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Modelling and Control System design to
control Water temperature in Heat Pump
Modellering och reglersystemdesign för att styra vattentemperaturen i
värmepump
Md Mafizul Islam
Md Abdul Salam
Faculty of Health, Science and Technology
Master’s Program in Electrical Engineering
Degree Project of 15 credit points
Supervisor: Jorge Solis (Karlstad University), Jonas Andersson (Hetvägg AB)
Examiner: Arild Moldsvor (Karlstad University)
Date: 09th December 2013
Serial number:
I
Abstract
The thesis has been conducted at Hetvägg AB and the aim is to develop a combined PID and
Model Predictive Controller (MPC) controller for an air to water heat pump system that
supplies domestic hot water (DHW) to the users. The current control system is PLC based but
because of its big size and expensive maintenance it must be replaced with a robust controller
for the heat pump. The main goal of this project has been to find a suitable improvement
strategy. By constructing a model of the system, the control system has been evaluated. First a
model of the system is derived using system identification techniques in Matlab-Simulink;
since the system is nonlinear and dynamic a model of the system is needed before the
controller is implemented. The data has been estimated and validated for the final selection of
the model in system identification toolbox and then the controller is designed for the selected
model. The combined PID and MPC controller utilizes the obtained model to predict the
future behavior of the system and by changing the constraints an optimal control of the system
is achieved. In this thesis work, first the PID and MPC controller are evaluated and their
results are compared using transient and frequency response plots. It is seen that the MPC
obtained better control action than the PID controller, after some tuning the MPC controller is
capable of maintaining the outlet water temperature to the reference or set point value. Both
the controllers are combined to remove the minor instabilities from the system and also to
obtain a better output. From the transient response behavior it is seen that the combined MPC
and PID controller delivered good output response with minimal overshoot, rise time and
settling time.
II
Acknowledgments
First of all we would like to thank our Supervisor Jonas Andersson at Hetvägg for his
support, suggestions and also for giving the facilities needed in completion of this thesis work
We would like to give special thanks to our supervisor Jorge Solis at Karlstad University for
his valuable guidance and advice in key situations of the project. Without his suggestions this
thesis would not have been possible.
We are very thankful to our Examiner Arild Moldsvor at Karlstad University for giving us an
opportunity to do this thesis work.
Finally, we are thankful to entire faculty at Karlstad University, Swapan Chatterjee and all
those people who have been involved in this thesis project. We are deeply indebted to our
parents for their encouragement and moral support through our entire studies.
III
Table of Contents
Abstract ....................................................................................................................................... I
Acknowledgments ..................................................................................................................... II
Nomenclature ........................................................................................................................... VI
List of Figures ....................................................................................................................... VIII
List of Tables ............................................................................................................................. X
1 Introduction ............................................................................................................................ 1
1.1 Overview ........................................................................................................................... 1
1.2 Background ....................................................................................................................... 1
1.3 Problem formulation ......................................................................................................... 3
1.4 Purposes of master’s thesis ............................................................................................... 3
1.5 Thesis Contribution ........................................................................................................... 4
2 System description ................................................................................................................. 5
2.1 Overview of the heat pump system ................................................................................... 5
2.2 Heat transfer of the system ................................................................................................ 6
2.3 Outside air temperature of the system ............................................................................... 7
2.4 Refrigerant of the system .................................................................................................. 8
2.5 Discharge of the heat exchanger ....................................................................................... 9
3 Modeling of the system ........................................................................................................ 11
3.1 System Identification Introduction .................................................................................. 11
3.2 Model structure for identification method ...................................................................... 11
3.3 Model quality and experimental design .......................................................................... 11
3.4 System identification principle ........................................................................................ 12
3.5 System identification loop ............................................................................................... 13
3.6 System identification method .......................................................................................... 14
3.7 Data Examination ............................................................................................................ 14
3.8 Model structure selection ................................................................................................ 15
3.9 Model Estimation ............................................................................................................ 17
3.9.1 Estimation of the ARX model structure ................................................................... 17
3.9.2 Estimation of the ARMAX model structure ............................................................ 18
3.10 Model Validation ........................................................................................................... 18
3.10.1 Residuals analysis .................................................................................................. 19
3.10.2 Pole-Zero analysis .................................................................................................. 20
3.11 Fitting model for controller design ................................................................................ 22
4 Controller design .................................................................................................................. 24
IV
4.1 Controllers of a system .................................................................................................... 24
4.1.1 Proposed controllers ................................................................................................. 24
4.2 PID Controller ................................................................................................................. 25
4.2.1 PID controller Theory .............................................................................................. 25
4.2.2 Proportional term ...................................................................................................... 25
4.2.3 Integral term ............................................................................................................. 26
4.2.4 Derivative Term ....................................................................................................... 26
4.3 PID Controller for the heat pump .................................................................................... 27
4.4 PID controller tuning rules .............................................................................................. 27
4.4.1 Ziegler Nichols Tuning ............................................................................................ 27
4.4.2 Traditional Z-N tuning Method ................................................................................ 28
4.4.3 Modified Z-N Tuning Method ................................................................................. 28
4.5 PID tuning for the system ................................................................................................ 29
4.6 Transient response specifications .................................................................................... 30
4.6.1 Traditional Ziegler-Nichols response ....................................................................... 30
4.6.2 Modified Ziegler-Nichols response .......................................................................... 31
4.7 Pole-Zero analysis of the PID Controller ........................................................................ 31
5 MPC controller design ......................................................................................................... 33
5.1 MPC Introduction ............................................................................................................ 33
5.2 MPC Model ..................................................................................................................... 33
5.3 MPC Theory .................................................................................................................... 33
5.3.1 MPC Internal model ................................................................................................. 34
5.3.2 Constraints ................................................................................................................ 34
5.3.3 Cost funciton ............................................................................................................ 35
5.3.4 Output prediction ...................................................................................................... 35
5.4 MPC Tuning .................................................................................................................... 36
5.4.1 Prediction horizon Np ............................................................................................... 36
5.4.2 Control horizon Nu ................................................................................................... 37
5.4.3 Weighting matrices .................................................................................................. 37
5.5 MPC controller response ................................................................................................. 38
5.6 Pole-Zero analysis of the MPC Controller ...................................................................... 39
5.7 PID-MPC controller response ......................................................................................... 39
6 Results analysis and Discussion ........................................................................................... 41
6.1 Simulation result analysis ................................................................................................ 41
6.2 Analysis of the model selection results ........................................................................... 41
6.3 Analysis of the PID controller result ............................................................................... 42
V
6.4 Analysis of the MPC and PID-MPC result ..................................................................... 42
6.5 Results comparison with previous work ......................................................................... 44
7 Conclusion and Future work ................................................................................................ 45
7.1 Conclusion ....................................................................................................................... 45
7.2 Future Work .................................................................................................................... 45
Bibliography ............................................................................................................................. 47
Appendix A .............................................................................................................................. 51
A.1 Constant coefficient for air to water .............................................................................. 51
A.2 constant coefficient for water to outside air .................................................................. 51
A.3 Water inside the condenser ............................................................................................ 52
A.4 Outlet temperature and area........................................................................................... 52
A.5 Minimum and Maximum ambient temperature effect .................................................. 53
A.6 P-h diagram for refrigerant R-134a ............................................................................... 54
Appendix B .............................................................................................................................. 55
B.1 System identification toolbox processor ........................................................................ 55
B.2 ARMAX2422 model specifications .............................................................................. 55
B.3 ARX791 model specifications ....................................................................................... 56
B.4 ARX221 model specifications ....................................................................................... 56
B.5 ARX611 model specifications ....................................................................................... 57
B.6 OE221 model specifications .......................................................................................... 57
Appendix C .............................................................................................................................. 58
C.1 simulation model without controller .............................................................................. 58
C.2 Simulation model with PID controller ........................................................................... 58
C.3 Simulation model with MPC controller ......................................................................... 59
C.4 Simulation model with PID-MPC controller ................................................................. 59
Appendix D .............................................................................................................................. 59
D.1 Bode plot of the PID controller scheme ........................................................................ 59
D.2 Bode plot of the MPC controller scheme ...................................................................... 60
D.3 Bode plot of the PID-MPC controller scheme .............................................................. 60
VI
Nomenclature Abbreviations
MPC Model Predictive Control
PID Proportional Integral and Derivative
COP Coefficient of performance
deg.C Degree Celsius
AR Autoregressive
ARX AR models with Extra Regressors
ARMAX ARMA models with Extra Regressors
ARMA Autoregressive moving average
BJ Box–Jenkins
OE Error Estimation
PI Proportional Integral
PD Proportional Derivative
MV Manipulated Variable
Z-N Ziegler Nichols
CHR Chien Hrones Reswick
TSP Temperature setpoint
Td Traditional method
Mod Modified method
Mathematical Symbols
𝑁𝑝 Prediction horizon
𝑁𝑐 Control horizon
∆T Change of temperature
TV Coolant temperature
TS Evaporation temperature
EC Electronically Commutated/ Brushless DC electric motor
R134a Refrigerant type
Total change of energy
Change of time
Constant depends on the refrigerant flow
The refrigerant flow rate
Constant depends on the water temperature
VII
The temperature of the refrigerant flowing inside the tube
Initial Energy
Constant which depends on outside temperature
Outside temperature
Constant depends on water and refrigerant
na Order of the polynomial A(q)
nb Order of the polynomial B(q) + 1
nc Order of the polynomial C(q)
nk Input-output delay expressed as fixed leading zeros of the B polynomial
( ) The rational transfer function
( ) The rational transfer function
( ) The cross covariance function
( ) Input autocorrelation
( ) Output autocorrelation
Kp Proportional gain
Ki Integral gain
Kd Derivative gain
Ti Integral time
Td Derivative gain
umin Minimum input flow rate
umax Maximum input flow rate
xmin Minimum state
xmax Maximum state
Tmin Minimum temperature
Tmax Maximum temperature
Mp Maximum overshoots
Mu Maximum undershoots
Tr Rise time
Tp Peak time
Ts Settling time
VIII
List of Figures
Figure 2.1 Overview of the heat pump system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Figure 2.2 Block diagram of the heat exchanger/condenser . . . . . . . . . . . . . . . . . . . . . . 6
Figure 2.3 Outside air temperatures during autumn season . . . . . . . . . . . . . . . . . . . . . . . 7
Figure 2.4 Comparison of the refrigerant coefficient of performance . . . . . . . . . . . . . . . 8
Figure 2.5 Outlet refrigerant temperatures from heat exchanger . . . . . . . . . . . . . . . . . . . 9
Figure 2.6 Outlet water temperatures from the heat exchanger . . . . . . . . . . . . . . . . . . . . 9
Figure 3.1 The system identification loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Figure 3.2 The data set time plot of the heat pump system . . . . . . . . . . . . . . . . . . . . . . . 14
Figure 3.3 Estimation data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Figure 3.4 Validation data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Figure 3.5 Step response plot for different model structure . . . . . . . . . . . . . . . . . . . . . . 16
Figure 3.6 Frequency response for different model structure . . . . . . . . . . . . . . . . . . . . . 16
Figure 3.7 The ARX model estimated output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Figure 3.8 The ARMAX model estimated output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Figure 3.9 Residual analysis of the ARX model structure . . . . . . . . . . . . . . . . . . . . . . . 19
Figure 3.10 Residual analysis of ARMAX model structure . . . . . . . . . . . . . . . . . . . . . . 20
Figure 3.11 Pole-Zero for the arx791 model structure . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Figure 3.12 Pole-Zero for the amx2422 model structure . . . . . . . . . . . . . . . . . . . . . . . . . 21
Figure 4.1 Block diagram of the PID-MPC controller scheme . . . . . . . . . . . . . . . . . . . . 24
Figure 4.2 Block diagram of PID controller for the condenser . . . . . . . . . . . . . . . . . . . . 27
Figure 4.3 Response curve for Ziegler Nichols method . . . . . . . . . . . . . . . . . . . . . . . . 28
Figure 4.4 The transient response specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Figure 4.5 Response curve using Traditional Ziegler-Nichols method . . . . . . . . . . . . . . 30
Figure 4.6 Response curve using Modified Ziegler Nichols method . . . . . . . . . . . . . . 31
Figure 4.7 Pole-Zero plot of the PID Controller scheme . . . . . . . . . . . . . . . . . . . . . . . . . 32
Figure 5.1 Block diagram of the MPC controller scheme . . . . . . . . . . . . . . . . . . . . . . . . 34
Figure 5.2 Prediction horizons tuning of the MPC controller . . . . . . . . . . . . . . . . . . . . . 37
Figure 5.3 Input weight tuning of the MPC controller . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Figure 5.4 Outlet water temperature using MPC controller . . . . . . . . . . . . . . . . . . . . . . 38
Figure 5.5 Poles and Zeros plot of MPC controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Figure 5.6 Outlet water temperature using PID-MPC controller . . . . . . . . . . . . . . . . . . . 39
IX
Figure 5.7 Poles and zeros plot of PID-MPC controller . . . . . . . . . . . . . . . . . . . . . . . . . 40
Figure 6.1 The outcome by Td. and Mod. PID tuning method . . . . . . . . . . . . . . . . . . . . 42
Figure 6.2 Outlet water temperature using PID and MPC controller . . . . . . . . . . . . . . . 42
Figure 6.3 Result comparison of PID, MPC and PID-MPC controller . . . . . . . . . . . . . . 43
X
List of Tables
Table 2.1 Transient response specifications for the system . . . . . . . . . . . . . . . . . . . . . . . 10
Table 3.1 ARX model structure specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Table 3.2 ARMAX model structure specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Table 3.3 The Pole-Zero locations of the arx791 model structure . . . . . . . . . . . . . . . . . . 21
Table 3.4 The Pole-Zero locations of the amx2422 model structure . . . . . . . . . . . . . . . . 21
Table 4.1 Ziegler-Nichols Tuning first (Traditional) method . . . . . . . . . . . . . . . . . . . . . 28
Table 4.2 Modified Ziegler-Nichols Tuning (CHR) method . . . . . . . . . . . . . . . . . . . . . 28
Table 4.3 Traditional Ziegler Nichols tuning method result . . . . . . . . . . . . . . . . . . . . . 29
Table 4.4 Modified Ziegler Nichols tuning method result . . . . . . . . . . . . . . . . . . . . . . . 29
Table 4.5 Comparison of controller parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Table 4.6 Transient responses of the Traditional Z-N tuning method . . . . . . . . . . . . . . . 31
Table 4.7 Transient responses of the Modified Z-N tuning method . . . . . . . . . . . . . . . . 31
Table 5.1 MPC tuning parameters value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Table 5.2 The transient response specifications of the MPC controller . . . . . . . . . . . . . 38
Table 5.3 The transient response specifications of the PID-MPC controller . . . . . . . . . . 40
Table 6.1 Experimental result for ARX and ARMAX models . . . . . . . . . . . . . . . . . . . . 41
Table 6.2 Transient response specifications comparison . . . . . . . . . . . . . . . . . . . . . . . . . 43
XI
Keywords
Water temperature control, System identification, system identification toolbox, Proportional
integral derivative (PID), Model predictive control (MPC), water flow control, Heat pump
control system.
