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An optimal control strategy for an integrated solar thermal hydronic system with a heat pump Hosseinirad, Sara 2018

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An Optimal Control Strategy for an Integrated SolarThermal Hydronic System with a Heat PumpbySara HosseiniradB.S. Aerospace Engineering and Physics, Sharif University of Technology, 2015A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFMaster of Applied ScienceinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Mechanical Engineering)The University of British Columbia(Vancouver)August 2018c© Sara Hosseinirad, 2018The following individuals certify that they have read, and recommend to theFaculty of Graduate and Postdoctoral Studies for acceptance, a thesis entitled:An Optimal Control Strategy for an Integrated Solar Thermal Hydronic Systemsubmitted bySara Hosseiniradin partial fulfillment of the requirements forthe degree ofMaster of Applied ScienceinMechanical EngineeringCommittee:SupervisorDr. Ryozo NagamuneMechanical EngineeringExaminerDr. Bhushan GopaluniChemical and Biological EngineeringExaminerDr. Steven RogakMechanical EngineeringiiAbstractIn this thesis, a new control strategy is proposed for an Integrated Solar ThermalHydronic System (ISTHS) to optimize the system performance. The ISTHS uti-lizes two sources of energy which are solar and electrical to provide the domestichot water. The ISTHS performance can be optimized by reducing the consumedelectricity and retaining the hot water demand temperature under disturbances suchas solar radiation, ambient temperature, and how water demand flow rate. For theperformance optimization, the proposed control strategy employs three techniquesthat are optimization, feedback control, and feedforward control.Required for designing the proposed controller, the ISTHS model is obtainedby applying heat transfer and state-space modeling techniques. Using the state-space model of the ISTHS, the control structure can be designed. The controlstructure consists of four sub-controllers described as off-line, STC-Side, feedback,and robust feedforward controllers. By a combination of logic based switches andfour sub-controllers, the final control inputs are robust against the predicted distur-bances (Off-line), the actual disturbances (STC-Side and robust feedforward), andthe model uncertainties (feedback). The off-line controller applies an optimizationmethod to compute the control inputs one day ahead. The STC-Side controllerperforms an optimization method to manage some of the control inputs which af-fect the stored solar energy. The feedback controller keeps the hot water temper-ature within an allowable range. By using the robust feedforward controller, theconsumed electricity is reduced by adjusting the control inputs which affects theamount of the transformed electricity to the thermal energy. For examining the ef-fectiveness of the proposed robust feedforward controller, another controller namedsimple feedforward controller is developed and separately added to the overall con-iiitroller. Both controllers are designed such that the impacts of deviated disturbancesfrom predicted values on the system’s output are eliminated. Unlike the robustfeedforward controller, the simple feedforward controller does not reduce the con-sumed electricity. Finally, by making some comparisons through simulations, theeffectiveness of the proposed control structure is demonstrated.ivLay SummaryThe use of renewable energy sources has been steadily growing in recent years dueto increasing concerns regarding the destructive effects of traditional sources ofenergy on the environment. Solar power is one of the most popular sources of re-newable energy due to its clean and low-cost operation. Much effort has been madetoward developing affordable and efficient solar systems for domestic applications,particularly for the off-grid communities. With the absence of a central grid, ac-cess to power is limited primarily to diesel based electricity generation, which isexpensive, causes pollution, and requires the regular supply of fuel which can bechallenging in more remote communities. Control techniques are suitable tools forimproving the system’s operation with less configuration modification costs. Inthis thesis, the proposed control structure efficiently utilizes information on vary-ing environmental conditions to reduce the amount of electricity consumed by theauxiliary heater of the ISTHS.vPrefaceThis thesis is an original intellectual property of the author, Sara Hosseinirad. Thethesis work was conducted under the supervision of Dr. Ryozo Nagamune.A preliminary result of the thesis results was published in:• S. Hosseinirad and R. Nagamune, and V. Grebenyuk. Simultaneous opti-mization of configuration and controller parameters in an integrated solarthermal hydronic system. In the Proceeding of Americal Control Confer-ence (ACC), Seattle, WA, USA, May 24-26, 2017, pp. 2378–5861.viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xixDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Solar Thermal System . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3.1 Component Design Review . . . . . . . . . . . . . . . . 61.3.2 Control Design Review . . . . . . . . . . . . . . . . . . . 61.4 Research Objective . . . . . . . . . . . . . . . . . . . . . . . . . 9vii1.5 Research Outline . . . . . . . . . . . . . . . . . . . . . . . . . . 92 System Description and Modelling . . . . . . . . . . . . . . . . . . . 112.1 Solar Thermal Hydronic System Description . . . . . . . . . . . . 112.1.1 Operation . . . . . . . . . . . . . . . . . . . . . . . . . . 122.1.2 Instrumentation . . . . . . . . . . . . . . . . . . . . . . . 132.2 Nonlinear Model . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2.1 Energy Conservation . . . . . . . . . . . . . . . . . . . . 152.2.2 Solar Thermal Collector . . . . . . . . . . . . . . . . . . 172.2.3 Heat Exchanger . . . . . . . . . . . . . . . . . . . . . . . 182.2.4 Thermal Storage Tank . . . . . . . . . . . . . . . . . . . 202.2.5 Air to Water Heat Pump . . . . . . . . . . . . . . . . . . 222.2.6 Pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.2.7 Water Pump . . . . . . . . . . . . . . . . . . . . . . . . . 232.3 Disturbance Signals Modeling . . . . . . . . . . . . . . . . . . . 242.3.1 Solar Radiation . . . . . . . . . . . . . . . . . . . . . . . 242.3.2 Hot Water Load . . . . . . . . . . . . . . . . . . . . . . . 252.3.3 Ambient Temperature . . . . . . . . . . . . . . . . . . . 273 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.1 Control Objective . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2 Control Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.3 Off-Line Controller . . . . . . . . . . . . . . . . . . . . . . . . . 323.3.1 Optimization Problem . . . . . . . . . . . . . . . . . . . 333.3.2 Optimization Method . . . . . . . . . . . . . . . . . . . . 343.4 On-Line Controller . . . . . . . . . . . . . . . . . . . . . . . . . 383.4.1 STC-Side . . . . . . . . . . . . . . . . . . . . . . . . . . 383.4.2 HP-Side . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.5 Simple Feedforward Controller . . . . . . . . . . . . . . . . . . . 423.6 Robust Feedforward Controller . . . . . . . . . . . . . . . . . . . 463.7 Feedback Controller . . . . . . . . . . . . . . . . . . . . . . . . . 50viii4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.1 Simulation Setting . . . . . . . . . . . . . . . . . . . . . . . . . 544.1.1 Assumption . . . . . . . . . . . . . . . . . . . . . . . . . 544.1.2 Disturbance Setting . . . . . . . . . . . . . . . . . . . . . 564.1.3 Parameter Setting . . . . . . . . . . . . . . . . . . . . . . 584.2 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.2.1 Off-Line Controller . . . . . . . . . . . . . . . . . . . . . 604.2.2 Feedback Controller . . . . . . . . . . . . . . . . . . . . 614.2.3 Feedback and Off-Line Controller . . . . . . . . . . . . . 644.2.4 Simple Feedforward Controller . . . . . . . . . . . . . . 674.2.5 Robust Feedforward Controller . . . . . . . . . . . . . . 704.3 Simulation Conclusion . . . . . . . . . . . . . . . . . . . . . . . 715 Conclusion and Future Works . . . . . . . . . . . . . . . . . . . . . 745.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.2 Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.3 Future Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77A Supporting Materials . . . . . . . . . . . . . . . . . . . . . . . . . . 82A.1 Nonlinear State-Space Model . . . . . . . . . . . . . . . . . . . . 82A.1.1 Specific Heating Capacity (cp) . . . . . . . . . . . . . . . 84A.1.2 Overall Heat Transfer Coefficient (U) . . . . . . . . . . . 84ixList of TablesTable 4.1 Hypothetical continuous measured disturbance parameters de-viations from predictions . . . . . . . . . . . . . . . . . . . . 58Table 4.2 Modeling parameters in simulations . . . . . . . . . . . . . . . 58Table 4.3 Optimization parameters . . . . . . . . . . . . . . . . . . . . . 59Table 4.4 OL Controller’s performance comparison under different dis-turbances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Table 4.5 FB controller’s performance comparison under different distur-bances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65Table 4.6 FB + OL controller’s performance comparison under differentdisturbances . . . . . . . . . . . . . . . . . . . . . . . . . . . 67Table 4.7 Simple feedforward controller’s performance comparison underdifferent Disturbances . . . . . . . . . . . . . . . . . . . . . . 69Table 4.8 Robust feedforward controller’s performance comparison underDifferent Disturbances . . . . . . . . . . . . . . . . . . . . . . 71Table 4.9 Performance comparison under worst case disturbances . . . . 72Table 4.10 Performance comparison under random case disturbances . . . 72Table 4.11 Performance comparison under best case disturbances . . . . . 72Table A.1 Constant values for calculating cp . . . . . . . . . . . . . . . . 84Table A.2 Typical h values for air and water . . . . . . . . . . . . . . . . 84xList of FiguresFigure 1.1 Canada’s energy consumption . . . . . . . . . . . . . . . . . 2Figure 1.2 Various applications of solar thermal systems . . . . . . . . . 4Figure 2.1 Schematic view of ISTHS . . . . . . . . . . . . . . . . . . . 12Figure 2.2 Structure of the state-space model . . . . . . . . . . . . . . . 15Figure 2.3 Schematic view of small control volume within components . 15Figure 2.4 Evacuated tube solar collectors . . . . . . . . . . . . . . . . . 18Figure 2.5 Schematic view of a heat exchanger element . . . . . . . . . . 19Figure 2.6 Schematic view of j th element of storage tank . . . . . . . . 20Figure 2.7 Schematic view of the vapor-compression cycle for HP . . . . 22Figure 2.8 Sun’s position determined by altitude and azimuth angles . . . 25Figure 2.9 The solar radiation on the first day of May in Vancouver . . . 25Figure 2.10 Barriers in front of solar radiation . . . . . . . . . . . . . . . 26Figure 2.11 Average daily DHW consumption for Canada in liters per hour 26Figure 2.12 Local temperature in Vancouver for the year of 2012 . . . . . 27Figure 3.1 Example of the desired temperature range . . . . . . . . . . . 30Figure 3.2 Control design procedure of designing the overall control struc-ture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Figure 3.3 Off-line controller block diagram . . . . . . . . . . . . . . . 33Figure 3.4 How to calculate Tcost . . . . . . . . . . . . . . . . . . . . . . 35Figure 3.5 Flowchart for GA . . . . . . . . . . . . . . . . . . . . . . . . 36Figure 3.6 Flowchart for PSO . . . . . . . . . . . . . . . . . . . . . . . 37Figure 3.7 Considered model for STC-Side control design . . . . . . . . 39xiFigure 3.8 STC-Side controller block diagram . . . . . . . . . . . . . . 39Figure 3.9 Control design procedure of designing the HP-Side controlstructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Figure 3.10 Simple feedforward controller structure . . . . . . . . . . . . 43Figure 3.11 Simple feedforward controller block diagram . . . . . . . . . 45Figure 3.12 Simple example of feedforward controller block diagram . . . 45Figure 3.13 Robust feedforward controller structure . . . . . . . . . . . . 47Figure 3.14 LFT form for designing robust feedforward controller . . . . . 48Figure 3.15 Robust feedforward controller block diagram . . . . . . . . . 48Figure 3.16 Robust feedforward controller block diagram with weightingfunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Figure 3.17 Example of feedforward controller allowable temperature range 51Figure 3.18 Feedback controller block diagram . . . . . . . . . . . . . . . 51Figure 4.1 Domestic hot water temperature as a function of two Controlvariables: u3 and u4 . . . . . . . . . . . . . . . . . . . . . . . 55Figure 4.2 Three disturbances scenarios . . . . . . . . . . . . . . . . . . 57Figure 4.3 Control design procedure of designing the OL control structure 60Figure 4.4 Iteration convergence of the PSO method . . . . . . . . . . . 61Figure 4.5 Off-line controller results . . . . . . . . . . . . . . . . . . . . 62Figure 4.6 Control design procedure of designing the FB control structure 63Figure 4.7 Feedback controller results . . . . . . . . . . . . . . . . . . . 64Figure 4.8 Control design procedure of designing the FB + OL controlstructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65Figure 4.9 Combination of FB and Off-line controllers results . . . . . . 66Figure 4.10 The results of the overall controller with the simple feedfor-ward controller . . . . . . . . . . . . . . . . . . . . . . . . . 68Figure 4.11 The results of the overall controller with the robust feedfor-ward controller . . . . . . . . . . . . . . . . . . . . . . . . . 70Figure 5.1 Solar thermal collectors and their stand: research setup . . . . 76xiiList of SymbolsASTC - Gross area of STC panels [m2]Ae f f - Effective area of STC panels [m2]ACoil - Cross section area of HX coil [m2]AShell - Cross section area of HX shell [m2]Apipes - Cross section area of pipes [m2]c0 - Constant term in the solar thermal collector efficiency equationc1 - First coefficient of the solar thermal collector efficiencyc2 - Second coefficient of the solar thermal collector efficiencycp - Specific heat capacity [ kJkg·K ]d - Disturbance vectorD(s) - Laplace transform of variation of disturbance vectorDT ST - Outer diameter of the tank [m]dp - Predicted disturbance vectorδd - Variation of disturbance vectore - System’s output errorE˙WP - Electrical energy consumption of water pump [kW ]E˙con - Overall electrical energy consumed [kW ]δe - Variation of ehair - Convective heat transfer coefficient of air [ Wm2·K ]hwater - Convective heat transfer coefficient of water [ Wm2·K ]hshell - Convective heat transfer coefficient of the fluid inside HX shell[ Wm2·K ]hcoil - Convective heat transfer coefficient of the fluid inside HX coil[ Wm2·K ]xiiihi - Convective heat transfer coefficient of the fluid inside the i th pipe[ Wm2·K ]I - Total solar radiation flux [W/m2]J - Cost FunctionKP - Proportional Gain of the feedback controllerKατ - Incident angle modifier of STC [ Wm2·K ]kcoil - Thermal conductivity of the HX coil material [W/(m ·K)]ki - Thermal conductivity of the i th pipe material [W/(m ·K)]kT ST - Thermal conductivity of the tank insulation [W/(m ·K)]L - Latitude of location [◦]M - Mass inside each control volume [kg]m˙load - Mass flow of water leaving thermal storage tank [kg/s]m˙STC - Mass flow of fluid circulating in STC [kg/s]m˙HP - Mass flow of fluid circulating in HP [kg/s]m˙shell - Mass flow of fluid circulating in the shell of HX [kg/s]m˙coil - Mass flow of fluid circulating in the coil of HX [kg/s]m˙STC−T ST - Mass flow rate in STC-TST loop [kg/s]m˙T ST−HP - Mass flow rate in TST-HP loop [kg/s]np - Number of particles in PSOPelec - Overall electrical energy consumption [kW ]PHP - Electrical energy consumption of heat pump [kW ]PWP - Electrical energy consumption of water pump [kW ]PGly - The volumetric percentage of the glycol in mixture [%]Q˙internal - rate of change of the fluid internal energy of the control volume[kW ]Q˙inlet - Inlet fluid energy rate [kW ]Q˙outlet - Outlet fluid energy rate [kW ]Q˙in - Rate of incoming thermal energy [kW ]Q˙out - Rate of outgoing thermal energy [kW ]Q˙STC - Heating capacity of solar thermal collectors [kW ]δq - Measured output of the LTI systemr - Desirable output rangeRComp - HP compressor speed ratioxivrout - Outer radius of HX coil [m]rin - Inner radius of HX coil [m]Tallowable - Allowable temperature rangeTmix - Temperature of the glycol and water mixture [◦C]Tcoil,in - Temperature of the fluid entering the HX coil [◦C]Tcoil,out - Temperature of the fluid leaving the HX coil [◦C]Tcost - Temperature cost [◦C]TDHW - Temperature of DHW [◦C]THP,out - Outlet temperature of HP [◦C]Tinside - The temperature of the inside the houseTlallowable - lower limit of the allowable temperature range [◦C]Tlower - lower limit of the desired temperature range [◦C]Tshell,in - Temperature of the fluid entering the HX shell [◦C]Tshell,out - Temperature of the fluid leaving the HX shell [◦C]TSTC,out - Temperature of the fluid leaving the STC [◦C]Tuallowable - Upper limit of the allowable temperature range [◦C]Tupper - Upper limit of the desired temperature range [◦C]ti - Thickness of i th pipe wall [m]tT ST - Thickness of the tank insulation [m]∆THX1 - Temperature difference between the inlet and the outlet tempera-ture of