[IEEE 2013 IV International Conference on Power Engineering, Energy and Electrical Drives (POWERENG)...

The Effect of Implementing of Dynamic Cross Saturation on Modeling

of Synchronous Generator

Marija Miroševi Department of Electrical Engineering

and Computing, University of Dubrovnik,

Dubrovnik, Croatia [email protected]

Zlatko Maljkovi Faculty of Electrical Engineering and

Computing, University of Zagreb, Zagreb, Croatia,

[email protected]

Mato Miškovi HEP Hydro-power Dubrovnik,

Dubrovnik, Croatia [email protected]

Abstract—The aim of this paper is to explore the effect of implementing of dynamic cross saturation in modeling of synchronous generator. The mathematical model of a saturated salient pole synchronous generator is developed and two cases are analyzed. In the first case the cross-magnetizing phenomenon in the saturated synchronous machine (the magnetic coupling between its direct and quadrature axis) is included, while in the second case the cross-saturation is neglected.

Keywords-synchronous generato: saturation; cross saturation

I. INTRODUCTION Modeling of synchronous generators is traditionally based

on the d-q reference frame and constant coefficients are used to simplify the calculations. To improve the modeling and simulation of synchronous generators the model of a saturated synchronous generator is proposed. However, the electromagnetic nonlinear effects should be considered.

At first, in mathematical model of salient pole synchronous machines the saturation effect has been taken into consideration along the poles, in the d-axis only. Then, the saturation effect is introduced in both d-axis and perpendicular to the poles, along q-axis. The air gap of the salient pole synchronous machines is non-uniform, however the reluctance is changed. As the magnetic fluxes along and perpendicular to the poles are changing, the degree of saturation along these two axes varying as well. Further, it has been recognized that magnetic coupling between d- and q- axis play an important role in analysis of saturated synchronous machines [1] - [4]. This paper presents the effect of implementing of dynamic cross saturation on modeling of salient pole synchronous generator. In this paper stand alone synchronous generator is considered, thus system under study consist of: a prime mover and speed control, a synchronous generator and voltage controller and passive load fed directly from the generator-unite terminals. The autonomous operation of the synchronous generator is characterized by a change in steady state which causes a change in voltage and frequency, which in turn affects the quality of electric power systems. Thus, to study the dynamics and transient performance of such isolated electrical system a detailed description of a salient pole synchronous

generator model is required. The aggregate under study is generator-unit with a diesel engine of 46.4 kW and a synchronous generator of 40 kVA. The voltage controller is modeled as PI and is implemented on the model. The prime mover, diesel engine, is modeled by differential equations of diesel engine and speed controller. A more detailed description of the model is given in [5]. It is assumed that the engine torque is directly proportional to the fuel consumption [6], [7].

The investigation was performed by the application of the software package Matlab/Simulink.

II. SYNCHRONOUS MACHINE AND LOAD MODELING

A. Saturated Salient Pole Synchronous Generator Model The salient pole synchronous generator under consideration

is assumed to have three armature windings, one field winding and damper windings. One damper winding is located along direct-axis (D), and one along the quadrature-axis (Q). Mathematical model of machine is derived from relationships between the machine terminal voltages, currents and flux linkages [8]. Two axis theory of electrical machines and Park’s transformation are applied.

The basic of the mathematical model is set of differential voltage equations of the synchronous generator in the standard dq-axis form:

d

d d q

qq q d

du r idt

du r i

dt

ψ ω ψ

ψω ψ

− = ⋅ + − ⋅

− = ⋅ + + ⋅ (1)

where u, i, r, and ψ denote voltage, current, resistance and flux respectively.

The voltage equations (1) are written in source convention system in which synchronous machines are usually represented, while equations of excitation are written in load convention system:

The results are carried out through the researches within scientific projects„New structures for hydro generating unit dynamic stability improvement“and “Revitalization and operating of hydro generator” supported by Ministryof Science, Education and Sports in the Republic of Croatia.