Page:- 1
Chapter 1
Introduction The aim of this chapter is to present the introduction of the project and overview of the topics
presented in this report. This chapter will also cover the background, objectives and purposes
of the master’s thesis.
1.1 Overview
The Heating system is a system with a very high thermal inertia so a good control of the
system is always a challenge. The system is complex and dynamic so accurate control of the
system is difficult to realize. In the heat pump system energy is drawn from the surrounding
air and sun which is used to heat water stored in a conventional water tank. Heat pump water
heaters can be designed for installation as either an integral part of the water heater tank [1].
When water flow through heat exchanger/condenser, they give up or gain energy. Thus, the
driving temperature varies through the exchanger [2]. On the other hand if the water in the
tank is cold it has to be heated up, so a good control strategy is needed to maintain exact
temperature of water before it is filled in tank. The purpose of the air to water source heat
pump is to utilize the energy stored in the air or renewable energy sources so as to get a lesser
heating cost.
When controlling the heat pump we need to see the amount of power consumption and also
the user comfort must not be affected [3]. In any control system, the designing of the control
system is the most important thing. There are different types of controllers, which can be
conventional or intelligent. A controller measure and control the supply of water [4] to the
condenser. All heat pumps require a control system either to control water level in the tank or
the outlet water temperature from the condenser. This thesis presents a strategy of designing
the control system for the heat pump that maintains the temperature of domestic hot water
(DHW) supply with the help of PID and MPC controller. PID and MPC controllers are
selected for the reason that it gives good control action, more robustness and simplicity. Each
heat pump uses the hot refrigerant from the compressor to heat the water inside the condenser.
As the water temperature in our system is varying the goal of the controller will be to obtain
the control over the flow of the refrigerant to get a constant domestic hot water temperature.
An accurate model of the system is needed for the proper designing of the controller; it is
shown that the model can be obtained using system identification toolbox techniques where
the estimation and validation of the model is done.
The traditional and modified Ziegler Nichols tuning is studied and compared for the selection
of better achievement of the control action, the PID and MPC results are studied and from the
transient response behavior of both the controllers it is seen that PID and MPC combined
scheme performs better than only using the PID and MPC controller Therefore Model
Predictive Control (MPC) and PID control is the best advanced technique that will help in
obtaining the control of water temperature for heat pump.
1.2 Background
The heat pump consists of four main parts: a condenser, evaporator, compressor and an
expansion valve. When the compressed or hot refrigerant is passed from the condenser, heat is
transferred from a hot medium to a cold medium. In ground source heat pump, heat is
extracted from a bore hole and transferred to the refrigeration medium by a heat exchanger
called evaporator. When the pressure on refrigerant increases its temperature also increases
which develops heat.
Page:- 2
Heat is then transferred from the refrigerant medium to the water by an exchanger called
condenser. After the refrigerant transfers heat it is passed through expansion valve where the
pressure and the temperature are lowered. To minimize the power consumption we are using
air to water heat pump because as the temperature of water increases in heating the consumed
electrical power also increases. Therefore air to water heat pump is a good alternative for
saving electrical energy. The main idea in this thesis is to control the water temperature for
which the refrigerant flow towards condenser must be controlled. To achieve the best output
of the system before designing the controller the system need linear modeling. The model
based design describes the system identification procedure which is used to identify the
system. The purpose of system identification is to establish a mathematical model and use the
results of system identification to resolve practical problems by developing a controller [5].
System identification is used in the process of formulating the mathematical model of system
using the measurement data [6].
There are several steps used for identification procedure which include the model selection [7]
model estimation, validation and error analysis [8]. This wide variety of model structures and
identification methods provides the investigator with an extensive toolkit [9]. The residual,
correlations analysis [10] is very important to validate the design model. The PID and MPC
controller are used for the reason that it gives efficient and faster results closer to equilibrium
or the set point. One type of controller which is most widely used these days is the PID
controller. In practice PID controller gives good performance although its tuning is a bit
complex task but it gives accurate results. MPC is advanced controlling method among all
strategies. Model predictive control is used to predict the impact of certain control signals to
improve the performance of the system. At each sampling instant, information about the real
plant is gathered through measurements which then are used as input data for the internal
plant model. An algorithm of PID based on the Model Predictive control methods is derived.
The three parameters of PID namely KP, KI and KD are tuned to achieve better closed loop
performance. Depending on this algorithm for time delay system will enhance the real time
performance and reliability of the process control system.
On analyzing the three parameters it seen that the effect is not ideal so a new structure is
developed in this paper which can effectively solve this problem by introducing a feedback
from the actuator output to the controller. This structure provides an effective way for
modeling and control of the process [11]. A PID controller is selected for controlling the
temperature of the heat pump system. The comparison performances are done between the
PID controller and conventional on-off controller. Both the controllers are designed and
evaluated using Matlab Simulink software. The comparison of simulation results showed the
effectiveness of PID controller in maintaining inner refrigerator temperature than
conventional controller [12].
A PID controller of Heat Exchanger system is done in this thesis paper. In heat exchanger the
temperature control of outlet water is very important. Due to the disadvantages of the
conventional controller a model based PID controller is designed in this system to control the
outlet water temperature. With the implementation of designed model based PID controller
the temperature of the outlet fluid reaches the desired set point in the shortest time irrespective
of the disturbances. The transient response behavior of the system has shown improvement in
overshoot and settling time [13]. The Application of Nonlinear PID controller in main steam
temperature control is discussed. The fixed parameters of the PID lead to poor performance.
The ideal change between the error of the control object and control parameters are evaluated
and nonlinear PID is formed to remove the error. The parameters of nonlinear PID controller
are tuned using NCD block set in Simulink and it performs better than the traditional linear
PID controller [14]. A Hybrid PID-fuzzy control scheme is developed for managing energy
resources in buildings. A parallel structure of either combination of PID and Fuzzy controller
Page:- 3
is selected or Fuzzy supervision of PID controller. The simulations of the controlled scheme is
tested in a mock building set up and finally a criteria describing the way energy is used and
controlled is evaluated using the proposed controlled scheme [15].
In application of model predictive controller in agricultural processes the main aim is to
achieve temperature control of the greenhouse. In this work a real time model predictive
controller is designed to control the nonlinear system with constrained manipulated variables.
The linearized model is obtained at each sample instant and optimal control is achieved [16].
MPC controller is compared with an adaptive PID controller in terms of energy, economic
savings and transparency.
A predictive control is implemented to control the temperature of a batch reactor. First a
cascade control structure is implemented according to the heating or cooling system and the
differences in the sub unit’s dynamics are also considered [17]. Predictive functional control
is implemented for the temperature control of the exothermic chemical reactor. Its differences
with the MPC controller are studied. The results describe the performance of the cascade
control structure in maintaining the temperature of the batch reactor.
We studied from the previous work that the MPC controller is suitable for heating systems
and no other controllers like optimal or adaptive controls. The selection of the controller is
mainly depended on the type of the system and the predicted results. As the heating system is
dynamic and for the system like DHW heat pump the temperature is abruptly changing so an
advanced controller is needed that can adapt and control the fast variations of the temperature.
MPC controller predicts the future behavior of the system and gives control action in advance
so it is selected for heat pump. The controller checks and calculates the errors and quickly
gives the control action.
1.3 Problem formulation
The control of water temperature is an important factor in the operation of the heat pump
system. The heat pump in this thesis works on air to water energy and it is a complex and
dynamic system therefore the outlet temperature of water from the condenser keeps changing
constantly. The water is used for domestic purposes therefore a good control of water
temperature is needed. The outlet temperature of water from the condenser must be around
600C when it is filled in the tank for domestic use.
Therefore the main aim of the thesis is to control the outlet water temperature from the
condenser/heat exchanger of the heat pump. In doing so we control the refrigerant flow as it
plays a vital role in heating of water in the condenser. A good approach would be to
implement a PID controller along with model based controller for the system. The suggested
control system is small size and relatively less expensive than previous controller. The
construction and evaluation of a new control scheme will require a model of the condenser.
So another objective of this project is the modeling of the system in order to use it as a base
for the controller design.
1.4 Purposes of master’s thesis
The main focus of this master thesis is to design a controller to control the outlet water
temperature from the condenser. In order to obtain the control of the water temperature our
primary task would be to control the refrigerant flow. In doing so we need to completely
analyze how the system behaves in different conditions. An accurate model of the system is to
be obtained using mathematical modeling and also system identification methods and later on
a PID and model based controller would be designed to achieve better control action of the
system.
The final results of this project thesis will indicate what types of controller/controlling
techniques will be best for similar systems. This thesis project also has additional education
Page:- 4
purposes to finalization of the master’s degree and can be viewed as to apply the theoretical
knowledge into the real life engineering problem and also gets the deeper insights of the real
system modeling and controlling techniques.
1.5 Thesis Contribution
Throughout the master’s thesis we have worked together, however there are some tasks that
are contributed mostly by individual in below:
Analyzing and calculating the mathematical expression and design the PID controller ,
tuning of the controller and adjust the controller for the system. Studying the behavior
of the system (Md. Abdul Salam).
Modeling the Heat Pump System identification techniques, studying various models
structure behaviors that are suitable for the system. Studying the working of system in
different temperature conditions and its effects (Md Mafizul Islam).
Comparing the tuning methods for PID controller, Programming for the Controller in
MATLAB-Simulink, Design and simulation of Model Predictive controller (Md. Abdul
Salam).
Combining the model with controller for the control of water temperature. Model
Predictive Controller (MPC) design and tuning (Md. Mafizul Islam).
Page:- 5
Chapter 2
System description The aim of this chapter is to give an indication of the system at Hetvägg prototype 8; it also
provide a subterranean look of the system sections specially the system dynamics related to
the heat pump. The structure presented in this chapter will be the foundation for the
simulation.
2.1 Overview of the heat pump system
In this project an air source heat pump is used to heat the water temperature. In the heat pump
section the refrigerant in the evaporator is passing through the compressor. The compressor
compresses the refrigerant and it gets superheated which is the input of the condenser of the
heat pump and passes through a copper tube inside the condenser. The system shown in figure
2.1 is a heat pump that works to heat the cold water in the tank. The cold water from the
outside source is filled in the tank. The thermostat detects the temperature of the water and if
the temperature is below 450C the circulating pump starts working and it pumps the cold
water into the condenser for heating.