first heat exchangeru - Input (or control) vectoru - Vector of maximum control inputsU5 - Laplace transform of δu5Upipe - Overall heat transfer coefficient of pipe [ Wm2·K ]uSTC - STC-Side control inputδu5 - Variation of u5δu5w - Weighted δu5VT ST - Total volume of the tank [L]vair - Speed of air [m/s]vwater - Speed of water [m/s]W - Weighting FunctionwP - Electrical power weight coefficient [−]xvwSTC - Solar thermal energy weight coefficient [−]wT - Temperature weight coefficient [−]x - State vectorxSTC - State vector of the STC side of systemY - Laplace transform of δyδy - Variation of system’s outputy - system’s outputηSTC - Solar thermal collector efficiency [−]δ - Solar declination angle [◦]θ - Solar incident angle [◦]xviGlossaryAC Adaptive ControlDHW Domestic Hot WaterFB Feedback ControllerFF Feedforward ControllerFLC Fuzzy Logic ControlGA Genetic AlgorithmGS Gain-SchedulingHD Heat DissipatorHX Heat ExchangerHP Air-to-Water Heat PumpISTHS Integrated Solar Thermal Hydronic SystemLFT Linear Fractional TransformationLQG Linear Quadratic Gaussian ControlMPC Model Predictive ControlNC Non-linear ControlNNC Neural Network ControlxviiNSERC Natural Sciences and Engineering Research CouncilOL Off-Line ControllerPID Proportional-Integral-DerivativePCM Phase-Change MaterialsPSO Particle Swarm OptimizationRC Robust ControlTST Thermal Storage TankSTC Solar Thermal CollectorWP Water PumpxviiiAcknowledgmentsI would like to thank my supervisor Dr. Ryozo Nagamune for his tremendous sup-port and guidance during my master studies at the University of British Columbia.To Vladimir Grebenyuk, the industrial partner of the integrated solar thermal hy-dronic system project, I thank for your constant support and supervision. I wouldlike to express my gratitude to Dr. Bhushan Gopaluni and Dr. Steven Rogak fortheir invaluable suggestions to improve my thesis.The research was financially supported by the Natural Sciences and Engineer-ing Research Council (NSERC) of Canada through the Collaborative Research andDevelopment Grant, as well as Ascent Systems Technologies. It was conductedat the UBC Control Engineering Laboratory in the UBC Institute for Computing,Information and Cognitive Systems.Furthermore, I am eternally grateful that I studied and researched alongside myfriendly and supportive research lab-mates: Pan Zhao, Mohammadreza Rostam,Jihoon Lim, Ali Cherom Kheirabadi, Chenlu Han, Anderson Soares, ChristopherCortes, Eduardo Escobar, Marcin Mirski, and Ran Fan in the Control EngineeringLaboratory.xixDedicationI would like to dedicate my thesis to my family and all my friends.xxChapter 1IntroductionThe sun’s radiation has warmed up Earth and has provided food for all lives onit. Humans also have taken many other advantages of solar energy throughouthistory. Although we believe using the renewable energy for various human needsis a recent issue, many ancient societies were knowledgeable about solar energyfeatures and how to apply them for their living purposes. Dwellers around theglobe lived in the caves which had the opening towards the southeast direction.These openings provided them warmer caves in the morning. Additionally, ancientGreeks and Romans developed more sophisticated technology for their houses tobenefit from the solar energy.The solar hot water system was invented in the late 1800s. The first solar waterheater was designed in California in 1891 which was named “Climax”. The systemwas a tank which could absorb solar energy directly through glass-side of its body.From 1891 to 1970s, several techniques have been employed to develop solar wa-ter heaters. For instance, the “flat plate” collector was the first collector which hasbeen designed to capture the solar energy systematically. In cold weather, watermight freeze inside the pipes, connecting solar thermal collectors to the tank. Re-garding the mentioned issue, a heat exchanger was integrated with the solar waterheater. Consequently, the water circulating inside collectors and pipes replacedwith another fluid which has the lower freezing point. Meanwhile, the politicianshad not, however, felt the urgency of replacing fossil fuels, especially oil, withrenewable energy until the big oil crisis occurred [1].1The interest in renewable energy, such as solar energy, experienced the first sig-nificant impulse after the big oil crisis during the 1970s. At that time, the primaryfactor in increasing the use of renewable energy was the high price of hydrocarbonfuels [2]. In recent years, however, the emphasis has shifted toward environmentalconsiderations, e.g., pollution and potentially irreversible climate change.A common concern of engineers is that whether enough solar energy is avail-able for various applications. Global power demand is approximately 16 TW, andthe sun transfers 150000 TW thermal power to Earth. Despite decreasing the ac-cessible solar radiation due to the reflection and atmosphere absorption, the solarenergy is the most abundant and sustainable energy, and it is still enough for meet-ing all human needs [3]. Nevertheless, in many countries, solar thermal systemsare not commercialized as it is expected.For instance, Figure 1.1 depicts differentenergy consumption sectors in Canada. In Figure 1.1 (a), it is shown how much(a) Canada’s energy consumption in different sectors(b) Canada’s energy consumption in residential sectorFigure 1.1: Canada’s energy consumption (https://www.nrcan.gc.ca/energy)2various sectors consumed energy in Canada in 2013. In the same year, Canadiansused 63% and 19% of the domestic energy,respectively, for space and water heatingas shown in Figure 1.1 (b). Although 18.1% of Canada’s consumed power comesfrom the renewable sources, the solar thermal systems provide the small portionof 0.08% of the renewable energy 1. In this thesis, a methodology is representedto improve the performance of the solar thermal hot water system such that theconsumed electricity by the system’s components is reduced.Various types of solar technologies have been developed, and their efficiencydramatically increased over the last several years. In places with abundant solarradiation, the solar energy is commonly used as clean energy, on various scales,from the domestic to the industrial applications. In the next section, the varioussolar thermal systems are generally introduced.1.1 Solar Thermal SystemThere is an extensive application range of solar thermal systems such as solar ther-mal power plants and solar domestic hot water systems, as depicted in Figure 1.2.The main common components among these systems are the solar thermal collec-tor and the thermal storage tank. Solar thermal collectors absorb the solar thermalenergy. This energy is stored inside the thermal storage tank for further and futureusages.In this research, we are focusing on solar domestic hot water systems. Severaltypes and configurations of the solar water heating system have been developed.Numerous components are also integrated into the system to enhance its operatingperformance such as a heat exchanger and a heat pump. Regarding using a heatexchanger, there are two types of solar water heating systems, in example openloop and closed loop. The system without a heat exchanger is called “open-loop”in which the residential water enters the solar thermal collector and is warmed updirectly. In contrast, the system integrated with a heat exchanger is named “closed-loop”. The reason for employing the heat exchanger is to separate the fluid insidethe collectors and the residential water. Closed-loop systems are advantageous ifboth freezing point and corrosion effect of the circulating fluid inside solar thermal1https://www.nrcan.gc.ca/energy3Figure 1.2: Various applications of solar thermal systems. Theleft figure is a domestic hot water solar thermal system(www.jacobskachels.nl), and the right one is a 20 MW solar powertower (https://www.abengoasolar.com)collectors are adequately low. In order to circulate fluid inside the pipes, solarwater heating systems can utilize the aid of either water pumps or the gravity. Theformer approach is called active method whereas the latter is called the passivemethod. Despite the advantages of environmentally friendly solar thermal energy,all solar-based systems have a common drawback which is the difference in theamount of the available energy during the day and night. Furthermore, clouds andprecipitation can have a remarkable effect on solar energy production. In order totackle this problem, therefore, another auxiliary thermal energy source is neededin solar thermal systems to provide the required energy when there is a lack ofsolar radiation. Electrical heaters and heat pumps are two main auxiliary thermalsources used in solar thermal systems.The various types of solar thermal collectors and thermal storage tanks canbe employed such that the overall efficiency of the system increases. The config-uration of the solar water heating system is not the main focus of this research.Therefore, a conventional configuration which has reasonable efficiency is consid-ered. In this research, for a solar thermal system with a specific configuration, amathematical modeling is provided. Also, an efficient control algorithm for suchconfiguration is proposed.41.2 Problem StatementHumans have always been seeking a source of energy which is inexpensive, abun-dant in nature, less destructive, and usable for several applications. The first threefeatures can be found in different kinds of renewable energy; however, using thesekinds of energy as a source of power for domestic applications is not convenient.As it is shown in Figure 1.1, solar thermal systems have a tiny contribution in pro-viding energy for residences in Canada. It is the same for many other countries.The problem is if the amount of available solar energy is enough for common ap-plications, why are not solar thermal systems commercialized as it should be?Existing solar systems are not efficient to compete with other conventional wa-ter heating systems, and generally speaking, the cost of solar thermal energy isrelatively high. One approach to increase the overall efficiency and reduce the to-tal cost of these systems is to improve the system’s operation. By doing so, theauxiliary heater needed in these systems can be replaced with a cheaper, smallerone.In this research, the focus is on improving the solar thermal hot water systems’operation by developing an optimal control structure. Using control techniques fordesigning a cost-effective system is tricky since many non-manipulating param-eters such as the solar radiation and the hot water demand influence the systemperformance [3]. Therefore, predicting these non-manipulating parameters plays acritical role in controlling solar thermal systems. If both the information on envi-ronmental conditions and the model describing the solar thermal system are accu-rate, a simple controller can manage the solar thermal system. However, there arealways errors either in the model or the predictions. By having both forecast andthe real-time measured data along with a proper control method, one can enhancethe system’s performance in a practical manner.1.3 Literature ReviewIn buildings equipped with solar thermal systems, their high performance may beachieved by two mechanisms [4]. The first mechanism is related to componentdesign. Let us interpret this mechanism by an example of the thermal storage tank.One of the efficient technique for operating the system is to capture all available so-5lar radiation and to store them in a well-insulated thermal storage tank. Recruitinga well-designed thermal storage tank is the goals of many studies on solar thermalsystems. The second mechanism for improving solar thermal systems performanceis to optimize the overall operation according to both real-time and predicted en-vironmental and demand conditions. Thus, we need an advanced control strategywhich would allow the adaptation of system operations to varying external condi-tions and demand profiles [4]. The focus of this research is on the latter mechanismwhich is improving overall system operation by developing an advanced controltechnique. Next, after providing a quick review on the component design, the stud-ies on control design of solar thermal systems are analyzed in detailed.1.3.1 Component Design ReviewOne of the main components of the solar thermal system is the thermal storagetank. Proper thermal storage has to be integrated into the system to efficiently storethe maximum amount of thermal energy which is captured by solar collectors.The extra solar energy absorbed and stored in the thermal storage tank will beconsumed later when the solar irradiation is not available. Some studies have beenperformed to increase the thermal storage capacity by using special materials tocapture more thermal energy, such as Phase-Change Materials (PCM) [5], or byinstalling more thermal insulation to prevent thermal loss. In [6], the proposedsmart tank is equipped with an auxiliary heater element and allows for a variablewater volume, and hence the thermal performance enhances. As a preliminaryresult of this thesis, a simultaneous optimization is performed to improve bothsystem and thermal storage tank operations [7]. In this thesis, we assumed that thesystem and components are designed and configured correctly.1.3.2 Control Design ReviewMany efforts have been made to implement new control techniques to improve theoperational performance of solar thermal systems. The goal of this part is to inves-tigate the advantages and disadvantages of each control method and to introducethe contribution of this thesis which is designing an optimal and efficient controlstructure by removing drawbacks in the previous techniques. One can classify the6studies on control design in two sectors, basic and advanced methods.Some earlier studies considered basic control methods such as feedforwardand Proportional-Integral-Derivative (PID) controllers. A feedforward controllermanages the system based on the real-time disturbances signal. The feedforwardcontroller is widely used for controlling solar thermal plants [8]. Camacho et al.proposed both parallel and series combinations of feedforward and feedback con-trollers to implement in a solar thermal plant [9]. These controllers were de-signed such that the outlet temperature of the solar plants reaches the desired tem-perature precisely by manipulating the flow rate inside a series solar collectors.The proposed feedforward controller was implemented in several studies including[10], [11], [12], and [3]. The proposed feedforward controller, however, needs ex-tremely accurate models for both disturbances and the plant. Otherwise, it can notoperate properly. In addition, those studies just dealt with the solar radiation as ameasured disturbance and did not consider any varying user loads. Later, a feedfor-ward controller was designed in the frequency domain so that all disturbances andcontrol inputs were considered [13]. Silva et al. [14] proposed a combination offeedback and feedforward control methods in the form of a dual adaptive controllerfor solar collector field applications.As for advanced control methods, strategies which have been applied to so-lar thermal systems include Adaptive Control (AC), Gain-Scheduling (GS), opti-mal control (Linear Quadratic Gaussian Control (LQG)), Non-linear Control (NC),Model Predictive Control (MPC), Robust Control (RC), Fuzzy Logic Control (FLC),and Neural Network Control (NNC). Among these control methods, MPC and AChave shown a better performance and become popular for solar plants. For ex-ample, in [15] and [16], using model predictive control methods, researchers con-trolled a solar thermal field in the presence of uncertainty in systems’ model. Like-wise, in [17] and [18], adaptive control methods are employed to adjust the controlsystem considering time-varying models and disturbances. In another research, anadaptive control was integrated with a PID to form a cascade control system. Thecascade control system has two loops, named the inner and the outer loops. Theinner loop suppresses the effect of disturbances on the system, while the outer loopcontrols the output process [19].Many studies showed that using forecast data can improve the performance7of controllers in solar thermal systems. For instance, in [20], data on measureddisturbances during a day was used for the next day. As a result, the electrical con-sumption of the backup heating component was reduced by employing a simplelogic controller. Parte et al. [21] employed a predictive sliding mode technique asa robust controller for a solar thermal plant. LeBreux et al. [22] proposed usingboth a fuzzy logic and a feedforward controller to estimate the amount of thermalenergy for the following day, and to indicate the electricity consumption profileof the heating element, respectively. Prud’homme et al. [23] validated a predictivecontrol strategy both to estimate disturbances and to calculate an optimal controllerfor a solar system. Favre et al. employed an optimization-based control strategyusing dynamic programming to control the temperature of a building [24]; how-ever, the designed controller is difficult to implement due to the requirement of theaccurate weather forecast. Gain-scheduling control techniques are another well-known methods for controlling solar power plant. Many studies have been doneto show the benefits and drawbacks of gain-scheduling techniques on solar powerplants [25], [26].To reduce the electrical cost of both water pumps and auxiliary heater, duringthe 1970s, There are several research which considered optimal control approachesfor solar thermal systems for heating applications. Orbach et al., in [27] and [28]introduced a bang-bang optimal control strategy with a constant flow rate by focus-ing on both the dynamic of both solar collector and thermal storage tank. Mean-while, Badescu [29] and Nhut et al. [30] indicated that using an optimal controlstrategy with variable flow rates improves system’s performance compared to aconstant flow rate. Roberta Padovan et. al [5] used both genetic optimization andsimplex optimization methods in order to optimize the energy consumption andthe volume of the thermal storage tank. In [31], the particle swarm optimizationmethod was combined with the generalized predictive control. An optimal controltechnique also was used in [32] for determining the heat loss coefficient of compo-nents. Despite their satisfactory performances, almost all optimal control methodshave time-consuming calculations due to the optimization steps. Therefore, thereare not suitable for practical implementations.There are two issues with the mentioned studies which have not been addressedso far. First, in all previous studies, they faced a tracking control problem. In other8words, they assumed there is a specific temperature for the output to follow. Thisassumption works for power plants in which the output temperature is set consid-ering the operation conditions. Yet in domestic hot water applications, typically,there is no particular temperature for the hot water. Second, most of the optimalcontrol design methods have been employed to deal with uncertainties in the sys-tems model while the uncertainties for disturbances are negligible. Almost noneof them, therefore, is able to provide an efficient method for producing optimalcontrol inputs by considering measured disturbances for domestic applications.1.4 Research ObjectiveThe objective of this research is to enhance the performance of an integrated so-lar thermal hydronic system by diminishing the consumed electricity while the hotwater temperature meets the demand. Unlike the previous studies, the feedbackcontroller retains the hot water demand temperature within an allowable rang, andthe measured disturbances are fed as signals into the optimal feedforward con-troller. In fact, if this goal can be reached by designing a proper and practical con-trol algorithm, the overall cost of existing systems will be reduced. A secondarybenefit to reducing the electrical consumption is that it would permit a smallerheat pump, which is the alternative heater, and a smaller integrated solar thermalsystem overall. Such a reduction in size would facilitate transportation to off-gridcommunities.For designing this controller, the uncertainties in the model’s parameters do nottake into account. In this research, the focus is on dealing with the uncertainties indisturbances which are mainly environmental conditions, the solar radiation, andthe domestic hot water usage. For practical domestic applications, the procedureof obtaining control inputs should be quick and straightforward.1.5 Research OutlineThe thesis is organized as follows. In Chapter 2, it is illustrated how the systemoperates and what are the main components of the system. The system is a solarthermal hot water system which is integrated with an air-to-water heat pump. Inthis chapter, the state-space model is also developed with the clarification of states,9control inputs, and disturbances. In Chapter 3, the procedure of designing thecontroller for the system is explained. After introducing the control objective andoverall control structure, different methods which are employed for forming eachsub-controller of the overall structure are illustrated. The high efficiency of theproposed controller is demonstrated in simulations in Chapter 4. The system’sresponse to different sub-controllers and disturbances scenarios are also analyzedin this chapter. Last, the research and conclusions are summarized in Chapter 5,and the possible future work is proposed.10Chapter 2System Description andModellingOne of the requirements of designing an advanced controller is to develop the dy-namical model of the system. In this chapter, we are seeking the nonlinear state-space system which is generally shown in Equation (2.1).x˙ = f (x,u,d). (2.1)Here, the vectors x, u, and d are respectively the state vector, the input vector, andthe disturbance vector. In this chapter, systems components are described, and theoperation of the system is clarified. A dynamic mathematical model using heattransfer information leads us to determine the systems’ parameters and variablesand to form the control problem in the state-space environment.Next, the system’s operation is explained. By introducing the energy conserva-tion law for this system, we go through the mathematical model of each componentone by one and derive the state-space model.2.1 Solar Thermal Hydronic System DescriptionFigure 2.1 illustrates the ISTHS that we consider in this thesis. The main com-ponents are Solar Thermal Collector (STC), Air-to-Water Heat Pump (HP), Ther-mal Storage Tank (TST), Heat Dissipator (HD), Heat Exchanger (HX), Water Pump11Figure 2.1: Schematic view of ISTHS(WP), and pipes. As indicated in Figure 2.1, this system contains four major loops:1) STC, 2) STC-TST, 3) TST-HP, and 4) HP loop. The fluid flow rate in each ofthe loops is controlled by a corresponding WP. The main parameters which arenot manipulative are the solar radiation, the Domestic Hot Water (DHW), and theambient temperature.2.1.1 OperationThe energy of solar radiation is captured by absorbers inside the vacuum tubes andtransferred through the manifold to the liquid circulating in the solar loop. Notethat ISTHS adopts evacuated tube collectors because, unlike flat plate collectors,they are designed to have minimum thermal losses and maximum solar radiationabsorptions. Due to gravity and pressure difference, the fluid moves toward the HXin the lower level of the STC loop. In the HX, a portion of the STC fluid’s thermalenergy is passed to the STC-TST loop. After transferring heat through the HX, the12STC loop fluid flows back to the STC with an assist of a WP. Meanwhile, a smallportion of thermal energy is lost through highly insulated pipes. If the amount ofsolar energy absorbed by the collector is more than what it can be safely stored inTST, HD releases excess energy into the environment.The cool water from the bottom of the storage tank is circulated by a WP inthe STC-TST loop. In the STC-TST HX, the cool water extracts energy from thehot fluid in the STC loop. The hot water enters the top of the TST where the hotwater is sent to the end-user. The bottom of the tank is connected to the inlet waterin order to compensate the water usage from the storage tank. For simplifying thegeneral model, the rate of entering tap water is equal to the hot water demand flowrate. This assumption ensures the constant water level inside the thermal storagetank. 1.Another component of the ISTHS considered in this paper is an air-to-waterHP. Thermal energy produced by the HP is transferred to the TST-HP HX in theHP loop. In this loop, the temperature of cool water from the bottom of the tankis raised by absorbing energy from the hot fluid in the HP loop. The hot fluid isdelivered to the top of the TST, with an assist of another WP.2.1.2 InstrumentationWith the proposed configuration in Figure 2.1, one can use a thermometer at theinlet and the outlet of each component and a flow-meter inside each loop to senseand measure temperatures and flow rates, respectively. Other thermometers canbe useful for measuring the temperature at each level of the TST and the ambienttemperature. In order to record the domestic hot water flow rates, a flow-meter canbe implemented. In addition a solar radiation sensor is needed to detect the amountof available solar radiation. For safety reason, other required instrumentations areair valves and expansion tanks. Air valves are used since the air must be removedfrom the piping for proper flow to occur. Air vents are usually installed at highpoints. As the heat transfer fluid temperature increases, so does its volume. Thusan expansion tank should also be used to capture the fluid as it is pushed out of thepiping loop.1The case of the varying water level, i.e., the case when the flow rate of the tap water and that ofthe hot water demand has been considered in [7].132.2 Nonlinear ModelIn this section, we first present a model of the system components. These compo-nents are STC (Section 2.2.2), Heat Exchanger (Section 2.2.3), TST (Section 2.2.4),HP (Section 2.2.5), and pipes (Section 2.2.6). We do not consider HD in the state-space modeling. Then, we will gather all these models into one state-space model.The state variables are the temperature distributions in the whole system. Themanipulative variables are four flow rates in each loop and the HP energy produc-tion which can be controlled by the assist of four WPs and the HP compressorspeed ratio, respectively. Three non-manipulative variables are the solar radiation,the ambient temperature, and the hot water load’s flow rate. These information aidsus to form the nonlinear state-space equation as followsx˙ = f (x,u,d), (2.2)wherex :=x1...x15 , u :=m˙STCm˙STC−T STm˙T ST−HPm˙HPRComp , d := ITambm˙load , y := x9. (2.3)where xi, i= 1, . . . ,15 denote the temperature distributions in all STC, HP, and TSTloops. The vector of I, Tamb, and m˙load notes the disturbances of the system whichare the solar radiation, the ambient temperature, and the hot water mass flow rate.The control inputs which are the manipulative variables are the mass flow rates infour different loops and RComp which is the HP compressor speed ratio. The outputof the system y is one of the system’s state - the temperature of the tank’s top layer.Figure 2.2 illustrates the location of each state xi, control variables ui, anddisturbances di. In this specific case, the states are the inlet and outlet fluid’s tem-perature of each component, except the TST which is divided into three layers,and three states are allocated to represent the temperature at different levels of the14Figure 2.2: Structure of the state-space modelTST. Here, the total number of states is 15. The output of this case is x9 whichshows both the temperature of the upper layer of the TST and domestic hot water.Next, the equation of the temperature variation inside each component is derived,recalling the energy conservation law.2.2.1 Energy ConservationBefore explaining the modeling of each component in Figure 2.2, let us considera schematic control volume inside a component shown in Figure 2.3. Here, Q˙in,Figure 2.3: Schematic view of small control volume within componentsQ˙out , Q˙inlet , Q˙outlet and Q˙internal are the rate of incoming energy absorbed from the15environment through the walls of the control volume, the rate of outgoing energy(dissipated through the walls of the element to the environment) from the controlvolume, the inlet fluid energy rate, the outlet fluid energy rate and the rate of changeof the fluid internal energy of the control volume, respectively, and measured inkW .The energy conservation equation for the fundamental control volume abovecan be written asQ˙internal = Q˙inlet − Q˙outlet + Q˙in− Q˙out , (2.4)where unit energies Q˙internal , Q˙inlet and Q˙outlet are given byQ˙internal = Mcp(T )T˙ , (2.5)Q˙outlet − Q˙inlet = ∆H. (2.6)Since the control volume is relatively small, the inlet and outlet pressures of thecontrol volume are considered to be the same. Thus, ∆H is written as∆H = m˙cp(T )(T −Tinlet).Here, M [kg] is mass of the fluid inside the element, T [K] is the temperature ofthe control volume, Tinlet [K] is the inlet fluid temperature, m˙ [kg/s] is the massflow rate, and cp [kJ/(kgK)] is the specific heat capacity of the control volumewhich is a function of temperature2. Since the chosen control volume is smallenough, the specific heat capacity can be assumed constant across it. In this way,we can assume the enthalpy variation is a constant pressure process. Thus, it canbe computed as Equation(2.6).In the following, the nature of Q˙in and Q˙out inside each component will beclarified. The major examples of these two rates of energy are the absorbed solarenergy, heat loss, and the heat generation by the HP. In the following text, we usethe notations U [W/m2K] and A [m2] to designate the overall heat transfer coef-ficient and the heat transfer area, respectively, and their subscript (e.g., UShell and2Engineering ToolBox. Specific heat capacity of water-antifreeze solutionshttp://homepage.usask.ca/ llr130/physics/HeatCapcityOfAntiFreeze.html16AShell) indicates the component (e.g., the shell of the heat exchanger). The overallheat transfer coefficient in each component is a function of convective heat transferof fluid and air and the thermal conductivity of the component’s material.3. Also,the convective heat transfer coefficient of any fluid depends on the fluid volumetricflow rate4.2.2.2 Solar Thermal CollectorThe primary component of the ISTHS is the solar thermal collector which absorbsthe thermal energy from the sun. Among different kinds of solar thermal collectors,although evacuated tube solar collectors are high-priced, it is efficient and capableof retaining the heat long [33]. This collector consists of a number parallel vacuumtubes which have been connected to each other by a manifold. The outer layerof each tube made out of Borosilicate glass coated by special layer in the waythat more solar energy passes through it and less heat loss happens. As shown inFigure 2.4, inside the tube, there is a heat pipe which is surrounded by vacuumedspace in order to reduce the heat loss. The fluid inside the heat pipe vaporizeswhile it is absorbing solar energy. The energy is transferred to the fluid inside themanifold at the top of each tube. Then, the fluid inside the heat pipe is condensed,and due to gravity, it flows downward.The STC system can be modeled as a static system. According to the datasheet,the STC generated energy rate is given by:Q˙STC = Ae f f IKατηSTC, (2.7)where Ae f f [m2] is the STC effective area which is normal to the direct solar radi-ation, I [W/m2] is the solar intensity which can be computed as a function of thesun position and the STC orientation, and Kατ is the incident angle modifier. ηSTCis the efficiency of STC computed asηSTC = c0+ c1Tin,STC−TambI+ c2(Tin,STC−Tamb)2I, (2.8)3web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node131.html4I. Martinez. Heat and mass convection. boundary layer flow. http://webserver.dmt.upm.es5http://www.apricus.com17Figure 2.4: Evacuated tube solar collectors5and other notations denote as follows. Parameters Ae f f , I and Kατ are computed asdescribed in [34], Tin and Tamb [K] are the STC inlet temperature and ambient tem-perature, respectively. In addition, as described in [34] c0, c1, and c2 are efficiencyparameters of the STC given by the manufacturer. The STC outlet temperature canbe computed asTout,STC = Tin,STC +Q˙STCm˙STCcp. (2.9)2.2.3 Heat ExchangerIn solar domestic hot water systems, in order to transfer the thermal energy betweentwo different fluids which must not be mixed, liquid to liquid heat exchangers areemployed. In Figure 2.5, the schematic view of the heat exchanger is displayed.It consists of a coil and a shell. The hot fluid flows in the coil and the energy isexchanged between the coil and shell. Since the hot water has less contact with theouter layer, the heat loss inside the coil is less than the heat loss inside the shell.Thus, inside the coil, Q˙in is zero and Q˙out is given byQ˙out :=UCoilACoil(T −TShell), (2.10)18where T and TShell [K] are respectively the temperatures in each control volumeinside the coil and the temperature of the corresponded element in the shell. InEquation (2.10), UCoil is the overall heat transfer coefficient of the HX’s coil. Thiscoefficient is a function of the thermal conductivity of the inner layer of HX, andthe convective heat transfer coefficient between the fluids inside the shell and thecoil. Since these fluids flow inside the shell and the coil, the convective heat trans-fer coefficient is a function of their flow rates. For example, for the HX1, thiscoefficient is a function of u1 and u2, and for the HX2, it is a function of u3 and u4.A more mathematical explanation is provided in the Appendix. ACoil is the areaof the inner cylinder, which is the area through which the thermal energy transfersbetween two fluids inside the shell and coil. For example, if the length of the coil’selement is L, and the diameter of the coil is r, ACoil equals to 2pirL.Figure 2.5: Schematic view of a heat exchanger elementFor every control volume in the shell, Q˙in and Q˙out are given byQ˙in :=UCoilACoil(Tcoil−T ), (2.11)Q˙out :=UShellAShell(T −Tamb), (2.12)where T and Tcoil are respectively the temperature inside each control volume ofthe shell and the corresponded volume in the coil. UShell is the overall heat losscoefficient between the fluid inside the shell and the ambient air. In this case, thecoefficient is computed with the thermal conductivity of the outer layer of HX, theconvective heat transfer coefficient between the air and the shell’s fluid, and some19configuration parameters such as the thickness of the HX outer layer and the lengthof the HX