Modeling and control of a District Heating System. 10 th SEMESTER PROJECT, AALBORG UNIVERSITY 2010 THE DEPARTMENT OF ELECTRONIC SYSTEMS

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1 Modeling and control of a District Heating System th SEMESTER PROJECT, AALBORG UNIVERSITY 2 THE DEPARTMENT OF ELECTRONIC SYSTEMS

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3 Control Engineering th Semester Fredrik Bajers Vej 7 Telephone Synopsis: Title: Modeling and control of a District Heating System Theme: Intelligent Autonomous systems Project period: Spring 2 Project group: gr939 Group members: Anders Jochumsen Supervisor: Associate Professor Klaus Trangbæk Publications: 9 Pages: 78 (7 including appendix) Finished: June 3 rd, 2 The purpose of this project is to design a model based control system, allowing district heating systems to follow a novel, energy saving, infrastructure. The control objective is to track a constant differential pressure reference for multiple end users, with varying consumptions. Throughout the project a laboratory model provided by Grundfoss A/S, of a scaled district heating system, is used as platform. A first principles model is derived and its parameters fitted by use of parameter estimation, through measurements on the laboratory model. Based on the derived model, Linear and Linear Parameter Varying (LPV) -state space descriptions are found. An Extenden Kalman Filter (EKF) is designed for estimation of flows and consumptions, based on differential pressure measurements. The performance of the EKF is concluded satisfactory, enabling independence of consumption sensors. In the project, two overall control methods are investigated. The first method is standard Linear Quadratic (LQ) control, based on the linear state space description. An LQ controller are designed and implemented, showing satifactory performance. The second class of control methods investigated are those referred to as LPV methods. With basis in the LPV model and the EKF, two LPV methods are planned applied. Due to massive numerical problems in the solving of Linear Matrix Inequalities, neither of the methods were implemented.

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5 Preface This thesis concerns the modeling and control of a scaled district heating system. The project is composed during the period from September 3 rd 29 to June 3 nd 2, as part of a th semester project at the department of Intelligent Autonomous Systems (IAS), at Aalborg University. Group 39, Aalborg University Anders Jochumsen

6 Contents Introductory 2. Introduction Description of scaled laboratory model Objectives Outline of the thesis I Modelling 7 2 Derivation of model 2. Pipe model Pump model Valve model Piecing the components together Parameter estimation 6 3. Input signal Estimation and verification of parameters Estimation of valve friction as a function of valve position Linearisation Linearisation Method Deriving a linear state space model Discrete time state space model Pump delay II Control 3 5 Linear Quadratic controller Optimal control Introducing reference and integral action Implementation of the LQ controller Simulation of the LQ controller Observer Design 52

7 6. Design of Kalman filter Design of Extended Kalman Filter Observer verification Linear Parameter Varying Control 6 7. LPV background and analysis Identification of an LPV model LPV control design III Epilogue 7 8 Conclusion 75 IV Appendices 79 A Matrices 8 A. Matrices used in section A.2 Matrices used in section A.3 Matrices used in section A.4 Matrices used in section B Measurement journals 87 B. Analysing the consumer valves B.2 Pump delays B.3 Measurement of pipe frictions proportions C Simulations of the LQ controller 94

8 Chapter Introductory This chapter will introduce the reader to the project, which this thesis concerns. The first section is a general introduction to the problem at hand. Then follows a section with a brief description of the test environment. This is followed by a presentation of the scope of the project and an outline of the content of the thesis.. Introduction The motivation behind this project originates in the context of the planning of a new housing sector. According to a technical report from Energistyrelsen [], a new housing sector calls for a redesign of the whole district heating system, the primary reason being, that too much heat are lost along the pipelines of the current network. In order to describe the problem with the current network, the basic idea of the current design are depicted in fig.. Figure.: The structure of current district heating systems. (figure from [2, p.3]) At the left side of the figure, there is a thermal plant, which, via a feeding pipe, provides hot water to the consumers, at the right side. The water is moved to the consumers by a single (large) pump. According to [3, p.2] The consumer flow rate is best controlled when the differential pressures across the heat exchangers are constant. In a one pump configuration the pressure are highest right after the pump and decreases as a function of the distance to the pump. To be able to deliver a sufficient pressure difference at the consumers, the diameter of the main pipe has to be relatively large. Cooling of an object is directly proportional to its surface area, and

9 Introductory 3 calculations performed in [, p.33-48] shows that the heat loss from a pipe decreases quite dramatically when the diameter (i.e the surface) of the pipe are made smaller. In order to reduce the pipe surface a novel approach to district heating system infrastructure are proposed in e.g [] [4]. The basic structure of the proposed infrastructure are depicted on fig.2. This method utilizes multiple pumps along the feeding pipe and a pump at each consumer. First of all, this allows the main pipe to be considerably smaller than in the current infrastructure, because a larger pressure drop can be overcome by the increased number of pumps. Secondarily the pumps introduced at the consumers gives an increased controllability allowing a constant differential pressure for each consumer, and with pumps operating in series, district heating systems can be made somewhat robust to defect pumps. Figure.2: The proposed structure of a new district heating system. (figure from [2, p.3]).2 Description of scaled laboratory model To investigate different control approaches for the new kind of district heating system a scaled laboratory model has been designed by Grundfoss A/S. This model will serve as test environment throughout the project. A picture of the scaled laboratory model are displayed at fig.3 and the hydraulic network of the model are shown on fig.4. The system consists of one main pipe and four branches. Each branch contains a computer controlled valve, simulating the primary side of a heat exchanger that would be present at a real life consumer. The input to the system are the speed of six pumps, two, at the main pipe an one in each branch. Relative pressure sensors are placed on each side of the " main pipe" pumps and prior to the consumer pumps. In addition to these, differential pressure sensors are placed over each of the consumer valves. The pumps, valves and sensors are connected to a computer with two I/O boards of type NI673(DAC) and NI624E(ADC) respectively. The ADC collects sensor data, whilst the DAC generates analog control signals (-V) to control the pumps and valves. The control signals for the pumps are led through an analog to PWM converter, such that the actual control signals received by the pumps, are PWM signals. On the laboratory computer the two I/O-boards has been installed with Comedi drivers providing an interface to Simulink Real Time Workshop. The dimensions of the laboratory model has been designed, such that the input-output behaviour, except for a time scale factor, corresponds to that of a realistic district heating system.

