Modelling of a Hybrid Power-Temperature Control System for

International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:19 No:06 69

191506-8383-IJMME-IJENS © December 2019 IJENS I J E N S

Abstract-- Different requirements from the gas turbine engines

have to be fulfilled by modern control systems, like unburden the

operator of any engine specific control, limiting tasks, reduce the

engine fuel consumption, maximize the mechanical power, and

follow the operator's commands as fast as possible with safe and

reliable operation. Two control system models were developed to

control the behaviors of the gas turbine engine. The first controller

model is the power controller model, which is responsible on

governor the net mechanical power created from the engine and

make the system reaches its steady state region as soon as possible

without cares for any critical exceed points, like the temperature

overshooting, transient acceleration, and uncontrolled pressure

rising. The second controller model has a temperature controller

loop, which is added to the power control loop to control the

temperature rising due to the power controller action within a

limiting value. The temperature controller loop programmed to

operate only at the abnormal operating conditions, i.e. when the

engine temperature exceeds the allowable critical point. The

power-temperature controller model has Multi Input- Single

Output (MISO) type, where the input to the controller is the

feedback power and temperature singles, while the fuel valve

position is the single output signal. The control system changes the

fuel flow according to the engine demands by changing the value

of the fuel valve angle. The auto selector was added to pass critical

controller signal and ignore the other. Mathematical modeling of

real-life systems is very much needed for simulation and controller

design. The linearized thermodynamic equations are based on the

first engineering principles utilizing algebraic equations for the

thermodynamic processes and differential equations of motion to

model the dynamic nature of the system. The two controller

models are going to simulate by Matlab\Simulink software under

different types and values of excitation. The responses of the main

effective parameters of gas turbine engines and the relationship

between these parameters and the fuel flow rate will present later.

Index Term— gas turbine unit, Haptic control system, Power

controller, Temperature controller, temperature overshooting,

Mathematical modeling.

I. INTRODUCTION

Recently, there has been a growing need for control systems

of gas turbine engines because of the importance of their

application areas (aviation, electric power trade, gas transport,

and military application), in addition, because of the demands

on their operation and the safety requirements of them [1]. In

fact, the domain of using gas turbines has been expanded.

However, more control accuracy becomes highly required, such

as tracking of the characteristics with maximum efficiency.

Fortunately, the studies in the field of control design

methodology were amended at the same time. Furthermore,

several solution techniques for the nonlinear models' problem

are rising as compensating techniques to the traditional linear

methods [2]. At the dynamic modeling point of view, gas

turbines can be regarded as mixed thermo-dynamical-mechanical systems. The dynamic conservation balance and the

algebraic equations of the model are nonlinear [3]. Therefore,

this represents a complex problem for the system analysis and

control point of view. Indeed, the most common way to perform

dynamic analysis and control design is to apply a locally

linearized model around an operating point instead of

considering a nonlinear model [4]. On the other hand, new

mathematical, control methods, and tools have recently come to

the front of the research, which can handle nonlinear systems

and models using special nonlinear techniques. Accordingly,

the results of the dynamic analysis of nonlinear models can be properly applied to nonlinear control design.

Gas turbines have been utilized to produce power for many

years. They are the main source of power for jet aircraft and can

be used to create industrial power in gas turbine power plants.

The general concept is similar to that of a combustion engine,

which is to convert the chemical energy of a fuel into

mechanical energy. The fluid cycle is similar to a combustion

engine. A working fluid, usually air, is compressed. Then, the

fuel is added and the mixture is ignited to initiate combustion.

The combustion releases energy and the fluid expands moving

a physical barrier. The moving of the barrier is the mechanical work out of the cycle. A portion of this mechanical energy is

then used to compress the fluid in the next cycle [5]. The

difference between the gas turbine and a combustion engine is

that the gas turbine cycle runs continuously instead of in

iterative cycles (one after the other). The basic components of

a gas turbine are a compressor, combustor or heat exchanger,

and a turbine. The compressor is typically an axial flow or

centrifugal design [6]. The working fluid flows through the

compressor and the pressure is increased. Heat energy is then

added to the fluid via combustion or a heat exchanger. The fluid

then expands through a turbine to create energy. The turbine is

used to run the compressor. The difference between the power it takes to run the compressor and the total power out of the

turbine is the net power produced by the cycle. Gas turbine

power plants can be designed for a multitude of cycles using

multiple compressors and turbines as well as heat exchangers

and throttling described [7].

Muayad M. Maseer1, Mohammed Najeh Nemah1,2, Cheng Yee Low1*, Hayfaa J. Jebur3

Modelling of a Hybrid Power-Temperature

Control System for Gas Turbine Unit

1Faculty of Mechanical and Manufacturing Engineering, University Tun

Hussein Onn Malaysia, 86400, Parit Raja, Batu Pahat, Johor, Malaysia

([email protected]). 2Engineering Technical College-Najaf, Al-Furat Al-Awsat Technical

University, 54001, Najaf, Iraq ([email protected]). 3General Company for the production of electric power / Southern, Iraqi

Ministry of Electricity, 64001, Dhi Qar, Iraq.

mailto:[email protected]



Increase in the turbine’s temperature and pressure ratio lead

to increases in thermal efficiency as well as power output from

the gas turbine system. On the other side, the metallurgical

properties of the gas turbine’s materials have appointed

limitation, however, the temperature inside the gas turbine must

be controlled within the allowable critical point of the overall the system. An increase in the turbine’s temperature beyond the

allowable critical point causes severe damage to the hot gas path

components such as gas turbine blades. To avoid such severe

phenomenon to occur, the gas turbine temperature needs to be

organized with some suitable control system [8].

The gas turbine unit is a complex nonlinear system. Its

operation and efficiency depend on different types of

parameters, such as the input fuel flow rate, rotational shaft

speed, rotational shaft acceleration, exhaust gas temperature,

the signal of the compressor discharge pressure, thrust force,

nozzle area, and variable (inlet) guide vanes (IGV). However,

it is very difficult to design a nonlinear control system for the gas turbine unit has the ability to adjust all these parameters at

the same operation time. The best solution to this problem is to

design the power controller model, as a first step, to control the

engine’s power by controlling the amount of inlet air-fuel

mixture. Subsequently, the power controller model modifies

with extra safety controller loops to control the abnormal values

of the other parameters. In this study, the gas turbine nonlinear

mathematical model was driven based on the principle

equations of power, the conservation balance of the total mass

flow rate, conservation balance of mechanical energy, and the

internal energy. Then, the Matlab\Simulation program was utilized to design a gas turbine’s power controller model and

faced the challenge of controlling of the turbine' output power

with high response and keep the turbine’s temperature within

the allowable range at the same time. This point was achieved

by adding an extra temperature control loop to control the gas

turbine unit at the abnormal case only, i.e. when the turbine

temperature exceeds the turbine allowable operating

temperature.

This work aim to model and stimulate a hybrid power-thermal

controller for gas turbine power unit. The objectives are: (i) to

design a gas turbine power controller model, which is in charge

on controlling the net mechanical power created from the engine and makes the system reaches its steady state region as

fast as possible, (ii) to modify power controller model with a

temperature controller loop, in order to limit the temperature

rising due to the power controller action within the allowable

critical point, and (iii) to validate and investigate the two

proposed controller models to prove the functionality of the

system.

The scope of this research includes the following:

a. Recognizing the configuration of each component of the

engine.

b. Estimating and investigate each parameter that have to be taken into consideration in the development the control

system model.

c. Driving the linearizing thermodynamic equations based

on the first engineering principles utilizing algebraic

equations for the thermodynamic processes and

differential equations of motion to model the dynamic

nature of the system.

d. Adding auto low-value selector between the power and

temperature controller outputs to select and pass the

instant critical signal.

e. Simulating the two controller models by using

Matlab\Simulink software.

f. Satisfying a useful type of controller relating to the operation requirement.

g. Tuning the controller gains by comparing the model

output response with a previous work.

h. Comparing the simulation result of the current controller

model with the simulation results of the previous works,

in order to prove the effectiveness of the current model.

i. Evaluating the functionality of the designed controller

model against different types and amplitudes of

excitations.

II. METHODOLOGY

The control system is the heart of a gas turbine engine. Gas

turbines require very precise control because of their natural

tendencies towards self-destruction. Generally, either it

happened due to compressor surge or excessive turbine blade

temperatures. Thus, these issues pose the greatest control

system challenges. The high rate of acceleration may induce compressor surge while the addition of too much fuel may

cause turbine temperatures to exceed their limits [9].