Figure 2.1 Overview of the heat pump system
The condenser transfers the heat energy from the compressed refrigerant flowing inside the
copper tube to the cold water and the resultant hot water is again filled back in the tank for the
use age. A compressor is used to increase the pressure of the refrigerant [18]. An immersion
heater which is operated by electric energy is placed at the bottom of the water tank and a
thermostat is placed 1/3 of the total height from the bottom of the tank. When the water
temperature goes down i.e. below 600C it increases the chance of legionella growth. So the
water temperature should not be less than 600C inside the tank. If the temperature decreases
below 450C thermostat reads this value and it gives signal to immersion heater and heater start
working for heating water in emergency conditions.
The evaporator sends the low pressure liquefied coolant to the compressor. The expansion
valve controls the high pressure of liquefied coolant which is streaming towards evaporator.
The behavior of the expansion valve can be studied by the calculating the temperature
difference at the inlet and outlet of the evaporator [19].
∆T = TV – Ts (2.1)
Page:- 6
where Ts = Evaporation temperature at the outlet of evaporator and Tv = Coolant temperature
at the inlet of evaporator. To reduce the electric consumption we have to keep running
immersion heater as less as possible. For better understanding of the system we are dividing
the whole system into two different systems that is primary system included the heat pump
section and the secondary section include the boiler section shown in figure 2.1. The heat
pump section includes the evaporation, compression, expansion and condensation parts and
the boiler section includes the water tank with placed immersion heater and thermostat inside
and circulation pump outside of the tank.
2.2 Heat transfer of the system
The condenser of the heat pump is used to transfer the heat energy from hot media to cold
media. It acts as a heat exchanger for the system [20]. It has two copper tubes with same
dimensions i.e. one is for air source heat pump and other one is for solar source heat pump
which is not used in this system.
Figure 2.2 Block diagram of the heat exchanger/condenser
From figure 2.2, the block diagram of the condenser where the input is the refrigerant and the
cold water. The output of the condenser is the hot water which is getting heat energy from
superheated refrigerant. The amount of energy transferred from copper tube to water with a
unit of time i.e. the total energy is directly proportional to the refrigerant flow rate and the
temperature of the air.
( ) ( ) ( ) (2.1)
where represent the total Energy of the system, is the constant value which depends
on the metal of the tube, is the refrigerant flow rate, is the constant value which
depends on water temperature and is the temperature of the refrigerant flowing inside
the tube. By taking the differential in equation (2.1) with respect to time we find the small
quantity of energy transferred,
( ) ( ) ( )
( ) ( ) ( ) (2.2)
To find the total energy transferred of the system if we take integration in equation (2.2) from
0 to t, we have
∫ ( )
∫ ( )
(2.3)
where, is the initial energy containing in the water. The system is not perfectly isolated so
the system will leave some temperature to the outside which will affect the system.
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If the outside temperature or disturbance is included in equation (2.3) we obtain the heat
transfer equation for the system shown in equation (2.4)
( ) ∫ ( )
∫ ( )
( ) ∫
(2.4)
where, is a constant. It depends on the outside surface and is the outside temperature
or room temperature. The constant value is directly proportional to the difference
between refrigerant and water temperature.
( ) (2.5)
In equation (2.4), the term is the heat transfer constant between water and refrigerant
[21].
The output water temperature from the condenser depends on the input water to the
condenser. The mass of input water is inversely proportional to the output water temperature
and directly proportional to the temperature getting from the heat transfer of the system.
2.3 Outside air temperature of the system
The outside air temperature or ambient temperature varies with time and it also depends on
weather conditions. The outside air temperature during winter and summer time is different.
In summer season the outside temperature is high so the outside air temperature also gives the
higher values compare to the winter season. In this heat pump water heating system have
heavy plastic condenser whose heat transfer coefficient is very less and also it is covered by a
case so the outside air temperature will affect the system. The simulation has been run in this
three days and the numbers of experimental result can be found with longer duration but for
simplicity the 98 samples of time has been taken which is the equal number of the samples of
discrete time system.
The heat transfer coefficient for the condenser is less as it is covered by a plastic case, see
appendix A2. Hence the ambient temperature will affect the system. In figure 2.3 the sample
is chosen with largest variations of the system during autumn season.
Figure 2.3 Outside air temperatures during autumn season
The detected outside air temperature from the sensor during autumn season is shown in figure
2.3. It can be seen that the minimum temperature is noted as 15.340C and the maximum
temperature fluctuation is 27.010C. In autumn operation it gave 1.01
0C higher peak than
winter maximum value and also the minimum value is 2.590C lesser than the winter operation
minimum value. Therefore the ambient temperature of autumn is chosen for the system
Page:- 8
modeling because of the large temperature variance for the real process. The range of the
ambient temperature that could affect the system performance is in between 1.060C~1.87
0C
which is found from the autumn operation of the system. It depends on the covers and the
tube materials of the heat exchanger. In this system, plastic cover and copper tube are used
and the experimental value found for autumn operation time shown in appendix A.4 by
considering the PVC plastic and copper tube heat transfer coefficients and their dimensions.
2.4 Refrigerant of the system
The main input to the condenser is the refrigerant flows and the cold water shown in block
diagram of the condenser in figure 2.2. The input refrigerant temperature depends on
refrigerant flows. The ambient temperature of condenser or outside surface temperature is the
output disturbance of the system. The cold water (100C) flows through the condenser to heat it
up. The flow rate of the inlet water to the condenser depends on the usage of the warm water
by the end user. The more warm water is used by the end user the more cold water needs to be
heated up and the flow rate of inlet water to the condenser will be high but the water inside
the condenser remain unchanged which is 3.902 kg can be seen in appendix A.3.
Modeling of the control system does not depend on the flow rate of inlet water it depends on
the amount of water contained in the condenser. The heat pump technology is very popular to
heat water for industrial and domestic purpose. However, the efficiency ratio of heat pump
water heaters is methodically related to the refrigerant used in the heat pump system. The
refrigerant R134a has been widely applied for industrial and domestic heat pump system. It
can be seen from figure 2.4 that it is really not a matter what kind of refrigerant is used, the
COP gradually declines with the decrease of the outside temperature/ambient temperature. To
find the best refrigerant for the system it is a need to evaluate the performance of R600a,
R290 (propane), R134a, and other refrigerants type in an optimized finned-tube air-to-
refrigerant evaporator and analyze its effect on the system coefficient of performance [22]
The increase of inlet water temperature of the copper tube condenser and the influence of
outside air temperature on the COP is 4.71%～8.33% greater than other refrigerant [23]. The
coefficient of performance can be found from the P-h diagram of the refrigerants. The P-h
diagram of the refrigerant R134a is shown in appendix A.6 with the description.
Figure 2.4 Comparison of the refrigerant coefficient of performance
The dynamic system’s refrigerant flow varies with time and with the varying evaporating
temperature. The refrigerant flow for the system is determined from the data analysis. The
refrigerant flow rate to the compressor depends on the evaporating temperature. When the
evaporating temperature reaches its minimum value which is -150C for this system, the
refrigerant flow reached its minimum value 9.40 kg/h. The Refrigerant flow from the
Page:- 9
evaporation meet the compressor before it passes through the condenser tube, where it is
compressed by the compressor and passed through a pressure switch which allows only high
pressure i.e. the flow rate is 27.51 kg/h～34.06 kg/h depending on the various condensation
and evaporation temperature.
2.5 Discharge of the heat exchanger
The cold water passes through the condenser to get heated. The heat transfer of the system
happens between cold media and the hot media when refrigerant passes through copper tube.
The cold media gains heat and the hot media loses heat i.e. the refrigerant lose energy and it
passes from condenser to expansion valve. The refrigerant temperature after losing heat is
shown in figure 2.5 and it is a continuous process. The input refrigerant flow has been taken
from the data analysis of the plant.
Figure 2.5 Outlet refrigerant temperatures from heat exchanger
The water temperature is the output response of the simulation design model with ambient
temperature using the condensing temperature range 350C ~55
0C and the evaporating
temperature range -150C ~ +10
0C. The water temperature is increasing from its initial values
to the maximum values. The condenser output water temperature is varying with change in
some parameter of the system such as input refrigerant and cold water temperature and the
outside air temperature or ambient temperature. The output water temperature from the
condenser is given in figure 2.6. From the above description of the system it is clear that the
system need accurate modeling and design of controller to improve the performance.
Figure 2.6 Outlet water temperatures from the heat exchanger without controller
Page:- 10
The condenser outlet water temperature without any controller shown in figure 2.6 gives the
response specifications shown in table 2.1. At time 8 minutes there is no change in the
response of the system due to the startup process. As the system runs it takes some time for
the refrigerant to reach the condenser and as the compressed refrigerant flows through the
condenser the water starts gaining heat and there is a change in the output response.
Table 2.1 Transient response specifications without controller
Response
specifications Overshoot Rise Time
Settling time Undershoot Peak time
values 6.22 16 26 7.78 55
The transient response specifications of the system are found from the Matlab for figure 2.6
when the system runs without a controller. It shows there is a high rise in overshoot, rise time
and settling time from the required range of temperature (600C). Therefore a controller is
needed to control the given range of temperature by minimizing the values in overshoot, rise
time and settling time and for the steady behavior of the system.
Page:- 11
Chapter 3
Modeling of the system The aim of this chapter is to describe the system identification procedure, modeling and
validations of the system. This chapter also describes and analyzes the dynamic system
behaviors.
3.1 System Identification Introduction
System identification is the procedure to find the model from data sets. In a dynamic system it
is very important to know the identity of the system. It is the science of building mathematical
models of dynamic systems from observed input-output data. The fundamental element in
science is to construct the models from observed data set for the system. System identification
is a very large topic especially for dynamic system with different techniques that depend on
the character of the models to be estimated. It is an iterative process and sometimes need to go
back to the previous steps and repeat it.
3.2 Model structure for identification method
The system input and output at sample k is given by u(k) and y(k) respectively. The dynamics
of the discrete time process is described by the following transfer function:
( )
( )
It is equivalent to the linear discrete time differential equation [56] is following
( ) ( ) ( ) ( ) ( ) (3.1)
The system’s input and output are chosen in discrete time, so that the observed data are
always collected in samples. In equation (3.1), the sampling interval is one time unit which is
not necessary but it makes the notation easier. The equation (3.1) can be written as a way of
determining the next output value given previous observations.
( ) ( ) ( ) ( ) ( )
The vector notation form is following
( ) ( ) ( ) ( ) ( )
Using the above vector notation the equation (3.1) can be rewritten as
( ) ( ) (3.2)
3.3 Model quality and experimental design
By taking n=0 in equation (3.1), the observe data for the process can be written as
( ) ( ) ( ) ( ) (3.3)
where e(k) is the white noise sequences with variances . So the equation (3.2) can be written
as
( ) ( ) ( ) (3.4)
The input sequences u(k) = 1,2,3,…..m. by replacing y(k) in equation (3.3) the obtained
expressions are
Page:- 12
(𝑁) [∑ ( ) ( ) ∑ ( ) ( )
]
(𝑁) ∑ ( ) ( )
(𝑁) ∑ ( ) ( )
The mathematical expectation of the system is following
(𝑁) ∑ ( ) ( ) (𝑁) ∑ ( ) ( )
(3.5)
The parameter error of the system can be defined as
(𝑁) ∑ ( ) ( )
( ) ( ) (𝑁)
(𝑁)
where e is a sequence of independent variables so that
( ) ( ) ( ) ( ) ( )
Thus the computed covariance matrix of the estimate is determined by the input properties
R(N) and the noise level .
𝑁 (𝑁)
The covariance matrix of the input of the ith
and jth
elements is
𝑁∑ ( ) ( )
If R is nonsingular the covariance of the parameter [44] is approximately given by
(3.6)
From equation (3.6), it can be seen that the covariance is proportional to the noise variance
and inversely proportional to the input power. The covariance does not depend on the input’s
or noise signal.
3.4 System identification principle
The system identification core [25] of estimating the model revolves around the following
concepts
Model: Model is the relationship between observed parameters. It allows the prediction of
properties or behaviors of the object.
True description: It is the description of the model which is the same character of the above
topic model but it covers more description and complexity of the system.
Model class: Model class is the set or collection of the models.
Estimation: It is the process of selecting a model. The data used for selecting the model is
commonly called Estimation data.