control volume. Additionally, AShell is the area of the outer cylinder.2.2.4 Thermal Storage TankDue to the limited availability of solar radiation, we need an energy storage compo-nent in order to save the excess harnessed thermal energy from the sun. The systemunder consideration uses a cylindrically shaped vertically oriented tank with fluidas a thermal storage medium. In this research, we consider fluid in the TST tobe stratified in layers, and the temperature is gradually decreasing from the toplayer to the bottom. Each layer is assumed to be a separate element with the fluidfully mixed inside each element such that we can assume the temperature is con-stant across the element. Figure 2.6 shows an element in the thermal storage tank.Here, for mass conservation and heat transfer modeling purposes, we hypotheti-cally assume three major flows pass through each element: m˙STC−T ST , m˙T ST−HP,and m˙load . Another assumption is that the hot water demand flow rate is equal tothe inlet water flow rate from the tap water which makes for the constant waterlevel inside the thermal storage tank.Figure 2.6: Schematic view of j th element of storage tankAs shown in Figure 2.6, the rate of energy transferred through three incoming20flows and three outgoing flows in each element of the TST is given byQ˙outlet − Q˙inlet := cp(Tj)(m˙STC-T ST + m˙T ST-HP)(Tj−Tj+1)+cp(Tj+1)m˙load(Tj+1−Tj), (2.13)where m˙STC−T ST , m˙T ST−HP, and m˙load are respectively mass flow rate in STC-TSTand TST-HP loop, and the mass flow rate of hot water demand. Tj and Tj+1 arerespectively the temperature of j th and ( j+ 1) th elements. For j th element, therate of energy Q˙in is zero, but two kind of heat loss occurs for each element:Q˙lossaand Q˙lossc . The first heat loss represents the heat transfers between the TST waterand the ambient air, and the second one shows the thermal energy which exchangesbetween the next layers. Thus, Q˙out is computed asQ˙out := Q˙lossa + Q˙lossc ,Q˙lossa := UaAa(Tj−Tamb),Q˙lossc := UcAc(2Tj−Tj+1−Tj−1).Here, Ua is the overall heat loss coefficient between the water inside the thermalstorage tank and the ambient air. Since we assume the TST is steady, we alsoassume the water inside the TST is stagnate. Thus, the convective heat transfercoefficient is the function of constant parameters such as air and water convectiveheat transfer coefficients and the TST’s wall thickness. Uc is the overall heat losscoefficient between the fluid inside jth element and two other fluids inside j+1thand j−1th elements. Similar to Ua, Uc is also a constant coefficient. Let us clarifythe difference between Aa and Ac with an example. If the length of each elementis L and the TST diameter is r, then respectively, Aa and Ac equal to 2pirL andpir2. These areas represent the area through which the thermal energy exchanges.Moreover, it is worth noting the top layer temperature of the thermal storage tankis the hot water temperature which is shown as TDHW ,TDHW = Ttop, (2.14)this temperature is also the output of the state-space model.212.2.5 Air to Water Heat PumpAs is discussed in Section 2.1.1, in order to provide an uninterrupted supply of en-ergy without the need for external supply an auxiliary heater is needed. For warmerregions in Canada, employing air to water HPs can reduce the heating cost up to 50percent 6.The benefits of integration of solar heating system and air-to-water heatpump is discussed widely in [35] and [36].An air-to-water heat pump can be used for either space or water heating. Inthe first case, the thermal energy extracted from outside air will warm up the insideair, and in the second case, it will warm up the domestic water. The HP suited forbuilding water or space heating operates on the vapor-compression cycle which isshown in Figure 2.7 [37].Figure 2.7: Schematic view of the vapor-compression cycle for HPIn [34], Zandinia et al. fitted a second-order surface to form the COP data pro-vided by the manufacturer and formulated the energy generation of an air-to-waterheat pump as a function of the ambient temperature and the inlet water tempera-ture of HP 2.15. In this equation Pii is representing one of the coefficients of thisformulation. The similar equation has been used for modeling the electrical power6https://www.nrcan.gc.ca/energy/publications/efficiency/heating-heat-pump/683122consumed by the HP.Q˙HP(Ta,THPi) = P00+P10Ta+P01THPi +P20T2a +P02T2HPi +P11TaTHPi (2.15)2.2.6 PipesThe most important feature that the pipes should have is low thermal conductivity.With proper insulation, we can reduce the heat loss. Hence, for every element inthe pipes, we assume Q˙in is zero and Q˙out is the rate of thermal energy loss.Q˙out :=UpipesApipes(T −Tamb). (2.16)Here, T and Tamb are respectively the control volume and ambient tempera-tures. Upipes is the overall heat loss coefficient between the inside fluid and theambient air which is also a function of the fluid flow rate. For lowering this co-efficient, the thickness of the pipes’ wall must increase; however, it might not bepossible due to manufacturing and cost issues. Additionally, Apipes is the area ofeach cylindrically shaped control volume of the pipe.2.2.7 Water PumpThere are two kinds of solar systems: passive and active system. In passive sys-tems, the fluid circulates through pipes by the aid of gravity; however, in active sys-tems, WPs produce the pressure differential which makes the fluid move. Changingflow rates in pipes affect the temperature inside the thermal storage tank. Moreover,one of the research goals in this thesis is to reduce the electrical cost of thermal en-ergy generation. In this case, by employing WPs in each loop of the system, onecan manipulate not only the thermal energy production but also the electrical cost.See Figure 2.2. Four WPs are used in this system to circulate water inside fourseparate loops. In [34], for mathematical modeling of a WP and fitting the curve,the data sheets from Wilo Canada. The mathematical model has a polynomial formwhich is a function of the flow rate. It means that by knowing the flow rate, we areable to calculate the electrical consumption of the WP. (Provide more detail aboutthe formulation and the datasheet). Similar to the WP modeling, the formula for23the electrical consumption of the WP isE˙WP(m˙) = P0+P1m˙+P2m˙2. (2.17)2.3 Disturbance Signals ModelingIn this section the non-manipulative signals will be mathematically modeled. Thesolar radiation model can be exploited from the sun’s position in the sky basedon a yearly pattern, in Section 2.3.1. For modeling both hot warer load and am-bient temperature, some real-data is available on Internet. Using curve fittingtechniques, the mathematical patterns of these two signals can be derived (Sec-tions 2.3.2 and 2.3.3).2.3.1 Solar RadiationThe sun’s relative position in the sky can be mathematically modeled such asin [38]. Based on the movement of the sun in the sky, the amount of solar radi-ation which is changing along a day can be modeled.As it is shown in Figure 2.8, from the perspective of the observer on the Earththe position of the sun is determined by two parameters, namely, altitude angle (α),and azimuth angle (θs) [38]. Given the latitude of the location, the declination, andthe hour angle, the altitude and azimuth angle are calculated bysinα = sinLsinδ + cosLcosδcosω, (2.18)sinθ =cosδ sinωcosα, (2.19)where L is the latitude of the location, δ is the declination calculated based on thedate, and ω is the hour angle given by the local time. Since, the solar irradiance hasa constant value, by calculating the altitude and azimuth angle, and the sun distancefrom the Earth based on the date, the solar maximum available solar radiation iscomputable. For example, in Figure 2.9 the solar radiation for the first day of Mayis shown.The main problem here is that the solar radiation is not always at the maximumlevel due to clouds, dust, shading from obstacles and other reasons, ass shown in24Figure 2.8: Sun’s position determined by altitude and azimuth angles [38]0 5 10 15 20Time (h)02004006008001000I (W/m2 s)Figure 2.9: The solar radiation on the first day of May in VancouverFigure 2.10. By taking them into account, a more realistic value for solar radiationcan be obtained. Although all of these calculations are an approximation for pre-dicting the solar radiation, the real-time solar radiation can be measured by a solarradiation detector.2.3.2 Hot Water LoadPeople need hot water for many purposes such as showering, washing, and spaceheating. The hot water which is provided with the ISTHS will be stored at theupper layer of the thermal storage tank. The hot water is connected to the building25Figure 2.10: Barriers in front of solar radiationpiping net. Since, the building pipes have a fixed size, by knowing one parameterwhich is the hot water flow rate, we can model the hot water load. Therefore, theload flow rate represents the hot water load.The data which were used for modeling DHW consumption were obtained fromthe report [39] and are shown in Figure 2.11. According to the report, Canadianshave two peaks in consuming the domestic hot water, one is around 8 am andanother one is around 8 pm.Figure 2.11: Average daily DHW consumption for Canada in liters per hour262.3.3 Ambient TemperatureTemperature inside many components of the ISTHS such as STC, HP, and TST isaffected by the ambient temperature. For more accurate modeling of this distur-bances, from a historical database for 2012, the hourly ambient temperature havebeen collected. This data which is depicted in Figure 2.12 is used as predicted am-bient temperature for simulation purpose. The actual ambient temperature can bemeasured by using a simple thermostat.0 50 100 150 200 250 300 350Time (Day)260270280290300310T amb(K)Figure 2.12: Local temperature in Vancouver for the year of 201227Chapter 3Controller DesignAs discussed briefly in Section 2.2, the critical fact about the solar thermal systemis that the disturbances of the system have both predictable and uncertain aspects.For instance, meteorologists can predict a sunny day for tomorrow, but there isstill a high chance that some parts of the day will be cloudy. In this case, thesolar radiation on a cloudy day can drop to the fifty percent of a sunny day. Fromthis example, it can be concluded that if the controller is designed just based onthe disturbances’ prediction, the amount of hot water probably does not meet thedemand adequately. On the other hand, despite some conventional methods fordisturbance rejection, we are not desired to reject the effect of all disturbances onthe system. For example, if the solar radiation is higher than expectations, controlinputs must be adjusted such that the maximum solar energy is captured, ratherthan dissipating the high solar radiation. The point of this chapter is to design aproper controller with the utmost efficiency which can deal with both predictableand uncertain parts of disturbances and provide enough amount of hot water.In this chapter, the control objective of the control system is presented. Acontrol structure involving feedforward and feedback controllers is proposed toachieve the control objective. Methods for designing these controllers are men-tioned subsequently.283.1 Control ObjectiveFor the system described in Section 2.1, the control objective is to meet the varyinghot water demand with the minimized amount of energy consumed over a chosentime, under disturbances in solar radiation and ambient temperature. This objectiveis to be achieved by manipulating volumetric flow rates in the four loops of thesystem with four water pumps, and the adequate heating capacity of the HP bychanging the compressor ratio. The solar radiation is a disturbance, but we desire totake the most advantage of its energy, instead of rejecting its effects. By extractingall available solar energy, the system operates at the optimum point which is theobjective of this research. In this research, the focus is to design a controller whichcan optimize the system’s operation in the presence of disturbances while the effectof uncertainties inside the model is ignored.The user requires to receive the hot water as much as it is needed at the desiredtemperature. In fact, in designing the controller for the domestic thermal system,the control problem is not a tracking problem, because the temperature can bewithin the desired range. In Figure 3.1, an example of the desired temperaturerange for the DHW is shown. Assume the desired temperature for the domesticusage is 60◦C. Since the human body cannot distinguish 5 to 6◦C temperaturedifference between two fluids, the minimum temperature of the hot water demandcan be 54◦C. In Figure 3.1, the lower red line represents the lower temperaturelimit. On the other hand, the maximum temperature of hot water demand can behigher than 60◦C as long as the solar thermal collector provides the excess thermalenergy. If the temperature of the hot water is too high for the domestic usage, theuser can mix it with cold water and reach the desired temperature. For defining themaximum temperature, we face another limitation. As mentioned in Section 2.1,the hot water is provided by the warmed water at the upper layer of the thermalstorage tank. The TST material can tolerate up to a specific temperature such as80◦C which is the upper limit and is shown with the high red line in Figure 3.1.One of the solution of the control design problem for solar thermal systems isto predict disturbances one day ahead and to optimize the flow rates and the com-pressor ratio. This method which is an off-line controller is ideal for the case thatboth system’s model and disturbances’ prediction are reasonably accurate. How-29Figure 3.1: Example of the desired temperature rangeever, this situation is not realistic for solar thermal systems. Therefore, an on-linecontroller is required. Additionally, using off-line controller results provide ben-eficial information for designing the on-line controller. In the next section, thecombination of the off-line and on-line controllers is introduced.3.2 Control StrategyThe control strategy consists of off-line and on-line parts, in the terms of controlinputs calculations. In the off-line part, the disturbances are assumed to be com-pletely predictable. By knowing disturbances, the optimal control inputs can becomputed. In the on-line side, the assumption has been put aside. Now, in the con-trol design, the measured disturbances take on the major role. The control designprocedure for designing the overall structure is depicted in Figure 3.2. The inputsignals are shown with the red color. For the off-line and on-line controllers, theinput signals are respectively the predicted and measured disturbances. The purplecolor distinguishes the feedback signal from the other signals. The output of eachcontroller has the blue color, and the methods, employed for designing controllers,are depicted with the green color. As shown in Figure 3.2, the on-line controller isthe combination of HP-Side and STC-Side controllers. In the STC-Side controller,30the STC side flow rates, m˙STC, and m˙STC−T ST , are computed and optimized by mea-suring the actual disturbances. The HP-Side controller generates the other controlinputs which are the HP side flow rates, m˙T ST−HP, and m˙HP, and the HP com-pressor ratio, RComp. This controller is a logical combination of feedforward andfeedback controllers. As you can see in the figure, the HP-Side controller utilizesthe information from other controllers. Subsequently, the procedure of designingeach part of the overall structure is concisely introduced.Figure 3.2: Control design procedure of designing the overall control struc-tureThe off-line controller is an open-loop controller, and in its designing proce-dure, the unknown variables are control inputs. By knowing the disturbances oneday ahead and correctly modeling the system, an appropriately chosen optimizationmethod can solve the control problem. The time consumption and computationalload are not any concern if the forecast data are available ahead, and if a standardpersonal computer can do the required amount of computation.The main issue that leads us to design the on-line controller is that the actualdisturbances may be different from the predicted ones. To increase the absorbedsolar energy and store it correctly, the proposed control algorithm for computingthe flow rates of STC side is separated from the flow rates of the HP side and the31compressor ratio of the HP. The STC-Side controller is designed so that systemtakes full advantage of the available solar radiation. In developing the STC-Sidecontroller, it is assumed that the HP is turned off. This assumption causes the re-duction in the number of both states and control inputs; thus, the considered controlinputs are the STC-Side flow rates. Since disturbances are measured, the optimalflow rates can be computed in an optimization procedure. In this procedure, anadequate initial guess of the solution can reduce the computing time. Hence, theresult of the off-line controller is set as the initial solution guess for designing theSTC-Side controller.The HP-Side controller is proposed as a logical combination of feedback andfeedforward controllers to reduce the electrical costs by rejecting the adverse ef-fect of disturbances on the electrical consumption. In designing this controller, theresults of both off-line and STC-Side controllers are used, as shown in Figure 3.2.The optimal STC flow