In fact, from the earliest gas turbine engines to advanced

modern engines; gas turbine control system primarily varies the

fuel flow rate. For an engine of fixed geometry in specified

conditions, the operating state of the engine is determined

completely by fixing the fuel flow [10]. Accordingly, the terms

control system and fuel control became fundamental for gas

turbines design. However, the control system became

responsible for more than just fuel control because of the

increasing complexity of modern gas turbines. Besides, the major requirements to operate the engines more closely to their

performance limits. This has necessitated more sophisticated

and precise control systems.

Typically, the development of an engine control system

requires mathematical modeling, engine model derivation,

model order reduction, controller design, simulation, and

experimental verification. In this study, the gas turbine control

system was designed in two main loops, as shown in Figure 1.

The first loop represents the power controller loop, which is

the main loop, designed to work at the normal operation

conditions. While the second loop is the temperature controller

system, which is responsible for controlling the gas turbine temperature at the abnormal case, i.e. when the gas turbine

temperature exceeds the allowable design temperature. The

operator of the gas turbine has to adjust the required power and

temperature set points. Then, the controllers should be follow

these two set points. The model was designed to set the lowest

controller signal value as the critical value. However, a low-

value selector was inserted after the power and temperature

controller to pass only the lowest controllers value. The

manipulated controller value will adjust the position of the fuel

valve, in order to inject proportional values of the air-fuel

mixture. Thus, the gas turbine will response to the injected air-fuel mixture and produce the thrust gases.



Fig. 1. The overall methodology flow chart.

III. THE DYNAMIC MODEL DEVELOPMENT

The nonlinear state equations have to derive from the laws of

conservation principles. Dynamic equations come from the

conservation balances constructed for the overall mass (m), internal energy (U), and the conservation balances of

mechanical energy (Inertia energy). Thus, these dynamic

equations have to be transformed into the intensive variable

form to contain measurable quantities. Therefore, the set of

differential balances include the dynamic mass balance for the

combustion chamber. The pressure form of the state equation

derived from the internal energy balance for the combustion

chamber and the intensive form of the overall mechanical

energy balance expressed for the number of revolutions (n).

In present work, the dynamic model was depended upon the

data of the J58 engine, which was described by Pratt and

Whitney [11-13]. The specifications of this engine are heavy-

duty, single shaft gas turbine of turbojet type with maximum

rate power about (20 MW), and high operating speed. The

engine usually uses as the main engine of the airplane. The

structure of the J58 engine described in Figure 2. Section (o)

represents the inlet duct, at which the air at atmospherically

pressure and temperature is metering and supplying to the compressor inlet section (1). The compressor of the axial type,

which fabricates from rotational and stationary blades, is going

to compress the air in multi-stage with no heat transfer until

reach to the maximum compression point (2). This process

depends on the compressor ratio of the compressor itself. Then,

the compressed air mix with the sufficient amount of fuel

supply to the engine, which depends on the power requirement.

The mixing occurs in the combustion chamber of the spherical

type. During the engine’s thermodynamic cycle, the maximum

values of pressure and temperature of the hot gases usually

satisfy at the exit point of the combustion chamber, i.e. section

(3). Finally, the hot gases inlet to the turbine of the axial type. Inside the turbine, the hot gases expansion with no heat transfer

to create mechanical power. Consequently, the hot gases exit

from the turbine at the point (4). Then reach to the gas -

deflector (nozzle) to create thrust at section (5). Table 1 displays

the J58 engine parameters measured through the operation state

at 12 MW as a net power created from the engine.

Table I

Test operating parameters of J58 turbojet engine [11-13].

Parameter Symbol Unit Value

Inlet compressor

temperature T1 K 300

Outlet compressor

temperature T2 K 402.1069

Inlet turbine temperature T3 K 1156.3396

Outlet turbine

temperature T4 K 613.1775

Inlet compressor

pressure P1 kpa 101.325

Outlet compressor

pressure P2 kpa 242.2498

Inlet turbine pressure P3 kpa 128.1908

Outlet turbine pressure P4 kpa 7.2054

Fuel mass flow rate �̇�fuel kg/s 0.985

Compressor flow air

flow rate �̇�𝐶 kg/s 10

Turbine flow air flow

rate �̇�T kg/s 10

Number of revolutions N rpm 6500

Engine power Power kW 12000

loading torque τ kN.m 17.58

Mass to combustion

chamber mcomb. kg 0.1158

Volume of combustion

chamber V m3 0.3

Inertial moment J kg.m2 17

lower thermal value of

fuel Qf MJ/kg 43.12



Fig. 2. The structure of the proposed gas turbine model.

IV. DESIGN ASSUMPTIONS

To simplify driving the dynamic model from the physical processes, i.e. the laws of conservation of energy and mass and

Newton's laws of motion, several assumptions were taken into

account, as follow:

1. Constant physico-chemical properties for gas are

assumed in each main part of the gas turbine, such as

specific heat at constant pressure, at constant volume,

and adiabatic expansion.

2. Heat loss (heat conduction, heat radiation) is neglected.

3. In the inlet duct (0) a constant pressure loss coefficient

(𝜎0 =𝑃1

𝑃0) was assumed. That means the total pressure

loss in the inlet duct of the engine is fixed as a percentage

of its inlet total pressure (𝑃0).

4. At the inlet and outlet of the compressor, sections 1 and

2, it has been assumed as:

a. The mass flow rate is constant, (�̇�1 = �̇�2 = �̇�𝐶 ).

b. There is no energy storage effect, (𝑈2=constant).

5. At the combustion chamber, it has been assumed as:

a. A constant pressure loss coefficient (𝜎𝑐𝑜𝑚𝑏. =𝑃3

𝑃2).

b. A constant efficiency of combustion (𝜂𝑐𝑜𝑚𝑏.).

c. The enthalpy of fuel was neglected.

6. At the inlet and outlet of the turbine, sections 3 and 4, it

has been assumed as:

a. The mass flow rate is constant, (�̇�3 = �̇�4 = �̇�𝑇 ).

b. There is no energy storage effect, (𝑈4=constant).

7. In the gas-deflector, section 5, a constant pressure loss

coefficient (𝜎5 =𝑃5

𝑃4) was assumed. That means the total

pressure loss in the gas-deflector duct of the turbine is

fixed as a percentage of its turbine outlet pressure (𝑃4).

V. CONSERVATION BALANCE OF THE TOTAL MASS AND THE

INTERNAL ENERGY

The conservation balance of the total mass flow rate at any part of the gas turbine engine described [14] as:

𝑑𝑚

𝑑𝑡= �̇�𝑖𝑛 − �̇�𝑜𝑢𝑡 (1)

Conservation balance of the internal energy, where the heat

energy flows and the work terms are also taken into account

described by P Ailer et. al. [15] as:

𝑑𝑢

𝑑𝑡= �̇�𝑖𝑛 𝑈𝑖𝑛 − �̇�𝑜𝑢𝑡 𝑈𝑜𝑢𝑡 + 𝑄 + 𝑊 (2)

The differential equation of the energy conservation can be

created in another way by considering the dependence of the

internal energy on the measurable temperature [16] as:

𝑈 = 𝑚𝐶𝑇 (3)

Take a derivative of equation 3.3 related to the time:

𝑑𝑢

𝑑𝑡= 𝐶𝑣

𝑑

𝑑𝑡 (𝑇 𝑚) = 𝐶𝑣 𝑇

𝑑𝑚

𝑑𝑡+ 𝐶𝑣 𝑚

𝑑𝑇

𝑑𝑡 (4)

Where:

𝑑𝑚

𝑑𝑡=

∆𝑚

∆𝑡= �̇�𝑖𝑛 − �̇�𝑜𝑢𝑡 (5)

Then another relation was obtained from equations 2 and 4 as:

�̇�𝑖𝑛 𝑈𝑖𝑛 − �̇�𝑜𝑢𝑡 𝑈𝑜𝑢𝑡 + 𝑄 + 𝑊 = 𝐶𝑣 𝑇 (�̇�𝑖𝑛 − �̇�𝑜𝑢𝑡) +

𝐶𝑣 𝑚 𝑑𝑇

𝑑𝑡 (6)