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Validation: This is the process to ensure the model that the model is not useful for estimation
data, also for data sets of interest.
Model fit: This is the measurement of the particular model that should be able to fit to a
particular data set. The best fit of the model is identified by getting error signal of the system.
3.5 System identification loop
The order of the steps in the loop does not only define the sequential order in which the tasks
are executed, but also how they influence each other. The system identification loop used to
implement and identify the dynamics is shown in figure 3.1.
Figure 3.1 The system identification loop
At the first step in the system identification procedure it is very important to state the purpose
of the model. Now days there are a huge variety of model applications, for example, the
model could be used for signal processing, control design, simulation and error detection.
Identification methods and experimental conditions depend on the purpose of the model so it
should therefore be clearly stated. If the model is used for control design, it is important to
have an accurate model around the desire choices. The identification experiment design
consists of a number of choices like which signal to manipulate or measure and how to
manipulate or measure. It also includes some practical aspects. The experimental data can be
changed only by a new experimental data [54]. The identification experimental designs are
done in mainly two steps. In the first step, preliminary identification experiment to get
primary knowledge about important system characteristics.
The step response, impulse responses are performed in this step. The information obtained
from the first step is then used to find the suitable experiments for the main experiments.
Some system characteristics of the preliminary experiments include time invariant, linearity,
transient response and frequency response analysis. In the main experiments, especially the
input signal is discussed. The identification gives the accurate model where the estimation
errors are lesser.
Page:- 14
3.6 System identification method
The above core concept for system identification will be described in details for the system in
below. The system identification toolbox with different models of the system are shown in
appendix B.1, in the toolbox state t=heating system, u=input refrigerant and y=output
temperature. By using the system identification toolbox, the refrigerant flows (u) used as
input data sets and water temperature (y) used as output data sets.
Figure 3.2 The data set time plot of the heat pump system
Figure 3.2 shows the input and output data sets used in the system identification are 98
samples of time plot which are found from the simulation of the mathematical equations
without effect of outside temperature and using the compressor performance check point data
at standard operating or testing conditions, the plotted input and output data are shown in
figure 3.2. The system identification procedure is executed using the data examination, model
structure selection, model estimation and model validation. These four steps are described in
sections 3.7-3.10 in details.
3.7 Data Examination
The input and output data sets sequence without any disturbance effect and standard testing
operating condition of the compressor are used to detect the data. The input and output data
sequences shown in figures 3.3 & 3.4 are divided into estimation and validation data sets. The
estimation and validation data are used to test the model characteristics, as it defines the
fitting percentages of the model and the errors associated with the design model. The total 98
samples input and output data sets are used of the time plot to identify the model where the
first 50% i.e. from 0 to 49 of the input and output data sets are used for estimation purpose
and the rest 49 to 98 samples of the data sets are used for validation purpose.
Figure 3.3 Estimation data Figure 3.4 Validation data
The select ranges option in the system identification toolbox processor is used to define the
boundaries for estimation and validation data and, then the data set was split into two separate
Page:- 15
parts. The first part of the separated data is used for estimation or identification and the
remaining part of the data is used for validation as shown in above figures. After estimation
and validation of the data sets it is required to check outliers, aliasing effects and the trends.
The outliers are the observations that are separated in some manner from the rest of the data.
According to their location the outliers may have moderate to severe effects on the regression
model [26].
It seen from the data observations that no outliers are obtained for the system. If there are any
aliasing effects in the experimental data sets, it can be improved by increasing the sampling
rate. In this case, the sampling time 1 second is used to get the best signals without aliasing.
The mean of the input and output signals are removed from the experimental data sets to
detect the linear trends of the input and output data.
3.8 Model structure selection
The model estimation is performed to determine the model structure set. It can be a very
simple model set such as the static gain K mapping the input to the output. The simple static
gain mapping for discrete time model is ( ) ( ). The model structure can be complex
which can affect the accuracy of the model to approximate the real process. In some cases the
simple models can be well approximated by using the simple model similar to discrete model
as seen above. The most common model structure in discrete time domain form used for
system identification process is given by
( ) ( ) ( )
( ) ( )
( )
( ) ( ) (3.7)
where u and y is the input and output sequences respectively, e(k) is a white noise with zero
mean. The polynomials A, B, C, D and F are defined as
( )
( )
( ) 𝑐 𝑐
(3.8)
( )
( )
The system model can be divided into AR, ARX, ARMAX, BJ, and OE [25]. The form of
model structure with one or more polynomials are identified as following
AR model
( ) ( ) ( ) (3.9)
ARX model
( ) ( ) ( ) ( ) ( )) (3.10)
ARMAX model
( ) ( ) ( ) ( ) ( ) ( ) (3.11)
Box-Jenkins (BJ) model
( ) ( )
( ) ( )
( )
( ) ( ) (3.12)
Output-error (OE) model
( ) ( )
( ) ( ) ( ) (3.13)
Page:- 16
The models shown in equations (3.9-3.13) are implemented from the system identification
toolbox shown in appendix B.1 and some test on analysis is done to select the best model
structure for the system. The choice of model structure depends on the estimation of the input
and output data sequences. It is not always necessary that a model structure with more
parameters and more polynomials is better. The best model is a matter of choosing a suitable
structure in combination with the number of parameters using the poles as less as possible for
lower orders. The estimated step response plot of the ARX, ARMAX, BJ and OE models are
shown in figure 3.5 as a reference model to check the response of the model.
Figure 3.5 Step response plot for different model structure
The step response analysis gives information on stationary gain, dominating time constant and
time delay. An indication of the disturbances acting on the system is also obtained from the
step response. The step response signals from input to output for ARX, ARMAX and OE
models are responding with time but BJ model is not responding with time. The frequency
responses of the ARX, ARMAX, BJ and OE models used as reference models are given
below in figure 3.6.
Figure 3.6 Frequency responses for different model structure
In figure 3.6, the ARX, ARMAX and OE structured model gives the frequency response
curve of the dynamic system. The frequency response for the system is used for the
quantitative measure of the out spectrum of the system and it is used to characterize the
dynamics of the system. The ARX model shows large phase offset because of the polynomial
difference and e(k) of the system..
The BJ model structure is not responding for the heat pump system. A sufficient condition for
the predictor to be stable is that the C(q)and F(q) are stable for all (Lemma 4.1). The
ARX, ARMAX and OE model with different polynomial orders are following these
conditions of stability whereas the BJ model for any polynomial orders does not follow that
Page:- 17
condition. It is well known from the system identification textbook [24] that in the prediction
error structure the predictors needs to be stable. When the ARX and ARMAX model
structures are used this isn’t a problem because the dynamic model and the noise model share
denominator polynomials and when the predictors are formed it cancel the polynomials. But
for BJ model it’s not the case and if the underlying system is unstable, the predictors will
basically be unstable and this makes the model structure inapplicable for the system. When
parameter estimation algorithm is implemented for the Box-Jenkins case, typically we should
secure stability in every iteration of the algorithm projecting the parameter vector into the
region of stability. For the system, this process of course leads to erroneous results [27].
The above analysis of the step and frequency response it can be seen that if the ARX and
ARMAX models compute in different orders or ways it can give the accurate models and it
contain fundamental characteristics of the true process.
3.9 Model Estimation
Model estimation is a procedure for fitting a model with a specified model structure given in
equations (3.9-3.13). Modeling errors are not to be considered systematic errors in the
observations [28]. The models have different structure such as ARX, ARMAX, BJ models.
3.9.1 Estimation of the ARX model structure The computed ARX models are to find suitable orders and delays the following equation
(3.14) & used to estimate for different polynomials orders.
( ) ( ) ( ) ( ) ( )(3.14)
where na nb and nk are in the range from 1 to 10. For each estimated model, the prediction
errors and sum of squares are computed. In figure 3.7, the best fit two ARX model are
presented by considering the prediction error and percentages of fitting the model with the
estimated data. In this figure, y axis represents the approximate water temperature of the
system of the estimation data. The measured and simulated output validation data from the
system id toolbox are presented in figure 3.7 using the ARX model structure.
Figure 3.7 The ARX models estimated output
The following table 3.1 shows the computed final prediction & mean square error for different
polynomial orders of the ARX model structure
Table 3.1 ARX model structure specifications
Model FPE MSE Fit (%)
arx791 0.0002 0.0001 99.96
arx611 0.54 0.45 96.64
arx221 0.17 0.16 97.61
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From the ARX models shown in appendix B.1, the following models arx791, arx611 and
arx221 model structure shows the less prediction and mean square errors compare to further
polynomial ARX model structure. So the ARX models shown in table 3.1 have been
considered for validation test.
3.9.2 Estimation of the ARMAX model structure The computed ARMAX models are to find suitable orders and delays the following equation
(3.15) & used to estimate for different polynomials orders.
( ) ( ) ( ) ( ) ( )
𝑐 ( ) 𝑐 ( ) (3.15)
with 𝑐( ) 𝑐 𝑐
(3.16)
For each estimated ARMAX model, the prediction errors and sum of squares are computed.
In the figure 3.8, the best fit two ARMAX model are presented by considering the prediction
error and percentages of fitting the model with the estimated data. The measured and
simulated model output plots from the system id toolbox are presented in figure 3.8 using the
ARMAX model structure.
Figure 3.8 The ARMAX models estimated output
The computed prediction and mean square errors for different polynomial orders of the
ARMAX models are given in table 3.2
Table 3.2 ARMAX model structure specifications
Model FPE MSE Fit (%)
amx4422 0.04 0.08 98.63
amx6422 0.03 1.84 91.88
amx2422 0.04 0.10 98.39
From the ARMAX models shown in appendix B.1, the following models amx4422, amx6422
and amx2422 models shows the less prediction and mean square errors compare to other
ARMAX models. Therefore the ARMAX models shown in table 3.2 are selected for final
validation.
3.10 Model Validation
The obtained model can validate in a variety of ways. In a typical identification all of these
are used to confirm an accurate model structure.
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3.10.1 Residuals analysis The residual analysis for different models of a system is very important to get the best model.
It is the analysis of a signal that describes the quantity of signal contain at the end of the
process [29]. The parametric model describes in section 3.8 is in the form
( ) ( ) ( ) ( ) ( ) (3.17)
where ( ) and ( ) are the rational transfer function. The residuals are computed from
the input output data as
( ) ( ) ( ) ( ) ( ) (3.18)
The residuals are computed based on the data used for the identification and the identified
model and, then ideally the residuals should be white and independent of the input signals.
The residuals analysis can be done in several ways such as the autocorrelation of the input
output signals for the residuals, the cross-correlation between the residuals and the input and
distribution of residual zero crossings. The covariance function is estimated as
( )
∑ ( ) ( )
(3.19)
where ( ) represent the cross covariance or cross-correlation of the input and output
signals [54]. Similarly, the auto-covariance or autocorrelation function ( ) and ( ) are
respectively. The impulse response estimate can be derived using the relationship
( ) ∑ ( ) ( ) (3.20)
The simplified form of the equation (3.20) is given below when u is the white noise sequence.
( )
( ) (3.21)
Correlation function is rather elusive when it’s being measured. Extreme care must be taken
to ensure that the measurement method itself does not introduce large errors. The problem
associated with the accuracy has been examined carefully and also another problem that has
not received the same degree of attention [29].
Figure 3.9 Residual analysis of the ARX model structure
arx791
arx221
arx611
Page:- 20
The residual analysis are best fit two ARX models shown in figure 3.9 with autocorrelation of
residuals for the output of the validation data and the cross correlation for input and the output
residuals of the validation data [30]. In the figure 3.10 the residual analysis of different order
ARMX models residuals are following
Figure 3.10 Residual analysis of ARMAX model structure
It is seen from analysis of figure 3.9 and 3.10 the models pass whiteness and independence
and it shows significant correlation between past inputs and the residuals. The stability is the
key concept in control system design. It is very important for the dynamic system to be stable.
The system can be input output stable if and only if its poles are inside the unit circle [31].
3.10.2 Pole-Zero analysis
The poles and zeros are the properties of a system. A system is characterized by its poles and
zeros. The poles and zeros plot is represented graphically by plotting their locations on the
complex z-plane. The plots variable z represents the axes which have imaginary and real
values. The location of the poles are usually marked by a cross (×) and zeros location are
marked by a circle (◦). The poles and zeros location provide qualitative insights of the
response characteristics of the system. The poles and zeros location for the ARX model is
shown in figure 3.11.
Figure 3.11 Pole-Zeros for the arx791 model structure
amx2422
amx4422
amx6422
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From figure 3.11 it is clear that the arx791 model has 9 poles and 8 zeros. The poles and zeros
location are shown in table 3.3. However the order of the model is the number of poles [32].