978-1-4673-6392-1/13/$31.00 ©2013 IEEE1477

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1 1

1

dq q

x dE er dt

ψ= + ⋅ , (2)

while equations for damper windings are:

0

0

DD D

QQ Q

dr idt

dr i

dt

ψ

ψ

= ⋅ +

= ⋅ +. (3)

It is assumed that main flux saturation and leakage flux saturation may be treated independently [9]. So, each flux linkage is presented as a sum of appropriate leakage and main flux:

1 1 1

d d dm q q qm

dm

D D D dm Q Q Q qm

L i L i

L iL i L i

ψ ψ ψ ψψ ψψ ψ ψ ψ

= + = +

= += + = +

(4)

where ψdm and ψqm are magnetizing flux components respectively:

( )( ) ( ) 2

dm dm dm dm

qmqm qm qm qm

dm

L i L i i

LL i L i L i m i

L

μ μ

μ μ μ μ

ψ

ψ

= =

= = = (5)

where Ldm and Lqm are magnetizing inductance in the direct and quadrature axes, while magnetizing current d-q axis components, idm and iqm are:

1dm d D

qm q Q

i i i ii i i

= + += +

(6)

During the process of model derivation, according [10] and [11] time derivative of Lμ is required. However, the saturation dependent inductances are:

2 2cos sinddL L Lμμ μ= ⋅ + ⋅ (7)

( )2 2 2sin cosqqL m L Lμμ μ= ⋅ ⋅ + ⋅ (8)

( ) cos sindq qdL L m L Lμ μ μ= = ⋅ − ⋅ ⋅

(9)

where the magnetizing inductance Lμ

and

the dynamic

inductance L respectively are:

Li

μμ

μ

ψ=

dL

diμ

μ

ψ=

cosμ=idm/iμ, sinμ=iqm/iμ and m is the saliency factor [12].

In the case when 0dq qdL L= = , dynamic inductance L is equal to Lμ so ddL Lμ= and

2qqL m Lμ= dynamics cross-

saturation is neglected. In that way main flux saturation is taken into account in an approximate way.

B. Mathematical Model of RL- load Three phase symmetric load is modeled as system of

voltage equations in dq axes form, where the resistance Rl and reactance xl refers to the passive consumer:

dldl l dl l l ql

qlql l ql l l dl

diu R i x x i

dtdi

u R i x x idt

= ⋅ + − ⋅ ⋅

= ⋅ + + ⋅ ⋅ (10)

Load is connected directly to a synchronous generator, what means that they are on the same voltage as the generator terminals: d d q q, u u u u− = − = and the current relationship between supplying and receiving elements can be expressed as: d d q q, i i i i= = .

Besides that, the equation of motion of mechanical parts of aggregate is included in the model:

DM eSGdJ T Tdtω = − (11)

where: TDM represents prime mover torque – mechanical torque of the diesel engine, TeSG is air-gap torque of the synchronous generator and J is moment of inertia of rotating parts of aggregate – the mass of generator rotor and rotating mass of diesel engine (a single rotating mass is considered).

III. STUDY CASES AND COMPUTATION RESULTS The synchronous generator is initially in a steady state

unloaded condition operating at the rated voltage (1 p.u.) and stator currents being zero. The rotation speed of the diesel generator-unit ( ) is equal to the rated speed of 1 p.u. The passive RL- load (Z=1∠arcos0.8) is connected directly to the terminals of the synchronous generator.

Two cases are analyzed. In the first case, the cross-magnetizing phenomenon (the magnetic coupling between its direct and quadrature axes) in mathematical model of saturated salient pole synchronous generator is included. After, results are compared with those at the second case that are obtained from numerical calculation when cross-saturation is neglected. Before impact of the load generator is idle, the voltage at the terminals of synchronous generator in autonomous operation is equal to the nominal, current of the stator windings are equal to zero and excitation current has corresponding nominal value.

In Figures 1 and 2 the stator current during impact RL- load, for the both cases are presented.

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Figure 1. Stator current during impact RL- load

i* - cross-saturation is neglected i- cross-saturation is included

Figure 2. Stator current during impact RL- load

i* - cross-saturation is neglected i- cross-saturation is included the beginning of the transients from Figure 1

The degree of saturation determinates the magnetizing current. As the field winding is in d-axis it means that the magnetizing current iμ, at idle, is equal to the component of the magnetizing current in d-axis idm, as in q-axis current component is equal to zero. At the moment of impact load stator currents begin to flow and generator voltage is changes. The component of the magnetizing current in d-axis idm is now equal to the vector sum of the current in d-axis, while the component the magnetizing current in q-axis iqm is equal to the vector sum of the currents in q-axis. Total magnetizing current iμ, which is determined by the degree of saturation of a synchronous generator is equal to the vector sum of the currents in both axes (d-q)

Main magnetic flux is the result of magnetic coupling between d- and q- axes (dynamic cross saturation). Saturation

of the main magnetic circuit affects at all system variables. When the synchronous generator is loaded the magnetizing current increased and the operating point moves deeper into the saturation region. Higher saturation means less magnetizing as well as dynamic inductance. During the transients, changes of instantaneous value of the magnetizing current results in changes of the saturation dependent reactance. This results in damped oscillations in the response of characteristic values of synchronous generators.