10 4.2 Description of scaled laboratory model Figure.3: Picture of the scaled laboratory model pressure sensor "long" pipe (R) pump (P) valve (V) V V 2 V 3 V 4 q q 2 q q P P P 2 3 P R 2 R 3 R 22 R 23 R 32 R 33 R 42 R 43 P m P m R R 2 R 3 R 4 q m q m2 q m3 q m4 R 4 R 24 R 34 R 44 Figure.4: The hydraulic network of the scaled laboratory model.

11 Introductory 5.3 Objectives The overall scope of the project is to apply control methods, capable of tracking a constant differential pressure reference for varying consumptions, utilizing a minimal number of sensors. Two control methods will be investigated and compared. The first method tested is standard Linear Quadratic (LQ) Control, which does not take consumption disturbances into account. The second method is Linear Parameter Varying (LPV) Control, which if cast properly, can take disturbances known on-line, into account, with guaranteed stability. Utilizing a minimal number of sensors, calls for the use of an observer. Consequently an observer will be designed for estimation of flows and consumption, by use of Extended Kalman Filtering. The controllers will be tested at a scenario resembling operation of a typical district heating system. In [2] an analysis are made of hot water consumption at real life end users. It is concluded that the dynamics of the consumption, almost exclusively consists of steps and that since heating exchangers functions as low pass filters, disturbances can be approximated by letting the position of the consumer valves change as ramp signals. [2] furthermore judges. [bar] as a reasonable differential pressure references, considering the proportions of the laboratory model. These conclusions are adopted in this thesis. Given a fixed differential pressure reference of. [bar], the range of valve positions, over which the designed control systems should function, can be analysed. This analysis is performed in Appendix B.. Based on the analysis it is found that the controllers designed for the system, are to operate in the range of valve positions displayed in table.. Name Valve position(s) [%] Valve -.67 Valve Valve Valve Table.: The range of valve positions, over which a differential pressure of. bar should be sustainable..4 Outline of the thesis The content of the thesis is divided into a number of chapters, which are outlined in this section to provide some overview of the thesis. Chapter 2: Derivation of Model In this chapter, a first principles model of the scaled district heating system are derived. This is done by first deriving the equations linking flow and pressure, for the pumps, pipes and valves, respectively. Based on these equations the total system description are obtained by use of Kirchoffs st and 2nd law. Chapter 3: Parameter Estimation In this chapter the parameters of the first principles model are estimated. First an optimal set of input signals are designed by minimizing the ratio between the maximal and minimal parameter sensitivity. Then the estimation of the majority of the parameters are performed by use of a Gauss Newton search algorithm. The rest of the parameters are measured in appendix B.3

12 6.4 Outline of the thesis Chapter 4: Linearisation In this chapter a linear discrete time state space description are derived. The linearisation is performed using first order Taylor approximations. At the end of the chapter, some considerations are made regarding the delays of the system, resulting in an augmentation of the state space description. Chapter 5: Linear Quadratic Control In this chapter an introduction to optimal control is given. Then a Linear Quadratic controller is designed based on the discrete time state space description, introducing reference and integral action through augmentation of the state space description. The chapter concludes with some thoughts regarding implementation, and simulations of the controller. Chapter 6: Observer Design This chapter starts with a discussion of different observer strategies, deciding for an Extended Kalman Filter (EKF). The EKF are designed to estimate flows and consumption, based on measurements of differential pressures over the consumer valves. The chapter concludes with a comparison between the EKF and a Normal Kalman Filter, with access to consumption measurements. Chapter 7: Linear Parameter Varying Control This chapter starts with a discussion of application of the LPV design methods presented in [5] and [6], in the context of the system and the EKF. Then part of the system is suggested Linear Parameter Varying and an LPV model are fitted, using the frictions of the consumer valves as parameter vector. After this a design based on [5] are presented and sought implemented. Chapter 8: Conclusion This chapter contains summaries of, and concluding remarks to, the subjects treated in the thesis

13 Part I Modelling

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15 Table of Contents 2 Derivation of model 2. Pipe model Pump model Valve model Piecing the components together Parameter estimation 6 3. Input signal Estimation and verification of parameters Estimation of valve friction as a function of valve position Linearisation Linearisation Method Deriving a linear state space model Discrete time state space model Pump delay

16 TABLE OF CONTENTS

17 Chapter 2 Derivation of model This part concerns the modeling of the scaled district heating system. An analogy between electrical and hydraulic circuits, exists, such that voltages and currents can be replaced with, respectively, pressures and flows. The hydraulic network of the district heating system consists entirely of three basic components: pipelines, pumps and valves. This means that equations describing the entire system can be made based on equations linking flows and pressure drops of the individual components. The aims of the next 3 sections will therefore be to find equations linking flows and pressure drops for the pipelines, pumps and valves, respectively. 2. Pipe model The pipe model is derived under the assumption that the flow is uniformly distributed along a cross section of the pipe, and that the flow is turbulent. Given a turbulent flow, the following pipe model applies [7, p.59-6]: Where: J dq dt = H pipe H fric (2.) H pipe is the pressure across the pipe q is the flow J are a constant parameter of the pipe H fric are a the pressure loss due to friction The pressure loss caused by friction H fric are, under the condition of a turbulent flow, according to [7, p.59-6] given as the following: Where H fric =.36 lρv2 Re.25 2d (2.2) where l is the length of the pipe