Then, the state equation for the temperature as state variable

was obtained from the above equation as:

𝑑𝑇

𝑑𝑡=

(�̇�𝑖𝑛 𝑈𝑖𝑛−�̇�𝑜𝑢𝑡 𝑈𝑜𝑢𝑡+𝑄+𝑊)−𝐶𝑣 𝑇(�̇�𝑖𝑛−�̇�𝑜𝑢𝑡)

𝐶𝑣 𝑚 (7)

Equation 3.7 represents the rate of temperature at any part of

the gas turbine engine. Thus, when applied equation 7 at the

turbine exit region and used the assumption (6). Then the rate

temperature at the turbine exit region was:

𝑑𝑇4

𝑑𝑡=

�̇�𝑇 𝐶𝑝 (𝑇3−𝑇4)

𝐶𝑣 𝑚𝑐𝑜𝑚𝑏. (8)

Where:

𝑚𝑐𝑜𝑚𝑏. = 𝜌𝑚 ∗ 𝑉𝑐𝑜𝑚𝑏. (9)

From the ideal gas equation (𝑃 𝑣 = 𝑚 𝑅 𝑇) [17], it has been

represented the temperature at the exit region of the combustion

chamber (point 3) as:

𝑇3 =𝑃3 𝑉𝑐𝑜𝑚𝑏.

𝑚𝑐𝑜𝑚𝑏. 𝑅 (10)

Re compensates equation 10 in equation 8 to get:

𝑑𝑇4

𝑑𝑡=

�̇�𝑇 𝐶𝑝 𝑉𝑐𝑜𝑚𝑏.

𝐶𝑣 𝑚𝑐𝑜𝑚𝑏.2 𝑅

∗ 𝑃3 −�̇�𝑇 𝐶𝑝

𝐶𝑣 𝑚𝑐𝑜𝑚𝑏.∗ 𝑇4 (11)

Then, assume:

𝐴11 = −�̇�𝑇 𝐶𝑝

𝐶𝑣 𝑚𝑐𝑜𝑚𝑏. (12)

𝐴12 =�̇�𝑇 𝐶𝑝 𝑉𝑐𝑜𝑚𝑏.

𝐶𝑣 𝑚𝑐𝑜𝑚𝑏.2 𝑅

(13)

Equation (11) becomes: 𝑑𝑇4

𝑑𝑡= 𝐴11 𝑇4 + 𝐴12 𝑃3 (14)



Equation 14 represents temperature rate at the turbine exit

region which depends on the temperature at the exit turbine (𝑇4)

and the maximum pressure in the cycle at the exit of the

combustion chamber (𝑃3).

Then to get the pressure rate at the exit of the combustion

chamber (𝑑𝑃3

𝑑𝑡), Substitute the ideal gas equation at the exit of

combustion chamber section as follow:

𝑃3 =𝑚𝑐𝑜𝑚𝑏. 𝑅

𝑉𝑐𝑜𝑚𝑏.∗ 𝑇3 (15)

Take the derivative of equation 15 to obtain:

𝑑𝑃3

𝑑𝑡=

𝑚𝑐𝑜𝑚𝑏. 𝑅

𝑉𝑐𝑜𝑚𝑏.∗

𝑑𝑇3

𝑑𝑡 (16)

Apply equation 7 at the exit of combustion chamber region

(point 3) to get (𝑑𝑇3

𝑑𝑡). Then insert the result in equation 16 to get:

𝑑𝑃3

𝑑𝑡=


𝑉𝑐𝑜𝑚𝑏. [

(�̇�𝐶 𝐶𝑝 𝑇2−�̇�𝑇 𝐶𝑝 𝑇3+𝑄+𝑊)

𝐶𝑣 𝑚𝑐𝑜𝑚𝑏.] (17)

The heat flow generated (𝑄) in the combustion chamber was

described by C. Mircea and B. Malvina [18], as follow:

𝑄 = 𝑄𝑓 𝜂𝑐𝑜𝑚𝑏. �̇�𝑓𝑢𝑒𝑙 (18)

There is no mechanical work applied or created in this region

(i.e. W=0). So, equation 17 becomes:

𝑑𝑃3

𝑑𝑡=

𝑅

𝐶𝑣 𝑉𝑐𝑜𝑚𝑏. [ �̇�𝐶 𝐶𝑝 𝑇2 − �̇�𝑇 𝐶𝑝 𝑇3 + 𝑄𝑓 𝜂𝑐𝑜𝑚𝑏. �̇�𝑓𝑢𝑒𝑙 ]

(19)

At the compressor, the temperature of exit region (point 2) can be described by the ideal gas equation as:

𝑇2 = 𝑃2 𝑣2

𝑚2 𝑅 (20)

Where (𝜌 =𝑚

𝑣 ), then equation 20 becomes:

𝑇2 =𝑃2

𝜌 𝑅 (21)

Depending upon the assumption (5.a), the total pressure at the

exit compressor was:

𝑃2 =𝑃3

𝜎𝑐𝑜𝑚𝑏. (22)

Then, the temperature at exit compressor was obtained from

equations 21 and 22 as:

𝑇2 =𝑃3

𝜌 𝑅 𝜎𝑐𝑜𝑚𝑏. (23)

Re compensates equation 10 and 23 in equation 19 to get the

pressure rate at exit of combustion chamber: 𝑑𝑃3

𝑑𝑡=

𝑅

𝐶𝑣 𝑉𝑐𝑜𝑚𝑏.

[�̇�𝐶 𝐶𝑝 ∗𝑃3

𝜌 𝑅 𝜎𝑐𝑜𝑚𝑏.

− �̇�𝑇 𝐶𝑝 ∗𝑃3𝑉𝑐𝑜𝑚𝑏.


+ 𝑄𝑓 𝜂𝑐𝑜𝑚𝑏. �̇�𝑓𝑢𝑒𝑙] (24)

Arrangement the above equation as:

𝑑𝑃3

𝑑𝑡= [

𝑅

𝐶𝑣 𝑉𝑐𝑜𝑚𝑏. (

�̇�𝐶 𝐶𝑝

𝜌 𝑅 𝜎𝑐𝑜𝑚𝑏.−

�̇�𝑇 𝐶𝑝 𝑉𝑐𝑜𝑚𝑏.

𝑚𝑐𝑜𝑚𝑏. 𝑅)] ∗ 𝑃3 +

(𝑅 𝑄𝑓 𝜂𝑐𝑜𝑚𝑏.

𝐶𝑣 𝑉𝑐𝑜𝑚𝑏.) ∗ �̇�𝑓𝑢𝑒𝑙 (25)

Then, assume:

𝐴22 = [𝐶𝑝

𝐶𝑣 𝑉𝑐𝑜𝑚𝑏. (

�̇�𝐶

𝜌 𝜎𝑐𝑜𝑚𝑏.−

�̇�𝑇 𝑉𝑐𝑜𝑚𝑏.

𝑚𝑐𝑜𝑚𝑏.)] (26)

𝐵2 =𝑅 𝑄𝑓 𝜂𝑐𝑜𝑚𝑏.

𝐶𝑣 𝑉𝑐𝑜𝑚𝑏. (27)

Equation 25 becomes:

𝑑𝑃3

𝑑𝑡= 𝐴22 𝑃3 + 𝐵2 �̇�𝑓𝑢𝑒𝑙 (28)

Equation 28 represents pressure rate at exit region of the

combustion chamber. It dependents upon the pressure at the exit

of the combustion chamber (𝑃3) and the fuel flow rate supply to

the combustion chamber (�̇�𝑓𝑢𝑒𝑙).

