Design and simulation of a steam turbine generator using observer based and lqr controllers

Article title: Design and Simulation of a Steam Turbine Generator using Observer Based and LQR Controllers
Authors: mustefa jibril[1], Messay Tadese[2], Eliyas Alemayehu[3]
Affiliations: School of Electrical and Computer Engineering[1], Faculty of Electrical and Computer Engineering[2]
Orcid ids: 0000-0002-3165-2410[1]
Contact e-mail: mustefazinet1981@gmail.com
License information: This work has been published open access under Creative Commons Attribution License
http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, distribution, and reproduction in any
medium, provided the original work is properly cited. Conditions, terms of use and publishing policy can be found at
https://www.scienceopen.com/.
Preprint statement: This article is a preprint and has not been peer-reviewed, under consideration and submitted to
ScienceOpen Preprints for open peer review.
DOI: 10.14293/S2199-1006.1.SOR-.PPESF8J.v1
Preprint first posted online: 31 May 2020
Keywords: Steam turbine, Generator, Observer based controller, linear quadratic regulator, LQR

1
Design and Simulation of a Steam Turbine Generator using
Observer Based and LQR Controllers
Mustefa Jibril1
, Messay Tadese2
, Eliyas Alemayehu Tadese3
1
Msc, School of Electrical & Computer Engineering, Dire Dawa Institute of Technology, Dire Dawa,
Ethiopia
2
Msc, School of Electrical & Computer Engineering, Dire Dawa Institute of Technology, Dire Dawa,
Ethiopia
3
Msc, Faculty of Electrical & Computer Engineering, Jimma Institute of Technology, Jimma, Ethiopia
Abstract
Steam turbine generator is an electromechanical system which converts heat energy to electrical
energy. In this paper, the modelling, design and analysis of a simple steam turbine generator have
done using Matlab/Simulink Toolbox. The open loop system have been analyzed to have an
efficiency of 76.92 %. Observer based & linear quadratic regulator (LQR) controllers have been
designed to improve the generating voltage. Comparison of this two proposed controllers have
been done for increasing the performance improvement to generate a 220 Dc volt. The simulation
result shows that the steam turbine generator with observer based controller has a small percentage
overshoot with minimum settling time than the steam turbine generator with LQR controller and
the open loop system. Finally, the steam turbine generator with observer based controller shows
better improvement in performance than the steam turbine generator with LQR controller.
Keywords: Steam turbine, Generator, Observer based controller, linear quadratic regulator
1. Introduction
A Steam Turbine is a mechanical system that takes thermal energy from pressurized steam and
converted it into mechanical output. Because the turbine generates rotational speed motion, it is
particularly used to drive electrical generators. a steam turbine gain power from a steam . As hot,
steam passes through the turbine ' the blades will spinning. Steam expands and cools, convert most
of the energy it contains. This steam continuously spins the blades. The blades converts the
potential energy of the steam into kinetic energy. Steam turbines is used to drive a generator in
order to produce electricity. In this paper, a steam turbine is designed to run a series wound DC
generator. The shaft of the turbine is connected to the shaft of the generator with two gears at each
side with a gear ratio between them.
2. Mathematical Modelling
Figure 1 shows the diagram of the steam turbine with a DC generator interconnection unit.

2
Figure 1 Steam turbine generator
The mechanical power developed by the steam turbine expressed by the equation:
     1in out
dW d
V F t F t
dt dt

  
Where:
W weight of steam in the steam turbine [g];
V – Volume of steam turbine [m3];
 – Density of steam [g/cm3];
F – Steam mass flow rate [g/s];
t – Time [sec.].
Assuming that the output mass flow rate of the steam turbine is proportional to the pressure in the
turbine system:
 0
0
2out
F
F P
P

Where:
P – Steam pressure in the system [kPa];
P0 – rated pressure;
F0 – rated output mass flow of the system
Assuming constant temperature in the system:

3
 3
d dP
dt dt P
 
 

From equations (1) to (3), it results:
    out out
in out T
dF dFdP
F t F t V V
dt P P dt dt
 

 
      
 
Where
T V
P



 

- Time constant
Then Equation (1) become
 
     4out
T out in
dF t
F t F t
dt
  
The steam mass flow rate is
     in outF t F t F t 
The turbine speed is integrally related to the steam flow rate:
          1 5in outt F t F t F t dt    
Substituting Equation (5) for  outF t into Equation (4) yields the transfer function between the
input steam mass flow and the output turbine torque.
 