18 2 2.2 Pump model ρ is the density of the fluid v is the velocity of the fluid d is the diameter of the pipe Re is the Reynolds number In [7, p.59] the Reynolds number are given as: where Re = v d h v k (2.3) v is the velocity of the fluid d h is the hydraulic diameter of the pipe v k is the kinematic viscosity Incertion of eq 2.3 into eq 2.2 gives the following expression for the pipe friction: where H fric =.36 lρv 2 ( v d h v k ).25 2d H fric =.36 lρ( q A )2 ( q A d h v k ).25 2d H fric = K pipe q.75 (2.4) K pipe are a constant parameter of the pipe 2.2 Pump model The pumps of the system are centrifugal pumps, which according to [8, p. 47] can be modeled as equation 2.5: n H pump = P max (( ) 2 q ( ) 2 ) (2.5) n max q max Where: H pump is the pressure across the pump P max is the maximal pressure the pump can deliver n is revolution speed n max is the maximal revolution speed q is the flow through the pump q max is the maximal flow through the pump

19 Derivation of model 3 Figure 2.: The relasioship between control signal and pump speed (from [9]) As described in section.2, the control signals to the pumps are PWM signals. The pumps each have an internal controller, which based on the duty cycle of the PWM signal controls the revolution speed as depicted on fig 2.. From the figure it is seen that the term n n max of equation 2.5 can be replaced by a variable U, defined as: Where U = for D < % 4 3 D +.33 for D >= and D <= 85% for D > 85% (2.6) D is the duty cycle of the control signal Insertion of U into 2.5 gives the final pump equation: H pump = P max U 2 P max q 2 max q 2 (2.7) 2.3 Valve model According to [, p.32] valves are normally viewed as pipe fittings. This means that the pressure across the valve, at a fixed valve position, can be modeled by a quadratic relationship with the flow through the valve: Where H valve = K v q 2 (2.8) H valve is the pressure across the valve K v is a valve constant, which changes with the position of the valve q is the flow through the valve

20 4 2.4 Piecing the components together 2.4 Piecing the components together As stated at the beginning of the chapter the hydraulic network of the scaled district heating system consists entirely of pipelines, pumps and valves. Equations linking pressure and flow, for each of the three, has been found in the previous section, and now the components must be pieced together into state space form, suitable for model based controller design in later chapters. In the following, the analogy between electric circuits and hydraulic networks are exploited, in the sense that flow can be viewed as current and pressure can be viewed as voltage. Applying Kirchoff s st and 2nd law to the system, displayed for convenience in fig 2.2, gives the following equations: Kirchoff s st law: q m = q + q 2 + q 3 + q 4 (2.9) q m2 = q 2 + q 3 + q 4 (2.) q m3 = q 3 + q 4 (2.) q m4 = q 4 (2.2) Kirchoff s 2nd law: H Pm H R H R2 + H P H V H R3 H R4 = (2.3) H Pm H R H R2 H R22 + H P2 H V2 H R23 (2.4) H R24 H R4 = (2.5) H Pm H R H R2 + H Pm2 H R3 H R32 + H Pm H V3 H R33 H R34 H R24 H R4 = (2.6) H Pm H R H R2 + H Pm2 H R3 H R4 H R42 + H P4 H V4 H R43 H R44 H R34 H R24 H R4 = (2.7) Insertion of the pipe, pump and valve models (eqs. 2., 2.7 and 2.8) along with the flows of eqs , yields the following set of equations: (J + J 4 ) q m + (J 2 + J 3 ) q = P max (Um 2 + U 2 ) ( Pmax Qmax )(q 2 2 m + q) 2 K v q 2 (K + K 4 ) qm.75 (K 2 + K 3 ) q.75 (2.8) (J + J 4 ) q m + (J 2 + J 24 ) q m2 + (J 22 + J 23 ) q 2 = P max (Um 2 + U2 2 ) ( Pmax Qmax )(q 2 2 m + q2) 2 K v2 q2 2 (K + K 4 ) qm.75 (K 2 + K 24 ) qm2.75 (K 22 + K 23 ) q2.75 (2.9) (J + J 4 ) q m + (J 2 + J 24 ) q m2 + (J 3 + J 34 ) q m3 + (J 32 + J 33 ) q 3 = P max (Um 2 + Um2 2 + U3 2 ) ( Pmax Qmax )(q 2 2 m + qm3 2 + q3) 2 K v3 q3 2 (K + K 4 ) qm.75 (K 2 + K 24 ) qm2.75 (K 3 + K 34 ) qm3.75 (K 32 + K 33 ) q3.75 (2.2)

