Comparative Analysis of Turbo Parameters at Variable Speed in Off Line Mode

Comparative Analysis of Turbo Parameters at Variable Speed in Off Line Mode

M. Olubiwe

Dept of Electrical and Electronic Engineering, Federal University of Technology

Owerri, Nigeria

Abstract – The design of a robust coupling for study, for dynamic performance of a synchronous generator requires an accurate dynamic model of the driving and the driven part of the entire design. In this paper, a new procedure for identifying the parameters under varying speed condition of a synchronous generator has been developed. It will be shown that the parameters of this model can be easily identified from offline mode of the generator. The validity of the theoretical model has been verified by comparing results from simulated oscillograms with analytical method from well known equations. The design was simulated with rubber, steel and rigid coupling respectively. The result of the variable speed simulation and that of constant speed were also compared. The results showed that with constant speed simulation at the time of sudden short circuit (ssc) the subtransient and the transient decayed rapidly to steady state condition, while under the influence of variable speed, it took longer time before the subtransient and transient short circuit current to come to steady state, thereby causing heavy oscilation in the machine. This paper has shown that with good coupling design, parameters of a turbo generator can be captured at a variable speed condition and compared

Key Words: Coupling, Dynamic, Parameters, Synchronous generator, Variable-speed

I INTRODUCTION

Knowledge of the operational parameters of synchronous machines is vital for reliable stability studies and post

during generator commissioning, leading to substantial differences between the actual and simulated dynamic behavior. AS power systems become more interconnected and complicated, analysis of dynamic performance of such systems becomes more important. Synchronous generators play a very important role in the stability of power systems. A valid model for synchronous generators is essential for a valid analysis of stability and dynamic performance. Almost three quarters of a century, after the first publications in modelling synchronous generators [3], [4], the subject is still a challenging and attractive research topic.

The first and traditional methods of modelling of synchronous generators are well specified in IEEE standards [5]. These methods assume a known structure for the synchronous machine, using well-established theories like Park transformation. The drive system consist of two DC motors each with per unit rotor inertia JDC and the driven synchronous generator with rotary inertia JGEN. Both motors drive a gear train with ratio K, which in turn drives the generator. The generator is coupled to the gear train with an elastic coupling of torsion spring coefficient C in between them.

Before the analysis of variable speed, it is important to recall the basic d-q-0 equations to aid in the simulations.

a

a

a a

a a

m

m

mortem analyses. Having accurate models of dynamic

dia ( ) 1 V

• r i

( ) K

( )

(1)

elements (including synchronous machines and other power plant controllers) is very important for optimal

decision making, planning, and efficient operation of the

d la

power grid. Parameters of generators may differ from those

dm ( ) Kia ( ) TSH ( )

(2)

in the utilitys database due to aging processes, magnetic saturation, or changes of temperature during machine operation [1], [2]. As a matter of fact, many utilities around the world still use machine model parameters calculated

d

Where

J EFF DC

() and ()are respectively, the armature current and rotor speed.

is per unit time.

and are the machine reactance and inductance in p.u.

is the armature flux.

K is the constant determined by the design of the armature winding.

JEFF-DC is the per unit moment of inertia of the entire moving mass comprising the DC motor rotor and the rotor of the synchronous generator as seen from the motor side.

Hence, the flux linkages in equation (8) becomes:

X md I fr

Vds rs Ids

dds

d r qs

d

(3)

ds dr

r

X fr I fr

(11)

V r I

(4)

fr

qs s qs d qs r ds

d

r

qs qr 0

Vfr rfr I fr d fr

(5)

d

d

d

d

With equation (11), equation (3) now become

V

V

(6)

ds

d ds

X md

r

d I fr 0

(since Ifr

is a dc current )

Where the flux linkages are

(7)

(12)

However, under steady-state no-load condition the terminal voltage Vs is 1.0p.u. and is given by

qs X qr O

X mq O

O 1qs

Vs

V 2 V 2 1 V = 1.0 p.u. (13)

ds qs qs

ds 1 O X ds

O X md

X md 1ds

X O

Xqr O

O 1

(8)