rates which are computed by the STC-Side controller are de-termined for designing the HP-Side controller. Furthermore, the off-line controllerresults are adopted for linearizing the system’s model in the feedforward controldesign.3.3 Off-Line ControllerIf we assume that the disturbances can be predicted precisely, and if the uncertaintyof the plant model can be ignored, we can design an off-line controller. Accordingto the weather forecast and the available data for the hot water demand profile, it ispossible to predict disturbances one day ahead with precision. It is not correct toassume the disturbances predictions are accurate; however, the results of the off-line controller can be useful for designing the various parts of the on-line controller.We can design an optimal controller using an optimization algorithm. The off-linecontroller structure is shown in Figure 3.3, where dp is the vector of the predicteddisturbances. r represents the desired temperature range which is [Tup,Tlow]. Tupand Tlow are respectively the upper and lower temperature limits. e is the errorof the system output which is non-zero if the DHW temperature is outside thedesired range. As it mentioned in Chapter 2, the output of the system is the DHWtemperature.32Figure 3.3: Off-line controller block diagramIn Section 2.1, some methods were employed to model the predicted distur-bances mathematically on some level of accuracy. In this case, the problem can berepresented as an optimization problem. In this section, first the optimization prob-lem is defined, then the methods which are used to solve the problem is explained.3.3.1 Optimization ProblemAccording to the control objective, the goal of designing the optimal controller isto reduce the electrical energy consumption as well as to increase capturing the so-lar thermal energy, besides the requirement that the hot water demand temperatureshould be desirable. The mentioned goal can formulate the cost function. If thedisturbances of the system are known, the number of the nonlinear model equa-tions are the same as the number of unknown variables which are the states. Thenonlinear state-space equations form the constraints for the optimization problem.Besides, each actuator works in the specific operating range, which determinesother limitations of the problem. Considering both cost function and constraints,one can define the optimization problem in Equation (3.1).minimizeuJ(u)subject to x˙ = f (x,u,dp),0≤ ui ≤ ui, i = 1, . . . ,5,(3.1)33whereJ(u) =∫ tL0{wPPelec−wSTCQ˙STC +wT Tcost}dt, (3.2)andPelec = PHP+PWP, Q˙STC = m˙STCcP∆THX1 . (3.3)ui is the maximum value for i th component of the control input vector. Keep inmind that all states, control inputs, and disturbances are the function of time. Forthe practical implementation aspect, it is more convenient to discretize these valuesin time. Pelec is the overall electrical power consumed by four water pumps (PWP)and the HP (PHP). Q˙STC is the solar thermal energy, captured by the STCs andtransferred to the STC-TST loop per second. Equation (3.3) shows how to calculateboth Pelec and Q˙STC. The term ∆THX1 is the temperature difference between the inletand outlet of the first HX in the STC loop. This temperature difference describeshow much the temperature of the fluid inside the STC-TST loop increases due tothe transferred solar thermal energy. The Q˙STC can be calculated as a function ofthe STC efficiency, the flow rate in the STC loop, and the solar radiation. Last,Tcost represents how much the hot water temperature deviates from the desiredrange. For example, the defined desirable temperature range is the summation ofshaded areas in Figure 3.4. In Equation (3.2), by adding the first term which is theelectrical power consumption, the less consumed electricity is assured. The goalof maximizing the captured solar radiation can be achieved by adding the secondterm to the cost function. The last term of this function represents the hot watertemperature limits.3.3.2 Optimization MethodBy analyzing the cost function formulation briefly, it will be clear that this func-tion is non-convex, which means the optimization problem is also non-convex.Among the optimization algorithms, the global optimization solvers are the practi-cal choices for solar thermal systems. Two methods which are implemented in thisthesis are Genetic Algorithm (GA) and Particle Swarm Optimization (PSO). Next,both GA and PSO are reviewed; then the quality of their solution are compared, byconcentrating on the application of solar thermal systems.34Figure 3.4: How to calculate TcostGenetic AlgorithmThe genetic algorithm is an evolutionary algorithm whose logic is based on thenatural choice and survival of the fittest. Since the genetic algorithm is globallysearching the optimal solution over the feasible set, it is practical for non-convexoptimization problems. The GA’s flowchart is shown in Figure 3.5.GA starts with choosing the initial population. This set might contain randomsolutions. Every iteration, the fitness of solutions are evaluated. In GA, it is as-sumed that solutions have the same feature of the chromosome. Thus, differentanswers have different rankings based on their fitness values. Through three ge-netic operators: selection, crossover, and mutation, the solutions with the higherranking are more probable to be the parent of the next generation, which meansfuture generations may be similar to the solution with best fitness values.Particle Swarm OptimizationSimilar to GA, PSO is also an evolutionary and global searching algorithm whichdeals with non-convex problems adequately. This method mimics the motion ofswarms of birds. The regarded flow chart is illustrated in Figure 3.6.PSO algorithm starts with generating the initial population of particles. Eachparticle represents a solution. The initial population may be chosen randomly.35Figure 3.5: Flowchart for GA [40]Next, cost functions of particles evaluate, and the global and personal best costsare updated. Depending on the stopping criteria, the program can stop here. Basedon the updated information, the new population generates, and the velocity andposition of particles update.GA and PSO Comparison for ISTHS ApplicationIn this research, two optimization problems must be solved. They are defined in theprocedure of designing the off-line and STC-Side controllers. Two fundamentaldifferences between these problems are their sizes and time limitations. In off-line control design, the number of optimization variables is high. For example,consider that the control time step for the ISTHS is five minutes. In this case,noting five control inputs, the number of the variables are going to be 1440 forone day optimization time horizon. In contrast, the only optimization variable ofSTC-Side control design is the STC loop flow rate; however, the time horizon ofoptimization equals to the control time step. Thus, the STC-Side control problem36Figure 3.6: Flowchart for PSO [40]must be solved within the control time step. For off-line control problem, the timeconsumed for computation is not an issue. Additionally, the solution quality fordesigning the off-line controller is more critical than the STC-Side controller. Thereason is that the optimization of off-line control design happens once for a day.Therefore, if the solution is not qualified, this will affect the daily performance ofthe system; however, for designing the STC-Side controller, one error in findingthe best solution is just propagated in one-time step, which means a mistake doesnot have too much effect on the system performance.Conclusively, the critical features of the optimization solver for the off-linecontroller and STC-Side controller are respectively the solution quality and com-putational cost. The quality of the PSO algorithm’s solution is relatively higherthan GA for the continuous optimization variables [41]. Thus, PSO is a betterchoice for the off-line controller. By increasing the size of the population, thecomputational time for GA decreases exponentially [40]. Since the number of thevariable of the STC-Side controller is one, it is possible to increase the number of37chromosomes to reach faster computations. Therefore, for designing the STC-Sidecontroller, GA algorithm is used.3.4 On-Line ControllerSince disturbances are not entirely predictable, the on-line controller is essential.The on-line controller contains two major parts, STC-Side and HP-Side, whichutilizes different strategies. The STC-Side controller determines the optimal flowrates in STC loops; while, the HP-Side generates the optimal HP compressor ratioand the HP flow rates.One of the objectives of this research is to design a controller which can man-age the significant uncertainties in disturbances. One of the goals of developing theon-line controller is to extract the solar thermal energy, a non-controllable sourceof energy, as much as it is accessible. Let us assume the HP is off; then the onlyenergy source for raising the hot water temperature is the solar energy. By perform-ing an optimization method, the optimal flow rates of the STC side are calculated.Since by applying the computed STC side flow rates, the maximum solar radiationcan be absorbed, computing them in other parts of the control structure is not rea-sonable. Thus, the flow rates in the STC side are considered as the pre-computedcontrol inputs for designing the HP-Side controller.As for the HP-Side, both feedback and feedforward controllers are combinedlogically. A logical condition is defined, with which the HP-Side controller switchesbetween feedback and feedforward controllers. Two different methods are appliedto design the feedforward parts of HP-Side controller. One of them is a conven-tional simple feedforward controller which have been used for several solar thermalplants. The other one is the robust feedforward method which can both minimizethe electrical consumption and provide the hot water within the desired tempera-tures range.3.4.1 STC-SideThe main assumption in designing the STC-Side controller is that the HP is off. InFigure 3.7, the model of the system for designing this controller is shown. Thereare two control variables, u1 and u2. Proper sensors measure all three disturbances.38The number of states is reduced to ten. The block diagram of this controller isshown in Figure 3.8. Here, d is the measured disturbances. Similar to off-linecontroller, the reference range r is the desired temperature range. In this case, tofind the optimal flow rates, u1 and u2, an optimization problem can be solved.Figure 3.7: Considered model for STC-Side control designFigure 3.8: STC-Side controller block diagram39The optimization problem is defined as follows.minimizeuSTCJ(uSTC)subject to ˙xSTC = f (xSTC,uSTC,d),0≤ uSTCi ≤ uSTCi , i = 1, . . . ,5,(3.4)whereJ(uSTC) =∫ tL0{wPPelec−wSTCQ˙STC +wT Tcost}dt, (3.5)uSTC :=[m˙STCm˙STC−T ST], Pelec = PWP (3.6)The control input vector uSTC has two elements which represent the flow rates inthe STC and STC-TST loop (Figure 3.7). PWP is the electrical power consumed bythe first and second water pumps (u1 and u2). Q˙STC is computed by Equation (3.3),and Tcost is the same as this term in the off-line control design.In this problem, the optimization is solved each time at the beginning of eachmeasurement time step. For example, if the time step for measurement is fiveminutes, tL is five minutes, and every five minutes, a new optimization problemmust be solved. Control inputs and disturbances are the functions of time; however,they are considered to be constant in the measuring time interval. Thus, they are theconstant optimization variables and parameters in each STC-Side’s optimizationproblem. Therefore, the only time-varying values are states.Challenges of choosing a method for solving these optimization problems arethe computational time and the accuracy of the final solution. As was mentionedearlier, the optimization problem is not convex, and for having precision in theoptimal solution, applying a global optimization solver is essential. However, forpractical implementation purposes, the optimization problem must be solved in theless amount of time, compared with the control time interval. Control inputs arecomputed for each measurement time step which means they are invalid for thenext time step. The high computational time issue prevents us from solving theoptimization problem for all control inputs and the full configuration. A globaloptimization solver can calculate the STC-Side optimization problem less than themeasurement time interval. Here, the chosen method for solving the problem is40GA.3.4.2 HP-SideThe HP-Side controller computes two flow rates in the HP part, u3 and u4, and theHP compressor ratio, u5. Both feedback and feedforward strategies are includedin this HP-Side controller. In fact, the feedback method operates as a backupcontroller once the information on disturbances is not sufficient for designing acontroller to retain the hot water demand temperature. The structure, shown inFigure 3.9 illustrates how the HP-Side controller is working.Figure 3.9: Control design procedure of designing the HP-Side control struc-tureThe procedure, shown in Figure 3.9 happens every measurement time step. Inother words, the HP flow rates and compressor ratio are constant during each step.In this figure, the allowable range, which will be introduced in Section 3.7 is thesmaller temperature range than the desirable range, defined in Section 3.3.1. Thereason for using the allowable instead of the desirable range in the HP-Side controldesign is to consider a safety margin by which we can assure that the hot watertemperature is within the desirable range. Thus, if the hot water temperature iswithin the allowable range, and if the difference between measured and predicted41disturbances is zero, u5 is considered as the optimal HP compressor ratio, computedby the off-line controller. If this difference is not zero, the feedback controller isactivated to generate u5. The case that hot water temperature is not in the allowablerange is considered as the emergency case; therefore, other controllers are off,and the feedback controller returns the temperature into the allowable range byadjusting u5. Finally, u3 and u4 are generated based on the status of u5.The feedback controller will be introduced in Section 3.7. Two feedforwardcontrollers are defined in this research, simple and robust controllers. A simplefeedforward controller has been used in controlling solar thermal plants severaltimes. This controller which will be described in Section 3.5. The point of design-ing the simple feedforward controller is to compare it with the robust feedforwardcontroller. To this extent, we can analyze the performance of the robust feedfor-ward controller. The procedure of designing the robust feedforward controller willbe explained in Section Simple Feedforward ControllerA feedforward controller which was introduced in [13] is designed for controllingsolar thermal plants. In this section, the controller is modified to be compatiblewith the integrated thermal hydronic system. The feedforward controller is neces-sary for the solar thermal systems since the response of thermal systems are slow.Thus, using only the feedback controller is not adequate for these systems. Thetemperature in the upper layer of the tank can be manipulated through a controlcommand with a delay. If the only considered controller is the feedback controller,a significant delay for the system response happens from the time that disturbancesare changing to the time that the hot water temperature returns to the desired value.In fact, for building applications, users may not tolerate more than a minute de-lay. For solving delay issue, a feedforward controller has been proposed in [13]which retains the system behavior under the impact of varying disturbances. Thiscontroller receives a direct signal from measured disturbances. Therefore, the con-troller can change the command according to disturbance changes immediately,and less delay in the system’s response can happen.The simple feedforward controller structure is shown in Figure 3.10. Respec-42tively, dp and d are the predicted and measured disturbances; consequently, δd isthe error of the anticipated disturbances which is the input signal of the feedfor-ward controller. dp and δd are respectively the input signal for the off-line andthe simple feedforward controllers. r is the reference temperature, and e is the de-viation of the DHW temperature from the reference point. Here, r is the optimaltemperature profile which is computed by the off-line controller. As it is shown inFigure 3.9, the HP flow rates are determined based on the condition that if the HPis ON or OFF. Moreover, the STC flow rates were previously determined by theSTC-Side controller. Thus, the only designed control input by this method is theHP compressor ratio which is u5.The simple feedforward controller generates δu5which is the adjustment of u5, according to measured disturbances. As shown inFigure 3.10, the actual u5 is the summation of δu5 and u5opt which is precomputedby off-line controller.Figure 3.10: Simple feedforward controller structureThe feedforward controller rejects the effect of the disturbances on the outputof the system. The first step of designing this controller is to linearize the nonlinearsystem. For this purpose, the nonlinear system of equations is linearized around theoptimal condition, (xopt ,uopt ,dp), which is computed