21 Derivation of model 5 pressure sensor "long" pipe (R) pump (P) valve (V) V V 2 V 3 V 4 q q 2 q q P P P 2 3 P R 2 R 3 R 22 R 23 R 32 R 33 R 42 R 43 P m P m R R 2 R 3 R 4 q m q m2 q m3 q m4 R 4 R 24 R 34 R 44 Figure 2.2: Diagram of the scaled laboratory models hydraulic system (J + J 4 ) q m + (J 2 + J 24 ) q m2 + (J 3 + J 34 ) q m3 + (J 4 + J 44 ) q m4 + (J 42 + J 43 ) q 4 = P max (Um 2 + Um2 2 + U4 2 ) ( Pmax Qmax )(q 2 2 m + qm3 2 + q4) 2 K v4 q 2 (K + K 4 ) qm.75 (K 2 + K 24 ) qm2.75 (K 3 + K 34 ) qm3.75 (K 4 + K 44 ) qm4.75 (K 42 + K 43 ) q4.75 (2.2) Eqs can be written into one compact matrix equation as follows: J q q q 2 q 3 4 = K v q 2 q2 2 q3 2 q4 2 + R Um 2 U 2 U2 2 Um 2 2 U3 2 U4 2 S q 2 q2 2 q3 2 q4 2 qm 2 qm 2 2 qm 2 3 qm 2 4 T q.75 q.75 2 q.75 3 q.75 4 q.75 m q.75 m 2 q.75 m 3 q.75 m 4 (2.22) Where J,K v, R, S and T are coeficint matrices (displayed in appendix A) 2.4. Chapter Summary This chapter has described the scaled district heating system mathematically. This where done by first setting up equations linking flow and pressure for the systems individual components (namely: pumps, pipes and valves), and then combine these equations by use of Kirchoffs st and 2nd law. The model description of eq resulted. Eq constitutes the model to be used in the following chapters and the aim of the next chapter, will be to determine its parameters.

22 Chapter 3 Parameter estimation The purpose of this chapter is to estimate the parameters of equation The estimation is performed with a Matlab toolbox called SENSTOOLS developed by Morten Knudsen at AAU []. The toolbox utilizes a Gauss-Newton search algorithm on a specified simulation model to minimize the cost function defined as: Where errn = Σ N k= (y(k) y sim(k)) 2 Σ N k= (y(k)2 ) % (3.) y sim (k) are the simulated output y(k) are the measured output Given the fact that the valve constants K v, K v2, K v3, K v4 are functions of the valve position, which for time being is assumed unknown, the parameter estimation is performed with fixed valve position. Based on the analysis of appendix B. the tests are performed with consumer valve positions of respectively, 45%, 45%, 4% and 23%. Since the system has no flow sensors, the model will contain a scale factor that cannot be determined. Therefore the estimations will be performed with parameter locked and at the end of the section it will, through the cost function of eq. 3., be verified whether the error is unaffected when performing another estimation with the parameter locked to a different value. Since parameter has to be locked, it is chosen to assign the pipe friction K the value e5, because this gives flows that are considered of a physically realistic magnitude. 3. Input signal In order to obtain as accurate parameter estimates, as possible, an appropriate input signal are designed for the test. The input signal are designed for the output of the model to have high relative sensitivity to changes in the parameters, estimated. The measure used for sensitivity are the one presented in [, p.52-53], where the relative parameter sensitivities are calculated based on the normed relative Hessian matrix H rn defined as in

23 Parameter estimation 7 eq Where H rn = y RMS L H L (3.2) y RMS = N y N 2 (k) (3.3) k= L = diag(θ N ) (3.4) H are the Hessian of the model in the provided starting guess of parameters Θ are the parameters, to be estimated N are the number of parameters, to be estimated For nonlinear systems the amplitude distribution are described to be important in [, p.73], and an input class of square wave signals superimposed by ramp functions, are recommended. Following this recommendation, reduces the task of designing the input signal to a problem of choosing amplitude and frequency for a square ramp signal based on the model description derived in section 2.4. To help choosing amplitudes and frequencies the sensitivity ratio R are introduced R = S max λmax = (3.5) S min λmin Where λ max is the largest eigenvalue of the normed Hessian (Hrn) λ min is the smallest eigenvalue of the normed Hessian (Hrn) Having a small sensitivity ratio is desirable, because the larger the sensitivity ratio is, the larger the correlation between some of the parameters are, making the parameter estimates less precise. In order to obtain an input signal with a small sensitivity ratio, the sensitivity ratio are calculated for " square ramp waves" where the frequency are varied whilst the amplitude is fixed at.5. During calculations it is found that the sensitivity for the Q max parameter are particularly low. As a consequence, it will in the next section, be investigated whether the terms including Q max are redundant. The sensitivity ratios calculated, excluding Q max, are plotted in figure 3. and it is seen to have a minimum at around.65 Hz. The increase in sensitivity ratio, when moving away from the minimum frequency are relatively flat, meaning that frequencies in the area of.2-2 Hz all should be applicable as fundamental frequencies in a good input signal. Having analysed the sensitivity ratio with regards to frequency and found that it should be some where around.65 Hz, the sensitivity ratio are now calculated with varying amplitudes and a constant frequency of.65 Hz. Amplitude in this context is the difference from wave top to.5, which is 5% of the maximum input signal. The calculated sensitivity ratios are plotted in figure 3.2 where the sensitivity ratio can be seen to be smallest when the amplitude are largest at.5. From.2 and up the decay are quite slow, meaning that amplitudes between.2 and.5 should be used in the input signal. The sensitivity ratios of fig 3. and 3.2 are calculated with the M-files optimalt_input_frequency.m and optimalt_input_amplitude.m, on the attached CD. Based on the analysis of sensitivity ratios the input signal of fig 3.3 has been chosen. The input signals are

24 8 3. Input signal 2 5 R=Smax/Smin Frequency [Hz] Figure 3.: Sensitivity ratio as a function of frequency 2 5 R=Smax/Smin Amplitude of input square waves Figure 3.2: Sensitivity ratio as a function of amplitude