Applying these initial conditions to equation (4) under steady

qr mq

qr

state open-circuit conditions gives.

r O X

O X X

1

dr

fr

Where

O

md

X md

dr

O X md

md

X fr

dr

1 fr

Vqs

= rds

or Ifr =

Vqs X md

1

X md

(14)

Similarly, equation (5) becomes

Xqs =Xmq + Xls

Xds

=Xmd

+ Xls

Vfr

rfr I fr

Vqsrfr

X md

rfr X md

(15)

Xqr =Xmq + Xlqr Xdr = Xmq + Xldr Xfr = Xmd + Xlfr

The speed and torque are related by

d TEX TL

The above equations were recalled to simulate for constant speed that could be compared with variable speed condition where different damping components were applied.

1. DERIVING MODELS FOR VARIABLE SPEED SIMULATION

   It is easy to see that the total inertia of the driving motors is only 2JDC since both are rigidly coupled together. If the speed

   d r

   The torque is given by

   JGEN

   (9)

   of the DC motor combination is DC and that of the driven synchronous generator is gen, then the ratio of the two speeds is proportional to the gear ratio:

   3 P

   K GEN

   (16)

   TEX

   2 2 ds Iqs qs Ids

   (10)

   DC

   All parameters have their meanings in p.u. of the rated base conditions.These equations are complete, there is need to deduce the state variables which can easily be obtained by noting that at no-load condition, the rotor is operating at rated synchronous speed, the rotor and stator currents Idr, Iqr, Iqs and Ids are nominally zero.

   If frictional losses in the transmission are neglected, and it is assumed that the inertia of the gear system is negligible, the kinetic energy due to the equivalent inertia seen by the DC motors JEFF_DC must be equal to the total kinetic energy of the various moving parts. This implies

   1

   2 J EFF

   _ DC DC

   1 (2J 2

   )

   )

   2 DC DC

   • 1 J

     2

     GEN

     2

     GEN

     Where the resonant frequency o, now appearing in equation (25), is given by

     2

     2

     (17)

     Combinin equation (14) and (15), the effective inertia of the

     o

     C (2J DC K J

     2

     2

     GEN )

     (26)

     entire drive system referred to the DC motor side is

     (2J DC K

     JGEN )

     JEFF_DC = 2JDC + K2 JGEN (18)

     In a similar manner, the effective inertia referred to the generator side would be

     Finally, a damping term (in the coupling) can now be included and then the resulting dynamic equation of the effective shaft angle becomes

     d 2 ( )

     d ( ) 2

     T ( )

     JEFF_GEN = 2K2 JDC + JGEN (19)

     d 2

   • h d

   w0 ( ) EX

   2J

   (27)

   The inertia of the two rotating masses (DC motor set and synchronous machine) and the torsional spring constant of the connecting shaft (and the flexible coupling) have a natural frequency of vibration o at which the two masses tend to oscillate in opposition to each other. The stresses in

   DC

   Where h 2Dc

   o

   the shaft are twisting stresses that load the shaft in shear. To evaluate the Oscillating shaft torque TSH at the flexible coupling, the torque equation for both rotating masses with

   D 0.01

   (2 )

   for steel coupling and

   their oscillation angles

   DC

   and

   GEN

   are related by two

   D = 0.1/(2) for rubber coupling. [7]. The solution of

   second order differential equations. [6].

   equation (27) will ultimately depend on the nature of the disturbance (exciting torque).

   2 d 2 ( )

   2

   2

   K JGEN GEN TSH ( ) 0

   d

   (20)

   A Simulation of the System

   It is known that the frequency of the exciting torque should be far from the resonant frequency 0 in order to avoid

   d 2 ( )

   torsional resonance of the drive system, which is capable of

   2

   2

   2J DC DC TSH ( ) TEX ( )

   d

   Where the shaft torque is given by

   (21)

   creating very high shaft torque that could break the couplings. For this reason, the study will consider the value

   of 0 = . However, the response of shaft torque due to

   TSH () =C (DC() GEN() (22)

   Tex appearing in equation (21) is the exciting torque, which, for the purpose of this study, is the synchronous machine short-circuit torque given in equation (10), and C is the spring constant.