under predicted disturbances,(dp). The optimal condition is the system response to the off-Line control inputs,43(uopt). The analytical linearization is computed as follows.A =∂ f∂x|(xopt ,uopt ,dp) Bu =∂ f∂u|(xopt ,uopt ,dp) Bd =∂ f∂d|(xopt ,uopt ,dp) (3.7)The linearized system isδ x˙ = Aδx+Buδu+Bdδd, (3.8)δy = Cδx.Moreover, the error is computed byδe = δy−δ r= δy.Since the reference point is not changing in a specific time step, the error varia-tion equals the output variation. The output of the system is the hot water tempera-ture. In this case, C is a unit horizontal vector by which the corresponded state, thetemperature of the top layer of the TST is indicated. Note that δd is the deviationof measured disturbances from the prediction. The transfer function of the wholesystem can be derived from the linearized model in Equation (3.8). In this case,one can draw the block diagram as shown in Figure 3.11.Respectively, D(s), U5(s),and Y (s) are the Laplace transform of δd, δu5, and δy. In this figure, the transferfunctions are generated by the following equations.GFF1(s) =U5(s)I(s), GI(s) =Y (s)I(s) ,GFF2(s) =U5(s)Tamb(s), GTamb(s) =Y (s)Tamb(s),GFF3(s) =U5(s)Uload(s), Guload (s) =Y (s)Uload(s),Gu5(s) =Y (s)U5(s).The summation of the outputs of the transfer functions, GFF1(s), GFF1(s), andGFF1(s), produces the simple feedforward control input which is δu5. Subse-44quently, the outputs of these three transfer functions are derived.Figure 3.11: Simple feedforward controller block diagramThe logic behind the simple feedforward control method is to diminish theimpact of the disturbances changes on the output of the system through the HPcompressor ratio. In fact, this controller’s goal is to make this effect zero. Forillustrating this logic, consider one disturbance, I(s). The block diagram, shownin Figure 3.11 changes to a simpler one like Figure 3.12. Based on this logic,the output deviation from the operating point, δy, and its Laplace transform, Y (s),should be zero. By doing some mathematical manipulation, we can derive theformula for the feedforward controller which is calculated byFigure 3.12: Simple example of feedforward controller block diagram45GFF(s) = −GI(s)Gu(s) . (3.9)In the original problem with three disturbances, depicted in Figure 3.11, thefeedforward control input, u5, is written asU5(s) =− GI(s)Gu5(s)I(s)− GTamb(s)Gu5(s)Tamb(s)− Guload (s)Gu5(s)Uload(s). (3.10)The order of the control system derived in Equation (3.10) is relatively high (e.g.of 20th order), which is significantly complicated for the implementation. For sim-plifying the controller, the transfer functions in both numerator and denominatorreplaced by their DC gain which means the feedforward controller changes to aconstant gain. In this case, the final feedforward controller amounts toδu5(t) =−K( GI(0)Gu5(0)δ I(t)− GTamb(0)Gu5(0)δTamb(t)− Guload (0)Gu5(0)δuload(t)). (3.11)In this equation, the coefficient, K, will adjust the magnitude of the controller suchthat the actuator does not saturate. Since the feedforward controller rejects distur-bances immediately, the control commands have large values. Adding K reducesthe amount of the control input. This coefficient is added manually. The othersolution for adjusting the amount of control input is to use the robust feedforwardcontroller. In designing procedure of this controller, the value of δu5 is optimized.3.6 Robust Feedforward ControllerIn the proposed simple feedforward controller, since there is no systematic pro-cedure to calculate the optimal solution for u5, there is no guarantee that thiscontroller generates the optimal control inputs. As one of the objectives of thisresearch is to diminish the consumed electricity by the HP compressor, anotherfeedforward controller is needed. To achieve this objective, a robust feedforwardcontroller is proposed in this section. Among studies on controlling solar thermalsystems, many optimal feedback controllers are proposed such as [29] and [30],but almost none of them benefits from disturbances’ signals efficiently. For thecontrol method which receives signals from the measured disturbances, the time46spent on calculating control inputs should be less than the measurement time step.Since the computational time of some advanced control method such as MPC andAC is long, these methods may not be adequate for controlling ISTHS.As for the implementation, the block diagram of the robust feedforward con-troller is the same as the simple feedforward shown in Figure 3.13. δd is theFigure 3.13: Robust feedforward controller structureerror of disturbances predictions which is the difference between the actual andpredicted disturbances. According to the similar logic mentioned in designing thesimple feedforward controller, δe equals δy which is the deviation of the systemoutput from the response of the system to the off-line controller, considering thepredicted disturbances. This response is the optimal behavior of the system; thus,one of the objectives of the robust feedforward controller should be to keep the hotwater temperature around the off-line controller system response. δu5 is the outputof the robust feedforward controller which is the difference between the on-lineand off-line HP compressor ratio, u5opt and u5, respectively.For designing the robust feedforward controller, Linear Fractional Transforma-tion (LFT) is formed in Figure 3.14. Since the robust control approach is usuallyused for the feedback control design, we need to modify the linear model of theoriginal system, ISTHS, to apply this approach for designing a robust feedfor-ward controller. In Figure 3.14, the shown LTI is the modified linearized model ofISTHS. The LTI system has three outputs, δq, δe, and δu5. δq is the measured47output signal of the LTI system which equals δd. Both δe and δu5 are the perfor-mance outputs of the LTI system. δu5 is also the input signal of the LTI. The inputFigure 3.14: LFT form for designing robust feedforward controllersignal of the robust feedforward controller is δd, disturbance deviations, and theoutput of this controller is δu. Since the system is linearized around the optimaloperating points, (xopt ,uopt ,dopt), by reducing the effect of δd on δe as shown inFigure 3.14, the optimal behavior of the system is retained. Additionally, as theimpact of δd on δu5 decreases, the HP compressor is consumed less electricitycompared to the simple feedforward controller. The next steps are to determinethe LTI system and to derive the robust feedforward controller by using H∞ controlmethod.As for designing purposes, the block diagram of the robust feedforward con-troller is demonstrated in Figure 3.15. The LTI model can be computed based onFigure 3.15: Robust feedforward controller block diagram48the linearized model mentioned in (3.8). The LTI state-space model is written asδ˙x = Aδx+Buδu+Bdδd, (3.12)[δeδu5]= C1δx+D11δu,δq = D22δd.In Equation (3.12), the performance outputs are the hot water temperature error,δe, and the HP compressor ratio variation, δu5. The measured output of the LTIsystem is δq which equals δd. The matrices in this equation areC1 =[C0], D11 =[0e5], D22 = I.Here, I is the identity matrix which dimension is the number of disturbances (three).e5 is the horizontal unit vector, and its nonzero element is the fifth one. One cansee that the defined LTI system is compatible with the block diagram shown inFigure 3.15.The robust feedforward controller is designed such that both optimal controlinputs and optimal trajectory of the hot water temperature are reserved. This goalachieves by defining the performance outputs as Equation (3.12). Since the controlinput of the system, u5, may saturate, the performance outputs, δu5 is multipliedby the weighting function W as shown in Figure 3.16). In this figure, C is a matrixFigure 3.16: Robust feedforward controller block diagram with weightingfunctionthat transforms the vector of all control inputs to the single control input, u5. u5w is49the weighted control input number 5, u5. Since the order of the robust feedforwardcontrol system is relatively high, for simplicity, the weighting function is consid-ered as a constant. By applying W to the control system, the magnitude of controlinput can be manipulated to diminish the electrical cost.Based on the block diagram in Figure 3.16, and the LTI state-space model inEquation (3.12), one can write the multi-objective problem asminimizeK∈K||Tδd→(δe,δu5w )(K)||∞subject to K stablizes the closed-loop system(3.13)K is a space of real rational transfer matrices K(s), called controller space. Thesolution is a member of this space. Tδd→(δe,δu5w ) is the transfer function fromδd to the vector of the performance outputs (δe,δu5w). The robust feedforwardcontroller transfer function which generates the optimal HP compressor ratio isderived by solving this problem. Since the LTI model is computed based on thesystem’s response of off-line controller, the robust feedforward controller transferfunction can be computed off-line. However, since the inputs of this controller aremeasured disturbances, the optimal HP compressor ratio is calculated on-line.The robust feedforward controller may not retain the hot water temperaturewithin the allowable range. Consequently, another controller is required to guar-antee the desired hot water temperature. The designing procedure of the feedbackcontroller is described in the next section.3.7 Feedback ControllerThe feedback controller is employed as a backup for both off-line and feedforwardcontrollers. This controller is necessary both for retaining the DHW temperatureand for rejecting the uncertainties in model design for practical purposes. Thedesigned feedback controller is a simple proportional controller which retains thetemperature within an acceptable range instead of tracking the desired value. Thistemperature range is named the allowable range, and it is narrower than the desiredtemperature range as depicted in Figure 3.17. In this thesis, the allowable temper-ature range is from 56 to 70◦C; however, the desired temperature for domestic hot50water usage is from 54 to 80 ◦C. Thus, if the domestic hot water temperature isoutside the allowable temperature range, the feedback controller will become ac-tive, and adjust the control input to return the temperature into the allowable range.Figure 3.17: Example of feedforward controller allowable temperature rangeThe feedback controller block diagram is shown in Figure 3.18. The inputFigure 3.18: Feedback controller block diagramsignal of the controller is the error of the output temperature, e. The system outputis the hot water temperature, TDHW . This controller does not receive any signalfrom disturbances. The control input, u5, and the error of the system output, e, canbe calculated by the following algorithm.if TDHW > Tuallowable then51e = TDHW −Tuallowableu5 = KP(TDHW −Tuallowable)elseif TDHW < Tlallowable thene = Tlallowable−TDHWu5 = KP(Tlallowable−TDHW )elsee = 0u5 = 0end ifend ifRespectively, Tuallowable and Tlallowable determine the upper and the lower limits of theallowable temperature range. KP is the proportional gain of the feedback controller.In this chapter, the control structure designed for controlling the ISTHS wasintroduced. The procedure of designing each part of this structure was explained.In the next chapter, the effectiveness of the overall structure will be examined.52Chapter 4Simulation ResultsTo demonstrate the requirement of sub-controllers such as Off-Line Controller(OL), Feedback Controller (FB), and Feedforward Controller (FF) which have beenused to form the overall control structure, the sub-controllers and the overall struc-ture are implemented in a simulation environment, and the results are compared.For this purpose, all controllers are programmed in MATLAB environment, andtheir performances are tested under three environmental conditions. In Section 4.1,the simulation settings are determined. The simulation assumptions and the pa-rameters assumed for both modeling and controller design will be introduced. Theadvantages and disadvantages of different parts of the overall structure will bedemonstrated if the performance of each sub-controller is separately examined.In Section 4.2, the combinations of the sub-controllers will be formed to illustratethe benefits and deficiencies of each part of the overall structure. In Chapter 3, twodesign procedures named simple and robust feedforward techniques were deter-mined for designing the feedforward part of the overall structure. In Sections 4.2.4and 4.2.5, two feedforward controllers are considered in the overall control struc-ture separately, and the system’s responses of them are compared. Finally, in Sec-tion 4.3, the several simulation results will be compared.534.1 Simulation SettingUsing different sources such as the existing data, the mathematical models, andthe curve fitting technique, the model of each component, the required parameters,and the predicted disturbances will be determined. In this section, the parametersvalues for modeling and controlling purposes will be given. In Section 4, applyingthese values, we will simulate the model by implementing different controllers andevaluate their efficiency.4.1.1 AssumptionThe optimization problem which was introduced in Section 3.3.1 is quite com-plicated, and it requires a complex computation to obtain the solution which isnot suitable for practical purposes. In addition to the off-line controller, the com-putational time of solving the feedforward part of the overall structure is long,compared with the measurement and the control time step. Three assumptions areconsidered to reduce the complexity of solving the optimization problem and thetime consumption of the control design. The first and the second assumptions aremade based on the fundamental analysis. For clarifying this analysis, a hypotheti-cal situation which the solar thermal collector is eliminated from the system is held,and the effect of the HP side of the system is only considered. For the specific dis-turbance conditions and HP compressor ratio, the impact of the HP side flow rates,u3 and u4, on the domestic hot water temperature is analyzed. For example, inFigure 4.1, the temperature of domestic hot water at the steady state, which is afunction of u3 and u4 is plotted for the fully-rated HP and the disturbances condi-tions at noon on the first day of July. The same analysis is performed for the STCside. In this case, the HP side is ignored, and the effect of the STC flow rates,u1 and u2, on the hot water supply is analyzed for the particular environmentalcondition.First, the flow rates of the STC side of the ISTHS are assumed to be the same,and the flow rates of the HP side are equal. Thus, the first assumption isu1 = u2 & u3 = u4. (4.1)54Figure 4.1: Domestic hot water temperature as a function of two Control vari-ables: u3 and u4Based on the fundamental analysis, it has been discovered that u1, the flow rate inthe first loop of the ISTHS shown in Figure 2.1, can be considered the same as theflow rate in the second loop (u2), since, in a specific environmental condition, themaximum temperature happens when these two control inputs are the same. Thisanalysis means, in the optimal point, u1 and u2 are equal; therefore, they can beconsidered as the same optimization and control variables from the beginning ofsolving the problems. It is the same story for the third and fourth control inputs. InFigure 4.1, for a particular temperature of domestic hot water, the minimum valuesof u3 and u4 are almost the same.Second, the third and fourth water pumps are assumed to be ON and OFF ac-tuators. In Figure 4.1, one can see that the maximum temperature in the thermalstorage tank is generated by the HP when the flow rates in the third and fourth loopsare the same and at the maximum level. In fact, the electrical energy consumed bythe system is mainly spent on the HP compressor. This statement means if waterpumps work at the maximum level, their electrical consumptions are not compara-ble with the HP compressor. Therefore, this assumption will not affect the researchgoal of minimizing the electrical cost. In both simulation and optimization, thethird and fourth control inputs are set to be on and off when the heat pump is onand off, respectively.55Third, for solving the optimization problem, the time interval is a day which isdivided into 12 time-steps. It means each control input is considered to be constantfor two hours. With the first and second assumptions above, the essential numberof control inputs is two (u1 and u5), and thus the number of optimization variablesamounts to 2×12 = Disturbance SettingHere, we used some methods and scenarios to imitate the noisy disturbances. Forthis purpose, some assumptions are considered. In fact, a significant disturbancedeviation from prediction can happen in practical problems. For example, in theexistence of cloud and dust, the solar radiation can reduce to the half of its max-imum level, and in the case of emergency needs of the hot water, the hot waterdemand value can increase by 200% rate [42]. To cover all possible disturbancecases, Figure 4.2 is introducing three scenarios of disturbances. In Figure 4.2, theCases (a) and (c) are showing the worst and best cases in the electricity consump-tion point of view respectively; Case (b) is introducing a randomized situation.From now on, green, yellow, and red colors represent the worst, random, and bestcases of the measured