VI. CONSERVATION BALANCE OF THE MECHANICAL ENERGY

OF THE COMPRESSOR-TURBINE SHAFT

The mechanical energy and speed rate of the compressor-

turbine shaft were reported by P Ailer et. al. [15], as follow:

𝑑𝐸𝑠ℎ𝑎𝑓𝑡

𝑑𝑡= �̇�𝑇 𝐶𝑃(𝑇3 − 𝑇4)𝜂𝑚𝑒𝑐ℎ. − �̇�𝐶 𝐶𝑃 (𝑇2 − 𝑇1) −

2𝜋 3

5 𝑛 𝜏 (29)

𝑑𝑛

𝑑𝑡=

1

4𝜋2 𝐼 𝑛𝑜 (�̇�𝑇 𝐶𝑃 (𝑇3 − 𝑇4) 𝜂𝑚𝑒𝑐ℎ. − �̇�𝐶 𝐶𝑃 (𝑇2 − 𝑇1) −

2𝜋 3

5 𝑛 𝜏) (30)

Barna Pongrácz [19] was described the total temperature

after the compressor by using the isentropic efficiency (𝜂𝐶) as:

𝑇2 = 𝑇1 [1 +1

𝜂𝑐 [ (

𝑃2

𝑃1 )

( 𝐾−1

𝐾 )

− 1 ]] (31)

From equations 22 and 31, the temperature of inlet process

cycle (atmospheric temperature) was

𝑇1 =1

[1+1

𝜂𝑐 [ (

𝑃3𝜎𝑐𝑜𝑚𝑏.𝑃1

)(

𝐾−1𝐾

)−1 ]]

∗ 𝑇2 (32)

From equation 20, the temperature at the exit of compressor

was:

𝑇2 =𝑃2 𝑉𝐶

𝑚𝐶 𝑅 (33)

The combustion process at the combustion chamber was done at a constant pressure loss coefficient (assumption 5). So,

equation 33 becomes:



𝑇2 =𝑃3 𝑉𝐶

𝑚𝐶 𝑅 𝜎𝑐𝑜𝑚𝑏. (34)

The main equation of the isentropic efficiency (𝜂𝑇) was

driven at the previous work [19]. Accordingly, the turbine’s

output temperature can be calculated as:

𝑇4 = 𝑇3 [ 1 − 𝜂𝑇 [1 − ( ( 𝑃4

𝑃3 )

( 𝐾−1

𝐾 )

)]] (35)

From the assumptions 3 and 7, the total pressure at point 4

was:

𝑃4 =𝑃1

𝜎0 𝜎5 (36)

Substitute equation 36 in equation 35 to get:

𝑇3 =1

[ 1− 𝜂𝑇 [1−( ( 𝑃1

𝜎0𝜎5𝑃3 )

( 𝐾−1𝐾

) )]]

∗ 𝑇4 (37)

Now substitute equation 10, 32, 34, and 37 in equation 30 to

get:

𝑑𝑛

𝑑𝑡=

1

4𝜋2 𝐼 𝑛𝑜

[

[

�̇�𝑇 𝐶𝑃

(

1

[1−𝜂𝑇(1−(𝑃1

𝜎0 𝜎5 𝑃3)(𝐾−1𝐾 )

)]

− 1

)

∗

𝜂𝑚𝑒𝑐ℎ.

]

∗ 𝑇4 −

[

�̇�𝐶 𝐶𝑃


(

1 −1

[1+1

𝜂𝑐 [ (


)(

𝐾−1𝐾

)−1 ]]

)

]

∗

𝑃3 − 2𝜋 ∗3

5 𝑛 𝜏

]

(38)

Then, assume:

𝐴31 =1


[

�̇�𝑇 𝐶𝑃

(

1

[1−𝜂𝑇(1−(𝑃1

𝜎0 𝜎5 𝑃3)(𝐾−1𝐾 )

)]

− 1

)

∗

𝜂𝑚𝑒𝑐ℎ.

]

(39)

𝐴32 =−1


[

�̇�𝐶 𝐶𝑃


(

1 −1

[1+1

𝜂𝑐 [ (


)(

𝐾−1𝐾

)−1 ]]

)

]

(40)

𝐴33 =−1

2𝜋 𝐼 𝑛𝑜∗

3

5∗ 𝜏 (41)

Equation 33 becomes:

𝑑𝑛

𝑑𝑡= 𝐴31 ∗ 𝑇4 + 𝐴32 ∗ 𝑃3 + 𝐴33 ∗ 𝑛 (42)

Equation 3.42 represents the differential equation of the turbine shaft speed. it depends upon the temperature at the exit

turbine (𝑇4), the pressure at the exit of the combustion chamber

(𝑃3), and compressor-turbine shaft rotation speed (𝑛).

VII. MODELING OF GAS TURBINE CONTROL SYSTEM

The gas turbine engines for both aircraft and industrial

applications have the ability to change its power or load

(increasing or decreasing), by mean of its control system. The

main controller task is to govern several engine variable

parameters, at the same operation time, by controlling the fuel flow that injected to the engine. The control performance

dependents upon the ambient pressure, ambient temperature,

and the engine's load requirements. Thus, the main job of the

control system is:

1. To enable the system to follow the demand of the

operator as fast as possible.

2. To decrease the time delay between the order and the

order execution.

3. To protect the engine from the damage by preventing the

engine to exceed fixed points such as surge point, over

speed point, and over temperature point.

However, the control system for a gas turbine engine should be designed in a different way to do multi-operator

requirements simultaneously. Thus, it is depending upon the

important engine parameters and the boundary condition of the

gas turbine engine.

In this study, two models of the gas turbine control system

were developed to control the power of the engine by changing

a power set point. The controller doing this action by set the

fuel demand, which is proportional to the power requirement

from the engine. The first model has a power control loop only

to make the engine tracking the order of the operator without

limited its action by any one of a critical point, like a compressor surge point, over-speed point, or over-temperature

point. The second model designed to do the same controller

requirement but without exceeding a critical over temperature

point.

Control systems engineers used block diagrams extensively in

system analysis and design. Block diagrams were provided with

two major benefits to the control system engineer. They

provided a clear and concise way of describing the behavior and



structure of the system. Besides, it provided formal methods for

analyzing system behavior [20].

Figure 3 represents a primary block diagram of the speed

controller. The transfer function 𝐺𝑛(𝑠) represents the rate of the

turbine shaft rotational speed to the fuel flow rate injected to the

combustion chamber. 𝐺𝑇4(𝑠) symbolizes the transfer function

of the turbine exit temperature to the engine fuel flow. 𝐺𝑉(𝑠) represents the transfer function of fuel flow to the controller

output signals.

Fig. 3. A block diagram of gas turbine speed controller.

The rotational speed of the compressor-turbine shaft

measured and scaled by the mechanical or electrical speed

sensor (tachometer transducer). The sensor signal was

subtracted to a reference (set point) to obtain an error signal.

The speed deviation signal is the only input to the controller.

The speed controller produces the manipulated signal, which

represents the order of the controller to the hydraulic valve (electrical valve in the small engine). In turns, the valve meters

and supplies a sufficient amount of fuel to the engine, which is

proportional to the controller order signal. In this time, the

engine steady state conditions are changing and the operation

travel from the steady state to the transient state. When the

system returns to the steady state, the rotational speed of the

turbine shaft will become equal to the speed’s set point.

In case of extra mechanical power is required from the engine,

the controller will increase the fuel flow supplying to the

combustion chamber. The increase in the injection fuel amount

causes increasing for combustion in the chamber. As this reason, the hot gases temperature was rising. If this temperature

is rise higher than the fixed turbine blades temperature damage

will occur. Thus, this rising in the temperature should be

controlled to prevent the damage.

The modern block diagram of the speed controller and

temperature controller of the gas turbine control system is

presented in Figure 4. This design can sense the turbine output

temperature by mean of the temperature sensor (thermocouple

transducer). The sensing signal feedbacks to the controller, to

compare the instant temperature with the required temperature

set point (constant set point). Indeed, it is equal to the limit

degree of the engine thermal ability. In the controller, the error signal is creating and entering to the temperature controller to

create the manipulated signal. This signal is proportional to the

engine fuel required to keep the temperature under the fixed

limit value.

The temperature limit value depends on the ambient

temperature. In case of ambient temperature increases, the

exhaust temperature will tend to increase too and the action of

the temperature control loop will reduce the amount of fuel consumption of the gas turbine. On the other hand, when

ambient temperatures decrease the exhaust temperature will

tend to decrease in turn. In this case, the load - frequency control

loop becomes the active control loop.

The temperature controller signal and the signal coming from

the speed controller inlet to the auto switch, named low-value

selector, to select the lowest instant value. The output

manipulated signal represents the fuel system inlet signal,

which is responsible for the amount of fuel injection to the

system. This amount proportional to the fuel requirement at the

instant of operation.

Fig. 4. A block diagram of gas turbine speed - temperature controller.

The mechanical power development from the engine indicates

the engine thrust more than the values of the turbine shaft speed.