   Equations (20) and (21) can be combined and written as

   variations in 0 and the spring constant c will be studied. An SSC performance can be observed by setting Vqs = Vds = 0 after the output voltage has settled to 1.0 p.u. The short circuit is performed at a point when the voltage waveform is at the maximum, the so-called best case. [8]. This ensures that the shortcircuit currents do not contain DC components, a situation that eases the procedure of transient parameter extraction from the short-circuit current waveforms. The simulations are performed using the commercial software

   d 2 ( ) d 2 ( ) 1 1

   MATLAB Simulink® maths work Inc 2006. Two broad

   DC GEN

   TSH ( )

   conditions, viz, constant speed and variable speed, are

   2J

   2J

   d 2

   d 2

   DC

   K 2 J

   GEN

   considered. The variable-speed condition clearly emphasizes

   T ( )

   the inaccuracies arising from not keeping the speed constant

   + EX

   2J DC

   (23)

   during the period in which the SSC current and torque waveforms are captured.

   If the effective shaft angle is represented as

   () = DC () – GEN () (24)

   Then equation (3.44) can be written as

   B Variable Speed Simulation

   The practicable case in which the generator is driven by the DC motors is now considered. Prior to commencement of the numerical solution in this case, the mechanical equation of the DC machine

   d 2 ( ) 2

   d 2 0

   ( )

   TEX ( )

   2J

   (25)

   DC

   Where,

   P = Generator pole pairs

   Tem= The driving turbine torque J = Equivalent moment of inertia

   = Rotor lag angle

   Field Current T SSC

   7

   6

   5

   Ifr

   Ifr

   4

   3

   2

   1

   has to be modified to :

   (28)

   0

   2.3 2.32 2.34 2.36 2.38 2.4 2.42

   Time in p.u

   Figure 2 Ifr at constant speed when SSC occured

   5

   x 10

   Where TBGEN and TBDC are, respectively, the rated base torques of the generator and the combined DC motor. There is no electrical connection between the DC motor and the generator, so Eq. (1) does not need any modification.

   All of the synchronous generator, DC motor, and drive system equations are solved together. The solution of Eq.

   6

   SSC current Ias ' P.U.W

   SSC current Ias ' P.U.W

   4

   2

   0

   -2

   -4

   2.335 2.34 2.345 2.35 2.355 2.36 2.365

   (27) with the generator torque TEX ( ) gives the shaft angle

   Time Per Unit Time

   5

   x 10

   ) with which the shaft torque TSH can be computed using Eq. (22). The resulting shaft torque is then used in the

   Figure 3 Effect of short circuit on Ias when the system is rigidly coupled

   Effect of SSC on Ifr when the generator is rigidly coupled

   solution of Eqs. (1) and (28) to obtain the speed of the DC 7

   6

   6

   motor that now drives the generator. The following four conditions are investigated for the variable-speed

   SSC Current Ifr, PU

   SSC Current Ifr, PU

   simulations. 5

2. WHEN THE SYSTEM IS RIGIDLY COUPLED

   A The generator is rigidly coupled to the DC machine through the gear only.