 
 1
6
1
T
in T
s
F s s
 



2.1 Modelling of DC Generator
The generated voltage is directly proportional to the generator input speed
 1 2 7ge k 
Where: k1 is a speed proportional constant.
The generator input speed is the product of the gear ratio and the steam turbine output speed
 2 1 8n 
Therefore Equation (7) becomes
 1 1 9ge k n

4
Then the transfer function between  inF s and  ge s become
 
 
1
1
g T
in T
e s k n
F s s




The equation of the generator is
     
     10a
g f a f a L a
di t
e t L L R R R i t
dt
    
Now substituting Equation (9) in to Equation (10) yields
     
     1 1 11a
f a f a L a
di t
k n t L L R R R i t
dt
     
Taking the Laplace transform and obtaining the transfer function the input speed and the output
current yields
 
     
 1
1
12a
f a f a L
I s nk
s s L L R R R

   
The output voltage is simply the product of load resistance times the armature current
     13L aV s R I s
Now substituting Equation (13) in to Equation (12) yields
 
     
 1
1
14L
f a f a L
V s nR k
s s L L R R R

   
Finally combining Equation (6) to Equation (14) yields to the transfer function between the input
steam mass flows to the output generated voltage
 
        
1
1
L T
in T f a f a L
V s nR k
F s s s L L R R R



    
The parameters of the system is shown in Table 1 below
Table 1 System parameters
No Parameters Symbol Value
1 Time constant T 2 second
2 Speed proportional constant. 1k 4 Vs/rad
3 Gear ratio n 4
4 Field winding inductance fL 3 H

5
5 Field winding resistance fR 2 ohm
6 Armature winding inductance aL 4 H
7 Armature winding resistance aR 4 ohm
8 Load resistance LR 8 ohm
Numerically the transfer function become
 
  2
18
2.5 1in
V s
F s s s

 
The state space form of the transfer function is
 
2.5 1 1
1 0 0
0 18
x x u
y x
    
    
   

3. The Proposed Controllers Design
3.1 Observer-Based Controller Design
The deal with the general case where only a subset of the states, or linear combinations of them,
are obtained from measurements and are available to our controller. Such a guidelines is referred
to as the output feedback problem.
The output is of the form
 15y Cx Du 
We shall examine a class of output feedback controllers constructed in two stages:
1. Contracting an observer | a system dynamics that is driven by the inputs and the outputs of
the system, and yield an deliberation of its state variables;
2. Using the estimated state instead of the actual state in a estate response scheme.
The block diagram of the steam turbine generator system with the observer-based controller is
shown in Figure 2 bellow.

6
Figure 2 Block diagram of the steam turbine generator system with the observer-based controller
The controller Gc(s) can be further derived in the following form:
     
1
16cG s I K sI A BK HC B

    
With its state space realization
   17c
A BK HC B
G s
K I
  
   
The controller Gc(s) in Equation (17) is called the observer-based controller, since the structural
idiot of the observer is implicitly reflected within the controller.
Where the state space model of the plant, G, the state feedback gain vector K, and the observer
gain vector H are then returned, respectively.
We select the weighting matrix Q and R as
8 0
1
0 8
Q and R
 
  
 
And we select the observer gain vector as
2
0.5
H
 
  
 
And we obtain the state feedback gain vector K as
 1.7720 2K 
The observer-based controller state space representation become
 
4.272 39 2
1 9 0.5
1.772 2
o o
o o
x x u
y x
    
    
   


