Dynamic Modeling of a Single Shaft Gas Turbine

Copyright © IFAC Control of Power Plants and Power Systems. Munich. Germany. 1992

DYNAMIC MODELING OF A SINGLE SHAFT GAS TURBINE A. Hussaln* and H. Selfi** ·Universiti Kebangsaan. Malaysia ·.University ofTarbiat·Modarres. Tehran. Iran

Abstract AB a basic source of power generation, gas turbines are widely used in power grids. AB it is a part of the power system, it affects overall system behavior during dynamic conditions so that its dynamic model is of prime importance. The present paper endeavor to establish a theoretical way of finding a suitable dynamic model of a gas turbine, especially for power system stability studies.

KEYWORDS: MODELING, GAS TURBINES, TURBINES, DYNAMIC RESPONSE, POWER PLANT MODELING

need to be made to arrive at a simple dynamic model which can yet provide a sufficiently accurate prediction of a gas turbine behavior. With this in mind, the present paper finds a theoretical way of establishing the dynamic model of a single shaft gas turbine.

1=. INTRODUCTION Gas turbines are widely used as sources of power generation especially during peak load conditions. Some features such as ease of installation and maintenance, high reliability l.2 and quick response has made it an attractive means of producing mechanical energy.

The derived model is then tested by simulation studies and its performance checked against the performance of a manufacturer's model and also against the actual measured responses obtained during factory testing. The manufacturer's model was available form RUSTONS LTD .. This model was developed through empirical method of modeling. Starting from a very complex model, by making inferences and observations and based on expenence, the manufacturer has arrived at a reduced and simple transfer function model suitable for power system studies.

On the other hand, as a basic element in an electric power network, a gas turbine affects overall system behavior during dynamic/ transient conditions so that its accurate dynamic model is of definite requirement. Although dynamic models for steam and hydroturbines have received special attention in the literature 3 • 4, no specific reference is made to gas turbines so that the task of finding a suitable model is of prime importance in power system stability studies.

2- SYSTEM DESCRIPTION The single shaft arrangement under consideration is shown in figure 1 which consists of a compressor, a combustion chamber and a turbine. This arrangement is especially suitable for operation at a fixed load and a fixed speed conditions such as peak load power generation

A fully dynamic model in which all dynamic processes are catered for can, however, be extremely complex, yet not so required especially in the time span of power system stability studies. Therefore, some simplifying assumptions

43

schemes.

T 2= temperature of turbine inlet

3- MODEL DERJV ATION

3) Flow Equation The flow equation in nozzles is given by

For the system shown in figure 1, an equivalent model is depicted in figure 2. Nozzle 1 represents the compressor part, nozzle 2 represents the turbine part and the rectangular block represents the combustion chamber unit.

minlet~ A Pinlet

27 .1 {(P

7-1 R

outlet)2 Pinlet

/7_

1 P outlet)7 7

+t}1/2

( P inlet

(6) Considering small variations and assuming the ambient temperature to remain constant during the processes, the tedious job of linearization leads to:

3-1 Basic Relationshipsl

1) Mass Balance Equation The mass balance equation is of the form given by:

(7)

dM . . dt=mrm2

(1) where

Considering small variations and assuming the mass flow rate, m l , to be constant ( i.e., LlriI I = 0), the equation reduces to : (in per unit) -

S

.6.M =

1

-T

KI =1+0.5 x

(-217)

.

(2)

.6.m 2

where bar represents per unit and is the gas turbine constant.

T=

(

p7 )2/7+ (7+117) (Pp7 )7+t/7

p

Mo 1 mo

(8) 2) Energy Equation The energy balance equation applied to gas turbine internal volumes is given by:

tt [

(3)

M h2 ]

= Internal Energy Balance

Q - Ws = where

tt [

4) Gas Equation The well known gas equation states that: pV=MRT

M h 2 ] - mlh l - m 2 h2

Again considering small variations and by using the relationship of enthalpy with temperature, the following per unit equation is obtained:

Q = Heat input rate Ws= Power Output and Losses mIh l Transported Energy in and out, m 2 h2 Respectively

(10) 3-2 Determination of Power Output

Considering small variations and assuming that power output and losses( Ws ), mass flow rate into the gas turbine system ( m l ) and enthalpy into the system (hI) to be constant, ( i.e . .6.W s ' .6.m l and MI are all zero ), some lengthy manipulation leads to the following peru nit equation which relates the energy to the enthalpy of the gas turbine system.