Hence, if the feedback signal becomes the output mechanical

power of the engine; the design of the engine’s controller will become easier. However, the block diagram, which is presented

in Figure 3, can be modified to the more modern design by

addition the block. Where represents the transfer function of

the net mechanical power to the turbine shaft rotational speed.

Accordingly, the model was modified by making the shaft

speed signal input to the speed to power transfer function , to

create a power output signal. Then, the power signal will sense

and compare with the instant variable power set point required

from the operator to develop the error signal. This error signal

enters to the power controller and complete the cycle. The block

diagram of the modified gas turbine power controller is

presented Figure 5. Regarding the same purpose described above, the block

diagram of the gas turbine speed - temperature controller

presented in Figure 6 has been modified to gas turbine power -

temperature controller model, as shown in Figure 4. This



modification included changing the speed controller loop to the

power controller loop.

Fig. 5. A block diagram of gas turbine power controller.

Fig. 6. A block diagram of gas turbine power - temperature controller.

VIII. NONLINEAR STATE SPACE MODEL OF GAS TURBINE

The transfer functions 𝐺𝑛(𝑠) and 𝐺𝑇4(𝑠) can be created by

utilizing the state space technique, which is one technique of

the modern control system theory. In present work, the control

system has single input-multi output (SIMO) parameters. However, many transfer functions (equal to the output

parameters) were derived by using the state space to transfer

function transformation method, which was developed from the

general state differential equation and the output equation

described [21-23], as follow:

�̇� = [𝐴]𝑋 + [𝐵] 𝑈 (43)

𝑌 = [𝐶] 𝑋 + [𝐷] 𝑈 (44)

Where:

𝑋 : (𝑛 ∗ 1) represent the state vector, �̇� : (𝑛 ∗ 1) represent

the time derivative of state vector, 𝑈 : (𝑚 ∗ 1) represent the

input vector, 𝑌 : (𝑟 ∗ 1) represent the output vector, [𝐴] : (𝑛 ∗𝑛) represent the dynamic matrix, [𝐵] : (𝑛 ∗ 𝑚) represent the

input matrix, [𝐶] : (𝑟 ∗ 𝑛) represent the output matrix, [𝐷] : (𝑟 ∗

𝑚) represent the direct transfer or feed forward matrix, n :

Number of state variables, m : Number of input variables, and

r : Number of output variables.

Substitute the dynamic model equations (14), (28) and (42) in equation (43) in term of the state variables where:

𝑋1 = 𝑇4 , 𝑋2 = 𝑃3 , 𝑋3 = 𝑛 and 𝑈 = �̇�𝑓𝑢𝑒𝑙

[

�̇�1

�̇�2

�̇�3

] = [𝐴11 𝐴12 00 𝐴22 0

𝐴31 𝐴32 𝐴33

] ∗ [𝑋1

𝑋2

𝑋3

] + [0𝐵2

0]𝑈 (45)

The fuel flow was chosen as the input of the dynamic model.

While, the compressor-turbine rotation shaft speed and the

turbine exit temperature were selected as the output of the

dynamic model. So, equation (44) becomes:

[𝑌1

𝑌2] = [1 0 0

0 0 1] ∗ [

𝑋1

𝑋2

𝑋3

] (46)

Where: (Y1) and (Y2) represent the rotational shaft speed (n)

and turbine exist temperature (T4).

Lastly, the Matlab comment (ss2tf) was used to convert a

state-space representation of a system into an equivalent

transfer function, as shown:

𝐺𝑇4(𝑠) =𝑌1(𝑠)

𝑈(𝑠)=

𝐴12(𝑠−𝐴33) 𝐵2

(𝑠−𝐴11)(𝑠−𝐴22)(𝑠−𝐴33) (47)

𝐺𝑛(𝑠) =𝑌2(𝑠)

𝑈(𝑠)=

[𝐴32(𝑠−𝐴11)+𝐴12𝐴31] 𝐵2

(𝑠−𝐴11)(𝑠−𝐴22)(𝑠−𝐴33) (48)

IX. DRIVE THE TRANSFER FUNCTION 𝐺𝑃(𝑠)

The development power is the more important parameter of

the gas turbine engine. So, to describe this parameter in the gas

turbine control system, the relationship between the power and

the rotational shaft speed must be found. The previous study

[24] described the torque conservation balance equation as:

𝐽 𝑑𝑊

𝑑𝑡= 𝜏𝑇 − 𝜏𝐶 (49)

But in general:

𝑃 = 𝜏 ∗ 𝑤 (50)

Substitute equation (50) in equation (49) to get:

𝐽 𝑑𝑊

𝑑𝑡=

𝑃𝑇−𝑃𝐶

𝑊° (51)

𝐽 𝑑𝑊

𝑑𝑡=

𝑃𝑛𝑒𝑡

𝑊° (52)

Take Laplace transform to equation (52) get:

𝐽 𝑊° ∗ 𝑠 ∗ 𝑊(𝑠) = 𝑃𝑛𝑒𝑡 (𝑠) (53)

But:



𝑊° =2𝜋

60∗ 𝑛° (54)

Substitute equation (54) in equation (53) to get:

𝐽 2𝜋

60∗ 𝑛° ∗ 𝑠 ∗

2𝜋

60∗ 𝑁(𝑠) = 𝑃𝑛𝑒𝑡 (𝑠) (55)

Arrangement the equation above to get speed to power

transfer function:

𝐺𝑃(𝑠) =𝑁(𝑠)

𝑃𝑛𝑒𝑡 (𝑠)=

1

[𝐽 (2𝜋

60)2∗𝑛°]𝑆

(56)

X. SIMULATION RESULTS AND DISCUSSION

To simulate the behavior of control system models; all the

necessary model’s parameters (e.g., geometry, material

properties, and heat transfer coefficients) must be available.

The simulation models described both the power control model

and power - temperature control model. Many different excitations, constant, step, ramp, and stairs, were performed to

closely simulate the real engine operation. The main

comparison specifications adopted to evaluate the models’ time

responses are the time delay, rise time, settling time, and the

response overshoot. The overall simulation depended on the

Matlab 2019a program. Moreover, the important parameters of

J58 turbojet gas engine were utilized as described in Table I.

A. Verification of the power controller model

The steady state relationship between mechanical power,

turbine shaft rotational speed, and turbine exhaust temperature

against the fuel flow rate are represented in Figures 7, 8, and 9,

respectively. It can be clearly noted that the relationships are

practically linear, which means the dynamic system has a

constant power deviation relative to the fuel flow at the steady

state region from engine response, i.e. ( ∆𝑃𝑛𝑒𝑡

∆𝑓𝑢𝑒𝑙= 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 ). On

the other side, there are other parameters have linearly increasing the relationship with the created power from the

engine. Therefore, these parameters have the same power

deviation behavior relative to engine fuel flow, i.e. ( ∆𝑛

∆𝑓𝑢𝑒𝑙=

∆𝑇4

∆𝑓𝑢𝑒𝑙= 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 ).

Fig. 7. Steady state relationship between fuel flow and the engine’s output

power.

Fig. 8. Steady state relationship between fuel flow and engine’s shaft

rotational speed.

Fig. 9. Steady state relationship between fuel flow and turbine output

temperature.

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

3.5

4x 10

4

Fuel Flow ( kg/S )

Pow

er

( kW

)

( Relationship Between Fuel Flow And Power )

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

4

Fuel Flow ( kg/S )

Speed (

rpm

)

( Relationship between Fuel flow and Speed )

0 0.5 1 1.5 2 2.5 3200

400

600

800

1000

1200

1400

1600

1800

2000

2200( Relationship Between Fuel Flow And Temperature )

Tem

pera

ture

( K

)

Fuel Flow ( kg/S )



Figure 10 shows the engine power response at 40 seconds as

a simulation time due to step acceleration from ground idle to

take-off power (12000 kW). Therefore, the power response

curve starts from the zero power level and grow up until reach

its steady state level after 6.2 seconds, which represents the

transient region of the power response behavior. The power response developed from the power controller model of a

present work compared with the theoretical result of the model

described in a previous work presented by Youhong [25]. The

results show an acceptable matching between both outputs.