25 Parameter estimation 9 Pump m Pump Pump Pump m2 Pump 3 Pump Figure 3.3: The input signals applied to the pumps calculated with the M-file input_til_parameter_estimering.m and consists of square waves with frequency.65 Hz and random amplitude, superimposed by a ramp signal with the same frequency and random slope. Although the signals were called random in the above, the inputs for pump m, and 2 has been biased a little in order to get differential pressures, that are in the same order of magnitude for end user valves,2 and 3, Estimation and verification of parameters In this section, the parameters of eq are estimated from measurements made on the laboratory model, with the input signal designed in the previous section, applied. The parameters of eq are summarized below: Pipe frictions: K, K 2, K 3, K 4, K 2, K 22, K 23, K 24 K 3, K 32, K 33, K 34, K 4, K 42, K 43, K 44 Pump coeficient: P max, Q max Inertia coeficients: J, J 2, J 3, J 4, J 2, J 22, J 23, J 24 J 3, J 32, J 33, J 34, J 4, J 42, J 43, J 44 Valve coeficients: K v, K v2, K v3, K v Reduction of parameters for estimation Before the part of the estimation, which utilizes SENSTOOLS, are performed, some simplifications are made. First of all it is noticed that some of the pipes have identical dimensions and are difficult to tell apart through

26 2 3.2 Estimation and verification of parameters measurements, with the available sensors. Because of this the equalities of table 3. will be assumed for the rest the report. It is noticed that the coefficient that are set equal in table 3. always appears in pairs in the model, meaning that the predictions of the respective differential pressures are invariant to the size of the individual coefficients, as long as the sum of the pairs are correct. K =K 4 K 2 =K 3 K 2 =K 24 K 22 =K 23 K 3 =K 34 K 32 =K 33 K 4 =K 42 K 4 =K 43 K 4 =K 44 J =J 4 J 2 =J 3 J 2 =J 24 J 22 =J 23 J 3 =J 34 J 32 =J 33 J 4 =J 42 J 4 =J 43 J 4 =J 44 Table 3.: Assumptions regarding friction and inertia coefficients of the pipes Second of all, it is chosen to determine the proportion between the pipe frictions by use of the pressure sensors on the laboratory model, independently from the estimation of the rest of the parameters. The reason is primarily that the placement of these sensors makes it easy and straightforward and that the robustness of SEN- STOOLS tends to be quite poor for larger number of parameters and measurements. The proportions between the pipe frictions are determined in Appendix B.3 and summarized below: K 2 = 5.3 K K 2 =.58 K K 22 = 6.98 K K 3 =.89 K K 32 = 5.2 K K 4 = 2.75 K (3.6) Estimation of the remaining parameters The remaining parameters are estimated by use of the M-file estparametre.m, which can be found on the attached CD. Fig 3.4 shows the measured output vs a simulated, using the estimated parameters. The model seems to be predicting the actual system behaviour decently with an average error of 2.77% (with the error being defined by eq. 3.). The average error of 2.77% are obtained, for a model structure excluding the terms where Q max appears. The M-file estparametre_s.m (also to be found on the CD) performs the same estimation for a model structure where the terms with Q max are included. An average error of 2.74% are obtained. Since the average errors are close to identical, the terms including the Q max parameter are concluded redundant. In order to validate the estimated parameters, a new independent measurement are made with an input signal made by the same M-file as the original input signal, but being different because the step sizes and slopes are decided by a random number generator. The measured outputs are displayed in figure 3.6, along with a simulated response using the parameters estimated before. The error between estimate and measurement are in this case %, which are considered about the same. The simulated response of figure 3.6 looks quite

27 Parameter estimation 2 similar to the fitted response of fig 3.4 and beacause of this the quality of the parameter estimates are concluded sufficient. At the beginning of the chapter it was claimed that the system has a scale factor that cannot be determined without flow sensors, and hence the pipe friction K, where " locked" at e5. In order to verify that this was a legitimate decision the parameter estimation is performed on the same data set with the pipe friction K locked at e7, which is different by a factor of. The fit using these parameters are presented in fig 3.5 and results in an average error of 2.79%, which is very close to the average error obtained for K=e5. The similarities between the fits of fig 3.4 and 3.5 are also pronounced and hence it is concluded that the system does have a scale factor, which cannot be determined from the differential pressure sensors alone. The parameter estimates, obtained are presented below: J = 86 J 2 = 67 J 2 = 93.6 J 22 = 53 J 3 = J 32 = 934 J 4 = P max =.4 K = e5 Kv (.45) = 2.37e6 Kv 2 (.45) = 2.36e6 Kv 3 (.4) = 3.2e6 Kv 4 (.23) = 3.7e6

28 Estimation and verification of parameters.5 Differential Pressure.5 Differential Pressure2 Pressure [Bar] Pressure [Bar] Pressure [Bar] Differential Pressure3 5 Pressure [Bar] Differential Pressure4 5 Figure 3.4: The measured and simulated outputs of the data set that the parameter estimation is performed on. The estimation performed with K=e5.

29 Parameter estimation 23.5 Differential Pressure.5 Differential Pressure2 Pressure [Bar] Pressure [Bar] Pressure [Bar] Differential Pressure3 5 Pressure [Bar] Differential Pressure4 5 Figure 3.5: The measured and simulated outputs of the data set that the parameter estimation is performed on. The estimation performed with K=e7.

30 Estimation of valve friction as a function of valve position Differential Pressure Differential Pressure 2 Pressure [bar] Time [s] Differential Pressure 3 Pressure [bar] Time [s] Differential Pressure 4 Pressure [bar] Time [s] Pressure [bar] Time [s] Figure 3.6: The measured and simulated outputs of an independent set of data. 3.3 Estimation of valve friction as a function of valve position The parameter estimation performed in the previous section, were with valve positions fixed at respectively, 45%, 45%, 4% and 23%. The purpose of this section is to find a desription of how the respective valve frictions relates to the corresponding valve position. In order to do this, tests are made where the input signals designed in sec. 3. are applied to the system, whilst the valve positions, one at a time, are fixed at a different position than the ones used in the previous section (45%, 45%, 4% and 23%). For each set of valvepositions the new valve friction are estimated by use of the method used in the previous sections, with the estimated parameters of sec. 3.2, incerted. The estimated valve coefficients, are displayed in table Chapter Summary In this chapter, the parameters of the system description derived in chapter 2, has been determined. The number of parameters were first reduced through observations of symmetry, sensitivity calculations and by expressing the pipe frictions as constant functions of one another through measurements (see appendix B.3). Then the remaining parameters were estimated by use of a Gauss Newton search algorithm for a fixed set of consumer valve positions. The Gauss Newton search algorithm were applied to a data set, consisting of measurements obtained with an optimal set of input signals applied to the laboratory model. The input signals were designed to minimize the ratio between the maximal and minimal parameter sensitivity. A description of the frictions of

