Modelling and Control System design to control Water temperature in Heat Pump

Transcription

1 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: 09 th December 2013 Serial number:

2 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. I

3 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. II

4 Table of Contents Abstract... I Acknowledgments... II Nomenclature... VI List of Figures... VIII List of Tables... X 1 Introduction Overview Background Problem formulation Purposes of master s thesis Thesis Contribution System description Overview of the heat pump system Heat transfer of the system Outside air temperature of the system Refrigerant of the system Discharge of the heat exchanger Modeling of the system System Identification Introduction Model structure for identification method Model quality and experimental design System identification principle System identification loop System identification method Data Examination Model structure selection Model Estimation Estimation of the ARX model structure Estimation of the ARMAX model structure Model Validation Residuals analysis Pole-Zero analysis Fitting model for controller design Controller design III

5 4.1 Controllers of a system Proposed controllers PID Controller PID controller Theory Proportional term Integral term Derivative Term PID Controller for the heat pump PID controller tuning rules Ziegler Nichols Tuning Traditional Z-N tuning Method Modified Z-N Tuning Method PID tuning for the system Transient response specifications Traditional Ziegler-Nichols response Modified Ziegler-Nichols response Pole-Zero analysis of the PID Controller MPC controller design MPC Introduction MPC Model MPC Theory MPC Internal model Constraints Cost funciton Output prediction MPC Tuning Prediction horizon N p Control horizon N u Weighting matrices MPC controller response Pole-Zero analysis of the MPC Controller PID-MPC controller response Results analysis and Discussion Simulation result analysis Analysis of the model selection results Analysis of the PID controller result IV

6 6.4 Analysis of the MPC and PID-MPC result Results comparison with previous work Conclusion and Future work Conclusion Future Work Bibliography Appendix A A.1 Constant coefficient for air to water A.2 constant coefficient for water to outside air A.3 Water inside the condenser A.4 Outlet temperature and area A.5 Minimum and Maximum ambient temperature effect A.6 P-h diagram for refrigerant R-134a Appendix B B.1 System identification toolbox processor B.2 ARMAX2422 model specifications B.3 ARX791 model specifications B.4 ARX221 model specifications B.5 ARX611 model specifications B.6 OE221 model specifications Appendix C C.1 simulation model without controller C.2 Simulation model with PID controller 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 D.2 Bode plot of the MPC controller scheme D.3 Bode plot of the PID-MPC controller scheme V

7 Nomenclature Abbreviations MPC PID COP deg.c AR ARX ARMAX ARMA BJ OE PI PD MV Z-N CHR TSP Td Mod Np Nc T T V T S EC R134a Model Predictive Control Proportional Integral and Derivative Coefficient of performance Degree Celsius Autoregressive AR models with Extra Regressors ARMA models with Extra Regressors Autoregressive moving average Box Jenkins Error Estimation Proportional Integral Proportional Derivative Manipulated Variable Ziegler Nichols Chien Hrones Reswick Temperature setpoint Traditional method Modified method Mathematical Symbols Prediction horizon Control horizon Change of temperature Coolant temperature Evaporation temperature Electronically Commutated/ Brushless DC electric motor 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 VI

8 n a 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 Order of the polynomial A(q) n b Order of the polynomial B(q) + 1 n c n k Order of the polynomial C(q) 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 K p K i K d T i T d u min u max x min x max T min T max M p M u T r T p T s Proportional gain Integral gain Derivative gain Integral time Derivative gain Minimum input flow rate Maximum input flow rate Minimum state Maximum state Minimum temperature Maximum temperature Maximum overshoots Maximum undershoots Rise time Peak time Settling time VII

9 List of Figures Figure 2.1 Overview of the heat pump system Figure 2.2 Block diagram of the heat exchanger/condenser Figure 2.3 Outside air temperatures during autumn season Figure 2.4 Comparison of the refrigerant coefficient of performance Figure 2.5 Outlet refrigerant temperatures from heat exchanger Figure 2.6 Outlet water temperatures from the heat exchanger Figure 3.1 The system identification loop Figure 3.2 The data set time plot of the heat pump system Figure 3.3 Estimation data Figure 3.4 Validation data Figure 3.5 Step response plot for different model structure Figure 3.6 Frequency response for different model structure Figure 3.7 The ARX model estimated output Figure 3.8 The ARMAX model estimated output Figure 3.9 Residual analysis of the ARX model structure Figure 3.10 Residual analysis of ARMAX model structure Figure 3.11 Pole-Zero for the arx791 model structure Figure 3.12 Pole-Zero for the amx2422 model structure Figure 4.1 Block diagram of the PID-MPC controller scheme Figure 4.2 Block diagram of PID controller for the condenser Figure 4.3 Response curve for Ziegler Nichols method Figure 4.4 The transient response specifications Figure 4.5 Response curve using Traditional Ziegler-Nichols method Figure 4.6 Response curve using Modified Ziegler Nichols method Figure 4.7 Pole-Zero plot of the PID Controller scheme Figure 5.1 Block diagram of the MPC controller scheme Figure 5.2 Prediction horizons tuning of the MPC controller Figure 5.3 Input weight tuning of the MPC controller Figure 5.4 Outlet water temperature using MPC controller Figure 5.5 Poles and Zeros plot of MPC controller Figure 5.6 Outlet water temperature using PID-MPC controller VIII