They have similar settling time, zero overshoot, and the same

steady state error. Nevertheless, there is different by delay time

at the acceleration (transient) period. The delay time of the

present work less than the other previous study. In addition, the

acceleration of the engine in the current work is higher than the

work in reference [25]. It has been shown at 6.2 seconds; the

power response reaches the steady state region and fixed on this

value. Figure 11 explains the engine rotational speed response at 40

seconds as a simulation time. The excitation of the power

controller model is the same value used in the comparison of

Figure 10. When the engine changes its power level from the

zero to 12000 kW; the torque on the turbine shaft must be kept

as constant as possible. Therefore, when the engine power

increases, in turn, the turbine shaft speed must be increase from

zero rotational speed to 6500 rpm. This was depended upon the

steady state relationship between the rotational speed and the

fuel flow rate, which was 0.98 kg/s, as shown in Figure 8. This

increased occurs during 6.2 seconds as a transient period. The rotational speed response of current work compared with other

rotational speed responses curves of previous theoretical results

presented by Youhong [25] and Jasim [26]. The competition

results present a good agreement and reasonable matching

between the outputs of the different studies. Furthermore, the

curve of the current work is more smoothly than the other

curves. This depends upon many factors, such as the method

used in the developing of the dynamic model, modelling

assumptions, the program of the controller model, and the

around error of the model software.

Figure 12 represents the turbine exit temperature response at

the same controller excitation value and simulation time. When the engine changes its power level from the zero to 12000 kW;

transient combustion must be occurring to generate the hot

gases. The hot gases temperature will increase the atmospheric

temperature (300 K) until fixed at the 916 K, this represents the

steady state value as shown in figure (5-3) with 0.98 kg/s fuel

flow rate to the combustion chamber. The turbine exit

temperature of the present work compares with the other turbine

exit temperature responses of previous theoretical results

presented by Merry [27] and Jasim [26]. The response delay

time and rise time of both works have excellent matching. The

response of the present work has a small value of overshoot lower than that of references [26] and [27]. That means the

present work has a good design because the high overshoot not

desirable in the controller design because it has very dangers

thermal shock. The temperature overshoot was pointed at the

time interval between 1.2 to 6.2 seconds; while reasonable

matching was presented for the other time levels.

The fuel flow rate response at the same controller excitation

value and simulation time was described in Figure 13. The fuel

flow response of the present work compares with the fuel flow

response of Youhong [25]. The figure shows that the two output

curves have the similar response time. While, the steady state

fuel consumption of present work is less than the steady state

fuel consumption of the model presented in reference [25].

Furthermore, the figure demonstrates that the fuel flow rate response curve of current study is higher than the curve of

reference [25] at the same acceleration period time. This

different occur due to the high acceleration (transient power

required) of a gas turbine engine at the present work need fuel

flow rate more than the work of reference [25] at the same

transient period.

Fig. 10. Power response for step (12 MW) power increase.

Fig. 11. Speed response for step (12 MW) power increase.



Fig. 12. Temperature response for step (12 MW) power increase.

Fig. 13. Fuel flow response for step (12 MW) power increase.

B. The power controller model

The proposed power controller model of the gas turbine

engine was excited with multi-steps excitation input to prove its

effectiveness. The power controller model confronts the multi-

steps power level to describe the behavior of the engine at all

the operation range. This occurs when the variable set point is

excited from ground idle to take-off power, i.e. it is raised from zero power level to 4000, 8000, 12000, 16000 and 20000 kW

power levels at 40 seconds simulation time, as shown in Figure

14.

However, the time responses of the gas turbine mechanical

power are presented in Figure 15. When the power set point

increases, the mechanical power created from the engine is

going to increase at the same time due to the controller action.

It was obvious that the mechanical power responses of the first

three power levels have no significant value of overshooting,

approximately zero value of overshoot, and the response of the

power’s curves reaches smoothly to the steady state value at 6.7 seconds. Moreover, when the value of power set point

increased; the value of overshooting has very little grow up (it

is possible to ignore this overshooting) and the response of this

parameter reaches efficiently to the steady state value. The time

response specifications (delay time and rise time) have the

behavior opposite to the behavior of the maximum overshoot,

i.e. its values decrease with the power set point increases. On

the other hand, the settling time increases with increasing the power set point. The reason beyond this behavior is the engine

design model tries to specify approximately the same response

time for all excitations values. These behaviors occur because

of the engine itself, the transient increase in the excitation

values make the engine to increase its load and torque sharply.

This behavior of the simulation control model is the goal of

design the maneuver pilot. This types of pilots need to change

its speed or height level in a very short time.

Figure 16 represents the time responses behavior of the

compressor - turbine shaft speed due to the same variable

excitation set points. The behavior of the time response of this

parameter shows that there is no overshooting and the response reaches smoothly to the steady state condition. The reason

beyond that behavior, i.e. the system under damping behavior

at different values of power variable set point, is that the main

factor that effects on the response of this behavior is the inertia

of the rotating shaft. Since this value is approximately constant

throughout operating the one expects that the overshoot absent

in the first value of valve angle. This value still absent in the

forward angles and the zero overshoot accompany the response

[27]. The settling time of the responses was decreasing with the

fuel valve and rotational speed increasing. This behavior occurs

due to the time constant of the fuel pump and the combustion chamber dead time, which decreases with the increases of the

shaft speed and the fuel valve angles. The other specifications

of the time response (i.e. the delay time and the rise time) gave

the sharpness increase relative to set points.

The time responses behavior of the turbine shaft acceleration

due to multi steps excitations were presented in Figure 17. The

shaft acceleration started from zero values, increased sharply to

its maximum values, and then returned to the zero levels. This

due to that the shaft speed starts from the zero value (i.e. zero

acceleration) and grow up to the specific values. Therefore, the

acceleration had sharpness increasing to enable taking off the

shaft from the idle state. Then, the shaft deceleration until reaches to the zero values again, i.e. when the shaft behavior

reaches the steady state region, the shaft speed had constant

deviation with the time, i.e. ∆𝑛

∆𝑡= 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡.

On the other side, Figure 18 demonstrates the time responses

behavior of the turbine exit temperature (𝑇4), at 40 seconds as a simulation time, at the same variable excitations. At the

acceleration period, the temperature response exhibited a

significant value of overshoot, and then the value of overshoot

increased with the increasing of the excitation variable power

set points. The behavior of the overshooting is due to main

reasons: (i) the accumulative heat energy attendant with the

combustion of access fuel, and (ii) relatively low heat

dissipation that lead a suddenly rising of temperature. The time

delay and the rise time shows a good sharpness increasing with

the increasing of the engine fuel flow rate, the response of the

system reaches the steady state conditions in a reasonable time. Indeed, the value of the turbine exit temperature was increased



with increasing of the fuel valve angle, i.e. the fuel mass flow

rate.

Figure 19 represents the time responses behavior of the

engine’s fuel flow rate. When increasing the power level set

points, the control system tends to increase the fuel valve angle,

in other words, increase the fuel flow rate feeding to the gas turbine engine. The response of the fuel flow rate has no

significant value of overshooting (approximately zero value of

overshoot). Besides, the response of the fuel flow rate’s curves

reaches smoothly to the steady state value. The delay time and

rise time are increasing with the increase of fuel flow rate.

Fig. 14. The behavior of multi power step input set points.

Fig. 15. Power responses at multi power step input set points.

Fig. 16. Speed responses at multi power step input set points.

Fig. 17. Acceleration responses at multi power step input set points.

Fig. 18. Temperature responses at multi power step input set points.

0 5 10 15 20 25 30 35 400

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2x 10

4

Pow

er

(kW

)

Time (s)

(Step Power Set Point)

P=4000 kW

P=8000 kW

P=12000 kW

P=16000 kW

P=20000 kW

0 5 10 15 20 25 30 35 400

0.5

1

1.5

2

x 104

Pow

er

(kW

)

Time ( S )

( Power Response )

P=4000 kW

P=8000 kW

P=12000 kW

P=16000 kW

P=20000 kW

0 5 10 15 20 25 30 35 400

2000

4000

6000

8000

10000

12000

Time ( S )

Speed (

rpm

)

( Speed Response )

At P=4000 kW

At P=20000 kW

At P=16000 kW

At P=12000 kW

At P=8000 kW

0 5 10 15 20 25 30 35 40-1000

0

1000

2000

3000

4000

5000

6000

7000

Time ( S )

Accala

ration (

rad/S

2 )

( Accalaration Response )

At P=4000 kW

At P=8000 kW

At P=12000 kW

At P=16000 kW

At P=20000 kW

0 5 10 15 20 25 30 35 40200

400

600

800

1000

1200

1400

1600( Temperature Response )

Time ( S )

Tem

pera

ture

( K

)

At P=4000 kW

At P=20000 KW

At P=16000 kW

At P=12000 kW

At P=8000 kW



Fig. 19. Fuel flow response at multi power step input set points.