The arx791 model structure characterizes 9th
order of the system.
Table 3.3 The Pole-Zero locations of the arx791 model structure
No.of pole-zero Poles location Zeros location
1 0 -1.86
2 0 1.14 + 1.17i
3 0.45 + 1.78i 1.14 - 1.17i
4 0.45 - 1.78i 0.9
5 0.89 0.26 + 1.01i
6 0.04 + 0.37i 0.26 - 1.01i
7 0.04 - 0.37i -0.37 + 0.37i
8 0.45 + 0.07i -0.37 - 0.37i
9 0.45 - 0.07i
The poles and zeros for amx2422 model structure are given in table 3.4. In figure 3.12, shows
the amx2422 has 4 poles and 3 zeros.
Figure 3.12 Pole-Zero for the amx2422 model structure
The location of the poles and zeros are presented in table 3.4. The order of the amx2422
model is 4 which is less compared to the arx791 model as the amx2422 model structure has
less number of poles and zeros
Table 3.4 The Pole-Zero locations of the amx2422 model structure
No of poles/zeros Poles location Zeros location
1 0 0.17 + 1.36i
2 0 0.17 - 1.36i
3 0.55 + 0.20i -0.37
4 0.55 - 0.20i
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The all poles location for the amx2422 model is inside the unit circle with double pole at
location 0 of the z- plane. The amx2422 model has 4 poles so the order of amx2422 model is
4. Due to the fact that all poles are located inside the unit circle the system is stable and the
response is decaying. The less number of poles and zeros give lesser order of the system that
synchronizes well with controller design.
3.11 Fitting model for controller design
The autoregressive (AR) moving average (MA) independent variable (x variable) ARMAX
model is same as ARX model with additional part moving average c(z)e(t). The numerical
numbers of the ARMAX (2,4,2,2) model represents the polynomial orders. The order of the
polynomial A(z), na equals 2, the order of the polynomial B(z)+1, nb equals 4, The order of
the polynomial C(z), nc equals 2 and nk equals 2 is the input output delay. In this system the
ARMAX (2,4,2,2) is best fit because it passes validation test successfully and it also has less
poles and zeros compared to the other model. The numerical calculations of the ARMAX
(2,4,2,2) model are
( ) ( ) ( ) (3.22)
( ) ( ) ( ) ( ) ( ) (3.23)
( ) ( ) ( )
( ) ( ) ( )
The amx2422 model is attained in the standard state space form
(3.25)
The linear state space model can be written in discrete form as
( ) ( ) ( )
( ) ( ) (3.26)
By taking the Laplace transform in equation (3.25) the output equation ( ) can be written as
( ) ( ) ( ) (3.27)
Here, D matrix is zero because the horizon control where the present information of the plant
model is important for prediction and control. As a consequence of this it is considered that
the input cannot affect the output at the same time i.e. D=0.
The matrices A, B, C and D are calculated from the state space form of the plant model.
The matrix [
], [
] and [
]
The frequency response of a system is the computable measure of the output spectrum of a
system in response to a stimulus. It is used to distinguish the dynamics of the system. It is the
measuring of the magnitude and phase response of the output as a function of frequency. The
frequency response can also be described using the Bode plot [33]. The frequency response
can be written as the transfer function.
( )) ( ))
( ))
Page:- 23
( )) ∑ ( ) ( ) ( )
( )) ∑ ( ) ( ) ( )
The frequency response can be defined using the pole -zero plot of the system except for the
arbitrary gain constant. The gain margin of the amx2422 model structure is 0.16 dB with the
gain frequency 1.9 rad/sec where the phase margin represents the infinite value.
From the residual and pole-zero analysis of system identification shown in section 3.10 the
amx2422 model is the most appropriate model. The amx2422 model is suitable for designing
the controller of the system because model shows less order compared to other model
(arx791). Also all poles of the amx2422 model are inside of the unit circle. So the amx2422
can be the most perfect model in designing the proposed controller (PID and MPC) for the
system.
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Chapter 4
Controller design The aim of this chapter is to present the design and evaluating process of the PID controller.
Due to the scope of this thesis project the design should be fairly simple and should be seen as
the good results for the specified system.
4.1 Controllers of a system
Controllers are a tool for regulating the dynamical systems so that desirable behavior is
obtained. The goal is to create the output signal from the system which is close to the set point
value and to minimize the overshoots and undershoots from the system. There are several
controllers that can be used for controlling the system like PI, PD, PID, LQG, Fuzzy Logic,
MPC etc. In this thesis PID and MPC are used to design the control system for heat pump. In
this chapter, PID controller design and evaluating process will be discussed in details. First a
description of control is given and later we will build the whole control system in Matlab-
Simulink.
4.1.1 Proposed controllers The armax2422 model found in chapter 3 will be used to design controllers. There are two
controllers (PID and MPC) are proposed for controlling the outlet temperature. In practice,
PID controller methods is widely used because of its good performance although its tuning
makes bit complex for that we will work on PID controller method and its tuning for the
system to get output more accurate. The MPC is the advanced prediction based controlling
method. In this project PID controller and Model predictive controller (MPC) will be used
where the MPC is used to predict the impact of a certain control signal to improve the
performance of the system.
In this thesis the PID and MPC controller are used to design the control system for heat pump
because it has been seen from the previous work related to the temperature control of heat
pump, PID and MPC shows better performance compare to the optimal or any other
controllers.
Figure 4.1 shows the flow chart of the system flow in designing the proposed PID-MPC
controller.
Figure 4.1 Block diagram of the PID-MPC controller scheme
The model is used to calculate the future response of the plant which in turn is used to
optimize the control signal. The control optimization is dependent mainly on the prediction
horizon (Np) and the control horizon (Nc) and internal model sends out the new control signal
to the system. Additional tuning and modifications needs to be performed before the
controller performance can be deemed satisfactory.
Page:- 25
4.2 PID Controller
A proportional-integral-derivative controller (PID) is a feedback controller that is widely used
in many control applications. The main function of PID controller is to minimize the error, A
PID calculates the error between the measured process variable and desired set point and then
gives a corrective action to adjust the process according to the set point and to keep the error
as low as possible. The proportional term gives reaction based on the current error; the
integral value determines the action based on the sum of recent errors and derivative value
gives the reaction based on the rate at which error changes. The combine action of these three
parameters helps in generating a control signal to adjust the process to the desired value. The
equation of the PID is given as
( ) ( ) ∫ ( )
( )
( )
By tuning the three parameters in the PID controller algorithm we can obtain a control action
based on the process requirements. The response of the controller is dependent on the
responsiveness of controller towards the error, the degree at which the controller overshoots
the set point and the system oscillations. The use of PID algorithm does not guarantee the
optimal control of the system or system stability.
In some control applications only two modes of parameters are required to achieve the control
of the process. This can be done by setting the gain of undesired control outputs to zero.
Depending on the absence of the control actions a PID controller is called a P, PI, PD or I
controller. The most common type of PID controller used in industries is a PI controller and
the reason for the absence of the derivative term is because it is sensitive to noise. The
performance of PID may affect where the systems are too complex [34]. The system will not
reach its target value if the integral term is neglected. Therefore the combination of PI
controller is the most common form. The applications of PID is very vast, it can be
implemented on a system with minimal information [35].
4.2.1 PID controller Theory The PID controller gives the manipulated variable (MV) which is the weighted sum of its
three correcting terms namely proportional, integral and the derivative.
( ) ( )
where the sum of gives the total output from the PID controller from each of
its parameters.
4.2.2 Proportional term The proportional term depends on the current error value by making a change to the output
that is proportional to the current error value. The proportional response term could be
adjusted by multiplying the error by a constant Kp, which is proportional gain and is given by
( )
where, output of the proportional term, proportional gain and Error.
A high proportional gain will lead to a large change in the output for a given change in the
error. A large proportional gain can make a system unstable and a small proportional gain will
lead to a less responsive or sensitive controller due to which the control action may be too
small while responding to system disturbances. When there are no disturbances, the
proportional term will retain a steady state error which is a function of proportional and
process gains. In PID controller it is mainly the proportional term that makes a major
contribution to the output change.
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Proportional gain ( )
Larger values give faster response. When KP becomes too large there is a possibility of system
getting unstable and if it is too small the system response will be sluggish [37]. If the error is
large that means proportional term compensation is also large. Process instability can occur
with an excessively large proportional term.
4.2.3 Integral term The output of the integral term is proportional to magnitude and the duration of error. The
integral mode will continuously increment or decrement the controller output to reduce the
error as long as there is an error present in the system. When the error is large, the integral
output will increment or decrement the controller output fast and if it is small the changes will
be slower. Also when the integral time (Ti) is large the response of the controller is slower and
when it is small the response is faster. Integration of error gives the accumulated offset that is
multiplied by the integration gain and added to controller output. The magnitude of the overall
contribution of the integral term is determined by the integral gain .
∫ ( )
where, = Integral output, = Integral gain, e = error and t is the instantaneous time.
The combination of integral term with the proportional term will give the output closer to the
set point and eliminates the residual steady state error that occurs only with the proportional
controller. The integral term responds to the accumulated errors from the past so it may cause
the present value to overshoot the set point, therefore a combination of PI controller gives a
better output.
Integral gain ( )
With large values of integral gain steady state error is eliminated faster but the outcome is
large overshoot. Any negative value of error integrated during the transient response must be
integrated by the positive error before reaching steady state error.
4.2.4 Derivative Term The rate of change of process error is calculated by determining the derivative of error with
respect to time and the contribution of derivative term is given by derivative gain .
( )
where, is derivative output, is derivative gain, e is error and t is instantaneous time
The main function of the derivative term shows the rate of change of controller output and its
effect is seen close to the set point.
The function of the derivative term is to reduce the overshoot caused by the integral term and
to improve the combined performance of the controller. As we know the differentiation of the
signal amplifies the noise and this term is highly sensitive to noise and larger values of
derivative gain which could lead to an unstable system. The differential control is mainly used
to suppress the noise caused by the derivative [36]
Derivative gain ( )
With large values of derivative gain overshoot created by integral term could be reduced but
could also lead to signal noise amplification with the differentiation of the error.
The discretized form of the PID controller is following
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All the three values of proportional, integral, derivatives are combined to calculate the output
of the PID controller. Defining ( ) as combined controller output which could be given as
( ) ( )
∑ ( )
( ( ) ( ))
( )
The control signal is calculated with reference to a base level, uo
4.3 PID Controller for the heat pump
In design PID controller some steps have to be followed. In figure 4.2 shows the block
diagram of the process flow for designing the PID controller.
Figure 4.2 Block diagram of PID controller for the condenser
Figure above shows the block diagram of closed loop PID controller for condenser to control
the water temperature. Here the set point of the controller is the desired water temperature. By
controlling the refrigerant flow the outlet water temperature from the condenser is controlled.
After controlling the refrigerant flow and temperature, PID sends control signal( ) to the
plant or condenser to get the output closer to the set value for the outlet temperature of water
from condenser.
4.4 PID controller tuning rules
There are a variety of techniques for the tuning of PID controller depending on the
information about the controlled process. Many tuning methods depend on the model of the
process and from the parameters of the model the controller parameters can be found
according to some rule. One technique that can be implemented to identify the properties of
the plant is the reaction curve method or step response method. From the given technique,
properties like static gain, overshoot, settling time and dominating time constants can be
obtained [38]. Also the tuning method is the best technique among all possible tuning for PID.
4.4.1 Ziegler Nichols Tuning The Ziegler Nichols is a heuristic method of tuning PID controller. After conducting a lot of
experiments Ziegler Nichols proposed the rules for tuning the controller and finding values of
KP, KI and KD based on transient step response of the plant.
The Ziegler Nichols proposed numerous methods for tuning but we use two methods in this
thesis, the Traditional method and Modified method of tuning. It applies to the plant whose
unit step response is an S-shaped curve with no overshoot. This S-shaped curve is also called
as reaction curve. In this thesis among the two methods we used the approach which gave us
the best possible results in obtaining the control of the system.
This method is most suitable for tuning PID controllers that uses proportional, derivative and
integral actions. This approach tests the open loop reaction of the process to a change in
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control variable output [36]. Ziegler Nichols derived the following control parameters based
on this model. The following model is also known as unit step response curve of plant model.