_ .6.h 2

1

Cq

(9)

The net power output can be calculated as:

(11) Taking small variations into consideration and in per unit, the following equation is obtained:

(12)

_

+ ST .6.Q

(4)

where

By appropriate substitution from the basic equations obtained so far, in equation (12) , we have:

(5) in which; hI = enthalpy value of compressor outlet h2= enthalpy value of turbine inlet T I = temperature of compressor outlet

(13) 44

that one can not expect identical. However, one considered is the format the same in both cases. validate the model by simulation analysis as subsection.

where

For power system stability studies, it is usual to express the steady state gain of the turbine model as unity. Normalizing the gain by letting: ~u yields:

= Cq~Q

the two models to be condition that can be of the model which is The next step is to performing computer shown in the next

(14) 4-2 Comparative Tests

(1 + sr")

-

~P out = (

1

" + sr,,1 + sr' )

The system to be studied is a single machine connected to an infinite busbar. Block diagram representation of the governing system and turbine is shown in figure 5. Other information is available fnm reference 5. Various disturbances were applied and the results with the derived model ( with various time constants, r ), the manufacturer's model and the actual plant are summarized in table 11. Also shown are simulation time responses for similar tests on derived model, manufacturer's model and the actual plant in figures (6,7), (8,9) and (10,11), respectively.

A-

.u.

U

(15)

where ~u is the input to the transfer function and it is such that one perunit change in the input yields one per unit change in the output. The overall block diagram representation of a gas turbine is then shown in figure 3 or equivalently in figure 4. 4- MODEL VALIDITY ASSESSMENT

4-3 DiscUBBion of the Results Referring to the results tabulated in table Il, it can be seen that the derived model is quite valid. By looking at the performances of its dynamic characteristics, it is clear that the derived model was able to perform as well as the manufacturer's model. When the disturbance (in power) is much larger, the results deviate a bit more. This was not solely due to the gas turbine model but it is in fact mainly due to the governing system. Since the scale of the plots is very small, it is really difficult to actually get an accurate assessment.

Before any simulation study to be performed, a suitable value of I should be chosen. For Pr = (Pao/Po) = 0.2, (see equation 8), the effect of changing r on the overall transfer function is shown in table I. As the transfer function available from the manufacturer is as shown below:

A-P

_

(

(1 + 0.65 s) + 1.5 s X1 + 0.1 s )

.u.

out -

(r

= 0.1 sec.,

1

AU

(16)

.u.

r'=1.5 sec., r"= 0.65 sec.) From the dynamic performances shown for the derived model, it can be stated that this simple representation of a single shaft gas turbine is valid for power system stability studies. Despite of the different time constants used, the derived model was able to approximate quite closely the measured test results. The time constants used in simulation studies were 0.1, 0.2 and 0.3 seconds. The fact that these time constants do not significantly affect the performance of the derived gas turbine model when compared to the manufacturer's model, is due to the fact that the governor system has the rolling effect.

it is found that as the value of I decreases, the transfer function becomes more and more similar to the manufacturer's model. It can be also seen from table I that when I = 1.4, the time constant r">r' whereas in the manufacturer's model, the time constant r"
5- CONCLUSIONS

The paper established a theoretical way of finding a suitable model of a single shaft gas turbine, mainly for power system stability 45

studies. Various studies made clearly demonstrated the validity of the derived model in comparison to a manufacturer model and factory test results. The authors are in the process of performing a similar task for twin shaft arrangements of gas turbines.