10 Figure 5.7 Poles and zeros plot of PID-MPC controller Figure 6.1 The outcome by Td. and Mod. PID tuning method Figure 6.2 Outlet water temperature using PID and MPC controller Figure 6.3 Result comparison of PID, MPC and PID-MPC controller IX

11 List of Tables Table 2.1 Transient response specifications for the system Table 3.1 ARX model structure specifications Table 3.2 ARMAX model structure specifications Table 3.3 The Pole-Zero locations of the arx791 model structure Table 3.4 The Pole-Zero locations of the amx2422 model structure Table 4.1 Ziegler-Nichols Tuning first (Traditional) method Table 4.2 Modified Ziegler-Nichols Tuning (CHR) method Table 4.3 Traditional Ziegler Nichols tuning method result Table 4.4 Modified Ziegler Nichols tuning method result Table 4.5 Comparison of controller parameters Table 4.6 Transient responses of the Traditional Z-N tuning method Table 4.7 Transient responses of the Modified Z-N tuning method Table 5.1 MPC tuning parameters value Table 5.2 The transient response specifications of the MPC controller Table 5.3 The transient response specifications of the PID-MPC controller Table 6.1 Experimental result for ARX and ARMAX models Table 6.2 Transient response specifications comparison X

12 Keywords Water temperature control, System identification, system identification toolbox, Proportional integral derivative (PID), Model predictive control (MPC), water flow control, Heat pump control system. XI

13 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:- 1

14 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 K P, K I and K D 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:- 2

15 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 60 0 C 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:- 3

16 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:- 4

17 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 45 0 C 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 60 0 C it increases the chance of legionella growth. So the water temperature should not be less than 60 0 C inside the tank. If the temperature decreases below 45 0 C 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 = T V T s (2.1) Page:- 5

18 where T s = Evaporation temperature at the outlet of evaporator and T v = 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. Page:- 6

19 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 C and the maximum temperature fluctuation is C. In autumn operation it gave C higher peak than winter maximum value and also the minimum value is C lesser than the winter operation minimum value. Therefore the ambient temperature of autumn is chosen for the system Page:- 7

20 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 C~ C 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 (10 0 C) 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 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-torefrigerant 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 C for this system, the refrigerant flow reached its minimum value 9.40 kg/h. The Refrigerant flow from the Page:- 8

21 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 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 35 0 C ~55 0 C and the evaporating temperature range C ~ C. 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:- 9

22 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 Overshoot Rise Time Settling time Undershoot Peak time specifications values 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 (60 0 C). 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:- 10

23 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 3.3 Model quality and experimental design ( ) ( ) (3.2) 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:- 11

24 (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 i th and j th 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. Page:- 12

25 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:- 13

26 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 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:- 14

27 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 ARMAX model Box-Jenkins (BJ) model Output-error (OE) model ( ) ( ) ( ) ( ) ( )) (3.10) ( ) ( ) ( ) ( ) ( ) ( ) (3.11) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (3.12) ( ) ( ) (3.13) Page:- 15

28 The models shown in equations ( ) 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:- 16

29 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 ( ). Modeling errors are not to be considered systematic errors in the observations [28]. The models have different structure such as ARX, ARMAX, BJ models 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 n a n b and n k 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 (%) arx arx arx Page:- 17

30 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 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 (%) amx amx amx 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 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. Page:- 18

31 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]. arx611 arx221 arx791 Figure 3.9 Residual analysis of the ARX model structure Page:- 19

32 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 amx6422 amx2422 amx4422 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] 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 Figure 3.11 Pole-Zeros for the arx791 model structure Page:- 20

33 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 9 th order of the system. Table 3.3 The Pole-Zero locations of the arx791 model structure No.of pole-zero Poles location Zeros location i i i i i i i i i i i i 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 i i i i Page:- 21

34 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 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), n a equals 2, the order of the polynomial B(z)+1, n b equals 4, The order of the polynomial C(z), n c equals 2 and n k 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 The linear state space model can be written in discrete form as ( ) ( ) ( ) (3.25) ( ) ( ) (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:- 22