31 Parameter estimation 25 Valve position K v K v2 K v3 K v4 5.47e6 5.98e6 9.72e6 9.57e6. 5.2e6 6.26e6 9.48e6.8e e6 6.35e6 8.4e6 5.57e e6 5.32e6 5.54e6.2e6.4 3.e6 3.26e6 3.2e6 3.88e5.5.76e6.66e6.49e6.38e e5 8.2e5 7.53e5 4.7e e5 3.67e5 4.6e5 3.e4.8.69e5 8.92e4 9.93e4 7.94e3.9.5e4.33e3 -.e4.6e4.24e3-2.98e2 -.89e4 5.4e3 Table 3.2: The friction coefficients of the valves at different valve positions the consumer valves as function of the positions of the consumer valves, were obtained at the end of the chapter through estimations performed on different sets of consumer valve positions. The aim of the next chapter is to obtain a linear discrete time state space description, for control design, based on the model description that has been derived in the current and previous -chapter.

32 Chapter 4 Linearisation The purpose of this chapter is to linearise the nonlinear system description derived in the previous chapters, obtaining a linear discrete time state space description, suitable for model based linear control in later chapters. First a brief presentation of the linearisation method, will be made. Then the linearisation method will be applied to the system. After this the chapter will conclude with some thoughts on discretization, and pump delays. 4. Linearisation Method The system will be linearised using first order Taylor approximations. In a first order Taylor approximation, each variable are substituted with an operating point value (steady-state value), and a small signal gain. A given function are then approximated by its value in the operating point, plus a contribution from the small signal gain(s), which depends on the functions first derivative in the operating point. In the one variable case, the first order Taylor approximation are described as in eq. 4. [2, p.77]. Where: f(x) f( x) + δf( x) x (4.) δx f(x) is a function of the variable x. x is the operating point value. x is (x x). In case of multiple variables, eq. 4. can be extended to: f(x, x 2,..., x n ) f( x, x 2,..., x n ) + δf(x, x 2,..., x n ) δx x + x= x... δf(x, x 2,..., x n ) δx 2 x 2 + δf(x, x 2,..., x n ) x2= x 2 δx n x n (4.2) xn= x n

33 Linearisation Deriving a linear state space model In this section, the nonlinear system description will be linearised in accordance to the method presented above and put on state space form, with pump signals as input and the differential pressures over the consumer valves as output. The equation detailing the dynamics of the system were derived in chapter 2.4 and its parameters were fitted in chapter 3, resulting in a minor reduction of the model. After this reduction, the non-linear model are written as follows: J q q q 2 q 3 4 = K v q 2 q2 2 q3 2 q4 2 + R Um 2 U 2 U2 2 Um 2 2 U3 2 U4 2 T q.75 q.75 2 q.75 3 q.75 4 q.75 m q.75 m 2 q.75 m 3 q.75 m 4 Calculating the derivatives with respect to pump input (U) and flows (q), and insertion of operating points and small signal gains gives the following st order approximation: (4.3) J q q q 2 q 3 4 = K v 2 q q 2 q 2 q 2 2 q 3 q 3 2 q 4 q 4 + R 2 Ūm Ũm 2 Ū Ũ 2 Ū2 Ũ2 2 Ūm 2 Ũm 2 2 Ū3 Ũ3 2 Ū4 Ũ4 T.75 q.75 q.75 q 2.75 q 2.75 q 3.75 q 3.75 q 4.75 q 4.75 q m.75 q m.75 q.75 m 2.75 q.75 m 3.75 q.75 m 4 q m2 q m3 q m4 (4.4) Multiplying anything, that is not a small signal gain into the coefficient matrices and replacing q m, q m2, q m3 and q m4 with their equivalents: q m = q + q 2 + q 3 + q 4 q m2 = q 2 + q 3 + q 4 q m3 = q 3 + q 4 q m4 = q 4 Gives the following: J q q q 2 q 3 4 = K v lin q q 2 q 3 q 4 + R lin Ũ m Ũ Ũ 2 Ũ m2 Ũ 3 T lin q q 2 q 3 q 4 (4.5) Ũ 4 Where K vlin,r lin, T lin are constant matrices, displayed in appendix A

34 Discrete time state space model Inspection of equation 4.5 shows that it can be rewritten into state space form: ẋ = Ax + Bu (4.6) y = Cx (4.7) Where x = q 2 (4.8) 3 q 4 u = Ũ m Ũ Ũ 2 Ũ m2 Ũ 3 (4.9) Ũ 4 A = J ( K vlin T lin ) (4.) B = J R lin (4.) C = K vlin (4.2) 4.3 Discrete time state space model The linear state space representation of equation 4.6, 4.7 is in continuous time. In order to implement a model based controller in practice, a discrete representation of the form in equation 4.3, 4.4 is needed. x(k + ) = Φx(k) + Γu(k) (4.3) y(k) = Hx(k) (4.4) A method for discretization of a linear state space system are derived in [3, p ]. The derivation takes a starting point in the solution to the continuous state space model which is written in equation 4.5. x(t) = e A(t t) x(t ) + t t e A(t τ) Bu(τ)dτ (4.5) When making a discretization the continuous time variable t is replaced with the sample number k. In this derivation t = kt + T and t = kt where T is the sampling time. The solution over the k-th sample is shown in 4.6 kt +T x(kt + T ) = e AT x(kt ) + e A(kT +T τ) Bu(τ)dτ (4.6) kt In this discretization the Zero Order Hold method is used, meaning that the input u(τ) is kept constant throughout a sample period. Introducing the help variable η: u(τ) = u(kt ) kt τ < kt + k (4.7) η = kt + T τ (4.8)