Figure 4.3 Response curve for Ziegler Nichols method [39]
4.4.2 Traditional Z-N tuning Method This method is applied on the step response of the plant and it is also called as reaction curve
or step response method. This technique is characterized by two constants delay time (L) and
time constant (T) [40]. These constants are obtained by drawing a tangent on the point of
inflection of the curve and then finding the intersections of the tangent line with the steady
state line and the time axis as shown in figure 4.3. The model of the plant [37] is therefore
( )
(4.4)
After getting the parameters L & T we can set the values of according to the
formula given in the table 4.1. The following obtained values of will help in the
tuning of the controller and will give an output response for our system
Table 4.1 Ziegler-Nichols Tuning first (traditional) method [39]
Controller KP Ti Td
P T/L 0 0
PI 0.9T/L L/0.3 0
PID 1.2T/L 2L 0.5L
4.4.3 Modified Z-N Tuning Method In Modified Ziegler Nichols Technique we use Chien-Hrones-Reswick (CHR) tuning
algorithm which emphasis on set point regulation [25]. The CHR method uses the time
constant T of the plant to determine Ti and Td compared to the traditional Ziegler Nichols
tuning formula. This is more dependent on the set point of the system [40]. The CHR PID
controller tuning formulas are given in the table 4.2 below
Table 4.2 Modified Ziegler-Nichols Tuning (CHR) method
Controller Type KP Ti Td
P 0.7/a 0 0
PI 0.6/a T 0
PID 0.95/a 1.4T 0.47T
In a real time process a number of plants are modeled by the above transfer function. If there
is no possibility of deriving the system model then it could be possible to extract the
parameters of the system. For example if the step response of the plant is obtained, the
parameters K, L, T or a ( ⁄ ) can be obtained by the Ziegler Nichols technique
Page:- 29
approach [36] shown in figure 4.2. The controller parameters are obtained by the formulas
shown in table 4.1 and table 4.2. The Modified technique is different from the traditional
technique in the way that in Modified we consider the set point or the desired value ( ⁄ ) to get the output more closely to the set point or the target value.
The integral gain (KI) and Derivative gain (KD) can be found by using the formulas, and for both traditional and modified techniques.
4.5 PID tuning for the system
The tuning of the PID controller is done based on the traditional Ziegler Nichols tuning rules
and also on the Modified technique for obtaining the necessary parameter values needed for
the evaluation of the PID parameters.
The step response of the plant model (condenser) will give the two main parameters needed to
get the PID parameters. The L (delay time parameter) and T (time constant) are computed by
drawing tangents at its point of inflection on the step response curve shown in figure 4.3. The
inflections points are basically the point of intersections of the vertical axis which is
correlated with the steady state value and horizontal time axis. The horizontal trace of the
tangent line is ‘T’. The coordinate formed by the point of interception of the two lines (a, T)
for our system is (60, 39). Where ‘a’ is the set point value 60 degC water temperature.
L = 3, a = 60
T = T-L = 36.
After getting the L, T and a value from the above plot, we will find the values of the gain
parameters according to the table 4.1 and 4.2. So the updated parameters according to
Traditional and Modified Ziegler- Nichols method is as follows
Table 4.3 Traditional Ziegler Nichols tuning method result
Controller Type KP Ti Td
PID 14.4 6 1.5
After applying Traditional Ziegler Nichols Tuning approach the following parameters for KP,
KI and KD are obtained.
KP = 14.4, KI = 2.4, KD = 21.6
Table 4.4 Modified Ziegler Nichols tuning method result
Controller Type Kp Ti Td
PID 0.19 50.4 16.92
The following controller parameters are obtained when we apply Modified Ziegler Nichols
Tuning approach.
KP = 0.19, KI = 0.0037, KD = 3.214
Table 4.5 Comparison of controller parameters
PID KP KI KD
Traditional Z-N 14.4 2.4 21.6
Modified Z-N 0.19 0.004 3.21
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4.6 Transient response specifications
The transient response is one of the most significant characteristics for control system. The
desired performance characteristics of control system design can be given in terms of transient
response specifications of the system.
The transient response of a practical control system often displays damped oscillations before
it reach to a steady state position. In specifying the transient response characteristics of a
control system it is common to name the following
Figure 4.4 The transient response specifications
The overshoot Mp is the values from the desire setpoint to peak value, the undershoot is Mu
calculated from the setpoint to the lowest value of the response curve after reaching the
setpoint value, the tolerance range is (±10C) from the setpoint, the rise time tr represents for
the rise 0% to 100% for 4th
order system. The peak time tp is the time value which is
calculated from 0 to the time need to reach its peak value and the settling times ts is the time
required for the response curve to reach and stay within the tolerance range of the final desire
value. The settling time is the largest time constant of the system. The transient response
requirement for the system using the PID controller is that it should not exceed the values
given in table 2.1.
4.6.1 Traditional Ziegler-Nichols response After applying the controller parameters obtained from the Traditional Ziegler-Nichols tuning
rules we obtained the following response with a considerable amount of change in the output.
Figure 4.5 Response curve using Traditional Ziegler-Nichols method
The output reaches the set point with minimum settling time and rise time. The transient
response of the controller with the Traditional Ziegler-Nichols tuning method is also given to
analyze which technique gives best results. The transient response behaviors from the
Traditional Ziegler-Nichols tuning method are given in table 4.6.
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Table 4.6 Transient responses of the Traditional Z-N tuning method
Method Maximum
Overshoot
Rise
Time
Settling
Time
Undershoot Peak
time
Traditional Z-N tuning 4 4 22 0.08 16
Using Traditional Ziegler-Nichols tuning method after tuning the parameter gains of the
controller it could be seen that the water temperature is around the desired set point
temperature.
4.6.2 Modified Ziegler-Nichols response
We apply the controller parameters obtained from Modified Ziegler-Nichols tuning technique
so as to get the best possible output response for our system and to achieve a good control.
The response of the Modified Ziegler-Nichols tuning is shown in figure 4.6.
After tuning the PID controller with Modified tuning there is a substantial improvement in the
rise time, settling time and overshoot. However the water temperature is not closer to the
desired set point temperature. Therefore we implement a MPC controller to achieve better
results for the system.
Figure 4.6 Response curve using Modified Ziegler Nichols method
The transient response behaviors from the Modified Ziegler-Nichols tuning method is given
in table 4.7
Table 4.7 Transient responses of the Modified Z-N tuning method
Method Maximum
Overshoot
Rise
Time
Settling
time
Undershoot Peak
time
Modified Z-N tuning 3.017 4 19 0 16
The final output temperature from this controller is 61.780C. After tuning the PID controller
using the Modified Ziegler-Nichols tuning method it could be seen that the water temperature
is closer to the desired temperature and a considerable improvement in the transient response
of the system compared to the traditional method. A combined output of both traditional and
modified approach is given in figure 6.1.
4.7 Pole-Zero analysis of the PID Controller
It is clear from the above transient response specifications of the traditional Ziegler-Nichols
response and Modified Ziegler-Nichols response that the modified PID tuning techniques
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gives less values compare to the traditional PID tuning techniques. So modified can give the
output closer to the desired set point value.
As we know many properties of a system can be obtained from the PID controller or feedback
control system. The PID control behavior can be obtained from a few dominant poles of the
closed loop system.
Figure 4.7 Pole-Zero plot of the PID Controller scheme
The poles and zeros for a typical feedback system can differ significantly [41]. The PID
controller is designed for the ARMAX 2422 model. In ARMAX 2422 model of this system
we have 4 poles and 3 zeros and all poles are inside the unit circle which means the system is
in stable condition. A PID controller is implemented on this model we obtained 2 poles and 2
zeros. The poles are on the unit circle in z plane which shows the system is marginally stable.
Figure 4.7 shows the poles and zeros of the PID controller. The distance of the poles from the
origin determines the envelope of the sinusoidal signal, and the angle with the real positive
axis [33]. As the poles are on the unit circle the system is marginally stable but if the poles
goes outside the unit circle the response becomes unbounded and unstable [58].
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Chapter 5
MPC controller design The aim of this chapter is to present the design and evaluating process of the MPC controller
which is suggested in chapter 4. Also describes the response of the combination of PID and
MPC controller. The basic idea and purpose to use a controller in a process is described in
chapter 4 section 4.1.
5.1 MPC Introduction
Model predictive control is the advanced controller [42] method of control that has been used
in many industries such as chemical plants, oil refineries and process control. Model
predictive control is developed based on the dynamic model of the process, mostly the linear
empirical models obtained by system identification.
The applied models are used to interpret the behavior of complex dynamical system. The
models must reimburse for the impact of non-linearity. Hence models are used to predict the
behavior of the dependent variables or outputs of the dynamical system with respect to change
in the process independent variables or inputs.
The model predictive controller uses the model and current plant measurements to calculate
future behavior in the independent variable that will result in the control of plant. MPC sends
the set of independent variable moves to the corresponding regulatory controller set points to
be implemented in the process.
5.2 MPC Model
The performance of the Model Predictive Controller depends mainly on the accuracy of the
internal model structure. The model used in a system may be different which depends on the
information of the plant. The system identification is used to approximate the model for the
plant. In this thesis an existing Simulink model will be used to design the controller.
5.3 MPC Theory
MPC is based on the iterative finite horizon optimization of the plant model. At time‘t’ the
current state of the plant is sampled and cost minimizing control strategy is computed (via a
numerical minimization algorithm) for a relatively short time horizon in the future [t, t+T].
MPC is based on:
A model of the system
Measured data from the system
A cost function which restrains undesirable behavior
Constraints which represent physical system limit
By using MPC one can predict future output signals from the system based on the current
measured data and mathematical model. These predictions can be described as a function of
future input signal sequence implemented on the system [43]. In developing the MPC
controller for the system using Matlab-Simulink, there are some important steps to be taken.
Page:- 34
Figure 5.1 shows the block diagram of MPC controller when combined with the plant model.
Figure 5.1 Block diagram of the MPC controller scheme
The first sequence of input signal is applied to the system and new measured data is obtained.
After this again the procedure of calculating new signal starts. The steps of MPC can be
summarized as.
1. Obtain measured data from the system.
2. Use present data and model of the system to find future output signals as a function of
future input signals
3. Minimize the cost function with respect to future input signals
4. Apply the first input signal in the obtained optimal input signal sequence.
5. Repeat the steps until we achieve the required control output.
5.3.1 MPC Internal model The minimized interval of the control law by considering the tracking reference equal to zero
can be written as
∑ ‖ ( )‖
∑ ‖ ( )‖
(5.1)
In equation (5.1), is the system states weighting matrix and is the input weighting
matrix. The system constraints are represented by A and b which are organized by the matrix
form. The internal model is used to predict future states in the real system. The internal model
will in this thesis be described as a state-space model. The internal model uses present and
future inputs to calculate present and future outputs. The inputs to the internal model are the
present and future control actions and the present and future disturbances.
5.3.2 Constraints The constraints are very important for the optimization problem which indicates the state
space description of the given system and it’s able to predict the future states. It is the
operational and physical limitation of the controlled system [44]. Constraints are used in
optimization problem so that the input and output and the states are kept in within this
boundaries. The constraints can be the minimum and the maximum refrigerant flow rate to
the condenser which is the input manipulated variables constraints and the minimum and
maximum water temperature which is the output variables constraints.
The constraints can be described into two categories [45] these are hard and soft constraints.
The hard constraints must be satisfied by the solution of the MPC controller. These
constraints never get violated as it represents the fixed values such as input and state
constraints. The other type of constraints called the soft constraints which can be changed. In
this system the hard constraints are input refrigerant flow rate and the state constraints and
soft constraints is the water temperature. If necessary these constraints values can vary. The
difference in implementation of these two constraints is hard constraints are used in the
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optimization as hard limitations to a state or while the soft constraints are used as a slack
variables. The slack variables are the variables which represents the non-zero values only if
the constraints are violated [46].
The constraints are used to the MPC optimization problem by setting the conditions. In
equations (5.2-5.4) showing the input manipulated variable constraints, state constraint and
the output variable constraints conditions respectively.
(5.2)
(5.3)
(5.4)
The operational constraints on system input and states can be incorporated into the
optimization procedure in the usual method [47]. Here the umin is the minimum refrigerant
flow rate 0.1567 kg/m, umax is the maximum refrigerant flow rate 0.567 kg/m, xmin and xmax are
the minimum and maximum state constraints respectively which is fixed 4 states, Tmin is the
minimum water temperature 100C and Tmax is the maximum water temperature 60
0C.
5.3.3 Cost function The cost function of a system using MPC controller can be expressed in different ways. The
rate of change of inputs and prediction control error is penalized [48]. The rate of change of
input expressed as
( ) ( ) ( )
Prediction control error:
( ) ( ) ( )
where ( ) is the reference and the ( ) is the predicted outputs at sample k.
