xl. xdq and xfq express the magnetic coupling of

the d axis and the field with the q axis and will not occur

when the permeance is at a maximum on the d axis and a

minimum on the q axis or when the permeance is at a

minimum on the d axis and a maximum on the q axis. On

the other hand, in the case of no saliency, the flux level is

the same, and if there are positions intermediate between

the d and q axes where the permeance is maximum or

minimum, the magnetic coupling of the d axis or the field

with the q axis will reach a maximum. Here, it can be

understood that since the saliency of the turbine generator

is relatively small and the calculations of Fig. 6 are based

on the condition of constant terminal voltage, the flux level

inside the generator differs only by virtue of the amount of

voltage drop due to the armature leakage reactance, and the

vector sums of fd and fq in all calculation cases are about

the same; xdq and xfq will become maximum when fd and

fq are about equal.

But in Fig. 6, we see that xdq and xfq have their

maximum value in the regions where fd is a little larger than

fq. Even in a cylindrical rotor such as a turbine generator,

the iron ratio in the magnetic circuit is high on the d axis

because there is a magnetic pole portion, while the iron ratio

is low on the q axis. Namely, the more prominent effect of

the magnetic saturation in the rotor is on the d-axis side;

and even considering the magnetic circuit as a whole, the

d-axis side is more easily affected by magnetic saturation

than the q-axis side. Therefore, if the flux level is about the

same, xdq and xfq are likely to be larger in the region where

fd is somewhat larger than fq.

3.3 Variations of reactance at constant output

Table 5 calculates the reactances at constant output.

The calculation case shown is one in which the voltage is

constant at 1.00 p.u. and the power factor is changed from

a leading power factor of 0.90 p.u. to a lagging power factor

of 0.85 p.u. such that the output becomes 0.85 p.u.

Figure 7 shows the variations of the reactances and

linkage fluxes of the armature winding when the reactive

Fig. 6. Variations of mutual reactances and linkage

fluxes under constant power factors.

80

power is changed. This figure is the case shown in Table 5.

Figure 7(a) plots the synchronous reactances xd, xq, the

equivalent synchronous reactances xdeq, xqeq, and the leak-

age reactance xl. The trend of the variations of the reac-

tances due to the reactive power Wx is that xd becomes

smaller, but xq and xl become larger when the armature

current increasingly lags the terminal voltage. Moreover,

xdeq and xqeq do not change at a leading power factor when

the flux level is low; however, they become smaller with an

increase of the reactive power when a lagging power factor

occurs. When xd, xq and xdeq, xqeq are compared, both

xdeq, xqeq are smaller than xd, xq; in particular, for xq and

xqeq, the trends of variation accompanying the increase of

the reactive power are different.

Figure 7(b) shows the mutual reactances between the

d and q axes and and between the field and q axis, as well

as the armature linkage fluxes fd and fq. The d-axis arma-

ture linkage flux fd becomes larger as the power factor

becomes increasingly lagging, while in contrast, the q-axis

armature linkage flux fq becomes smaller. This is because

the demagnetizing action becomes larger as the power

factor lags to an increasing extent and the ratio of the field

magnetomotive force to the total magnetomotive force be-

comes larger, so that the effect of the armature reaction on

the fluxes inside the generator becomes smaller. Thus, in

the range of lagging power factors, magnetic saturation

occurs only on the d-axis component, and xd becomes

smaller but xq becomes larger when the reactive power

becomes larger. On the other hand, the mutual reactances

xdq and xfq between the d and q axes and between the field

and q axis agree roughly, and reach their maximum values

when the reactive power Wx is about 0.2 p.u. with a lagging

power factor. As shown in Fig. 6, this result also implies

that xdq and xfq are at about their maximum values when

fd @ fq in the case of constant armature voltage.

4. Conclusions

Using a 592-MVA turbine generator of standard de-

sign, we have analyzed the variations of the synchronous

reactance xd, xq, and so forth under load conditions, taking

into account the magnetic coupling typical of saturation;

and we have defined the equivalent reactances xdeq, xqeq,

taking into account the magnetic coupling typical of satu-

ration and have summarized the results of analyses of the

variations due to the loads. The results obtained are as

follows.

(1) The equivalent reactances xdeq, xqeq considering

magnetic coupling typical of saturation are smaller than the

reactances xd, xq based on the original definition in all

regions.

(2) In operation under constant voltage, the reac-

tances xdq, xfq expressing the coupling typical of saturation

become maximum when the d-axis flux and the q-axis flux

become about equal; these reactances are then also about

equal to the leakage reactance xl.

Table 5. Calculation of reactances at constant active

power

Active power (p.u.) 0.85

Reactive power (p.u.) 0.41 (leading)�0.53 (lagging)

Armature voltage (p.u.) 1.00

Power factor (p.u.) 0.90 (leading)�0.85 (lagging)

Fig. 7. Variation of reactances and linkage fluxes under

constant active power.