C. The power-temperature controller model

In general, the high power demand from the gas turbine

engine causes increasing in the value of the hot gases

temperature. Thus, if this increasing reach higher than the

turbine thermal ability, the damage will occur. Therefore, the

temperature controller added to the power controller model to control the system near of this critical operation point. The

simulation results show that, at the low power excitation, the

temperature control loop does not exceed any clear change in

the behavior of the controller responses. This identity occurs

due to the step power excitation less than 16 MW located in the

limit of the operation work, i.e. the hot gases temperature do not

exceed the turbine blades limiting temperature. Therefore, the

temperature control loop is expected to activate when the step

power excitation exceeding the 16 MW required power.

The mechanical power response due to 16 MW step power

excitation was displayed in Figure 20. The power-temperature

controller model excited during 50 seconds of simulation time. The response of the power-temperature controller model

compares with the same power response of the power controller

model to verify the response behavior of the two models. The

figure evinces that the engine changes its power level from zero

to 16 MW during 6.23 seconds. In addition, the engine satisfies

50% of its final value during the first 1.89 seconds, which is

mean; the engine has high-efficiency acceleration. The major

reason bound this behavior to return to the controller action. On

the other side, the response specification (delay time and raise

time) of the power controller model lower than the response

specification of power-temperature controller model and the steady state error of both models responses are similar. Also,

from the figure notes that the power response has overshoot

from 6.55 to 16.76 seconds and reach to 3.6875% as maximum

overshoot value at the 11.47 seconds. The maximum percentage

of the overshoot is often represented the maximum value to the

final value of the step response. The overshoot in the

temperature response curve occurs in a very short time and has

very low value compares with the value of the steady state

temperature. Therefore, the controller model shows a small

overshoot never mean has a bad design because the

overshooting occurs in the operation range limit.

Figure 21 displays the rotational shaft speed response of both

power and power-temperature controller at the same excitation

value, through 50 seconds as a simulation time. The figure

shows that sharp increase in the shaft speed occurs during 6.23

seconds after the controller excitation. This increasing is taken

place to match the sharp increase in the engine’s power because the engine never changes its power without change the

compressor and the turbine rotational speed. Thus, the speed

response of the power-temperature controller reaches the steady

state value during 17.17 seconds, at the time, the shaft rotational

speed becomes 8740 rpm. The figure indicates that the speed

response of the power-temperature controller has clearly

overshooting during the period from 5.85 to 17.17 seconds and

satisfy maximum overshoot equal to 3.123% at 11.49 seconds.

While the speed response of the power controller model reaches

smoothly to the steady state value during 7.3 seconds. This

behavior occurs due to the important demand needs from the

power controller model is to satisfy the lowest response time (the lowest delay time and rise time). Therefore, the power

controller model operates without worry about thermal

overshooting.

On the other side, the exit turbine temperature value has very

sharp increase from the atmospheric temperature degree to

1118 K at 1.65 seconds as displayed in Figure 22. This transient

rising time is very important to produce the net power and make

the engine to accelerate from the idle state to its specific

rotational speed. The response of the power-temperature

controller model has 10.82% as a percent overshoot, this

overshooting greater than the overshoot of the power response controller model of value 9.928%. That is mean; there was a

reasonable matching between the temperature response of the

power-temperature controller model and the response of the

power controller model except that the overshoot of the power-

temperature model is higher than the second response. Lastly,

the exit turbine temperature reaches its steady state value at 15

seconds and continues on this stability value.

Figure 23 illustrates the engine fuel flow rate response

injection to the engine during the simulation times to make the

engine creates the power of the same behavior as described in

figure 20. The figure shows that the fuel flow rate response of

the power controller reaches the steady state value during 5.4 seconds. While the fuel flow rate response of the power-

temperature controller reaches the steady state value during

18.2 seconds. Therefore, this difference in the responses occurs

because when the excitation happens; the temperature

controller starts to command the engine. Thus, the temperature

controller tries to reach the fixed turbine exit temperature of

1250 K as soon as possible. The engine develops 16 MW power

at the 1118 K exit temperature, not at 1250 K. So, the power

controller sensing this error and take the leader of the engine

and decreases the temperature until reaches the steady state

temperature at 1118 K, which was proportional to 16 MW mechanical power.

The output signals of both power and power-temperature

controller models due to the 16 MW power increase at 50

seconds of simulation time were demonstrated in Figure 24.

The fuel valve’s signal outputs from the auto switch (the low-

value selector) and feeds to the fuel flow system, which also

described in the same figure. The figure shows that the

temperature controller loop leads the engine at the first 11.19

0 5 10 15 20 25 30 35 400

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Fuel F

low

(kg/s

)( Fuel Flow Response )

Time ( S )

At P=4000 kW

At P=20000 KW

At P=16000 kW

At P=12000 kW

At P=8000 kW



seconds after handing the leader to the power controller loop.

The auto switches subordinate this behavior because when the

excitation occurs; the engine creates sharpness acceleration.

This sharpness acceleration causes sharpness increasing in the

value of the hot gases temperature. After a few time the

acceleration rate decreases, so the turbine exit temperature decreases too and the power control loop has the ability to

command the gas turbine engine now.

Fig. 20. Power response at 16 MW step excitation.

Fig. 21. Speed response at 16 MW step excitation.

Fig. 22. Temperature response at 16 MW step excitation.

Fig. 23. Fuel flow response at 16 MW step excitation.

Fig. 24. The outputs of power control and temperature control at 16 MW step

excitation.

0 5 10 15 20 25 30 35 40 45 500

2000

4000

6000

8000

10000

12000

14000

16000

18000

Time ( S )

Pow

er

(kW

)

( Power Response )

Simulated without

temperature controller

Simulated with


0 5 10 15 20 25 30 35 40 45 500

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Time ( S )

Speed (

rpm

)

( Speed Response )

Simulated without


Simulated with


0 5 10 15 20 25 30 35 40 45 50300

400

500

600

700

800

900

1000

1100

1200

1300

Time ( S )

Tem

pera

ture

( K

)

( Temperature Response )

Simulated without


Simulated with


0 5 10 15 20 25 30 35 40 45 500

0.2

0.4

0.6

0.8

1

1.2

1.4

Time ( S )

( Fuel Flow Response )

Fuel F

low

( K

g/S

)

Simulated without


Simulated with


0 5 10 15 20 25 30 35 40 45 50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

4

Time ( S )

contr

olle

r sig

nals

Power controller signal

Temperature controller signal

Fuel valve signal



Figure 25 displays the power response due to the 17 MW step

power excitation. This power response compares with the same

response for the power controller model at the same excitation.

The figure shows that the power response of the power-

temperature controller reaches the steady state region at the

time 45 second. While the power response of the power controller satisfies its steady state value at 6.7 seconds.

Accordingly, there is a big difference between the two

controllers responses because the engine was operating near the

maximum limit. However, the operation time of the

temperature controller loop was increased. Moreover, the figure

demonstrates that the power response of the power-temperature

controller model has 6.059% maximum overshoot at the 18030

kW power value and the response time equal to 37.14 seconds.

Continually, the rotational shaft speed response of both power

and power-temperature controller at the same excitation and

simulation time described above were displayed in Figure 26.

The speed response of the power-temperature controller reaches to the steady state value during 45 seconds, and the shaft

rotational speed at this time becomes 9243 rpm. Furthermore,

the speed response of the power-temperature controller has

clearly overshooting during the period from 19.65 to 45 seconds

and satisfy maximum overshoot equal to 6.026% at 36.81

seconds. While the speed response of the power controller

model reaches smoothly to the steady state value during 7.074

second. The reason bound this different that the power-

temperature controller model designed to be anxious about the

probability of the turbine damage due to the thermal

overshooting. Therefore, the oscillation in the behavior of the power-temperature controller response occurs to disappearance

the rabid temperature increasing in the turbine.