35 ( ) ) ( ) ( ) ( ) ( ) ) ( ) ( ) ( ) 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. Page:- 23

36 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 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 (N p ) and the control horizon (N c ) 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:- 24

37 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] 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 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 K p, 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. ( ) Page:- 25

38 Proportional gain ( ) Larger values give faster response. When K P 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 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 (T i ) 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 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 Page:- 26

39 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, u o 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 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 K P, K I and K D 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 Page:- 27

40 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] 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 formula given in the table 4.1. The following obtained values of tuning of the controller and will give an output response for our system Table 4.1 Ziegler-Nichols Tuning first (traditional) method [39] according to the will help in the Controller K P Ti Td P T/L 0 0 PI 0.9T/L L/0.3 0 PID 1.2T/L 2L 0.5L 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 K P 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:- 28

41 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 (K I ) and Derivative gain (K D ) 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 K P Ti Td PID After applying Traditional Ziegler Nichols Tuning approach the following parameters for K P, K I and K D are obtained. K P = 14.4, K I = 2.4, K D = 21.6 Table 4.4 Modified Ziegler Nichols tuning method result Controller Type K p T i T d PID The following controller parameters are obtained when we apply Modified Ziegler Nichols Tuning approach. K P = 0.19, K I = , K D = Table 4.5 Comparison of controller parameters PID K P K I K D Traditional Z-N Modified Z-N Page:- 29

42 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 M p is the values from the desire setpoint to peak value, the undershoot is M u calculated from the setpoint to the lowest value of the response curve after reaching the setpoint value, the tolerance range is (±1 0 C) from the setpoint, the rise time t r represents for the rise 0% to 100% for 4 th order system. The peak time t p is the time value which is calculated from 0 to the time need to reach its peak value and the settling times t s 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 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. Page:- 30

43 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 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 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 The final output temperature from this controller is C. 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 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 Page:- 31

44 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]. Page:- 32

45 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 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:- 33

46 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 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 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 Page:- 34

47 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 ( ) 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 u min is the minimum refrigerant flow rate kg/m, u max is the maximum refrigerant flow rate kg/m, x min and x max are the minimum and maximum state constraints respectively which is fixed 4 states, T min is the minimum water temperature 10 0 C and T max is the maximum water temperature 60 0 C 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 ( ) ( ) ( ) ( ) ( ) 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 N u -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). ( ) Page:- 35

48 [ ( ) ( ) ( 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 blockdiagonal 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 Prediction horizon N p 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 Page:- 36

49 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 Control horizon N u 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 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 n p -1. ( ) and ( ) Page:- 37

50 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 Prediction Control Horizon Input weight Output Tuning horizon (N P ) (N u ) ( ) weight ( ) parameters Tuning value 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 The output signal start from the initial water temperature 10 0 C and the delay time is 1 minutes. It s taking 7 minutes rise time to reach the set point. Its reaches peak value C in 8 th minute. The system using MPC controller is going to be stable at 18 th minute with the final value C. Page:- 38

51 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 60 0 C but with an overshoot. Page:- 39

52 Table 5.3 The transient response specifications of the PID-MPC controller Overshoot Undershoot Rise time Peak time Settling time 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 C. 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:- 40

53 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 60 0 C 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 arx arx amx 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 and the final prediction error is 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:- 41

54 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 (t p ), Rise time (t r ), Settling time (t s ) and overshoot (M p ). 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 C. 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 C. 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:- 42

55 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 MPC PID-MPC 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 C, C and C 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:- 43

56 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 The phase margin of the PID controller scheme is at rad/sec which is the difference between the phase of the response and 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 at 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 K P = 100, K I = 10 and K D = 3 for 30 C [12] and the best tune obtained for PID in controlling the heat pump system with modified technique is K P = 0.19, K I = and K D = With the Modified technique in tuning PID the water temperature of the system was C 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 ( C) 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 C (for MPC) to C (for PID-MPC) where the desire set point is 60 0 C and also with the small improvement of the rise time, settling time, and peak time. Page:- 44

57 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 60 0 C with ±1 0 C 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 59 0 C~61 0 C. 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:- 45