The cost function can be written as
( ) ∑ ( ) ( ) ( )
( )
5.3.4 Output prediction In equation (3.27) given only one step of the system, if we take more steps for the system we
get the equation given in equation (5.5). In equation (3.27) the first row says that the state
vector at sampling time one, i.e. x(k+1) can be calculated using Ax(k)+Bu(k). To solve for the
next state vector, i.e. x(k+2) the first row in (3.27) can thus be used recursively as
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ∑ ( )
(5.5)
where Nu-1 is the future control signals. By using the structure of vector and matrix form with
the prediction horizon as upper limit we can simplify the equation (5.5).
( )
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[
( ) ( )
( 𝑁𝑝)
] [
( ) ( )
( 𝑁𝑝)
]
(
)
(
)
The optimization problem given in equation (5.1) is the minimization problem which is
solved at each control interval. By introducing new characters and representing block-
diagonal matrices of time respectively as
(
) and (
) (5.6)
We can rewrite equation (5.1) in the term U and X as
∑‖ ( )‖ ∑‖ ( )‖
( ( ) ) ( ( ) )
The values of the matrices H and S, the state vector and variables can be found from the given
matrices above.
5.4 MPC Tuning
The MPC controller needs tuning to get it work in a satisfactory way. The MPC technique has
been familiar as efficient approach to improve profitability and efficiency [49,50, 53]. We can
tune many parameters of the MPC controller such as prediction and control horizons, the
control time steps and the values of the weighted matrices. Unfortunately there are no specific
methods to tune the model predictive controller [51]. It is always better to start tuning from
horizons then the input and output weighting matrices.
5.4.1 Prediction horizon Np The prediction horizon of the system should be large enough to cover the settling time. In
figure 5.2, the effect of changing the prediction horizon of the system output is shown. In the
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figure it is clear that a short prediction horizon gives less performance compared to the long
prediction horizon.
Figure 5.2 Prediction horizons tuning of the MPC controller
In our case the sampling time is a fairly large of 1 second because the settling time is quite
large. The prediction horizon for this system of 70 samples gives the output closer to the
setpoint which is suitable for the implementation.
5.4.2 Control horizon Nu The control horizon for different system should be different. It depends on the output signal of
the system. In the most cases the control horizon should be large enough to get the reasonable
stabilize output signal of the system. The long control horizon is required to improve the
performance [42]. The tuning of the control horizon used by the controller for this system is
45 which gives a reasonably fast response while not inducing oscillations.
5.4.3 Weighting matrices
The input and output weighted matrices common formula is given in equation (5.6). The best
result of MPC controller for our system is shown in figure 5.3 after tuning the controller.
Figure 5.3 Input weight tuning of the MPC controller
For weighting matrix of the controller, the tuning obtained for the input weighted matrix
=3 and output weighted matrix = 0.15 with both matrices in dimension np-1.
(
) and (
)
Page:- 38
In discrete time model the weighting function are the positive function. The input weighted
matrix function influences the input of the system. For our system we adjust input weighted
value ( ) for the MPC controller is 3. The increasing of the input weighted values gives the
input function more weight which influences the output water temperature and it goes down
from the set point value. The MPC controller parameter adjusted values are given in table 5.1.
Table 5.1 MPC tuning parameters value
Tuning
parameters
Prediction
horizon (NP)
Control Horizon
(Nu)
Input weight
( )
Output
weight ( )
Tuning value 70 45 3 0.15
The adjusted output variable weighted matrix ( ) value is 0.15 that gives the result
closer to the set point. By increasing the output variable weighted matrix ( ) from 0.15
does not influence the output of the system and when the value is lesser than 0.15 the
output decreases.
5.5 MPC controller response
The output water temperature using MPC controller follows in figure 5.4. The MPC controller
tuning parameters are adjusted by using the steps given in MPC tuning section. The adjusted
parameter values has been taken from table 5.1
Figure 5.4 Outlet water temperature using MPC controller
From the analysis of the transient response shown in previous chapter figure 4.3 the transient
specifications are given in table 5.2
Table 5.2 The transient response specifications of the MPC controller
Overshoot Undershoot Rise time Peak time Settling time
1.70 0 7 8 10
The output signal start from the initial water temperature 100C and the delay time is 1
minutes. It’s taking 7 minutes rise time to reach the set point. Its reaches peak value 61.700C
in 8th
minute. The system using MPC controller is going to be stable at 18th
minute with the
final value 60.40C.
Page:- 39
5.6 Pole-Zero analysis of the MPC Controller
The location of the poles and zeros provide approximate insights in the output response of the
system [41]. Figure 5.5 shows the poles and zeros location for the discrete time closed loop
system of MPC controller.
Figure 5.5 Poles and Zeros plot of MPC controller
The MPC controller is designed for the ARMAX 2422 model obtained from the modelling of
the system. The open loop system of ARMAX 2422 model gives 4 poles and 3 zeros. All
poles of the system lie inside the unit circle in the z plane that confirms the stability of the
system [58]. The MPC controller implemented for this system gives 6 poles and 3 zeros for
the discrete time closed loop system. All the poles of the closed loop system are inside the
unit circle in z plane that shows the system is stable.
5.7 PID-MPC controller response
The proposed method of combination of PID and MPC controller block diagram is shown in
figure 4.1 are implemented in Matlab-Simulink for the amx2422 model using the same
configuration of PID and MPC controller shown in chapter 4 and 5 respectively. The
simulation diagram of the scheme using PID/MPC controller is shown in appendix C.4. The
simulation output response using PID-MPC controller is shown in figure 5.6.
Figure 5.6 Outlet water temperature using PID-MPC controller
From figure 5.6, it is seen that the overshoot of the system increases compared to the MPC
transient response shown in table 5.2. The outlet temperature reaches the set point value 600C
but with an overshoot.
Page:- 40
Table 5.3 The transient response specifications of the PID-MPC controller
Overshoot Undershoot Rise time Peak time Settling time
3.90 0 4 7 9
The MPC controller estimates the constraints in evolving prediction horizon and computes
optimal increments on a control horizon. The values of the two horizons (prediction and
control) are 70 minutes and 45 minutes respectively. The PID-MPC controller acts well for
this system. The final stable output temperature from this controller is 60.10C. The
specification value using the PID-MPC controller is shown in table 5.3. It takes less time
compared to the scheme using only PID and MPC controller to reach the set point and the
peak value.
5.8 Pole-Zero analysis of the PID-MPC Controller The characteristic behavior of the signal depends on the location of the poles and zeros
according to the region where they lie inside the unit circle. As said before if the poles are
outside the unit circle then the system is unstable bacause the signal continues to increase[42].
In figure 5.7 it shows the poles and zeros plot for the PID-MPC controller.
Figure 5.7 Poles and zeros plot of PID-MPC controller
The PID-MPC controller is designed for the ARMAX 2422 model. The order of this model is
less due to less number of poles. The ARMAX 2422 consists of 4 poles and 3 zeros. The
implementation of PID-MPC controller on this model gives 7 poles and 4 zeros. In a discrete
time system for the system to be stable all poles must lie inside the unit circle [58]. The poles
of PID-MPC controller are inside unit circle in the z plane and the system behavior is
improved.
To summarize the real poles and complex conjugate poles which are inside the unit circle are
always bounded in amplitude [24]. The overall system behavior is improved due to the
location of the poles in the unit circle and the system response becomes better damped.
Page:- 41
Chapter 6
Results analysis and Discussion In this chapter we will discuss about the results of the modeling and the controller designing
scheme and also we will compare the results.
6.1 Simulation result analysis
The simulation model inputs for the system are collected from the real plant and the output of
the system is obtained from the simulation model. The output for system is presented in figure
2.6 with enormous oscillations. The simulation results are not meeting the desired output 600C
temperature for most of the time period. It’s showing several overshoots and undershoots
which makes the system unstable.
6.2 Analysis of the model selection results
The simulation model for the system showed a good resemblance to data collected from the
real plant. For identification of the model and to achieve best performances from the
controllers the model should be identified in a good way. The system identification toolbox is
very popular and well known way to identify the nonlinear model.
There are however a few things that is to be simplified to attain the better performance from
the system identification toolbox to obtain the best model. For simplification of the
identification procedure the ambient temperature of the system is neglected.
The oscillations created from the ambient temperature would be the worst case to identify the
model. Several sample models has been analyzed and tests are shown in appendix B.1.To find
the best fit model the system inputs and outputs need to be analyzed and test the signals are
shown in chapter 3 (section 3.10) in detail. For the selection of the model, the final prediction
error (FPE), loss function & number of poles are the properties considered for the given
system.
The three models (i.e. arx221, arx791 and amx2422) successfully passed the tests, giving less
error (see table 3.1 and 3.2) also can be seen in appendix B and the best estimation data fitting
percentage. The means square errors or loss function and the final prediction error are
different for different model. The better performance of the real plant depends on the loss
function and final prediction error.
Table 6.1 Experimental result for ARX and ARMAX models
Model Loss function FPE Poles
arx221 0.1604 0.172 2
arx791 0.0001054 0.000154 8
amx2422 0.104 0.03523 4
Akaike’s final prediction error [31] criterion provides the measure of the models quality.
According to Akaike’s theory the most accurate model represent the smallest final prediction
error.The loss function for the ARMAX (2,4,2,2) model is 0.104 and the final prediction error
is 0.03523 which is less compare to arx2422 and arx221 models but higher to arx791 model.
The arx791 model is giving less loss function and FPE but this model has more poles and
zeros compare to the amx2422 model which makes the system higher order and complex for
further process. The final model (amx2422) has been chosen considering the less errors and
less poles-zeros.
Page:- 42
6.3 Analysis of the PID controller result
The PID design and implementation for the control system could be improved to remove the
overshoots and undershoots to achieve the output closer to the desired set point value. The
Traditional Ziegler Nichols Tuning is most common and reliable PID tuning method where
the step response curve is used to find and adjust the controller parameter values shown in
figure 4.2. The controller parameters namely proportional, integral and derivative gains are
calculated from that curve. This PID tuning method works well for our system and it sets the
output temperature closer to the desired value. The comparison result using traditional Ziegler
Nichols tuning and Modified Ziegler Nichols tuning are shown in figure 6.1 below.
Figure 6.1 The outcome of Td. and Mod. PID tuning method
After Simulation we have found that the controller has different values for the transient
response specifications such as Peak time (tp), Rise time (tr), Settling time (ts) and overshoot
(Mp). In the analysis we have seen that the more accurate results came with the Modified
Ziegler Nichols technique. The final output value of the Modified Ziegler Nichols techinique
attain 61.80C. The table 4.6 and 4.7 shows the transient response specifications for Traditional
and Modified Z-N tuning method respectively. It can be seen that there is a considerable
amount of change in the rise time, settling time and overshoot in using both techniques. The
Modified Ziegler Nichols is the best controller tuning approach and gave satisfactory results
i.e., minimum rise time, settling time and overshoot.
6.4 Analysis of the MPC and PID-MPC result
The Model predictive control design and implementation gives the temperature much closer
to the set point. The MPC controller final output value is 60.40C. The result shows that the
model predictive controller can improve the system performance and water temperature
variations.
Figure 6.2 Outlet water temperature using PID and MPC controller
Page:- 43
The table 6.2 shows the transient response comparison for Modified PID controller and MPC
controller.
Table 6.2 Transient response specifications comparison
Controller type overshoot undershoot Rise time Peak time Settling time
PID 3.02 0.08 4 16 19
MPC 1.70 0 7 8 10
PID-MPC 3.90 0 4 7 9
The objectives of the controller for the condenser water temperature met in the case of system
stability and it achieves the desired temperature level. The most essential improvement would
be to practice a nonlinear model to base the model predictive control predictions.
The MPC controller gives less overshoot and also undershoots is improved. The rise time is
increasing from 4 to 7 minutes but the sum of the rise and delay time is decreased from 15
minutes to 8 minutes which represent that from initial value the PID controller takes total 15
minutes to reach the set point where the MPC controller takes only 8 minutes.
Figure 6.3 Result comparison of PID, MPC and PID-MPC controller
The settling time and peak time are also less for MPC compared to the modified PID
controller. The PID-MPC controller scheme gives more overshoot shown in table 6.2 but its
showing less rise time, settling time and the outlet water temperature reaches the set point
within 7 minutes to the peak value, when using only the PID and MPC controller give the
peak time 16 and 8 minutes respectively. The PID-MPC controller scheme gives less settling
time compare to the only PID and MPC controller separately. The PID controller shows
oscillating output whereas the MPC and PID-MPC controller schemes set output without any
oscillation after a certain period of time.
The main goal for the proposed PID-MPC design process and implementation was to see if
the control system could be improved while still maintaining a good temperature level. The
results from the PID-MPC scheme show that the system performance can be improved even
though the improvements are fairly small. The performance of the controller is determined by
the value of the condenser outlet water temperature.