I 4

I dP

6- REFERENCES 1. Cohen, H., Rogers, G.F.C., Saravanamuttoo, H.I.H., (1987), Gas Turbine Theory, Wiley, NY

out

Turbine l -Compressor --------

2. Shepherd, D.G., (1960), Introduction to Gas Turbines, Whiterfriars, London

Figure 1 Single Shaft Gas Turbine System

3. IEEE Committee, (1973), Dynamic Models for Steam and Hydroturbines, IEEE P AS-92, 1904-1915

Combustion Chamber

4. IEEE Committee, (1991), Dynamic Models for Fossil Fueled Steam Units in Power System Studies, IEEE PWRS, 753-761

I

L

Energy

Input

P.V P

H.T

a

5. Hussain, A., (1988), Development of Simplified Gas Turbine Simulation Models, Msc Dissertation, UMIST

Pa

Nozzle 2

Nozzle 1

NOMENCLA TURE Figure 2 Single Shaft Gas Turbine Equivalent Model

p p& V M :rh

: Gas pressure : Atmospheric pressure : Volume of combustion chamber : Mass of gas in combustion chamber : Gas mass flow rate Pm : Mechanical power output 1 : Specific heat ratio T : Temperature Q : Heat flow rate H : Enthalpy o : Steady state value 1 : Compressor inlet condition 2 : Compressor outlet condition 3 : Turbine inlet condition .( : Turbine outlet condition & : Ambient condition

~J

T

lip I::

Figure 3 Gas Turbine Model

-

r GOVERNOR SYSTEM

lIu I

r--I I..-

-

-

- - -

1 1 + ST )

(

-

-

-

-,

( 1 + ST"

)

( 1 + s't"

)

I

f-

- -

-

_

I --.J

CAS TURBINE Jot:lDEL

Figure.( Transfer function of & Gas Turbine

46

liPout

~

I

CAS TURBINE I1)OE!.

,

r----------- 1r-----, UV"":I "

limit

....

7 . 338 HII

~. Y ' IIt1

IrSI

I ~.· ~ • •

It . I..

J: • 18 . 136

'31'1

I :

•... 11·

Lower 1I.1t -I. 379 HII H . W.

Rate lI.lt

. ..,

128.6 HIlls ___ _

I

J

~" t o'

Ii i 2..

i" '" ,

1. 5

i

cl ~ I", cl, 2.5

i"

I ~-ni

J . fa

3 5

,.

1

l.B

3.5

4. 1

4. S

:;, •

-.""

Figure 5 System Block Diagram

-. 18

~ ... ~

B. 8

2:

S" fURB .

POI«:R

7 .8

6.8 ~.

'.8

e . O~

3.8

2 .• .. 8 ~"

. !a

I.•

1. 5

2. "

;!.s

J. a

3 .5

.. . ,

".S

S. I 8.S

.. I

1.5

2.8

2.5

5.1

.1. - .OS

- 2.

-.1.

Figure 6 Simulation Test Response for Power Change from 2 to .( MW Ulling Derived Model with T= 0 .1 sec.

e.

....

Table I

,

~ -------

//

l. 2.

"i" 1.

I

,

Results for 'Y effect study

c =-p

C

p

( l+ s T ")

r

v

K

I

T

T"

1

( 1+ 5 T)( 1+5"["')

11 1

I

Ov erall T. f (1.1. 94s'r)

I. 40

0. 2

0 . 5299

I. 89T

I. 94T

(,1, , '" 1.S I"" 2.I"11 " 2 '.5" "",,, I" "I" . "EI " " I". S "I • . S "I 1.8 3. ' 3.5 S. I

(l+sT){ 1+1. 89sT)

-I.

(I-I. 98sT) I. 35

- 2.

0.2

0 . 5093

I. 96T

I. 98T

O-ST)( I-I. 96ST) (I-2 . 025T)

Figure 7 Simulation Test Response for Power Change from o to .( MW Ulling Derived Model with T= 0 .1 Sec.