35 Linearisation 29 Reduces Equation 4.6 to the following equation. ( ) T x(kt + T ) = e AT x(kt ) + e Aη dη Bu(kT ) (4.9) From which the discrete time state space matrices can be identified as: Φ = e AT (4.2) ( ) T Γ = e Aη dη B (4.2) H = C (4.22) The method just described are implemented in the Matlab function c2d(sysc,ts, zoh ), where sysc are the continuous state space description, Ts the sample time and zoh refers to the discretization method. 4.4 Pump delay This section addresses a problem with the pumps, which are not accounted for in the system representation so far. The problem is as the title suggests that there exist a significant delay from the input signals are sent from the Matlab Real Time Workshop until, the pumps actuate accordingly. The reason for this delay are so far unknown and will not be investigated in this project. Appendix B.2 documents the delays which are found to be as presented in table 4.. Name Delay [s] Pump m.7 Pump.7 Pump 2.7 Pump m2.7 Pump 3.5 Pump 4.6 Table 4.: The delays of the respective pumps It is seen that the delays are quite substantial compared to the dynamics of the system (the time constants of the differential pressures over the consumer valves seems to be around or below s). In addition to being relatively large, the delays also seems to vary from measurement to measurement, ranging from.5 s to. s. As mentioned in the introduction of the thesis, the laboratory model has been constructed with the intention of resembling time scaled version of a realistic district heating system. It is noted that the delays are not made intentionally and that the size of the delays are unrealistically large. A method for taking these delays into account, when implementing on the laboratory model, can be to choose the sample time as approximately the length of delays. By doing this a discrete time state space representation which takes the delays into account, can be made by augmenting the state vector with the last input signals, as in eq. 4.23: [ ] [ ] [ ] [ ] x(k + ) Φ Γ x(k) = + u(k) u(k) u(k ) I (4.23)

36 3 4.4 Pump delay 4.4. Chapter Summary In this chapter a linear discrete time state space description has been derived by use of first order Taylor approximations. In addition, severe pump delays has been documented and inserted in the state space description by augmentation of the state vector, with prior input signals. In the next chapter, a Linear Quadratic controller will be designed on the background of the linear discrete time state space description of eq

37 Part II Control

38 32

39 Table of Contents 5 Linear Quadratic controller Optimal control Introducing reference and integral action Implementation of the LQ controller Simulation of the LQ controller Observer Design Design of Kalman filter Design of Extended Kalman Filter Observer verification Linear Parameter Varying Control 6 7. LPV background and analysis Identification of an LPV model LPV control design

40 34 TABLE OF CONTENTS

41 TABLE OF CONTENTS 35 This part will concern the design of two different controllers for the scaled district heating system. The first are constructed by use of standard LQ-design. The second will be designed by transforming the system into LPV form and then apply the LPV methods described in [5] and [6]. Furthermore an Extended Kalman filter are designed for estimation of the flows through the consumer valves and for estimation of the friction of the consumer valves. This is done in order to reduce the amount of sensors on which the controllers depends and because e.g. the LQ-controller uses state feedback of the flows. Because of the unrealistically large pump delays documented in Appendix B.2 it is chosen to perform the majority of the control design on a model where the delays has been reduced. The majority of the tests performed will consequentially be made as simulations on the non-linear model with reduced delays.

42 Chapter 5 Linear Quadratic controller The aim of this chapter is to design a Linear Quadratic (LQ) controller for the scaled district heating system, capable of keeping the differential pressures over the consumer valves at a constant level of. [Bar]. First a general introduction to optimal control is presented. Then a LQ feedback controller is designed with reference and integral action. After the design has been described, the chapter will be concluded by simulations and a short description of the implementation 5. Optimal control The concept of optimal control is to find an input signal, which minimizes some performance function. In this project, the performance function relates to the states and input signals of a linearized model of the district heating system, and it is chosen to be quadratic with respect to these. This gives a performance function with the form of eq.5., which is a special case of optimal control referred to as LQ control. Where I = N H(x(k), u(k)) = k= N x T (k)q x(k) + u T (k)q 2 u(k) (5.) k= N is the finite time horizon Q is the weighting matrix of the states Q 2 is the weighting matrix of the inputs x(k) is the state vector u(k) is a vector with the input signals The performance function can be viewed as a way of weighting the size of control signals versus state sizes. In a system with n states and p inputs the performance function for a given system can be expanded to: I = N ( x T (k)q x(k) + u T (k)q 2 u(k) ) + x T (N)Q N x(n) (5.2) k=