Figure 27 describes a comparison between the turbine exit

temperature responses of two models due to 17 MW step power

excitation. The figure displays that there is a sharp increase in

the exit turbine temperature during the first 2 seconds after the

excitation. The rising time and the delay time of the power-

temperature controller response are faster than the rising time

of the power controller response. Also, the figure shows that the

power-temperature controller response has two overshoots, the

first one of value 6.073% at the time 3.01 seconds and the

second one of value 4.87% at the time 36.43 seconds. The figure notes that the temperature of the power controller model

response reaches to the 1287 K at the time 2.52 seconds, that’s

mean the temperature of this controller exceed the turbine

thermal limit of value 1250 K. On the other hand, the

temperature controller of the power-temperature model saves

the temperature under the critical value at all the simulation

time.

The comparison between the fuel flow rate response of the

power controller model with the same parameter for the power-

temperature controller model, due to 17 MW step power

increase was explained in Figure 28. The figure shows that the fuel flow response reaches the steady state value at the time 45

second and the response curve has overshoot at the time 36.77

seconds of value 5.98%.

Finally, Figure 29 shows the power controller, power-

temperature controller, and fuel system signals at all the

simulation time. The signal outputted from the selector and fed

to the fuel valve system is the temperature control loop signal

at the period from zero to 36 seconds. In this condition, the gas

turbine was operating above the limits and the control loop that

commands the response of the gas turbine is the power-

temperature control. After the transient load demand from the

engine was decreased, the output of the temperature controller

becomes higher than the output of the power controller. At this

moment, the power control loop commands the gas turbine engine and reducing its output power.


Fig. 26. Speed response at 17 MW step excitation.

0 5 10 15 20 25 30 35 40 45 500

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

4

Time ( S )

Pow

er

(kW

)

( Power Response )

Simulated without


Simulated with


0 5 10 15 20 25 30 35 40 45 500

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Time ( S )

Speed (

rpm

)

( Speed Response )

Simulated without


Simulated with







excitation.

Figure 30 explains the power response of both the power-

temperature controller model and power controller model, due

to 18 MW step power excitation. The figure describes when the

excitation occurs; the power response of the power-temperature

controller model increasing at a slow state from zero power level to 18 MW during 36.5 seconds. This response was

increasing without any overshooting but with clear steady state

error of percent 5.55%. The time response of the power-

temperature controller model has very large value compares

with the time response of the power controller model of

transient period 7.721 seconds. Although, the very small

response time of controller model is the one of important

consideration takes in account in the design specification, the

power controller model still not the favorite design at this high

power levels because the engine operates out the working

range. Therefore, the power-temperature controller model is the

favorite design because it is saving the engine from the thermal damage.

The comparison between the speed response of the two

controller models, at the same excitation and simulation time

described in Figure 31. At first five seconds after the excitation,

the speedy response of the power-temperature controller model

has very large acceleration rate and the shaft rotational speed

satisfying increasing of percent 86.47% from the final speed

value, i.e. satisfying 8500 rpm from 9830 rpm final value.

Meanwhile, at the first five seconds, the engine operates

normally until the temperature reaches 1100 K; the engine

enters the critical thermal region and became to decrease the rate of the changing speed.

Figure 32 compares the temperature response of the power-

temperature controller model with the same response for the

power controller model. Accordingly, the temperature of the

power controller model was exceeding more than the critical

thermal point and reached 1345 K. while the power-temperature

controller model kept its temperature under the critical thermal

point and recorded the value 1240 K as the maximum

temperature value.

Figure 33 demonstrates the comparison between fuel flow

responses. The speed response of the power-temperature

controller model shows that the behavior similar to the speed response behavior, i.e. the fuel response satisfying 86.45% of

its final value at the first five seconds. This behavior occurs to

make the engine created a large amount of power at the smallest

time when its work under the critical range.

Lastly, Figure 34 shows the responses of the power control

signal, the temperature controller signal and the net signal

feeding through the selector to the fuel valve system. From the

figure notes that the signal going to the fuel valve system

similar to the temperature controller signal. Thus, the

temperature controller commands the engine during all the

simulation time because of the high step power supply to the engine. In this critical point, the interference of the power

controller at any time makes the engine exceed the critical

temperature degree.

0 5 10 15 20 25 30 35 40 45 50300

400

500

600

700

800

900

1000

1100

1200

1300

Tem

pera

ture

( K

)

Time ( S )


Simulated without


Simulated with


0 5 10 15 20 25 30 35 40 45 500

0.5

1

1.5

Time ( S )

Fuel F

low

( K

g/S

)


Simulated without


Simulated with


0 5 10 15 20 25 30 35 40 45 50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

x 104

Time ( S )

contr

olle

r sig

nals

Power controller signal


Fuel valve signal




Fig.31. Speed response at 18 MW step excitation.




excitation.

XI. CONCLUSIONS

The performance of a single shaft gas turbine engine has been

studied in this work. However, two controller models were

designed to govern the operation of the gas turbine engine,

which are named power control model and the power-

temperature control model. The effectiveness, functionality, and the operability were chosen as the main standards to

evaluate the performance of the two proposed models.

Therefore, the two controller models were evaluated by

subjected to different of the input excitations. Moreover, the

performance of the two models were compared with the

performance of the previous designs presented in the previous

studies.

The temperature controller loop was added to the power

controller model to control the abnormal temperature rising due

to the power controller action. The governor between the two

controller loops is the low-value selector. It is working on

passing the critical controller signal and ignoring another one. The two controller models were simulated by using Matlab \

0 5 10 15 20 25 30 35 40 45 500

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

4 ( Power Response )

Pow

er

(kW

)

Time ( S )

Simulated without


Simulated with


0 5 10 15 20 25 30 35 40 45 500

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

11000

Time ( S )

Speed (

rpm

)

( Speed Response )

Simulated without


Simulated with


0 5 10 15 20 25 30 35 40 45 50200

400

600

800

1000

1200

1400

Time ( S )

Tem

pera

ture

( K

)


Simulated without


Simulated with


0 5 10 15 20 25 30 35 40 45 500

0.5

1

1.5

Time ( S )


Fuel F

low

( K

g/S

)

Simulate without


Simulate with


0 5 10 15 20 25 30 35 40 45 50

0.5

1

1.5

2

2.5

3

3.5x 10

4

Time ( S )

contr

olle

r sig

nals

Load - frequency controller signal


Fuel valve signal



Simulink software under different types of excitations.

Depending on the previous studies, the fuel flow rate that

injection to the combustion chamber was found as the main

parameter that effectively possible to utilize for controlling the

overall engine performance. However, it used as the main

actuating parameter in the two proposed control models. The PID controller is a suitable controller for the gas turbine

engines because of its ability to give good response

specifications at the transient period, i.e. affects the engine to

change its power levels and track the input set points as soon as

possible. Besides, the PID controller can keep the engine's

power on the steady state region with low oscillation and steady

state error.

When the power controller model subjected to the multi steps

power excitation, it is obvious that the delay time of the

responses of the output mechanical power, the rotational turbine

shaft speed, exhaust turbine temperature, and the fuel flow rate

decrease with the increasing of the power level position, i.e. increasing the fuel valve angle. Furthermore, the maximum

overshoot of the exhaust turbine temperature response increases

with the increasing of the fuel valve angle. This is due to the

increase in the cumulative energy inside the combustion

chamber because of increasing the amount of the injection air-

fuel mixture. Finally, the peak point on the acceleration

response curve increases with the increasing of the excitation

power level. This is due to when the power increase, the turbine

shaft speed will increase too, at the same transient time.

On the other side, when the power-temperature controller

model subjected to the step power excitation, it is obvious that the low power required from the engine, the temperature

controller never works and the power response of the power

controller model and power-temperature controller model

become the same. Therefore, the temperature controller usually

operates at the abnormal operation conditions only. At any

input excitation, the power controller and the temperature

controller operate and create manipulated signals at the same

time, but the low-value selector passes only one controller

signal, which is the lowest value, and ignore the other controller

signal. When the engine operates near the maximum thermal

operation limit, the temperature controller commands the

engine at the entire operation time. The main reason behind this behavior is, at this specific operation point, the power control

model leads to an increase in the temperature more than the

thermal limit whenever it takes the lead.

Expand the current design to a multi inputs- multi outputs gas

turbine control system (MIMO systems) to govern the entire

engine’s variable are suggested as a future work.

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Documents

Modelling of a Hybrid Power-Temperature Control System for