   In this case, there is no spring coupling. The shaft torque

   4

   3

   2

   1

   0

   2.32 2.34 2.36 2.38 2.4 2.42 2.44

   then equals the short circuit torque (TSH in Eq. (28) is

   Time in per unit

   5

   x 10

   replaced by TEX for this case). The mechanical equation of shaft coupling dynamics in Eq. (27) is not included in the model equations to be solved in this case. The various oscilogram or the envelop under the rigid coupling condition is shown below

   Armature current at instance of SSC

   Figure 4 Effect of short circuit on Ifr when the generator is rigidly coupled

   SSC Current on Vas' P

   SSC Current on Vas' P

   1

   0.5

   0

   6

   -0.5

   4 -1

   0 1 2 3 4 5

   2 Time Per Unit P.U.

   5

   x 10

   Ias

   Ias

   0

   0

   Figure 5 Effect of short circuit on generators armature voltage when the system is rigidly coupled

   -2

3. THE GENERATOR IS COUPLED TO THE DC

   -4

   MACHINE WHEN THE DAMPING IS ZERO(0.0)

   -6

   2.32 2.34 2.36 2.38 2.4 2.42 2.44

   Time in pu

   Figure 1 Ias at constant speed at SSC

   5

   x 10

   The gnerator is coupled to the DC machine through a gear, but no damping is included. All the equation sets are solved with D = 0. This modifies the nature of Eq. (27). The various envelops or oscilograms are shown below.

   Effect of SSC on Ias when the damping is 0.0

   6

   4

   SSC current Ias, PU

   SSC current Ias, PU

   2

   0

   -2

   Effect of SSC on Ifr when the damper is 0.1

   7

   6

   SSC Current, Ifr PU

   SSC Current, Ifr PU

   5

   4

   3

   2

   1

   -4

   -6

   2.33 2.34 2.35 2.36 2.37 2.38 2.39

   Time in per unit PU

   5

   x 10

   0

   2.33 2.335 2.34 2.345 2.35 2.355 2.36

   Time in per unit PU

   5

   x 10

   Figure 10 Ifr current at short circuit when the damper is

   Figure 6 Effect of short circuit on Ias when the damping is 0.0

   7

   6

   SSC Current Ifr, P.U.

   SSC Current Ifr, P.U.

   5

   4

   3

   2

   1

   0

   2.325 2.33 2.335 2.34 2.345 2.35 2.355 2.36 2.365 2.37

   1.08

   1.06

   1.04

   Speeed w

   Speeed w

   1.02

   1

   0.98

   0.96

   0.94

   0.92

   Speed departure at 0.1

   Per Uniit Time

   5

   x 10

   0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

   Figure 7 Effect of sudden short circuit on Ifr when the damping is 0.0

   1.02

   Time in p.u

   Figure 11 Speed characteristics when the damping is

   5

   x 10

   1.015

   Shaft Speed P.U.

   Shaft Speed P.U.

   1.01

   1.005

   1

   0.995

   VI THE GENERATOR IS COUPLED TO THE DC MACHINE WHEN THE DAMPING IS 0.01

   The rubber coupling is at , and all equations are

   0.99

   0.985

   0.98

   1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6

   also solved.when the generator is coupled to the DC motors with the damping set at the following

   Per Unit Time

   5

   x 10

   characteristics were obtained sudden short circuit.Figure 12

   Figure 8 Speed departure from synchronism at short circuit when the damper is 0.0

4. THE GENERATOR IS COUPLED TO THE DC MACHINE WHEN THE DAMPING IS 0.1

The steel coupling is set that and all equations are solved. The corresponding oscillograms after simulation are shown below