disturbances respectively.In Case (a), when the available solar radiation is not at the maximum level, andif the hot water demand and the ambient temperature are respectively higher andlower than our expectation, the required thermal energy from the HP side increases,which means the higher electrical power is consumed. In contrast, in Case (c), forthe maximum level of both solar radiation and ambient temperature, and for thelower hot water demand, less thermal energy from the heat pump is needed. Thiscase is the best electrical energy consumption case. For simulating unpredictablecircumstances, we consider noisy randomized disturbances which are shown inCase (b). In Table 4.1, the continuous deviation of assumed measured disturbancesfrom their prediction values are mentioned.In addition to continuous parameters deviation from predictions, two patternsin measured disturbances for the solar radiation and the DHW load are assumed.As depicted in Figure 4.2, for solar radiation, a cloud is supposed to be in the skyaround 10 a.m. to 12 p.m., which reduces the solar radiation by the rate of 50%.560 5 10 15 20012 10-30 5 10 15 202602803000 5 10 15 2005001000 PredictionAssumedMeasurement(a) Worst case0 5 10 15 20012 10-30 5 10 15 202602803000 5 10 15 2005001000PredictionAssumedMeasurement(b) Random case0 5 10 15 20012 10-30 5 10 15 202602803000 5 10 15 2005001000 PredictionAssumedMeasurement(c) Best caseFigure 4.2: Three disturbances scenarios57Table 4.1: Hypothetical continuous measured disturbance parameters devia-tions from predictionsDisturbance Worst Random BestDHW load 30% [−30 30]% −30%Ambient Temperature +30K [−15 15]K −15KSolar Radiation 20% [0 20]% 0%For the DHW load, between 15 p.m. to 18 p.m., in the worst and the random cases,an unpredicted increase in the hot water load is considered which is 200% higherthan the prediction values. For the best case, during the mentioned time, the DHWdemand drops by 100% rate which means it becomes zero.4.1.3 Parameter SettingFor simulation purpose, a hypothetical, but practical, ISTHS is employed. Parame-ters are set by the aid of the available datasheets for each element. For example, inthis research, a 100 liters thermal storage tank is considered which diameter is 0.5meters, and the required information for the STC part extracted from the datasheetof Sunda Seido 1-16 vacuum tube collector. Other parameters are provided in Ta-ble 4.2.Table 4.2: Modeling parameters in simulationsSymbol [unit] Value Symbol [unit] Value Symbol [unit] ValueDT ST [m] 0.47 AShell [cm2] 29.84 Acoil [cm2] 2.84VT ST [m3] 0.1 Apipes [m2] 0.12 Tinside [◦C] 25ASTC [m2] 4 c0 N/A 0.526 c1 N/A 1.3253c2 N/A 0.0042Furthermore, for the optimization purposes, some other parameters listed in 4.3are required to be determined. Due to the actuator saturation, control inputs oper-ates within an operating band. u1, u2, u3, u4, and u5 are the maximum level offour flow rates and the HP compressor ratio, respectively. The number of parti-58cles of the particle swarm optimization method, noted as np, is given. The desiredtemperature range is also another optimization parameter which is required to bedetermined. Respectively, the higher and lower levels of the desired temperaturerange are shown by Tupper and Tlower.Table 4.3: Optimization parametersSymbol [unit] Value Symbol [unit] Valueu1 [kgs ] 1 u5 % 1u2 [kgs ] 1 np N/A 10u3 [kgs ] 1 Tupper [◦C] 80u4 [kgs ] 1 Tlower [◦C] 544.2 SimulationThrough various simulations, the performances of all designed controllers are stud-ied one by one. In the end, the performance of the overall structure with the ro-bust feedforward controller is compared with the proposed combination of sub-controllers. The three disturbance cases are applied to examine the robustness ofeach sub-controller.First, the off-line controller without the on-line controller is implemented (Sec-tion 4.2.1); then, the feedback controller joined with the STC-Side controller isapplied in Section 4.2.2. This controller is combined with the off-line controller inSection 4.2.3 to benefit from the predicting information. Then, the overall con-troller in Figure 3.2 considered with the simple feedforward controller in Sec-tion 4.2.4. Last, to assess the effect of the proposed optimal control structure,the simple feedforward controller in the overall control structure is replaced by therobust feedforward controller. From now on, the overall controller with simplefeedforward controller and roust feedforward controller are named Simple FF andRobust FF controllers, respectively. Additionally, the off-line, feedback, and thecombination of off-line and feedback controllers are denoted respectively by OL,FB, and OL + FB. Note that PHP, PWP, and Pelec are respectively the daily averagepower consumptions of HP, WP, and ISTHS, which dimensions are KW .594.2.1 Off-Line ControllerLet us omit the on-line part of the overall controller. The corresponding structureis designed based on the control design procedure shown in Figure 4.3. In thisway, we can analyze the advantage and disadvantage of using only predictions todesign controller. As for the off-line controller, we employed the particle swarmoptimization, as mentioned in Section 3.3.2. By solving the optimization problemdefined in Section 3.3.1, the optimal control inputs are obtained. Figure 4.4 showshow PSO is converging. The computational time is around 12 hours which is ac-ceptable since, practically, the off-line control inputs are computed one day ahead.Figure 4.3: Control design procedure of designing the OL control structureThe results of the off-line controller are shown in Figure 4.5. ConsideringFigure 4.5 (b), one noticeable feature of the mass flow rate in the STC loop isthat this control input is zero during the night time. Additionally, in the daytime,the HP compressor ratio is partly near to zero. According to the assumption pro-vided in Section 4.1.1, the flow rate inside the third and fourth loops are set basedon whether the HP is on or off. The off-line controller temperature response topredicted disturbances, the best, random, and worst scenarios are depicted in Fig-6020 40 60 80 100 120Iteration3456789101112Best CostFigure 4.4: Iteration convergence of the PSO methodure 4.5 (a). The blue line representing the predicted disturbances’ response, re-mains close to the lower boundary of the desired temperature range, all day exceptthe time when adequate solar radiation is accessible. In the best disturbance sce-nario, the obtained optimal controller generates thermal energy more than needed,and in the worst case, this controller cannot retain the temperature. These resultsindicate that control adjustments of the solar thermal system are required.In Table 4.4, the performances of system responses to the off-line controllerin the presence of three different disturbances’ scenarios are summarized. Twocritical parameters in this table are the electrical and temperature costs which areintroduced in Section 3.3.1. For different disturbance scenarios, although the elec-trical costs are relatively the same, temperature drops dramatically through a day,in the worst case.4.2.2 Feedback ControllerIn this part, the considered simple feedback controller is the same as the controllerintroduced in Section 3.7. Same as the overall controller, the feedback part pro-duces the HP compressor ratio, and other control inputs are produced either byanother sub-controller named STC-Side or from the logic whether the HP is on oroff. The corresponding structure is designed based on the control design proce-610 5 10 15 204550556065OL+Predicted DisOL+Best DisOL+Random DisOL+Worst DisLower Limitation(a) Temperature0 5 10 15 2000.20.40 5 10 15 2000.510 5 10 15 2000.20.4(b) Control inputsFigure 4.5: Off-line controller resultsdure shown in Figure 4.6. As is shown in this figure, both off-line and feedforwardcontrollers are omitted from the overall structure. In this case, the effectiveness ofthe feedback part can be examined. In these simulations, the allowable tempera-ture range is considered to be between 56 to 75◦C. The system’s response to thefeedback controller is provided in Figure 4.7, where the temperature response (Fig-ure 4.7a) and the control inputs of the designed feedback controller (Figure 4.7b)are depicted. Furthermore, the performance of the system’s response to the feed-back controller with different disturbances options is compared in Table 4.5.62Table 4.4: OL Controller’s performance comparison under different distur-bancesPHP PWP Pelec TcostPredicted Disturbances 0.0397 0.000555 0.0403 0Worst Disturbances 0.0317 0.000555 0.0323 31.325Random Disturbances 0.0387 0.000555 0.0393 6.955Best Disturbances 0.0474 0.000555 0.048 0Figure 4.6: Control design procedure of designing the FB control structureThe designed feedback controller is the simple one; whereas, it can deal withthree disturbance cases. The problem with this controller is that it is probablyan expensive strategy which consumes excessive electricity. Since the impacts ofdisturbances cause changes slowly in the thermal system by using feedback con-trollers, the system generates extra energy in the HP side to maintain the tempera-ture.One of the differences between the overall structure and the feedback controlleris that the feedback controller does not receive any signal from the measured distur-bances. Therefore, the issue mentioned in the feedback controller results suggests630 5 10 15 204550556065FB + Best DisFB + Random DisFB + Worst DisLower Limitation(a) Temperature0 5 10 15 2000.5 Off-LineFB + Best DisFB + Random DisFB + Worst Dis0 5 10 15 2000.510 5 10 15 2000.51(b) Control inputsFigure 4.7: Feedback controller resultsthe usage of the feedforward controller which exploits the disturbance information,rather than the feedback signal, to adjust the control inputs. Hence, an optimalfeedforward controller is required.4.2.3 Feedback and Off-Line ControllerThe feedback controller does not receive any input signal from disturbances. Somepractical data provided by measurement and forecasts are valuable for controllingthe HP compressor. In this section, the system operation benefits from predicteddisturbances. The off-line controller is combined with the feedback controller suchthat the feedback controller put less effort into adjusting the temperature. Thefeedback controller is the one introduced in Section 4.2.2. The control structure isdesigned based on the control design procedure shown in Figure 4.8. This structure64Table 4.5: FB controller’s performance comparison under different distur-bancesPHP PWP Pelec TcostWorst Disturbances 0.1079 0.000457 0.1084 0Random Disturbances 0.0610 0.000311 0.0613 0Best Disturbances 0.0285 0.0001985 0.0287 0is a parallel combination of both feedback and off-line controllers which means thesummation of control outputs of both feedback and off-line controllers forms themain control inputs of the system. The difference between this controller and theoverall structure is that the feedforward part is omitted which means the measureddisturbances are ignored for controlling the HP-Side.Figure 4.8: Control design procedure of designing the FB + OL control struc-tureThe control inputs computed by this method is pictured in Figure 4.9b. Thesimulation is done for three disturbances scenarios, and the corresponding domes-tic hot water temperatures are shown in Figure 4.9a. The temperature responsesof the FB controller usually are higher than the FB + OL controllers. This com-650 5 10 15 204550556065FB + OL + Best DisFB + OL + Random DisFB + OL + Worst DisLower Limitation(a) Temperature0 5 10 15 2000.5 Off-LineFB + OL + Best DisFB + OL + Random DisFB + OL + Worst Dis0 5 10 15 2000.510 5 10 15 2000.51(b) Control inputsFigure 4.9: Combination of FB and Off-line controllers resultsparison illustrates that the solar thermal energy is absorbed more during day-timeusing FB + OL controller. The captured energy can be consumed in the evening.In the worst scenario of disturbances, by implementing FB + OL controller, theHP is consumed less electricity than the HP managed by the FB controller. Theperformances can be compared in Table 4.6.The comparison between Tables 4.5 and 4.6 illustrates the electrical consump-tion by the HP compressor is less for the feedback controller in the best and randomcases. However, the predicted data is useful in the worst case scenario, for control-ling the system under other disturbance cases, it is not a sufficient signal. As aresult, one can conclude that more information is required. Real-time disturbancesignals can be useful.66Table 4.6: FB + OL controller’s performance comparison under different dis-turbancesPHP PWP Pelec TcostWorst Disturbances 0.1013 0.000492 0.1017 0Random Disturbances 0.0687 0.000381 0.0691 0Best Disturbances 0.0451 0.000297 0.0454 04.2.4 Simple Feedforward ControllerBy far, two kinds of data, the feedback signal of DHW demand temperature andpredicted disturbances, are employed for controlling the domestic hot water sys-tem. Now, the performance of the overall structure with simple feedforward con-troller is compared with the previously examined sub-controllers to indicate thebenefits of using the measured disturbances. In the next step, the performance ofSimple FF controller is compared with the Robust FF controller. By implement-ing these controllers, one can demonstrate whether the robust controller is flexiblefor different disturbance situations and if the HP managed by Robust FF controllerconsumes less electrical power.The control structure is designed based on the control design procedure isshown in Figure 3.2. The feedforward part of the overall controller is replacedwith the simple feedforward controller. Here, in Figure 4.10 (a), the system’s re-sponse to the Simple FF controller is shown. The temperature is entirely within thedesirable range. For all three disturbances conditions, the temperature responsesare relatively similar. The control inputs computed by this controller is shown inFigure 4.10 (b). In this figure, control inputs are compared with optimal controlinputs which are shown with blue lines. Some notes can be mentioned in this figureby comparing it with the previously examined sub-controllers.• In Figure 4.10b (b), the red line in the third plot which represents the powerconsumption of the HP is often at the lowest level which means when thereis sufficient solar energy, and the domestic hot water demand is not high,there is usually no need to turn on the HP.670 5 10 15 204550556065Simple FF + Best DisSimple FF + Random DisSimple FF + Worst DisLower Limitation(a) Temperature0 5 10 15 2000.5 Off-LineSimple FF + Best DisSimple FF + Random DisSimple FF + Worst Dis0 5 10 15 2000.510 5 10 15 2000.51(b) Control inputsFigure 4.10: The results of the overall controller with the simple feedforwardcontroller• Let us compare the control inputs of the FB controller with the Simple FFcontroller under the worst disturbance scenario, shown by the green line.In the feedback controller shown in Figure 4.7 (b), during the high demandfor the domestic hot water, the HP compressor ratio is at maximum. In thesimple feedforward controller, at the beginning of the day, the HP is on;therefore, there is no need to consume excessive electricity around the peakof domestic hot water usages, as depicted in Figure 4.10 (b).• The water pumps inside HP loops are acting on and off which means whenthe HP compressor is on, the flow rates are at maximum level, and when theHP compressor is off, the water pumps are also off.68• In Figure 4.10 (a), noting the HP operation for all three disturbances from 3p.m. to midnight, we can see the effect of using a feedback controller in theoverall structure which retrains the temperature within the allowable range.• In the first plot of Figure 4.10 (b), the possible reason for the jittery behaviorof the STC flow rate is that there are two close solutions of the optimizationproblem. This situation happens when the non-convexity of the problemis complicated, and the final answer to the optimization problem changesfrom one to another. For the best disturbance case, the gap between the twosolutions becomes significant, and the higher STC flow rate is the dominantsolution for most of the time.In Table 4.7, the performance of the Simple FF controller is shown. The tem-perature cost is zero. The overall, HP, and WP electrical costs are decreasing fromthe worst case disturbance to the best case disturbance. These values can be com-pared with the results of the Robust FF controller to see the effectiveness of theoverall structure.Table 4.7: Simple feedforward controller’s performance comparison underdifferent DisturbancesPHP PWP Pelec TcostWorst Disturbances 0.0994 0.000537 0.0999 0Random Disturbances 0.0633 0.000373 0.0637 0Best Disturbances 0.0251 0.000165 0.0253 0The simple FF controller is designed such that the effect of disturbances onthe output of the system rejects, which seems it is not efficient in some disturbancecases. Thus, the Robust FF controller is required such that the ISTHS operates withless electrical costs. The performance of the Robust FF controller is examined inthe next part.694.2.5 Robust Feedforward ControllerHere, the feedforward part of the overall control structure which is designed basedon the procedure shown in Figure 3.2, is replaced by robust feedforward controller.Like previously examined sub-controllers, the temperature response of the RobustFF controller which depicted in Figure 4.11 (a) is acceptable for the domestic us-age. By comparing the temperature responses of three disturbances scenarios, wecan see that they have similar profiles which determine the optimal temperatureresponse to the uncertain disturbances. The robust control inputs illustrated in Fig-ure 4.11 (b). In the following, some features of this figure are listed.0 5 10 15 204550556065Robust FF + Best DisRobust FF + Random DisRobust FF + Worst DisLower Limitation(a) Temperature0 5 10 15 2000.5Off-LineRobust FF + Best DisRobust FF + Random DisRobust FF + Worst Dis0 5 10 15 2000.510 5 10 15 2000.51(b) Control inputsFigure 4.11: The results of the overall controller with the robust feedforwardcontroller• The HP compressor ratio amounts to values around the optimal control input70which are shown with the blue lines, except for the amounts of this controlinput around 8 p.m.