The closer the value of outlet water temperature from the condenser using controller to the set
point temperature value i.e. 60°C, means that the smaller error has been generated and the
controller performance is better. The PID, MPC and PID-MPC controller final temperature
are 61.80C, 60.4
0C and 60.1
0C respectively. So the PID –MPC controller generated small
error and the performance is better to the other controller results. The simulation results also
confirm that the PID-MPC controller outcomes were closer to the set point value compared to
the PID and MPC controller.
Page:- 44
The frequency response analysis of the PID, MPC and PID-MPC controller scheme shows the
gain margin and phase margin for the PID controller output to input is infinite, which is the
amount of gain of a system to increase or decrease required to make the loop gain unity at a
gain margin frequency where the phase angle is -1800. The phase margin of the PID controller
scheme is -1010 at 0.316 rad/sec which is the difference between the phase of the response
and -1800 when the loop gain is 1.0.
The MPC controller scheme characterizes the gain margin 18.1dB at 1.81 rad/sec and phase
margin -89.30 at 0.0002 rad/sec. The PID-MPC controller scheme characterizes the gain
margin 81.5 dB at 1.32 rad/sec and the phase margin is infinite value. The frequency response
experimental results for the all three controller schemes are shown in appendix D.
6.5 Results comparison with previous work
The result obtained in modeling for the actuator servo system [7] the arx331 model is selected
using the system identification techniques with 95.79% fit for the system. In heat pump
system we found the amx2422 model with 98.39% fit for the system which shows that less
FPE and MSE are generated.
In designing the PID controller it is seen the effectiveness of the control action given in order
to control the water temperature. From the previous work we can relate the PID controller
used to control the temperature in refrigeration system [11]. The PID performs better than
ON-OFF controller; the performance of the controller is judged based on the value of the
temperature, the closer the value to the set point means less is the error [12].
It is seen that the PID controller works efficiently in maintaining temperatures in heat pumps.
The results obtained from tuning PID in controlling temperature is KP = 100, KI = 10 and KD
= 3 for 30°C [12] and the best tune obtained for PID in controlling the heat pump system with
modified technique is KP = 0.19, KI = 0.037 and KD = 3.214. With the Modified technique in
tuning PID the water temperature of the system was 61.80C which is not so close to the set
point; therefore model based controller was used. The MPC controller used for heat pump to control the temperature is discussed in [43] shows
that the MPC controller maintains the mean DHW temperature lower compared to the
conventional controller. In our case the results obtained from the MPC controller is lower
(60.40C) than the conventional PID controller.
The combination of PID and MPC controller results are shown in [52,55] the obtain result
improved although it is fairly small and the overshoot increases. In this thesis the PID-MPC
controller shows the improvement of the results. With the combination of the PID and MPC
controller the condenser outlet water temperature is improved from 60.40C (for MPC) to
60.10C (for PID-MPC) where the desire set point is 60
0C and also with the small
improvement of the rise time, settling time, and peak time.
Page:- 45
Chapter 7
Conclusion and Future work This chapter discusses about the conclusion of the thesis and the work that can be done in
future.
7.1 Conclusion
The main aim of the thesis is to implement a combined MPC and PID controller to control the
outlet water temperature for the heat pump and to select the best model using system
identification techniques. Here in this thesis the MPC and PID controller improved the
performance to a great extent. The PID controller helped in obtaining the water temperature to
a reasonable extent but there was still some instability in the system. To obtain better
performance and stability we implemented MPC controller along with PID which helped in
achieving a greater control action for our system.
The MPC controller helped in obtaining better results for our system with minimum
overshoot, rise time and settling time. For a non-linear or a dynamic system where the system
response is not stable MPC could help in obtaining better control action. One of our goals in
this thesis was to maintain the outlet temperature at 600C with ±1
0C tolerance range, which
was achieved with the help of PID-MPC controller together. As we can see from the obtained
results the change in the output responses from both the controllers in rise time, settling time,
overshoot.
The combined MPC, PID controller gave minimum rise time, settling time and overshoot. The
outlet water temperature from the condenser should be in between 590C~61
0C. If the
temperature crosses this range it is not good for the system. One of the most probable reasons
for the implementation of the MPC controller is that it does not produce oscillations as with
the conventional controller. With MPC the output reaches set point quickly and remains stable
throughout the process. In MPC and PID controller it is easier to set the reference temperature
and the change in the ambient temperature is analyzed with the sensor and effectively
controlled by the controller. We can conclude from the given results that the combined action
of PID and MPC controller gives us a good output response that helped in achieving a
constant water temperature of the air to water heat pump.
7.2 Future Work
Future work could be described as
Development of the controller for Solar heat pump
By using a model of an air source heat pump instead of a ground source heat pump the
controller could be adapted to work for air source heat pumps as well. For the future work
perspective we can develop a strategy for using a model of solar source heat pump and fix it
as an alternate choice for the end users. Depending on weather conditions the controller
should be able to utilize the solar energy and heat the water which would lower the heating
costs. Depending on the consumption of hot water, for example the user only consumes hot
water during the morning and evening hours.
So during the work days it is beneficial to let the heat pump work as less as possible, this will
help in increasing the compressor life. Also it is good to add ON and OFF feature of the
compressor in the controller so the compressor should run for some time and stop. Also
further development of the controller for the current model to minimize the overshoot and
decrease the rise time.
Page:- 46
Implementation of the controller
There is a lot of work needed in the implementation for the PID-MPC controller on the real
system. For implementing MPC controller it would be necessary to have an online parameter
estimation of heating system since it can differ for a lot conditions like the refrigerant flow
rate, temperature, water flow rate and also the circulating pump.
Page:- 47
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Appendix A
A.1 Constant coefficient for air to water %%%%%%%%%%*********hetvagg***********%%%%%%%% %%%%%%%%%***constant coefficient(ksa)******%%%%%%%
r1=0.004;%% tube radious inner side [meter(m)] r2=.0125;%% tube radious outer side [m] k=401;%% copper conductivity [W/mK] Tw=10;%% input water temperature [degrees] Ta=60;%% input air temperature [degrees] T1=60;%%copper tube inner side temperature[degrees] T2=59.8;%% copper tube outer side temperature[degrees] N=10;%% total length of the copper tube [m]
Q=(2*pi*k*N*(T1-T2)/log(r2/r1))/1000;%% heat energy flow equation [kw] %Aa=2*3.1416*r1*N;%%%% area of tube inner side Aw=2*pi*r2*N; %%% area water side %ha=Q/(Aa*(Ta-T1));%% air heat transer coefficient %hw=Q/(Aw*(T2-Tw));%% water heat transfer coefficeint %c=1/(r2/ha)*r1+(1/hw)+r2*log(r2/r1)/k;%%% constant value
ks=Q/(Aw*(Ta-Tw))%% heat transfer coeff in this system[kw/Km^2] const=ks*60 %% constant value [kj/mKm^2]
A.2 constant coefficient for water to outside air %%%%%%%%%%*********hetvagg***********%%%%%%%%5 %%%%%%%%%***constant value finding (kso)******%%%%%%%
r1=0.0125;%%%%% tube radious inner side [meter(m)] r2=0.0150;%%%%% tube radious outer side [meter(m)] k=0.19;%% PVC plastic conductivity [W/mK] Tw=60;%% input water temperature [degrees] Ts=22;%% input air temperature [degrees] N=10;%% total length of the copper tube [m]
Q=(2*pi*k*N*(Tw-Ts)/log(r2/r1))/1000;%% heat energy flow equation [kw] %Aa=2*3.1416*r1*N;%%%% area of tube inner side Aw=2*pi*r2*N; %%% area water side %ha=Q/(Aa*(Ta-T1));%% air heat transfer coefficient %hw=Q/(Aw*(T2-Tw));%% water heat transfer coefficeint %c=1/(r2/ha)*r1+(1/hw)+r2*log(r2/r1)/k;%%% constant value
k_dist=Q/(Aw*(Tw-Ts))%% heat transfer coeff in this system[kw/Km^2]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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A.3 Water inside the condenser %%%%%%%%*** AMOUNT OF WATER INSIDE THE CONDENSER*****%%%%%%%%% %%%%%%%%%%%%%% ********VOLUME OF CONDENSER******%%%%%%%%
%%% given values from the company h=10;%% condenser length[meter] d=25/1000;%%condenser diameter[meter] dat=8/1000;%%air tube diameter[meter] dst=dat;%%solar tube diameter[meter]
hat=h;%%air tube length[meter] hst=hat;%%solar tube length[meter]
p=999.7026;%% water density[kg/m^3]at 10 degreeC cp_water=4.184;%% cp values of water kj/kg
total_v=pi*d^2*h/4;%% total volume of the condenser with tubes vat=pi*dat^2*hat/4;%% volume of air tube vst=pi*dst^2*hst/4;%% volume of solar tube v_frees=total_v-(vat+vst);%% volume of condenser free space filed with
water
amount_wa=p*v_frees%% the exact amount of water inside the condenser m=amount_wa*cp_water; m_inverse=1/m; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
A.4 Outlet temperature and area
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%***system output ******%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%
%%% put all values [qa,wa,ca,cp,tc1,th1]
qa= 34.01;%air flow rate kg/h wc= 10;%water flow rate kg/h ca=1.009;%specific heat of air cp=4.184;%%4.184;%specific heat of water tc1= 10;%input water temperature to condenser k=2200;%%K wm2.oC th=25; th1=90;%air input temperature
%%%find the outgoing air temperature from condenser th2= tc1+(th-tc1);%%(th2-tc1)%air output temperature
%%% find the output water temperature tc2=((qa*ca)/(wc*cp)*(th1-th2))+tc1; %output water temperature
p=(qa*ca*((th1-th2)/3600)); %the transfer heat flux(amount of energy), kW
%%% logarithemic avg value to find the require area
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lt=((th2-tc1)-(th1-tc2)/log(th2-tc1)/(th1-tc2));%Logarithemic average temp.
water A=(p*1000)/(k*lt); %the require area (m2)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
A.5 Minimum and Maximum ambient temperature effect
%%%%%%%%%%*********hetvagg***********%%%%%%%%
%%%%%%%%%***ambient temperature effect******%%%%%%%
%%%%%%%%%%% autumn operation condition%%%%%%%%%%%%%
r1=0.0125;%%%%% tube radious inner side [meter(m)] r2=0.0150;%%%%% tube radious outer side [meter(m)] k=0.19;%% PVC plastic conductivity [W/mK] Tw=60;%% input water temperature [degrees] Ts=22;%% input air temperature [degrees] N=10;%% total length of the copper tube [m]
tmin=15.3456; % autumn time condenser outside minimum temperature tmax=27.0100;% autumn time condenser outside maximum temperature
Q=(2*pi*k*N*(Tw-Ts)/log(r2/r1))/1000;%% heat energy flow equation [kw] %Aa=2*3.1416*r1*N;%%%% area of tube inner side Aw=2*pi*r2*N; %%% area water side %ha=Q/(Aa*(Ta-T1));%% air heat transfer coefficient %hw=Q/(Aw*(T2-Tw));%% water heat transfer coefficeint %c=1/(r2/ha)*r1+(1/hw)+r2*log(r2/r1)/k;%%% constant value
k_dist=Q/(Aw*(Tw-Ts));%% heat transfer coeff in this system[kw/Km^2] amb_effectmin=tmin*k_dist; %%% amb temp minimum effect of the system output amb_effectmax=tmax*k_dist; %%% amb temp maximum effect of the system output
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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A.6 P-h diagram for refrigerant R-134a
A pressure enthalpy (P-H) diagram is a technique to show changes in system pressure and
energy changes. The PH diagram of refrigerant shows the refrigeration cycle for R-134a
refrigerant and also pressure and energy changes.
It is seen from above p-h diagram that compressor compresses the refrigerant from 0.04 bar to
1 bar. The suction pressure is therefore 0.04 bar. The delivery pressure is 1 bar. The work
input to compressor is 50 kj/kg. The compressor work is calculated from Wcomp = Mref (h2-h1)
[57].
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Appendix B
B.1 System identification toolbox processor
B.2 ARMAX2422 model specifications
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B.3 ARX791 model specifications
B.4 ARX221 model specifications
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B.5 ARX611 model specifications
B.6 OE221 model specifications
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Appendix C
C.1 simulation model without controller
C.2 Simulation model with PID controller
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C.3 Simulation model with MPC controller
C.4 Simulation model with PID-MPC controller
Appendix D
D.1 Bode plot of the PID controller scheme
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D.2 Bode plot of the MPC controller scheme
D.3 Bode plot of the PID-MPC controller scheme