Effect of short circuit on Ias when the damper is at

Effect of SSC onIfr when the damping is 0.01

7

6

SSC Current Ifr, PU

SSC Current Ifr, PU

5

4

3

2

SSC Current Ias, P.U

SSC Current Ias, P.U

1

0

0

5 Time in per unit 5

2.34 2.35 2.36 2.37 2.38 2.39 2.4

x 10

Figure 13 Ifr Characteristic at short circuit when the damper is

0

-5

2.34 2.35 2.36 2.37

Per Unit Time

5

x 10

1.15

1.1

1.05

Speed w

Speed w

1

Speed departure when coupling is 0.01

Figure 9 Effect of short circuit on Ias when the damping is

0.95

0.9

0.85

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Time in p.u

5

x 10

Figure 14 Speed wm deviation at short circuit when the damper is

1.02

1.01

1

speed w

speed w

0.99

0.98

0.97

0.96

0.95

Speed Departure at Rigid Coupling

1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4

Time in p.u x 105

VIII EFFECT OF SUDDEN SHORT CIRCUIT ON THE SPEED OF THE SYSTEM

Figures 8, 11, 14 and 15 show the extent of speed departures from synchronous conditions following SSC on the terminals of the generator. The ideal case of constant speed is not represented in table 2 since there is no speed departure. The speed departure for the steel coupling appears to be the average of that of the undamped case a short while after the SSC, even though the corresponding current curves are

Figure 15 Speed departure at SSC when the machine is rigidly coupled.

VII ARMATURE CURRENT OF THE GENERATOR (IAS)

The stator current waveforms for the constant-speed operation of figure 1, shows that at the point of the short circuit, the current rose spontaneously and decayed rapidly. This point is the sub-transient region which degenerated to the transient period of the oscillogram. The various regions in which the transients occurred are recorded in the section that discussed the extraction of the parameters. The rubber coupling of Figure 9 and figure 12 look alike. However, some differences exist between these and can only be appreciated when the transient parameters have been extracted from each curve and compared. At the sub transient region, the current decayed fastly and at the transient region, the current tends to rise before coming to the steady state condition. The parameter extraction has shown the major difference between the two different envelop. Figures 3 and 6 indicate that these oscillated very well before settling to final values. Under this condition, it is difficult to ascertain the transient region from the oscillograph. However, their steady state and subtransient regions were captured.

A Field Currents (Ifr)

Figures 2, 4, 7, 10 and 13 depicts the rotor field current when sudden short circuit occurred at the terminals of the generator driven by the two DC motors. The five oscillograms or envelop looks alike but the envelop captured during the de excitation of the machine comparatively analyze their differences. Between the subtransient and the transient region, the oscillogram shows a sharp rise in short circuit current before dropping to steady state condition.

Careful observation shows that the rigid coupling and when the damper is 0.01 decayed faster to steady state than the other envelops.

Table 1 shows the field current at different time intervals. The table also reveal the closeness of the field current at various condition of the machine.

Table 1 Field current during short circuit captured at various conditions

remarkably similar. A general observation is that for even very low speed departure, significant shaft torques result; the farther the speed deviates from synchronous, the longer it takes the system to attain steady-state mode, and the more the current waveforms oscillate. In all the cases, the short circuit is arranged to occur when the machine voltage has built up to 1.0 p.u. in steady state and at a time when the voltage wave is a maximum. The table below shows the various point of departure from synchronism.

Table 2 Speed departure characteristics from synchronism at the point of short circuit.

S/N	STATE OF THE SYSTEM	SPEED W (pu)
1	Constant speed	–
2	D = 0.0	0.9929
3	D =	0.9963
4	D =	0.9866
6	Rigid Coupling	0.9779

IX EXTRACTED PARAMETERS AT CONSTANT AND VARIABLE SPEED CONDITIONS

The conditions at which the speed deviates from constancy or out of synchronism during short circuit of the generator in which different coupling methods were adopted were studied and parameters of the generator were extracted at the conditions of sub transients, transients and steady state as shown in the table below.

S/N	Condition of the Generator	Ifr at Normal (Amps)
1	Constant Speed	5.885
2	D = 0.0	5.960
3	D = 0.1	5.828
4	D = 0.01	5.899
5	Rigid Coupling	5.910

Table 3 Summary of Extracted Parameters

Extracted Values
Parameters	Consta nt Speed	Rigid coupli ng	D = 0	D =	D =
	2.1978	2.181	2.197 8	2.197	2.1973
	0.217	–	–	0.215 6	–
	0.1923	0.1854	1.910 5	0.192 6	0.1917
	0.021 sec	–	–	0.109 sec	–
	0.001	0.018s ec	0.018 sec	0.019 sec	0.018 sec
Ias (Final)	0.4548	0.4585	0.455	0.455 1	0.4551
(Transient)	4.600	–	–	4.638	–
(SusubTra nsient)	5.200	5.393	5.234	5.192	5.216

A Analysis of the Waveforms.