43 Linear Quadratic controller 37 Q N is introduced to perform a separate weighting of the last value of the states. Q N will have dimensions (n n), Q 2 will have dimensions p p and Q will have dimensions n n. All three matrices are positive semi definite[4, p. 9]. It can be found that for a first order system, the inputs, u(k), are proportional to the current state, x(k), for each step. Thus the performance function can be rewritten as a quadratic function of the current state x(k) for the samples k... N: u = Kx(k) (5.3) J N k (x(k)) = x T (k)s(k)x(k) (5.4) where the matrices K(k) and S(k) describes the linear properties from u(k) to x(k) and the quadratic solution to x(k) respectively. K(k) is of dimension (p n) and S(k) is of dimension (n n). The general dynamic expression for the quadratic solution of x(k) can be described as [4, p. ]: J N k (x(k)) = min u(k) [H(x(k), u(k)) + J N k+(x(k + ))] (5.5) which describes the iterative properties of the solution to the quadratic form of x(k). Inserting the expressions for H(x(k), u(k)) and Jk+ N (x(k + )) and solving this for the control signal u(k), the equation will yield the minimum performance[4, p. ]: J N k (x(k)) = min u(k) [xt (k)q x(k) + u T (k)q 2 u(k) + x T (k + )S(k + )x(k + )] (5.6) = min u(k) [xt (k)q x(k) + u T (k)q 2 u(k) + (Φx(k) + Γu(k)) T S(k + )(Φx(k) + Γu(k))] To find the optimal control signal u, which minimizes the performance function at time k, it is differentiated and put equal to zero[4, p. ]: (5.7) δj N k (x(k)) δu(k) = 2Q 2 u + 2Γ T S(k + )(Φx(k) + Γu(k)) = (5.8) If [Q 2 + Γ T S(k + )Γ] is invertible, it can be solved with respect to the optimal control signal u (k): u (k) = [Q 2 + Γ T S(k + )Γ] Γ T S(k + )Φx(k) (5.9) From 5.9, the proportional gain K(k) can then be recognised as: K(k) = [Q 2 + Γ T S(k + )Γ] Γ T S(k + )Φ (5.) If Equation 5. is inserted into Equation 5.6 S(k) can be recursively expressed as: S(k) = Q + K T (k)q 2 K(k) + ((Φ ΓK(k)) T S(k + )(Φ ΓK(k))) (5.) 5.. Stationary LQ Control It is chosen to use the values of K() and S() determined for N, in calculating the controller gains. This yields, that the K(k) and S(k) of eqs. 5. and 5. will go toward steady state, when the performance index s sum tends toward infinity[4, p. 9]. As N goes toward infinity, K() = K(k) = K(k + ) = K and S() = S(k) = S(k + ) = S, are solutions to the steady state Riccati equations: K = [Q 2 + Γ T SΓ] Γ T SΦ (5.2) S = Q + Φ T SΦ Φ T SΓ[Q 2 + Γ T SΓ] Γ T SΦ (5.3)

44 Introducing reference and integral action In eqs. 5.2 and 5.3 the design parameters are the Q and Q 2 matrices. It is easily seen that the number of design parameters increases exponentially (r 2 + n 2 ) with the number of states (n) and inputs (r). In order to reduce the number of design parameters, the Q and Q 2 matrices could be chosen as diagonal, reducing the number of design parameters to r + n. In this case, the diagonal elements of Q (i, i) and Q 2 (j, j) can be interpreted as relative punishment of x i (k) and u j (k). This is not done, since relative punishment of flows and pump speeds, not makes much sense when the purpose is to track a constant differential pressure reference. It is however noted that punishment of control error and change in control signal would be meaningful tuning parameters. Punishing control error rather than state size, falls naturally in the next section, where Q (i, i) will be exposed to a simple transformation, as part of the method for introducing a reference signal. In order to punish change in pump inputs rather than the absolute size of the pump inputs, the system representation of eq are rearranged such that the control signal are changed from pump input to the difference between current and last pump input. Where [ ] x(k + ) u(k) = [ Φ Γ I ] [ ] x(k) u(k ) [ ] + [u(k) u(k )] I (5.4) Φ is the state matrix of eq Γ is the input matrix of eq Introducing reference and integral action Calculating the feedback control law with the LQ method, presented in the section above, based on the system representation of eq. 5.4, will not make sense, since eq. 5.4 alone contains no information of the reference. Apart from the problem of no reference, eq. 5.4 will inevitably have some error in its prediction of the system behaviour, due to linearisation and general modeling errors, which will be causing a steady state control error. To combat these two problems, the system description of eq. 5.4 will in the following two subsections be augmented, making reference and integral error part of the system description Introducing the reference The reference can be related to the system through an output error defined as: Where e(k) = r(k) y(k) (5.5) e(k) is the error, r(k) is the reference y(k) are the actual output

45 Linear Quadratic controller 39 As stated in the previous section it is desirable(necesarry) to punish control error rather than state size. This calls for the performance function to have a form of eq. 5.6 I = N (e T (k)q e e(k) + u T (k)q 2 u(k)) (5.6) k= In order to minimise 5.6, the system state vector x s (k) is augmented with the reference x r (k), giving the augmented system, described by [4, p. 3]: [ ] [ ] [ ] [ ] x s (k + ) Φ s x s (k) Γ s = + u(k) (5.7) x r (k + ) Φ r x r (k) Where e(k) = [ H s H r ] [ x s (k) x r (k) ] (5.8) H r is the reference model output matrix H s is the old output matrix Φ r is the state matrix of the reference model Φ s is the state matrix of eq. 5.4 Γ s is the input matrix of eq. 5.4 It is chosen to model the reference as a step, giving the following H r and Φ r H r = Φ r = (5.9) With the augmented system description at hand the performance function can be rewritten as: Where I = = = = N (e T (k)q e e(k) + u T (k)q 2 u(k)) (5.2) k= N ((Hx(k)) T Q e (Hx(k)) + u T (k)q 2 u(k)) (5.2) k= N (x T (k)h T Q e Hx(k) + u T (k)q 2 u(k)) (5.22) k= N (x T (k)q x(k) + u T (k)q 2 u(k)) (5.23) k= Q = [ ] H T [ ] s H T Q e H s H r r (5.24) Since eq has the same structure, as the original performance function in eq. 5., the method described in section 5. can be applied directly to find a feedback gain matrix K(k). The feedback matrix will be divided