• By comparing Figure 4.10 (b) with 4.11 (b), for all disturbance scenarios,one can see that the HP compressor ratio of the Robust FF controller seemsto be lower than the Simple FF controller compressor ratio.• Since the STC flow rates are computed separately from the flow rates ofthe HP side, they do not change by different sub-controllers, except for theoff-line controller.The performance of the Robust feedforward controller for different disturbancesscenarios are compared in Table 4.8. With a quick comparison between this ta-Table 4.8: Robust feedforward controller’s performance comparison underDifferent DisturbancesPHP PWP Pelec TcostWorst Disturbances 0.0977 0.000551 0.0983 0Random Disturbances 0.0586 0.000508 0.0591 0Best Disturbances 0.0248 0.000355 0.0251 0ble and Table 4.7, we see that the electrical costs of Robust feedforward controllerare less than the simple feedforward controller. The temperature cost is the sameand zero. In the next part, all simulation results are summarized in three tables(4.9, 4.10, and 4.11). These results are compared to demonstrate the optimal con-trol structure.4.3 Simulation ConclusionIn this part, for one disturbance case, the performance of examined control struc-tures is compared to determine the benefits and deficiency of each sub-controllersin Tables 4.9, 4.10, and 4.11. The examined control structures are OL, FB, FB +OL, Simple FF, and Robust FF. In each table, the overall, HP, and WP electricalcosts are compared as well as the temperature cost which shows whether the tem-71perature profile is within the desirable range. In these simulations, four interestingpoints are itemized as follow.Table 4.9: Performance comparison under worst case disturbancesPHP PWP Pelec TcostOL 0.0317 0.000555 0.0323 31.325FB 0.1079 0.000457 0.1087 0OL + FB 0.1013 0.000492 0.1017 0Simple FF 0.0994 0.000537 0.0999 0Robust FF 0.0977 0.000551 0.0983 0Table 4.10: Performance comparison under random case disturbancesPHP PWP Pelec TcostOL 0.0387 0.000555 0.0393 6.955FB 0.0610 0.000311 0.0613 0OL + FB 0.0687 0.000381 0.0691 0Simple FF 0.0633 0.000373 0.0637 0Robust FF 0.0586 0.000508 0.0591 0Table 4.11: Performance comparison under best case disturbancesPHP PWP Pelec TcostOL 0.0474 0.000555 0.048 0FB 0.0285 0.0001985 0.0287 0OL + FB 0.0451 0.000297 0.0454 0Simple FF 0.0251 0.000165 0.0253 0Robust FF 0.0248 0.000355 0.0251 0First, for the worst and random cases shown in Tables 4.9 and 4.10, the OLcontroller has the lowest electrical costs; however, the temperature cost of thesetwo cases is not zero which is not acceptable. Moreover, in the best case, the OL72controller has the highest electrical cost which is not desirable. In this case, sinceboth the solar energy and the hot water demand are respectively higher and lowerthan their predicted values, the optimal control inputs produce extra energy in theHP side which is not desirable.Second, let us compare the results of FB and FB + OL controllers. In Table 4.9,for the worst case scenario, the overall electrical power consumption of OL + FBcontroller is less than the FB controller by the rate of 6.43%. However, for therandom and best cases, the FB controller is more efficient than FB controller. Thus,for different disturbances scenarios, the predicted data can be useful or not.Third, comparing the simple FF controller with both FB and FB + OL con-trollers reveals some advantages and disadvantages of this controller. In the worstand best cases, the simple FF controller outperforms both FB and FB + OL con-trollers; however, in the random case, the FB controller is consumed less electricitythan Simple FF by the rate of 3.4%. In the best case, the Simple FF controller per-forms 11.84% more efficient than the FB, and in the worst case, simple FF overallelectrical cost is lower than the FB + OL controller by the rate of 1.8%. Thesecomparisons demonstrate the benefits of exploiting feedforward signals from dis-turbances just for two disturbance cases.Fourth, the Robust FF controller is designed such that the electrical cost re-duces while the measured disturbance signals are applied. In the worst and the bestscenarios, this controller is compared with the simple FF, while In the random case,it is compared with FB controller. Respectively, in the worst, random, and best dis-turbance cases, the consumed electrical power of Robust FF controller is 1.63%,3.72%, and 0.8% lower than the best one among previously examined controllers.If one compares the Robust FF controller with the FB, FB + OL, and simple FFcontrollers for all disturbance cases, the Robust FF controller is consumed between1 to 45% less electrical power. In fact, the winner of all examined controllers isthe Robust FF controller which is the overall control structure with the robust con-troller in the feedforward part. These simulation results demonstrate the potentialof the proposed Robust FF controller in the solar thermal water heating application.73Chapter 5Conclusion and Future WorksIn Section 5.1, a summary of the methods and findings of this thesis are provided.Next, the contribution of this research is stated in Section 5.2. Finally, the potentialfuture works are discussed in Section SummaryIn this thesis, a control strategy was developed to optimize the performance ofan integrated solar thermal hydronic system. The research goal was to reduce theelectrical cost of the system while the system’s output meets the domestic hot waterdemand. Reaching this goal was a necessary step for reducing the overall cost ofthe solar system. For this purpose, a state-space model was derived in Chapter 2;then, a new control strategy was developed to design an overall controller in Chap-ter 3. The overall control structure is composed of some sub-controllers named asoff-line, STC-Side, feedback, and feedforward controllers. By exploiting the pre-dictable profiles of the disturbances, the off-line controller generates the optimalcontrol inputs one-day ahead by means of an optimization method. The STC-Sidecontroller also employs an optimization approach under the measured disturbances,and it produces the control inputs of the STC side. The feedback controller wasdesigned such that the hot water temperature is maintained within a specific allow-able range. Two types of feedforward controllers were developed, i.e., simple androbust controllers. In the simple feedforward controller, the effects of the devia-74tion of the measured disturbances from the predicted values on the output of thesystem are rejected by adjusting the HP compressor ratio. The robust feedforwardcontroller was designed by using H∞ control method, and besides rejecting theeffect of disturbances, this controller generates the optimal HP compressor ratio.In Chapter 3, this controller was claimed to be efficient in the sense of electricalconsumption. This claim was verified in Chapter 4 through simulation studies. Inthis chapter, comparisons were made between different sub-controllers. By omit-ting some part of the overall control structure, the advantages and disadvantages ofeach employed sub-controller were examined. These comparisons demonstratedthe overall control structure with the robust feedforward controller is an optimalcontroller for solar domestic hot water systems.The overall control structure with the robust feedforward controller uses up to45% less electrical power under three disturbance cases relative to other controllers.The temperature responses of this controller are within the desirable range, whichmeans the domestic hot water will meet the demand for all disturbance conditions.In addition to the robust feedforward controller, among other sub-controllers, thefeedback controller performs more efficiently under random disturbances, and withthe simple feedforward controller, less electrical power is consumed in the presenceof both best and worst case disturbances. These comparisons illustrated to reach theresearch goal, using all sub-controllers in the overall control structure is necessary.5.2 ContributionThe contribution of this research was to introduce a control structure employingthree techniques, i.e., optimization, feedback, feedforward controllers. The controlstructure optimally deals with the uncertainties in disturbances, such that the elec-trical cost is reduced. Instead of tracking a particular temperature, an allowabletemperature range is introduced, and the proposed controller retains the hot watertemperature within this range. Unlike previous studies, the measured disturbances,which might be different from the predicted ones, were fed as a signal into the opti-mal feedforward controller, and the robust control design approach was employedto design the optimal feedforward controller.755.3 Future WorksPotential future work and extensions of this topic may be categorized as follows:• In this thesis, the uncertainties in modeling were ignored. To make themodel-based controllers applicable to real solar thermal systems which can-not be modeled precisely, the advanced controller design will be required.• For validating the proposed controller performance practically, the researchsetup shown in Figure 5.1 is under construction on the top of the CIRS build-ing 1 at UBC. The real-data that will be collected from this setup will aid usto improve the performance of the controller.Figure 5.1: Solar thermal collectors and their stand: research setup1Center for Interactive Research on Sustainability76Bibliography[1] B. Ramlow, Solar Water Heating: a Comprehensive Guide to Solar Waterand Heating Systems. New Society Publishers, 2010. → pages 1[2] J. Lemos, R. Neves-Silva, and J. Igreja, Adaptive Control of Solar EnergyCollector Systems. Advances in Industrial Control, Springer InternationalPublishing, 2014. → pages 2[3] E. Camacho, M. Soria, F. Rubio, and D. Martı´nez, Control of Solar EnergySystems. Advances in Industrial Control, Springer London, 2012. → pages2, 5, 7[4] Y. Ma, A. Kelman, A. Daly, and F. Borrelli, “Predictive control for energyefficient buildings with thermal storage: Modeling, stimulation, andexperiments,” IEEE Control Systems, vol. 32, pp. 44–64, Feb 2012. → pages5, 6[5] R. Padovan and M. Manzan, “Genetic optimization of a pcm enhancedstorage tank for solar domestic hot water systems,” Solar Energy, vol. 103,pp. 563–573, May 2014. → pages 6, 8[6] S. Furbo, E. Andersen, S. Knudsen, N. K. Vejen, and L. J. 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Rossiter, “Gain scheduling dual mode mpc for asolar thermal power plant,” IFAC-PapersOnLine, vol. 49, pp. 128–133, Jan.2016. → pages 5681Appendix ASupporting MaterialsA.1 Nonlinear State-Space ModelIn this section, the complete non-linear model of the ISTHS is provided by recall-ing Figure 2.2.M1cp(x1)dx1dt= u1cp(x1)(TSTC,out − x1)−U1A1(x1−d2), (A.1)M2cp(x2)dx2dt= u1cp(x2)(x3− x2)−U2A2(x2−d2),M3cp(x3)dx3dt= u1cp(x3)(x4− x3)−UCoil1ACoil1(x3− x5),M4cp(x4)dx4dt= u2cp(x4)(x5− x4)−U4A4(x4−d2),M5cp(x5)dx5dt= u2cp(x5)(x6− x5)−UShell1AShell1(x5−d2)−UCoil1ACoil1(x5− x3),M6cp(x6)dx6dt= u2cp(x6)(x7− x6)−U6A6(x6−d2),M7cp(x7)dx7dt= cp(x7)(u2+u3)(x8− x7)+d3cp(x7)(Tsource− x7),−U7aA7a(x7−d2)−U7cA7c(x7− x8),82M8cp(x8)dx8dt= cp(x8)(u2+u3)(x9− x8)+d3cp(x8)(x7− x8),−U8aA8a(x8−d2)−U8cA8c(2x8− x7− x9),M9cp(x9)dx9dt= u2cp(x9)(x4− x9)+u3cp(x9)(x10− x9)+d3cp(x9)(x8− x9),−U9aA9a(x9−d2)−U9cA9c(x9− x8),M10cp(x10)dx10dt= u3cp(x10)(x11− x10)−U10A10(x10−d2).M11cp(x11)dx11dt= u3cp(x11)(x12− x11)−UShell2AShell2(x11−d2)−UCoil2ACoil2(x11− x14).M12cp(x12)dx12dt= u3cp(x12)(x7− x12)−U12A12(x12−d2).M13cp(x13)dx13dt= u4cp(x13)(x14− x13)−U13A13(x13−d2).M14cp(x14)dx14dt= u4cp(x14)(x15− x14)−UCoil2ACoil2(x14− x11).M15cp(x15)dx15dt= u4cp(x15)(THP,out − x15)−U15A15(x15−d2).where d1, d2, and d3, respectively, are the three disturbances, i.e., solar radiation,ambient temperature, and the domestic hot water flow rate. u1, ..., u4, and u5 arefive control inputs which introduced in Equation (2.3). Here, TSTC,out and THP,outare computed asTSTC,out = x2+Q˙STCu1cp(x2), (A.2)THP,out = x13+Q˙HPu4cp(x13). (A.3)Q˙STC is the rate of the absorbed solar energy calculated as Equation (2.7), and Q˙HPis the thermal power produced by the HP which is derived as Equation (2.15). Miis the mass inside the i th control volume shown in Figure 2.3, which is derived asMi = ρpiDiLi, (A.4)83where ρ is the density of the fluid inside the control volume, and Di and Li arerespectively the diameter and length of the control volume. Next, the formulationand procedure of designing other parameters such as cp and U are given.A.1.1 Specific Heating Capacity (cp)A 2nd-order polynomial surface (A.5) is used to determine the specific heat ca-pacity of the mixture, cp, as a function of the mixture temperature, Tmix, and thevolumetric percentage of the glycol, PGly.cp(Tmix) = P00+P10PGly+P01Tmix+P20(PGly)2+P11PGlyTmix+P02(Tmix)2 (A.5)The corresponding coefficients are listed in Table A.1.Constant ValueP00 4.265P10 -0.01813P01 0.001249P20 −1.375×10−5P11 3.852×10−5P02 −4.977×10−7Table A.1: Constant values for calculating cpA.1.2 Overall Heat Transfer Coefficient (U)For calculating U , convective and conductive heat transfer coefficients of each el-ement should be specified. The convective heat transfer coefficient of a fluid ishighly dependent on the fluid flowing speed, v, and Table A.2 is formed. Using theConfiguration Typical value of h, [ Wm2◦C ]Natural convection in air (v < 1m/s) 10Forced convection in air (v > 5m/s) 50Natural convection in water (v < 0.1m/s) 200Forced convection in water (v > 0.5m/s) 5000Table A.2: Typical h values for air and water84data in Table A.2, one can conduct linear regression to write convective heat trans-fer coefficients of air and water, hair and hwater as a function of the flowing speed.In the equations (A.6) and (A.7), “sat”, meaning saturation, is used to indicate theupper and lower limit of the h value.hair = sat5010(10 · vair) (A.6)hwater = sat5000200 (12000 · vwater−1000) (A.7)The fluid flowing speed inside each element can be computed from the volumetricflow rate of the fluid which is derived as a function of mass flow rates of the ITHSloops representing by four control inputs, ui, i = {1,2, ,3,4}. In this thesis, theair speed is considered to be always under 1m/s. The conductive heat transfercoefficient of each element k is considered as a constant value which is acceptablefor domestic applications. The amount of this coefficient depends on how muchthe corresponding component is insulated.PipesThe overall heat loss coefficient of all pipes, Ui, i = {1,2, ...,15}, are written as1Ui=1hi+tiki+1hair(A.8)hi can be calculated using equation (A.7) and the corresponding mass flow rate(control input), the convective heat transfer coefficient of the air outside the pipe,hair can be calculated using equation (A.6), the thickness of the i th pipe, ti and thethermal conductivity of the pipe material is ki.Heat exchangersAs for Ushell1 and Ushell2 , it is needed to replace the parameters used in Equa-tion (A.8) with the parameters corresponding to the first and the second HXs’ shellrespectively. hshell1 is the convective heat transfer coefficient of the shell of the firstHX which is a function of u2. Likewise, hshell2 is the function of u3.The overall heat transfer coefficient, UCoil1 , is determined by the convective85heat transfer coefficient of the coil and shell fluid, hCoil1 , as a function of and hShell1 ,the thermal conductivity of the coil material (stainless steel in this project), kcoil ,the inner and outer radius of the coil, rin and rout , as1UCoil1=routrinhCoil1+routkcoilln(routrin)+1hShell1(A.9)In HX coil, the convective heat transfer occurs between two fluids. Thus, the con-vective heat transfer coefficient, hCoil1 can be calculated using equation (A.7) aftercalculating the flow speed, vcoil and vshell knowing first and second control inputs,u1 and u2. Similar to UCoil1 , UCoil2 is computed with parameters corresponding tothe second HX. Additionally, hCoil2 is a function of u3 and u4.Thermal storage tankThe overall heat loss coefficient of the TST elements, Uia , i = {7,8,9}, is a func-tion of the convective heat transfer coefficient of the water inside the tank and theair outside, the wall thickness of the TST, tT ST and the thermal conductivity ofthe tank wall material, kT ST . Since the water inside the TST does not move, theconvective heat transfer coefficient of the TST elements is constant and equal thenatural convective coefficient of water, hwater.1UT ST=1hwater+tT STkT ST+1hair(A.10)Since the heat transfer between two layers of TST is just convection, the overallheat transfer coefficient between two elements of TST, Uic equals hwater.86


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