The stator current waveforms for the constant-speed operation of figure 1, shows that at the point of the short circuit, the current rose spontaneously and decayed rapidly. This point is the sub-transient region which degenerated to the transient period of the oscillogram. The various regions in which the transients occurred are recorded in the section that discussed the extraction of the parameters. The rubber coupling of Figure 9 and figure 12 look alike. However, some differences exist between these and is appreciated at table 3 where transient parameters extraction from each curve have treated and compared. At the sub transient region, the current decayed fastly and at the transient region, the current tends to rise before coming to the steady state condition. The parameter extraction has shown the major difference between the two different envelop. Figures 3 and 6 indicate that these oscillated very well before settling to final values. Under this condition, it is difficult to ascertain the transient region from the oscillograph. However, their steady state and subtransient regions were captured. Comparing table 3, it is noticed that when the speed is constant, at subtransient level, the current rise spontaneously and decay spontaneously to transient condition and finally to steady state condition. Unlike the constant speed conditions, the variable speed, where different coupling methods were adopted, this impacted deeply on the speed of the rotor. This is attested from the oscillogram of the captured envelops at the time of sudden short circuit on the machine. This resulted in the longer time it took the machine to settle to steady state position otherwise known as the recovery condition of the machine.

X CONLUSION

In spite of synchronous generators long history of operation, the research work presented hitherto have not put forward fully sufficient comparative studies with regard to sudden short circuits of the machine. This can be due to the fact that short circuit faults occurring during operation cannot offer sufficient information with a view to fault correct assessment, simply because of the various circumstances in which these faults do practically occur. Using a number of basic building blocks easily found in the Mathlab Simulink® library, it is possible to simulate the fault analysis and study its dynamic response using the graphical displays through the oscillogram from the scope block. The graphical results from the time the fault occurred was then observed and the parameters were extracted from the graph analysis. The simulink models parameters extraction of the synchronous machine are then verified as the results from the oscillograms studied can be used to extract accurate parameters for the synchronous machine. The case in which a rubber coupling is used gives the best results here. Under that condition, the shear stresses on the shaft are minimal as the rubber provides sufficient flexibility. The speed is nearly constant, even after short circuit.

XI REFERENCES

1. E. Kyriakides, G. T. Heydt, and V. Vittal, On-line estimation of synchronous generator parameters using a damper current observer and a graphic user interface, IEEE Trans. Energy Convers., vol. 19, no. 3, pp. 499507, Sep. 2004.

2. E. Kyriakides, G. T. Heydt, and V. Vittal, On-line parameter estimation of round rotor synchronous generators including magnetic saturation,IEEE Trans.Energy Convers.,vol. 20, no.3,pp.529537, Sep. 2005

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4. S. H. Wright, Determination of synchronous machine constants by test, AIEE Trans., vol. 50, pp. 13311351, Dec. 1931

5. Test Procedures for Synchronous Machines, Part I cceptance and Performance Testing, Part IITest Procedures and Parameter Determination for Dynamic Analysis.

6. E. F. Merrill, Dynamics of ac electrical machines, IEEE Trans. Industry Appl., Vol. 30, 350 No. 2, pp. 277285, 1994.

7. G. K. Dubey, Fundamentals of Electrical Drives, New Delhi: Narosa Publishing House, PP506, 2001.

8. B. Adkins, The General Theory of Electrical Machines,

   London: Chapman and Hall, P36, 1964

9. E.S Obe, Dynamics of a Turbo genrator Driven by DC Motors during Off line Three-phase Sudden Short Circuit. Electric Power Components and Systems, 39:1828-1844, Taylor & Francis Group, Informa Ltd LLC Office, Mortimer House, 37-41 Mortimer street, London WIT 3JH, UK, 2011.