Synchronous generator d-, q- axis stator current are presented in Figures 3 and 4 and results are given for the cross-saturation is neglected as well as cross-saturation is included.

Figure 3. Synchronous generator d-, q- axis current

id*, iq* - cross-saturation is neglected id, iq - cross-saturation is included

Figure 4. Synchronous generator d-, q- axis current id*, iq* - cross-saturation is neglected id, iq - cross-saturation is included the beginning of the transients from Figure 3

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Accordingly, the air-gap torque is determined by the character of currents and fluxes of the machine.

As one can see in Figure 5 and Figure 6, at the beginning of the transient phenomenon, damped oscillations are presented in the response of air-gap torque of synchronous generator.

Figure 5. The air-gap torque (TeSG) of the synchronous generator

TeSG * - cross-saturation is neglected TeSG - cross-saturation is included

Figure 6. The air-gap torque of the synchronous generator(TeSG), the beginning of the transients from Figure 1

In Fig. 7 the time variation of the synchronous generator current (cross-saturation is included) during sudden impact load is presented.

Figure 7. Time variation of the synchronous generator stator

current during sudden impact load

Figure 8. Time variation of the synchronous generator stator voltage during sudden impact load

IV. CONCLUSION The influence of cross-saturation on dynamics of a

synchronous generator in autonomous operation is analyzed. The mathematical model of a saturated salient pole synchronous generator is developed. The cross-magnetizing phenomenon in the saturated synchronous machine (the magnetic coupling between its direct and quadrature axis) is included. The degree of saturation determinates the magnetizing current. During the transients, changes of instantaneous value of the magnetizing current results in changes of the saturation dependent reactance. This results in damped oscillations in the response of characteristic values of synchronous generators. Neglected the cross-saturation main flux saturation is taken into account in an approximate way, so this approach can be used in related studies where simplification is allowed.

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REFERENCES [1] I. Boldea, S. A. Nasar, “A General Equivalent circuit (GEC) of Electric

Machines Including Crosscoupling Saturation and Frequency Effect”, IEEE Transactions On Energy Conversion, Vol. 3, N0 3, September 1998.

[2] A. M. El-Serafi, A. S. Abdallah, M. K. El-Sherbiny, E. H. Badaway, “Experimental Study of the Saturation and Cross-magnetizing Phenomenon in Saturated Synchronous Machines”, IEEE Transactions On Energy Conversion, Vol. EC-3, pp. 815-823, 1988.

[3] A. M. El-Serafi, J. Wu, “Determination of the Parameters Presenting the Cross-Magnetizing Effect in Saturated Synchronous Machines“, IEEE Transactions On Energy Conversion, Vol. 8, No 3, September 1993.

[4] Brown, J. E.,. Kovacs, K.P., Vas, P.: A Method of Including the Effect of Main Flux Path Saturation in the Generalized Equations of A.C. Machines, IEEE Transactions on Power apparatus and Systems, Vol. PAS 102 N01 January 1983.

[5] M. Miroševi , D. Sumina, N. Buli , “Influence of Time Gap between Impact Loads on Torsional Dynamics of Generator Units”, International Review of Electrical Engineering, Vol. 5, No. 3, pp. 1012-1021, 2010.

[6] V. I. Krutov, “Dvigatelj vnutrennego sgorania kak reguliruemij object, Mašinostroenie”, Moskva 1978.

[7] V. I. Tolšin, E. S. Kovalevskij, “Prethodnie procesi u dizel generatorov”, Mašinostroenie, Leningrad 1977.

[8] Vas, P. “Electrical Machines and Drives A Space vector theory Approach”, Oxford Science Publications, 1996.

[9] E. Levi, “Saturation Modelling in D-Q Axis Models of Salient Pole Synchronous Machines”, IEEE Transactions On Energy Conversion, Vol. 14, N0 1, pp 44-50, March 1999.

[10] E. Levi, “State - space d-q axis models of saturated salient pole synchronous machines”, IEE Proc.-Electr. Power Appl., Vol. 145, No 3, pp 206-216, May 1998.

[11] E. Levi, V.A. Levi, “Impact of Dynamic Cross-Saturation on Accuracy of Saturated Synchronous Machine Models“, IEEE Transactions On Energy Conversion, Vol. 15, N0 2, June 2000.

[12] Iglesias, I. Garcia Tabares, L., Tamarit, J, “A D-Q Model for the Self-Commutated Synchronous Machine Considering the Effect of Magnetic Saturation“, IEEE Transactions On Energy Conversion, Vol. 7, N0 4, December 1992.

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