7
3.2 LQR Controller Design
LQR is a controller that gives the best possible achievement with respect to some given measure
of performance. The realization measure is a quadratic function composed of state vector and
manipulated input. The block diagram of the steam turbine generator with LQR controller is shown
in Figure 2 below.
Figure 3 Block diagram of the steam turbine generator with LQR controller
Here we choose Q and R matrixes as
3 0
5
0 3
Q and R
 
  
 
The value of obtained feedback gain matrix K of LQR is given by
 0.2166 0.2649K 
4. Result and Discussion
In this section, the open loop simulation of the generated voltage and power, the simulation of the
performance comparison of the proposed controllers for generating a desired voltage is discussed
below.
4.1 Simulation of the open loop steam turbine generator voltage and power
The open loop voltage and power generated for a steam mass flow of 5 g/s input is shown in Figure
4 and Figure 5 respectively.

8
Figure 4 Open loop generated voltage
Figure 5 Open loop output power
As Figure 4 and Figure 5, in the first 10 seconds, the voltage increases from 0 to 90 volts and it
reaches is steady state value of 90 volt and the output power is 1000 W. The induced voltage in
the armature circuit and the input power to the generator simulation result is shown in Figure 6
and Figure 7 respectively.

9
Figure 6 the induced voltage in the armature circuit
Figure 7 Input power
The efficiency of the system become
1000
100% 100% 76.92%
1300
Output power W
Input power W
   
4.2 Comparison of the Steam Turbine Generator with Observer Based & LQR
Controllers for Generating a Desired Voltage
The simulation result of the performance of the steam turbine generator with observer based &
LQR controllers for generating a desired voltage of 220 Dc volt is shown in Figure 8 below.

10
Figure 8 Performance of the steam turbine generator with observer based & LQR controllers for
generating a desired voltage of 220 Dc volt
As seen from Figure 8, the steam turbine generator with observer based controller has a small
percentage overshoot and a settling time of 7 second which is better than the steam turbine
generator with LQR controller and the open loop system.
5. Conclusion
In this paper, a simple steam turbine generator is modelled and designed using a series wound Dc
generator. The open loop system is simulated for generated voltage, input and output power. For
a steam mass flow of 5 g/s the steam turbine generator generates a 90 volt with an output power
of 1000 W. The generator induced armature circuit voltage is simulated with an input power of
1300 W. The efficiency of the simple steam turbine generator is 76.92 %. Comparison of the steam
turbine generator with observer based & LQR controllers for generating a desired voltage of 220
Dc volt have been simulated and the simulation result shows that the steam turbine generator with
observer based controller has a small percentage overshoot and less settling time than the steam
turbine generator with LQR controller and the open loop system. Finally the comparison and
analysis result prove the effectiveness of the proposed steam turbine generator with observer based
controller.
Reference
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Applied Sciences, Vol. 13, Issue 1, pp. 162-164, 2018.
[2].P. I. Onwuamaeze “Improving Steam Turbine Efficiency: An Appraisal” Research Journal
of Mechanical Operations, Vol. 1, No. 1, pp. 24-30, 2018.
[3].Hosham S. et al. “Enhancement of Performance for Steam Turbine in Thermal Power
Plants using Artificial Neural Network and Electric Circuit Design” Applied
Computational Intelligence and Soft Computing, Vol. 2018, Article ID. 8042498, 9 pages,
2018.

11
[4].V I Brezgin et al. “Improvement of Steam Turbine Operational Performance and Reliability
with using Modern Information Technologies” Journal of Physics: Conference Series, Vol.
891, 2017.
[5].A. Sinan “Performance Analysis of a Steam Turbine Power Plant at Part Load Conditions”
Journal of Thermal Engineering, Vol. 3, Issue 2, pp. 1121-1128, 2017.
[6].Li Sun et al. “Performance Modelling of New and Existing Steam Turbines” Ind. Eng.
Chem. Res, Vol. 54, Issue 6, pp. 1908-1915, 2015.
[7].G. Aravie et al. “Performance Modelling of Steam Turbine Performance using Fuzzy Logic
Membership Functions” Journal of Applied Sciences and Environmental Management,
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