58 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:- 46

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61 [31] F.Yarman and B. W. Dickinson, Autoregredon Estimation Using Final Rediction Error, volume 70, Princeton University, Princeton, NJ, Proceedings of the IEEE, Aug [32] Torkel Glad & Lennart Ljung, Control Theory: Multivariable and Nonlinear Methods, Taylor &Francis, USA and Canada, [33] Prof. David Trumper, Analysis and Design of Feedback Control Systems institute of Understanding Poles and Zeros, Department of mechanical engineering, Massachusetts Technology. [34] Robert A. Paz, The Design of the PID Controller, Klipsch School of Electrical and Computer Engineering, June [35] Jingqing Han, From PID to Active Disturbance Rejection Control, IEEE Transactions on Industrial Electronics, March [36] A. Bazoune, System Dynamics and Control, King Fahad University of Petroleum and Minerals. [37] Aiping Shi, Maoli Yan, Jiangyong Li, Weixing Xu, Yunyang Shi, The Research of FUZZY PID Control Application in DC Motor of Automatic Doors, Zhenjiang, China, [38] Oludaya John Oguntoyinbo, PID Control of Brushless DC Motor and Robot Trajectory Planning, [39] P.M. Meshram and Rohit G. Kanojiya, Tuning of PID Controller using Ziegler- Nichols Method for speed control of DC Motor, March [40] K. Astrom and T. Hagglund, PID controller: Theory, Design and Tuning, 2 nd Ed, Library of congress cataloging-in-publication data, [41] A/Prof. Valeri Ougrinovski, Control Theory 1: Effects of poles and zeros on performance of control systems, University of New South Wales, Australia. [42] Antonio Balsemin, Applications Oriented Input design for MPC: An analysis of a Quadruple water tank process, KTH Electrical Engineering degree project, Stockholm, Sweden August [43] Markus Sundbrandt, Control of a Ground Source Heat Pump using Hybrid Model Predictive Control, Department of Electrical Engineering, Linköpings universitet, SE Linköping, Sweden, [44] V. T. Chow, D. R. Maidment, and L. W. Mays, Applied Hydrology, McGraw-Hill, [45] J. Rossiter, Model-based predictive control: A Practical Approach, CRC Press LLC, Page:- 49

62 [46] E. C. Kerrigan and J. M. Maciejowski, Soft constraints and exact penalty functions in Model predictive control, in Proc. UKACC International Conference (Control 2000). [47] Jing Zeng 1, 2, Ding-Yu Xue De-Cheng Yuan 2, The Study of Multiple Models Predictive Control in waste water treatment processes. [48] Danielle D, Doug C, A practical multiple model adaptive strategy for single-loop MPC, Control Engineering Practice, Vol.11, No.2, pp , [49] Van Antwerp J G, Braatz R D, Model predictive control of large scale process, Journal of Process Control, Vol.10, No.1, pp1-8, [50] Muhammad Usman Khalid, Muhammad Bilal Kadri, Liquid Level Control of Nonlinear Coupled Tanks System using Linear Model Predictive Control, Pakistan Navy Engineering College, Karachi, Pakistan, [51] Fredrik Gabrielsson, Model Predictive Control of Skeboå Water system, KTH Electrical Engineering degree project, Stockholm, Sweden [52] Carl W. Ressel, Modelling and Control of Feed Water Systems in a Pressurized Water Reactor, Department of Signals and Systems, Division of Automatic Control, Automation and Mechatronics, Charlmers University of Technology, Göteborg, Sweden, [53] Zerina SehiC, Drago Matko, Zenan SehiC, An Example of Self tuning Controllers on Distributed System, Proceedings of the American Control Conference, San Diego, California, June 1999 [54] Sanjit K.Mitra, Digital Signal Processing: A computer-based approach, third edition, McGraw-Hill international edition, [55] Benjamin Paris, Julien Eynard, Stéphane Grieu, Thierry Talbert, Monique Polit, Heating control schemes for energy management in buildings, ELIAUS Laboratory, University of Perpignan Via Domitia, 52 Av. Paul Alduy, Perpignan, France, [56] Kyung Hwan Ryu, Kyung Su Kim, Ho Suk Kang, Si Nae Lee, Jun Young Chom, Jietae Lee, Su Whan Sung, Frequency Response Model Identification Method for Discretetime Processes with Final Cyclic-Steady-State, Department of Chemical Engineering, Kyungpook National University, Daegu , Korea. [57] Thomas T.S. Wan, P-H Diagram Refrigeration Cycle Analysis & Refrigeration Flow Diagram, Sep [58] Katsuhiko Ogata, Discrete-Time Control Systems, 2 nd Ed, University of Minnesota, New Jersey, Prentice Hall, Page:- 50

63 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] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Page:- 51

64 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= ;%% 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 Page:- 52

65 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= ; % autumn time condenser outside minimum temperature tmax= ;% 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 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Page:- 53

66 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 W comp = M ref (h 2 -h 1 ) [57]. Page:- 54

67 Appendix B B.1 System identification toolbox processor B.2 ARMAX2422 model specifications Page:- 55

68 B.3 ARX791 model specifications B.4 ARX221 model specifications Page:- 56

69 B.5 ARX611 model specifications B.6 OE221 model specifications Page:- 57

70 Appendix C C.1 simulation model without controller C.2 Simulation model with PID controller Page:- 58

71 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 Page:- 59