81

(3) xd becomes smaller but xq becomes larger when

the voltage and output are constant and the power factor is

changed from leading side to lagging side.

(4) Regarding (3), the trends of the variation of xdand xdeq are the same; however, the trends of xq and xqeq are

completely different in that xq becomes larger but xqeqbecomes smaller when the power factor is changed from

leading side to lagging side.

Saturation of reactances has been recognized quali-

tatively for a long time and attempts at modeling have also

been made; however, there are few examples which show

quantitatively the variations of the actual reactances in

large-capacity generators. The results presented in this pa-

per show quantitatively the variations of the reactances with

allowance for the magnetic coupling typical of saturation.

They may be useful numerical data for improvement of

characteristics under load conditions.

Acknowledgments

The authors thank Mr. Motoya Ito, Senior Researcher

at the Hitachi Research Laboratory of Hitachi Ltd., for his

guidance in this research.

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Japan, Vol. 104-B, No. 1 (1984).

3. Tamura and Takeda. A consideration on expression

of saturation in synchronous machine model for

power system simulation. Trans. I.E.E. Japan, Vol.

113-B, No. 2, pp. 1413�1421 (1993).

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Trans. Power Appar. Syst., Vol. PAS-100, No. 1

(1981).

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modeling of large turbine generators: Calculations

versus load test data. IEEE Trans. Power Appar. Syst.,

Vol. PAS-100, No. 8 (1981).

6. Ide et al. Synchronous machine field current calcula-

tions using dq-axes equations considering magnetic

saturation. Trans. I.E.E. Japan, Vol. 116-D, No. 3, pp.

328�336 (1996).

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Tech. Rep. (Part II) I.E.E. Japan, No. 405 (1990).

APPENDIX

The equations for determining the d�q axis armature

linkage fluxes from the magnetic field analysis are given

below. The following fluxes are used with the transforma-

tion into per-unit system in this paper.

The respective magnetic vector potentials of the up-

per and lower coils of the armature obtained for every coil

section and the fundamental components of the distribu-

tions for the respective circumferential coordinates q are

taken as aSU(q) [Wb/m] and aSL(q) [Wb/m]. Then aSU(q)

and aSL(q) are given by

where q is the circumferential coordinate value with d axis

as reference (electrical angle).

Therefore, the d- and q-axis armature linkage fluxes

Fd [Wb] and Fd [Wb] are respectively

where Nspp is the number of slots per pole per phase of

stator, fwd is the distribution factor of armature winding,

fws is the short-pitch factor of armature winding, P is the

number of poles, nY is the number of parallel circuits of

armature winding, and Lc is the axial length of stator core

(m).

However, it is assumed that Eqs. (A3) and (A4) are

for double-layer windings and that there is one parallel

circuit in the winding section per layer.

(A1)

(A2)

(A3)

(A4)

82

AUTHORS (from left to right)

Shin�ichi Wakui (member) completed the master�s course in electrical engineering at the Mining Research Division of

the Graduate School of Akita University in 1992. He then joined Hitachi Ltd., and is now at the Hitachi Research Laboratory.

He is engaged in research on rotating electric machines. He received a Paper Award of the IEE Japan in 1996.

Kazumasa Ide (member) completed the first half of the doctoral course in electrical and communication engineering at

the Engineering Research Division of the Graduate School of Tohoku University in 1988. He then joined Hitachi Ltd., and is

now in the Electric Power and Machine Development Group. He is engaged in research on rotating electric machines. He has

a D.Eng. degree. He is a member of the Japan Society of Applied Magnetism and IEEE. He received a Paper Award of the IEE

Japan in 1993.

Miyoshi Takahashi (member) graduated from the Department of Electrical Engineering of the Faculty of Mining of Akita

University in 1972. He then joined Hitachi Ltd., and is now working in the Electric Power and Machine Development Group.

He is engaged in research on rotating electric machines. He is currently a Senior Researcher. He has a D.Eng. degree. He is a

member of the Japan Society of Applied Magnetism, Cryogenic Association of Japan, and IEEE. He received a Paper Award of

the IEE Japan in 1989 and 1996.

Gustavo Montoya (nonmember) completed the master�s course in electrical engineering in the Engineering Research

Division of the Graduate School of Muroran Institute of Technology in 1991. He then joined Hitachi Ltd., and is now at Hitachi

Works. He is engaged in the design and development of turbine generators.

Futoshi Hiyama (nonmember) joined Hitachi Ltd. in 1970 and is now at Hitachi Works. He is engaged in the design and

development of turbine generators. He is currently a Senior Engineer.

Yasuomi Yagi (member) completed the master�s course at the University of Tokyo in 1979. He then joined Hitachi Ltd.,

and is now at Hitachi Works. He is engaged in the design and development of superconducting magnets, nuclear fusion

equipment, turbine generators, and superconducting generators. In 1985, he completed the master�s course at the University of

Michigan. He is currently a Senior Engineer. He received a Paper Award of the IEE Japan in 1996.

83