I. 30

0.2

0 . 4885

2.05T

2.02T (1-5T) (1-2. 055T) (1-2 . 08sT)

I. 25

0.2

0 . 4627

2 . 16T

2 . 0ST (1-5T)( 1-2.165T) (I-2 . 105T)

I. 20

0. 2

0 . 4377

2.30T

2 . lOT (l+FiT)( 1+2 . 30"T)

0-2 . 2IST) I. 15

0.2

0 . 4117

2 . 43T

2 . 21T (I-ST) (1-2. 435T) (I-I. 70ST)

I. 10

0. 2

0 . 3764

2 . 70T

1. 70T (1 +s't) ( 1 +2. 105T)

47

I I

n

Table

Computer Simulation Results oC the Sinpe Shalt Model

Predicled Peak Values: S peed And Tlm. of Occuranc e

I I I

I(

Aclual Plan\. MW)

Speed

Ti_

(RPM)

(Sec)

Speed (RPM)

Ti_ ($ec)

I

1 . 44

!

0 - 2

-40

1. 40

- 30

2 - 4

-40

1 . 40

-30

40

1. 40

30

40

1. 40

30

-eo

1. 40

80

1. 40

-120 120

I

I

4 - 2

M. lit

T

C

2 - 0

Speed (RPM)

.

0 . 2 sec

T

~

0 . 3 sec

•

Ti_ (Sec)

Tl_

Speed (RPM)

(Sec)

Speed (RPM)

- 36

1 . 45

- 30

1. 45

- 36

1. 45

- 30

1. 45

30

1. 45

30

1 . 45

30

1. 45

30

1. 45

TI ..... ( Sec)

- 3(1

1 . 45

1.441 1. 44 I 1.44 I

1.45

30

1. 45

30

1. 45

-72

1. 44

- 70

1 . 45

- 70

1. 45

- 70

1. 45

76

1. 78

76

1 . 78

76

1. 78

76

1. 0 0

2 . 00

- 108

1.85

- 110

1 . 05

- 108

1. 00

- 1 ::>8

1. 8 5

2 . 00

118

2 . 20

1 22

2 . 18

1 22

2 . 18

12 4

2. 2 0

11 . 1"

I

I I I

to.- ...... I :

I . .... .

1 . 85

. .....+.-......+rr"...+rrTrI-nn-r+rrnTl-n"...l-nn-rl " .5 L ' 1.5 2 . 1 2.5 l. 1 3. 5

•. _+naThTrrtTnThn~,,~IT'n"T,hIrT'n,,~I~"~'T'hIrT'n,,~ln"n'T,hl'T,rrl,rI

1.'

l. 5

2.1

2. 5

J. '

J

$

l'

1

'5.

'5.,

- _85

-

- . SI

-

.

T

-30

';,re 511 r

,;,. ..... I :

0. 1 sec

liP

I

I

, .i e v d Mod e 1 Wilt. y • 1 D

Manufacturers Model

e_" -

G.. a.ph 2:

Sft TURB.

.. . .

4. 5

5.1

....

4.50

S.I

.... 0

J'

e_

PDiLR

" I " " I TnT!

..

k .... 2: 5" TUAa.

IPDI«A

5.

5.

4.

l+rnTtt.~Tn.-.l-,.,.....+r~+-rn- . 1 1. 5

1.1

1.5

2. 1

2. 5

, " , I, " , I, , " I " , rI

3.1

3.5

.. . .

4. 5

I. S

5. 1

1.1

1.5

2.1

2. 5

' 01

3.5

- 2.

Figure 8 Simulation Test Response for Power Change from 2 to .. MW using Manufacturer's Model

Figure 9 Simulation Test Response Cor Power Cb.ange from 0 to .. MW using Manufacturer's Model

I

:. : ._-==::t

:. y

__

~ :::::::~:=-:-f:::.-

~

.: _.:..:=..:._._ _____

:. -~ -" :-~::- -f-- ---- ':

..

! - --_ .. _-

-- .--.:.-.----=

j'=' :~':::"~ ---.:=----::~ - ~~ -----..::.

:.-===::..:::

:~::---: :~~ -:=~~-~ :::.:... -- -- -~:: - - --~--=-= - - .==:t:::= ~-

~ - - - ~---

------- -- -

Figure 10 Actual }<'adory Test Response Cor Power

-

~e

from 0 to" MW

Figure 11 Actual Factory Test Response Cor Power Change from 2 to 4 MW

48